MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.     

Periodic optimal control of nonlinear constrained systems using economic model predictive control

In this paper, we consider the problem of periodic optimal control of nonlinear systems subject to online changing and periodically time-varying economic performance measures using model predictive control (MPC). The proposed economic MPC scheme uses an online optimized artificial periodic orbit to ensure recursive feasibility and constraint satisfaction despite unpredictable changes in the economic performance index. We demonstrate that the direct extension of existing methods to periodic orbits does not necessarily yield the desirable closed-loop economic performance. Instead, we carefully revise the constraints on the artificial trajectory, which ensures that the closed-loop average performance is no worse than a locally optimal periodic orbit. In the special case that the prediction horizon is set to zero, the proposed scheme is a modified version of recent publications using periodicity constraints, with the important difference that the resulting closed loop has more degrees of freedom which are vital to ensure convergence to an optimal periodic orbit. In addition, we detail a tailored offline computation of suitable terminal ingredients, which are both theoretically and practically beneficial for closed-loop performance improvement. Finally, we demonstrate the practicality and performance improvements of the proposed approach on benchmark examples.     

Periodic event‐triggered control for distributed networked multiagents with asynchronous communication: A predictive control approach

This paper investigates the periodic event‐triggered control problem for distributed networked multiagent systems with interconnected nonlinear dynamics subject to asynchronous communication. A method of state trajectory estimation for the interconnected neighboring agents over each prediction horizon with guaranteed error bounds is addressed to handle the asynchronous communication. Based on it, a distributed robust model predictive control (MPC) is proposed with a distributed periodic event‐triggered scheme for each agent. According to this algorithm, each subsystem generates presumed state trajectories for all its upstream neighbors and computes its own control locally. By checking the designed triggering condition periodically, the optimization problem of MPC will be implemented and solved when the local error of the subsystem exceeds a specified threshold. Then, the optimized control input will be determined and applied until the next time instant when the triggering condition is invoked. Moreover, sufficient condition for ensuring feasibility of the designed algorithm is conducted, along with the analysis of asymptotic stabilization of the closed‐loop system. The illustrative example for a set of coupled Van der Pol oscillators is reported to verify the effectiveness of the proposed approach.     

基于输入状态稳定的离散广义系统预测控制

针对一类具有持续扰动和输入约束的离散广义系统, 研究其鲁棒预测控制器的设计问题. 将输入状态稳定的概念引入广义系统预测控制, 在quasi-min-max 性能指标下, 提出了广义系统双模鲁棒预测控制器的设计方法, 证明了基于双模鲁棒预测控制器的闭环广义系统输入状态稳定, 且具有正则、因果性. 数值仿真结果验证了所提出方法的有效性.

     

Feedback noncausal model predictive control of wave energy converters

In this paper, a novel feedback noncausal model predictive control (MPC) strategy for sea wave energy converters (WECs) is proposed, where the wave prediction information can be explicitly incorporated into the MPC strategy to improve the WEC control performance. The main novelties of the MPC strategy proposed in this paper include: (i) the recursive feasibility and robust constraints satisfaction are guaranteed without a significant increase in the computational burden; (ii) the information of short-term wave prediction is incorporated into the feedback noncausal MPC method to maximise the potential energy output; (iii) the sea condition for the WEC to safely operate in can be explicitly calculated. The proposed feedback noncausal MPC algorithm can also be extended to a wide class of control design problems, especially to the energy maximisation problems with constraints to be satisfied and subject to persistent but predictable disturbances. Numerical simulations are provided to show the efficacy of the proposed feedback noncausal MPC.     

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.     

Least-restrictive move-blocking model predictive control

Move-blocking lowers the computational complexity of model predictive control (MPC) problems by reducing the number of optimization variables. However, this may render states close to constraints infeasible. Thus move-blocking generally results in control laws that are restrictive; the controller domains may be unacceptably and unnecessarily small. Furthermore, different move-blocking strategies may result in controller domains of different sizes, all other factors being equal. In this paper an approach is proposed to design move-blocking MPC control laws that are least-restrictive, i.e. the controller domain is equal to the maximum controlled invariant set. The domains of different move-blocking controllers are then by design equal to each other. This allows comparison of differing move-blocking strategies based on cost performance only, without needing to consider domain size also. Thus this paper is a step towards being able to derive optimal move-blocking MPC control laws.     

