Enlarging the domain of attraction of MPC controllers

This paper presents a method for enlarging the domain of attraction of nonlinear model predictive control (MPC). The usual way of guaranteeing stability of nonlinear MPC is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function. The domain of attraction of the controller depends on the size of the terminal region and the control horizon. By increasing the control horizon, the domain of attraction is enlarged but at the expense of a greater computational burden, while increasing the terminal region produces an enlargement without an extra cost.In this paper, the MPC formulation with terminal cost and constraint is modified, replacing the terminal constraint by a contractive terminal constraint. This constraint is given by a sequence of sets computed off-line that is based on the positively invariant set. Each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one-step set. This property allows us to use one-step approximations with a trade off between accuracy and computational burden for the computation of the sequence. This strategy guarantees closed loop-stability ensuring the enlargement of the domain of attraction and the local optimality of the controller. Moreover, this idea can be directly translated to robust MPC.     

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.     

On Stability of Constrained Receding Horizon Control with Finite Terminal Weighting Matrix

In this paper, a new stabilizing receding horizon control, based on a finite input and state horizon cost with a finite terminal weighting matrix, is proposed for time-varying discrete linear systems with constraints. We propose matrix inequality conditions on the terminal weighting matrix under which closed-loop stability is guaranteed for both cases of unconstrained and constrained systems with input and state constraints. We show that such a terminal weighting matrix can be obtained by solving a linear matrix inequality (LMI). In the case of constrained time-invariant systems, an artificial invariant ellipsoid constraint is introduced in order to relax the conventional terminal equality constraint and to handle constraints. Using the invariant ellipsoid constraints, a feasibility condition of the optimization problem is presented and a region of attraction is characterized for constrained systems with the proposed receding horizon control.     

MPC for tracking piecewise constant references for constrained linear systems

In this paper, a novel model predictive control (MPC) for constrained (non-square) linear systems to track piecewise constant references is presented. This controller ensures constraint satisfaction and asymptotic evolution of the system to any target which is an admissible steady-state. Therefore, any sequence of piecewise admissible setpoints can be tracked without error. If the target steady state is not admissible, the controller steers the system to the closest admissible steady state.These objectives are achieved by: (i) adding an artificial steady state and input as decision variables, (ii) using a modified cost function to penalize the distance from the artificial to the target steady state (iii) considering an extended terminal constraint based on the notion of invariant set for tracking. The control law is derived from the solution of a single quadratic programming problem which is feasible for any target. Furthermore, the proposed controller provides a larger domain of attraction (for a given control horizon) than the standard MPC and can be explicitly computed by means of multiparametric programming tools. On the other hand, the extra degrees of freedom added to the MPC may cause a loss of optimality that can be arbitrarily reduced by an appropriate weighting of the offset cost term.     

Economic optimization using model predictive control with a terminal cost

In the standard model predictive control implementation, first a steady-state optimization yields the equilibrium point with minimal economic cost. Then, the deviation from the computed best steady state is chosen as the stage cost for the dynamic regulation problem. The computed best equilibrium point may not be the global minimum of the economic cost, and hence, choosing the economic cost as the stage cost for the dynamic regulation problem, rather than the deviation from the best steady state, offers potential for improving the economic performance of the system. It has been previously shown that the existing framework for MPC stability analysis, which addresses to the standard class of problems with a regulation objective, does not extend to economic MPC. Previous work on economic MPC developed new tools for stability analysis and identified sufficient conditions for asymptotic stability. These tools were developed for the terminal constraint MPC formulation, in which the system is stabilized by forcing the state to the best equilibrium point at the end of the horizon. In this work, we relax this constraint by imposing a region constraint on the terminal state instead of a point constraint, and adding a penalty on the terminal state to the regulator cost. We extend the stability analysis tools, developed for terminal constraint economic MPC, to the proposed formulation and establish that strict dissipativity is sufficient for guaranteeing asymptotic stability of the closed-loop system. We also show that the average closed-loop performance outperforms the best steady-state performance. For implementing the proposed formulation, a rigorous analysis for computing the appropriate terminal penalty and the terminal region is presented. A further extension, in which the terminal constraint is completely removed by modifying the regulator cost function, is also presented along with its stability analysis. Finally, an illustrative example is presented to demonstrate the differences between the terminal constraint and the proposed terminal penalty formulation.     

Reconfigurable predictive control for redundantly actuated systems with parameterised input constraints

A method is proposed for on-line reconfiguration of the terminal constraint used to provide theoretical nominal stability guarantees in linear model predictive control (MPC). By parameterising the terminal constraint, its complete reconstruction is avoided when input constraints are modified to accommodate faults. To enlarge the region of feasibility of the terminal control law for a certain class of input faults with redundantly actuated plants, the linear terminal controller is defined in terms of virtual commands. A suitable terminal cost weighting for the reconfigurable MPC is obtained by means of an upper bound on the cost for all feasible realisations of the virtual commands from the terminal controller. Conditions are proposed that guarantee feasibility recovery for a defined subset of faults. The proposed method is demonstrated by means of a numerical example.     

