1746

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

A Simple Solution to the Optimal Eigenvalue Assignment Problem

application of the proposed method in a real problem, an illustrative example from the power system control field is used.

D. P. Iracleous and A. T. Alexandridis

II. MAIN RESULTS Consider the time-invariant multi-input linear system

Abstract—The problem of the optimal eigenvalue assignment for multiinput linear systems is considered. It is proven that for an -order system with independent inputs, the problem is split into the following two sequential stages. Initially, the eigenvalues are assigned using an -order system. This assignment is not constrained to satisfy optimality criteria. Next, an -order system is used to assign the remaining eigenvalues in such a way that linear quadratic optimal criteria are simultaneously satisfied. Therefore, the original -order optimal eigenvalue assignment problem is reduced to an -order optimal assignment problem. Moreover, the structure of the equivalent -order system permits further simplifications which lead to solutions obtained by inspection.

m

n0m m

m

n

n0m

m

n m

Index Terms— Eigenvalue assignment, linear systems, multivariable systems, optimal control.

x _

=

Ax

+ Bu;

u

(1)

x0

=

(2)

Kx

we obtain the following closed-loop form: x _

=

where

Ac x;

Ac

=

A

+ BK:

(3)

Assume that (A; B ) is a completely controllable pair and B is of full rank, i.e., rank(B ) = m where m < n. Then, there always exists an n 2 (n 0 m) arbitrary constant matrix Bc such that the matrix [B Bc ] is invertible. Let S1

Manuscript received June 19, 1997. Recommended by Associate Editor, M. Dahleh. The authors are with the Department of Electrical and Computer Engineering, University of Patras, Rion 26500, Patras, Greece. Publisher Item Identifier S 0018-9286(99)06248-0.

=

where x 2 Rn ; u 2 Rm and A; B are constant matrices with dimensions n 2 n and n 2 m, respectively. Applying on system (1) the state feedback control law

I. INTRODUCTION The problem of designing a feedback gain matrix which satisfies eigenvalue assignment demands and optimal control criteria has received considerable attention, especially in the case of multiinput linear systems. The problem has been generally studied as a kind of inverse optimal control problem [1], [2]. Necessary and sufficient conditions [1] as well as methods and algorithms have been proposed [3]–[11]. The main idea of these methods is to calculate an appropriate state weighting matrix, for a given input weighting matrix, so that the resulting optimal linear quadratic (LQ) regulator assigns the closed-loop eigenspectrum of the system at some desired locations [3]. The existing methods result in the optimal solution by assigning simultaneously the closed-loop eigenvalues either in a prescribed region on the left complex plane [4]–[6] or exactly at preselected stable positions [7]–[11]. To determine the solution, recursive as well as nonrecursive methods have been developed. However, the main drawback of all these methods is complexity which may create computational problems, especially for large scale systems. In this paper, a very simple method for the optimal assignment of the closed-loop eigenvalues of a multi-input linear system is proposed. It is proven that for an n-order system with m independent inputs (n > m), the n-order optimal eigenvalue assignment problem can be reduced to an m-order optimal eigenvalue assignment problem where the remaining n 0 m eigenvalues are assigned by any common technique. This significantly simplifies the complexity of the problem. However, the structure of the equivalent m-order optimal control problem offers a great possibility of further simplifications. Among many possible solutions a simple diagonal gain-matrix, obtained by inspection, is proposed. Particularly, in the simplest case, the n 0 m eigenvalues are assigned exactly at any desired stable positions to provide the performance characteristics of the system while the m eigenvalues are assigned to ensure optimality. Practically, these m eigenvalues determine the nondominant poles of the system. To demonstrate the

x(0)

= [B

S2

Bc ]

01

(4)

where S1 and S2 are m 2 n and (n 0 m) 2 n constant matrices. Under these assumptions we next propose a different approach for the eigenvalue assignment by state feedback which leads to a simple optimal eigenvalue assignment procedure. A. Eigenvalue Assignment: An Alternative Procedure We start our approach by presenting the following theorem. Theorem 2.1: There exists a state feedback gain matrix K which assigns the entire set of the n eigenvalues of the closed-loop system (3) exactly at the same positions where: 1) the arbitrary m 2 (n 0 m) matrix L assigns the n 0 m eigenvalues of the matrix A11 + A12 L where A11 = S2 ABc and A12 = S2 AB and 2) the arbitrary m 2 m matrix F assigns the m eigenvalues of the matrix A22 + F where A22 = S1 AB 0 LS2 AB . Furthermore, this gain-matrix K is determined by the expression K = F S1 + GS2 where G = LA11 0 A21 0 A22 L 0 F L and A21 = S1 ABc . Proof: Let M be the n 2 n matrix M

=

0m

0

S2

L

Im

S1

In

0

=

S2

S1

0

(5)

LS2

where Im ; In0m are the m and (n 0 m)-identity matrices, respectively, and L is an arbitrary m 2 (n 0 m) constant matrix. Then, the inverse of M is M

01 = [Bc

B]

0m

In

L

0 Im

= [Bc + BL

B ]:

(6)

Transforming the closed-loop matrix Ac = A + BK by using the similarity transformation M Ac M 01 we arrive at M Ac M

01 =

S1 Ac [Bc

[S1

0

+ BL] + BL]

LS2 ]Ac [Bc

S2 Ac B

[S1

0

LS2 ]Ac B

:

(7)

After some simple algebraic manipulations and taking into account (4), which implies that S2 B = 0 and S1 B = Im , the last expression results in M Ac M

