Controller Design in Frequency Domain

ECSE 4440

Control System Engineering

Fall 2001

Project 3

Controller Design in Frequency Domain

TA

1. Abstract

2. Introduction

3. Controller design in Frequency domain

4. Experiment

1.

1.

1.

1. Abstract

Abstract

Abstract

Abstract

In this project, controller design in frequency domain will be studied. Not only the performance of a controller but the stability and robustness will be considered in frequency domain. There are many useful graphical tools available in frequency domain such that bode plot and Nyquist plot. Those tools make it possible to design a lead-lag controller in frequency domain.

2.

2.

2.

2. Introduction

Introduction

Introduction

Introduction

So far, the performance of a control system has been measured and analyzed by its time-domain response characteristics. In practice, however, many signals to be processed are sinusoidal or can be represented by sinusoidal components. And the difficulties of the time domain design lie in the fact that there are no unified methods of arriving at a designed system given the time-domain specifications, such as peak overshoot, rise time, delay time, settling time, and so on. And for system with poorly known or changing high-frequency resonance, time domain design is not efficient.

Therefore the design of feedback control systems is probably carried out using frequency-response methods more often than any other. The frequency domain design method gives designer various advantages over the time domain method. The first advantage is the stability margins and robustness of a controller for noise or measurement error. And there are several handy graphical methods available in the frequency domain, suitable for the analysis and design of control systems. And it is important to realize that once the analysis and design are carried out in the frequency domain, the time-domain properties of the system can be interpreted based on the relationships that exist between the time-domain and the frequency-time-domain characteristics. Therefore, we may consider that the primary motivation of conducting control systems analysis and design in the frequency domain is because of convenience and the availability of the existing analytical tools.

2.1 Stability Margin

In the previous project, a system becomes unstable if the gain or time delay in input and feedback loop increases past a certain critical point. In other words, there are some margins in gain and phase from stable state to unstable state.

The stability margins can be described by phase margin and gain margin. The gain margin(GM) is the factor by which the gain is less than the neutral stability value and the phase margin(PM) is the amount by which the phase of

G

(

j

ω

)

exceeds

−

180

°

when

1

)

(

j

ω

=

KG

(figure1).

Figure 1

And the PM determines the time specifications of the controlled system such as damping

coefficient

ς

and the overshoot

M

_p. So, PM is sometimes used directly to specify control system performance. Based on the standard second order systems (with no zero), there are rules of thumb as follows;

100

PM

≈

ς

and

ω ≈

_n

ω

_cg (1)

where

ω

_cg is gain cross over freq and

ω

_n is natural freq.. If we recall the rules of the relationship between

ω

_n

,

ς

and the specifications, the time specifications can be achieved by using (1).

2.2 Sensitivity

Frequency domain methods enable a designer to make a control system as insensitive in operating frequency range. One of the main objectives of the control system is to keep the tracking error(e=r-y) small for a reference input excitation and to keep the output y small for a disturbance input d in figure 2.

Figure 2

Special functions have been defined to show the sensitivity of a closed-loop control system. The sensitivity function is defined as

1

))

(

)

(

1

(

)

(

s

=

+

G

s

K

s

−

S

(2)

The sensitivity function is the transfer function of –n to y and d to y (in lecture note). And the complementary sensitivity function is defined as

)

(

)

(

1

)

(

)

(

)

(

s

K

s

G

s

K

s

G

s

T

+

=

(3)

The complementary sensitivity function is the transfer function of r to e and d to u.

Therefore the control objective is to keep

S

(s

)

small over bandwidth of r and d and to keep

T

(s

)

small over bandwidth of n while maintaining closed loop stability.

The loop gain(

G

(

s

)

K

(

s

)

) shaping can achieve the control objective if we note that

1

,

1

1

0

,

1

1

≈

≈

⇒

<<

≈

≈

⇒

>>

T

S

GK

S

T

GK

(4)

2.3 Steady State Error

If input is a step function, the steady state error will be

(

0

)

)

0

(

)

0

(

1

1

d

K

G

⋅

+

from figure 2.

