An LMI approach to stabilization of linear discrete-time periodic systems

The problem of stabilization of linear discrete-time periodic systems is considered. LMI based conditions for stabilization via static periodic state feedback as well as via static periodic output feedback are presented. In the case of state feedback, the conditions are necessary and sufficient whereas for output feedback the result is only sufficient as it depends on the particular state-space representation used to describe the system. The problem of quadratic stabilization in the presence of either norm-bounded or polytopic parameter uncertainty is also treated. As an application of the output feedback stabilization technique, we consider the problem of designing a stabilizing (respectively, quadratically stabilizing) static periodic output feedback controller for linear time-invariant discrete-time systems which are not stabilizable (respectively, quadratically stabilizable) by static constant output feedback.     

Satisfactory control of discrete-time linear periodic systems

In this paper satisfactory control for discrete-time linear periodic systems is studied. Based on a suitable time-invariant state sampled reformulation, periodic state feedback controller has been designed such that desired requirements of steady state covariance, H-infinity rejection bound and regional pole assignment for the periodic system are met simultaneously. By using satisfactory control theory, the problem of satisfactory periodic controller can be transformed into a linear programming problem subject to a set of linear matrix inequalities （LMIs）, and a feasible designing approach is presented via LMI technique. Numeric example validates the obtained conclusion.     

Strong stabilization with infinite multivariable gain margin through linear periodic control

The problem of the strong stabilization with infinite gain margin, with the additional requirement of a prescribed rate of convergence of the free responses, is addressed for linear time-invariant discrete-time multivariable plants in the case when unknown different scalar gains act either on the inputs or on the outputs. Necessary and sufficient conditions for the solvability of the problem by means of a stable linear periodic discrete-time output feedback dynamic controller are derived. Algorithmic procedures are given for designing the proposed periodic controllers.     

Robust control of linear time-invariant plants using periodic compensation

This paper considers the use and design of linear periodic time-varying controllers for the feedback control of linear time-invariant discrete-time plants. We will show that for a large class of robustness problems, periodic compensators are superior to time-invariant ones. We will give explicit design techniques which can be easily implemented. In the context of periodic controllers, we also consider the strong and simultaneous stabilization problems. Finally, we show that for the problem of weighted sensitivity minimization for linear time-invariant plants, time-varying controllers offer no advantage over the time-invariant ones.     

Optimal and robust controllers for periodic and multirate systems

The problem of optimal rejection of bounded persistent disturbances is solved in the case of linear discrete-time periodic systems. The solution consists of solving an equivalent time-invariant standard l¹ optimization problem subject to an additional constraint. This constraint assures the causality of the resulting periodic controller. By the duality theory, the problem is shown to be equivalent to a linear programming problem, which is no harder than the standard l¹ problem. Also, it is shown that the method of solution presented applies exactly to the problem of disturbance rejection in the case of multirate sampled data systems. Finally, the results are applied to the problem of robust stabilization of periodic and multirate systems     

Descent algorithms for optimal periodic output feedback control

Periodic output feedback is investigated in the context of linear-quadratic regulation for finite-dimensional time-invariant linear systems. Discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control. The optimal solution is obtained in two steps by separating the continuous-time from the discrete-time structure. First, the optimal pole placement problem under periodic output feedback is solved explicitly under the assumption that the behavior at the sample times has been specified in terms of a gain matrix G. Then the minimum value, which depends on G, is substituted into the overall objective. This results in a finite-dimensional nonlinear programming problem over all admissible gain matrices G. The solution defines the optimal periodic output feedback control via the formulas of the optimal pole placement problem. A steepest descent and a direct iterative method for solving this problem are formulated and compared. Numerical examples show that the performance using periodic output feedback is almost equivalent to that using optimal continuous-state feedback     

Stability and stabilization of discrete-time periodic linear systems with actuator saturation

This paper is concerned with the problems of stability and stabilization for discrete-time periodic linear systems subject to input saturation. Both local results and global results are obtained. For local stability and stabilization, the so-called periodic invariant set is used to estimate the domain of attraction. The conditions for periodic invariance of an ellipsoid can be expressed as linear matrix inequalities (LMIs) which can be used for both enlarging the domain of attraction with a given controller and synthesizing controllers. The periodic enhancement technique is introduced to reduce the conservatism in the methods. As a by-product, less conservative results for controller analysis and design for discrete-time time-invariant systems with input saturation are obtained. For global stability, by utilizing the special properties of the saturation function, a saturation dependent periodic Lyapunov function is constructed to derive sufficient conditions for guaranteeing the global stability of the system. The corresponding conditions are expressed in the form of LMIs and can be efficiently solved. Several numerical and practical examples are given to illustrate the theoretical results proposed in the paper.     

Output feedback receding horizon control of constrained systems

This paper considers output feedback control of linear discrete-time systems with convex state and input constraints which are subject to bounded state disturbances and output measurement errors. We show that the non-convex problem of finding a constraint admissible affine output feedback policy over a finite horizon, to be used in conjunction with a fixed linear state observer, can be converted to an equivalent convex problem. When used in the design of a time-varying robust receding horizon control law, we derive conditions under which the resulting closed-loop system is guaranteed to satisfy the system constraints for all time, given an initial state estimate and bound on the state estimation error. When the state estimation error bound matches the minimal robust positively invariant (mRPI) set for the system error dynamics, we show that this control law is time-invariant, but its calculation generally requires solution of an infinite-dimensional optimization problem. Finally, using an invariant outer approximation to the mRPI error set, we develop a time-invariant control law that can be computed by solving a finite-dimensional tractable optimization problem at each time step that guarantees that the closed-loop system satisfies the constraints for all time.     