=

01

A21

A11 + A12 L + K Bc + K BL LA11 + A22 L

0018–9286/99$10.00 1999 IEEE

0

A12 A22

+ KB

(8)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

where A11 = S2 ABc ; A12 = S2 AB; A21 = S1 ABc ; A22 = S1 AB 0 LS2 AB . Defining, however, the matrices F and G as follows: F = KB

and G = KBc

(9)

where G is constrained by the equation G = LA11

0A 0A 21

22 L

0 F L:

(10)

The similarity transformation of Ac , given by (8), results in M Ac M

01 =

A11 + A12 L 0

A12 A22 + F

(11)

where from (9) we have by definition [F G] = K [B Bc ] and, therefore, the state feedback gain matrix K which satisfies (11) for arbitrary L; F; and G given by (10), is determined as follows: K = F S1 + GS2 :

(12)

Equation (11) shows that the feedback gain matrix K given by (12) assigns all the closed-loop eigenvalues of Ac (which is similar to M Ac M 01 ) at the n 0 m eigenvalues of A11 + A12 L and the m eigenvalues of A22 + F . Theorem 2.1 reveals that the assignment of the closed-loop eigenvalues of the system (3) can be achieved into two sequential stages. First stage: The n 0 m eigenvalues are assigned by selecting an appropriate matrix L. As indicated by the form of the matrix A11 + A12 L, this selection of L is equivalent to the solution of an n 0 m reduced-order state feedback eigenvalue-assignment problem. Therefore, the assignment of the n 0 m eigenvalues is possible if the pair (A11 ; A12 ) is a completely controllable pair [12]. This is true for system (1), in accordance with the following lemma [12]. Lemma 2.1: The pair (A11 ; A12 ) = (S2 ABc ; S2 AB ) is a completely controllable pair if and only if the pair (A; B ) is a completely controllable pair. Second stage: Using the L determined from the first stage, we construct the matrix A22 = S1 AB 0 LS2 AB . Then, the remaining m eigenvalues are assigned as the eigenvalues of A22 + F by selecting an appropriate matrix F . This is equivalent to the solution of an m reduced-order state feedback eigenvalue-assignment problem. However, since in this case the input matrix is the m-order identity matrix Im , the assignment of the m eigenvalues is always possible since it obviously holds that the pair (A22 ; Im ) is a completely controllable pair for any A22 [6]. Matrix G is then obtained from (10) with L and F determined in the two previous stages. Consequently, the gain-matrix K is determined from (12).

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in a simple procedure for the solution of the optimal eigenvalue assignment problem. Lemma 2.2: For the closed-loop system (3) the state feedback gain-matrix K is optimal and the corresponding AMRE has a positive definite solution for some symmetric state weighting matrix Q, providing that the input weighting matrix R > 0 is given, if and only if: 1) Re (A + BK ) < 0 and 2) the matrix 0RKB is a positive definite symmetric matrix. Proof—Necessity: If K is optimal, then it minimizes a quadratic performance index [14] of the form

1

1

J =

2

T

T

[x Qx + u Ru] dt =

0

1

1 2

T

T

x [Q + K RK ]x dt

0

(13) and there exists a constant symmetric matrix P > 0 such that the optimal K can be expressed as follows: 01 T K = 0R B P (14) where P satisfies the AMRE T T P A + A P 0 K RK + Q = 0:

(15)

However, since (13) is minimized by this K , then the second-order variation of J satisfies the inequality [14] 2

J =

1

1 2

T

T

[x (t)Qx(t) + u (t)Ru(t)] dt

0

0:

(16)

Completing the square in (16) (see [14] and [16]) we obtain 2

J =

1 2 +

T

x (0)P x(0)

1

1 2

0

[u(t)

0 21 xT (1)P x(1)

0 u3 (t)]T R[u(t) 0 u3 (t)] dt

(17)

where u3 (t) = 0R01 B T P x(t). For R > 0; P > 0 and for any arbitrary x(0), condition 2 J 0 yields that limt!1 x(t) = 0, i.e., the closed-loop matrix Ac is asymptotically stable. This can be proved by contradiction, i.e., if Ac was unstable, then there exists at least one unstable eigenvalue. Let u and vu be an unstable eigenvalue-eigenvector of Ac . Selecting x(0) = cvu (c is a scalar constant) we get x(t) = eA t x(0) = ceA t vu = ce (t) vu . Therefore, x(1) = limt!1 ce t vu = 1 and hence the term of (17) 12 xT (1)P x(1) ! +1 which yields 2 J = 01. Constructing, now, the matrix 0RKB with K given by (14), we have 0RKB = BT P B > 0 (18)

B. Eigenvalue Assignment by LQ Regulator

since P is symmetric positive definite. Sufficiency: If the matrix 0RKB is symmetric positive definite, then there always exists a suitable positive definite symmetric matrix Sn0m such that the matrix T S1 0RKB T 0RKBc S1 P = (19) S2 (0RKBc ) Sn0m S2

In view of the inverse optimal control problem, i.e., the problem of recognizing when a given state feedback gain-matrix minimizes 1 an LQ criterion J = 12 0 [xT Qx + uT Ru] dt [1], [2], we present Lemma 2.2. Generalizing some results presented in [1], Lemma 2.2 defines the conditions under which a state feedback gain-matrix K is optimal for a given input weighting matrix R. It provides a class of positive definite solutions of the corresponding algebraic matrix Riccati equation (AMRE) and consequently determines suitable state weighting matrices Q. In the next section, combining Lemma 2.2 with Theorem 2.1 we easily arrive at Theorem 2.2 which in turn results

is a symmetric positive definite constant matrix. Constructing the matrix 0R01 B T P by using P from (19) and T 01 T making use of B T [ S S ] = [Im 0] we confirm that 0R B P = S 0 1 0R [0RKB 0RKBc ][ S ] = K [BS1 + Bc S2 ] = K . Furthermore, for this P we can find a symmetric Q so that T T P Ac + Ac P + K RK + Q = 0: (20) 0 1 T Substituting Ac = A+BK = A0BR B P , we obtain the AMRE 01 T T (21) P A + A P 0 P BR B P + Q = 0:

