On-line optimization of batch reactors with two manipulated inputs

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

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

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

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.

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. >

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

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.

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

The problem of reinforcement learning (RL) in an unknown nonlinear dynamical system is equivalent to the search for an optimal feedback law utilizing only data from the simulations/rollouts of the dynamical system. Most RL techniques search over a complex global nonlinear feedback parametrization making them suffer from high training times as well as variance. Instead, we advocate searching over a local feedback representation consisting of an open-loop sequence, and an associated optimal linear feedback law completely determined by the open-loop. We show that this alternate approach results in highly efficient training, the answers obtained are globally optimum, repeatable with negligible variance, and hence reliable, and the resulting closed performance is superior to global state-of-the-art RL techniques. Finally, if we replan, whenever required, which is feasible due to the fast and reliable local solution, it allows us to recover the optimal global feedback law.

Robust Trajectory Tracking for Underactuated VTOL Aerial Vehicles: Extended for Adaptive Disturbance Compensation

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.

The paper aims to discuss the issues of actuator saturation and external disturbance in the magnetic levitation (maglev) system. The proposed technique is composed of composite nonlinear feedback (CNF) and super-twisting algorithm (STA)-based integral sliding mode (ISM) control to tackle the problem of actuator saturation and external disturbances simultaneously. The composite nonlinear feedback scheme comprises of linear feedback law which provides stability and fast response whereas the nonlinear feedback law takes care of input saturation and reduces the overshoot. The super-twisting algorithm (STA)-based integral sliding mode (ISM) controller is designed for disturbance rejection. A super-twisting algorithm-based approach is applied on ISM scheme to eliminate the chattering effect and make it continuous in nature for its direct implementation to the physical maglev system. The designed scheme is successfully tested on real-time feedback instruments model of the maglev system.

A cooperative dynamic hybrid control method for multimotion robotic mechanisms handling a single object whose motion is constrained by environment is discussed. This method takes the manipulator dynamics and object dynamics into consideration and is for controlling the motion of the object as well as for controlling the constraint force and the internal force. The following results are obtained: (1) the force and kinematic relations are established for arms grasping and manipulating an object using various types of end effector; and (2) using the first result, a nonlinear state feedback law for the joint driving force that linearizes and decouples the robot system with respect to the object motion, constraint force, and internal force is given. A basic structure of the dynamic hybrid control system with a servo compensator is presented. The effectiveness of the proposed approach has been examined by a simple experiment. These results are useful for the fine manipulation of an object, in the sense that one can specify the controller characteristics independently for the individual specification of the object motion, the interaction between the object and its environment, and the grasping of the object. >

On global external stochastic stabilization of linear systems with input saturation

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