Uncertain Discrete-time Linear Systems Research Articles

Robust H∞ observer-based model predictive controller for uncertain linear discrete-time systems due to external disturbances

This article investigates an observer-based robust model predictive control (RMPC) design to control the uncertain discrete-time linear systems with disturbances. To make a more practical scheme, it is supposed that the uncertain system has been faced with unknown disturbance and input constraints. The proposed RMPC approach is based on a state feedback control design that ensures the [Formula: see text] performance criterion to attenuate disturbance affections. Furthermore, in view of practical application, the control law is constructed based on the estimated states obtained from the Luenberger observer. Based on Lyapunov’s theory, the input to state practically stability (ISPS) of the closed-loop system is ensured. Appropriate conditions for the ISPS of the closed-loop system and the estimation error are obtained in terms of online linear matrix inequalities (LMIs) which lead to obtaining the time-varying gain matrices of both controller and observer. Finally, to validate the obtained results, the proposed approach is applied to a numerical example and it is compared with the existing control scheme and the superiority is proved.

Set-Valued State Estimator With Sparse and Delayed Measurements for Uncertain Discrete-Time Linear Systems

Set-Valued State Estimator With Sparse and Delayed Measurements for Uncertain Discrete-Time Linear Systems

Robust Optimal Control for Discrete-Time LTI Systems Over Multiple Additive White Gaussian Noise Channels

Unlike the traditional point-to-point control structure, control system design through network media introduces additional constraints due to limited capacities of the used communication channels. In this paper, we address the optimal performance control problem for linear uncertain discrete-time systems over multiple additive white Gaussian noise (AWGN) channels, where mean power constraints are imposed on the actuator side. The desired controller is aimed to robustly stabilize the plant, to minimize the linear quadratic (LQ) cost, and to satisfy diverse pre-specified mean power limits associated with individual elements of the control law, simultaneously. Within the framework of system level synthesis, the performance objective and power constraints, together with relevant upper bounds, over all admissible uncertainties are characterized in terms of system responses of the closed-loop system and statistics of the communication channels. Based on this characterization, numerically tractable algorithms are proposed for controller synthesis. The counterpart for deterministic systems is also discussed as a special case with corresponding algorithms being provided. Moreover, the relative performance loss incurred by the size of model uncertainties and the finite truncation is explicitly established. An example is also included to show the effectiveness of the present results.

Robust regulation of discrete-time systems subject to parameter uncertainties and state delay

This paper deals with the robust recursive regulation problem of uncertain discrete-time linear systems subject to unknown state delays. The variation rate between two consecutive delays is considered bounded. The parameter matrices are affected by norm-bounded uncertainties. Applying the lifting method and modeling the delay as a Markov chain, systems with state delays are converted to augmented delay-free Markovian jump linear systems. Then, a robust recursive linear quadratic regulator is obtained, solving an optimization problem through robust regularized least-squares approaches. The solution is given in terms of algebraic Riccati equations. We assess the proposed regulator through a numerical example and compare its performance with other robust control approaches.

State and unknown input set-valued estimation for uncertain linear discrete-time systems

Set-valued algorithms are designed in this work to compute rigorous bounds of the unknown inputs of uncertain discrete-time linear systems. First, based on the classical full-order and reduced-order unknown input observers, non-conservative numerical schemes are proposed to characterize the reachable set of the estimation error. Then, by using these results, set-membership estimators of the unknown inputs are presented. Moreover, the results of these estimators can be merged together via set-intersection operators to obtain more accurate estimates.

Robust peak-to-peak gain analysis using integral quadratic constraints

This work provides a framework to compute an upper bound on the robust peak-to-peak gain of discrete-time uncertain linear systems using integral quadratic constraints (IQCs). Such bounds are of particular interest in the computation of reachable sets and the l1-norm, as well as when safety-critical constraints need to be satisfied pointwise in time. The use of p-hard IQCs with a terminal cost enables us to deal with a wide variety of uncertainty classes, for example, we provide p-hard IQCs with a terminal cost for the class of parametric uncertainties. This approach unifies, generalizes, and significantly improves state-of-the-art methods, which is also demonstrated in a numerical example.

Robust Output Regulation for Uncertain Discrete-Time Linear Systems under the Effect of a Sinusoidal Disturbance

This paper studies the robust output regulation problem of uncertain single-input single-output (SISO) discrete-time linear systems. To reject the effect of a sinusoidal disturbance that only the frequency information is known prior, a novel strategy based on adaptive feedforward control (AFC) is developed. Compared with existing regulators for uncertain discrete-time systems, neither the knowledge on the sign of the real part or the imaginary part of the transfer function at the frequency of interest (the so-called strictly positive real (SPR)-like condition), nor persistence of excitation condition is required in this approach. Stability of the closed-loop system is rigorously analyzed using small-gain theorem and Lyapunov-based stability theory. Essentially, the proposed scheme guarantees that all signals of closed-loop system are bounded while the output of system asymptotically converges to zero, which is demonstrated by a numerical example.

