Optimal pole placement with prescribed eigenvalues for continuous systems

Optimal Pole Placement with Prescribed Eigenualues for Continuous Systems ~~ABDUL-RAZZAQSARAR

Department KS 67208.

of Electrical U.S.A.

and MAHMOUDESAWAN Engineering,

Wichita

State

University,

Wichita,

ABSTRACT: A recursive method jOr determining the state weighting matrix qf a linear quadratic regulator problem in order to shift the open loop poles to the desired locations is presented. This method is capable of shifiing the real and imaginary parts.for continuous time systems. Aggregation is used in each step of the recursive process. Therefore each time the order of the system is reduced tojirst- or second-order, a constrained minimization problem with linear and nonlinear constraints has to be solved in order to find the state weighting matrix qf' the reduced-order system that will shift the open loop poles to the desired locations. An example is given to illustrate the theory.

I. Introduction In the literature of control theory, concerning the optimal linear quadratic regulator for multivariable systems, there have been different methods for selecting the weighting matrices of the performance index to achieve the desired closed-loop behavior. At the beginning, in order to achieve some acceptable transient response behavior, these weighting matrices have been selected by trial and error, and are assumed to be diagonal to simplify this process, which limited the generality of the solution. The pioneers in studying the relationship between the weighting matrices in the quadratic cost function and the closed-loop eigenspectrum are Solheim (1) and Graupe (2). In Van De Vegte et al. (3), Maki et al. (4) and Sebakhy et al. (5), the optimal control of a linear time invariant system, with respect to a given quadratic cost function, is investigated when the closed-loop eigenvalues are specified. Recently, a way of selecting the state weighting matrix for the linear quadratic (LQ) problem has been considered by a number of authors, such as Amin (6) and Medanic et al. (7). The problem is basically that of calculating an appropriate state weighting matrix, for a fixed choice of the control weighting matrix, so that the resulting optimal control law would place the closed-loop eigenspectrum of the system at some desired locations. These approaches are based on mirror-image property of the symplectic Hamiltonian matrix, and are capable of placing only the real part of the eigenvalues. Paul et al. (8) and Rousan et al. (9) studied this method for discrete systems and succeeded in shifting the open-loop poles by a proportion of I/CC(CX> 1).

c'The Franklinlnst~tutc OOl&OO32/93 $6.00+000

985

A.-R. S. Arm and M. E. Suwun Most recently, Saif (10) presented a new technique to shift both imaginary and real parts of the open-loop poles of a continuous system to be at the desired locations. However, as mentioned in his paper, some times his method fails to shift a complex pair to be at two real locations. Also, in his recursive method, and at each step one has to deal with a two- or four-dimensional matrix equation. In this paper, a design technique which combines optimal control and pole placement methods to obtain a minimization problem subject to linear and nonlinear constraints is considered. This new method is more efficient than those previously mentioned. It is capable of finding the state weighting matrix of the quadratic performance index that shifts the real and imaginary parts of the openloop poles of the continuous system to any desired locations, it even shifts a complex pair to be at any two real locations. In this recursive method, one has to deal with a first- or second-dimensional matrix equation regardless of the order of the original system.

II. Problem Statement Consider

the controllable

linear system z?(t) = Ax(t) +Bu(r)

(1)

where A E W”“‘, BE FYX”, x(t) E R” is the state vector vector. It is required to design a control u(t) = Lx(t),

LEWX”

(2)

using the inverse method, by finding the state weighting index given by the following equation J=i

“’ (x’(t)Q~(z, i‘

and u(r) E W’ is the input

+u’(t)Ru(t))

matrix of the performance

dt

(3)

such that the open-loop poles (2,) will be shifted to the desired locations (p,). In this paper, and without loss of generality, the input weighting matrix R will be equal to the identity matrix (11).

III. Design Procedure In this method, our ob_jective is to fmd the state weighting optimal state feedback u(t) = -B’Kx(t)

matrix,

such that the

(4)

can shift the open-loop poles (2,) to the desired locations at (,u,), where K is the solution of the following Algebraic Riccati Equation (ARE) A’-KfKAAKBB’KfQ 986

= 0.

