Control Theory
**************

The Octave Control Systems Toolbox (OCST) was initially developed by Dr. A. Scottedward Hodel a.s.hodel@eng.auburn.edu with the assistance of his students

  * R. Bruce Tenison btenison@dibbs.net,

  * David C. Clem,

  * John E. Ingram John.Ingram@sea.siemans.com, and

  * Kristi McGowan.

This development was supported in part by NASA's Marshall Space Flight Center as part of an in-house CACSD environment.  Additional important contributions were made by Dr. Kai Mueller mueller@ifr.ing.tu-bs.de and Jose Daniel Munoz Frias (place.m).

An on-line menu-driven tutorial is available via DEMOcontrol; beginning OCST users should start with this program.

 -- Function File: DEMOcontrol
     Octave Control Systems Toolbox demo/tutorial program.  The demo allows the user to select among several categories of OCST function:

          octave:1> DEMOcontrol
          O C T A V E    C O N T R O L   S Y S T E M S   T O O L B O X
          Octave Controls System Toolbox Demo

            [ 1] System representation
            [ 2] Block diagram manipulations
            [ 3] Frequency response functions
            [ 4] State space analysis functions
            [ 5] Root locus functions
            [ 6] LQG/H2/Hinfinity functions
            [ 7] End

     Command examples are interactively run for users to observe the use of OCST functions.

Demo Programs: bddemo.m, frdemo.m, analdemo.m, moddmeo.m, rldemo.m

System Data Structure
=====================

The OCST stores all dynamic systems in a single data structure format that can represent continuous systems, discrete-systems, and mixed (hybrid) systems in state-space form, and can also represent purely continuous/discrete systems in either transfer function or pole-zero form. In order to provide more flexibility in treatment of discrete/hybrid systems, the OCST also keeps a record of which system outputs are sampled.

Octave structures are accessed with a syntax much like that used by the C programming language.  For consistency in use of the data structure used in the OCST, it is recommended that the system structure access m-files be used (see sysinterface). Some elements of the data structure are absent depending on the internal system representation(s) used.  More than one system representation can be used for SISO systems; the OCST m-files ensure that all representations used are consistent with one another.

 -- Function File: sysrepdemo
     Tutorial for the use of the system data structure functions.

Variables common to all OCST system formats
-------------------------------------------

The data structure elements (and variable types) common to all  system representations are listed below; examples of the initialization and use of the system data structures are given in subsequent sections and in the online demo DEMOcontrol.

n
nz
     The respective number of continuous and discrete states in the system (scalar)

inname
outname
     list of name(s) of the system input, output signal(s). (list of strings)

sys
     System status vector.  (vector)

     This vector indicates both what representation was used to initialize the system data structure (called the primary system type) and which other representations are currently up-to-date with the primary system type (see structaccess).

     The value of the first element of the vector indicates the primary system type.

     0
          for tf form (initialized with tf2sys or fir2sys)

     1
          for zp form (initialized with zp2sys)

     2
          for ss form (initialized with ss2sys)

     The next three elements are boolean flags that indicate whether tf, zp, or ss, respectively, are ``up to date" (whether it is safe to use the variables associated with these representations).  These flags are changed when calls are made to the sysupdate command.

tsam
     Discrete time sampling period  (nonnegative scalar). TSAM is set to 0 for continuous time systems.

yd
     Discrete-time output list (vector)

     indicates which outputs are discrete time (i.e., produced by D/A converters) and which are continuous time. yd(ii) = 0 if output ii is continuous, = 1 if discrete.

The remaining variables of the  system data structure are only present if the corresponding entry of the sys vector is true (=1).

tf format variables
-------------------

num
     numerator coefficients   (vector)

den
     denominator coefficients   (vector)

zp format variables
-------------------

zer
     system zeros   (vector)

pol
     system poles    (vector)

k
     leading coefficient   (scalar)

ss format variables
-------------------

a
b
c
d
     The usual state-space matrices. If a system has both continuous and discrete states, they are sorted so that continuous states come first, then discrete states

     Note some functions (e.g., bode, hinfsyn) will not accept systems with both discrete and continuous states/outputs

stname
     names of system states   (list of strings)

System Construction and Interface Functions
===========================================

Construction and manipulations of the OCST system data structure (see sysstruct) requires attention to many details in order to ensure that data structure contents remain consistent.  Users are strongly encouraged to use the system interface functions in this section.  Functions for the formatted display in of system data structures are given in sysdisp.

Finite impulse response system interface functions
--------------------------------------------------

 -- Function File: fir2sys (NUM, TSAM, INNAME, OUTNAME)
     construct a system data structure from FIR description

     Inputs

     num
          vector of coefficients [c0, c1, ..., cn] of the SISO FIR transfer function C(z) = c0 + c1*z^(-1) + c2*z^(-2) + ... + cn*z^(-n)

     tsam
          sampling time (default: 1)

     inname
          name of input signal;  may be a string or a list with a single entry.

     outname
          name of output signal; may be a string or a list with a single entry.

     Output

     sys
          system data structure

     Example

          octave:1> sys = fir2sys([1 -1 2 4],0.342,\
          > "A/D input","filter output");
          octave:2> sysout(sys)
          Input(s)
                  1: A/D input

          Output(s):
                  1: filter output (discrete)

          Sampling interval: 0.342
          transfer function form:
          1*z^3 - 1*z^2 + 2*z^1 + 4
          -------------------------
          1*z^3 + 0*z^2 + 0*z^1 + 0

 -- Function File: [C, TSAM, INPUT, OUTPUT] = sys2fir (SYS)

     Extract FIR data from system data structure; see fir2sys for parameter descriptions.

fir2sys

State space system interface functions
--------------------------------------

 -- Function File: ss (A, B, C, D, TSAM, N, NZ, STNAME, INNAME, OUTNAME, OUTLIST)
     Create system structure from state-space data.   May be continous, discrete, or mixed (sampled data)

     Inputs

     a
     b
     c
     d
          usual state space matrices.

          default: D = zero matrix

     tsam
          sampling rate.  Default: tsam = 0 (continuous system)

     n
     nz
          number of continuous, discrete states in the system

          If TSAM is 0, n = rows(A), nz = 0.