1748

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

Assuming, now, that the closed-loop system is asymptotically stable, i.e., Re (Ac ) < 0, so that x(t) vanishes with time, it is verified from (17) that the second-order variation of J is greater than zero. T d Additionally, since xT (P Ac + ATc P )x = dt (x P x), the closed-loop performance index results in

J=

1

1 2

0 1

x (Q + K RK )x dt = 1 x T

T

T

2

(0)Px(0)

0 2 lim !1 x(t) P x(t) T

(22)

t

which converges to the positive optimal value

J = 1x

T

2

(0)Px(0):

the other hand, the optimality condition 2) of Lemma 2.2 is satisfied since 0RKB = 0RF = aR > 0 for any arbitrary R > 0. In practical applications, however, the n0m eigenvalues which can be assigned exactly in any desired stable positions are the dominant eigenvalues of the closed-loop system. The m eigenvalues which are manipulated to provide optimality and which are constrained to be located more to the left than the eigenvalues of A22 are obviously the nondominant closed-loop eigenvalues. Moreover, if one can exploit the degrees of freedom of the gainmatrix L, which assigns the first n 0 m eigenvalues, in such a way that the m eigenvalues of A22 = S1 AB 0 LS2 AB are in locations distance a from the desired locations, i.e.,

(A22 ) = desired (A

c)

Remark 2.1: As indicated by the sufficiency of Lemma 2.2, for given R and optimal K , an infinite number of suitable P can be obtained. Therefore, the same optimal K results in an infinite number of different Q. In the case where Q can be expressed as Q = DDT , then the pair (A; D) is completely observable since the following lemma [4] holds. Lemma 2.3: The pair (A; D) is completely observable if and only if P is positive definite. Now, to proceed with our approach we use Lemma 2.2 to establish the following theorem. Theorem 2.2: Let K be a state feedback gain matrix which assigns n stable eigenvalues of the closed-loop system (3) in accordance with Theorem 2.1. Then, this K is optimal for some Q with given R > 0, if and only if the feedback gain matrix F of the subsystem z_ = A22 z + u~, where u~ = F z , is optimal. Proof: Following the procedure concluded by Theorem 2.1, a K which assigns n stable eigenvalues of Ac is determined. Then, F = KB assigns m stable eigenvalues of the subsystem

z_ = A22 z + u~;

u~ = F z:

(23)

Now, if this K is optimal for (1), it holds true that 0RKB > 0 or is also optimal for (23), and vice versa. Theorem 2.2 clearly shows that the problem of the optimal assignment of n eigenvalues can be reduced to a problem of the optimal assignment of m eigenvalues while the remaining n 0 m eigenvalues are simply assigned in the desired locations by using any well-known technique [12], [13]. Therefore, many methods which result in exact optimal pole-placement [7]–[10], or in optimal pole-placement in a specified region as a whole [4]–[6], can be applied on the m reducedorder subsystem (23) instead of the original nth-order system. This clearly minimizes the computational effort. Furthermore, the structure of the m reduced-order subsystem (23) leads to more simple optimal solutions as explained in the following.

0RF > 0, and from Lemma 2.2 this implies that F

D. A Simple Design Algorithm The procedure described above leads to the following algorithm: Initialize: (i) System data A; B ; input weighting matrix R; (ii) Select Bc ; calculate S1 ; S2 from (4); calculate A11 ; A12 ; A21 as defined in Theorem 2.1. Step 1: Derive L, (use a pole-placement technique to assign n 0 m eigenvalues of the subsystem (A11 ; A12 )). Step 2: Calculate A22 = S1 AB 0 LS2 AB and find its eigenvalues. Step 3: Determine F = 0aIm (a is selected so that the poles of A22 + F lie in desired locations); determine G from (10). Step 4: Determine the optimal gain-matrix K from (12); calculate the solution P of the AMRE from (19); calculate the weighting matrix Q from either (20) or (21). Terminate algorithm. III. ILLUSTRATIVE EXAMPLE The linear dynamical model of a two area power system, taken from [15], is used to demonstrate the application of the proposed method. The system is a seventh-order system with two inputs. The state vector is x = [f1 Pt1 Ptie f2 Pt2 Pu1 Pu2 ]T where f1 ; f2 are the frequencies, Pt1 ; Pt2 are the turbine powers, Pu1 ; Pu2 are the governor states at area 1 and 2 correspondingly, and Ptie is the transferred power between the two areas. The system matrices are

00:05

A=

0aI ; m

a>0

(24)

where the scalar a is suitably selected to ensure stability and assignment of the eigenvalues in a desired region. The m eigenvalues of A22 + F which are also the m closed-loop eigenvalues of Ac are, then, located at

(A22 + F ) = (A22 ) 0

0 0 0:45 0 0:521 0

(25)

i.e., more to the left on the complex plane than the eigenvalues of A22 = S1 AB 0 LS2 AB which have been determined by L. On

B=

0 0 0 0 0 12:5 0

6

03:33

0

In this section a further simple solution is proposed. Particularly, since the input matrix of (23) is the identity matrix Im the feedback gain-matrix F is an m 2 m square matrix. Therefore, we can consider =

(26)

then, exact optimal pole-placement can be achieved since (25) and (26) obviously lead to desired (Ac ) = (A22 + F ).

