Robust constrained predictive control using comparison model

This paper proposes a quadratic programming (QP) approach to robust model predictive control (MPC) for constrained linear systems having both model uncertainties and bounded disturbances. To this end, we construct an additional comparison model for worst-case analysis based on a robust control Lyapunov function (RCLF) for the unconstrained system (not necessarily an RCLF in the presence of constraints). This comparison model enables us to transform the given robust MPC problem into a nominal one without uncertain terms. Based on a terminal constraint obtained from the comparison model, we derive a condition for initial states under which the ultimate boundedness of the closed loop is guaranteed without violating state and control constraints. Since this terminal condition is described by linear constraints, the control optimization can be reduced to a QP problem.     

Robust nonlinear MPC without terminal constraint for tracking changing setpoints in the presence of non-additive unknown disturbance

This paper develops a novel robust tracking model predictive control (MPC) without terminal constraint for discrete-time nonlinear systems capable to deal with changing setpoints and unknown non-additive bounded disturbances. The MPC scheme without terminal constraint avoids difficult computations for the terminal region and is thus simpler to design and implement. However, the existence of disturbances and/or sudden changes in a setpoint may lead to feasibility and stability issues in this method. In contrast to previous works that considered changing setpoints and/or additive slowly varying disturbance, the proposed method is able to deal with changing setpoints and non-additive non-slowly varying disturbance. The key idea is the addition of tightened input and state (tracking error) constraints as new constraints to the tracking MPC scheme without terminal constraints based on artificial references. In the proposed method, the optimal tracking error converges asymptotically to the invariant set for tracking, and the perturbed system tracking error remains in a variable size tube around the optimal tracking error. Closed-loop input-to-state stability and recursive feasibility of the optimization problem for any piece-wise constant setpoint and non-additive disturbance are guaranteed by tightening input and state constraints as well as weighting the terminal cost function by an appropriate stabilizing weighting factor. The simulation results of the satellite attitude control system are provided to demonstrate the efficiency of the proposed predictive controller.     

Superposition in efficient robust constrained predictive control

A stabilizing control method, which does not require on-line optimizations, is developed for linear systems with polytopic model uncertainties and hard input constraints. This work is motivated by the constrained robust MPC (CRMPC) approach (IEEE Trans. Automat. Control 45 (2000a) 1765) which adopts the dual mode prediction strategy (i.e. free control moves and invariant set) and minimizes a worst case performance criterion. Based on the observation that, a feasible control sequence for a particular state can be found as a linear combination of feasible sequences for other states, we suggest a stabilizing control algorithm providing sub-optimal and feasible control sequences using pre-computed optimal sequences for some canonical states. The on-line computation of the proposed method reduces to simple matrix multiplication.     

Output feedback model predictive control for LPV systems based on quasi-min–max algorithm

This paper proposes a robust output feedback model predictive control (MPC) scheme for linear parameter varying (LPV) systems based on a quasi-min–max algorithm. This approach involves an off-line design of a robust state observer for LPV systems using linear matrix inequality (LMI) and an on-line robust output feedback MPC algorithm using the estimated state. The proposed MPC method for LPV systems is applicable for a variety of systems with constraints and guarantees the robust stability of the output feedback systems. A numerical example for an LPV system subject to input constraints is given to demonstrate its effectiveness.     

终端滑模约束的时滞系统模型预测控制

终端约束区域和终端代价项在模型预测控制中起着关键的作用,针对输入受限的时滞系统,提出了终端滑模约束的模型预测控制．将满足输入约束的滑模面作为终端约束区域,使得终端约束区域扩大,有效缩短预测时域,减少计算量,有利于在线应用．最后通过仿真验证了所提方法的有效性．     

Predictive metamorphic control

In spite of its easy implementation, ability to handle constraints and nonlinearities, etc., model predictive control (MPC) does have drawbacks including tuning difficulties. In this paper, we propose a refinement to the basic MPC strategy by incorporating a tuning parameter such that one can move smoothly from an existing controller to a new MPC strategy. Each change of this tuning parameter leads to a new stabilising control law, therefore, allowing one to gradually move from an existing control law to a new and better one. For the infinite horizon case without constraints and for the general case with state and input constraints, stability results are established. We also examine the practical applicability of the proposed approach by employing it in the nominal prediction model of the tube-based output feedback robust MPC method. The merits of the proposed method are illustrated by examples.     

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.     

Input to state stability of min-max MPC controllers for nonlinear systems with bounded uncertainties

Min-max model predictive control (MPC) is one of the control techniques capable of robustly stabilize uncertain nonlinear systems subject to constraints. In this paper we extend existing results on robust stability of min-max MPC to the case of systems with uncertainties which depend on the state and the input and not necessarily decaying, i.e. state and input dependent bounded uncertainties. This allows us to consider both plant uncertainties and external disturbances in a less conservative way.It is shown that the input-to-state practical stability (ISpS) notion is suitable to analyze the stability of worst-case based controllers. Thus, we provide Lyapunov-like sufficient conditions for ISpS. Based on this, it is proved that if the terminal cost is an ISpS-Lyapunov function then the optimal cost is also an ISpS-Lyapunov function for the system controlled by the min-max MPC and hence, the controlled system is ISpS. Moreover, we show that if the system controlled by the terminal control law locally admits certain stability margin, then the system controlled by the min-max MPC retains the stability margin in the feasibility region.     

