Modern Control Engineering

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206 Chapter 5 / Transient and Steady-State Response Analyses

Referring to Equation (5–52), the state x(t)is given by Thus,

(5–55)

The solution of Equations (5–54) and (5–55) gives the response to the initial condition.

Summarizing, the response of Equation (5–49) to the initial condition x(0)is obtained

by solving the following state-space equations:

where

MATLAB commands to obtain the response curves, where we do not specify the time

vectort(that is, we let the time vector be determined automatically by MATLAB), are

given next.

% Specify matrices A and B

[x,z,t] = step(A,B,A,B);

x1 = [1 0 0 ... 0]*x';

x2 = [0 1 0 ... 0]*x';







xn = [0 0 0 ... 1]*x';

plot(t,x1,t,x2, ... ,t,xn)

If we choose the time vector t(for example, let the computation time duration be

fromt= 0 to t = tpwith the computing time increment of ), then we use the following

MATLAB commands:

t = 0: Δt: tp;

% Specify matrices A and B

[x,z,t] = step(A,B,A,B,1,t);

x1 = [1 0 0 ... 0]*x';

x2 = [0 1 0 ... 0]*x';







xn = [0 0 0 ... 1]*x';

plot(t,x1,t,x2, ... ,t,xn)

(See, for example, Example 5–9.)

¢t

B=x(0), u=1(t)

x=Az+Bu

z# =Az+Bu

x=z# =Az+Bu

z#(t).

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