Intensive insulin therapy for critically ill subjects based on direct data-driven model predictive control

Patients in the intensive care units (ICU) can suffer from stress-induced hyperglycemia, which can result in negative outcomes and even death. Recent studies show that, regulation of blood glucose (BG) brings in improved outcomes. In this study, a novel direct data-driven model predictive control (MPC) strategy is developed to tightly regulate BG concentration in the ICU. The effectiveness of the proposed direct data-driven MPC strategy is validated on 30 virtual ICU patients, and the in silico results demonstrate the proposed method's excellent robustness with respect to intersubject variability and measurement noises. In addition, the mean percentage values in A-zone of the control variability grid analysis (CVGA) plots are 14% under the Yale protocol, 67% under the combination of particle swarm optimization (PSO) and MPC method (for short, termed as PSO–MPC method), and 90% under the proposed method. In summary, as a good candidate for full closed-loop glycemic control algorithm, the proposed method has superior performance to the nurse-driven Yale protocol and the closed-loop PSO–MPC method.     

Tube-based robust economic model predictive control

In this paper, we develop a tube-based economic MPC framework for nonlinear systems subject to unknown but bounded disturbances. Instead of simply transferring the design procedure of tube-based stabilizing MPC to an economic MPC framework, we rather propose to consider the influence of the disturbance explicitly within the design of the MPC controller, which can lead to an improved closed-loop average performance. This will be done by using a specifically defined integral stage cost, which is the key feature of our proposed robust economic MPC algorithm. Furthermore, we show that the algorithm enjoys similar properties as a nominal economic MPC algorithm (i.e., without disturbances), in particular with respect to bounds on the asymptotic average performance of the resulting closed-loop system, as well as stability and optimal steady-state operation.     

Synthesis of Mixed Objective Output Feedback Robust Model Predictive Control

Aiming at the constrained polytopic uncertain system with energy‐bounded disturbance and unmeasurable states, a novel synthesis scheme to design the output feedback robust model predictive control(MPC)is put forward by using mixed H₂/H_∞ design approach. The proposed scheme involves an offline design of a robust state observer using linear matrix inequalities(LMIs)and an online output feedback robust MPC algorithm using the estimated states in which the desired mixed objective robust output feedback controllers are cast into efficiently tractable LMI‐based convex optimization problems. In addition, the closed‐loop stability and the recursive feasibility of the proposed robust MPC are guaranteed through an appropriate reformulation of the estimation error bound (EEB). A numerical example subject to input constraints illustrates the effectiveness of the proposed controller.     

A model predictive control approach to the periodic implementation of the solutions of the optimal dynamic resource allocation problem

This paper proposes a model predictive control (MPC) approach to the periodic implementation of the optimal solutions of a class of resource allocation problems in which the allocation requirements and conditions repeat periodically over time. This special class of resource allocation problems includes many practical energy optimization problems such as load scheduling and generation dispatch. The convergence and robustness of the MPC algorithm is proved by invoking results from convex optimization. To illustrate the practical applications of the MPC algorithm, the energy optimization of a water pumping system is studied.     

周期时变系统的鲁棒自适应重复控制

针对周期时变系统,提出一种鲁棒自适应重复控制方法.该方法利用周期学习律估计周期时变参数,并结合鲁棒自适应方法处理非周期不确定性.与现有重复控制不同的是,在控制器设计中引入了新变量—周期数,利用周期系统的重复特性,使界的逼近误差随周期数的增加而逐渐减少,保证了系统的全局渐近稳定性.同时将该方法应用于一类非线性参数化系统,使系统在非参数化扰动的情形下,输出误差仍能收敛于0,倒立摆模型的仿真验证了此结果.该设计方法适用于消除神经网络逼近误差对重复控制系统的影响,理论证明了基于神经网络的鲁棒自适应重复控制系统中所有变量的有界性和输出误差的渐近收敛性,关于机械臂模型的仿真结果验证了受控系统具有良好的跟踪性能.     

Robust distributed model predictive control of linear systems with structured time-varying uncertainties

In this work, synthesis of robust distributed model predictive control (MPC) is presented for a class of linear systems subject to structured time-varying uncertainties. By decomposing a global system into smaller dimensional subsystems, a set of distributed MPC controllers, instead of a centralised controller, are designed. To ensure the robust stability of the closed-loop system with respect to model uncertainties, distributed state feedback laws are obtained by solving a min–max optimisation problem. The design of robust distributed MPC is then transformed into solving a minimisation optimisation problem with linear matrix inequality constraints. An iterative online algorithm with adjustable maximum iteration is proposed to coordinate the distributed controllers to achieve a global performance. The simulation results show the effectiveness of the proposed robust distributed MPC algorithm.     

