Moodle 8 - Controller Actions PDF

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Control Engineering...

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CHAPTER 8 Controller Actions 8.1 Introduction Consider an automatic control system whose block diagram is shown in Fig. 8.1. The output is maintained within prescribed limits by means of a control action. A deviation of the controlled variable from the reference signal is detected by an error detector. The error thus detected is converted into a control signal by the controller.

E(s) R(s) + Error Reference signal or command signal

Controller U(s) Gc(s) Control signal

Plant Gp(s)

C(s) Controlled (output) signal

Fig. 8.1 Block diagram of an automatic control system The controller is typically expected to ensure that the closed-loop system meets the following criteria: • Stability; • Fast response to changes in command signal; • Reasonable damping; • Zero steady-state error to prescribed command inputs; and • Rejection of disturbance effects. By far the most popular controller in industry is the so-called three-term or PID controller. The controller generates a control signal which is a combination of signals that are proportional to, the integral of, and the derivative of the error signal, hence the term proportional plus integral plus derivative (PID) controller. Its transfer function is ideally given by U ( s)

1 = G ( s ) = K [1 + + τ s] D c c τ s E ( s) i

(8.1)

where Kc = controller gain τi = integral time τD = derivative time Although most controllers use Kc as the proportional setting, a term, called proportional band (PB), could also be used.

PB is defined as follows:

PB =

1 × 100% Kc

(8.2)

In the following sections, the effects of the individual control actions, namely, proportional, integral and derivative are briefly investigated on the basis of a simple second-order plant. Also the use of derivative feedback is considered. Let the plant transfer function be

ω n2 G p (s ) = s ( s + 2ζωn )

(8.3)

where ωn = undamped natural frequency and ζ = damping ratio of the plant. 8.2 Proportional Control Action (P) As shown in Fig. 8.2, the controller transfer function is Gc ( s ) = K c . The system open-loop transfer function is

GH ( s) =

K Cω n2 s( s + 2ζω n )

(8.4)

Plant E(s)

R(s) +

Controller U(s) Kc

-

ω n2 s( s + 2ζω n )

C(s)

Fig. 8.2 Block diagram of an automatic control system with proportional control action. The velocity error constant is given by

Kv = lim sGH ( s) = s →0

K cω n 2ζ

(8.5)

Thus the steady-state error due to a unit ramp input signal is

ess =

1 2ζ = K v K cω n

(8.6)

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The closed-loop transfer function is

C ( s) K c ωn2 ωˆ n2 = ≡ R( s) s 2 + 2ζω n s + K cω n2 s 2 + 2ζˆωˆ n s + ωˆ n2

(8.7)

where ωˆ n , the new closed-loop undamped natural frequency, and ζˆ , the new damping ratio, are, by comparing terms in eqn (8.7), defined as

ωˆ n = ωn K c ζ ζˆ = Kc

Therefore, increasing the proportional gain, Kc, has the following effects: 1. Increase in Kv, and thus a reduction of the steady-state error 2. Decrease in ζ, and hence increase in peak overshoot 3. Increase in undamped natural frequency, and hence faster response. However, (1) and (2) are contradictory and a compromise has to be made when selecting the value of Kc. In other words, for satisfactory performance of a control system, a convenient adjustment has to be made between the maximum overshoot and the steady-state error. Note that when Kc = 1, we have ζˆ = ζ , ωˆ n = ω n , K v =

2ζ ωn and ess = . ωn 2ζ

Therefore, for the sake of revealing the effects of the remaining control actions, the value of Kc = 1 will be used. 8.3 Derivative Action (D) The block diagram of a system with derivative control action is shown in Fig. 8.3. The controller transfer function is (8.8) Gc = 1 + τ D s The open-loop transfer function is

[1 + sτ ]ω GH ( s) = D

2 n

s (s + 2ζω n )

(8.9)

Hence the velocity error constant is

K v = lim sGH (s ) = s→ 0 and

ess =

1 2ζ = Kv ωn

ωn 2ζ

(8.10)

(8.11)

129

τD s R(s) +

E(s)

+ +

U(s)

-

ωn2 s( s + 2ζω n )

C(s)

Fig. 8.3 Block diagram of an automatic control system with derivative control action. The closed-loop transfer function is

C ( s) 1 + sτ D ]ω n2 [ = R( s) s 2 + [ωn2τ D + 2ζω n ]s + ω n2

(8.12)

The characteristic equation is

s 2 + [ω n2τ D + 2ζω n2 ]s + ω n2 ≡ s 2 + 2ζˆωˆ n2 s + ωˆ n2 = 0

(8.13)

