Class Of Linear Time-varying Systems Research Articles

Smooth Switching Tracking Control for a Class of Linear Time-Varying Systems with Input Quantization and Input Saturation

Smooth Switching Tracking Control for a Class of Linear Time-Varying Systems with Input Quantization and Input Saturation

Synchronisation control of composite chaotic systems

ABSTRACTSynchronisation conditions are studied for composite chaotic systems with complex compound structure and the signum function based on the theorem of zero-solution stability for a class of linear time-varying systems with countable discontinuous points. The synchronisation controller and its gain range are deduced according to the stability theorem, where the gain of the controller can speed synchronisation. Numerical simulation further proves the control theory and the validity of the synchronisation controller. The proposed controller can be widely applied in those chaotic systems with switch functions or other hybrid chaotic systems.

A new class of reducible linear time-varying systems and its relation to the optimal control problem

The previously described classes of linear time-varying systems that can be reduced to time-invariant systems are enumerated, and new constructively reducible first- and second-order systems are introduced. A theorem regarding the effective necessary and sufficient conditions for reducibility is formulated and proved for second-order linear time-varying systems. For first-order linear time-varying systems of the new class, a comparison is made with the systems of equations that appear when the optimal control problem with a quadratic quality criterion is solved.

On the Uniform Exponential Stability of Time-Varying Systems Subject to Discrete Time-Varying Delays and Nonlinear Delayed Perturbations

This paper addresses the problem of stability analysis of systems with delayed time-varying perturbations. Some sufficient conditions for a class of linear time-varying systems with nonlinear delayed perturbations are derived by using an improved Lyapunov-Krasovskii functional. The uniform global asymptotic stability of the solutions is obtained in terms of convergence toward a neighborhood of the origin.

A modified highly precise direct integration method for a class of linear time-varying systems

It is always a bottleneck to design an effective algorithm for linear time-varying systems in engineering applications. For a class of systems, whose coefficients matrix is based on time-varying polynomial, a modified highly precise direct integration (VHPD-T method) was presented. Through introducing new variables and expanding dimensions, the system can be transformed into a time-invariant system, in which the transfer matrix can be computed for once and used forever with a highly precise direct integration method. The method attains higher precision than the common methods (e.g. RK4 and power series) and high efficiency in computation. Some numerical examples demonstrate the validity and efficiency of the method proposed.

Robust stabilizer design for linear time-varying internal model based output regulation and its application to an electrohydraulic system

This paper focuses on the design of a low order robust stabilizer for the tracking/disturbance rejection problem based on the internal model principle in the time-varying setting and its application to the hydraulic pressure tracking with varying frequency. The problem of this kind known as output regulation generally consists of two major parts: internal model unit construction and stabilizer design. While the construction of the time-varying internal model unit is non-trivial by itself and a very recent research outcome enables its synthesis for a class of linear time-varying systems, the effective stabilization of the augmented system (internal model unit and plant) for practical applications remains a challenge. This is due to the need to stabilize the high order time-varying augmented system using a low order stabilizer in a robust fashion and with desirable transient performance. While directly applying the stabilization approaches for a general LTV system will result in a high order stabilizer, a new method is proposed in this paper that overcomes this bottleneck by taking advantage of the unique structure of the internal model based control system. Instead of using a dynamic stabilizer with high order, this approach uses a sequence of time-varying gains that are directly injected into the internal model unit. A critical issue addressed is how to avoid the non-convex optimization associated with the time-varying gain synthesis and then convert the stabilizer design into a series of Linear Matrix Inequalities (LMIs). The proposed control approach is then demonstrated on an electrohydraulic system.

Non-asymptotic state estimation for a class of linear time-varying systems with unknown inputs

In this paper, we extend the modulating functions method to estimate the state and the unknown input of a linear time-varying system defined by a linear differential equation. We first estimate the unknown input by taking a truncated Jacobi orthogonal series expansion with unknown coefficients which can be estimated by the modulating functions method. Then, we estimate the state by using extended modulating functions and the estimated input. Both input and state estimators are given by exact integral formulae involving modulating functions and the noisy output. Hence, estimations at different instants can be non-asymptotically obtained using a sliding window of finite length. Numerical results are given to show the accuracy and the robustness of the proposed estimators against corrupting noises.

