/usr/share/octave/packages/control-3.0.0/doc-cache is in octave-control 3.0.0-5.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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Anderson
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 62
Frequency-weighted coprime factorization controller reduction.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 62
Frequency-weighted coprime factorization controller reduction.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
BMWengine
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1277
-- Function File: SYS = BMWengine ()
-- Function File: SYS = BMWengine ("SCALED")
-- Function File: SYS = BMWengine ("UNSCALED")
Model of the BMW 4-cylinder engine at ETH Zurich's control
laboratory.
OPERATING POINT
Drosselklappenstellung alpha_DK = 10.3 Grad
Saugrohrdruck p_s = 0.48 bar
Motordrehzahl n = 860 U/min
Lambda-Messwert lambda = 1.000
Relativer Wandfilminhalt nu = 1
INPUTS
U_1 Sollsignal Drosselklappenstellung [Grad]
U_2 Relative Einspritzmenge [-]
U_3 Zuendzeitpunkt [Grad KW]
M_L Lastdrehmoment [Nm]
STATES
X_1 Drosselklappenstellung [Grad]
X_2 Saugrohrdruck [bar]
X_3 Motordrehzahl [U/min]
X_4 Messwert Lamba-Sonde [-]
X_5 Relativer Wandfilminhalt [-]
OUTPUTS
Y_1 Motordrehzahl [U/min]
Y_2 Messwert Lambda-Sonde [-]
SCALING
U_1N, X_1N 1 Grad
U_2N, X_4N, X_5N, Y_2N 0.05
U_3N 1.6 Grad KW
X_2N 0.05 bar
X_3N, Y_1N 200 U/min
# name: <cell-element>
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Model of the BMW 4-cylinder engine at ETH Zurich's control laboratory.
# name: <cell-element>
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# length: 9
Boeing707
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 389
-- Function File: SYS = Boeing707 ()
Creates a linearized state-space model of a Boeing 707-321 aircraft
at V=80 m/s (M = 0.26, GA0 = -3 deg, ALPHA0 = 4 deg, KAPPA = 50
deg).
System inputs: (1) thrust and (2) elevator angle.
System outputs: (1) airspeed and (2) pitch angle.
*Reference*: R. Brockhaus: 'Flugregelung' (Flight Control),
Springer, 1994.
# name: <cell-element>
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Creates a linearized state-space model of a Boeing 707-321 aircraft at
V=80 m/s
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# length: 9
MDSSystem
# name: <cell-element>
# type: sq_string
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# length: 154
Robust control of a mass-damper-spring system. Type 'which MDSSystem'
to locate, 'edit MDSSystem' to open and simply 'MDSSystem' to run the
example file.
# name: <cell-element>
# type: sq_string
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Robust control of a mass-damper-spring system.
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# type: sq_string
# elements: 1
# length: 9
Madievski
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Demonstration of frequency-weighted controller reduction. The system
considered in this example has been studied by Madievski and Anderson
[1] and comprises four spinning disks. The disks are connected by a
flexible rod, a motor applies torque to the third disk, and the angular
displacement of the first disk is the variable of interest. The
state-space model of eighth order is non-minimumphase and unstable. The
continuous-time LQG controller used in [1] is open-loop stable and of
eighth order like the plant. This eighth-order controller shall be
reduced by frequency-weighted singular perturbation approximation (SPA).
The major aim of this reduction is the preservation of the closed-loop
transfer function. This means that the error in approximation of the
controller K by the reduced-order controller KR is minimized by
min ||W (K-Kr) V||
Kr inf
where weights W and V are dictated by the requirement to preserve (as
far as possible) the closed-loop transfer function. In minimizing the
error, they cause the approximation process for K to be more accurate at
certain frequencies. Suggested by [1] is the use of the following
stability and performance enforcing weights:
-1 -1
W = (I - G K) G, V = (I - G K)
This example script reduces the eighth-order controller to orders
four and two by the function call 'Kr = spaconred (G, K, nr, 'feedback',
'-')' where argument NR denotes the desired order (4 or 2). The
key-value pair ''feedback', '-'' allows the reduction of negative
feedback controllers while the default setting expects positive feedback
controllers. The frequency responses of the original and reduced-order
controllers are depicted in figure 1, the step responses of the closed
loop in figure 2. There is no visible difference between the step
responses of the closed-loop systems with original (blue) and fourth
order (green) controllers. The second order controller (red) causes
ripples in the step response, but otherwise the behavior of the system
is unaltered. This leads to the conclusion that function 'spaconred' is
well suited to reduce the order of controllers considerably, while
stability and performance are retained.
*Reference*
[1] Madievski, A.G. and Anderson, B.D.O. 'Sampled-Data Controller
Reduction Procedure', IEEE Transactions of Automatic Control, Vol. 40,
No. 11, November 1995
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Demonstration of frequency-weighted controller reduction.
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VLFamp
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-- Function File: VLFamp
-- Function File: RESULT = VLFamp (VERBOSE)
Calculations on a two stage preamp for a multi-turn, air-core
solenoid loop antenna for the reception of signals below 30kHz.
The Octave Control Package functions are used extensively to
approximate the behavior of operational amplifiers and passive
electrical circuit elements.
This example presents several 'screen' pages of documentation of
the calculations and some reasoning about why. Plots of the
results are presented in most cases.
The process is to display a 'screen' page of text followed by the
calculation and a 'Press return to continue' message. To proceed
in the example, press return. ^C to exit.
At one point in the calculations, the process may seem to hang,
but, this is because of extensive calculations.
The returned transfer function is more than 100 characters long so
will wrap in screens that are narrow and appear jumbled.
# name: <cell-element>
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Calculations on a two stage preamp for a multi-turn, air-core solenoid
loop ante
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# type: sq_string
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# length: 12
WestlandLynx
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1260
-- Function File: SYS = WestlandLynx ()
Model of the Westland Lynx Helicopter about hover.
INPUTS
main rotor collective
longitudinal cyclic
lateral cyclic
tail rotor collective
STATES
pitch attitude theta [rad]
roll attitude phi [rad]
roll rate (body-axis) p [rad/s]
pitch rate (body-axis) q [rad/s]
yaw rate xi [rad/s]
forward velocity v_x [ft/s]
lateral velocity v_y [ft/s]
vertical velocity v_z [ft/s]
OUTPUTS
heave velocity H_dot [ft/s]
pitch attitude theta [rad]
roll attitude phi [rad]
heading rate psi_dot [rad/s]
roll rate p [rad/s]
pitch rate q [rad/s]
*References*
[1] Skogestad, S. and Postlethwaite I. (2005) 'Multivariable
Feedback Control: Analysis and Design: Second Edition'. Wiley.
<http://www.nt.ntnu.no/users/skoge/book/2nd_edition/matlab_m/matfiles.html>
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Model of the Westland Lynx Helicopter about hover.
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# type: sq_string
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# length: 6
append
# name: <cell-element>
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-- Function File: SYS = append (SYS1, SYS2, ..., SYSN)
Group LTI models by appending their inputs and outputs.
# name: <cell-element>
# type: sq_string
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Group LTI models by appending their inputs and outputs.
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# type: sq_string
# elements: 1
# length: 3
arx
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1953
-- Function File: [SYS, X0] = arx (DAT, N, ...)
-- Function File: [SYS, X0] = arx (DAT, N, OPT, ...)
-- Function File: [SYS, X0] = arx (DAT, OPT, ...)
-- Function File: [SYS, X0] = arx (DAT, 'NA', NA, 'NB', NB)
Estimate ARX model using QR factorization.
A(q) y(t) = B(q) u(t) + e(t)
*Inputs*
DAT
iddata identification dataset containing the measurements,
i.e. time-domain signals.
N
The desired order of the resulting model SYS.
...
Optional pairs of keys and values. ''key1', value1, 'key2',
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
SYS
Discrete-time transfer function model. If the second output
argument X0 is returned, SYS becomes a state-space model.
X0
Initial state vector. If DAT is a multi-experiment dataset,
X0 becomes a cell vector containing an initial state vector
for each experiment.
*Option Keys and Values*
'NA'
Order of the polynomial A(q) and number of poles.
'NB'
Order of the polynomial B(q)+1 and number of zeros+1. NB <=
NA.
'NK'
Input-output delay specified as number of sampling instants.
Scalar positive integer. This corresponds to a call to
function 'nkshift', followed by padding the B polynomial with
NK leading zeros.
*Algorithm*
Uses the formulae given in [1] on pages 318-319, 'Solving for the
LS Estimate by QR Factorization'. For the initial conditions,
SLICOT IB01CD is used by courtesy of NICONET e.V.
(http://www.slicot.org)
*References*
[1] Ljung, L. (1999) 'System Identification: Theory for the User:
Second Edition'. Prentice Hall, New Jersey, USA.
# name: <cell-element>
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Estimate ARX model using QR factorization.
# name: <cell-element>
# type: sq_string
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# length: 8
augstate
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# length: 264
-- Function File: AUGSYS = augstate (SYS)
Append state vector x of system SYS to output vector y.
. .
x = A x + B u x = A x + B u
y = C x + D u => y = C x + D u
x = I x + O u
# name: <cell-element>
# type: sq_string
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Append state vector x of system SYS to output vector y.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
augw
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# type: sq_string
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# length: 2976
-- Function File: P = augw (G, W1, W2, W3)
Extend plant for stacked S/KS/T problem. Subsequently, the robust
control problem can be solved by h2syn or hinfsyn.
*Inputs*
G
LTI model of plant.
W1
LTI model of performance weight. Bounds the largest singular
values of sensitivity S. Model must be empty '[]', SISO or of
appropriate size.
W2
LTI model to penalize large control inputs. Bounds the
largest singular values of KS. Model must be empty '[]', SISO
or of appropriate size.
W3
LTI model of robustness and noise sensitivity weight. Bounds
the largest singular values of complementary sensitivity T.
Model must be empty '[]', SISO or of appropriate size.
All inputs must be proper/realizable. Scalars, vectors and
matrices are possible instead of LTI models.
*Outputs*
P
State-space model of augmented plant.
*Block Diagram*
| W1 | -W1*G | z1 = W1 r - W1 G u
| 0 | W2 | z2 = W2 u
P = | 0 | W3*G | z3 = W3 G u
|----+-------|
| I | -G | e = r - G u
+------+ z1
+---------------------------------------->| W1 |----->
| +------+
| +------+ z2
| +---------------------->| W2 |----->
| | +------+
r + e | +--------+ u | +--------+ y +------+ z3
----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |----->
^ - +--------+ +--------+ | +------+
| |
+----------------------------------------+
+--------+
| |-----> z1 (p1x1) z1 = W1 e
r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u
| |-----> z3 (p3x1) z3 = W3 y
u (mx1) ----->| |-----> e (px1) e = r - y
+--------+
+--------+
r ----->| |-----> z
| P(s) |
u +---->| |-----+ e
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
*References*
[1] Skogestad, S. and Postlethwaite I. (2005) 'Multivariable
Feedback Control: Analysis and Design: Second Edition'. Wiley.
See also: h2syn, hinfsyn, mixsyn.
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# type: sq_string
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Extend plant for stacked S/KS/T problem.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
bode
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-- Function File: bode (SYS)
-- Function File: bode (SYS1, SYS2, ..., SYSN)
-- Function File: bode (SYS1, SYS2, ..., SYSN, W)
-- Function File: bode (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: [MAG, PHA, W] = bode (SYS)
-- Function File: [MAG, PHA, W] = bode (SYS, W)
Bode diagram of frequency response. If no output arguments are
given, the response is printed on the screen.
*Inputs*
SYS
LTI system. Must be a single-input and single-output (SISO)
system.
W
Optional vector of frequency values. If W is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell '{wmin, wmax}' specifies a frequency
range, where WMIN and WMAX denote minimum and maximum
frequencies in rad/s.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
MAG
Vector of magnitude. Has length of frequency vector W.
PHA
Vector of phase. Has length of frequency vector W.
W
Vector of frequency values used.
See also: nichols, nyquist, sigma.
# name: <cell-element>
# type: sq_string
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Bode diagram of frequency response.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
bodemag
# name: <cell-element>
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# elements: 1
# length: 1189
-- Function File: bodemag (SYS)
-- Function File: bodemag (SYS1, SYS2, ..., SYSN)
-- Function File: bodemag (SYS1, SYS2, ..., SYSN, W)
-- Function File: bodemag (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: [MAG, W] = bodemag (SYS)
-- Function File: [MAG, W] = bodemag (SYS, W)
Bode magnitude diagram of frequency response. If no output
arguments are given, the response is printed on the screen.
*Inputs*
SYS
LTI system. Must be a single-input and single-output (SISO)
system.
W
Optional vector of frequency values. If W is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell '{wmin, wmax}' specifies a frequency
range, where WMIN and WMAX denote minimum and maximum
frequencies in rad/s.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
MAG
Vector of magnitude. Has length of frequency vector W.
W
Vector of frequency values used.
See also: bode, nichols, nyquist, sigma.
# name: <cell-element>
# type: sq_string
# elements: 1
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Bode magnitude diagram of frequency response.
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# type: sq_string
# elements: 1
# length: 9
bstmodred
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# type: sq_string
# elements: 1
# length: 6085
-- Function File: [GR, INFO] = bstmodred (G, ...)
-- Function File: [GR, INFO] = bstmodred (G, NR, ...)
-- Function File: [GR, INFO] = bstmodred (G, OPT, ...)
-- Function File: [GR, INFO] = bstmodred (G, NR, OPT, ...)
Model order reduction by Balanced Stochastic Truncation (BST)
method. The aim of model reduction is to find an LTI system GR of
order NR (nr < n) such that the input-output behaviour of GR
approximates the one from original system G.
BST is a relative error method which tries to minimize
-1
||G (G-Gr)|| = min
inf
*Inputs*
G
LTI model to be reduced.
NR
The desired order of the resulting reduced order system GR.
If not specified, NR is chosen automatically according to the
description of key 'ORDER'.
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
GR
Reduced order state-space model.
INFO
Struct containing additional information.
INFO.N
The order of the original system G.
INFO.NS
The order of the ALPHA-stable subsystem of the original
system G.
INFO.HSV
The Hankel singular values of the phase system
corresponding to the ALPHA-stable part of the original
system G. The NS Hankel singular values are ordered
decreasingly.
INFO.NU
The order of the ALPHA-unstable subsystem of both the
original system G and the reduced-order system GR.
INFO.NR
The order of the obtained reduced order system GR.
*Option Keys and Values*
'ORDER', 'NR'
The desired order of the resulting reduced order system GR.
If not specified, NR is the sum of NU and the number of Hankel
singular values greater than 'MAX(TOL1,NS*EPS)'; NR can be
further reduced to ensure that 'HSV(NR-NU) > HSV(NR+1-NU)'.
'METHOD'
Approximation method for the H-infinity norm. Valid values
corresponding to this key are:
'SR-BTA', 'B'
Use the square-root Balance & Truncate method.
'BFSR-BTA', 'F'
Use the balancing-free square-root Balance & Truncate
method. Default method.
'SR-SPA', 'S'
Use the square-root Singular Perturbation Approximation
method.
'BFSR-SPA', 'P'
Use the balancing-free square-root Singular Perturbation
Approximation method.
'ALPHA'
Specifies the ALPHA-stability boundary for the eigenvalues of
the state dynamics matrix G.A. For a continuous-time system,
ALPHA <= 0 is the boundary value for the real parts of
eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1
represents the boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and 1 for
discrete-time systems.
'BETA'
Use '[G, beta*I]' as new system G to combine absolute and
relative error methods. BETA > 0 specifies the
absolute/relative error weighting parameter. A large positive
value of BETA favours the minimization of the absolute
approximation error, while a small value of BETA is
appropriate for the minimization of the relative error. BETA
= 0 means a pure relative error method and can be used only if
rank(G.D) = rows(G.D) which means that the feedthrough matrice
must not be rank-deficient. Default value is 0.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of reduced system. For model reduction,
the recommended value of TOL1 lies in the interval [0.00001,
0.001]. TOL1 < 1. If TOL1 <= 0 on entry, the used default
value is TOL1 = NS*EPS, where NS is the number of ALPHA-stable
eigenvalues of A and EPS is the machine precision. If 'ORDER'
is specified, the value of TOL1 is ignored.
'TOL2'
The tolerance for determining the order of a minimal
realization of the phase system (see METHOD) corresponding to
the ALPHA-stable part of the given system. The recommended
value is TOL2 = NS*EPS. TOL2 <= TOL1 < 1. This value is used
by default if 'TOL2' is not specified or if TOL2 <= 0 on
entry.
'EQUIL', 'SCALE'
Boolean indicating whether equilibration (scaling) should be
performed on system G prior to order reduction. Default value
is true if 'G.scaled == false' and false if 'G.scaled ==
true'. Note that for MIMO models, proper scaling of both
inputs and outputs is of utmost importance. The input and
output scaling can *not* be done by the equilibration option
or the 'prescale' function because these functions perform
state transformations only. Furthermore, signals should not
be scaled simply to a certain range. For all inputs (or
outputs), a certain change should be of the same importance
for the model.