By comparing terms in eqn (8.13), we have

ωˆ n = ωn

and

ζˆ = ζ +

ωnτ D 2

Therefore, derivative control action 1. Has no effect on the steady-state error, and 2. Increases the closed-loop damping ratio, and hence decreases the maximum overshoot. However, in spite of the increase in ζ, the rise time is faster due to the presence of a zero in the closed-loop transfer function. Hence derivative control action results in a faster response. The value of τD may be selected to prescribe the desired ζ and hence Mp. Note that the derivative term is never used alone due to its tendency to amplify high frequency noise signals. 8.4 Integral Control Action (I) In this case, the controller’s transfer function is

G c (s ) = 1 +

1 τ is

(8.14)

and the system’s block diagram is shown in Fig. 8.4

130

1 τi s R(s) +

+ +

E(s) -

U(s)

ω n2 s (s + 2ζωn )

C(s)

Fig. 8.4 Block diagram of an automatic control system with integral control action. The open-loop transfer function is

GH ( s) =

(τ s + 1)ω i

2

n

s 2τ i (s + 2ζω n )

The velocity error constant is

(τ i s + 1)ω n2 K v = lim sGH (s ) = lim =∞ s →0 s →0 s τ i ( s + 2ζωn ) Therefore,

ess =

1 =0 Kv

The closed-loop transfer function is

C ( s) = R( s)

(τ i s + 1)

ω n2 τi

ωn2 s +2ζωn s + ω s + τi 3

2

2 n

It may therefore be observed that integral control action 1. Eliminates the velocity steady-state error 2. Increases the order of the system. Higher-order systems are prone to instability 3. Reduces speed of response.

131

8.5 PID Action and Controller Parameter Tuning The combination of the P, I and D terms is popular because it exploits the advantages of each of the terms. The controller transfer function is given by eqn (8.1), i.e.,

1 + τ s] G ( s) = K [1 + c c τ s D i The tuning of the parameters, Kc, τi and τD does not require a mathematical model of the plant. This is one of the reasons that PID controller is widely used in industry. Two of the widely accepted techniques for tuning PID controllers are due to Ziegler and Nichols. 8.5.1 Ziegler-Nichols Continuous Cycling Method 1. Turn off the integral and derivative controls by setting τi = ∞ and τD = 0. This leaves the proportional gain in the controller for adjustment.

E(s)

R(s) +

K

U(s)

Gp(s)

C(s)

-

2. Adjust K until the system just bursts into continuous oscillations and record the value of K = Ko for this condition. Record also the output signal and measure the period of oscillation τu, known as the ultimate period. 3. The recommended Ziegler-Nichols controller settings are as tabulated below.

CONTROL ACTION P

Kc

τi

τD

0.5Ko

∞

0

PI

0.45Ko

τu/1.2

0

PID

0.6Ko

τu/2

τu/8

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8.5.2 Ziegler-Nichols Reaction Curve Method 1. Break the loop as shown below.

Plant R(s) +

Controller -

∆u

Gp(s)

C(s)

2. Apply a step input of size ∆u to the plant and record the step response as shown below. This is the step response of the plant; it is also known as the reaction curve of the plant. c(t)

L

SLOPE = N

NL

0

t

3. Measure the delay, L, the slope, N, and the length, NL. 4. The recommended Ziegler-Nichols controller settings are as tabulated below.

133

CONTROL ACTION P PI PID

Kc

τi

τD

∆u NL ∆u 0.9 NL ∆u 1.2 NL

∞

0

L/0.3

0

2L

0.5L

8.6 Realization of Controllers Implementation of controllers may be based on the operational amplifier which is characterized by 1. Very high input impedance 2. Very high open-loop gain 3. Very wide bandwidth 4. Very low output impedance 8.6.1 Proportional Controller R2 R e

R1

-

R

-

+

u +

The transfer function is

u R2 = =K c e R 1

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8.6.2 Proportional + Derivative (PD) Controller R2 R

C1 e

R1

R

u

+ +

The transfer function is

u R2 [1 + sτ D ] = K [1 + sτ D ] = c e R 1 R where τ = R C and K = 2 D c R 1 1 1 8.6.3 Proportional + Integral (PI) Controller R2

C2 R

e

R1

-

R

-

+

u +

The transfer function is

⎡ u R2 ⎡ 1 ⎤ 1⎤ = ⎢1 + ⎥ = K c ⎢1 + s ⎥ e R ⎣ sτi ⎦ τi ⎦ ⎣ 1 R where τ = R C and K = 2 i c R 2 2 1 135

8.7 Derivative Feedback Control Action The derivative feedback control is also known as either rate feedback control or tachometer feedback control. The control signal is obtained as a difference between the proportional error signal and a derivative of the output signal as shown in the figure below.

