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This is Info file octave, produced by Makeinfo-1.64 from the input file octave.tex. START-INFO-DIR-ENTRY * Octave: (octave). Interactive language for numerical computations. END-INFO-DIR-ENTRY Copyright (C) 1996, 1997 John W. Eaton. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions of this manual under the conditions for verbatim copying, provided that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. Permission is granted to copy and distribute translations of this manual into another language, under the above conditions for modified versions. File: octave, Node: numerical, Next: sysprop, Prev: blockdiag, Up: Control Theory Numerical Functions =================== - Function File: are (A, B, C, OPT) Solve the algebraic Riccati equation a' * x + x * a - x * b * x + c = 0 *Inputs* for identically dimensioned square matrices A NxN matrix. B NxN matrix or NxM matrix; in the latter case B is replaced by b:=b*b'. C NxN matrix or PxM matrix; in the latter case C is replaced by c:=c'*c. OPT (optional argument; default = `"B"'): String option passed to `balance' prior to ordered Schur decomposition. *Outputs* X: solution of the ARE. *Method* Laub's Schur method (IEEE Transactions on Automatic Control, 1979) is applied to the appropriate Hamiltonian matrix. - Function File: dare (A, B, C, R, OPT) Return the solution, X of the discrete-time algebraic Riccati equation a' x a - x + a' x b (r + b' x b)^(-1) b' x a + c = 0 *Inputs* A N by N. B N by M. C N by N, symmetric positive semidefinite, or P by N. In the latter case c:=c'*c is used. R M by M, symmetric positive definite (invertible). OPT (optional argument; default = `"B"'): String option passed to `balance' prior to ordered QZ decomposition. *Outputs* X solution of DARE. *Method* Generalized eigenvalue approach (Van Dooren; SIAM J. Sci. Stat. Comput., Vol 2) applied to the appropriate symplectic pencil. See also: Ran and Rodman, "Stable Hermitian Solutions of Discrete Algebraic Riccati Equations," Mathematics of Control, Signals and Systems, Vol 5, no 2 (1992) pp 165-194. - Function File: X = dlyap (A, B) Solve the discrete-time Lyapunov equation *Inputs* A N by N matrix B Matrix: N by N, N by M, or P by N. *Outputs* X: matrix satisfying appropriate discrete time Lyapunov equation. Options: * B is square: solve `a x a' - x + b = 0' * B is not square: X satisfies either a x a' - x + b b' = 0 or a' x a - x + b' b = 0, whichever is appropriate. *Method* Uses Schur decomposition method as in Kitagawa, `An Algorithm for Solving the Matrix Equation X = FXF' + S', International Journal of Control, Volume 25, Number 5, pages 745-753 (1977). Column-by-column solution method as suggested in Hammarling, `Numerical Solution of the Stable, Non-Negative Definite Lyapunov Equation', IMA Journal of Numerical Analysis, Volume 2, pages 303-323 (1982). - Function File : M = gram (A, B) Return controllability grammian M of the continuous time system dx/dt = a x + b u. M satisfies a m + m a' + b b' = 0 . - Function File: lyap (A, B, C) - Function File: lyap (A, B) Solve the Lyapunov (or Sylvester) equation via the Bartels-Stewart algorithm (Communications of the ACM, 1972). If A, B, and C are specified, then `lyap' returns the solution of the Sylvester equation a x + x b + c = 0 If only `(a, b)' are specified, then `lyap' returns the solution of the Lyapunov equation a' x + x a + b = 0 If B is not square, then `lyap' returns the solution of either a' x + x a + b' b = 0 or a x + x a' + b b' = 0 whichever is appropriate. Solves by using the Bartels-Stewart algorithm (1972). - Loadable Function: pinv (X, TOL) Return the pseudoinverse of X. Singular values less than TOL are ignored. If the second argument is omitted, it is assumed that tol = max (size (X)) * sigma_max (X) * eps, where `sigma_max (X)' is the maximal singular value of X. - Function File : X = qzval (A, B) Compute generalized eigenvalues of the matrix pencil (A - lambda B). A and B must be real matrices. *Note* `qzval' is obsolete; use `qz' instead. - Function File : Y = zgfmul(A,B,C,D,X) Compute product of zgep incidence matrix F with vector X. Used by zgepbal (in zgscal) as part of generalized conjugate gradient iteration. - Function File : x = zgfslv(N,M,P,B) solve system of equations for dense zgep problem - Function File : zz = zginit(A,B,C,D) construct right hand side vector zz for the zero-computation generalized eigenvalue problem balancing procedure called by zgepbal - Function File : retsys = zgpbal(Asys) used internally in `tzero'; minimal argument checking performed implementation of zero computation generalized eigenvalue problem balancing method (Hodel and Tiller, Allerton Conference, 1991) Based on