In the last decade there has been an increasing interest in developing numerical algorithms for the analysis and design of linear periodic discrete-time control systems of the form

where A_{k}, B_{k}, C_{k} and D_{k} are N-periodic matrices. Note that for discrete-time models the state-, input- and output-vectors x_{k}, u_{k} and y_{k}, respectively, are allowed to have time-varying dimensions. The matrices of discrete-time periodic systems are stored as cell arrays of length N. 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, control of rotating machineries like helicopters, multirate systems). 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 some of recently developed numerical algorithms for the analysis and design of periodic systems see [1].Some of open problems formulated in [1] have been in the meantime solved by developing numerically satisfactory algorithms.

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 for constant dimensions and in RASP-PERIODIC library (see bellow) for time-varying dimensions. These routines underlie the implementation of robust software available in RASP-PERIODIC to solve periodic Lyapunov equations with time-varying dimensions.

Order reduction

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]. An alternative method for minimal realization has been proposed in [12] and relies on periodic Kalman reachability and observability forms computed using orthogonal similarity transformations. The underlying algorithms generalize the orthogonal staircase form algorithms for standard systems.

Periodic system analysis

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. 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. Recently, a general algorithm to compute Kronecker-like forms of periodic pairs has been proposed in [14]. This algorithm has many applications, allowing to solve periodic system inversion problems without building lifted representation [15] or computing left or right annihilators of periodic systems with immediate applications to the fault detection filter design for periodic systems [16].

Model conversions

The computation of poles and zeros underlies the new algorithm of [11] 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 [12] represent the basis of reliable conversions between state-space and input-output representations of periodic systems. The new algorithm of [10] 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.

Periodic stabilization techniques

Algorithms for basic design procedures for periodic systems have been developed for the periodic output feedback control [7], robust pole assignment via periodic state-feedback [8], and periodic linear-quadratic (LQ) optimisation techniques [17]. The first two 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.The solution of the periodic LQ optimisations problem relies on solving periodic Riccati equations for which new, general algorithms have been proposed in [17].

Satellite control applications

Periodic control techniques have been employed to design controllers for satellite positioning by means of magnetic actuators.The synthesis method proposed in [7] relies on using an optimal periodic output feedback control law which minimizes an associated quadratic cost function. A gradient search based optimization approach, specially developed for large order problems, has been used to compute the periodic output feedback matrices. The evaluation of the cost function and its gradient is based on analytic expressions derived in [7]. Each function and gradient evaluation involves the numerical solution of a pair of discrete-time periodic Lyapunov equations. A more realistic application has been addressed in [18], where a periodic fixed structure state-feedback controller has been. The problem to be solved is equivalent to determine a constant output feedback controller for a periodic system and has been solved usingthe approach proposed in [7]. The resulting controller has practically the same dynamic closed-loop performance as an optimal (unrestricted) periodic state-feedback controller. The main advantage of the new approach is that the fixed-structure periodic state-feedback depends only of the measured magnetic field components, and thus is very easy to implement using magnetic field measurements.

Numerical software for periodic systems

High quality numerical software for periodic systems has been implemented in RASP-PERIODIC, a collection of Fortran routines from RASP (the control library of DLR) 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], for computing gradients for solving periodic optimal output feedback problems [7], and for computing reachability and observability Kalman decompositions [12]. These routines underlie the implementation of an user-friendly PERIODIC SYSTEMS Toolbox [-> pertool.doc] for MATLAB [19]. A first application of this toolbox was the solutionof an optimal discrete-time magnetic attitude control of satellite of a fixed structure periodic state-feedback control problem [18].

Related publications:

[1] Varga, A., Van Dooren, P.:

Computational methods for periodic systems - an overview. Prepr. of 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, Brüssel, 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 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., P. Van Dooren:

Computing the zeros of periodic descriptor systems. Systems & Control Letters, vol. 50, pp. 371–381, 2003.

[10] Varga, A.:

Computation of minimal periodic realizations of transfer-function matrices. IEEE Transactions on Automatic Control, vol. 46, pp. 146-149, 2004.

[11] Varga, A.:

Computation of transfer function matrices of periodic systems. International Journal of Control, vol. 76, pp. 1712–1723, 2003.

[12] Varga, A.:

Computation of Kalman decompositions of periodic systems. European Journal of Control, vol. 10, pp. 1-8, 2004.

[13] Varga, A.:

Strongly stable algorithm for computing periodic system zeros. Proc. of CDC, Maui, Hawaii, 2003.

[14] Varga, A.:

Computation of Kronecker-like forms of periodic matrix pairs. Proc. of MTNS, Leuven, Belgium, 2004.

[15] Varga, A.:

Computation of generalized inverses of periodic systems. Proc. of CDC'2004, Paradise Island, Bahamas, 2003.

[16] Varga, A.:

Design of fault detection filters for periodic systems. Proc. of CDC'2004, Paradise Island, Bahamas, 2003.

[17] Varga, A.:

On solving discrete-time periodic Riccati equations. Proc. of IFAC'05 World Congress, Prague, Czech Republik, 2005.

[18] Lovera, M., Varga, A.:

Optimal discrete-time magnetic attitude control of satellites. Proc. of IFAC'05 World Congress, Prague, Czech Republik, 2005.

[19] Varga, A.:

A PERIODIC SYSTEMS Toolbox for MATLAB. Proc. of IFAC'05 World Congress, Prague, Czech Republik, 2005.