ROBUST CONTROLLER DESIGN - Shodhganga : a …shodhganga.inflibnet.ac.in/bitstream/10603/4988/13/13_chapter4.pdf · ∞ Optimal Robust H Controller for an Integrating Process with

Optimal Robust H∞ Controller for an Integrating Process with Dead Time

CHAPTER 4

ROBUST CONTROLLER DESIGN

4.1 Introduction

The robust controller (H∞

Controller) has been designed and its robustness is checked

with the help of µ synthesis for the unstable processes with dead time. It is important to consider

the various issues like disturbance rejection and the robustness of the controller due to the

uncertainties present in the system. While designing the controller, the weighting functions are

chosen such that the system could meet the performance requirements such as the peak value of

the µ plot should be less than one [12]. The D-K iteration is also used to improve the

performance of the robust controller. Once the D(s) and D-1

(s) were approximated, the plant is

scaled appropriately and the H-infinity design is synthesized for the scaled plant. This procedure

is repeated until the “µ” calculation for robust performance yields a value less than “1” for all

frequencies.

4.2 Design of H-Infinity Controller.

Dan Dai and Roy Smith [11] have discussed about the application of robust control

theory for the cart-spring pendulum system with uncertainties and disturbances. The objective is

to design a controller that meets the specified robust performance criteria. In [12] the authors

have used the “hinfsyn” command to find the controller transfer function by defining all the

input and output parameters. The partitioned matrix of the plant has to be known before using the

70

Optimal Robust H Controller for Unstable Processes with Dead Time

71

“hinfsyn” command. Through the D-K iteration method, plant has been scaled properly and the

controller is found for the synthesized plant. Controller’s robust performance has improved by

using the D-K iteration.

Roy Smith [12]has viewed into the problem of designing a controller to achieve a

performance specification for all plants, G(s), in a set of plants G. Fig. 4.1 represents the generic

synthesis interconnection structure. The lower half of this figure is the same as that for the h-

infinity design procedure. The problem is to find the C(s) such that for all ∆ ε B∆,K(s) stabilizes

Fu(G(s),∆) and ║Fu(Fl(G(s),C(s)),∆║∞ ≤ 1.Unfortunately, this problem is not yet been solved,

except in few special cases. The current approach to this problem, known as D-K iteration,

involves the iterative application of the h-infinity design technique and the upper bound µ

calculation.

Fig 4.1 The generic interconnection structure for synthesis.


72

The transfer function model of the distillation column is considered for the design of

robust controller for the unstable process with dead time [1]. Partitioned matrix of the plant is

obtained by considering the performance weight, disturbance weight along with the uncertainties

present in the system Fig. 4.2 which is given by

-------(4.1)

From eq. (4.1), we can define the number of control inputs and number of measurand, which

help to find the H-infinity controller for the plant using “hinfsyn” command.

Fig.4.2 Robust Controller design with weighting functions.


73

4.2.1 Robust Control Design Methodology

1. To have the basic idea of designing/selecting lead, lag, lead–lag compensator (It helps

to select the weighting function properly).

2. To model the expected uncertainty into the bound. (│∆│< 1) [41-43]

3. The robust performance of the controller needs to be checked in the presence of the

uncertainty in the system.

4. D-K iteration (D-scaling) method can be used to improve the performance of the

H-infinity controller design [12] for the system.

5. The weighting functions have to be selected based on the systems input requirements.

By understanding the concepts of lead-lag compensator design, the selection of

weighting functions can be made easily.

6. Robust performance can be analyzed in two ways [13]

I. ║│W1S│+│W2T│║∞< 1.

II. Peak value of the µ (D-K iteration) bound should be less than one.

7. Loop shaping method :

a) This section focuses on the tracking of a reference signal.

b) If L denotes the loop transfer function, as L=GC, then the transfer

function from reference input r to tracking error e is given by

S = L1

1 called the sensitivity function, for the specified plant

The condition for therobust performance specification is ║W1S║∞< 1.


74

8. The following conditions are used to check for Robust Stability, Nominal Performance,

Robust Performance

The phrases robust stability, nominal performance, and robust performance of the plant.

(i) Nominal Performance: The closed-loop system achieves nominal performance if the

performance objective is satisfied for the nominal plant model, Gnom. In this problem, it is

equivalent to: Nominal Performance ||WP(I + GnomC)–1

||∞< 1

(ii) Robust Stability:The closed-loop system achieves robust stability, if the closedloop system is

internally stable for all of the possible plant models

G .In this problem it is equivalent to a simple

norm test on a particularnominal closed-loop transfer function.

Robust Stability ||WdelCGnom(I + CGnom)–1

||∞< 1

(iii) Robust Performance:The closed-loop system achieves robust performance if theclosed-loop

system is internally stable for all

G , and in addition to that, theperformance objective,||WP(I +

GC)–1

||∞< 1,is satisfied for every

G . The property of robust performance is equivalentto a

structured singular value test (a generalization of the two H∞ norm testsin the previous

conditions) on a particular, nominal closed- loop transferfunction.


