Robust Sequential Design Procedure for Multivariable Control Systems with Parameters Bounded by Intervals

Copyright © IFAC Large Scale Systems. London. UK. 1995

ROBUST SEQUENTIAL DESIGN PROCEDURE FOR MULTIV ARIABLE CONTROL SYSTEMS WITH PARAMETERS BOUNDED BY INTERVALS

Winnie

H. Y. Ng, L. F. Yeung,

Department of Electronic Engineering City University ofHong Kong Tat Chee Avenue. Kowloon Hong Kong

Abstract: In this paper, a sequential design method based on the factorization technique is extended to systems with parameters bounded by intervals. Here, the transfer function of a system is modeled by a polynomial with complex variable s and its coefficients are represented by a parameter vector bounded by a hyper-cube. Intervals algorithm is first developed in this paper and then is applied to investigate the robust stability problem existing in the sequential design procedure. Keywords : Robustness, Sequential, Factorization, Multivariable Topics: Decomposition, Stability, Decentralized control

List of Main Symbols:

P(s,A,B)

I

N[f(s),x]

an interval set of real number. I x x I y is a closed region on a

C

complex plane, where

z A

DN

GE GJ

STF G(s) R (i )

(s)

E

C ~ Re(z)

E

Ix,Im(z)

the polynomial with the parameters included by the intervals A and B. winding number of a complex function about x. which maps the D N contour onto the complex plane.

Ely'

1. INTRODUCTION

the hyper-cube whose vertices is defined by the extreme points of interval. the Nyquist contour Gauss elimination Gauss-Jordan elimination the sequential transfer function the nominal plant. the return difference matrix of a system with the first i loops closed the return difference interval matrix

In resent years, the introduction of the interval arithmetic has been used to study the robust stability with various success; for instance, the application of the Kharitonov theorem for robust stability analysis, and the computation of the Horuwitz bound for interval plants [1-3], etc. This new approach generates an expanding research interest in the frequency domain methods. This method focuses on the analysis of the system ' s characteristic polynomial (CP). Recently a number of restrictions has been resolved, for instance the coefficients of the system's CP are no longer required to be independent. Some research work is still required in order to enable this useful method to be used as a practical controller synthesis too\. Here, a Nyquist type design method

the transfer function of a system with the first i loops closed

763

N(s,A) P(s,A,B) = D(s,B)

and 0 e BI2 + B; in the case of division. Then the range produced by the operations of two complex regions are:

(2. 1)

where N(s,A) and D(s,B) are interval polynomials N(s,A)

= a o +a ls+a 2 s + .+a.s·

D(s,B)

=bo +bls +b 2 s 2 + ·+bmsm

Property 2.2 a) ZI + Z2 = AI + BI + j(Az + B2 )

2

=

= [a l

-

j

-

... E

that,

2)

+ j (AI B2 +

Az B.)

Az B

2

)/(BI' + B;) + 2

AI B2 )/(B I + B;)

+ j[y, y]

l~, d]

(2.5) where

(2.2) - min {a 2b2, a2~' a 2 b2, a2~}

---

B2 are real intervals. Let us adopt the means

Az B

~ = min{alb" al~ ' aA , al~}

BI = [QI' bl ], B2 = [Q2' b2]

ZI*Z2

-

(Az BI -

= jeu

[g] = AI + j. A2 ' where Al = [£!I' al ], A2 = [£!2' a 2]

notations: and VZ2 E

2

--

[~, x]

and a fixed (() , we have shown that [g] and [f] are rectangle as follows :

~, ~,

a +~]

bp 211 - bl J + J [a, - b 2 , 212 - b 2 J

d) ZI/Z2 = (AI BI +

Given two polynomials g(s,A), f(s,B) and their parameter vectors are bounded by hyper-cubes AI &

AI'

+~]+ j[a 2 +b2 ,

= [~, u] + j[~, \/]

2.2. The Interval Arithmetic

-

l

-

-

c) ZI' Z2 = (AI BI -

I

s

a

b) ZI - Z2 = AI - BI + j CA, - B2 )

whose coefficients can be represented by vectors A and B respectively. The parameter spaces which bound the range of these vectors are hyper-cubes AI and A2 defined by intervals g j ~ a j ~ aj and h- I ~ h,. !(. ii, respectively.

