Linear Periodic Discrete-time Systems Research Articles

Some Convergence Properties of Two Iterative Algorithms for Discrete Periodic Lyapunov Equations

In this paper, some convergence conditions are investigated for the multiple tuning parameters iterative algorithm (MIA) and the single tuning parameter iterative algorithm (SIA), which are proposed to solve the discrete periodic Lyapunov matrix equations related to discrete-time linear periodic systems. First, when all the tuning parameters are selected in the interval (0,1] and the initial conditions are arbitrarily given, it is proven that the MIA is convergent if and only if the discrete-time linear periodic system is asymptotically stable. In particular, when the coefficient matrices of the considered matrix equations are nonnegative, it is shown that the convergence rate of the MIA increases with the tuning parameter increases from 0 to 1. Moreover, the above convergence results derived for the MIA are extended to the SIA. Furthermore, the searching interval of the optimal tuning parameter for the SIA to achieve the fastest convergence rate is narrowed. Finally, two numerical examples are provided to demonstrate the correctiveness of the proposed theoretical results.

Decay estimates of Green's matrices for discrete-time linear periodic systems

We study periodic Lyapunov matrix equations for a general discrete-time linear periodic system , where the matrix coefficients and can be singular. The block coefficients of the inverse operator of the system are referred to as the Green matrices. We derive new decay estimates of the Green matrices in terms of the spectral norms of special solutions to the periodic Lyapunov matrix equations. The study is based on the periodic Schur decomposition of matrices.

On the stability of delayed linear discrete-time systems with periodic coefficients

Stability estimates are obtained for delayed linear periodic discrete-time systems. Bounds on the decay of the solution are derived via a suitable Lyapunov–Krasovskii-type functional and the solvability of some periodic discrete-time Lyapunov equations.

Uniform Global Asymptotic Stabilization of Semilinear Periodic Discrete-Time Systems

Semilinear discrete-time control systems with periodic coefficients are considered. The problem of uniform global asymptotic stabilization of the zero equilibrium of the closed-loop system by state feedback is studied. It is assumed that the free dynamic system has a Lyapunov stable zero equilibrium. The method for constructing a damping control is extended from time-invariant systems to time varying periodic semilinear discrete-time systems. By using this approach, sufficient conditions for uniform global asymptotic stabilization for those systems are obtained. Moreover, the converse Lyapunov theorem on Lyapunov (nonasymptotic) stability is proved for complex and real linear periodic discrete-time systems. Finally, examples of using the obtained results are presented.

A dual reduction strategy for reduce-order modeling of periodic control system

Model order reduction (MOR) of periodic systems using the Krylov subspace methods received lots of interest in last few decades. In this paper, a structured Krylov subspace based model reduction for linear discrete-time periodic (LDTP) control system has been proposed using the corresponding lifted form. The proposed model reduction strategy first finds a reduced order Lyapunov equation via a projection where the projection subspace is computed by using the periodic Arnoldi method. Then, by applying the balanced truncation model reduction approach, it forms a reduced order model of the original system. In both cases, the periodic structure is ensured in the solutions and systems’ dynamics. This paper also presents numerical results to confirm the efficiency and accuracy of the proposed algorithm.

Design of an optimal preview controller for linear discrete-time periodic systems

In this paper, the preview tracking control problem for linear discrete-time periodic systems is considered. First, to overcome the difficulty arising from periodicity of the system, the linear discrete-time periodic system is transformed into an ordinary time-invariant system by lifting method. Secondly, the difference between a system state and its steady-state value is used to derive an augmented system instead of the usual difference between system states. Then, the preview controller for the augmented system is proposed by the preview control theory, which solves the preview tracking control problem for the periodic systems. Moreover, an integrator is introduced to ensure that the output can track the reference signal without static error. Finally, the obtained results are illustrated by the simulation examples.

