Asymptotic Hyperstability 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.

Asymptotic Hyperstability and Input–Output Energy Positivity of a Single-Input Single-Output System Which Incorporates a Memoryless Non-Linear Device in the Feed-Forward Loop

This paper visualizes the role of hyperstable controllers in the closed-loop asymptotic stability of a single-input single-output system subject to any nonlinear and eventually time-varying controller within the hyperstable class. The feed-forward controlled loop (or controlled plant) contains a strongly strictly positive real transfer function in parallel with a non-linear and memory-free device. The properties of positivity and boundedness of the input–output energy are examined based on the “ad hoc” use of the Rayleigh energy theorem on the truncated relevant signals for finite time intervals. The cases of minimal and non-minimal state-space realizations of the linear part are characterized from a global asymptotic stability (asymptotic hyperstability) point of view. Some related extended results are obtained for the case when the linear part is both positive real and externally positive and for the case of incorporation of other linear components which are stable but not necessarily positive real.

DC Drive Adaptive Speed Controller Based on Hyperstability Theory

The scope of this research is to develop a hyperstable adaptive control system of a direct current (DC) drive speed for effective load torque attenuation. The proposed speed controller is based on the model reference adaptive control framework and integrated into the conventional DC drive cascade control system. Its main features are as follows: (1) the boundedness of the control action signal, as well as the armature current control loop non-stationarities, are taken into consideration with the help of the reference model hedging technique; (2) its inputs include only measurable signals, thus there is no need to use any kind of state estimators; (3) it attenuates the disturbances, which are matched with its control action signal, particularly, the inertia moment non-stationarity and load torque. The asymptotic hyperstability of the obtained DC drive control system is proven with the help of Lyapunov’s theorems and Popov’s criterion. The numerical experiments corroborate the obtained results. They include the demonstration of disadvantages of the conventional cascade control system under conditions of the drive parameters’ non-stationarity and advantages of the proposed solution for different disturbance types and amplitudes.

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.

On the Asymptotic Hyperstability of Linear Time-Invariant Continuous-Time Systems under a Class of Controllers Satisfying Discrete-Time Popov’s Inequality

This paper is concerned with the property of asymptotic hyperstability of a continuous-time linear system under a class of continuous-time nonlinear and perhaps time-varying feedback controllers belonging to a certain class with two main characteristics; namely, (a) it satisfies discrete-type Popov’s inequality at sampling instants and (b) the control law within the intersample period is generated based on its value at sampling instants being modulated by two design weighting auxiliary functions. The closed-loop continuous-time system is proved to be asymptotically hyperstable, under some explicit conditions on such weighting functions, provided that the discrete feed-forward transfer function is strictly positive real.

About a Class of Positive Hybrid Dynamic Linear Systems and an Associate Extended Kalman-Yakubovich-Popov Lemma

This paper formulates an “ad hoc” robust version under parametrical disturbances of the discrete version of the Kalman-Yakubovich-Popov Lemma for a class of positive hybrid dynamic linear systems which consist of a continuous-time system coupled with a discrete-time or a digital one. An extended discrete system, whose state vector contains both the digital one and the discretization of the continuous-time one at sampling instants, is a key analysis element in the formulation. The hyperstability and asymptotic hyperstability properties of the studied class of positive hybrid systems under feedback from any member of a nonlinear (and, eventually, time-varying) class of controllers, which satisfies a Popov’s-type inequality, are also investigated as linked to the positive realness of the associated transfer matrices.

On the asymptotic hyperstability of switched systems under integral-type feedback regulation Popovian constraints

This paper deals with the asymptotic hyperstability of switched time-varying dynamic systems involving switching actions among linear time-invariant parametrizations in the feed-forward loop for any feedback regulator controller potentially being also subject to switching through time while being within a class which satisfies a Popov-type integral inequality. Asymptotic hyperstability is proved to be achievable under very generic switching laws if at least one of the feed-forward parametrization possesses a strictly positive real transfer function, a minimum residence time interval is respected for each activation time interval of such a parametrization and a maximum allowable residence time interval is simultaneously maintained for all active parametrizations which are not positive real, if any. In the case where all the feed-forward parametrizations possess a common Lyapunov function, asymptotic hyperstability of the switched closed-loop system is achieved under arbitrary switching.

