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For the control of discrete event systems, the notion of directed control refines that of supervisory control. A directed controller is one that selects at most one controllable event to be enabled at any state (without disabling any uncontrollable event), which is in fact how a discrete event control is implemented. In contrast, a supervisory controller computes a maximal allowable set of controllable events at each state, leaving undecided exactly which controllable event should be enabled. In previous works, we developed a framework for the computation of optimal directed controllers and a polynomial synthesis algorithm for acyclic plants. In this paper, we present a novel synthesis approach for general plants, i.e., plants with or without cycles, thus providing a complete solution to the optimal directed control problem. The complexity of the approach remains polynomial in the size of plant. Index Terms--Automata, directed control, director, discrete event system, formal language, nonblocking, optimal control, supervisor, supervisory control.

Hysteresis hinders the effectiveness of smart materials in sensors and actuators. It is a challenging task to control the systems with hysteresis. This note discusses the adaptive control for discrete time linear dynamical systems preceded with hysteresis described by the Prandtl-Ishlinskii model. The time delay and the order of the linear dynamical system are assumed to be known. The contribution of the note is the fusion of the hysteresis model with adaptive control techniques without constructing the inverse hysteresis nonlinearity. Only the parameters (which are generated from the parameters of the linear system and the density function of the hysteresis) directly needed in the formulation of the controller are adaptively estimated online. The proposed control law ensures the global stability of the closed-loop system, and the output tracking error can be controlled to be as small as required by choosing the design parameters. Simulation results show the effectiveness of the proposed algorithm. Index Terms--Adaptive control, discrete time linear systems, hysteresis, Prandtl-Ishlinskii model.

Iterative learning control (ILC) is a learning technique used to improve the performance of systems that execute the same task multiple times. Learning transient behavior has emerged as an important topic in the design and analysis of ILC systems. In practice, the learning control is often low-pass filtered with a "Q-filter" to prevent transient growth, at the cost of performance. In this note, we consider linear time-invariant, discrete-time, single-input single-output systems, and convert frequency-domain uncertainty models to a time-domain representation for analysis. We then develop robust monotonic convergence conditions, which depend directly on the choice of the Q-filter and are independent of the nominal plant dynamics. This general result is then applied to a class of linear time-varying Q-filters that is particularly suited for precision motion control. Index Terms--Iterative learning control (ILC), monotonic convergence, motion control, robust, time-varying filters.

A periodic adaptive control approach is proposed for a class of nonlinear discrete-time systems with time-varying parametric uncertainties that are periodic, and the only prior knowledge is the periodicity. The new adaptive controller updates the parameters and the control signal periodically in a point-wise manner over one entire period, and in the sequel, achieves the asymptotic tracking convergence. The result is further extended to a scenario with mixed time-varying and time-invariant parameters, and a hybrid classical and a periodic adaptation law is proposed to handle the scenario more appropriately. The extension of the periodic adaptation to systems with unknown input gain, higher order dynamics, and tracking problems is also discussed. Index Terms--Adaptive control, discrete-time system, nonlinear process, periodic adaptation, time-varying parameters.

This paper investigates the evaluation of infinitesimal perturbation analysis (IPA) estimates that have been derived based on a stochastic fluid model (SFM) using data observed from the sample path of a discrete event system (DES). First, we show that a straightforward implementation of the SFM-based IPA estimates may yield biased estimates when the data are obtained from the actual DES. Then, in order to better approximate the sample path of the DES, we propose a special case of SFM where the arrival and service processes are modeled by piecewise constant ON/OFF sources. The proposed SFM violates some of the assumptions made in [1]-[4], and, as a result, the sample derivatives no longer exist. However, using the proposed SFM, we obtain the left and right sided sample derivative estimates. As shown in this paper, the sided sample derivatives are much better in approximating the required derivatives compared to the straightforward implementation of the SFM-based IPA estimates. Index Terms--Discrete event systems (DESs), perturbation analysis, stochastic fluid models (SFMs).

