Robust Controller Design for Uncertain Linear Time Invariant SISO Plants

Robust Controller Design for Uncertain Linear Time Invariant SISO Plants

Copyright © 1FAC I1 th Triennial \I'orld Congress. Tallinn. Estonia. l ·SS R. 1 ~ 190 ROBUST CONTROLLER DESIGN FOR UNCERT AIN LINEAR TIME INVARIANT S...

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Copyright © 1FAC I1 th Triennial \I'orld Congress. Tallinn. Estonia. l ·SS R. 1 ~ 190

ROBUST CONTROLLER DESIGN FOR UNCERT AIN LINEAR TIME INVARIANT SISO PLANTS A. Cavallo, G. Ce lentano and G. De Maria Diparlilllenlo di InJonnatica e Sislelllistica, Universila degli Sludi di Xapoli , via Claudio 21 , 80125 Xapoli, Italy

Abstract. This paper deals with the problem of designing a controller which assures that the closed-loop system remains stable with respect to a specified unconnected domain V of the complex plane in the face of plant parameter uncertainties. The problem is handled in transfer function domain by using two different concepts . The former is the parametrization of all V-stabilizing compensators, not exceeding a specified maximum order, for the given plant. The latter is concerned with the determination of the largest V-stability domain in plant parameter space, given a poly topic family of closed-loop characteristic polynomial in coefficient space. By combining the above concepts, it is shown that a robust controller can be designed via parametric optimization. Keywords . Robustness; robust control; control system design; feedback control; synthesis methods.

I NTRODUCTI ON

The second necessary tool is a procedure to perform V-stabi I i ty test in plant parameter space. In the last years control literature has grown rich with new methodologies for robust stability analysis, mainly due to the work of Kharitonov (1979). Starting from Kharitonov's theorem, many authors have proposed several procedures in order to solve the problem of robust stability and V- stability analysis, in the case of structured perturbations, and particularly when characteristic-polynomial coefficients depend I inearly on physical parameters (Tesi and Vicino, 1988; Barmish, 1989).

The goal of any practical motivated feedback design is that the feedback system should preserve stabi I i ty and loop performance requirements in the face of plant parameter uncertainties, 1. e. stabi I i ty robustness and performance robustness respectively. In the last years a lot of papers has been written on this subject both in the case of unstructured and structured uncertainties, we refer to Derato (1987), Bhattacharyya (1987), Keel, Bhattacharyya and Howze (1988) and De Gaston and Safonov (1988) for a very clear treatment of the subject and for an updated bibliography.

Recently the authors have proposed a procedure very efficient from a. computational point of view in the case of linearly dependent coefficient perturbations (Cavallo, Celentano and De Maria, 1989) .

This paper deals with the problem of performance robustness of feedback control systems with uncertain linear time-invariant SISO plants. Performance requirements are expressed in terms of robust closed-loop pole location, i. e. in terms of stability of the feedback control system with respect to a specified unconnected domain V of the complex plane (V-stability). A suitable robustness index which meets this requirement is assumed to be the measure of the largest V-stability domain in the plant physicalparameter space . Then the controller must be designed in order to guarantee the assigned nominal closed- loop pole location and to enlarge the V-stabi I i ty region in the plant parameter space to the largest extent possible.

In this paper by jointly using the results of Celentano and De Maria (1989a, 1989b) and Caval 10, Celentano and De Maria (1989), and parametric optimization, we give a procedure to design, in the case of linearly dependent coefficient perturbations, a compensator of suitable orde r which gives the largest V-stability domain in plant parameter space for assigned closed-loop pole location. PRELI HI NARI ES

Consider a linear time-invariant SISO plant of order n, and suppose that a number p of physical

