Ch. 9 Application to Control

9.1 Introduction to Control

• Consider a causal linear time-invariant system with input x(t) and output y(t).

• Y(s) = Gp(s)X(s) where Y(s) is the Laplace transform of y(t) and X(s) is the Laplace transform of x(t).

• H(s) = Gp(s) and the p stands for “plant”.

9.1 (cont.)

• The Tracking Problem• In many applications, x(t) is a reference

signal (say r(t)) and we want the output of the “plant” to be the same as the reference signal.

• x(t) is called a control input, and often x(t) or r(t)=ro, where ro is a constant or “set point”, which should also be the value of y(t) as t goes to infinity.

Open Loop Control

• Ideal tracking would have Y(s) = R(s) .

• The open loop control is then found from X(s) = R(s)/Gp(s).

• The above expression is called “model inversion” since Gp(s) is in the denominator.

• For setpoint control, r(t) = ro and so R(s)=ro/s

Problems with Setpoint Control

• Suppose Gp(s) = Bp(s)/Ap(s).• Suppose the degree of Bp(s) is M.• Suppose the degree of Ap(s) is N.• If N> M+1, then X(s) has a numerator with a polynomial

that is a degree larger than the denominator.• The inverse would result in an impulse or a derivative of

an impulse, which cannot be generated.• Also, the zeros of Gp(s) might be in the open right hand

plane, and when inverted the result would be poles in the open right hand plan (which again means x(t) cannot be generated.)

Nonideal Control Solution

• For setpoint control, a non ideal control x(t) could be found such that y(t) is asymptotic to ro; then y(t)→ ro as t→∞.

• Let Gc(s) be some rational function of s. • Let X(s) =( ro /s)Gc(s)• Then Y(s) =( ro /s)Gc(s)Gp(s).• So, zeros of Gc(s) could be implemented

to cancel out slow but stable poles of Gp(s).

Closed-Loop Control

• Figure 9.1a shows an open loop control system with addition disturbance provided by d(t).

• This disturbance could come from modeling errors and other disturbances.

• A control should be more robust that open loop control, and be able to tolerate disturbances.

Closed-Loop Control

• Closed-loop (or feedback) control will improve robustness, but requires that y(t) be measureable (using sensors) and then compared to r(t).

• This means the tracking error can be computed as e(t)=r(t)-y(t).

• So e(t) can be applied to the controller or “compensator” as in Figure 9.1b.

Proportional Control

• Simplest controller would be a constant, and so Gc(s) = Kp, where the p here stands for “proportional”.

• In figure 9.1b, X(s)=Gc(s) E(s) = Gc(s) [R(s)-Y(s)]• Here R(s) is the transform of the reference and E(s) is

the transform of the tracking error.• Taking the inverse transform gives x(t)=Kpe(t)=Kp{r(t) –

y(t)}• Thus the control signal, x(t), is “proportional” to the

tracking error.• So, Y(s)/R(s) = KpGp(s)/{1 + KpGp(s)}• Fig. 9.2 shows some example curves—note the rise of

each curve and the resulting steady state value.

Other Types of Control

• 9.2 Tracking Control– The simple proportional control constant becomes a

rational function of s, Gc(s).

• 9.2.1 Tracking a Step Reference– Proportional plus integral (PI) controller

• Gc(s) = KP + KI/s =(KPs + KI)/s

• Note: x(t) will have an integral of the tracking error.

– Proportional plus derivative (PD) controller• Gc(s) = KP + KDs

• Note: x(t) will have a derivative of the tracking error.

Other Concepts

• 9.3 Root Locus– Consists of all poles that satisfies the “angle

criterion”—this is related to the stability of the closed loop transfer function.

• 9.4 Applications to Control System Design– Root Locus can be used in the design of

controllers.– Of concern is the steady state error, settling

time and the percent of overshoot.