Optimal control of nonlinear systems with input constraints using linear time varying approximations

We propose a new method to solve input constrained optimal control problems for autonomous nonlinear systems affine in control. We then extend the method to compute the bang-bang control solutions under the symmetric control constraints. The most attractive aspect of the proposed technique is that it enables the use of linear quadratic control theory on the input constrained linear and nonlinear systems. We illustrate the effectiveness of our technique both on linear and nonlinear examples and compare our results with those of the literature

Optimal scheduling and control for constrained multi-agent networked control systems

In this paper, we study optimal control and communication schedule co-design for multi-agent networked control systems, with assuming shared parallel communication channels and uncertain constrained linear time-invariant discrete-time systems. To that end, we specify the communication demand for each system using an associated robust control invariant set and reachability analysis. We use these communication demands and invariant sets to formulate tube-based model predictive control and offline/online communication schedule co-design problems. Since the scheduling part includes an infinite dimension integer problem, we propose heuristics to find suboptimal solutions that guarantee robust constraints satisfaction and recursive feasibility. The effectiveness of our approach is illustrated through numerical simulations

Optimal scheduling and control for constrained multi-agent networked control systems

In this paper, we study optimal control and communication schedule co-design for multi-agent networked control systems, with assuming shared parallel communication channels and uncertain constrained linear time-invariant discrete-time systems. To that end, we specify the communication demand for each system using an associated robust control invariant set and reachability analysis. We use these communication demands and invariant sets to formulate tube-based model predictive control and offline/online communication schedule co-design problems. Since the scheduling part includes an infinite dimension integer problem, we propose heuristics to find suboptimal solutions that guarantee robust constraints satisfaction and recursive feasibility. The effectiveness of our approach is illustrated through numerical simulations

Shooting methods for numerical solutions of control problems constrained by linear and nonlinear hyperbolic partial differential equations

We consider shooting methods for computing approximate solutions of control problems constrained by linear or nonlinear hyperbolic partial differential equations. Optimal control problems and exact controllability problems are both studied, with the latter being approximated by the former with appropriate choices of parameters in the cost functional. The types of equations include linear wave equations, semilinear wave equations, and first order linear hyperbolic equations. The controls considered are either distributed in part of the time-space domain or of the Dirichlet type on the boundary. Each optimal control problem is reformulated as a system of equations that consists of an initial value problem (IVP) for the state equations and a terminal value problem for the adjoint equations. The optimality systems are regarded as a system of an IVP for the state equation and an IVP for the adjoint equations with unknown initial conditions. Then the optimality system is solved by shooting methods, i.e. we attempt to find adjoint initial values such that the adjoint terminal conditions are met. The shooting methods are implemented iteratively and Newton\u27s method is employed to update the adjoint initial values. The convergence of the algorithms are theoretically discussed and numerically verified. Computational experiments are performed extensively for a variety of settings: different types of constraint equations in 1-D or 2-D, distributed or boundary controls, optimal control or exact controllability

Exact and Approximate Relaxation Techniques for Computational Guidance

The focus of this dissertation is in the development and application of relaxation techniques that enable efficient and real-time solution of complex computational guidance problems. Relaxations transform a non-convex constraint into a convex constraint and provides proof that the optimal solutions to the relaxed problem are optimal for the original problem. Unique contributions of this work include: 1) a relaxation technique for solving fixed final time problems between fixed points, 2) a performance analysis on the application of computational guidance for the Mars Ascent Vehicle, and 3) establishment of sufficient conditions for non-singularity of optimal control for problems on a smooth manifold with mixed constraints. The first result states that for annularly constrained linear systems, controllability is a sufficient condition for the problem to be solvable as a sequence of convex programs. The second result applies relaxations to an ascent problem. The third result is the most general result to date for problems with mixed constraints. It uses a minimum principle on manifolds with mixed constraints to analyze the problem in a geometric framework, and shows that strong observability of the dual system is sufficient for non-singularity