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Backward reachability analysis is essential to synthesizing controllers that ensure the correctness of closed-loop systems. This article is concerned with developing scalable algorithms that underapproximate the backward reachable sets, for discrete-time uncertain linear and nonlinear systems. Our algorithm sequentially linearizes the dynamics and uses constrained zonotopes for set representation and computation. The main technical ingredient of our algorithm is an efficient way to underapproximate the Minkowski difference between a constrained zonotopic minuend and a zonotopic subtrahend, which consists of all possible values of the uncertainties and the linearization error. This Minkowski difference needs to be represented as a constrained zonotope to enable subsequent computation, but, as we show, it is impossible to find a polynomial-size representation for it in polynomial time. Our algorithm finds a polynomial-size underapproximation in polynomial time. We further analyze the conservatism of this underapproximation technique and show that it is exact under some conditions. Based on the developed Minkowski difference technique, we detail two backward reachable set computation algorithms to control the linearization error and incorporate nonconvex state constraints. Several examples illustrate the effectiveness of our algorithms.

Optimal discrete-time sliding-mode control based on recurrent neural network: a singular value approach

In this paper, a strategy involving the combination of optimal discrete-time sliding-mode control and recurrent neural networks is proposed for a class of uncertain discrete-time linear systems. First, a performance index based on the reaching law and the control signal is defined. Then, the constrained quadratic programming problem is formulated considering the limitations on the control signal as the static constraint. The dynamic and algebraic model of the neural network is derived based on the optimization conditions of the quadratic problem and their relationship with the projection theory. The proposed method prevents the chattering by selecting proper parameters of the twisting reaching law. The convergence of the neural network is analysed using the Lyapunov stability theory. A singular value-based analysis is employed for robustness of the proposed method. The stability conditions of the discrete-time closed-loop system are analysed by studying eigenvalues of the closed-loop matrix using the singular value approach. The performance of the proposed algorithm is assessed in simulated example in terms of chattering elimination, solution feasibility, and encountering uncertainties and is compared with the recently proposed DSMC methods in the literature.

Decentralised controller design for a large-scale linear discrete-time polytopic uncertain systems

One of the great challenges to control theory is to design controllers for the increasing size of uncertain large-scale systems. Such problems arise in many real-word applications and the systems are too large and too complex to be centralised controlled. For these reasons, the complex system is split into several interconnected subsystems and it is controlled in a decentralized fashion. In this article, a novel, original approach to the control of large-scale uncertain linear discrete-time systems by a decentralised controller is presented. In the existing references, such systems are divided into two groups: complex systems with strong or weak interactions. However, the new proposed approach depends on whether the system is stable or not. The design procedure consists of two steps: In the first step, the dynamic properties of the closed-loop decentralised controlled subsystems are determined in such a way that the stability, robustness, and performance of the closed-loop subsystems and the entire system are guaranteed. In the second step, a decentralised control algorithm must be designed that ensures the required properties of the subsystems that have been obtained in the first step. The advantage of this approach is that the decentralized controller design is performed on the subsystem level.

Reduced Order Positive Filter Design for Positive Uncertain Discrete-Time Linear Systems

The problems of reduced order \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> positive filter design for positive uncertain discrete-time linear systems are investigated in this letter. Due to the positivity constraint on the matrices of the filter, optimal \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> filters cannot be obtained through standard linear matrix inequality (LMI) methods, even in the context of full order filtering for precisely known systems. Therefore, new sufficient LMI conditions are proposed for \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> positive filter design for positive discrete-time linear systems, having as main advantage the fact that the filter matrices are variables of the problem. In this case, no structural constraints on the optimization variables (source of conservativeness) are needed to ensure positivity. Thanks to a relaxation in the stability of the filter, an iterative algorithm with a feasible initial condition is proposed, allowing the search for positive filters that assure an \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> or \pmb <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> guaranteed attenuation level for the filtered system. The conditions can deal with full or reduced order filtering, polytopic type uncertainty and structural constraints. Examples inspired on models borrowed from the literature illustrate the results.

Robust Kalman filter for systems subject to parametric uncertainties

State estimation plays a fundamental role in control systems that rely on the knowledge of the underlying system state, especially when it is not readily available. The Kalman filter is among the most popular techniques in this matter. Nonetheless, one of its well-known shortcomings is the assumption that an exact system model is available. The difficulty in fulfilling this premise justifies the demand for estimation strategies that limit the effect of model uncertainties. In this paper, we propose a robust Kalman filter for uncertain linear discrete-time systems. We assume a general setting in which all matrices of both the system and measurement models are subject to norm-bounded parametric uncertainties. By adopting a purely deterministic viewpoint and applying the penalty function method, we formulate a robust regularized least-squares estimation problem. We further analyze how the penalty parameter influences the estimation performance. The resulting filter is presented both in a symmetric matrix arrangement and as explicit algebraic expressions in a Kalman-like structure that is suitable for online applications. Under reasonable conditions, we show that the steady-state filter is stable and, for quadratically stable systems, guarantees a bounded error variance. With a numerical example, we evaluate the proposed filter and compare its performance with other robust filtering approaches.

