Studies in the approximate control of flexible structures

An identification approach for linear and nonlinear time-variant structural systems via harmonic wavelets

Periodic Motions for Estimation of the Attraction Domain in the Wheeled Robot Stabilization Problem

Chang transformation was introduced for decoupling the slow and fast dynamics of a singularly perturbed Linear Time Invariant (LTI) system, and it was subsequently extended to Linear Time-Varying (LTV) systems under the slowly-varying assumption by way of frozen-time eigenvalues. This paper extends Chang transformation from slowly-varying LTV systems to LTV systems, when the singularly perturbed system has a semi-proper coefficient matrix A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">22</sub> (t) and LTV system ẇ(t) = A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">22</sub> (t)w(t) is exponentially stable. Instead of using frozen-time eigenvalues based on slowly time-varying conditions, PD-eigenvalues for LTV systems are employed to characterize the exponential stability, thereby circumventing the slowly-varying constraint. Our results provide a larger bound on epsilon, which is a gauge on the validity of the Chang transformation, in some situations than using previous techniques. We have also recast the original Chang transformation so as to provide more insight into the decoupled subsystems. The insight will be useful in subsequent investigation on estimate of the Singular Perturbation Margin (SPM) for LTV systems. An equivalence relationship has recently been established for LTI system between the SPM and the Phase Margin (PM). The new results in this paper will facilitate the development of a PM-type stability margin metric for LTV systems, and for nonlinear, time-varying systems in the future.

In this paper, adaptive optimal control is proposed for linear discrete time-varying (LDTV) systems subject to unknown system dynamics. The idea of the method is a direct application of the Q-learning adaptive dynamic programming for time-varying systems. In order to derive the optimal control policy, an actor-critic structure is constructed and the time-varying least square method is adopted for parameter adaptation. The derived control policy robustly stabilizes the time-varying system and guarantees an optimal control performance. As no particular system information is required throughout the process, the proposed method provides a feasible solution to a large variety of applications. The validity of the proposed method is verified through simulation studies.

In this paper a new systematic switching control approach to adaptive stabilization of linear time-varying (LTV) discrete-time systems is presented.A feature of the localization-based method is its high model falsification capability, which in the case of LTI systems is manifested as the rapid convergence of the switching controller.We believe that the proposed method may help pave the way for design of practical adaptive switching controllers applicable to a wide range of linear time-invariant and timevarying systems.

This paper studies data-driven learning-based methods for the finite-horizon optimal control of linear time-varying discrete-time systems. First, a novel finite-horizon Policy Iteration (PI) method for linear time-varying discrete-time systems is presented. Its connections with existing infinite-horizon PI methods are discussed. Then, both data-driven off-policy PI and Value Iteration (VI) algorithms are derived to find approximate optimal controllers when the system dynamics is completely unknown. Under mild conditions, the proposed data-driven off-policy algorithms converge to the optimal solution. Finally, the effectiveness and feasibility of the developed methods are validated by a practical example of spacecraft attitude control.

In the paper we consider time-varying systems with coefficients depending mero-morphically on time. In differential operator representations these systems are described by matrices over a skew polynomial ring with coefficients in the field of real meromorphic functions. Different kinds of indices (controllability, minimal, geometric and dynamical) are introduced and it is proved that they essentially coincide. The input module and the formal transfer matrix are defined and used for an algebraic description of time-varying systems. A characterization of system equivalence is given in these terms and also a complete list of invariants of similarity for time-varying state-space systems.

Global stabilization for linear continuous time-varying systems

A physical parameter identification method for the linear time-varying system is presented in this paper. For an arbitrarily time-varying system, a series of Hankel matrices are established directly from the combined measurements of displacement, velocity, and acceleration together with the input excitations from a single experiment with the system. The time-varying equivalent state-space system matrices of the structure at each time instance are then identified by singular value decomposition using the free or random responses. A time-varying transformation matrix for transforming the identified equivalent system matrices into the physical system matrices is developed to determine the mass, stiffness, and damping matrices. Details of the proposed approach are investigated with a two degrees-of-freedom spring-mass-damper system with three kinds of time-varying parameters. Numerical results show that the proposed method is accurate and effective in identifying time-varying physical parameters.

The paper deals with the construction of generalized frozen-time (FT) models-for linear time-varying continuous-time systems. The generalized FT models are higher-order models generated from a given n-dimensional state model by applying a stable differential operator to the state equation. This results in FT models whose coefficients depend on the derivatives of the time-varying system coefficient matrices. The accuracy of the higher-order FT models over local time intervals is investigated in terms of truncated Taylor series. In the last part of the paper, the application of the higher-order FT models to state feedback control is considered.

It is well known that nonlinear systems, as well as linear time-varying systems, are characterized by non-stationary response signals. In this sense, they both show natural frequencies that are not constant over time; this variation has however different origins: for a time-varying system the mass, and possibly the stiffness distributions, are changing over time, while for a nonlinear system the natural frequencies are amplitude-dependent. An interesting case of time-varying system occurs when analyzing the transit of a train over a railway bridge, easily simulated by the crossing of a moving load over a beam. In this case, the presence of a nonlinearity in the beam behaviour can cause a significant alteration of the modal parameters extracted from the linearized model, such that the contributions of the two effects are no more distinguishable.

ABSTRACTIn this paper, we describe a constructive test to decide whether a given linear time-varying (LTV) differential system admits a stabilising compensator for the control tasks of tracking, disturbance rejection or model matching and construct and parametrise all of them if at least one exists. In analogy to the linear time-invariant (LTI) case, the ring of stable rational functions, noncommutative in the LTV situation, and the Kučera–Youla parametrisation play prominent parts in the theory. We transfer Blumthaler's thesis from the LTI to the LTV case and sharpen, complete and simplify the corresponding results in the book ‘Linear Time-Varying Systems’ by Bourlès and Marinescu.

A method is presented for reducing the sensitivity of time-varying and non-linear control systems. The procedure is an extension of a method proposed recently for controllable linear time-invariant systems.

Structural system identification using Least Mean Square (LMS) adaptive technique

Finite-time stability and stabilization of linear discrete time-varying stochastic systems