Piecewise Constant Control Research Articles

The Stabilization Problem for Certain Class of Ecological Systems

The stabilization problem for dynamical systems is to find control providing stability of some sets in phase space. This problem is not solved in general for nonlinear systems. The ecological systems are nonlinear, as usual, with restrictions on phase coordinates and controls. In this paper we consider piecewise constant control stabilization. Some general results are obtained. A new class of population dynamical systems is introduced, for which the constructive method of global stabilization and controllability is given. The set of attraction is found.

Obtaining smoother singular arc policies using a modified iterative dynamic programming algorithm

Dynamic programming is a very powerful technique for the optimization of dynamic systems. With the ready availability of high-speed computers and the development of the iterative dynamic programming (IDP) algorithm, a feasible alternative to the calculus of variations approach to the optimal control problem is now available. Inherent in the IDP algorithm is the application of piecewise constant discretized controls. This often leads to singular optimal control policies that are highly active. The aim of this research is to modify the IDP algorithm to reduce the control energy variations that are typical of IDP while at the same time developing policies that are very close to the true optimal control. This is achieved by including a filter within the IDP procedure. Two types of filtering schemes are considered: a median filter and a first-order filter. Application of this modified algorithm to two bioreactor systems that yield singular optimal control profiles is presented and the usefulness of this scheme is demonstrated.

Mathematical modelling and optimal control of solar dryers

A mathematical model of the solar dryers designed and used in Cuba is presented and discussed. The model consists of a set of ordinary nonlinear differential equations which describe the behaviour of the main parts of the dryer. Using the model, an optimal control problem is formulated and solutions within the class of piecewise constant controls are numerically obtained. It is shown that controls which are constant over intervals of 24 hours have the best properties for application to drying plants.

The optimal control of the linear wave equation

We consider in this paper the existence and numerical estimation of an optimal boundary control for the linear wave equation. The problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted to a finite-dimensional linear programming problem. The solution to this problem is used to construct a piecewise-constant control. Some numerical examples are considered to illustrate the procedure.

Use of Piecewise Linear Continuous Optimal Control for Time-Delay Systems

By using piecewise linear continuous control, rather than piecewise constant control, excellent control can be obtained with the use of a relatively small number of stages. For nonlinear systems, the use of a multipass method in iterative dynamic programming (IDP) is convenient and effective in obtaining convergence to the optimum. However, a good initial control policy, which can be readily obtained by the use of Taylor series approximation and piecewise constant control, may be necessary. Computational experience with a two-stage CSTR system shows the viability of using IDP for typical chemical engineering systems with time delays

Duality of hybrid optimal regulator and hybrid optimal filter

This note shows that the problems of optimal control of linear, continuous-time, deterministic systems with piecewise constant control and optimal filtering of linear, continuous-time stochastic systems using sampled filtered observations are dual to each other.

Hierarchical Sub-Optimal Control of Batch Distillation with Slop Recycling

Slop recycle increases the production capacity of batch distillation. It avoids exhaustion of the mixture or decreases the difficulty of separation. A hierarchical control approach is proposed. At scheduling level predictions of the required component production times and the ratio between product and slop cut quantities are made. These predictions are based on the degree of difficulty of separation and the mean feed quality and quantity. At conditioning level sub-optimal operation under constant quality control is applied. The production capacity approximates the capacity under dynamic optimal control, and is up to 20% higher than under piecewise constant reflux control.

The Semi-Lattice of Piecewise Constant Controls for Non-Linear Systems: A Possible Foundation for Fuzzy Control

An examination of the literature on fuzzy control systems shows that the design of fuzzy controllers depends upon systematic refinement of a sequence of piecewise constant control laws rather than the use of results of the subject going under the title of fuzzy logic. In this paper, the lattice structure of piecewise constant stabilizing controls, with its natural refinement operation induced by a partial ordering ≤ and an associated meet operation π, is proposed as a foundation for the selection of control laws in fuzzy controllers.

Adaptive sliding mode control in discrete-time systems

Discrete-time sliding mode control is considered. The case of known parameters is first analyzed, leading to a new definition of the so-called ‘equivalent control’ as the piecewise-constant control that reduces to zero, in finite time, the distance of the system state from the sliding manifold. In the presence of bounded parametric uncertainties, an adaptive control scheme that guarantees the asymptotic satisfaction of the same control objective is presented. The main feature of this approach is the reduction of the order of the relevant error equation, and the possibility of dealing with the nonmatched uncertainties introduced by the sampling process.

Reachability of Interior States by Piecewise Constant Controls

Reachability of Interior States by Piecewise Constant Controls

Remarks on the Relay Controllability of Control Systems

For a control process governed by a differential equation a special controllability property is studied. We consider relay controls, that is, piecewise constant scalar controls whose absolute value is constant, and we give sufficient conditions for the controllability by means of relay controls with a minimum number of switches or with a constant amplitude.