The first necessary tool to achieve the above requirement is the parametrization of all compensators that V-stabilize a given plant. The first complete characterization of stabilizing compensators was obtained by Youla, Jabr and Bongiorno (1976). Later an algebraic formulation was given by Desoer et alii (1980). A very clear treatment of this subject can be found in Vidyasagar (1985) . However, in using the above parametrization, it is difficult to characterize the class of stabilizing compensators of the same order. In order to overcome this drawback, a parametrization of stabilizing compensators of order not exceeding a specified maximum one has been proposed (Celentano and De Maria, 1989a, 1989b)

parameters are uncertain. Let rr=(rr ... rr 1

vector of uncertain parameters

p

)T

belonging

be the to a

compact subset ITc~p. By allowing rr to take on arbi trary values in IT we obtain a whole fami ly of plants described by b(s,rr) (1)

p(s,rr) a(s,rr) where

165

w (s) :=

(1 S

..

n

sn)T

(2)

a(· ), b(· ): TT -) IR n+1, are real affine functions of the parameter vector n; b(n):=bO+Hbn,

b ER + 1, n

°

a(n): =a +H n, aoE

°

!R n + 1

H elR(n+1 )xp,

b H elR(n+Ux p ,

,

a

a

g1

0 g1

vectorial S

I,

n

(g):

=

(3)

neTT neTT.

(4)

g On

g

0

0

n gelR .

(9)

n gn

Now we state the following lemma, whose proof can be found in the same work (Celentano and De Maria, 1989a).

In the following we will assume:

Lenwna (5)

If

1.

a

n+1

(no);
and

the

polynomials

a(s,rrO) and b(s,no) are coprime, then

°

rank S(rr )=min(n,v+1)+v+1.

where nO = (nO .. nO)TeIRP is the nominal para1

0

0 81

A

P

meter vector and p a suitable real positive number;

( 10)



Such a result is an extension of Silvester's Resultant Theorem (Kailath, 1980). Indeed it can be easily verified that the latter follows from Lemma 1 by putting v=n-1.

2) the denominator polynomial a(s,n) to be a manic one, i.e. a (n)=l. n+1

LINEAR PARAHETRIZATION OF ALL STABILIZING COMPENSATORS OF MAXIMUM ORDER v.

Now consider the SISO feedback control system in Fig. 1,

The aim of this section is to present a procedure for designing ~-stabilizing compensators of order not exceeding a specified order v. The domain ~ in which one wishes to confine closed-loop poles is the union of msn+v compact connected domains ~h in the complex plane, with ~/"\~J=0,

Fig. 1 - Feedback control system. where c(s)

(3(s)

(6)

o:(s)

Then the transfer function of the feedback system is given by

Theorem 1. order n,

wT

(s)n(n)

wT

(s)d(n)

n+V

n+V

(7)

In a recent work (Celentano and De Maria, 1989a) it is shown that d (rr) can be expressed as (a(rr))

~

.

S

V+1,n+1

( b(rr ))) [

the boundaries of

0:

{3

Let pes, rr) be a fami ly of plants of

and p(s,rro) the nominal plant. Suppose

wT(s)(B d(rro)+B z)

~=

1 TO

1)

{

W

v

1.I+1,n+1

a~h

a(s, rro) and b(s, rro) coprime, and v an integer greater than n-1. Then the set ~ of all stabilizing compensators c(s) of order 11, n-lsllsv which assure that the poles of the nominal feedback system lie in a specified stability region~, is given by:

b(s,n){3(s) a(s,n)o:(s) + b(s,n){3(s)

d(rr)= (S

denote by

Recalling that the design objective is to locate nominal closed-loop poles inside the domain ~ and to enlarge the ~-stability region to the largest extent possible, a compensator of order v>n-1 is required. Such a compensator can be designed by means of the following theorem, whose proof can be , derived from the one of Theorem 3 in Celentano and De Maria (1989a).

is any compensator of order v.

n(s) des)

lIi;
such domains.