BST is often suitable to perform model reduction in order to obtain
low order design models for controller synthesis.
Approximation Properties:
* Guaranteed stability of reduced models
* Approximates simultaneously gain and phase
* Preserves non-minimum phase zeros
* Guaranteed a priori error bound
*Algorithm*
Uses SLICOT AB09HD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Model order reduction by Balanced Stochastic Truncation (BST) method.
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btaconred
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-- Function File: [KR, INFO] = btaconred (G, K, ...)
-- Function File: [KR, INFO] = btaconred (G, K, NCR, ...)
-- Function File: [KR, INFO] = btaconred (G, K, OPT, ...)
-- Function File: [KR, INFO] = btaconred (G, K, NCR, OPT, ...)
Controller reduction by frequency-weighted Balanced Truncation
Approximation (BTA). Given a plant G and a stabilizing controller
K, determine a reduced order controller KR such that the
closed-loop system is stable and closed-loop performance is
retained.
The algorithm tries to minimize the frequency-weighted error
||V (K-Kr) W|| = min
inf
where V and W denote output and input weightings.
*Inputs*
G
LTI model of the plant. It has m inputs, p outputs and n
states.
K
LTI model of the controller. It has p inputs, m outputs and
nc states.
NCR
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically according
to the description of key 'ORDER'.
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
KR
State-space model of reduced order controller.
INFO
Struct containing additional information.
INFO.NCR
The order of the obtained reduced order controller KR.
INFO.NCS
The order of the alpha-stable part of original controller
K.
INFO.HSVC
The Hankel singular values of the alpha-stable part of K.
The NCS Hankel singular values are ordered decreasingly.
*Option Keys and Values*
'ORDER', 'NCR'
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically such that
states with Hankel singular values INFO.HSVC > TOL1 are
retained.
'METHOD'
Order reduction approach to be used as follows:
'SR', 'B'
Use the square-root Balance & Truncate method.
'BFSR', 'F'
Use the balancing-free square-root Balance & Truncate
method. Default method.
'WEIGHT'
Specifies the type of frequency-weighting as follows:
'NONE'
No weightings are used (V = I, W = I).
'LEFT', 'OUTPUT'
Use stability enforcing left (output) weighting
-1
V = (I-G*K) *G , W = I
'RIGHT', 'INPUT'
Use stability enforcing right (input) weighting
-1
V = I , W = (I-G*K) *G
'BOTH', 'PERFORMANCE'
Use stability and performance enforcing weightings
-1 -1
V = (I-G*K) *G , W = (I-G*K)
Default value.
'FEEDBACK'
Specifies whether K is a positive or negative feedback
controller:
'+'
Use positive feedback controller. Default value.
'-'
Use negative feedback controller.
'ALPHA'
Specifies the ALPHA-stability boundary for the eigenvalues of
the state dynamics matrix K.A. For a continuous-time
controller, ALPHA <= 0 is the boundary value for the real
parts of eigenvalues, while for a discrete-time controller, 0
<= ALPHA <= 1 represents the boundary value for the moduli of
eigenvalues. The ALPHA-stability domain does not include the
boundary. Default value is 0 for continuous-time controllers
and 1 for discrete-time controllers.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of the reduced controller. For model
reduction, the recommended value of TOL1 is c*info.hsvc(1),
where c lies in the interval [0.00001, 0.001]. Default value
is info.ncs*eps*info.hsvc(1). If 'ORDER' is specified, the
value of TOL1 is ignored.
'TOL2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given controller.
TOL2 <= TOL1. If not specified, ncs*eps*info.hsvc(1) is
chosen.
'GRAM-CTRB'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
'STANDARD'
Choice corresponding to standard Enns' method [1].
Default method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
Enns' method of [2].
'GRAM-OBSV'
Specifies the choice of frequency-weighted observability
Grammian as follows:
'STANDARD'
Choice corresponding to standard Enns' method [1].
Default method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
Enns' method of [2].
'EQUIL', 'SCALE'
Boolean indicating whether equilibration (scaling) should be
performed on G and K prior to order reduction. Default value
is false if both 'G.scaled == true, K.scaled == true' and true
otherwise. Note that for MIMO models, proper scaling of both
inputs and outputs is of utmost importance. The input and
output scaling can *not* be done by the equilibration option
or the 'prescale' function because these functions perform
state transformations only. Furthermore, signals should not
be scaled simply to a certain range. For all inputs (or
outputs), a certain change should be of the same importance
for the model.
*Algorithm*
Uses SLICOT SB16AD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Controller reduction by frequency-weighted Balanced Truncation
Approximation (BT
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btamodred
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-- Function File: [GR, INFO] = btamodred (G, ...)
-- Function File: [GR, INFO] = btamodred (G, NR, ...)
-- Function File: [GR, INFO] = btamodred (G, OPT, ...)
-- Function File: [GR, INFO] = btamodred (G, NR, OPT, ...)
Model order reduction by frequency weighted Balanced Truncation
Approximation (BTA) method. The aim of model reduction is to find
an LTI system GR of order NR (nr < n) such that the input-output
behaviour of GR approximates the one from original system G.
BTA is an absolute error method which tries to minimize
||G-Gr|| = min
inf
||V (G-Gr) W|| = min
inf
where V and W denote output and input weightings.
*Inputs*
G
LTI model to be reduced.
NR
The desired order of the resulting reduced order system GR.
If not specified, NR is chosen automatically according to the
description of key 'ORDER'.
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
GR
Reduced order state-space model.
INFO
Struct containing additional information.
INFO.N
The order of the original system G.
INFO.NS
The order of the ALPHA-stable subsystem of the original
system G.
INFO.HSV
The Hankel singular values of the ALPHA-stable part of
the original system G, ordered decreasingly.
INFO.NU
The order of the ALPHA-unstable subsystem of both the
original system G and the reduced-order system GR.
INFO.NR
The order of the obtained reduced order system GR.
*Option Keys and Values*
'ORDER', 'NR'
The desired order of the resulting reduced order system GR.
If not specified, NR is chosen automatically such that states
with Hankel singular values INFO.HSV > TOL1 are retained.
'LEFT', 'OUTPUT'
LTI model of the left/output frequency weighting V. Default
value is an identity matrix.
'RIGHT', 'INPUT'
LTI model of the right/input frequency weighting W. Default
value is an identity matrix.
'METHOD'
Approximation method for the L-infinity norm to be used as
follows:
'SR', 'B'
Use the square-root Balance & Truncate method.
'BFSR', 'F'
Use the balancing-free square-root Balance & Truncate
method. Default method.
'ALPHA'
Specifies the ALPHA-stability boundary for the eigenvalues of
the state dynamics matrix G.A. For a continuous-time system,
ALPHA <= 0 is the boundary value for the real parts of
eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1
represents the boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and 1 for
discrete-time systems.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of the reduced model. For model
reduction, the recommended value of TOL1 is c*info.hsv(1),
where c lies in the interval [0.00001, 0.001]. Default value
is info.ns*eps*info.hsv(1). If 'ORDER' is specified, the
value of TOL1 is ignored.
'TOL2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given model. TOL2
<= TOL1. If not specified, ns*eps*info.hsv(1) is chosen.
'GRAM-CTRB'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
'STANDARD'
Choice corresponding to a combination method [4] of the
approaches of Enns [1] and Lin-Chiu [2,3]. Default
method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
combination method of [4].
'GRAM-OBSV'
Specifies the choice of frequency-weighted observability
Grammian as follows:
'STANDARD'
Choice corresponding to a combination method [4] of the
approaches of Enns [1] and Lin-Chiu [2,3]. Default
method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
combination method of [4].
'ALPHA-CTRB'
Combination method parameter for defining the
frequency-weighted controllability Grammian. abs(alphac) <=
1. If alphac = 0, the choice of Grammian corresponds to the
method of Enns [1], while if alphac = 1, the choice of
Grammian corresponds to the method of Lin and Chiu [2,3].
Default value is 0.
'ALPHA-OBSV'
Combination method parameter for defining the
frequency-weighted observability Grammian. abs(alphao) <= 1.
If alphao = 0, the choice of Grammian corresponds to the
method of Enns [1], while if alphao = 1, the choice of
Grammian corresponds to the method of Lin and Chiu [2,3].
Default value is 0.
'EQUIL', 'SCALE'
Boolean indicating whether equilibration (scaling) should be
performed on system G prior to order reduction. This is done
by state transformations. Default value is true if 'G.scaled
== false' and false if 'G.scaled == true'. Note that for MIMO
models, proper scaling of both inputs and outputs is of utmost
importance. The input and output scaling can *not* be done by
the equilibration option or the 'prescale' function because
these functions perform state transformations only.
Furthermore, signals should not be scaled simply to a certain
range. For all inputs (or outputs), a certain change should
be of the same importance for the model.
Approximation Properties:
* Guaranteed stability of reduced models
* Lower guaranteed error bound
* Guaranteed a priori error bound
*References*
[1] Enns, D. 'Model reduction with balanced realizations: An error
bound and a frequency weighted generalization'. Proc. 23-th CDC,
Las Vegas, pp. 127-132, 1984.
[2] Lin, C.-A. and Chiu, T.-Y. 'Model reduction via
frequency-weighted balanced realization'. Control Theory and
Advanced Technology, vol. 8, pp. 341-351, 1992.
[3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. 'New results
on frequency weighted balanced reduction technique'. Proc. ACC,
Seattle, Washington, pp. 4004-4009, 1995.
[4] Varga, A. and Anderson, B.D.O. 'Square-root balancing-free
methods for the frequency-weighted balancing related model
reduction'. (report in preparation)
*Algorithm*
Uses SLICOT AB09ID by courtesy of NICONET e.V.
(http://www.slicot.org)
# name: <cell-element>
# type: sq_string
# elements: 1
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Model order reduction by frequency weighted Balanced Truncation
Approximation (B
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# type: sq_string
# elements: 1
# length: 4
care
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# type: sq_string
# elements: 1
# length: 1626
-- Function File: [X, L, G] = care (A, B, Q, R)
-- Function File: [X, L, G] = care (A, B, Q, R, S)
-- Function File: [X, L, G] = care (A, B, Q, R, [], E)
-- Function File: [X, L, G] = care (A, B, Q, R, S, E)
Solve continuous-time algebraic Riccati equation (ARE).
*Inputs*
A
Real matrix (n-by-n).
B
Real matrix (n-by-m).
Q
Real matrix (n-by-n).
R
Real matrix (m-by-m).
S
Optional real matrix (n-by-m). If S is not specified, a zero
matrix is assumed.
E
Optional descriptor matrix (n-by-n). If E is not specified,
an identity matrix is assumed.
*Outputs*
X
Unique stabilizing solution of the continuous-time Riccati
equation (n-by-n).
L
Closed-loop poles (n-by-1).
G
Corresponding gain matrix (m-by-n).
*Equations*
-1
A'X + XA - XB R B'X + Q = 0
-1
A'X + XA - (XB + S) R (B'X + S') + Q = 0
-1
G = R B'X
-1
G = R (B'X + S')
L = eig (A - B*G)
-1
A'XE + E'XA - E'XB R B'XE + Q = 0
-1
A'XE + E'XA - (E'XB + S) R (B'XE + S') + Q = 0
-1
G = R B'XE
-1
G = R (B'XE + S)
L = eig (A - B*G, E)
*Algorithm*
Uses SLICOT SB02OD and SG02AD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: dare, lqr, dlqr, kalman.
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# length: 55
Solve continuous-time algebraic Riccati equation (ARE).
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# elements: 1
# length: 8
cfconred
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# elements: 1
# length: 4360
-- Function File: [KR, INFO] = cfconred (G, F, L, ...)
-- Function File: [KR, INFO] = cfconred (G, F, L, NCR, ...)
-- Function File: [KR, INFO] = cfconred (G, F, L, OPT, ...)
-- Function File: [KR, INFO] = cfconred (G, F, L, NCR, OPT, ...)
Reduction of state-feedback-observer based controller by coprime
factorization (CF). Given a plant G, state feedback gain F and full
observer gain L, determine a reduced order controller KR.
*Inputs*
G
LTI model of the open-loop plant (A,B,C,D). It has m inputs, p
outputs and n states.
F
Stabilizing state feedback matrix (m-by-n).
L
Stabilizing observer gain matrix (n-by-p).
NCR
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically according
to the description of key 'ORDER'.
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
KR
State-space model of reduced order controller.
INFO
Struct containing additional information.
INFO.HSV
The Hankel singular values of the extended system?!?.
The N Hankel singular values are ordered decreasingly.
INFO.NCR
The order of the obtained reduced order controller KR.
*Option Keys and Values*
'ORDER', 'NCR'
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically such that
states with Hankel singular values INFO.HSV > TOL1 are
retained.
'METHOD'
Order reduction approach to be used as follows:
'SR-BTA', 'B'
Use the square-root Balance & Truncate method.
'BFSR-BTA', 'F'
Use the balancing-free square-root Balance & Truncate
method. Default method.
'SR-SPA', 'S'
Use the square-root Singular Perturbation Approximation
method.
'BFSR-SPA', 'P'
Use the balancing-free square-root Singular Perturbation
Approximation method.
'CF'
Specifies whether left or right coprime factorization is to be
used as follows:
'LEFT', 'L'
Use left coprime factorization. Default method.
'RIGHT', 'R'
Use right coprime factorization.
'FEEDBACK'
Specifies whether F and L are fed back positively or
negatively:
'+'
A+BK and A+LC are both Hurwitz matrices.
'-'
A-BK and A-LC are both Hurwitz matrices. Default value.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of the reduced system. For model
reduction, the recommended value of TOL1 is c*info.hsv(1),
where c lies in the interval [0.00001, 0.001]. Default value
is n*eps*info.hsv(1). If 'ORDER' is specified, the value of
TOL1 is ignored.
'TOL2'
The tolerance for determining the order of a minimal
realization of the coprime factorization controller. TOL2 <=
TOL1. If not specified, n*eps*info.hsv(1) is chosen.
'EQUIL', 'SCALE'
Boolean indicating whether equilibration (scaling) should be
performed on system G prior to order reduction. Default value
is true if 'G.scaled == false' and false if 'G.scaled ==
true'. Note that for MIMO models, proper scaling of both
inputs and outputs is of utmost importance. The input and
output scaling can *not* be done by the equilibration option
or the 'prescale' function because these functions perform
state transformations only. Furthermore, signals should not
be scaled simply to a certain range. For all inputs (or
outputs), a certain change should be of the same importance
for the model.
*Algorithm*
Uses SLICOT SB16BD by courtesy of NICONET e.V.
(http://www.slicot.org)
# name: <cell-element>
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Reduction of state-feedback-observer based controller by coprime
factorization (
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# elements: 1
# length: 5
covar
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# type: sq_string
# elements: 1
# length: 319
-- Function File: [P, Q] = covar (SYS, W)
Return the steady-state covariance.
*Inputs*
SYS
LTI model.
W
Intensity of Gaussian white noise inputs which drive SYS.
*Outputs*
P
Output covariance.
Q
State covariance.
See also: lyap, dlyap.
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Return the steady-state covariance.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
ctrb
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# type: sq_string
# elements: 1
# length: 379
-- Function File: CO = ctrb (SYS)
-- Function File: CO = ctrb (A, B)
Return controllability matrix.
*Inputs*
SYS
LTI model.
A
State matrix (n-by-n).
B
Input matrix (n-by-m).
*Outputs*
CO
Controllability matrix.
*Equation*
2 n-1
Co = [ B AB A B ... A B ]
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 30
Return controllability matrix.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
ctrbf
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# type: sq_string
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# length: 958
-- Function File: [SYSBAR, T, K] = ctrbf (SYS)
-- Function File: [SYSBAR, T, K] = ctrbf (SYS, TOL)
-- Function File: [ABAR, BBAR, CBAR, T, K] = ctrbf (A, B, C)
-- Function File: [ABAR, BBAR, CBAR, T, K] = ctrbf (A, B, C, TOL)
If Co=ctrb(A,B) has rank r <= n = SIZE(A,1), then there is a
similarity transformation Tc such that Tc = [t1 t2] where t1 is the
controllable subspace and t2 is orthogonal to t1
Abar = Tc \\ A * Tc , Bbar = Tc \\ B , Cbar = C * Tc
and the transformed system has the form
| Ac A12| | Bc |
Abar = |----------|, Bbar = | ---|, Cbar = [Cc | Cnc].
| 0 Anc| | 0 |
where (Ac,Bc) is controllable, and Cc(sI-Ac)^(-1)Bc =
C(sI-A)^(-1)B. and the system is stabilizable if Anc has no
eigenvalues in the right half plane. The last output K is a vector
of length n containing the number of controllable states.
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If Co=ctrb(A,B) has rank r <= n = SIZE(A,1), then there is a similarity
transfor
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# elements: 1
# length: 4
dare
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-- Function File: [X, L, G] = dare (A, B, Q, R)
-- Function File: [X, L, G] = dare (A, B, Q, R, S)
-- Function File: [X, L, G] = dare (A, B, Q, R, [], E)
-- Function File: [X, L, G] = dare (A, B, Q, R, S, E)
Solve discrete-time algebraic Riccati equation (ARE).
