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.     

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.     

Distributed model predictive control of linear systems with persistent disturbances

This article presents a new form of robust distributed model predictive control (MPC) for multiple dynamically decoupled subsystems, in which distributed control agents exchange plans to achieve satisfaction of coupling constraints. The new method offers greater flexibility in communications than existing robust methods, and relaxes restrictions on the order in which distributed computations are performed. The local controllers use the concept of tube MPC – in which an optimisation designs a tube for the system to follow rather than a trajectory – to achieve robust feasibility and stability despite the presence of persistent, bounded disturbances. A methodical exploration of the trades between performance and communication is provided by numerical simulations of an example scenario. It is shown that at low levels of inter-agent communication, distributed MPC can obtain a lower closed-loop cost than that obtained by a centralised implementation. A further example shows that the flexibility in communications means the new algorithm has a relatively low susceptibility to the adverse effects of delays in computation and communication.     

Approximate explicit constrained linear model predictive control via orthogonal search tree

Solutions to constrained linear model predictive control problems can be precomputed off-line in an explicit form as a piecewise linear state feedback on a polyhedral partition of the state-space, avoiding real-time optimization. We suggest an algorithm that can determine an approximate explicit piecewise linear state feedback by imposing an orthogonal search tree structure on the partition. This leads to a real-time computational complexity that is logarithmic in the number of regions in the partition, and the algorithm yields guarantees on the suboptimality, asymptotic stability and constraint fulfillment.     

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.     

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.     

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.     

Stochastic model predictive control for probabilistically constrained Markovian jump linear systems with additive disturbance

Model predictive control (MPC) for Markovian jump linear systems with probabilistic constraints has received much attention in recent years. However, in existing results, the disturbance is usually assumed with infinite support, which is not considered reasonable in real applications. Thus, by considering random additive disturbance with finite support, this paper is devoted to a systematic approach to stochastic MPC for Markovian jump linear systems with probabilistic constraints. The adopted MPC law is parameterized by a mode‐dependent feedback control law superimposed with a perturbation generated by a dynamic controller. Probabilistic constraints can be guaranteed by confining the augmented system state to a maximal admissible set. Then, the MPC algorithm is given in the form of linearly constrained quadratic programming problems by optimizing the infinite sum of derivation of the stage cost from its steady‐state value. The proposed algorithm is proved to be recursively feasible and to guarantee constraints satisfaction, and the closed‐loop long‐run average cost is not more than that of the unconstrained closed‐loop system with static feedback. Finally, when adopting the optimal feedback gains in the predictive control law, the resulting MPC algorithm has been proved to converge in the mean square sense to the optimal control. A numerical example is given to verify the efficiency of the proposed results.     

Model predictive control for constrained uncertain piecewise linear systems

For constrained piecewise linear (PWL) systems, the possible existing model uncertainty will bring the difficulties to the design approaches of model predictive control (MPC) based on mixed integer programming (MIP). This paper combines the robust method and hybrid method to design the MPC for PWL systems with structured uncertainty. For the proposed approach, as the system model is known at current time, a free control move is optimized to be the current control input. Meanwhile, the MPC controller uses a sequence of feedback control laws as the future control actions, where each feedback control law in the sequence corresponds to each partitions and the arbitrary switching technique is adopted to tackle all the possible switching. Furthermore, to reduce the online computational burden of MPC, the segmented design procedure is suggested by utilizing the characteristics of the proposed approach. Then, an offline design algorithm is proposed, and the reserved degree of freedom can be online used to optimize the control input with lower computational burden.     

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.     

Networked predictive control of constrained linear systems with input quantisation

This article investigates the problem of stabilising predictive control for constrained systems, wherein communication from the controller to the plant input is through a digital channel subject to quantisation and delay. A novel model with structured norm-bounded uncertainties is proposed to describe control system with input quantisation. Under a multirate scheme, a delay compensation strategy is presented. The networked predictive control synthesis approach is developed by solving a finite receding horizon optimisation problem with free control moves. It is shown that the proposed predictive controller not only efficiently reduces the negative effects of the quantisation and communication delays but also guarantees the closed-loop stability and satisfies constraints. Finally, a simulation example illustrates the effectiveness of the derived method.     