          If TSAM is greater than zero, n = 0, nz = rows(A)

          see below for system partitioning

     stname
          cell array of strings of state signal names

          default (STNAME=[] on input): x_n for continuous states, xd_n for discrete states

     inname
          cell array of strings of input signal names

          default (INNAME = [] on input): u_n

     outname
          cell array of strings of input signal names

          default (OUTNAME = [] on input): y_n

     outlist

          list of indices of outputs y that are sampled

          If TSAM is 0, outlist = [].

          If TSAM is greater than 0, outlist = 1:rows(C).

     Unlike states, discrete/continous outputs may appear in any order.

     sys2ss returns a vector YD where YD(OUTLIST) = 1; all other entries of YD are 0.

     Outputs OUTSYS = system data structure

     System partitioning

     Suppose for simplicity that outlist specified that the first several outputs were continuous and the remaining outputs were discrete.  Then the system is partitioned as

          x = [ xc ]  (n x 1)
              [ xd ]  (nz x 1 discrete states)
          a = [ acc acd ]  b = [ bc ]
              [ adc add ]      [ bd ]
          c = [ ccc ccd ]  d = [ dc ]
              [ cdc cdd ]      [ dd ]

              (cdc = c(outlist,1:n), etc.)

     with dynamic equations: d/dt xc(t)     = acc*xc(t)      + acd*xd(k*tsam) + bc*u(t)

     xd((k+1)*tsam) = adc*xc(k*tsam) + add*xd(k*tsam) + bd*u(k*tsam)

     yc(t)      = ccc*xc(t)      + ccd*xd(k*tsam) + dc*u(t)

     yd(k*tsam) = cdc*xc(k*tsam) + cdd*xd(k*tsam) + dd*u(k*tsam)

     Signal partitions

                  | continuous      | discrete               |
          ----------------------------------------------------
          states  | stname(1:n,:)   | stname((n+1):(n+nz),:) |
          ----------------------------------------------------
          outputs | outname(cout,:) | outname(outlist,:)     |
          ----------------------------------------------------

     where cout is the list of in 1:rows(P) that are not contained in outlist. (Discrete/continuous outputs may be entered in any order desired by the user.)

     Example

          octave:1> a = [1 2 3; 4 5 6; 7 8 10];
          octave:2> b = [0 0 ; 0 1 ; 1 0];
          octave:3> c = eye (3);
          octave:4> sys = ss (a, b, c, [], 0, 3, 0, {"volts", "amps", "joules"});
          octave:5> sysout(sys);
          Input(s)
                  1: u_1
                  2: u_2

          Output(s):
                  1: y_1
                  2: y_2
                  3: y_3

          state-space form:
          3 continuous states, 0 discrete states
          State(s):
                  1: volts
                  2: amps
                  3: joules

          A matrix: 3 x 3
             1   2   3
             4   5   6
             7   8  10
          B matrix: 3 x 2
            0  0
            0  1
            1  0
          C matrix: 3 x 3
            1  0  0
            0  1  0
            0  0  1
          D matrix: 3 x 3
            0  0
            0  0
            0  0

     Notice that the D matrix is constructed  by default to the correct dimensions.  Default input and output signals names were assigned since none were given.

 -- Function File: [A, B, C, D, TSAM, N, NZ, STNAME, INNAME, OUTNAME, YD] = sys2ss (SYS)
     Extract state space representation from system data structure.

     Input

     sys
          System data structure.

     Outputs

     a
     b
     c
     d
          State space matrices for SYS.

     tsam
          Sampling time of SYS (0 if continuous).

     n
     nz
          Number of continuous, discrete states (discrete states come last in state vector X).

     stname
     inname
     outname
          Signal names (lists of strings);  names of states, inputs, and outputs, respectively.

     yd
          Binary vector; YD(II) is 1 if output Y(II) is discrete (sampled); otherwise  YD(II) is 0.

     A warning massage is printed if the system is a mixed continuous and discrete system.

     Example

          octave:1> sys=tf2sys([1 2],[3 4 5]);
          octave:2> [a,b,c,d] = sys2ss(sys)
          a =
             0.00000   1.00000
            -1.66667  -1.33333
          b =
            0
            1
          c = 0.66667  0.33333
          d = 0

Transfer function system interface functions
--------------------------------------------

 -- Function File: tf2sys (NUM, DEN, TSAM, INNAME, OUTNAME)
     Build system data structure from transfer function format data.

     Inputs

     num
     den
          Coefficients of numerator/denominator polynomials.

     tsam
          Sampling interval; default: 0 (continuous time).

     inname
     outname
          Input/output signal names; may be a string or cell array with a single string entry.

     Output

     sys
          System data structure.

     Example

          octave:1> sys=tf2sys([2 1],[1 2 1],0.1);
          octave:2> sysout(sys)
          Input(s)
                  1: u_1
          Output(s):
                  1: y_1 (discrete)
          Sampling interval: 0.1
          transfer function form:
          2*z^1 + 1
          -----------------
          1*z^2 + 2*z^1 + 1

 -- Function File: [NUM, DEN, TSAM, INNAME, OUTNAME] = sys2tf (SYS)
     Extract transfer function data from a system data structure.

     See tf for parameter descriptions.