*Inputs*
A
Real matrix (n-by-n).
B
Real matrix (n-by-m).
Q
Real matrix (n-by-n).
R
Real matrix (m-by-m).
S
Optional real matrix (n-by-m). If S is not specified, a zero
matrix is assumed.
E
Optional descriptor matrix (n-by-n). If E is not specified,
an identity matrix is assumed.
*Outputs*
X
Unique stabilizing solution of the discrete-time Riccati
equation (n-by-n).
L
Closed-loop poles (n-by-1).
G
Corresponding gain matrix (m-by-n).
*Equations*
-1
A'XA - X - A'XB (B'XB + R) B'XA + Q = 0
-1
A'XA - X - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0
-1
G = (B'XB + R) B'XA
-1
G = (B'XB + R) (B'XA + S')
L = eig (A - B*G)
-1
A'XA - E'XE - A'XB (B'XB + R) B'XA + Q = 0
-1
A'XA - E'XE - (A'XB + S) (B'XB + R) (B'XA + S') + Q = 0
-1
G = (B'XB + R) B'XA
-1
G = (B'XB + R) (B'XA + S')
L = eig (A - B*G, E)
*Algorithm*
Uses SLICOT SB02OD and SG02AD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: care, lqr, dlqr, kalman.
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Solve discrete-time algebraic Riccati equation (ARE).
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# type: sq_string
# elements: 1
# length: 6
db2mag
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# length: 275
-- Function File: MAG = db2mag (DB)
Convert Decibels (dB) to Magnitude.
*Inputs*
DB
Decibel (dB) value(s). Both real-valued scalars and matrices
are accepted.
*Outputs*
MAG
Magnitude value(s).
See also: mag2db.
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Convert Decibels (dB) to Magnitude.
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# type: sq_string
# elements: 1
# length: 4
dlqe
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# type: sq_string
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# length: 1696
-- Function File: [M, P, Z, E] = dlqe (A, G, C, Q, R)
-- Function File: [M, P, Z, E] = dlqe (A, G, C, Q, R, S)
-- Function File: [M, P, Z, E] = dlqe (A, [], C, Q, R)
-- Function File: [M, P, Z, E] = dlqe (A, [], C, Q, R, S)
Kalman filter for discrete-time systems.
x[k] = Ax[k] + Bu[k] + Gw[k] (State equation)
y[k] = Cx[k] + Du[k] + v[k] (Measurement Equation)
E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S
*Inputs*
A
State transition matrix of discrete-time system (n-by-n).
G
Process noise matrix of discrete-time system (n-by-g). If G
is empty '[]', an identity matrix is assumed.
C
Measurement matrix of discrete-time system (p-by-n).
Q
Process noise covariance matrix (g-by-g).
R
Measurement noise covariance matrix (p-by-p).
S
Optional cross term covariance matrix (g-by-p), s = cov(w,v).
If S is empty '[]' or not specified, a zero matrix is assumed.
*Outputs*
M
Kalman filter gain matrix (n-by-p).
P
Unique stabilizing solution of the discrete-time Riccati
equation (n-by-n). Symmetric matrix.
Z
Error covariance (n-by-n), cov(x(k|k)-x)
E
Closed-loop poles (n-by-1).
*Equations*
x[k|k] = x[k|k-1] + M(y[k] - Cx[k|k-1] - Du[k])
x[k+1|k] = Ax[k|k] + Bu[k] for S=0
x[k+1|k] = Ax[k|k] + Bu[k] + G*S*(C*P*C' + R)^-1*(y[k] - C*x[k|k-1]) for non-zero S
E = eig(A - A*M*C) for S=0
E = eig(A - A*M*C - G*S*(C*P*C' + Rv)^-1*C) for non-zero S
See also: dare, care, dlqr, lqr, lqe.
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Kalman filter for discrete-time systems.
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# elements: 1
# length: 4
dlqr
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# type: sq_string
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# length: 1374
-- Function File: [G, X, L] = dlqr (SYS, Q, R)
-- Function File: [G, X, L] = dlqr (SYS, Q, R, S)
-- Function File: [G, X, L] = dlqr (A, B, Q, R)
-- Function File: [G, X, L] = dlqr (A, B, Q, R, S)
-- Function File: [G, X, L] = dlqr (A, B, Q, R, [], E)
-- Function File: [G, X, L] = dlqr (A, B, Q, R, S, E)
Linear-quadratic regulator for discrete-time systems.
*Inputs*
SYS
Continuous or discrete-time LTI model (p-by-m, n states).
A
State transition matrix of discrete-time system (n-by-n).
B
Input matrix of discrete-time system (n-by-m).
Q
State weighting matrix (n-by-n).
R
Input weighting matrix (m-by-m).
S
Optional cross term matrix (n-by-m). If S is not specified, a
zero matrix is assumed.
E
Optional descriptor matrix (n-by-n). If E is not specified,
an identity matrix is assumed.
*Outputs*
G
State feedback matrix (m-by-n).
X
Unique stabilizing solution of the discrete-time Riccati
equation (n-by-n).
L
Closed-loop poles (n-by-1).
*Equations*
x[k+1] = A x[k] + B u[k], x[0] = x0
inf
J(x0) = SUM (x' Q x + u' R u + 2 x' S u)
k=0
L = eig (A - B*G)
See also: dare, care, lqr.
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Linear-quadratic regulator for discrete-time systems.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
dlyap
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# type: sq_string
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# length: 525
-- Function File: X = dlyap (A, B)
-- Function File: X = dlyap (A, B, C)
-- Function File: X = dlyap (A, B, [], E)
Solve discrete-time Lyapunov or Sylvester equations.
*Equations*
AXA' - X + B = 0 (Lyapunov Equation)
AXB' - X + C = 0 (Sylvester Equation)
AXA' - EXE' + B = 0 (Generalized Lyapunov Equation)
*Algorithm*
Uses SLICOT SB03MD, SB04QD and SG03AD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: dlyapchol, lyap, lyapchol.
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# elements: 1
# length: 52
Solve discrete-time Lyapunov or Sylvester equations.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
dlyapchol
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# type: sq_string
# elements: 1
# length: 476
-- Function File: U = dlyapchol (A, B)
-- Function File: U = dlyapchol (A, B, E)
Compute Cholesky factor of discrete-time Lyapunov equations.
*Equations*
A U' U A' - U' U + B B' = 0 (Lyapunov Equation)
A U' U A' - E U' U E' + B B' = 0 (Generalized Lyapunov Equation)
*Algorithm*
Uses SLICOT SB03OD and SG03BD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: dlyap, lyap, lyapchol.
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Compute Cholesky factor of discrete-time Lyapunov equations.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 3
dss
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# length: 2196
-- Function File: SYS = dss (SYS)
-- Function File: SYS = dss (D, ...)
-- Function File: SYS = dss (A, B, C, D, E, ...)
-- Function File: SYS = dss (A, B, C, D, E, TSAM, ...)
Create or convert to descriptor state-space model.
*Inputs*
SYS
LTI model to be converted to state-space.
A
State matrix (n-by-n).
B
Input matrix (n-by-m).
C
Output matrix (p-by-n).
D
Feedthrough matrix (p-by-m).
E
Descriptor matrix (n-by-n).
TSAM
Sampling time in seconds. If TSAM is not specified, a
continuous-time model is assumed.
...
Optional pairs of properties and values. Type 'set (dss)' for
more information.
*Outputs*
SYS
Descriptor state-space model.
*Option Keys and Values*
'A', 'B', 'C', 'D', 'E'
State-space matrices. See 'Inputs' for details.
'STNAME'
The name of the states in SYS. Cell vector containing strings
for each state. Default names are '{'x1', 'x2', ...}'
'SCALED'
Logical. If set to true, no automatic scaling is used, e.g.
for frequency response plots.
'TSAM'
Sampling time. See 'Inputs' for details.
'INNAME'
The name of the input channels in SYS. Cell vector of length
m containing strings. Default names are '{'u1', 'u2', ...}'
'OUTNAME'
The name of the output channels in SYS. Cell vector of length
p containing strings. Default names are '{'y1', 'y2', ...}'
'INGROUP'
Struct with input group names as field names and vectors of
input indices as field values. Default is an empty struct.
'OUTGROUP'
Struct with output group names as field names and vectors of
output indices as field values. Default is an empty struct.
'NAME'
String containing the name of the model.
'NOTES'
String or cell of string containing comments.
'USERDATA'
Any data type.
*Equations*
.
E x = A x + B u
y = C x + D u
See also: ss, tf.
# name: <cell-element>
# type: sq_string
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# length: 50
Create or convert to descriptor state-space model.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
estim
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1085
-- Function File: EST = estim (SYS, L)
-- Function File: EST = estim (SYS, L, SENSORS, KNOWN)
Return state estimator for a given estimator gain.
*Inputs*
SYS
LTI model.
L
State feedback matrix.
SENSORS
Indices of measured output signals y from SYS. If omitted,
all outputs are measured.
KNOWN
Indices of known input signals u (deterministic) to SYS. All
other inputs to SYS are assumed stochastic (w). If argument
KNOWN is omitted, no inputs u are known.
*Outputs*
EST
State-space model of estimator.
*Block Diagram*
u +-------+ ^
+---------------------------->| |-------> y
| +-------+ + y | est | ^
u ----+--->| |----->(+)------>| |-------> x
| sys | ^ + +-------+
w -------->| | |
+-------+ | v
See also: kalman, lqe, place.
# name: <cell-element>
# type: sq_string
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Return state estimator for a given estimator gain.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
filt
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2804
-- Function File: SYS = filt (NUM, DEN, ...)
-- Function File: SYS = filt (NUM, DEN, TSAM, ...)
Create discrete-time transfer function model from data in DSP
format.
*Inputs*
NUM
Numerator or cell of numerators. Each numerator must be a row
vector containing the coefficients of the polynomial in
ascending powers of z^-1. num{i,j} contains the numerator
polynomial from input j to output i. In the SISO case, a
single vector is accepted as well.
DEN
Denominator or cell of denominators. Each denominator must be
a row vector containing the coefficients of the polynomial in
ascending powers of z^-1. den{i,j} contains the denominator
polynomial from input j to output i. In the SISO case, a
single vector is accepted as well.
TSAM
Sampling time in seconds. If TSAM is not specified, default
value -1 (unspecified) is taken.
...
Optional pairs of properties and values. Type 'set (filt)'
for more information.
*Outputs*
SYS
Discrete-time transfer function model.
*Option Keys and Values*
'NUM'
Numerator. See 'Inputs' for details.
'DEN'
Denominator. See 'Inputs' for details.
'TFVAR'
String containing the transfer function variable.
'INV'
Logical. True for negative powers of the transfer function
variable.
'TSAM'
Sampling time. See 'Inputs' for details.
'INNAME'
The name of the input channels in SYS. Cell vector of length
m containing strings. Default names are '{'u1', 'u2', ...}'
'OUTNAME'
The name of the output channels in SYS. Cell vector of length
p containing strings. Default names are '{'y1', 'y2', ...}'
'INGROUP'
Struct with input group names as field names and vectors of
input indices as field values. Default is an empty struct.
'OUTGROUP'
Struct with output group names as field names and vectors of
output indices as field values. Default is an empty struct.
'NAME'
String containing the name of the model.
'NOTES'
String or cell of string containing comments.
'USERDATA'
Any data type.
*Example*
3 z^-1
H(z^-1) = -------------------
1 + 4 z^-1 + 2 z^-2
octave:1> H = filt ([0, 3], [1, 4, 2])
Transfer function 'H' from input 'u1' to output ...
3 z^-1
y1: -------------------
1 + 4 z^-1 + 2 z^-2
Sampling time: unspecified
Discrete-time model.
See also: tf.
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Create discrete-time transfer function model from data in DSP format.
# name: <cell-element>
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# elements: 1
# length: 6
fitfrd
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# type: sq_string
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# length: 1169
-- Function File: [SYS, N] = fitfrd (DAT, N)
-- Function File: [SYS, N] = fitfrd (DAT, N, FLAG)
Fit frequency response data with a state-space system. If
requested, the returned system is stable and minimum-phase.
*Inputs*
DAT
LTI model containing frequency response data of a SISO system.
N
The desired order of the system to be fitted. 'n <=
length(dat.w)'.
FLAG
The flag controls whether the returned system is stable and
minimum-phase.
0
The system zeros and poles are not constrained. Default
value.
1
The system zeros and poles will have negative real parts
in the continuous-time case, or moduli less than 1 in the
discrete-time case.
*Outputs*
SYS
State-space model of order N, fitted to frequency response
data DAT.
N
The order of the obtained system. The value of N could only
be modified if inputs 'n > 0' and 'flag = 1'.
*Algorithm*
Uses SLICOT SB10YD by courtesy of NICONET e.V.
(http://www.slicot.org)
# name: <cell-element>
# type: sq_string
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# length: 54
Fit frequency response data with a state-space system.
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# type: sq_string
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# length: 10
fwcfconred
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# length: 3225
-- Function File: [KR, INFO] = fwcfconred (G, F, L, ...)
-- Function File: [KR, INFO] = fwcfconred (G, F, L, NCR, ...)
-- Function File: [KR, INFO] = fwcfconred (G, F, L, OPT, ...)
-- Function File: [KR, INFO] = fwcfconred (G, F, L, NCR, OPT, ...)
Reduction of state-feedback-observer based controller by
frequency-weighted coprime factorization (FW CF). Given a plant G,
state feedback gain F and full observer gain L, determine a reduced
order controller KR by using stability enforcing frequency weights.
*Inputs*
G
LTI model of the open-loop plant (A,B,C,D). It has m inputs, p
outputs and n states.
F
Stabilizing state feedback matrix (m-by-n).
L
Stabilizing observer gain matrix (n-by-p).
NCR
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically according
to the description of key 'ORDER'.
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
KR
State-space model of reduced order controller.
INFO
Struct containing additional information.
INFO.HSV
The Hankel singular values of the extended system?!?.
The N Hankel singular values are ordered decreasingly.
INFO.NCR
The order of the obtained reduced order controller KR.
*Option Keys and Values*
'ORDER', 'NCR'
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically such that
states with Hankel singular values INFO.HSV > TOL1 are
retained.
'METHOD'
Order reduction approach to be used as follows:
'SR', 'B'
Use the square-root Balance & Truncate method.
'BFSR', 'F'
Use the balancing-free square-root Balance & Truncate
method. Default method.
'CF'
Specifies whether left or right coprime factorization is to be
used as follows:
'LEFT', 'L'
Use left coprime factorization.
'RIGHT', 'R'
Use right coprime factorization. Default method.
'FEEDBACK'
Specifies whether F and L are fed back positively or
negatively:
'+'
A+BK and A+LC are both Hurwitz matrices.
'-'
A-BK and A-LC are both Hurwitz matrices. Default value.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of the reduced system. For model
reduction, the recommended value of TOL1 is c*info.hsv(1),
where c lies in the interval [0.00001, 0.001]. Default value
is n*eps*info.hsv(1). If 'ORDER' is specified, the value of
TOL1 is ignored.
*Algorithm*
Uses SLICOT SB16CD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Reduction of state-feedback-observer based controller by
frequency-weighted copr
# name: <cell-element>
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# length: 6
gensig
# name: <cell-element>
# type: sq_string
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# length: 802
-- Function File: [U, T] = gensig (SIGTYPE, TAU)
-- Function File: [U, T] = gensig (SIGTYPE, TAU, TFINAL)
-- Function File: [U, T] = gensig (SIGTYPE, TAU, TFINAL, TSAM)
Generate periodic signal. Useful in combination with lsim.
*Inputs*
SIGTYPE = "SIN"
Sine wave.
SIGTYPE = "COS"
Cosine wave.
SIGTYPE = "SQUARE"
Square wave.
SIGTYPE = "PULSE"
Periodic pulse.
TAU
Duration of one period in seconds.
TFINAL
Optional duration of the signal in seconds. Default duration
is 5 periods.
TSAM
Optional sampling time in seconds. Default spacing is tau/64.
*Outputs*
U
Vector of signal values.
T
Time vector of the signal.
See also: lsim.
# name: <cell-element>
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Generate periodic signal.
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# type: sq_string
# elements: 1
# length: 4
gram
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-- Function File: W = gram (SYS, MODE)
-- Function File: WC = gram (A, B)
'gram (SYS, "c")' returns the controllability gramian of the
(continuous- or discrete-time) system SYS. 'gram (SYS, "o")'
returns the observability gramian of the (continuous- or
discrete-time) system SYS. 'gram (A, B)' returns the
controllability gramian WC of the continuous-time system dx/dt = a
x + b u; i.e., WC satisfies a Wc + m Wc' + b b' = 0.
# name: <cell-element>
# type: sq_string
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'gram (SYS, "c")' returns the controllability gramian of the
(continuous- or dis
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h2syn
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# type: sq_string
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-- Function File: [K, N, GAMMA, INFO] = h2syn (P, NMEAS, NCON)
-- Function File: [K, N, GAMMA, INFO] = h2syn (P)
H-2 control synthesis for LTI plant.