Ward's balancing algorithm (SIAM J. Sci Stat. Comput., 1981) zgpbal computes a state/input/output weighting that attempts to reduced the range of the magnitudes of the nonzero elements of [a,b,c,d] The weighting uses scalar multiplication by powers of 2, so no roundoff will occur. zgpbal should be followed by zgpred - Function File : retsys = zgreduce(ASYS,MEPS) Implementation of procedure REDUCE in (Emami-Naeini and Van Dooren, Automatica, # 1982). - Function File : [NONZ, ZER] = zgrownorm (MAT, MEPS) returns NONZ = number of rows of MAT whose two norm exceeds MEPS ZER = number of rows of mat whose two norm is less than meps References: ZGEP: Hodel, "Computation of Zeros with Balancing," 1992, submitted to LAA Generalized CG: Golub and Van Loan, "Matrix Computations, 2nd ed" 1989 - Function File : [a ,b ] = zgsgiv(C,S,A,B) apply givens rotation c,s to row vectors A,B No longer used in zero-balancing (zgpbal); kept for backward compatibility References: *ZGEP* Hodel, "Computation of Zeros with Balancing," 1992, Linear Algebra and its Applications **Generalized CG** Golub and Van Loan, "Matrix Computations, 2nd ed" 1989 File: octave, Node: sysprop, Next: systime, Prev: numerical, Up: Control Theory System Analysis-Properties ========================== - Function File: [N, M, P] = abcddim (A, B, C, D) Check for compatibility of the dimensions of the matrices defining the linear system [A, B, C, D] corresponding to dx/dt = a x + b u y = c x + d u or a similar discrete-time system. If the matrices are compatibly dimensioned, then `abcddim' returns N The number of system states. M The number of system inputs. P The number of system outputs. Otherwise `abcddim' returns N = M = P = -1. Note: n = 0 (pure gain block) is returned without warning. See also: is_abcd - Function File : [Y, MY, NY] = abcddims (X) Used internally in `abcddim'. If X is a zero-size matrix, both dimensions are set to 0 in Y. MY and NY are the row and column dimensions of the result. - Function File : QS = ctrb(SYS {, B}) - Function File : QS = ctrb(A, B) Build controllability matrix 2 n-1 Qs = [ B AB A B ... A B ] of a system data structure or the pair (A, B). *Note* `ctrb' forms the controllability matrix. The numerical properties of `is_controllable' are much better for controllability tests. - Function File : RETVAL = h2norm(SYS) Computes the H2 norm of a system data structure (continuous time only) Reference: Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989 - Function File : [G, GMIN, GMAX] = hinfnorm(SYS{, TOL, GMIN, GMAX, PTOL}) Computes the H infinity norm of a system data structure. *Inputs* SYS system data structure TOL H infinity norm search tolerance (default: 0.001) GMIN minimum value for norm search (default: 1e-9) GMAX maximum value for norm search (default: 1e+9) PTOL pole tolerance: * if sys is continuous, poles with |real(pole)| < ptol*||H|| (H is appropriate Hamiltonian) are considered to be on the imaginary axis. * if sys is discrete, poles with |abs(pole)-1| < ptol*||[s1,s2]|| (appropriate symplectic pencil) are considered to be on the unit circle * Default: 1e-9 *Outputs* G Computed gain, within TOL of actual gain. G is returned as Inf if the system is unstable. GMIN, GMAX Actual system gain lies in the interval [GMIN, GMAX] References: Doyle, Glover, Khargonekar, Francis, "State space solutions to standard H2 and Hinf control problems", IEEE TAC August 1989 Iglesias and Glover, "State-Space approach to discrete-time Hinf control," Int. J. Control, vol 54, #5, 1991 Zhou, Doyle, Glover, "Robust and Optimal Control," Prentice-Hall, 1996 $Revision: 1.9 $ - Function File : QB = obsv (SYS{, C}) Build observability matrix | C | | CA | Qb = | CA^2 | | ... | | CA^(n-1) | of a system data structure or the pair (A, C). Note: `obsv()' forms the observability matrix. The numerical properties of is_observable() are much better for observability tests. - Function File : [ZER, POL]= pzmap (SYS) Plots the zeros and poles of a system in the complex plane. *Inputs* SYS system data structure *Outputs* if omitted, the poles and zeros are plotted on the screen. otherwise, pol, zer are returned as the system poles and zeros. (see sys2zp for a preferable function call) - Function File : RETVAL = is_abcd( A{, B, C, D}) Returns RETVAL = 1 if the dimensions of A, B, C, D are compatible, otherwise RETVAL = 0 with an appropriate diagnostic message printed to the screen. The matrices b, c, or d may be omitted. - Function File : [RETVAL, U] = is_controllable (SYS{, TOL}) - Function File : [RETVAL, U] = is_controllable (A{, B ,TOL}) Logical check for system controllability. *Inputs* SYS system data structure A, B N by N, N by M matrices, respectively TOL optional roundoff paramter. default value: `10*eps' *Outputs* RETVAL Logical flag; returns true (1) if the system SYS or the pair (A,B) is controllable, whichever was passed as input arguments. U U is an orthogonal basis of the controllable subspace. *Method* Controllability is determined by applying Arnoldi iteration with complete re-orthogonalization to obtain an orthogonal basis of the Krylov subspace span ([b,a*b,...,a^{n-1}*b]). The Arnoldi iteration is executed with `krylov' if the system has a single input; otherwise a block Arnoldi iteration is performed with `krylovb'. *See also* `is_observable', `is_stabilizable', `is_detectable', `krylov', `krylovb' - Function File : [RETVAL, U] = is_detectable (A, C{, TOL}) - Function File : [RETVAL, U] = is_detectable (SYS{, TOL}) Test for detactability (observability of unstable modes) of (A,C). Returns 1 if the system A or the pair (A,C)is detectable, 0 if not. *See* `is_stabilizable' for detailed description of arguments and computational method. Default: tol = 10*norm(a,'fro')*eps - Function File : [RETVAL, DGKF_STRUCT ] = is_dgkf (ASYS, NU, NY, TOL ) Determine whether a continuous time state space system meets assumptions of DGKF algorithm. Partitions system into: [dx/dt] = [A | Bw Bu ][w] [ z ] [Cz | Dzw Dzu ][u] [ y ] [Cy | Dyw Dyu ] or similar discrete-time system. If necessary, orthogonal transformations QW, QZ and nonsingular transformations RU, RY are applied to respective vectors W, Z, U, Y in order to satisfy DGKF assumptions. Loop shifting is used if DYU block is nonzero. *Inputs* ASYS system data structure NU number of controlled inputs NY number of measured outputs TOL threshhold for 0. Default: 200EPS *Outputs* RETVAL true(1) if system passes check, false(0) otherwise DGKF_STRUCT data structure of `is_dgkf' results. Entries: NW, NZ dimensions of W, Z A system A matrix BW (N x NW) QW-transformed disturbance input matrix BU (N x NU) RU-transformed controlled input matrix; *Note* B = [Bw Bu] CZ (NZ x N) Qz-transformed error output matrix CY (NY x N) RY-transformed measured output matrix *Note* C = [Cz; Cy] DZU, DYW off-diagonal blocks of transformed D matrix that enter Z, Y from U, W respectively RU controlled input transformation matrix RY observed output transformation matrix DYU_NZ nonzero if the DYU block is nonzero. DYU untransformed DYU block DFLG nonzero if the system is discrete-time `is_dgkf' exits with an error if the system is mixed discrete/continuous *References* *[1]* Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems," IEEE TAC August 1989 *[2]* Maciejowksi, J.M.: "Multivariable feedback design," - Function File : [RETVAL,U] = is_observable (A, C{,TOL}) - Function File : [RETVAL,U] = is_observable (SYS{, TOL}) Logical check for system observability. Default: tol = 10*norm(a,'fro')*eps Returns 1 if the system SYS or the pair (A,C) is observable, 0 if not. *See* `is_controllable' for detailed description of arguments and default values. - Function File : RETVAL = is_sample (TS) return true if TS is a legal sampling time (real,scalar, > 0) - Function File : RETVAL = is_siso (SYS) return nonzero if the system data structure SYS is single-input, single-output. - Function File : [RETVAL, U] = is_stabilizable (SYS{, TOL}) - Function File : [RETVAL, U] = is_stabilizable (A{, B ,TOL}) Logical check for system stabilizability (i.e., all unstable modes are controllable). Test for stabilizability is performed via an ordered Schur decomposition that reveals the unstable subspace of the system A matrix. Returns `retval' = 1 if the system, `a', is stabilizable, if the pair (`a', `b') is stabilizable, or 0 if not. `U' = orthogonal basis of controllable subspace. Controllable subspace is determined by applying Arnoldi iteration with complete re-orthogonalization to obtain an orthogonal basis of the Krylov subspace. span ([b,a*b,...,a^ b]). tol is a roundoff paramter, set to 200*eps if omitted. - Function File: FLG = is_signal_list (MYLIST) Returns true if mylist is a list of individual strings (legal for input to SYSSETSIGNALS). - Function File : RETVAL = is_stable (A{,TOL,DFLG}) - Function File : RETVAL = is_stable (SYS{,TOL}) Returns retval = 1 if the matrix A or the system SYS is stable, or 0 if not. *Inputs* TOL is a roundoff paramter, set to 200*EPS if omitted. DFLG Digital system flag (not required for system data structure): `DFLG != 0' stable if eig(a) in unit circle `DFLG == 0' stable if eig(a) in open LHP (default) File: octave, Node: systime, Next: sysfreq, Prev: sysprop, Up: Control Theory System Analysis-Time Domain =========================== - Function File : DSYS = c2d (SYS{, OPT, T}) - Function File : DSYS = c2d (SYS{, T}) *Inputs* SYS system data structure (may have both continuous time and discrete time subsystems) OPT