Invariance Conditions for Nonlinear Dynamical Systems

Recently, Horv\'ath, Song, and Terlaky [\emph{A novel unified approach to invariance condition of dynamical system, submitted to Applied Mathematics and Computation}] proposed a novel unified approach to study, i.e., invariance conditions, sufficient and necessary conditions, under which some convex sets are invariant sets for linear dynamical systems. In this paper, by utilizing analogous methodology, we generalize the results for nonlinear dynamical systems. First, the Theorems of Alternatives, i.e., the nonlinear Farkas lemma and the \emph{S}-lemma, together with Nagumo's Theorem are utilized to derive invariance conditions for discrete and continuous systems. Only standard assumptions are needed to establish invariance of broadly used convex sets, including polyhedral and ellipsoidal sets. Second, we establish an optimization framework to computationally verify the derived invariance conditions. Finally, we derive analogous invariance conditions without any conditions

Characterizations of robust and stable duality for linearly perturbed uncertain optimization problems

We introduce a robust optimization model consisting in a family of perturbation functions giving rise to certain pairs of dual optimization problems in which the dual variable depends on the uncertainty parameter. The interest of our approach is illustrated by some examples, including uncertain conic optimization and infinite optimization via discretization. The main results characterize desirable robust duality relations (as robust zero-duality gap) by formulas involving the epsilon-minima or the epsilon-subdifferentials of the objective function. The two extreme cases, namely, the usual perturbational duality (without uncertainty), and the duality for the supremum of functions (duality parameter vanishing) are analyzed in detail. © Springer Nature Switzerland AG 2020

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

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

Optimization Theory and Dynamical Systems: Invariant Sets and Invariance Preserving Discretization Methods

Invariant set is an important concept in the theory of dynamical systems and it has a wide range of applications in control and engineering. This thesis has four parts, each of which studies a fundamental problem arising in this field. In the first part, we propose a novel, simple, and unified approach to derive sufficient and necessary conditions, which are referred to as invariance conditions for simplicity, under which four classic families of convex sets, namely, polyhedral, polyhedral cones, ellipsoids, and Lorenz cones, are invariant sets for linear discrete or continuous dynamical systems. This novel method establishes a solid connection between optimization theory and dynamical systems. In the second part, we propose novel methods to compute valid or largest uniform steplength thresholds for invariance preserving of three classic types of discretization methods, i.e., forward Euler method, Taylor type approximation, and rational function type discretization methods. These methods enable us to find a pre-specified steplength threshold which preserves invariance of a set. The identification of such steplength threshold has a significant impact in practice. In the third part, we present a novel approach to ensure positive local and uniform steplength threshold for invariance preserving on a set when a discretization method is applied to a linear or nonlinear dynamical system. Our methodology not only applies to classic sets, discretization methods, and dynamical systems, but also extends to more general sets, discretization methods, and dynamical systems. In the fourth part, we derive invariance conditions for some classic sets for nonlinear dynamical systems. This part can be considered as an extension of the first part to a more general case

Robust control of uncertain systems: H2/H∞ control and computation of invariant sets

This thesis is mainly concerned with robust analysis and control synthesis of linear time-invariant systems with polytopic uncertainties. This topic has received considerable attention during the past decades since it offers the possibility to analyze and design controllers to cope with uncertainties. The most common and simplest approach to establish convex optimization procedures for robust analysis and synthesis problems is based on quadratic stability results, which use a single (parameter-independent) Lyapunov function for the entire uncertainty polytope. In recent years, many researchers have used parameter-dependent Lyapunov functions to provide less conservative results than the quadratic stability condition by working with parameterized Linear Matrix Inequalities (LMIs), where auxiliary scalar parameters are introduced. However, treating the scalar parameters as optimization variables leads to large computational complexity since the scalar parameters belong to an unbounded domain in general. To address this problem, we propose three distinct iterative procedures for H2 and H∞state feedback control, which are all based on true LMIs (without any scalar parameter). The first and second procedures are proposed for continuous-time and discrete-time uncertain systems, respectively. In particular, quadratic stability results can be used as a starting point for these two iterative procedures. This property ensures that the solutions obtained by our iterative procedures with one step update are no more conservative than the quadratic stability results. It is important to emphasize that, to date, for continuous-time systems, all existing methods have to introduce extra scalar parameters into their conditions in order to include the quadratic stability conditions as a special case, while our proposed iterative procedure solves a convex/LMI problem at each update. The third approach deals with the design of robust controllers for both continuous-time and discrete-time cases. It is proved that the proposed conditions contain the many existing conditions as special cases. Therefore, the third iterative procedure can compute a solution, in one step, which is at least as good as the optimal solution obtained using existing methods. All three iterative procedures can compute a sequence of non-increasing upper bounds for H2-norm and H∞-norm. In addition, if no feasible initial solution for the iterative procedures is found for some uncertain systems, we also propose two algorithms based on iterative procedures that offer the possibility of obtaining a feasible initial solution for continuous-time and discrete-time systems, respectively. Furthermore, to address the problem of analysis of H∞-norm guaranteed cost computation, a generalized problem is firstly proposed that includes both the continuous-time and discrete-time problems as special cases. A novel description of polytopic uncertainties is then derived and used to develop a relaxation approach based on the S-procedure to lift the uncertainties, which yields an LMI approach to compute H∞-norm guaranteed cost by incorporating slack variables. In this thesis, one of the main contributions is to develop convex iterative procedures for the original non-convex H2 and H∞ synthesis problems based on the novel separation result. Nonlinear and non-convex problems are general in nature and occur in other control problems; for example, the computation of tightened invariant tubes for output feedback Model Predictive Control (MPC). We consider discrete-time linear time-invariant systems with bounded state and input constraints and subject to bounded disturbances. In contrast to existing approaches which either use pre-defined control and observer gains or optimize the volume of the invariant sets for the estimation and control errors separately, we consider the problem of optimizing the volume of these two sets simultaneously to give a less conservative design.Open Acces

Conic Optimization Theory: Convexification Techniques and Numerical Algorithms

Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic programming. The recent advances in various topics of modern optimization have also been revamping the area of machine learning. Motivated by the crucial role of optimization theory in the design, analysis, control and operation of real-world systems, this tutorial paper offers a detailed overview of some major advances in this area, namely conic optimization and its emerging applications. First, we discuss the importance of conic optimization in different areas. Then, we explain seminal results on the design of hierarchies of convex relaxations for a wide range of nonconvex problems. Finally, we study different numerical algorithms for large-scale conic optimization problems.Comment: 18 page