     Example

          octave:1> sys=ss([1 -2; -1.1,-2.1],[0;1],[1 1]);
          octave:2> [num,den] = sys2tf(sys)
          num = 1.0000  -3.0000
          den = 1.0000   1.1000  -4.3000

Zero-pole system interface functions
------------------------------------

 -- Function File: zp2sys (ZER, POL, K, TSAM, INNAME, OUTNAME)
     Create system data structure from zero-pole data.

     Inputs

     zer
          Vector of system zeros.

     pol
          Vector of system poles.

     k
          Scalar leading coefficient.

     tsam
          Sampling period; default: 0 (continuous system).

     inname
     outname
          Input/output signal names (lists of strings).

     Output

     sys
          System data structure.

     Example

          octave:1> sys=zp2sys([1 -1],[-2 -2 0],1);
          octave:2> sysout(sys)
          Input(s)
                  1: u_1
          Output(s):
                  1: y_1
          zero-pole form:
          1 (s - 1) (s + 1)
          -----------------
          s (s + 2) (s + 2)

 -- Function File: [ZER, POL, K, TSAM, INNAME, OUTNAME] = sys2zp (SYS)
     Extract zero/pole/leading coefficient information from a system data structure.

     See zp for parameter descriptions.

     Example

          octave:1> sys=ss([1 -2; -1.1,-2.1],[0;1],[1 1]);
          octave:2> [zer,pol,k] = sys2zp(sys)
          zer = 3.0000
          pol =
            -2.6953
             1.5953
          k = 1

Data structure access functions
-------------------------------

 -- Function File: syschnames (SYS, OPT, LIST, NAMES)
     Superseded by syssetsignals.

 -- Function File: syschtsam (SYS, TSAM)
     This function changes the sampling time (tsam) of the system.  Exits with an error if sys is purely continuous time.

 -- Function File: [N, NZ, M, P, YD] = sysdimensions (SYS, OPT)
     return the number of states, inputs, and/or outputs in the system SYS.

     Inputs

     sys
          system data structure

     opt
          String indicating which dimensions are desired.  Values:

          "all"
               (default) return all parameters as specified under Outputs below.

          "cst"
               return N= number of continuous states

          "dst"
               return N= number of discrete states

          "in"
               return N= number of inputs

          "out"
               return N= number of outputs

     Outputs

     n
          number of continuous states (or individual requested dimension as specified by OPT).

     nz
          number of discrete states

     m
          number of system inputs

     p
          number of system outputs

     yd
          binary vector; YD(II) is nonzero if output II is discrete. yd(ii) = 0 if output II is continous

sysgetsignals and sysgettsam

 -- Function File: [STNAME, INNAME, OUTNAME, YD] = sysgetsignals (SYS)
 -- Function File: SIGLIST = sysgetsignals (SYS, SIGID)
 -- Function File: SIGNAME = sysgetsignals (SYS, SIGID, SIGNUM, STRFLG)
     Get signal names from a system

     Inputs

     sys
          system data structure for the state space system

     sigid
          signal id.  String.  Must be one of

          "in"
               input signals

          "out"
               output signals

          "st"
               stage signals

          "yd"
               value of logical vector YD

     signum
          index(indices) or name(s) or signals; see sysidx

     strflg
          flag to return a string instead of a cell array;  Values:

          0
               (default) return a cell array (even if signum specifies an individual signal)

          1
               return a string.  Exits with an error if signum does not specify an individual signal.

     Outputs

     If SIGID is not specified:

          stname
          inname
          outname
               signal names (cell array of strings);  names of states, inputs, and outputs, respectively.

          yd
               binary vector; YD(II) is nonzero if output II is discrete.

     If SIGID is specified but SIGNUM is not specified:

          sigid="in"
               SIGLIST is set to the cell array of input names.

          sigid="out"
               SIGLIST is set to the cell array of output names.

          sigid="st"
               SIGLIST is set to the cell array of state names.

               stage signals

          sigid="yd"
               SIGLIST is set to logical vector indicating discrete outputs; SIGLIST(II) = 0 indicates that output II is continuous (unsampled), otherwise it is discrete.

     If the first three input arguments are specified:
          SIGNAME is a cell array of the specified signal names (SIGID is "in", "out", or "st"), or else the logical flag indicating whether output(s) SIGNUM is(are) discrete (SIGVAL=1) or continuous (SIGVAL=0).

     Examples (From sysrepdemo)

          octave> sys=ss(rand(4),rand(4,2),rand(3,4));
          octave># get all signal names
          octave> [Ast,Ain,Aout,Ayd] = sysgetsignals(sys)
          Ast =
          (
            [1] = x_1
            [2] = x_2
            [3] = x_3
            [4] = x_4
          )
          Ain =
          (
            [1] = u_1
            [2] = u_2
          )
          Aout =
          (
            [1] = y_1
            [2] = y_2
            [3] = y_3
          )
          Ayd =

            0  0  0
          octave> # get only input signal names:
          octave> Ain = sysgetsignals(sys,"in")
          Ain =
          (
            [1] = u_1
            [2] = u_2
          )
          octave> # get name of output 2 (in cell array):
          octave> Aout = sysgetsignals(sys,"out",2)
          Aout =
          (
            [1] = y_2
          )
          octave> # get name of output 2 (as string):
          octave> Aout = sysgetsignals(sys,"out",2,1)
          Aout = y_2

 -- Function File: sysgettype (SYS)
     return the initial system type of the system

     Input

     sys
          System data structure.

     Output

     systype
          String indicating how the structure was initially constructed. Values: "ss", "zp", or "tf".

     FIR initialized systems return systype="tf".

 -- Function File: syssetsignals (SYS, OPT, NAMES, SIG_IDX)
     change the names of selected inputs, outputs and states.

     Inputs

     sys
          System data structure.

     opt
          Change default name (output).

          "out"
               Change selected output names.

          "in"
               Change selected input names.

          "st"
               Change selected state names.

