Stability Of Linear Time-varying Systems Research Articles

Nonconvex, Fully Distributed Optimization Based CAV Platooning Control Under Nonlinear Vehicle Dynamics

CAV platooning technology has received considerable attention, driven by the next generation smart transportation systems. This paper considers nonlinear vehicle dynamics and develops fully distributed optimization based CAV platooning control schemes via the platoon centered MPC approach for a possibly heterogeneous CAV platoon. The nonlinear vehicle dynamics leads to major difficulties in distributed algorithm development and control analysis. Specifically, the underlying MPC optimization problem is nonconvex and densely coupled. Further, the closed loop dynamics becomes a time-varying nonlinear system with non-vanishing external perturbations, making stability analysis rather complicated. To overcome these difficulties, we formulate the underlying MPC optimization problem as a locally coupled, albeit nonconvex, optimization problem and develop a sequential convex programming based fully distributed scheme for a general MPC horizon. Such a scheme can be effectively implemented for real-time computing using operator splitting methods. To analyze the closed loop stability, we apply various tools from global implicit function theorems, stability of linear time-varying systems, and Lyapunov theory for input-to-state stability to show that the closed loop system is locally input-to-state stable uniformly in all small coefficients pertaining to the nonlinear dynamic effects. Numerical tests on a heterogeneous CAV platoon in a real traffic condition illustrate the effectiveness of the proposed method.

Networked robust stability for linear time-varying systems

A networked control system (NCS) with the communication channels described by cascaded two-port networks for linear time-varying systems is studied. In this framework, a system is considered as a (possibly unbounded) linear operator that maps an input space to an output space. We consider the operator norm bounded uncertainties in the transmission matrices of two-port networks as well as the gap-type uncertainties in the plant and controller. A necessary and sufficient condition for robust stability of the NCS is generalized from the time-invariant case.

Finite-time stability of positive switched time-delay systems based on linear time-varying copositive Lyapunov functional

In this paper, we deal with the finite-time stability of positive switched linear time-delay systems. By constructing a class of linear time-varying copositive Lyapunov functionals, we present new explicit criteria in terms of solvable linear inequalities for the finite-time stability of positive switched linear time-delay systems under arbitrary switching and average dwell-time switching. As an important application, we apply the method to finite-time stability of linear time-varying systems with time delay.

Lyapunov differential equations and inequalities for stability and stabilization of linear time-varying systems

This paper studies exponential stability and stabilization of linear time-varying systems without assuming that the coefficient matrix is bounded, which is generally necessary in the existing literature. First, by introducing the notions of uniformly exponentially stable and uniformly exponentially expanding functions, some Lyapunov differential inequalities based characterizations for exponential stability of linear time-varying systems are established. Second, it is shown that the linear time-varying system is both uniformly exponentially stable and uniformly exponentially expanding if and only if the corresponding Lyapunov differential equation has a unique positive definite solution. Third, some degenerated Lyapunov differential equations, where the Q matrix is only positive semi-definite, are also re-examined. Finally, a weighted controllability Gramian, which contains a weighting time-varying function that can be properly designed to reveal a trade-off between the convergence rate/regulation time of the closed-loop system and the control effort, based stabilization approach is established to ensure that the closed-loop system is both uniformly exponentially stable and uniformly exponentially expanding. The effectiveness of the proposed approach is illustrated by the design of the spacecraft attitude control system with magnetic actuation.

Zero dynamics assignment and its applications to the stabilization of linear time-varying systems

An algorithm to select an output for desired vector relative degree, which is not restricted to be uniform (the same for all outputs), and zero dynamics in linear time-varying (LTV) multi-input multi-output (MIMO) systems is proposed. The proposed algorithm is developed under reasonable assumptions of uniform (with respect to time) and lexicography-fixed controllability (with some choice of fixed controllability indices) and is inspired by a generalization of the Ackermann’s formula. One application of the proposed algorithm is the context of higher-order sliding-mode control for perturbed LTV MIMO systems, achieving finite-time stability. The developed algorithm also provides an extension of the classical Ackermann–Utkin formula for outputs of uniform relative degree one and includes the design of flat outputs for LTV MIMO systems. Hence other application is the feedback controller design for trajectory tracking task. Finally, a complete HOSM methodology is provided and tested in an LTV model of the non-holonomic car.

Observation and stabilization of LTV systems with time-varying measurement delay

This work deals with observer-based output-feedback stabilization of linear time-varying (LTV) systems with time-varying measurement delay. The delay is modeled by a first-order hyperbolic partial differential equation (PDE), whose boundary condition evolves according to an ordinary differential equation (ODE), namely, the plant. The overall system is thus represented by an ODE–PDE cascade, which is also the structure of the proposed observer. The exponential stability of the estimation error is established for arbitrarily large time-varying delays. An observer-based controller is also introduced and the closed-loop stability is proved. The separation principle holds and the design of both the observer and controller gains can be done as if there was no delay. Some simulations illustrate the feasibility of this approach.

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.

Finite-time dynamic output feedback stabilization of linear time-varying systems by a piecewise constant method

The paper mainly deals with the problem of finite-time stabilization of linear time-varying systems. A dynamic output feedback controller is designed, which is able to stabilize the linear time-varying systems in finite time. By virtue of extended piecewise constant method, novel criteria for the existence of a dynamic output feedback controller is established in terms of linear matrix inequalities. Compared with the existing method, the proposed method is more efficient from a computational point of view. A simulation is given to illustrate the effectiveness of the obtained result.