约束非线性系统构造性模型预测控制

研究了连续时间约束非线性系统模型预测控制设计.利用控制Lyapunov函数离线构造单变量可调预测控制器,再根据性能指标在线优化可调参数,其中该参数近似于闭环系统的"衰减率".同时,控制Lyapunov函数保证了算法的可行性和闭环系统的稳定性.最后通过数值仿真验证了该算法的有效性.     

Robust constrained predictive control of uncertain norm-bounded linear systems

A novel robust predictive control algorithm is presented for uncertain discrete-time input-saturated linear systems described by structured norm-bounded model uncertainties. The solution is based on the minimization, at each time instant, of a semi-definite convex optimization problem subject to a number of LMI feasibility constraints which grows up only linearly with the control horizon length N. The general case of arbitrary N is considered. Closed-loop stability and feasibility retention over the time are proved and comparisons with robust multi-model (polytopic) MPC algorithms are reported.     

Adaptive model predictive control for a class of constrained linear systems based on the comparison model

This paper proposes an adaptive model predictive control (MPC) algorithm for a class of constrained linear systems, which estimates system parameters on-line and produces the control input satisfying input/state constraints for possible parameter estimation errors. The key idea is to combine the robust MPC method based on the comparison model with an adaptive parameter estimation method suitable for MPC. To this end, first, a new parameter update method based on the moving horizon estimation is proposed, which allows to predict an estimation error bound over the prediction horizon. Second, an adaptive MPC algorithm is developed by combining the on-line parameter estimation with an MPC method based on the comparison model, suitably modified to cope with the time-varying case. This method guarantees feasibility and stability of the closed-loop system in the presence of state/input constraints. A numerical example is given to demonstrate its effectiveness.     

Globally robust explicit model predictive control of constrained systems exploiting SVM‐based approximation

This paper presents a systematic method to address the reduction of online computational complexity and infeasibility problem of explicit model predictive control for constrained systems under external disturbance. In feasible state space, in order to avoid the expensive database searching procedure, support vector machine‐based approximation is proposed to yield a novel unified explicit optimal control law rather than a piecewise affine one developed by explicit model predictive control. In infeasible state space, through constructing finite maximum control invariant sets around fictitious equilibrium points, a reachable controller is devised to steer the infeasible state asymptotically to the feasible state space without violating the hard constraint. Consequently, global robustness is guaranteed by introducing a minimum robust positively invariant set by means of the tube‐based technique, despite the coexistence of external disturbance and training error. Finally, the performance of the presently proposed control law is evaluated through three groups of numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Fuzzy robust constrained model predictive control for nonlinear systems

This paper addresses robust constrained model predictive control (MPC) for a class of nonlinear systems with structured time‐varying uncertainties. First, the Takagi‐Sugeno (T‐S) fuzzy model is employed to represent a nonlinear system. Then, we develop some techniques for designing fuzzy control which guarantees the system stabilization subject to input and output constraints. Both parallel and nonparallel distributed compensation control laws (PDC and non‐PDC) are considered. Sufficient conditions for the solvability of the controller design problem are given in the form of linear matrix inequalities. A simulation example is presented to illustrate the design procedures and performances of the proposed methods. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

The explicit linear quadratic regulator for constrained systems

For discrete-time linear time invariant systems with constraints on inputs and states, we develop an algorithm to determine explicitly, the state feedback control law which minimizes a quadratic performance criterion. We show that the control law is piece-wise linear and continuous for both the finite horizon problem (model predictive control) and the usual infinite time measure (constrained linear quadratic regulation). Thus, the on-line control computation reduces to the simple evaluation of an explicitly defined piecewise linear function. By computing the inherent underlying controller structure, we also solve the equivalent of the Hamilton-Jacobi-Bellman equation for discrete-time linear constrained systems. Control based on on-line optimization has long been recognized as a superior alternative for constrained systems. The technique proposed in this paper is attractive for a wide range of practical problems where the computational complexity of on-line optimization is prohibitive. It also provides an insight into the structure underlying optimization-based controllers.     

Robust low complexity predictive control of constrained piecewise affine systems with bounded disturbances

A robust low complexity model predictive control (MPC) scheme, referred to as robust one-step control, is proposed for constrained piecewise affine (PWA) systems with bounded disturbances. First, the maximal robust stabilizable set is added into the MPC formulation to guarantee the robust feasibility and low complexity. Second, the robust stability is analyzed via linear matrix inequalities (LMI). Extensive numerical examples illustrate the low complexity of the proposed robust one-step control.