GNU Octave: Control Theory

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# 29. 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.

29.1 System Data Structure |  |
29.2 System Construction and Interface Functions |  |
29.3 System display functions |  |
29.4 Block Diagram Manipulations |  |
29.5 Numerical Functions |  |
29.6 System Analysis-Properties |  |
29.7 System Analysis-Time Domain |  |
29.8 System Analysis-Frequency Domain |  |
29.9 Controller Design |  |
29.10 Miscellaneous Functions (Not yet properly filed/documented) |  |

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## 29.1 System Data Structure

29.1.1 Variables common to all OCST system formats |  |
29.1.2 `tf` format variables |  |
29.1.3 `zp` format variables |  |
29.1.4 `ss` format variables |  |

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 section 29.2 System Construction and Interface Functions). 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.

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### 29.1.1 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 section 29.2.5 Data structure access functions).

  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).

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### 29.1.2 `tf` format variables

num
  numerator coefficients (vector)

den
  denominator coefficients (vector)

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### 29.1.3 `zp` format variables

zer
  system zeros (vector)

pol
  system poles (vector)

k
  leading coefficient (scalar)

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### 29.1.4 `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)

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## 29.2 System Construction and Interface Functions

Construction and manipulations of the OCST system data structure (see section 29.1 System Data Structure) 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 29.3 System display functions.

29.2.1 Finite impulse response system interface functions |  |
29.2.2 State space system interface functions |  |
29.2.3 Transfer function system interface functions |  |
29.2.4 Zero-pole system interface functions |  |
29.2.5 Data structure access functions |  |
29.2.6 Data structure internal functions |  |

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### 29.2.1 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 of the SISO FIR transfer function

    C(z) = c0 + c1*z^{-1} + c2*z^{-2} + ... + znz^{-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.

  Outputs 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.

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### 29.2.2 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 (sampeled-data)

  Inputs

  a
  b
  c
  d
    usual state space matrices.

    default: d = zero matrix

  tsam
    sampling rate. Default: (continuous system)

  n
  nz
    number of continuous, discrete states in the system

    If tsam is 0, , .

    If tsam is greater than zero, ,

    see below for system partitioning

  stname
    list of strings of state signal names

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

  inname
    list of strings of input signal names

    default (inname = [] on input): `u_n`

  outname
    list 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, .

    If tsam is greater than 0, .

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

  Note `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:

  Signal partitions

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

  where 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,list("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 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.

  Inputs 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) 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

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### 29.2.3 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.

  Outputs 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

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### 29.2.4 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)

  Outputs 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

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### 29.2.5 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. if output ii is continous

Function File: [stname, inname, outname, yd] = sysgetsignals (sys)
@deftypefnx{Function File}: siglist = sysgetsignals (sys, sigid)
@deftypefnx{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

  @bullet{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.

  @bullet{If sigid is specified but signum is not specified, then}
    `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.

  @bullet{if the first three input arguments are specified, then 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> [Ast,Ain,Aout,Ayd] = sysgetsignals(sys) i  # get all signal names
      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> Ain = sysgetsignals(sys,"in")   # get only input signal names
      Ain =
      (
        [1] = u_1
        [2] = u_2
      )
      octave> Aout = sysgetsignals(sys,"out",2)   # get name of output 2 (in cell array)
      Aout =
      (
        [1] = y_2
      )
      octave> Aout = sysgetsignals(sys,"out",2,1)  # get name of output 2 (as string)
      Aout = y_2

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

  Inputs sys: system data structure

  Outputs systype: string indicating how the structure was initially constructed: values: `"ss"`, `"zp"`, or `"tf"`

  Note 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", or "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.

function [systype, nout, nin, ncstates, ndstates] = minfo(inmat)

MINFO: Determines the type of system matrix. INMAT can be a varying(*), system, constant, and empty matrix.

