Cone-bounded Nonlinearity Research Articles

Sliding mode control of Lur'e nonlinear switched systems under dynamic schedule protocol

ABSTRACT In this work, the sliding mode control (SMC) problem is investigated for a class of Lur'e nonlinear switched systems subject to finite channels constraints, which may lead to data collision. To solve this phenomenon, the Weighted Try-Once-Discard (WTOD) protocol is proposed on the sensor-to-controller (S/C) channels to regulate which one sensor node is permitted to send data via the communication network at each instant. A key issue of this work is how to integrate the WTOD protocol into the design of observer and the sliding mode controller for the switched systems. To this end, a token-dependent observer is constructed to estimate the unmeasured states, and then an observer-based SMC law is properly designed. Moreover, by constructing a Lur'e-type Lyapunov function interconnected with a cone-bounded nonlinearity, some sufficient conditions are established such that the reachability with a sliding domain around the specified sliding surface is achieved, meanwhile, the exponential stability of the resultant closed-loop switched systems is guaranteed. Finally, some numerical simulation results are provided.

Cone-bounded feedback laws for linear second order systems

The strong stabilizability of linear second order equation is investigated in this paper. It is supposed that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. Then this control is subject to a cone-bounded nonlinearity. Well-posedness and stability results of the closed-loop system under such (nonlinear) control are stated. The first result is proven by using nonlinear semigroups techniques and the Schauder fixed-point theorem and the second one is based on the result of Marx et al. [12]. Our results are then applied to the particular damped and undamped wave equation. Simulation results are presented to validate the theoretical results. Note that some of the results of this paper apply for a large class systems.

A hidden Markov model based control for periodic systems subject to singular perturbations

This study analyzes the problem of hidden Markov model based control for periodic systems subject to singular perturbations and Lur’e cone-bounded nonlinearity. Different from the existing time-invariant fading channels, the fading channels are alleviated to be time-varying. Furthermore, to better depict the time-varying property of the fading channels, a novel periodic Markov process framework subject to mean and variance is developed. The highlight of this study lies that the hidden Markov model detector is forwarded to observe the fading channel mode, whose detection probabilities are generalized to be partially recognized. New techniques are developed in dealing with the stochastic Lyapunov functional, and sufficient conditions are gained to ensure the resulting dynamic is stochastically stable. In the sequel, the asynchronous controller parameters are further achieved to reflect the discrepancy between fading channel mode and its detected one. Finally, an application-oriented example is rendered to confirm the effectiveness and applicability of the developed control strategy.

Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces

This work studies the influence of some constraints on a stabilizing feedback law. It is considered an abstract nonlinear control system for which we assume that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable. This controller is then modified via a cone-bounded nonlinearity. A well-posedness and a stability theorems are stated. The first theorem is proved thanks to the Schauder fixed-point theorem, the second one with an infinite-dimensional version of LaSalle's Invariance Principle. These results are illustrated on a linear Korteweg-de Vries equation by some simulations and on a nonlinear heat equation.

Wave Equation With Cone-Bounded Control Laws

This paper deals with a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection. Two kinds of control are considered: a distributed action and a boundary control. It is supposed that the control signal is subject to a cone-bounded nonlinearity. This kind of feedback laws includes (but is not restricted to) saturating inputs. By closing the loop with such a nonlinear control, it is thus obtained a nonlinear partial differential equation, which is the generalization of the classical 1D wave equation. The well-posedness is proven by using nonlinear semigroups techniques. Considering a sector condition to tackle the control nonlinearity and assuming that a tuning parameter has a suitable sign, the asymptotic stability of the closed-loop system is proven by Lyapunov techniques. Some numerical simulations illustrate the asymptotic stability of the closed-loop nonlinear partial differential equations.

Distributed ${\mathcal{H}}_{\infty}$ Filtering for a Class of Discrete-Time Markov Jump Lur’e Systems With Redundant Channels

This paper investigates the problem of distributed H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> filtering for a class of discrete-time Markov jump Lur'e systems over sensor networks with stochastic switching topologies. As a first effort, the redundant channels are considered to use in the data transmission to strive against the fragility of networks commonly configured by a single channel in the communication networks subject to quantization and randomly occurring packet dropouts. By constructing a Lur'e-type Lyapunov function interconnected with a cone-bounded nonlinearity, the filtering performance analysis for the underlying systems is carried out first. Then, a new class of distributed filters, which can be used to reflect the mismatching characteristic of modes jumping between the target plants and the proposed filters, is designed such that the filtering error system is stochastically stable and achieves a prescribed average H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance index. A numerical example is provided to show the effectiveness of the proposed filtering scheme.

Stochastic stabilization and induced [formula omitted]-gain for discrete-time Markov jump Lur’e systems with control saturation

This paper addresses the problem of control synthesis for locally stabilizing and minimizing the finite ℓ2 gain for discrete-time Markov jump Lur’e systems with control saturation and exogenous ℓ2-type disturbance. We consider that the jump parameter defining the active mode for both the linear and the cone-bounded nonlinearity is governed by a finite state homogeneous Markov chain. The local stochastic stability is established by exploiting the invariant probability measure of the Markov chain to define the level set of the expected value of the stochastic Lur’e type Lyapunov function. Optimization problems for maximizing our estimate of the domain of stochastic stability and minimizing the induced ℓ2 gain with respect to exogenous disturbances are presented subject to LMI constraints. The paper is concluded with an academic example.