*Inputs*
P
Generalized plant. Must be a proper/realizable LTI model. If
P is constructed with 'mktito' or 'augw', arguments NMEAS and
NCON can be omitted.
NMEAS
Number of measured outputs v. The last NMEAS outputs of P are
connected to the inputs of controller K. The remaining
outputs z (indices 1 to p-nmeas) are used to calculate the H-2
norm.
NCON
Number of controlled inputs u. The last NCON inputs of P are
connected to the outputs of controller K. The remaining
inputs w (indices 1 to m-ncon) are excited by a harmonic test
signal.
*Outputs*
K
State-space model of the H-2 optimal controller.
N
State-space model of the lower LFT of P and K.
INFO
Structure containing additional information.
INFO.GAMMA
H-2 norm of N.
INFO.RCOND
Vector RCOND contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and estimates
of the reciprocal condition numbers of the Riccati equations
which have to be solved during the computation of the
controller K. For details, see the description of the
corresponding SLICOT routine.
*Block Diagram*
gamma = min||N(K)|| N = lft (P, K)
K 2
+--------+
w ----->| |-----> z
| P(s) |
u +---->| |-----+ v
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
+--------+
w ----->| N(s) |-----> z
+--------+
*Algorithm*
Uses SLICOT SB10HD and SB10ED by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: augw, lqr, dlqr, kalman.
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H-2 control synthesis for LTI plant.
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hinfsyn
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-- Function File: [K, N, GAMMA, INFO] = hinfsyn (P, NMEAS, NCON)
-- Function File: [K, N, GAMMA, INFO] = hinfsyn (P, NMEAS, NCON, ...)
-- Function File: [K, N, GAMMA, INFO] = hinfsyn (P, NMEAS, NCON, OPT,
...)
-- Function File: [K, N, GAMMA, INFO] = hinfsyn (P, ...)
-- Function File: [K, N, GAMMA, INFO] = hinfsyn (P, OPT, ...)
H-infinity control synthesis for LTI plant.
*Inputs*
P
Generalized plant. Must be a proper/realizable LTI model. If
P is constructed with 'mktito' or 'augw', arguments NMEAS and
NCON can be omitted.
NMEAS
Number of measured outputs v. The last NMEAS outputs of P are
connected to the inputs of controller K. The remaining
outputs z (indices 1 to p-nmeas) are used to calculate the
H-infinity norm.
NCON
Number of controlled inputs u. The last NCON inputs of P are
connected to the outputs of controller K. The remaining
inputs w (indices 1 to m-ncon) are excited by a harmonic test
signal.
...
Optional pairs of keys and values. ''key1', value1, 'key2',
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
K
State-space model of the H-infinity (sub-)optimal controller.
N
State-space model of the lower LFT of P and K.
INFO
Structure containing additional information.
INFO.GAMMA
L-infinity norm of N.
INFO.RCOND
Vector RCOND contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and estimates
of the reciprocal condition numbers of the Riccati equations
which have to be solved during the computation of the
controller K. For details, see the description of the
corresponding SLICOT routine.
*Option Keys and Values*
'METHOD'
String specifying the desired kind of controller:
'OPTIMAL', 'OPT', 'O'
Compute optimal controller using gamma iteration.
Default selection for compatibility reasons.
'SUBOPTIMAL', 'SUB', 'S'
Compute (sub-)optimal controller. For stability reasons,
suboptimal controllers are to be preferred over optimal
ones.
'GMAX'
The maximum value of the H-infinity norm of N. It is assumed
that GMAX is sufficiently large so that the controller is
admissible. Default value is 1e15.
'GMIN'
Initial lower bound for gamma iteration. Default value is 0.
GMIN is only meaningful for optimal discrete-time controllers.
'TOLGAM'
Tolerance used for controlling the accuracy of GAMMA and its
distance to the estimated minimal possible value of GAMMA.
Default value is 0.01. If TOLGAM = 0, then a default value
equal to 'sqrt(eps)' is used, where EPS is the relative
machine precision. For suboptimal controllers, TOLGAM is
ignored.
'ACTOL'
Upper bound for the poles of the closed-loop system N used for
determining if it is stable. ACTOL >= 0 for stable systems.
For suboptimal controllers, ACTOL is ignored.
*Block Diagram*
gamma = min||N(K)|| N = lft (P, K)
K inf
+--------+
w ----->| |-----> z
| P(s) |
u +---->| |-----+ v
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
+--------+
w ----->| N(s) |-----> z
+--------+
*Algorithm*
Uses SLICOT SB10FD, SB10DD and SB10AD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: augw, mixsyn.
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H-infinity control synthesis for LTI plant.
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hnamodred
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-- Function File: [GR, INFO] = hnamodred (G, ...)
-- Function File: [GR, INFO] = hnamodred (G, NR, ...)
-- Function File: [GR, INFO] = hnamodred (G, OPT, ...)
-- Function File: [GR, INFO] = hnamodred (G, NR, OPT, ...)
Model order reduction by frequency weighted optimal Hankel-norm
(HNA) method. The aim of model reduction is to find an LTI system
GR of order NR (nr < n) such that the input-output behaviour of GR
approximates the one from original system G.
HNA is an absolute error method which tries to minimize
||G-Gr|| = min
H
||V (G-Gr) W|| = min
H
where V and W denote output and input weightings.
*Inputs*
G
LTI model to be reduced.
NR
The desired order of the resulting reduced order system GR.
If not specified, NR is chosen automatically according to the
description of key "ORDER".
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
GR
Reduced order state-space model.
INFO
Struct containing additional information.
INFO.N
The order of the original system G.
INFO.NS
The order of the ALPHA-stable subsystem of the original
system G.
INFO.HSV
The Hankel singular values corresponding to the
projection 'op(V)*G1*op(W)', where G1 denotes the
ALPHA-stable part of the original system G. The NS
Hankel singular values are ordered decreasingly.
INFO.NU
The order of the ALPHA-unstable subsystem of both the
original system G and the reduced-order system GR.
INFO.NR
The order of the obtained reduced order system GR.
*Option Keys and Values*
'ORDER', 'NR'
The desired order of the resulting reduced order system GR.
If not specified, NR is the sum of INFO.NU and the number of
Hankel singular values greater than 'max(tol1,
ns*eps*info.hsv(1)';
'METHOD'
Specifies the computational approach to be used. Valid values
corresponding to this key are:
'DESCRIPTOR'
Use the inverse free descriptor system approach.
'STANDARD'
Use the inversion based standard approach.
'AUTO'
Switch automatically to the inverse free descriptor
approach in case of badly conditioned feedthrough
matrices in V or W. Default method.
'LEFT', 'V'
LTI model of the left/output frequency weighting. The
weighting must be antistable.
|| V (G-Gr) . || = min
H
'RIGHT', 'W'
LTI model of the right/input frequency weighting. The
weighting must be antistable.
|| . (G-Gr) W || = min
H
'LEFT-INV', 'INV-V'
LTI model of the left/output frequency weighting. The
weighting must have only antistable zeros.
|| inv(V) (G-Gr) . || = min
H
'RIGHT-INV', 'INV-W'
LTI model of the right/input frequency weighting. The
weighting must have only antistable zeros.
|| . (G-Gr) inv(W) || = min
H
'LEFT-CONJ', 'CONJ-V'
LTI model of the left/output frequency weighting. The
weighting must be stable.
|| V (G-Gr) . || = min
H
'RIGHT-CONJ', 'CONJ-W'
LTI model of the right/input frequency weighting. The
weighting must be stable.
|| . (G-Gr) W || = min
H
'LEFT-CONJ-INV', 'CONJ-INV-V'
LTI model of the left/output frequency weighting. The
weighting must be minimum-phase.
|| V (G-Gr) . || = min
H
'RIGHT-CONJ-INV', 'CONJ-INV-W'
LTI model of the right/input frequency weighting. The
weighting must be minimum-phase.
|| . (G-Gr) W || = min
H
'ALPHA'
Specifies the ALPHA-stability boundary for the eigenvalues of
the state dynamics matrix G.A. For a continuous-time system,
ALPHA <= 0 is the boundary value for the real parts of
eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1
represents the boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and 1 for
discrete-time systems.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of the reduced model. For model
reduction, the recommended value of TOL1 is c*info.hsv(1),
where c lies in the interval [0.00001, 0.001]. TOL1 < 1. If
'ORDER' is specified, the value of TOL1 is ignored.
'TOL2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given model. TOL2
<= TOL1 < 1. If not specified, ns*eps*info.hsv(1) is chosen.
'EQUIL', 'SCALE'
Boolean indicating whether equilibration (scaling) should be
performed on system G prior to order reduction. Default value
is true if 'G.scaled == false' and false if 'G.scaled ==
true'. Note that for MIMO models, proper scaling of both
inputs and outputs is of utmost importance. The input and
output scaling can *not* be done by the equilibration option
or the 'prescale' function because these functions perform
state transformations only. Furthermore, signals should not
be scaled simply to a certain range. For all inputs (or
outputs), a certain change should be of the same importance
for the model.
Approximation Properties:
* Guaranteed stability of reduced models
* Lower guaranteed error bound
* Guaranteed a priori error bound
*Algorithm*
Uses SLICOT AB09JD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Model order reduction by frequency weighted optimal Hankel-norm (HNA)
method.
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hsvd
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-- Function File: HSV = hsvd (SYS)
-- Function File: HSV = hsvd (SYS, "OFFSET", OFFSET)
-- Function File: HSV = hsvd (SYS, "ALPHA", ALPHA)
Hankel singular values of the stable part of an LTI model. If no
output arguments are given, the Hankel singular values are
displayed in a plot.
*Algorithm*
Uses SLICOT AB13AD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Hankel singular values of the stable part of an LTI model.
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impulse
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-- Function File: impulse (SYS)
-- Function File: impulse (SYS1, SYS2, ..., SYSN)
-- Function File: impulse (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: impulse (SYS1, ..., T)
-- Function File: impulse (SYS1, ..., TFINAL)
-- Function File: impulse (SYS1, ..., TFINAL, DT)
-- Function File: [Y, T, X] = impulse (SYS)
-- Function File: [Y, T, X] = impulse (SYS, T)
-- Function File: [Y, T, X] = impulse (SYS, TFINAL)
-- Function File: [Y, T, X] = impulse (SYS, TFINAL, DT)
Impulse response of LTI system. If no output arguments are given,
the response is printed on the screen.
*Inputs*
SYS
LTI model.
T
Time vector. Should be evenly spaced. If not specified, it
is calculated by the poles of the system to reflect adequately
the response transients.
TFINAL
Optional simulation horizon. If not specified, it is
calculated by the poles of the system to reflect adequately
the response transients.
DT
Optional sampling time. Be sure to choose it small enough to
capture transient phenomena. If not specified, it is
calculated by the poles of the system.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
Y
Output response array. Has as many rows as time samples
(length of t) and as many columns as outputs.
T
Time row vector.
X
State trajectories array. Has 'length (t)' rows and as many
columns as states.
See also: initial, lsim, step.
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Impulse response of LTI system.
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initial
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-- Function File: initial (SYS, X0)
-- Function File: initial (SYS1, SYS2, ..., SYSN, X0)
-- Function File: initial (SYS1, 'STYLE1', ..., SYSN, 'STYLEN', X0)
-- Function File: initial (SYS1, ..., X0, T)
-- Function File: initial (SYS1, ..., X0, TFINAL)
-- Function File: initial (SYS1, ..., X0, TFINAL, DT)
-- Function File: [Y, T, X] = initial (SYS, X0)
-- Function File: [Y, T, X] = initial (SYS, X0, T)
-- Function File: [Y, T, X] = initial (SYS, X0, TFINAL)
-- Function File: [Y, T, X] = initial (SYS, X0, TFINAL, DT)
Initial condition response of state-space model. If no output
arguments are given, the response is printed on the screen.
*Inputs*
SYS
State-space model.
X0
Vector of initial conditions for each state.
T
Optional time vector. Should be evenly spaced. If not
specified, it is calculated by the poles of the system to
reflect adequately the response transients.
TFINAL
Optional simulation horizon. If not specified, it is
calculated by the poles of the system to reflect adequately
the response transients.
DT
Optional sampling time. Be sure to choose it small enough to
capture transient phenomena. If not specified, it is
calculated by the poles of the system.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
Y
Output response array. Has as many rows as time samples
(length of t) and as many columns as outputs.
T
Time row vector.
X
State trajectories array. Has 'length (t)' rows and as many
columns as states.
*Example*
.
Continuous Time: x = A x , y = C x , x(0) = x0
Discrete Time: x[k+1] = A x[k] , y[k] = C x[k] , x[0] = x0
See also: impulse, lsim, step.
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Initial condition response of state-space model.
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isctrb
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-- Function File: [BOOL, NCON] = isctrb (SYS)
-- Function File: [BOOL, NCON] = isctrb (SYS, TOL)
-- Function File: [BOOL, NCON] = isctrb (A, B)
-- Function File: [BOOL, NCON] = isctrb (A, B, E)
-- Function File: [BOOL, NCON] = isctrb (A, B, [], TOL)
-- Function File: [BOOL, NCON] = isctrb (A, B, E, TOL)
Logical check for system controllability. For numerical reasons,
'isctrb (sys)' should be used instead of 'rank (ctrb (sys))'.
*Inputs*
SYS
LTI model. Descriptor state-space models are possible. If
SYS is not a state-space model, it is converted to a minimal
state-space realization, so beware of pole-zero cancellations
which may lead to wrong results!
A
State matrix (n-by-n).
B
Input matrix (n-by-m).
E
Descriptor matrix (n-by-n). If E is empty '[]' or not
specified, an identity matrix is assumed.
TOL
Optional roundoff parameter. Default value is 0.
*Outputs*
BOOL = 0
System is not controllable.
BOOL = 1
System is controllable.
NCON
Number of controllable states.
*Algorithm*
Uses SLICOT AB01OD and TG01HD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: isobsv.
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Logical check for system controllability.
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isdetectable
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-- Function File: BOOL = isdetectable (SYS)
-- Function File: BOOL = isdetectable (SYS, TOL)
-- Function File: BOOL = isdetectable (A, C)
-- Function File: BOOL = isdetectable (A, C, E)
-- Function File: BOOL = isdetectable (A, C, [], TOL)
-- Function File: BOOL = isdetectable (A, C, E, TOL)
-- Function File: BOOL = isdetectable (A, C, [], [], DFLG)
-- Function File: BOOL = isdetectable (A, C, E, [], DFLG)
-- Function File: BOOL = isdetectable (A, C, [], TOL, DFLG)
-- Function File: BOOL = isdetectable (A, C, E, TOL, DFLG)
Logical test for system detectability. All unstable modes must be
observable or all unobservable states must be stable.
*Inputs*
SYS
LTI system.
A
State transition matrix.
C
Measurement matrix.
E
Descriptor matrix. If E is empty '[]' or not specified, an
identity matrix is assumed.
TOL
Optional tolerance for stability. Default value is 0.
DFLG = 0
Matrices (A, C) are part of a continuous-time system. Default
Value.
DFLG = 1
Matrices (A, C) are part of a discrete-time system.
*Outputs*
BOOL = 0
System is not detectable.
BOOL = 1
System is detectable.
*Algorithm*
Uses SLICOT AB01OD and TG01HD by courtesy of NICONET e.V.
(http://www.slicot.org) See 'isstabilizable' for description of
computational method.
See also: isstabilizable, isstable, isctrb, isobsv.
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Logical test for system detectability.
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isobsv
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-- Function File: [BOOL, NOBS] = isobsv (SYS)
-- Function File: [BOOL, NOBS] = isobsv (SYS, TOL)
-- Function File: [BOOL, NOBS] = isobsv (A, C)
-- Function File: [BOOL, NOBS] = isobsv (A, C, E)
-- Function File: [BOOL, NOBS] = isobsv (A, C, [], TOL)
-- Function File: [BOOL, NOBS] = isobsv (A, C, E, TOL)
Logical check for system observability. For numerical reasons,
'isobsv (sys)' should be used instead of 'rank (obsv (sys))'.
*Inputs*
SYS
LTI model. Descriptor state-space models are possible.
A
State matrix (n-by-n).
C
Measurement matrix (p-by-n).
E
Descriptor matrix (n-by-n). If E is empty '[]' or not
specified, an identity matrix is assumed.
TOL
Optional roundoff parameter. Default value is 0.
*Outputs*
BOOL = 0
System is not observable.
BOOL = 1
System is observable.
NOBS
Number of observable states.
*Algorithm*
Uses SLICOT AB01OD and TG01HD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: isctrb.
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Logical check for system observability.
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# length: 8
issample
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-- Function File: BOOL = issample (TS)
-- Function File: BOOL = issample (TS, FLG)
Return true if TS is a valid sampling time.
*Inputs*
TS
Alleged sampling time to be tested.
FLG = 1
Accept real scalars TS > 0. Default Value.
FLG = 0
Accept real scalars TS >= 0.
FLG = -1
Accept real scalars TS > 0 and TS == -1.
FLG = -10
Accept real scalars TS >= 0 and TS == -1.
FLG = -2
Accept real scalars TS >= 0, TS == -1 and TS == -2.
*Outputs*
BOOL
True if conditions are met and false otherwise.
