Digital Controller Design - School Of Electrical & Electronic ...

Motivation and Overview 

Compensation 

DIGITAL CONTROL SYSTEM 

EEE354 

Digital Controller Design 

MUHAMMAD NASIRUDDIN MAHYUDDIN 

School of Electrical and Electronic Engineering, 

UNIVERSITI SAINS MALAYSIA 

week 13 

Semester II 2007/2008 

nasiruddin@eng.usm.my EEE 354 : DIGITAL CONTROLLER DESIGN

Outline 


Compensation 

week 13: 10/03/2008 - 15/03/2008 

1 Motivation and Overview 

2 Compensation 

Phase-Lag Compensation 

Phase-Lead Compensation 



Compensation 

Overview of system performance specification 

The desired characteristic or performance specification generally 

relate to: 

steady-state accuracy 

transient response 

relative stability 

sensitivity to change in system parameters 



Compensation 

Steady state accuracy 

The steady state accuracy is increased if poles at z=1 are added to 

the open-loop function and/or if the open loop gain is increased. 

However, added poles at z=1 in the open loop function introduce 

phase lag into the open loop frequency response, resulting in 

reduced stability margins. 

The control system design is usually a trade-off between steady state 

accuracy and acceptable relative stability. 



Compensation 

Transient Response 

Typical performance criteria are:- 

rise time t r 

peak overshoot or peak magnitude M p 

resonance peak M r 

time-to-peak overshoot t p 

settling time t s 



Compensation 

transient response 

The transient system is related to system bandwidth. 

To decrease the tise time and increase speed of response, the 

system bandwidth must be increased. However, if significant 

high-frequency noise sources are present in the system. 

A trade-off must be made between a fast rise time and an acceptable 

response. 



Compensation 

Characteristic of second-order system 



Compensation 

Characteristic of second-order system 

Given a second order LTI analog system with transfer function 

T(s) = 

2 

ω n 

s 2 +2ζω ns+ω 2 n 

which has unit step response, the following 

characteristic can be obtained: 

√ −ζπ 

M p = 1 + e 

1−ζ 2 

1 

M r = √ 

2ζ 1−ζ 

(1) 

2 

ω n t p 

π 

= √ 

1 

1−ζ 2 

Rule of thumb: resonance peak, M r is set at 2 dB in order to have 

suitable peak magnitude, M p . Peak magnitude will have an effect on 

the overshoot of the system. 



Compensation 

Relative stability 

The stability margins (gain margins and phase margins) are related to 

peak overshoot M p . 

The phase margin can be expressed as, 

[ 

] 

φ m = tan −1 q 

2ζ √ 

4ζ 4 +1−2ζ 2 


Sensitivity 


Compensation 

We prefer that the control system characteristics do not vary as 

certain parameters in the system vary due to change in temperature, 

humidity, altitude, age and so on. This is due to the fact that the 

system characteristic are a function of the system parameters. 

A closed-loop transfer function of a discrete system is given by 

T(s) = 

G(z) 

1+G(z) 

Sensitivity to a parameter a, is normally defined as a measure of the 

percentage change in T(z) to a percentage change in parameter a. 

∆T/T 

One such definition is sensitivity ≈ 

∆a/a = ∆T a 

∆a T 


Sensitivity 


Compensation 

∆T is the variation in T caused by ∆a, the variation in parameter a. 

Now, we are to find the sensitivity of T with respect to G for a discrete 

system: 

SG T = ∂T G 

∂G 

T = 1+G(z)−G(z) G(z) 

[1+G(z)] 2 G(z)/[1+G(z)] = 1 

1+G(z) 

At the frequency ω, we let z = e jωT , and 

S T G = 1 

1+G(e j ωT) 

For this sensitivity to be small within the system bandwidth, we 

require that G(e jωT >> 1 

Therefore, we can reduce the sensitivity of T to G by increasing 

the open loop gain. However, this action will cause stability 

problems. Again, a trade-off is there to decide. 



Compensation 

Disturbance Rejection 

Figure 2: Discrete Control System 

In the system shown in Figure 2, F(s) is a disturbance. Since R(s) is 

the control input, we design the system such that c(t) is approx equal 

to r(t). If F is zero, then 

C(z) = 

D(z)G 1G 2 (z) 

1+D(z)G 1 G 2 (z)R(z) 



Compensation 

Disturbance Rejection 

hence, in terms of frequency response, we require that 

D(e jωT )G 1 G 2 (e jωT ) >> 1 

over the desired system bandwidth. Then, C(e jωT ) ≈ R(e jωT ) 

If we consider only the disturbance input in Figure 2, then, 

C(z) = 

G 2 F(z) 

1+D(z)G 1 G 2 (z) 

hence, over the desired system bandwidth, 

C(e jωT ) ≈ 

G 2 F(e jωT ) 

D(e jωT )G 1 G 2 (e jωT ) 


Disturbance 


Compensation 

Referring to the previous slide, the disturbance response will be 

small since the denominator of the expression 

is large. Disturbance rejection for the system shown is good 

provided we have a high loop gain not occuring at the direct path 

between the disturbance input and the system output. 


Control Effort 


Compensation 

Any physical motor will have a maximum torque that can be 

developed. If we can this control effort (the torque) u(t), then |u(t)| will 

be bounded. Usually the system to be controlled is simulated without 

considering this limitation on the control effort. Then, worst-case 

condition is included in the simulation including considering the real 

physical characteristic of the motor (or any other actuator). 


