## MATLAB MODULE 5

Time Response Characteristics and LTI Viewer

## Example M5.1

Consider the position control system shown in Fig. M5.8.

.

### Fig. M5.8

(transfer function of motor, gears, and load)

(transfer function of power amplifier)

KA = preamplifier gain

sensitivity   of input and output potentiometers

Time response characteristics of the system   are generated by the following MATLAB script.

s = tf('s');

G1 = 100/(s+100);

G2 = 0.2083/(s*(s+1.71));

K = 1000;

G = series(K*G1,G2);

H = 1/pi;

M = series(feedback(G,H),H);

ltiview(M);

```
```

MATLAB response is shown in Fig. M5.9.

In the following we compare the time response characteristics of this system with the one obtained by replacing the power amplifier with a transfer function of unity (Fig. M5.10).

s = tf('s');

G1 = 1;

G2 = 0.2083/(s*(s+1.71));

K = 1000;

G = series(K*G1,G2);

H = 1/pi;

M1 = series(feedback(G,H),H);

ltiview(M1);

```

```

Fig. M5.9

Fig. M5.10

The second-order approximation seems to be quite reasonable.

The two responses in Figs M5.9 and M5.10 can be obtained in a single plot using the command ltiview(M,M1). Alternatively, choose Import... under the File menu in the LTI Viewer window and select M.

Example M5.2

Consider a unity feedback system shown in Fig. M5.11.

Fig. M5.11

We will consider the following three controllers:

D1(s) = 3; D2(s)=3+(15/ s ); and D3(s)=(3+(15/ s )+0.3 s )

The following MATLAB code generates the step responses shown in Fig. M5.12.

s = tf('s');

G = 59.292/(s^2+6.9779*s+15.12);

D1 = 3;

D2 = 3 + 15/s;

D3 = 3 + 15/s +0.3*s;

M1 = feedback(D1*G,1);

M2 = feedback(D2*G,1);

M3 = feedback(D3*G,1);

step(M1);

hold;

step(M2);

step(M3);

We need not access LTI viewer to find out the time response characteristics of the closed-loop system. The step command in MATLAB has this feature inbuilt in it as is seen in Fig. M5.12.

Fig. M5.12

Note that adding the integral term increases the oscillatory behaviour but eliminates the steady-state error, and that adding the derivative term reduces the oscillations, while maintaining zero steady-state error.