75

4.3 D-K Iteration for Robust Performance

1. To understand the basic concepts and design of lead-lag compensator (It helps for the

selection the weighting functions).

2. The H-infinity controller has been designed for the integrating process with dead time as

given in [1],by considering various weighting function(Wu, Wd and We).Wu is used to

reduce the error signal at low frequencies to improve tracking, while We limit the control

signal at high frequency to avoid saturation an dynamic range and bandwidth.

3. D-K iteration Algorithm :

The objective is to design a controller which minimizes the upper bound to µ for the

closed loop system ║D(ω)Fl(G(s),C(s)D(ω)-1

║∞.The major problem in doing this is that

the D-Scale that results from µ calculation is in the form frequency by frequency data

and the D-scale required above must be a dynamic system. This requires an

approximation to the upper bound D-Scale in the iteration. Looking into this issuemore

closely, it can be summarized as follows

Initialize procedure withKo(s).

Calculate the resulting closed loopFl(G(s),C(s))

Calculate D scale for µ upper bound

σmax║D(ω)Fl(G(s),C(s)D(ω)-1

║∞ . Approximate the frequency data D(ω),by

D (s) ϵRH∞ , with

D (jω) ≈ D(ω)

Design H∞ Controller for the scaled plant

D (s)G(s)

D-1

(s).


76

The notation D(ω) is used to emphasize that the D-Scale arises from frequency by frequency µ

analyses of G(jω)=Fl(G(jω),C(jω)) and therefore is a function of “ω”. Note that it is notthe

frequency response of some transfer function and D(jω) cannot be used as notation. The µ

analysis of the closed loop system is unaffected by the D-Scales. However the H∞ design

problem is strongly affected by scaling. The procedure aims at finding at D such that the upper

bound for the closed loop system is a closed approximation to µ for the closed loop system. At

each frequency, a scaling matrix, D(ω) , can be found such that σmax(D(ω)G(jω)D(ω)-1

) is a

closed upper bound to µ(G(jω)).

Another aspect of “µ” synthesis is to consider the iteration which approaches the optimal

“µ”value, the resulting controllers often have more and more response at high frequencies. The

above discussion used an H∞ controller to initialize the iteration. Actually any stabilizing

controller can be used. In higher order, lightly damped, interconnection structures are used, the

H-infinity design of Ko(s) may be badly conditioned. In such case the software may fail to

generate a controller, or may give controller which doesn’t stabilize the system. A different

controller can be used to get a stable closed loop system, and thereby obtain D scales.

Application of these D scales often results in a better conditioned H-infinity design problem and

the iteration can proceed.

The robust performance difference between the H∞ controller, K0(s), and C(s), can be

dramatic even after a sing D-K iteration. The H∞ problem is sensitive to the relative scaling

between v and w. The D-scale provides the significance better choice of relative scaling for


77

closed loop robust performance. Even the application of a constant D scale can have dramatic

benefits [12].

4.4 D-K Iteration results

The stability and robust performance measure of the plant G(s) = 60.0506 se

s

were analyzed and

those are given in the following figures Fig 4.3 to Fig 4.16.

D-K iteration attempts to minimize the quantity ║D(ω)Fl(G(s),C(s)D(ω)-1

║∞ by alternatively

minimizing this expression for the controller C(s) or the “µ” upper bound scaling D, while

holding the other constant. This process is carried out iteratively until a satisfactory controller is

constructed.

Fig 4.3 Bode plot of the plant with and without controller

The frequency roll off is increased with the presence of the controller.

10-5

10-4

10-3

10-2

10-1

100

101

102

-80

-60

-40

-20

0

20

40

60

80

Magnitude (

dB

)

Bode Diagram

Frequency (rad/sec)

Without controller

With controller


78

Fig. 4.4 Bode plot of the weighting functions.

Frequency response of the various weighting function is shown in the Figure 4.4.

Fig.4.5 Singular value plot for iteration1

10-3

10-2

10-1

100

101

102

-40

-30

-20

-10

0

10

20

30

40

Magnitu

de (

dB

)

Bode Diagram

Frequency (rad/sec)

w etf

w dtf

w utf

10-3

10-2

10-1

100

101

102

103

0

1

2

3

4

5

6SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE

FREQUENCY (rad/s)

MA

GN

ITU

DE


79

Fig.4.6. Mu bound without D Scaling

The peak value of the µ plot is about 4.4 after 1st iteration

Fig.4.7 Mu bound with D-K Scaling

10-3

10-2

10-1

100

101

102

103

2.5

3

3.5

4

4.5CLOSED-LOOP MU: CONTROLLER #1

FREQUENCY (rad/s)