C. A Simple Optimal Eigenvalue Assignment Solution

F

+

0 0 0 0 0 0 12:5

0 0 0 0 0

0 0 0:05 0:545 0 0 5:21

0 0 0

06 0 6 0 0 0 0

0

0 0 6 0 3:33 0 0

0 3:33 0 0 0 12:5 0

0

:

0 0 0 0 3:33 0 12:5

0

(27)

We select Bc = [ 12:05I ], where I5 is the fifth-order identity matrix. In this case we have n = 7; m = 2. The open-loop eigenvalues are

00:553 6 3:134i; 00:870 6 1:372i; 03:048; 012:588; 013:277:

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

0:9081

P

=

00:8634 01:2649 00:3567 00:9360 00:0292 00:0359

1749

00:8634 01:2649 00:3567 00:9360 00:0292 00:0359 5:1361 1:4990 04:6602 02:1411 00:1177 00:2131 1:4990 1:9964 00:1389 1:1989 0:0599 0:0347 04:6602 00:1389 14:0859 6:7031 0:1155 0:5354 02:1411 1:1989 6:7031 6:1236 0:1135 0:2256 00:1177 0:0599 0:1155 0:1135 0:1600 0 00:2131 0:0347 0:5354 0:2256 0 0:1600 (31)

We select the input weighting matrix R to be the 2 2 2 identity matrix. To assign the five (=n 0 m) closed-loop eigenvalues of Ac at the values: f00:7 6 1:5i; 01:5 6 0:2i; 04g, we determine the gain-matrix L of the subsystem (A11 ; A12 ) as follows:

L=

0:1164 0:1840

0:5375 0:9899

00:2593 00:4996 00:4973 00:3245 02:3176 01:0300 :

(28)

:29 1:66 Then, A22 = [ 0014 3:30 09:07 ] has the following eigenvalues: 010:517 and 012:843. In order to assign the nondominant eigenvalues of Ac more to the left on the complex plane, i.e., 025 from the existing values, we select a = 25 and, therefore, the F which assigns the remaining two eigenvalues at the desired positions 035.5 and 037.8 is

F

=

025I2 :

(29)

Now, matrix G is determined from (10) and the optimal state feedback gain matrix is determined from (12) as

K =

0:3648 0:4481

1:4710 2:6643

00:7491 01:4437 01:4186 02 0 00:4338 06:6927 02:8199 0 02 (30)

which assigns the closed-loop eigenvalues exactly at the desired locations. We note that a solution P > 0 of the AMRE can be calculated from (19) if one selects a suitable matrix Sn0m . Such a P is shown in (31), at the top of the page.

IV. CONCLUSION The problem of the optimal eigenvalue assignment of linear multiinput systems has been solved into two major steps. The first step assigns n 0 m arbitrary stable eigenvalues determining the feedback gain-matrix L of the system matrix A11 + A12 L (Theorem 2.1). The second step assigns the m remaining stable eigenvalues, using F given from (24), at the points (A22 ) 0 . Then, the feedback gain matrix K is determined from (10) and (12). As shown in Theorem 2.2, this solution meets optimal LQ criteria since it satisfies the conditions for optimality established in Lemma 2.2. REFERENCES [1] B. P. Molinari, “The stable regulator problem and its inverse,” IEEE Trans. Automat. Contr., vol. AC-18, pp. 454–459, 1973. [2] A. Jameson and E. Kreindler, “Inverse problem of linear optimal control,” SIAM J. Contr., vol. 11, no. 1, pp. 1–19, 1973.

[3] M. J. Grimble and M. A. Johnson, Optimal Control and Stochastic Estimation: Theory and Applications, vol. 1. New York: Wiley, 1988, ch. 4. [4] B. D. O. Anderson and J. B. Moore, Linear Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1971, ch. 4. [5] K. Furuta and S. B. Kim, “Pole assignment in a specified disc,” IEEE Trans. Automat. Contr., vol. 32, pp. 423–427, May 1987. [6] W. M. Haddad and D. S. Bernstein, “Controller design with regional pole constraints,” IEEE Trans. Automat. Contr., vol. 37, pp. 54–69, 1992. [7] M. C. Maki and J. Van de Vegte, “Optimization of multi-input systems with assigned poles,” IEEE Trans. Automat. Contr., vol. 19, pp. 130–133, 1974. [8] Y. B. Ness, “Optimal closed-loop pole assignment,” Int. J. Contr., vol. 27, no. 3, pp. 421–430, 1978. [9] A. T. Alexandridis and G. D. Galanos, “Optimal pole-placement for linear multi-input controllable systems,” IEEE Trans. Circuits Syst., vol. CAS-34, pp. 1602–1604, 1987. [10] M. H. Amin, “Optimal pole-shifting for continuous multivariable linear systems,” Int. J. Contr., vol. 41, pp. 701–707, 1985. [11] A. T. Alexandridis, “Optimal entire eigenstructure assignment of discrete-time linear systems,” IEE Proc.—Control Theory Appl., vol. 143, no. 3, pp. 301–304, 1996. [12] C. T. Chen, Linear System Theory and Design. New York: Holt, Rinehart and Wiston, 1984. [13] J. J. D’Azzo and C. H. Houpis, Linear Control System Analysis and Design. New York: McGraw-Hill, 1988. [14] F. L. Lewis and V. L. Syrmos, Optimal Control. New York: Wiley, 1995. [15] A. K. Mahalanabis, D. P. Kothari, and S. I. Ahson, Computer Aided Power Systems Analysis and Control. New York: McGraw-Hill, 1988, ch. 5. [16] M. Green and D. Limebeer, Linear Robust Control. Englewood Cliffs, NJ: Prentice-Hall, 1995.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

A Simple Solution to the Optimal Eigenvalue Assignment Problem

application of the proposed method in a real problem, an illustrative example from the power system control field is used.

