Distributed Model Predictive Control And Optimization For Constrained Systems

Constraints are ubiquitous in practical industry systems,including physical conditions,manufacturing processes,temperature,humidity,and so on.Controllers designed by the classical control theory and applied on practical industry systems generally cannot lead to satisfactory system performance.The reason is that the external constraints are not fully taken into account during the controller design.As a result,the controller designed according to the nominal system cannot achieve the optimal control performance.Model predictive control(MPC)has attracted increasing attention for its capability in handling the constraints explicitly.The scale of the practical industry systems is increasing rapidly in recent years.The main features of such systems are high dimension,multi-subsystems,multi-constraints,and multi-targets.Implementation of the traditional centralized MPC(CMPC)to such systems may result in the heavy computational burden,high communication width consumption,low system fault tolerance,etc.In distributed MPC(DMPC)strategies,the global optimization problem is decomposed into several sub-optimization problems and respectively assigned to each subsystem for solving them.This can reduce the difficulty of solving such optimization problems,alleviate the communication burden,and improve the fault tolerance of the system.The stability analysis and distributed solving of DMPC have been widely studied in recent years.However,it is still an open issue for large-scale complex systems subject to disturbances and communication time-delays.The consumption of the computation and communication of DMPC should be reduced.Moreover,to study the DMPC from the perspective of distributed optimization,there are lots of difficult and important problems to be solved.For instance,how to respectively solve the distributed optimization problem with or without assuming the separability of the globally coupled constraints;how to guarantee the convergence rate of the algorithm.This dissertation studies the DMPC and the distributed solving problem,respectively.Inspired by the principle of optimality,the invariant set theory,and the contraction theory,we propose two DMPC schemes for the large-scale complex systems subject to the additive disturbance and the communication time-delay.Moreover,for the globally coupled constraints in the distributed optimization problem of DMPC,we propose two distributed optimization algorithms with guaranteed convergence rates by using the contraction theory and the convex optimization theory.Especially,the two algorithms are developed under or not the assumption of the separability of the globally coupled constraints,respectively.The main results of this dissertation can be summarized in the following aspects.(1)Considering a group of nonlinear dynamically decoupled multi-agent systems(MAS),the self-triggered mechanism is introduced to reduce the computational burden of the centralized MPC.Thereafter,an event-triggered DMPC is studied over such systems cooperating through a coupled cost function subject to the input constraints.From the practical point of view,the additive disturbances are considered in the controller design.Using the contraction theory in the framework of Riemannian manifolds,a novel constraint is constructed in the DMPC optimization problem to reject the additive disturbances.Moreover,the event-triggered mechanism is introduced to reduce the computational and communicational burden.The event-triggering condition is designed by checking the Riemannian distance between the actual and optimal state trajectories.The stability and recursive feasibility are rigorously analyzed.The stability of the closed-loop system is verified based on the contraction theory,which distinguishes this work from the existing results using the conventional Lyapunov theory.It is shown that the recursive feasibility is guaranteed if the additive disturbances are bounded and the event-triggering condition is properly designed.(2)A min-max DMPC scheme is proposed for tracking consensus of linear multi-agent systems(MAS)subject to additive disturbances and time-varying communication delays.A terminal constraint set is constructed by Lyapunov-Razumikhin functional and a corresponding local controller is designed for each agent.Furthermore,the sufficient conditions for ensuring the terminal constraint set are provided in the form of linear matrix inequalities(LMIs).The recursive feasibility of the proposed algorithm is guaranteed based on the designed terminal constraint set,terminal cost,and local controller.Moreover,the closed-loop system is shown to be input-to-state stable(ISS).(3)A DMPC strategy is investigated for a class of discrete-time linear systems with consideration of globally coupled constraints.To reduce the computing burden,the constraint tightening technique is adopted for premature termination.The optimization problem of the proposed DMPC builds on a convex and smooth nonlinear objective function taking into account all subsystems.The dual form of such a problem is solved utilizing the primal-dual algorithm in a distributed manner using Laplacian consensus.In the context of discrete-time updating dynamics,we theoretically show the geometric convergence of the primal-dual gradient optimization by the contraction theory.Especially,the stopping number of iterations is bounded for each subsystem according to the convergence rate.Under some reasonable assumptions,the recursive feasibility of the proposed DMPC strategy and the stability of the closed-loop system can be established under the inexact solution.(4)Consider a distributed optimization problem with globally coupled constraints over networks.The global objective function is composed of the local convex and possibly non-smooth costs held by a group of subsystems,and the coupled constraint is the sum of the local linear equality constraints.We build up a distributed primal-dual algorithm by using the distributed ALM and dynamic average consensus.The optimization problem is converted to an unconstrained saddle-point seeking problem appealing to the augmented Lagrangian method(ALM).A consensus-based primal-dual algorithm is proposed to solve the reformulated problem by only requiring peer-to-peer communication and local computation.The local estimates of the globally coupled constraints are introduced into the proposed algorithm,with which the algorithm can be implemented without assuming the separability of the globally coupled constraints.We theoretically establish the non-ergodic convergence rate of O(1/k)in terms of the objective residual,where k is the iteration counter.In addition to the convex cases,we analyze an extension to nonconvex problems subject to convex constraints.Under some common assumptions in the analysis of nonconvex optimization methods,we show the convergence of the presented algorithm to a KKT(Karush-Kuhn-Tucker)point of the problem.