Discrete-time Polytopic Systems Research Articles

Stabilization of polytopic discrete-time varying systems with rate and magnitude saturating actuators and bounded disturbances

This study tackles the critical challenge of designing parameter-dependent controllers for the wide class of discrete-time polytopic systems under rate and magnitude saturating actuators. Such a class finds widespread use in diverse applications spanning from medicine to industrial engineering. Our central contribution lies in developing novel convex parameter-dependent state feedback controller synthesis conditions that ensure regional and input-to-state stability, effectively mitigating the adverse effects of rate and magnitude-saturating actuators. Our approach considers the ℓ2/ℓ∞ gain between disturbance input and controlled output, offering performance specifications and retaining validity for specific initial conditions and amplitude-limited input disturbances. Moreover, our methodology is highly adaptable and suitable for a broad spectrum of systems, including LPV/quasi-LPV and T–S fuzzy systems. We leverage the position-type feedback model with speed limitation (PMSL) and the generalized sector condition to derive our controller synthesis method, resulting in a parallel-distributed-compensation strategy under standard assumptions, ensuring practicality and applicability to diverse system requirements. To highlight the effectiveness of our approach, we present numerical examples for comparative evaluation concerning the existing literature. Furthermore, we validate our methodology through real-time experiments conducted on a nonlinear coupled tank system, providing concrete evidence of its efficacy and feasibility for real-world implementation.

Security-Based Resilient Robust Model Predictive Control for Polytopic Uncertain Systems Subject to Deception Attacks and RR Protocol

This article is concerned with the security-based resilient robust model predictive control (RMPC) problem for a class of discrete-time polytopic uncertain systems with deception attacks and the round-robin (RR) protocol. The well-known RR protocol is adopted with the hope to reduce the communication burden, where the control signal is transmitted only at the given transmission instant. By means of the concept of bounded energy and the independent Bernoulli-distributed (BD) white sequences, a representative attack model is given. A definition of mean-square (MS) security in <italic 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> -sense is employed to reasonably explain the dynamics process of the controlled systems. We aim at designing a set of resilient RMPC controllers so as to make the closed-loop system state under consideration of deception attacks and the RR protocol enter and stay in a certain bounded region. Furthermore, with the aid of stochastic analysis methods and inequality analysis techniques, sufficient conditions are obtained to satisfy the desirable security requirements. To tackle the nonconvex obstacles resulting from attacks and the RR protocol, the cone complementary linearization (CCL) method is exploited to cast them into a convex optimization problem (OP) for its solvability. Then, an online OP regarding a certain upper bound of the concerned objective is provided for the solvability and resilient RMPC-based controller gains are obtained. Finally, two examples, including a high-purity distillation one and a numerical one, are used to demonstrate the effectiveness and the validity of the proposed techniques.

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.

Robust pole placement and random disturbance rejection for linear polytopic systems with application to grid-connected converters

This paper deals with an application of robust state feedback pole placement for discrete-time polytopic systems with stochastic disturbance attenuation. A set of stochastic signals is described using differential entropy functional. In the paper the parameter-independent robust state-feedback controller design procedure for linear discrete-time polytopic systems with theoretic-information constraints on the set of external random disturbances is proposed. Such additional constraints allow to find a tradeoff between H2 and H∞ control with better control energy consumption and improved robustness against random disturbances. This approach is applied to PWM inverters with LCL filters connected to the grid under uncertain parameters. Simulation results indicates that proposed controller allows to improve frequency and transient responses with better performance than existing ones.

Resilient distributed MPC for systems under synchronous round-robin scheduling

This paper is concerned with the resilient dynamic output-feedback (DOF) distributed model predictive control (DMPC) problem for discrete-time polytopic uncertain systems under synchronous Round-Robin (RR) scheduling. In order to alleviate the computation burden and improve the system robustness against uncertainties, the global system is decomposed into several subsystems, where each subsystem under synchronous RR scheduling communicates with each other via a network. The RR scheduling is adopted to avoid data collisions, however the updating information at each time instant is unfortunately reduced, and the underlying RR scheduling of subsystems are deeply coupled. The main purpose of this paper is to design a set of resilient DOF-based DMPC controllers for systems under the consideration of polytopic uncertainties and synchronous RR scheduling, such that the desirable performance can be obtained at a low cost of computational time. A novel distributed performance index dependent of the synchronous RR scheduling is constructed, where the last iteration information from the neighbor subsystems is used to deal with various couplings. Then, by resorting to the distributed RR-dependent Lyapunov-like approach and inequality analysis technique, a certain upper bound of the objective is put forward to establish a solvable auxiliary optimization problem (AOP). Moreover, by using the Jacobi iteration algorithm to solve such a problem online, the distributed feedback gains are directly obtained to guarantee the convergence of system states. Finally, two examples including a distillation process example and a numerical example are employed to show the effectiveness of the proposed resilient DMPC strategy.