Returns: systype can be one of: varying, system, constant, and empty nout is the number of outputs of the system nin is the number of inputs of the system ncstates is the number of continuous states of the system ndstates is the number of discrete states of the system

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

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### 29.2.6 Data structure internal functions

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## 29.3 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 to the screen. x defaults to the string `"s"`

Function File: zpout (zer, pol, k, x)
  print formatted zero-pole form to the screen. x defaults to the string `"s"`

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## 29.4 Block Diagram Manipulations

See section 29.7 System Analysis-Time Domain.

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 it's 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 H2 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]` ==> new input 1 is old inpout 1 + output 2 - output 1, 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 alowed 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 alowed 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
    (1. row) connect input 1 (G) with output 2 (K). (2. row) connect input 2 (K) with neg. 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   |                    +----+
      ----|-------------+                   || GW   ||     => min.
          |             |                        vz   infty
          |    +---+    v      +----+
          *--->| G |--->O--*-->| W2 |---> v2
          |    +---+       |   +----+
          |                |
          |                v
         u                  y

  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: jet707 ()
  Creates linearized state space model of a Boeing 707-321 aircraft at v=80m/s. (M = 0.26, Ga0 = -3 deg, alpha0 = 4 deg, kappa = 50 deg) System inputs: (1) thrust and (2) elevator angle System outputs: (1) airspeed and (2) pitch angle Ref: R. Brockhaus: Flugregelung (Flight Control), Springer, 1994

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.
  Outputs outsys system data structure has representation with :

          /                                        \
          | / -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: 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; indicates a continuous output; indicates a discrete output.

  Outputs sys

         sys.b := [sys.b , b]
         sys.c := [sys.c  ]
                  [ c     ]
         sys.d := [sys.d | D12 ]
                  [D21   | D22 ]

  where , , and are the appropriate dimensioned blocks of the input parameter d.
  - The leading block 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))`

Function File: 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 is connected to the input specified by .
  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.

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

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.

Function File: 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))`.

  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)

Function File: sysgroup (asys, bsys)
  Combines two systems into a single system

  Inputs asys, bsys: system data structures

  Outputs

               __________________
               |    ________    |
      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: sysmult (asys, bsys)
  Compute (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).

Function File: 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,list("tx","ty","tz"), 4);

  Outputs retsys: resulting system

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

Function File: sysreorder (vlen, list)

  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.

Function File: 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.

Function File: syssub (gsys, hsys)
  Return .

  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.

Function File: 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 at low frequencies

  vh = Gain at high frequencies

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

Function File: parallel (asys, bsys)
  Forms the parallel connection of two systems.

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

Function File: [retsys, nc, no] = sysmin (sys, flg)
  return a minimal (or reduced order) system inputs: sys: system data structure flg: 0 [default] return minimal system; state names lost : 1 return 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)

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## 29.5 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 .
  c
    nxn matrix or pxm matrix; in the latter case c is replaced by .
  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, 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.

  b
    n by m.

  q
    n by n, symmetric positive semidefinite, or p by n. In the latter case 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: [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_{t_0}^{t_f} x' Q x + u' R u dt + x(t_f)' Qf x(t_f)

  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 { ii } is p(tvals(ii)).

  tvals

        is selected so that || Plist{ii} - Plist{ii-1} || < Ptol
        for ii=2:length(tvals)

Function File: 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: 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 , 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 grammian m of the continuous time system .

  m satisfies .

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.

  Note `qzval` is obsolete; use `qz` instead.

Function File: zgfmul (a, b, c, d, x)
  Compute product of zgep incidence matrix 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: 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: zgscal (f, z, n, m, p)
  Generalized conjugate gradient iteration to solve zero-computation generalized eigenvalue problem balancing equation ; 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: zgshsr (y)
  apply householder vector based on 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

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## 29.6 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.

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).

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

Function Fil: 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

Function File: 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: is_abcd (a, b, c, d)
  Returns retval = 1 if the dimensions of a, b, c, d are compatible, otherwise retval = 0 with an appropriate diagnostic message printed to the screen. The matrices b, c, or d may be omitted.

