A sufficient condition for null controllability of nonlinear control systems

The optimal control of nonlinear time-varying systems, particularly when the mathematical model of the system is unavailable or inexact, is an interesting and difficult control problem. This paper outlines a methodology for the design of an intelligent controller to perform optimal control of a nonlinear system adaptively, using emerging technologies of fuzzy logic (FL) and artificial neural networks (ANN). FL is utilized to incorporate the available knowledge into the control system, and ANN technology is applied to adaptively provide an optimal control strategy based on some performance criteria. The technique is tested on a system that consists of a DC motor (a linear time-invariant (LTI) system) and a thermal system (a time-varying nonlinear system). Performance criteria such as tracking accuracy, cost, robustness, are considered, and the results are presented in this paper. >

Self-triggering adaptive optimal control for nonlinear systems based on encoding mechanism

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

In this study, the problem of finding an optimal controller for nonlinear systems with one input and a reference tracking signal is approached. With the problem's formulation, any desired signal can be tracked instantly with a closed-loop controller without the need for integral terms. Presentation lies at the heart of optimal control. This study, however, does not consider the integral term, allowing tracking and stability to occur naturally. It has a broad scope with a wide range of applications, namely when dealing with affine nonlinear systems, which provide geometric control unification with asymptotic stability in some cases. A common scenario that comes from optimal control involves the minimization of integral cost functionals. Issues like asymptotic stability or even tracking to the desired reference signal have always been the main limitations. In this study, the main theorem allows the solution of optimal control problems with no-integral terms, in other words tracking problems with input/state constraints, providing closed-loop controllers. A DC motor with a pendulum in upright position is an example of an application for which singular optimal control is tested in this study. The results confirm both asymptotic stability and optimal tracking with an accuracy of 95%. The main contributions of this study include an optimal closed-loop controller with no mixed initial/final conditions, input/state constraints, asymptotic stability guarantee, a strong connection with geometric tools and finally the possibility to generalize to systems with multiple inputs. As a conclusion, general nonlinear control systems can be included in the optimal control methodology presented in this study including input/state constraints. Due to the lack of integral terms, the problem can be solved in closed form by using an optimal closed-loop controller.

LSTM-empowered reinforcement learning in Bi-level optimal control for nonlinear systems with uncertain dynamics.

This paper focuses on a class of unknown discrete-time nonlinear systems with dead-zone control constraints based on adaptive dynamic programming (ADP). The discrete-time Hamilton-Jacobi-Bellman (DTHJB) equation corresponding to the dead-zone control input is formulated. Based on ADP technique, a new cost function is proposed that solves the optimal control with dead zone constraints effectively. It shows that this algorithm allows the implementation of the optimal control without knowing nonlinear affine system model and dead-zone dynamics. Finally, a simulation example is provided to verify the effectiveness of the proposed iterative algorithm.

Theory of nonlinear control

International audience

Let a nonlinear control system having the state space\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\bar X} \subseteq R^n \) be governed by the vector differential equation $$\dot x\left( t \right) = f\left( {t, x\left( t \right), u\left( t \right)} \right),$$ wherex(0)=x0 and is the family of all bounded measurable functions from [0,T] intoU, a compact and convex subset ofR m . Letg:U→R m be a bounded measurable function such thatg(U) is compact and convex, and letF be a function from\(\left[ {0, T} \right] \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\bar X} \) intoR n×m . If, among other conditions, fori=1, ...,n, $$f^i \left( {t, x, u_1 } \right) - f^i \left( {t, x, u_2 } \right) \leqslant F^i \left( {t, x} \right)\left( {g\left( {u_1 } \right) - g\left( {u_2 } \right)} \right),$$ whereF i is theith row ofF, then the main result of the paper establishes the existence of a control which minimizes the cost functional $$I\left( u \right) = \int {_0^T } c\left( {t, x\left( t \right), u\left( t \right)} \right)dt,$$ wherec(t,x,u) is convex inu for each (t,x). An example is worked out in detail.

An adaptive system for linear systems with unknown parameters is a nonlinear system. The analysis of such adaptive systems requires similar techniques to analyse nonlinear systems. Therefore it is natural to treat adaptive control as a part of nonlinear control systems. Nonlinear and Adaptive Control Systems treats nonlinear control and adaptive control in a unified framework, presenting the major results at a moderate mathematical level, suitable for MSc students and engineers with undergraduate degrees. Topics covered include introduction to nonlinear systems; state space models; describing functions for common nonlinear components; stability theory; feedback linearization; adaptive control; nonlinear observer design; backstepping design; disturbance rejection and output regulation; and control applications, including harmonic estimation and rejection in power distribution systems, observer and control design for circadian rhythms, and discrete-time implementation of continuous-time nonlinear control laws.

In this paper, we study the use of state-of-the-art nonlinear system identification techniques for the optimal control of nonlinear systems. We show that the nonlinear system identification problem is equivalent to estimating the generalized moments of an underlying sampling distribution and is bound to suffer from ill-conditioning and variance when approximating a system to high order, requiring samples combinatorial-exponential in the order of the approximation. We show that the iterative identification of "local" linear time varying (LTV) models around the current estimate of the optimal trajectory, coupled with a suitable optimal control algorithm such as iterative LQR (ILQR), alleviates these issues and is sufficient to accurately solve the underlying optimal control problem.

In this paper, a model-free incremental adaptive optimal control scheme is proposed for nonlinear systems in presence of disturbance and input time-delay. The linear time-varying approximate model of the nonlinear system is obtained by incremental method, in which the relevant matrix parameters are identified by recursive least squares (RLS) estimation. A time-delay matrix function is constructed by neural network (NN) to eliminate the input time-delay, and the external disturbance of nonlinear system is dealt with by [Formula: see text] optimal control; incremental adaptive dynamic programming (IADP) algorithm is used to get the control law by solving Hamilton–Jacobi–Isaacs (HJI) equation. Convergence analysis of the proposed control scheme is provided. Simulations are given to verify the effectiveness of the proposed control scheme.

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.

A survey of the authors' recent contributions are given to a new multi-constrained and multi-criteria optimization approach to the design of optimal compensators for general MIMO nonlinear feedback control systems in several practical considerations including such as robust stabilization with the presence of uncertainty, tracking and model matching, and disturbance rejection problems. First, the general framework for nonlinear closed-loop feedback systems is described in a Banach space setting in the time domain. Then, several typical optimal feedback design problems are formulated. Moreover, existence, uniqueness and characteristics theorems are established. Finally, a convergent recursive algorithm for solving the general constrained optimization is included.

MTN optimal control of MIMO non-affine nonlinear time-varying discrete systems for tracking only by output feedback