Formulation of min-max model predictive control as a box-constrained robust least squares estimation problem

The purpose of this paper is to propose a new approach to the Min-Max Model Predictive Control (MMMPC) of Linear Time-Invariant Discrete-time Polytopic (LTIDP) systems. The purpose is to simplify the treatment of complex issues like stability and feasibility analysis of robust MPC as well as to reduce the complexity of the relative optimization procedure. The new approach is based on a two Degrees Of Freedom (2DOF) control scheme where the output r(k) of the feedforward Input Estimator (IE) is used as input forcing a stable closed-loop system Σf. Σf is the feedback connection of an LTIDP plant Σp with an LTI dynamic controller Σg. The task of Σg is to guarantee the quadratic stability of Σf, as well as the fulfillment of hard constraints on some physical variables of Σf for any input r(k) satisfying an ”a priori” determined admissibility condition. The input r(k) is computed by the feedforward IE through the on-line minimization of a worst case finite-horizon quadratic cost functional and is applied to Σf according to the usual receding horizon strategy. Rather than resorting to an ”ad hoc” software, the numerical complexity issue is here addressed reducing the number of both decision variables and constraints involved in the on-line constrained optimization procedure. This is obtained modeling r(k) as a B-spline function, which is known to be a universal approximator which also admits a parsimonious parametric representation. This allows us to reformulate the minimization of the worst case cost functional as a box-constrained Robust Least Squares (RLS) estimation problem which can be efficiently solved using Second Order Cone Programming (SOCP).

Output feedback preview tracking control for discrete-time polytopic time-varying systems

ABSTRACTThis paper focuses on the problem of static output feedback preview tracking control for discrete-time systems with time-varying parameters subject to a previewable reference signal. We develop a design method of a robust controller with integral and preview actions achieving robust tracking performance. First, an augmented error system including future information on reference signal is constructed by introducing two new related auxiliary variables to the original system state and input. This leads to a tracking problem being transformed into a regulator problem. Then, a previewable reference signal is fully utilised through reformulation of the output equation for the augmented error system while considering a static output feedback. Meanwhile, the static output preview control gains are solved explicitly by the proposed conditions. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.

Robust Disturbance Rejection by the Attractive Ellipsoid Method – Part II: Discrete-time Systems

Abstract This paper presents sufficient conditions for the robust stabilization of discrete-time polytopic systems subject to control constraints and unknown but bounded perturbations. The attractive ellipsoid method (AEM) is extended and applied to cope with this problem. To tackle the stabilization problem, new linear matrix inequality (LMI) conditions for robust state-feedback control are developed. These conditions ensure the convergence of state trajectories of the system to a minimal size ellipsoidal set despite the presence of non-vanishing disturbances. The developed LMI conditions for the AEM are extended to deal with the problem of gain-scheduled state-feedback control, where the scheduling parameters governing the time-variant dynamical system are unknown in advance but can be measured in real-time. A feature of the obtained conditions is that the state-space matrices and Lyapunov matrix are separated. The desired robust control laws are obtained by convex optimization. Numerical simulations are given to illustrate the feasibility of the proposed AEM for robust disturbance rejection.

Memory scheduling robust filter‐based fault detection for discrete‐time polytopic uncertain systems over fading channels

A novel memory scheduling robust H ∞ fault detection filter (FDF) is proposed for a class of discrete-time polytopic uncertain systems with fading channel communication networks. The main merit of this filter-based fault detection method is that it can significantly improve the robustness of FDF to attenuate influences from external disturbance, channel fading and model uncertainty on fault detection accuracy. Designing such FDF involves three main stages. First of all, a memory scheduling FDF structure is proposed based on the utilisation of weighted historical filter's states over interval instants, and a residual error system is formulated based on time partition and state augmented approaches. Then, the parameter-dependent Lyapunov method is further utilised to analyse the stochastic stability of the residual error system with the help of Finsler equivalent transformation. In the following, a two-stage optimisation algorithm combined with scalar parameters method is constructed to design memory scheduling FDF in a less conservative linear matrix inequality manner. Finally, a random numerical verification with 300 test systems and a case study of an industrial continuous-stirred tank reactor are exploited to show the effectiveness of obtained results.

