Solving Large Dynamical Systems by Constraint Sampling

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

Most dynamical systems in real-life applications are usually nonlinear and nonstationary, such as biomechanical systems, neuroscience, healthcare, economics, and engineering. Nowadays, the availability of data makes it possible to investigate and discover the hidden dynamics of target systems. However, numerous concerns arise in modeling and analysis of such systems, including complex system structure, data-inherent uncertainty in observations, poor interpretability, etc. To this end, we will investigate dynamical behaviors by proposing data-driven machine learning and dynamical system techniques. Specifically, we present various approaches for extracting characteristics or patterns that have good interpretability in various complex dynamical system applications.In the first study, we consider a pattern recognition problem in biomechanical system using multivariate human kinematic data and propose data-driven analytical methods to model human movement. We first perform recurrence quantification analysis on the phase space of the kinematic system to visualize and evaluate the recurrence behavior of the folded phase space trajectory. However, this method could miss out the relationship between variables. We therefore introduce a new modeling approach based on the self-expressive assumption to discover the interactions between variables and characterize the patterns of human motion. Our model measures correlations between variables by underlying linear subspaces, which can be further extended to nonlinear subspaces. Based on these correlations, our model improves the classification accuracy by 20% - 30% over other standard state-of-art correlation measurements and clustering methods. In the second study, we consider a specific neuroscience problem, seizure detection, to recognize abnormal brain states by learning from highly nonlinear electroencephalography. Based on dynamical systems theory, Koopman operator theory, and dynamic mode decomposition, we project the complex brain dynamics into an infinite-dimensional linear space, where the intrinsic linear dynamic shares the same dynamical behavior as the original nonlinear dynamical system. We propose a new system measurement to unify the projections in different Koopman spaces. Our measurement indicates the stability of the dynamical system, which represents the system state trajectory with respect to the corresponding attractor. We demonstrate the effectiveness of our approach with a seizure onset detection task. In this case study, our model gives high accuracy with a low false-positive rate. In the last study, we consider dynamical systems with multiple modalities. We design a Koopman-based deep learning network to identify and simplify the intrinsic nonlinear dynamics from different sources/observations of a multi-modal dynamical system. Since these different modalities describe the same system, they jointly share the same intrinsic dynamics. We develop an autoencoder framework with custom loss functions to find proper project functions/operators and to obtain the intrinsic Koopman coordinates. Our framework can reconstruct the original system states based on the joint dynamics. Once the intrinsic linear dynamics is learned, future system states can be predicted by the evolution in the Koopman space. This approach allows us to uncover and linearize the hidden nonlinear dynamics, which enables linear dynamical tools, for example, system control, spectrum analysis, to analyze complex dynamical systems.--Author's abstract

Steven L. Brunton, Marko Budišić, Eurika Kaiser, and J. Nathan Kutz are the authors of the Survey and Review paper in this issue, “Modern Koopman Theory for Dynamical Systems.” Koopman theory is a valuable formalism for the study of dynamical systems; it has gained popularity in recent years in connection with data-driven analysis, control theory, and other areas. The basic idea is simple. If $(d/dt) {x} = {f}({x})$ is a dynamical system in ${\mathbb R}^n$, its solution flow is the one-parameter family of maps ${F}^t: {\mathbb R}^n\rightarrow {\mathbb R}^n$ such that $t\mapsto {F}^t({x}_0)$ is the solution that at time $t=0$ takes the value ${x}_0$. In an alternative description, rather than looking at points in ${\mathbb R}^n$ being moved by the flow, one may consider real-valued functions $g({x})$ being transformed by the dynamical system as $g \mapsto {\cal K}^t(g)$, where the Koopman (or composition) operator ${\cal K}^t$ is defined by ${\cal K}(g)({x}) = g({F}^t({x}))$. Thus the Koopman operator acts on an infinite-dimensional space of functions $g$ (bad news), but it is linear, even if the original dynamical system is not (good news). The idea behind the operator ${\cal K}^t$ is particularly useful when the dynamical system is being studied by big data$/$machine learning techniques in cases where $f$ and ${F}^t$ are unknown but measurements $g_i({x}(t_j))$ along a solution ${x}(t)$ are available. While the use of composition operators is very much older, B. O. Koopman noted in a seminal 1931 paper that, in the particular case of conservative dynamics, the operator ${\cal K}^t$ will be unitary in a suitable $L^2$ space, an observation that allowed him to apply to classical mechanics the theory of Hilbert space operators being developed at the time to formulate mathematically quantum mechanics. Koopman's results have been extended in many directions to dissipative or conservative systems, in continuous or discrete time. The survey in this issue does not assume a previous knowledge of Koopman theory and reviews recent developments on Koopman operators, with particular emphasis on the dynamic mode decomposition (DMD) algorithm and its variants used in many real-life applications. The paper contains almost five hundred references, shows applications to fluid mechanics, epidemiology, neuroscience, plasma physics, finance, robotics, the power grid, and other fields, and discusses connections with many areas, including numerical linear algebra, control theory, statistics, model reduction, and uncertainty qualification. I believe it will be of interest to a wide range of readers.

