Bounded-input Bounded-output Research Articles

Input to Output Finite-Time Stabilization of Discrete-Time Linear Systems

Bounded–Input Bounded–Output (BIBO) stability is usually studied when only the input-output behavior of a dynamical system is of interest. The present paper investigates the analogous concept in the framework of Finite Time Stability (FTS), namely the Input–Output FTS (IO-FTS). FTS has been already investigated in several papers in terms of state boundedness, whereas in this work we deal with the characterization of the input-output behavior. A system is said to be input-output finite time stable if, assigned a class of input signals and some boundaries in the output signal space, the output never exceeds such boundaries over a prespecified (finite) interval of time. This paper provides some sufficient conditions for the analysis of IO–FTS and for the design of a static state feedback controller guaranteeing IO–FTS of the closed loop system. The effectiveness of the proposed results is eventually illustrated by means of two numerical examples.

Sufficient Conditions for Robust Input-Output Finite-Time Stability of Linear Systems in Presence of Uncertainties

AbstractThe concept of Bounded-Input Bounded-Output (BIBO) stability arises when one wants to focus on the study of the the input-output behavior of a dynamical system, as opposed to the classical Lyapunov stability. The present paper investigates the analogous concept in the framework of Finite Time Stability (FTS), namely the Input-Output FTS. A system is said to be IO finite time stable if, assigned a bounded input class and some boundaries in the output signal space, the output never exceeds such boundaries over a prespecified (finite) interval of time. IO-FTS has been already investigated in previous papers, whereas this is the first work dealing with the input-output behavior of an uncertain dynamical system in the FTS framework. Two sufficient conditions are given, concerning the class of 2 and ∞ input signals, for the analysis of robust IO-FTS. The applicability of the results is illustrated by means of a numerical examples.

New results on BIBO stability analysis for a class of neutral delay systems

Abstract This paper studies bounded input bounded output (BIBO) stability for a class of neutral systems with time-varying delays. Based on Lyapunov method and linear matrix inequalities, some new BIBO stability criteria are established. The numerical simulation is made to demonstrate the effectiveness of the theoretical results obtained in this paper.

Robust stability testing function and Kharitonov-like theorem for fractional order interval systems

This study deals with the subject of robust bounded-input bounded-output (BIBO)-stability of a family of fractional order interval systems. Employing the idea of ‘robust stability testing function’ and extending it to the case of intended systems, a simple graphical procedure for checking the robust BIBO-stability applicable to both commensurate and incommensurate orders is developed. Moreover, a Kharitonov-like theorem is provided that presents necessary and sufficient conditions for checking the mentioned stability of the fractional order interval systems with commensurate order α belonging to [1,2), but only sufficient conditions for commensurate order α in interval (0,1). Besides, lower and upper bounds applicable to both commensurate and incommensurate cases are provided which are useful for simulation purposes. Finally, three numerical examples are given to illustrate the results.

Robust BIBO Stabilization Analysis for Discrete-time Uncertain System

The discrete-time uncertain system with time delay is investigated for bounded input bounded output (BIBO). By constructing an augmented Lyapunov function, three different sufficient conditions are established for BIBO stabilization. These conditions are expressed in the form of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Two numerical examples are provided to demonstrate the effectiveness of the derived results. Keywords—Robust BIBO stabilization, delay-dependent stabilization, discrete-time system, time delay.

Passive learning and input-to-state stability of switched Hopfield neural networks with time-delay

In this paper, we propose a new passive weight learning law for switched Hopfield neural networks with time-delay under parametric uncertainty. Based on the proposed passive learning law, some new stability results, such as asymptotical stability, input-to-state stability (ISS), and bounded input-bounded output (BIBO) stability, are presented. An existence condition for the passive weight learning law of switched Hopfield neural networks is expressed in terms of strict linear matrix inequality (LMI). Finally, numerical examples are provided to illustrate our results.

BIBO stability of switched uncertain neutral control system

This paper is concerned with the problem of bounded input bounded output (BIBO) stability for a class of switched uncertain neutral system. The uncertainty is assumed to be of structured linear fractional from which includes the norm-bounded uncertainty as a special case. Firstly, by introducing the general variation-of-constants formula of neutral systems with perturbation, the BIBO stability property of general switched neutral systems with perturbation are established, which conquer the difficulties caused by the interaction between the switching rules and time delay. Subsequently, combined with the general variation-of-constants formula with stated-dependent switching scheme, the sufficient conditions of the BIBO stability are obtained in terms of the so-called Lyapunov-Metzler linear matrix inequalities (LMIs). Finally, the simulation examples are given to demonstrate the effectiveness and the potential of the proposed techniques in this paper.

BIBO stabilization of T-S fuzzy neutral systems by LMI approach

In this paper, the problem of bounded-input bounded-output (BIBO) stabilization for a class of Takagi-Sugeno (T-S) fuzzy neutral systems is considered. Based on the Lyapunov-Krasovskii functional, the descriptor model transformation and the linear matrix inequality (LMI) approach, sufficient conditions guaranteeing BIBO stabilization of the T-S fuzzy neutral systems are established in terms of LMIs. And an explicit expression of the desired controller is readily given when these LMIs are feasible. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed approach.