D. P. Iracleous and A. T. Alexandridis

II. MAIN RESULTS Consider the time-invariant multi-input linear system

Abstract—The problem of the optimal eigenvalue assignment for multiinput linear systems is considered. It is proven that for an -order system with independent inputs, the problem is split into the following two sequential stages. Initially, the eigenvalues are assigned using an -order system. This assignment is not constrained to satisfy optimality criteria. Next, an -order system is used to assign the remaining eigenvalues in such a way that linear quadratic optimal criteria are simultaneously satisfied. Therefore, the original -order optimal eigenvalue assignment problem is reduced to an -order optimal assignment problem. Moreover, the structure of the equivalent -order system permits further simplifications which lead to solutions obtained by inspection.

m

n0m m

m

n

n0m

m

n m

Index Terms— Eigenvalue assignment, linear systems, multivariable systems, optimal control.

x _

=

Ax

+ Bu;

u

(1)

x0

=

(2)

Kx

we obtain the following closed-loop form: x _

=

where

Ac x;

Ac

=

A

+ BK:

(3)

Assume that (A; B ) is a completely controllable pair and B is of full rank, i.e., rank(B ) = m where m < n. Then, there always exists an n 2 (n 0 m) arbitrary constant matrix Bc such that the matrix [B Bc ] is invertible. Let S1

Manuscript received June 19, 1997. Recommended by Associate Editor, M. Dahleh. The authors are with the Department of Electrical and Computer Engineering, University of Patras, Rion 26500, Patras, Greece. Publisher Item Identifier S 0018-9286(99)06248-0.

=

where x 2 Rn ; u 2 Rm and A; B are constant matrices with dimensions n 2 n and n 2 m, respectively. Applying on system (1) the state feedback control law

I. INTRODUCTION The problem of designing a feedback gain matrix which satisfies eigenvalue assignment demands and optimal control criteria has received considerable attention, especially in the case of multiinput linear systems. The problem has been generally studied as a kind of inverse optimal control problem [1], [2]. Necessary and sufficient conditions [1] as well as methods and algorithms have been proposed [3]–[11]. The main idea of these methods is to calculate an appropriate state weighting matrix, for a given input weighting matrix, so that the resulting optimal linear quadratic (LQ) regulator assigns the closed-loop eigenspectrum of the system at some desired locations [3]. The existing methods result in the optimal solution by assigning simultaneously the closed-loop eigenvalues either in a prescribed region on the left complex plane [4]–[6] or exactly at preselected stable positions [7]–[11]. To determine the solution, recursive as well as nonrecursive methods have been developed. However, the main drawback of all these methods is complexity which may create computational problems, especially for large scale systems. In this paper, a very simple method for the optimal assignment of the closed-loop eigenvalues of a multi-input linear system is proposed. It is proven that for an n-order system with m independent inputs (n > m), the n-order optimal eigenvalue assignment problem can be reduced to an m-order optimal eigenvalue assignment problem where the remaining n 0 m eigenvalues are assigned by any common technique. This significantly simplifies the complexity of the problem. However, the structure of the equivalent m-order optimal control problem offers a great possibility of further simplifications. Among many possible solutions a simple diagonal gain-matrix, obtained by inspection, is proposed. Particularly, in the simplest case, the n 0 m eigenvalues are assigned exactly at any desired stable positions to provide the performance characteristics of the system while the m eigenvalues are assigned to ensure optimality. Practically, these m eigenvalues determine the nondominant poles of the system. To demonstrate the

x(0)

= [B

S2

Bc ]

01

(4)

where S1 and S2 are m 2 n and (n 0 m) 2 n constant matrices. Under these assumptions we next propose a different approach for the eigenvalue assignment by state feedback which leads to a simple optimal eigenvalue assignment procedure. A. Eigenvalue Assignment: An Alternative Procedure We start our approach by presenting the following theorem. Theorem 2.1: There exists a state feedback gain matrix K which assigns the entire set of the n eigenvalues of the closed-loop system (3) exactly at the same positions where: 1) the arbitrary m 2 (n 0 m) matrix L assigns the n 0 m eigenvalues of the matrix A11 + A12 L where A11 = S2 ABc and A12 = S2 AB and 2) the arbitrary m 2 m matrix F assigns the m eigenvalues of the matrix A22 + F where A22 = S1 AB 0 LS2 AB . Furthermore, this gain-matrix K is determined by the expression K = F S1 + GS2 where G = LA11 0 A21 0 A22 L 0 F L and A21 = S1 ABc . Proof: Let M be the n 2 n matrix M

=

0m

0

S2

L

Im

S1

In

0

=

S2

S1

0

(5)

LS2

where Im ; In0m are the m and (n 0 m)-identity matrices, respectively, and L is an arbitrary m 2 (n 0 m) constant matrix. Then, the inverse of M is M

01 = [Bc

B]

0m

In

L

0 Im

= [Bc + BL

B ]:

(6)

Transforming the closed-loop matrix Ac = A + BK by using the similarity transformation M Ac M 01 we arrive at M Ac M

01 =

S1 Ac [Bc

[S1

0

+ BL] + BL]

LS2 ]Ac [Bc

S2 Ac B

[S1

0

LS2 ]Ac B

:

(7)

After some simple algebraic manipulations and taking into account (4), which implies that S2 B = 0 and S1 B = Im , the last expression results in M Ac M