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

In analogy to the Ku?era–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.     

Disturbance localization for left invertible linear periodic discrete-time systems

The disturbance localization problem for left invertible linear periodic discrete-time systems is solved using periodic state feedback controllers. The proposed technique is of algebraic nature and has the following two main characteristics: (i) It yields simple algebraic criteria for testing the solvability of the problem, as compared to known geometric criteria, which may not be so easy to check. (ii) It derives analytically the general expressions of all periodic controllers admissible for disturbance localization, as compared to known techniques, which lead to nonanalytic parametrizations of the admissible controllers via constructive procedures. Moreover, for the aforementioned class of periodic systems, the state feedback simultaneous disturbance localization and stabilization or pole placement problem is treated, and conditions for its solvability are established, on the basis of a decentralized control approach, that makes use of the equivalence between the above problem and the stabilization or pole placement problem of a general proper multichannel system by decentralized static output feedback.     

Robust performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.     

A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation

This paper is concerned with semi-global stabilization of discrete-time linear periodic systems subject to actuator saturation. Provided that the open loop characteristic multipliers are within the closed unit circle, a low gain feedback design approach is proposed to solve the problem by state feedback. Our approach is based on the solution to a parametric discrete-time periodic Lyapunov equation. The proposed approaches not only generalize the corresponding results for time-invariant systems to periodic systems, but also reveal some important intrinsic properties of this class of periodic matrix equations. A numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

A robust periodic pole assignment algorithm

In this note a robust periodic pole assignment algorithm is proposed for linear, time-invariant, discrete-time systems. The condition numbers of the eigenvector matrices of the closed-loop system are assumed as a robustness measure and a periodic state-feedback law is deduced by the minimization of the condition numbers associated to the eigenvectors of the monodromy matrix of the closed-loop system. The proposed periodic pole assignment algorithm has been tested on a number of examples, giving satisfactory results     

The model matching problem for periodic discrete-time systems

In this paper, the model matching problem via state feedback is studied for discrete-time periodic systems. It is shown that, under some zero-matching conditions, it is possible to assign a target input-output periodic system by means of a periodic state-feedback control law. Furthermore, this entails that a periodic system can be converted into a closed-loop time-invariant system via state feedback     

Cyclomonotonicity and stabilizability properties of solutions ofthe difference periodic Riccati equation

The Kalman filter associated with a discrete-time linear T-periodic system is tested. The problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first T values of the Kalman filter gain is stabilizing. Sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic Riccati equation. Potential applications are found in filter design, quasi-linearization techniques for the periodic Riccati equation, and the design of receding-horizon control strategies for periodic and multirate systems. When specialized to time-invariant systems, the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time-invariant Riccati equation and the existence of periodic stabilizing feedback     

Optimal state reconstruction algorithms for linear discrete-time systems

This paper deals with optimal time-invariant reconstruction of the state of a linear time-invariant discrete-time system from output measurements. The problem is analysed in two settings, depending on whether or not the present output measurement is available for the estimation of the present state. The results prove complete separation of observer and controller design for the optimal dynamic output feedback control with respect to a quadratic cost.     

Pole assignment for linear periodic systems by memoryless outputfeedback

We consider linear periodic discrete-time systems. We are interested in the problem of placing the poles of the monodromy map by means of periodic output feedback of the same or multiple period. It is well known that, in general, the poles of time-invariant systems cannot be assigned by constant output feedback. This is in contrast with what can be obtained in the context of time-variant systems. The main contribution of this paper is that periodic output feedback suffices for pole placement of periodic systems     

线性离散周期系统输出反馈参数化极点配置

采用周期输出反馈的途径, 考虑了线性离散时间周期系统的极点配置问题. 通过将闭环单值性矩阵进行一系列转化, 周期输出反馈律的求解问题可以转化为一类Sylvester矩阵方程的求解问题. 利用Sylvester矩阵方程的最新结果, 可以将输出反馈增益表示为参数化形式, 从而得到了能够实现极点配置的所有输出反馈控制器. 数值算例验证了方法的有效性.     

Model matching for discrete-time periodic systems with time-varying relative degree and order

This paper extends the well-known solution for the linear time invariant model matching problem to discrete-time periodic systems with time-varying relative degree and order. It is shown that a key step to the design of a periodic output feedback controller is to compute the stable inverse of the periodic system. Using input–output equations, this problem is solved and model matching is achieved with system internal stability.     

An integrated optimal control algorithm for discrete-time nonlinear stochastic system

Consider a discrete-time nonlinear system with random disturbances appearing in the real plant and the output channel where the randomly perturbed output is measurable. An iterative procedure based on the linear quadratic Gaussian optimal control model is developed for solving the optimal control of this stochastic system. The optimal state estimate provided by Kalman filtering theory and the optimal control law obtained from the linear quadratic regulator problem are then integrated into the dynamic integrated system optimisation and parameter estimation algorithm. The iterative solutions of the optimal control problem for the model obtained converge to the solution of the original optimal control problem of the discrete-time nonlinear system, despite model-reality differences, when the convergence is achieved. An illustrative example is solved using the method proposed. The results obtained show the effectiveness of the algorithm proposed.