Compensation 


Compensation 



We will learn how to design compensator limited to SISO 

system. The compensators to be learnt are the followings: 

Phase-Lag Compensator 

Phase-Lead Compensator 

Lead-Lag Compensator (sekiranya ada masa) 

In this chapter, Bode Plot will be used as a tool to design our digital 

compensators in frequency response. Therefore, my students are 

encouraged to review back steps/procedure in drawing bode plot 

correctly. Controller Design via root locus is not covered. 



Compensation 




The transfer function of the phase-lag controller in w-domain: 

D lag (w) = K α 

w+ω h 

w+ω l 

, 

ω l


Compensation 




Figure 3: phase-lag compensator 

ω l : low break frequency 

ω h : high break frequency. 



Compensation 



Procedure to design Phase-Lag Compensator 

1 Set the gain K, to the value that satisfies the steady-state error 

requirement. 

2 Obtain the bode plot using the value of the gain K as defined in 

Step 1. 

3 Calculate using the formula relating the phase margin with the 

transient response specifications. 

recap 

[ 

] 

φ m = tan −1 q √ 

2ζ 

4ζ 4 +1−2ζ 2 



Compensation 




4 Determine the required additional phase φ Mlag to be 

contributed by the phase-lag controller in order to satisfy 

Step 3 as follows: 

φ Mlag = φ Mcomp + φ 

φ is the correcting factor to compensate for the phase angle 

contribution of the lag compensator. Normally, the correcting factor 

within the region of 5 ◦ ≤ φ ≤ 12 ◦ is used. This is due to the fact that 

the phase angle contribution of the lag compensator may still 

contribute anywhere from −5 ◦ ≤ φ ≤ −12 ◦ of phase at the gain 

crossover frequency. 



Compensation 




5 Determine the gain crossover frequency f g (rad/s) from the bode 

plot which yield φ Mlag . At this frequency, f g , the magnitude plot 

must cross the 0 dB line. 

6 Determine the magnitude G M (dB) at the gain cross over 

frequency f g rad/s from the magnitude plot. Thus, the lag 

compensator must contribute −G M dB attenuation so that the 

magnitude plot will cross the 0 dB line at f g . 

7 Draw the high frequency asymptote of the lag compensator at 

−G M dB. Select the higher break frequency about one decade 

below the gain crossover frequency f g , that is, ω h = 0.1f g rad/s. 

Draw a -20 dB/decade line starting at the intersection of the 

higher break frequency ω h with the lag compensator’s high 

frequency asymptote until the 0dB line. 



Compensation 




8 Locate the lag’s compensator’s lower break frequency ω l at the 

intersection of the -20dB/decade line with the 0 dB line. Hence, 

the phase-lag compensator’s transfer function is given in the 

form of equation. 

D lag (w) = K α 

w+ω h 

w+ω l 

, 

ω l


Compensation 




Figure 4: Bode plot of uncompensated system and the phase-lag 

compensator 



Compensation 

Phase-Lead Compensator 



The transfer function of the phase-lead controller in w-domain will be 

of the form: 

D lead (w) = 1 β 

( 

w+ωl 

w+ ω l 

β 

) 

= 

( ) 

w+ω 

K l 

β w+ω h 

, ω l


Compensation 

Phase-Lead Bode plot 



Figure 5: Bode plot of the phase lead compensator 



Compensation 



Procedure to design phase-lead controller 

1 Set the gain K for the uncompensated system to a value 

satisfying the steady state error 

2 Draw the bode plot using the value of K above and determine 

the uncompensated system phase margin, φ Muncomp . 

3 Calculate the closed-loop bandwidth, ω b required to meet the 

transient response requirement using correct formula. 

4 Calculate the phase angle, φ Mcomp required to meet the damping 

ratio or percentage overshoot requirement. 

5 Determine the additional phase contribution φ Mlead required from 

the phase-lead compensator, in order to meet the desired phase 

angle, 

φ Mlead 

= φ Mcomp − φ Muncomp + φ 

φ is normally set at 10 ◦ , a correction factor to compensate for the lower uncompensated system’s phase angle at 

this higher phase-margin frequency. 



Compensation 




6 Determine the value of β from the phase lead compensator’s 

required phase margin as follows: 

[ ] [ ] 

φ Mlead = tan −1 1−β 

2 √ = sin −1 1−β 

β 

1+β 

7 Determine the compensator’s magnitude at the peak of the 

phase curve using the following formula: 

|G lead (jω max )| = 1 √ β 



Compensation 




8 Determine the new gain crossover frequency ω max on the Bode 

plot by finding where the uncompensated system’s curve is the 

negative value of |G lead (jω max )| as obtained above. At this 

frequency ω max , the uncompensated system’s magnitude plot 

must cross the 0 dB line. Since, the compensator’s magnitude 

contributes |G lead (jω max )|dB at this frequency, the uncompensated 

system’s magnitude curve must be −|G lead (jω max )|dB at this 

frequency. 

9 Calculate the phase-lead compensators break frequencies ω l 

and ω h using the following formula: 

ω max = ωl √ β 



Compensation 




10 The phase-lead compensator’s transfer function is given in the 

form of, 

( ) ( 

) 

D lead (w) = 1 w+ω l 

w+ω 

β w+ ω l = K l β w+ω 

β 

h 

, ω l

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