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Return true if TS is a valid sampling time.
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isstabilizable
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-- Function File: BOOL = isstabilizable (SYS)
-- Function File: BOOL = isstabilizable (SYS, TOL)
-- Function File: BOOL = isstabilizable (A, B)
-- Function File: BOOL = isstabilizable (A, B, E)
-- Function File: BOOL = isstabilizable (A, B, [], TOL)
-- Function File: BOOL = isstabilizable (A, B, E, TOL)
-- Function File: BOOL = isstabilizable (A, B, [], [], DFLG)
-- Function File: BOOL = isstabilizable (A, B, E, [], DFLG)
-- Function File: BOOL = isstabilizable (A, B, [], TOL, DFLG)
-- Function File: BOOL = isstabilizable (A, B, E, TOL, DFLG)
Logical check for system stabilizability. All unstable modes must
be controllable or all uncontrollable states must be stable.
*Inputs*
SYS
LTI system. If SYS is not a state-space system, it is
converted to a minimal state-space realization, so beware of
pole-zero cancellations which may lead to wrong results!
A
State transition matrix.
B
Input matrix.
E
Descriptor matrix. If E is empty '[]' or not specified, an
identity matrix is assumed.
TOL
Optional tolerance for stability. Default value is 0.
DFLG = 0
Matrices (A, B) are part of a continuous-time system. Default
Value.
DFLG = 1
Matrices (A, B) are part of a discrete-time system.
*Outputs*
BOOL = 0
System is not stabilizable.
BOOL = 1
System is stabilizable.
*Algorithm*
Uses SLICOT AB01OD and TG01HD by courtesy of NICONET e.V.
(http://www.slicot.org)
* Calculate staircase form (SLICOT AB01OD)
* Extract unobservable part of state transition matrix
* Calculate eigenvalues of unobservable part
* Check whether
real (ev) < -tol*(1 + abs (ev)) continuous-time
abs (ev) < 1 - tol discrete-time
See also: isdetectable, isstable, isctrb, isobsv.
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Logical check for system stabilizability.
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kalman
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-- Function File: [EST, G, X] = kalman (SYS, Q, R)
-- Function File: [EST, G, X] = kalman (SYS, Q, R, S)
-- Function File: [EST, G, X] = kalman (SYS, Q, R, [], SENSORS, KNOWN)
-- Function File: [EST, G, X] = kalman (SYS, Q, R, S, SENSORS, KNOWN)
Design Kalman estimator for LTI systems.
*Inputs*
SYS
Nominal plant model.
Q
Covariance of white process noise.
R
Covariance of white measurement noise.
S
Optional cross term covariance. Default value is 0.
SENSORS
Indices of measured output signals y from SYS. If omitted,
all outputs are measured.
KNOWN
Indices of known input signals u (deterministic) to SYS. All
other inputs to SYS are assumed stochastic. If argument KNOWN
is omitted, no inputs u are known.
*Outputs*
EST
State-space model of the Kalman estimator.
G
Estimator gain.
X
Solution of the Riccati equation.
*Block Diagram*
u +-------+ ^
+---------------------------->| |-------> y
| +-------+ + y | est | ^
u ----+--->| |----->(+)------>| |-------> x
| sys | ^ + +-------+
w -------->| | |
+-------+ | v
Q = cov (w, w') R = cov (v, v') S = cov (w, v')
See also: care, dare, estim, lqr.
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Design Kalman estimator for LTI systems.
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lqe
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-- Function File: [L, P, E] = lqe (SYS, Q, R)
-- Function File: [L, P, E] = lqe (SYS, Q, R, S)
-- Function File: [L, P, E] = lqe (A, G, C, Q, R)
-- Function File: [L, P, E] = lqe (A, G, C, Q, R, S)
-- Function File: [L, P, E] = lqe (A, [], C, Q, R)
-- Function File: [L, P, E] = lqe (A, [], C, Q, R, S)
Kalman filter for continuous-time systems.
.
x = Ax + Bu + Gw (State equation)
y = Cx + Du + v (Measurement Equation)
E(w) = 0, E(v) = 0, cov(w) = Q, cov(v) = R, cov(w,v) = S
*Inputs*
SYS
Continuous or discrete-time LTI model (p-by-m, n states).
A
State matrix of continuous-time system (n-by-n).
G
Process noise matrix of continuous-time system (n-by-g). If G
is empty '[]', an identity matrix is assumed.
C
Measurement matrix of continuous-time system (p-by-n).
Q
Process noise covariance matrix (g-by-g).
R
Measurement noise covariance matrix (p-by-p).
S
Optional cross term covariance matrix (g-by-p), s = cov(w,v).
If S is empty '[]' or not specified, a zero matrix is assumed.
*Outputs*
L
Kalman filter gain matrix (n-by-p).
P
Unique stabilizing solution of the continuous-time Riccati
equation (n-by-n). Symmetric matrix. If SYS is a
discrete-time model, the solution of the corresponding
discrete-time Riccati equation is returned.
E
Closed-loop poles (n-by-1).
*Equations*
.
x = Ax + Bu + L(y - Cx -Du)
E = eig(A - L*C)
See also: dare, care, dlqr, lqr, dlqe.
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Kalman filter for continuous-time systems.
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lqr
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-- Function File: [G, X, L] = lqr (SYS, Q, R)
-- Function File: [G, X, L] = lqr (SYS, Q, R, S)
-- Function File: [G, X, L] = lqr (A, B, Q, R)
-- Function File: [G, X, L] = lqr (A, B, Q, R, S)
-- Function File: [G, X, L] = lqr (A, B, Q, R, [], E)
-- Function File: [G, X, L] = lqr (A, B, Q, R, S, E)
Linear-quadratic regulator.
*Inputs*
SYS
Continuous or discrete-time LTI model (p-by-m, n states).
A
State matrix of continuous-time system (n-by-n).
B
Input matrix of continuous-time system (n-by-m).
Q
State weighting matrix (n-by-n).
R
Input weighting matrix (m-by-m).
S
Optional cross term matrix (n-by-m). If S is not specified, a
zero matrix is assumed.
E
Optional descriptor matrix (n-by-n). If E is not specified,
an identity matrix is assumed.
*Outputs*
G
State feedback matrix (m-by-n).
X
Unique stabilizing solution of the continuous-time Riccati
equation (n-by-n).
L
Closed-loop poles (n-by-1).
*Equations*
.
x = A x + B u, x(0) = x0
inf
J(x0) = INT (x' Q x + u' R u + 2 x' S u) dt
0
L = eig (A - B*G)
See also: care, dare, dlqr.
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Linear-quadratic regulator.
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lsim
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-- Function File: lsim (SYS, U)
-- Function File: lsim (SYS1, SYS2, ..., SYSN, U)
-- Function File: lsim (SYS1, 'STYLE1', ..., SYSN, 'STYLEN', U)
-- Function File: lsim (SYS1, ..., U, T)
-- Function File: lsim (SYS1, ..., U, T, X0)
-- Function File: [Y, T, X] = lsim (SYS, U)
-- Function File: [Y, T, X] = lsim (SYS, U, T)
-- Function File: [Y, T, X] = lsim (SYS, U, T, X0)
Simulate LTI model response to arbitrary inputs. If no output
arguments are given, the system response is plotted on the screen.
*Inputs*
SYS
LTI model. System must be proper, i.e. it must not have more
zeros than poles.
U
Vector or array of input signal. Needs 'length(t)' rows and
as many columns as there are inputs. If SYS is a single-input
system, row vectors U of length 'length(t)' are accepted as
well.
T
Time vector. Should be evenly spaced. If SYS is a
continuous-time system and T is a real scalar, SYS is
discretized with sampling time 'tsam = t/(rows(u)-1)'. If SYS
is a discrete-time system and T is not specified, vector T is
assumed to be '0 : tsam : tsam*(rows(u)-1)'.
X0
Vector of initial conditions for each state. If not
specified, a zero vector is assumed.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
Y
Output response array. Has as many rows as time samples
(length of t) and as many columns as outputs.
T
Time row vector. It is always evenly spaced.
X
State trajectories array. Has 'length (t)' rows and as many
columns as states.
See also: impulse, initial, step.
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Simulate LTI model response to arbitrary inputs.
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ltimodels
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-- Function File: test ltimodels
-- Function File: ltimodels
-- Function File: ltimodels (SYSTYPE)
Test suite and help for LTI models.
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Test suite and help for LTI models.
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lyap
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-- Function File: X = lyap (A, B)
-- Function File: X = lyap (A, B, C)
-- Function File: X = lyap (A, B, [], E)
Solve continuous-time Lyapunov or Sylvester equations.
*Equations*
AX + XA' + B = 0 (Lyapunov Equation)
AX + XB + C = 0 (Sylvester Equation)
AXE' + EXA' + B = 0 (Generalized Lyapunov Equation)
*Algorithm*
Uses SLICOT SB03MD, SB04MD and SG03AD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: lyapchol, dlyap, dlyapchol.
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Solve continuous-time Lyapunov or Sylvester equations.
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lyapchol
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-- Function File: U = lyapchol (A, B)
-- Function File: U = lyapchol (A, B, E)
Compute Cholesky factor of continuous-time Lyapunov equations.
*Equations*
A U' U + U' U A' + B B' = 0 (Lyapunov Equation)
A U' U E' + E U' U A' + B B' = 0 (Generalized Lyapunov Equation)
*Algorithm*
Uses SLICOT SB03OD and SG03BD by courtesy of NICONET e.V.
(http://www.slicot.org)
See also: lyap, dlyap, dlyapchol.
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Compute Cholesky factor of continuous-time Lyapunov equations.
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mag2db
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-- Function File: DB = mag2db (MAG)
Convert Magnitude to Decibels (dB).
*Inputs*
MAG
Magnitude value(s). Both real-valued scalars and matrices are
accepted.
*Outputs*
DB
Decibel (dB) value(s).
See also: db2mag.
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Convert Magnitude to Decibels (dB).
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margin
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-- Function File: [GAMMA, PHI, W_GAMMA, W_PHI] = margin (SYS)
-- Function File: [GAMMA, PHI, W_GAMMA, W_PHI] = margin (SYS, TOL)
Gain and phase margin of a system. If no output arguments are
given, both gain and phase margin are plotted on a bode diagram.
Otherwise, the margins and their corresponding frequencies are
computed and returned. A more robust criterion to assess the
stability of a feedback system is the sensitivity Ms computed by
function 'sensitivity'.
*Inputs*
SYS
LTI model. Must be a single-input and single-output (SISO)
system.
TOL
Imaginary parts below TOL are assumed to be zero. If not
specified, default value 'sqrt (eps)' is taken.
*Outputs*
GAMMA
Gain margin (as gain, not dBs).
PHI
Phase margin (in degrees).
W_GAMMA
Frequency for the gain margin (in rad/s).
W_PHI
Frequency for the phase margin (in rad/s).
*Algorithm*
Uses function 'roots' to calculate the frequencies W_GAMMA, W_PHI
from special polynomials created from the transfer function of SYS
as listed below in section <<Equations>>.
*Equations*
CONTINUOUS-TIME SYSTEMS
Gain Margin
_ _
L(jw) = L(jw) BTW: L(jw) = L(-jw) = conj (L(jw))
num(jw) num(-jw)
------- = --------
den(jw) den(-jw)
num(jw) den(-jw) = num(-jw) den(jw)
imag (num(jw) den(-jw)) = 0
imag (num(-jw) den(jw)) = 0
Phase Margin
|num(jw)|
|L(jw)| = |-------| = 1
|den(jw)|
_ 2 2
z z = Re z + Im z
num(jw) num(-jw)
------- * -------- = 1
den(jw) den(-jw)
num(jw) num(-jw) - den(jw) den(-jw) = 0
real (num(jw) num(-jw) - den(jw) den(-jw)) = 0
DISCRETE-TIME SYSTEMS
Gain Margin
jwT log z
L(z) = L(1/z) BTW: z = e --> w = -----
j T
num(z) num(1/z)
------ = --------
den(z) den(1/z)
num(z) den(1/z) - num(1/z) den(z) = 0
Phase Margin
|num(z)|
|L(z)| = |------| = 1
|den(z)|
L(z) L(1/z) = 1
num(z) num(1/z)
------ * -------- = 1
den(z) den(1/z)
num(z) num(1/z) - den(z) den(1/z) = 0
PS: How to get L(1/z)
4 3 2
p(z) = a z + b z + c z + d z + e
-4 -3 -2 -1
p(1/z) = a z + b z + c z + d z + e
-4 2 3 4
= z ( a + b z + c z + d z + e z )
4 3 2 4
= ( e z + d z + c z + b z + a ) / ( z )
See also: sensitivity, roots.
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Gain and phase margin of a system.
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mixsyn
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-- Function File: [K, N, GAMMA, INFO] = mixsyn (G, W1, W2, W3, ...)
Solve stacked S/KS/T H-infinity problem. Mixed-sensitivity is the
name given to transfer function shaping problems in which the
sensitivity function
-1
S = (I + G K)
is shaped along with one or more other closed-loop transfer
functions such as K S or the complementary sensitivity function
-1
T = I - S = (I + G K)
in a typical one degree-of-freedom configuration, where G denotes
the plant and K the (sub-)optimal controller to be found. The
shaping of multivariable transfer functions is based on the idea
that a satisfactory definition of gain (range of gain) for a matrix
transfer function is given by the singular values
of the transfer function. Hence the classical loop-shaping ideas
of feedback design can be generalized to multivariable systems. In
addition to the requirement that K stabilizes G, the closed-loop
objectives are as follows [1]:
1. For _disturbance rejection_ make
small.
2. For _noise attenuation_ make
small.
3. For _reference tracking_ make
4. For _input usage (control energy) reduction_ make
small.
5. For _robust stability_ in the presence of an additive
perturbation
make
small.
6. For _robust stability_ in the presence of a multiplicative
output perturbation
make
small.
In order to find a robust controller for the so-called stacked
S/KS/T H-infinity problem, the user function 'mixsyn' minimizes the
following criterion
| W1 S |
min || N(K) || N = | W2 K S |
K oo | W3 T |
'[K, N] = mixsyn (G, W1, W2, W3)'. The user-defined weighting
functions W1, W2 and W3 bound the largest singular values of the
closed-loop transfer functions S (for performance), K S (to
penalize large inputs) and T (for robustness and to avoid
sensitivity to noise), respectively [1]. A few points are to be
considered when choosing the weights. The weigths WI must all be
proper and stable. Therefore if one wishes, for example, to
minimize S at low frequencies by a weighting W1 including integral
action,
1
-
s
needs to be approximated by
1
----- where e << 1.
s + e
Similarly one might be interested in weighting K S with a
non-proper weight W2 to ensure that K is small outside the system
bandwidth. The trick here is to replace a non-proper term such as
1 + T1 s
1 + T1 s by --------, where T2 << T1.
1 + T2 s
[1, 2].
*Inputs*
G
LTI model of plant.
W1
LTI model of performance weight. Bounds the largest singular
values of sensitivity S. Model must be empty '[]', SISO or of
appropriate size.
W2
LTI model to penalize large control inputs. Bounds the
largest singular values of KS. Model must be empty '[]', SISO
or of appropriate size.
W3
LTI model of robustness and noise sensitivity weight. Bounds
the largest singular values of complementary sensitivity T.
Model must be empty '[]', SISO or of appropriate size.
...
Optional arguments of 'hinfsyn'. Type 'help hinfsyn' for more
information.
All inputs must be proper/realizable. Scalars, vectors and
matrices are possible instead of LTI models.
*Outputs*
K
State-space model of the H-infinity (sub-)optimal controller.
N
State-space model of the lower LFT of P and K.
INFO
Structure containing additional information.
INFO.GAMMA
L-infinity norm of N.
INFO.RCOND
Vector RCOND contains estimates of the reciprocal condition
numbers of the matrices which are to be inverted and estimates
of the reciprocal condition numbers of the Riccati equations
which have to be solved during the computation of the
controller K. For details, see the description of the
corresponding SLICOT routine.
*Block Diagram*
| W1 S |
gamma = min||N(K)|| N = | W2 K S | = lft (P, K)
K inf | W3 T |
+------+ z1
+---------------------------------------->| W1 |----->
| +------+
| +------+ z2
| +---------------------->| W2 |----->
| | +------+
r + e | +--------+ u | +--------+ y +------+ z3
----->(+)---+-->| K(s) |----+-->| G(s) |----+---->| W3 |----->
^ - +--------+ +--------+ | +------+
| |
+----------------------------------------+
+--------+
| |-----> z1 (p1x1) z1 = W1 e
r (px1) ----->| P(s) |-----> z2 (p2x1) z2 = W2 u
| |-----> z3 (p3x1) z3 = W3 y
u (mx1) ----->| |-----> e (px1) e = r - y
+--------+
+--------+
r ----->| |-----> z
| P(s) |
u +---->| |-----+ e
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
+--------+
r ----->| N(s) |-----> z
+--------+
Extended Plant: P = augw (G, W1, W2, W3)
Controller: K = mixsyn (G, W1, W2, W3)
Entire System: N = lft (P, K)
Open Loop: L = G * K
Closed Loop: T = feedback (L)
*Algorithm*
Relies on functions 'augw' and 'hinfsyn', which use SLICOT SB10FD,
SB10DD and SB10AD by courtesy of NICONET e.V.
