Robust model predictive control: robust control invariant sets and efficient implementation

Robust model predictive control (RMPC) is widely used in industry. However, the online computational burden of this algorithm restricts its development and application to systems with relatively slow dynamics. We investigate this problem in this thesis with the overall aim of reducing the online computational burden and improving the online efficiency. In RMPC schemes, robust control invariant (RCI) sets are vitally important in dealing with constraints and providing stability. They can be used as terminal (invariant) sets in RMPC schemes to reduce the online computational burden and ensure stability simultaneously. To this end, we present a novel algorithm for the computation of full-complexity polytopic RCI sets, and the corresponding feedback control law, for linear discrete-time systems subject to output and initial state constraints, performance bounds, and bounded additive disturbances. Two types of uncertainty, structured norm-bounded and polytopic uncertainty, are considered. These algorithms are then extended to deal with systems subject to asymmetric initial state and output constraints. Furthermore, the concept of RCI sets can be extended to invariant tubes, which are fundamental elements in tube based RMPC scheme. The online computational burden of tube based RMPC schemes is largely reduced to the same level as model predictive control for nominal systems. However, it is important that the constraint tightening that is needed is not excessive, otherwise the performance of the MPC design may deteriorate, and there may even not exist a feasible control law. Here, the algorithms we proposed for RCI set approximations are extended and applied to the problem of reducing the constraint tightening in tube based RMPC schemes. In order to ameliorate the computational complexity of the online RMPC algorithms, we propose an online-offline RMPC method, where a causal state feedback structure on the controller is considered. In order to improve the efficiency of the online computation, we calculate the state feedback gain offline using a semi-definite program (SDP). Then we propose a novel method to compute the control perturbation component online. The online optimization problem is derived using Farkas' Theorem, and then approximated by a quadratic program (QP) to reduce the online computational burden. A further approximation is made to derive a simplified online optimization problem, which results in a large reduction in the number of variables. Numerical examples are provided that demonstrate the advantages of all our proposed algorithms over current schemes.Open Acces

Control of Constrained Dynamical Systems with Performance Guarantees: With Application to Vehicle motion Control

In control engineering, models of the system are commonly used for controller design. A standard control design problem consists of steering the given system output (or states) towards a predefined reference. Such a problem can be solved by employing feedback control strategies. By utilizing the knowledge of the model, these strategies compute the control inputs that shrink the error between the system outputs and their desired references over time. Usually, the control inputs must be computed such that the system output signals are kept in a desired region, possibly due to design or safety requirements. Also, the input signals should be within the physical limits of the actuators. Depending on the constraints, their violation might result in unacceptable system failures (e.g. deadly injury in the worst case). Thus, in safety-critical applications, a controller must be robust towards the modelling uncertainties and provide a priori guarantees for constraint satisfaction. A fundamental tool in constrained control application is the robust control invariant sets (RCI). For a controlled dynamical system, if initial states belong to RCI set, control inputs always exist that keep the future state trajectories restricted within the set. Hence, RCI sets can characterize a system that never violates constraints. These sets are the primary ingredient in the synthesis of the well-known constraint control strategies like model predictive control (MPC) and interpolation-based controller (IBC). Consequently, a large body of research has been devoted to the computation of these sets. In the thesis, we will focus on the computation of RCI sets and the method to generate control inputs that keep the system trajectories within RCI set. We specifically focus on the systems which have time-varying dynamics and polytopic constraints. Depending upon the nature of the time-varying element in the system description (i.e., if they are observable or not), we propose different sets of algorithms.The first group of algorithms apply to the system with time-varying, bounded uncertainties. To systematically handle the uncertainties and reduce conservatism, we exploit various tools from the robust control literature to derive novel conditions for invariance. The obtained conditions are then combined with a newly developed method for volume maximization and minimization in a convex optimization problem to compute desirably large and small RCI sets. In addition to ensuring invariance, it is also possible to guarantee desired closed-loop performance within the RCI set. Furthermore, developed algorithms can generate RCI sets with a predefined number of hyper-planes. This feature allows us to adjust the computational complexity of MPC and IBC controller when the sets are utilized in controller synthesis. Using numerical examples, we show that the proposed algorithms can outperform (volume-wise) many state-of-the-art methods when computing RCI sets.In the other case, we assume the time-varying parameters in system description to be observable. The developed algorithm has many similar characteristics as the earlier case, but now to utilize the parameter information, the control law and the RCI set are allowed to be parameter-dependent. We have numerically shown that the presented algorithm can generate invariant sets which are larger than the maximal RCI sets computed without exploiting parameter information.Lastly, we demonstrate how we can utilize some of these algorithms to construct a computationally efficient IBC controller for the vehicle motion control. The devised IBC controller guarantees to meet safety requirements mentioned in ISO 26262 and the ride comfort requirement by design