          "yd"
               Change selected outputs from discrete to continuous or from continuous to discrete.

     names

          opt = "out", "in", "st"
               string or string array containing desired signal names or values.

          opt = "yd"
               To desired output continuous/discrete flag. Set name to 0 for continuous, or 1 for discrete.

     sig_idx
          indices or names of outputs, yd, inputs, or states whose respective names/values should be changed.

          Default: replace entire cell array of names/entire yd vector.

     Outputs

     retsys
          SYS with appropriate signal names changed (or YD values, where appropriate).

     Example

          octave:1> sys=ss([1 2; 3 4],[5;6],[7 8]);
          octave:2> sys = syssetsignals(sys,"st",str2mat("Posx","Velx"));
          octave:3> sysout(sys)
          Input(s)
                  1: u_1
          Output(s):
                  1: y_1
          state-space form:
          2 continuous states, 0 discrete states
          State(s):
                  1: Posx
                  2: Velx
          A matrix: 2 x 2
            1  2
            3  4
          B matrix: 2 x 1
            5
            6
          C matrix: 1 x 2
            7  8
          D matrix: 1 x 1
          0

 -- Function File: sysupdate (SYS, OPT)
     Update the internal representation of a system.

     Inputs

     sys:
          system data structure

     opt
          string:

          "tf"
               update transfer function form

          "zp"
               update zero-pole form

          "ss"
               update state space form

          "all"
               all of the above

     Outputs

     retsys
          Contains union of data in sys and requested data. If requested data in SYS is already up to date then RETSYS=SYS.

     Conversion to tf or zp exits with an error if the system is mixed continuous/digital.

tf, ss, zp, sysout, sys2ss, sys2tf, and sys2zp

 -- Function File: [SYSTYPE, NOUT, NIN, NCSTATES, NDSTATES] = minfo (INMAT)
     Determines the type of system matrix.  INMAT can be a varying, a system, a constant, and an empty matrix.

     Outputs

     systype
          Can be one of: varying, system, constant, and empty.

     nout
          The number of outputs of the system.

     nin
          The number of inputs of the system.

     ncstates
          The number of continuous states of the system.

     ndstates
          The number of discrete states of the system.

 -- Function File: sysgettsam (SYS)
     Return the sampling time of the system SYS.

Data structure internal functions
---------------------------------

System display functions
========================

 -- Function File: sysout (SYS, OPT)
     print out a system data structure in desired format

     sys
          system data structure

     opt
          Display option

          []
               primary system form (default)

          "ss"
               state space form

          "tf"
               transfer function form

          "zp"
               zero-pole form

          "all"
               all of the above

 -- Function File: tfout (NUM, DENOM, X)
     Print formatted transfer function n(s)/d(s) to the screen. X defaults to the string "s"

polyval, polyvalm, poly, roots, conv, deconv, residue, filter, polyderiv, polyinteg, and polyout

 -- Function File: zpout (ZER, POL, K, X)
     print formatted zero-pole form to the screen. X defaults to the string "s"

polyval, polyvalm, poly, roots, conv, deconv, residue, filter, polyderiv, polyinteg, and polyout

Block Diagram Manipulations
===========================

See systime.

Unless otherwise noted, all parameters (input,output) are system data structures.

 -- Function File: bddemo (INPUTS)
     Octave Controls toolbox demo: Block Diagram Manipulations demo.

 -- Function File: buildssic (CLST, ULST, OLST, ILST, S1, S2, S3, S4, S5, S6, S7, S8)

     Form an arbitrary complex (open or closed loop) system in state-space form from several systems. buildssic can easily (despite its cryptic syntax) integrate transfer functions from a complex block diagram into a single system with one call. This function is especially useful for building open loop interconnections for H-infinity and H-2 designs or for closing loops with these controllers.

     Although this function is general purpose, the use of sysgroup sysmult, sysconnect and the like is recommended for standard operations since they can handle mixed discrete and continuous systems and also the names of inputs, outputs, and states.

     The parameters consist of 4 lists that describe the connections outputs and inputs and up to 8 systems S1--S8. Format of the lists:

     clst
          connection list, describes the input signal of each system. The maximum number of rows of Clst is equal to the sum of all inputs of s1-s8.

          Example: [1 2 -1; 2 1 0] means that:  new input 1 is old input 1 + output 2 - output 1, and new input 2 is old input 2 + output 1. The order of rows is arbitrary.

     ulst
          if not empty the old inputs in vector ULST will be appended to the outputs. You need this if you want to ``pull out'' the input of a system. Elements are input numbers of S1--S8.

     olst
          output list, specifiy the outputs of the resulting systems. Elements are output numbers of S1--S8. The numbers are allowed to be negative and may appear in any order. An empty matrix means all outputs.

     ilst
          input list, specifiy the inputs of the resulting systems. Elements are input numbers of S1--S8. The numbers are allowed to be negative and may appear in any order. An empty matrix means all inputs.

     Example:  Very simple closed loop system.

          w        e  +-----+   u  +-----+
           --->o--*-->|  K  |--*-->|  G  |--*---> y
               ^  |   +-----+  |   +-----+  |
             - |  |            |            |
               |  |            +----------------> u
               |  |                         |
               |  +-------------------------|---> e
               |                            |
               +----------------------------+

     The closed loop system GW can be optained by

          GW = buildssic([1 2; 2 -1], 2, [1 2 3], 2, G, K);

     clst
          1st row: connect input 1 (G) with output 2 (K).

          2nd row: connect input 2 (K) with negative output 1 (G).

     ulst
          Append input of 2 (K) to the number of outputs.

     olst
          Outputs are output of 1 (G), 2 (K) and appended output 3 (from ULST).

     ilst
          The only input is 2 (K).

