Piecewise Constant Feedback Research Articles

Event-triggered aperiodic intermittent control for linear time-varying systems

This paper considers the aperiodic intermittent control (AIC) for linear time-varying systems (LTVSs), where the occurrence instants are determined by an event triggering mechanism based on Lyapunov functions. For LTVSs, most of the existing results are demanded that the feedback controls are exerted all the time. In fact, in many practical applications, the applied controls are unnecessary/impossible to be imposed all the time. Therefore, the event-triggered AIC is introduced in this paper for LTVSs, and the uniformly stability, globally asymptotic stability and finite-time stability are proposed for LTVSs with event-triggered AIC, respectively. In addition, by using the piecewise constant feedback control method, effective intermittent controllers are designed for LTVSs. Finally, we present two numerical examples to illustrate the efficacy of the derived results.

Global stabilizability of an anaerobic biodegradation process via piecewise constant feedback

SummaryWe consider a nonlinear model of an anaerobic digestion process with methane production. We propose a stabilizing control law involving piecewise constant feedback and study the asymptotic stability of the obtained closed‐loop system. An extremum seeking algorithm is applied to maximize the output. Numerical simulation results are also presented to illustrate the theoretical investigations.

Stability analysis and stabilization of linear continuous-time periodic systems by variation-of-constant discretization

By the variation-of-constant formula for solutions to differential equations, discretization remodeling of linear continuous-time periodic (LCP) systems is established, which provides us with variation-of-constant difference (VOCD) models for LCP systems with uniform convergence under mild assumptions. Involving neither Fourier analysis nor Floquet factorizations, the VOCD models are linear discrete-time periodic, and essentially time-invariant; this paves a way to apply analysis and synthesis techniques for linear discrete-time systems to LCP ones. More specifically, stability criteria for LCP systems are developed with their VOCD counterparts. Also, stabilizing LCP systems is addressed by periodically time-varying state feedback in the VOCD models via sequential pole assignment, which can be implemented as gain-scheduling and periodic piecewise constant state feedback in the continuous-time sense. Numerical examples are included to illustrate efficiency of the approach and applications.

Maxwell’s demon in the quantum-Zeno regime and beyond

The long-standing paradigm of Maxwell’s demon is till nowadays a frequently investigated issue, which still provides interesting insights into basic physical questions. Considering a single-electron transistor, where we implement a Maxwell demon by a piecewise-constant feedback protocol, we investigate quantum implications of the Maxwell demon. To this end, we harness a dynamical coarse-graining method, which provides a convenient and accurate description of the system dynamics even for high measurement rates. In doing so, we are able to investigate the Maxwell demon in a quantum-Zeno regime leading to transport blockade. We argue that there is a measurement rate providing an optimal performance. Moreover, we find that besides building up a chemical gradient, there can be also a regime where the feedback loop additionally extracts energy, which results from the energy non-conserving character of the projective measurement.

Finite-time stabilization of linear time-varying systems by piecewise constant feedback

Most existing methods for finite-time stabilizing controller design of linear time-varying systems involve solving differential linear matrix equations. Due to the non-convexity of the problem, it requires a high computational burden. This paper proposes a numerical method to solve finite-time stabilization problems. Successive approximations are performed to estimate the evolution of system states. Accordingly, a gain-switched state feedback controller can be obtained by solving a sequence of linear matrix inequalities (LMIs) based optimization problems. The proposed algorithm is used to design the mass–spring system and the autopilot system of the BTT missile. Comparison with existing methods is given and the simulation results show the effectiveness of the proposed method.

Piece-wise constant predictive feedback control of nonlinear systems

We investigate the problem of receding horizon control for a class of nonlinear processes. A computationally efficient method is developed to identify the optimal control action with respect to predefined performance criteria. Using Carleman linearization and assuming piece-wise constant control action, the state vector is discretized explicitly in time. The optimal control problem is then reformulated as a nonlinear optimization problem and is efficiently solved using analytically computed sensitivity functions and standard gradient-based algorithms.