string argument; conversion option (optional argument; may be omitted as shown above) `"ex"' use the matrix exponential (default) `"bi"' use the bilinear transformation 2(z-1) s = ----- T(z+1) FIXME: This option exits with an error if SYS is not purely continuous. (The `ex' option can handle mixed systems.) T sampling time; required if sys is purely continuous. *Note* If the 2nd argument is not a string, `c2d' assumes that the 2nd argument is T and performs appropriate argument checks. *Outputs* DSYS discrete time equivalent via zero-order hold, sample each T sec. converts the system data structure describing . x = Ac x + Bc u into a discrete time equivalent model x[n+1] = Ad x[n] + Bd u[n] via the matrix exponential or bilinear transform *Note* This function adds the suffix `_d' to the names of the new discrete states. - Function File : CSYS = d2c (SYS{,TOL}) - Function File : CSYS = d2c (SYS, OPT) Convert discrete (sub)system to a purely continuous system. Sampling time used is `sysgettsam(SYS)' *Inputs* SYS system data structure with discrete components TOL Scalar value. tolerance for convergence of default `"log"' option (see below) OPT conversion option. Choose from: `"log"' (default) Conversion is performed via a matrix logarithm. Due to some problems with this computation, it is followed by a steepest descent algorithm to identify continuous time A, B, to get a better fit to the original data. If called as `d2c'(SYS,TOL), TOL=positive scalar, the `"log"' option is used. The default value for TOL is `1e-8'. `"bi"' Conversion is performed via bilinear transform z = (1 + s T / 2)/(1 - s T / 2) where T is the system sampling time (see `sysgettsam'). FIXME: bilinear option exits with an error if SYS is not purely discrete *Outputs* CSYS continuous time system (same dimensions and signal names as in SYS). - Function File : [DSYS, FIDX] = dmr2d (SYS, IDX, SPREFIX, TS2 {,CUFLG}) convert a multirate digital system to a single rate digital system states specified by IDX, SPREFIX are sampled at TS2, all others are assumed sampled at TS1 = `sysgettsam(SYS)'. *Inputs* SYS discrete time system; `dmr2d' exits with an error if SYS is not discrete IDX list of states with sampling time `sysgettsam(SYS)' (may be empty) SPREFIX list of string prefixes of states with sampling time `sysgettsam(SYS)' (may be empty) TS2 sampling time of states not specified by IDX, SPREFIX must be an integer multiple of `sysgettsam(SYS)' CUFLG "constant u flag" if CUFLG is nonzero then the system inputs are assumed to be constant over the revised sampling interval TS2. Otherwise, since the inputs can change during the interval T in [k Ts2, (k+1) Ts2], an additional set of inputs is included in the revised B matrix so that these intersample inputs may be included in the single-rate system. default CUFLG = 1. *Outputs* DSYS equivalent discrete time system with sampling time TS2. The sampling time of sys is updated to TS2. if CUFLG=0 then a set of additional inputs is added to the system with suffixes _d1, ..., _dn to indicate their delay from the starting time k TS2, i.e. u = [u_1; u_1_d1; ..., u_1_dn] where u_1_dk is the input k*Ts1 units of time after u_1 is sampled. (Ts1 is the original sampling time of discrete time sys and TS2 = (n+1)*Ts1) FIDX indices of "formerly fast" states specified by IDX and SPREFIX; these states are updated to the new (slower) sampling interval TS2. *WARNING* Not thoroughly tested yet; especially when CUFLG == 0. - Function File : damp(P{, TSAM}) Displays eigenvalues, natural frequencies and damping ratios of the eigenvalues of a matrix P or the A-matrix of a system P, respectively. If P is a system, TSAM must not be specified. If P is a matrix and TSAM is specified, eigenvalues of P are assumed to be in Z-domain. See also: `eig' - Function File : GM = dcgain(SYS{, tol}) Returns dc-gain matrix. If dc-gain is infinite an empty matrix is returned. The argument TOL is an optional tolerance for the condition number of A-Matrix in SYS (default TOL = 1.0e-10) - Function File : [Y, T] = impulse (SYS{, INP,TSTOP, N}) Impulse response for a linear system. The system can be discrete or multivariable (or both). If no output arguments are specified, `impulse' produces a plot or the impulse response data for system SYS. *Inputs* SYS System data structure. INP Index of input being excited TSTOP The argument TSTOP (scalar value) denotes the time when the simulation should end. N the number of data values. Both parameters TSTOP and N can be omitted and will be computed from the eigenvalues of the A-Matrix. *Outputs* Y, T: impulse response - Function File : [Y, T] = impulse (SYS{, INP,TSTOP, N}) Step response for a linear system. The system can be discrete or multivariable (or