Abstract: - In this paper, inverse optimal control theory is used to design a state or an output feedback controller that both places the closed loop poles at desired ...
Simultaneous pole placement and optimal state or output feedback controller design D. P. Iracleous, N.E. Mastorakis Department of Computer Science Military Institutions of University Education Hellenic Naval Academy 18539,Terma Hatzikyriakou Piraeus, GREECE
Abstract: - In this paper, inverse optimal control theory is used to design a state or an output feedback controller that both places the closed loop poles at desired positions and satisfies a linear quadratic criterion. The design method consists of two steps, first n-m poles are precisely placed and second the remaining m poles are selected to satisfy optimality criteria, where n is the number of the system states and m is the number of system inputs. An illustrative example demonstrates the proposed algorithm. Key-Words: - Inverse optimal control, pole placement techniques, controller design, state, output feedback.
1 Introduction The problem of the state or output feedback design for continuous-time linear is one of the major problems in control theory and practice. Numerous methodologies and algorithms have been proposed [5]. The main goal of these methodologies is to stabilize the closed loop system. More goals are usually required, i.e. pole placement, eigenstructure assignment, optimal design, decoupling, disturbance rejection, robustness etc. In [6] the necessary and sufficient conditions were given that a given state feedback law is optimal. In this paper we extend the theorem to cover the case of output feedback control law. Furthermore, the twostage state feedback design methodology for continuous time systems [6] or for discrete time systems [7] is analyzed and suitably modified for output feedback controller. In the paper we will consider square n-order systems linear time invariant systems, i.e. the number of the independent inputs is equal to the number of independent outputs. If a state feedback controller is designed, (n-m) poles are arbitrary placed while the m eigenvalues are assigned to ensure optimality. Practically these m eigenvalues determine the non-dominant poles of the system. In the case of output feedback the (n-m) poles are restricted to certain areas to ensure the overall stability of the system.
To demonstrate the application of the proposed method in a real problem, an illustrative example is used.
2 Problem Formulation Consider the completely controllable and observable linear dynamical system x = Ax + Bu, y = Cx (1) where x ∈ R n , u ∈ R m , y ∈ R r and A, B, and C are constant matrices of appropriate dimensions with rank(B)=m and rank(C) = m. Proportional state feedback control law
u = Ksx
(2a)
or output feedback control law
u = Ko y
(2b)
is applied to the system (1). Thus, it takes on the following closed-loop form
x = Acl x, where Acl = A + BK s = A + BK o C . and K s ≡ K o C
(3)
(4) The feedback control law is applied to change the characteristics of the system.
Assume that (A, B) is a completely controllable pair and B is of full rank, i.e. rank(B) = m where m < n. In this paper, for simplicity sake, we can use
and
I B = m 0
(5a)
C = [C1 0]
(5b)
If this is not the case we can use suitable similarity transformation [8] to the system (1) so that it has the expected form.
2.1
Pole-placement regulator
providing that the input weighting matrix R > 0 is given, if and only if: (9) i) Reλi ( A+BK o C ) < 0 and (10) ii) − RK o C1 > 0 (positive definite symmetric matrix) Proof: Using lemma 2.2 in [5] and eq. (4), (5), then (9) and (10) are straightforward consequences. In this case matrix solution P of AMRE is given by
− RK 0C1 P= 0
0 S n − m
(11)
The gain matrix K that places the closed loop poles at desired positions is related to the eigenvector matrix V through the well-known Sylvester equation AV + BΨ = VΛ (6) where the mxn matrix Ψ is a free parameter matrix, Λ is the desired spectrum and V is the obtained eigenvector matrix. The gain matrix is given by the solution of Ψ = K sV .
where Sn-m any suitable positive definite symmetric matrix such that the matrix P is a symmetric positive definite constant matrix.
In the case of output feedback the closed loop system is similar to the following form [3] X Λ m (7) 0 A − LA 22 12
3 Optimal pole placement design
where L = C U r [CU r ] defined in [3] eq. 10. −1
2.2 Optimal output regulator In view of the inverse optimal control problem, i.e. the problem of recognizing when a given state feedback gain-matrix minimizes a LQ criterion [1,4]
Furthermore, for this P we can find a symmetric Q so that the AMRE (12) P Ac + AcT P +KTRK + Q = 0. is satisfied.