(5)

Optimal Pole Placement 3.1. Transformation Consider the following

transformation x,(t) = TTx(t),

TT E R’“”

(6)

where x,(t) E R”, and r < n. By applying the transformation given by Eq. (6) on the original following reduced-order system will be obtained a,.(t) = Wx,(t)+Gu(t), u(t) = Fx,(t),

the

WE[W”~~ GEIW’~‘”

(7)

FE R’““’

(8)

with TTA = WTT, G = TTB, L = FTT. Then the performance index of the discrete reduced-order J, = 4

system,

= (xT,(t)Qx,(t) s0

system is given by

+u’(t>u(t>> dt

(9)

with Q = TQ,.TT. 3.2. Shifting one eigenvalue An eigenvalue I* is to be shifted to a desired location at ~1 (6). Apply the transformation given by Eq. (6), with TT being the left eigenvector of A associated with 2, we get i,.(t) = Ax,(t)+Gu(t) and the closed-loop

(10)

system is given by p = &GGTM.

Then matrix M can be easily determined

by solving

1-P M=m and the state weighting

(11)

(12)

matrix that will shift 1, to be at ,u is given by Qr = -2AM+MGGTM.

(13)

3.3. Shifting a pair qf eigenvalues A complex conjugate pair is to be shifted to a new complex conjugate or real pair, or a real pair is to be shifted to a new complex conjugate or real pair Applying the following transformation : x,(t) = TTx(t)

(14)

on the original system, where x,(t) E R”, TT~ R’“” and r < n, we get the following reduced-order system : i;.,(t) = Wx,(t)

+Gu(t)

u(t) = Fx,.(t) Vol. 330, No. 5. pp. 985-994. 1993 Prmted in Great Bntam

(15) (16) 987

A.-R. S. Arur und A4. E. Su~wz with TTA = WTT, G = TTB, L = FTT, WER’~‘, GEIR”~“‘, FE[W”‘~‘, and TT being the matrix of the left eigenvectors of A associated with the two poles of 3., needed to be shifted. Then the performance index of the reduced-order system is given by J,.=+

’ (.u;(t)Q,.~,.(t)

+u’(t)tl(t))

dt

(17)

s0 with Q = TQ,TT. The open-loop matrix

and input matrix

of the reduced-order

system is given by

(18) where a = c if the pair is complex, state feedback gain matrix

else p = 0 and H # [. Then we need to find the F=

-G’M.

(19) form :

Now, assume that matrix M has the following

with a condition that M is positive semi-definite. should satisfy the following conditions :

Therefore

the elements

of M

x 3 0 q-_;’ in order to be positive following form :

semi-definite.

Then

3 0 the closed-loop

W, = (W-GGTM).

(21) matrix

will have the

(22)

In order for M to shift the eigenvalues of W to the desired locations, the characteristic polynomials of Eq. (14) and of the desired eigenvalues should be equal. Let the characteristic polynomial of the desired eigenvalues take the following form : s’+$,s+$2 and the characteristic

polynomial

of matrix S’+U,S+02

with

988

= 0

(23)

WC has the form = 0

(24)

Optimal Pole Placement f, =

1:-l,ls,

f2 = p12-g,,

f4 = pl,-al,-112-~1,, Then, by equating

f3 = -p1,-Ml, .fs =

p+dY.

(25)

(15) and (16), we get h,x+h*y+h3z+h4-$,

= 0

f,22-f‘lXy+f*X+f7y+f~Z+.fS-~Z = 0. However,

it is known

(26)

that

min (Jr) = min (tr M),

(27)

E[x,.(O)x~(O)] = I.

Now, to minimize the cost function, Eq. (19) should be satisfied, and Eq. (13) should be satisfied to guarantee that the solution of the Riccati equation is positive semi-definite. To shift the poles of the open-loop system to the desired locations, Eq. (18) should also be satisfied. Hence, we obtain the following constrained minimization problem : min (tr M) subject to x 3 0 .Xy--z’ > 0 h,x+&Y+h_iz+hq-$I .f,z’-.f,x~+.~~~+.f~~+f4Z+f5-~*

= 0 = 0.

