Nonlinear Time-varying Feedback Research Articles

Hyperstability of Linear Feed-Forward Time-Invariant Systems Subject to Internal and External Point Delays and Impulsive Nonlinear Time-Varying Feedback Controls

This paper investigates the asymptotic hyperstability of a single-input–single-output closed-loop system whose controlled plant is time-invariant and possesses a strongly strictly positive real transfer function that is subject to internal and external point delays. There are, in general, two controls involved, namely, the internal one that stabilizes the system with linear state feedback independent of the delay sizes and the external one that belongs to an hyperstable class and satisfies a Popov’s-type time integral inequality. Such a class of hyperstable controllers under consideration combines, in general, a regular impulse-free part with an impulsive part.

Global Stability of Positive Discrete-time Time-varying Nonlinear Feedback Systems

The global stability of positive discrete-time time-varying nonlinear systems with time-varying scalar feedbacks is investigated. Sufficient conditions for the asymptotic stability of discrete-time positive time-varying linear systems are given. The new conditions are applied to discrete-time positive time-varying nonlinear systems with time-varying feedbacks. Sufficient conditions are established for the global stability of the discrete-time positive time-varying nonlinear systems with feedbacks.

On the Design of Hyperstable Feedback Controllers for a Class of Parameterized Nonlinearities. Two Application Examples for Controlling Epidemic Models.

This paper studies the hyperstability and the asymptotic hyperstability of a single-input single-output controlled dynamic system whose feed-forward input-output dynamics is nonlinear and eventually time-varying consisting of a linear nominal part, a linear incremental perturbed part and a nonlinear and eventually time-varying one. The nominal linear part is described by a positive real transfer function while the linear perturbation is defined by a stable transfer function. The nonlinear and time-varying disturbance is, in general, unstructured but it is upper-bounded by the combination of three additive absolute terms depending on the input, output and input-output product, respectively. The non-linear time-varying feedback controller is any member belonging to a general class which satisfies an integral Popov’s-type inequality. This problem statement allows the study of the conditions guaranteeing the robust stability properties under a variety of the controllers designed for the controlled system and controller disturbances. In this way, set of robust hyperstability and asymptotic hyperstability of the closed-loop system are given based on the fact that the input-output energy of the feed-forward controlled system is positive and bounded for all time and any given initial conditions and controls satisfying Popov’s inequality. The importance of those hyperstability and asymptotic hyperstability properties rely on the fact that they are related to global closed-loop stability, or respectively, global closed-loop asymptotic stability of the same uncontrolled feed-forward dynamics subject to a great number of controllers under the only condition that that they satisfy such a Popov’s-type inequality. It is well-known the relevance of vaccination and treatment controls for Public Health Management at the levels of prevention and healing. Therefore, two application examples concerning the linearization of known epidemic models and their appropriate vaccination and/or treatment controls on the susceptible and infectious, respectively, are discussed in detail with the main objective in mind of being able of achieving a fast convergence of the state- trajectory solutions to the disease- free equilibrium points under a wide class of control laws under deviations of the equilibrium amounts of such populations.

Ultimate uniform bounded-stability of inertial coupling electromechanical system via nonlinear time-varying feedback

ABSTRACTThis paper concerns with the problem of stabilisation of inertial coupling electromechanical system. The concept of inertial coupling electromechanical system is an interesting topic from the control theory point of view and its universe of applications. This is because, for stabilisation proposes, the complete model of electromechanical system significantly reduces higher frequency functions in their dynamics caused by high gain controllers. Here, the system stabilisation is achieved by applying nonlinear time-varying controllers. The reason to use a time-varying controller is to give some learning or adaptation in the control process. Moreover, to get a good result for the closed-loop system, the structure of a storage function is based on the classical quadratic function in addition to quadratic function of system total energy. So, the main goal of this paper is obtain a controller that guarantees that the system state error arrives near the required equilibrium in exponential form. Hence, the Ultimate Uniform Bounded-Stability of the system trajectories of desired equilibrium is concluded. Finally, experimental results of the control algorithm developed in this paper are presented briefly.

A new revised desired compensation adaptive control for enhanced tracking: application to RA-PKMs

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Observer Design for Nonlinear Systems: A Dissipative Approach

A new method to design observers for a class of nonlinear systems is proposed. The main idea consists in decomposing the observer error system in a linear time invariant part and a nonlinear time-varying feedback. This last term contains the influence of the plant on the error equation. If the nonlinearity is dissipative with respect to a quadratic supply rate, then the linear part is designed to be dissipative with respect to a related supply rate, so that the closed loop is exponentially stable. This design allows for big flexibility in the possible nonlinearities, and includes and generalizes several standard observer design methods. The observer design is simple, since in many cases it reduces to the solution of Linear Matrix Inequalities (LMI).

