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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

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.

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 section 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.

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 Data structure access functions).
sys(0)
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)
sys(1:3)
boolean flags to 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 section 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 section System display functions.

Finite impulse response system interface functions

Function File : sys = fir2sys ( num{, tsam, inname, outname } )
construct a system data structure from FIR description Inputs:
num
vector of coefficients @math{[c_0 c_1 ... c_n]} 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 section Finite impulse response system interface functions for parameter descriptions.

State space system interface functions

Function File : sys = ss2sys (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: @math{tsam = 0} (continuous system)
n, nz
number of continuous, discrete states in the system default:
tsam = 0
@math{n = rows(a)}, @math{nz = 0}
tsam > 0
@math{ n = 0}, @math{nz = rows(a)} 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 default:
tsam = 0
@math{outlist = []}
tsam > 0
@math{outlist = 1:rows(c)}
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: @math{ d/dt xc(t) = acc*xc(t) + acd*xd(k*tsam) + bc*u(t)} @math{ xd((k+1)*tsam) = adc*xc(k*tsam) + add*xd(k*tsam) + bd*u(k*tsam)} @math{ yc(t) = ccc*xc(t) + ccd*xd(k*tsam) + dc*u(t)} @math{ 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 @math{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 = ss2sys(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 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. Inputs sys system data structure (See section 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

Transfer function system interface functions

Function File : sys = 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 list 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

Zero-pole system interface functions

Function File : sys = 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 section Zero-pole system interface functions for parameter descriptions. Example
 octave:1> sys=ss2sys([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 : retsys = syschnames (sys, opt, list, names)
Superseded by syssetsignals

Function File : retsys = 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. @math{yd(ii) = 0} if output ii is continous

Function File : systype = 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 : sys = 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.

Data structure internal functions

Function File : syschnamesl
used internally in syschnames item olist: index list old_names: original list names inames: new names listname: name of index list combines the two string lists old_names and inames

Function File : ioname = sysdefioname (n,str {,m})
return default input or output names given n, str, m. n is the final value, str is the string prefix, and m is start value used internally, minimal argument checking Example ioname = sysdefioname(5,"u",3) returns the list:
 ioname =
 (
   [1] = u_3
   [2] = u_4
   [3] = u_5
 )

Function File : stname = sysdefstname (n, nz)
return default state names given n, nz used internally, minimal argument checking

System display functions

Function File : y = polyout ( c{, x})
write formatted polynomial
    c(x) = c(1) * x^n + ... + c(n) x + c(n+1)

to string y or to the screen (if y is omitted) x defaults to the string "s"

See also: polyval, polyvalm, poly, roots, conv, deconv, residue, filter, polyderiv, polyinteg, polyout

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

Function File : outlist (lmat{, tabchar, yd, ilist })
Prints an enumerated list of strings. internal use only; minimal argument checking performed Inputs
lmat
list of strings
tabchar
tab character (default: none)
yd
indices of strings to append with the string "(discrete)" (used by sysout; minimal checking of this argument) @math{yd = [] } indicates all outputs are continuous
ilist
index numbers to print with names. default: 1:rows(lmat)

Outputs prints the list to the screen, numbering each string in order.

Block Diagram Manipulations

See section System Analysis-Time Domain

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

Function File : outputs = bddemo ( inputs )
Octave Controls toolbox demo: Block Diagram Manipulations demo

Function File : sys = buildssic(Clst, Ulst, Olst, Ilst, s1, s2, s3, s4, s5, s6, s7, s8)
Contributed by Kai Mueller. 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 : outsys = 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 see also: ord2 Contributed by Kai Mueller

See also: jet707 (MIMO example, Boeing 707-321 aircraft model)

Function File : sys = sysadd ( Gsys,Hsys)
returns sys = Gsys + Hsys.
           ________
      ----|  Gsys  |---
 u   |    ----------  +|         
 -----                (_)----> y
     |     ________   +|
      ----|  Hsys  |---
           --------

Function File : retsys = sysconnect (sys, out_idx,in_idx{,order, tol})
Close the loop from specified outputs to respective specified inputs Inputs
sys
system data structure
out_idx, in_idx
list of connections indices; @math{y(out_idx(ii))} is connected to @math{u(in_idx(ii))}.
order
logical flag (default = 0)
0
leave inputs and outputs in their original order
1
permute inputs and outputs to the order shown in the diagram below
tol
tolerance for singularities in algebraic loops default: 200eps

Outputs sys: resulting closed loop system. Method sysconnect internally permutes selected inputs, outputs as shown below, closes the loop, and then permutes inputs and outputs back to their original order

                  ____________________
  u_1       ----->|                  |----> y_1
                  |        sys       |
          old u_2 |                  |
 u_2* ---->(+)--->|                  |----->y_2 
 (in_idx)   ^     -------------------|    | (out_idx)
            |                             |
            -------------------------------

The input that has the summing junction added to it has an * added to the end of the input name.