Automatic Offline Formulation of Robust Model Predictive Control Based on Linear Matrix Inequalities Method

Two automatic robust model predictive control strategies are presented for uncertain polytopic linear plants with input and output constraints. A sequence of nested geometric proportion asymptotically stable ellipsoids and controllers is constructed offline first. Then the feedback controllers are automatically selected with the receding horizon online in the first strategy. Finally, a modified automatic offline robust MPC approach is constructed to improve the closed system's performance. The new proposed strategies not only reduce the conservatism but also decrease the online computation. Numerical examples are given to illustrate their effectiveness

A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

Design and Implementation of Model Predictive Control Approaches

Ph.DDOCTOR OF PHILOSOPH

Splitting Methods for Distributed Optimization and Control

This thesis contributes towards the design and analysis of fast and distributed optimization algorithms based on splitting techniques, such as proximal gradient methods or alternation minimization algorithms, with the application of solving model predictive control (MPC) problems. The first part of the thesis focuses on developing an efficient algorithm based on the fast alternating minimization algorithm to solve MPC problems with polytopic and second-order cone constraints. Due to the requirement of bounding the online computation time in the context of real-time MPC, complexity bounds on the number of iterations to achieve a certain accuracy are derived. In addition, a discussion of the computation of the complexity bounds is provided. To further improve the convergence speed of the proposed algorithm, an o-line pre-conditioning method is presented for MPC problems with polyhedral and ellipsoidal constraints. The inexact alternating minimization algorithm, as well as its accelerated variant, is proposed in the second part of the thesis. Different from standard algorithms, inexact methods allow for errors in the update at each iteration. Complexity upper-bounds on the number of iterations in the presence of errors are derived. By employing the complexity bounds, sufficient conditions on the errors, which guarantee the convergence of the algorithms, are presented. The proposed algorithms are applied for solving distributed optimization problems in the presence of local computation and communication errors, with an emphasis on distributed MPC applications. The convergence properties of the algorithms for this special case are analysed. Motivated by the complexity upper-bounds of the inexact proximal gradient method, two distributed optimization algorithms with an iteratively refining quantization design are proposed for solving distributed optimization problems with a limited communication data-rate. We show that if the parameters of the quantizers satisfy certain conditions, then the quantization error decreases linearly, while at each iteration only a fixed number of bits is transmitted, and the convergence of the distributed algorithms is guaranteed. The proposed methods are further extended to distributed optimization problems with time-varying parameters

Robust feedback model predictive control of norm-bounded uncertain systems

This thesis is concerned with the Robust Model Predictive Control (RMPC) of linear discrete-time systems subject to norm-bounded model-uncertainty, additive disturbances and hard constraints on the input and state. The aim is to design tractable, feedback RMPC algorithms that are based on linear matrix inequality (LMI) optimizations. The notion of feedback is very important in the RMPC control parameterization since it enables effective disturbance/uncertainty rejection and robust constraint satisfaction. However, treating the state-feedback gain as an optimization variable leads to non-convexity and nonlinearity in the RMPC scheme for norm-bounded uncertain systems. To address this problem, we propose three distinct state-feedback RMPC algorithms which are all based on (convex) LMI optimizations. In the first scheme, the aforementioned non-convexity is avoided by adopting a sequential approach based on the principles of Dynamic Programming. In particular, the feedback RMPC controller minimizes an upper-bound on the cost-to-go at each prediction step and incorporates the state/input constraints in a non-conservative manner. In the second RMPC algorithm, new results, based on slack variables, are proposed which help to obtain convexity at the expense of only minor conservatism. In the third and final approach, convexity is achieved by re-parameterizing, online, the norm-bounded uncertainty as a polytopic (additive) disturbance. All three RMPC schemes drive the uncertain-system state to a terminal invariant set which helps to establish Lyapunov stability and recursive feasibility. Low-complexity robust control invariant (LC-RCI) sets, when used as target sets, yield computational advantages for the associated RMPC schemes. A convex algorithm for the simultaneous computation of LC-RCI sets and the corresponding controller for norm-bounded uncertain systems is also presented. In this regard, two novel results to separate bilinear terms without conservatism are proposed. The results being general in nature also have application in other control areas. The computed LC-RCI sets are shown to have substantially improved volume as compared to other schemes in the literature. Finally, an output-feedback RMPC algorithm is also derived for norm-bounded uncertain systems. The proposed formulation uses a moving window of the past input/output data to generate (tight) bounds on the current state. These bounds are then used to compute an output-feedback RMPC control law using LMI optimizations. An output-feedback LC-RCI set is also designed, and serves as the terminal set in the algorithm.Open Acces