Robust Model Predictive Control of Networked Control Systems under Input Constraints and Packet Dropouts

This paper deals with the problem of robust model predictive control (RMPC) for a class of linear time-varying systems with constraints and data losses. We take the polytopic uncertainties into account to describe the uncertain systems. First, we design a robust state observer by using the linear matrix inequality (LMI) constraints so that the original system state can be tracked. Second, the MPC gain is calculated by minimizing the upper bound of infinite horizon robust performance objective in terms of linear matrix inequality conditions. The method of robust MPC and state observer design is illustrated by a numerical example.

Simultaneous controller design for a class of time-varying linear systems

Simultaneous controller design for a class of time-varying linear systems

The effect of dissipative and gyroscopic forces on the stability of a class of linear time-varying systems

It is shown that the effect of dissipative and gyroscopic forces on a certain class of potential linear time-varying system differs considerably from the effect of these forces on a time-invariant system. Examples are considered. In the first of these, the equations of motion of a disk, attached to a rotating weightless elastic shaft, are investigated, taking external friction into account. The results obtained are compared with the results obtained previously by others when considering this problem. In the second example, certain problems of the stability of rotation of a Lagrange top on a base subjected to vertical harmonic vibrations are investigated.

Controllability and Controller-Observer Design for a Class of Linear Time-Varying Systems

In this paper a class of linear time-varying control systems is considered. The time variation consists of a scalar time-varying coefficient multiplying the state matrix of an otherwise time-invariant system. Under very weak assumptions of this coefficient, we show that the controllability can be assessed by an algebraic rank condition, Kalman canonical decomposition is possible, and we give a method for designing a linear state-feedback controller and Luenberger observer.

Exact Output Regulation for Nonlinear Systems Described by Takagi–Sugeno Fuzzy Models

The exact output regulation for Takagi-Sugeno (T-S) fuzzy models depends on two conditions: 1) The local steady-state zero-error manifolds have to be the same for every local subsystem, and 2) the local input matrices have to be the same for every local subsystem included in the T-S fuzzy model. These conditions are difficult to satisfy in general. In this paper, those conditions are relaxed by solving the fuzzy regulation problem directly on the overall T-S fuzzy model, instead of constructing the fuzzy regulator on the basis of linear local controllers. By considering the fuzzy model as a special class of linear time-varying systems, existence conditions are rigorously derived. These new conditions, which can be solved by means of any mathematical software, depend on the solution of a set of symbolic simultaneous linear equations depending on the membership values of the plant and/or the exosystem. Two examples are given to illustrate the construction of the proposed regulator and to validate the improvement that is achieved with the proposed approach.

Stability Analysis for Linear Systems With Singular Second-Order Vector Differential Equations

This technical note is concerned with exponential stability analysis of linear time-varying systems represented by the second-order vector differential equations with the singular leading coefficient matrix. Using bounding techniques on the trajectories of the linear time-varying systems, the stability problem of the time-varying systems is transformed to that of the time-invariant systems and a new sufficient condition for exponential stability is derived. The obtained condition can be applied to the nonsingular case and it is proven that the condition for the nonsingular case is superior to a test presented in the recent literature. Analogously, the proposed method is also extended to the stability analysis for a class of linear time-varying systems with delay. By illustrative examples, better results are obtained by the proposed criteria as compared with some results in the literature.

Singular value decomposition for a class of linear time-varying systems with application to switched linear systems

The present paper considers singular value decomposition (SVD) for a class of linear time-varying systems. The class considered herein describes time-driven switched linear systems. Based on an appropriate input–output description, the calculation method of singular values and singular vectors is derived. The SVD enables us to characterize the dominant input–output signals using singular vectors, which form orthogonal systems in input and output spaces. The SVD is then applied to switched linear systems to improve the transient response. A numerical example is provided to demonstrate the proposed method.

