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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: ord2, Next: parallel, Prev: jet707, Up: blockdiag - Function File : OUTSYS = ord2 (NFREQ, DAMP{[, GAIN}) Creates a continuous 2nd order system with parameters: *Inputs* NFREQ: NATURAL FREQUENCY [HZ]. (NOT IN RAD/S) DAMP: DAMPING COEFFICIENT GAIN: DC-GAIN This is steady state value only for damp > 0. gain is assumed to be 1.0 if ommitted. *Outputs* OUTSYS system data structure has representation with `w = 2 * pi * nfreq': / \ | / -2w*damp -w \ / w \ | G = | | |, | |, [ 0 gain ], 0 | | \ w 0 / \ 0 / | \ / *See also* `jet707' (MIMO example, Boeing 707-321 aircraft model) File: octave, Node: parallel, Next: sysadd, Prev: ord2, Up: blockdiag - Function File : SYSP = parallel(ASYS, BSYS) Forms the parallel connection of two systems. ____________________ | ________ | u ----->|----> | Asys |--->|----> y1 | | -------- | | | ________ | |--->|----> | Bsys |--->|----> y2 | -------- | -------------------- Ksys File: octave, Node: sysadd, Next: sysappend, Prev: parallel, Up: blockdiag - Function File : SYS = sysadd ( GSYS,HSYS) returns SYS = GSYS + HSYS. * Exits with an error if GSYS and HSYS are not compatibly dimensioned. * Prints a warning message is system states have identical names; duplicate names are given a suffix to make them unique. * SYS input/output names are taken from GSYS. ________ ----| Gsys |--- u | ---------- +| ----- (_)----> y | ________ +| ----| Hsys |--- -------- File: octave, Node: sysappend, Next: sysconnect, Prev: sysadd, Up: blockdiag - Function File : RETSYS = sysappend (SYS,B{, C, D, OUTNAME, INNAME, YD}) appends new inputs and/or outputs to a system *Inputs* SYS system data structure B matrix to be appended to sys "B" matrix (empty if none) C matrix to be appended to sys "C" matrix (empty if none) D revised sys d matrix (can be passed as [] if the revised d is all zeros) OUTNAME list of names for new outputs INNAME list of names for new inputs YD binary vector; `yd(ii)=0' indicates a continuous output; `yd(ii)=1' indicates a discrete output. *Outputs* SYS sys.b := [sys.b , b] sys.c := [sys.c ] [ c ] sys.d := [sys.d | D12 ] [D21 | D22 ] where D12, D21, and D22 are the appropriate dimensioned blocks of the input parameter D. * The leading block D11 of D is ignored. * If INNAME and OUTNAME are not given as arguments, the new inputs and outputs are be assigned default names. * YD is a binary vector of length rows(c) that indicates continuous/sampled outputs. Default value for YD is: * SYS = continuous or mixed YD = `zeros(1,rows(c))' * SYS = discrete YD = `ones(1,rows(c))' File: octave, Node: sysconnect, Next: syscont, Prev: sysappend, Up: blockdiag - Function File : RETSYS = sysconnect (SYS, OUT_IDX,IN_IDX{,ORDER, TOL}) Close the loop from specified outputs to respective specified inputs *Inputs* SYS system data structure OUT_IDX, IN_IDX list of connections indices; `y(out_idx(ii))' is connected to `u(in_idx(ii))'. ORDER logical flag (default = 0) `0' leave inputs and outputs in their original order `1' permute inputs and outputs to the order shown in the diagram below TOL tolerance for singularities in algebraic loops default: 200EPS *Outputs* SYS: resulting closed loop system. *Method* `sysconnect' internally permutes selected inputs, outputs as shown below, closes the loop, and then permutes inputs and outputs back to their original order ____________________ u_1 ----->| |----> y_1 | sys | old u_2 | | u_2* ---->(+)--->| |----->y_2 (in_idx) ^ -------------------| | (out_idx) | | ------------------------------- The input that has the summing junction added to it has an * added to the end of the input name. File: octave, Node: syscont, Next: syscont_disc, Prev: sysconnect, Up: blockdiag - Function File: [CSYS, ACD, CCD] = syscont (SYS) Extract the purely continuous subsystem of an input system. *Inputs* SYS is a system data structure *Outputs* CSYS is the purely continuous input/output connections of SYS ACD, CCD: connections from discrete states to continuous states, discrete states to continuous outputs, respectively. returns CSYS empty if no continuous/continous path exists File: octave, Node: syscont_disc, Next: sysdisc, Prev: syscont, Up: blockdiag - Function File : [N_TOT, ST_C, ST_D, Y_C, Y_D] = syscont_disc(SYS) Used internally in syscont and sysdisc. *Inputs* SYS is a system data structure. *Outputs* N_TOT total number of states ST_C vector of continuous state indices (empty if none) ST_D vector of discrete state indices (empty if none) Y_C vector of continuous output indices Y_D vector of discrete output indices File: octave, Node: sysdisc, Next: sysdup, Prev: syscont_disc, Up: blockdiag - Function File : [DSYS, ADC, CDC] = sysdisc (SYS) *Inputs* SYS = system data structure *Outputs* DSYS purely discrete portion of sys (returned empty