In this paper, we propose a new approach to the theory of finite multichain Markov decision processes (MDPs) with different performance optimization criteria. We first propose the concept of nth-order bias; then, using the average reward and bias difference formulas derived in this paper, we develop an optimization theory for finite MDPs that covers a complete spectrum from average optimality, bias optimality, to all high-order bias optimality, in a unified way. The approach is simple, direct, natural, and intuitive; it depends neither on Laurent series expansion nor on discounted MDPs. We also propose one-phase policy iteration algorithms for bias and high-order bias optimal policies, which are more efficient than the two-phase algorithms in the literature. Furthermore, we derive high-order bias optimality equations. This research is a part of our effort in developing sensitivity-based learning and optimization theory. Index Terms--Average optimality, bias optimality, discrete-event systems, Markov decision processes (MDPs), nth-bias optimality, nth potentials, policy iteration.

Work on decentralized discrete-event control systems is extended to handle the case when, instead of always observing or never observing an event, a supervisor may observe only some occurrences of a particular event. Results include a necessary and sufficient condition for solving this version of the decentralized problem (which is analogous to the co-observability property used in the standard version of the problem) and a method for checking when this condition holds. In this paper, whether an event is observed by a given agent is dependent on that agent's state (or the string of events that agent has seen so far). This model of event observation is applicable to problems where a supervisor communicates observations of event occurrences to another supervisor to help the other one make control decisions. Index Terms--Decentralized control, discrete-event systems, partial observation, supervisory control.

In this paper, a linear matrix inequality technique for the state estimation of discrete-time, nonlinear switched descriptor systems is developed. The considered systems are composed of linear and nonlinear parts. An observer giving a perfect unknown input decoupled state estimation is proposed. Sufficient conditions of global convergence of observers are proposed. Numerical examples are given to illustrate this method. Index Terms--Hybrid systems, polyquadratic stability, switched descriptor systems, unknown input (UI) observers.

The traditional approach for sliding mode control design has been the design of a controller to achieve a predesigned sliding objective. However, not much research has been carried out on the design of the sliding surface. This note presents a technique for designing a sliding surface such that when confined to the surface, the closed-loop system has optimality in the linear quadratic sense. The paper also proposes a multirate-output-feedback-based controller that leads the system to the aforementioned optimal sliding mode. Index Terms--Optimal control, output feedback, sliding mode control, uncertain systems.

This paper addresses the problem of cooperative control of a team of distributed agents with decoupled nonlinear discrete-time dynamics, which operate in a common environment and exchange-delayed information between them. Each agent is assumed to evolve in discrete-time, based on locally computed control laws, which are computed by exchanging delayed state information with a subset of neighboring agents. The cooperative control problem is formulated in a receding-horizon framework, where the control laws depend on the local state variables (feedback action) and on delayed information gathered from cooperating neighboring agents (feedforward action). A rigorous stability analysis exploiting the input-to-state stability properties of the receding-horizon local control laws is carried out. The stability of the team of agents is then proved by utilizing small-gain theorem results. Index Terms--Constrained systems, cooperative control, model predictive control, nonlinear systems, receding-horizon control.

In this paper, we consider the use of nonlinear networks towards obtaining nearly optimal solutions to the control of nonlinear discrete-time (DT) systems. The method is based on least squares successive approximation solution of the generalized Hamilton-Jacobi-Bellman (GHJB) equation which appears in optimization problems. Successive approximation using the GHJB has not been applied for nonlinear DT systems. The proposed recursive method solves the GHJB equation in DT on a well-defined region of attraction. The definition of GHJB, pre-Hamiltonian function, HJB equation, and method of updating the control function for the affine nonlinear DT systems under small perturbation assumption are proposed. A neural network (NN) is used to approximate the GHJB solution. It is shown that the result is a closed-loop control based on an NN that has been tuned a priori in offline mode. Numerical examples show that, for the linear DT system, the updated control laws will converge to the optimal control, and for nonlinear DT systems, the updated control laws will converge to the suboptimal control. Index Terms--Generalized Hamilton-Jacobi-Bellman (BHJB) equation, neural network (NN), nonlinear discrete-time (DT) system.

In this note, we investigate the detectability problem in discrete event systems. We assume that we do not know initially which state the system is in. The problem is to determine the current and subsequent states of the system based on a sequence of observations. The observation includes partial event observation and/or partial state observation, which leads to four possible cases. We further define four types of detectabilities: strong detectability, (weak) detectability, strong periodic detectability, and (weak) periodic detectability. We derive necessary and sufficient conditions for these detectabilities. These conditions can be checked by constructing an observer, which models the estimation of states under different observations. The theory developed in this note can be used in feedback control and diagnosis. If the system is detectable, then the observer can be used as a diagnoser to diagnose the failure states of the system. Index Terms--Detectability, discrete event systems, observability, state estimation.

