Continuous-time Linear Time-varying Systems Research Articles

Event-triggered control for continuous-time linear systems with a delay in the input

We provide an event-triggered control technique for a family of linear time-varying continuous-time systems with a constant known pointwise delay in the input. We adopt a subpredictor based prediction technique, and we provide sufficient conditions that ensure that Zeno behavior does not occur. At each time, only delayed measurements are needed to implement the control. Also, the delay can be an arbitrarily large constant. We prove an input-to-state stability property for the closed-loop system, using the theory of cooperative systems. We apply our method to a gyroscopic control problem for a curve tracking dynamics arising in marine robotics.

Commutativity of Sixth-Order Time-Varying Linear Systems

This paper studies commutativity of sixth-order differential systems with zero and nonzero initial conditions (ICs). Given a sixth-order differential system $$A$$ , we find its commutative pair, that is a new differential system $$B$$ of order $$m \le 6$$ ; the obtained result will be used to show the input–output equivalency between the cascaded-connected systems $$AB$$ and $$BA$$ . Necessary and sufficient conditions for commutativity of sixth-order continuous-time linear time-varying systems (CTLTVSs) are derived. Based on our findings, we discovered that only a small group of sixth-order CTLTVSs has commutative pairs other than their feedback conjugates obtained by constant feedback and forward path gains. The explicit results obtained in this paper will help to fill the gap on commutativity theory, system behaviors which has significant application to science and play an important role in engineering. This research will also address the problems of sensitivity due to changes in ICs or parameters, and also disturbance as a result of external noise. The results are well verified by examples as well as validated by Matlab Simulink tool and Wolfram Mathematica 11.

Identification of Continuous-time Linear Time-varying Systems with Abrupt Changes in Parameters

Abstract In this paper, we propose an adaptive identification method for continuous-time systems with slowly time-varying parameters subject to infrequent abrupt changes using sampled data in the presence of measurement noise. The proposed estimator consists of two steps. First, the time-derivatives of the input and output signals are approximated through a combination of a Kalman filter and fixed-interval smoother. The parameters of the time-varying system are then estimated using the approximate time-derivatives by two Kalman filters running forward and backward in time to track the abrupt changes. The performance of the proposed estimator is verified through simulation studies where it is shown to perform better than an existing direct continuous-time estimator.

Robust Finite-Time Boundedness for Linear Time-Varying Systems

A method for the synthesis of time-varying state-feedback gains, capable of guaranteeing robust finite-time boundedness properties for linear time-varying continuous-time systems, is proposed in this paper. The strategy relies on the computation of discrete-time gains for a discretized version of the open-loop system. If some sufficient conditions are satisfied, the desired continuous-time gain is obtained. A numerical example illustrates the validity of the technique.

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.

Computing stabilising output-feedback gains for continuous-time linear time-varying systems through discrete-time periodic models

A methodology for the synthesis of time-varying static output-feedback gains, capable of stabilising continuous-time linear time-varying systems, is proposed in this paper. The method is based on the computation of output-feedback gains that stabilise a family of periodic discrete-time subsystems, related to the original model, followed by a transformation that yields the desired stabilising gains for the continuous-time system. Based on such strategy, three different approaches to compute stabilising periodic discrete-time gains, as well as the necessary transformations to generate the continuous-time output-feedback gains, are proposed. Numerical examples are presented in order to compare and analyze the efficiency of the proposed approaches.

Finite‐time distributed global optimal control for linear time‐varying multi‐agent systems: a dynamic output‐feedback perspective

This study presents a systematic procedure to solve the problem of finite-time distributed global optimal control for the leaderless and leader-follower homogeneous multi-agent systems (MASs). Each agent's dynamics is given in general form of continuous-time linear time-varying (LTV) systems. The communication network among the agents is also assumed to be undirected and directed under fixed topology. First, based on asymptotic stability principle, inverse optimal results of general LTV systems are constructed. Second, the necessary and sufficient conditions are given for the existence of the distributed optimal state feedback and optimal dynamic output feedback control that solves a global optimal control problem for MASs. Then, the sufficient condition for finite-time stability of both the state and the output feedback problems for MASs are stated. Finally, some numerical example results are provided to illustrate the effectiveness and applicability of the proposed scheme.