if there is no purely discrete path from inputs to outputs) ADC, CDC connections from continuous states to discrete states and discrete outputs, respectively. File: octave, Node: sysdup, Next: sysgroup, Prev: sysdisc, Up: blockdiag - Function File : RETSYS = sysdup (ASYS, OUT_IDX, IN_IDX) Duplicate specified input/output connections of a system *Inputs* ASYS system data structure (*Note ss2sys::) OUT_IDX,IN_IDX list of connections indices; duplicates are made of Y(OUT_IDX(II)) and U(IN_IDX(II)). *Outputs* RETSYS: resulting closed loop system: duplicated i/o names are appended with a `"+"' suffix. *Method* `sysdup' creates copies of selected inputs and outputs as shown below. u1/y1 is the set of original inputs/outputs, and u2,y2 is the set of duplicated inputs/outputs in the order specified in IN_IDX, OUT_IDX, respectively ____________________ u1 ----->| |----> y1 | Asys | u2 ------>| |----->y2 (in_idx) -------------------| (out_idx) File: octave, Node: sysgroup, Next: sysgroupn, Prev: sysdup, Up: blockdiag - Function File : SYS = sysgroup ( ASYS, BSYS) Combines two systems into a single system *Inputs* ASYS, BSYS: system data structures *Outputs* `sys = block diag(Asys,Bsys)' __________________ | ________ | u1 ----->|--> | Asys |--->|----> y1 | -------- | | ________ | u2 ----->|--> | Bsys |--->|----> y2 | -------- | ------------------ Ksys The function also rearranges the internal state-space realization of SYS so that the continuous states come first and the discrete states come last. If there are duplicate names, the second name has a unique suffix appended on to the end of the name. File: octave, Node: sysgroupn, Next: sysmult, Prev: sysgroup, Up: blockdiag - Function File : NAMES = sysgroupn (NAMES) Locate and mark duplicate names. Used internally in sysgroup (and elsewhere). File: octave, Node: sysmult, Next: sysprune, Prev: sysgroupn, Up: blockdiag - Function File : SYS = sysmult( ASYS, BSYS) Compute `sys = Asys*Bsys' (series connection): u ---------- ---------- --->| Bsys |---->| Asys |---> ---------- ---------- A warning occurs if there is direct feed-through from an input of Bsys or a continuous state of Bsys through a discrete output of Bsys to a continuous state or output in Asys (system data structure does not recognize discrete inputs). File: octave, Node: sysprune, Next: sysreorder, Prev: sysmult, Up: blockdiag - Function File : RETSYS = sysprune ( ASYS, OUT_IDX, IN_IDX) Extract specified inputs/outputs from a system *Inputs* ASYS system data structure OUT_IDX,IN_IDX list of connections indices; the new system has outputs y(out_idx(ii)) and inputs u(in_idx(ii)). May select as [] (empty matrix) to specify all outputs/inputs. *Outputs* RETSYS: resulting system ____________________ u1 ------->| |----> y1 (in_idx) | Asys | (out_idx) u2 ------->| |----| y2 (deleted)-------------------- (deleted) File: octave, Node: sysreorder, Next: sysscale, Prev: sysprune, Up: blockdiag - Function File : PV = sysreorder( VLEN, {varlist) *Inputs* VLEN=vector length, LIST= a subset of `[1:vlen]', *Outputs* PV: a permutation vector to order elements of `[1:vlen]' in `list' to the end of a vector. Used internally by `sysconnect' to permute vector elements to their desired locations. File: octave, Node: sysscale, Next: syssub, Prev: sysreorder, Up: blockdiag - Function File : SYS = sysscale (SYS, OUTSCALE, INSCALE{, OUTNAME, INNAME}) scale inputs/outputs of a system. *Inputs* sys: structured system outscale, inscale: constant matrices of appropriate dimension *Outputs* SYS: resulting open loop system: ----------- ------- ----------- u --->| inscale |--->| sys |--->| outscale |---> y ----------- ------- ----------- If the input names and output names (each a list of strings) are not given and the scaling matrices are not square, then default names will be given to the inputs and/or outputs. A warning message is printed if outscale attempts to add continuous system outputs to discrete system outputs; otherwise YD is set appropriately in the returned value of SYS. File: octave, Node: syssub, Prev: sysscale, Up: blockdiag - Function File : SYS = syssub (GSYS, HSYS) returns `sys = Gsys - Hsys' Method: GSYS and HSYS are connected in parallel The input vector is connected to both systems; the outputs are subtracted. Returned system names are those of GSYS. ________ ----| Gsys |--- u | ---------- +| ----- (_)----> y | ________ -| ----| Hsys |--- -------- - Function File : OUTSYS = ugain(n) Creates a system with unity gain, no states. This trivial system is sometimes needed to create arbitrary complex systems from simple systems with buildssic. Watch out if you are forming sampled systems since "ugain" does not contain a sampling period. See also: hinfdemo (MIMO H_infinty example, Boeing 707-321 aircraft model) - Function File : WSYS = wgt1o (VL, VH, FC) State space description of a