Guidance law for hypersonic gliders based on piecewise constant control

A midcourse guidance law is developed for the descent of a hypersonic glider, initially in level flight, to a fixed target on the ground. It is based on an optimal piecewise constant control (N intervals) obtained from an approximate physical model (flat Earth, exponential atmosphere, parabolic drag polar, etc). The resulting optimal control equations can be integrated either analytically or by quadrature, and the guidance algorithm requires the solution of 2N + nonlinear algebraic equations. The guidance law is implemented in a realistic glider simulation, the intercept is achieved, and final velocities within 14% of the true values are obtained for the downrange and crossranges considered.

Optimal control of batch reactors by iterative dynamic programming

Four batch reactor systems are chosen to examine the viability of using iterative dynamic programming (IDP) for highly nonlinear systems encountered by chemical engineers. The first system is mildly nonlinear and rapid convergence resulted with the use of only a single state grid point. The use of piecewise linear continuous control with 40 stages yielded better results that the use of 80 stages with piecewise constant control. The need for more than a single grid point for the other three systems led to a systematic study of the effects of the number of grid points, of the number of allowable values for control and of the region contraction factor on the convergence of IDP. In every case the global optimum could be obtained with reasonable computational effort, and no difficulties were encountered even with systems exhibiting several local optima. The use of stages of different length allowed a refined solution to be obtained with a reasonably small number of stages in the last example.

Discretized Energy Minimization in a Wave Guuide with point Sources

An anti-noise problem on a finite time interval is solved by minimization of a quadratic functional on the Hilbert space of square integrable controls. To this end, the one-dimensional wave equation with point sources and pointwise reflecting boundary conditions is decomposed into a system for the two propagating components of waves. Wellposedness of this system is proved for a class of data that includes piecewise linear initial conditions and piecewise constant forcing functions. It is shown that for such data the optimal piecewise constant control is the solution of a sparse linear system. Methods for its computational treatment are presented as well as examples of their applicability. The convergence of discrete approximations to the general optimization problem is demonstrated by finite element methods.

Linear control semigroups acting on projective space

Linear control semigroupsL⊂=Gl(d,R) are associated with semilinear control systems of the form whereA:R m → gl(d,R) is continuous in some open set containingU. The semigroupL then corresponds to the solutions with piecewise constant controls, i.e., L acts in a natural way onR d {0}, on the sphereS d−1, and on the projective spaceP d−1. Under the assumption that the group generated byL in Gl(d,R) acts transitively onP d−1, we analyze the control structure of the action ofL onP d−1: We characterize the sets inP d−1, where the system is controllable (the control sets) using perturbation theory of eigenvalues and (generalized) eigenspaces of the matrices g eL For nonlinear control systems on finitedimensional manifoldsM, we study the linearization on the tangent bundleTM and the projective bundleP M via the theory of Morse decompositions, to obtain a characterization of the chain-recurrent components of the control flow onU×PM. These components correspond uniquely to the chain control sets onP M, and they induce a subbundle decomposition ofU×TM. These results are used to characterize the chain control sets ofL acting onP d−1 and to compare the control sets and chain control sets.

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.

On the equivalency of piecewise-constant control with a known number of switchings and arbitrary amplitude bounded control in a terminal problem for a linear nonstationary system

It is shown that any amplitude-bounded control of a linear terminal nonstationary system with two control coordinates can be replaced by a piecewise-constant (relaying) control with a known number of switchings, and moreover, the procedure of determining the moments of switching is reduced to a known finite number of problems of nonlinear programming in a space of two variables, which are given in the paper.

Certain systems with piecewise constant feedback controls with a time delay

Certain systems with piecewise constant feedback controls with a time delay

Design of Piecewise Constant Gains for Optimal Control via Block-Pulse Functions

Using block-pulse functions, a method is presented to determine the piecewise constant feed-back controls for a finite linear regulator problem optimal for time multiplied quadratic performance indices. This method gives a complete solution to the time weighted finite and infinite quadratic regulator problem. This is illustrated with two examples.

Computation of the digital lqg regulator and tracker for time-varying systems

Digital optimal control problems, i.e. problems where a continuous-time system is controlled by a digital computer, are very often approximated by either discrete-time or continuous-time optimal control problems. A digital controller based on one of these approximations requires a small sampling time and constitutes only an approximate solution. The digital LQG regulator and tracker constitute solutions to real digital control problems which involve sampled-data, piecewise constant controls and integral criteria. Until now only the numerical computation of the digital LQG regulator in the case of time-invariant system and criterion matrices has been considered in the literature. The control of non-linear stochastic systems about state trajectories is very often performed by an LQG regulator based on the linearized dynamics about the trajectory, which constitute a time-varying system. We present a numerical procedure to compute the digital LQG regulator and tracker in the case where the system and criterion matrices are time-varying. Finally we present a numerical example.