(http://www.slicot.org)
*References*
[1] Skogestad, S. and Postlethwaite I. (2005) 'Multivariable
Feedback Control: Analysis and Design: Second Edition'. Wiley,
Chichester, England.
[2] Meinsma, G. (1995) 'Unstable and nonproper weights in
H-infinity control' Automatica, Vol. 31, No. 11, pp. 1655-1658
See also: hinfsyn, augw.
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Solve stacked S/KS/T H-infinity problem.
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mktito
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-- Function File: P = mktito (P, NMEAS, NCON)
Partition LTI plant P for robust controller synthesis. If a plant
is partitioned this way, one can omit the inputs NMEAS and NCON
when calling the functions 'hinfsyn' and 'h2syn'.
*Inputs*
P
Generalized plant.
NMEAS
Number of measured outputs v. The last NMEAS outputs of P are
connected to the inputs of controller K. The remaining
outputs z (indices 1 to p-nmeas) are used to calculate the
H-2/H-infinity norm.
NCON
Number of controlled inputs u. The last NCON inputs of P are
connected to the outputs of controller K. The remaining
inputs w (indices 1 to m-ncon) are excited by a harmonic test
signal.
*Outputs*
P
Partitioned plant. The input/output groups and names are
overwritten with designations according to [1].
*Block Diagram*
min||N(K)|| N = lft (P, K)
K norm
+--------+
w ----->| |-----> z
| P(s) |
u +---->| |-----+ v
| +--------+ |
| |
| +--------+ |
+-----| K(s) |<----+
+--------+
+--------+
w ----->| N(s) |-----> z
+--------+
*Reference*
[1] Skogestad, S. and Postlethwaite, I. (2005) 'Multivariable
Feedback Control: Analysis and Design: Second Edition'. Wiley,
Chichester, England.
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Partition LTI plant P for robust controller synthesis.
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moen4
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-- Function File: [SYS, X0, INFO] = moen4 (DAT, ...)
-- Function File: [SYS, X0, INFO] = moen4 (DAT, N, ...)
-- Function File: [SYS, X0, INFO] = moen4 (DAT, OPT, ...)
-- Function File: [SYS, X0, INFO] = moen4 (DAT, N, OPT, ...)
Estimate state-space model using combined subspace method: MOESP
algorithm for finding the matrices A and C, and N4SID algorithm for
finding the matrices B and D. If no output arguments are given, the
singular values are plotted on the screen in order to estimate the
system order.
*Inputs*
DAT
iddata set containing the measurements, i.e. time-domain
signals.
N
The desired order of the resulting state-space system SYS. If
not specified, N is chosen automatically according to the
singular values and tolerances.
...
Optional pairs of keys and values. ''key1', value1, 'key2',
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
SYS
Discrete-time state-space model.
X0
Initial state vector. If DAT is a multi-experiment dataset,
X0 becomes a cell vector containing an initial state vector
for each experiment.
INFO
Struct containing additional information.
INFO.K
Kalman gain matrix.
INFO.Q
State covariance matrix.
INFO.RY
Output covariance matrix.
INFO.S
State-output cross-covariance matrix.
INFO.L
Noise variance matrix factor. LL'=Ry.
*Option Keys and Values*
'N'
The desired order of the resulting state-space system SYS. S
> N > 0.
'S'
The number of block rows S in the input and output block
Hankel matrices to be processed. S > 0. In the MOESP theory,
S should be larger than N, the estimated dimension of state
vector.
'ALG', 'ALGORITHM'
Specifies the algorithm for computing the triangular factor R,
as follows:
'C'
Cholesky algorithm applied to the correlation matrix of
the input-output data. Default method.
'F'
Fast QR algorithm.
'Q'
QR algorithm applied to the concatenated block Hankel
matrices.
'TOL'
Absolute tolerance used for determining an estimate of the
system order. If TOL >= 0, the estimate is indicated by the
index of the last singular value greater than or equal to TOL.
(Singular values less than TOL are considered as zero.) When
TOL = 0, an internally computed default value, TOL =
S*EPS*SV(1), is used, where SV(1) is the maximal singular
value, and EPS is the relative machine precision. When TOL <
0, the estimate is indicated by the index of the singular
value that has the largest logarithmic gap to its successor.
Default value is 0.
'RCOND'
The tolerance to be used for estimating the rank of matrices.
If the user sets RCOND > 0, the given value of RCOND is used
as a lower bound for the reciprocal condition number; an
m-by-n matrix whose estimated condition number is less than
1/RCOND is considered to be of full rank. If the user sets
RCOND <= 0, then an implicitly computed, default tolerance,
defined by RCOND = m*n*EPS, is used instead, where EPS is the
relative machine precision. Default value is 0.
'CONFIRM'
Specifies whether or not the user's confirmation of the system
order estimate is desired, as follows:
TRUE
User's confirmation.
FALSE
No confirmation. Default value.
'NOISEINPUT'
The desired type of noise input channels.
'N'
No error inputs. Default value.
x[k+1] = A x[k] + B u[k]
y[k] = C x[k] + D u[k]
'E'
Return SYS as a (p-by-m+p) state-space model with both
measured input channels u and noise channels e with
covariance matrix RY.
x[k+1] = A x[k] + B u[k] + K e[k]
y[k] = C x[k] + D u[k] + e[k]
'V'
Return SYS as a (p-by-m+p) state-space model with both
measured input channels u and white noise channels v with
identity covariance matrix.
x[k+1] = A x[k] + B u[k] + K L v[k]
y[k] = C x[k] + D u[k] + L v[k]
e = L v, L L' = Ry
'K'
Return SYS as a Kalman predictor for simulation.
^ ^ ^
x[k+1] = A x[k] + B u[k] + K(y[k] - y[k])
^ ^
y[k] = C x[k] + D u[k]
^ ^
x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k]
^ ^
y[k] = C x[k] + D u[k] + 0 y[k]
*Algorithm*
Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Estimate state-space model using combined subspace method: MOESP
algorithm for f
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moesp
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-- Function File: [SYS, X0, INFO] = moesp (DAT, ...)
-- Function File: [SYS, X0, INFO] = moesp (DAT, N, ...)
-- Function File: [SYS, X0, INFO] = moesp (DAT, OPT, ...)
-- Function File: [SYS, X0, INFO] = moesp (DAT, N, OPT, ...)
Estimate state-space model using MOESP algorithm. MOESP:
Multivariable Output Error State sPace. If no output arguments are
given, the singular values are plotted on the screen in order to
estimate the system order.
*Inputs*
DAT
iddata set containing the measurements, i.e. time-domain
signals.
N
The desired order of the resulting state-space system SYS. If
not specified, N is chosen automatically according to the
singular values and tolerances.
...
Optional pairs of keys and values. ''key1', value1, 'key2',
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
SYS
Discrete-time state-space model.
X0
Initial state vector. If DAT is a multi-experiment dataset,
X0 becomes a cell vector containing an initial state vector
for each experiment.
INFO
Struct containing additional information.
INFO.K
Kalman gain matrix.
INFO.Q
State covariance matrix.
INFO.RY
Output covariance matrix.
INFO.S
State-output cross-covariance matrix.
INFO.L
Noise variance matrix factor. LL'=Ry.
*Option Keys and Values*
'N'
The desired order of the resulting state-space system SYS. S
> N > 0.
'S'
The number of block rows S in the input and output block
Hankel matrices to be processed. S > 0. In the MOESP theory,
S should be larger than N, the estimated dimension of state
vector.
'ALG', 'ALGORITHM'
Specifies the algorithm for computing the triangular factor R,
as follows:
'C'
Cholesky algorithm applied to the correlation matrix of
the input-output data. Default method.
'F'
Fast QR algorithm.
'Q'
QR algorithm applied to the concatenated block Hankel
matrices.
'TOL'
Absolute tolerance used for determining an estimate of the
system order. If TOL >= 0, the estimate is indicated by the
index of the last singular value greater than or equal to TOL.
(Singular values less than TOL are considered as zero.) When
TOL = 0, an internally computed default value, TOL =
S*EPS*SV(1), is used, where SV(1) is the maximal singular
value, and EPS is the relative machine precision. When TOL <
0, the estimate is indicated by the index of the singular
value that has the largest logarithmic gap to its successor.
Default value is 0.
'RCOND'
The tolerance to be used for estimating the rank of matrices.
If the user sets RCOND > 0, the given value of RCOND is used
as a lower bound for the reciprocal condition number; an
m-by-n matrix whose estimated condition number is less than
1/RCOND is considered to be of full rank. If the user sets
RCOND <= 0, then an implicitly computed, default tolerance,
defined by RCOND = m*n*EPS, is used instead, where EPS is the
relative machine precision. Default value is 0.
'CONFIRM'
Specifies whether or not the user's confirmation of the system
order estimate is desired, as follows:
TRUE
User's confirmation.
FALSE
No confirmation. Default value.
'NOISEINPUT'
The desired type of noise input channels.
'N'
No error inputs. Default value.
x[k+1] = A x[k] + B u[k]
y[k] = C x[k] + D u[k]
'E'
Return SYS as a (p-by-m+p) state-space model with both
measured input channels u and noise channels e with
covariance matrix RY.
x[k+1] = A x[k] + B u[k] + K e[k]
y[k] = C x[k] + D u[k] + e[k]
'V'
Return SYS as a (p-by-m+p) state-space model with both
measured input channels u and white noise channels v with
identity covariance matrix.
x[k+1] = A x[k] + B u[k] + K L v[k]
y[k] = C x[k] + D u[k] + L v[k]
e = L v, L L' = Ry
'K'
Return SYS as a Kalman predictor for simulation.
^ ^ ^
x[k+1] = A x[k] + B u[k] + K(y[k] - y[k])
^ ^
y[k] = C x[k] + D u[k]
^ ^
x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k]
^ ^
y[k] = C x[k] + D u[k] + 0 y[k]
*Algorithm*
Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Estimate state-space model using MOESP algorithm.
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n4sid
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-- Function File: [SYS, X0, INFO] = n4sid (DAT, ...)
-- Function File: [SYS, X0, INFO] = n4sid (DAT, N, ...)
-- Function File: [SYS, X0, INFO] = n4sid (DAT, OPT, ...)
-- Function File: [SYS, X0, INFO] = n4sid (DAT, N, OPT, ...)
Estimate state-space model using N4SID algorithm. N4SID: Numerical
algorithm for Subspace State Space System IDentification. If no
output arguments are given, the singular values are plotted on the
screen in order to estimate the system order.
*Inputs*
DAT
iddata set containing the measurements, i.e. time-domain
signals.
N
The desired order of the resulting state-space system SYS. If
not specified, N is chosen automatically according to the
singular values and tolerances.
...
Optional pairs of keys and values. ''key1', value1, 'key2',
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
SYS
Discrete-time state-space model.
X0
Initial state vector. If DAT is a multi-experiment dataset,
X0 becomes a cell vector containing an initial state vector
for each experiment.
INFO
Struct containing additional information.
INFO.K
Kalman gain matrix.
INFO.Q
State covariance matrix.
INFO.RY
Output covariance matrix.
INFO.S
State-output cross-covariance matrix.
INFO.L
Noise variance matrix factor. LL'=Ry.
*Option Keys and Values*
'N'
The desired order of the resulting state-space system SYS. S
> N > 0.
'S'
The number of block rows S in the input and output block
Hankel matrices to be processed. S > 0. In the MOESP theory,
S should be larger than N, the estimated dimension of state
vector.
'ALG', 'ALGORITHM'
Specifies the algorithm for computing the triangular factor R,
as follows:
'C'
Cholesky algorithm applied to the correlation matrix of
the input-output data. Default method.
'F'
Fast QR algorithm.
'Q'
QR algorithm applied to the concatenated block Hankel
matrices.
'TOL'
Absolute tolerance used for determining an estimate of the
system order. If TOL >= 0, the estimate is indicated by the
index of the last singular value greater than or equal to TOL.
(Singular values less than TOL are considered as zero.) When
TOL = 0, an internally computed default value, TOL =
S*EPS*SV(1), is used, where SV(1) is the maximal singular
value, and EPS is the relative machine precision. When TOL <
0, the estimate is indicated by the index of the singular
value that has the largest logarithmic gap to its successor.
Default value is 0.
'RCOND'
The tolerance to be used for estimating the rank of matrices.
If the user sets RCOND > 0, the given value of RCOND is used
as a lower bound for the reciprocal condition number; an
m-by-n matrix whose estimated condition number is less than
1/RCOND is considered to be of full rank. If the user sets
RCOND <= 0, then an implicitly computed, default tolerance,
defined by RCOND = m*n*EPS, is used instead, where EPS is the
relative machine precision. Default value is 0.
'CONFIRM'
Specifies whether or not the user's confirmation of the system
order estimate is desired, as follows:
TRUE
User's confirmation.
FALSE
No confirmation. Default value.
'NOISEINPUT'
The desired type of noise input channels.
'N'
No error inputs. Default value.
x[k+1] = A x[k] + B u[k]
y[k] = C x[k] + D u[k]
'E'
Return SYS as a (p-by-m+p) state-space model with both
measured input channels u and noise channels e with
covariance matrix RY.
x[k+1] = A x[k] + B u[k] + K e[k]
y[k] = C x[k] + D u[k] + e[k]
'V'
Return SYS as a (p-by-m+p) state-space model with both
measured input channels u and white noise channels v with
identity covariance matrix.
x[k+1] = A x[k] + B u[k] + K L v[k]
y[k] = C x[k] + D u[k] + L v[k]
e = L v, L L' = Ry
'K'
Return SYS as a Kalman predictor for simulation.
^ ^ ^
x[k+1] = A x[k] + B u[k] + K(y[k] - y[k])
^ ^
y[k] = C x[k] + D u[k]
^ ^
x[k+1] = (A-KC) x[k] + (B-KD) u[k] + K y[k]
^ ^
y[k] = C x[k] + D u[k] + 0 y[k]
*Algorithm*
Uses SLICOT IB01AD, IB01BD and IB01CD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Estimate state-space model using N4SID algorithm.
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ncfsyn
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-- Function File: [K, N, GAMMA, INFO] = ncfsyn (G, W1, W2, FACTOR)
Loop shaping H-infinity synthesis. Compute positive feedback
controller using the McFarlane/Glover loop shaping design procedure
[1]. Using a precompensator W1 and/or a postcompensator W2, the
singular values of the nominal plant G are shaped to give a desired
open-loop shape. The nominal plant G and shaping functions W1, W2
are combined to form the shaped plant, GS where 'Gs = W2 G W1'. We
assume that W1 and W2 are such that GS contains no hidden modes.
It is relatively easy to approximate the closed-loop requirements
by the following open-loop objectives [2]:
1. For _disturbance rejection_ make
large; valid for frequencies at which
2. For _noise attenuation_ make
small; valid for frequencies at which
3. For _reference tracking_ make
large; valid for frequencies at which
4. For _robust stability_ to a multiplicative output perturbation
small; valid for frequencies at which
.
Then a stabilizing controller KS is synthesized for shaped plant
GS. The final positive feedback controller K is then constructed
by combining the
H-infinity
controller KS with the shaping functions W1 and W2 such that 'K =
W1 Ks W2'. In [1] is stated further that the given robust
stabilization objective can be interpreted as a
H-infinity
problem formulation of minimizing the
H-infinity
norm of the frequency weighted gain from disturbances on the plant
input and output to the controller input and output as follows:
-1 -1 -1
min || N(K) || , N = | W1 | (I - K G) | W1 G W2 |
K oo | W2 G |
'[K, N] = ncfsyn (G, W1, W2, f)' The function 'ncfsyn' - the
somewhat cryptic name stands for _normalized coprime factorization
synthesis_ - allows the specification of an additional argument,
factor F. Default value 'f = 1' implies that an optimal controller
is required, whereas 'f > 1' implies that a suboptimal controller
is required, achieving a performance that is F times less than
optimal.
*Inputs*
G
LTI model of plant.
W1
LTI model of precompensator. Model must be SISO or of
appropriate size. An identity matrix is taken if W1 is not
specified or if an empty model '[]' is passed.
W2
LTI model of postcompensator. Model must be SISO or of
appropriate size. An identity matrix is taken if W2 is not
specified or if an empty model '[]' is passed.
FACTOR
'factor = 1' implies that an optimal controller is required.
'factor > 1' implies that a suboptimal controller is required,
achieving a performance that is FACTOR times less than
optimal. Default value is 1.
*Outputs*
K
State-space model of the H-infinity loop-shaping controller.
Note that K is a _positive_ feedback controller.
N
State-space model of the closed loop depicted below.
INFO
Structure containing additional information.
INFO.GAMMA
L-infinity norm of N. 'gamma = norm (N, inf)'.
INFO.EMAX
Nugap robustness. 'emax = inv (gamma)'.
INFO.GS
Shaped plant. 'Gs = W2 * G * W1'.
INFO.KS
Controller for shaped plant. 'Ks = ncfsyn (Gs)'.