=

01

A21

A11 + A12 L + K Bc + K BL LA11 + A22 L

0018–9286/99$10.00 1999 IEEE

0

A12 A22

+ KB

(8)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

where A11 = S2 ABc ; A12 = S2 AB; A21 = S1 ABc ; A22 = S1 AB 0 LS2 AB . Defining, however, the matrices F and G as follows: F = KB

and G = KBc

(9)

where G is constrained by the equation G = LA11

0A 0A 21

22 L

0 F L:

(10)

The similarity transformation of Ac , given by (8), results in M Ac M

01 =

A11 + A12 L 0

A12 A22 + F

(11)

where from (9) we have by definition [F G] = K [B Bc ] and, therefore, the state feedback gain matrix K which satisfies (11) for arbitrary L; F; and G given by (10), is determined as follows: K = F S1 + GS2 :

(12)

Equation (11) shows that the feedback gain matrix K given by (12) assigns all the closed-loop eigenvalues of Ac (which is similar to M Ac M 01 ) at the n 0 m eigenvalues of A11 + A12 L and the m eigenvalues of A22 + F . Theorem 2.1 reveals that the assignment of the closed-loop eigenvalues of the system (3) can be achieved into two sequential stages. First stage: The n 0 m eigenvalues are assigned by selecting an appropriate matrix L. As indicated by the form of the matrix A11 + A12 L, this selection of L is equivalent to the solution of an n 0 m reduced-order state feedback eigenvalue-assignment problem. Therefore, the assignment of the n 0 m eigenvalues is possible if the pair (A11 ; A12 ) is a completely controllable pair [12]. This is true for system (1), in accordance with the following lemma [12]. Lemma 2.1: The pair (A11 ; A12 ) = (S2 ABc ; S2 AB ) is a completely controllable pair if and only if the pair (A; B ) is a completely controllable pair. Second stage: Using the L determined from the first stage, we construct the matrix A22 = S1 AB 0 LS2 AB . Then, the remaining m eigenvalues are assigned as the eigenvalues of A22 + F by selecting an appropriate matrix F . This is equivalent to the solution of an m reduced-order state feedback eigenvalue-assignment problem. However, since in this case the input matrix is the m-order identity matrix Im , the assignment of the m eigenvalues is always possible since it obviously holds that the pair (A22 ; Im ) is a completely controllable pair for any A22 [6]. Matrix G is then obtained from (10) with L and F determined in the two previous stages. Consequently, the gain-matrix K is determined from (12).

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in a simple procedure for the solution of the optimal eigenvalue assignment problem. Lemma 2.2: For the closed-loop system (3) the state feedback gain-matrix K is optimal and the corresponding AMRE has a positive definite solution for some symmetric state weighting matrix Q, providing that the input weighting matrix R > 0 is given, if and only if: 1) Re (A + BK ) < 0 and 2) the matrix 0RKB is a positive definite symmetric matrix. Proof—Necessity: If K is optimal, then it minimizes a quadratic performance index [14] of the form

1

1

J =

2

T

T

[x Qx + u Ru] dt =

0

1

1 2

T

T

x [Q + K RK ]x dt

0

(13) and there exists a constant symmetric matrix P > 0 such that the optimal K can be expressed as follows: 01 T K = 0R B P (14) where P satisfies the AMRE T T P A + A P 0 K RK + Q = 0:

(15)

However, since (13) is minimized by this K , then the second-order variation of J satisfies the inequality [14] 2

J =

1

1 2

T

T

[x (t)Qx(t) + u (t)Ru(t)] dt

0

0:

(16)

Completing the square in (16) (see [14] and [16]) we obtain 2

J =

1 2 +

T

x (0)P x(0)

1

1 2

0

[u(t)

0 21 xT (1)P x(1)

0 u3 (t)]T R[u(t) 0 u3 (t)] dt

(17)

where u3 (t) = 0R01 B T P x(t). For R > 0; P > 0 and for any arbitrary x(0), condition 2 J 0 yields that limt!1 x(t) = 0, i.e., the closed-loop matrix Ac is asymptotically stable. This can be proved by contradiction, i.e., if Ac was unstable, then there exists at least one unstable eigenvalue. Let u and vu be an unstable eigenvalue-eigenvector of Ac . Selecting x(0) = cvu (c is a scalar constant) we get x(t) = eA t x(0) = ceA t vu = ce (t) vu . Therefore, x(1) = limt!1 ce t vu = 1 and hence the term of (17) 12 xT (1)P x(1) ! +1 which yields 2 J = 01. Constructing, now, the matrix 0RKB with K given by (14), we have 0RKB = BT P B > 0 (18)

B. Eigenvalue Assignment by LQ Regulator

since P is symmetric positive definite. Sufficiency: If the matrix 0RKB is symmetric positive definite, then there always exists a suitable positive definite symmetric matrix Sn0m such that the matrix T S1 0RKB T 0RKBc S1 P = (19) S2 (0RKBc ) Sn0m S2

In view of the inverse optimal control problem, i.e., the problem of recognizing when a given state feedback gain-matrix minimizes 1 an LQ criterion J = 12 0 [xT Qx + uT Ru] dt [1], [2], we present Lemma 2.2. Generalizing some results presented in [1], Lemma 2.2 defines the conditions under which a state feedback gain-matrix K is optimal for a given input weighting matrix R. It provides a class of positive definite solutions of the corresponding algebraic matrix Riccati equation (AMRE) and consequently determines suitable state weighting matrices Q. In the next section, combining Lemma 2.2 with Theorem 2.1 we easily arrive at Theorem 2.2 which in turn results

is a symmetric positive definite constant matrix. Constructing the matrix 0R01 B T P by using P from (19) and T 01 T making use of B T [ S S ] = [Im 0] we confirm that 0R B P = S 0 1 0R [0RKB 0RKBc ][ S ] = K [BS1 + Bc S2 ] = K . Furthermore, for this P we can find a symmetric Q so that T T P Ac + Ac P + K RK + Q = 0: (20) 0 1 T Substituting Ac = A+BK = A0BR B P , we obtain the AMRE 01 T T (21) P A + A P 0 P BR B P + Q = 0:

