ss __ Functions (Control System Toolbox™)

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Functions (Control System Toolbox) jar:file:///C:/Program%20Files/MATLAB/R2008a/help/toolbox/

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Control System Toolbox

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ssSpecify state-space models or convert LTI model to state space

Syntax

ss

sys = ss( a, b, c, d)sys = ss( a, b, c, d, Ts)sys = ss( d)sys = ss( a, b, c, d, l t i sys)sys_ss = ss( sys)

Description

ss is used to create real- or complex-valued state-space models (SS objects) or to convert transfer function orzero-pole-gain models to state space.

Creation o f State-Space Models

sys = ss( a, b, c, d) creates the continuous-time state-space model

For a model with Nx states, Ny outputs, and Nu inputs:

a is an Nx-by-Nx real- or complex-valued matrix.

b is an Nx-by-Nu real- or complex-valued matrix.

c is an Ny-by-Nx real- or complex-valued matrix.

d is an Ny-by-Nu real- or complex-valued matrix.

The output sys is an SS model that stores the model data (see "State-Space Models" on page 2-14). If , you cansimply set d to the scalar0 (zero), regardless of the dimension.

sys = ss( a, b, c, d, Ts) creates the discrete-time model

with sample timeTs (in seconds). SetTs = - 1 orTs = [ ] to leave the sample time unspecified.

sys = ss( d) specifies a static gain matrix and is equivalent to

sys = ss( [ ] , [ ] , [ ] , d)

sys = ss( a, b, c, d, l t i sys) creates a state-space model with generic LTI properties inherited from the LTI modell t i sys (including the sample time). See "Generic Properties" on page 2-26 for an overview of generic LTI properties.

See "Building LTI Arrays" on page 4-12 for information on how to build arrays of state-space models.

Any of the previous syntaxes can be followed by property name/property value pairs.

' Proper t yName' , Proper t yVal ue

Each pair specifies a particular LTI property of the model, for example, the input names or some notes on the modelhistory. See set and the example below for details. Note that

sys = ss(a, b, c, d, ' Pr oper t y1' , Val ue1, . . . , ' Pr oper t yN' , Val ueN)

is equivalent to the sequence of commands.


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sys = ss( a, b, c, d)set ( sys, ' Pr oper t y1' , Val ue1, . . . , ' Pr oper t yN' , Val ueN)

Conversion to State Space

sys_ss = ss( sys) converts an arbitrary TF or ZPK model sys to state space. The output sys_ss is an equivalentstate-space model (SS object). This operation is known as state-space realization.

sys_ss = ss( sys, ' mi ni mal ' ) produces a state-space realization with no uncontrollable or unobservable states.This is equivalent to sys_ss = mi nr eal ( ss( sys) ) .

Algori thm

In the case of TF to SS model conversion, ss( sys_t f ) returns a modified version of the controllable canonical form. Ituses an algorithm similar to t f 2ss, but further rescales the state vector to compress the numerical range in state matrixA and to improve numerics in subsequent computations.

In the case of ZPK to SS conversion, ss( sys_zpk) uses direct form II structures as defined in signal processing texts.See "Discrete-Time Signal Processing" by Oppenheim and Schafer for details.

For example, in the following code, A and sys. a differ by a diagonal state transformation:

n=[ 1 1] ;d=[ 1 1 10] ;[ A, B, C, D] =t f 2ss( n, d) ;sys=ss(t f ( n, d) ) ;A

A =

- 1 - 101 0

sys. a

ans =- 1 - 52 0

See the bal ance orssbal documentation for details.

Examples

Example 1

The command

sys = ss(A, B, C, D, 0. 05, ' st at ename' , {' posi t i on' ' vel oci t y' }, . . .' i nputname' , ' f orce' , . . .' not es' , ' Cr eat ed 10/ 15/ 96' )

creates a discrete-time model with matrices and sample time 0.05 second. This model has two states

labeled posi t i on and vel oc i ty , and one input labeled f orce (the dimensions of should be consistent

with these numbers of states and inputs). Finally, a note is attached with the date of creation of the model.

Example 2

Compute a state-space realization of the transfer function


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by typing

H = [ t f ( [ 1 1] , [ 1 3 3 2] ) ; t f ( [ 1 0 3] , [ 1 1 1] ) ] ;sys = ss( H) ;si ze( sys)St ate- space model wi t h 2 out put s, 1 i nput , and 5 st ates.

Note that the number of states is equal to the cumulative order of the SISO entries ofH(s).

To obtain a minimal realization ofH(s), type

sys = ss( H, ' mi n' ) ;s i ze( sys)St ate- space model wi t h 2 out put s, 1 i nput , and 3 st ates.

The resulting state-space model order has order three, the minimum number of states needed to represent H(s). This

can be seen directly by factoring H(s)as the product of a first order system with a second order one.

See Also

dss , f r d, get , set , ssdat a, t f, zpk

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sminreal ss2ss

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ss __ Functions (Control System Toolbox™)

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