     Here is a real example:

                                   +----+
              -------------------->| W1 |---> v1
          z   |                    +----+
          ----|-------------+
              |             |
              |    +---+    v      +----+
              *--->| G |--->O--*-->| W2 |---> v2
              |    +---+       |   +----+
              |                |
              |                v
             u                  y

          min || GW   ||
                   vz   infty

     The closed loop system GW from [z, u]' to [v1, v2, y]' can be obtained by (all SISO systems):

          GW = buildssic([1, 4; 2, 4; 3, 1], 3, [2, 3, 5],
                         [3, 4], G, W1, W2, One);

     where ``One'' is a unity gain (auxillary) function with order 0. (e.g. One = ugain(1);)

 -- Function File: SYS = jet707 ()
     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.

ord2

 -- Function File: 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.

     Output

     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)

 -- Function File: 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  |---
                    --------

 -- Function File: SYS = sysappend (SYST, B, C, D, OUTNAME, INNAME, YD)
     appends new inputs and/or outputs to a system

     Inputs

     syst
          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 := [syst.b , b]
                  sys.c := [syst.c  ]
                           [ c     ]
                  sys.d := [syst.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 is continuous or mixed YD = zeros(1,rows(c))

                 - SYS is discrete YD = ones(1,rows(c))

 -- Function File: CLSYS = 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
          Names or indices of signals to connect (see sysidx). The output specified by out_idx(ii) is connected to the input specified by 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

     clsys
          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.

 -- Function File: [CSYS, ACD, CCD] = syscont (SYS)
     Extract the purely continuous subsystem of an input system.

     Input

     sys
          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

 -- Function File: [DSYS, ADC, CDC] = sysdisc (SYS)

     Input

     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.

 -- Function File: RETSYS = sysdup (ASYS, OUT_IDX, IN_IDX)
     Duplicate specified input/output connections of a system

     Inputs

     asys
          system data structure

     out_idx
     in_idx
          indices or names of desired signals (see sigidx). duplicates are made of y(out_idx(ii)) and u(in_idx(ii)).

     Output

     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)

 -- Function File: SYS = sysgroup (ASYS, BSYS)
     Combines two systems into a single system.

     Inputs

     asys
     bsys
          System data structures.

     Output

     sys
          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.

 -- 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 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).

 -- 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
          Indices or signal names of the outputs and inputs to be kept in the returned system; remaining connections are ``pruned'' off. May select as [] (empty matrix) to specify all outputs/inputs.

               retsys = sysprune (Asys, [1:3,4], "u_1");
               retsys = sysprune (Asys, {"tx", "ty", "tz"}, 4);

     Output

     retsys
          Resulting system.

                     ____________________
          u1 ------->|                  |----> y1
           (in_idx)  |       Asys       | (out_idx)
          u2 ------->|                  |----| y2
            (deleted)-------------------- (deleted)

 -- Function File: PV = sysreorder (VLEN, LIST)

     Inputs

     vlen
          Vector length.

     list
          A subset of [1:vlen].

     Output

     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.

 -- Function File: RETSYS = sysscale (SYS, OUTSCALE, INSCALE, OUTNAME, INNAME)
     scale inputs/outputs of a system.

     Inputs

     sys
          Structured system.

     outscale
     inscale
          Constant matrices of appropriate dimension.

     outname
     inname
          Lists of strings with the names of respectively outputs and inputs.

     Output

     retsys
          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.

 -- Function File: SYS = syssub (GSYS, HSYS)
     Return 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: 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.

hinfdemo and jet707

 -- Function File: W = wgt1o (VL, VH, FC)
     State space description of a first order weighting function.

     Weighting function are needed by the H-2/H-infinity design procedure. These function are part of the augmented plant P (see hinfdemo for an application example).

     Inputs

     vl
          Gain at low frequencies.

     vh
          Gain at high frequencies.

     fc
          Corner frequency (in Hz, not in rad/sec)

     Output

     W
          Weighting function, given in form of a system data structure.

 -- Function File: KSYS = parallel (ASYS, BSYS)
     Forms the parallel connection of two systems.

                       --------------------
                       |      --------    |
              u  ----->|----> | asys |--->|----> y1
                  |    |      --------    |
                  |    |      --------    |
                  |--->|----> | bsys |--->|----> y2
                       |      --------    |
                       --------------------
                            ksys

 -- Function File: [RETSYS, NC, NO] = sysmin (SYS, FLG)
     Returns a minimal (or reduced order) system

     Inputs

     sys
          System data structure

     flg
          When equal to 0 (default value), returns minimal system, in which state names are lost; when equal to 1, returns system with physical states removed that are either uncontrollable or unobservable (cannot reduce further without discarding physical meaning of states).

     Outputs

     retsys
          Returned system.

     nc
          Number of controllable states in the returned system.

     no
          Number of observable states in the returned system.

     cflg
          is_controllable(retsys).

     oflg
          is_observable(retsys).

Numerical Functions
===================

 -- Function File: X = 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
          N by N matrix;

     b
          N by N matrix or N by M matrix; in the latter case B is replaced by b:=b*b';

     c
          N by N matrix or P by M 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.

     Output

     x
          solution of the ARE.

     Method Laub's Schur method (IEEE Transactions on Automatic Control, 1979) is applied to the appropriate Hamiltonian matrix.

balance and dare

 -- Function File: X = dare (A, B, Q, 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 + q = 0

     Inputs

     a
          N by N matrix;

     b
          N by M matrix;

     q
          N by N matrix, symmetric positive semidefinite, or a P by N matrix, In the latter case q:=q'*q 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.

