New techniques for the pole placement of singular systems

New techniques for the pole placement of singular systems

Department of Electrical Engineering, Rm 241

National Taiwan University Taipci, Taiwan 50617

R.O.C.

Email : frchang@ac.ee.ntu.edu.tw Lee-Chuang Hsu and Fan-Ren Chanig

Abstract

New techniques for pole placement problem of single input singular systems are proposed in this paper. These techniques provide different ways to approach the generalized Ackermann’s formula with better numerical properties and flexibility. Since the solution of the pole placement problem depend on the singularity of the matrix E. Two sets of recursive algorithms are presented separately corresponding to the matrix E i s singular and nonsingular respectively. These algorithms are verified and implemented in MATLAB program.

1. Preliminaries

Consider the single input singular linear system

where and

2

ER”“”,

h

E R n X ’ , Eis possibly singular matrix, and u(t) E R and x ( t ) E R ” are input and state vectors respectively. The problem of pole placement in singular systems is to find the state feedback control law u(t) = - h ( l ) + r ( l ) , where k E

X””

and r ( t ) E R

such that the closed-loop system has prescribed finite and infinite eigenvalues. To develop the generalized Ackerinann’s formula, the restricted equivalent transformation is performed as

follows EX(t) = &(t) + F u ( l ) (1) E =

(/LE

-

Z)-’

E

( 2 4 A = ( / L E

-

z)-’

z

(2b) b = ( p E - X ) % . (2c) Definition 1

singular system of system (2)

if system (3) is obtained from the (2a), (2b) and

0

The generalized system (1) is controllable

rank[sE -.

2

6-1

= n for all s

(4)

0

For the standard singular system satisfying the controllable condition of Lenima 1 is referred as standard controllable singular system. In Lemma 2, we define a indeterminate parameter p and explore the determinant relationship of the open-loop system and closed- loop system in terms olf p.

Lemma 2

The system (3) is called the standard

Ex(t) = A x @ )

+

bu(t) ₍₃₎ (2c). Lemma 1

[SI

if and only if Forthecaseof @ - A = Z a n d s # p 7 1 det(sE

-

A ) = (-)” det(pl - E )

P

( 5 ) det[sE-(A-bK)]=(-)ndet[p(Z+bk)-E] ( 6 ) 1 or s = p - - where p = - P - s P 1

P

1

0

For a standard controlllable generalized system, the following two theorems are derived.

Theorem 1 (Generaliiied Ackermann’s forimula$[ 11

Let E i ( t ) = A x ( t )

+

bu(t) be a standard controllable generakzed system, satisfying. Assume the state kedback control law is

897 0-7803-3636-4197 $10.00 0 1997 IEEE

u ( t ) = -kx(t)

+

r ( t ) and the desired closed-loop characteristics polynomial is A,,(P> =

(P)"

d e t W - ( A

4 0 1

I

/ = 2 , , - , p 1 kE = e ; , C - ' A d ( E ) (7) .\ I / - - - l = O P , where d, = del(-E) . l h e n where = [0

o

..-

o

13 C = [ b Eb ... E"-'b] A d ( E ) =

E

dn-, E' i = O

CT

Theorem 2 [l]

Let E ( t ) = A x ( f )

+

bu(t) , with E singular, be a standard controllable generalized system, satisfying pE - A = I . Assume the desired closed-loop characteristic polynomial is

Ad ( P ) =

(P)"

det(sE - ( A - bk)]( I = ?d,,pi , Then . s = p - ,=I P where Remark : n-l i=O det(p1- E ) = Ea,-,p'

+

p"

0

The Leverrier's algorithm [ 6 ] can be used to compute the coefficients of A,,(p) . To compute the coefficients of A,,(p)

,

the following procedure can be proceeded :

(1) Convert the desired eigenvalues sld to p,d 1

via formulap, = ~

P - Sld

(2) Compute A d

0

= 0,

-

P I ~

10-I

- PZ(/ ). +

SO,

- Pn d

1

(3) Compute the scale factor c=c&(-EJ/[(-l)"~pd], and adjust the A(, ( y ) = c A,, (17) to meet the

condition det(-E) = d,,

.