(28)

Then M can be found by solving the above problem, and the state weighting that will shift the poles to the desired locations is given by Q,. = -WTM-MW+MGGTM.

matrix

(29)

The relation between the eigenvalues of the full-order closed-loop system and the eigenvalues of the reduced-order closed-loop system and the full-order open-loop system is given by the following lemma. Lemma Eigenvalues of the matrix (A- BFTT) are the sum of the eigenvalues matrix (W-GF) and the n-r undisturbed eigenvalues of matrix A. Proof: For the proof see (12).

of the

n

IV. Shifting Several Eigenvalues Recursively (1) Separate the eigenvalues to sets, let N and n be the number number of the eigenvalues needed to be shifted respectively, then N = 42,

if n is even.

N = (n+ 1)/2, Vol. 330. No. 5, pp. 985-994, 1993 Printed m Great Britam

of sets and the

if n is odd. 989

A.-R. S. Avar and M. E. Suwan (2) Let S, be the sets of the eigenvalues needed to be shifted. D, be the sets of the corresponding desired eigenvalues, with i = 1, 2, 3, . . , N if n is even. i-1,2,3 ,..., N- 1 if n is odd. (3) Set the counter i = 1, and let A, = A. (4) Reduce the order of the system, by using the transformation given by Eq. (6), where TY being the left eigenvectors of A, associated with the eigenvalues of s,. (5) Find M, by solving the constrained minimization problem given by Eq. (20). (6) Find the state weighting matrix that will shift the poles of S, to be at the desired locations given by D,, which is given by

Q, = TQ,.,T,T

(30)

where Qr, can be found by solving Eq. (21). (7) Find the state feedback gain matrix L, = F,Tp F, = -G;rM,. Then the closed-loop

matrix

will be updated

(31)

by using the form

A r+ I = A,+BL,.

(32)

(8) Increment the counter i = i+ 1, then : ifnisoddandi=N-l,gotostep9; if i = N, go to step 10. Otherwise go to step 4. (9) Find the state feedback gain LIv and the state weighting matrix QN that will shift the single real eigenvalue to the desired location. (10) The state weighting matrix and the state feedback gain matrix that will shift all the eigenvalues to the desired locations are given by Q=

and the closed-loop

2 Q,,

Lx

i= I

$Li

(33)

,= I

system that has all the desired eigenvalues

is given by

A, = A+BL.

(34)

V. Example Consider

the controllable -1.5 0.0

i(t)

990

=

linear system 3.0

-1.0

0.5 3.0

2.5 -3.5

1

0

0

1

1

1

0

1

-2.o0.5

- 1.0

0.0

-2.5

0.0

1.0

- 1.0

3.0

-1.5

2.0

-2.0

_ - 1.0

3.0

-1.5

2.5

-2.5

x(t)+

u(t).

-1

1

Journal

of the Frankhn lnst~tute Pergamon Press Ltd

Optimal Pole Placement

The above open-loop system has poles at A,,* = - 1 fj3, 13,,, = - 1.5fjl and /2, = -0.5. It is desirable to shift them to the desired locations at P,,~ = -2 +j2, p3 = - 3, p4 = -4 and p5 = - 2. In order to solve the problem NPSOL software package was used. This software is made to solve a minimization problem with linear and nonlinear constraints. By solving this example, we get -0.9847