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 : [n_tot, st_c, st_d, y_c, y_d] = syscont_disc(sys)
Used internally in syscont and sysdisc. Inputs sys is a system data structure. Outputs
n_tot
total number of states
st_c
vector of continuous state indices (empty if none)
st_d
vector of discrete state indices (empty if none)
y_c
vector of continuous output indices
y_d
vector of discrete output indices

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 : sys = sysgroup ( Asys, Bsys)
Combines two systems into a single system Inputs Asys, Bsys: system data structures Outputs @math{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 : names = sysgroupn (names)
names = sysgroupn(names) Locate and mark duplicate names inputs: names: list of signal names kind: kind of signal name (used for diagnostic message purposes only) outputs: returns names with unique suffixes added; diagnostic warning message is printed to inform the user of the new signal name

used internally in sysgroup and elsewhere.

Function File : sys = sysmult( Asys, Bsys)
Compute @math{sys = Asys*Bsys} (series connection):
 u   ----------     ----------
 --->|  Bsys  |---->|  Asys  |--->
     ----------     ----------

A warning occurs if there is direct feed-through from an input of Bsys or a continuous state of Bsys through a discrete output of Bsys to a continuous state or output in Asys (system data structure does not recognize discrete inputs).

Function File : retsys = sysprune ( Asys, out_idx, in_idx)
Extract specified inputs/outputs from a system Inputs
Asys
system data structure
out_idx,in_idx
list of connections indices; the new system has outputs y(out_idx(ii)) and inputs u(in_idx(ii)). May select as [] (empty matrix) to specify all outputs/inputs.

Outputs retsys: resulting system

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

Function File : pv = sysreorder( vlen, {var{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 : sys = sysscale (sys, outscale, inscale{, outname, inname})
scale inputs/outputs of a system. Inputs sys: structured system outscale, inscale: constant matrices of appropriate dimension Outputs sys: resulting open loop system:
       -----------    -------    -----------
 u --->| inscale |--->| sys |--->| outscale |---> y
       -----------    -------    -----------

If the input names and output names (each a list of strings) are not given and the scaling matrices are not square, then default names will be given to the inputs and/or outputs. A warning message is printed if outscale attempts to add continuous system outputs to discrete system outputs; otherwise yd is set appropriately in the returned value of sys.

Function File : sys = syssub (Gsys, Hsys)
returns @math{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 : wsys = wgt1o (vl, vh, fc)
State space description of a first order weighting function. Weighting function are needed by the H2/H_infinity design procedure. These function are part of thye augmented plant P (see hinfdemo for an applicattion example). vl = Gain @ low frequencies vh = Gain @ high frequencies fc = Corner frequency (in Hz, *not* in rad/sec)

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 @math{b:=b*b'}.
c
nxn matrix or pxm matrix; in the latter case c is replaced by @math{c:=c'*c}.
opt
(optional argument; default = "B"): String option passed to balance prior to ordered Schur decomposition.

Outputs x: solution of the ARE. Method Laub's Schur method (IEEE Transactions on Automatic Control, 1979) is applied to the appropriate Hamiltonian matrix.

Function File: dare (a, b, c, r, opt)
Return the solution, x of the discrete-time algebraic Riccati equation
 a' x a - x + a' x b (r + b' x b)^(-1) b' x a + c = 0

Inputs

a
n by n.
b
n by m.
c
n by n, symmetric positive semidefinite, or p by n. In the latter case @math{c:=c'*c} is used.
r
m by m, symmetric positive definite (invertible).
opt
(optional argument; default = "B"): String option passed to balance prior to ordered QZ decomposition.

Outputs x solution of DARE. Method Generalized eigenvalue approach (Van Dooren; SIAM J. Sci. Stat. Comput., Vol 2) applied to the appropriate symplectic pencil. See also: Ran and Rodman, "Stable Hermitian Solutions of Discrete Algebraic Riccati Equations," Mathematics of Control, Signals and Systems, Vol 5, no 2 (1992) pp 165-194.

Function File: x = dlyap (a, b)
Solve the discrete-time Lyapunov equation Inputs
a
n by n matrix
b
Matrix: n by n, n by m, or p by n.

Outputs x: matrix satisfying appropriate discrete time Lyapunov equation. Options:

Method Uses Schur decomposition method as in Kitagawa, An Algorithm for Solving the Matrix Equation X = FXF' + S, International Journal of Control, Volume 25, Number 5, pages 745--753 (1977). Column-by-column solution method as suggested in Hammarling, Numerical Solution of the Stable, Non-Negative Definite Lyapunov Equation, IMA Journal of Numerical Analysis, Volume 2, pages 303--323 (1982).
Function File : m = gram (a, b)
Return controllability grammian m of the continuous time system @math{ dx/dt = a x + b u}. m satisfies @math{ a m + m a' + b b' = 0 }.
Function File: lyap (a, b, c)
Function File: lyap (a, b)
Solve the Lyapunov (or Sylvester) equation via the Bartels-Stewart algorithm (Communications of the ACM, 1972). If a, b, and c are specified, then lyap returns the solution of the Sylvester equation
       a x + x b + c = 0
If only (a, b) are specified, then lyap returns the solution of the Lyapunov equation
       a' x + x a + b = 0
If b is not square, then lyap returns the solution of either
       a' x + x a + b' b = 0
or
       a x + x a' + b b' = 0
whichever is appropriate. Solves by using the Bartels-Stewart algorithm (1972).
Loadable Function: pinv (x, tol)
Return the pseudoinverse of x. Singular values less than tol are ignored. If the second argument is omitted, it is assumed that
tol = max (size (x)) * sigma_max (x) * eps,
where sigma_max (x) is the maximal singular value of x.
Function File : x = qzval (A, B)
Compute generalized eigenvalues of the matrix pencil
 (A - lambda B).
A and B must be real matrices. Note qzval is obsolete; use qz instead.
Function File : y = zgfmul(a,b,c,d,x)
Compute product of zgep incidence matrix F with vector x. Used by zgepbal (in zgscal) as part of generalized conjugate gradient iteration.
Function File : x = zgfslv(n,m,p,b)
solve system of equations for dense zgep problem
Function File : zz = zginit(a,b,c,d)
construct right hand side vector zz for the zero-computation generalized eigenvalue problem balancing procedure called by zgepbal
Function File : retsys = zgpbal(Asys)
used internally in tzero; minimal argument checking performed implementation of zero computation generalized eigenvalue problem balancing method (Hodel and Tiller, Allerton Conference, 1991) Based on Ward's balancing algorithm (SIAM J. Sci Stat. Comput., 1981) zgpbal computes a state/input/output weighting that attempts to reduced the range of the magnitudes of the nonzero elements of [a,b,c,d] The weighting uses scalar multiplication by powers of 2, so no roundoff will occur. zgpbal should be followed by zgpred
Function File : retsys = zgreduce(Asys,meps)
Implementation of procedure REDUCE in (Emami-Naeini and Van Dooren, Automatica, # 1982).
Function File : [nonz, zer] = zgrownorm (mat, meps)
returns nonz = number of rows of mat whose two norm exceeds meps zer = number of rows of mat whose two norm is less than meps
References: ZGEP: Hodel, "Computation of Zeros with Balancing," 1992, submitted to LAA Generalized CG: Golub and Van Loan, "Matrix Computations, 2nd ed" 1989
Function File : [a ,b ] = zgsgiv(c,s,a,b)
apply givens rotation c,s to row vectors a,b No longer used in zero-balancing (zgpbal); kept for backward compatibility
References:
ZGEP
Hodel, "Computation of Zeros with Balancing," 1992, Linear Algebra and its Applications
Generalized CG
Golub and Van Loan, "Matrix Computations, 2nd ed" 1989

System Analysis-Properties

Function File: [n, m, p] = abcddim (a, b, c, d)
Check for compatibility of the dimensions of the matrices defining the linear system [A, B, C, D] corresponding to
 dx/dt = a x + b u
 y = c x + d u

or a similar discrete-time system. If the matrices are compatibly dimensioned, then abcddim returns

n
The number of system states.
m
The number of system inputs.
p
The number of system outputs.

Otherwise abcddim returns n = m = p = -1. Note: n = 0 (pure gain block) is returned without warning. See also: is_abcd

Function File : [y, my, ny] = abcddims (x)
Used internally in abcddim. If x is a zero-size matrix, both dimensions are set to 0 in y. my and ny are the row and column dimensions of the result.

Function File : Qs = ctrb(sys {, b})
Function File : Qs = ctrb(A, B)
Build controllability matrix
              2       n-1
 Qs = [ B AB A B ... A   B ]

of a system data structure or the pair (A, B). Note ctrb forms the controllability matrix. The numerical properties of is_controllable are much better for controllability tests.

Function File : retval = h2norm(sys)
Computes the H2 norm of a system data structure (continuous time only) Reference: Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989

Function File : [g, gmin, gmax] = hinfnorm(sys{, tol, gmin, gmax, ptol})
Computes the H infinity norm of a system data structure. Inputs
sys
system data structure
tol
H infinity norm search tolerance (default: 0.001)
gmin
minimum value for norm search (default: 1e-9)
gmax
maximum value for norm search (default: 1e+9)
ptol
pole tolerance:
  • if sys is continuous, poles with |real(pole)| < ptol*||H|| (H is appropriate Hamiltonian) are considered to be on the imaginary axis.
  • if sys is discrete, poles with |abs(pole)-1| < ptol*||[s1,s2]|| (appropriate symplectic pencil) are considered to be on the unit circle
  • Default: 1e-9

Outputs

g
Computed gain, within tol of actual gain. g is returned as Inf if the system is unstable.
gmin, gmax
Actual system gain lies in the interval [gmin, gmax]

References: Doyle, Glover, Khargonekar, Francis, "State space solutions to standard H2 and Hinf control problems", IEEE TAC August 1989 Iglesias and Glover, "State-Space approach to discrete-time Hinf control," Int. J. Control, vol 54, #5, 1991 Zhou, Doyle, Glover, "Robust and Optimal Control," Prentice-Hall, 1996 $Revision: 1.9 $

Function File : Qb = obsv (sys{, c})
Build observability matrix
      | C        |
      | CA       |
 Qb = | CA^2     |
      | ...      |
      | CA^(n-1) |

of a system data structure or the pair (A, C). Note: obsv() forms the observability matrix. The numerical properties of is_observable() are much better for observability tests.