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem. Index Terms--Constrained hybrid control, controllable processing times, flow shop, hierarchical decomposition, optimal control.

Currently, one of the most actively researched approaches regarding the design of deadlock avoidance policies for sequential resource allocation systems is based on concepts and techniques provided by the, so called, theory of regions, that addresses the broader problem of synthesizing PN models with prespecified behaviors. However, one limitation of the theory of regions and its aforementioned derivatives is that they cannot be applied when the target behavior has a nonconvex representation in the underlying state space. In this note, we show how this problem can be circumvented by appropriately generalizing the employed class of the candidate policies. Index Terms--Deadlock avoidance, discrete event systems (DES), mathematical programming, resource allocation systems (RAS), supervisory control.

This paper proposes a two-level hierarchical supervisory control scheme for a discrete event system (DES), where the low-level is represented in the standard Ramadge--Wonham framework and the high level is represented by a DES with flexible marking (DESFM). As shown, the flexibility in the definition of marking provided by a DESFM simplifies the modeling of the high-level marking behavior. This approach, when compared with other approaches, requires less events and states to model the high-level, while guaranteeing hierarchical consistency. Index Terms--Discrete event systems (DES), hierarchical control, supervisory control.

In this note, an output feedback based discrete-time sliding-mode control scheme is proposed. It incorporates a steady-state tracking requirement through the use of integral action. Previous work has shown that with an appropriate choice of sliding surface, discrete-time sliding-mode control can be applied to nonminimum phase systems. The original scheme employed static output feedback and this imposed restrictions on the class of systems to which it was applicable--specifically a certain "fictitious" subsystem was required to be output feedback stabilizable. The scheme proposed in this note includes a compensator which broadens the class of systems for which the results are applicable. In the presence of bounded matched disturbances, ultimate boundedness results are obtained. It is also shown that in the presence of a class of sector bounded uncertainty, asymptotic stability can be achieved. Index Terms--Discrete-time, linear matrix inequalities (LMIs), output feedback, sliding modes.

This paper presents a new approach to robust tracking control of the nonlinear sampled systems using a discrete-time fuzzy disturbance observer (DFDO). Novel update and control laws are proposed to guarantee that all the signals in the closed-loop control system are uniformly ultimately bounded (UUB) in a compact set. No persistence of excitation (PE) condition, nor the assumption on the slowness of the change of the fuzzy parameters, is required. In addition, a robustifying controller is designed to improve the tracking performance. Finally, a computer simulation example is presented to illustrate the effectiveness and the applicability of the suggested method. Index Terms--Discrete-time, filtered signal, fuzzy disturbance observer, projection operator, robustifying controller.

This paper presents a new approach for sliding-mode control (SMC) of discrete-time systems using the reaching law approach together with the fast output sampling (FOS) feedback technique. This method does not need the system states for feedback as it makes use of only the output samples for designing the controller. Thus, this methodology is more practical and easy to implement. Numerical example demonstrates the design technique. Simulation results show that the proposed FOS SMC technique produces the same results as obtained by state feedback SMC technique. Index Terms--Discrete-time system, output feedback, sliding-mode control.

The authors discuss supervisory control of probabilistic discrete-event systems in this paper. Supervisors for such systems will take probabilities of event occurrences into consideration in making control decisions. The authors introduce a new concept called "epsilon-containment of a given language" and subsequently study its algebraic properties. They discuss standard supervisor (with only enablement/disablement) synthesis problems and reset supervisor (with recovery mechanism) synthesis problems for both fully observed and partially observed systems. Index Terms: Containment, controllability, discrete-event systems, observability, probability.

This note presents the discrete-time version of the entropy formulation of optimal control problems developed by Saridis in [2]. Given a dynamical system, the uncertainty in the selection of the control is characterized by the probability distribution (density) function which maximizes the total entropy. We establish the equivalence between the optimal control problem and the optimal entropy problem and then decompose the total entropy into a term associated with the certainty equivalent control law, the entropy of estimation, and the so-called equivocation of the active transmission of information from the controller to the estimator. This provides a useful framework for studying the certainty equivalent and adaptive control laws.

A. HAURIE [1],[*] F. MORESINO [2] This paper proposes and tests an approximation of the solution of a class of piecewise deterministic control problems, typically used in the modeling of [...]