--- /dev/null
+# Created by Octave 3.6.2, Mon Jun 25 21:47:37 2012 UTC <root@t61>
+# name: cache
+# type: cell
+# rows: 3
+# columns: 71
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 8
+Anderson
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 64
+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: 1283
+ -- 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>
+# type: sq_string
+# elements: 1
+# length: 70
+Model of the BMW 4-cylinder engine at ETH Zurich's control laboratory.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+Boeing707
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 392
+ -- 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>
+# type: sq_string
+# elements: 1
+# length: 80
+Creates a linearized state-space model of a Boeing 707-321 aircraft at
+V=80 m/s
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+MDSSystem
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 156
+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
+# elements: 1
+# length: 46
+Robust control of a mass-damper-spring system.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+Madievski
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 42
+Frequency-weighted controller reduction.
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 40
+Frequency-weighted controller reduction.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 12
+WestlandLynx
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1288
+ -- 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]
+
+ Reference:
+ Skogestad, S. and Postlethwaite I.
+ Multivariable Feedback Control: Analysis and Design
+ Second Edition
+ Wiley 2005
+ http://www.nt.ntnu.no/users/skoge/book/2nd_edition/matlab_m/matfiles.html
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 50
+Model of the Westland Lynx Helicopter about hover.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+augw
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 3059
+ -- 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) |<----+
+ +--------+
+
+ Reference:
+ Skogestad, S. and Postlethwaite I.
+ Multivariable Feedback Control: Analysis and Design
+ Second Edition
+ Wiley 2005
+ Chapter 3.8: General Control Problem Formulation
+
+ See also: h2syn, hinfsyn, mixsyn
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 40
+Extend plant for stacked S/KS/T problem.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+bode
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 890
+ -- 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.
+
+ *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
+# elements: 1
+# length: 35
+Bode diagram of frequency response.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+bodemag
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 831
+ -- 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.
+
+ *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
+# length: 45
+Bode magnitude diagram of frequency response.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+bstmodred
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6097
+ -- 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 command `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' command 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)
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 69
+Model order reduction by Balanced Stochastic Truncation (BST) method.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+btaconred
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6173
+ -- 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 command `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' command 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 Balanced Truncation
+Approximation (BT
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+btamodred
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7326
+ -- 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 command `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' command
+ 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
+# length: 80
+Model order reduction by frequency weighted Balanced Truncation
+Approximation (B
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+care
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1627
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 55
+Solve continuous-time algebraic Riccati equation (ARE).
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 8
+cfconred
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4348
+ -- 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 command `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' command 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>
+# type: sq_string
+# elements: 1
+# length: 80
+Reduction of state-feedback-observer based controller by coprime
+factorization (
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 5
+covar
+
+
+# name: <cell-element>
+# 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 35
+Return the steady-state covariance.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+ctrb
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 390
+ -- Function File: CO = ctrb (SYS)
+ -- Function File: CO = ctrb (A, B)
+ Return controllability matrix.
+
+ *Inputs*
+ SYS
+ LTI model.
+
+ A
+ State transition 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
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# 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.
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+If Co=ctrb(A,B) has rank r <= n = SIZE(A,1), then there is a similarity
+transfor
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+dare
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1786
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 53
+Solve discrete-time algebraic Riccati equation (ARE).
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+dlqe
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1695
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 40
+Kalman filter for discrete-time systems.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+dlqr
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 53
+Linear-quadratic regulator for discrete-time systems.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 5
+dlyap
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 527
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 52
+Solve discrete-time Lyapunov or Sylvester equations.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+dlyapchol
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 478
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 60
+Compute Cholesky factor of discrete-time Lyapunov equations.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 3
+dss
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 935
+ -- 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 transition matrix (n-by-n).
+
+ B
+ Input matrix (n-by-m).
+
+ C
+ Measurement 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.
+
+ *Equations*
+ .
+ E x = A x + B u
+ y = C x + D u
+
+ See also: ss, tf
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# 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: 654
+ -- 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. If argument
+ KNOWN is omitted, no inputs u are known.
+
+ *Outputs*
+ EST
+ State-space model of estimator.
+
+ See also: kalman, place
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 50
+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: 1602
+ -- 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.
+
+ *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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 69
+Create discrete-time transfer function model from data in DSP format.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+fitfrd
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1168
+ -- 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
+# elements: 1
+# length: 54
+Fit frequency response data with a state-space system.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 10
+fwcfconred
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 3221
+ -- 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 command `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)
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+Reduction of state-feedback-observer based controller by
+frequency-weighted copr
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+gensig
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# 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>
+# type: sq_string
+# elements: 1
+# length: 25
+Generate periodic signal.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+gram
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 462
+ -- 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
+# elements: 1
+# length: 80
+`gram (SYS, "c")' returns the controllability gramian of the
+(continuous- or dis
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 5
+h2syn
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1997
+ -- Function File: [K, N, GAMMA, RCOND] = h2syn (P, NMEAS, NCON)
+ H-2 control synthesis for LTI plant.