Function File : [zer, pol]= pzmap (sys)
Plots the zeros and poles of a system in the complex plane. Inputs sys system data structure Outputs if omitted, the poles and zeros are plotted on the screen. otherwise, pol, zer are returned as the system poles and zeros. (see sys2zp for a preferable function call)

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

Function File : [retval, U] = is_controllable (sys{, tol})
Function File : [retval, U] = is_controllable (a{, b ,tol})
Logical check for system controllability. Inputs
sys
system data structure
a, b
n by n, n by m matrices, respectively
tol
optional roundoff paramter. default value: 10*eps

Outputs

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

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

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

The Arnoldi iteration is executed with krylov if the system has a single input; otherwise a block Arnoldi iteration is performed with krylovb. See also is_observable, is_stabilizable, is_detectable, krylov, krylovb

Function File : [retval, U] = is_detectable (a, c{, tol})
Function File : [retval, U] = is_detectable (sys{, tol})
Test for detactability (observability of unstable modes) of (a,c). Returns 1 if the system a or the pair (a,c)is detectable, 0 if not. See is_stabilizable for detailed description of arguments and computational method. Default: tol = 10*norm(a,'fro')*eps

Function File : [retval, dgkf_struct ] = is_dgkf (Asys, nu, ny, tol )
Determine whether a continuous time state space system meets assumptions of DGKF algorithm. Partitions system into:
 [dx/dt] = [A  | Bw  Bu  ][w] 
 [ z   ]   [Cz | Dzw Dzu ][u]
 [ y   ]   [Cy | Dyw Dyu ]

or similar discrete-time system. If necessary, orthogonal transformations Qw, Qz and nonsingular transformations Ru, Ry are applied to respective vectors w, z, u, y in order to satisfy DGKF assumptions. Loop shifting is used if Dyu block is nonzero. Inputs

Asys
system data structure
nu
number of controlled inputs
ny
number of measured outputs
tol
threshhold for 0. Default: 200eps

Outputs

retval
true(1) if system passes check, false(0) otherwise
dgkf_struct
data structure of is_dgkf results. Entries:
nw, nz
dimensions of w, z
A
system A matrix
Bw
(n x nw) Qw-transformed disturbance input matrix
Bu
(n x nu) Ru-transformed controlled input matrix; Note @math{B = [Bw Bu] }
Cz
(nz x n) Qz-transformed error output matrix
Cy
(ny x n) Ry-transformed measured output matrix Note @math{C = [Cz; Cy] }
Dzu, Dyw
off-diagonal blocks of transformed D matrix that enter z, y from u, w respectively
Ru
controlled input transformation matrix
Ry
observed output transformation matrix
Dyu_nz
nonzero if the Dyu block is nonzero.
Dyu
untransformed Dyu block
dflg
nonzero if the system is discrete-time

is_dgkf exits with an error if the system is mixed discrete/continuous References

[1]
Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems," IEEE TAC August 1989
[2]
Maciejowksi, J.M.: "Multivariable feedback design,"

Function File : [retval,U] = is_observable (a, c{,tol})
Function File : [retval,U] = is_observable (sys{, tol})
Logical check for system observability. Default: tol = 10*norm(a,'fro')*eps Returns 1 if the system sys or the pair (a,c) is observable, 0 if not. See is_controllable for detailed description of arguments and default values.

Function File : retval = is_sample (Ts)
return true if Ts is a legal sampling time (real,scalar, > 0)

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

Function File : [retval, U] = is_stabilizable (sys{, tol})
Function File : [retval, U] = is_stabilizable (a{, b ,tol})
Logical check for system stabilizability (i.e., all unstable modes are controllable). Test for stabilizability is performed via an ordered Schur decomposition that reveals the unstable subspace of the system A matrix. Returns retval = 1 if the system, a, is stabilizable, if the pair (a, b) is stabilizable, or 0 if not. U = orthogonal basis of controllable subspace. Controllable subspace is determined by applying Arnoldi iteration with complete re-orthogonalization to obtain an orthogonal basis of the Krylov subspace.
   span ([b,a*b,...,a^   b]).

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

Function File: flg = is_signal_list (mylist)
Returns true if mylist is a list of individual strings (legal for input to syssetsignals).

Function File : retval = is_stable (a{,tol,dflg})
Function File : retval = is_stable (sys{,tol})
Returns retval = 1 if the matrix a or the system sys is stable, or 0 if not. Inputs
tol
is a roundoff paramter, set to 200*eps if omitted.
dflg
Digital system flag (not required for system data structure):
dflg != 0
stable if eig(a) in unit circle
dflg == 0
stable if eig(a) in open LHP (default)

System Analysis-Time Domain

Function File : dsys = c2d (sys{, opt, T})
Function File : dsys = c2d (sys{, T})
Inputs
sys
system data structure (may have both continuous time and discrete time subsystems)
opt
string argument; conversion option (optional argument; may be omitted as shown above)
"ex"
use the matrix exponential (default)
"bi"
use the bilinear transformation
     2(z-1)
 s = -----
     T(z+1)
FIXME: This option exits with an error if sys is not purely continuous. (The ex option can handle mixed systems.)
T
sampling time; required if sys is purely continuous. Note If the 2nd argument is not a string, c2d assumes that the 2nd argument is T and performs appropriate argument checks.

