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.

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

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.

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).

29.1.2 tf format variables

num

numerator coefficients (vector)

den

denominator coefficients (vector)

29.1.3 zp format variables

zer

system zeros (vector)

pol

system poles (vector)

k

leading coefficient (scalar)

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)

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

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.

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

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

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

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.

29.2.6 Data structure internal functions

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"

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)

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

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)

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.

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.

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)

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.