This paper presents a stability analysis tool for model predictive control (MPC) where control action is generated by optimising a cost function over a finite horizon. Stability analysis of MPC with a limited horizon but without terminal weight is a well known challenging problem. We define a new value function based on an auxiliary one-step optimisation related to stage cost, namely optimal one-step value function (OSVF). It is shown that a finite horizon MPC can be made to be asymptotically stable if OSVF is a (local) control Lyapunov function (CLF). More specifically, by exploiting the CLF property of OSFV to construct a contractive terminal set, a new stabilising MPC algorithm (CMPC) is proposed. We show that CMPC is recursively feasible and guarantees stability under the condition that OSVF is a CLF. Checking this condition and estimation of the maximal terminal set are discussed. Numerical examples are presented to demonstrate the effectiveness of the proposed stability condition and corresponding CMPC algorithm.
Ensuring nominal asymptotic stability of the Non-linear Model Predictive Control (NMPC) controller is not trivial. Stabilizing ingredients such as terminal penalty term and terminal region are crucial in establishing the asymptotic stability. Approaches available in the literature provide limited degrees of freedom for the characterization of the terminal region for the discrete time Quasi Infinite Horizon NMPC (QIH-NMPC) formulation. Current work presents alternate approaches namely arbitrary controller based approach and LQR based approach, which provide large degrees of freedom for enlargin…
This paper presents a complementary approach to establish stability of finite receding horizon control with a terminal cost. First a new augmented stage cost is defined by rotating the terminal cost. Then a one-step optimisation problem is defined based on this augmented stage cost. It is shown that a slightly modified Model Predictive Control (MPC) algorithm is stable if the value function of the augmented one-step cost (OSVF) is a control Lyapunov function. The proposed stability condition is completely complementary to the existing terminal cost based MPC stability conditions in the sense t…
Model predictive control (MPC) schemes are commonly designed with fixed, i.e., time-invariant, horizon length and cost functions. If no stabilizing terminal ingredients are used, stability can be guaranteed via a sufficiently long horizon. A suboptimality index can be derived that gives bounds on the performance of the MPC law over an infinite-horizon (IH). While for time-invariant schemes such index can be computed offline, less attention has been paid to time-varying strategies with adapting cost function which can be found, e.g., in learning-based optimal control. This work addresses the pe…
Stability under model predictive control (MPC) schemes is frequently ensured by terminal ingredients. Employing a (control) Lyapunov function as the terminal cost constitutes a common choice. Learning-based methods may be used to construct the terminal cost by relating it to, for instance, an infinite-horizon optimal control problem in which the optimal cost is a Lyapunov function. Value iteration, an approximate dynamic programming (ADP) approach, refers to one particular cost approximation technique. In this work, we merge the results of terminally unconstrained predictive control and approx…
Ensuring nominal asymptotic stability of the Nonlinear Model Predictive Control controller is not trivial. Stabilizing ingredients such as terminal penalty term and terminal region are crucial in establishing the asymptotic stability. Current work presents alternate approaches namely arbitrary controller based approach and linear quadratic regulator based approach, which provide larger degrees of freedom for enlarging the terminal region as against conservative approaches from the literature. Efficacy of the proposed approaches is demonstrated using benchmark two state continuous stirrer tank …
Addressing Terminal Constraints in Data-Driven Demand Response Scheduling
Sargent Centre for Process Systems Engineering, Imperial College London, London, SW7 2AZ, UK（过程系统工程中心，伦敦帝国理工学院，伦敦，SW7 2AZ，英国）；Department of Computer Science, University of Alberta, Edmonton, AB, Canada（计算机科学系，阿尔伯塔大学，埃德蒙顿，AB，加拿大）；Department of Computing, Imperial College London, London, SW7 2AZ, UK（计算系，伦敦帝国理工学院，伦敦，SW7 2AZ，英国）
Predictive control is frequently used for control problems involving constraints. Being an optimization based technique utilizing a user specified so-called stage cost, performance properties, i.e., bounds on the infinite horizon accumulated stage cost, aside closed-loop stability are of interest. To achieve good performance and to influence the region of attraction associated with the prediction horizon, the terminal cost of the predictive controller's optimization objective is a key design factor. Approximate dynamic programming refers to one particular approximation paradigm that pursues it…
We provide a stability and performance analysis for nonlinear model predictive control (NMPC) schemes subject to input constraints. Given an exponential stabilizability and detectability condition w.r.t. the employed state cost, we provide a sufficiently long prediction horizon to ensure asymptotic stability and a desired performance bound w.r.t. the infinite-horizon optimal controller. Compared to existing results, the provided analysis is applicable to positive semi-definite (detectable) cost functions, provides tight bounds using a linear programming analysis, and allows for a seamless inte…
This chapter compares different formulations for Economic nonlinear Model Predictive Control (EMPC) which are all based on an established dissipativity assumption on the underlying Optimal Control Problem (OCP). This includes schemes with and without stabilizing terminal constraints, respectively, or with stabilizing terminal costs. We recall that a recently proposed approach based on gradient correcting terminal penalties implies a terminal constraint on the adjoints of the OCP. We analyze the feasibility implications of these dual/adjoint terminal constraints and we compare our findings to a…
In this paper, we provide a stability and performance analysis of model predictive control (MPC) schemes based on finite-tail costs. We study the MPC formulation originally proposed by Magni et al. (2001) wherein the standard terminal penalty is replaced by a finite-horizon cost of some stabilizing control law. In order to analyse the closed loop, we leverage the more recent technical machinery developed for MPC without terminal ingredients. For a specified set of initial conditions, we obtain sufficient conditions for stability and a performance bound in dependence of the prediction horizon a…