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure or displays chaotic dynamics. In particular, we prove that the differential equations for the expansion coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. Additionally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

A high order multiplication perturbation method for linear time-varying dynamic system is presented. Firstly the time domain is dispersed with small time interval and the coefficient matrix of the dynamic system is decomposed into a “large amount” and a “small amount” in each time interval. One order perturbation system is then presented from the original dynamic system by a variable transformation which is also a linear time-varying dynamic system. The coefficient matrix of the new system is also decomposed into a “large amount” and a “small amount” and then higher order perturbation system can be obtained by another variable transformation. The final perturbation system can be solved exactly after abandoning the “small amount” of the coefficient matrix in the final system, and then the answer to the original problem can be determined through a series of inverse transform. Since the transfer matrix is the product of a series of exponential matrix which can be calculated accurately by the precise time integration method, so the proposed method has fine accuracy, efficiency and stability. The proposed method actually is a high order symplectic conservative perturbation method for the Hamiltonian system. The examples show that the proposed method can also give good results even though a large time step selected, and with the increase of the perturbation number, the perturbation solutions can tend to exact solutions rapidly.

We present a multifidelity uncertainty quantification numerical method that leverages the benefits of both sampling and surrogate modeling, while mitigating their downsides, for enabling rapid computation in complex dynamical systems such as automotive propulsion systems. In particular, the proposed method utilizes intrusive generalized polynomial chaos to quickly generate additional information that is highly correlated with the original nonlinear dynamical system. We then leverage a Monte Carlo-based control variate to correct the bias caused by the surrogate approximation. In contrast to related works merging adaptive surrogate approximation and sampling in a multifidelity setting, the intrusive generalized polynomial chaos (gPC) surrogate is selected because it avoids statistical errors by design by providing analytical estimates of output statistics. Moreover, it enables theoretical contributions that provide an estimator design strategy that optimally balances the computational efforts allocated to sampling and to gPC construction. We deploy our approach to multiple numerical examples including simulations of hybrid-electric propulsion systems, where the proposed estimator is shown to achieve orders-of-magnitude reduction in mean squared error of statistics estimation under comparable costs of purely sampling or purely surrogate approaches.

Many applications concerning physical and technical processes employ dynamical systems for simulation purposes. The increasing demand for a more accurate and detailed description of realistic phenomena leads to high dimensional dynamical systems and hence, simulation often yields an increased computational effort. An approximation, e.g. with model order reduction techniques, of these large-scale systems becomes therefore crucial for a cost efficient simulation. This paper focuses on a model order reduction method for linear time in-variant (LTI) systems based on modal approximation via dominant poles. There, the original large-scale LTI system is projected onto the left and right eigenspaces corresponding to a specific subset of the eigenvalues of the system matrices, namely the dominant poles of the system's transfer function. Since these dominant poles can lie anywhere in the spectrum, specialized eigenvalue algorithms that can compute eigentriplets of large and sparse matrices are required. The Jacobi-Davidson method has proven to be a suitable and competitive candidate for the solution of various eigenvalue problems and hence, we discuss how it can be incorporated into this modal truncation approach. Generalizations of the reduction technique and the application of the algorithms to second-order systems are also investigated. The computed reduced order models obtained with this modal approximation can be combined with the ones constructed with Krylov subspace or balanced truncation based model order reduction methods to get even higher accuracies.

