Continuous Time-varying Systems Research Articles

Delay-Dependent Stability Condition in Finite Frequency Domain: Input-Output Approach

Over finite frequency ranges, this study analyzes delay-dependent stability criteria for time-varying delay systems. To develop new delay-dependent stability conditions in finite frequency domains for the continuous time-varying delay systems, we construct an input-output technique for delay-dependent systems utilizing the two-term transformation model paired with Scaled-Small Gain (SSG) approach. Sufficient conditions are obtained and described in terms of linear matrix inequalities (LMIs) constraints based on a novel delay-dependent finite frequency inequality. As an illustration, a numerical example has been supplied to test the correctness of the current approach.

Novel negative-definiteness conditions on the quadratic polynomial function with application to stability analysis of continuous time-varying delay systems

When analyzing the stability of time-varying delay systems in view of the Lyapunov–Krasovskii functional, a quadratic polynomial function with regard to time-varying delay is always generated. And it is particularly crucial to determine the negativeness of the matrix of such a quadratic form function for obtaining an analysis result expressed in linear matrix inequalities. This paper proposes a method of tangent intersection in the delay interval segmentation, producing the generalized quadratic convex conditions by further utilizing the cross point between every two tangent lines. It reduces the conservatism of the existing conditions remarkably without requiring unexplainable adjustable parameters and additional decision variables. Benefiting from the newly proposed quadratic convex conditions, the novel stability conditions are derived, the superiority of which is demonstrated through several widely used numerical instances and single area power system PI control example.

Behavior near time infinity of solutions of nonautonomous systems with unbounded perturbations

Abstract This paper derives convergence results for a class of nonnegative absolutely continuous functions that satisfy some differential inequality. These results are shown to be advantageous in studying the behavior at infinity of a class of unboundedly perturbed continuous and discontinuous time-varying systems that is frequently encountered in Lyapunov theory. In particular, sufficient conditions are given to ensure the positivity, the asymptotic stability and the instability of solutions of unidimensional and multidimensional systems. To illustrate the proposed results, several examples are simulated.

Proportional local assignability of dichotomy spectrum of one-sided continuous time-varying linear systems

We consider a local version of the assignment problem for the dichotomy spectrum of linear continuous time-varying systems defined on the half-line. Our aim is to show that uniform complete controllability is a sufficient condition to place the dichotomy spectrum of the closed-loop system in an arbitrary position within some Hausdorff neighborhood of the dichotomy spectrum of the free system using an appropriate time-varying linear feedback. Moreover, we assume that the norm of the matrix of the linear feedback should be bounded from above by the Hausdorff distance between these two spectra with some constant multiplier.

Design of [formula omitted]-based sampled-data control for fuzzy Markov jump systems with stochastic sampling

This paper concentrates on analyzing the stability problem for fuzzy Markov jump systems (FMJSs) via stochastic sampling and H∞-based sampled-data control scheme. With the help of input-delay technique, the probabilistic sampling intervals are reconstructed into a continuous time-varying system under stochastic parameters in the system matrices. Subsequently, a set of sufficient conditions ensuring the stochastic stability of the addressed FMJS with an allowable H∞ disturbance attenuation value is formulated in the form of linear matrix inequalities (LMIs) by choosing a proper Lyapunov–Krasovskii functional involving triple integral terms and utilizing Wirtinger-based integral inequality. The desired sampled-data control gain can be determined in terms of the solution to the obtained LMIs. Lastly, the developed theoretical results and the benefits of the desired controllers are verified through numerical examples including a single link robot arm system.

Necessary and sufficient conditions for assignability of dichotomy spectra of continuous time-varying linear systems

We consider a version of the pole placement problem for continuous time-varying linear systems. Our purpose is to prove that uniform complete controllability is equivalent to possibility of arbitrary assignment of the dichotomy spectrum. The main ingredients of the proof are the reduction of system to upper triangular form and the use of the concept of uniform complete stabilization. To illustrate the theoretical result, we consider scalar continuous time-varying control systems. For these systems we provide a simple necessary and sufficient condition for uniform complete controllability and if this condition holds, then we construct an explicit control to assign the dichotomy spectrum.

