Class Of Nonlinear Time-varying Systems Research Articles

The Minimal Time-Varying Realization of a Nonlinear Time-Invariant System

Abstract The state realization is called minimal if it is either accessible and observable or its state dimension is minimal. In the linear case those two definitions are equivalent, but not for nonlinear time-invariant systems. It is shown that definitions remain equivalent in case one is searching for minimal realization in a larger class of nonlinear time-varying systems. First, nonlinear realization theory is recasted for time-varying nonlinear systems. A necessary and sufficient realizability condition is given in terms of integrability of certain subspace. The mathematical tools used for this purpose are the algebraic approach of differential forms and the theory of the skew polynomial rings; these tools are again extended from time-invariant to time-varying systems.

Recursive design of finite-time convergent observers for a class of time-varying nonlinear systems

This paper considers the problem of designing globally finite-time convergent observers for a class of nonlinear systems with time-varying and output-dependent coefficients, which make the existing design approaches inapplicable. To solve this problem, a bottom-up design approach is first employed to recursively construct a finite-time convergent observer with time-varying coefficients for the nominal system. Then, using the homogeneous domination approach, we scale the finite-time convergent observer with an appropriate choice of gain for the original nonlinear system satisfying a Hölder condition. In addition, we show that the Hölder condition imposed on the nonlinearities can be removed for nonlinear systems with bounded trajectories.

Interval state observer for nonlinear time varying systems

This paper is devoted to the design of interval observers for Linear Time Varying (LTV) systems and a class of nonlinear time-varying systems in the output canonical form. An interval observer design is feasible if it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable. It is shown that under some mild conditions the cooperativity of an LTV system can be ensured by a static linear transformation of coordinates. The efficiency of the proposed approach is demonstrated through numerical simulations.

Global Bounded Synchronization of General Dynamical Networks With Nonidentical Nodes

This note addresses the problem of synchronization for general dynamical networks with nonidentical nodes. The coupling strength, outer coupling configuration and inner connection in such networks are all time varying. Neither an equilibrium for each node nor a synchronization manifold is assumed to exist. An estimate of the convergence domain for a general class of time-varying nonlinear systems is given. By introducing the average dynamics of all nodes and based on this estimate, a criterion of global synchronization in the sense of boundedness of the maximum state deviation between nodes is developed. An explicit bound of the maximum state deviation between nodes is obtained by the maximum difference between each node dynamics and the average dynamics. The proposed criterion is an extension of several related synchronization criteria for the case of identical nodes to the case of nonidentical nodes.

Discrete‐time Adaptive ILC for Non‐parametric Uncertain Nonlinear Systems with Iteration‐Varying Trajectory and Random Initial Condition

AbstractThis paper presents a new discrete‐time adaptive iterative learning control approach (AILC) for a class of time‐varying nonlinear systems with nonparametric uncertainties and non‐repeatable external disturbances by incorporating a novel iterative estimate scheme. A major distinct feature of the presented approach is that uncertainties can be completely compensated for, using only I/O data. Another distinct feature is that the pointwise convergence is achieved over a finite time interval without requiring the matching condition on initial states and reference trajectory. Rigorous mathematical analysis is developed, and simulation results illustrate the effectiveness of the proposed approach.

Probability-guaranteed [formula omitted] finite-horizon filtering for a class of nonlinear time-varying systems with sensor saturations

In this paper, the probability-guaranteed H∞ finite-horizon filtering problem is investigated for a class of nonlinear time-varying systems with uncertain parameters and sensor saturations. The system matrices are functions of mutually independent stochastic variables that obey uniform distributions over known finite ranges. Attention is focused on the construction of a time-varying filter such that the prescribed H∞ performance requirement can be guaranteed with probability constraint. By using the difference linear matrix inequalities (DLMIs) approach, sufficient conditions are established to guarantee the desired performance of the designed finite-horizon filter. The time-varying filter gains can be obtained in terms of the feasible solutions of a set of DLMIs that can be recursively solved by using the semi-definite programming method. A computational algorithm is specifically developed for the addressed probability-guaranteed H∞ finite-horizon filtering problem. Finally, a simulation example is given to illustrate the effectiveness of the proposed filtering scheme.

Finite-time stabilization by state feedback control for a class of time-varying nonlinear systems

In this paper, finite-time stabilization is considered for a class of nonlinear systems dominated by a lower-triangular model with a time-varying gain. Based on the finite-time Lyapunov stability theorem and dynamic gain control design approach, state feedback finite-time stabilization controllers are proposed with gains being tuned online by two dynamic equations. Different from many existing finite-time control designs for lower-triangular nonlinear systems, the celebrated backstepping method is not utilized here. It is observed that our design procedure is much simpler, and the resulting control gains are in general not as high as those provided by the backstepping method. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.