Step (3) can be skipped when theorem 2 is

I

applied. Since the condition det(-E) = d , is reached as matrix E is singular matrix.

2. Main Results

Prom prcvious scction, it is clcar that thc solution of feedback gain k is depend on the singularity of matrix E. In subsection, algorithms are proposed separate respect to the singularity of matrix E.

There are two major problem been criticized about the Ackermann's formula (1) the inverse of controllability matrix C might cause numerical error due to the ill-conditioning of C (2) the numerical error computing feedback gain due to the multiple multiplication of matrix E . For the first problem, technique to solve this problem i s recommended by [2] which is listed as follows :

Algorithm 1 : (Computation of

CO

=

eiC-')

rl = b for i = 1 : n n, = norm(r, ) r, = r, I n, r,,, = Er, end e, = e, / n n ,

X =

[rl r2 r,] CO= ( ~ - ~ ' e , ) ' ' n ,=I

In algorithm 1, the controllability matrix is normalized and the scaling adjustment is made as the computation of Co. Algorithm 1 has been verified and claimed by [2] that have the advantage of numerical robustness.

2.1 Matrix E is nonsingular

Let A,,(p) = ?d,,-,p' be the closed-loop polynomial in terms of p with d,, = det(-E). There are two ways to approach the computation of C0Ad ( E ) . First, we can factorize A,[ ( p ) in

terms of closed-loop eigenvalues pd,l as follows :

1=0

A d ( p ) = do$ + d l p n - l + * . + d , - l p + d , ,

= do(P- P,,l)(P-P,,,)...(P-I'd,)

= dOii(P-Pd.;) i=l

rt

Since d , = ( - l ) " d ,

n

pd,r = det(-E)

,

we have do = det(-E) / [(-1)"

n

p d , ; ] . Hence, i=l II ;=I CO&

( E )

= doCO

ir

( E

-

P d , ,

0

, = I where do = det(-E) / [(-1)"

fi

ptl,r ]

Next, we will develop a recursive procedure to compute CoAd ( E ) in terms of the

;=I

coefficients of closed-loop systems di

.

CO

Ad ( E ) = doC0E"

+

dl CoE"-I

+-

-

-tdn-,

CO

E

+

dnCo

= doln

+

dIl,,-l +***+dn-lll +d,,Zo = i d n - r l l

r=O

where Z, = Zr-l E and Zo =

CO

From remark 2 and above derivation, k can be computed as

k=t;,C"4/(@E'

=[%1,]P

=4[~fl(E-p(/,I)]El (9)

ra -1

Algorithm 2 and 3 are provided to implement the approaching methods discussed above. Algorithm 2 : (Use desired closed-loop

Compute

CO

as Algorithm 1 eigenvalues) fo = (det(-E) / [(-1)" f i p r l>C0

J;

= f , ( E - P d , r o r=l for i = l : n - 1 end k = J,E-'

Algorithm 3 : (Use the coefficients of the closed-loop system) Compute CO as Algorithm 1

k

= dnlo for i = l : n lo

= C O

1, = L I E It = k

+

dnJi end k = kE-' 2.2 Matrix E is singular

From theorem

2,

the solution of feedback gain

k

need no( only the information of closed- loop system but also open-loop system. From last section, we see that the information of closed-loop system can be expressed in either eigenvalues cif closed-loop systems or coefficients of closed-loop systems. Therefore, as matrix E is singular, there are four possible combination algorithms.

From eq. (8), we can separate the solution k into two parts as follows :

k = q[(dn-l -awl)1+(4,-2 - a w J 3 - *-+(do -1)E'-'I

=q(&

I

+ ~ J c +

*

+,E-')

- ~ ( % 1

I

+qn

E+

+IT'

) =

CO&

( E ) -

CO&,

(

E )

Coxc ( E ) . represents the information of closed- loop systems while Coxf) ( E ) represents the information of open-loop systems.