2 . dE IR~n+lJ+l ZE IR~lJ+l -n ' D •

} (11)

(s) (A den )+A z) 1

2

where

]= (8)

lR~n+V+l denotes the set of the (n+v+l) - vectors D

d(rro) = (dT(rrO) OT)T, with d (rro)elR n+Il +1 such 11 11 that the roots of polynomials wT (s) d (rr ) n+1l 11 are inside the domain ~;

°

where S

I,n

( g) denotes the real (n+i -1) xi matrix

1R~IJ+l-n denotes the set of the (v+l-n) - vectors z=(zT OT)T, with z elR ll +1- n ; 11 11 and the matrices A B IRlv+llxln+v+1) and I" lE , A2' B2 elRlV+llxlV+1-n) are given by

166

~ea7l , h

( 12)

i=1, .. ,le.,

h=1, .. ,m and that d

ots inside the domain 7l

on,lI+1-n

-S

I

0

(b(1t )

1)+1-n, n+1

)

n+I1+1

(lI)"O

vlIen. Then the family of polynomials d(s,lI) has Ih ro-

sup I'B(d ~ea7l

(13)

vi

h

h=1, .. ,m if and only if

,

(~»)-'B(d

vJ

(~») l
(21)

h

B=arg(d

1/+1-n

vi

(~»,Vi,J=1, .. ,Ie.,i>J,Vh=1, .. ,m,

where where

Be[-lI , lI), denotes arg(ze- JB ); T=

I

n+V+l

(14)

]

arg(z) denotes (arg(z)e[ -lI,lI»;

[

the

main

argument

of

zeC

°V+l-n ,n +1I+1



and at least one polynomial of the family (20) has Ih roots lying inside the domain 7l ,

0



V=S(1t )T

Re.mrk 1.

vector /3

n+1

h

(15)

h=1, .. ,m, where

Note that the /1+1-n components of the z/1 coincide with the components

, .. ,/3

O
of the vector /3 associated to the

1J+1

;~lh=n+v . •

(22)

I

numerator polynomial /3(s) of the compensator . Being the closed-loop zeros coinciding with the zeros of the polynomials b(s,lt) and /3(s), then z/1

The proof of this theorem can be found in a recent work of the authors (Cavallo, Celentano and De Maria, 1989).

can determine the location of /1+1-n zeros of the closed-loop transfer function.

Theorem 2 ensures that V-stability can be tested by simply checking if the vertex polynomials of poly topic family (20) verify condition (21)

ROBUST V-STABILITY TEST

"

In this section, starting from the preliminary assumptions on the family of plants p(s,It), we present a procedure to test the robust V-stability of families of characteristic polynomials d(s , It).

Vseav

From hypothesis (3) and (7) can be expressed as

The core of the algorithm consists of the following three steps:

(4)

the vector d(lt)

h

,

h=1, .. ,m. This test can be carried out by

using the following algorithm. Algorithm 1.

in

Step 1. m (~):=d

Compute

1

vi

(~)d' (~), vI

i=2, .. ,It,

where

dyl(S) are given by (19), and, by examining the

where

sig~

d =S o

n+l , V+l

H =S d

n+l.V+l

(a)a +S 0

(a)H

a

+S

n+l , V+l

n+l.V+l

(/3)b

of the real and imaginary part of" each ml(s), find the quadrant to which each ml(s) be-

( 17)

0

longs, or exit unsuccessfully if m (~)=O for some

(/3)H .

1.

(18)

b

I

Step 2. Since d(lt) is an affine vectorial function in It and n a box in the parameter space, the image den) is a polytope in the coefficient space

Exit unsuccessfully if quadrants II and III are both occupied, i. e . if at least a couple

Rn+v+1 wi th a number It of vert ices v .

the second quadrant and m (~) to the third.

(ml(s),mJ(s)) " " exists such that ml(s) " belongs to

I

It is well known that the entire family of polynomials des , It) can be generated by means of a convex combination of the exposed vertex polynomials d

vi

(s)=wT

n+V

(s)v, i=1, .. ,It.

