Output feedback receding horizon control of constrained systems

This paper considers output feedback control of linear discrete-time systems with convex state and input constraints which are subject to bounded state disturbances and output measurement errors. We show that the non-convex problem of finding a constraint admissible affine output feedback policy over a finite horizon, to be used in conjunction with a fixed linear state observer, can be converted to an equivalent convex problem. When used in the design of a time-varying robust receding horizon control law, we derive conditions under which the resulting closed-loop system is guaranteed to satisfy the system constraints for all time, given an initial state estimate and bound on the state estimation error. When the state estimation error bound matches the minimal robust positively invariant (mRPI) set for the system error dynamics, we show that this control law is time-invariant, but its calculation generally requires solution of an infinite-dimensional optimization problem. Finally, using an invariant outer approximation to the mRPI error set, we develop a time-invariant control law that can be computed by solving a finite-dimensional tractable optimization problem at each time step that guarantees that the closed-loop system satisfies the constraints for all time.     

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.     

A general dissipativity constraint for feedback system design,with emphasis on MPC

A “general dissipativity constraint” (GDC) is introduced to facilitate the design of stable feedback systems. A primary application is to MPC controllers when it is preferred to avoid the use of “stabilising ingredients” such as terminal constraint sets or long prediction horizons. Some very general convergence results are proved under mild conditions. The use of quadratic functions, replacing GDC by “quadratic dissipativity constraint” (QDC), is introduced to allow implementation using linear matrix inequalities. The use of QDC is illustrated for several scenarios: state feedback for a linear time‐invariant system, MPC of a linear system, MPC of an input‐affine system, and MPC with persistent disturbances. The stability that is guaranteed by GDC is weaker than Lyapunov stability, being “Lagrange stability plus convergence.” Input‐to‐state stability is obtained if the control law is continuous in the state. An example involving an open‐loop unstable helicopter illustrates the efficacy of the approach in practice.     

基于鲁棒一步集的Tube不变集鲁棒模型预测控制

针对一类干扰有界的输入和状态受约束线性离散系统,提出了一种基于鲁棒一步集的Tube不 变集鲁棒模型预测控制方法.首先采用多面体不变集离线设计方法得到基于多面体不变集序列的扩 展终端约束集;然后为了扩大鲁棒模型预测控制的初始状态允许区域,并提高系统的鲁棒性,在扩展终端约束集的基础上,通过引入鲁棒一步集并借助Tube不变集控制策略,设计了基于鲁棒一步集的鲁棒模型预测控制方法,并给出了算法的存在性和稳定性证明. 该方法不仅极大地扩大了初始状态允许区域,而且对有界干扰具有有效的抑制作用,使得受扰系统收敛到以原点为中心的最小鲁棒正不变集内.最后仿真验证了算法的有效性.     

Design of switched linear systems in the presence of actuator saturation and L-infinity disturbances

This paper considers the problem of disturbance tolerance/rejection of a switched system resulting from a family of linear systems subject to actuator saturation and L-infinity disturbances. For a given set of linear feedback gains, a given switching scheme and a given bound on the L-infinity norm of the disturbances, conditions are established, in terms of linear or bilinear matrix inequalities, under which a set of a certain form is invariant for a given switched linear system in the presence of actuator saturation and L-infinity disturbances, and the closed-loop system possesses a certain level of disturbance rejection capability. With these conditions, the design of feedback gains and switching scheme can be formulated and solved as constrained optimization problems. Disturbance tolerance is measured by the largest bound on the disturbances for which the trajectories starting from a given set remain bounded. Disturbance rejection is measured either by the L-infinity norm of the system output or by the system’s ability to steer its state into and/or keep it within a small neighborhood of the origin. In the event that all systems in the family are identical, the switched system reduces to a single system under a switching feedback law. Simulation results show that such a single system under a switching feedback law could have stronger disturbance tolerance/rejection capability than a single linear feedback law can.     

Constrained linear system with disturbance: Convergence under disturbance feedback

This paper proposes a disturbance-based control parametrization under the Model Predictive Control framework for constrained linear discrete time systems with bounded additive disturbances. The proposed approach has the same feasible domain as that obtained from parametrization over the family of time-varying state feedback policies. In addition, the closed-loop system is stable in the sense that the state converges to a bounded set that has a characterization determined by a feedback gain.     

Model predictive control of constrained LPV systems

This article considers robust model predictive control (MPC) schemes for linear parameter varying (LPV) systems in which the time-varying parameter is assumed to be measured online and exploited for feedback. A closed-loop MPC with a parameter-dependent control law is proposed first. The parameter-dependent control law reduces conservativeness of the existing results with a static control law at the cost of higher computational burden. Furthermore, an MPC scheme with prediction horizon ‘1’ is proposed to deal with the case of asymmetric constraints. Both approaches guarantee recursive feasibility and closed-loop stability if the considered optimisation problem is feasible at the initial time instant.     

Dynamic Output Feedback Robust MPC With one Free Control Move for LPV Model With Bounded Disturbance

For a linear parameter‐varying (LPV) model which is a convex combination of several linear time invariant sub‐models, this paper considers the case when the combining coefficients are unknown (except being nonnegative and their sum being one). For this model with norm‐bounded unknown disturbance, an output feedback robust model predictive control (MPC) is proposed by parameterizing the infinite horizon control moves and estimated states into one free control move, one free estimated state (i.e., one control move and one estimated state as degrees of freedom for optimization) and a dynamic output feedback law. This is the first endeavour to apply the free control move and free estimated state in the output feedback MPC for this model. The algorithm is shown to be recursively feasible and the system state is guaranteed to converge to the neighborhood of the equilibrium point. A numerical example verifies the effectiveness of the proposed algorithm.     