both). If no output arguments are specified, `impulse' produces a plot or the step response data for system SYS. *Inputs* SYS System data structure. INP Index of input being excited TSTOP The argument TSTOP (scalar value) denotes the time when the simulation should end. N the number of data values. Both parameters TSTOP and N can be omitted and will be computed from the eigenvalues of the A-Matrix. *Outputs* Y, T: impulse response When invoked with the output paramter y the plot is not displayed. - Function File : [y, t] = stepimp(SITYPE,SYS[, INP, TSTOP, N]) Impulse or step response for a linear system. The system can be discrete or multivariable (or both). This m-file contains the "common code" of step and impulse. Produces a plot or the response data for system sys. Limited argument checking; "do not attempt to do this at home". Used internally in `impulse', `step'. Use `step' or `impulse' instead. File: octave, Node: sysfreq, Next: cacsd, Prev: systime, Up: Control Theory System Analysis-Frequency Domain ================================ *Demonstration/tutorial script* - Function File : frdemo ( ) Octave Controls toolbox demo: Frequency Response demo - Function File : [MAG, PHASE, W] = bode(SYS{,W, OUT_IDX, IN_IDX}) If no output arguments are given: produce Bode plots of a system; otherwise, compute the frequency response of a system data structure *Inputs* SYS a system data structure (must be either purely continuous or discrete; see is_digital) W frequency values for evaluation. if SYS is continuous, then bode evaluates G(jw) where G(s) is the system transfer function. if SYS is discrete, then bode evaluates G(`exp'(jwT)), where * T=`sysgettsam(SYS)' (the system sampling time) and * G(z) is the system transfer function. * Default* the default frequency range is selected as follows: (These steps are NOT performed if W is specified) 1. via routine bodquist, isolate all poles and zeros away from W=0 (JW=0 or `exp'(jwT)=1) and select the frequency range based on the breakpoint locations of the frequencies. 2. if SYS is discrete time, the frequency range is limited to jwT in [0,2 pi /T] 3. A "smoothing" routine is used to ensure that the plot phase does not change excessively from point to point and that singular points (e.g., crossovers from +/- 180) are accurately shown. OUT_IDX, IN_IDX the indices of the output(s) and input(s) to be used in the frequency response; see `sysprune'. *Outputs* MAG, PHASE the magnitude and phase of the frequency response G(jw) or G(`exp'(jwT)) at the selected frequency values. W the vector of frequency values used *Notes* 1. If no output arguments are given, e.g., bode(sys); bode plots the results to the screen. Descriptive labels are automatically placed. Failure to include a concluding semicolon will yield some garbage being printed to the screen (`ans = []'). 2. If the requested plot is for an MIMO system, mag is set to ||G(jw)|| or ||G(`exp'(jwT))|| and phase information is not computed. - Function File : [WMIN, WMAX] = bode_bounds (ZER, POL, DFLG{, TSAM }) Get default range of frequencies based on cutoff frequencies of system poles and zeros. Frequency range is the interval [10^wmin,10^wmax] Used internally in freqresp (`bode', `nyquist') - Function File : [F, W] = bodquist (SYS, W, OUT_IDX, IN_IDX) used internally by bode, nyquist; compute system frequency response. *Inputs* SYS input system structure W range of frequencies; empty if user wants default OUT_IDX list of outputs; empty if user wants all IN_IDX list of inputs; empty if user wants all RNAME name of routine that called bodquist ("bode" or "nyquist") *Outputs* W list of frequencies F frequency response of sys; f(ii) = f(omega(ii)) *Note* bodquist could easily be incorporated into a Nichols plot function; this is in a "to do" list. Both bode and nyquist share the same introduction, so the common parts are in bodquist. It contains the part that finds the number of arguments, determines whether or not the system is SISO, and computes the frequency response. Only the way the response is plotted is different between the two functions. - Function File : [REALP, IMAGP, W] = nyquist (SYS{, W, OUT_IDX, IN_IDX, ATOL}) - Function File : nyquist (SYS{, W, OUT_IDX, IN_IDX, ATOL}) Produce Nyquist plots of a system; if no output arguments are given, Nyquist plot is printed to the screen. Compute the frequency response of a system. *Inputs* (pass as empty to get default values) SYS system data structure (must be either purely continuous or discrete; see is_digital) W frequency values for evaluation. if sys is continuous, then bode evaluates G(jw) if sys is discrete, then bode evaluates G(exp(jwT)), where T=sysgettsam(SYS) (the system sampling time) DEFAULT the default frequency range is selected as follows: (These steps are NOT performed if W is specified) 1. via routine bodquist, isolate all poles and zeros away from W=0 (JW=0 or exp(JWT)=1) and select the frequency range based on the breakpoint locations of the frequencies. 