Nonlinear optimal control: alternatives to Hamilton-Jacobi equation

Nonlinear matrix inequality approach based heading control for an autonomous underwater vehicle with experimental realization

A state-space characterization of robustness analysis and synthesis for nonlinear uncertain systems is proposed. The robustness of a class of nonlinear systems subject to L/sub 2/-bounded structured uncertainties is characterized in terms of a nonlinear matrix inequality (NLMI), which yields a convex feasibility problem. As in the linear case, scalings are used to find a Lyapunov or storage function that give sufficient conditions for robust stability and performances. Sufficient conditions for the solvability of robustness synthesis problems are represented in terms of NLMIs as well. With the proposed NLMI characterizations, it is shown that the computation needed for robustness analysis and synthesis is not more difficult than that for checking Lyapunov stability; the numerical solutions for robustness problems are approximated by the use of finite element methods or finite difference schemes, and the computations are reduced to solving linear matrix inequalities. Unfortunately, while the development in this paper parallels the corresponding linear theory, the resulting computational consequences are, of course, not as favourable.

The stability and performance robustness analysis for a class of uncertain nonlinear systems with bounded structured uncertainties are characterized in terms of various types of nonlinear matrix inequalities (NLMIs). As in the linear case, scalings or multipliers are used to find Lyapunov functions that give sufficient conditions, and the resulting NLMIs yield convex feasibility problem. For these problems, robustness analysis is essentially no harder than stability analysis of the system with no uncertainty. Sufficient conditions for the solvability of related robust synthesis problems are developed in terms of NLMIs as well. >

This paper mainly discusses the residual generation problem for discrete-time polynomial nonlinear systems with unknown inputs. By employing state-dependent polynomial Lyapunov functions, sufficient conditions for fault detection using lscr2-gain performance indices are presented in terms of nonlinear matrix inequalities (NMIs). We show that those NMIs can be solved by the recently developed technique named sum of squares (SOS), which essentially is an LMI feasibility problem.

A new model transformation method and its application to extending a class of stability criteria of neutral type systems

SUMMARYThis paper investigates the state feedback robustH ∞ control problem of a class of discrete‐time singular systems with norm‐bounded uncertainties and interval time‐varying delays in state and input. A new bounded real lemma for discrete‐time singular systems with a pair of time‐varying interval state delays is first investigated. Mathematical comparisons of the new bounded real lemma and two existing ones are presented. Then, on the basis of the bounded real lemma proposed here, a sufficient condition in the form of nonlinear matrix inequality, such that the considered state feedback robustH ∞ control problem is solvable, is given. In order to solve the nonlinear matrix inequality, a cone complementarity linearization algorithm is offered. Several numerical examples are presented to show the applicability of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.

A nonlinear matrix inequality is derived as a stabilizability condition of linear uncertain time-delay systems. This inequality is seen as a less conservative one as well as efficient for numerical computation than the existing results as seen when solving by cone-complementary algorithm. Copyright © 2010 John Wiley & Sons, Ltd.

This paper focuses on the problem of H/sub /spl infin// control for neutral linear parameter-varying systems with mixed time-varying delays. A delay-dependent bounded real lemma is described in terms of parameterized linear matrix inequalities. Based on the bounded real lemma obtained, sufficient conditions for the existence of H/sub /spl infin// control are presented in terms of nonlinear matrix inequalities. To solve these nonlinear matrix inequalities, an iterative algorithm involving convex optimisation is proposed. Numerical example is provided to illustrate the applicability of the new theory.

This paper considers an optimization model and a solution method for the design of two-dimensional mechanical mechanisms. The mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously. The underlying mechanical analysis model is based on a truss representation allowing for large displacements. For mechanisms undergoing large displacements elastic stability is of major concern. We derive conditions, modeled by nonlinear matrix inequalities, which guarantee that a stable equilibrium is found and that buckling is prevented. The feasible set of the design problem is described by nonlinear differentiable and non-differentiable constraints as well as nonlinear matrix inequalities.To solve the mechanism design problem a branch and bound method based on convex relaxations is developed. To guarantee convergence of the method, two different types of convex relaxations are derived. The relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub-problems. Encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality.

Observer-based control for time-delayed quasi-one-sided Lipschitz nonlinear systems under input saturation

A discrete-time jump fuzzy system with two Markov chains is proposed to portray the asymmetric network characteristic of a class of nonlinear NCSs with random but bounded communication delays and packets dropout. The less conservative dual-mode-depend guaranteed cost controller is designed base on the model. The design approach can be cast into a set of nonlinear matrix inequalities (NMIs) solved by a homotopy- based iterative algorithm. Simulation examples are carried out to show the effectiveness of the proposed approaches.

A REFERENCE MODEL BASED ROBUST ℋ∞ FILTERING APPROACH TO FAULT DETECTION IN UNCERTAIN SYSTEMS

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton–Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded.

This study presents a fault-tolerant synthesis controller design scheme for a class of non-linear systems with minimised performance index. A radial basis function (RBF) neural network with the online-updated centre and the width vector of Gaussian function is utilised to approximate the unknown non-linear dynamics. An updating rule is designed to estimate actuator failure. Based on estimated non-linear function and fault knowledge, a synthesis controller is developed with an optimal control action and a sliding-mode control action that is used to eliminate the effect of neural network approximation error. The sufficient condition for the optimal performance is given in terms of non-linear fault-dependent quadratic matrix inequality. Then, the solution of matrix inequality is reduced to solve off-line a fault-free Riccati equation. The feasibility of proposed method is demonstrated by a spacecraft model.

The objective of this paper is to design linear and nonlinear Hinfin controllers, and compare them. For linear systems, the Hinfin controller is obtained either by solving Riccati equations using iterative procedures or by linear matrix inequalities. However, for nonlinear systems, the task is more complicated. In the particular case of nonlinear affine-input systems, the solution of the Hinfin control problem is derived from the Hamilton-Jacobi Equations (HJE). The exact explicit solutions of this type of equations are, in general, unknown or hard to compute. Approximate solutions are obtained from an iterative procedure. In this paper, we propose a nonlinear matrix inequalities approach to design a nonlinear Hinfin controller. Simulations of the closed-loop synchronous motors responses for both nonlinear and linear Hinfin controllers will be performed and compared. It is found that the nonlinear Hinfin controller has better performances and robustness than the linear controller.