In the following the problem of the optimal assignment of n eigenvalues is reduced to a problem of the simple assignment n-m eigenvalues in the desired locations by using any well-known technique and then, the optimal assignment of remaining m eigenvalues. To present the proposed algorithm we state again the closed loop system matrix Acl = A + BK oC
A1 A3
=
∞
J=
1 [ x T Qx + u T Ru ]dt ∫ 20
(8)
And the optimal state regulator [6], we present lemma 1. Generalizing some results presented in [1], lemma 1 defines the conditions under which an output feedback gain-matrix Ko 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. Lemma 1.: For the closed-loop system (3) the output feedback gain-matrix Ko is optimal and the corresponding AMRE has a positive definite solution for some symmetric state weighting matrix Q,
A2 I K [C 0] + A4 0 o 1
A1 + K o C1
=
A3
A2 A4
(13)
Further similarity transformation is used, namely
I 0 −1 I 0 T = , T = − L I L I
(14)
giving
TAcl T −1 =
A1 + K o C1 − A2 L LA + LK C + A − LA L − A L o 1 3 2 4 1
A2 A4 + LA2 (15)
If a suitable output feedback controller is used so that (16) LKoC= -( LA1 + A3 − LA2 L − A4 L ). then, the poles of the closed loop system are the poles of A4+LA2 and A1+KoC-A2L. Giving a specific value to L we place m poles at almost arbitrary stable positions independently. Then, we can select a Ko so that a) satisfies eq. (10), b) satisfies eq. (16) and c) stabilizes A1+KoC-A2L. The gain matrix Ko is an output feedback controller that places exactly n-m poles, the remaining in a stable area and finally is an optimal controller.
4 Illustrative example Let a 3-state, 2-input and 2-output system with the following system matrices
1 2 0 A = − 2 3 1 0 1 − 1 1 0 B = 0 1 0 0
(17)
(18)
and
2 0 0 C= 0 3 0
(19)
For this system we can assign exactly n-m=3-2=1 closed loop poles. We will place the dominant pole at –5. To this end we select
L = [0 − 4]
(20) problem
solving the state feedback sub eig(A4+LA2)=-5. Then a suitable feedback matrix that satisfies the three requirements (a), (b) and (c) is
2/3 − 31 / 12 K0 = − 31 / 12 1
(21)
The poles of the closed loop system A+BKoC are located at –5, -0.75 and -4,1667. A matrix solution for the AMRE is given by eq. (11).
5 Conclusions A simultaneous optimal and pole placement controller design method is proposed. The method consists of two discrete steps, is easy to use and applies to any controllable linear time invariant system with independent inputs.
References: [1] A. Jameson and E. Kreindler, "Inverse problem of linear optimal control", SIAM J. Control, Vol. 11, No. 1, 1973, pp. 1-19. [2] A.T. Alexandridis and G.D. Galanos, “Optimal pole-placement for linear multi-input controllable systems”, IEEE Trans. Circ. and Syst., Vol. CAS-34, 1987, pp.1602-1604. [3] A.T. Alexandridis, P.N. Paraskevopoulos, "A new approach to eigenstructure assignment by output feedback", IEEE Trans. Automat. Contr., Vol. AC41, No. 7, 1996, pp. 1046-1050. [4] B.P. Molinari, "The stable regulator problem and its inverse", IEEE Trans. Automat. Contr., Vol. AC-18, No. 8, 1973, p.p. 454-459. [5] C.T. Chen, Linear System Theory and Design, Holt, Rinehart and Wiston Inc., 1984. [6] D.P. Iracleous, A.T. Alexandridis, "A simple solution to the optimal eigenvalue assignment problem", IEEE Trans. Automat. Contr., Vol. 44, No. 9, Sept. 1999, pp. 1746-1749. [7] D.P. Iracleous, A.T. Alexandridis, "New results to the inverse optimal control problem for discrete-time linear systems", Journal of Applied Mathematics and Computer Science, Vol.8, No. 3, 1998, pp. 517-528. [8] F.L. Lewis and V.L. Syrmos, Optimal Control, John Wiley and Sons, 1995. [9] Kyung-Soo Kim, Youngjin Park, Equivalence between two solvability conditions for a static output feedback problem, IEEE Trans. On Autom. Contr., Vol. 45, No. 10, Oct. 2000, p. 1877. [10] M.C. Maki and J. Van de Vegte, "Optimization of multi-input systems with assigned poles", IEEE Trans. Automat. Contr., Vol. 19, 1974, pp. 130-133. [11] W.M. Haddad and D.S. Bernstein, "Controller design with regional pole constraints", IEEE Trans. Automat. Contr., Vol. 37, No 1, 1992, pp. 54-69. [12] Y.Bar Ness, "Optimal closed-loop pole assignment", Int. J. Control, Vol. 27, No. 3, 1978, pp. 421-430.