Function File: [retval, u] = is_controllable (sys, tol)
Function File: [retval, u] = is_controllable (a, b, tol)
  Logical check for system controllability.

  Inputs

  sys
    system data structure
  a
  b
    n by n, n by m matrices, respectively
  tol
    optional roundoff paramter. default value: `10*eps`

  Outputs

  retval
    Logical flag; returns true (1) if the system sys or the pair (a,b) is controllable, whichever was passed as input arguments.
  U
    U is an orthogonal basis of the controllable subspace.

  Method Controllability is determined by applying Arnoldi iteration with complete re-orthogonalization to obtain an orthogonal basis of the Krylov subspace

      span ([b,a*b,...,a^{n-1}*b]).

  The Arnoldi iteration is executed with `krylov` if the system has a single input; otherwise a block Arnoldi iteration is performed with `krylovb`.

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.

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 matrix
    bw
      (n x nw) qw-transformed disturbance input matrix
    bu
      (n x nu) ru-transformed controlled input matrix;

      Note

    cz
      (nz x n) Qz-transformed error output matrix
    cy
      (ny x n) ry-transformed measured output matrix

      Note

    dzu
    dyw
      off-diagonal blocks of transformed system 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: is_digital (sys)
  Return nonzero if system is digital; inputs: sys: system data structure eflg: 0 [default] exit with an error if system is mixed (continuous and discrete components) : 1 print a warning if system is mixed (continuous and discrete) : 2 silent operation outputs: DIGITAL: 0: system is purely continuous : 1: system is purely discrete : -1: system is mixed continuous and discrete 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.

  Default: tol = 10*norm(a,'fro')*eps

  Returns 1 if the system sys or the pair (a,c) is observable, 0 if not.

  See `is_controllable` for detailed description of arguments and default values.

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

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

This file is part of Octave.

Octave is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2, or (at your option) any later version.

Octave is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Octave; see the file COPYING. If not, write to the Free Software Foundation, 59 Temple Place, Suite 330, Boston, MA 02111 USA.

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 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)

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## 29.7 System Analysis-Time Domain

Function File: c2d (sys, opt, t)
Function File: 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.

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

  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: d2c (sys, tol)
Function File: 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)`, with tol positive scalar, the `"log"` option is used. The default value for tol is `1e-8`.

    `"bi"`
      Conversion is performed via bilinear transform where 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
    indices or names of states with sampling time `sysgettsam(sys)` (may be empty); see `listidx`
  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 , 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 -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.

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 -Matrix in sys (default tol = 1.0e-10)

Function File: [y, t] = impulse (sys, inp, tstop, n)
  Impulse response for a linear system. The system can be discrete or multivariable (or both). If no output arguments are specified, `impulse` produces a plot or the impulse response data for system sys.

  Inputs

  sys
    System data structure.
  inp
    Index of input being excited
  tstop
    The argument tstop (scalar value) denotes the time when the simulation should end.
  n
    the number of data values.

    Both parameters tstop and n can be omitted and will be computed from the eigenvalues of the A-Matrix.

  Outputs y, t: impulse response

Function File: [y, t] = 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, t: impulse response

  When invoked with the output paramter y the plot is not displayed.

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## 29.8 System Analysis-Frequency Domain

Demonstration/tutorial script

Function File: frdemo ()
  Octave Controls toolbox demo: Frequency Response demo

Function File: [mag, phase, w] = bode (sys, w, out_idx, in_idx)
  If no output arguments are given: produce Bode plots of a system; otherwise, compute the frequency response of a system data structure

  Inputs

  sys
    a system data structure (must be either purely continuous or discrete; see is_digital)
  w
    frequency values for evaluation.

    if sys is continuous, then bode evaluates where is the system transfer function.

    if sys is discrete, then bode evaluates G(`exp`(jwT)), where

    - is the system sampling time
    - 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 =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 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",list("u_1","u_4");

  Outputs
  mag
  phase
    the magnitude and phase of the frequency response or 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 or 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: ltifr (a, b, w)
Function File: ltifr (sys, w)
  Linear time invariant frequency response of single input systems Inputs
  a
  b
    coefficient matrices of
  sys
    system data structure
  w
    vector of frequencies
  Outputs out

                                 -1
                  G(s) = (jw I-A) B

  for complex frequencies .