     Output

     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.

balance and are

 -- Function File: [TVALS, PLIST] = dre (SYS, Q, R, QF, T0, TF, PTOL, MAXITS)
     Solve the differential Riccati equation

            -d P/dt = A'P + P A - P B inv(R) B' P + Q
            P(tf) = Qf

     for the LTI system sys.  Solution of standard LTI state feedback optimization

            min int(t0, tf) ( x' Q x + u' R u ) dt + x(tf)' Qf x(tf)

     optimal input is

            u = - inv(R) B' P(t) x

     Inputs

     sys
          continuous time system data structure

     q
          state integral penalty

     r
          input integral penalty

     qf
          state terminal penalty

     t0
     tf
          limits on the integral

     ptol
          tolerance (used to select time samples; see below); default = 0.1

     maxits
          number of refinement iterations (default=10)

     Outputs

     tvals
          time values at which P(T) is computed

     plist
          list values of P(T); PLIST { I } is P(TVALS(I))

     TVALS is selected so that:

          || Plist{i} - Plist{i-1} || < Ptol

     for every I between 2 and length(TVALS).

 -- Function File: dgram (A, B)
     Return controllability gramian of discrete time system

            x(k+1) = a x(k) + b u(k)

     Inputs

     a
          N by N matrix

     b
          N by M matrix

     Output

     m
          N by N matrix, satisfies M (N by N) satisfies

                a m a' - m + b*b' = 0

 -- Function File: 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.

     Output

     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 = F X F' + 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: gram (A, B)
     Return controllability gramian 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).

 -- Function File: qzval (A, B)
     Compute generalized eigenvalues of the matrix pencil

          (A - lambda B).

     A and B must be real matrices.

     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: 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: zgreduce (SYS, MEPS)
     Implementation of procedure REDUCE in (Emami-Naeini and Van Dooren, Automatica, # 1982).

 -- Function File: [NONZ, ZER] = zgrownorm (MAT, MEPS)
     Return NONZ = number of rows of MAT whose two norm exceeds MEPS, and ZER = number of rows of mat whose two norm is less than MEPS.

 -- Function File: X = zgscal (F, Z, N, M, P)
     Generalized conjugate gradient iteration to solve zero-computation generalized eigenvalue problem balancing equation fx=z; called by zgepbal.

 -- 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.

 -- Function File: X = zgshsr (Y)
     Apply householder vector based on e^(m) 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.

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.

is_abcd

 -- Function File: ctrb (SYS, B)
 -- Function File: 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).

     ctrb forms the controllability matrix. The numerical properties of is_controllable are much better for controllability tests.

 -- Function File: h2norm (SYS)
     Computes the H-2 norm of a system data structure (continuous time only).

     Reference: Doyle, Glover, Khargonekar, Francis, State-Space Solutions to Standard H-2 and H-infinity 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 value: 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 H-2 and H-infinity control problems, IEEE TAC August 1989; Iglesias and Glover, State-Space approach to discrete-time H-infinity control, Int. J. Control, vol 54, no. 5, 1991; Zhou, Doyle, Glover, Robust and Optimal Control, Prentice-Hall, 1996.

 -- Function File: obsv (SYS, C)
 -- Function File: obsv (A, C)
     Build observability matrix:

               | C        |
               | CA       |
          Qb = | CA^2     |
               | ...      |
               | CA^(n-1) |

     of a system data structure or the pair (A, C).

     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.

     Input

     sys
          System data structure.

     Outputs

     pol

     zer
          if omitted, the poles and zeros are plotted on the screen. otherwise, POL and 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.

abcddim

 -- 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.

size, rows, columns, length, ismatrix, isscalar, isvector is_observable, is_stabilizable, is_detectable, krylov, and krylovb

 -- Function File: RETVAL = is_detectable (A, C, TOL, DFLG)
 -- Function File: RETVAL = 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, and -1 if the system has unobservable modes at the imaginary axis (unit circle for discrete-time systems).

     See is_stabilizable for detailed description of arguments and computational method.

is_stabilizable, size, rows, columns, length, ismatrix, isscalar, and isvector

 -- 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
          threshold for 0; default: 200*eps.

     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;

               B = [Bw Bu]

          cz
               (NZ x N) Qz-transformed error output matrix

          cy
               (NY x N) RY-transformed measured output matrix

               C = [Cz; Cy]

          dzu

          dyw
               off-diagonal blocks of transformed system 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 H-2 and H-infinity Control Problems, IEEE TAC August 1989.

     [2]
          Maciejowksi, J.M., Multivariable Feedback Design, Addison-Wesley, 1989.

 -- Function File: DIGITAL = is_digital (SYS, EFLG)
     Return nonzero if system is digital.

     Inputs

     sys
          System data structure.

     eflg
          When equal to 0 (default value), exits with an error if the system is mixed (continuous and discrete components); when equal to 1, print a warning if the system is mixed (continuous and discrete); when equal to 2, operate silently.

     Output

     digital
          When equal to 0, the system is purely continuous; when equal to 1, the system is purely discrete; when equal to -1, the system is mixed continuous and discrete.

     Exits with an error if 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.

     Default: tol = 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.

size, rows, columns, length, ismatrix, isscalar, and isvector

 -- Function File: is_sample (TS)
     Return true if TS is a valid sampling time (real, scalar, > 0).

 -- Function File: is_siso (SYS)
     Returns nonzero if the system data structure SYS is single-input, single-output.

 -- Function File: RETVAL = is_stabilizable (SYS, TOL)
 -- Function File: RETVAL = is_stabilizable (A, B, TOL, DFLG)
     Logical check for system stabilizability (i.e., all unstable modes are controllable). Returns 1 if the system is stabilizable, 0 if the the system is not stabilizable, -1 if the system has non stabilizable modes at the imaginary axis (unit circle for discrete-time systems.

     Test for stabilizability is performed via Hautus Lemma. If DFLG!=0 assume that discrete-time matrices (a,b) are supplied.

size, rows, columns, length, ismatrix, isscalar, isvector is_observable, is_stabilizable, is_detectable

 -- Function File: is_signal_list (MYLIST)
     Return true if MYLIST is a list of individual strings.