Q, =

-0.1590

_I

A2 =

-0.8257

0.9847

1.1743

0.1896

0.9847

- 1.1743

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

5.8347

- 2.6726

8.5073

- 5.8347

0.0000

- 2.6726

- 1.0151

- 1.6575

2.6726

0.0000

8.5073

- 1.6575

10.1648

- 8.5073

0.0000

- 5.8347

2.6726

- 8.5073

5.8347

1

- 1.5000

3.9847

0.6590

3.3257

- 2.9847

0.0000

- 2.1743

2.8104

- 4.4847

1.6743

-0.1590

1.1896

- 1.oooo

-0.1896

- 2.5306

- 1.oooo

1.8257

- 1.6896

1.0153

-0.8257

- 1.oooo

2.8104

- 1.5306

2.3410

-2.3104

where Q, is the state weighting matrix that shifts A,,2 of A, to be at p,,*, and will leave the rest of the poles unchanged. The closed-loop matrix of open-loop matrix A, associated with the state feedback gain matrix L, is given by A2 = A,+BL, where the poles of A2 are p ,,2, I,., and Jb5.To shift the next two poles A3+ we get 1.9354

L2 =

-0.2915

-

Q2 =

A3 =

6.5686

- 0.4526 0.0682 - 1.2237

- 1.2237

0.2131

10.7324

10.7324

-2.1126

- 1.4828

0.3182

0.2234

-6.8135

-5.3449

1

- 1.9515

1.2524

1.0106

- 1.9515

17.3812

- 11.0779

- 8.7809

-6.8135

1.2524

- 11.0779

7.0482

5.5612

- 5.3449

1.0106

- 8.7809

5.5612

4.3344

- 3.4354

4.4373

- 2.7999

0.2915

- 2.2425

3.3314

0.1948

- 5.4684

i -2.6439

5.4383 - 4.8029 1.6354

- 1.5018

1.7575

- 1.1686

0.6971

- 1.0491

- 2.6439

3.1948

- 4.4684

4.1354

- 1.0509

Vol.330,No.5,pp.985-994, 1993

matrix

1

1.4509 2.4491 1 ,

-0.7085

where Q2 is the state weighting

Printed in Great Britam

3.4588 -0.5210

that shifts A,,., of A2 to be at p3 and p4,

991

A.-R. S. Arm and 114.E. Smun while keeping the remaining poles in their locations. The closed-loop matrix open-loop matrix A2 associated with state feedback gain L, is given by

of

Al = A>+BL, with ,u,,~, Pi, p4 and i, being the poles of Aj. To shift the last pole jVsof A? to be at ,Llj

L3=

0.7886

1.4535

0.5154 3.9637

-

-0.2796

0.7143

-2.1505

-2.1505

Q3 =

-2.0143

1.1668

- 4.0470

4.8003

1.4350

1.7021

1

5.4932

11.0364

- 13.0904

- 2..9804

- 5.9879

7.1024

5.4932

- 2.9804

7.6128

15.2950

- 18.1418

1 I .0364

- 5.9879

15.2950

30.7294

- 36.4489

- 13.0905

7.1024

- 18.1418

- 36.4489

43.2330

A3 =

I-

- 1.9819

3.6487

- 0.7855

9.4853

- 0.2238

- 1.9628

2.6 I72

- 6.2380

- 1.7058

-0.3142

-4.1684

- 1.2238

2.0372

- 1.8828

I .7058

2.6858

-3.1684

-6.3021 3.1530

4.2474

1

-0.6491

-0.7380

,

0.6530

6.7474

-4.1491

1

where Q3 is the state weighting matrix that shifts lbs of A, to the location at ,us without changing the rest of the poles. The closed-loop matrix of the open-loop matrix A, associated with the state feedback gain matrix L, is given by A, = AJfBL, where the desired poles are the poles of A?. To shift all the poles of matrix the desired locations given in the example by one step we use

L=

0.4819

-0.6487

0.2238

0.9628

1.2855

- 6.9853

0.3828

2.7380

TABLE

A to

4.302 1 -2.6530

1

I ..