Outputs dsys discrete time equivalent via zero-order hold, sample each T sec. converts the system data structure describing

 .
 x = Ac x + Bc u

into a discrete time equivalent model

 x[n+1] = Ad x[n] + Bd u[n]

via the matrix exponential or bilinear transform Note This function adds the suffix _d to the names of the new discrete states.

Function File : csys = d2c (sys{,tol})
Function File : csys = d2c (sys, opt)
Convert discrete (sub)system to a purely continuous system. Sampling time used is sysgettsam(sys) Inputs
sys
system data structure with discrete components
tol
Scalar value. tolerance for convergence of default "log" option (see below)
opt
conversion option. Choose from:
"log"
(default) Conversion is performed via a matrix logarithm. Due to some problems with this computation, it is followed by a steepest descent algorithm to identify continuous time A, B, to get a better fit to the original data. If called as d2c(sys,tol), tol=positive scalar, the "log" option is used. The default value for tol is 1e-8.
"bi"
Conversion is performed via bilinear transform @math{z = (1 + s T / 2)/(1 - s T / 2)} where T is the system sampling time (see sysgettsam). FIXME: bilinear option exits with an error if sys is not purely discrete

Outputs csys continuous time system (same dimensions and signal names as in sys).

Function File : [dsys, fidx] = dmr2d (sys, idx, sprefix, Ts2 {,cuflg})
convert a multirate digital system to a single rate digital system states specified by idx, sprefix are sampled at Ts2, all others are assumed sampled at Ts1 = sysgettsam(sys). Inputs
sys
discrete time system; dmr2d exits with an error if sys is not discrete
idx
list of states with sampling time sysgettsam(sys) (may be empty)
sprefix
list of string prefixes of states with sampling time sysgettsam(sys) (may be empty)
Ts2
sampling time of states not specified by idx, sprefix must be an integer multiple of sysgettsam(sys)
cuflg
"constant u flag" if cuflg is nonzero then the system inputs are assumed to be constant over the revised sampling interval Ts2. Otherwise, since the inputs can change during the interval t in @math{[k Ts2, (k+1) Ts2]}, an additional set of inputs is included in the revised B matrix so that these intersample inputs may be included in the single-rate system. default cuflg = 1.

Outputs

dsys
equivalent discrete time system with sampling time Ts2. The sampling time of sys is updated to Ts2. if cuflg=0 then a set of additional inputs is added to the system with suffixes _d1, ..., _dn to indicate their delay from the starting time k Ts2, i.e. u = [u_1; u_1_d1; ..., u_1_dn] where u_1_dk is the input k*Ts1 units of time after u_1 is sampled. (Ts1 is the original sampling time of discrete time sys and Ts2 = (n+1)*Ts1)
fidx
indices of "formerly fast" states specified by idx and sprefix; these states are updated to the new (slower) sampling interval Ts2.

WARNING Not thoroughly tested yet; especially when cuflg == 0.

Function File : damp(p{, tsam})
Displays eigenvalues, natural frequencies and damping ratios of the eigenvalues of a matrix p or the A-matrix of a system p, respectively. If p is a system, tsam must not be specified. If p is a matrix and tsam is specified, eigenvalues of p are assumed to be in z-domain. See also: eig

Function File : gm = dcgain(sys{, tol})
Returns dc-gain matrix. If dc-gain is infinite an empty matrix is returned. The argument tol is an optional tolerance for the condition number of A-Matrix in sys (default tol = 1.0e-10)

Function File : [y, t] = impulse (sys{, inp,tstop, n})
Impulse response for a linear system. The system can be discrete or multivariable (or both). If no output arguments are specified, impulse produces a plot or the impulse response data for system sys. Inputs
sys
System data structure.
inp
Index of input being excited
tstop
The argument tstop (scalar value) denotes the time when the simulation should end.
n
the number of data values. Both parameters tstop and n can be omitted and will be computed from the eigenvalues of the A-Matrix.

Outputs y, t: impulse response

Function File : [y, t] = impulse (sys{, inp,tstop, n})
Step response for a linear system. The system can be discrete or multivariable (or both). If no output arguments are specified, impulse produces a plot or the step response data for system sys. Inputs
sys
System data structure.
inp
Index of input being excited
tstop
The argument tstop (scalar value) denotes the time when the simulation should end.
n
the number of data values. Both parameters tstop and n can be omitted and will be computed from the eigenvalues of the A-Matrix.

Outputs y, t: impulse response When invoked with the output paramter y the plot is not displayed.

Function File : [y, t] = stepimp(sitype,sys[, inp, tstop, n])
Impulse or step response for a linear system. The system can be discrete or multivariable (or both). This m-file contains the "common code" of step and impulse. Produces a plot or the response data for system sys. Limited argument checking; "do not attempt to do this at home". Used internally in impulse, step. Use step or impulse instead.

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 @math{G(jw)} where @math{G(s)} is the system transfer function. if sys is discrete, then bode evaluates G(exp(jwT)), where
  • T=sysgettsam(sys) (the system sampling time) and
  • @math{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 @math{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 @math{jwT} in [0,2 pi /T]
  3. A "smoothing" routine is used to ensure that the plot phase does not change excessively from point to point and that singular points (e.g., crossovers from +/- 180) are accurately shown.
out_idx, in_idx
the indices of the output(s) and input(s) to be used in the frequency response; see sysprune.

