Discrete-time Periodic Systems Research Articles

Uniform Local and Global Asymptotic Stabilization of Nonlinear Periodic Discrete-Time Systems by State Feedback

Uniform Local and Global Asymptotic Stabilization of Nonlinear Periodic Discrete-Time Systems by State Feedback

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.

Stabilization of networked periodic piecewise linear systems under asynchronous switching with packet dropouts

In this paper, the networked stabilization of discrete-time periodic piecewise linear systems under transmission package dropouts is investigated. The transmission package dropouts result in the loss of control input and the asynchronous switching between the subsystems and the associated controllers. Before studying the networked control, the sufficient conditions of exponential stability and stabilization of discrete-time periodic piecewise linear systems are proposed via the constructed dwell-time dependent Lyapunov function with time-varying Lyapunov matrix at first. Then to tackle the bounded time-varying packet dropouts issue of switching signal in the networked control, a continuous unified time-varying Lyapunov function is employed for both the synchronous and asynchronous subintervals of subsystems, the corresponding stabilization conditions are developed. The state-feedback stabilizing controller can be directly designed by solving linear matrix inequalities (LMIs) instead of iterative optimization used in continuous-time periodic piecewise linear systems. The effectiveness of the obtained theoretical results is illustrated by numerical examples.

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.

Preview control for discrete-time periodic systems with state delay

State feedback and output feedback control strategies are developed for discrete-time periodic systems with state delay via periodic preview control (PPC) technique in this article. To utilize future reference information for the controller design, a periodic augmented error system (PAES) with delay is constructed using the state augmentation technique and difference operator approach. Then, the PPC problem is converted into a stabilization problem. Furthermore, the state feedback and static output preview controllers are designed and sufficient conditions that guarantee the closed-loop stability of PAES are derived via the linear matrix inequality (LMI) approach. Finally, two examples are presented to illustrate the effectiveness of the proposed method.

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.

Reduced order dynamic controller synthesis conditions for LPV periodic discrete-time systems subject to input saturation

A method for the synthesis of reduced-order dynamic output-feedback controllers for periodic discrete-time LPV systems, subject to input saturations and external noises, is proposed in this paper. The controllers are designed to guarantee the robustness of the closed-loop system to exogenous noises, achieved through the minimization of a bound for the L2 gain. The proposed method is based on a two-stage procedure, where a state-feedback gain is computed in the first stage and then applied on the second step to generate the desired controller. A third stage is then developed to synthesize an anti-windup compensation to deal with the input saturation, but since the controller is not fixed in this step, the control system dynamics can be redesigned to better aggregate the desired robustness to the external noises and input saturation, decreasing the conservativeness of the technique. The validity of the proposed method is validate through a numerical example.

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.

Posture Stabilization Control of a One-Legged Hopping Robot Based on Periodic Discrete-Time System

Posture Stabilization Control of a One-Legged Hopping Robot Based on Periodic Discrete-Time System

Stability analysis and robust performance of periodic discrete-time uncertain systems via structured Lyapunov functions

This paper is concerned with stability and H2 analysis of periodic discrete-time uncertain systems. New conditions based on structured Lyapunov functions are provided in the form of parameter-dependent Linear Matrix Inequalities (LMIs). The Lyapunov functions make use of the periodic dynamic of the system to provide less conservative results in terms of scalar decision variables, LMI rows and H2 guaranteed cost. The proposed techniques are necessary and sufficient conditions to certify the stability of periodic discrete-time uncertain systems. Numerical experiments illustrate the improvement granted by the methods proposed in this paper over existing techniques.

Model reduction for discrete-time periodic systems with dissipativity

ABSTRACTThis paper is concerned with the model reduction problem for discrete-time periodic systems with time-varying delay. By utilising an adjusted reciprocally convex combination approach, some sufficient criteria are established to guarantee that the discrete-time periodic system with time-varying delay is asymptotically stable. Then, the reduced-order model is designed to approximate the original system. In the design, both the dissipativity and the Hankel norm performances are considered to test the model approximation, and correspondingly, the conditions are also established, which guarantee that the approximation error system is asymptotically stable with the dissipativity performance and the Hankel norm property, respectively. The convex linearisation and the projection are presented to design the reduced-order model, respectively. Finally, numerical examples are provided to verify the effectiveness of the design scheme.

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.

Observer-based state-feedback control design for LPV periodic discrete-time systems

This paper proposes new linear matrix inequality conditions to the observer-based control design problem for Linear Parameter Varying (LPV) periodic discrete-time systems. The time-varying parameters are considered to be available online (measured or estimated). In this way, periodic gain scheduling observers and controllers can be designed. Less conservative results in terms of the H∞ performance can be obtained if the variation rates of the parameters are known, otherwise arbitrarily fast variation can be considered. The H∞ guaranteed cost from the disturbance input to the error signal has been used for the observer design, and the state feedback control design considers the minimization of the H∞ bound from both error signal and input noise to the output of the controlled system. In this way, the designed controller provides robustness against inaccurate estimation. The analysis of the augmented system composed by the observer and controller is performed and numerical experiments illustrate the efficacy of the proposed method.

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.

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.

Linear estimation for discrete-time periodic systems with unknown measurement input and missing measurements

This paper considers the estimation problem for periodic systems with unknown measurement input and missing measurements. The missing measurements phenomenon is described by an independent and identically distributed Bernoulli process. The quality of the estimation achieved by an admissible filter is measured by a performance criterion described by the Cesaro limit of the mean square of the deviation between the remote signal and the estimated signal. By employing the minimum variance unbiased estimation technique, the periodic unbiased estimator is obtained, where the estimator gain is designed in terms of the unique periodic solution of a Lyapunov equation together with the periodic stabilizing solution of a Riccati equation. Finally, a numerical example is provided to show the effectiveness of the proposed estimation approach.

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.