H∞ model predictive control for constrained discrete‐time piecewise affine systems

This paper investigates stability analysis for piecewise affine (PWA) systems and specifically contributes a new robust model predictive control strategy for PWA systems in the presence of constraints on the states and inputs and with l₂ or norm‐bounded disturbances. The proposed controller is based on piecewise quadratic Lyapunov functions. The problem of minimization of the cost function for model predictive control design is changed to minimization of the worst case of the cost function. Then, this objective is reduced to minimization of a supremum of the cost function subject to a terminal inequality by considering the induced l₂‐norm. Finally, the predictive controller design problem is turned into a linear matrix inequality feasibility exercise with constraints on the input signal and state variables. It is shown that the closed‐loop system is asymptotically stable with guaranteed robust performance. The validity of the proposed method is verified through 3 well‐known examples of PWA systems. Simulation results are provided to show good convergence properties along with capability of the proposed controller to reject disturbances.     

Stochastic model predictive control for constrained discrete-time Markovian switching systems

In this paper we study constrained stochastic optimal control problems for Markovian switching systems, an extension of Markovian jump linear systems (MJLS), where the subsystems are allowed to be nonlinear. We develop appropriate notions of invariance and stability for such systems and provide terminal conditions for stochastic model predictive control (SMPC) that guarantee mean-square stability and robust constraint fulfillment of the Markovian switching system in closed-loop with the SMPC law under very weak assumptions. In the special but important case of constrained MJLS we present an algorithm for computing explicitly the SMPC control law off-line, that combines dynamic programming with parametric piecewise quadratic optimization.     

Distributed model predictive control for constrained linear systems

We address the distributed model predictive control (MPC) for a set of linear local systems with decoupled dynamics and a coupled global cost function. By the decomposition of the global cost function, the distributed control problem is converted to the MPC for each local system associated with a cost involving neighboring system states and inputs. For each local controller, the infinite horizon control moves are parameterized as N free control moves followed by a single state feedback law. An interacting compatibility condition is derived, disassembled and incorporated into the design of each local control so as to achieve the stability of the global closed‐loop system. Each local system exchanges with its neighbors the current states and the previous optimal control strategies. The global closed‐loop system is shown to be exponentially stable provided that all the local optimizers are feasible at the initial time. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust model predictive control for constrained continuous-time nonlinear systems

In this paper, a robust model predictive control (MPC) is designed for a class of constrained continuous-time nonlinear systems with bounded additive disturbances. The robust MPC consists of a nonlinear feedback control and a continuous-time model-based dual-mode MPC. The nonlinear feedback control guarantees the actual trajectory being contained in a tube centred at the nominal trajectory. The dual-mode MPC is designed to ensure asymptotic convergence of the nominal trajectory to zero. This paper extends current results on discrete-time model-based tube MPC and linear system model-based tube MPC to continuous-time nonlinear model-based tube MPC. The feasibility and robustness of the proposed robust MPC have been demonstrated by theoretical analysis and applications to a cart-damper springer system and a one-link robot manipulator.     

Robust explicit model predictive control for linear systems via interpolation techniques

This paper presents a novel interpolation‐based model predictive control (IMPC) for constrained linear systems with bounded disturbances. The idea of so‐called ‘pre‐stabilizing’ MPC is extended by making interpolation among several ‘pre‐stabilizing’ MPC controllers, through which the domain of attraction can be magnificently enlarged. Compared with the standard ‘pre‐stabilizing’ MPC, the proposed approach has the advantage of combining the merits of having a large domain of attraction and a good behavior. Furthermore, such an IMPC problem can be solved off‐line by multi‐parametric programming. The optimal solution is given in an explicitly piecewise affine form. A simple algorithm for the implementation of the explicit MPC control laws is also proposed. Copyright © 2009 John Wiley & Sons, Ltd.     

A new approach to constrained state estimation for discrete‐time linear systems with unknown inputs

This paper addresses the problem of estimating the state for a class of uncertain discrete‐time linear systems with constraints by using an optimization‐based approach. The proposed scheme uses the moving horizon estimation philosophy together with the game theoretical approach to the filtering to obtain a robust filter with constraint handling. The used approach is constructive since the proposed moving horizon estimator (MHE) results from an approximation of a type of full information estimator for uncertain discrete‐time linear systems, named in short ‐MHE and –full information estimator, respectively. Sufficient conditions for the stability of the ‐MHE are discussed for a class of uncertain discrete‐time linear systems with constraints. Finally, since the ‐MHE needs the solution of a complex minimax optimization problem at each sampling time, we propose an approximation to relax the optimization problem and hence to obtain a feasible numerical solution of the proposed filter. Simulation results show the effectiveness of the robust filter proposed.     