Stabilization of Discrete-time Nonlinear Systems subject to Input Saturations: a New Lyapunov Function Class

AbstractThis paper addresses the problem of stabilization of discrete-time systems including a cone-bounded nonlinearity and a saturating actuator. In the sense of Lyapunov stability, we introduce a new candidate Lyapunov function which takes nonlinearity behavior into account. The local stability criterion is formulated as a set of Bilinear Matrix Inequalities (BMI) conditions. We present an optimization problem in order to guarantee the closed-loop stability aiming the largest basin of attraction, which may be nonconvex, and/or, nonconnected. Furthermore, a simple iterative algorithm is proposed in order to solve our BMI problem. Some numerical examples are presented to highlight the relevance of the new Lyapunov function in regard to the classical quadratic function.

Control design for a class of nonlinear continuous-time systems

This paper addresses the control design problem for a certain class of continuous-time nonlinear systems subject to actuator saturations. The system under consideration consists of a system with two nested nonlinearities of different type: saturation nonlinearity and cone-bounded nonlinearity. The control law investigated for stabilization purposes depends on both the state and the cone-bounded nonlinearity. Constructive conditions based on LMIs are then provided to ensure the regional or global stability of the system. Different points, like other approaches issued from the literature, are quickly discussed. An illustrative example allows to show the interest of the approach proposed.

Observer-based strict positive real (SPR) feedback control system design

This paper presents theory for stability analysis and design for a class of observer-based feedback control systems. Relaxation of the controllability and observability conditions imposed in the Yakubovich–Kalman–Popov (YKP) lemma can be made for a class of nonlinear systems described by a linear time-invariant system (LTI) with a feedback-connected cone-bounded nonlinear element. It is shown how a circle-criterion approach can be used to design an observer-based state feedback control which yields a closed-loop system with specified robustness characteristics. The approach is relevant for design with preservation of stability when a cone-bounded nonlinearity is introduced in the feedback loop. Important applications are to be found in nonlinear control with high robustness requirements.

Observer-Based Strict Positive Real (SPR) Feedback Control System Design

This paper presents theory for stability analysis and design for a class of observer-based feedback control systems. Relaxation of the controllability and observability conditions imposed in the Yakubovich-Kalman-Popov (YKP) lemma can be made for a class of nonlinear systems described by a linear time-invariant system (LTI) with a feedback-connected cone-bounded nonlinear element. It is shown how a circle-criterion approach can be used to design an observer-based state feedback control which yields a closed-loop system with specified robustness characteristics. The approach is relevant for design with preservation of stability when a cone-bounded nonlinearity is introduced in the feedback loop. Important applications are to be found in robotics (force control) and nonlinear control in hybrid control.

On designing controllers for a class of uncertain sampled-data nonlinear systems

The subject of the paper is the robust control of a class of sampled-data nonlinear systems with real time-varying parameter uncertainty and unknown cone-bounded nonlinearity. We consider the problems of robust stabilisation and robust performance in an H∞ sense, and attention is focused on the design of linear dynamic sampled-data output feedback controllers. A control design methodology is proposed for solving both problems by recasting them into ‘scaled’ H∞ control problems for a related system which involve neither parameter uncertainty nor unknown nonlinearity.

Synthesis of discrete-time compensators with specified gain and phase margins

This paper extends the results of Lee and Lee (1986) to an output feedback compensator that yields a closed-loop system with asymptotical stability and specified robustness characteristics. This robustness can be interpreted as gain and phase margins when a cone-bounded non-linearity is introduced in the loop of an equivalent system represented in state space form.

Synthesis of control laws with specified gain and phase margin and poles region

The paper presents a method of synthesising a state-feedback control law which yields a closed-loop system with specified robustness characteristics and meanwhile ensures all closed-looppoles inside a circular region, centred at (e, 0) with radius r, where |e| + r ≤ 1. This robustness can be interpreted as gain and phase margins when a cone-bounded nonlinearity is introduced in the loop. The condition imposed on the robustness sector, so that the feedback control will be optimal with respect to a quadratic performance index, is also obtained.

Gain and phase margins for discrete-time systems

This paper presents a method of synthesizing a state-feedback control law which yields a closed-loop system with specified robustness characteristics. This robustness can be interpreted as gain and phase margins when a cone-bounded non-linearity is introduced in the loop. The condition imposed on the robustness sector that the feedback control will meanwhile minimize a quadratic performance index is also obtained.

Synthesis of state feedback control laws with a specified gain and phase margin

It is shown how one may use a "circle criterion" philosophy to design a state feedback control law which yields a closed-loop system with specified robustness characteristics. This robustness is most immediately given in terms of preservation of stability when a cone-bounded nonlinearity is introduced in the loop, but may also be interpreted in terms of gain and phase margin and gain reduction tolerance.