MU

10-3

10-2

10-1

100

101

102

103

2.5

3

3.5

4

4.5MU bnds (solid) and ||D*M*D-1|| (dashed): ITERATION 2


80


Fig.4.9. Mu bound without D Scaling

The peak value of the µ plot is about 4.05 after 2nd

iteration

10-3

10-2

10-1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE

FREQUENCY (rad/s)

MA

GN

ITU

DE

10-3

10-2

10-1

100

101

102

103

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2CLOSED-LOOP MU: CONTROLLER #2

FREQUENCY (rad/s)

MU


81



10-3

10-2

10-1

100

101

102

103

2.6

2.8

3

3.2

3.4

3.6

3.8

4

4.2MU bnds (solid) and ||D*M*D-1|| (dashed): ITERATION 3

10-3

10-2

10-1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE

FREQUENCY (rad/s)

MA

GN

ITU

DE


82

Fig.4.12 Mu bound without D Scaling

The peak value of the µ plot is about 4 after 3rd

iteration


10-3

10-2

10-1

100

101

102

103

2.5

3

3.5

4CLOSED-LOOP MU: CONTROLLER #3

FREQUENCY (rad/s)

MU


83


Fig.4.15 Mu bound without D Scaling

The peak value of the µ plot is about 3.9 after 4th

iteration.

Since the peak value of “µ” is not less than 1, the robust performance is not ensured for the

controller.

10-3

10-2

10-1

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5

4SINGULAR VALUE PLOT: CLOSED-LOOP RESPONSE

FREQUENCY (rad/s)

MA

GN

ITU

DE

10-3

10-2

10-1

100

101

102

103

2.5

3

3.5

4CLOSED-LOOP MU: CONTROLLER #4

FREQUENCY (rad/s)

MU


84

4.5Stability Analysis

The condition used to analyze the robust stability was │K(jω)S(jω)│< │1

aW │

Fig.4.16. The robust stability analysis.

4.6 Robust PID controller satisfying the H-Infinity Principles

4.6.1 Introduction

The design of robust controller helps to satisfy the robust stability and robust

performance criteria as given in [13] under the plant parameter changes. Most of the time the

order of the robust controller (H∞) will be of higher order when compared to the process transfer

function. In this thesisan attempt has been made to design the low order controller design (PID

Controller with H-Infinity Principles for aPure Integrating Process with Dead Time (PIPDT)).

The controller design parameters such as Proportional gain, Derivative gain and Integral gain are

fixed using the principle of Hurwitz Criteria. The weighing functions of the system needs to be

selected, based on the low pass or high pass requirements of the input signal and disturbance

rejection. After finding the range of Derivative gain and the Integral gain, by sweeping the

proportional gain, we can find the various set of admissible layers of PID Controller values.


85

Finally, the three dimensional plots of the admissible set of PID controller values for the PIPDT

need to be plotted. Once we obtain the admissible set of PID controller settings, that satisfies the

robust performance conditions [13], it can also be easily implemented in the industries as a lower

order H-Infinity controller.

4.6.2.Robust PID Controller Design

Most of the chemical and process industries are encountered with the unstable and higher

order systems. We can model the higher order systems to FOPDT, for the purpose of controller

design. For example, in the study of the distillation column, the resultant FOPDT will have a

very large time constant, which may lead the process to settle after a very long time dead time

instants [1]. For those cases we can treat the model as a pure integrating process for the design of

controller. Once the controller has been designed, the same can be implemented for higher order

systems.

The mathematical model of the distillation column is considered for simulation studies.

Model free PID controller design is more suitable for the integrating process with various

uncertainties. The common problem with the ordinary conventional controller is that, the PID

values which are tuned will give better response only for one particular transfer function. The

tuned controller value may lead to the poor time domain response or unstable output under some

additional uncertainties added into the system. A robust controller may find solution to these

kinds of problems, the only disadvantage with the design of H-Infinity is that the order of the

controller will be very high. In order to reduce the order of the controller and to ensure the robust

performance, an H-Infinity based PID controller [10, 13, 25, 54, 55, 56, ]are proposed for the


86

PIPDT processes. Also the admissible set of PID controller which is found using the Hurwitz

Criterion satisfies the robust performance condition. The uncertainty is considered in the form of

dead time in PIPDT process and the PID tuning values are found for uncertain system and

plotted as a 3D plot.

4.6.3 Design Approach

Consider the mathematical model of the distillation column as 60.0506

( )se

G ss

for the

controller design [4, 59-62, 68]. The general structure of the PID controller is given by

2

( )i p dK K s K s

C ss

. Our objective is to find the values ,p iK K and dK using the Hurwitz

Criterion.