INFO.RCOND
Estimates of the reciprocal condition numbers of the Riccati
equations and a few other things. For details, see the
description of the corresponding SLICOT routine.
*Block Diagram of N*
^ z1 ^ z2
| |
w1 + | +--------+ | +--------+
----->(+)---+-->| Ks |----+--->(+)---->| Gs |----+
^ + +--------+ ^ +--------+ |
| w2 | |
| |
+-------------------------------------------------+
*Algorithm*
Uses SLICOT SB10ID, SB10KD and SB10ZD by courtesy of NICONET e.V.
(http://www.slicot.org)
*References*
[1] D. McFarlane and K. Glover, 'A Loop Shaping Design Procedure
Using H-infinity Synthesis', IEEE Transactions on Automatic
Control, Vol. 37, No. 6, June 1992.
[2] S. Skogestad and I. Postlethwaite, 'Multivariable Feedback
Control: Analysis and Design: Second Edition'. Wiley, Chichester,
England, 2005.
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Loop shaping H-infinity synthesis.
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nichols
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-- Function File: nichols (SYS)
-- Function File: nichols (SYS1, SYS2, ..., SYSN)
-- Function File: nichols (SYS1, SYS2, ..., SYSN, W)
-- Function File: nichols (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: [MAG, PHA, W] = nichols (SYS)
-- Function File: [MAG, PHA, W] = nichols (SYS, W)
Nichols chart of frequency response. If no output arguments are
given, the response is printed on the screen.
*Inputs*
SYS
LTI system. Must be a single-input and single-output (SISO)
system.
W
Optional vector of frequency values. If W is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell '{wmin, wmax}' specifies a frequency
range, where WMIN and WMAX denote minimum and maximum
frequencies in rad/s.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
MAG
Vector of magnitude. Has length of frequency vector W.
PHA
Vector of phase. Has length of frequency vector W.
W
Vector of frequency values used.
See also: bode, nyquist, sigma.
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Nichols chart of frequency response.
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nyquist
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-- Function File: nyquist (SYS)
-- Function File: nyquist (SYS1, SYS2, ..., SYSN)
-- Function File: nyquist (SYS1, SYS2, ..., SYSN, W)
-- Function File: nyquist (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: [RE, IM, W] = nyquist (SYS)
-- Function File: [RE, IM, W] = nyquist (SYS, W)
Nyquist diagram of frequency response. If no output arguments are
given, the response is printed on the screen.
*Inputs*
SYS
LTI system. Must be a single-input and single-output (SISO)
system.
W
Optional vector of frequency values. If W is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell '{wmin, wmax}' specifies a frequency
range, where WMIN and WMAX denote minimum and maximum
frequencies in rad/s.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
RE
Vector of real parts. Has length of frequency vector W.
IM
Vector of imaginary parts. Has length of frequency vector W.
W
Vector of frequency values used.
See also: bode, nichols, sigma.
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Nyquist diagram of frequency response.
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obsv
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-- Function File: OB = obsv (SYS)
-- Function File: OB = obsv (A, C)
Return observability matrix.
*Inputs*
SYS
LTI model.
A
State matrix (n-by-n).
C
Measurement matrix (p-by-n).
*Outputs*
OB
Observability matrix.
*Equation*
| C |
| CA |
Ob = | CA^2 |
| ... |
| CA^(n-1) |
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Return observability matrix.
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obsvf
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-- Function File: [SYSBAR, T, K] = obsvf (SYS)
-- Function File: [SYSBAR, T, K] = obsvf (SYS, TOL)
-- Function File: [ABAR, BBAR, CBAR, T, K] = obsvf (A, B, C)
-- Function File: [ABAR, BBAR, CBAR, T, K] = obsvf (A, B, C, TOL)
If Ob=obsv(A,C) has rank r <= n = SIZE(A,1), then there is a
similarity transformation Tc such that To = [t1;t2] where t1 is c
and t2 is orthogonal to t1
Abar = To \\ A * To , Bbar = To \\ B , Cbar = C * To
and the transformed system has the form
| Ao 0 | | Bo |
Abar = |----------|, Bbar = | --- |, Cbar = [Co | 0 ].
| A21 Ano| | Bno |
where (Ao,Bo) is observable, and Co(sI-Ao)^(-1)Bo = C(sI-A)^(-1)B.
And system is detectable if Ano has no eigenvalues in the right
half plane. The last output K is a vector of length n containing
the number of observable states.
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If Ob=obsv(A,C) has rank r <= n = SIZE(A,1), then there is a similarity
transfor
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optiPID
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Numerical optimization of a PID controller using an objective function.
The objective function is located in the file 'optiPIDfun'. Type 'which
optiPID' to locate, 'edit optiPID' to open and simply 'optiPID' to run
the example file. In this example called 'optiPID', loosely based on
[1], it is assumed that the plant
1
P(s) = -----------------------
(s^2 + s + 1) (s + 1)^4
is controlled by a PID controller with second-order roll-off
1 1
C(s) = Kp (1 + ---- + Td s) -------------
Ti s (tau s + 1)^2
in the usual negative feedback structure
L(s) P(s) C(s)
T(s) = -------- = -------------
1 + L(s) 1 + P(s) C(s)
The plant P(s) is of higher order but benign. The initial values for
the controller parameters Kp, Ti and Td are obtained by applying the
Astroem and Haegglund rules [2]. These values are to be improved using
a numerical optimization as shown below. As with all numerical methods,
this approach can never guarantee that a proposed solution is a global
minimum. Therefore, good initial guesses for the parameters to be
optimized are very important. The Octave function 'fminsearch'
minimizes the objective function J, which is chosen to be
inf
J(Kp, Ti, Td) = mu1 INT t |e(t)| dt + mu2 (||y(t)|| - 1) + mu3 ||S(jw)||
0 inf inf
This particular objective function penalizes the integral of
time-weighted absolute error
inf
ITAE = INT t |e(t)| dt
0
and the maximum overshoot
y - 1 = ||y(t)|| - 1
max inf
to a unity reference step in the time domain. In the frequency
domain, the sensitivity
Ms = ||S(jw)||
inf
is minimized for good robustness, where S(jw) denotes the
_sensitivity_ transfer function
1 1
S(s) = -------- = -------------
1 + L(s) 1 + P(s) C(s)
The constants mu1, mu2 and mu3 are _relative weighting factors_ or
<<tuning knobs>> which reflect the importance of the different design
goals. Varying these factors corresponds to changing the emphasis from,
say, high performance to good robustness. The main advantage of this
approach is the possibility to explore the tradeoffs of the design
problem in a systematic way. In a first approach, all three design
objectives are weigthed equally. In subsequent iterations, the
parameters mu1 = 1, mu2 = 10 and mu3 = 20 are found to yield
satisfactory closed-loop performance. This controller results in a
system with virtually no overshoot and a phase margin of 64 degrees.
*References*
[1] Guzzella, L. 'Analysis and Design of SISO Control Systems', VDF
Hochschulverlag, ETH Zurich, 2007
[2] Astroem, K. and Haegglund, T. 'PID Controllers: Theory, Design and
Tuning', Second Edition, Instrument Society of America, 1995
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Numerical optimization of a PID controller using an objective function.
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optiPIDctrl
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===============================================================================
optiPIDctrl Lukas Reichlin February 2012
===============================================================================
Return PID controller with roll-off for given parameters Kp, Ti and Td.
===============================================================================
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===============================================================================
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optiPIDfun
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===============================================================================
optiPIDfun Lukas Reichlin July 2009
===============================================================================
Objective Function
Reference: Guzzella, L. (2007) Analysis and Synthesis of SISO Control Systems.
vdf Hochschulverlag, Zurich
===============================================================================
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===============================================================================
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# length: 7
options
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# length: 918
-- Function File: OPT = options ('KEY1', VALUE1, 'KEY2', VALUE2, ...)
Create options struct OPT from a number of key and value pairs.
For use with order reduction and system identification functions.
Option structs are a way to avoid typing the same key and value
pairs over and over again.
*Inputs*
KEY, PROPERTY
The name of the property.
VALUE
The value of the property.
*Outputs*
OPT
Struct with fields for each key.
*Example*
octave:1> opt = options ("method", "spa", "tol", 1e-6)
opt =
scalar structure containing the fields:
method = spa
tol = 1.0000e-06
octave:2> save filename opt
octave:3> # save the struct 'opt' to file 'filename' for later use
octave:4> load filename
octave:5> # load struct 'opt' from file 'filename'
# name: <cell-element>
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Create options struct OPT from a number of key and value pairs.
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# elements: 1
# length: 3
pid
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# length: 371
-- Function File: C = pid (KP)
-- Function File: C = pid (KP, KI)
-- Function File: C = pid (KP, KI, KD)
-- Function File: C = pid (KP, KI, KD, TF)
Return the transfer function C of the PID controller in parallel
form with first-order roll-off.
Ki Kd s
C(s) = Kp + ---- + --------
s Tf s + 1
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Return the transfer function C of the PID controller in parallel form
with first
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# elements: 1
# length: 6
pidstd
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-- Function File: C = pidstd (KP)
-- Function File: C = pidstd (KP, TI)
-- Function File: C = pidstd (KP, TI, TD)
-- Function File: C = pidstd (KP, TI, TD, N)
Return the transfer function C of the PID controller in standard
form with first-order roll-off.
1 Td s
C(s) = Kp (1 + ---- + ----------)
Ti s Td/N s + 1
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Return the transfer function C of the PID controller in standard form
with first
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# elements: 1
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place
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# length: 1997
-- Function File: F = place (SYS, P)
-- Function File: F = place (A, B, P)
-- Function File: [F, INFO] = place (SYS, P, ALPHA)
-- Function File: [F, INFO] = place (A, B, P, ALPHA)
Pole assignment for a given matrix pair (A,B) such that 'p = eig
(A-B*F)'. If parameter ALPHA is specified, poles with real parts
(continuous-time) or moduli (discrete-time) below ALPHA are left
untouched.
*Inputs*
SYS
Continuous- or discrete-time LTI system.
A
State matrix (n-by-n) of a continuous-time system.
B
Input matrix (n-by-m) of a continuous-time system.
P
Desired eigenvalues of the closed-loop system state-matrix
A-B*F. 'length (p) <= rows (A)'.
ALPHA
Specifies the maximum admissible value, either for real parts
or for moduli, of the eigenvalues of A which will not be
modified by the eigenvalue assignment algorithm. 'alpha >= 0'
for discrete-time systems.
*Outputs*
F
State feedback gain matrix.
INFO
Structure containing additional information.
INFO.NFP
The number of fixed poles, i.e. eigenvalues of A having real
parts less than ALPHA, or moduli less than ALPHA. These
eigenvalues are not modified by 'place'.
INFO.NAP
The number of assigned eigenvalues. 'nap = n-nfp-nup'.
INFO.NUP
The number of uncontrollable eigenvalues detected by the
eigenvalue assignment algorithm.
INFO.Z
The orthogonal matrix Z reduces the closed-loop system state
matrix 'A + B*F' to upper real Schur form. Note the positive
sign in 'A + B*F'.
*Note*
Place is also suitable to design estimator gains:
L = place (A.', C.', p).'
L = place (sys.', p).' # useful for discrete-time systems
*Algorithm*
Uses SLICOT SB01BD by courtesy of NICONET e.V.
(http://www.slicot.org)
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Pole assignment for a given matrix pair (A,B) such that 'p = eig
(A-B*F)'.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
pzmap
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 679
-- Function File: pzmap (SYS)
-- Function File: pzmap (SYS1, SYS2, ..., SYSN)
-- Function File: pzmap (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: [P, Z] = pzmap (SYS)
Plot the poles and zeros of an LTI system in the complex plane. If
no output arguments are given, the result is plotted on the screen.
Otherwise, the poles and zeros are computed and returned.
*Inputs*
SYS
LTI model.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
P
Poles of SYS.
Z
Invariant zeros of SYS.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Plot the poles and zeros of an LTI system in the complex plane.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
ramp
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1694
-- Function File: ramp (SYS)
-- Function File: ramp (SYS1, SYS2, ..., SYSN)
-- Function File: ramp (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: ramp (SYS1, ..., T)
-- Function File: ramp (SYS1, ..., TFINAL)
-- Function File: ramp (SYS1, ..., TFINAL, DT)
-- Function File: [Y, T, X] = ramp (SYS)
-- Function File: [Y, T, X] = ramp (SYS, T)
-- Function File: [Y, T, X] = ramp (SYS, TFINAL)
-- Function File: [Y, T, X] = ramp (SYS, TFINAL, DT)
Ramp response of LTI system. If no output arguments are given, the
response is printed on the screen.
r(t) = t * h(t)
*Inputs*
SYS
LTI model.
T
Time vector. Should be evenly spaced. If not specified, it
is calculated by the poles of the system to reflect adequately
the response transients.
TFINAL
Optional simulation horizon. If not specified, it is
calculated by the poles of the system to reflect adequately
the response transients.
DT
Optional sampling time. Be sure to choose it small enough to
capture transient phenomena. If not specified, it is
calculated by the poles of the system.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
Y
Output response array. Has as many rows as time samples
(length of t) and as many columns as outputs.
T
Time row vector.
X
State trajectories array. Has 'length (t)' rows and as many
columns as states.
See also: impulse, initial, lsim, step.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 28
Ramp response of LTI system.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
repsys
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 353
-- Function File: RSYS = repsys (SYS, M, N)
-- Function File: RSYS = repsys (SYS, [M, N])
-- Function File: RSYS = repsys (SYS, M)
Form a block transfer matrix of SYS with M copies vertically and N
copies horizontally. If N is not specified, it is set to M.
'repsys (sys, 2, 3)' is equivalent to '[sys, sys, sys; sys, sys,
sys]'.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Form a block transfer matrix of SYS with M copies vertically and N
copies horizo
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
rlocus
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 910
-- Function File: rlocus (SYS)
-- Function File: [RLDATA, K] = rlocus (SYS, INCREMENT, MIN_K, MAX_K)
Display root locus plot of the specified SISO system.
*Inputs*
SYS
LTI model. Must be a single-input and single-output (SISO)
system.
INCREMENT
The increment used in computing gain values.
MIN_K
Minimum value of K.
MAX_K
Maximum value of K.
*Outputs*
RLDATA
Data points plotted: in column 1 real values, in column 2 the
imaginary values.
K
Gains for real axis break points.
*Block Diagram*
u + +---+ +------+ y
------>(+)----->| k |----->| SISO |-------+------->
^ - +---+ +------+ |
| |
+---------------------------------+
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Display root locus plot of the specified SISO system.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
sensitivity
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1880
-- Function File: [MS, WS] = sensitivity (L)
-- Function File: [MS, WS] = sensitivity (P, C)
-- Function File: [MS, WS] = sensitivity (P, C1, C2, ...)
Return sensitivity margin MS. The quantity MS is simply the
inverse of the shortest distance from the Nyquist curve to the
critical point -1. Reasonable values of MS are in the range from
1.3 to 2.
Ms = ||S(jw)||
inf
If no output arguments are given, the critical distance 1/Ms is
plotted on a Nyquist diagram. In contrast to gain and phase margin
as computed by function 'margin', the sensitivity MS is a more
robust criterion to assess the stability of a feedback system.
*Inputs*
L
Open loop transfer function. L can be any type of LTI system,
but it must be square.
P
Plant model. Any type of LTI system.
C
Controller model. Any type of LTI system.
C1, C2, ...
If several controllers are specified, function 'sensitivity'
computes the sensitivity MS for each of them in combination
with plant P.
*Outputs*
MS
Sensitivity margin MS as defined in [1]. Scalar value. If
several controllers are specified, MS becomes a row vector
with as many entries as controllers.
WS
The frequency [rad/s] corresponding to the sensitivity peak.
Scalar value. If several controllers are specified, WS
becomes a row vector with as many entries as controllers.
*Algorithm*
Uses SLICOT AB13DD by courtesy of NICONET e.V.
(http://www.slicot.org) to calculate the infinity norm of the
sensitivity function.
*References*
[1] Astro"m, K. and Ha"gglund, T. (1995) PID Controllers: Theory,
Design and Tuning, Second Edition. Instrument Society of America.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 29
Return sensitivity margin MS.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
sigma
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1374
-- Function File: sigma (SYS)
-- Function File: sigma (SYS1, SYS2, ..., SYSN)
-- Function File: sigma (SYS1, SYS2, ..., SYSN, W)
-- Function File: sigma (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: [SV, W] = sigma (SYS)
-- Function File: [SV, W] = sigma (SYS, W)
Singular values of frequency response. If no output arguments are
given, the singular value plot is printed on the screen.
*Inputs*
SYS
LTI system. Multiple inputs and/or outputs (MIMO systems)
make practical sense.
W
Optional vector of frequency values. If W is not specified,
it is calculated by the zeros and poles of the system.
Alternatively, the cell '{wmin, wmax}' specifies a frequency
range, where WMIN and WMAX denote minimum and maximum
frequencies in rad/s.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
SV
Array of singular values. For a system with m inputs and p
outputs, the array sv has 'min (m, p)' rows and as many
columns as frequency points 'length (w)'. The singular values
at the frequency 'w(k)' are given by 'sv(:,k)'.