2. if SYS is discrete time, the frequency range is limited to JWT in [0,2p*pi] 3. A "smoothing" routine is used to ensure that the plot phase does not change excessively from point to point and that singular points (e.g., crossovers from +/- 180) are accurately shown. outputs, inputs: the indices of the output(s) and input(s) to be used in the frequency response; see sysprune. *Inputs* (pass as empty to get default values) ATOL for interactive nyquist plots: atol is a change-in-slope tolerance for the of asymptotes (default = 0; 1e-2 is a good choice). This allows the user to "zoom in" on portions of the Nyquist plot too small to be seen with large asymptotes. *Outputs* REALP, IMAGP the real and imaginary parts of the frequency response G(jw) or G(exp(jwT)) at the selected frequency values. W the vector of frequency values used If no output arguments are given, nyquist plots the results to the screen. If ATOL != 0 and asymptotes are detected then the user is asked interactively if they wish to zoom in (remove asymptotes) Descriptive labels are automatically placed. Note: if the requested plot is for an MIMO system, a warning message is presented; the returned information is of the magnitude ||G(jw)|| or ||G(exp(jwT))|| only; phase information is not computed. - Function File : ZR = tzero2 (A, B, C, D, BAL) Compute the transmission zeros of a, b, c, d. bal = balancing option (see balance); default is "B". Needs to incorporate `mvzero' algorithm to isolate finite zeros; use `tzero' instead. File: octave, Node: cacsd, Next: misc, Prev: sysfreq, Up: Control Theory Controller Design ================= - Function File : dgkfdemo ( ) Octave Controls toolbox demo: H2/Hinfinity options demos - Function File: hinfdemo () H_infinity design demos for continuous SISO and MIMO systems and a discrete system. The SISO system is difficult to control because it is non minimum phase and unstable. The second design example controls the "jet707" plant, the linearized state space model of a Boeing 707-321 aircraft at v=80m/s (M = 0.26, Ga0 = -3 deg, alpha0 = 4 deg, kappa = 50 deg). Inputs: (1) thrust and (2) elevator angle outputs: (1) airspeed and (2) pitch angle. The discrete system is a stable and second order. SISO plant s - 2 G(s) = -------------- (s + 2)(s - 1) +----+ -------------------->| W1 |---> v1 z | +----+ ----|-------------+ || T || => min. | | vz infty | +---+ v y +----+ u *--->| G |--->O--*-->| W2 |---> v2 | +---+ | +----+ | | | +---+ | -----| K |<------- +---+ W1 und W2 are the robustness and performance weighting functions MIMO plant The optimal controller minimizes the H_infinity norm of the augmented plant P (mixed-sensitivity problem): w 1 -----------+ | +----+ +---------------------->| W1 |----> z1 w | | +----+ 2 ------------------------+ | | | | v +----+ v +----+ +--*-->o-->| G |-->o--*-->| W2 |---> z2 | +----+ | +----+ | | ^ v u (from y (to K) controller K) + + + + | z | | w | | 1 | | 1 | | z | = [ P ] * | w | | 2 | | 2 | | y | | u | + + + + DISCRETE SYSTEM This is not a true discrete design. The design is carried out in continuous time while the effect of sampling is described by a bilinear transformation of the sampled system. This method works quite well if the sampling period is "small" compared to the plant time constants. The continuous plant 1 G (s) = -------------- k (s + 2)(s + 1) is discretised with a ZOH (Sampling period = Ts = 1 second): 0.199788z + 0.073498 G(s) = -------------------------- (z - 0.36788)(z - 0.13534) +----+ -------------------->| W1 |---> v1 z | +----+ ----|-------------+ || T || => min. | | vz infty | +---+ v +----+ *--->| G |--->O--*-->| W2 |---> v2 | +---+ | +----+ | | | +---+ | -----| K |<------- +---+ W1 and W2 are the robustness and performancs weighting functions - Function File: [L, M, P, E] = dlqe (A, G, C, SIGW, SIGV, Z) Construct the linear quadratic estimator (Kalman filter) for the discrete time system x[k+1] = A x[k] + B u[k] + G w[k] y[k] = C x[k] + D u[k] + w[k] where W, V are zero-mean gaussian noise processes with respective intensities `SIGW = cov (W, W)' and `SIGV = cov (V, V)'. If specified, Z is `cov (W, V)'. Otherwise `cov (W, V) = 0'. The observer structure is z[k+1] = A z[k] + B u[k] + k (y[k] - C z[k] - D u[k]) The following values are returned: L The observer gain, (A - ALC). is stable. M The Riccati equation solution. P The estimate error covariance after the measurement update. E The closed loop poles of (A - ALC). - Function File: [K, P, E] = dlqr (A, B, Q, R, Z) Construct the linear quadratic regulator for the discrete time system x[k+1] = A x[k] + B u[k] to minimize the cost functional J = Sum (x' Q x + u' R u) Z omitted or J = Sum (x' Q x + u' R u + 2 x' Z u) Z included. The following values are returned: K The state feedback gain, (A - BK) is stable. P The solution of algebraic Riccati equation. E The closed loop poles of (A - BK). *References* 1. Anderson and Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, 1990, pp. 56-58 2. Kuo, Digital Control Systems, Harcourt Brace Jovanovich, 1992, section 11-5-2. - Function File : [K , GAIN, KC, KF, PC, PF] = h2syn(ASYS, NU, NY, TOL) Design H2 optimal controller per procedure in Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989 Discrete time control per Zhou, Doyle, and Glover, ROBUST AND OPTIMAL CONTROL, Prentice-Hall, 1996 *Inputs* input system is passed as either ASYS system data structure (see ss2sys, sys2ss) * controller is implemented for continuous time systems * controller is NOT implemented for discrete time systems NU number of controlled inputs NY number of measured outputs TOL threshhold for 0. Default: 200*eps *Outputs* K system controller GAIN optimal closed loop gain KC full information control (packed) KF state estimator (packed) PC ARE solution matrix for regulator subproblem PF ARE solution matrix for filter subproblem - Function File : K = hinf_ctr(DGS, F, H, Z, G) Called by `hinfsyn' to compute the H_inf optimal controller. *Inputs* DGS data structure returned by `is_dgkf' F, H feedback and filter gain (not partitioned) G final gamma value *Outputs* controller K (system data structure) Do not attempt to use this at home; no argument checking performed. - Function File : [K, G, GW, XINF, YINF] = hinfsyn(ASYS, NU, NY, GMIN, GMAX, GTOL{, PTOL, TOL}) *Inputs* input system is passed as either ASYS system data structure (see ss2sys, sys2ss) * controller is implemented for continuous time systems * controller is NOT implemented for discrete time systems (see bilinear transforms in `c2d', `d2c') NU number of controlled inputs NY number of measured outputs GMIN initial lower bound on H-infinity optimal gain GMAX initial upper bound on H-infinity optimal gain GTOL gain threshhold. Routine quits when gmax/gmin < 1+tol PTOL poles with abs(real(pole)) < ptol*||H|| (H is appropriate Hamiltonian) are considered to be on the imaginary axis. Default: 1e-9 TOL threshhold for 0. Default: 200*eps GMAX, MIN, TOL, and TOL must all be postive scalars. *Outputs* K system controller G designed gain value GW closed loop system XINF ARE solution matrix for regulator subproblem YINF ARE solution matrix for filter subproblem 1. Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems," IEEE TAC August 1989 2. Maciejowksi, J.M., "Multivariable feedback design," Addison-Wesley, 1989, ISBN 0-201-18243-2 3. Keith Glover and John C. Doyle, "State-space formulae for all stabilizing controllers that satisfy and h-infinity-norm bound and relations to risk sensitivity," Systems & Control Letters 11, Oct. 1988, pp 167-172. - Function File: [RETVAL, PC, PF] = hinfsyn_chk(A, B1, B2, C1, C2, D12, D21, G, PTOL) Called by `hinfsyn' to see if gain G satisfies conditions in Theorem 3 of Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989 *Warning* Do not attempt to use this at home; no argument checking performed. *Inputs* as returned by `is_dgkf', except for: G candidate gain level PTOL as in `hinfsyn' *Outputs* RETVAL 1 if g exceeds optimal Hinf closed loop gain, else 0 PC solution of "regulator" H-inf ARE PF solution of "filter" H-inf ARE Do not attempt to use this at home; no argument checking performed. - Function File: [K, P, E] = lqe (A, G, C, SIGW, SIGV, Z) Construct the linear quadratic estimator (Kalman filter) for the continuous time system dx -- = a x + b u dt y = c x + d u where W and V are zero-mean gaussian noise processes with respective intensities sigw = cov (w, w) sigv = cov (v, v) The optional argument Z is the cross-covariance `cov (W, V)'. If it is omitted, `cov (W, V) = 0' is assumed. Observer structure is `dz/dt = A z + B u + k (y - C z - D u)' The following values are returned: K The observer gain, (A - KC) is stable. P The solution of algebraic Riccati equation. E The vector of closed loop poles of (A - KC). See also: h2syn, lqe, lqr - Function File: [K, P, E] = lqr (A, B, Q, R, Z) construct the linear quadratic regulator for the continuous time system dx -- = A x + B u dt to minimize the cost functional infinity / J = | x' Q x + u' R u / t=0 Z omitted or infinity / J = | x' Q x + u' R u + 2 x' Z u / t=0 Z included. The following values are returned: K The state feedback gain, (A - BK) is stable and minimizes the cost functional P The stabilizing