Periodic event-triggered state-feedback control for discrete-time linear systems

In this work we are interested in the stability and ℒ2-gain of hybrid systems with linear flow dynamics, periodic time-triggered jumps and nonlinear possibly set-valued jump maps. This class of hybrid systems includes various interesting applications such as periodic event-triggered control. In this paper we also show that sampled-data systems with arbitrarily switching controllers can be captured in this framework by requiring the jump map to be set-valued. We provide novel conditions for the internal stability and ℒ2-gain analysis of these systems adopting a lifting-based approach. In particular, we establish that the internal stability and contractivity in terms of an ℒ2-gain smaller than 1 are equivalent to the internal stability and contractivity of a particular discrete-time set-valued nonlinear system. Despite earlier works in this direction, these novel characterisations are the first necessary and sufficient conditions for the stability and the contractivity of this class of hybrid systems. The results are illustrated through multiple new examples.

In this paper, we were concerned with the periodic event-triggered asynchronous stabilization of a class of hybrid stochastic systems driven by continuous-time Markov chain and Brownian motion, where the measurements of state and mode were available only at sampling instants, and the control was diffusion-dependent. Static and dynamic periodic event-triggered control (PETC) strategies were proposed with a guaranteed minimum interevent time for every sample path solution. Different from the well-known input-to-state stability framework for stability and synthesis of event-triggered control systems, a comparison system approach was developed to show that if the hybrid stochastic system under continuous-time feedback control was $ p $th-moment exponentially stable, then there existed a small sampling period and event-triggering parameters such that the resulting event-triggered control hybrid stochastic system was almost surely exponentially stable. Particularly, the proposed PETC strategies could integrate the beneficial impacts of stochastic noises, which distinguished them from previous results. Two numerical examples were provided to illustrate the efficiency of the theoretical results.

In this paper, a novel event-triggered control (ETC) strategy is proposed by striking a balance between periodic sampled-data control and ETC. This leads to so-called periodic event-triggered control (PETC), in which the advantage of reduced resource utilisation is preserved on the one hand, while, on the other hand, the conditions that trigger the events still have a periodic character. The latter aspect has the advantage that the event-triggering condition has to be verified only at the periodic sampling times, instead of continuously, as in conventional ETC. To analyse the stability and the L 2 -gain properties of the resulting PETC systems, two different approaches will be presented based on (i) piecewise linear systems, and (ii) impulsive systems, respectively. Moreover, the advantages and disadvantages of each of the methods will be highlighted. The developed theory will be illustrated using a numerical example.

In this paper we study the stability and L 2 — gain properties of periodic event-triggered control (PETC) systems including time-varying delays. We introduce a general framework that captures these PETC systems and encompasses a class of hybrid systems that exhibit linear flow, aperiodic time-triggered jumps (possibly with different deadlines) and arbitrary nonlinear time-varying jump maps. New notions on the stability and contractivity (L 2 -gain strictly smaller than 1) from the beginning of the flow and from the end of the flow are introduced and formal relationships are deduced between these notions, revealing that some are stronger than others. Inspired by ideas from lifting, it is shown that the internal stability and contractivity in L 2 -sense of a continuous-time hybrid system in the framework is equivalent to the stability and contractivity in l 2 -sense of an appropriate time-varying discrete-time nonlinear system. These results recover existing works in the literature as special cases and indicate that analysing different discrete-time nonlinear systems (of the same level of complexity) than in existing works yield stronger conclusions on the L 2 -gain. At the end of the paper we indicate several extensions of the framework, which even include the possibility of the interjump times depending on the state, such that, for instance, self-triggered control systems can also be included allowing their stability and contractivity analysis. A numerical example is presented showing how stability and contractivity analyses are carried out for PETC systems with delays.

To estimate the growth rate of matrix products $A_{n}··· A_{1}$ with factors from some set of matrices $\mathscr{A}$, such numeric quantities as the joint spectral radius $ρ(\mathscr{A})$ and the lower spectral radius $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality $ρ(\mathscr{A})<1$ serves as a criterion for the stability of a system, and the inequality $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})<1 $ as a criterion for stabilizability.Given a set $\mathscr{A}$ of $N×M$ matrices and a set $\mathscr{B}$ of $M×N$ matrices. Then, for matrix products $A_{n}B_{n}··· A_{1}B_{1}$ with factors $A_{i}∈\mathscr{A}$ and $B_{i}∈\mathscr{B}$, we introduce the quantities $μ(\mathscr{A},\mathscr{B})$ and $η(\mathscr{A},\mathscr{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathscr{A},\mathscr{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}··· A_{1}B_{1}$ over all sets of matrices $A_{i}∈\mathscr{A}$ and the minimal growth rate over all sets of matrices $B_{i}∈\mathscr{B}$. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.