Q-Learning Algorithms in Control Design of Discrete-Time Linear Periodic Systems by Lifting Technique

Q-Learning Algorithms in Control Design of Discrete-Time Linear Periodic Systems by Lifting Technique

Multi‐parametric iterative algorithms for discrete periodic Lyapunov matrix equations

Two multi-parametric iterative algorithms are developed to solve the forward discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. An important feature of one of the proposed algorithms is that the information in the current and the last steps is used to update the iterative sequence. Necessary and sufficient conditions for these two algorithms to be convergent are provided in terms of the roots of a set of polynomial equations. Based on these conditions, a two-dimensional section method is established to search suboptimal tuning parameters for these two algorithms. In addition, convergence properties are also analysed for some special cases of the obtained algorithms. Finally, numerical examples are provided to illustrate the effectiveness of the proposed algorithms. It can be found that the presented algorithms have better convergence performance than some existing iterative algorithms by choosing proper tuning parameters.

State-space identification of switching linear discrete time-periodic systems with known scheduling signals

State-space identification of switching linear discrete time-periodic systems with known scheduling signals

A robust reduced-order observers design approach for linear discrete periodic systems

This paper investigates the problem of designing reduced-order observers for linear discrete-time periodic (LDP) systems. In case that the linear discrete-time periodic system is observable, an algebraic equivalent system is obtained by non-singular linear transformation, and the partial states to be observed are separated simultaneously. Then the considered problem is transformed into the problem of solving a class of periodic Sylvester matrix equation and an iterative algorithm for periodic reduced-order state observers design is derived. In addition, robust consideration based on periodic reduced-order state observers for LDP systems is also conducted. At last, one numerical example is worked out to illustrate the effectiveness of the proposed approaches.

Preview control for a class of linear discrete-time periodic systems

ABSTRACT This paper considers the problem of preview tracking control for linear discrete-time periodic systems subject to previewable reference and disturbance signals. First, the difference operator approach is introduced to derive an augmented error system that includes future information on the previewable signals. As a result, the problem of preview tracking control of the original system is transformed into a stabilisation problem via a periodic static output feedback. Second, sufficient conditions that guarantee the closed-loop stability of the augmented error systems are derived, and meanwhile, the design of a periodic controller with preview actions is proposed through the linear matrix inequalities approach. Finally, a numerical simulation example is presented to illustrates the effectiveness of the proposed method.

Convergence characterisation of an iterative algorithm for periodic Lyapunov matrix equations

ABSTRACTThis paper is concerned with convergence characterisation of an iterative algorithm for a class of reverse discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. Firstly, a simple necessary condition is given for this algorithm to be convergent. Then, a necessary and sufficient condition is presented for the convergence of the algorithm in terms of the roots of polynomial equations. In addition, with the aid of the necessary condition explicit expressions of the optimal parameter such that the algorithm has the fastest convergence rate are provided for two special cases. The advantage of the proposed approaches is illustrated by numerical examples.

Optimal Preview Control for Linear Discrete-Time Periodic Systems

In this paper, the optimal preview tracking control problem for a class of linear discrete-time periodic systems is investigated and the method to design the optimal preview controller for such systems is given. Initially, by fully considering the characteristic that the coefficient matrices are periodic functions, the system can be converted into a time-invariant system through lifting method. Then, the original problem is also transformed into the scenario of time-invariant system. Later on, the augmented system is constructed and the preview controller of the original system is obtained with the help of existing preview control method. The controller comprises integrator, state feedback, and preview feedforward. Finally, the simulation example shows the effectiveness of the proposed preview controller in improving the tracking performance of the close-loop system.

Robust Performance and Observer Based Control for Periodic Discrete-Time Uncertain Systems

This paper presents an approach for the synthesis of observer-based controllers for discrete-time periodic linear systems. The H2 performance criterion has been employed to design both the observer and the controller. For the periodic observer design, two conditions in the form of Linear Matrix Inequalities (LMIs) are proposed, which stem from the Lyapunov Theory applied over the dynamics of the estimation error. The LMI condition obtained for the periodic state-feedback controller results from the application of the duality principle over the periodic system, under the assumption that only the estimated states are available to be used in the state-feedback compensation. Numerical experiments illustrate the potential of the proposed observer-based control technique.

