Optimal Eigenvalue Placement Linear Quadratic Performance

Abstract

A new design algorithm is presented for clustering eigenvalues of a dynamic system within a sector in the left hand side of the complex s-plane while minimizing a LinearQuadratic (La) performance index. The development of the algorithm is based on a simple transformation of the system into a higher order associated dynamic system. The proposed algorithm explicitly deals with the relative damping and stability performance requirements. A numerical example is also given to illustrate and compare the design method. INTRODUCTION It is well-known that the time and frequency response of a linear system is directly influenced by the relative locations of the closed loop system eigenvalues. Hence, quite often, the main goal of the control system designer is not merely to stabilize a given plant, but to shape the dynamic response by placing the closed loop eigenvalues in some pre-specified region of the left hand s-plane. In practice, the following two important timedomain requirements are often included in the performance specifications: l-Response must be sufficiently fast and smooth. 2-Response must not exhibit excessive overshoot and oscillations. The first requirement places a bound on the settling time, whereas the second one gives rise to a bound on the damping ratio, C . By enforcing these bounds, a designer can achieve a uniform degree of damping and stability. For this reason, the shaded area of Fig.(l) has been widely accepted in the control systems community as a suitable design sector. Fig(1) Design sector with damping and stability specifications Because direct optimal pole placement in the shaded area is a very difficult problem to solve, a multitute of approximate regions have been proposed in the literature [1 ] [6 ] . Some of the approximations involve circular, elliptical, parabolic, and hyperbolic as well as horizontal and vertical strips. In order to contrast and compare the contributions of this paper, two of the most related root clustering methods will be briefly discussed below. The method developed in [ 5 ] is based on the preservation of the eigenvalues and eigenvectors of the quadratic and linear functions of a matrix. For example, if the system matrix A has I = x+yj as an eig nvalue, t en the corresponding eigenvalue of A’-(r21 implies that is The asymptotic stability of A2-a I 1 9 -a2. Re(A2-a2) < 0 or x2-y2 < a2 The above inequality defines a sector to the left of a hyperbola. Clearly, by making A2-a21 stable, it is possible to place all the eigenvalues of A in the shaded region of Fig. (1) . The design procedure of [ 51 involves an iterative technique to update the quadratic weights of the LQ performance index to move the eigenvalues into the desired region. The main disadvantage of this approach is that the asym totes of the hyperbola are fixed at e=45 g . A different class of root clustering algorithms can be developed by using the following theorem; The RELATIVE STABILITY THEOREM: eigenvalues of the matrix A lie within the shaded stability region of Fig.(2), if and only if the eigenvalues of the 2NX2N matrix have negative real parts. The angle @ is given by @= x/2-6 and A is defined as A-a1 . PROOF: see [7f and [ 8 ] . Recently, a design procedure based on the above theorem appeared in [ 4 ] . The major problem associated with the straight forward implementation of this theorem, however, is