The PID values are selected by satisfying the following condition

1] ( , , , )p i ds K K K (4.2)

2] ( , , , , , )p i ds K K K (4.3)

3]1 2( ) ( ) 1W s S W s T

(4.4)

The controller has been designed for the plant with uncertainty, the uncertainty is

considered in the form of various dead time in the process. The perturbed plant with various dead

times are given below (after the pade approximation), for which the controller needs to be

designed.

Case-1: With dead time Td=6sec.

2

0.0506 0.0168( )

0.33

sG s

s s

(4.5)


87


2

0.0506 0.0253( )

0.5

sG s

s s

(4.6)


ss

ssG

0

0506.00506.0)(

2

(4.7)

The root locus plot helps to get the approximate values of Kp for all the three cases.


88

Fig.4.17. Root locus plot for Case 1, the maximum value of the gain is 6.63.

Root Locus

Real Axis

Imagin

ary A

xis

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.220.440.62

0.76

0.85

0.92

0.965

0.992

0.220.440.62

0.76

0.85

0.92

0.965

0.992

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

System: s

Gain: 6.63

Pole: 0.00231 + 0.332i

Damping: -0.00695

Overshoot (%): 102

Frequency (rad/sec): 0.332


89

Fig.4.18. Root locus plot for Case 2, the maximum value of the gain is 9.85.

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.20.40.58

0.72

0.83

0.91

0.96

0.99

0.20.40.58

0.72

0.83

0.91

0.96

0.99

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

System: s

Gain: 9.85

Pole: 0.00109 + 0.497i

Damping: -0.00218

Overshoot (%): 101


Root Locus

Real Axis

Imagin

ary A

xis


90

Fig.4.19.Root locus plot for Case 3, the maximum value of the gain is 19.8.

The weighting functions W1(s) and W2(s) are quite sensitive and should be chosen carefully for

all the three cases based on the frequency inputs of the plants.Let T(s) and S(s) be the

complementary sensitivity function and the sensitivity function respectively.

)()(1

)()()(

sCsG

sCsGsT

(4.8)

)()(1

1)(

sCsGsS

(4.9)

-1 -0.5 0 0.5 1 1.5 2 2.5 3-1.5

-1

-0.5

0

0.5

1

1.5

0.220.42

0.6

0.74

0.84

0.92

0.965

0.99

0.220.42

0.6

0.74

0.84

0.92

0.965

0.99

0.2

0.4

0.6

0.8

1

1.2

1.4

0.2

0.4

0.6

0.8

1

1.2

1.4

System: s

Gain: 19.8

Pole: 0.00359 + 0.996i

Damping: -0.0036

Overshoot (%): 101


Root Locus

Real Axis

Imagin

ary A

xis


91

Also for the case 1, the weighting functions are selected as

5

2)(1

ssW (4.10)

2

15)(2

s

ssW (4.11)

By substituting the proposed control structure C(s) and the plant transfer function P(s) eq. 4.5 in

eq. 4.8 and eq. 4.9 gives the following expressions

3 2

3 2

0.0506 (0.0168 0.0506 ) (0.0168 0.0506 ) 0.0168( )

(1 0.0506 ) (0.333 0.0168 0.0506 ) (0.0168 0.0506 ) 0.0168

d p p i i

d d p i i

Kds K K s K K s KT s

K s K K s Kp K s K

3

3 2

0.333( )

(1 0.0506 ) (0.333 0.0168 0.0506 ) (0.0168 0.0506 ) 0.0168d d p i i

s sS s

K s K K s Kp K s K

Especially, we consider the problem of disturbance rejection for the plant with multiplicative

uncertainty. This problem can be formulated as the following robust performance condition

1|(s)T(s) W| +|(s)S(s) W| 21

This condition can be converted into a simultaneous polynomial stabilization by considering the

following lemma.


92

Lemma 1:

Convolute the transfer function of the W1(s) and the transfer function of the Sensitivity

function “S” to form the single transfer function ( )

( )

A s

B s , similarly convolute the W2(s) and

complementary sensitivity function “T” to form the single transfer function ( )

( )

C s

D s

Let

0 11

0 1

...( )( ) ( )

( ) ...

x

x

x

x

a a s a sA sW s S s

B s b b s b s

and

0 12

0 1

...( )( ) ( )

( ) ...

x

x

x

x

c c s c sC sW s T s

D s d d s d s

The transfer function of the( )

( )

A s

B s should be stable and proper rational functions with

and d x not equal to zero. Then,

( ) ( )

1( ) ( )

A s C s

B s D s

If and only if:

a) 1x x

x x

a c

b d

b) ( ) ( ) ( ) ( ) ( ) ( )j jB s D s e A s D s e C s B s is Hurwitz for all θ and ϕ Є [0,2π].

