An optimal linear control design for nonlinear systems

International audience

This paper presents a polynomial chaos-based framework for designing optimal linear feedback control laws for nonlinear systems with stochastic parametric uncertainty. The spectral decomposition of the original stochastic dynamical model in an orthogonal polynomial basis, prescribed by the Wiener–Askey scheme, provides a deterministic model from which the optimal linear control law is designed. Optimality of the proposed control law is proved by solving the Hamilton–Jacobi–Bellman equation, and asymptotic stability of the controlled nonlinear systems is guaranteed in the Lyapunov sense. We are especially interested in synchronization of chaotic systems. For this reason, the control strategy is applied in the trajectory tracking of periodic orbits for the Duffing oscillator and the Rossler system with uncertain stochastic parameters and initial conditions. The results are verified with Monte Carlo simulations.

The work deals finite frequency H<sub>∞</sub> control design for continuous time nonlinear systems, we provide sufficient conditions, ensuring that the closed-loop model is stable. Simulations will be gifted to show level of attenuation that a H<sub>∞</sub> lower can be by our method obtained developed where further comparison.

Theproblem ofapproximatedecoupling of nonlinear systemsby outputfeedback is considered. The structure of the controller is high-gain proportional plus integral. Linearization of the nonlinear closed-loop equations results in the standard singular perturbation form. Based on the singular perturbation concept, a control law is found that accomplishes approximate decoupling with stability for nonlinear systems. Application oftheproposed design schemeto thefullsix-degree-of-freedomnonlinearaircraftdynamicsmodelispresented,demonstrating itsvalidity. I. Introduction A SYSTEM is said to be decoupled if each component of the output depends only on the corresponding component of the input. The problem of decoupling a multivariable system has been extensively treated in the literature in the past 30 years. The necessary and sufe cient conditions were developed for linear 1;2 and nonlinear 3i6 systems. However, no systematic procedures for the controller design were considered. Nonlinear system controller design is typically approached by linearizing the system about a nominal operating point and applying linear system design methods. This will provide a linear control law that yields approximate noninteracting control and satisfy design objectives in the neighborhood of the constant operating point. For the purpose of this work, the design objectives for this highly coupled system of e ight dynamics are to closely track individual desired outputs, with zero steady-state response from others. However, the problem of computing the decoupling feedback control laws for a nonlinear system is quite dife cult. Approximate decoupling control for nonlinear systems is based onsingularperturbationconcepts.Thestructureofcontrollerishighgain output feedback with proportional plus integral (PI) control.It will be shown that the linearization of nonlinear closed-loop system about the nominal operating point yields the standard singular perturbation equation form when a scalar gain weighting g ! 1. Then,thesingularperturbationconcept,i.e.,necessaryandsufe cient conditions, is utilized to e nd a decoupling control law. This approach is a new, conceptually simple, but practically useful design method for approximate noninteracting control law design with stability for nonlinear systems. The results developed are easy to follow and are easily applicable to aircraft control decoupling problems. This paper introduces the application of this design method to approximate noninteracting e ight control system design.

In this paper, an application of the Principal Component Analysis (PCA) technique to data-driven approach is addressed. The proposed scheme is applied to OFSP-based adaptive output feedback control systems. It is well recognized that an adaptive control system based on OFSP conditions can achieve asymptotic stability via a static output feedback. However, these conditions are very restrictive on most realistic systems. Therefore, a parallel feedforward compensator (PFC) is introduced to alleviate such restrictions, but remains a steady state error which is removed by utilizing the feedforward input. Besides, the data-driven approach is one of effective control strategies for nonlinear systems in the process control, and the control parameters can be updated rapidly at each equilibrium point. However, there is no research work that had been done in terms of applying the data-driven approach in an adaptive control system based on OFSP. Therefore, once the robust stability of the adaptive control system based on OFSP can be guaranteed, the adaptive gains can be optimized by the data-driven approach such that the output performance is able to be improved. Moreover, the PCA technique is considered to imporove the data-driven approach in terms of computional cost. The proposed scheme is verified through a numerical example, the results of which demonstrate the effectiveness.

Robust velocity and height control problem is dealt with for hypersonic vehicles subject to highly nonlinearity and uncertainties. The linear robust controller design for nonlinear systems suffers from the conservation problem, because the nonlinear dynamics are considered as uncertainties. In this paper, a feedback linearization controller is firstly applied to achieve the dynamic decoupling of the velocity and height channels. Then, robust controllers are designed for each channel to restrain the effects of the parametric uncertainties, external disturbances, and nonlinear dynamics, which cannot be counteracted accurately by the feedback linearization technique. This method can guarantee the robust stability and robust tracking properties of the closed-loop control system, while the conservation of the robust controller design can be reduced because the nonlinear information of dynamic system is used. Simulation comparisons are given to show the advantages of the proposed robust nonlinear control method.

The problem of robust velocity and height control is dealt with for hypersonic vehicles subject to high nonlinearity and uncertainties. Robust linear controller design for nonlinear systems suffers from the problem of conservation because the nonlinear dynamics are considered as uncertainties. In this paper, a feedback linearization controller is firstly applied to achieve dynamic decoupling of the velocity and height channels. Robust controllers are then designed for each channel to achieve the desired tracking and restrain the effects of parametric uncertainties, external disturbances, and nonlinear dynamics, which cannot be accurately counteracted by the feedback linearization technique. This method can guarantee the robust stability and robust tracking properties of the closed-loop control system. Moreover, the conservation of the robust controller design can be reduced because the nonlinear information of the dynamic system is used. Simulation results are given to show the advantages of the proposed robust nonlinear control method compared with its linear counterpart.