Disturbance decoupling of linear time-varying singular systems

The problem of disturbance decoupling by state feedback is defined for a linear time-varying singular system. It is required that the closed-loop system has a unique impulse-free solution and its output is not affected by disturbances. An algorithm, namely disturbance decoupling algorithm, is proposed. It is proved that the feasibility of the disturbance decoupling algorithm is invariant under any regular feedback control law. Based on the disturbance algorithm, a constructive method is provided to design a disturbance decoupling feedback control law. Sufficient conditions for the solvability of the disturbance decoupling problem are derived. It is proved that one of the sufficient conditions is also necessary provided that other conditions are satisfied     

Conditions under which suboptimal nonlinear MPC is inherently robust

We address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control (MPC), i.e., when a suboptimal solution of the (generally nonconvex) optimization problem, rather than an element of the optimal solution set, is used for the control. The suboptimal control law is then a set-valued map, and consequently, the closed-loop system is described by a difference inclusion. Under mild assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium, and with a continuity assumption on the feasible input set, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. These results are obtained by showing that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system, written as a difference inclusion, and that recursive feasibility is implied by such (nominal) exponential cost decay. These novel robustness properties for suboptimal MPC are inherited also by optimal nonlinear MPC. We conclude the paper by showing that, in the absence of state constraints, we can replace the terminal constraint with an appropriate terminal cost, and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances. The somewhat surprising and satisfying conclusion of this study is that suboptimal MPC has the same inherent robustness properties as optimal MPC.     

Composite quadratic Lyapunov functions for constrained control systems

A Lyapunov function based on a set of quadratic functions is introduced in this paper. We call this Lyapunov function a composite quadratic function. Some important properties of this Lyapunov function are revealed. We show that this function is continuously differentiable and its level set is the convex hull of a set of ellipsoids. These results are used to study the set invariance properties of continuous-time linear systems with input and state constraints. We show that, for a system under a given saturated linear feedback, the convex hull of a set of invariant ellipsoids is also invariant. If each ellipsoid in a set can be made invariant with a bounded control of the saturating actuators, then their convex hull can also be made invariant by the same actuators. For a set of ellipsoids, each invariant under a separate saturated linear feedback, we also present a method for constructing a nonlinear continuous feedback law which makes their convex hull invariant.     

Stabilizing polynomial approximation of explicit MPC

A given explicit piecewise affine representation of an MPC feedback law is approximated by a single polynomial, computed using linear programming. This polynomial state feedback control law guarantees closed-loop stability and constraint satisfaction. The polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage.     

Multi-objective performance optimisation for model predictive control by goal attainment

This article proposes an approach for performance tuning of model predictive control (MPC) using goal-attainment optimisation of the cost function weighting matrices. The approach is developed for three formulations of the control problem: (i) minimal and (ii) non-minimal design based on the same cost function and (iii) a non-minimal MPC approach with an explicit integral-of-error state variable and modified cost function. This approach is based on earlier research into multi-objective optimisation for proportional-integral-plus control systems. Simulation experiments for a 3-input, 3-output Shell heavy oil fractionator model illustrate the feasibility of MPC goal attainment for multivariable decoupling and attainment of a specific output response. For this example, the integral-of-error state variable offers improved design flexibility and hence, when it is combined with the proposed tuning method, yields an improved closed-loop response in comparison to minimal MPC.     

Optimization over state feedback policies for robust control with constraints

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.     

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.     

Enlargement of polytopic terminal region in NMPC by interpolation and partial invariance

This paper uses the concept of partial invariance to derive a sequence of linear programs in order to maximize offline the volume of an invariant polytopic set with an arbitrary predefined number of vertices subject to a bound on closed-loop performance. Interpolation techniques are used to determine a nonlinear control law which is optimal with respect to a closed-loop cost bound through the on-line solution of a linear program. The invariant polytope is also used to define a receding horizon control law through an appropriate terminal constraint and cost.     

MPC for tracking with optimal closed-loop performance

In the recent paper [Limon, D., Alvarado, I., Alamo, T., & Camacho, E.F. (2008). MPC for tracking of piece-wise constant references for constrained linear systems. Automatica, 44, 2382-2387], a novel predictive control technique for tracking changing target operating points has been proposed. Asymptotic stability of any admissible equilibrium point is achieved by adding an artificial steady state and input as decision variables, specializing the terminal conditions and adding an offset cost function to the functional.In this paper, the closed-loop performance of this controller is studied and it is demonstrated that the offset cost function plays an important role in the performance of the model predictive control (MPC) for tracking. Firstly, the controller formulation has been enhanced by considering a convex, positive definite and subdifferential function as the offset cost function. Then it is demonstrated that this formulation ensures convergence to an equilibrium point which minimizes the offset cost function. Thus, in case of target operation points which are not reachable steady states or inputs for the constrained system, the proposed control law steers the system to an admissible steady state (different to the target) which is optimal with relation to the offset cost function. Therefore, the offset cost function plays the role of a steady-state target optimizer which is built into the controller. On the other hand, optimal performance of the MPC for tracking is studied and it is demonstrated that under some conditions on both the offset and the terminal cost functions optimal closed-loop performance is locally achieved.     

一类不确定广义系统的非脆弱保性能控制

研究了一类时变时滞不确定广义系统的非脆弱保性能控制问题。在系统矩阵、滞后状态矩阵、输入矩阵和滞后输入矩阵都具有不确定性时,带有反馈增益扰动的控制器能够保证闭环系统是正则、无脉冲、指数稳定的;且性能函数存在上界。对于出现的两个不确定项乘积的问题,采用两个线性矩阵不等式,两次运用Schur补引理的方法得以解决。控制器的增益可通过求解一系列严格线性矩阵不等式得到。数值仿真实例证明了所提方法的有效性。