Uncertain Discrete Systems Research Articles

Memoryless Dynamic Output-Feedback Stabilization for Discrete-Time Closed-Loop Robot Systems With Nonlinear Uncertainties and Multiple Time-Delays

The dynamic output-feedback control problem is addressed for a class of nonlinear discrete uncertain systems with multiple time-delays. First, the system is decomposed into two subsystems based on the output and input matrix. Second, the compensator is designed for first the subsystem, and the output feedback controller is designed based on the second subsystem and compensator. Then, by choosing a Lyapunov–Krasovskii functional, we show that the developed controller makes the solutions of the closed-loop system exponentially convergent to a ball. Compared with previous work, the developed controller only depends on the system output. The design conditions of the controller are relaxed because of the proposed dynamic compensator. Furthermore, the results are extended to a general nonlinear system and a robot system. Finally, numerical examples are included to show the effectiveness of the theoretical results.

Multi-constrained fuzzy intelligent control for uncertain discrete systems with complex noises: an application to ship steering systems

ABSTRACTIn this paper, a performance constrained fuzzy controller design is investigated for the nonlinear uncertain discrete-time ship steering system with complex noises. The performance constraints considered in this paper include individual state variance constraint and passivity constraint. The nonlinear discrete-time ship steering system considered in this paper is represented by a discrete-time Takagi–Sugeno fuzzy model with perturbations, additional noise and multiplicative noise. According to this complex nonlinear discrete-time stochastic system, a fuzzy controller design approach is investigated based on Lyapunov theory, passivity theory and covariance control theory. Some sufficient conditions are derived to satisfy stability constraint, individual state variance constraint and passivity constraint, simultaneously. These sufficient conditions are constructed as linear matrix inequality forms that can be solved by the convex optimal programming algorithm. Finally, some simulation results are provided to verify the validity and applicability of the proposed fuzzy control approach with multiple performance constraints.

Robust sliding mode control for uncertain discrete singular systems with time-varying delays and external disturbances

In this paper, a sliding mode control (SMC) of uncertain discrete singular systems with external disturbances and time-varying delays is under consideration. By use of the free weighting matrices and the Lyapunov–Krasovskii functional, a delay-dependent sufficient condition is given in strict linear matrix inequality (LMI) format to guarantee the sliding mode dynamics to be admissible (regular, causal and stable). Furthermore, a proposed SMC law and an adaptive SMC law are synthesized to make sure that the trajectories of system can be driven to a region near equilibrium point in finite time. Finally, a numerical example is designed to display the effectiveness of the control scheme. All these results are expected to propose a new approach for the research on SMC of discrete time-delay singular systems.

New classes of stabilizable uncertain systems

Consideration was given to the new classes of continuous and discrete uncertain systems where the elements of the control plant matrix represent physically realizable arbitrary functionals with only the boundaries of their variation known. The stabilizing controls robust in the elements of the control plant matrix were constructed using special quadratic Lyapunov functions.

Robust sliding mode control for uncertain discrete singular systems with time-varying delays

ABSTRACTThis paper investigates robust sliding mode control (SMC) for discrete singular systems which include time-varying delays, parameter uncertainties and nonlinear perturbations. An appropriate discrete sliding surface function is constructed such that the corresponding sliding mode dynamics are gained. By using some free-weighting matrices, a linear matrix inequality constraint is established to make sure that the closed-loop system is regular, causal and stable. Furthermore, in consideration of the improved discrete reaching condition, an SMC law is synthesised for reaching motion and the chattering can be weakened, while few existing papers focus on how to employ it to study the SMC problem for the discrete singular systems with time-varying delays. At last, the designed law is tested through an example.

A fault-tolerant approach to the control of a battery assembly system

The paper concerns fault-tolerant control of a real battery assembly system which is under a pilot implementation at RAFI GmbH Company (one of the leading electronic manufacturing service providers in Germany). The proposed framework is based on an interval analysis approach, which along with max-plus algebra, allows describing uncertain discrete event system such as the production one being considered in this paper. Having a mathematical system description, a model predictive control-based fault tolerant strategy is developed which can cope with both processing, transportation and mobile robot faults. In particular, it enables tolerating (up to some degree) the influence of these faults on the overall system performance. As a result, a novel robust predictive fault-tolerant strategy is developed that is applied to the advanced battery assembly system. The final part of the paper shows the implementation and experimental validation of the proposed strategy. The proposed approach is tested against single as well as simultaneous faults concerning processing, transportation and mobile robots.