An analytic approach to the design of nonlinear time-varying control systems

This paper addresses a controller design approach for a specific class of nonlinear time-varying systems based on an appropriate choice of a nonlinear time-varying dynamic feedback. A general form of the nonlinear time-varying dynamic controller is proposed. Lyapunov function theory was extensively used to prove global stability of closed systems. Conditions of local and global asymptotic stability of the nonlinear time-varying system are given. An analytic algorithm of controller design is presented. It is shown that the proposed design technique can successfully be applied to the specific class of the systems, the state variables of which are not fully observed. Simulation results show that the proposed approach to the nonlinear time-varying controller design is effective, and the closed-loop system is robust to small parametrical variations.

Simultaneous Stabilization of Linear and Nonlinear Systems by Means of Nonlinear State Feedback

The simultaneous stabilization (resp., asymptotic stabilization) of a countable family of control systems consists of finding a control which stabilizes (resp., asymptotically stabilizes) all the systems in the family. In this paper, we introduce a new method which enables us to show that, given any countable family of stabilizable nonlinear systems, there exists a continuous state feedback law which simultaneously stabilizes (not asymptotically) the family. Then, by enriching this method, we prove that any finite family of stabilizable linear time invariant (LTI) systems can be simultaneously exponentially stabilized by means of nonlinear time-varying state feedback. We also derive sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of LTI systems. Finally, sufficient conditions for the simultaneous asymptotic stabilizability of finite families of nonlinear systems are provided and used for the simultaneous asymptotic stabilization of certain pairs of nonlinear homogeneous systems.

Relative stability of a linear time-varying process with first-order nonlinear time-varying feedback

A method of establishing sufficient conditions for the stability of a system consisting of a time-varying linear process and a first-order nonlinear time-varying feedback is presented. Rate of decay of unforced response and BIBO stability can also be determined. The governing equation of the system is represented in the form of an integro-differential equation, so that a modified form of the Lyapunov-Razumikhin-type condition can be applied. The first part of the paper deals with stability of vector functional equations. The results obtained are applied to feedback systems. An example is worked out to show the applicability of the results obtained. >

A globally convergent pole placement indirect adaptive controller

A pole displacement indirect adaptive control algorithm is discussed for discrete-time linear deterministic plants with arbitrary zeros. The global convergence of the resulting closed-loop control system is achieved subject to the assumptions that the plant order and a nonzero lower bound on its degree of controllability are known. The problem of controllability of the plant model estimate is handled by using both a parameter correction and time-varying nonlinear feedback. A key property of the algorithm is that the plant estimate reaches a reasonable degree of controllability in a finite time, after which the parameter correction and the nonlinear feedback are no longer used.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Converse hölder inequality and the Lp-instability of nonlinear time-varying feedback systems

WE DEAL with the Lp-instability of a nonlinear time-varying feedback system governed by the pair of equations: v(t) = x(t) - k(t)cp (y(t)),y(t) = (qJv)(t) (1) := ~ g i v ( t - ri) + g( O v ( t - r) dr i=1 for all t I> 0, where x, v, and y are respectively the input to the system, the error signal and output of the system; N is a time-invariant linear operator; qv is a first and third quadrant continuous, memoryless (monotone) nonlinearity; and k is a time-varying gain. For assumptions on the components of (1), see Section 2 below. We derive Lp-instability conditions in terms of the frequency response of ~3 and a general causal + anticausal multiplier function. The derivation is based on the converse H61der inequality and the energy balance argument as used in [1].The problem of instability of feedback systems with a single time-varying nonlinearity was initially considered by Brockett and Lee [2] who, in a Lyapunov-Chetaev setting, derived an instability version of the circle criterion [3, 4] under certain assumptions on the related linear time-invariant systemwith a constant gain in the feedback loop. It is found that this instability theorem is conservative [5, Section 5]. See [5] for references to other instability results. As far as the analysis of instability of feedback systems by functional methods is concerned,different types of results are available. Willems [6] extends the domain of operators and imbedsthe system causal operator in a noncausal operator in an attempt to prove the noncausality of the inverse of the original operator by contradiction. This technique has been explicitly used [7] but there do exist certain unresolvable difficulties [8; 9, Chapter 7]. Similar difficulties are encountered in the results of Takeda and Bergen [10, 11] and Steding and Bergen [12], who consider a subclass of inputs over which the linear time-invariant part of the system is well- behaved (and satisfies some additional conditions), and prove instability by contradiction. See [9, Chapter 7] for a complete analysis of these contributions in which, more importantly, the assumptions made on the linear time-invariant part of the system are too severe.