 -- Function File: is_stable (A, TOL, DFLG)
 -- Function File: is_stable (SYS, TOL)
     Returns 1 if the matrix A or the system SYS is stable, or 0 if not.

     Inputs

     tol
          is a roundoff parameter, set to 200*eps if omitted.

     dflg
          Digital system flag (not required for system data structure):

          DFLG != 0
               stable if eig(a) is in the unit circle

          DFLG == 0
               stable if eig(a) is in the open LHP (default)

size, rows, columns, length, ismatrix, isscalar, isvector is_observable, is_stabilizable, is_detectable, krylov, and krylovb

System Analysis-Time Domain
===========================

 -- Function File: c2d (SYS, OPT, T)
 -- Function File: c2d (SYS, T)

     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.

     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.)

          "matched"
               Use the matched pole/zero equivalent transformation (currently only works for purely continuous SISO systems).

     t
          sampling time; required if SYS is purely continuous.

          Note: if the second argument is not a string, c2d() assumes that the second argument is T and performs appropriate argument checks.

     Output

     dsys
          Discrete time equivalent via zero-order hold, sample each T sec.

     This function adds the suffix  _d to the names of the new discrete states.

 -- Function File: d2c (SYS, TOL)
 -- Function File: d2c (SYS, OPT)
     Convert a discrete (sub)system into a purely continuous one. The 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), with 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

     Output

     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
          indices or names of states with sampling time sysgettsam(SYS) (may be empty); see cellidx

     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 the discrete time system 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.

eig

 -- Function File: 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 the 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
          Values of the impulse response.

     t
          Times of the impulse response.

step, __stepimp__

 -- Function File: [Y, T] = step (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, step 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
          Values of the step response.

     t
          Times of the step response.

     When invoked with the output parameter Y the plot is not displayed.

impulse and __stepimp__

System Analysis-Frequency Domain
================================

Demonstration/tutorial script
 -- Function File: frdemo ()
     Octave Control 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 is the system sampling time

            * 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 names or indices of outputs and inputs to be used in the frequency response.  See sysprune.

          Example

               bode(sys,[],"y_3", {"u_1","u_4"});

     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

      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: freqchkw (W)
     Used by __freqresp__ to check that input frequency vector W is valid. Returns boolean value.

 -- 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

     Output

     out
          frequency response, that is:

                                     -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.

     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 is 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.

     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: [ZER, GAIN] = tzero (A, B, C, D, OPT)
 -- Function File: [ZER, GAIN] = tzero (SYS, OPT)
     Compute transmission zeros of a continuous system:

          .
          x = Ax + Bu
          y = Cx + Du

     or of a discrete one:

          x(k+1) = A x(k) + B u(k)
          y(k)   = C x(k) + D u(k)

     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.

Controller Design
=================

 -- Function File: dgkfdemo ()
     Octave Controls toolbox demo: H-2/H-infinity 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=80 m/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   |                    +----+
                    ----|-------------+
                        |             |
                        |    +---+    v   y  +----+
                      u *--->| G |--->O--*-->| W2 |---> v2
                        |    +---+       |   +----+
                        |                |
                        |    +---+       |
                        -----| K |<-------
                             +---+

               min || T   ||
                       vz   infty

          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                       y (to K)
                        (from 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(z) = --------------------------
                           (z - 0.36788)(z - 0.13534)

                                             +----+
                        -------------------->| W1 |---> v1
                    z   |                    +----+
                    ----|-------------+
                        |             |
                        |    +---+    v      +----+
                        *--->| G |--->O--*-->| W2 |---> v2
                        |    +---+       |   +----+
                        |                |
                        |    +---+       |
                        -----| K |<-------
                             +---+

               min || T   ||
                       vz   infty

          W1 and W2 are the robustness and performance 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] + v[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|k] = z[k|k-1] + L (y[k] - C z[k|k-1] - D u[k])
          z[k+1|k] = A z[k|k] + B 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).

 -- Function File: [LP, LF, P, Z] = dkalman (A, G, C, QW, RV, S)
     Construct the linear quadratic estimator (Kalman predictor) 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] + v[k]

     where W, V are zero-mean gaussian noise processes with respective intensities QW = cov (W, W) and RV = cov (V, V).

     If specified, S is cov (W, V).  Otherwise cov (W, V) = 0.

     The observer structure is

          x[k+1|k] = A x[k|k-1] + B u[k] + LP (y[k] - C x[k|k-1] - D u[k])
          x[k|k] = x[k|k-1] + LF (y[k] - C x[k|k-1] - D u[k])

     The following values are returned:

     Lp
          The predictor gain, (A - LP C) is stable.

     Lf
          The filter gain.

     P
          The Riccati solution.

          P = E [(x - x[n|n-1])(x - x[n|n-1])']

     Z
          The updated error covariance matrix.

          Z = E [(x - x[n|n])(x - x[n|n])']

 -- Function File: [K, GAIN, KC, KF, PC, PF] =  h2syn (ASYS, NU, NY, TOL)
     Design H-2 optimal controller per procedure in Doyle, Glover, Khargonekar, Francis, State-Space Solutions to Standard H-2 and H-infinity Control Problems, IEEE TAC August 1989.

     Discrete-time control per Zhou, Doyle, and Glover, Robust and optimal control, Prentice-Hall, 1996.

     Inputs

     asys
          system data structure (see ss, 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
          threshold 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-infinity optimal controller.

     Inputs

     dgs
          data structure returned by is_dgkf

     f
     h
          feedback and filter gain (not partitioned)

     g
          final gamma value

     Outputs

     K
          controller (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 ss, 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 threshold.  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
          threshold 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.