solution of appropriate algebraic Riccati equation. E The vector of the closed loop poles of (A - BK). *Reference* Anderson and Moore, OPTIMAL CONTROL: LINEAR QUADRATIC METHODS, Prentice-Hall, 1990, pp. 56-58 - Function File : lsim (SYS, U, T{,X0}) Produce output for a linear simulation of a system Produces a plot for the output of the system, sys. U is an array that contains the system's inputs. Each column in u corresponds to a different time step. Each row in u corresponds to a different input. T is an array that contains the time index of the system. T should be regularly spaced. If initial conditions are required on the system, the x0 vector should be added to the argument list. When the lsim function is invoked with output parameters: [y,x] = lsim(sys,u,t,[x0]) a plot is not displayed, however, the data is returned in y = system output and x = system states. - Function File : K = place (SYS, P) Computes the matrix K such that if the state is feedback with gain K, then the eigenvalues of the closed loop system (i.e. A-BK) are those specified in the vector P. Version: Beta (May-1997): If you have any comments, please let me know. (see the file place.m for my address) Written by: Jose Daniel Munoz Frias. File: octave, Node: misc, Prev: cacsd, Up: Control Theory Miscellaneous Functions (Not yet properly filed/documented) =========================================================== - Function File : AXVEC = axis2dlim (AXDATA) determine axis limits for 2-d data(column vectors); leaves a 10% margin around the plots. puts in margins of +/- 0.1 if data is one dimensional (or a single point) *Inputs* AXDATA nx2 matrix of data [x,y] *Outputs* AXVEC vector of axis limits appropriate for call to axis() function - Function File : outputs = moddemo ( inputs ) Octave Controls toolbox demo: Model Manipulations demo Written by David Clem August 15, 1994 - Function File : outputs = prompt ( inputs ) function prompt([str]) Prompt user to continue str: input string. Default value: "\n ---- Press a key to continue ---" Written by David Clem August 15, 1994 Modified A. S. Hodel June 1995 - Function File : outputs = rldemo ( inputs ) Octave Controls toolbox demo: Root Locus demo - Function File : outputs = rlocus ( inputs ) [rldata, k] = rlocus(sys[,increment,min_k,max_k]) Displays root locus plot of the specified SISO system. ----- --- -------- --->| + |---|k|---->| SISO |-----------> ----- --- -------- | - ^ | |_____________________________| inputs: sys = system data structure min_k, max_k,increment: minimum, maximum values of k and the increment used in computing gain values Outputs: plots the root locus to the screen. rldata: Data points plotted column 1: real values, column 2: imaginary values) k: gains for real axis break points. - Function File : outputs = ss2tf ( inputs ) [num,den] = ss2tf(a,b,c,d) Conversion from tranfer function to state-space. The state space system . x = Ax + Bu y = Cx + Du is converted to a transfer function num(s) G(s)=------- den(s) used internally in system data structure format manipulations - Function File : outputs = ss2zp ( inputs ) Converts a state space representation to a set of poles and zeros. [pol,zer,k] = ss2zp(a,b,c,d) returns the poles and zeros of the state space system (a,b,c,d). K is a gain associated with the zeros. used internally in system data structure format manipulations - Function File : outputs = susball ( inputs ) - Function File : [A, B, C, D] = zp2ss (ZER, POL, K) Conversion from zero / pole to state space. *Inputs* ZER, POL vectors of (possibly) complex poles and zeros of a transfer function. Complex values must come in conjugate pairs (i.e., x+jy in zer means that x-jy is also in zer) K real scalar (leading coefficient) *Outputs* A, B, C, D The state space system . x = Ax + Bu y = Cx + Du is obtained from a vector of zeros and a vector of poles via the function call `[a,b,c,d] = zp2ss(zer,pol,k)'. The vectors `zer' and `pol' may either be row or column vectors. Each zero and pole that has an imaginary part must have a conjugate in the list. The number of zeros must not exceed the number of poles. `k' is `zp'-form leading coefficient. - Function File : [POLY, RVALS] = zp2ssg2 (RVALS) Used internally in `zp2ss' Extract 2 values from RVALS (if possible) and construct a polynomial with those roots. - Function File : [NUM, DEN] = zp2tf (ZER, POL, K) Converts zeros / poles to a transfer function. *Inputs* ZER, POL vectors of (possibly complex) poles and zeros of a transfer function. Complex values should appear in conjugate pairs K real scalar (leading coefficient) `[num,den] = zp2tf(zer,pol,k)' forms the transfer function `num/den' from the vectors of poles and zeros. % DO NOT EDIT! Generated automatically by munge-texi.