Reliability and time‐to‐failure bounds for discrete‐time constrained Markov jump linear systems

This paper presents a methodology to obtain a guaranteed‐reliability controller for constrained linear systems, which switch between different modes according to a Markov chain (Markov jump linear systems). Inside the classical maximal robust controllable set, there is 100% guarantee of never violating constraints at future time. However, outside such set, some sequences might make hitting constraints unavoidable for some disturbance realisations. A guaranteed‐reliability controller based on a greedy heuristic approach was proposed in an earlier work for disturbance‐free, robustly stabilisable Markov jump linear systems. Here, extensions are presented by, first, considering bounded disturbances and, second, presenting an iterative algorithm based on dynamic programming. In non‐stabilisable systems, reliability is zero; therefore, prior results cannot be applied; in this case, optimisation of a mean‐time‐to‐failure bound is proposed, via minor algorithm modifications. Optimality can be proved in the disturbance‐free, finitely generated case. Copyright © 2016 John Wiley & Sons, Ltd.     

通讯信息约束下具有全局稳定性的分布式系统预测控制

本文针对一类由状态相互耦合的子系统组成的分布式系统, 提出了一种可以处理输入约束的保证稳定性的非 迭代协调分布式预测控制方法(distributed model predictive control, DMPC). 该方法中, 每个控制器在求解控制率时只与 其它控制器通信一次来满足系统对通信负荷限制; 同时, 通过优化全局性能指标来提高优化性能. 另外, 该方法在优化 问题中加入了一致性约束来限制关联子系统的估计状态与当前时刻更新的状态之间的偏差, 进而保证各子系统优化问 题初始可行时, 后续时刻相继可行. 在此基础上, 通过加入终端约束来保证闭环系统渐进稳定. 该方法能够在使用较少 的通信和计算负荷情况下, 提高系统优化性能. 即使对于强耦合系统同样能够保证优化问题的递推可行性和闭环系统的 渐进稳定性. 仿真结果验证了本文所提出方法的有效性.     

Self‐triggered model predictive control for networked control systems based on first‐order hold

In this work, a new self‐triggered model predictive control (STMPC) algorithm is proposed for continuous‐time networked control systems. Compared with existing STMPC algorithms, the proposed STMPC is implemented based on linear interpolation (first‐order hold) rather than the standard zero‐order hold, which helps further reduce the difference between the self‐triggered control signal and the original time‐triggered counterpart and thus reduce the rate of triggering. Based on the first‐order hold implementation, a self‐triggering condition is derived and the corresponding theoretical properties of the closed‐loop system are analyzed. Finally, the comparison between the proposed algorithm and the zero‐order hold–based STMPC is carried out through both theoretical analysis and a simulation example to illustrate the effectiveness of the proposed method.     

Stochastic model predictive control: Insights and performance comparisons for linear systems

In this paper, we define several instances of model predictive control (MPC) for linear systems, including both deterministic and stochastic formulations. We show by explicit computation of the associated control laws that, under certain conditions, different formulations lead to identical results. This paper provides insights into the performance of stochastic MPC. Amongst other things, it shows that stochastic MPC and traditional MPC can give identical results in special cases. In cases where the solutions are different, we show that the explicit formulation of the problem can give insight into the performance gap.     

A model predictive control framework for constrained uncertain positive systems

In this paper, a new model predictive control framework is proposed for positive systems subject to input/state constraints and interval/polytopic uncertainty. Instead of traditional quadratic performance index, simple linear performance index, linear Lyapunov function, cone invariant set with linear form and linear computation tool are first adopted. Then, a control law that can handle the constraints and robustly stabilise the systems is proposed. The advantages of the new framework lie in the following facts: (1) an equivalent linear problem is formulated that can be easily solved than other problems including the quadratic ones, (2) simple linear index and linear tool can be used based on the essential property of positive systems to achieve the desired control performance and (3) a general model predictive control law without sign restriction is designed. Finally, an attempt of application on mitigating viral escape is provided to verify the effectiveness of the proposed approach.     

Model predictive control of directed‐graph constrained systems

This paper addresses a new type of model predictive control problem for a hybrid system that consists of a continuous‐time linear system and a temporal/spatial directed graph, called a directed‐graph constrained system. Motivated by the obstacle avoidance problem, the problem is newly formulated, where the continuous‐time control input and the waypoints of the state are simultaneously optimized under a temporal/spatial directed graph as well as input/state linear constraints, and a method for efficiently solving this problem is developed. Numerical examples are presented to verify that the proposed approach is effective. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust output feedback model predictive control of constrained linear systems

This paper provides a solution to the problem of robust output feedback model predictive control of constrained, linear, discrete-time systems in the presence of bounded state and output disturbances. The proposed output feedback controller consists of a simple, stable Luenberger state estimator and a recently developed, robustly stabilizing, tube-based, model predictive controller. The state estimation error is bounded by an invariant set. The tube-based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the ‘center’ of which is the solution of a nominal (disturbance-free) system and the ‘cross-section’ of which is also invariant. Satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system. The complexity of the resultant controller is similar to that required for nominal model predictive control.     