Use the same technique as (9) in previous section, we have n-l r=O r=l n-1 r+O 1 x 1

CO&(E)

= ~ i d ~ ~ l - l l r = CO

rI(E

-PJ)

c,x,,(E) = ~ a n . . r - l ~ l =- c,%'(E - p ~ ) where 1, = lr.$ p',,; : desired eigenvalues

pr : eigenvalues of open-loop system

U,, d, : cciefficients of open-loop systems

and closed-loop systems respectively. Lemma 3 141

The closed-loop system

has

at most

rank E finite poles fca any feedback control.

0

From lemma 3, there are only rank E eigenvalues can be atsigned. Hence, there are n -.rank E closed-loop poles

( P ~ , ~ , , i = rankE

+

l , - - - , i t ) being assigned as 0. Four algor.ithms corresponding to different

combination of the information of open-loop system and closed-loop system are given as following.

Algorithm 4 : (Use the coefficients of the closed-loop system and open-loop system)

Compute CO as Algorithm 1 lo = CO k =

( 4 - 1

- - % I ) ~ O 1, = l1J k = k + (dn+ - q,-J for i = l : n - 1 end

Algorithm 5 : (Use eigenvalues of closed-loop system and open-loop system) Compute CO as Algorithm 1 for i = l : n - 1 Io =

f,

= CO

4

= 1 1 - 1

( E

- P d , l I ) f ; =

“6-1

( E

- PI 1) end k = f n - l -/,-I

Algorithm 6 : (Use the coefficients of closed- loop system and eigenvalues of open-loop system) Compute

CO

as Algorithm 1 lo = f o

= C O

k = dnJo 2, = lf-l E f , = f ; - , t E - P , I ) k = k + dn-.,-lll for i = 1 : n - 1 end k = k - f n - l

Algorithm 7 : (Use the eigenvalues of closed- loop system and coefficients of open-loop system) Compute

CO

as Algorithm 1 10 =

.fo

= CO k = an-lEo for i = l : n - 1

I,

= 11-1 E f, =

JL

( E - Pl.dO k = k

+

an-i-lZi end k = f n - l -

k

3. Conclusio~i

Algorithms to solve the pole placement problem of the singular systems utilizing the

generalized Ackermann’s formula are proposed. The weakness of numerical properties of generalized Ackermann’s formula are improved by these algorithms. Since the open-loop characteristic polynomial can be obtained by the Leverrier’s algorithm and the desired closed- loop poles are more straightforward needed, algorithm 7 is better than the others (algorithm 4

to 6 ) . The algorithm 4 has the least computation

flops in the four algorithms of section 2.2. All of

the proposed algorithms in this paper are implemented and verified by MATLAB software package.

References .

[ 1 ] L.C. Hsu and F.R. Chang, The Generalized Ackermam’s formula for singular Systems, System & Control Letters 27, 1996, pp. 117- 123.

[2] M. Valasok and N. Olgac, Efficient pole placement technique for linear time-variant SISO systems, IEE Pt. D, Vol. 142, No. 5,

[3 J J. Ackermann, Der Entwurf linear Regelung- ssysteme im Zurstandraum, Regel. tech. Proz- Pantenverarb., 1972, 7, pp. 297-300.

[4] L. Dai, ’Singular Control Systems’, Lecture notes in control and information science series, vol. 188, Springer-Verlag, New York

1989.

[5J J.

D.

Cobb, Controllability,

observability

and

duality in singular systems, IEEE Trans. Automatic Control 29, 1984; pp. 1076-1 082.

[6] D. K. Fadeev and V. N. Faddeva, Computati- onal methods of Linear Algebra ,Freeman, San Francisco, 1963

1995, pp. 45 1- 458.