Distributed Kalman–Bucy Filter With Embedded Dynamic Averaging Algorithm

This paper focuses on distributed estimation using networked sensor agents with local measurements and local communication. A distributed Kalman–Bucy filter implementation is proposed and analyzed for the general case of continuous-time linear time-varying systems as compared to linear time-invariant systems. The sensor agents employ a distributed average tracking algorithm and its extension that accounts for bounded noise to estimate the averages of certain time-varying signals by communicating with their local neighbors in the network. Such estimates are then used to recover certain information in the implementation of the distributed Kalman–Bucy filter. It is shown that using the proposed distributed filter, in the absence of measurement noise, the distributed local estimates approach the centralized filter's estimate asymptotically. In the presence of bounded measurement noise, the distributed local estimates asymptotically approach the centralized filter's estimate within some bound. Simulation results illustrate the good performance of the distributed filter.

Dual-loop iterative optimal control for the finite horizon LQR problem with unknown dynamics

Achieving optimal performance over a finite-time horizon has gained a lot of attention in many engineering applications. Among them, the Finite Horizon Linear Quadratic Regulator (FHLQR) formulation for continuous-time linear time-varying systems has been well studied, with an optimal solution characterised by the Differential Riccati Equation (DRE). The solution of the DRE requires that the exact system dynamics are known. However, this assumption may not always hold, as the plant model might not be completely known or may change over time due to wear and tear. This paper proposes a dual-loop iterative algorithm to find the optimal solutions of the FHLQR formulation for continuous-time LTV systems. The inner loop utilises input trajectories based on an estimate of the optimal control gain with the addition of some excitation noise, and produces measured state trajectories. The outer loop improves the estimate of the optimal control gain utilising these measured state trajectories. It is shown in this work that with appropriate selection of the discretisation parameter T and the set of excitation signals, the proposed dual-loop iterative algorithm can converge to an arbitrarily small neighbourhood of the optimal solution. A simulation example demonstrates the effectiveness of the proposed method.

Stabilization of linear time-varying systems using proportional-derivative state feedback

This paper presents the stabilization approach for linear time-varying continuous-time systems using proportional-derivative (PD) state feedback control. The solvability conditions for the problem are considered. The general analytical expressions for the PD controller gains are derived, which describe the available degrees of freedom offered by PD state feedback. The non-uniqueness of the controller gains is utilized to obtain closed-loop systems with small gain elements. Two numerical examples are introduced to demonstrate the effectiveness of the proposed approach.

Continuous‐time linear time‐varying system identification with a frequency‐domain kernel‐based estimator

A novel estimator for the identification of continuous-time linear time-varying systems is presented in this paper. The estimator uses kernel-based regression to identify the time-varying coefficients of a linear ordinary differential equation, based on noisy samples of the input and output signals. The estimator adopts a mixed time- and frequency-domain formulation, which allows it to be formulated as the solution of a set of algebraic equations, without relying on finite differences to approximate the time derivatives. Since a kernel-based approach is used, the model complexity selection of the time-varying parameters is formulated as an optimisation problem with continuous variables. Variance and bias expressions of the estimate are derived and validated on a simulation example. Also, it is shown that, in highly noisy environments, the proposed kernel-based estimator provides more reliable results than an ‘Oracle’-based estimator which is deprived of regularisation.

Synthesis of controllers for finite-time stabilisation of LTV continuous-time systems using a discrete-time approach

ABSTRACTA methodology for the synthesis of state-feedback controllers, guaranteeing the finite-time stability of linear time-varying continuous-time systems, is proposed in this paper. The main approach consists of obtaining a controller for a discretised version of the original system and, by applying a transformation on the obtained controller, computing the desired continuous-time state-feedback gain. The transformation is derived from a condition, also presented in this paper, that relates the finite-time stability of the discretised system, with a predefined sampling rate, to the original continuous-time representation. Such condition, if feasible, also guarantees the desired behaviour of the original system for the given sampling rate. Numerical experiments illustrate the validity of the approach.