L Ald vl (s),

I =I

Step 3. If the first (second) quadrant and the third (fourth) are not both occupied exit successfully.

(19)

1

Else if a pair of complex numbers (m (~), m (~» le.

le.

d(s,It)=

In

J

this case, in fact , condition (21) of Theorem 2 is not verified.

AI~O

"

LA I =1

(20)

J

urth) quadrant, m.(~) to the third (second) and

1=1

J

Jm m (~)/?1e m (~) '" Jm m (~)/?1e m (~)

Now we can state the following theorem:

I

Theorem 2 . Let the polynomial d(s,lt) be a poly tope in the coefficient space IRn+V+I. Consider m"'n+v compact connected domains V

h

the complex plane, with VIA V =0, Vi*j. J Suppose that the vertex polynomials

I

exists such that ml(s) belongs to the first (fo-

d

vi

I

J

J

(23)

exit unsuccessfully. Otherwise exit successfully .

in In order to carry out the test repeat the above three steps sweeping ~ along the boundaries av , h=1, .. ,m. h

(~)"O

Hi7

The above algorithm can be used in order to estimate the largest V-stability domain in plant parameter space, i . e ., referring to (5), the maximum value of p such that all poles of the closedloop system belong to V. This can be accomplished by using algorithm 1 iteratively and bisection algorithm.

-[Z~l"V+1-n

Z k

-- En

can be computed

The function cS (.), z

by using algorithm 1.

Example.

The goal of robust controller design is to find a

Consider a perturbed polynomial

vector Zk which maximizes p(s,n)=(30+n +n )+(29+n )s+(lO+n )s2+s3 . 121

(24)

2

r

max

=2v'2,

minimum and

6

and IDln

maximum

=±lOo,

6

max

where

6

Proposition 1. Given a nominal plant p(s,no) of order n, and the set of ~-order D-stabil1zing

is

compensators, let cS

be the V-stability margin of k the feedback system with the compensator c(s,z ) . k

Suppose that the domain V includes intervals of the real axis, then

2

By using a VAX-Station VS2000 we have obtained the following result : tolet'ance =

Denote with cS k is now proposed as

The following proposition holds .

evaluated with respect to the real negative semiaxis. The remaining real zero is required to have real part ~-S . Referring to (5) assume that w =w =1. 1

. ).

k a D-stability margin for the feedback system with the controller c(s,zk) '

radii

=±4S0,

~k (

such a maximum. The value cS

The nominal polynomial p(s,O) has zeros at s=-2±J and s=-6. We require that the complex zeros remain within rectangles in polar coordinates (r,6) having

,% En .,k ~

0

cS

1.00000 E- 03

;,: cS

k+1

(26)

k

Denote

Proof.

the

feedback

characteristic polynomial of degree n+~ obtained with the compensator C(S,zk)' and with ak(s) and ~k (s)

RhO

= 3.33244E-01

the denominator and numerator polynomials

of c(s,zk) respectively . Let

Parameter interval s:

d p( p (

1):

[

-0 .333

0.333

2):

[

-0 .333

0.333

total execution time:

The maximum value of V-stability margin.

k+1

(s,no)=a(s)a

k+1

(s)+b(s)~

k+l

(s)=

(27)

2590 roi lli"ec p

can

be

assumed

as

since expression (11) represents a complete parametrization of V-stabilizing compensators for the

a

plant pes, nO), it is easy to verify that a compensator of order ~+l~v which satisfies (27) is

Due to the very short execution time, the above procedure will be used in the next section to design,via parametric optimization, a robust D-stabil izing compensator which enlarges D-stabi li ty margin to the largest extent possible.

c(s,Z

k+1

) = c(s,z )(S+7)/(S+7). k

(28)

With this compensator we obtain

ROBUST CONTROLLER DESIGN cS

By using the tools presented in the previous sections, it is possible to design a set of compensators of order v>n-1 which locate nominal closed-loop poles at specified points of the complex plane, and to select the compensator in this set which gives the largest V-stability margin by suitably selecting the values of the v-n+1 free parameters Zl'

k+1

(z

k+l

) = cS

(29)

k

Then (26) follows from (29).



