Class Of Linear Time-varying Systems Research Articles

Approximation-Based Approach to Adaptive Control of Linear Time-Varying Systems

An adaptive state-feedback control system is proposed for a class of linear timevarying systems represented in the controller canonical form. The adaptation problem is reduced to the one of Taylor series-based first approximations of the ideal controller parameters. The exponential convergence of identification and tracking errors of such an approximation to an arbitrarily small and adjustable neighbourhood of the equilibrium point is ensured if the condition of the regressor persistent excitation with a sufficiently small time period is satisfied. The obtained theoretical results are validated via numerical experiments.

Distributed exponential state estimation of linear systems over jointly connected switching networks

Recently, the distributed state estimation problem for continuous-time linear systems over jointly connected switching networks was solved. Due to the use of the generalized Barbalat’s Lemma, it can only show that the estimation errors asymptotically converge to the origin. By a completely different approach, this paper further studies the distributed state estimation problem with two new features. First, the asymptotic convergence is strengthened to the exponential convergence. This strengthened result not only offers a guaranteed convergence rate, but also renders the error system total stability and thus the capability to withstand small disturbances. Second, the coupling gains of our local observers can be different from each other and thus offers greater design flexibility, while the coupling gains in the existing result were required to be identical. These two new features are achieved by establishing exponential stability for two classes of linear time-varying systems, which may have other applications.

Finite-Time State Observer for a Class of Linear Time-Varying Systems With Unknown Inputs

A finite-time observer for a class of linear time-varying systems with bounded unknown inputs is presented in this note. The design of the observer exploits structural properties of the system and, through a linear operator, allows applying the robust exact differentiator in a block form without requiring additional stabilizing terms. The proposed observer provides an exact estimate of the states after a finite transient time despite the possible instability of the system and the effects of bounded unknown inputs.

Negative Imaginary Theory for a Class of Linear Time-Varying Systems

This letter introduces the notion of linear time-varying (LTV) negative imaginary systems. LTV negative imaginary systems are defined using a time-domain dissipative supply rate $w$ ( $u,\dot {y}$ ) that depends on input to the system ( $u$ ), time-derivative of the system’s output ( $\dot {y}$ ) and an index $\delta \geq 0$ . For $\delta > 0$ , it gives rise to a strict subclass within the LTV negative imaginary systems, termed as LTV output strictly negative imaginary systems. For characterizing the proposed class of systems, a set of linear differential matrix inequality conditions is derived based on the given state-space realization. Subsequently, LTV negative imaginary theory is specialized to linear parameter-varying (LPV) cases for which, the differential matrix inequality conditions can easily be avoided by considering the rate of variation of the uncertain parameters as independent LMI variables. Finally, a set of sufficient conditions is derived which ensures that the origin is a globally asymptotically stable equilibrium point of an unforced positive feedback interconnection of two uniformly asymptotically stable LTV negative imaginary systems.

Set-Membership Filtering Subject to Impulsive Measurement Outliers: A Recursive Algorithm

This paper is concerned with the set-membership filtering problem for a class of linear time-varying systems with norm-bounded noises and impulsive measurement outliers. A new representation is proposed to model the measurement outlier by an impulsive signal whose minimum interval length (i.e., the minimum duration between two adjacent impulsive signals) and minimum norm (i.e., the minimum of the norms of all impulsive signals) are larger than certain thresholds that are adjustable according to engineering practice. In order to guarantee satisfactory filtering performance, a so-called parameter-dependent set-membership filter is put forward that is capable of generating a time-varying ellipsoidal region containing the true system state. First, a novel outlier detection strategy is developed, based on a dedicatedly constructed input-output model, to examine whether the received measurement is corrupted by an outlier. Then, through the outcome of the outlier detection, the gain matrix of the desired filter and the corresponding ellipsoidal region are calculated by solving two recursive difference equations. Furthermore, the ultimate boundedness issue on the time-varying ellipsoidal region is thoroughly investigated. Finally, a simulation example is provided to demonstrate the effectiveness of our proposed parameter-dependent set-membership filtering strategy.

Sequential covariance intersection‐based Kalman consensus filter with intermittent observations

This paper investigates the distributed state estimation for a class of linear time-varying systems with intermittent observations in sensor networks. Unlike the existing studies in distributed state estimation, this work considers the scenario where the cross-covariances between different sensors are unavailable and the measurements for state estimation encounter intermittent observations and/or random losses. For this practical scenario, a new sequential covariance intersection-based Kalman consensus filer (SCIKCF) is then developed. We show that, with the proposed SCIKCF, each sensor can achieve consensus estimates regardless of the order of fusion. Furthermore, the stability of the SCIKCF as well as the boundedness of the estimation error and the corresponding error covariances are analysed. Finally, three examples are performed to verify the effectiveness of the proposed SCIKCF.