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 if sys is discrete, then bode evaluates , where 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 ) and select the frequency range based on the breakpoint locations of the frequencies.
  2. if sys is discrete time, the frequency range is limited to jwT in [0,2p*pi]
  3. A "smoothing" routine is used to ensure that the plot phase does not change excessively from point to point and that singular points (e.g., crossovers from +/- 180) are accurately shown.
  outputs, inputs: names or indices of the output(s) and input(s) to be used in the frequency response; see sysprune.

  Inputs (pass as empty to get default values)

  atol
    for interactive nyquist plots: atol is a change-in-slope tolerance for the of asymptotes (default = 0; 1e-2 is a good choice). This allows the user to "zoom in" on portions of the Nyquist plot too small to be seen with large asymptotes.
  Outputs
  realp
  imagp
    the real and imaginary parts of the frequency response or 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: 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.

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## 29.9 Controller Design

Function File: dgkfdemo ()
  Octave Controls toolbox demo: H2/Hinfinity options demos

Function File: hinfdemo ()

  H_infinity design demos for continuous SISO and MIMO systems and a discrete system. The SISO system is difficult to control because it is non minimum phase and unstable. The second design example controls the "jet707" plant, the linearized state space model of a Boeing 707-321 aircraft at v=80m/s (M = 0.26, Ga0 = -3 deg, alpha0 = 4 deg, kappa = 50 deg). Inputs: (1) thrust and (2) elevator angle outputs: (1) airspeed and (2) pitch angle. The discrete system is a stable and second order.

  SISO plant

                        s - 2
             G(s) = --------------
                    (s + 2)(s - 1)

                                      +----+
                 -------------------->| W1 |---> v1
             z   |                    +----+
             ----|-------------+                   || T   ||     => min.
                 |             |                       vz   infty
                 |    +---+    v   y  +----+
               u *--->| G |--->O--*-->| W2 |---> v2
                 |    +---+       |   +----+
                 |                |
                 |    +---+       |
                 -----| K |<-------
                      +---+

    W1 und W2 are the robustness and performance weighting functions

  MIMO plant
    The optimal controller minimizes the H_infinity norm of the augmented plant P (mixed-sensitivity problem):

             w
              1 -----------+
                           |                   +----+
                       +---------------------->| W1 |----> z1
             w         |   |                   +----+
              2 ------------------------+
                       |   |            |
                       |   v   +----+   v      +----+
                    +--*-->o-->| G  |-->o--*-->| W2 |---> z2
                    |          +----+      |   +----+
                    |                      |
                    ^                      v
                     u (from                 y (to K)
                       controller
                       K)

                          +    +           +    +
                          | z  |           | w  |
                          |  1 |           |  1 |
                          | z  | = [ P ] * | w  |
                          |  2 |           |  2 |
                          | y  |           | u  |
                          +    +           +    +

  DISCRETE SYSTEM
    This is not a true discrete design. The design is carried out in continuous time while the effect of sampling is described by a bilinear transformation of the sampled system. This method works quite well if the sampling period is "small" compared to the plant time constants.