Hubungan antara Keterobservasian dan Keterkonstruksian Sistem Linier Kontinu Bergantung Waktu

Continuous time-varying linier system is a model which many peoples to find in any applications. This paper to teaching connection of observability and constructibility for continuous time-varying linear system. With algebra method we can proof any theorems to result connection of observability and constructibility for continuous time-varying linear system. Futhermore, we gave any examples for illustration to be valid this theorems and this proof.

Ellipsoidal reachable sets of linear time-varying continuous and discrete systems in control and estimation problems

The paper is devoted to reachable sets of linear time-varying continuous or discrete systems under uncertain initial states and disturbances with a bounded uncertainty measure. The uncertainty measure is the sum of a quadratic form of the initial state and the integral/sum over the finite-time interval from a quadratic form of the disturbance. It is shown that the reachable set of the system under this assumption is an evolving ellipsoid with a matrix being a solution to the linear matrix differential or difference equation. This result is used to synthesize optimal observers and estimators providing the minimal ellipsoidal sets as the estimates of the system state and unknown parameters, respectively, as well as optimal controllers steering the system state into a final target ellipsoidal set or keeping the entire system trajectory in a prescribed ellipsoidal tube under all admissible initial states and disturbances. The relationships between the optimal ellipsoidal observer and the Kalman filter as well as between the optimal ellipsoidal estimator and the recursive least weighted squares algorithm are established. Numerical modeling with the Mathieu equation for parametric vibrations of a linear oscillator illustrates the results.

Control and Filtering Problems for Linear Time-Varying Systems Based on Ellipsoidal Reachable Sets

The paper is devoted to reachable sets of linear time-varying continuous systems under uncertain initial states and disturbances with a bounded uncertainty measure. The uncertainty measure is the sum of a quadratic form of the initial state and the integral over the finite-time interval from a quadratic form of the disturbance. It is shown that the reachable set of the system under this assumption is an evolving ellipsoid with a matrix being a solution to the linear matrix differential equation. This result is used to synthesize the optimal observer providing the minimal ellipsoidal set as the estimate of the system state, as well as optimal controllers steering the system state into a final target ellipsoidal set or keeping the entire system trajectory in a prescribed ellipsoidal tube under all admissible initial states and disturbances. The relationship between the optimal ellipsoidal observer and the Kalman filter are established. Numerical modeling with the Mathieu equation for parametric vibrations of a linear oscillator illustrates the results.

Adaptive iterative learning control for unknown linear time-varying continuous systems

This paper presents a novel model reference adaptive iterative learning control (ILC) for unknown continuous-time linear time-varying systems. The unknown time-varying parameters of the system are neither required to vary slowly nor to have known bounds. The system is not required to be minimum-phase, stable, controllable or observable. The input of the system is determined by a differentiator-free control law. The used reference model is time-invariant and first order and thus choosing its parameters is easily possible, even though, the system under control is high order and time variant. Almost all of the components of the system initial condition can be iteration variant. By introducing a novel kind of Lyapunov function the convergence of the proposed adaptive ILC (AILC) and achieving asymptotic tracking are proved. Also, by rigorous mathematical analysis and with the help of some mathematical key techniques such as Bellman-Gronwall lemma, it is shown that all signals and quantities in the closed-loop system are bounded in the sense of at least one norm. Finally, the effectiveness of the proposed method is verified by two simulation examples.

