Signals and Systems - Electrical Engineering

364 C H A P T E R 6: Application to Control and Communications

Hc(s)

x(t) e(t) y(t)

η(t)

+ G(s) +

−

Hc(s)

x(t) y(t)

η(t)

G(s) +

(a)(b)

FIGURE 6.5

(a) Closed- and (b) open-loop control systems. The transfer function of the plant isG(s)and the transfer function

of the controller isHc(s).

let us call itG(s). The controller, with a transfer functionHc(s), is designed to make the output of

the overall system perform in a specified way. For instance, in a cruise control the plant is the car,

and the desired performance is to automatically set the speed of the car to a desired value. There are

two possible ways the controller and the plant are connected: in open-loop or in closed-loop (see

Figure 6.5).

Open-Loop Control

In theopen-loopapproach the controller is cascaded with the plant (Figure 6.5(b)). To make the

outputy(t)follow the reference signal at the inputx(t), we minimize an error signal

e(t)=y(t)−x(t)

Typically, the output is affected by a disturbanceη(t), due to modeling or measurement errors. If we

assume initially no disturbance,η(t)=0, we find that the Laplace transform of the output of the

overall system is

Y(s)=L[y(t)]=Hc(s)G(s)X(s)

and that of the error is

E(s)=Y(s)−X(s)=[Hc(s)G(s)−1]X(s)

To make the error zero, so thaty(t)=x(t), it would require thatHc(s)= 1 /G(s)or the inverse of the

plant, making the overall transfer function of the systemHc(s)G(s)unity.

Remarks

Although open-loop systems are simple to implement, they have several disadvantages:

n The controller Hc(s)must cancel the poles and the zeros of G(s)exactly, which is not very practical. In

actual systems, the exact location of poles and zeros is not known due to measurement errors.

n If the plant G(s)has zeros on the right-hand s-plane, then the controller Hc(s)will be unstable, as its poles

are the zeros of the plant.

n Due to ambiguity in the modeling of the plant, measurement errors, or simply the presence of noise, the

output y(t)is typically affected by a disturbance signalη(t)mentioned above (η(t)is typically random—

we are going to assume for simplicity that it is deterministic so we can compute its Laplace transform).