Recursive filtering for time-varying systems under duty cycle scheduling based on collaborative prediction

In this paper, a novel recursive filtering scheme combined with collaborative prediction (CP) method is proposed for a class of linear time-varying systems under duty cycle communication scheduling. The communication between the sensor nodes and the remote filter is implemented through wireless networks. In order to save energy even further, the working states of sensor nodes alternate between activation and dormancy depending on the preset duty cycle. The aim of this paper is to design a usable recursive filtering scheme for linear time-varying system subject to the duty cycle scheduling (DCS) in the case of limited energy. The DCS is modeled according to the corresponding scheduling rule. The unsent measurement outputs due to sensors being dormancy state are predicted by CP algorithm, based on which a recursive filtering scheme is developed. Also, the filter gain is calculated by minimizing the trace of error covariance. Subsequently, the boundedness of the designed filtering algorithm is analyzed. Finally, a numerical simulation example is presented to illustrate the effectiveness of the proposed recursive filtering approach based on CP algorithm subject to the DCS.

Satellite Control Using Magnetic Moments: Controllability and Stabilization Algorithms

The stabilization of the relative equilibrium of a satellite moving in a circular orbit, which is equipped with magnetic orientation system. It is shown that the linearized system of equations of motion belongs to a special class of linear time-varying systems, reducible to time-invariant ones. For obtained time-invariant system, controllability is analyzed, and the optimal stabilization algorithm based on the LQR method is constructed. The results of numerical simulation demonstrate the efficiency of the proposed approach.

A new design of adaptive model following control system for lateral motion of aircraft

In this paper, a new design of an adaptive model following control (AMFC) system is proposed and analysed for a class of multi-input multi-output (MIMO) linear time-varying plants with unknown bounded parameters, and sufficiently slowly varying with time. We then show that the new approach based on combined Lyapunov method and hyper-stability criteria tries to overcome the problems of stability and zero tracking error. The main difficulty of Lyapunov methods is in finding an appropriate Lyapunov control function for the closed-loop system, therefore different choices of Lyapunov functions may result in different control structures and control performance. We show that the design of AMFC for the class of linear time-varying systems could be composed of a stable linear block in the feed forward path which turns out to be easily solvable by Lyapunov equation and a nonlinear time-varying block in the feedback path can be handled with using adaptive control techniques, such that the closed-loop dynamics of the system approaches the reference model. This structure results in greater flexibility in the choice of the adaptive laws. Finally, a typical numerical example to an aircraft control problem is analysed.

A new design of adaptive model following control system for lateral motion of aircraft

In this paper, a new design of an adaptive model following control (AMFC) system is proposed and analysed for a class of multi-input multi-output (MIMO) linear time-varying plants with unknown bounded parameters, and sufficiently slowly varying with time. We then show that the new approach based on combined Lyapunov method and hyper-stability criteria tries to overcome the problems of stability and zero tracking error. The main difficulty of Lyapunov methods is in finding an appropriate Lyapunov control function for the closed-loop system, therefore different choices of Lyapunov functions may result in different control structures and control performance. We show that the design of AMFC for the class of linear time-varying systems could be composed of a stable linear block in the feed forward path which turns out to be easily solvable by Lyapunov equation and a nonlinear time-varying block in the feedback path can be handled with using adaptive control techniques, such that the closed-loop dynamics of the system approaches the reference model. This structure results in greater flexibility in the choice of the adaptive laws. Finally, a typical numerical example to an aircraft control problem is analysed.

On Boundedness of Error Covariances for Kalman Consensus Filtering Problems

In this paper, the uniform bounds of error covariances for several types of Kalman consensus filters (KCFs) are investigated for a class of linear time-varying systems over sensor networks with given topologies. Rather than the traditional detectability assumption, a new concept called collectively uniform detectability (CUD) is proposed to address the detectability issues over sensor networks with relaxed restrictions. By using matrix inequality analysis techniques, the conditions for the newly proposed CUD concept are established, and then, the explicit expressions of the uniform upper/lower bounds are derived for error covariances of several commonly used KCF algorithms. Consequently, a comparison is conducted between the obtained bounds so as to reveal their relationships. Finally, a numerical example is provided to calculate and further compare the bounds of interest in order to demonstrate the practical usefulness of the developed theory.

Predictor-based control design for UAVs: robust stability analysis and experimental results

This paper proposes predictor-based controllers for a class of linear time-varying systems with input delay, model parameter uncertainty, and delay uncertainty. Sufficient conditions for global asymptotic stability are given in terms of linear matrix inequalities. The class of systems considered corresponds to two problems relating to the control of unmanned aerial vehicles. An inner–outer-loop control structure is assumed with the delay appearing in the outer-loop variables involving the vehicle's translational degrees-of-freedom. The first control problem considered is output tracking for the position of the vehicle. The second problem is visual servoing where camera measurements are used to control the vehicle's pose relative to a visual target. The effectiveness of the proposed controllers are shown via simulation. Experimental validation is performed using an indoor quadrotor platform at the Applied Nonlinear Control Lab (ANCL), University of Alberta.