Robust model predictive control of constrained linear systems with bounded disturbances

This paper provides a novel solution to the problem of robust model predictive control of constrained, linear, discrete-time systems in the presence of bounded disturbances. The optimal control problem that is solved online includes, uniquely, the initial state of the model employed in the problem as a decision variable. The associated value function is zero in a disturbance invariant set that serves as the ‘origin’ when bounded disturbances are present, and permits a strong stability result, namely robust exponential stability of the disturbance invariant set for the controlled system with bounded disturbances, to be obtained. The resultant online algorithm is a quadratic program of similar complexity to that required in conventional model predictive control.     

Explicit improved vertex control for uncertain,time‐varying linear discrete‐time systems with state and control constraints

A technique is presented to compute an explicit state feedback solution to the regulation problem for uncertain and/or time‐varying linear discrete‐time systems with state and control constraints. A piecewise affine control law is provided that not only guarantees recursive feasibility and robust asymptotic stability but is also optimal for a region of the state space containing the origin. Copyright © 2015 John Wiley & Sons, Ltd.     

Decomposition principle in model predictive control for linear systems with bounded disturbances

Considering a constrained linear system with bounded disturbances, this paper proposes a novel approach which aims at enlarging the domain of attraction by combining a set-based MPC approach with a decomposition principle. The idea of the paper is to extend the “pre-stabilizing” MPC, where the MPC control sequence is parameterized as perturbations to a given pre-stabilizing feedback gain, to the case where the pre-stabilizing feedback law is given as the linear combination of a set of feedback gains. This procedure leads to a relatively large terminal set and consequently a large domain of attraction even when using short prediction horizons. As time evolves, by minimizing the nominal performance index, the resulting controller reaches the desired optimal controller with a good asymptotic performance. Compared to the standard “pre-stabilizing” MPC, it combines the advantages of having a flexible choice of feedback gains, a large domain of attraction and a good asymptotic behavior.     

Min-max model predictive control for constrained nonlinear systems via multiple LPV embeddings

A min-max model predictive control strategy is proposed for a class of constrained nonlinear system whose trajectories can be embedded within those of a bank of linear parameter varying (LPV) models. The embedding LPV models can yield much better approximation of the nonlinear system dynamics than a single LTV model. For each LPV model, a parameter-dependent Lyapunov function is introduced to obtain poly-quadratically stable control law and to guarantee the feasibility and stability of the original nonlinear system. This approach can greatly reduce computational burden in traditional nonlinear predictive control strategy. Finally a simulation example illustrating the strategy is presented. Supported by the National Natural Science Foundation of China (Grant Nos. 60774015, 60825302, 60674018), the National High-Tech Research & Development Program of China (Grant No. 2007AA041403), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060248001), and partly by Shanghai Natural Science Foundation (Grant No. 07JC14016)     

Model reference adaptive control of discrete‐time piecewise linear systems

This article presents a switched model reference adaptive controller for discrete‐time piecewise linear systems. In the spirit of the work by Landau in the late seventies, proof of asymptotic stability of the closed‐loop error system is obtained, recasting its dynamics as a feedback system and showing the feedforward and the feedback paths are both passive. The challenge is that both paths can be piecewise linear. Numerical results show excellent performance of the proposed controller even in the face of sudden variations of the plant parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Enhanced information reconfiguration for distributed model predictive control for cyber‐physical networked systems

This paper considers a class of cyber‐physical networked systems, which are composed of many interacted subsystems, and are controlled in a distributed framework. The operating point of each subsystem changes with the varying of working conditions or productions, which may cause the change of the interactions among subsystems correspondingly. How to adapt to this change with good closed‐loop optimization performance and appropriate information connections is a problem. To solve this problem, the impaction of a subsystem's control action on the performance of related closed‐loop subsystems is first deduced for measuring the coupling among subsystems. Then, a distributed model predictive control (MPC) for tracking, whose subsystems online reconfigure their information structures, is proposed based on this impaction index. When the operating points changed, each local MPC calculates the impaction indices related to its structural downstream subsystems. If and only if the impaction index exceeds a defined bound, its behavior is considered by its downstream subsystem's MPC. The aim is to improve the optimization performance of entire closed‐loop systems and avoid the unnecessary information connections among local MPCs. Besides, contraction constraints are designed to guarantee that the overall system converges to the set points. The stability analysis is also provided. Simulation results show that the proposed impaction index is reasonable along with the efficiency of the proposed distributed MPC.