Page 146 - MATLAB an introduction with applications

Control Systems ——— 131

we obtain
z = Az + x(0) 1(t) = Az + B u ...(3.11)
where
B = x(0) and u = 1(t) ...(3.12)
Since x = z , Eq. (3.9) becomes
y = Cz ...(3.13)
From Eqs. (3.11) and (3.13), we obtain
y = C (Az + Bu ) = CAz + CBu ...(3.14)
The response of the system is obtained from the Eqs. (3.11) and (3.14) to a given initial condition
The following MATLAB commands may be used to obtain the response curves:

[y, z, t ] = step (A, B, C*A, C*B);
y = [1 0 0 …0] * y’ ;
1
y = [0 1 0 …0] * y’ ; ...(3.15)
2
. .
. .
ym = [0 0 0 …1] * y’ ;

plot (t, y1 t, y2,........, t, ym).
,

3.11 SECOND-ORDER SYSTEMS
The standard form of a second-order system is defined by

ω 2
G(s) = n ...(3.16)
2
s +ξω n s + ω 2 n
2
where
ξ is the damping ratio of the system and ω is the undamped natural frequency of the system.
n
The dynamic behaviour of the second order system is then described in terms of two parameters ξ and ù .
n
If 0 < ξ < 1, the closed loop poles are complex conjugates and lie in the left-half plane. The system is
called underdamped, and the transient response is oscillatory. If ξ = 0, the transient response does not die
out. If ξ = 1, the system is called critically damped. Overdamped system corresponds to ξ = 1.
Given ω and ξ, then the MATLAB command
n
printsys (num, den)
or
printsys(num, den, s)
prints the num/den as a ratio of polynomials in s.
The unit-step response of the transfer-function system using MATLAB is obtained with the use of step-
response commands with left-hand arguments.
c = step ( num, den, t )
or
[y, x, t] = step (num, den, t).

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