A2 respectively. With

[a l +~,

for

'
E ZI

Z2' Z = (ZI'" Z2) E ZI ... Z2 where {+,-,x,+} . For real number intervals, we obtain

the following relations easily [5]: +max{aA, a2~'

Property 2.1

a)

A+B=[g+Q,a+b],

b)

A- B

c)

A·B=[min{ab, CJ!; , aq" ab},

--

=[g - b, a - Q],

A+B=[g, a] · [lIb, 1Iq,] .

aA, a2~}

+ min {a 2b2, a2~ ' a 2b2 , a2~} (2.3)

x= max{alb" al~' alb .. al ~}

--

max{gq" CJ!;, aq" ab}], d)

-

-

~=min{a2 b" a2~' aA. a,~}

o

where A = [g,a],B = [Q,b] and in the case of division, 0 e B. Hence, the number of calculation in finding the resultant range of these operations is reduced.

y=max{a 2b.. a2~ ' a,b .. a2~}

For the case of complex regions, let:

~ = min{b I , b1 } + min {b,2. b;}

= [g] = AI + j Az Z2 =[f] = BI + j B2 ZI

where

-ma\:{alb" a l~' alb" alE;}

--

(2 .6)

2

(2.4)

d=

ma"({b I2 , b12 } + ma\:{b; . b; }

o

al]' ~ =[~" aJ _ _ BI = [QI' hi]' B, = [Q2' b2 ]

AI

2

=[~I'

Note that the computed ranges based on relation (2.5c) and (2 .5d) are bounds of the operations {x, +} between two complex regions only. The advantage of such a choice is that their bounds are simple 765

the first loop design. The system matrix \\;th parameter uncertainty is given as follows : (3 .6)

Equation (3 .6) shows that elements in the above matrix involve four fundamental interval operations which has been introduced in section 2. The robust stability can then be defined in the following sense:

where a~l = [-1.1, - 0.9]; bill = [3.6,4.4] ;

=[2 .7, 3.3] , a~2 =[1.8, 2.2]; b~l

Definition (Robust Stability) : Let Q(s) and its range [Q] have the same number of unstable modes. Then the system is stable if a family ofpolynomials [Q] are all stable.

stability test of the STF

rY-

The GJ design procedure [4] for nominal systems can be applied to this example as follows :

(jw) . Each column of

the precompensator can be tuned till the STF meets the stability and the performance criteria. When this GJ design method is applied to plants with uncertain parameters, the robust stability becomes a family of Nyquist stability test. In addition, if the parameters of a plant are bounded by intervals, the robust stability tests reduce to a finite number of tests that can be derived from the vertices of the hyper-cube which bounds the parameter vector, hence the number of computation is drastically reduced Theorem 1 can be restated for systems with parameters bounded by intervals as follows :

Stage 1: With KO(s)=J; [G o(s)]=[G(s»); [S O(s)] = [S(s»); . [R (O) (s)] = [R(s)] . Stage 2: By Equation (1.2) and the above comment, the extra degree of freedom in selecting a column of K(s) , the non-minimum phase problem in loop 1 can be removed. An initial attempt found that the first

=[

-1 1] T column of the precompensator K ~ I (s) is sufficient to resolve this problem. The next aim is to reduce the effect of the cross-coupling terms

Corollary 1: Let [R(s)]=l+[G(s)]·K(s) be the family of the return difference matrix of the compensated feedback system with parameters bounded by intervals. If (i) All the members of interval matrix [R(s) ] are invertible over DN and

[g~1 (s)]

and

to

improve

the

closed

loop

performance of the first loop. The following K I (s) is adequate:

r

0.1 1-8(1+-)

~ &[lj),-l)(s)],O J+ Po = O;s E DN

(ii)

=[2.7,

3.3], a~l = [0.9, 1.1]; b;l = [1.8,2.2] , a o22 = [ 2.7, 3.3]; b022 = [l.8, 2.2] ,

Recall Equation(1.2), the design of the precompensator K(s) is based on the Nyquist 1 )

b~2

KI (s) = 1

0.1

1

01 1

l 8(1 +~) I J

then the closed-loop system is robust stable; where Po is the number of unstable modes of the systelll.

Stage 3: Compute the partial closed-loop transfer function

4. EXAMPLE

[HI(s») . All [Hi,( s) ] in the Nyquist plane should be

Let the nominal system matrix G(s) be :

r G(s)

=1

l and Q(s)

2

(s - I)

(s+ I)}S+ 3)

(s + 2)

=G(s)K(s) .

~ 3)

as close to I I I as possible, [Hi2 (s») and [H~ I (s)] should be as small as possible. Figure 6 shows the transfer function of the system with the first loop closed. If the performance criteria are unsatisfactory, KI (s) should be redesigned by repeating Stage 2; otherwise proceed to the next stage.

1

i, (s ~ 2) J (s

Stage 4: We observed that g~, has high uncertain part and the stability margins of its nominal part arc low, g~, has \'ery small uncertain part and good stability margins. therefore. the second column of K(s) is chosen to be K:~(s) = [0 15]T We yield the following precompensator

In this case. the element

g~1 (s) is non-minimum phase although G(s) is minimum phase. Since the sequential design is equivalent to a series of multi-input single-output (MISO) designs, a column of K(s) can be chosen in order to avoid the non-minimum phase problem in 767