An algorithm for non-linear space—Time nuclear reactor control

A novel multivariable control algorithm for non-linear space-time nuclear reactor dynamics is proposed in this paper. The multivariable control algorithm is based on a mathematical model of the nuclear reactor which includes: a single energy group of neutrons, delayed neutron precursors, iodine, xenon and thermal-hydraulic feedback. The multivariable control algorithm is composed of non-linear time-varying feedforward and feedback control signals, a reference model of the nuclear reactor and a dynamic observer. The non-linear proportional plus integral feedback controller forces the nuclear reactor to follow the response of the reference model. The dynamic observer estimates the unmeasurable state variables. The feedforward and feedback control signals are determined in a novel approach by specifying the form of the closed-loop response of the neutron density variables. By virtue of the multivariable control algorithm the closed-loop differential equations are linear and time-varying. A linear stability analysis for base-load and load-cycle operation indicates that the closed-loop system is stable provided that the thermal-hydraulic subsystem is inherently stable. The simulated dynamic response indicates that the multivariable control algorithm provides excellent response characteristics.

L2stability of a special type of non-linear time-varying (NLTV) feedback system

The well known positive-operator theory is made use of in deriving interesting stability criterion for a special type of non-linear time-varying (NLTV) feedback system.

On the dynamic response and stability of sampled-data non-linear time-varying feedback systems

Time-domain analysts is used to examine the stability of sampled-data feedback systems comprising a time-varying plant and a time-varying non-linearity. Three aspects of the systems dynamic response are given a unified treatment : bounded-ness. unboundedness, and asymptotic decay. In addition to qualitative criteria, the results provide quantitative bounds on the system response, and indicate its explicit dependence on the sampling rate.

Instability Criteria for Time-Varying Nonlinear Functional Differential Systems

Instability criteria are derived for feedback systems containing a time-invariant linear element in the forward branch, and a time-varying nonlinear feedback amplifier. The transfer function of the linear element is not restricted to rational functions. The results are obtained using Lyapunov techniques in function space. They are considerably stronger than those in early papers on the subject, which are confirmed as special cases of the present ones.

Lp-Stability (1 ≤p≤ ∞) of multivariable non-linear time-varying feedback systems that are open-loop unstable†

This paper considers a class of multivariable, non-linear time-varying feedback systems with an unstable convolution subsystem as feedforward and a time-varying non-linear gain as feedback. The impulse response of the convolution subsystem is the sum of (i) a finite number of increasing exponentials multiplied by non-negative powers of the time t, (ii) a term that is absolutely integrable and (iii) an infinite series of delayed impulses. The main result of the paper is theorem 1. It essentially states that (i) if the unstable convolution subsystem can be stabilized by a constant feedback gain F and (ii) if the incremental gain of the difference between the nonlinear gain function and F is sufficiently small, then the non-linear system is Lp instable for any p∊[l, ∞]; furthermore, the solutions of the non-linear system depend continuously on the inputs in any Lp -norm. The fixed point theorem is crucial in deriving the above theorem.

Stability of non-linear feedback systems with time-varying linear sub-system (plant)

A new approach is used to examine the stability of feedback systems comprising a time-varying linear subsystem and a time-varying non-linearity. The paper examines three stability aspects of the systems dynamic response: boundedness, unboundedness, and asymptotic decay. In addition to qualitative criteria, the results provide explicit quantitative bounds on the system response. The distinct time-domain method of analysis presented stipulates rather weak a priori restrictions on the nature of the linear and non-linear parts of the system, thus admitting a relatively broad class of systems. In the general time-varying non-linear feedback system considered, the non-linearity may exhibit hysteresis, and the linear subsystem may be non-causal, and may comprise any number of feed-forward differentiators of any order. The criteria obtained are given a simple graphical interpretation. A transformation which trades time-varying gains between plant and non-linearity is introduced. A number of examples demonstrate t...

A Criterion for the Bounded-Input Bounded-Output Stability of Time- Varying Nonlinear Systems

A new approach is used to derive sufficient conditions for the boundedness of a broad class of time-varying nonlinear feedback systems. The stability problem is defined and treated independently from the problem of well-posedness of the feedback system. The a priori restrictions stipulated on the nature of the system are relatively unrestrictive. In the general time-dependent system considered, time-variation may be exhibited by linear as well as the nonlinear portions of the system. The type of linear subsystem admitted includes the class of systems with feed-forward differentiators. The stability criterion is derived via a time-domain method of analysis that makes no reference to Lyapunov functions, transform relationships, or normed spaces. A number of examples demonstrate that the present criterion can predict stability information which is not readily obtainable for other comparable criteria.

Time-domain criteria for the&amp;lt;tex&amp;gt;L_{2}&amp;lt;/tex&amp;gt;-stability of nonstationary feedback systems

Criteria for the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -stability of linear and nonlinear time-varying feedback systems are given. These are conditions in the time domain involving the solution of certain associated matrix Riccati equations and permitting the use of a very general class of L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -operators as multipliers.

On the stability of nonlinear time-varying systems

Concerning the absolute stability of a nonlinear time-varying feedback system, an important correction to the results of the author [1] is presented.