Robust economic model predictive control based on a periodicity constraint

This paper proposes robust economic model predictive control based on a periodicity constraint for linear systems subject to unknown‐but‐bounded additive disturbances. In this economic MPC design, a periodic steady‐state trajectory is not required and thus assumed unknown, which precludes the use of enforcing terminal state constraints as in other standard economic formulations. Instead, based on the desired periodicity of system operation, we optimize the economic performance over a set of periodic trajectories that include the current state. To achieve robust constraint satisfaction, we use a tube‐based technique in the economic MPC formulation. The mismatches between the nominal model and the closed‐loop system with perturbations are limited using a local control law. With the proposed robust tube‐based strategy, recursive feasibility is guaranteed. Moreover, under a convexity assumption, the closed‐loop convergence of the closed‐loop system is analyzed, and an optimality certificate is provided to check if the closed‐loop trajectory reaches a neighborhood of the optimal nominal periodic steady trajectory using Karush‐Kuhn‐Tucker optimality conditions. Finally, through numerical examples, we show the effectiveness of the proposed approach.     

Spatial periodic adaptive control approach for rotary systems in sampled time

A periodic adaptive control approach is proposed for a class of nonlinear sampled‐time systems with varying parametric uncertainties that are periodic with respect to angular displacement, and the only prior knowledge is the periodicity. The new adaptive controller updates the parameters and the control signal periodically in a pointwise manner between two consecutive spatial cycles and in the sequel achieves the asymptotic tracking convergence. Rigorous analysis are presented along with a numerical example to show the effectiveness of the proposed controller. Copyright © 2012 John Wiley & Sons, Ltd.     

Event‐switched control design with guaranteed performances

In this paper, a new event‐switched control method is presented for controlling discrete‐time linear systems subject to bounded disturbances. The main advantage of the proposed method is that the nominal performance of the controlled system with periodic control updates is kept in a framework that do not require to periodically update the control law. The feedback control loop can be opened as long a state‐dependent event condition is satisfied. This condition is obtained using set theory approaches. In particular, the concept of robustly positively invariant sets is used to calculate the nominal performance and the event condition. The simulation presented in this paper confirms the efficiency of the present approach. A reduction of the numerical complexity of the approach is also proposed. Copyright © 2016 John Wiley & Sons, Ltd.     

ILC strategy for progress improvement of economic performance in industrial model predictive control systems

A novel approach to progress improvement of the economic performance in model predictive control (MPC) systems is developed. The conventional LQG based economic performance design provides an estimation which cannot be done by the controller while the proposed approach can develop the design performance achievable by the controller. Its optimal performance is achieved by solving economic performance design (EPD) problem and optimizing the MPC performance iteratively in contrast to the original EPD which has nonlinear LQG curve relationship. Based on the current operating data from MPC, EPD is transformed into a linear programming problem. With the iterative learning control (ILC) strategy, EPD is solved at each trial to update the tuning parameter and the designed condition; then MPC is conducted in the condition guided by EPD. The ILC strategy is proposed to adjust the tuning parameter based on the sensitivity analysis. The convergence of EPD by the proposed ILC has also been proved. The strategy can be applied to industry processes to keep enhancing the performance and to obtain the achievable optimal EPD. The performance of the proposed method is illustrated via an SISO numerical system as well as an MIMO industry process.     

Receding horizon output feedback control for constrained uncertain systems using periodic invariance

In this article, we consider a receding horizon output feedback control (RHOC) method for linear discrete-time systems with polytopic model uncertainties and input constraints. First, we derive a set of estimator gains and then we obtain, on the basis of the periodic invariance, a series of state feedback gains stabilising the augmented output feedback system with these estimator gains. These procedures are formulated as linear matrix inequalities. An RHOC strategy is proposed based on these state feedback and state estimator gains in conjunction with their corresponding periodically invariant sets. The proposed RHOC strategy enhances the performance in comparison with the case in which static periodic gains are used, and increases the size of the stabilisable region by introducing a degree of freedom to steer the augmented state into periodically invariant sets.     

Adaptive control Lyapunov function based model predictive control for continuous nonlinear systems

A design of adaptive model predictive control (MPC) based on adaptive control Lyapunov function (aCLF) is proposed in this article for nonlinear continuous systems with part of its dynamics being unknown at the starting time. Specifically, to guarantee the convergence of the closed-loop system with online predictive model updating, a stability constraint is designed. It limits the aCLF of the system under the MPC to be less than that under an online updated auxiliary adaptive control. The auxiliary adaptive control which implements in a sampling-hold fashion can guarantee the convergence of the controlled system. The sufficient conditions that guarantee the states to be steered to a small region near the equilibrium by the proposed MPC are provided. The calculation of the proposed algorithm does not depend on the model mismatch at the starting time. And it does not require the Lyapunov function of the state of the real system always to be reduced at each time. These provide the potential to improve the performance of the closed-loop system. The effectiveness of the proposed method is illustrated through a chemical process example.