Decentralized tube‐based model predictive control of uncertain nonlinear multiagent systems

This paper addresses the problem of decentralized tube‐based nonlinear model predictive control (NMPC) for a general class of uncertain nonlinear continuous‐time multiagent systems with additive and bounded disturbance. In particular, the problem of robust navigation of a multiagent system to predefined states of the workspace while using only local information is addressed under certain distance and control input constraints. We propose a decentralized feedback control protocol that consists of two terms: a nominal control input, which is computed online and is the outcome of a decentralized finite horizon optimal control problem that each agent solves at every sampling time, for its nominal system dynamics; and an additive state‐feedback law which is computed offline and guarantees that the real trajectories of each agent will belong to a hypertube centered along the nominal trajectory, for all times. The volume of the hypertube depends on the upper bound of the disturbances as well as the bounds of the derivatives of the dynamics. In addition, by introducing certain distance constraints, the proposed scheme guarantees that the initially connected agents remain connected for all times. Under standard assumptions that arise in nominal NMPC schemes, controllability assumptions, communication capabilities between the agents, it is guaranteed that the multiagent system is input‐to‐state stable with respect to the disturbances, for all initial conditions satisfying the state constraints. Simulation results verify the correctness of the proposed framework.     

Stabilization and second‐order optimization for multimodule impulsive switched linear systems

In this work, we investigate stabilization and optimization issues for a class of multimodule impulsive switched linear systems. First, we establish a necessary and sufficient criterion on asymptotic stabilizability via a pathwise state‐feedback scheme, which achieves the merits of both time‐driven and state‐feedback mechanisms. Then, we present an impulse/switching instant optimization problem that usually arises in finite‐horizon optimal control. To reduce the computational burden, we propose a novel method via developing efficiently computable expressions for the cost function, the gradient vector, and the Hessian matrix. Next, we design a second‐order optimization algorithm searching for the optimal impulse/switching instants. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.     

Optimal control of continuous-time switched affine systems

This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an optimal control problem. The first approach iterates between a procedure that finds an optimal switching sequence of modes, and a procedure that finds the optimal switching instants. The second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur, therefore providing a state feedback control law.     

约束非线性系统的一个准无限时域预测控制方案

提出了一个具有准无限预测时域的模型预测控制方案,该方案可用于输入和状态的非线性系统的控制,用优化柔性状态约束条件代替了硬性状态约束,以避免优化问题的不可解,开环优化问题含有附加的终端代价项和终的约束条件,这样预测时域延伸至准无限,而需在线优化获得的控制函数仅为有限时域,如果在最初时刻优化问题有解,则闭环系统具有保证稳定性。     

A COMPARATIVE STUDY ON COMPUTATIONAL SCHEMES FOR NONLINEAR MODEL PREDICTIVE CONTROL

This paper briefly reviews development of nonlinear model predictive control (NMPC) schemes for finite horizon prediction and basic computational algorithms that can solve the stable real‐time implementation of NMPC in space state form with state and input constraints. In order to ensure stability within a finite prediction horizon, most NMPC schemes use a terminal region constraint at the end of the prediction horizon — a particular NMPC scheme using a terminal region constraint, namely quasi‐infinite horizon, that guarantees asymptotic closed‐loop stability with input constraints is presented. However, when nonlinear processes have both input and state constraints, difficulty arises from failure to satisfy the state constraints due to constraints on input. Therefore, a new NMPC scheme without a terminal region constraint is developed using soften state constraints. A brief comparative simulation study of two NMPC schemes: quasi‐infinite horizon and soften state constraints is done via simple nonlinear examples to demonstrate the ability of the soften state constraints scheme. Finally, some features of future research from this study are discussed.     

Adaptive horizon economic nonlinear model predictive control

In this paper, we present a computationally efficient economic NMPC formulation, where we propose to adaptively update the length of the prediction horizon in order to reduce the problem size. This is based on approximating an infinite horizon economic NMPC problem with a finite horizon optimal control problem with terminal region of attraction to the optimal equilibrium point. Using the nonlinear programming (NLP) sensitivity calculations, the minimum length of the prediction horizon required to reach this terminal region is determined. We show that the proposed adaptive horizon economic NMPC (AH-ENMPC) has comparable performance to standard economic NMPC (ENMPC). We also show that the proposed adaptive horizon economic NMPC framework is nominally stable. Two benchmark examples demonstrate that the proposed adaptive horizon economic NMPC provides similar performance as the standard economic NMPC with significantly less computation time.     