  The continuous plant

                           1
             G (s) = --------------
              k      (s + 2)(s + 1)

    is discretised with a ZOH (Sampling period = Ts = 1 second):

                       0.199788z + 0.073498
             G(s) = --------------------------
                    (z - 0.36788)(z - 0.13534)

                                      +----+
                 -------------------->| W1 |---> v1
             z   |                    +----+
             ----|-------------+                   || T   ||     => min.
                 |             |                       vz   infty
                 |    +---+    v      +----+
                 *--->| G |--->O--*-->| W2 |---> v2
                 |    +---+       |   +----+
                 |                |
                 |    +---+       |
                 -----| K |<-------
                      +---+

    W1 and W2 are the robustness and performancs weighting functions

Function File: [l, m, p, e] = dlqe (a, g, c, sigw, sigv, z)
  Construct the linear quadratic estimator (Kalman filter) for the discrete time system

      x[k+1] = A x[k] + B u[k] + G w[k]
        y[k] = C x[k] + D u[k] + 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 H2 optimal controller per procedure in Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989

  Discrete time control per Zhou, Doyle, and Glover, ROBUST AND OPTIMAL CONTROL, Prentice-Hall, 1996

  Inputs input system is passed as either

  asys
    system data structure (see 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
    threshhold for 0. Default: 200*eps

  Outputs

  k
    system controller
  gain
    optimal closed loop gain
  kc
    full information control (packed)
  kf
    state estimator (packed)
  pc
    ARE solution matrix for regulator subproblem
  pf
    ARE solution matrix for filter subproblem

Function File: hinf_ctr (dgs, f, h, z, g)
  Called by `hinfsyn` to compute the H_inf optimal controller.

  Inputs

  dgs
    data structure returned by `is_dgkf`
  f
  h
    feedback and filter gain (not partitioned)
  g
    final gamma value
  Outputs controller (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 threshhold. Routine quits when gmax/gmin < 1+tol
  ptol
    poles with abs(real(pole)) < ptol*||H|| (H is appropriate Hamiltonian) are considered to be on the imaginary axis. Default: 1e-9
  tol
    threshhold for 0. Default: 200*eps

    gmax, min, tol, and tol must all be postive scalars.

  Outputs
  k
    system controller
  g
    designed gain value
  gw
    closed loop system
  xinf
    ARE solution matrix for regulator subproblem
  yinf
    ARE solution matrix for filter subproblem

  1. Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems," IEEE TAC August 1989

  2. Maciejowksi, J.M., "Multivariable feedback design," Addison-Wesley, 1989, ISBN 0-201-18243-2

  3. Keith Glover and John C. Doyle, "State-space formulae for all stabilizing controllers that satisfy and h-infinity-norm bound and relations to risk sensitivity," Systems & Control Letters 11, Oct. 1988, pp 167-172.

Function File: [retval, pc, pf] = hinfsyn_chk (a, b1, b2, c1, c2, d12, d21, g, ptol)
  Called by `hinfsyn` to see if gain g satisfies conditions in Theorem 3 of Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989

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

  Inputs as returned by `is_dgkf`, except for:

  g
    candidate gain level
  ptol
    as in `hinfsyn`

  Outputs

  retval
    1 if g exceeds optimal Hinf closed loop gain, else 0
  pc
    solution of "regulator" H-inf ARE
  pf
    solution of "filter" H-inf ARE
  Do not attempt to use this at home; no argument checking performed.

Function File: [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`, `listidx`)

    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

Function File: [k, p, e] = lqr (a, b, q, r, z)
  construct the linear quadratic regulator for the continuous time system

      dx
      -- = A x + B u
      dt

  to minimize the cost functional

            infinity
            /
        J = |  x' Q x + u' R u
           /
          t=0

  z omitted or

            infinity
            /
        J = |  x' Q x + u' R u + 2 x' Z u
           /
          t=0

  z included.

  The following values are returned:

  k
    The state feedback gain, (a - bk) is stable and minimizes the cost functional

  p
    The stabilizing solution of appropriate algebraic Riccati equation.

  e
    The vector of the closed loop poles of (a - bk).

  Reference Anderson and Moore, OPTIMAL CONTROL: LINEAR QUADRATIC METHODS, Prentice-Hall, 1990, pp. 56-58

Function File: lsim (sys, u, t, x0)
  Produce output for a linear simulation of a system

  Produces a plot for the output of the system, sys.