Outputs

mag, phase
the magnitude and phase of the frequency response @math{G(jw)} or @math{G(exp(jwT))} at the selected frequency values.
w
the vector of frequency values used

Notes

  1. If no output arguments are given, e.g.,
     bode(sys);
    
    bode plots the results to the screen. Descriptive labels are automatically placed. Failure to include a concluding semicolon will yield some garbage being printed to the screen (ans = []).
  2. If the requested plot is for an MIMO system, mag is set to @math{||G(jw)||} or @math{||G(exp(jwT))||} and phase information is not computed.

Function File : [wmin, wmax] = bode_bounds (zer, pol, dflg{, tsam })
Get default range of frequencies based on cutoff frequencies of system poles and zeros. Frequency range is the interval [10^wmin,10^wmax] Used internally in freqresp (bode, nyquist)

Function File : [f, w] = bodquist (sys, w, out_idx, in_idx)
used internally by bode, nyquist; compute system frequency response. Inputs
sys
input system structure
w
range of frequencies; empty if user wants default
out_idx
list of outputs; empty if user wants all
in_idx
list of inputs; empty if user wants all
rname
name of routine that called bodquist ("bode" or "nyquist")

Outputs

w
list of frequencies
f
frequency response of sys; @math{f(ii) = f(omega(ii))}

Note bodquist could easily be incorporated into a Nichols plot function; this is in a "to do" list.

Both bode and nyquist share the same introduction, so the common parts are in bodquist. It contains the part that finds the number of arguments, determines whether or not the system is SISO, and computes the frequency response. Only the way the response is plotted is different between the two functions.

Function File : [realp, imagp, w] = nyquist (sys{, w, out_idx, in_idx, atol})
Function File : nyquist (sys{, w, out_idx, in_idx, atol})
Produce Nyquist plots of a system; if no output arguments are given, Nyquist plot is printed to the screen. Compute the frequency response of a system. Inputs (pass as empty to get default values)
sys
system data structure (must be either purely continuous or discrete; see is_digital)
w
frequency values for evaluation. if sys is continuous, then bode evaluates @math{G(jw)} if sys is discrete, then bode evaluates @math{G(exp(jwT))}, where @math{T=sysgettsam(sys)} (the system sampling time)
default
the default frequency range is selected as follows: (These steps are NOT performed if w is specified)
  1. via routine bodquist, isolate all poles and zeros away from w=0 (jw=0 or @math{exp(jwT)=1}) and select the frequency range based on the breakpoint locations of the frequencies.
  2. if sys is discrete time, the frequency range is limited to jwT in [0,2p*pi]
  3. A "smoothing" routine is used to ensure that the plot phase does not change excessively from point to point and that singular points (e.g., crossovers from +/- 180) are accurately shown.

outputs, inputs: the indices of the output(s) and input(s) to be used in the frequency response; see sysprune. Inputs (pass as empty to get default values)

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

Outputs

realp, imagp
the real and imaginary parts of the frequency response @math{G(jw)} or @math{G(exp(jwT))} at the selected frequency values.
w
the vector of frequency values used

If no output arguments are given, nyquist plots the results to the screen. If atol != 0 and asymptotes are detected then the user is asked interactively if they wish to zoom in (remove asymptotes) Descriptive labels are automatically placed. Note: if the requested plot is for an MIMO system, a warning message is presented; the returned information is of the magnitude ||G(jw)|| or ||G(exp(jwT))|| only; phase information is not computed.

Function File : zr = tzero2 (a, b, c, d, bal)
Compute the transmission zeros of a, b, c, d.

bal = balancing option (see balance); default is "B".

Needs to incorporate mvzero algorithm to isolate finite zeros; use tzero instead.

Controller Design

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

Function File: hinfdemo ()

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

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

	                         +----+
	    -------------------->| W1 |---> v1
	z   |                    +----+
	----|-------------+                   || T   ||     => min.
	    |             |                       vz   infty
	    |    +---+    v   y  +----+
	  u *--->| G |--->O--*-->| W2 |---> v2
	    |    +---+       |   +----+
	    |                |
	    |    +---+       |
	    -----| K |<-------
	         +---+
W1 und W2 are the robustness and performance weighting functions
MIMO plant
The optimal controller minimizes the H_infinity norm of the augmented plant P (mixed-sensitivity problem):
      w
       1 -----------+
                    |                   +----+
                +---------------------->| W1 |----> z1
      w         |   |                   +----+
       2 ------------------------+
                |   |            |
                |   v   +----+   v      +----+
             +--*-->o-->| G  |-->o--*-->| W2 |---> z2
             |          +----+      |   +----+
             |                      |
             ^                      v
              u (from                 y (to K)
                controller
                K)

                   +    +           +    +
                   | z  |           | w  |
                   |  1 |           |  1 |
                   | z  | = [ P ] * | w  |
                   |  2 |           |  2 |
                   | y  |           | u  |
                   +    +           +    +
DISCRETE SYSTEM
This is not a true discrete design. The design is carried out in continuous time while the effect of sampling is described by a bilinear transformation of the sampled system. This method works quite well if the sampling period is "small" compared to the plant time constants.
The continuous plant
	              1
	G (s) = --------------
	 k      (s + 2)(s + 1)

is discretised with a ZOH (Sampling period = Ts = 1 second):
 