Disturbance Rejection for Uncertain Discrete-Time Linear Systems Through Event-Triggered Control

Disturbance Rejection for Uncertain Discrete-Time Linear Systems Through Event-Triggered Control

Invariant cover: Existence, cardinality bounds, and computation

An invariant cover quantifies the information needed by a controller to enforce an invariance specification. This paper investigates some fundamental problems concerning existence and computation of an invariant cover for uncertain discrete-time linear control systems subject to state and control constraints. We develop necessary and sufficient conditions on the existence of an invariant cover for a polytopic set of states. The conditions can be checked by solving a set of linear programs, one for each extreme point of the state set. Based on these conditions, we give upper and lower bounds on the minimal cardinality of the invariant cover, and design an iterative algorithm with finite-time convergence to compute an invariant cover. We further show in two examples how to use an invariant cover in the design of a coder–controller pair that ensures invariance of a given set for a networked control system with a finite communication data rate.

Robust control Minkowski–Lyapunov functions

This article introduces, and studies, the robust control Minkowski–Lyapunov inequality, which is a natural generalization of the recently introduced control Minkowski–Lyapunov inequality. The generalization considers the setting of parametrically uncertain linear discrete time systems. The necessary and sufficient conditions for the characterization and existence of Minkowski functions that verify the robust control Minkowski–Lyapunov inequality are derived. The article also characterizes, and establishes topological properties of, the corresponding robustly stabilizing set-valued control map.

Reachability-Based Approach to Design Interval State Observers

This work aims at showing the advantage to consider the design problem of interval observers for uncertain discrete-time linear systems as a reachability problem. Based on the classical equations of the Lunenberger observer, a reachability algorithm is proposed to compute tight outer approximations of the reachable set of the system. Moreover, based on the symmetric property of interval vectors a more convenient version of this reachability algorithm is introduced. Through numerical examples, the simulation results of the proposed reachability approach are compared to those obtained by interval observers.

An auxiliary system discretization approach to Takagi-Sugeno fuzzy models

This paper proposes a new procedure for discretizing nonlinear systems described by Takagi-Sugeno fuzzy models. The discretization procedure consists of obtaining a linear auxiliary system that approximates the Takagi-Sugeno model over a sampling instant. By discretizing this auxiliary system, a norm bounded uncertain linear discrete-time system is found, which is capable of representing the fuzzy model. This auxiliary system, as well as the norm bounded uncertainty, is found by solving an optimization problem with Linear Matrix Inequality (LMI) constraints. To illustrate the discretization procedure, a constant state observer is synthesized based on simple LMI conditions and then applied to a real nonlinear Chua's circuit. Additionally, a state-feedback controller based on our discretization approach is synthesized and we obtain larger maximum sampling periods than other tested strategies from the literature.

Finite-frequency memory filter design for uncertain linear discrete-time systems: A polynomially parameter-dependent approach

This paper investigates the robust filtering problem for linear uncertain systems affected by noises in restricted frequency intervals. Different from traditional filter schemes, a finite-frequency memory filter is designed to generalize conventional memoryless ones in such a way that a sequence of latest output measurements are employed for current estimation. To be specific, a memory filter is sought which ensures that for all admissible uncertainties, the filtering error system is asymptotically stable with a prescribed noise-attenuation level in the restricted frequency range. To accomplish this, the finite-frequency specification is characterized by the generalized Kalman–Yakubovich–Popov (KYP) lemma, aiming at improving the capability of noise-attenuation over the given frequency range. Moreover, the homogeneous polynomially parameter-dependent technique is adopted to facilitate filter design and reduce conservatism further. Based on the proposed scheme, we prove rigorously that additional past output measurements contribute to less conservative results. Finally, a quarter-car model with an active-suspension system is used to validate the proposed scheme.

Robust set‐membership state estimator against outliers in data

Based on interval computation, a set-membership state estimator capable to manage a certain type of outliers in measurements is proposed for uncertain discrete-time linear systems. To achieve this purpose, two set-valued filtering techniques are implemented in the presented state estimation algorithm. The setting up of these techniques offers two main advantages. On one hand, the convergence of the estimated state enclosures width is guaranteed and, on the other hand, the algorithm robustness against outliers in data is ensured. That is, unlike former methods, the proposed set-valued state estimator preserves the framing property despite the presence of some false values in available sensors data. To show the efficiency and the performance of the introduced set-valued state estimator, it is compared, through two numerical examples, with an optimal interval observer selected from the literature for its high performance.

On the Reachable Set of Uncertain Closed Loop Discrete-Time Linear Systems

This work tackles the reachability problem of uncertain discrete-time linear systems controlled by estimated state feedback regulators. First an augmented system is considered in order to include the effect of the estimation error on the behavior of the closed loop system. Then, based on interval analysis, an interval predictor is proposed to compute tight trajectory tubes that contain in a guaranteed way the actual state trajectory of the augmented system. Moreover, under the standard controllability and observability assumptions of linear systems, the convergence of the width of these trajectory tubes is shown.