422 J. GUIDANCE VOL. 10, NO. 5 A Homotopy Approach to the Feedback Stabilization of Linear Systems Pierre T. Kabamba* University of Michigan, Ann Arbor, Michigan and Richard W. Longmant and Sun Jian-Guo$ Columbia University, New York, New York Constant-gain, fixed-order controllers for linear time-invariant systems are considered. A closed-form necessary and sufficient condition for the stabilizability of such a system by a controller of chosen order is established. This criterion is obtained by solving a constrained optimization problem and results in a system of nonlinear matrix equations. A method based on homotopy is proposed and studied to solve this system of nonlinear equations. The numerical implementation of the homotopy is discussed and various properties of the optimizing feedback control as a function of the homotopy parameter are established. Introduction R ICHARD Bellman often talked of "the curse of dimen- sionality." In the same spirit, we can identify "the curse of modern control theory." The most general and powerful modern control design methods, the linear-quadratic theory and the pole placement methods, require knowledge of the full state; in most cases, this implies using a full-order observer or filter in real-time operation—something that can easily invoke a curse of dimensionality on such designs. This is in stark con- trast to classical design methods having low-dimensional controllers. There is a need for effective fixed-order controller design methods to obtain controllers of prescribed intermediate dimension that can obtain some of the advantages of modern control theory without incurring the full-dimensionality "curse." One way to pose such problems is to consider the linear system with a quadratic cost problem, but to add the stipulation that the control must be obtained from the measurements via a dynamic feedback system whose dimen- sion is prescribed in advance. Considerable attention has been given to this problem over the years and a few of the papers are cited here. 1 ' 10 Reference 1 presents a fairly com- prehensive overview of the possible ways in which the quadratic cost, fixed-order controller problem can be posed. It also gives the first simple treatment of the finite-time ver- sions of these problems. Extensive use of the design ap- proach has been made by Ly 2 employing his computer pro- gram SANDY. Hyland and Bernstein 7 ' 10 showed that the necessary conditions for optimality of a fixed-order con- troller are an elegant generalization of the well-known LQG design procedure obtained by introducing optimal oblique projections. Their results led to new computational pro- cedures 8 ' 10 and extensions to stochastic systems. 11 Experience with fixed-order controller design has shown that often there is surprisingly little penalty paid for using controllers of dimension less than that of the system. Also, there are some indications that the lower-dimension con- Received March 20,, 1986; revision received Oct. 4, 1986. Copyright © American Institute of Aeronautics and Astronautics, Inc., 1987. All rights reserved. * Assistant Professor of Aerospace Engineering. tProfessor of Mechanical Engineering. ^Visiting Scholar (on leave from Nanjing Aeronautical Institue, Nanjing, China). trollers can be more robust than their full-order counter- parts. However, the design of these controllers can be dif- ficult, being plagued with problems of local minima and in- volving use of a gradient method that can have painfully slow convergence. A still more fundamental difficulty occurs in designing controllers for unstable systems. To start the iterations of the gradient method, it is necessary to prescribe a starting value for the controller parameters that succeeds at stabilizing the system; otherwise, the gradient equations are invalid. Ex- perience has shown that it can be very difficult to discover an initial stabilizing controller. The primary purpose of this paper is to examine the fun- damental question of existence of a stabilizing controller of specified order for a given system. This question is answered by the closed-form criterion of theorem 1. This result is ob- tained by solving a constrained optimization problem. The criterion requires solving a system of nonlinear matrix equa- tions. The secondary purpose of the paper is to study a homotopy method for solving this system of nonlinear matrix equations. The homotopy is performed on the param- eter a when the system matrix A is replaced by A + ol. The numerical implementation of the homotopy is discussed and various properties of the optimizing feedback controller as a function of the homotopy parameter are established. Problem Formulation and Preliminary Results We consider a finite-dimensional linear time invariant sys- tem with minimal representation x s =A s x s +B s u y=c s x s (i) where # 5 € \R"s is the system state vector, u£ \R m * the input vector, y^ \R P * the output vector, and A s> B s , C s real matrices of appropriate dimensions, B s and C 5 having full rank. The controller has the form x c =A c x c +B c y (2) where jc c € \R n c is the controller state vector and A c , B c , C c , D c real matrices of appropriate dimensions. The order n c of the controller ranges from 0 for direct output feedback to n s for full-order controller.