Eigenvalues A, AZ A3 A,

A,

992

A A,+BL, A2+BL, &+BL, A+BL

_ I *.;3, -2+.j2, -2kj2, -2kj2, -2+j2,

- l.s+.jl, -0.5 ~ 1.5&J, -0.5 -3, -4. -0.5 -3, -4, -2 - 3, -4, -2

Weighting matrix

I :,

Q3

Q = Q,+Q?+Qj

Feedback gain

L, L2

L3

L = L,+Lz+L, Journalofthe Frankhn

Institute Perpamon Prew Ltd

Optimal Pole Placement -

Q=

10.5323

- 3.3742

16.2255

4.2228

- 18.4355

- 3.3742

7.2146

- 7.6045

3.7717

2.2784

16.2255

- 7.6045

23.9789

2.5596

- 24.2500

4.2228

3.7717

2.5596

47.9424

- 39.3951

- 18.4355

2.2784

- 24.2500

- 39.3951

53.4020

Q = Q, +Qz+Qj L = L, +Lz+L,, where Q is the needed state weighting matrix. The closed-loop the desired eigenvalues is A, which is given by

matrix

that has all

A, = A+BL. The solution

of the example

is summarized

in Table I.

VI. Conclusion This paper presents a recursive method to choose the state weighting matrix of an optimal linear quadratic regulator. By choosing the weighting on the state the resulting optimal controller minimizes the performance index while the eigenvalues of the open-loop system are shifted to the desired locations. An attractive feature of this method is that it gives the freedom to shift the real and imaginary parts. Another feature is that each time only a pair of poles or one pole is shifted. Therefore in our computations we deal with either a second- or first-order model. Finally, the state weighting matrix that minimizes the performance index and shifts the eigenvalues of the system does not have to be positive semi-definite; as shown in the example, it can be indefinite.

References

(1) 0. A. Solhem, “Design of optimal control systems with prescribed (2) (3) (4) (5)

(6) (7) (8)

eigenvalues”, ht. J. Control, Vol. 15, No. 1, pp. 143.-160, 1972. D. Graupe, “Derivation of weighting matrices towards satisfying eigenvalues requirement”, ht. J. Control, Vol. 16, No. 5, pp. 881-888, 1972. J. Van De Vegte and M. C. Maki, “Optimization of systems with assigned poles”, ht. .I. Contro/,Vol. 18,No. 5, pp. 1105-1112, 1973. M. C. Maki and J. Van De Vegte, “Optimization of multiple-input systems with assigned poles”, IEEE Trans. Adorn. Control, pp. 130-l 33, April 1974. 0. A. Sebakhy and N. N. Sorial “Optimization of linear multivariable systems with prespecified closed-loop eigenvalues”, IEEE Trans. Autorn. Control, Vol. AC-24, No. 2, pp. 355-357, April 1979. M. H. Amin, “Optimal pole shifting for continuous multivariable linear systems”, In,?. J. Control, Vol. 41, No. 3, pp. 701-707, 1985. J. Medanic, H. S. Tharp and W. R. Perkins, “Pole placement by performance criterion modification”, IEEE Truns. Autom. Control, Vol. 33, No. 5, pp. 4699472, May 1988. P. K. Paul and M. E. Sawan, “Pole placement by performance criterion modification”, Proc. 32nd Midwest Symposium on Circuits and Systems, pp. 128-131, 1989.

Vol. 330. No. 5. pp. 9X5-994, Printed in Great Brilam

1993

993

A.-R.

S. Arar and M. E. Sawan

(9) N. Rousan and M. E. Sawan, “Optimal pole shifting for discrete multivariable systems”, ht. J. Syst. Sci., Vol. 23, No. 5, pp. 7999806, 1992. (10) Mehrdad Saif, “Optimal linear regulator pole-placement by weight selection”, ht. J. Control, Vol. 50, No. 1, pp. 3999414, 1989. (11) Clyde Martin, “Equivalence of quadratic performance criteria”, ht. J. Control, Vol. 17, No. 3, pp. 653-658, 1973. (12) S. Vittal Rao and S. S. Lamba, “Eigenvalue assignment in linear optimal-control systems via reduced-order models”, Proc. IEE, Vol. 122, No. 2, pp. 1977201, Feb. 1975. Received : 30 September 1992 Accepted : 10 February 1993

Journal

994

of the Franklin Institute Pergamon Press Ltd