Vibrational stabilization and the Brockett problem

The present paper deals with the exposition of methods for solving the Brockett problem on the stabilization of linear control systems by a nonstationary feedback. The paper consists of two parts. We consider continuous linear control systems in the first part and discrete systems in the second part. In the first part, we consider two approaches to the solution of the Brockett problem. The first approach permits one to obtain low-frequency stabilization, and the second part deals with high-frequency stabilization. Both approaches permit one to derive necessary and sufficient stabilization conditions for two-dimensional (and three-dimensional, for the first approach) linear systems with scalar inputs and outputs. In the second part, we consider an analog of the Brockett problem for discrete linear control systems. Sufficient conditions for low-frequency stabilization of linear discrete systems are obtained with the use of a piecewise constant periodic feedback with sufficiently large period. We obtain necessary and sufficient conditions for the stabilization of two-dimensional discrete systems. In the second part, we also consider the control problem for the spectrum (the pole assignment problem) of the monodromy matrix for discrete systems with a periodic feedback.

Continuous-Time Limit of a Shift Map Yielding an Exactly Solvable Chaotic Differential Equation

A chaotic nonlinear differential equation possessing an exact analytic solution is derived from a continuous-time limit of an iterated shift map. The resulting equation consists of an unstable harmonic oscillator forced by a nonlinear piecewise-constant feedback. The continuous time solution exhibits a shift dynamic at integer return times, and the waveform is described completely in terms of the symbols of the contained shift map. Significantly, this exactly solvable nonlinear system provides the first example of a chaotic ordinary differential equation possessing an exact symbolic dynamics.

A mathematical framework for the control of piecewise-affine models of gene networks

This article introduces results on the control of gene networks, in the context of piecewise-affine models. We propose an extension of this well-documented class of models, where some input variables can affect the main terms of the equations, with a special focus on the case of affine dependence on inputs. Some generic control problems are proposed, which are qualitative, respecting the coarse-grained nature of piecewise-affine models. Piecewise constant feedback laws that solve these control problems are characterized in terms of affine inequalities, and can even be computed explicitly for a subclass of inputs. The latter is characterized by the condition that each state variable of the system is affected by at most one input variable. These general feedback laws are then applied to a two-dimensional example, consisting in two genes inhibiting each other. This example has been observed in real biological systems, and is known to present a bistable switch for some parameter values. Here, the parameters can be controlled, allowing to express feedback laws leading to various behaviours of this system, including bi-stability as well as situations involving a unique global equilibrium.

Control of sampled-data systems with variable sampling rate

This paper addresses stability and performance of sampled-data systems with variable sampling rate, where the change between sampling rates is decided by a scheduler. A motivational example is presented, where a stable continuous time system is controlled with two sampling rates. It is shown that the resulting system could be unstable when the sampling changes between these two rates, although each individual closed-loop system is stable under the designed controller that minimizes the same continuous loss function. Two solutions are presented in this paper. The first solution is to impose restrictions on switching sequences such that only stable sequences are chosen. The second solution presented is more general, where a piecewise constant state feedback control law is designed which guarantees stability for all possible variations of sampling rate. Furthermore, the performance defined by a continuous time quadratic cost function for the sampled-data system with variable sampling rate can be optimized using the proposed synthesis method.

A computational method for optimal control problem of time-varying state-delayed systems by Haar wavelets

We present a computational method using Haar wavelets to determine the piecewise constant feedback controls for a finite-time linear optimal control problem of a time-varying state-delayed system. The method is simple and computationally advantageous. The approximate optimal trajectory and optimal control are calculated using the Haar wavelet integral, product, and delay operational matrices. An illustrative example is included to demonstrate the validity and applicability of the technique.

AN ASYMPTOTIC SECOND‐ORDER SMOOTH SLIDING MODE CONTROL

ABSTRACTPresented is a method of smooth sliding mode control design to provide for an asymptotic second‐order sliding mode on the selected sliding surface. The control law is a nonlinear dynamic feedback that in absence of unknown disturbances provides for an asymptotic second‐order sliding mode. Application of the second‐order disturbance observer in a combination with the proposed continuous control law practically gives the second‐order sliding accuracy in presence of unknown disturbances and discrete‐time control update. The piecewise constant control feedback is “smooth” in the sense that its derivative numerically taken at sampling rate does not contain high frequency components. A numerical example is presented.