It is easy to prove that if the domain V does not include intervals of the real axis, (26) can be written as cS

Let us recall that (11) allows to design all D-stabi 1 izing compensators of any order ~ not exceeding a specified one v.

k+2

;,: cS

k

(30)

Now we present an algorithm which allows to select among the D-stabilizing compensators of maximum order v the one which gives the largest D-stability margin.

Denote with cS (.): Il v + 1 - n -+ R a function such that k

where zk =(Z~OT) T is a value of the free k parameters vector zeIl V + 1 - n in (11) and k=~-n+1, gives the measure, according to an assigned metric, of the V-stability domain in the plant parameter space of the feedback system in Fig. 1. The compensator c(s) is given by cS (zk)'

wT(s)(B d(nO)+B Z ) v 1 2 k TO' w (s)(A den )+A Z ) v 1 2 k

Algorithm 2. Step 1. Ini tialization: compute the matrices AI' A , B , B2 2 1 for the given maximum order v of the compensator according to eqs. (12), (13). Select a nominal D-stable closed-loop characteristic polynomial of degree 2n-l. Compute the

(25)

D-stability margin 80 '

168

Step 2. jw

If the domain 'D includes intervals of the real axis then build the current nominal closed-loop characteristic

polynomial

d (s , rro) k

by adding a jS

real zero to the pre v ious one, i.e .

-- --40 __

otherwise add a couple zeros belonging to 'D .

of

complex

conjugate

-4

-2

a

1)

Step 3. Compute the ~-order compensator which 'D-stabilizes the nominal feedback system . Consider the perturbed closed-loop characteristic polynomial whose coefficient vector d(rr) is given by (8). Step 4. Compute the stability margin ok associated to the

Fig. 2 - Specified 'D-stability domain

compensator, and the corresponding parameter vector zk ' Then, if ~ < v and a significant ~- order

Select the compensator maximum order v=5. With this select ion the matrices AI' A , B , B2 are the 2 l following :

improvement of the stability margin has been achieved, increase ~ and go to step 2 . In the abo ve problems arise: -

algorithm

the function 0k(zk)

some

computational

is generally not convex,

hence there is no guarantee that a global maximum will be found . An usual procedure in such cases is to start the optimization procedure from different points zk and to select the best local maximum; - sinc e the free parameter vector zk cannot be v

swept over IR + l - n but must vary in the cartesian product of bounded intervals, we choose. such a product as a neighbourhood of the point zk given

000 1 -7 37 -185 821 -4597 o 000 -7 37 -185 821 o 000 0 -7 37 -185 A= 1 o0 0 0 0 0 -7 37 1 -7 0 0 0 0 0 0 0 1 o0 0 0 0 0 0 0 1

(34)

100 -10 10 -370 1850 -9210 45970 010 -12 74 -374 1850 -9202 45945 001 -7 37 -185 921 -4597 22971 B = 1 000 0 0 0 0 0 0 000 0 0 0 0 0 0 000 0 0 0 0 0 0

(35)

-1 7 -37 0 -1 7 0 0 -1 A= 2 0 0 0 0 0 0 0 0 0

(36)

10 -70 370 12 -74 374 7 -37 185 B= 2 1 0 0 0 1 0 0 0

(37)

by

(32)

where e

1

de notes the vect or (1 0 . . 0) T .

This f ormula can be easil y obtained from the numerat or of (28) and by c onsidering the particular structure of matri ces B , B2 in (12) and (13) . A l

similar r e s u lt c a n be obtained when 'D does not include interva ls of the real ax is.

