Swing-up and stabilization of inverted pendulum by nonlinear optimal state feedback law

This work develops a numerical algorithm for the calculation of an optimal nonlinear state feedback law for nonlinear systems. A quadratic performance index is used which contains quadratic error terms, and quadratic input penalty terms. The optimization problem is solved using the Hamilton-Jacobi equations, which determine the optimal nonlinear state feedback law. A Newton-Kantorovich iteration is developed for the solution of the pertinent Hamilton-Jacobi equations, which involves solving a Zubov partial differential equation, at each step of the iteration, using a power series method. At step N of the iteration, the method generates the (N + 1)-th order truncation of the Taylor series expansion of the optimal state feedback function. The method is applied to the problem of controlling a system of two non-isothermal continuous stirred tank reactors (CSTR), where an exothermic reaction takes place

State feedback synthesis for on-line optimization of batch reactors with multiple manipulated inputs

On-line optimization of batch reactors with two manipulated inputs

Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization

This paper presents a method for designing nonlinear state feedback laws for systems with input and state constraints. The objective is to achieve practical stabilization with large stability region and strong disturbance rejection. Two invariant sets will be constructed within the state constraints: the outer one for stabilization and the inner one for asymptotic disturbance rejection. The nonlinear feedback law is designed such that all trajectories starting from the outer invariant set will enter the inner invariant set and stay there. Both invariant sets will be constructed by using the convex hull function, a recently introduced non-quadratic Lyapunov function. Since the invariant sets are convex hull of ellipsoids, they are able to incorporate the input and state constraints more effectively than simple ellipsoids, thus promising larger stability region within state constraint and stronger disturbance rejection capability. Since the convex hull functions are constructed from quadratic functions, the optimization problems can be treated with LMI-based method. Numerical examples demonstrate the effectiveness of the design methods.

On global external stochastic stabilization of linear systems with input saturation

Based on the feedback linearization theory of differential geometry theory and the state-space-averaging model of Buck converter working on continuous conduction mode (CCM), the nonlinear state feedback law is proposed. Further, a linear quadratic performance index is presented based on the passivity-based control theory. The state feedback coefficients are optimized using the optimal control theory. The controller is robust to the load variation and variety of parameters. Key waveforms based on simulated extensive work show that the control system has good static and dynamic performance.

We propose an algorithm for nonlinear receding-horizon output feedback control by combining a real-time nonlinear optimal state feedback control algorithm and a real-time nonlinear optimal state estimation algorithm. The algorithms of state feedback control and state estimation are obtained by minimizing receding-horizon performance indexes. Application of the stabilized continuation method results in algorithms that do not involve any successive approximation methods. Simulation results of a 2-wheeled car clarify characteristics of the output feedback control algorithm.

Drug Administration in Cancer Treatment via Optimal Nonlinear State Feedback Gain Matrix Design

In this paper, the transient stabilization of power systems will be enhanced using optimal nonlinear feedback control. To this end, a partial differential equation called Hamilton-Jacobi-Belman (HJB), should be solved, which does not have a closed form solution in general. Nevertheless, there are some approximate solutions to solve HJB equations. In this paper, by using a Taylor Series expansion, a sub-optimal nonlinear control law is obtained. As a result, a transient controller for a single-machine infinite-bus power system is designed. Also, the superiority of the designed nonlinear controller in some grounds like increasing the region of asymptotic stability of the system or in other words, increasing the domain of validity is shown. Simulation results reveal the effectiveness of the proposed approach.

Composite nonlinear control with state and measurement feedback for general multivariable systems with input saturation

The authors provide an alternative solution to the problem of semiglobal stabilization of a class of minimum phase nonlinear systems that was considered by A.R. Teel (1991). The method used yields a stabilizing linear state feedback law in contrast to a nonlinear state feedback law proposed by Teel. The peaking phenomenon is eliminated by inducing a specific time-scale structure in the linear part of the closed-loop system. This structure consists of a very slow and a very fast time scale. The crucial component in the method is the relation between the slow and the fast time-scales. The proposed linear state feedback control law has a single tunable gain parameter that allows for local asymptotic stability and regulation to the origin for any initial condition in some a priori given (arbitrarily large) bounded set. >

This paper introduces a framework for solving time-autonomous nonlinear infinite horizon optimal control problems, under the assumption that all minimizers satisfy Pontryagin’s necessary optimality conditions. In detail, we use methods from the field of symplectic geometry to analyze the eigenvalues of a Koopman operator that lifts Pontryagin’s differential equation into a suitably defined infinite dimensional symplectic space. This has the advantage that methods from the field of spectral analysis can be used to characterize globally optimal control laws. A numerical method for constructing optimal feedback laws for nonlinear systems proceeds by computing the eigenvalues and eigenvectors of a matrix that is obtained by projecting the Pontryagin–Koopman operator onto a finite dimensional space. We illustrate the effectiveness of this approach by computing accurate approximations of the optimal nonlinear feedback law for a Van der Pol control system, which cannot be stabilized by a linear control law.

This paper presents a nonlinear control design method for robust stabilization and robust performance of linear differential inclusions. A recently introduced non-quadratic Lyapunov function, the convex hull quadratic function, is used for the construction of nonlinear state feedback laws. Design objectives include stabilization with maximal convergence rate, disturbance rejection with minimal reachable set and least /spl Lscr//sub 2/ gain. Conditions for stabilization and performances are derived in terms of bilinear matrix inequalities (BMIs), which cover the existing linear matrix inequality (LMI) conditions as special cases. Optimization problems with BMI constraints are formulated and solved effectively by combining the path-following algorithm and the direct iterative algorithm. The design results are guaranteed to be at least as good as the existing results obtained from LMI conditions. In most examples, significant improvements on system performances have been achieved, which demonstrate the advantages of using nonlinear feedback control over linear feedback control for linear differential inclusions. It is also observed through numerical computation that nonlinear control strategies may help to reduce control effort substantially.

On improvement of transient performance in tracking control for a class of nonlinear systems with input saturation