Piecewise Constant Control Law Research Articles

Sampled-data in space nonlinear control of scalar semilinear parabolic and hyperbolic systems

The paper describes a novel method of sampled-data in space (spatial variable) nonlinear control of scalar semilinear parabolic and hyperbolic systems with unknown parameters, distributed disturbances and finite number of measurements along the spatial variable. Differently from recent results based on piecewise constant control laws, the proposed one is used piecewise nonlinear functions choosing by designer for providing some properties in the closed-loop system. In particular, we propose several types of functions providing reduced control. The gain design in the control law is found as a solution of linear matrix inequalities with minimum ultimate bound guarantee. The simulations confirm theoretical results and show the efficiency of the proposed control scheme compared with some existing ones.

Control strategies for sustained oscillations in a disrupted biological clock

As many disorders have been correlated with dysfunctional biophysical rhythms, new therapies based on the control of clock functions are emerging for the slowdown of disease progression. In this context, a general disrupted biological clock is modeled by the canonical form of a genetic negative feedback loop. When the unique fixed point of this N-dimensional non-linear differential system is stable, the model reproduces accurately the damped oscillations observed in a damaged oscillator. First, a synthetic modification of the network is proved to generate sustained oscillations and allow to recover a functional clock. The desired periodic trajectories are obtained by destabilizing the fixed point of the model and monotone properties are applied for global results. In a limit case, this modification of the loop is shown to be equivalent to an external piecewise constant control law, supporting the conjecture that simple qualitative control strategies may be able to guarantee sustained oscillations. From the perspective of a biological implementation, this result is promising as these types of control are well adapted to experimental constraints. To support this theoretical work, the methods are applied to the disrupted circadian clock observed in human cancer cells.

Poincaré maps design for the stabilization of limit cycles in non-autonomous nonlinear systems via time-piecewise-constant feedback controllers with application to the chaotic Duffing oscillator

Abstract In this paper, a design of Poincare maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period–m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincare map from the structure of several local maps. An expression of the generalized controlled Poincare map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT–piecewise–constant control law, a T–piecewise–constant control law and a T n –piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period–m limit cycle and hence the chaotic motion is controlled.

A new qualitative control strategy for the genetic Toggle Switch

Genetic positive feedback loops are essential for cell decision making and cell differentiation. They are traditionally described by a two-dimensional smooth non-linear differential system composed of Hill functions. The bistability of this classic model properly captures the decision properties of these biological motifs. This paper designs a new control strategy based on the measurement and control of a unique gene within the loop, in order to stabilize the system around its unstable fixed point. The quantized nature of genetic measurements and the new synthetic control approaches available in biology encourage the use of a piecewise constant control law. A specific partitioning of the state space and the study of successive repelling regions allow to prove global convergence and global stability for the resulting system. The same strategy is shown to be efficient as well in the more realistic context in which measurements are considered possibly uncertain. This new control strategy is compared with a real biological experiment that was implemented for the Toggle Switch with the same stabilization objective, but for which both genes were measured and controlled.

Practical encoders for controlling nonlinear systems under communication constraints

We introduce a new class of dynamic encoders for continuous-time nonlinear control systems which update their parameters only at discrete times. We prove that the information reconstructed from the encoded feedback can be used to deliver a piece-wise constant control law which yields semi-global practical stability.

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.