A computational method for simultaneous LQ optimal control design via piecewise constant output feedback

The paper is concerned with simultaneous linear-quadratic (LQ) optimal control design for a set of LTI systems via piecewise constant output feedback. First, the discrete-time simultaneous LQ optimal control design problem is reduced to solving a set of coupled matrix inequalities and an iterative LMI algorithm is presented to compute the feedback gain. Then, simultaneous stabilization and simultaneous LQ optimal control design of a set of LTI continuous-time systems are considered via periodic piecewise constant feedback gain. It is shown that the design of a periodic piecewise constant feedback gain simultaneously minimizing a set of given continuous-time performance indexes can be reduced to that of a constant feedback gain minimizing a set of equivalent discrete-time performance indexes. Explicit formulas for computing the equivalent discrete-time systems and performance indexes are derived. Examples are used to demonstrate the effectiveness of the proposed method.

Feedback stabilization of discrete-time homogeneous semi-linear systems

For discrete-time semi-linear systems satisfying an accessibility condition asymptotic null controllability is equivalent to exponential feedback stabilizability using a piecewise constant feedback. A constructive procedure that yields such a feedback is presented.

Piecewise-Constant Stabilization

With the help of topological necessary conditions for continuous stabilization it is shown that, in general, in order to stabilize continuous- and discrete-time systems one has to use time-dependent or discontinuous feedback controls. On the other hand, the criterion of stabilization in the class of piecewise-constant feedbacks is established. In the context of this paper a piecewise-constant feedback is associated with a piecewise-constant function of the form u = u(x), where $x \in {\rm R}^n_x .$ The piecewise-constant feedback synthesis outlined here has several attractive features. First, it can be effectively applied to design feedback stabilizers subjected to control constraints. Second, the designed feedback laws do not cause sliding mode or chattering behavior in the closed loop system; i.e., on a finite interval of time the control in the closed loop system may have only a finite number of jump discontinuities.

Stabilization of Uncertain Control Systems through Piecewise Constant Feedback

This paper investigates the problem of stabilization of nonlinear systems submitted to a deterministic disturbance known by its bounds. The stabilization property is considered in terms of exponential decay of a Lyapunov-like function. As such systems cannot be stabilized by continuous state feedback laws, we consider an approximate stabilization problem with finite horizon. The main result of the paper is the existence of a—discontinuous—piecewise constant feedback control law, achieving the approximate stabilization property. Moreover, in deriving explicitly this feedback law, we show that the number of discontinuity jumps can be related to both precision of the approximate stabilization and time horizon of the control problem considered.

Asymptotic stabilization of the bilinear time-invariant system via piecewise-constant feedback

Sufficient conditions for the asymptotic stabilization in a preassigned domain of the nonhomogeneous bilinear system (BLS) via piecewise-constant feedback are given, based on those for an auxillary BLS with additional control in the drift term. Application to the swing equation is also discussed.

Certain systems with piecewise constant feedback controls with a time delay

Certain systems with piecewise constant feedback controls with a time delay

Noise and critical behavior of the pupil light reflex at oscillation onset.

We have induced oscillations in the human pupil light reflex using two different kinds of external electronic feedback: smooth negative feedback (SNF) and piecewise constant negative feedback (PCNF). The behavior of the mean amplitude and period at oscillation onset are shown to be in good agreement with a model of this neural system incorporating the external feedback. The critical behavior displayed through amplitude and period fluctuations is different in each case. The observation that amplitude fluctuations are larger (smaller) than period fluctuations for SNF (PCNF) is explained theoretically and by numerical integration of a stochastic delay-differential equation with additive and multiplicative colored noise. We find that both types of noise postpone the Hopf bifurcation in SNF by an amount proportional to the noise intensity and inversely proportional to the correlation time. The implications for analyzing bifurcations in neural systems are discussed.

A new optimal multirate control of linear periodic and time-invariant systems

The optimal multirate design of linear, continuous-time, periodic and time-invariant systems is considered. It is based on solving the continuous linear quadratic regulation (LQR) problem with the control being constrained to a certain piecewise constant feedback. Necessary and sufficient conditions for the asymptotic stability of the resulting closed-loop system are given. An explicit multirate feedback law that requires the solution of an algebraic discrete Riccati equation is presented. Such control is simple and can be easily implemented by digital computers. When applied to linear time-invariant systems, multirate optimal feedback optimal control provides a satisfactory response even if the state is sampled relatively slowly. Compared to the classical single-rate sampled-data feedback in which the state is always sampled at the same rate, the multirate system can provide a better response with a considerable reduction in the optimal cost. In general, the multirate scheme offers more flexibility in choosing the sampling rates.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>