# Periodic Control Systems: Analysis, Efficient Model Reduction and Development of Numerical Algorithms

Periodic systems arise in various scientific fields such as aerospace realm, wind turbines, pumps, fans, control of industrial processes and communication systems, modeling of periodic time-varying filters and networks, wave motion in periodic media such as ultra-cold atoms in an optical lattice, phenomenon of parametric resonance in forced oscillators and nonlinear systems linearized about a periodic trajectory.

Real life problems where the mathematical models demand periodic (descriptor) settings can be described by the example figures. The figure on the left shows the controlled orbital periodic motion of TOPEX/Poseidon satellite in space. It was launched in 1992 by a joint satellite mission between NASA and CNES (the French space agency) which made it possible for the first time to determine the patterns of ocean circulation by observing how heat stored in the ocean moves from one place to another. The attitude dynamics of such a (rigid) spacecraft is nonlinear and can be described by means of possible parameterizations. Linearizing the model around the external disturbance torques that come from the disturbing resources (i.e., gravity gradient, aerodynamics, solar radiation), results a linear periodic model [Lovera et al. '2002]. Figure in the middle shows a digital filter used for super sound quality in digital audio industries. This digital filter is a prime component of multirate data sampling. The linearized model is a periodic model [P.P. Vaidyanathan'1990].

Wave motion in periodic media such as ultracold atoms in an optical lattice which is shown in the figure on the right corner. By interfering optical laser beams an optical lattice is formed, creating a spatially periodic polarization pattern. It may trap ultracold gases of bosons and fermions. Storing such gases in artificial periodic potential of light has opened innovative manipulation and control possibilities and creating new structures.

The goal of this project is to analyze the periodic systems and develop numerical algorithms for model reduction. We investigate the following two cases while studying the periodic systems and the applicabilities of model reduction for problems coming from different fields.

The first part of this project is to analyze periodic descriptor systems both in continuous and discrete-time case. Simulations and analysis of such periodic systems can be unacceptably expensive and time-consuming when the systems are very large. Hence, model reduction is an efficient tool which helps scientists and engineers to replace such a large periodic model by a smaller model which is amenable to fast and efficient simulation and still preserve the input-output behavior of the original large model as good as possible. A core part of this project work is to develop numerical algorithms for efficient model reduction for these periodic descriptor systems. Mainly the projection based model reduction approaches are considered in this project work.

We develop efficient implementations of Krylov subspace based projection methods for model order reduction of problems that result from the linearization of nonlinear circuit problems and the resulting models are linear periodic time-varying (LPTV). The algorithmic realization of the method employs recycling techniques for shifted Krylov subspaces and their invariance properties. The efficiency and accuracy of the developed algorithm is illustrated by real-life numerical examples and compared to other Krylov based projection methods used for model reduction.

Balancing based projection technique is applied for model reduction of these large sparse discrete-time LPTV descriptor systems. The projected periodic Lyapunov equations that appear in the stability analysis and in model reduction of these periodic descriptor systems are first solved analytically (using direct solvers). For large scale systems, these projected periodic Lyapunov equations are solved using iterative techniques, e.g., ADI method, Smith method. The main task of these iterative computations is to preserve the cyclic structure of system matrices in each iteration steps of the periodic solutions. Our proposed algorithms analyze the cyclic structure of the matrices arising in the iterative computations of the periodic solutions of the projected discrete-time periodic Lyapunov equations. Low-rank version of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations. A further development of these algorithms is in progress.

In this part of the project we analyze ordinary differential equations (ODEs). These systems have a special periodic structure, which in the linear case is the periodicity of the coefficients, i.e. ẋ = A(t)x, where A(t)=A(t+T) is T-periodic. Calculating bounds on the solution for suitable norms are part of this project as well as stability analysis. In the stability analysis we investigate the stability of solutions near to a point of equilibrium. The stability result is of major concern while modeling dynamical systems to ensure that the model fulfills the observations.