A Periodic Observers Synthesis Approach for LDP Systems Based on Iteration

The problem of state observer design for the linear discrete-time periodic (LDP) system and its robust consideration are discussed in this paper. Applying the lifting technique and algebraic operations based on the well-known CG-algorithm, an iterative algorithm for periodic observer gain can be generated. By optimizing the free parameter matrix in the proposed algorithm, an algorithm on the minimum norm and robust observer design for the LDP systems is presented. One numerical example is worked out to illustrate the effect of the proposed approaches.

An iterative algorithm for discrete periodic Lyapunov matrix equations

In this paper, a novel iterative algorithm with a tuning parameter is developed to solve the forward discrete periodic Lyapunov matrix equation associated with discrete-time linear periodic systems. An important feature of the proposed algorithm is that the information in the current and the last steps is used to update the iterative sequence. The convergence rate of the algorithm can be significantly improved by choosing a proper tuning parameter. It is shown that the sequence generated by this algorithm with zero initial conditions monotonically converges to the unique positive definite solution of the periodic Lyapunov matrix equation if the tuning parameter is within the interval(0,1]. In addition, a necessary and sufficient convergence condition is given for the proposed algorithm in terms of the roots of a set of polynomial equations. Also, a method to choose the optimal parameter is developed such that the algorithm has the fastest convergence rate. Finally, numerical examples are provided to illustrate the effectiveness of the proposed algorithm.

An efficient LQR design for discrete-time linear periodic system based on a novel lifting method

This paper proposes a novel lifting method which converts the standard discrete-time linear periodic system to an augmented linear time-invariant system. The linear quadratic optimal control is then based on the solution of the discrete-time algebraic Riccati equation associated with the augmented linear time-invariant model. An efficient algorithm for solving the Riccati equation is derived by using the special structure of the augmented linear time-invariant system. It is shown that the proposed method is very efficient, compared to the ones that use algorithms for discrete-time periodic algebraic Riccati equation. The efficiency and effectiveness of the proposed algorithm is demonstrated by the simulation test for the design problem of spacecraft attitude control using magnetic torques.

Fault Detection and Isolation of Linear Discrete-Time Periodic Systems Using the Geometric Approach

This technical note solves the problem to detect and isolate faults in linear periodic discrete-time systems. A residual generator is designed through a suitable unknown input observer with the requirement that each residual should be sensitive only to one fault and, simultaneously, insensitive to the other faults that can affect the system, and with the additional requirement of ρ-detectability, that is the transition matrix of its relative error system has all eigenvalues smaller than ρ, in modulus, for a given positive p ≤ 1. The solution is based on geometric tools, and it makes use of the outer observable subspace notion.

Current‐estimation‐based iterative algorithms for solving periodic Lyapunov matrix equations

In this study, two novel iterative algorithms are presented to solve the Lyapunov matrix equations appearing in discrete-time periodic linear systems. In both algorithms, a weighted combination of the estimation in the last and the current steps is used to update the estimation of the unknown matrices. It is shown that the sequences generated by the proposed algorithms with zero initial conditions monotonically converge to the unique positive definite solution of the periodic Lyapunov matrix equation if the associated system is asymptotically stable. Finally, a numerical example is used to illustrate the effectiveness of the proposed algorithms.

Robust stabilization of discrete-time periodic linear systems for tracking and disturbance rejection

In analogy to the Kucera–Youla parametrization, we construct and parametrize all stabilizing controllers of a stabilizable linear periodic discrete-time input/output system, the plant. We establish a necessary and sufficient algebraic condition for the existence of controllers among these for which the output of the plant tracks a given reference signal in spite of disturbance signals on the input and the output of the plant. With a minor additional assumption, the tracking stabilizing controllers are robust. As in the linear time-invariant (LTI) case, the reference and disturbance signals are assumed to be generated by an autonomous system. Our results are the analogs for periodic behaviors of the corresponding LTI results of Vidyasagar. A completely different approach to stabilization and control of discrete periodic systems was developed by Bittanti and Colaneri. We derive a categorical duality between periodic behaviors over the time-axis of natural numbers and finitely generated modules over a suitable noncommutative ring of difference operators and use this for the proof of the main stabilization and control results. Morita’s theory of equivalences between module categories is employed as an essential algebraic tool. All results of the paper are constructive.