Continuous Discrete Sequential Observers for Time-Varying Systems Under Sampling and Input Delays

We provide continuous-discrete observers for a large class of time-varying linear systems where the inputs and outputs have sampling and delays, and where the systems and outputs contain uncertainties. We allow the delays in the output and input to differ and to be arbitrarily long. We use the observers to design controls that ensure input-to-state stability, under delays and sampling. The observers and controls have no distributed terms. We illustrate our work in a dc motor model.

Sampled-data control for a class of linear time-varying systems

This paper is concerned with the sampled-data control for a class of linear time-varying system. A classic Halanay inequality is first extended to the time-varying sampled-data system. Then based on the comparison principle and the extended Halanay inequality, new criteria for globally uniformly exponential stability and globally uniformly asymptotic stability of the corresponding closed-loop system are derived. Furthermore, an algorithm is presented to solve the gain synthesis problem. Finally, one example is given to show the effectiveness of the obtained results.

Gap Metrics for Linear Time-Varying Systems

The relationship between two gap metrics is investigated for a class of time-varying linear systems. Specifically, systems are considered to be causal linear maps between finite-energy continuous-time signals of nonuniformly lower-bounded support, and by definition, the corresponding input-output graphs admit normalized so-called strong-right and strong-left representations. First, a time-varying generalization of Vinnicombe's $\nu$-gap is revisited to rectify an omission in the original developments and to verify compliance with the axioms of a metric. Subsequently, it is shown that this generalized $\nu$-gap metric induces the same topology as an apposite adaptation of Feintuch's time-varying gap metric for discrete-time systems defined over finite-energy signals of uniformly lower-bounded support. The two generalized metrics are therefore qualitatively equivalent in robust stability analysis for linear time-varying feedback interconnections. Quantitatively, the $\nu$-gap between a given pair of systems is never larger in value than the Feintuch gap.

Robust optimal filtering for linear time‐varying systems with stochastic uncertainties

In this study, the robust optimal filtering problem is investigated for a class of linear time-varying systems with stochastic uncertainties. A new model is presented that accounts for both stochastic parameter uncertainties and noise. On the basis of the new model, a robust optimal filter is proposed. It takes stochastic uncertainties into full consideration, but does not depend on any specific structure of uncertainties. By resorting to linear system theory and utilising stochastic analysis methods, the filter gain is designed such that the estimation error covariance is minimised at each time step. The authors' developed filtering algorithm is recursive, and therefore suitable for real-time online applications. In the end, two simulation studies are performed to illustrate the effectiveness and applicability of their proposed strategy.

Design and Analysis of Fractional Order PDα-Type Iterative Learning Control

This paper is concerned with fractional-order iterative learning control (ILC) for the tracking problems of a class of linear time-varying systems. Both open-loop and closed-loop PDα-type iterative learning updating laws are considered. Sufficient conditions for the convergence of the tracking errors are obtained. Finally the efficiency of the methods are verified by numerical simulations.

Strict Lyapunov functions for time-varying systems with persistency of excitation

We study the stability of the origin for a class of linear time-varying systems with a drift that may be divided in two parts. Under the action of the first, a function of the trajectories is guaranteed to converge to zero; under the action of the second, the solutions are restricted to a periodic orbit. Hence, by assumption, the system’s trajectories are bounded. Our main results focus on two generic case studies that are motivated by common nonlinear control problems: model-reference adaptive control, control of nonholonomic systems, tracking control problems, to name a few. Then, based on the standing assumption that the system’s dynamics is persistently excited, we construct a time-dependent Lyapunov function that has a negative definite derivative. Our main statements may be regarded as off-the-shelf tools of analysis for linear and nonlinear time-varying systems.

Generalized predictive control of linear time-varying systems

We study the problem of predictive control of linear time-varying systems and develop a generalized predictive controller for optimal control of a standard single-input single-output controlled autoregressive moving average model. Linear time-varying output filtering and incremental of control is included for flexibility for applications. The controller is applicable to a large class of linear time-varying systems.

New stability conditions for a class of linear time-varying systems

This paper considers the problem of asymptotic stability for linear time-varying systems of the form ẋ(t)=A(t)x(t). Some new stability conditions are proposed. First, two stability conditions for nonlinear time-varying systems are given by using non-monotonic Lyapunov functions. Then the results obtained are extended to the linear case, two stability conditions with infinite integral are derived. Furthermore, by using the top-floor function, a linear matrix inequalities (LMI) condition and an eigenvalue criterion for asymptotic stability of systems are presented. Comparing with the existing results, the conditions obtained allow both A(t) and Ȧ(t) are unbounded at t=+∞. Some numerical examples are provided to show the effectiveness of the theoretical results.