Variance-constrained multiobjective control and filtering for nonlinear stochastic systems: A survey

The multiobjective control and filtering problems for nonlinear stochastic systems with variance constraints are surveyed. First, the concepts of nonlinear stochastic systems are recalled along with the introduction of some recent advances. Then, the covariance control theory, which serves as a practical method for multi-objective control design as well as a foundation for linear system theory, is reviewed comprehensively. The multiple design requirements frequently applied in engineering practice for the use of evaluating system performances are introduced, including robustness, reliability, and dissipativity. Several design techniques suitable for the multi-objective variance-constrained control and filtering problems for nonlinear stochastic systems are discussed. In particular, as a special case for the multi-objective design problems, the mixed H 2 / H ∞ control and filtering problems are reviewed in great detail. Subsequently, some latest results on the variance-constrained multi-objective control and filtering problems for the nonlinear stochastic systems are summarized. Finally, conclusions are drawn, and several possible future research directions are pointed out

Steady-state and periodic exponential turnpike property for optimal control problems in hilbert spaces

First Published in SIAM Journal on Control and Optimization in Volume 56, Issue 2, 2018, Pages 1222-1252, published by the Society for Industrial and Applied Mathematics (SIAM)In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking termsThe authors acknowledge the nancial support by the grant FA9550-14-1-0214 of the EOARD-AFOSR. The second author was partially supported by the National Natural Science Foundation of China under grants 11501424 and 11371285. The third author was partially supported by the Advanced Grant DYCON (Dynamic Control) of the European Research Council Executive Agency, FA9550-15-1-0027 of AFOSR, the MTM2014-52347 and MTM2017-92996 grants of the MINECO (Spain), and ICON of the French AN

A structure-preserving doubling algorithm for Lur'e equations

We introduce a numerical method for the numerical solution of the Lur'e equations, a system of matrix equations that arises, for instance, in linear-quadratic infinite time horizon optimal control. We focus on small-scale, dense problems. Via a Cayley transformation, the problem is transformed to the discrete-time case, and the structural infinite eigenvalues of the associated matrix pencil are deflated. The deflated problem is associated with a symplectic pencil with several Jordan blocks of eigenvalue 1 and even size, which arise from the nontrivial Kronecker chains at infinity of the original problem. For the solution of this modified problem, we use the structure-preserving doubling algorithm. Implementation issues such as the choice of the parameter γ in the Cayley transform are discussed. The most interesting feature of this method, with respect to the competing approaches, is the absence of arbitrary rank decisions, which may be ill-posed and numerically troublesome. The numerical examples presented confirm the effectiveness of this method

Dissipative systems theory : analysis and synthesis

Finite L2 gain and passivity (or positive real) methods have recently played an important role in a large number of robust, high performance engineering designs for both nonlinear and linear systems. This has renewed interest in the classical concept of dissipative systems. In particular, in various finite gain or passivity system synthesis methods in the literature, one studies a relevant dissipation inequality and looks for an appropriate solution to it. When such a solution exists, one then constructs the desired system by using this solution. The main theme of the thesis is the development of a framework for general dissipative systems analysis and synthesis. We firstly present a numerical method for testing dissipativity of a given system. We characterize a dissipative system in terms of a weak (viscosity) solution to a partial differential inequality (PDI) which is the relevant dissipation inequality for the system being considered and develop a finite-difference based discretization method that results in a partial difference inequality approximating the PDI. We then propose two iterative methods to solve the partial difference inequality. We report a number of computational experiment results to demonstrate the utility of the method. Under certain circumstances, strict dissipativity is of the main concern. We provide characterization of a strongly stable, strictly quadratic dissipative nonlinear system in terms of a solution to a PDI or a solution to a partial differential equation (PDE), in the viscosity sense. When the solution to the PDE is smooth, then it also has a stabilizing (in some sense) property. These results generalize the strict bounded real lemma in the linear H control literature. We also provide characterization of a stable, strictly quadratic dissipative linear system in terms of a stabilizing solution to an algebraic Riccati equation (ARE). Connections between quadratic dissipative systems and finite gain related systems are given. In the thesis, we propose a synthesis method for a general dissipative control problem for nonlinear and linear systems with state feedback. We express the solution to the roblem in terms of a solution to a Hamilton-Jacobi-Isaacs (HJI) PDI/PDE in the non­ linear systems case (algebraic Riccati equation/inequality in the linear systems case). In particular, in the case of nonlinear systems with a general quadratic supply rate, we show that whenever there exists a static state feedback control that renders the closed loop system dissipative, then there exists a solution to the Hamilton-Jacobi-Isaacs PDI/PDE in the viscosity solution. This extends and generalizes a number of synthesis results in the nonlinear H control literature. We then consider a general dissipative output feedback control problem and propose a solution by employing the recently developed information state method. We formulate an information state and then convert the original output feedback problem into a new full state one in which the information state provides the appropriate state. The dynamics of the information state takes the form of a controlled PDE. We then solve the new problem by using game theoretic methods leading to a (infinite dimensional) HJI PDI. This is the relevant (ontrolled) dissipation inequality for the output feedback problem at hand. The solution is then specialized to bilinear and linear systems yielding finite dimensional solutions. As a by product, we formulate and solve a general dissipativity filtering problem for nonlinear and linear systems. The problem takes the nonlinear H filtering as a special case. As in the control case, the solution to the filtering problem is expressed in terms of a controlled PDE describing the dynamics of the corresponding information state and a(infinite dimensional) HJI PDI. When specialized to linear systems with a general quadratic supply rate, the solution reduces to new finite dimensional linear filters with the (central) linear H filter appearing as a special one. Finally, we propose application of general dissipativity control methods to two stabilization problems. In the first problem we look for a controller that stabilizes linear systems possesing sector bounded nonlinearities at their inputs and outputs. In the second one, we look for a controller that stabilizes an uncertain nonlinear systenfconsisting of a nonlinear nominal model and an unknown nonlinear model belonging to a class of general dissipative systems described in terms of a specific suppply rate function. In either case, we pose the stabilization problem as a dissipativity control synthesis one for a related system

Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation

In this paper, two methods for approximating the stabilizing solution of the Hamilton–Jacobi equation are proposed using symplectic geometry and a Hamiltonian perturbation technique as well as stable manifold theory. The first method uses the fact that the Hamiltonian lifted system of an integrable system is also integrable and regards the corresponding Hamiltonian system of the Hamilton–Jacobi equation as an integrable Hamiltonian system with a perturbation caused by control. The second method directly approximates the stable flow of the Hamiltonian systems using a modification of stable manifold theory. Both methods provide analytical approximations of the stable Lagrangian submanifold from which the stabilizing solution is derived. Two examples illustrate the effectiveness of the methods.

Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation

We study the existence of positive and negative semidefinite solutions of algebraic Riccati equations (ARE) corresponding to linear quadratic problems with an indefinite cost functional. The problem to formulate reasonable necessary and sufficient conditions for the existence of such solutions is a long-standing open problem. A central role is played by certain two-variable polynomial matrices associated with the ARE. Our main result characterizes all unmixed solutions of the ARE in terms of the Pick matrices associated with these two-variable polynomial matrices. As a corollary of this result we obtain that the signatures of the extremal solutions of the ARE are determined by the signatures of particular Pick matrices