In this technical note, we propose a new iterative learning control (ILC) scheme for nonlinear systems with parametric uncertainties that are temporally and iteratively varying. The time-varying characteristics of the parameters are described by a set of unknown basis functions that can be any continuous functions. The iteratively varying characteristics of the parameters are described by a high-order internal model (HOIM) that is essentially an auto-regression model in the iteration domain. The new parametric learning law with HOIM is designed to effectively handle the unknown basis functions. The method of composite energy function is used to derive convergence properties of the HOIM-based ILC, namely the pointwise convergence along the time axis and asymptotic convergence along the iteration axis. Comparing with existing ILC schemes, the HOIM-based ILC can deal with nonlinear systems with more generic parametric uncertainties that may not be repeatable along the iteration axis. The validity of the HOIM-based ILC under identical initialization condition (i.i.c.) and the alignment condition is also explored.

This paper proposes an adaptive sliding mode fault tolerant control design for lipschitz nonlinear system subject to simultaneous actuator and sensor faults. First, in order to estimate the system states, actuator and sensor faults, only one observer with an adaptive nonlinear gain is required where the minimum phase condition is relaxed to delectability and the matching condition is weakened to become a condition related to the system dimensions. Furthermore, our approach is applicable where there are faults greater than outputs. Next, a virtual sensor technique is developed to replace a failed, missing or corrupted sensor to provide a signal having the same effect of the nominal sensor. Then, this study provides an adaptive sliding mode fault tolerant control to achieve an optimal interaction between observer, virtual sensor and controller models. Thus, using the LMI technique with multiobjective optimization performance, sufficient conditions are derived to ensure the closed-loop Lipschitz nonlinear system stability and guarantee that the Lipschitz set of the adaptive sliding mode observer is a subset of the adaptive sliding mode control Lipschitz set. Two illustrative examples the first is the robotic manipulator model and the second is the longitudinal dynamics of the following vehicle system were performed to verify the effectiveness of the proposed approach.

The goal of this book is to present different methodologies of the adaptive-approximation-based control in a unified context to facilitate discussions and comparison. Some of the topics covered include: a review of typical terminology used in nonlinear system control; approximation theory; approximation structures; learning methodologies used for adaptive function approximation; the design of nonlinear control systems; analysis and synthesis of nonlinear adaptive-approximation-based control systems; higher-order systems; and the design and analysis of adaptive-approximation-based controllers as applied to a fixed-wing aircraft. The book includes two appendices, with the first giving a background knowledge for stability and control of nonlinear systems, and the second providing a useful guide for implementing and debugging adaptive-approximation-based controllers. The book is written in plain language and flows from chapter to chapter in a natural way. Each chapter concludes with a summary as well as the exercises and design problems. The book constitutes an excellent reference for scientists and engineers interested in the subject matter. It would also serve well as a text for senior undergraduate or first year graduate courses in the field.

It has been a common consensus that general techniques for stabilization of nonlinear systems are available only for some special classes of nonlinear systems. Control design for nonlinear systems with uncertain components is usually carried out on a per system basis, especially when physical control constraints, and certain control performance measures such as optimum time control are imposed. Elegant adaptive control techniques are difficult to apply to this type of problems. A new neural network based control design is proposed and presented in this paper to deal with a special class of uncertain nonlinear systems with multiple inputs. The desired system dynamics are analyzed and utilized in the process of the proposed intelligent control design. The theoretical results are provided to justify the design procedures. The simulation study is conducted on a second-order bilinear system with two inputs and uncertainties on its parameters. The simulation results indicate that the proposed design approach is effective.

Interval Criterion of Robust Stability for Nonlinear System and Its Application to Flight Control

Direct feedback control design for nonlinear systems

SUMMARY The development of control strategies for controllable passive dampers — ‘semiactive’ damping devices — is complicated by the nonlinear and dissipative nature of the devices and the nonlinear nature of the closed-loop system with any form of parametric feedback control. Control design for nonlinear systems is often achieved by designing a control for a linearized model since strategies for linear systems are straightforward. One such approach is clipped-optimal control in which the desired damper forces are determined from an optimal controller (e.g., linear quadratic regulator, linear quadratic Gaussian, H2 and H∞), which is designed assuming that the damping devices are fully linear actuators that can exert any forces (dissipative or nondissipative), and a secondary bang–bang controller commands the controllable damper to exert forces as close as possible to the desired forces. However, designs using any linearized model generally result in suboptimal (and sometimes very poor) performance because the linear actuator assumption differs from the actual implementation with a dissipative damping device. Thus, one must generally resort to a large-scale parameter study in which the nonlinear system is simulated many times to determine control strategies that are actually optimal for the nonlinear controlled closed-loop system. This paper demonstrates how an approach developed by the authors, which can reduce the model of a system with local modifications to a lower order set of Volterra integral equations in the modification forces, can significantly decrease the computational burden of a complex control design study for controllable dampers. The approach is first adapted to use a state feedback control with a bang–bang on–off secondary controller. Then, the structural and damping device models are detailed. The optimal semiactive control design is demonstrated via both a parameter study and an optimization algorithm. Numerical examples demonstrate that the approach is effective and, compared with a conventional simulation approach, can reduce computation time by two orders of magnitude. Copyright © 2014 John Wiley & Sons, Ltd.

On a switching control scheme for nonlinear systems with ill-defined relative degree