Distributed Stochastic Model Predictive Control

Distributed model predictive control(DMPC) has been widely used in industrial applications due to its good control performance, capability of handling constraints and the?exibility and error tolerance characterizes. Over the past years, many important results of the deterministic systems have been proposed, and the guaranteed stability and the online computation burden of DMPC have been particularly addressed. However, for systems with model uncertainty and stochastic disturbances, the research of DMPC is still in an embryonic stage, and lots of di?cult but rather important problems still remain to be solved. For example, how to coordinate e?orts to ensure that the distributed decisions lead to coupled constraint satisfaction, and to reduce the conservativeness of constraint handling; how to make use of the probabilistic information on uncertainty to design the distributed controllers which ensure the recursive feasibility and closed-loop stability, and to reduce online computational complexity; how to utilize the relationship between the global objective and the independent decision-making of subsystems to achieve a coordinated response and good control performance of the entire system.Motivated by this, this dissertation focuses on distributed stochastic model predictive control(DSMPC) of a group of linear subsystems subject to uncertainty, which are dynamically-decoupled but share coupled probabilistic constraints. Three kinds of uncertainty in practical applications are taken into account, namely: stochastic disturbances,measurement noise or estimation errors, and stochastic model uncertainty or parametric uncertainty. By employing optimization theory, probability theory, invariant set theory and generalized polynomial chaos expansions(GPCEs) theory, four schemes of DSMPC are proposed and some theoretical results with more practical value are obtained. The main contents and results in this dissertation are summarized as follows:1. A DSMPC scheme is proposed for a team of linear subsystems with stochastic disturbances and probabilistic constraints. By making explicit use of the probabilistic distribution of the disturbances, and permitting a single subsystem to optimize at each time step, coupled probabilistic constraints are converted into a set of deterministic constraints for the predictions of nominal models non-conservatively. Then the problem discussed can therefore be cast as a quadratic program(QP), which can be solved reliably and e?ciently.Under the explicit assumption that the uncertainty is ?nitely supported, the property of recursive feasibility is guaranteed with respect to both local and coupled probabilistic constraints. A form of the quadratic stability of the overall closed-loop system is guaranteed for any choice of update sequence despite the action of stochastic disturbances. Two numerical examples, including homogeneous subsystems and heterogeneous subsystems,are given to show the e?cacy of the proposed algorithm.2. After that, we further investigate the cooperative DSMPC problem for multiple dynamically decoupled subsystems with stochastic disturbances and coupled probabilistic constraints, for which states are not measurable. Cooperation between subsystems is promoted by a scheme in which a local subsystem designs hypothetical plans for others in some cooperating set, and considers the weighted costs of these subsystems in its objective. By using the lifting technique and the probabilistic information on disturbances, measurement noise and the state estimation error, a set of local deterministic constraints is constructed.Then the online optimization reduces to a QP problem which can be solved easily online.Recursive feasibility with respect to both local and coupled constraints is guaranteed and quadratic stability for any choice of update sequence and any structure of cooperation is ensured. Numerical examples illustrate the e?cacy of the proposed algorithm. Simulation results show that the proposed cooperative method leads to a more coordinated response and better global performance than the non-cooperative one.3. Based on GPCEs theory, a novel approach to design DSMPC is proposed for a team of linear subsystems in the presence of stochastic parametric uncertainties and coupled probabilistic constraints. To obtain a deterministic formulation for the DSMPC problem, ?rstly GPCEs theory is applied to approximate uncertain subsystem matrices and the state trajectory by polynomial chaos. Utilizing the Galerkin projection, stochastic dynamical systems are then transformed into equivalent deterministic dynamical systems in higher-dimensional space. Furthermore, both local and coupled probabilistic constraints can be reformulated as the deterministic convex second-order cone constraints over the coe?cient of GPCEs, and the stochastic optimization problem which cannot be solved online is therefore reduced to a convex optimization problem. The proposed GPCEs-based DSMPC algorithm guarantees probabilistic constraint satisfaction and recursive feasibility with respect to both local and coupling constraints, and also ensures asymptotically stable in all the moments. Finally, a numerical example illustrates the e?ectiveness of the proposed algorithm.4. Based on the investigations above, a DSMPC strategy is proposed for multiple linear subsystems with both polytopic uncertainty and stochastic disturbances. To handle local and coupled probabilistic constraints, the system can be divided into a nominal dynamics and an uncertain dynamics. The uncertain dynamics is further decomposed into two parts:the ?rst of which is constrained to lie in probabilistic tubes that are calculated o?ine with the explicit use of the disturbance distribution information, whereas the second part is constrained to lie in polytopic tube invariant sets with bounding facets of ?xed orientation whose distances from the origin are optimized online. After that, probabilistic constraints are reduced into a set of linear constraints, and the online optimization is transformed into a convex problem that can be performed e?ciently. A tailored invariant terminal set is investigated to ensure the recursive feasibility with respect to both local and coupled probabilistic constraints and the recursive feasibility in turn enables the proof of stability of the DSMPC algorithm. Though delays are not explicitly considered for the new algorithm developed, its single-update nature implicitly allows time for communications after each optimization, and no instantaneous inter-subsystem exchanges of information are assumed.Hence, the new algorithm o?ers greater ?exibility in communication, and has a relatively low susceptibility to the adverse e?ects of delays in computation and communication.At the end of this dissertation, the main results are concluded and the problems to be solved in the future are presented.