Formation control of multiple mecanum-wheeled mobile robots with physical constraints and uncertainties

Aiming at the formation control of multiple Mecanum-wheeled mobile robots (MWMRs) with physical constraints and model uncertainties, a novel robust control scheme that combines model predictive control (MPC) and extended state observer-based adaptive sliding mode control (ESO-ASMC) is proposed in this paper. First, a linear MPC strategy is proposed to address the motion constraints of MWMRs, which can transform the robot formation model based on leader-follower into a constrained quadratic programming (QP) problem. The QP problem can be solved iteratively online by a delay neural network (DNN) to obtain the optimal control velocity of the follower robot. Then, to address the input saturation constraints, model uncertainties and unknown disturbances in the dynamic model, an improved ESO-ASMC is proposed and compared with the robust adaptive terminal sliding mode control (RATSMC) and the conventional sliding mode control (SMC) to prove the effectiveness. The proposed scheme, considering the optimal control velocity obtained by the kinematics controller as the given desired velocity of the dynamics controller, can implement precise formation control, while solving various physical constraints of the robot, and eliminating the effects of model uncertainties and disturbances. Finally, through a comparative simulation case, the effectiveness and robustness of the proposed method are verified.

     

Design of robust model predictive control with input constraints

This article addresses the problem of designing a robust output feedback model predictive control (MPC) with input constraints, which ensures a parameter-dependent quadratic stability and guaranteed cost for the case of linear polytopic systems. A new heuristic method is introduced to guarantee input constraints for the MPC. To reject disturbances and maintain the process at the optimal operating conditions or setpoints, the integrator is added to the controller design procedure. Finally, some numerical examples are given to illustrate the effectiveness of the proposed method.     

Robust output-feedback model predictive control for systems with unstructured uncertainty

In this paper, we present novel results that parameterize a broad class of robust output-feedback model predictive control (MPC) policies for discrete-time systems with constraints and unstructured model uncertainty. The MPC policies we consider employ: (i) a linear state estimator, (ii) a pre-determined feedback gain (iii) a set of “tighter constraints” and (iv) a quadratic cost function in the degrees of freedom and the estimated state. Contained within the class, we find both well-known control policies and policies with novel features. The unifying aspect is that all MPC policies within the class satisfy a robust stability test. The robust stability test is suited to synthesis and incorporates a novel linear matrix inequality (LMI) condition which involves the parameters of the cost function. The LMI is shown to always be feasible under an appropriate small-gain condition on the pre-determined feedback gain and the state estimator. Moreover, we show, by means of both theoretical and numerical results, that choosing the cost function parameters subject to the proposed condition often leads to good nominal performance whilst at the same time guaranteeing robust stability.     

Real‐time model predictive control based on dual gradient projection: Theory and fixed‐point FPGA implementation

This paper proposes a method to design robust model predictive control (MPC) laws for discrete‐time linear systems with hard mixed constraints on states and inputs, in case of only an inexact solution of the associated quadratic program is available, because of real‐time requirements. By using a recently proposed dual gradient‐projection algorithm, it is proved that the discrepancy of the optimal control law as compared with the obtained one is bounded even if the solver is implemented in fixed‐point arithmetic. By defining an alternative MPC problem with tightened constraints, a feasible solution is obtained for the original MPC problem, which guarantees recursive feasibility and asymptotic stability of the closed‐loop system with respect to a set including the origin, also considering the presence of external disturbances. The proposed MPC law is implemented on a field‐programmable gate array in order to show the practical applicability of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Non-fragile observer-based output feedback control for polytopic uncertain system under distributed model predictive control approach

In this paper, a non-fragile observer-based output feedback control problem for the polytopic uncertain system under distributed model predictive control (MPC) approach is discussed. By decomposing the global system into some subsystems, the computation complexity is reduced, so it follows that the online designing time can be saved.Moreover, an observer-based output feedback control algorithm is proposed in the framework of distributed MPC to deal with the difficulties in obtaining the states measurements. In this way, the presented observer-based output-feedback MPC strategy is more flexible and applicable in practice than the traditional state-feedback one. What is more, the non-fragility of the controller has been taken into consideration in favour of increasing the robustness of the polytopic uncertain system. After that, a sufficient stability criterion is presented by using Lyapunov-like functional approach, meanwhile, the corresponding control law and the upper bound of the quadratic cost function are derived by solving an optimisation subject to convex constraints. Finally, some simulation examples are employed to show the effectiveness of the method.     

