Lab Practice 1

MATLAB Script:

% Automatic Control, Laboratory Practice Session 1
% Objective: Solve LTI Continuous-time dyn. systems using SS equations
% and/or transfer function

% Problem 1:
% Given the LTI system
% x'(t) = Ax(t) + Bu(t) - State Equation

A = [5,8; 1,3]; % State Matrix
B = transpose([4,1]); % Input Matrix
% y(t) = Cx(t) + Du(t) - Output Equation
C = [3, -4]; % Output matrix
D = [7];     % I/O "matrix", scalar, but we'll represent it as a single-component vector
			 % because MATLAB likes creating problems out of everything.

sys = ss(A,B,C,D)

% a) Compute the analytical expressions of state and output responses, with
% initial conditions
x0 = [3; -3];

% and null input
ut = 0;
us = 0; % The laplace transform of zero is zero

% We define the laplace variable (s, for solving in the s-domain.)
s = tf('s');

% Solve for Xzi,Xs:
xzi = zpk(minreal((inv(s*eye(2)-A)*x0),1e-3));
xzs = zpk(minreal((inv(s*eye(2)-A)*B*us),1e-3)); % zero-state response is zero

% Find PFE coef. via tfdata
[numx1,denx1] = tfdata(xzi(1),'v');
[rx1,px1] = residue(numx1,denx1)

[numx2,denx2] = tfdata(xzi(2),'v');
[rx2,px2] = residue(numx2,denx2)

% Zeroes of first component: -2, 5
% Poles of first component: 7.0, 1.0

% Zeroes of second component: -0.5, -2.5
% Poles of second component: 7.0, 1.0

% We now determine output response
Y = zpk(minreal((C*inv(s*eye(2)-A))*(x0+(B*us)),1e-3));

% One component. Let us determine zeroes and poles.
% PFE
[numy,deny] = tfdata(Y,'v');
[ry,py] = residue(numy,deny)

% Y zeroes and poles, approximated to nearest integer
% Zeroes: -4, 25
% Poles: 7, 1

eigvalues = eig(A)
% eigenvalues: 7,1

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