Optimal Control of Switching Times in Switched Stochastic Systems

In this paper, we solve an optimal control problem for switched stochastic systems using calculus of variations, where the objective is to minimize a cost functional defined on the state and the switching times are the sole control variables. In particular, we focus on the problem in which a pre‐specified sequence of active subsystems is given. For one switching time case, the derivative of the cost functional with respect to the switching time is derived, which has an especially simple form and can be directly used in gradient descent algorithms to locate the optimal switching instant. Then, we propose an approach to deal with the problem with multi‐switching times case. Finally, two numerical examples are given, highlighting the viability and advantages of the proposed methodology.     

Reduced order model predictive control for constrained discrete‐time linear systems

A reduced order model predictive control (MPC) is discussed for constrained discrete‐time linear systems. By employing a decomposition method for finite‐horizon linear systems, an MPC law is obtained from a reduced order optimization problem. The decomposition enables us to construct pairs of initial state and control sequence which have large influence on system responses, and it also characterizes the standard LQ control. The MPC law is obtained based on a combination of the LQ control and dominant input sequences over the prediction horizon. The proposed MPC method is illustrated with numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.     

Active modes and switching instants identification for linear switched systems based on Discrete Particle Swarm Optimization

In this paper, a methodology for identifying switching sequences and switching instants of switched linear systems (SLS) is derived. The identification problem of a SLS is a challenging and non-trivial problem. In fact, it involves interaction between binary, discrete and real-valued variables. A SLS switches many times over a finite time horizon and thus estimating the sequence of activated modes and the switches locations is a crucial problem for both control and Fault Detection and Isolation (FDI). The proposed methodology is based on the Discrete Particle Swarm Optimization (DPSO) technique. The identification problem is formulated as an optimization problem involving noisy data (system inputs and outputs). Both a set of binary variables corresponding to each sub-model before and after each switch, and the corresponding switching instants are iteratively adjusted by the DPSO algorithm. Thus, the DPSO algorithm has to classify which sub-system has generated which data. The efficiency of the proposed approach is illustrated through a numerical example and a physical one. The numerical example is a Switched Auto-Regressive eXogenous (SARX) system and the physical one is a buck–boost DC/DC converter.     

Discrete‐Valued Model Predictive Control Using Sum‐of‐Absolute‐Values Optimization

In this paper, we propose a new design method of discrete‐valued model predictive control for continuous‐time linear time‐invariant systems based on sum‐of‐absolute‐values (SOAV) optimization. The finite‐horizon discrete‐valued control design is formulated as an SOAV optimal control, which is an expansion of L¹ optimal control. It is known that under the normality assumption, the SOAV optimal control exists and takes values in a fixed finite alphabet set if the initial state lies in a subset of the reachable set. In this paper, we analyze the existence and discreteness property for systems that do not necessarily satisfy the normality assumption. Then, we extend the finite‐horizon SOAV optimal control to infinite‐horizon model predictive control (MPC). We give sufficient conditions for the recursive feasibility and the stability of the MPC‐based feedback system in the presence of bounded noise. Simulation results show the effectiveness of the proposed method.     

Event-driven optimization-based control of hybrid systems with integral continuous-time dynamics

In this paper we introduce a class of continuous-time hybrid dynamical systems called integral continuous-time hybrid automata (icHA) for which we propose an event-driven optimization-based control strategy. Events include both external actions applied to the system and changes of continuous dynamics (mode switches). The icHA formalism subsumes a number of hybrid dynamical systems with practical interest, e.g., linear hybrid automata. Different cost functions, including minimum-time and minimum-effort criteria, and constraints are examined in the event-driven optimal control formulation. This is translated into a finite-dimensional mixed-integer optimization problem, in which the event instants and the corresponding values of the control input are the optimization variables. As a consequence, the proposed approach has the advantage of automatically adjusting the attention of the controller to the frequency of event occurrence in the hybrid process. A receding horizon control scheme exploiting the event-based optimal control formulation is proposed as a feedback control strategy and proved to ensure either finite-time or asymptotic convergence of the closed-loop.     

Necessary and sufficient conditions for constraint satisfaction in switched systems using switch‐robust control invariant sets

This paper studies the control of constrained systems whose dynamics and constraints switch between a finite set of modes over time according to an exogenous input signal. We define a new type of control invariant sets for switched constrained systems, called switch–robust control invariant (switch‐RCI) sets, that are robust to unknown mode switching and exploit available information on minimum dwell‐time and admissible mode transitions. These switch‐RCI sets are used to derive novel necessary and sufficient conditions for the existence of a control‐law that guarantees constraint satisfaction in the presence of unknown mode switching with known minimum dwell‐time. The switch‐RCI sets are also used to design a recursively feasible model predictive controller (MPC) that enforces closed‐loop constraint satisfaction for switched constrained systems. We show that our controller is nonconservative in the sense that it enforces constraints on the largest possible domain, ie, constraints can be recursively satisfied if and only if our controller is feasible. The MPC and switch‐RCI sets are demonstrated on a vehicle lane‐changing case study.     

Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control problems.     

Multirate sliding mode disturbance compensation for model predictive control

In this paper, a novel hierarchical multirate control scheme for nonlinear discrete‐time systems is presented, consisting of a robust nonlinear model predictive controller (NMPC) and a multirate sliding mode disturbance compensator (MSMDC). The proposed MSMDC acts at a faster rate than the NMPC in order to keep the system as close as possible to the nominal trajectory predicted by NMPC despite model uncertainties and external disturbances. The a priori disturbance compensation turns out to be very useful in order to improve the robustness of the NMPC controller. A dynamic input allocation between MSMDC and NMPC allows to maximize the benefits of the proposed scheme that unites the advantages of sliding mode control (strong reduction of matched disturbances, low computational burden) to those of NMPC (optimality, constraints handling). Sufficient conditions required to guarantee input‐to‐state stability and constraints satisfaction by the overall scheme are also provided. Copyright © 2014 John Wiley & Sons, Ltd.     

A convex approach for NMPC based on second order Volterra series models

In model predictive control (MPC), the input sequence is computed, minimizing a usually quadratic cost function based on the predicted evolution of the system output. In the case of nonlinear MPC (NMPC), the use of nonlinear prediction models frequently leads to non‐convex optimization problems with several minimums. This paper proposes a new NMPC strategy based on second order Volterra series models where the original performance index is approximated by quadratic functions, which represent a lower bound of the original performance index. Convexity of the approximating quadratic cost functions can be achieved easily by a suitable choice of the weighting of the control increments in the performance index. The approximating cost functions can be globally minimized by convex optimization techniques in order to compute the input sequence. The minimization of the performance index is carried out by an iterative optimization procedure, which guarantees convergence to the solution. Furthermore, for a nominal prediction model, asymptotic stability for the proposed NMPC strategy can be shown. In the case of considering an estimation error in the prediction model, input‐to‐state practical stability is assured. The control performance of the NMPC strategy is illustrated by experimental results. Copyright © 2014 John Wiley & Sons, Ltd.     

Adaptive Control of Piecewise Linear Systems with State Feedback for Output Tracking

A piecewise linear system consists of a set of linear time‐invariant (LTI) subsystems, with a switching sequence specifying an active subsystem at each time instant. This paper studies the adaptive control problem of single‐input, single‐output (SISO) piecewise linear systems. By employing the knowledge of the time instant indicator functions of system parameter switches, a new controller structure parametrization is proposed for the development of a stable adaptive control scheme with reduced modeling error in the estimation error signal used for parameter adaptive laws. This key feature is achieved by the new control scheme's ability to avoid a major parameter swapping term in the error model, with the help of indicator functions whose knowledge is available in many applications. A direct state feedback model reference adaptive control (MRAC) scheme is presented for such systems to achieve closed‐loop signal boundedness and small output tracking error in the mean square sense, under the usual slow system parameter switching condition. Simulation results on linearized NASA GTM models are presented to demonstrate the effectiveness of the proposed scheme.     

On the stabilization of switched linear stochastic systems with unobservable switching laws

1 Introduction Consider the following switched linear stochastic system dx(t) = Aθtx(t)dt +Bθtu(t)dt + Fθtdwt, E x0 2 < ∞ (1) where x0 = x(0). The system state x(t) ∈ Rn is completely observable and u(t) ∈ Rr is the input. The system noise {wt} is an l-dimensional standard Wiener process, which is independent of {x(s),s t}. Aθt, Bθt, Fθt are coefficient matrices with suitable dimensions. The switching law (or signal) θt : [0, ∞) → Θ is a piecewise constant function of time. I…     

On the stabilization of switched linear stochastic systems with unobservable switching laws

This paper is concerned with the stabilization problem of switched linear stochastic systems with unob- servable switching laws. In this paper the system switches among a finite family of linear stochastic systems. Since there are noise perturbations, the switching laws can not be identified in any finite time horizon. We prove that if each individual subsystem is controllable and the switching duration uniformly has a strict positive lower bound, then the system can be stabilized by using a controller that uses online state estimation.     

Enlarging the terminal region of quasi-infinite horizon NMPC based on T-S fuzzy model

The paper presents a method for enlarging the terminal region of quasi-infinity horizon nonlinear model predictive control (NMPC) for nonlinear systems with constraints. The main technique builds on the fact that terminal controllers are fictitious and never applied to the system in the quasi-infinite horizon NMPC [1]. Based on T-S fuzzy models of nonlinear systems, we show that a parameter-dependent state feedback law exists such that the corresponding value function and its level set can be served as terminal cost and terminal region. The problem of maximizing the terminal region is formulated as a convex optimization problem based on linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed method.