Stabilization of Large Scale Dynamical Systems

In this paper we discuss the stabilization of large scale linear time invariant dynamical systems via feedback. Efficient schemes based on the Discrete Riccati Difference Equation are presented. The SQR variant is described in detail and a more efficient version, CSQR, is motivated. 1 The Problem In this paper, we focus on the stabilization of a discrete-time system (1) where and are and real matrices which are known, and and are vectors of dimension and respectively. The stabilization of the system requires the computation of a feedback matrix such that all eigenvalues of are inside the unit circle and therefore the system defined by replacing with ! " is stable. For small and moderate values of , can be computed via pole placement or the solution of a matrix equation, e.g., Riccati or Lyapunov equations. The computational requirements for standard algorithms for these approaches, however, is prohibitive for large values of . Fortunately, when is large and $#%#& , the system matrix and/or input matrix are typically very sparse. Algorithms for such problems must therefore exploit this structure in order to efficiently compute a stabilizing feedback. 2 Saad’s Approach A major contribution to solving large scale stabilization problems with a few unstable eigenvalues is Y. Saad’s projection method [1]. In this algorithm, stabilization or eigenvalue assignment is only imposed on a small invariant subspace that contains the unstable invariant subspace of . Such an approach is often effective, but it can have convergence difficulties and the need for a basis of the invariant subspace can cause excess space requirements for very large systems. In this paper, we discuss efficient alternatives that address the convergence difficulties. We will also motivate and algorithm that avoids the need for an explicitly formed basis of the invariant subspace. The latter will be explored in detail in a forthcoming paper. Details on all of the algorithms can be found in [6]. In Saad’s projection algorithm, a left invariant subspace ')( of (with presumably small dimension), that contains the left unstable invariant subspace of is computed. There are two major classes of methods that can be used. The first computes the unstable eigenvalues and recovers their eigenvectors by some form of inverse iteration. The second class computes the basis directly by subspace iteration-like methods. The low-order projected system * '+( ' ')( , is then stabilized and the reduced feedback /. is lifted back to form a stabilizing feedback /. ')( of the original system * + 0 -, . Methods in the first class benefit from years of sparse eigenvalue algorithm research but often require very high accuracy in the eigenvalues in order to produce the basis and hence result in more computation than necessary for stabilization. Effective convergence is one of the main issues of the second class of methods. The convergence of subspace iterationlike (SSI) methods which generate the sequence of approximations to the invariant subspace starting from initial subspace '21 and updating ' by extracting an orthogonal basis of (3' 34 is usually consistent with the separation between desired eigenvalues and undesired eigenvalues in absolute value. In practice, it is often difficult to tune the parameters of such methods to converge even this quickly. They can be accelerated and some parameter sensitivity mitigated by the use of Stewart’s SRR (Schur-Rayleigh-Ritz) refinement [4]. The acceleration is achieved by enlarging the size of initial subspace ' 1 , extracting the Schur vectors 5 corresponding to largest (or unstable) eigenvalues of ')( (6' and combining