Therefore, large DC gain leads the small steady state error. Similarly, if the input is a sinusoidal function having a certain frequency

ω

_o , the steady state error will be

d

u

-r

e K(s) G(s)

y

n

)

(

)

(

1

1

)

(

o o o

jw

K

jw

G

jw

s

+

=

. Again large

K

(

jw

o

)

leads the steady state error at

ω

o

small. The large

K

(

jw

o

)

, however, make the system less robust. So, proper gain

should be chosen for the given specifications.

2.4 Time Delay

The pure time delay adds a phase shift of

−

ω

T

because the frequency response of the delay is given by the magnitude and phase of

ω j s sT

e

=

− _{. Therefore for a Given PM,}

maximum time delay can be expected by

cg

PM

T

ω

=

max .

3.

3.

3.

3. Controller Design

Controller Design

Controller Design

Controller Design in

in

in

in Frequency Domain

Frequency Domain

Frequency Domain

Frequency Domain

3.1 Control Objective

In the previous projects, there are useful rules between

ς

ω

_nand the time specification as follows in addition to (1). s n

t

8

.

1

≥

ω

(5) 2 2

1 b

b

+

=

ς

where

b

=

ln

M

_p

/

( )

−

π

(6) s n

t

6

.

4

≥

=

ςω

σ

(7)

To satisfy the specification with the above rules, the below conditions should be satisfied.

°

>

70

PM

(8)

20

5

<

ω

_cg

<

(9)

Once the condition (9) becomes satisfied, the spec for

S

(s

)

and

T

(s

)

will be achieved. So, target

ω

_cg

=

7rad

/

sec

is chosen in the range of (9).

0064

.

)

0

(

)

0

(

1

)

0

(

57

.

1

sgn

01

.

)

0

(

)

0

(

1

)

0

(

<

+

=

<

=

<

+

•

K

G

G

I

F

I

F

d

d

K

G

G

c c c c

θ

₍₁₀₎

Therefore the objective open loop gain

G

(

s

)

K

(

s

)

should be shaped like

Figure 3 3.2 Lead Controller

A lead filter can be used to add phase lead and increase bandwidth with the form of

1

,

1

1

<

+

+

=

α

α

Ts

Ts

K

K

lead . There are 3 design parameters

K

,

α

,

T

. To choose the

parameters, a systematic method has been studied in class. 5<wcg<20 PM>70o -180o Arg(K(jw)G(jw)) 0dB Wcg |K(jw)G(jw)| K(0)G(0)>44dB

For the given

)

(

1

)

(

c v

I

F

s

s

s

G

+

=

, the target gain crossover frequency

ω

_cg has chosen to

be 7 rad/sec. From bode plot of

G

(s

)

,

K

=

14

.

791

(

23

.

4

dB

)

is chosen to make

ω

_cg of

)

(s

KG

3.5 rad/sec (7/2) in figure 4.

Figure 4

And the PM of

KG

(s

)

is

34

.

87

°

. So,

70

°

+

5

°

−

34

.

87

°

≈

43

°

(

5

°

is extra pad.) phase lead is needed to achieve

PM

>

70

°

.

.

189

43

sin

1

43

sin

1

=

°

+

°

−

=

α

(-7.235dB). the

sec

/

57

.

5

max

=

rad

ω

should be chosen at the the

−

α

(dB) in figure 5. Finally,

413

.

)

189

.

*

51

.

5

/(

1

=

=

T

Figure 5 With the decided parameters, the lead filter is

1

08177

.

1

413

.

791

.

14

+

+

=

s

s

K

lead (11)

The PM and BW of

G

(

s

)

K

lead

(

s

)

should be checked. In figure 6, the PM and BW don’t

Figure 6

The procedure should be iterated from adding extra margin. After several tries, the extra

pad=

30

°

is found to be enough.

70

°

+

30

°

−

34

.

87

°

≈

65

°

.

0486

65

sin

1

65

sin

1

=

°

+

°

−

=

α

(-13.13dB),

ω

max

=

7

.

99

,

T

=

1

/(

7

.

99

*

.

0486

)

=

.

5677

So, the overall filter is

1

0277

.