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

Assuming, now, that the closed-loop system is asymptotically stable, i.e., Re (Ac ) < 0, so that x(t) vanishes with time, it is verified from (17) that the second-order variation of J is greater than zero. T d Additionally, since xT (P Ac + ATc P )x = dt (x P x), the closed-loop performance index results in

J=

1

1 2

0 1

x (Q + K RK )x dt = 1 x T

T

T

2

(0)Px(0)

0 2 lim !1 x(t) P x(t) T

(22)

t

which converges to the positive optimal value

J = 1x

T

2

(0)Px(0):

the other hand, the optimality condition 2) of Lemma 2.2 is satisfied since 0RKB = 0RF = aR > 0 for any arbitrary R > 0. In practical applications, however, the n0m eigenvalues which can be assigned exactly in any desired stable positions are the dominant eigenvalues of the closed-loop system. The m eigenvalues which are manipulated to provide optimality and which are constrained to be located more to the left than the eigenvalues of A22 are obviously the nondominant closed-loop eigenvalues. Moreover, if one can exploit the degrees of freedom of the gainmatrix L, which assigns the first n 0 m eigenvalues, in such a way that the m eigenvalues of A22 = S1 AB 0 LS2 AB are in locations distance a from the desired locations, i.e.,

(A22 ) = desired (A

c)

Remark 2.1: As indicated by the sufficiency of Lemma 2.2, for given R and optimal K , an infinite number of suitable P can be obtained. Therefore, the same optimal K results in an infinite number of different Q. In the case where Q can be expressed as Q = DDT , then the pair (A; D) is completely observable since the following lemma [4] holds. Lemma 2.3: The pair (A; D) is completely observable if and only if P is positive definite. Now, to proceed with our approach we use Lemma 2.2 to establish the following theorem. Theorem 2.2: Let K be a state feedback gain matrix which assigns n stable eigenvalues of the closed-loop system (3) in accordance with Theorem 2.1. Then, this K is optimal for some Q with given R > 0, if and only if the feedback gain matrix F of the subsystem z_ = A22 z + u~, where u~ = F z , is optimal. Proof: Following the procedure concluded by Theorem 2.1, a K which assigns n stable eigenvalues of Ac is determined. Then, F = KB assigns m stable eigenvalues of the subsystem

z_ = A22 z + u~;

u~ = F z:

(23)

Now, if this K is optimal for (1), it holds true that 0RKB > 0 or is also optimal for (23), and vice versa. Theorem 2.2 clearly shows that the problem of the optimal assignment of n eigenvalues can be reduced to a problem of the optimal assignment of m eigenvalues while the remaining n 0 m eigenvalues are simply assigned in the desired locations by using any well-known technique [12], [13]. Therefore, many methods which result in exact optimal pole-placement [7]–[10], or in optimal pole-placement in a specified region as a whole [4]–[6], can be applied on the m reducedorder subsystem (23) instead of the original nth-order system. This clearly minimizes the computational effort. Furthermore, the structure of the m reduced-order subsystem (23) leads to more simple optimal solutions as explained in the following.

0RF > 0, and from Lemma 2.2 this implies that F

D. A Simple Design Algorithm The procedure described above leads to the following algorithm: Initialize: (i) System data A; B ; input weighting matrix R; (ii) Select Bc ; calculate S1 ; S2 from (4); calculate A11 ; A12 ; A21 as defined in Theorem 2.1. Step 1: Derive L, (use a pole-placement technique to assign n 0 m eigenvalues of the subsystem (A11 ; A12 )). Step 2: Calculate A22 = S1 AB 0 LS2 AB and find its eigenvalues. Step 3: Determine F = 0aIm (a is selected so that the poles of A22 + F lie in desired locations); determine G from (10). Step 4: Determine the optimal gain-matrix K from (12); calculate the solution P of the AMRE from (19); calculate the weighting matrix Q from either (20) or (21). Terminate algorithm. III. ILLUSTRATIVE EXAMPLE The linear dynamical model of a two area power system, taken from [15], is used to demonstrate the application of the proposed method. The system is a seventh-order system with two inputs. The state vector is x = [f1 Pt1 Ptie f2 Pt2 Pu1 Pu2 ]T where f1 ; f2 are the frequencies, Pt1 ; Pt2 are the turbine powers, Pu1 ; Pu2 are the governor states at area 1 and 2 correspondingly, and Ptie is the transferred power between the two areas. The system matrices are

00:05

A=

0aI ; m

a>0

(24)

where the scalar a is suitably selected to ensure stability and assignment of the eigenvalues in a desired region. The m eigenvalues of A22 + F which are also the m closed-loop eigenvalues of Ac are, then, located at

(A22 + F ) = (A22 ) 0

0 0 0:45 0 0:521 0

(25)

i.e., more to the left on the complex plane than the eigenvalues of A22 = S1 AB 0 LS2 AB which have been determined by L. On

B=

0 0 0 0 0 12:5 0

6

03:33

0

In this section a further simple solution is proposed. Particularly, since the input matrix of (23) is the identity matrix Im the feedback gain-matrix F is an m 2 m square matrix. Therefore, we can consider =

(26)

then, exact optimal pole-placement can be achieved since (25) and (26) obviously lead to desired (Ac ) = (A22 + F ).