Input–output finite time stabilization of linear systems

Bounded Input Bounded Output (BIBO) stability is usually studied when only the input–output behavior of a dynamical system is of concern. The present paper investigates the analogous concept in the framework of Finite Time Stability (FTS), namely the Input–Output FTS (IO-FTS). FTS has been already investigated in several papers in terms of state boundedness, whereas in this work we deal with the characterization of the input–output behavior. Sufficient conditions are given, concerning the class of L 2 and L ∞ input signals, for the analysis of IO-FTS and for the design of a static state feedback controller, guaranteeing IO-FTS of the closed-loop system. The effectiveness of the proposed results is eventually illustrated by means of some numerical examples.

Augmented Lyapunov Method for BIBO Stabilization of Discrete System

The problem of bounded-input bounded-output (BIBO) stabilization for discrete-time uncertain system with time delayis investigated. By constructing an augmented Lyapunov function, some sucient conditions guaranteeing BIBO stabilizationand robust BIBO stabilization are established. These conditions are expressed in the forms of linear matrixinequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Two numerical examples areprovided to demonstrate the e ectiveness of the derived results.

Architecture, performance and stability analysis of a formula-based fuzzy I − fuzzy P − fuzzy D controller

In this paper, a formula-based fuzzy integral minus fuzzy proportional minus fuzzy derivative (FI − FP − FD) controller is proposed based on the modified conventional parallel PID controller, i.e. parallel integral minus proportional minus derivative (I − P − D) controller, in order to avoid some practical industrial process problems. It freezes the linear structure of conventional parallel I − P − D controller, with analytical formulas. The final shape of the controller is a discrete-time fuzzy version of conventional parallel I − P − D controller. Computer simulation is performed to evaluate the performance of FI − FP − FD controller for setpoint tracking and load-disturbance rejection for some complex processes, such as, first- and second-order process with delay, inverse response process with and without delay and higher order processes. The simulation is done using National Instrument™ software (LabVIEW™). The response of FI − FP − FD controller is compared with the conventional parallel I − P − D controller, tuned with the Ziegler–Nichols tuning technique. It is observed that the FI − FP − FD controller performed much better than conventional I − P/I − P − D controller. Simulation results demonstrate the effectiveness and usefulness of proposed formula-based FI − FP − FD controller. Also, the bounded-input bounded-output (BIBO) stability of the overall fuzzy control system is established with sufficient conditions, using the famous “Small Gain Theorem”.

A Novel Diversity-Controlled Genetic Algorithm for Optimization of BIBO Stable Digital IF Filters Over CSD Multiplier Coefficient Space

This paper presents a novel diversity-controlled (DC) genetic algorithm (GA) for the optimization of digital Intermediate Frequency (IF) filters over the (finite-precision) canonical signed-digit (CSD) multiplier coefficient space. This optimization exploits the bilinear-lossless-discrete- integrator (bilinear-LDI) lattice digital filter design approach for the realization of the required infinite-precision seed digital IF filter chromosome. A look-up table (LUT) approach is proposed to ensure that the finite-precision CSD digital IF filter chromosomes generated in the course of DCGA optimization are guaranteed to be bounded-input bounded-output (BIBO) stable. The salient feature of DCGA optimization is that it permits external control over the population diversity (i.e. the parent selection pressure) to achieve a high convergence speed. This feature is illustrated through the application of the proposed DCGA optimization to the design of a pair of practical digital IF filters satisfying different design specifications. It is observed that, for both digital IF filter designs, the DCGA optimization results in around an order of magnitude improvement in the convergence speed as compared to a conventional GA optimization.

Disturbance attenuation analysis of state feedback Nash strategy for two-player Linear Quadratic Sequential Games

There is limited formal mathematical analysis of one type of games — dynamic sequential games with large, or even infinitely large, planning horizons, from the point view of system controls. In this paper, we use a zero-sum game theoretical approach to address the disturbance attenuation analysis of state feedback Nash strategies for Dynamic Linear Quadratic Sequential Games (LQSGs) with uncertainties or disturbances. Based on the assumption that the disturbance will do the worst to the normal game players, we provide a simultaneous zero-sum game formulation between nature and each player, and a non-zero sum formulation among the players. For finite-horizon LQSGs, we first provide state feedback Nash strategies with optimal attenuation levels. Then we extend the approach to infinite-horizon LQSGs. We prove that the feedback system is Bounded Input Bounded Output (BIBO) stable with respect to the disturbances.