Lyapunov Functions for Persistently-Excited Cascaded Time-Varying Systems: Application to Consensus

We present some results on stability of linear time-varying systems with particular structures. Such systems appear in diverse problems, which include the analysis of adaptive systems, persistently-excited observers and consensus of systems interconnected through time-varying links. The originality of our statements rely in the fact that we provide smooth strict Lyapunov functions hence, our proofs are constructive and direct. Moreover, we establish uniform global exponential stability with explicit stability and decay estimates. For illustration, we address a brief but representative case-study of consensus of Lagrangian systems interconnected through unreliable links.

Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach

We provide a new sequential predictors approach for the exponential stabilization of linear time-varying systems. Our method circumvents the problem of constructing and estimating distributed terms in the control laws, and allows arbitrarily large input delay bounds, pointwise time-varying input delays, and uncertainties. Instead of using distributed terms, our approach to handling longer delays is to increase the number of predictors. We obtain explicit formulas to find lower bounds for the number of required predictors. The formulas involve bounds on the delays and on the derivatives of the delays. We illustrate our method in three examples.

Dynamic Stability Analysis of Linear Time-varying Systems via an Extended Modal Identification Approach

The problem of linear time-varying(LTV) system modal analysis is considered based on time-dependent state space representations, as classical modal analysis of linear time-invariant systems and current LTV system modal analysis under the “frozen-time” assumption are not able to determine the dynamic stability of LTV systems. Time-dependent state space representations of LTV systems are first introduced, and the corresponding modal analysis theories are subsequently presented via a stability-preserving state transformation. The time-varying modes of LTV systems are extended in terms of uniqueness, and are further interpreted to determine the system’s stability. An extended modal identification is proposed to estimate the time-varying modes, consisting of the estimation of the state transition matrix via a subspace-based method and the extraction of the time-varying modes by the QR decomposition. The proposed approach is numerically validated by three numerical cases, and is experimentally validated by a coupled moving-mass simply supported beam experimental case. The proposed approach is capable of accurately estimating the time-varying modes, and provides a new way to determine the dynamic stability of LTV systems by using the estimated time-varying modes.

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.

Stabilization of linear time-varying systems with state and input constraints using convex optimization

The stabilization problem of linear time-varying systems with both state and input constraints is considered. Sufficient conditions for the existence of the solution to this problem are derived and a gain-switched(gain-scheduled) state feedback control scheme is built to stabilize the constrained timevarying system. The design problem is transformed to a series of convex feasibility problems which can be solved efficiently. A design example is given to illustrate the effect of the proposed algorithm.

Finite-time stabilization of linear time-varying systems by piecewise constant feedback

Most existing methods for finite-time stabilizing controller design of linear time-varying systems involve solving differential linear matrix equations. Due to the non-convexity of the problem, it requires a high computational burden. This paper proposes a numerical method to solve finite-time stabilization problems. Successive approximations are performed to estimate the evolution of system states. Accordingly, a gain-switched state feedback controller can be obtained by solving a sequence of linear matrix inequalities (LMIs) based optimization problems. The proposed algorithm is used to design the mass–spring system and the autopilot system of the BTT missile. Comparison with existing methods is given and the simulation results show the effectiveness of the proposed method.

On asymptotic stability of linear time-varying systems

This paper is concerned with asymptotic stability analysis of linear time-varying (LTV) systems. With the help of the notion of stable functions, some differential Lyapunov inequalities (DLIs) based necessary and sufficient conditions are derived for testing asymptotic stability, exponential stability and uniformly exponential stability of general LTV systems. With the help of the concept of (non-uniformly) exponential stability, a class of upper-triangular LTV systems is carefully investigated based on the developed stability analysis approaches. A couple of numerical examples with some of them borrowed from the literature is provided to illustrate the effectiveness of the proposed theoretical results.

Instance of hidden instability traps in intermittent transition of moving masses along a flexible beam

The problem of an elastic beam under the periodic loading of successive moving masses is investigated as a pragmatic case for studying dynamic stability of linear time-varying systems. This model serves to highlight the odds of multi-solutions coexistence, a form of hidden instability which reveals dangerous as it may be precipitated by the slightest disturbance or variation in the model. Since no engineering model perfectly represents a physical system, such situations for which Floquet theory naively predicts stability are potentially inevitable. The harmonic balancing method is used in order to thoroughly explore the stability diagrams for detecting these instability gaps. Although this phenomenon has also been described in other physical systems, it has not been addressed for beam–moving mass systems. This result may find particular importance in applications involving self-induced vibrations of elastic structures and hence also appears of practical relevance.

An integral function approach to the exponential stability of linear time-varying systems

This paper studies the exponential stability of linear time-varying (LTV) systems using the recent proposed integral function. By showing the properties of the integral function and applying the Bellman-Gronwall Lemma, a sufficient and necessary condition for the exponential stability of LTV systems is derived. Furthermore, the exponential decay rate of the system trajectories can be obtained by computing the radii of convergence of integral function. The algorithm for computing the integral function is also developed and two classical examples are given to illustrate the proposed approach.

Finite-Time Stability of Linear Time-Varying Systems: Analysis and Controller Design

The note deals with the finite-time analysis and design problems for continuous-time, time-varying linear systems. Necessary and sufficient conditions and a sufficient condition for finite-time stability are devised. Moreover, sufficient conditions for the solvability of both the state and the output feedback problems are stated. Such results require the feasibility of optimization problems involving Differential Linear Matrix Inequalities. Some numerical examples illustrate the effectiveness of the proposed approach.

Delay feedback control in exponential stabilization of linear time-varying systems with input delay

Delay feedback control in exponential stabilization of linear time-varying systems with input delay

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.