     References:

      1. Doyle, Glover, Khargonekar, Francis, State-Space Solutions to Standard H-2 and H-infinity 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 an 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 H-2 and H-infinity 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-infinity ARE

     pf
          solution of ``filter'' H-infinity ARE

     Do not attempt to use this at home; no argument checking performed.

 -- Function File: [XINF, X_HA_ERR] = hinfsyn_ric (A, BB, C1, D1DOT, R, PTOL)
     Forms

          xx = ([bb; -c1'*d1dot]/r) * [d1dot'*c1 bb'];
          Ha = [a 0*a; -c1'*c1 - a'] - xx;

     and solves associated Riccati equation. The error code X_HA_ERR indicates one of the following conditions:

     0
          successful

     1
          XINF has imaginary eigenvalues

     2
          HX not Hamiltonian

     3
          XINF has infinite eigenvalues (numerical overflow)

     4
          XINF not symmetric

     5
          XINF not positive definite

     6
          R is singular

 -- 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).

 -- Function File: [K, Q1, P1, EE, ER] = lqg (SYS, SIGW, SIGV, Q, R, IN_IDX)
     Design a linear-quadratic-gaussian optimal controller for the system

          dx/dt = A x + B u + G w       [w]=N(0,[Sigw 0    ])
              y = C x + v               [v]  (    0   Sigv ])

     or

          x(k+1) = A x(k) + B u(k) + G w(k)       [w]=N(0,[Sigw 0    ])
            y(k) = C x(k) + v(k)                  [v]  (    0   Sigv ])

     Inputs

     sys
          system data structure

     sigw
     sigv
          intensities of independent Gaussian noise processes (as above)

     q
     r
          state, control weighting respectively.  Control ARE is

     in_idx
          names or indices of controlled inputs (see sysidx, cellidx)

          default: last dim(R) inputs are assumed to be controlled inputs, all others are assumed to be noise inputs.

     Outputs

     k
          system data structure format LQG optimal controller (Obtain A, B, C matrices with sys2ss, sys2tf, or sys2zp as appropriate).

     p1
          Solution of control (state feedback) algebraic Riccati equation.

     q1
          Solution of estimation algebraic Riccati equation.

     ee
          Estimator poles.

     es
          Controller poles.

h2syn, lqe, and 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: [Y, X] = 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 row in U corresponds to a different time step.  Each column 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 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)

Miscellaneous Functions (Not yet properly filed/documented)
===========================================================

 -- Function File: axis2dlim (AXDATA)
     Determine axis limits for 2-D data (column vectors); leaves a 10% margin around the plots. Inserts margins of +/- 0.1 if data is one-dimensional (or a single point).

     Input

     axdata
          N by 2 matrix of data [X, Y].

     Output

     axvec
          Vector of axis limits appropriate for call to axis function.

 -- Function File: moddemo (INPUTS)
     Octave Control toolbox demo: Model Manipulations demo.

 -- Function File: prompt (STR)
     Prompt user to continue

     Input

     str
          Input string. Its default value is:

               \n ---- Press a key to  continue ---

 -- Function File: rldemo (INPUTS)
     Octave Control toolbox demo: Root Locus demo.

 -- Function File: [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
          Minimum value of K

     max_k
          Maximum value of K

     increment
          The increment used in computing gain values

     Outputs

     Plots the root locus to the screen.

     rldata
          Data points plotted: in column 1 real values, in column 2 the imaginary values.

     k
          Gains for real axis break points.

 -- Function File: [YY, IDX] = sortcom (XX[, OPT])
     Sort a complex vector.

     Inputs

     xx
          Complex vector

     opt
          sorting option:

          "re"
               Real part (default);

          "mag"
               By magnitude;

          "im"
               By imaginary part.

          if OPT is not chosen as "im", then complex conjugate pairs are grouped together, a - jb followed by a + jb.

     Outputs

     yy
          Sorted values

     idx
          Permutation vector: yy = xx(idx)

 -- Function File: [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: [POL, ZER, K] = ss2zp (A, B, C, D)
     Converts a state space representation to a set of poles and zeros; K is a gain associated with the zeros.

     Used internally in system data structure format manipulations.

 -- Function File: starp (P, K, NY, NU)

     Redheffer star product or upper/lower LFT, respectively.

                         +-------+
               --------->|       |--------->
                         |   P   |
                    +--->|       |---+  ny
                    |    +-------+   |
                    +-------------------+
                                     |  |
                    +----------------+  |
                    |                   |
                    |    +-------+      |
                    +--->|       |------+ nu
                         |   K   |
               --------->|       |--------->
                         +-------+

     If NY and NU ``consume'' all inputs and outputs of K then the result is a lower fractional transformation. If NY and NU ``consume'' all inputs and outputs of P then the result is an upper fractional transformation.

     NY and/or NU may be negative (i.e. negative feedback).

 -- Function File: [A, B, C, D] = tf2ss (NUM, DEN)
     Conversion from tranfer function to state-space. The state space system:

                .
                x = Ax + Bu
                y = Cx + Du

     is obtained from a transfer function:

                          num(s)
                    G(s)=-------
                          den(s)

     The vector DEN must contain only one row, whereas the vector NUM may contain as many rows as there are outputs Y of the system. The state space system matrices obtained from this function will be in controllable canonical form as described in Modern Control Theory, (Brogan, 1991).

 -- Function File: [ZER, POL, K] = tf2zp (NUM, DEN)
     Converts transfer functions to poles-and-zero representations.

     Returns the zeros and poles of the SISO system defined by NUM/DEN. K is a gain associated with the system zeros.

 -- 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). The number of zeros must not exceed the number of poles.

     k
          Real scalar (leading coefficient).

     Outputs

     A
     B
     C
     D
          The state space system, in the form:

                    .
                    x = Ax + Bu
                    y = Cx + Du

 -- 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 must appear in conjugate pairs.

     k
          Real scalar (leading coefficient).