Asymptotic Hyperstability of a Class of Linear Systems under Impulsive Controls Subject to an Integral Popovian Constraint

This paper is focused on the study of the important property of the asymptotic hyperstability of a class of continuous-time dynamic systems. The presence of a parallel connection of a strictly stable subsystem to an asymptotically hyperstable one in the feed-forward loop is allowed while it has also admitted the generation of a finite or infinite number of impulsive control actions which can be combined with a general form of nonimpulsive controls. The asymptotic hyperstability property is guaranteed under a set of sufficiency-type conditions for the impulsive controls.

On Bounded Strictly Positive Operators of Closed Range and Some Applications to Asymptotic Hyperstability of Dynamic Systems

The problem discussed is the stability of two input-output feedforward and feedback relations, under an integral-type constraint defining an admissible class of feedback controllers. Sufficiency-type conditions are given for the positive, bounded and of closed range feed-forward operator to be strictly positive and then boundedly invertible, with its existing inverse being also a strictly positive operator. The general formalism is first established and the linked to properties of some typical contractive and pseudocontractive mappings while some real-world applications and links of the above formalism to asymptotic hyperstability of dynamic systems are discussed later on.

Asymptotic Hyperstability of Dynamic Systems with Point Delays

It is proved that a linear time-invariant system with internal point delays is asymptotically hyperstable independent of the delays if an associate delay-free system is asymptotically hyperstable and the delayed dynamics are sufficiently small.

Asymptotic hyperstability under unstructured and structured modeling deviations from the linear behavior

This paper deals with the asymptotic hyperstability of nominally asymptotic hyperstable linear systems in the presence of unstructured modeling errors. It is assumed that the nominal plant is linear, time-invariant and of strictly positive real transfer function with the feedback loop satisfying a Popov's type input–output time integral inequality for all time and the combination resulting in an asymptotically hyperstable closed-loop system. The current plant is assumed to be subjected, in general, unstructured deviations from its nominal behavior but then asymptotic hyperstability results are also obtained for particular structured modeling errors like time-varying linear dynamics, bilinear or delay-dependent dynamics. The key technique used for obtaining the results is to guarantee that a measure of the input/output energy of the forward current dynamics is positive and uniformly bounded for all time for certain amounts of modeling errors provided that the nominal one exhibits the same property.

Feedback error learning and nonlinear adaptive control

In this paper, we present our theoretical investigations of the technique of feedback error learning (FEL) from the viewpoint of adaptive control. We first discuss the relationship between FEL and nonlinear adaptive control with adaptive feedback linearization, and show that FEL can be interpreted as a form of nonlinear adaptive control. Second, we present a Lyapunov analysis suggesting that the condition of strictly positive realness (SPR) associated with the tracking error dynamics is a sufficient condition for asymptotic stability of the closed-loop dynamics. Specifically, for a class of second order SISO systems, we show that this condition reduces to K D 2> K P, where K P and K D are positive position and velocity feedback gains, respectively. Moreover, we provide a ‘passivity’-based stability analysis which suggests that SPR of the tracking error dynamics is a necessary and sufficient condition for asymptotic hyperstability. Thus, the condition K D 2> K P mentioned above is not only a sufficient but also necessary condition to guarantee asymptotic hyperstability of FEL, i.e. the tracking error is bounded and asymptotically converges to zero. As a further point, we explore the adaptive control and FEL framework for feedforward control formulations, and derive an additional sufficient condition for asymptotic stability in the sense of Lyapunov. Finally, we present numerical simulations to illustrate the stability properties of FEL obtained from our mathematical analysis.

On the asymptotic hyperstability of dynamic systems with point delays

It is proved that a linear time-invariant system with internal point delays is asymptotically hyperstable independent of the delays if an associate delay-free system is asymptotically hyperstable and the delayed dynamics is sufficiently small.

