Model reduction of discrete-time switched linear systems over finite-frequency ranges

Nonlinear Dynamics - This paper is concerned with the model reduction problem of a class of discrete-time switched linear systems over finite-frequency ranges. By the aid of generalized...     

Identification of discrete-time bilinear systems through equivalent linear models

This paper presents a novel method for identification of discrete-time, time-invariant state-space models of bilinear dynamical systems using the steady-state portion of a single input/multiple output time-history measurements. These measurements are recorded by exciting the system with a linear combination of sine and cosine functions of user-selected frequencies enriched by a subtle amount of random component. The proposed method relies on conversion of the bilinear system into an equivalent linear model (ELM) by an accurate approximation of the state in the bilinear term using a set of sine and cosine basis functions whose frequencies are obtained as combinations of the input frequencies. Observer/Kalman Filter Identification (OKID), a?linear time invariant (LTI) system identification algorithm, is used to identify the aforementioned ELM from which the original bilinear model is recovered. A?numerical example is also provided.     

Analysis and control of a class of uncertain linear periodic discrete-time systems

Feedback control problems for linear periodic systems （LPSs） with interval- type parameter uncertainties are studied in the discrete-time domain. First, the stability analysis and stabilization problems are addressed. Conditions based on the linear matrices inequality （LMI） for the asymptotical stability and state feedback stabilization, respec-tively, are given. Problems of L2-gain analysis and control synthesis are studied. For the L2-gain analysis problem, we obtain an LMI-based condition such that the autonomous uncertain LPS is asymptotically stable and has an L2-gain smaller than a positive scalar γ. For the control synthesis problem, we derive an LMI-based condition to build a state feedback controller ensuring that the closed-loop system is asymptotically stable and has an L2-gain smaller than the positive scalar γ. All the conditions are necessary and sufficient.     

Control of fractional periodic discrete-time linear systems by parametric state feedback matrices

Nonlinear Dynamics - In this article stabilization of fractional-order periodic discrete-time linear system by a complete nonlinear parametric approach for pole assignment with respect to stability...     

First order piecewise linear systems with random parametric excitation

The Fokker-Planck equation is used to develop a general method for finding the spectral density and other properties of first order systems governed by stochastic differential equations of the form

dx/dt + f(x)[1 + m(t)] = n(t)

, where f(x) is piecewise linear and m(t) and n(t) represent stationary Gaussian white noise. The method is similar to one used by the authors to deal with the case m(t) = 0, but is complicated by the possible existence of irregular (singular) points of the Fokker-Planck equation. Graphical results for some special cases are presented.     

Design and analysis of a first order time-delayed chaotic system

The present paper reports the design and analysis of a new time-delayed chaotic system and its electronic circuit implementation. The system is described by a first-order nonlinear retarded type delay differential equation with a closed form mathematical function describing the nonlinearity. We carry out stability and bifurcation analysis to show that with the suitable delay and system parameters the system shows sustained oscillation through supercritical Hopf bifurcation. It is shown through numerical simulations that the system depicts bifurcation and chaos for a certain range of the system parameters. The complexity and predictability of the system are characterized by Lyapunov exponents and Kaplan?CYork dimension. It is shown that, for some suitably chosen system parameters, the system shows hyperchaos even for a small or moderate delay. Finally, we set up an experiment to implement the proposed system in electronic circuit using off-the-shelf circuit elements, and it is shown that the behavior of the time delay chaotic electronic circuit agrees well with our analytical and numerical results.     

Model order reduction for dynamical systems: A geometric approach

The aim of this paper is to ask the question as whether it is possible, for a given dynamical system defined by a vector field over a finite dimensional inner product space, to construct a reduced-order model over a finite dimensional manifold. In order to give a positive answer to this question, we prove that if the manifold under consideration is an immersed submanifold of the vector space, considered as ambient manifold, then it is possible to construct explicitly a reduced-order vector field over this submanifold. In particular, we found that the reduced-order vector field satisfies the variational principle of Dirac–Frenkel and that we can formulate the Proper Orthogonal Decomposition under this framework. Finally, we propose a local-point estimator of the time-dependent error between the original vector field and the reduced-order one.     

Energy determines multiple stability in time-delayed systems

Nonlinear Dynamics - Infinite dimensions always challenge the analysis of multiple stability in nonlinear time-delayed systems, as the computation and visualization of conventional basin of...     

Control of linear instabilities by dynamically consistent order reduction on optimally time-dependent modes

Nonlinear Dynamics - Identification and control of transient instabilities in high-dimensional dynamical systems remain a challenge because transient (non-normal) growth cannot be accurately...     

Order reduction of structural dynamic systems with static piecewise linear nonlinearities

A technique for order reduction of dynamic systems in structural form with static piecewise linear nonlinearities is presented. By utilizing two methods which approximate the nonlinear normal mode (NNM) frequencies and mode shapes, reduced-order models are constructed which more accurately represent the dynamics of the full model than do reduced models obtained via standard linear transformations. One method builds a reduced-order model which is dependent on the amplitude (initial conditions) while the other method results in an amplitude-independent reduced model. The two techniques are first applied to reduce two-degree-of-freedom undamped systems with clearance, deadzone, bang-bang, and saturation stiffness nonlinearities to single-mode reduced models which are compared by direct numerical simulation with the full models. It is then shown via a damped four-degree-of-freedom system with two deadzone nonlinearities that one of the proposed techniques allows for reduction to multi-mode reduced models and can accommodate multiple nonsmooth static nonlinearities with several surfaces of discontinuity. The advantages of the proposed methods include obtaining a reduced-order model which is signal-independent (doesn’t require direct integration of the full model), uses a subset of the original physical coordinates, retains the form of the nonsmooth nonlinearities, and closely tracks the actual NNMs of the full model.     