C. A Simple Optimal Eigenvalue Assignment Solution

F

+

0 0 0 0 0 0 12:5

0 0 0 0 0

0 0 0:05 0:545 0 0 5:21

0 0 0

06 0 6 0 0 0 0

0

0 0 6 0 3:33 0 0

0 3:33 0 0 0 12:5 0

0

:

0 0 0 0 3:33 0 12:5

0

(27)

We select Bc = [ 12:05I ], where I5 is the fifth-order identity matrix. In this case we have n = 7; m = 2. The open-loop eigenvalues are

00:553 6 3:134i; 00:870 6 1:372i; 03:048; 012:588; 013:277:

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 44, NO. 9, SEPTEMBER 1999

0:9081

P

=

00:8634 01:2649 00:3567 00:9360 00:0292 00:0359

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00:8634 01:2649 00:3567 00:9360 00:0292 00:0359 5:1361 1:4990 04:6602 02:1411 00:1177 00:2131 1:4990 1:9964 00:1389 1:1989 0:0599 0:0347 04:6602 00:1389 14:0859 6:7031 0:1155 0:5354 02:1411 1:1989 6:7031 6:1236 0:1135 0:2256 00:1177 0:0599 0:1155 0:1135 0:1600 0 00:2131 0:0347 0:5354 0:2256 0 0:1600 (31)

We select the input weighting matrix R to be the 2 2 2 identity matrix. To assign the five (=n 0 m) closed-loop eigenvalues of Ac at the values: f00:7 6 1:5i; 01:5 6 0:2i; 04g, we determine the gain-matrix L of the subsystem (A11 ; A12 ) as follows:

L=

0:1164 0:1840

0:5375 0:9899

00:2593 00:4996 00:4973 00:3245 02:3176 01:0300 :

(28)

:29 1:66 Then, A22 = [ 0014 3:30 09:07 ] has the following eigenvalues: 010:517 and 012:843. In order to assign the nondominant eigenvalues of Ac more to the left on the complex plane, i.e., 025 from the existing values, we select a = 25 and, therefore, the F which assigns the remaining two eigenvalues at the desired positions 035.5 and 037.8 is

F

=

025I2 :

(29)

Now, matrix G is determined from (10) and the optimal state feedback gain matrix is determined from (12) as

K =

0:3648 0:4481

1:4710 2:6643

00:7491 01:4437 01:4186 02 0 00:4338 06:6927 02:8199 0 02 (30)

which assigns the closed-loop eigenvalues exactly at the desired locations. We note that a solution P > 0 of the AMRE can be calculated from (19) if one selects a suitable matrix Sn0m . Such a P is shown in (31), at the top of the page.

IV. CONCLUSION The problem of the optimal eigenvalue assignment of linear multiinput systems has been solved into two major steps. The first step assigns n 0 m arbitrary stable eigenvalues determining the feedback gain-matrix L of the system matrix A11 + A12 L (Theorem 2.1). The second step assigns the m remaining stable eigenvalues, using F given from (24), at the points (A22 ) 0 . Then, the feedback gain matrix K is determined from (10) and (12). As shown in Theorem 2.2, this solution meets optimal LQ criteria since it satisfies the conditions for optimality established in Lemma 2.2. REFERENCES [1] B. P. Molinari, “The stable regulator problem and its inverse,” IEEE Trans. Automat. Contr., vol. AC-18, pp. 454–459, 1973. [2] A. Jameson and E. Kreindler, “Inverse problem of linear optimal control,” SIAM J. Contr., vol. 11, no. 1, pp. 1–19, 1973.

[3] M. J. Grimble and M. A. Johnson, Optimal Control and Stochastic Estimation: Theory and Applications, vol. 1. New York: Wiley, 1988, ch. 4. [4] B. D. O. Anderson and J. B. Moore, Linear Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1971, ch. 4. [5] K. Furuta and S. B. Kim, “Pole assignment in a specified disc,” IEEE Trans. Automat. Contr., vol. 32, pp. 423–427, May 1987. [6] W. M. Haddad and D. S. Bernstein, “Controller design with regional pole constraints,” IEEE Trans. Automat. Contr., vol. 37, pp. 54–69, 1992. [7] M. C. Maki and J. Van de Vegte, “Optimization of multi-input systems with assigned poles,” IEEE Trans. Automat. Contr., vol. 19, pp. 130–133, 1974. [8] Y. B. Ness, “Optimal closed-loop pole assignment,” Int. J. Contr., vol. 27, no. 3, pp. 421–430, 1978. [9] A. T. Alexandridis and G. D. Galanos, “Optimal pole-placement for linear multi-input controllable systems,” IEEE Trans. Circuits Syst., vol. CAS-34, pp. 1602–1604, 1987. [10] M. H. Amin, “Optimal pole-shifting for continuous multivariable linear systems,” Int. J. Contr., vol. 41, pp. 701–707, 1985. [11] A. T. Alexandridis, “Optimal entire eigenstructure assignment of discrete-time linear systems,” IEE Proc.—Control Theory Appl., vol. 143, no. 3, pp. 301–304, 1996. [12] C. T. Chen, Linear System Theory and Design. New York: Holt, Rinehart and Wiston, 1984. [13] J. J. D’Azzo and C. H. Houpis, Linear Control System Analysis and Design. New York: McGraw-Hill, 1988. [14] F. L. Lewis and V. L. Syrmos, Optimal Control. New York: Wiley, 1995. [15] A. K. Mahalanabis, D. P. Kothari, and S. I. Ahson, Computer Aided Power Systems Analysis and Control. New York: McGraw-Hill, 1988, ch. 5. [16] M. Green and D. Limebeer, Linear Robust Control. Englewood Cliffs, NJ: Prentice-Hall, 1995.