Variance-Constrained ${\cal H}_{\infty}$ Filtering for a Class of Nonlinear Time-Varying Systems With Multiple Missing Measurements: The Finite-Horizon Case

This paper is concerned with the robust H ∞ finite-horizon filtering problem for a class of uncertain nonlinear discrete time-varying stochastic systems with multiple missing measurements and error variance constraints. All the system parameters are time-varying and the uncertainty enters into the state matrix. The measurement missing phenomenon occurs in a random way, and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in the interval . The stochastic nonlinearities under consideration here are described by statistical means which can cover several classes of well-studied nonlinearities. Sufficient conditions are derived for a finite-horizon filter to satisfy both the estimation error variance constraints and the prescribed H ∞ performance requirement. These conditions are expressed in terms of the feasibility of a series of recursive linear matrix inequalities (RLMIs). Simulation results demonstrate the effectiveness of the developed filter design scheme.

An all-coefficient adaptive control method for a class of nonlinear time-varying systems

In this paper, the extension of the all-coefficient adaptive control method to nonlinear time-varying systems is studied. A novel discretizing method is first proposed to derive the discrete-time model for a class of nonlinear time-varying systems. The characteristics of the coefficients of the discrete-time model are derived by this method, based on which the all-coefficient adaptive control method is given for the class of nonlinear time-varying systems. Sufficient conditions on the closed-loop stability are given. Simulations show the purposed adaptive control method can achieve satisfying performance. Finally further discussion on applying low-order all-coefficient adaptive control method to high order nonlinear time-varying system is given.

Design of Variable Structure Control System With Nonlinear Time-Varying Sliding Sector

A variable structure control (VSC) algorithm is proposed with a nonlinear time-varying (NTV) sliding sector for a class of nonlinear time-variant systems represented in a state-dependent linear time-variant form. The NTV sliding sector, a subset of the state space is designed by the state-dependent differential Riccati equation (SDDRE). A VSC law is designed such that the system enters the sliding sector in finite time and a Lyapunov function candidate decreases inside the NTV sliding sector. An NTV control and a sliding mode control algorithms are proposed based on the solution of the SDDRE as well.

Time-varying observers for a class of nonlinear systems

For a class of time-varying nonlinear systems, including those whose dynamics have triangular structure, the global observer design problem is investigated. First, a general result is presented providing sufficient conditions for the existence of both causal and noncausal Luenberger time-varying observers for a class of time-varying nonlinear systems with no inputs. This result together with a forwarding induction procedure is used to derive sufficient conditions for the solvability of the observer design problem for a wide class of triangular time-varying systems. The corresponding methodology extends some ideas of earlier author’s works on same subject.

PD-TYPE ON-LINE LEARNING CONTROL FOR SYSTEMS WITH STATE UNCERTAINTIES AND MEASUREMENT DISTURBANCES

In this paper, a PD-type on-line learning control (OLC) is proposed for the tracking problem in a class of nonlinear time-varying systems with state uncertainties and measurement disturbances. In this proposed OLC algorithm, we use a combination of PD feedback control and feedforward control that is based on the previous control input profiles in an iterative manner. The proposed control scheme has a simple and straightforward structure that leads to its easy implementation. The uniform boundary of the tracking error is established based on the convergence analysis. It is demonstrated that the proposed control algorithm is robust in coping with the state uncertainties and measurement disturbances that result from the system modeling inaccuracies and the initial state errors. Further, one unique feature of the proposed PD-type on-line learning control is that the controller allows a wide range of control gains to be selected to improve its convergence rate. Simulation studies demonstrate the effectiveness of the proposed control algorithm.

Synthesis of switching controllers: A fuzzy supervisor approach

In this paper the synthesis of multiple controllers for a class of time-varying nonlinear systems is considered. The proposed hybrid control scheme is based on a fuzzy supervisor which manages the combination of controllers of two types: with sliding mode control (SMC) and H ∞ control. A convex formulation of the two controllers leads to a structure which benefits from the advantages of both controllers to (i) ensure a good tracking performance in both the transient state (SMC) and the steady state ( H ∞ ), (ii) provide a fast dynamic response to enlarge the stability limits of the system, and (iii) efficiently reduce the chattering phenomena induced by the SMC. The stability analysis uses the Lyapunov technique, inspired from switching system theory, to prove that the system with the proposed controller remains globally stable despite the configuration changing. Some simulation results and conclusions end the paper.