Lemma 2:

For a plant, the control found must satisfy

δ( s,Kp,Ki) is Hurwitz if and only if

( ( , , , ) *( )) ( ( ( ) ( ( ))i p i ds K K K M s n l N s r N s


93

The task now is to find the sets of (Kp,Ki,Kd) for which the formula in lemma 2 holds. By

considering the even and odd parts of N(s) and using them to find M*(s), then substitute the

result to find ( , , , ) *( )p i ds K K K M s

Analysis

Using the Lemma 1,

The necessity of condition (b) is established in the following way:

||

| |

||

1x x

x x

a c

b d

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

j j

j je A j D j e C j B j

B s D s e A s D s e C s B sB j D j

( ) ( ) ( ) ( ) ( ) ( )j jB s D s e A j D j e C j B j

( ) ( )

( ) ( )

A j C j

B j D j

0 0 0 00 0

0 0 0 0

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )

A j D j C j B jA j C j

B j D j B j D j

0 0( ) ( )

( ) ( ) ( ) ( )1

( ) ( )

j j

B j D j

e A j D j e C j B j

B j D j

It follows that

( ) ( ) ( ) ( ) ( ) ( )j jB j D j e A j D j e C j B j

Because B(s) and D(s) are Hurwitz, using Rouch’s Theorem we conclude that,


94

( ) ( ) ( ) ( ) ( ) ( )j jB s D s e A s D s e C s B s is Hurwitz for all ϴ&Φ Є [0,2π].

Sufficiently proceeding by contradiction we assume that conditions (a) and (b) are true but,

( ) ( )

1( ) ( )

A s C s

B s D s

a) Since ( ) ( )

( ) ( )

A s C s

B s D s is continuous function of ω and

(||

| |

||) |

| |

|

Then there must exist at least one ω0 Є R such that

(( ) ( )

( ) ( )

A s C s

B s D s ) 1x x

x x

a c

b d = 1

Therefore, it implies that there exists ϴ and Φ Є [0,2π] such that

(a) ( ) ( ) ( ) ( ) ( ) ( ) 0j jB s D s e A s D s e C s B s and it obviously contradicts condition

(b). Designing of PID controllers such that the robust performance condition

1|(s)T(s) W| +|(s)S(s) W| 21

holds. Based on lemma 1, The problem of synthesizing PID

controllers for robust performance[9] can be converted into the problem of determining values

of (Kp, Ki, Kd) for which the condition (4.2),(4.3) & (4.4) must hold

Where2( , , , ) ( ) ( ) ( )p i d i p ds K K K sD s K K s K s N s

For condition (4.4) to hold,

|(s)T W| +|(s)SW| 2 1 |

| , Kd must lie in

within the range (-4.9407 , 3.2938 ).

Let :


95

2( , , , ) (1 0.0506 (0.333 0.0168 0.0506 ) (0.0168 0.0506 ) 0.0168p i d d p p i is K K K Kd K K s K K s K

5 4( , , , , , ) (1 0.0257844950 ) (8.9713 1.34626 0.025784490 )p i d d d ps K K K K s K K s

3

2

(16.153154 0.23784 1.34626 0.0257844950) (4.42110780 0.2368086

0.23784 1.34626 )

(0.2368086 0.23784 )

0.2368086

d p d

p i

p i

i

K K s K

K K s

K K s

K

After the range of Kd is fixed, the next step is to fix the range of Kp such that the robust

performance condition holds good and also ρ and ψ must be Hurwitz within this range.

Let[ We are defining from [13] ]

N(s)=Ne(s2)+s No(s

2) (4.12)

D(s)=De(s2)+s Do(s

2) (4.13)

Using this decomposition of N(s) and D(s) into even and odd part will simplify the calculation

that should be used for fixing the range of Kp.

Using (4.12) and (4.13), M*(s) can be redefined as

M*(s)=N(-s)= Ne(s2)-s No(s

2)

The closed loop characteristics polynomial is

2( , , , ) ( ) ( ) ( ) ( )p i d i d ps K K K sD s K K s N s K sN s

Let n, m be the degree of ( , , , )p i ds K K K and M*(s) respectively. Multiplying ( , , , )p i ds K K K

by M*(s) and examining the resulting polynomial, we get

( , , , ) *( ) ( ( , , , ) *( )) ( ( , , , ) *( ))dp i p i d p i ds K K K M s l s K K K M s r s K K K M s

= ( ( , , , )) ( ( , , , )) ( ( )) ( ( ))l s Kp Ki Kd r s Kp Ki Kd lN s r N s


96

( , , , )p i ds K K K of degree n is Hurwitz if and only if

( ( , , , ))p i dl s K K K n

and

( ( , , , )) 0p i dr s K K K

Now, we consider this theorem

Let 𝜹(s)be a given real polynomial of degree n. then

Moreover, ( , , , )p i ds K K K is Hurwitz if and only if .