	          0.199788z + 0.073498
	G(s) = --------------------------
	       (z - 0.36788)(z - 0.13534)

	                         +----+
	    -------------------->| W1 |---> v1
	z   |                    +----+
	----|-------------+                   || T   ||     => min.
	    |             |                       vz   infty
	    |    +---+    v      +----+
	    *--->| G |--->O--*-->| W2 |---> v2
	    |    +---+       |   +----+
	    |                |
	    |    +---+       |
	    -----| K |<-------
	         +---+
W1 and W2 are the robustness and performancs weighting functions

Function File: [l, m, p, e] = dlqe (a, g, c, sigw, sigv, z)
Construct the linear quadratic estimator (Kalman filter) for the discrete time system
 x[k+1] = A x[k] + B u[k] + G w[k]
   y[k] = C x[k] + D u[k] + w[k]

where w, v are zero-mean gaussian noise processes with respective intensities sigw = cov (w, w) and sigv = cov (v, v). If specified, z is cov (w, v). Otherwise cov (w, v) = 0. The observer structure is

 z[k+1] = A z[k] + B u[k] + k (y[k] - C z[k] - D u[k])

The following values are returned:

l
The observer gain, (a - alc). is stable.
m
The Riccati equation solution.
p
The estimate error covariance after the measurement update.
e
The closed loop poles of (a - alc).

Function File: [k, p, e] = dlqr (a, b, q, r, z)
Construct the linear quadratic regulator for the discrete time system
 x[k+1] = A x[k] + B u[k]

to minimize the cost functional

 J = Sum (x' Q x + u' R u)

z omitted or

 J = Sum (x' Q x + u' R u + 2 x' Z u)

z included. The following values are returned:

k
The state feedback gain, (a - bk) is stable.
p
The solution of algebraic Riccati equation.
e
The closed loop poles of (a - bk).

References

  1. Anderson and Moore, Optimal Control: Linear Quadratic Methods, Prentice-Hall, 1990, pp. 56-58
  2. Kuo, Digital Control Systems, Harcourt Brace Jovanovich, 1992, section 11-5-2.

Function File : {[K}, gain, Kc, Kf, Pc, Pf] = h2syn(Asys, nu, ny, tol)
Design H2 optimal controller per procedure in Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989 Discrete time control per Zhou, Doyle, and Glover, ROBUST AND OPTIMAL CONTROL, Prentice-Hall, 1996 Inputs input system is passed as either
Asys
system data structure (see ss2sys, sys2ss)
  • controller is implemented for continuous time systems
  • controller is NOT implemented for discrete time systems
nu
number of controlled inputs
ny
number of measured outputs
tol
threshhold for 0. Default: 200*eps

Outputs

K
system controller
gain
optimal closed loop gain
Kc
full information control (packed)
Kf
state estimator (packed)
Pc
ARE solution matrix for regulator subproblem
Pf
ARE solution matrix for filter subproblem

Function File : K = hinf_ctr(dgs, F, H, Z, g)
Called by hinfsyn to compute the H_inf optimal controller. Inputs
dgs
data structure returned by is_dgkf
F, H
feedback and filter gain (not partitioned)
g
final gamma value

Outputs controller K (system data structure) Do not attempt to use this at home; no argument checking performed.

Function File : [K, g, GW, Xinf, Yinf] = hinfsyn(Asys, nu, ny, gmin, gmax, gtol{, ptol, tol})
Inputs input system is passed as either
Asys
system data structure (see ss2sys, sys2ss)
  • controller is implemented for continuous time systems
  • controller is NOT implemented for discrete time systems (see bilinear transforms in c2d, d2c)
nu
number of controlled inputs
ny
number of measured outputs
gmin
initial lower bound on H-infinity optimal gain
gmax
initial upper bound on H-infinity optimal gain
gtol
gain threshhold. Routine quits when gmax/gmin < 1+tol
ptol
poles with abs(real(pole)) < ptol*||H|| (H is appropriate Hamiltonian) are considered to be on the imaginary axis. Default: 1e-9
tol
threshhold for 0. Default: 200*eps gmax, min, tol, and tol must all be postive scalars.

Outputs

K
system controller
g
designed gain value
GW
closed loop system
Xinf
ARE solution matrix for regulator subproblem
Yinf
ARE solution matrix for filter subproblem

  1. Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems," IEEE TAC August 1989
  2. Maciejowksi, J.M., "Multivariable feedback design," Addison-Wesley, 1989, ISBN 0-201-18243-2
  3. Keith Glover and John C. Doyle, "State-space formulae for all stabilizing controllers that satisfy and h-infinity-norm bound and relations to risk sensitivity," Systems & Control Letters 11, Oct. 1988, pp 167-172.