Synthesis of a robust controller with reduced dimension by the Loop Shaping Design Procedure and decomposition based on Laguerre functions

In this paper, we present the synthesis of a robust controller for uncertain discrete systems. The synthesis method of such a robust controller is the generalization of the Loop Shaping Design Procedure (LSDP) approach of McFarlane and Glover in the discrete case based on the work of Gu et al. We exploit the bilinear transform known as Tustin’s method in order to formulate the discrete loop shaping technique. A discrete weighting filter and a shaped discrete plant result from this technique. By taking into account the coprime factor uncertainty representation for the resulting shaped plant and by applying the small gain theorem, we define the concept of the robust stabilization of the discrete LSDP approach. This concept is based on the resolution of an optimization problem characterized by the maximum stability margin for the synthesis of the robust controller. To calculate the robust controller we transform this problem to a standard robust [Formula: see text] controller design based on the resolution of the Riccati equations. Also, we present the gap metric theory to characterize the controller’s robustness. We note that the resulting final controller is the combination of the discrete weighting filter and the robust controller. We propose then to exploit the recent work of Bouzrara et al. in order to develop a reduced robust controller by expanding the final controller on two independent Laguerre orthonormal bases. The discrete LSDP and the reduced controller approaches were validated on a Continuous Stirred Tank Reactor chemical reactor for a set of different equilibrium points in order to take into account the nonlinearities.

Robust multiobserver design for discrete uncertain nonlinear systems with time-varying delay

In this paper, a robust multiobserver is proposed for the state estimation of discrete-time uncertain nonlinear systems with time-varying delay. The designed multiobserver is based on the decoupled multimodel approach. Unlike the classically used multimodel structures, the decoupled multimodel provides a flexibility of modelling. Indeed, the partial models’ structures can be adapted to the complexity of the system in each operating regime, thus the partial models can be with different dimensions. Delay-dependent sufficient conditions for the synthesis of a robust multiobserver against norm-bounded parametric uncertainties and in the presence of measurement noise are established in terms of linear matrix inequalities. A simulation example is given to illustrate the effectiveness of the designed multiobserver.

Guaranteed Cost Control of Uncertain Discrete Time Systems Subjected to Actuator Saturation via Homogenous Polynomial Lyapunov Function

Estimation of an optimal controller is a fundamental problem in control engineering and is widely known as Optimization. Numerous computation and numerical techniques have been evolved during the past years for the estimation of the optimal solution. Optimal control of a discrete-time system is concerned with optimizing a given objective function using “Homogenous Polynomial Lyapunov Function (HPLF)”. This research focuses upon the design of optimal guaranteed cost controller (GCC) for discrete-time uncertain system using HPLF. The uncertainties are assumed to be norm bounded uncertainties. The effect of actuator saturation is also incorporated in the system. Sufficient conditions for the existence of HPLF are derived in term of Linear Matrix Inequalities (LMI). The LMI approach has the advantage that it can be solved efficiently using Convex Optimization. LMI’s combined with HPLF helps to design the guaranteed cost controller which minimizes the cost by minimizing cost function. Furthermore, the state trajectories and the invariant set is also shown to observe the overall performance.

Robust output feedback control of 2D discrete systems with actuator saturation and time-varying delay

An observer-based dynamic output feedback H∞ controller is proposed for a class of two-dimensional (2D) uncertain discrete systems described by the Roesser model with actuator saturation, time-varying state delay and external disturbances. First, a delay-dependent Lyapunov stability condition is derived in linear matrix inequality (LMI) form which uses the reciprocal convex approach and H∞ disturbance attenuation performance is also analysed. Secondly, a convex hull is adopted to represent the saturation nonlinearity. The H∞ control synthesis for uncertain 2D discrete systems is described by a Roesser model subjected to actuator saturation and external disturbances using an observer-based dynamic output feedback approach. Some practical examples are provided to highlight the usefulness of the presented results.

Robust Kalman Filtering for Uncertain Discrete Time-Varying System with State-Delay

In this paper, a robust finite-horizon kalman filter for discrete time-varying uncertain system with constant state-delay both in state and output matrices. The system parameters uncertainties are expected in the state, output and white noise covariance matrices. The upper bound on the estimation error covariance is given and minimized. In addition, the filter parameters state, state delay and gain matrices are optimized to give a minimal upper bound on the estimation error covariance for all admissible uncertainties. A numerical example is presented to illustrate the applicability of this method.

Stability criterion for uncertain 2-$D$ discrete systems with interval-like time-varying delay employing quantization/overflow nonlinearities

This paper considers the problem of global asymptotic stability of a class of two-dimensional (2-$D$) uncertain discrete systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model under the influence of various combinations of quantization/overflow nonlinearities and interval-like time-varying delay in the state. The systems under consideration involve parameter uncertainties that are assumed to be deterministic and norm-bounded. A delay-dependent stability criterion is established by bounding the forward difference of the 2-$D$ Lyapunov functional using the reciprocally convex approach. The criterion is compared with a recently reported criterion.

Robust admissibility and admissibilisation of uncertain discrete singular time-delay systems

abstractThis paper is concerned with the characterisation of robust admissibility and admissibilisation for uncertain discrete-time singular system with interval time-varying delay. Considering the norm-bounded uncertainty and the interval time-varying delay, a new comparison model is introduced to transform the original singular system into two connected subsystems. After this transformation, a singular system without uncertainty and delay can be handled by the Lyapunov–Krasovskii functional method. By virtue of the scaled small gain theorem, an admissibility condition of the original singular system is proposed in terms of linear matrix inequalities. Moreover, the problem of robust admissibilisation of uncertain discrete singular time-varying system is also studied by iterative linear matrix inequality algorithm with initial condition optimisation. Several numerical examples are used to illustrate that the results are less conservative than existing ones.