Optimal Control with Information Pattern Constraints

Despite the abundance of available literature that starts with the seminal paper of Wang and Davison almost forty years ago, when dealing with the problem of decentralized control for linear dynamical systems, one faces a surprising lack of general design methods, implementable via computationally tractable algorithms. This is mainly due to the fact that for decentralized control configurations, the classical control theoretical framework falls short in providing a systematic analysis of the stabilization problem, let alone cope with additional optimality criteria. Recently, a significant leap occurred through the theoretical machinery developed in Rotkowitz and Lall, IEEE-TAC, vol. 51, 2006, pp. 274-286 which unifies and consolidates many previous results, pinpoints certain tractable decentralized control structures, and outlines the most general known class of convex problems in decentralized control. The decentralized setting is modeled via the structured sparsity constraints paradigm, which proves to be a simple and effective way to formalize many decentralized configurations where the controller feature a given sparsity pattern. Rotkowitz and Lall propose a computationally tractable algorithm for the design of H2 optimal, decentralized controllers for linear and time invariant systems, provided that the plant is strongly stabilizable. The method is built on the assumption that the sparsity constraints imposed on the controller satisfy a certain condition (named quadratic invariance) with respect to the plant and that some decentralized, strongly stablizable, stabilizing controller is available beforehand. For this class of decentralized feedback configurations modeled via sparsity constraints, so called quadratically invariant, we provided complete solutions to several open problems. Firstly, the strong stabilizability assumption was removed via the so called coordinate free parametrization of all, sparsity constrained controllers. Next we have addressed the unsolved problem of stabilizability/stabilization via sparse controllers, using a particular form of the celebrated Youla parametrization. Finally, a new result related to the optimal disturbance attenuation problem in the presence of stable plant perturbations is presented. This result is also valid for quadratically invariant, decentralized feedback configurations. Each result provides a computational, numerically tractable algorithm which is meaningful in the synthesis of sparsity constrained optimal controllers

Overlapping Schwarz Decomposition for Constrained Quadratic Programs

We present an overlapping Schwarz decomposition algorithm for constrained quadratic programs (QPs). Schwarz algorithms have been traditionally used to solve linear algebra systems arising from partial differential equations, but we have recently shown that they are also effective at solving structured optimization problems. In the proposed scheme, we consider QPs whose algebraic structure can be represented by graphs. The graph domain is partitioned into overlapping subdomains (yielding a set of coupled subproblems), solutions for the subproblems are computed in parallel, and convergence is enforced by updating primal-dual information in the overlapping regions. We show that convergence is guaranteed if the overlap is sufficiently large and that the convergence rate improves exponentially with the size of the overlap. Convergence results rely on a key property of graph-structured problems that is known as exponential decay of sensitivity. Here, we establish conditions under which this property holds for constrained QPs (as those found in network optimization and optimal control), thus extending existing work that addresses unconstrained QPs. The numerical behavior of the Schwarz scheme is demonstrated by using a DC optimal power flow problem defined over a network with 9,241 nodes

Optimal control problems constrained by hyperbolic conservation laws.

Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This thesis deals with the solutions of optimal control problems constrained by hyperbolic conservation laws. Such problems pose significant challenges for mathematical analysis and numerical simulations. Those challenges are mainly because of the discontinuities that occur in the solutions of non-linear systems of conservation laws and become more acute when dealing with the multidimensional case. The problem is formulated as the minimisation of a flow matching cost functional constrained by multi-dimensional hyperbolic conservation laws. The control variable is the initial condition of the partial differential equations. In our analysis of the problem, we review extensively the constraints equation and we consider successively the one-dimensional and the multi-dimensional cases. In all the cases, we derive the optimality conditions in the adjoint approach at the continuous level, which are then discretised to arrive at a numerical algorithm for the solution. In the derivation of the optimality conditions, we replace the non-linear conservation laws either by the relaxation equation or the Lattice Boltzmann equation. We illustrate our findings on examples related to the multi-dimensional Burger and the Euler equations

Nonlinear robust H∞ control.

A new theory is proposed for the full-information finite and infinite horizontime robust H∞ control that is equivalently effective for the regulation and/or tracking problems of the general class of time-varying nonlinear systems under the presence of exogenous disturbance inputs. The theory employs the sequence of linear-quadratic and time-varying approximations, that were recently introduced in the optimal control framework, to transform the nonlinear H∞ control problem into a sequence of linearquadratic robust H∞ control problems by using well-known results from the existing Riccati-based theory of the maturing classical linear robust control. The proposed method, as in the optimal control case, requires solving an approximating sequence of Riccati equations (ASRE), to find linear time-varying feedback controllers for such disturbed nonlinear systems while employing classical methods. Under very mild conditions of local Lipschitz continuity, these iterative sequences of solutions are known to converge to the unique viscosity solution of the Hamilton-lacobi-Bellman partial differential equation of the original nonlinear optimal control problem in the weak form (Cimen, 2003); and should hold for the robust control problems herein. The theory is analytically illustrated by directly applying it to some sophisticated nonlinear dynamical models of practical real-world applications. Under a r -iteration sense, such a theory gives the control engineer and designer more transparent control requirements to be incorporated a priori to fine-tune between robustness and optimality needs. It is believed, however, that the automatic state-regulation robust ASRE feedback control systems and techniques provided in this thesis yield very effective control actions in theory, in view of its computational simplicity and its validation by means of classical numerical techniques, and can straightforwardly be implemented in practice as the feedback controller is constrained to be linear with respect to its inputs