Using Lemma 2, the values of Kp, Ki & Kd needs to be found out

2 2 2 2 2

0( , , , ) *( ) [ ( ( ) ( ) ( ) ( ))p i d e e os K K K M s s N s D s D s N s

2 2 2 2 2 2 2 2( )( ( ) ( ) ( ) ( ))] ( ( ) ( )i d e e o o e eK s K N s N s S N s N S s D s N s

2 2 2 2 2 2 2 2( ) ( ) ( ( ) ( ) ( ) ( ))]o o p e e o os D s N s K N s N s s N s N s

Substituting (s=jw):

( , , , ) *( ) ( , , ) ( , )p i d i d ps K K K M s p K K jq K

Where

2

1 2( , , ) ( ) ( ) ( )i d i dp K K p K s K p

1( , ) ( ) ( )p pq K q K q

2 2 2 2 2

1( ) ( ( ) ( ) ( ) ( ))e o e op N D D N

2 2 2 2

2( ) ( ) ( ) ( 2) ( )e e o op N N N N

2 2 2 2 2

1( ) [ ( ) ( ) ( ) ( )]e e o oq D N D N

2 2 2 2 2

2( ) [ ( ) ( ) ( ) ( )e e o oq N N N N

Also define


97

After computing the equations above, the next step to is to fix the range kp. This can be done

using root locus ideas as following.

Let

Define

as a function of then find the derivative of

with respect to

(

)

by setting

(

) , then from the real zeros of the equation a corresponding Kp values that

are distinct and finite producing either real breakaway points or a root at the origin can be found

from .

The next step is to find the real root distributions of with respect to the

origin.

The necessary condition for fixing the range of Kp is that q( ,Kp) has at least

{

( )

( ) ( )

} (4.14)

real, nonnegative, distinct roots of odd multiplicity. These ranges of Kp satisfying this condition

are called allowable.

Fixing a value for Kp in within the range and evaluate q( Kp) at this value. The real, non-

negative values, distinct finite zeros of are the zeros of q( ,Kp). Let these zeros be

0= 0< 1< 2< … …. < l-2< l-1

Then we have to construct a sequence of numbers as follows:

{

}

Where 𝜶 Є {-1, 1}

For t= 1,2,3,…, l-1:


98

{

}

{

}

With defined in this way, we define the string as the following sequence

of numbers: . This admissible string must satisfy the following condition:

( )

For an admissible string , the set of (Ki,Kd) values can be determined from the following string

of linear inequalities:

For which M*(j t)≠0.

This procedure has to be repeated for all admissible strings to obtain the corresponding

admissible (Ki,Kd) sets. The set of all stabilizing (Ki,Kd) values corresponding to the fixed Kp is

then given by

Now we apply this procedure to determine the admissible set of (Kp,Ki,Kd) values. First, let’s

consider the transfer function in case 1.

N(s) and D(s) can be decomposed into even odd parts as following:

* 2 4 2

2 3 3

( , , , ) ( ) 0.0055944 0.0506 ( )

(0.00028224 0.00256036 ) 0.0336498 (0.00028224 0.00256036 )

p i d i d

p

s K K K M s s s K s K

s s K a s


99

Evaluating the above equation at s=j :

* 2 4 2 2) 0.0055944 0.0506 ( )(0.00028224 0.( , , , 002560) 3( 6 )i dp i d K Ks K K K M j

3 3(0.00028224( 0.033649 0.002560368 ))j Kp

* 2

1 2 1 2, , , ) ( ) [ ( ) ( ) ( )]( [ ( ) ( )]p i d i d pK K K M j p K K p j q K qj

P1( ), P2( ), q1( ) and q2( ) are found to be:

2 4

2

2

3

1

1

3

2

) 0.0055944 0.0506

( ) (0.00028224 0.00256036 )

( ) 0.0336498

( ) 0.00028224 0.00256036

(

P

q

q

P

q(ω,Kp) can be written in the q( ,Kp)= [ U( ) + Kp V( ) ] form as:

2 2, ) [ 0.0336498 (0.00028224 0.0( 0256036 )]pKq Kp

Where

2

2

) 0.0336498

( ) 0.00028224 0.002

(

56036

U

V

Let

2

2

( 0.00028224 0.00256036

( 0.03364

)

) 98

V

U

2 2

( 0.00001899463910

( ( 0.0336498

)

) )

d V

d U

2 2

0.000018994639100

( 0.0336498 )

(4.15)


100

The only real zero eq. (4.15) is =0. Substituting this value in , the

corresponding value w is Kp =0. This gives that the range of Kp can be:

(-∞,0] or [0,∞). Examining the distribution of the roots for Kp will give a more accurate range

for Kp.

Since n+m=4 (even), there must be at least ( )

real, non-negative distinct finite

zeros in the range of Kp. In this case there should be at least

. By choosing any

arbitrary value for Kp in each range, the exact range of Kp can be determined. For this case in our

hand the range of Kp is [0, 13.115].