EXAMPLE Consider the plant p(s, rr)

We require that the nominal closed-loop characte-

(33)

ristic polynomial d (s,rro ) (~=2) has roots at:

s3+(5rr +2rr )s2+(10rr +2rr )s+(3rr +7rr ) 1 2 1 2 1 2

°

SI,2=-2±J2, s3=-12, s4=-15, s5=-20. whose nominal parameter vector is rro=(1 l)T. We require that closed-loop poles remain in the domain depicted in Fig. 2.

Referring to (5) let w =1, w =2 . The 'D-stability

° =0.119 has been obtained . •

margin

1

2

0

By increasing the compensator obtained the following results :

order

by adding a closed-loop pole at s6=-25 margin has increased to further pole at s7=-30 has been achieved .

169

° =0.3087;

1 (~=4)

we (~=3),

have the

by adding a •

the margin 02=0.3184

By further increasing the compensator order no significant improvement has been obtained. The

vector

z

z

which

assures

the

Cavallo A. , Celentano G. and De Maria G. (1989) . Robust stabi I i ty analysis of polynomials with linearly dependent coefficients perturbations. Submitted to IEEE Trans. Autom . Contr . Celentano G. and De Maria G. (1989a) . A new linear parametrization of all stabilizing compensators for single input single output plants . To appear on Proc. lEE part D. Celentano G. and De Maria G. (1989b) . Effective stability margin with respect to specified stability regions . ICCON '89, WA-2-2 . De Gaston R. E. and Safonov M. G. (1988) . Exact calculation of the multiloop stability margin. IEEE Trans. Automat. Contr., AC33, 156-171. Desoer C. A. , Li u R. W., Murray J. and Saeks R. (1980) . Feedback system design : the fractional representation approach to analysis and synthesis. IEEE Trans. Automat. Contr., AC-25, 399-412. Dorato P. (1987). Robust control. IEEE Reprint Volume. P. Dorato editor . Kallath T. (1980). Linear systems. Prentlce- Hall, Englewood Cliffs . Keel L.H., Bhattacharyya S.P . and Howze J.W. (1988) . Robust Control with structured perturbations . IEEE Trans. Automat. Contr., 33, 68-78. Kharitonov V. L. (1979). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations . Differential Equations, 11, 1483-1485. Tesi A. and Vicino A. (1988) . Robustness analysis of uncertain dynamical systems with structured perturbations. Proc. of the 27th CDC, 519-525. Vidyasagar M. (1985) . Control system synthesis: a factorization approach . M. A. M.I.T. Press, Cambridge. Youla D. C., Jabr H. A. and Bongiorno J. J. (1976). Modern Wiener-Hopf design of optimal controllers, part 11. IEEE Trans . Automat. Contr., AC-21, 319-338.

V-stability

margin ~z is Zz=(253081 16660 O)T.

CONCLUSIONS In this paper a procedure to design robust controllers for uncertain linear time-invariant SISO plants has been presented . Such controllers are robust in the sense that they assure closed-loop performances in the face of plant parameter uncertainties. Closed-loop performances are specified in terms of closed-loop pole location, by assigning a stability domain, generally unconnected, in the complex plane. Firstly we have presented a procedure to design the set of all compensators which allow to achieve the design specifications, providing the designer with suitable free parameters. Then a robust stability test for polynomials which are poly topic in coefficient space has been described. By Jointly using these procedures and parametric optimization, a robust controller has been obtained for given plants whose physical uncertain parameters vary independently in intervals, and whose numerator and denominator polynomial coefficients depend linearly on physical parameters . Further researches on this topic concern the optimal closed- loop pole location in the specified stability region in order to further enlarge the stability domain in the plant parameter space.

REFERENCES Barmish B.R. (1989). A generalization of Kharitonov's four polynomials concept for robust stability problems with linearly dependent coefficient perturbations . IEEE Trans . Autom. Contr . , 34, 157-165. Bhattacharyya S. P. (1987). Robust stabilization against structured perturbations. In Lecture Notes in Control and Information Sciences, 99, Springer-Verlag, New York.

170