Periodic Systems

In the last few years there has been an increasing interest in developing numerical algorithms for the analysis and design of linear periodic discrete-time control systems. Such models arise usually by the discretization of linear continuous-time periodic models which are the primary mathematical descriptions encountered in several practical applications (e.g. satellite attitude control based on the periodicity of the earth magnetic field; or control of rotating machinery). The numerical advantage of using discrete-time models instead of continuous-time ones is the possibility to develop and to use efficient computational algorithms which completely parallel those for standard discrete-time systems. For an overview of recently developed numerical algorithms for the analysis and design of peridic systems see [1].

A computational problem with many applications is the efficient solution of periodic Lyapunov equations. These equations have several important applications in the analysis and design of linear periodic control systems, e.g. the analysis of controllability/observability/minimality, balancing and balancing-related model reduction, stabilization with periodic state feedback and with output feedback. Efficient, numerically reliable algorithms based on the periodic Schur decomposition have been developed recently for the solution of several types of periodic Lyapunov equations [2-4]. The new algorithms are extensions of the direct solution methods for standard discrete-time Lyapunov equations for the cases of indefinite as well as of nonnegative definite solutions both with constant and time-varying dimensions. Efficient implementations for computing periodic Hessenberg and periodic Schur decompositions are available in the SLICOT library. These routines underlie the implementation of robust software available in RASP to solve periodic Lyapunov equations.

Minimal realization and model reduction problems of periodic systems can be solved using balancing-related computational approaches. Order reduction algorithms extending the square-root and balancing-free techniques for standard systems have been derived in [4,5] for systems with time-varying dimensions. A bound for the approximation error generalizing that for standard systems has been derived in [5]. The main computation in the algorithms of [4,5] is the solution of nonnegative definite periodic Lyapunov equations with time-varying dimensions directly for the Cholesky factors of the solutions. The basic computational ingredient in these algorithms is an extension of the periodic real Schur form of a square product of rectangular matrices introduced in [4]. The model reduction algorithm of [5] has been recently extended to handle the reduction of unstable periodic systems in [6].

The computation of zeros of a periodic system represents a universal system analysis tool. By computing particular types of zeros various properties of periodic systems can be studied. The stability of a periodic system can be assessed by determining the system poles (or characteristic multipliers) defined as the zeros of a system without inputs and outputs (i.e., the eigenvalues of the monodromy matrix). The reachability/stabilizability or observability/detectability properties can be analyzed by computing the input or output decoupling zeros of the periodic system, respectively. The system zeros defined in terms of the standard lifted representation can be used to assess properties like minimum-phase or the existence of stable and proper inverses. A numerically stable algorithm to compute the zeros of periodic systems has been recently proposed in [9]. This algorithm has low computational complexity and can address even systems with time-varying dimensions. The computation of zeros relies on structure exploiting reduction of an appropriate system pencil and the computation of zeros of a reduced order linear pencil using tools available in the DESCRIPTOR SYSTEMS Toolbox. A strongly stable algorithm for computation of finite zeros of periodic descriptor systems has been proposed in [13]. The algorithm relies on a structure preserving orthogonal reduction of the lifted system pencil. It can be shown that the zeros computed by this algorithm in the presence of rounding errors are exact for a periodic system having nearby system matrices to those of the original system. This algorithm is computationally efficient and fulfills all requirements for a satisfactory algorithm for periodic systems. A similar strongly stable algorithm using exclusively orthogonal transformations has been also developed to compute periodic Kalman decompositions of standard periodic systems [12]. This algorithm generalizes to periodic systems well established controllability staircase algorithms for standard systems. The algorithms for periodic reachability and observability Kalman decompositions can serve to compute efficiently minimal realizations of periodic systems with time-varying state vector dimension.

The computation of poles and zeros underlies the new algorithm of [10] to compute the transfer-function matrix of lifted representations of a periodic system. This algorithm together with minimal realization procedures based either on balancing techniques [4] or the recently developed numerically stable approach of [11] represent the basis of reliable conversions between state-space and input-output representations of periodic systems. The new algorithm of [11] determines minimal order realizations of transfer-function matrices using exclusively orthogonal rank revealing decompositions. This algorithm can be easily employed in conjunction with subspace identification tools to determine minimal realizations of periodic systems starting from input and output measurement data.

Algorithms for basic design procedures for periodic systems have been developed for the periodic **output feedback** control [7] and **robust pole assignment** via periodic state-feedback [8]. Both approaches relies on a parametric optimization based reformulation of the synthesis problems. To solve the potentially high dimensional optimization problems, gradient-based methods, as for example the limited-memory BFGS quasi-Newton method, are well suited. The main computations in both approaches are the efficient evaluations of function and gradients. Each function and gradient evaluation involves in the case of algorithm of [7] the solution of two periodic Lyapunov equations, while in the case of algorithm of [8] the solution of two periodic Sylvester equations.

High quality **numerical software** for periodic systems is available in RASP-PERIODIC, a collection of Fortran routines from the RASP library for solving computational problems appearing in the context of analysis and design of periodic systems. RASP-PERIODIC includes routines for reduction of a square periodic matrix to the periodic Hessenberg and periodic real Schur forms, solution of periodic Lyapunov equations [2], and for computing gradients for solving periodic optimal output feedback problems [7]. The RASP-PERIODIC collection of routines has been partly included in the freely available SLICOT library. An user-friendly software, representing a prototype of a PERIODIC SYSTEMS Toolbox for MATLAB is in preparation. For the preliminary contents of this toolbox and planned implementations see the Contents.m file.

References

[1] Varga, A., Van Dooren, P.:**Computational methods for periodic systems - an overview**. Prepr. IFAC Workshop on Periodic Control Systems, Como, Italy, 2001.

[2] Varga, A.:

** Periodic Lyapunov equations: some applications and new algorithms**. Int. J. Control, vol. 67, pp. 69-87, 1997.

[3] Varga, A.:

** Solution of positive periodic discrete Lyapunov equations with applications to the balancing of periodic systems**. Proc. of European Control Conference, ECC'97, BrĂ¼ssel, 1997.

[4] Varga, A.:**Balancing related methods for minimal realization of periodic systems**. Systems & Control Letters, vol. 36, pp. 339-349, 1999.

[5] Varga, A.:**Balanced truncation model reduction of periodic systems**. Proc. of IEEE Conference on Decision and Control, CDC'2000, Sydney, Australia, 2000.

[6] Varga, A.:**On balancing and order reduction of unstable periodic systems**. Prepr. IFAC Workshop on Periodic Control Systems, Como, Italy, 2001.

[7] Varga, A., Pieters, S.:** Gradient-based approach to solve optimal periodic output feedback control problems**. Automatica, vol. 34, pp. 477-481, 1998.

[8] Varga, A.: ** Robust and minimum norm pole assignment with periodic state feedback**. IEEE Transaction on Automatic Control, vol. 45, pp. 1017-1022, 2000.

[9] Varga, A., Van Dooren, P.:**On computing the zeros of periodic systems**. Proc. of CDC2002, Las Vegas, NV, 2002.

[10] Varga, A.:

**Computation of transfer function matrices of periodic systems**. Proc. of CDC2002, Las Vegas, NV, 2002.

[11] Varga, A.:

**Computation of minimal periodic realizations of transfer-function matrices**. ACC2003, Denver, CO, 2003.

[12] Varga, A.:**Computation of Kalman decompositions of periodic systems**. Submitted to ECC2003, Cambridge, UK, 2003.

[13] Varga, A.:**Strongly stable algorithm for computing periodic system zeros**. Submitted to CDC2003, Maui, Hawaii, 2003.