A Novel Internal Model-Based Tracking Control for a Class of Linear Time-Varying Systems

Abstract This paper provides a novel method of constructing an internal model-based design of reference tracking and/or disturbance rejection for a class of linear time-varying plants with a known linear time invariant (LTI) exosystem. It is shown how the realization of an appropriate time-varying internal model can be constructed by means of a novel feedback mechanism. The design of the internal model consists of two ingredients: (1) a time-varying system immersion of the exosystem, and (2) an automatic generation of the desired control input to render the error-zeroing subspace invariant, based on the complete knowledge of the plant model. The important features of the proposed method lie in that the tracking problem setup and the proposed feedback mechanism allow us to avoid explicitly calculating the desired input, which keeps the regulated error identically at zero. Moreover the time-varying immersion is guaranteed to hold for the class of plant models under consideration. These features significantly broaden the range of applications of the proposed method, and simplify the control implementation process.

Relaxed Conditions for the Exponential Stability of a Class of Linear Time-Varying Systems

This technical note states new sufficient conditions for the exponential stability of linear time-varying (LTV) systems of the form aeligmacr(ldr) =A(t)aelig(t). The approach proposed derives and uses the notion of perturbed frozen time (PFT) form that can be associated to any LTV system. Exploiting the Bellman-Gronwall lemma, relaxed stability conditions are then stated in terms of ldquoaveragerdquo parameter variations. Salient features of the approach are: pointwise stability of A(ldr) is not required, ||Amacr(ldr)|| may not be bounded, the stability conditions also apply to uncertain systems. The approach is illustrated by numerical examples.

Improved exponential stability for time-varying systems with nonlinear delayed perturbations

In this paper, a new sufficient delay dependent exponential stability condition for a class of linear time-varying systems with nonlinear delayed perturbations is derived by using an improved Lyapunov–Krasovskii functional. The proposed exponential stability conditions are formulated in terms of the solution of Lyapunov differential equations. The approach allows for computation of the bounds that characterize the exponential stability rate of convergence of the solution. Compared with existing results, our conditions are shown to be less conservative. Numerical examples are given to illustrate the effectiveness of the obtained conditions.

Discrete Time-Frequency Characterizations of Dispersive Linear Time-Varying Systems

A class of linear time-varying systems can be characterized by dispersive signal transformations, such as nonlinear shifts in the phase of the propagating signal, causing different frequencies to be shifted in time by different amounts. In this paper, we propose a discrete time-frequency model to decompose the dispersive system output into discrete dispersive frequency shifts and generalized time shifts, weighted by a smoothed and sampled version of the dispersive spreading function. The discretization formulation is obtained from the discrete narrowband system model through a unitary warping relation between the narrowband and dispersive spreading functions. This warping relation depends on the nonlinear phase transformations induced by the dispersive system. In order to demonstrate the effectiveness of the proposed discrete characterization, we investigate acoustic transmission over shallow water environments that suffers from severe degradations as a result of modal frequency dispersions and multipath fading. Using numerical results, we demonstrate that the discrete dispersive model can lead to a joint multipath-dispersion diversity that we achieve by properly designing the transmitted waveform and the reception scheme to match the dispersive environment characteristics

Stability of linear time-varying delay systems and applications to control problems

This paper deals with the stability of a class of linear time-varying systems with multiple delays. Using the Lyapunov function method, we give sufficient delay-dependent conditions for the exponential stability with a given convergence rate, which are described in terms of linear matrix inequalities (LMI) and the solution of Riccati differential equations (RDE). The results are applied to the problem of stabilization of linear time-varying control systems with multiple delays. Numerical examples are given to illustrate the results.

On the Stability of a Certain Class of Linear Time-Varying Systems

This study firstly considers the exponential stability of unforced linear systems of slowly time-varying dynamics. Possible switchings of the system structure to unstable dynamics during certain finite time intervals are admitted. The maintenance of global exponential stability does not necessarily require at most a finite number of switchings in the dynamics while infinitely many switches can also lead to stability. The mechanism to achieve stability under infinitely many switches in the dynamics is to maintain the system in the stable region during time intervals of sufficient large length without switches provided that the system dynamics evolves at a sufficiently small rate with time. Special attention is paid to the robust tolerance for a class of state disturbances and to the case of time-varying matrix of dynamics that possess either piecewise constant or constant eigenvalues. The obtained results can be relevant for their use in stability issues for the cases of multimodel non- adaptive and adaptive control with improved transient performances.