W
Vector of frequency values used.
See also: bodemag, svd.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 38
Singular values of frequency response.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
spaconred
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6245
-- Function File: [KR, INFO] = spaconred (G, K, ...)
-- Function File: [KR, INFO] = spaconred (G, K, NCR, ...)
-- Function File: [KR, INFO] = spaconred (G, K, OPT, ...)
-- Function File: [KR, INFO] = spaconred (G, K, NCR, OPT, ...)
Controller reduction by frequency-weighted Singular Perturbation
Approximation (SPA). Given a plant G and a stabilizing controller
K, determine a reduced order controller KR such that the
closed-loop system is stable and closed-loop performance is
retained.
The algorithm tries to minimize the frequency-weighted error
||V (K-Kr) W|| = min
inf
where V and W denote output and input weightings.
*Inputs*
G
LTI model of the plant. It has m inputs, p outputs and n
states.
K
LTI model of the controller. It has p inputs, m outputs and
nc states.
NCR
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically according
to the description of key 'ORDER'.
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
KR
State-space model of reduced order controller.
INFO
Struct containing additional information.
INFO.NCR
The order of the obtained reduced order controller KR.
INFO.NCS
The order of the alpha-stable part of original controller
K.
INFO.HSVC
The Hankel singular values of the alpha-stable part of K.
The NCS Hankel singular values are ordered decreasingly.
*Option Keys and Values*
'ORDER', 'NCR'
The desired order of the resulting reduced order controller
KR. If not specified, NCR is chosen automatically such that
states with Hankel singular values INFO.HSVC > TOL1 are
retained.
'METHOD'
Order reduction approach to be used as follows:
'SR', 'S'
Use the square-root Singular Perturbation Approximation
method.
'BFSR', 'P'
Use the balancing-free square-root Singular Perturbation
Approximation method. Default method.
'WEIGHT'
Specifies the type of frequency-weighting as follows:
'NONE'
No weightings are used (V = I, W = I).
'LEFT', 'OUTPUT'
Use stability enforcing left (output) weighting
-1
V = (I-G*K) *G , W = I
'RIGHT', 'INPUT'
Use stability enforcing right (input) weighting
-1
V = I , W = (I-G*K) *G
'BOTH', 'PERFORMANCE'
Use stability and performance enforcing weightings
-1 -1
V = (I-G*K) *G , W = (I-G*K)
Default value.
'FEEDBACK'
Specifies whether K is a positive or negative feedback
controller:
'+'
Use positive feedback controller. Default value.
'-'
Use negative feedback controller.
'ALPHA'
Specifies the ALPHA-stability boundary for the eigenvalues of
the state dynamics matrix K.A. For a continuous-time
controller, ALPHA <= 0 is the boundary value for the real
parts of eigenvalues, while for a discrete-time controller, 0
<= ALPHA <= 1 represents the boundary value for the moduli of
eigenvalues. The ALPHA-stability domain does not include the
boundary. Default value is 0 for continuous-time controllers
and 1 for discrete-time controllers.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of the reduced controller. For model
reduction, the recommended value of TOL1 is c*info.hsvc(1),
where c lies in the interval [0.00001, 0.001]. Default value
is info.ncs*eps*info.hsvc(1). If 'ORDER' is specified, the
value of TOL1 is ignored.
'TOL2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given controller.
TOL2 <= TOL1. If not specified, ncs*eps*info.hsvc(1) is
chosen.
'GRAM-CTRB'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
'STANDARD'
Choice corresponding to standard Enns' method [1].
Default method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
Enns' method of [2].
'GRAM-OBSV'
Specifies the choice of frequency-weighted observability
Grammian as follows:
'STANDARD'
Choice corresponding to standard Enns' method [1].
Default method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
Enns' method of [2].
'EQUIL', 'SCALE'
Boolean indicating whether equilibration (scaling) should be
performed on G and K prior to order reduction. Default value
is false if both 'G.scaled == true, K.scaled == true' and true
otherwise. Note that for MIMO models, proper scaling of both
inputs and outputs is of utmost importance. The input and
output scaling can *not* be done by the equilibration option
or the 'prescale' function because these functions perform
state transformations only. Furthermore, signals should not
be scaled simply to a certain range. For all inputs (or
outputs), a certain change should be of the same importance
for the model.
*Algorithm*
Uses SLICOT SB16AD by courtesy of NICONET e.V.
(http://www.slicot.org)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Controller reduction by frequency-weighted Singular Perturbation
Approximation (
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
spamodred
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7196
-- Function File: [GR, INFO] = spamodred (G, ...)
-- Function File: [GR, INFO] = spamodred (G, NR, ...)
-- Function File: [GR, INFO] = spamodred (G, OPT, ...)
-- Function File: [GR, INFO] = spamodred (G, NR, OPT, ...)
Model order reduction by frequency weighted Singular Perturbation
Approximation (SPA). The aim of model reduction is to find an LTI
system GR of order NR (nr < n) such that the input-output behaviour
of GR approximates the one from original system G.
SPA is an absolute error method which tries to minimize
||G-Gr|| = min
inf
||V (G-Gr) W|| = min
inf
where V and W denote output and input weightings.
*Inputs*
G
LTI model to be reduced.
NR
The desired order of the resulting reduced order system GR.
If not specified, NR is chosen automatically according to the
description of key 'ORDER'.
...
Optional pairs of keys and values. '"key1", value1, "key2",
value2'.
OPT
Optional struct with keys as field names. Struct OPT can be
created directly or by function 'options'. 'opt.key1 =
value1, opt.key2 = value2'.
*Outputs*
GR
Reduced order state-space model.
INFO
Struct containing additional information.
INFO.N
The order of the original system G.
INFO.NS
The order of the ALPHA-stable subsystem of the original
system G.
INFO.HSV
The Hankel singular values of the ALPHA-stable part of
the original system G, ordered decreasingly.
INFO.NU
The order of the ALPHA-unstable subsystem of both the
original system G and the reduced-order system GR.
INFO.NR
The order of the obtained reduced order system GR.
*Option Keys and Values*
'ORDER', 'NR'
The desired order of the resulting reduced order system GR.
If not specified, NR is chosen automatically such that states
with Hankel singular values INFO.HSV > TOL1 are retained.
'LEFT', 'OUTPUT'
LTI model of the left/output frequency weighting V. Default
value is an identity matrix.
'RIGHT', 'INPUT'
LTI model of the right/input frequency weighting W. Default
value is an identity matrix.
'METHOD'
Approximation method for the L-infinity norm to be used as
follows:
'SR', 'S'
Use the square-root Singular Perturbation Approximation
method.
'BFSR', 'P'
Use the balancing-free square-root Singular Perturbation
Approximation method. Default method.
'ALPHA'
Specifies the ALPHA-stability boundary for the eigenvalues of
the state dynamics matrix G.A. For a continuous-time system,
ALPHA <= 0 is the boundary value for the real parts of
eigenvalues, while for a discrete-time system, 0 <= ALPHA <= 1
represents the boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.
Default value is 0 for continuous-time systems and 1 for
discrete-time systems.
'TOL1'
If 'ORDER' is not specified, TOL1 contains the tolerance for
determining the order of the reduced model. For model
reduction, the recommended value of TOL1 is c*info.hsv(1),
where c lies in the interval [0.00001, 0.001]. Default value
is info.ns*eps*info.hsv(1). If 'ORDER' is specified, the
value of TOL1 is ignored.
'TOL2'
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given model. TOL2
<= TOL1. If not specified, ns*eps*info.hsv(1) is chosen.
'GRAM-CTRB'
Specifies the choice of frequency-weighted controllability
Grammian as follows:
'STANDARD'
Choice corresponding to a combination method [4] of the
approaches of Enns [1] and Lin-Chiu [2,3]. Default
method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
combination method of [4].
'GRAM-OBSV'
Specifies the choice of frequency-weighted observability
Grammian as follows:
'STANDARD'
Choice corresponding to a combination method [4] of the
approaches of Enns [1] and Lin-Chiu [2,3]. Default
method.
'ENHANCED'
Choice corresponding to the stability enhanced modified
combination method of [4].
'ALPHA-CTRB'
Combination method parameter for defining the
frequency-weighted controllability Grammian. abs(alphac) <=
1. If alphac = 0, the choice of Grammian corresponds to the
method of Enns [1], while if alphac = 1, the choice of
Grammian corresponds to the method of Lin and Chiu [2,3].
Default value is 0.
'ALPHA-OBSV'
Combination method parameter for defining the
frequency-weighted observability Grammian. abs(alphao) <= 1.
If alphao = 0, the choice of Grammian corresponds to the
method of Enns [1], while if alphao = 1, the choice of
Grammian corresponds to the method of Lin and Chiu [2,3].
Default value is 0.
'EQUIL', 'SCALE'
Boolean indicating whether equilibration (scaling) should be
performed on system G prior to order reduction. Default value
is true if 'G.scaled == false' and false if 'G.scaled ==
true'. Note that for MIMO models, proper scaling of both
inputs and outputs is of utmost importance. The input and
output scaling can *not* be done by the equilibration option
or the 'prescale' function because these functions perform
state transformations only. Furthermore, signals should not
be scaled simply to a certain range. For all inputs (or
outputs), a certain change should be of the same importance
for the model.
*References*
[1] Enns, D. 'Model reduction with balanced realizations: An error
bound and a frequency weighted generalization'. Proc. 23-th CDC,
Las Vegas, pp. 127-132, 1984.
[2] Lin, C.-A. and Chiu, T.-Y. 'Model reduction via
frequency-weighted balanced realization'. Control Theory and
Advanced Technology, vol. 8, pp. 341-351, 1992.
[3] Sreeram, V., Anderson, B.D.O and Madievski, A.G. 'New results
on frequency weighted balanced reduction technique'. Proc. ACC,
Seattle, Washington, pp. 4004-4009, 1995.
[4] Varga, A. and Anderson, B.D.O. 'Square-root balancing-free
methods for the frequency-weighted balancing related model
reduction'. (report in preparation)
*Algorithm*
Uses SLICOT AB09ID by courtesy of NICONET e.V.
(http://www.slicot.org)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Model order reduction by frequency weighted Singular Perturbation
Approximation
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
step
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1661
-- Function File: step (SYS)
-- Function File: step (SYS1, SYS2, ..., SYSN)
-- Function File: step (SYS1, 'STYLE1', ..., SYSN, 'STYLEN')
-- Function File: step (SYS1, ..., T)
-- Function File: step (SYS1, ..., TFINAL)
-- Function File: step (SYS1, ..., TFINAL, DT)
-- Function File: [Y, T, X] = step (SYS)
-- Function File: [Y, T, X] = step (SYS, T)
-- Function File: [Y, T, X] = step (SYS, TFINAL)
-- Function File: [Y, T, X] = step (SYS, TFINAL, DT)
Step response of LTI system. If no output arguments are given, the
response is printed on the screen.
*Inputs*
SYS
LTI model.
T
Time vector. Should be evenly spaced. If not specified, it
is calculated by the poles of the system to reflect adequately
the response transients.
TFINAL
Optional simulation horizon. If not specified, it is
calculated by the poles of the system to reflect adequately
the response transients.
DT
Optional sampling time. Be sure to choose it small enough to
capture transient phenomena. If not specified, it is
calculated by the poles of the system.
'STYLE'
Line style and color, e.g. 'r' for a solid red line or '-.k'
for a dash-dotted black line. See 'help plot' for details.
*Outputs*
Y
Output response array. Has as many rows as time samples
(length of t) and as many columns as outputs.
T
Time row vector.
X
State trajectories array. Has 'length (t)' rows and as many
columns as states.
See also: impulse, initial, lsim.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 28
Step response of LTI system.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
strseq
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 248
-- Function File: STRVEC = strseq (STR, IDX)
Return a cell vector of indexed strings by appending the indices
IDX to the string STR.
strseq ("x", 1:3) = {"x1"; "x2"; "x3"}
strseq ("u", [1, 2, 5]) = {"u1"; "u2"; "u5"}
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Return a cell vector of indexed strings by appending the indices IDX to
the stri
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
sumblk
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 820
-- Function File: S = sumblk (FORMULA)
-- Function File: S = sumblk (FORMULA, N)
Create summing junction S from string FORMULA for name-based
interconnections.
*Inputs*
FORMULA
String containing the formula of the summing junction, e.g.
'e = r - y + d'
N
Signal size. Default value is 1.
*Outputs*
S
State-space model of the summing junction.
*Example*
octave:1> S = sumblk ('e = r - y + d')
S.d =
r y d
e 1 -1 1
Static gain.
octave:2> S = sumblk ('e = r - y + d', 2)
S.d =
r1 r2 y1 y2 d1 d2
e1 1 0 -1 0 1 0
e2 0 1 0 -1 0 1
Static gain.
See also: connect.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 78
Create summing junction S from string FORMULA for name-based
interconnections.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 12
test_control
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1134
-- Script File: test_control
Execute all available tests at once. The Octave control package is
based on the SLICOT (http://www.slicot.org) library. SLICOT needs
BLAS and LAPACK libraries which are also prerequisites for Octave
itself. In case of failing tests, it is highly recommended to use
Netlib's reference BLAS (http://www.netlib.org/blas/) and LAPACK
(http://www.netlib.org/lapack/) for building Octave. Using ATLAS
may lead to sign changes in some entries of the state-space
matrices. In general, these sign changes are not 'wrong' and can
be regarded as the result of state transformations. Such state
transformations (but not input/output transformations) have no
influence on the input-output behaviour of the system. For better
numerics, the control package uses such transformations by default
when calculating the frequency responses and a few other things.
However, arguments like the Hankel singular Values (HSV) must not
change. Differing HSVs and failing algorithms are known for using
Framework Accelerate from Mac OS X 10.7.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 36
Execute all available tests at once.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
tfpoly2str
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 315
-- Function File: STR = tfpoly2str (P)
-- Function File: STR = tfpoly2str (P, TFVAR)
Return the string of polynomial vector P with string TFVAR^-1 as
variable. Note that there is an almost identical function for the
'tfpoly' class which returns a string with TFVAR (not TFVAR^-1) as
variable.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 74
Return the string of polynomial vector P with string TFVAR^-1 as
variable.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
tfpolyones
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 57
Return (pxm) cell of tfpoly([1]). For internal use only.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 33
Return (pxm) cell of tfpoly([1]).
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
tfpolyzeros
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 57
Return (pxm) cell of tfpoly([0]). For internal use only.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 33
Return (pxm) cell of tfpoly([0]).
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
thiran
# name: <cell-element>
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-- Function File: SYS = thiran (TAU, TSAM)
Approximation of continuous-time delay using a discrete-time
allpass Thiran filter.
Thiran filters can approximate continuous-time delays that are
non-integer multiples of the sampling time (fractional delays).
This approximation gives a better matching of the phase shift
between the continuous- and the discrete-time system. If there is
no fractional part in the delay, then the standard discrete-time
delay representation is used.
*Inputs*
TAU
A continuous-time delay, given in time units (seconds).
TSAM
The sampling time of the resulting Thiran filter.
*Outputs*
SYS
Transfer function model of the resulting filter. The order of
the filter is determined automatically.
*Example*
octave:1> sys = thiran (1.33, 0.5)
Transfer function 'sys' from input 'u1' to output ...
0.003859 z^3 - 0.03947 z^2 + 0.2787 z + 1
y1: -----------------------------------------
z^3 + 0.2787 z^2 - 0.03947 z + 0.003859
Sampling time: 0.5 s
Discrete-time model.
octave:2> sys = thiran (1, 0.5)
Transfer function 'sys' from input 'u1' to output ...
1
y1: ---
z^2
Sampling time: 0.5 s
Discrete-time model.
See also: absorbdelay, pade.
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Approximation of continuous-time delay using a discrete-time allpass
Thiran filt
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zpk
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-- Function File: S = zpk ('S')
-- Function File: Z = zpk ('Z', TSAM)
-- Function File: SYS = zpk (SYS)
-- Function File: SYS = zpk (K, ...)
-- Function File: SYS = zpk (Z, P, K, ...)
-- Function File: SYS = zpk (Z, P, K, TSAM, ...)
-- Function File: SYS = zpk (Z, P, K, TSAM, ...)
Create transfer function model from zero-pole-gain data. This is
just a stop-gap compatibility wrapper since zpk models are not yet
implemented.
*Inputs*
SYS
LTI model to be converted to transfer function.
Z
Cell of vectors containing the zeros for each channel. z{i,j}
contains the zeros from input j to output i. In the SISO
case, a single vector is accepted as well.
P
Cell of vectors containing the poles for each channel. p{i,j}
contains the poles from input j to output i. In the SISO
case, a single vector is accepted as well.
K
Matrix containing the gains for each channel. k(i,j) contains
the gain from input j to output i.
TSAM
Sampling time in seconds. If TSAM is not specified, a
continuous-time model is assumed.
...
Optional pairs of properties and values. Type 'set (tf)' for
more information.
*Outputs*
SYS
Transfer function model.
See also: tf, ss, dss, frd.
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Create transfer function model from zero-pole-gain data.
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