Adaptive Control Strategy for Time-Varying Power System Based on Continuous Markov Model

Concerning the variation of power system operating condition caused by time-varying factors such as wind generator output power fluctuation and load change, an adaptive control strategy for time-varying power system based on continuous Markov model is proposed in this paper. First, the Lyapunov functional containing the continuous Markov time-varying power system model is constructed, and the robust stochastic stability theorem with H∞ norm bound γ is derived using the Dynkin lemma. On this basis, according to the Schur complement theorem, the robust stochastic stability LMI (linear matrix inequality) which satisfies the disturbance attenuation degree γ and the minimum variance constraint is derived considering the system operating conditions steady-state variance constraint. And then, through transforming the LMI to the minimization problem of linear objective function, the matching controller of each operating condition is solved. Finally, the stochastic gradient method is used to identify the system operating condition and determine the weighting coefficient of each controller, thus the adaptive control strategy is established. Time-domain simulation tests show that, the proposed control strategy could effectively suppress cascading disturbances as well as avoid the problem of mismatch between current operating condition and the fixed fault set. Besides, the impact on power system caused by controller switching is reduced.

Sampled-Data Synchronization of Complex Networks With Partial Couplings and T–S Fuzzy Nodes

This paper is concerned with the synchronization of complex networks subject to partial couplings and Takagi–Sugeno (T-S) fuzzy nodes, by adopting the aperiodic sampled-data strategy. A decoupling method is utilized to handle the partial couplings among connected nodes, which enables us to investigate each channel of complex networks independently. On the basis of the input-delay approach, the hybrid system about each channel is reformulated to a continuous time-varying delay system. Then, the free-weighting matrix approach and a novel continuous Lyapunov functional are adopted to capture the information of sampling pattern. Sufficient conditions are obtained to ensure that the complex networks achieves synchronization with the target node. Furthermore, the proposed strategy is extended to more general case, in which there exist constant transmission time delays in the local interactions of connected nodes. Based on Wirtinger's inequality, a simplified and efficient synchronization strategy is proposed. Moreover, the corresponding optimization problem about the maximal sampling interval upper bound is addressed as well. Finally, the communication frequency reduction potential of the proposed synchronization strategy is well demonstrated via a numerical example.

Synthesis of State Unknown Inputs Observers for Nonlinear Lipschitz Systems with Uncertain Disturbances

We propose methods to synthesize observers for the state and unknown input influences that ensure that estimation error is finite time bounded with respect to given sets of initial states and admissible trajectories or suppress initial deviations and uncertain bounded in L∞-norm external disturbances for time-varying continuous Lipschitz systems. Here gain coefficients of the observers depend on time and are determined based on numerical solutions of optimization problems with differential linear matrix inequalities or numerical solutions of the corresponding matrix comparison system. With the example of an electric drive system with elastic transmission of motion we show that their application for state estimation and unknown inputs for time-invariant systems proves to be more efficient (with respect to convergence time and accuracy of the resulting estimates) compared to observers with constant coefficients obtained based on numerical solutions of optimization problems with linear matrix inequalities.

Robust finite-time non-fragile sampled-data control for T-S fuzzy flexible spacecraft model with stochastic actuator faults

In this paper, the problem of robust finite-time non-fragile sampled-data control is investigated for uncertain flexible spacecraft model with stochastic actuator faults based on Takagi-Sugeno (T-S) fuzzy model approach. Specifically, the existence of stochastic actuator faults are described by using the Bernoulli distribution. On the basis of the input-delay approach, the sampled-data system is reformulated to a continuous time-varying delay system. Further, based on Lyapunov functional approach and linear matrix inequality technique, sufficient conditions are derived for the existence of the desired state feedback controller ensuring the stochastic finite-time bounded with prescribed H∞ performance index. Finally, numerical simulations are provided for the practical flexible spacecraft control system to verify the effectiveness and applicability of the proposed control design.

State and Unknown Inputs Finite Time Estimation for Time-Varying Nonlinear Lipschitz Systems with Uncertain Disturbances

The continuous time-varying systems with Lipschitz nonlinearities subject to bounded by L2 norm external disturbances and unknown inputs are considered. There suggested ways of observers synthesis for state and unknown inputs, providing H∞ rejection of initial deviations and bounded external disturbances simultaneously. Thus the observers gain depend on a time and is defined on the basis of the numerical solving of optimization problem with differential linear matrix inequalities or the numerical solving of corresponding matrix comparison system. The proposed observers are used to state and unknown inputs estimation of Rössler system.