A new control scheme for nonlinear systems with disturbances

A new learning control scheme, based on a nonlinear disturbance observer (NDO) coupled with a sliding-mode fuzzy neural network (SFNN) with a feedback-error-learning (FEL) strategy, is proposed for a class of time-varying nonlinear systems with unknown disturbances. The proposed controller, referred to as NDOFEL, involves two steps for obtaining an estimate of the time-varying lumped disturbance d(t) for improving the precision of the tracking control. The NDO is initially applied to estimate d(t), but an observer error does not converge to zero since d/spl dot/(t)/spl ne/0. The SFNN is then presented to estimate the observer error such that the output of systems follows a desired trajectory. The proposed NDOFEL has stable on-line learning ability, maintains high control performance in the presence of disturbance, and guarantees the stability of closed-loop systems on the basis of the Lyapunov theorem. The effectiveness and robustness of the proposed NDOFEL is demonstrated through simulation results obtained for the tracking control during wing rock phenomena. The results suggest that the proposed controller can significantly enhance the tracking performance of aircraft.

Global stabilization for linear continuous time-varying systems

In this paper, stabilization problem via static output feedback controls for linear time-varying systems is investigated. Based on the Lyapunov function techniques, we show that for linear time-varying systems the global null-controllability guarantees the output feedback stabilization. The verifiable stabilizability conditions and output feedback control design are stated. The result can be applicable to the output feedback stabilizability of a class of nonlinear time-varying systems. Numerical examples illustrated the conditions are given.

Further remarks on strict input-to-state stable Lyapunov functions for time-varying systems

We study the stability properties of a class of time-varying non-linear systems. We assume that non-strict input-to-state stable (ISS) Lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given Lyapunov functions which guarantee the existence of strict ISS Lyapunov functions for our systems. Next, we provide simple direct constructions of explicit strict ISS Lyapunov functions for our systems by applying an integral smoothing method. We illustrate our constructions using a tracking problem for a rotating rigid body.

Enhanced iterative learning using wavelet transform for a class of nonlinear systems

A novel wavelet-based iterative learning control (WILC) algorithm is proposed for a class of nonlinear time-varying systems performing repetitive task with initial state error, input disturbance and output measurement noise. This novel learning control algorithm include three main terms: a feedforward high order P-type learning term (P-type ILC) is updated by more than one past control and error data in the previous trials, a feedback control term with current cycle error (CCR), and the wavelet transform term of the previous cycle error (PCR). The effectiveness of the proposed WILC is validated through numerical simulations. It is shown that this novel WILC algorithm can improve the learning rate significantly.

Feedback linearisation of time-varying nonlinear systems via time-varying diffeomorphism

A control scheme is proposed to stabilise a class of time-varying nonlinear systems. The scheme is based on the feedback-linearisation scheme and is developed via the newly proposed time-varying diffeomorphism concept, which results in the transformation of time-varying systems into equivalent time-invariant linear systems. As a result, feedback linearisation of time-varying nonlinear systems is achieved and it provides the stabilisation of a class of nonlinear time-varying systems. In addition, the approximate feedback LTIsation concept is introduced for its practicability.

Observer-based iterative learning control for a class of time-varying nonlinear systems

In this brief, we propose an observer-based iterative learning control (ILC) scheme for the tracking problem of a class of time-varying nonlinear systems. First, a state observer is derived for the system under consideration, and sufficient conditions for the boundedness and the convergence to zero of the estimation error are given. Thereafter, an iterative learning rule - based on the proposed state observer - ensuring the boundedness of the tracking error is derived. Moreover, it is shown that if the initial state variables are known, it is possible to obtain a perfect convergence to zero, over a finite tracking horizon, when the number of iterations tends to infinity. By associating a state observer with the ILC scheme it is possible to avoid the use of state and output time-derivative measurements which are generally necessary in contraction mapping based ILC design for nonlinear systems without zero relative degree.

On persistently exciting observers and a non-linear separation principle: Application to the stabilization of a generator

We provide a separation principle for a class of non-linear time-varying systems. The only assumption we make on the state feedback control law is that it does not vanish as the states grow unboundedly. For the observer, we propose a design method based on the assumption that the system is transformable into a form affine in the unmeasured variable and a property of persistency of excitation. This assumption covers but is not restricted to the common observability condition that the vector field multiplying the unmeasured variable is separated from zero for all state values. Our separation principle relies on stability results for cascades systems.