Function File: [retval, Pc, Pf] = hinfsyn_chk(A, B1, B2, C1, C2, D12, D21, g, ptol)
Called by hinfsyn to see if gain g satisfies conditions in Theorem 3 of Doyle, Glover, Khargonekar, Francis, "State Space Solutions to Standard H2 and Hinf Control Problems", IEEE TAC August 1989 Warning Do not attempt to use this at home; no argument checking performed. Inputs as returned by is_dgkf, except for:
g
candidate gain level
ptol
as in hinfsyn

Outputs

retval
1 if g exceeds optimal Hinf closed loop gain, else 0
Pc
solution of "regulator" H-inf ARE
Pf
solution of "filter" H-inf ARE

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

Function File: [k, p, e] = lqe (a, g, c, sigw, sigv, z)
Construct the linear quadratic estimator (Kalman filter) for the continuous time system
 dx
 -- = a x + b u
 dt
 
 y = c x + d u

where w and v are zero-mean gaussian noise processes with respective intensities

 sigw = cov (w, w)
 sigv = cov (v, v)

The optional argument z is the cross-covariance cov (w, v). If it is omitted, cov (w, v) = 0 is assumed. Observer structure is dz/dt = A z + B u + k (y - C z - D u) The following values are returned:

k
The observer gain, (a - kc) is stable.
p
The solution of algebraic Riccati equation.
e
The vector of closed loop poles of (a - kc).

See also: h2syn, lqe, lqr

Function File: [k, p, e] = lqr (a, b, q, r, z)
construct the linear quadratic regulator for the continuous time system
 dx
 -- = A x + B u
 dt

to minimize the cost functional

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

z omitted or

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

z included. The following values are returned:

k
The state feedback gain, (a - bk) is stable and minimizes the cost functional
p
The stabilizing solution of appropriate algebraic Riccati equation.
e
The vector of the closed loop poles of (a - bk).

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

Function File : lsim (sys, u, t{,x0})
Produce output for a linear simulation of a system Produces a plot for the output of the system, sys. U is an array that contains the system's inputs. Each column in u corresponds to a different time step. Each row in u corresponds to a different input. T is an array that contains the time index of the system. T should be regularly spaced. If initial conditions are required on the system, the x0 vector should be added to the argument list. When the lsim function is invoked with output parameters: [y,x] = lsim(sys,u,t,[x0]) a plot is not displayed, however, the data is returned in y = system output and x = system states.

Function File : K = place (sys, P)
Computes the matrix K such that if the state is feedback with gain K, then the eigenvalues of the closed loop system (i.e. A-BK) are those specified in the vector P. Version: Beta (May-1997): If you have any comments, please let me know. (see the file place.m for my address) Written by: Jose Daniel Munoz Frias.

Miscellaneous Functions (Not yet properly filed/documented)

@deftypefn{Function File }: axvec = axis2dlim (axdata)
determine axis limits for 2-d data(column vectors); leaves a 10% margin around the plots. puts in margins of +/- 0.1 if data is one dimensional (or a single point) Inputs axdata nx2 matrix of data [x,y] Outputs axvec vector of axis limits appropriate for call to axis() function

Function File : outputs = moddemo ( inputs )
  Octave Controls toolbox demo: Model Manipulations demo
  Written by David Clem August 15, 1994
 

Function File : outputs = prompt ( inputs )
  function prompt([str])
  Prompt user to continue
  str: input string. Default value: "\n ---- Press a key to continue ---"
  Written by David Clem August 15, 1994
  Modified A. S. Hodel June 1995
 
 

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

Function File : outputs = rlocus ( inputs )
  [rldata, k] = rlocus(sys[,increment,min_k,max_k])
  Displays root locus plot of the specified SISO system.
  
        -----   --     -------- 
    --->| + |---|k|---->| SISO |----------->
        -----   --     --------        | 
        - ^                             | 
          |_____________________________|  
 
 inputs: sys = system data structure
         min_k, max_k,increment: minimum, maximum values of k and
                the increment used in computing gain values
  Outputs: plots the root locus to the screen.  
    rldata: Data points plotted column 1: real values, column 2: imaginary
            values)
    k: gains for real axis break points.
 
 

Function File : outputs = ss2tf ( inputs )
  [num,den] = ss2tf(a,b,c,d)
  Conversion from tranfer function to state-space.
  The state space system
       . 
       x = Ax + Bu
       y = Cx + Du
 
  is converted to a transfer function
 
                 num(s)
           G(s)=-------
                 den(s)
 
  used internally in system data structure format manipulations
 
 

Function File : outputs = ss2zp ( inputs )
  Converts a state space representation to a set of poles and zeros.
 
  [pol,zer,k] = ss2zp(a,b,c,d) returns the poles and zeros of the state space 
  system (a,b,c,d).  K is a gain associated with the zeros.
 
  used internally in system data structure format manipulations
 
 

Function File : outputs = susball ( inputs )


Function File : [A, B, C, D] = zp2ss (zer, pol, k)
Conversion from zero / pole to state space. Inputs
zer, pol
vectors of (possibly) complex poles and zeros of a transfer function. Complex values must come in conjugate pairs (i.e., x+jy in zer means that x-jy is also in zer)
k
real scalar (leading coefficient)

Outputs A, B, C, D The state space system

 .
 x = Ax + Bu
 y = Cx + Du

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

Function File : [poly, rvals] = zp2ssg2 (rvals)
Used internally in zp2ss Extract 2 values from rvals (if possible) and construct a polynomial with those roots.

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

[num,den] = zp2tf(zer,pol,k) forms the transfer function num/den from the vectors of poles and zeros.

% DO NOT EDIT! Generated automatically by munge-texi.


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