Pointwise frequency responses framework for stability analysis in periodically time-varying systems

ABSTRACTThis paper explicates a pointwise frequency-domain approach for stability analysis in periodically time-varying continuous systems, by employing piecewise linear time-invariant (PLTI) models defined via piecewise-constant approximation and their frequency responses. The PLTI models are piecewise LTI state-space expressions, which provide theoretical and numerical conveniences in the frequency-domain analysis and synthesis. More precisely, stability, controllability and positive realness of periodically time-varying continuous systems are examined by means of PLTI models; then their pointwise frequency responses (PFR) are connected to stability analysis. Finally, Nyquist-like and circle-like criteria are claimed in terms of PFR's for asymptotic stability, finite-gain Lp-stability and uniformly boundedness, respectively, in linear feedbacks and nonlinear Luré connections. The suggested stability conditions have explicit and direct matrix expressions, where neither Floquet factorisations of transition matrices nor open-loop unstable poles are involved, and their implementation can be graphical and numerical. Illustrative studies are sketched to show applications of the main results.

$$H_{\infty }$$ H ∞ Performance Analysis of 2D Continuous Time-Varying Delay Systems

This paper deals with the problem of $$H_{\infty }$$H? performance analysis of 2D continuous time-varying delay systems described by Roesser model. Using a simple Lyapunov---Krasovskii functional, a new delay-dependent stability criterion is derived in terms of linear matrix inequalities. The obtained result is then extended to the problem of $$H_{\infty }$$H? performance analysis. Several examples are provided in order to illustrate the effectiveness of our results.

PREVIEW CONTROL FOR A CLASS OF LINEAR CONTINUOUS TIME-VARYING SYSTEMS

A design method of an optimal preview controller for a class of linear continuous time-varying systems is presented in this paper. By differentiating both the output error signal and the two sides of the state equation, an augmented error system is constructed, which contains the error signal, the state vector and the derivative of the state vector. Then the performance index function is introduced and the control problem is solved by optimal control theory. Thus the optimal preview control input for the original time-varying system is found. The proposed controller includes the previewable future signal, so the response speed is accelerated. Meanwhile, the integrators in the controller make the output vector track the target signal better. The numerical simulations illustrate the validity of the proposed method.

Robust fault-tolerant control against time-varying actuator faults and saturation

A set invariance condition for a piecewise continuous time-varying system is developed and applied to the fault-tolerant control problem against the actuator effectiveness loss for linear systems subject to system uncertainties and actuator saturation in this study. Linear matrix inequality conditions are proposed to compute invariant ellipsoidal sets. Discussions on system performance optimisation are given including reduction on computational complexity, maximisation of the domain of attraction and disturbance rejection. The proposed design techniques are applied to an unstable multi-input and multi-output system and to the flight control of a non-linear unmanned aerial vehicle model subject to the control effectiveness loss fault in the elevator.

Switched state-feedback control for continuous time-varying polytopic systems

This article deals with switched state-feedback ℋ2 control design of continuous time-varying polytopic systems. More specifically, the main goal is to determine, simultaneously, a set of state-feedback gains and a switching rule to orchestrate them, rendering the closed-loop system globally asymptotically stable for all time-varying uncertain parameter under consideration and assuring a guaranteed ℋ2 cost. A contribution of the present switched control technique compared to the gain scheduling, widely used in the literature, is that the online measurement of the uncertain parameter is not required and no assumption on its time derivative is imposed. The conditions are based on modified Lyapunov–Metzler inequalities and can be solved by line search coupled with LMIs. An academic example illustrates the theoretical results and compares the present technique with other techniques from literature.