Enlarging the domain of attraction of stable MPC controllers, maintaining the output performance

This work presents an alternative way to formulate the stable Model Predictive Control (MPC) optimization problem that allows the enlargement of the domain of attraction, while preserving the controller performance. Based on the dual MPC that uses the null local controller, it proposed the inclusion of an appropriate set of slacked terminal constraints into the control problem. As a result, the domain of attraction is unlimited for the stable modes of the system, and the largest possible for the non-stable modes. Although this controller does not achieve local optimality, simulations show that the input and output performances may be comparable to the ones obtained with the dual MPC that uses the LQR as a local controller.     

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.     

An efficient model predictive controller with pole placement

For model predictive control (MPC) of constrained systems, enlarging the feasible region is usually in conflict with improving the dynamic performance. To resolve the conflict, we proposed an efficient model predictive controller with pole placement for a class of discrete-time linear systems. By specifying a group of circular regions that contain the desired closed-loop poles, appropriate terminal weighting matrices and local controllers are calculated to construct a time-varying terminal convex set, which is a significant constraint for the online optimization problem. During the online optimization, the size of the terminal convex set can adjust itself according to the actual state at each sampling time. In this way, a large initial feasible region can be achieved while maintaining the good dynamic performance. An illustrative example is used to show the effectiveness of the proposed approach.     

Optimal MPC for tracking of constrained linear systems

Model predictive control (MPC) is one of the few techniques which is able to handle constraints on both state and input of the plant. The admissible evolution and asymptotic convergence of the closed-loop system is ensured by means of suitable choice of the terminal cost and terminal constraint. However, most of the existing results on MPC are designed for a regulation problem. If the desired steady-state changes, the MPC controller must be redesigned to guarantee the feasibility of the optimisation problem, the admissible evolution as well as the asymptotic stability. Recently, a novel MPC has been proposed to ensure the feasibility of the optimisation problem, constraints satisfaction and asymptotic evolution of the system to any admissible target steady-state. A drawback of this controller is the loss of a desirable property of the MPC controllers: the local optimality property. In this article, a novel formulation of the MPC for tracking is proposed aimed to recover the optimality property maintaining all the properties of the original formulation.     

Output feedback MPC with guaranteed robust stability

This paper focuses on the issues of robust stability of model predictive control (MPC). The control problem is formulated as linear matrix inequalities (LMI) optimization problem. A suboptimal solution for the output feedback control problem is proposed. The size of the resulting MP controller is reduced by using a suitable state-space representation of the process. Guaranteed stability conditions for the output feedback MPC are enforced via a Lyapunov type constraint. An iterative algorithm is developed resulting in a pair of coupled LMI optimization problems which provide a robustly stable output feedback gain. Model uncertainties are considered via a polytopic set of process models. The methodology is illustrated with the simulation of the control problem of two chemical processes. The results show that the proposed strategy eliminates the need to detune the MP controller improving the performance for most of the cases considered.     

Constrained linear MPC with time-varying terminal cost using convex combinations

Recent papers (IEEE Transactions on Automatic Control 48(6) (2003) 1092-1096, Automatica 38 (2002) 1061-1068, Systems and Control Letters 48 (2003) 375-383) have introduced dual-mode MPC algorithms using a time-varying terminal cost and/or constraint. The advantage of these methods is the enlargement of the admissible set of initial states without sacrificing local optimality of the controller, but this comes at the cost of a higher computational complexity. This paper delivers two main contributions in this area. First, a new MPC algorithm with a time-varying terminal cost and constraint is introduced. The algorithm uses convex combinations of off-line computed ellipsoidal terminal constraint sets and uses the associated cost as a terminal cost. In this way, a significant on-line computational advantage is obtained. The second main contribution is the introduction of a general stability theorem, proving stability of both the new MPC algorithm and several existing MPC schemes (IEEE Transactions on Automatic Control 48(6) (2003) 1092-1096, Automatica 38 (2002) 1061-1068). This allows a theoretical comparison to be made between the different algorithms. The new algorithm using convex combinations is illustrated and compared with other methods on the example of an inverted pendulum.     

Enlarged terminal sets guaranteeing stability of receding horizon control

The purpose of this paper is to relax the terminal conditions typically used to ensure stability in model predictive control, thereby enlarging the domain of attraction for a given prediction horizon. Using some recent results, we present novel conditions that employ, as the terminal cost, the finite-horizon cost resulting from a nonlinear controller u=−sat(Kx) and, as the terminal constraint set, the set in which this controller is optimal for the finite-horizon constrained optimal control problem. It is shown that this solution provides a considerably larger terminal constraint set than is usually employed in stability proofs for model predictive control.     