In this paper, stabilizing problems in control design are addressed for linear discrete-time systems, reflecting equality constraints tying together some state variables. Based on an enhanced representation of the bounded real lemma for discretetime systems, the existence of a state feedback control for such conditioned stabilization is proven, and an LMI-based design procedure is provided. The control law gain computation method used circumvents generally an ill-conditioned singular design task. The principle, when compared with previously published results, indicates that the proposed method outperforms the existing approaches, guarantees feasibility, and improves the steady-state accuracy of the control. Furthermore, better performance is achieved with essentially reduced design effort. The approach is illustrated on simulation examples, where the validity of the proposed method is demonstrated using one state equality constraint.

In this paper, we present an approach for designing structural constrained controllers for discrete-time linear systems, based on a new stabilizability property of the Riccati equation solution. First, the feedback stabilization problem under a general structural constraint is considered and a simple numerical procedure to solve it is presented. Special attention is given to the output feedback stabilization problem, for which sufficient conditions for the existence and convergence toward a stabilizing matrix are provided. Some examples are solved and comparisons with other methods available in the literature are made.

Periodic event-triggered control for networked control systems based on non-monotonic Lyapunov functions

This paper is concerned with the structural controllability analysis for discrete-time linear control systems with time-delay. By adding virtual delay nodes, the linear systems with time-delay are transformed into corresponding linear systems without time-delay, and the structural controllability of them is equivalent. That is to say, the time-delay does not affect or change the controllability of the systems. Several examples are also presented to illustrate the theoretical results.

While a large amount of research over the past two decades has focused on discrete abstractions of infinite-state dynamical systems, many structural and algorithmic details of these abstractions remain unknown. To clarify the computational resources needed to perform discrete abstractions, this paper examines the algorithmic properties of an existing method for deriving finite-state systems that are bisimilar to linear discrete-time control systems. We explicitly find the structure of the finite-state system, show that it can be enormous compared to the original linear system, and give conditions to guarantee that the finite-state system is reasonably sized and efficiently computable. Though constructing the finite-state system is generally impractical, we see that special cases could be amenable to satisfiability based verification techniques.

In this paper, the stability problem for event-triggered nonlinear control systems is addressed in the presence of stochastic denial of service (DoS) attacks on the communication network. Dynamic event-triggered control (DETC) and periodic event-triggered control (PETC) are considered under the stochastic attacks. The stochastic attacks are modeled by a stochastic hybrid timer and a logic variable which is used to identify whether the network is attacked or not. Then the entire closed-loop system can be modeled into the framework of stochastic hybrid systems. Uniform global asymptotic stability in probability is concluded for the resulting stochastic system by proving that the equilibrium set is uniformly globally stable in probability and each neighborhood of the equilibrium set is uniformly globally recurrent. A nonlinear system example is provided to illustrate the effectiveness of the proposed results.

In this paper, the periodic event-triggered suboptimal control (PETSOC) method is developed for continuous-time linear systems. Different from event-triggered control, where the triggering condition is monitored continuously, the developed PETSOC method only verifies the triggering condition periodically at sampling instants, which further reduces computational resources. First, the control gain of the PETSOC is designed based on the algebraic Riccati equation. Subsequently, the periodic event-triggering condition is proposed for the suboptimal control method, which is only verified at sampling instants periodically. The sampling period is determined and analyzed based on the continuous form of the triggering condition. Moreover, the stability and the performance upper bound of the closed-loop system with the PETSOC are proved. Finally, the effectiveness of the developed PETSOC is validated through simulation on an unstable batch reactor.

A Note on Stabilizing and Time-Optimal Controllers for Discrete-Time Linear Systems with Positive Controls

A method for computing time-varying dynamic output feedback controllers for discrete-time linear systems subject to input saturation is proposed. The method is based on a locally valid polytopic representation of the saturation term. From this representation, it is shown that, at each sampling time, the matrices of the stabilising time-varying controller can be computed from the current system output and from constant matrices obtained as a solution of some matrix inequalities. Linear matrix inequality-based optimisation problems are therefore proposed in order to compute the controller aiming at the maximisation of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of ℒ2 disturbance tolerance and rejection.

ABSTRACTIn this article, to tackle with the iteration-varying trail lengths and random initial state shifts, an average operator-based PD-type iterative learning control (ILC) law is firstly presented for linear discrete-time multiple-input multiple-output (MIMO) systems with vector relative degree. The proposed PD-type ILC law includes an initial rectifying action against initial state shifts, and pursues the reference trajectory tracking beyond the initial time points. As special cases of the PD-type ILC law, P-type and D-type ILC laws are then introduced. It is proved that for linear discrete-time MIMO systems with vector relative degree, the three proposed ILC laws can drive the varying trail lengths-based ILC tracking errors to zero in mathematical expectation beyond the initial time points. A numerical example is used to illustrate the effectiveness of the proposed ILC laws.