first order weighting function. Weighting function are needed by the H2/H_infinity design procedure. These function are part of thye augmented plant P (see hinfdemo for an applicattion example). vl = Gain @ low frequencies vh = Gain @ high frequencies fc = Corner frequency (in Hz, *not* in rad/sec) 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 : M = dgram ( A, B) Return controllability grammian of discrete time system x(k+1) = a x(k) + b u(k) *Inputs* A N by N matrix B N by M matrix *Outputs* M (N by N) satisfies a m a' - m + b*b' = 0 - 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 and B must be real matrices. *Note* `qzval' is obsolete; use `qz' instead. - Function File : zgfmul - Function File : zgfslv - Function File : zginit - Function File : RETSYS = zgpbal (ASYS) - Function File : zgreduce - Function File : [NONZ, ZER] = zgrownorm (MAT, MEPS) - Function File : x = zgscal (F, Z, N, M, P) - Function File : zgsgiv ( ) - Function File : X = zgshsr( Y) Used internally by `tzero'. Minimal argument checking performed. Details involving major subroutines: `zgpbal' Implementation of zero computation generalized eigenvalue problem balancing method. `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' `zgreduce' Implementation of procedure REDUCE in (Emami-Naeini and Van Dooren, Automatica, 1982). `zgrownorm' 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 `zgscal' Generalized conjugate gradient iteration to solve zero-computation generalized eigenvalue problem balancing equation `fx=z'; called by `zgepbal' `zgsgiv' apply givens rotation c,s to column vector a,b `zgshsr' Apply Householder vector based on `e^(m)' (all ones) to (column vector) Y. Called by `zgfslv'. 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 : analdemo ( ) Octave Controls toolbox demo: State Space analysis demo - 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: outputs = synKnames (inputs) Return controller signal names based in plant signal names and dimensions - 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. - 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. - 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 = is_digital ( SYS) Return nonzero if system is digital; Exits with an error of sys is a mixed (continuous and discrete) system - Function File : [RETVAL,U] = is_observable (A, C{,TOL}) - Function File : [RETVAL,U] = is_observable (SYS{, TOL}) Logical check for system observability. 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). *See* `is_controllable' for description of inputs, outputs. Test for stabilizability is performed via an ordered Schur decomposition that reveals the unstable subspace of the system A matrix. - 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 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 - Function File : [Y, T] = step (SYS{, INP,TSTOP, N}) Step response of a linear system; calling protocol is identical to `impulse'. - Function File : stepimp ( ) Used internally in `impulse', `step'. File: octave, Node: sysfreq, Next: cacsd, Prev: systime, Up: Control Theory System Analysis-Frequency Domain ================================ *Demonstration/tutorial script* - Function File : frdemo ( ) - 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. - Function File : RETVAL = freqchkw ( W ) Used by `freqresp' to check that input frequency vector W is legal. Returns boolean value. - Function File : OUT = freqresp (SYS,USEW{,W}); Frequency response function - used internally by `bode', `nyquist'. minimal argument checking; "do not attempt to do this at home" *Inputs* SYS system data structure USEW returned by `freqchkw' OPTIONAL must be present if USEW is true (nonzero) *Outputs* OUT vector of finite `G(j*w)' entries (or `||G(j*w)||' for MIMO) W vector of corresponding frequencies - Function File : OUT = ltifr (A, B, W) - Function File : OUT = ltifr (SYS, W) Linear time invariant frequency response of single input systems *Inputs* A, B coefficient matrices of `dx/dt = A x + B u' SYS system data structure W vector of frequencies *Outputs* OUT -1 G(s) = (jw I-A) B for complex frequencies `s = jw'. - 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. Arguments are identical to `bode' with exceptions as noted below: *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: tzero (A, B, C, D{, OPT}) - Function File: tzero (SYS{,OPT}) Compute transmission zeros of a continuous . x = Ax + Bu y = Cx + Du or discrete x(k+1) = A x(k) + B u(k) y(k) = C x(k) + D u(k) system. *Outputs* ZER transmission zeros of the system GAIN leading coefficient (pole-zero form) of SISO transfer function returns gain=0 if system is multivariable *References* 1. Emami-Naeini and Van Dooren, Automatica, 1982. 2. Hodel, "Computation of Zeros with Balancing," 1992 Lin. Alg. Appl. - 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.