Interpolation-based Off-line Robust MPC for Uncertain Polytopic Discrete-time Systems

In this paper, interpolation-based off-line robust MPC for uncertain polytopic discrete-time systems is presented. Instead of solving an on-line optimization problem at each sampling time to find a state feedback gain, a sequence of state feedback gains is pre-computed off-line in order to reduce the on-line computational time. At each sampling time, the real-time state feedback gain is calculated by linear interpolation between the pre-computed state feedback gains. Three interpolation techniques are proposed. In the first technique, the smallest ellipsoids containing the measured state are approximated and the corresponding real-time state feedback gain is calculated. In the second technique, the pre-computed state feedback gains are interpolated in order to get the largest possible real-time state feedback gain while robust stability is still guaranteed. In the last technique, the real-time state feedback gain is calculated by minimizing the violation of the constraints of the adjacent inner ellipsoids so the real-time state feedback gain calculated has to regulate the state from the current ellipsoids to the adjacent inner ellipsoids as fast as possible. As compared to on-line robust MPC, the proposed techniques can significantly reduce on-line computational time while the same level of control performance is still ensured.

An Approach to Distributed Robust Model Predictive Control of Discrete-Time Polytopic Systems

A suboptimal approach to distributed robust MPC for uncertain systems consisting of polytopic subsystems with coupled dynamics subject to both state and input constraints is proposed. The approach applies the dynamic dual decomposition method and reformulates the original centralized robust MPC problem into a distributed robust MPC problem. It is based on distributed on-line optimization by applying an accelerated gradient method. The suggested approach is illustrated on a simulation example of an uncertain system consisting of two interconnected polytopic subsystems.

Periodic Memory State-Feedback Controller: New Formulation, Analysis, and Design Results

This paper proposes a unified setup for robust stability and performance analysis and synthesis for periodic polytopic discrete-time systems. Relying on a general formulation for state-feedback periodic memory controllers and a new time-lifting, new sufficient LMI conditions for the existence of robust stability certificates and H2 guaranteed cost control laws are derived. Comparisons of the efficiency of different controller structures illustrate these developments on a numerical example.

A Polyhedral Off-Line Robust MPC Strategy for Uncertain Polytopic Discrete-Time Systems

In this paper, an off-line synthesis approach to robust constrained model predictive control for uncertain polytopic discrete-time systems is presented. Most of the computational burdens are moved off-line by pre-computing a sequence of state feedback control laws that corresponds to a sequence of polyhedral invariant sets. The state feedback control laws computed are derived by minimizing the nominal performance cost in order to improve control performance. At each sampling instant, the smallest polyhedral invariant set containing the currently measured state is determined. The corresponding state feedback control law is then implemented to the process. The controller design is illustrated with two examples in chemical processes. The proposed algorithm is compared with an ellipsoidal off-line robust model predictive control algorithm derived by minimizing the worst-case performance cost and an ellipsoidal off- line robust model predictive control algorithm derived by minimizing the nominal performance cost. The results show that the proposed algorithm can achieve better control performance. Moreover, a significantly larger stabilizable region is obtained.

An off-line robust MPC algorithm for uncertain polytopic discrete-time systems using polyhedral invariant sets

In this paper, an off-line synthesis approach to robust model predictive control (MPC) using polyhedral invariant sets is presented. Most of the computational burdens are moved off-line by computing a sequence of state feedback control laws corresponding to a sequence of polyhedral invariant sets. At each sampling time, the smallest polyhedral invariant set that the currently measured state can be embedded is determined. The corresponding state feedback control law is then implemented to the process. The controller design is illustrated with two examples. Comparisons between the proposed algorithm and an ellipsoidal off-line robust MPC algorithm have been undertaken. The proposed algorithm yields a substantial expansion of the stabilizable region. Therefore, it can achieve less conservative result as compared to an ellipsoidal off-line robust MPC algorithm.