基于可达集的鲁棒模型预测控制

设计了一种基于可达集的鲁棒模型预测控制算法.首先确定了一个鲁棒不变集,并将此不变集用作模型预测控制的终端约束集;接着采用终端约束集对可达集的包含度作为优化指标;最后,采用预测时域逐渐减小的控制策略以保证在线优化存在可行解.从理论上证明了吸引域内的任意点在有限时域内都会被引导至终端约束集并始终停留在此集之内,并由仿真算例验证了本文所设计鲁棒模型预测控制算法的可行性.     

Unconstrained receding-horizon control of nonlinear systems

It is well known that unconstrained infinite-horizon optimal control may be used to construct a stabilizing controller for a nonlinear system. We show that similar stabilization results may be achieved using unconstrained finite horizon optimal control. The key idea is to approximate the tail of the infinite horizon cost-to-go using, as terminal cost, an appropriate control Lyapunov function. Roughly speaking, the terminal control Lyapunov function (CLF) should provide an (incremental) upper bound on the cost. In this fashion, important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers. The absence of constraints allows a significant speedup in computation. Furthermore, it is shown that in order to guarantee stability, it suffices to satisfy an improvement property, thereby relaxing the requirement that truly optimal trajectories be found. We provide a complete analysis of the stability and region of attraction/operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class. It is shown that the guaranteed region of operation contains that of the CLF controller and may be made as large as desired by increasing the optimization horizon (restricted, of course, to the infinite horizon domain). Moreover, it is easily seen that both CLF and infinite-horizon optimal control approaches are limiting cases of our receding horizon strategy. The key results are illustrated using a familiar example, the inverted pendulum, where significant improvements in guaranteed region of operation and cost are noted     

小型无人直升机的模型预测控制算法研究

针对无人直升机在阵风干扰环境中的姿态控制精度低的问题.本文将非线性刚体动力学模型在悬停点应用小扰动理论得到了线性化数学模型.考虑系统输入输出和控制量约束,采用模型预测控制将控制器的设计问题转化为每个采样时刻求解一个带不等式和等式约束的凸二次规划问题.通过设计终端状态约束解决了有限时域模型预测控制(model predictive control, MPC)算法的稳定性问题,并通过引入松弛变量使得约束优化问题更容易求解.随机和常值阵风干扰下无人机悬停仿真验证了本文MPC预测控制器具有幅度不超过0.25 m/s的良好干扰抑制能力,性能明显优于线性二次型调节器(linear-quadratic regulator, LQR).     

Nominally Robust Model Predictive Control With State Constraints

In this paper, we present robust stability results for constrained discrete-time nonlinear systems using a finite-horizon model predictive control (MPC) algorithm for which we do not require the terminal cost to have any particular properties. We introduce a definition that attempts to characterize the robustness properties of the MPC optimization problem. We assume the systems under consideration satisfy this definition (for which we give sufficient conditions) and make two further assumptions. These are that the value function is bounded by a K_infin function of a state measure (related to the distance from the state to some target set) and that this measure is detectable from the stage cost used in the MPC algorithm. We show that these assumptions lead to stability that is robust to sufficiently small disturbances. While in general the results are semiglobal and practical, when the detectability and upper bound assumptions are satisfied with linear K_infin functions, the stability and robustness are either semiglobal or global (with respect to the feasible set). We discuss algorithms employing terminal inequality constraints and also provide a specific example of an algorithm that employs a terminal equality constraint.     

Robust stabilization of switched discrete-time systems with actuator saturation

This paper studies the robust stabilization problem of switched discrete-time linear systems subject to actuator saturation. New switched saturation-dependent Lyapunov functions are exploited to design a robust stabilizing state feedback controller that maximizes an estimation of the domain of attraction. The design problem of controller （coefficient matrices） is then reduced to an optimization problem with linear matrix inequality （LMI） constraints. A numerical example is given to show the effectiveness of the proposed method.     

基于终端凸集约束的新MPC 控制器

针对一类离散系统,研究了带有终端约束凸集的MPC控制问题.通过离线设计一组椭圆不变集,并将其组合成一个终端约束凸集,其中凸集参数作为在线优化变量.在线运算时,根据实际的终端状态即时地选择合适的终端不变集,从而有效地扩大了系统的可行域.分别给出了设计MPC控制器的离线和在线算法,仿真实例说明了该方法的有效性.     

Suboptimal stochastic H-two/H-infinity design with spectrum constraint

This paper studies the problem of suboptimal state-feedback H-two/H-infinity control of stochastic systems with spectrum constraint. Concretely speaking, a mixed suboptimal H-two/H-infinity controller synthesis together with placing the spectrum into a strip region is considered, which is achieved via solving a convex optimization problem.