ω 2n

U(s)

E(s) +

R(s) +

s (s + 2ζωn )

-

-

C(s)

sKt

The open-loop transfer function is

GH ( s) =

ω n2 s 2 + (ωn2K t + 2ζωn )s

The velocity error constant

K v = lim sGH ( s) = →0 s

Therefore,

ess =

=

ω n2 s[s + (ω n2 K t + 2ζω n )]

ωn ω n K t + 2ζ

1 ω n K t + 2ζ = Kv ωn

The closed-loop transfer function is

C ( s) ωn2 = 2 R (s ) s + (ω n2K t + 2ζω n )s + ω n2 Thus the following observations can be made about derivative feedback control: 1. It has no effects on ωn , i.e., the speed of response is not affected 2. It increases the steady-state error 3. It increases the system’s damping and hence reduces the peak overshoot. Note that the new damping ratio for the system is given by

ζˆ = ζ +

ωn K t 2

136

Examples 1. a) Show that the steady-state error to a unit ramp input in Fig. 1a is given by ess = 2 ζ

ωn .

b) Find the new steady-state error when a proportional + derivative control action at the input is used as shown in Fig. 1b. Also, obtain the value of K that will eliminate this error.

ω 2n s (s + 2ζω n )

E(s)

R(s) + -

C(s)

Fig. Q1a

R(s)

1 + Ks

E(s)

+ -

ω n2 s (s + 2ζω n )

C(s)

Fig. Q1b Solution:

ωn2 a) GH ( s) = s( s + 2ζω n ) ⇒ K v = lim sGH ( s) = → s

ess =

0

ωn and 2ζ

1 2ζ = K v ωn

C ( s) (1 + Kv )ω n2 = b) R( s) s 2 + 2ζω n s + ω n2 and

137

s 2 + 2ζωn s − ω n2 Ks E (s ) = R (s ) − C (s ) = 2 R ( s) s + 2ζω n s + ω n2 Since

R (s ) =

1 , therefore s2

s2 + 2 ζωn s − ωn2 K s E( s) = 2 2 and s (s + 2ζω n s + ω n2 ) 2ζω n − ωn2 K ess = lim sE ( s) = 2 s→0

ωn

ess equals zero, i.e.,

2ζω n − ω 2n K

ω n2

=0

iff

2ζω n − ω2n K = 0 2ζ ⇒ K=

ωn

2. The open-loop transfer function of a unit feedback control system is given by

G (s ) =

25 s( s + 25)

a) Calculate the natural frequency, the damped natural frequency, damping factor, damping ratio, and the peak overshoot for a unit step input. b) Obtain the steady-state error for a unit ramp input. c) If the damping ration is to be 0.75 using a tachometer feedback, find the tachometer constant and the peak overshoot.

138

Solution: R(s) +

25 s( s + 5)

E(s) -

C(s)

C ( s) 25 = 2 . R( s) s + 5 s + 25 Then

s 2 + 5s + 25 ≡ s 2 + 2ζω n s + ω n2 a) ω n = 25 = 5rad/s 2ζω n = 5 ⇒ ζωn = 2.5 The damping factor is 2.5 The damping ratio is ζ =

2.5

ωn

= 0.5

The damped natural frequency is

ω d = ω n 1 − ζ 2 = 5 1 − 0.5 2 = 4.3rad/s The peak overshoot is

⎡ − ζπ M p = exp ⎢ 2 ⎣ 1−ζ 2ζ = 0.2rad b) ess =

⎤ ⎥ × 100% = 16.3% ⎦

ωn

c) The block diagram for this condition is shown below. The required damping ration is 0.75.

25 ωn2 C ( s) = ≡ R (s ) s 2 + (25K t + 5)s + 25 s 2 + 2ζω ns + ω n2 ∴ 2ζω n 25K t + 5 ⇒ 2 × 0.75 × 5 = 25 Kt + 5

Kt = 0.1V/(rad/s) and

139

⎡ − ζπ ⎤ M p = exp⎢ ×100% = 2.8% 2⎥ − 1 ζ ⎦ ⎣

U(s)

E(s) +

R(s) + -

-

25 s (s + 5)

C(s)

sKt

140...