Taken any arbitrary value for Kp in within the specified range, for instance let Kp =6.5, and

substituting the value in q( ,Kp)gives:

3,6.5) 0.017007460 0.001834 0( 56q

The real zeros of q( , 6.5) with odd multiplicity are ω0=0 and ω1=0.32843. Also define 2=∞.

The next step is to find the admissible strings

Define

2 4

1 2 2

( 0.0055944 0.0506 )( )

(1 )P

Since M*(s) has no zero at the origin and M*(j ωt)≠0 for t=, =𝜶. also will have the value of

𝜶 because n+m is even number.

Since m+n=4, which is even, and M*(s) has no jω-axis roots, then the set A(6.5) becomes

{

}

Since ( ) ( ) and , then

every admissible string must satisfy the condition


101

The only string which satisfies the condition above is . It follows that the

stabilizing set (Ki,Kd) corresponding to Kp=5 must satisfy the string of inequalities:

{

Substituting for in the above expression, we obtain the set S1(5):

{

The graph of this set of (Ki,Kd) values is shown in figure 5. By sweeping over different kp values

within the range [0, 13.115] and repeating the above procedure at each stage, the set of

(Kp,Ki,Kd) values can be generated. The set of (Kp,Ki,Kd) values is shown in figure 6.

Fig.4.20 The stabilization set of (Ki,Kd) values for a fixed Kp=5.

In order to find such a common set of (Kp,Ki,Kd) values that hold both ρ and ψ to be Hurwitz and

satisfy the robust performance condition, Routh table is used. The first column of Routh tables of

ρ and ψ should be positive, because for any system to be stable the first column in Routh table

must be positive. The ranges of (Kp,Ki,Kd) values that makes it positive are shown in the table

below.

Table 4.1 The stabilizing ranges of (Kp,Ki,Kd) values for G(s) of case 1.

Case 1 Kd Kp Ki

Ρ Kd<19.76 For Kd =2 For Kd =2, Kp =5

Kd

Ki


102

Kp<7.24 0 <Ki< 0.458

Ψ Kd< 3.878 For Kd = 1.5

Kp<26.96

For Kd =1.5, Kp =5

0 <Ki< 77.433

But in order to find a common set of values that satisfied both ρ and ψ and also satisfied the

robust performance condition, the values in the above table have to be modified slightly. One of

the sets that can satisfy the three conditions is in the table below:

Table 4.2 Optimal robust set of (Kp,Ki,Kd) values.

Kd Kp Ki

Case 1 0.5 5 0.2

Note: These are the common values of Kp,Ki and Kd which are satisfying the condition ρ and ψ.

Then, values obtained were substituted in the original equations of ρ and ψ to ensure they will

make the system stable. Table 4.3 &4.4 below shows the first column of ψ and ρ with the values

of (Kp,Ki,Kd) respectively.

Table 4.3Routh table for the condition 2 (Eq. 4.3) - ( , , , , , )p i ds K K K

Routh-Hurwitz: 1st Column is TF Zeros are

s5: 0.87108 -6.4888

s4:7.0089 -1.2726

s3:8.8684 -0.11902-0.35566i

s2: 2.1875 -0.11902+0.35566i

s1:0.93858 -0.046809

s0:0.047362

Table 4.4. Routh table for the condition 1 (Eq. 4.2) – ( , , , )p i ds K K K

Routh-Hurwitz: 1st Column is TF Zeros are

s3:0.9747 -0.021974-0.27067i

s2: 0.0884 -0.021974+0.27067i

s1:0.036833 -0.046746

s0:0.00336


103

From Table 4.3 and 4.4 above that the system is completely stable for the set of (Kp,Ki,Kd)

values. All the zeros of ρ and ψ are in the left side of the plane. By sweeping over different

Kpvalues within the interval [0, 13.115) and repeating the above procedure at each stage, the set

of stabilizing (Kp,Ki,Kd) values can be generated.

Fig.4.21. The stabilizing set of (Kp,Ki,Kd) values.

The design of the Robust PID controller satisfying the robust performance for case1 was

completed. The same procedure is to be applied for the other cases. The resulting stabilization set

of (Ki,Kd) and the 3D stabilizing set of (Kp,Ki,Kd) are shown for the case 2 (Plant with

uncertainty) transfer function.

For CASE 2 Transfer Function:

For the plant transfer function of case 2, ρ and ψ equations are as follow:

3 2( , , , ) (1 0.0506 ) (0.5 0.0253 0.0506 ) (0.0253 0.0506 )p i d d d p p is K K K K s K K s K K s

0.0253 iK

0.1

0.2

0.3

0.4

0.5

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

5

5.05

5.1

5.15

5.2

5.25

5.3

5.35

5.4

5.45

5.5

kp

kd

ki

Kp

Kd

Ki


104

5 4( , , , ) ( 0.257844950 1) ( 1.30295 0.257844950 9.138)p i d d d ps K K K K s K K s

3( 1.30295 0.002692 0.257844950 17.59575)p d iK K K s

2( 1.30295 0.356622475 0.002692 6.6383)i d pK K K s

(0.35662247 0.002692 ) 0.356622475p i iK K s K

4.7 Robust Performance Condition

For the robust performance condition to be hold:

| |

||

For this condition to be hold, Kd must lies in the range (-4.9407, 3.2938).