Many chemical processes exhibit diverse timescale dynamics with a strong coupling between timescale sensitive variables. Model predictive control with a non-uniformly spaced optimisation horizon is an effective approach to multi-timescale control and offers opportunities for reduced computational complexity. In such an approach the fast, moderate and slow dynamics can be included in the optimisation problem by implementing smaller time intervals earlier in the prediction horizon and increasingly larger intervals towards the end of the prediction. In this paper, a reference-flexible condition i…
Rollout approaches are an effective tool to address the problem of co-designing the transmission schedule and the corresponding input values, when the controller is connected to the plant via a resource-constrained communication network. These approaches typically employ an MPC, activated at multiples of the period length of a base transmission schedule. Using multi-step invariant terminal regions and a prediction horizon longer than the base period, stability can be ensured. This strategy, however, suffers from intrinsic shortcomings, such as a high computational complexity and low robustness…
In this paper, we present a quasi infinite horizon nonlinear model predictive control (MPC) scheme for tracking of generic reference trajectories. This scheme is applicable to nonlinear systems, which are locally incrementally stabilizable. For such systems, we provide a reference generic offline procedure to compute an incrementally stabilizing feedback with a continuously parameterized quadratic quasi infinite horizon terminal cost. As a result we get a nonlinear reference tracking MPC scheme with a valid terminal cost for general reachable reference trajectories without increasing the onlin…
In this paper, model predictive control (MPC) strategies are proposed for dead-beat control of linear systems with and without state and control constraints. In unconstrained MPC, deadbeat performance can be guaranteed by setting the control horizon to the system dimension, and adding an terminal equality constraint. It is proved that the unconstrained deadbeat MPC is equivalent to linear deadbeat control. The proposed constrained deadbeat MPC is designed by setting the control horizon equal to the system dimension and penalizing only the terminal cost. The recursive feasibility and deadbeat p…
Improving greenhouse fruit-production control by integrating reinforcement learning into short-horizon model predictive control
Wageningen University & Research（瓦赫宁根大学与研究）；Delft Center for Systems and Control (DCSC), Delft University of Technology（代尔夫特理工大学系统与控制中心）；Wageningen Research（瓦赫宁根研究）
In this paper, we investigate the problem of Model Predictive Control (MPC) of dynamic systems for high-level specifications described by Signal Temporal Logic (STL) formulae. Recent works show that MPC has the great potential in handling logical tasks in reactive environments. However, existing approaches suffer from the heavy computational burden, especially for tasks with large horizons. In this work, we propose a computationally more efficient MPC framework for STL tasks based on time interval decomposition. Specifically, we still use the standard shrink horizon MPC framework with Mixed In…
A finite horizon optimal tracking problem is considered for linear dynamical systems subject to parametric uncertainties in the state-space matrices and exogenous disturbances. A suboptimal solution is proposed using a model predictive control (MPC) based explicit dual control approach which enables active uncertainty learning. A novel algorithm for the design of robustly invariant online terminal sets and terminal controllers is presented. Set membership identification is used to update the parameter uncertainty online. A predicted worst-case cost is used in the MPC optimization problem to mo…
In this paper, it is shown that a simple formulation of Economic Model Predictive Control can be used which possesses two features that are generally viewed as mutually exclusive, namely, a rather short prediction horizon (reachability-compatible) on one side, and the absence of final constraint on the other side. The result holds for discrete-time models that are originated from continuous-time models by means of discretization. Practical stability at a small neighborhood of the optimal unknown steady-state pair is shown when some parameters increase and sampling time decreases. A commonly us…
To provide robustness of distributed model predictive control (DMPC), this work proposes a robust DMPC formulation for discrete-time linear systems subject to unknown-but-bounded disturbances. Taking advantage of the structure of certain classes of distributed systems seen in applications with interagent coupling like vehicle platooning, a novel robust DMPC is formulated. The proposed approach is characterised by separable terminal costs and locally robust terminal sets, with the latter sets adaptively estimated in the online optimisation problem. A constraint tightening approach based on a se…
This work presents a suboptimality study of a particular model predictive control with a stage cost shaping based on the ideas of reinforcement learning. The focus of the suboptimality study is to derive quantities relating the infinite-horizon cost function under the said variant of model predictive control to the respective infinite-horizon value function. The basis control scheme involves usual stabilizing constraints comprising of a terminal set and a terminal cost in the form of a local Lyapunov function. The stage cost is adapted using the principles of Q-learning, a particular approach …
We consider an input-constrained differential-drive robot with actuator dynamics. For this system, we establish asymptotic stability of the origin on arbitrary compact, convex sets using Model Predictive Control (MPC) without stabilizing terminal conditions despite the presence of state constraints and actuator dynamics. We note that the problem without those two additional ingredients was essentially solved beforehand, despite the fact that the linearization is not stabilizable. We propose an approach successfully solving the task at hand by combining the theory of barriers to characterize th…
This paper proposes a stabilising model predictive control (MPC) scheme with preview information of disturbance for nonlinear systems. The proposed MPC algorithm is able to not only reject disturbance by making use of disturbance preview information as necessary, but also take advantage of the disturbance if it is good for a control task. This is realised by taking into account both the task (e.g. reference trajectory) and disturbance preview in the prediction horizon when performing online optimisation. Conditions are established to ensure recursive feasibility and stability under the disturb…