Stability Margin Determination of Linear Discrete‐Time Systems Described By Fuzzy Convex Sets Uncertainty

AbstractThis paper proposes a new method for determining the stability margin of uncertain linear discrete‐time systems with uncertainty described by fuzzy convex sets. Nonsymmetrical triangular membership functions describing the uncertainty of characteristic polynomial coefficients are considered. This method is based on transformation of the uncertain discrete systems to the continuous time area and generalization of Kharitonov theorem for determining the uncommon minimum confidence level that guarantees stability of the system.

Uncertain discrete systems: Stability, instability, and an attractor

The system xk + 1 = A(k)xk is considered, where A(k) є ℝn × ℝn is a matrix coefficients aij(k) of which are nonanticipating functionals of any nature. It is assumed that the elements of the first to (p + 1)st columns above the main diagonal and the elements of the (p + 1)st to (n–1)st columns below the main diagonal satisfy the condition \(\mathop {\sup }\limits_k |{a_{ij}}(k)| \leqslant a.\) For the other elements, the estimate \(\mathop {\sup }\limits_k |{a_{ij}}(k)| \leqslant \delta \) holds. An upper bound for the values of the parameter δ at which system (1) is globally exponentially stable if \(\mathop {\sup }\limits_k |{a_{ii}}(k)| 1\) is obtained. In the case where the elements on the main diagonal are functions of xk and satisfy the conditions \(\mathop {\sup }\limits_{|{x_k}| > R} |{a_{ii}}({k_|})| 1\)\(|(i \in \overline {1,n} )\) for 0 < r< R, system (1) has a global attractor.

Sliding Mode Control for Discrete Uncertain System with Time Delay

This paper considers the sliding mode control for discrete uncertain system with time delay. First, the discrete uncertain system is transformed into a simplified form; then some sufficient conditions of guaranteeing the asymptotic stability of the uncertain system are given; finally the effectiveness of the proposed method is confirmed by the simulations.

A New Delay-Dependent Stability Criterion for Uncertain 2-D Discrete Systems Described by Roesser Model Under the Influence of Quantization/Overflow Nonlinearities

This paper considers the problem of global asymptotic stability of a class of two-dimensional uncertain discrete systems described by the Roesser model under the influence of various combinations of quantization/overflow nonlinearities and interval-like time-varying delay in the state. The systems under consideration involve parameter uncertainties that are assumed to be deterministic and norm-bounded. A delay-dependent stability criterion is derived by estimating the forward difference of the Lyapunov functional based on the reciprocally convex approach. Examples are provided to illustrate the effectiveness of the proposed criterion.

Observer-Based Bounded Control for Discrete Time-Delay Uncertain Nonlinear Systems

A bounded controller is proposed for a class of uncertain discrete time-delay systems with nonlinearity and disturbance based on state estimator and disturbance observer technique. A state estimator is developed to estimate the unmeasured system state vector. Suppose that the disturbance is generated by an exogenous system; a disturbance observer is designed to estimate the unknown disturbance. The parameters of the state estimator and the disturbance observer are calculated by solving linear matrix inequalities (LMIs). By applying the outputs of the state estimator and the disturbance observer, the sufficient condition for the existence of the bounded controller is derived based on an appropriate Lyapunov function candidate. Under the developed bounded controller, the stability of the closed-loop system can be guaranteed. Simulation examples are provided to show the effectiveness of the proposed bounded control scheme.

A learning-based synthesis approach to decentralized supervisory control of discrete event systems with unknown plants

In this paper, we consider the problem of automatic synthesis of decentralized supervisor for uncertain discrete event systems. In particular, we study the case when the uncontrolled plant is unknown a priori. To deal with the unknown plants, we first characterize the conormality of prefix-closed regular languages and propose formulas for computing the supremal conormal sublanguages; then sufficient conditions for the existence of decentralized supervisors are given in terms of language controllability and conormality and a learning-based algorithm to synthesize the supervisor automatically is proposed. Moreover, the paper also studies the on-line decentralized supervisory control of concurrent discrete event systems that are composed of multiple interacting unknown modules. We use the concept of modular controllability to characterize the necessary and sufficient conditions for the existence of the local supervisors, which consist of a set of local supervisor modules, one for each plant module and which determines its control actions based on the locally observed behaviors, and an on-line learning-based local synthesis algorithm is also presented. The correctness and convergence of the proposed algorithms are proved, and their implementation are illustrated through examples.

An LMI approach to non-fragile robust optimal guaranteed cost control of 2D discrete uncertain systems

This paper addresses the problem of non-fragile robust optimal guaranteed cost control for a class of two-dimensional discrete systems described by the general model with norm-bounded uncertainties. Based on Lyapunov method, a new linear matrix inequality (LMI)-based criterion for the existence of non-fragile state feedback controller is established. Furthermore, a convex optimization problem with LMI constraints is formulated to select a non-fragile robust optimal guaranteed cost controller, which minimizes the upper bound of the closed-loop cost function. The merit of the proposed criterion in aspect of conservativeness over a recently reported criterion is demonstrated with the help of illustrative examples.