Robust Constrained MPC based on Nominal Performance Cost with Applications in Chemical Processes

Abstract In this paper, a synthesis approach to robust constrained model predictive control (MPC) for uncertain polytopic discrete-time systems is presented. An overall algorithm is derived by using parameter-dependent Lyapunov function. The nominal model of the plant is included in the controller design in order to improve control performance. The optimization problem at each time step is formulated as the convex optimization problem involving linear matrix inequalities (LMI). Thus, the algorithm is computationally tractable. The algorithm is proved to guarantee robust stability. The controller design is illustrated with an example of continuous stirred-tank reactor.

An ellipsoidal off-line robust model predictive control strategy for uncertain polytopic discrete-time systems

In this paper, an off-line robust model predictive control strategy for uncertain polytopic discrete-time systems is presented. The on-line computational burdens are reduced by precomputing off-line the sequence of state feedback gains corresponding to the sequences of nested ellipsoids. The number of sequences of nested ellipsoids constructed is equal to the number of vertices of polytope. At each sampling instant, the smallest ellipsoid containing the currently measured state is determined in each sequence of ellipsoids. The real-time state feedback gain is then calculated by solving a numerically low-demanding optimization problem. The nominal model of the plant is included in the controller design in order to improve control performance. An overall algorithm is proved to guarantee robust stability.

Parameter-varying state feedback control for discrete-time polytopic systems

This article provides sufficient stability conditions for time-varying discrete-time polytopic systems along with H ∞ performance optimisation. First, the homogeneous polynomially parameter-dependent matrix function (HPP-DMF for abbreviation) is introduced. Then, based on the HPP-DMF, new parameter-varying state feedback controller and Lyapunov functions are designed. By applying the controller and Lyapunov functions, some new conditions, which are expressed in terms of linear matrix inequalities (LMIs) and admit more freedom in guaranteeing the stability of closed-loop systems, are obtained. In addition, it is shown that as the degree increases the conditions become less conservative. Although the number of LMIs increases, each LMI is simple and easy to be fulfilled. Furthermore, the determination of guaranteed disturbance attenuation is addressed. Finally, two examples are given to demonstrate the applicability of the proposed approach.

Dilated LMI conditions for time-varying polytopic descriptor systems: the discrete-time case

This article addresses the admissibility analysis and state-feedback control synthesis problems of discrete-time polytopic descriptor systems with possibly time-varying parameters. First, we present a new necessary and sufficient strict linear matrix inequality (LMI) condition for the admissibility analysis of linear time-invariant (LTI) descriptor systems. Then, based on the concept of poly-quadratic admissibility, introduced in this article, we extend this result to the admissibility analysis of possibly time-varying parameter-dependent descriptor systems. This extension, which applies to both uncertain and measurable parameters, uses parameter-dependent Lyapunov functions and employs two slack variables. The extended conditions are also expressed as strict LMI conditions which are easily tractable numerically compared to the non-strict ones often encountered when dealing with descriptor systems. Note that we have proposed two separate admissibility analysis conditions: one directly exploitable for state estimation and the other for state-feedback control. The need for different admissibility analysis conditions for each synthesis problem is motivated by the fact that the duality between state estimation and state-feedback control, which hold in the case of measurable parameters, does not hold when dealing with uncertain ones. In our approach, we overcome the absence of such duality by considering a dilation only on the dynamical part of the descriptor system. The application of our analysis result to both robust state-feedback control and polytopic state-feedback control are also presented in this article. Our analysis results extend to descriptor systems, some existing results developed for regular systems.

Necessary and sufficient condition for robust stability of discrete-time descriptor polytopic systems

This study investigates the robust stability of discrete-time descriptor polytopic systems. Necessary and sufficient conditions for robust stability expressed as parameterised linear matrix inequalities (LMIs) are obtained. Furthermore, the parameterised LMIs are reduced to parameter-independent and finite-dimensional LMIs. Finally, two numerical examples are given, illustrating the effectiveness of the proposed method.

Robust stabilization of polytopic discrete-time systems with time-varying state delay: A convex approach

Abstract Convex conditions, expressed as linear matrix inequalities (LMIs), for stability analysis and robust design of uncertain discrete-time systems with time-varying delay are presented in this paper. Delay-dependent and delay-independent convex conditions are given. This paper is particularly devoted to the synthesis case where convex conditions are proposed to consider maximum allowed delay interval. It is also presented some relaxed LMIs that yield less conservative conditions at the expense of increasing the computational burden. Extensions to cope with decentralized control and output feedback control are discussed. Numerical examples, including real world motivated models, are presented to illustrate the effectiveness of the proposed approach.