POD-identification reduced order model of linear transport equations for control purposes

Intrusive reduced order modeling techniques require access to the solver's discretization and solution algorithm, which are not available for most computational fluid dynamics codes. Therefore, a nonintrusive reduction method that identifies the system matrix of linear fluid dynamical problems with a least-squares technique is presented. The methodology is applied to the linear scalar transport convection-diffusion equation for a 2D square cavity problem with a heated lid. The (time-dependent) boundary conditions are enforced in the obtained reduced order model (ROM) with a penalty method. The results are compared and the accuracy of the ROMs is assessed against the full order solutions and it is shown that the ROM can be used for sensitivity analysis by controlling the nonhomogeneous Dirichlet boundary conditions.     

Generalized heterochronous synchronization in coupled time-delayed chaotic systems

A new kind of generalized heterochronous synchronization phenomenon is reported. Different kinds of generalized synchronous states (including generalized anticipated, isochronous and lag projective synchronization) coexist among different state variables between two unidirectionally coupled time-delayed chaotic systems. The analytical conditions for generalized heterochronous synchronization are obtained. We also find that the synchronization conditions are independent of the delay times in the original time-delayed system. The theoretical results are well confirmed by the numerical simulations and electronic circuit experiments.     

Nonlinear active observer-based generalized synchronization in time-delayed systems

When two different chaotic oscillators are coupled, generalized synchronization can occur. It may imply a very complicated relation between the states of drive and response systems. We propose a method that can be used to detect and characterize the generalized synchronization in modulated time-delayed systems. Using Krasovskii–Lyapunov theory, sufficient condition for generalized synchronization is derived. The proposed technique has been applied to synchronize prototype and Ikeda models by numerical simulation.     

The synchronization between two discrete-time chaotic systems using active robust model predictive control

In this paper, we consider the synchronization of two discrete-time chaotic systems. A novel active robust model predictive strategy is proposed to guarantee the synchronization of two discrete-time systems in the presence of model uncertainty. The proposed approach reduces the synchronization to a convex optimization involving linear matrix inequalities. The numerical simulations illustrate the effectiveness of the proposed method.     

Fast data-driven model reduction for nonlinear dynamical systems

We present a fast method for nonlinear data-driven model reduction of dynamical systems onto their slowest nonresonant spectral submanifolds (SSMs). While the recently proposed reduced-order modeling method SSMLearn uses implicit optimization to fit a spectral submanifold to data and reduce the dynamics to a normal form, here, we reformulate these tasks as explicit problems under certain simplifying assumptions. In addition, we provide a novel method for timelag selection when delay-embedding signals from multimodal systems. We show that our alternative approach to data-driven SSM construction yields accurate and sparse rigorous models for essentially nonlinear (or non-linearizable) dynamics on both numerical and experimental datasets. Aside from a major reduction in complexity, our new method allows an increase in the training data dimensionality by several orders of magnitude. This promises to extend data-driven, SSM-based modeling to problems with hundreds of thousands of degrees of freedom.

     

Invariant spectral foliations with applications to model order reduction and synthesis

Nonlinear Dynamics - The paper introduces a technique that decomposes the dynamics of a nonlinear system about an equilibrium into low-order components, which then can be used to reconstruct the...     

Periodic solutions of linear degenerate systems of ordinary differential equations of the second order

We propose sufficient conditions for the existence of a periodic solution of a system of linear ordinary differential equations of the second order with a degenerate symmetric matrix in the coefficient of the second-order derivative in the case of an arbitrary periodic inhomogeneity.     

Output feedback controller for hysteretic time-delayed MIMO nonlinear systems

In this paper, an H ^?? output feedback controller is developed for a class of time-delayed MIMO nonlinear systems, containing backlash as an input nonlinearity. Particularly, a state observer is proposed to estimate unmeasurable states. The control law can be divided into two elements: An adaptive interval type-2 fuzzy part which approximates the uncertain model. The second part is an H ^??-based controller, which attenuates the effects of external disturbances and approximation errors to a prescribed level. Furthermore, the Lyapunov theorem is used to prove stability of proposed controller and its robustness to external disturbance, hysteresis input nonlinearity, and time varying time-delay. As an example, the designed controller is applied to address the tracking problem of 2-DOF robotic manipulator. Simulation results not only verify the robust properties but also in comparison with an existing method reveal the ability of the proposed controller to exclude the effects of unknown time varying time-delays and hysteresis input nonlinearity.     

A new model reduction method for nonlinear dynamical systems

The present research work proposes a new systematic approach to the problem of model-reduction for nonlinear dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear invariance partial differential equations (PDEs), and a rather general explicit set of conditions for solvability is derived. In particular, within the class of analytic solutions, the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution to the above system of singular PDEs is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration exponentially attracting all dynamic trajectories. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow manifold enables the development of a series solution method, which allows the polynomial approximation of the system dynamics on the slow manifold up to the desired degree of accuracy and can be easily implemented with the aid of a symbolic software package such as MAPLE. Finally, the proposed approach and method is evaluated through an illustrative biological reactor example.