Complete Characterization of Stable Bandlimited Systems Under Quantization and Thresholding

In this paper, we analyze the approximation behavior of sampling series, where the sample values-taken equidistantly at Nyquist rate-are disturbed either by the nonlinear threshold operator or the nonlinear quantization operator. We perform the analysis for several spaces of bandlimited signals and completely characterize the spaces for which an approximation is possible. Additionally, we study the approximation of outputs of stable linear time-invariant systems by sampling series with disturbed samples for signals in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PW</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pi</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> . We show that there exist stable systems that become unstable under thresholding and quantization and that the approximation error is unbounded irrespective of how small the quantization step size is chosen. Further, we give a necessary and sufficient condition for the pointwise and the uniform convergence of the series. Surprisingly, this condition is the well-known condition for bounded-input bounded-output (BIBO) stability. Finally, we discuss the special case of finite-impulse-response (FIR) filters and give an upper bound for the approximation error.

Robust stability analysis of fractional order interval polynomials

The paper deals with the robust stability analysis of a Fractional Order Interval Polynomial (FOIP) family. Some new results are presented for testing the Bounded Input Bounded Output (BIBO) stability of dynamical control systems whose characteristic polynomials are fractional order polynomials with interval uncertainty structure. It is shown that the Kharitonov theorem is not applicable for this type of polynomial. A procedure is given for computation of the value set of FOIP. Based on the value set, an algorithm is presented for testing the stability of FOIP. The results presented in the paper are useful for the analysis and design of Fractional Order Interval Control Systems (FOICS). Examples are given to show how the proposed method can be used to assess the effects of parametric variations on the stability in feedback loops with fractional order interval transfer functions.

The Stability and Continuity Behavior of the Spectral Factorization in the Wiener Algebra With Applications in Wiener Filtering

This paper investigates the stability and the continuity behavior of the spectral factorization and of the Wiener filter in the bounded-input bounded-output (BIBO) stability norm. It shows that if the minimum of the given spectra becomes not smaller than half its norm, there exist uniform upper bounds on the stability norm of the spectral factor and Wiener filter. If on the other hand, the minimum becomes smaller than a quarter of its norm, no such upper bounds can exist. In the second part, it is shown that every BIBO-stable spectral density is a continuity point of the spectral factorization. From this, it is derived that the Wiener filter always depends continuously on the data in the BIBO-norm. These results are compared with energy stable systems. It turns out that every continuous spectrum is a discontinuous point for the spectral factorization in the energy norm. It follows that the Wiener filter depends not continuous on the data in this norm.

BIBO stabilisation of Takagi-Sugeno fuzzy systems under persistent perturbations using fuzzy output-feedback controllers

Fuzzy output-feedback controllers are designed to achieve a bounded input–bounded output (BIBO) stabilisation of Takagi-Sugeno (T-S) fuzzy systems under persistent perturbations. The designed fuzzy controllers are composed of a fuzzy observer and a static state-feedback controller, both based in the parallel distributed computation method. The design is carried out minimising the induced 1-norm of the T-S system from perturbations to controlled outputs. This design can be formulated in terms of linear matrix inequalities (LMIs) and bilinear matrix inequalities (BMIs), which are solved using two proposed iterative LMI (ILMI) algorithms. The designed controllers assume that all the premise variables are measurable. Finally, this method is applied to an example.

Design and Stability Discussion of a Hybrid Intelligent Controller

In this paper, the pitch angle control of a laboratory model helicopter is discussed. The control has some specific features. As a main feature, it is observed that the steady state control command is completely dependent on the setpoint, so error-based controller design is not applicable to this case. Moreover, the system has a highly oscillating dynamics. In order to solve this control problem, two controllers are designed, an artificial neural network, whose input is the setpoint, is used to provide the steady state control command, and a fuzzy inference system, whose input is the error of the system, is used to provide the transient control command. The total control command is the sum of the aforementioned two control commands. It is proved that both ANN and FIS are bounded-input bounded-output (BIBO) systems.

The simplest fuzzy two-term controllers: mathematical models and stability analysis

This paper reveals mathematical models for some simplest fuzzy two-term (PI/PD) controllers which employ two symmetric fuzzy sets for each of the two input variables and three symmetric fuzzy sets for the output variable. The basic constituents of these models are Γ-function type and L-function type membership functions for the input variables and triangular membership functions for the output variable, minimum triangular norm, algebraic sum triangular conorm, different inference methods and Centre of Sums (COS) defuzzification method. The properties of these models are studied to examine their suitability for control application. Using the well-known small-gain theorem, sufficient condition for Bounded-Input Bounded-Output (BIBO) stability of a feedback system, containing any fuzzy controller as a subsystem, is established. Finally, some numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PI/PD controllers.

Analytical structure and stability analysis of a fuzzy PID controller

Analytical structure for a fuzzy PID controller is introduced by employing two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. This structure is derived via left and right trapezoidal membership functions for inputs, trapezoidal membership functions for output, algebraic product triangular norm, bounded sum triangular co-norm, Mamdani minimum inference method, and center of sums (COS) defuzzification method. Conditions for bounded-input bounded-output (BIBO) stability are derived using the Small Gain Theorem. Finally, two numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controller.