Ki

Fig.4.22. The stabilizing set of (Ki,Kd) values when Kp =6.

Table 4.5 The ranges of (Kp,Ki,Kd) values for G(s) of case 2.

Case 2 Kd Kp Ki

Ρ Kd<19.76 For Kd =2

Kp<10.88

For Kd =2, Kp =6

0< Ki<1.062

Ψ Kd<3.878 For Kd =2

Kp<25.334

For Kd =2, Kp =6

0< Ki<69.07

Kd


105

Table 4.6.Checked set of (Kp,Ki,Kd) values.

Kd Kp Ki

Case 1 0.5 6 0.5

Fig.4.23. Admissible set of PID controller values for the case2.

4.8Analysis of Various Combination of Pade Approximation

The plant delay function is approximated to a rational transfer function using different

Pade approximations. The plant transfer function is considered with different approximations of

6se. In case 4, first order Pade approximation in numerator and second order Pade

approximation in the denominator is considered. In case 5, approximated first order in the

numerator (( 1 0a s a )) which is obtained by comparing the magnitude of “S” and “T” of the

higher order controller and the and second order Pade approximation in the denominator is

considered for the analysis. The analysis of various combination of PADE approximation has

been carried out to check whether there is any improvement in the time domain response.

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

6

6.05

6.1

6.15

6.2

6.25

6.3

6.35

6.4

6.45

6.5kp

kd

ki

Kp

Kd Ki


106

Case 4:

60.0506( ) sG s e

s

0.0506 1

2

st

nds Order Pade approximation of

6se

2

0.0506 0.333

0.333

s

s s s

Thus, case 4 transfer function is:

3 2

0.0506 0.0169( )

0.333

sG s

s s

Case 5:

60.0506( ) sG s e

s

1 0

2

0.0506

0.333

a s a

s s s

Thus, case 5 transfer function is:

3 2

1.6955 0.0169( )

0.333

sG s

s s s

By designing the robust PID controller for the Case 4 and Case 5 transfer functions the following

servo and regulatory responses are obtained.

(5.19)

(5.20)


107

Fig.4.24. Servo response of all three cases.

Fig.4.25. Regulatory response of all three cases.

0 50 100 150 200 250 300-5

-4

-3

-2

-1

0

1

2

3

4

Time(Sec)

Am

p

Servo Response

Case1

Case4

Case5

0 50 100 150 200 250 300-3

-2

-1

0

1

2

3

4

5

6

Time(Sec)

Am

p

Regulatory Response

Case1

Case4

Case5


108

It is observed that the first order PADE approximation gives the better result for the servo and

regulatory response when compared to the other combination of PADE approximation.

4.9 Chapter Conclusion:

Robust controller has been designed for the unstable process with uncertainty. The

analysis of robust performance has been done using µ synthesis (D-K scaling). The weighing

functions have to be adjusted to minimize the peak of the µ plot to be less than one, to maintain

the robust performance of the system. After finding the uncertainty del(∆), the weighing function

for the uncertainty has to be selected as a covering function of the magnitude plot of the del(∆),

to assure the robust stability.The controller bandwidth has been increased with the perturbed

system to maintain the controller stability. When analyzing the robust stability criteria with

respect to bode magnitude plot, it does not hold in the frequency range between 0.00008 rad/sec

to 1.2 rad/sec as shown in Fig.4.16. Therefore the controller is not robustly stable with respect to

the additive uncertainty set. Also the order of the resultant robust controller is found to be

seventh order controller.

In this chapter the robust controller has been designed using the µ synthesis has resulted

with the higher order controller and it finds difficulty in the real time implementation. The need

is to reduce the order of the robust controller and also by preserving the H-Infinity principles.

This leads to the design of robust PID controller.The admissible sets of robust PID controller

values have been found using the Hurwitz criterion for the Pure Integrating Process with dead

time. The PID Controller values obtained satisfy the robust performance condition for the

system. The Controller has been designed for the plant with and without uncertainty in the dead


109

time and also the admissible set of 3D plot has been plotted.From the obtained admissible set of

PID values, the narrowed down values of controller parameters need to be identified which will

give the good time domain specifications.

Fig. 4.17, Fig. 4.18 and Fig. 4.19 shows the open loop gain of the cases (1),(2) and (3).

Fig. 4.20 and Fig. 4.22 shows the range of Ki&Kd of Case(1) and Case(2) for the fixed

value of Kp

Fig.4.21 and Fig. 4.23 shows the 3D plot of the admissible set of Robust PID values.

******

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