MATLAB Control Theory
State Space Control

State Space Basics

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DEFINITIONS & DERIVATION:

Housekeeping

close all; clear; clc; s = tf('s');

DEFINITIONS & DERIVATION:

Given a system provided by the state equation and the output equation the system can be represented in state-space by the equations

and

where:

State Space.PNG
A = [0 1; 1 0];
B = [0; 1];
C = [0 1];
D = [0];
E = [0];
F = [0];
sys = ss(A,B,C,D)
sys = A = x1 x2 x1 0 1 x2 1 0 B = u1 x1 0 x2 1 C = x1 x2 y1 0 1 D = u1 y1 0 Continuous-time state-space model. Model Properties

CONTROLLABILITY & OBSERVABILITY

A matrix is said to be controllable if the Controllability Matrix is full range; i.e.

A matrix is said to be observable if the Observability Matrix is full range; i.e.

A matrix is said to be output-controllable if the Output Controllability Matrix is full range; i.e.

CM = ctrb(sys);
OM = obsv(sys);
OCM = [C*B C*A*B];
disp("The Controllability Matrix = "+mat2str(CM)+" and is rank "+string(rank(CM))+newline+newline...
+"The Observability Matrix = "+mat2str(OM)+" and is rank "+string(rank(OM))+newline+newline...
+"The Output Controllability Matrix = "+mat2str(OCM)+" and is rank "+string(rank(OCM)))
The Controllability Matrix = [0 1;1 0] and is rank 2 The Observability Matrix = [0 1;1 0] and is rank 2 The Output Controllability Matrix = [1 0] and is rank 1

Transfer Function Matrix

A system in state space form can be converted to a transfer function (or a matrix of transfer functions in the case of multiple inputs or outputs) via the equation:

[num, den] = ss2tf(A,B,C,D);
sys_TF = tf(num,den)
sys_TF = s ------- s^2 - 1 Continuous-time transfer function. Model Properties

Solution of State Equations

Consider the state equation with as an input [forcing] function:

We can take the Laplace Transform and simplify:

which can be simplified to:

and by taking the inverse Laplace Transform we get: where