+
+ *Inputs*
+ P
+ Generalized plant. Must be a proper/realizable LTI model.
+
+ 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.
+
+ GAMMA
+ H-2 norm of N.
+
+ 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 algorithm.
+
+ *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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 36
+H-2 control synthesis for LTI plant.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+hinfsyn
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 2269
+ -- Function File: [K, N, GAMMA, RCOND] = hinfsyn (P, NMEAS, NCON)
+ -- Function File: [K, N, GAMMA, RCOND] = hinfsyn (P, NMEAS, NCON, GMAX)
+ H-infinity control synthesis for LTI plant.
+
+ *Inputs*
+ P
+ Generalized plant. Must be a proper/realizable LTI model.
+
+ 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.
+
+ 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.
+
+ *Outputs*
+ K
+ State-space model of the H-infinity (sub-)optimal controller.
+
+ N
+ State-space model of the lower LFT of P and K.
+
+ GAMMA
+ L-infinity norm of N.
+
+ 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 algorithm.
+
+ *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 and SB10DD by courtesy of NICONET e.V.
+ (http://www.slicot.org)
+
+ See also: augw, mixsyn
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 43
+H-infinity control synthesis for LTI plant.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+hnamodred
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6621
+ -- 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 command `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' command 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)
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 77
+Model order reduction by frequency weighted optimal Hankel-norm (HNA)
+method.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+hsvd
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 403
+ -- 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)
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 58
+Hankel singular values of the stable part of an LTI model.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+impulse
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1247
+ -- 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.
+
+ *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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 31
+Impulse response of LTI system.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+initial
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1546
+ -- 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.
+
+ *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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 48
+Initial condition response of state-space model.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+isctrb
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1096
+ -- 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.
+
+ A
+ State transition matrix.
+
+ B
+ Input matrix.
+
+ E
+ Descriptor matrix. 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 41
+Logical check for system controllability.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 12
+isdetectable
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1509
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 38
+Logical test for system detectability.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+isobsv
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1094
+ -- 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 transition matrix.
+
+ C
+ Measurement matrix.
+
+ E
+ Descriptor matrix. 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 39
+Logical check for system observability.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 8
+issample
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 614
+ -- 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.
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 43
+Return true if TS is a valid sampling time.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 14
+isstabilizable
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1786
+ -- 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.
+
+ 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 41
+Logical check for system stabilizability.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+kalman
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1539
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 40
+Design Kalman estimator for LTI systems.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 3
+lqe
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1696
+ -- 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 transition 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 42
+Kalman filter for continuous-time systems.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 3
+lqr
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1353
+ -- 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 transition 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 27
+Linear-quadratic regulator.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+lsim
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1744
+ -- 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)
+ -- Function File: [Y, T, X] = lsim (SYS, U, T, [], METHOD)
+ -- Function File: [Y, T, X] = lsim (SYS, U, T, X0, METHOD)
+ 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.
+
+ METHOD
+ Discretization method for continuous-time models. Default
+ value is zoh (zero-order hold). All methods from `c2d' are
+ supported.
+
+ *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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 48
+Simulate LTI model response to arbitrary inputs.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 9
+ltimodels
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 144
+ -- Function File: test ltimodels
+ -- Function File: ltimodels
+ -- Function File: ltimodels (SYSTYPE)
+ Test suite and help for LTI models.
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 35
+Test suite and help for LTI models.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+lyap
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 527
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 54
+Solve continuous-time Lyapunov or Sylvester equations.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 8
+lyapchol
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 479
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 62
+Compute Cholesky factor of continuous-time Lyapunov equations.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+margin
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 2742
+ -- 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.
+
+ *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).
+
+ *Equations*
+ CONTINUOUS 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 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: roots
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 34
+Gain and phase margin of a system.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+mixsyn
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4424
+ -- Function File: [K, N, GAMMA, RCOND] = mixsyn (G, W1, W2, W3, ...)
+ Solve stacked S/KS/T H-infinity problem. Bound the largest
+ singular values of S (for performance), K S (to penalize large
+ inputs) and T (for robustness and to avoid sensitivity to noise).
+ In other words, the inputs r are excited by a harmonic test signal.
+ Then the algorithm tries to find a controller K which minimizes
+ the H-infinity norm calculated from the outputs z.
+
+ *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.
+
+ GAMMA
+ L-infinity norm of N.
+
+ 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 algorithm.
+
+ *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)
+
+ Reference:
+ Skogestad, S. and Postlethwaite I.
+ Multivariable Feedback Control: Analysis and Design
+ Second Edition
+ Wiley 2005
+ Chapter 3.8: General Control Problem Formulation
+
+ *Algorithm*
+ Relies on commands `augw' and `hinfsyn', which use SLICOT SB10FD
+ and SB10DD by courtesy of NICONET e.V. (http://www.slicot.org)
+
+ See also: hinfsyn, augw
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 40
+Solve stacked S/KS/T H-infinity problem.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 6
+ncfsyn
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 2310
+ -- Function File: [K, N, GAMMA, INFO] = ncfsyn (G, W1, W2, FACTOR)
+ Loop shaping H-infinity synthesis. Compute positive feedback
+ controller using the McFarlane/Glover normalized coprime factor
+ (NCF) loop shaping design procedure.