1

57

.

791

.

14

+

+

=

s

s

Figure 7 3.3 Lag Controller

A lag filter can be used to boost DC gain to reduce steady state error but can reduce phase

margin. Its transfer function is

,

1

1

1

)

(

>

+

+

=

α

α

α

Ts

Ts

s

K

lag . There are two parameters for

the filter

α

,

T

. From (10),

d

s

s

s

T

s

T

S

T

s

T

K

s

s

d

s

G

s

K

s

G

lag lag lag lead lead lead lag lead

)

(

1

1

1

1

1

1

)

(

1

)

(

)

(

1

)

(

β

α

α

α

β

+

+

+

+

+

+

+

=

+

where •

=

=

θ

β

,

sgn

Ic

F

d

I

F

_c c v _{. Set}

0

=

s

for the steady state error, the above equation is

reduced to

157

57

.

1

,

01

.

<

=

⇒

>

<

lead lag c c lag lead

K

I

F

d

K

d

α

α

(12)

The lag compensation can ruin the system performance satisfying the time response spec.

So, the zero of the lag filter

(

1

)

lag

T

should be chosen one decade below than zero of lead

filter

(

1

)

lead

T

.

So, the lag filter is

1

44

.

67

83

.

11

44

.

67

)

(

+

+

=

s

s

s

K

(13) 3.4 Lead-Lag Controller

The lead controller and lag controller defined in previous section control the system in the range of the given specification. The step function of the lead-lag controlled system is in figure 8. The time response specs have been satisfied.

Figure 8

margin and the BW spec have been satisfied again.

3.5 Sensitivity

For the sensitivity spec, the sensitivity function and the complementary sensitivity function is given in figure 10. As mentioned in previous section,

S

(

s

),

T

(

s

)

functions falls down around the gain crossover frequency

ω

_cg. Therefore the properly chosen

ω

_cg should achieve the sensitivity spec and bandwidth spec.

Figure 10

3.6 Time Delay

The time delay in feedback loop is allowed up to PM since the time delay adds a phase shift

of

−

ω

_cg

T

. For the PM= 1.409(rad) of the compensated system,

sec

1747

.

0645

.

8

409

.

1

max

=

=

=

cg

PM

T

ω

.

In the linear simulation, it can be verified.

……… % Time delay

Tmax=pm_rd/wcp

[n d]=pade(Tmax,3);Gd=tf(n,d); max(real(pole(feedback(Gd*K*G,1))))

………. real(max_pole) = 2.3280e-006 for Tmax = 0.1747

real(max_pole) = .0017 for Tmax=0.1748

3.6 Nonlinear Simulation

The nonlinear simulation controlled by same lead-lag compensator has the step response in figure 11.

Figure 11

Most of the spec has been achieved. But the settling time is almost twice of the given spec. It might be caused the disturbance term (Coulomb Friction).

Figure 12

For the sinusoidal inputs with freq=6 rad/sec and 10 rad/sec, the responses are in figure 13 and figure 14.

Figure 14

As we can see in figure 13 and figure 14, the magnitude of output is decreased as input frequency is increased. In figure 14, the magnitude is reduced substantially. So, the

overall system is a lowpass filter with the cutoff frequency=

ω

_cg. 3.7 Discrete Simulation

Using Euler’s method(forward difference), descretization of the continuous transfer

function can be done with the relationship

s

T

z

s

=

−

1

. The discretized system is in figure 15.

Figure 15

The quantizer in feedback loop should be pointed out here. Usually, a encoder is used for the measurement of the output for feedback. A encoder transforms the continuous output to discrete value. Better encoder gives better measurement values. measurement error, however, can not be avoidable in figure 16.

Figure 16

4.

4.

4.

5.

5.

5.

5. Conclusion

Conclusion

Conclusion

Conclusion

In this project, the frequency domain design method has been studied. It is more general and comprehensive than the time domain design. There are graphical tools for frequency domain such as bode plot and nyquist plot. For the 2nd order system, lead-lag compensator has been designed with the help of the graphical tools. The performance of the controlled system has met with the given specification in time domain and frequency domain.