Modern Control Engineering

Example Problems and Solutions 699

This is the MATLAB representation of the following state-space equations:

A–9–6. Consider the system defined by

where

Obtain the response of the system to each of the following inputs:

(a) The rcomponents of uare impulse functions of various magnitudes.

(b) The rcomponents of uare step functions of various magnitudes.

(c) The rcomponents of uare ramp functions of various magnitudes.

Solution.

(a)Impulse response:Referring to Equation (9–43), the solution to the given state equation is

Substitutingt 0 =0–into this solution, we obtain

Let us write the impulse input u(t)as

wherewis a vector whose components are the magnitudes of rimpulse functions applied at

t=0. The solution of the state equation when the impulse input d(t)wis given at t=0is

(9–95)

(b)Step response:Let us write the step input u(t)as

u(t)=k

=eAt x(0-)+eAt Bw

x(t)=eAt x(0-)+

3

t

0 -

eA(t-t) Bd(t) wdt

u(t)=d(t) w

x(t)=eAt x(0-)+

3

t

0 -

eA(t-t) Bu(t)dt

x(t)=eAAt-t^0 B xAt 0 B+

3

t

t 0

eA(t-t) Bu(t)dt

B=n*r constant matrix

A=n*n constant matrix

u=control vector (r-vector)

x=state vector (n-vector)

x# =Ax+Bu

y =[0 25.04 5.008]C

x 1

x 2

x 3

S +[0]u

C

x# 1

x# 2

x

3

S =C

- 5.0325

1

0

- 25.1026

0

1

- 5.008

0

0

SC

x 1

x 2

x 3

S +C

1

0

0

Su