+
+ *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.
+
+ N
+ State-space model of the closed loop depicted below.
+
+ GAMMA
+ L-infinity norm of N. `gamma = norm (N, inf)'.
+
+ INFO
+ Structure containing additional information.
+
+ 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 algorithm.
+
+ *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)
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 34
+Loop shaping H-infinity synthesis.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+nichols
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 894
+ -- 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.
+
+ *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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 36
+Nichols chart of frequency response.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+nyquist
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 901
+ -- 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.
+
+ *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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 38
+Nyquist diagram of frequency response.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 4
+obsv
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 459
+ -- Function File: OB = obsv (SYS)
+ -- Function File: OB = obsv (A, C)
+ Return observability matrix.
+
+ *Inputs*
+ SYS
+ LTI model.
+
+ A
+ State transition matrix (n-by-n).
+
+ C
+ Measurement matrix (p-by-n).
+
+ *Outputs*
+ OB
+ Observability matrix.
+
+ *Equation*
+ | C |
+ | CA |
+ Ob = | CA^2 |
+ | ... |
+ | CA^(n-1) |
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 28
+Return observability matrix.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 5
+obsvf
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 925
+ -- 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.
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+If Ob=obsv(A,C) has rank r <= n = SIZE(A,1), then there is a similarity
+transfor
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+optiPID
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 235
+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.
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 71
+Numerical optimization of a PID controller using an objective function.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 11
+optiPIDctrl
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 397
+ ===============================================================================
+ optiPIDctrl Lukas Reichlin February 2012
+ ===============================================================================
+ Return PID controller with roll-off for given parameters Kp, Ti and Td.
+ ===============================================================================
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+ ===============================================================================
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 10
+optiPIDfun
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 464
+ ===============================================================================
+ optiPIDfun Lukas Reichlin July 2009
+ ===============================================================================
+ Objective Function
+ Reference: Guzzella, L. (2007) Analysis and Synthesis of SISO Control Systems.
+ vdf Hochschulverlag, Zurich
+ ===============================================================================
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 80
+ ===============================================================================
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 7
+options
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 791
+ -- 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 commands.
+
+ *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>
+# type: sq_string
+# elements: 1
+# length: 63
+Create options struct OPT from a number of key and value pairs.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 5
+place
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1978
+ -- 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
+ LTI system.
+
+ A
+ State transition 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)
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 74
+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: 415
+ -- Function File: pzmap (SYS)
+ -- 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.
+
+ *Outputs*
+ P
+ Poles of SYS.
+
+ Z
+ Transmission 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: 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.
+
+ MIN_K
+ Minimum value of K.
+
+ MAX_K
+ Maximum value of K.
+
+ INCREMENT
+ The increment used in computing gain values.
+
+ *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: 5
+sigma
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1643
+ -- Function File: [SV, W] = sigma (SYS)
+ -- Function File: [SV, W] = sigma (SYS, W)
+ -- Function File: [SV, W] = sigma (SYS, [], PTYPE)
+ -- Function File: [SV, W] = sigma (SYS, W, PTYPE)
+ 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.
+
+ PTYPE = 0
+ Singular values of the frequency response H of system SYS.
+ Default Value.
+
+ PTYPE = 1
+ Singular values of the frequency response `inv(H)'; i.e.
+ inversed system.
+
+ PTYPE = 2
+ Singular values of the frequency response `I + H'; i.e.
+ inversed sensitivity (or return difference) if `H = P * C'.
+
+ PTYPE = 3
+ Singular values of the frequency response `I + inv(H)'; i.e.
+ inversed complementary sensitivity if `H = P * C'.
+
+ *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: 6224
+ -- 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 command `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' command 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: 7167
+ -- 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 command `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' command 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: 1235
+ -- 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.
+
+ *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: 250
+ -- 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: 12
+test_control
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1089
+ -- 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 a LAPACK library which is also a prerequisite for Octave
+ itself. In case of failing test, it is highly recommended to use
+ Netlib's reference LAPACK (http://www.netlib.org/lapack/) for
+ building Octave. Using ATLAS may lead to sign changes in some
+ entries in 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: 158
+ -- Function File: STR = tfpoly2str (P)
+ -- Function File: STR = tfpoly2str (P, TFVAR)
+ Return the string of a polynomial with string TFVAR as variable.
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 64
+Return the string of a polynomial with string TFVAR as variable.
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 10
+tfpolyones
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 59
+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: 59
+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: 3
+zpk
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 1373
+ -- 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
+
+
+
+
+
+# name: <cell-element>
+# type: sq_string
+# elements: 1
+# length: 56
+Create transfer function model from zero-pole-gain data.
+
+
+
+
+
git://git.creatis.insa-lyon.fr / CreaPhase.git / blobdiff