BMI-Based Stabilization of Linear Uncertain Plants With Polynomially Time Varying Parameters

This technical note considers the robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics. Each mode is characterized by a theoretically unbounded dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. Using a parameter dependent Lyapunov function polynomially depending on time, the stabilizing controller for each single mode is directly obtained by the solution of some BMI's, which become LMI's by fixing two positive scalars. The stability conditions of the switching closed loop system are derived defining a switched Lyapunov function and involving the permanence time interval of the switching plant over each single mode. A salient feature of the technical note is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.

Robust stabilization of linear uncertain plants with polynomially time varying parameters

The robust stabilization of uncertain linear time-varying continuous-time systems with a mode-switch dynamics is considered. Each mode is characterized by a dynamical matrix containing elements whose time behavior over bounded time intervals is sufficiently smooth to be well described by interval polynomials of arbitrary degree. The stability conditions of the switching closed-loop system are derived defining a switched Lyapunov function and involving the dwell time of the system over each single mode. An important feature of the paper is that, unlike all the other existing methods, each plant mode can be stabilized over arbitrarily large uncertain domains of parameters and their derivatives.

Kalman Canonical Decomposition of Linear Time-Varying Systems

In this paper, a Kalman canonical decomposition of finite-dimensional linear time-varying continuous-time systems is revisited. Two types of Kalman canonical decompositions are considered; one is collectively dubbed the intervalwise Kalman canonical decomposition and the other is collectively dubbed the global Kalman canonical decomposition. Each type is classified into subtypes resulting from variants of controllabilities (i.e., controllability, reachability, and influenceability) and observabilities (i.e., observability, determinability, and visibility). This paper first investigates the invariance properties of subspaces resulting from variants of controllabilities and observabilities. This paper then gives a self-contained exposition of each Kalman canonical decomposition on an interval. The main result is to prove the existence of intervals on each of which each Kalman canonical decomposition exists. Moreover, the relations among different Kalman canonical decompositions are extensively studied and the inconsistency between the works by Youla and Weiss is resolved.

On iterative learning control with high‐order internal models

AbstractIn this work we focus on iterative learning control (ILC) for iteratively varying reference trajectories, which are described by a high‐order internal models (HOIM) that can be formulated as a polynomials between two consecutive iterations. The classical ILC with iteratively invariant reference trajectories, on the other hand, is a special case of HOIM where the polynomial renders to a first‐order internal model with a unity coefficient. By incorporating HOIM into the ILC law, and designing appropriate learning control gains, the learning convergence in the iteration axis can be guaranteed for continuous‐time linear time‐varying systems. The initial resetting condition, P‐type and D‐type ILC, and possible extension to nonlinear cases are also explored in this work. Copyright © 2010 John Wiley & Sons, Ltd.

Finite-time stability of linear time-varying systems with jumps

This paper deals with the finite-time stability problem for continuous-time linear time-varying systems with finite jumps. This class of systems arises in many practical applications and includes, as particular cases, impulsive systems and sampled-data control systems. The paper provides a necessary and sufficient condition for finite-time stability, requiring a test on the state transition matrix of the system under consideration, and a sufficient condition involving two coupled differential–difference linear matrix inequalities. The sufficient condition turns out to be more efficient from the computational point of view. Some examples illustrate the effectiveness of the proposed approach.

On strong stabilization for linear continuous time time-varying systems

This paper deals with the strong stabilization problem for linear time-varying continuous time systems. The main result is that for such systems internally stabilizable systems may not be strongly stabilizable. This follows from the fact that the algebra of stable systems has infinite (Bass) stable rank.

Orthogonal Lyapunov transformations with some applications to stability studies

For continuous-time linear time-varying systems, this paper shows how to determine pointwise orthogonal Lyapunov transformations, reveals their advantage over standard Lyapunov transformations for the nonperiodic case, and outlines some potential applications to stability assessment for both linear and nonlinear systems.

Stability robustness of linear discrete-time systems under time-varying perturbations

Abstract The stability robustness of an asymptotically stable linear time-invariant discrete-time nominal system subject to linear time-varying perturbations is analysed. Based on the stability theory of linear time-varying continuous-time systems and the matrix norm theory, both the norm bounds and the element bounds on linear time-varying perturbations are obtained for the nominal system. The new bounds are simple to implement and they have less conservative results when the condition number of the transformation matrix is small.