Modified Optimal Control: Robust Stabilization of Nonlinear Uncertain Systems

This contribution focuses on the problem of controller design for nonlinear uncertain systems. Our solution is based on a new nonlinear controller design approach, called modified optimal control. In addition to achieving global asymptotic stability and appropriate closed loop system dynamics this method represents a systematic and powerful approach for stabilizing uncertain nonlinear systems. Various types of uncertainties, e.g. matched and unmatched ones, are considered and a new condition for asymptotic hyperstability of uncertain nonlinear systems is presented. In order to demonstrate the robustness property, a nonlinear torque control of a spark ignition engine is designed.

On the hyperstability of a class of hybrid systems

Sufficient conditions of the hyperstability of a class of single-input single-output hybrid systems, composed of interconnected continuous and discrete subsystems, are given. Hyperstability is guaranteed by that of a discrete extended system which is built by incorporating samples of the continuous substrate. Asymptotic hyperstability is guaranteed by that of one discrete system, together with that of a continuous one. Both systems are related to an augmented state representation of the linear part of the overall system for all the nonlinear output-feedback time-varying actuators that satisfy generalized Popov-type inequalities.

Adaptive model-following controller for robotic manipulators

A nonlinear state-space model representing the robot dynamics and containing a disturbance term due to gravitational loading is presented. An adaptive model-following control problem satisfying the matching conditions is formulated using a suitable linear time-invariant reference model. The control input is designed to have two components: a non-adaptive linear component to do the task of model-following and a nonlinear unit-vector component based on hyperstability theory to do the adaptive task. An additional integral feedback term is further added and then the overall asymptotic hyperstability is established. Simulation experiments on the first three joints of a PUMA 560 robot manipulator have indicated the potential of our design approach.

A design approach to adaptive model-following control of robotic manipulators

In this paper, the robot dynamics are represented by a nonlinear state-space model containing a disturbance term due to gravitational loading. Using a suitable linear time-invariant reference model, an adaptive model-following control problem is formulated which satisfies the matching conditions. The control input is designed to have two components: a nonadaptive linear component to do the task of model-following and a nonlinear unit-vector component based on hyperstability theory to do the adaptive task. An additional integral feedback term is further superimposed and then the overall asymptotic hyperstability is established. The simulation results on the first three joints of PUMA 560 robot manipulator indicate the potential of our design approach.

Robust strict positive real stabilization and asymptotic hyperstability robustness

Abstract This paper is devoted to the study of robust strict positive real stabilization of SISO uncertain systems, i.e. finding a single controller that not only robustly stabilizes a plant with uncertainties under unity-feedback but also makes the closed-loop system robustly strictly positive real. It is proved that there must exist a stable controller achieving robust strict positive real stabilization of uncertain systems with both parameter variations and unstructured uncertainties under some reasonable assumptions. The propemess of the controller is guaranteed when the largest relative degree of an uncertain system is retricted. By applying these results to nonlinear systems, some significant conclusions on asymptotic hyperstability robustness are given.

Robustness of Asymptotic Hyperstability

Abstract The robustness of asymptotic hyperstabiIity for the perturbation caused by parasitic dynamics is analyzed by using the singular perturbation method. Since the strictly positive realness of a transfer function is not robust for such perturbations, it is required that some kind of limitations must be set to the hyperstabiIity problem. In this paper we introduce the limited hyperstabiIity problem which has Iimited feedback blocks in the feedback loop. By considering the Iimited hyperstable problem, we can give a limitation on the hyperstabiIity problem and analyze the robustness of the asymptotic hyperstabiIity.

A Novel Series-Parallel Identifier for Microcomputer Process Control

A series-parallel or SP adaptive identifier [1-4] with enhanced tracking capabilities and transient response characteristics is highlighted. The a posteriori tracking error is incorporated into the identification algorithm. Enchanced performance is achieved by passing the process inputs and outputs through high-pass digital filters and then applying these filtered signals to the adaptation algorithm. The asymptotic hyperstability of this SP identifier with variable relaxation gains, which are positive functions of the tracking error between the process and identifier outputs, has been proved theoretically and demonstrated experimentally. Computational considerations for implementing SP identifiers on microcomputers are noted and on-line computational requirements are indicated.