  U is an array that contains the system's inputs. Each 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 with output parameters: [y,x] = lsim(sys,u,t,[x0]) a plot is not displayed, however, the data is returned in y = system output and x = system states.

Function File: 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)

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## 29.10 Miscellaneous Functions (Not yet properly filed/documented)

@deftypefn{Function File}: axis2dlim (axdata)
  determine axis limits for 2-d data(column vectors); leaves a 10% margin around the plots. puts in margins of +/- 0.1 if data is one dimensional (or a single point)

  Inputs axdata nx2 matrix of data [x,y]

  Outputs axvec vector of axis limits appropriate for call to axis() function

Function File: moddemo (inputs)
  Octave Controls toolbox demo: Model Manipulations demo

Function File: prompt (inputs)

      function prompt([str])
      Prompt user to continue
      str: input string. Default value: "\n ---- Press a key to continue ---"

Function File: rldemo (inputs)
  Octave Controls toolbox demo: Root Locus demo

Function File: rlocus (inputs)

      [rldata, k] = rlocus(sys[,increment,min_k,max_k])
      Displays root locus plot of the specified SISO system.

             -----   ---     --------
         --->| + |---|k|---->| SISO |----------->
             -----   ---     --------        |
             - ^                             |
               |_____________________________|

      inputs: sys = system data structure
      min_k, max_k,increment: minimum, maximum values of k and
      the increment used in computing gain values
      Outputs: plots the root locus to the screen.
      rldata: Data points plotted column 1: real values, column 2: imaginary
      values)
      k: gains for real axis break points.

Function File: sortcom (inputs)

      [yy,idx] = sortcom(xx[,opt]): sort a complex vector
      xx: complex vector
      opt: sorting option:
       "re": real part (default)
       "mag": by magnitude
       "im": by imaginary part

      if opt != "im" then complex conjugate pairs are grouped together,
      a - jb followed by a + jb.
      yy: sorted values
      idx: permutation vector: yy = xx(idx)

Function File: ss2tf (inputs)

      [num,den] = ss2tf(a,b,c,d)
      Conversion from tranfer function to state-space.
      The state space system
            .
            x = Ax + Bu
            y = Cx + Du

      is converted to a transfer function

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

      used internally in system data structure format manipulations

Function File: ss2zp (inputs)

      Converts a state space representation to a set of poles and zeros.

      [pol,zer,k] = ss2zp(a,b,c,d) returns the poles and zeros of the state space
      system (a,b,c,d).  K is a gain associated with the zeros.

      used internally in system data structure format manipulations

Function File: 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 (= negative feedback)

Function File: tf2ss (inputs)

      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)

      via the function call [a,b,c,d] = tf2ss(num,den).
      The vector 'den' must contain only one row, whereas the vector 'num'
      may contain as many rows as there are outputs of the system 'y'.
      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: tf2zp (inputs)
  Converts transfer functions to poles / zeros.

  [zer,pol,k] = tf2zp(num,den) 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)
  k
    real scalar (leading coefficient)
  Outputs a, b, c, d The state space system

      .
      x = Ax + Bu
      y = Cx + Du

  is obtained from a vector of zeros and a vector of poles via the function call `[a,b,c,d] = zp2ss(zer,pol,k)`. The vectors ``zer`' and ``pol`' may either be row or column vectors. Each zero and pole that has an imaginary part must have a conjugate in the list. The number of zeros must not exceed the number of poles. ``k`' is `zp`-form leading coefficient.

Function File: [num, den] = zp2tf (zer, pol, k)
  Converts zeros / poles to a transfer function. Inputs
  zer
  pol
    vectors of (possibly complex) poles and zeros of a transfer function. Complex values should appear in conjugate pairs
  k
    real scalar (leading coefficient)
  `[num,den] = zp2tf(zer,pol,k)` forms the transfer function `num/den` from the vectors of poles and zeros.

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This document was generated by John W. Eaton on March, 12 2004 using texi2html