Inherent robustness properties of quasi-infinite horizon nonlinear model predictive control

We consider inherent robustness properties of model predictive control (MPC) for continuous-time nonlinear systems with input constraints and terminal constraints. We show that MPC with a nominal prediction model and persistent but bounded disturbances has some degree of inherent robustness when the terminal control law and the terminal penalty matrix are chosen as the linear quadratic control law and the related Lyapunov matrix, respectively. We emphasize that the input constraint sets can be any compact set rather than convex sets, and our results do not depend on the continuity of the optimal cost function or of the control law in the interior of the feasible region.     

Model predictive control for linear parameter varying constrained systems using ellipsoidal set prediction

This paper proposes a new model predictive control (MPC) method for linear parameter varying systems with bounded parameter variation subject to input constraints. The method adopts closed-loop prediction and constructs ellipsoidal sets to predict the future states with reasonable computational effort. Then the information on the parameter variation rate is exploited to improve the accuracy of the prediction. Furthermore, a relaxed terminal condition, which guarantees the stability for infinite horizon, is introduced to enlarge the stabilizable region. It is shown that the feasibility of the MPC problem at the initial step ensures the stability of the closed-loop system. Finally, a simulation result illustrates the effectiveness of the proposed method.     

一类离散不确定系统的Tube不变集离线鲁棒模型预测控制

针对一类输入和状态受限的离散线性不确定系统,提出了一种基于Tube不变集的离线鲁棒模型预测控制方法.首先针对输入和状态约束线性时不变标准系统,设计了改进的基于多面体不变集的离线模型预测控制算法,并证明了稳定性.其次对于存在未知有界干扰的实际不确定系统,引入了Tube不变集策略,通过设计对应标准模型的最优控制序列和状态轨迹,给出了实际不确定系统的离线Tube不变集控制策略,保证系统状态鲁棒渐近稳定,并收敛于终端干扰不变集.仿真结果验证了该控制方法的有效性.     

LMI-based robust model predictive control and its application to an industrial CSTR problem

In this paper, robust model predictive control (MPC) is studied for a class of uncertain linear systems with structured time-varying uncertainties. This general class of uncertain systems is useful for nonlinear plant modeling in many circumstances. The controller design is characterizing as an optimization problem of the “worst-case” objective function over infinite moving horizon, subject to input and output constraints. A sufficient state-feedback synthesis condition is provided in the form of linear matrix inequality (LMI) optimizations, and will be solved on-line. The stability of such a control scheme is determined by the feasibility of the optimization problem. To demonstrate its usefulness, this robust MPC technique is applied to an industrial continuous stirred tank reactor (CSTR) problem with explicit input and output constraints. Its relative merits to conventional MPC approaches are also discussed.     

Constrained nonlinear MPC for accelerated tracking piece-wise references and its applications to thermal systems

In this paper, we propose a model predictive control (MPC) strategy for accelerated offset-free tracking piece-wise constant reference signals of nonlinear systems subject to state and control constraints. Some special contractive constraints on tracking errors and terminal constraints are embedded into the tracking nonlinear MPC formulation. Then, recursive feasibility and closed-loop convergence of the tracking MPC are guaranteed in the presence of piece-wise references and constraints by deriving some sufficient conditions. Moreover, the local optimality of the tracking MPC is achieved for unreachable output reference signals. By comparing to traditional tracking MPC, the simulation experiment of a thermal system is used to demonstrate the acceleration ability and the effectiveness of the tracking MPC scheme proposed here.     

Exponential stability of constrained receding horizon control withterminal ellipsoid constraints

In this paper, state- and output-feedback receding horizon controllers are proposed for linear discrete time systems with input and state constraints. The proposed receding horizon controllers are obtained from the finite horizon optimization problem with the finite terminal weighting matrix and the artificial invariant ellipsoid constraint, which is less restrictive than the conventional terminal equality constraint. Both hard constraints and mixed constraints are considered in the state-feedback case, and mixed constraints are considered in the output-feedback case. It is shown that all proposed state- and output-feedback receding horizon controllers guarantee the exponential stability of closed-loop systems for all feasible initial sets using the Lyapunov approach     

A tool to analyze robust stability for model predictive controllers

A strategy based on Nonlinear Programming (NLP) sensitivity is developed to establish stability bounds on the plant/model mismatch for a class of optimization-based Model Predictive Control (MPC) algorithms. By extending well-known nominal stability properties for these controllers, we derive a sufficient condition for robust stability of these controllers. This condition can also be used to assess the extent of model mismatch that can be tolerated to guarantee robust stability. In this derivation we deal with MPC controllers with final time constraints or infinite time horizons. Also for this initial study we concentrate only on discrete time systems and unconstrained state feedback control laws with all of the states measured. To illustrate this approach we give two examples: a linear first-order dynamic system and a nonlinear SISO system involving a first order reaction. ©