We propose a novel approach to discovering latent structures from multimodal time series. We view a time series as observed data from an underlying dynamical system. In this way, analyzing multimodal time series can be viewed as finding latent structures from dynamical systems. In light this, our approach is based on the concept of generating partition which is the theoretically best symbolization of time series maximizing the information of the underlying original continuous dynamical system. However, generating partition is difficult to achieve for time series without explicit dynamical equations. Different from most previous approaches that attempt to approximate generating partition through various deterministic symbolization processes, our algorithm maintains and estimates a probabilistic distribution over a symbol set for each data point in a time series. To do so, we develop a Bayesian framework for probabilistic symbolization and demonstrate that the approach can be successfully applied to both simulated data and empirical data from multimodal agent-agent interactions. We suggest this unsupervised learning algorithm has a potential to be used in various multimodal datasets as first steps to identify underlying structures between temporal variables.

An identification method of nonlinear dynamic system is studied in this paper. First, suppose that the original nonlinear dynamic system is described by Hammerstein model. Then, an intermediate model is generated. Next, Parameters of the intermediate model are obtained using a Bacterial Chemotaxis Optimization (BCO) approach. Finally, through the relationships of the parameters of intermediate model and those of Hammerstein model, we derive the parameters of the system. Consequently, the original nonlinear dynamic system is identified. The feasibility and efficiency of the presented algorithm are demonstrated using numerical simulations.

Considering the problem for inverse system method in EMU AC induction traction motor linear decoupling, the zero dynamics subsystem will be separated from the original dynamic system through coordinate transformation. Firstly, a getting method for zero dynamics of the multiple input multiple output nonlinear system is discussed when γ＜ n. Second, the zero dynamics analysis for five order nonlinear model of asynchronous traction motor which base on the stationary coordinate system is given by using inverse decoupling method. The analysis results show that if the stability of the zero dynamics can be ensured, then the entire linearization of original nonlinear system is not necessary, need only partial linearization which effect on the external dynamic portion. The inverse decoupling process of asynchronous traction motor can be simplified by this conclusion. DOI : http://dx.doi.org/10.11591/telkomnika.v12i1.3354

This paper presents a dynamic model of the equivalent Great Britain (GB) 36-zone power system, which can be used for reliable and realistic assessment of emerging load frequency control mechanisms. Flexible architecture of the presented dynamic test system permits a broad range of security of supply and small-signal stability studies for design of future power grids. It can be particularly useful for academic research, but also for undertaking feasibility studies in power industries. The proposed dynamic test system, which is obtained through network reduction of the original full-scale GB transmission power system developed by National Grid Electricity System Operator (NGESO) Company, provides detailed information about the GB power system. In this regard, the required data and modelling approaches to develop the 36-zone system are provided in detail. The presented dynamic test system represents the system topology, impedance characteristics and electromechanical oscillations of the original GB power system however, it is not an exact equivalent of the master GB system. Illustrative dynamic models of the key system components, including synchronous generators, automatic voltage regulators, power system stabilizers, hydro and steam turbines models along with speed governing systems are presented. Dynamic behavior of 36-zone test system in response to infeed loss contingencies is investigated. Particularly, the impact of changes in the system inertia on the system electromechanical modes is examined using the modal analysis approach. In this context, the mode shape concept is employed to determine dominant generators and contribution of different zones in the low frequency oscillations. Moreover, time-domain simulations are undertaken to validate the modal analysis results. Additionally, the condition of different zones from the viewpoint of frequency nadir and maximum rate of change of frequency for various contingencies and extreme cases are examined.

In this exploratory study, we present a new method of approximating a large system of ODEs by one with fewer equations, while attempting to preserve the essential dynamics of a reduced set of variables of interest. The method has the following key elements: (i) put a (simple, ad-hoc) probability distribution on the phase space of the ODE; (ii) assert that a small set of replacement variables are to be unknown linear combinations of the not-of-interest variables, and let the variables of the reduced system consist of the variables-of-interest together with the replacement variables; (iii) find the linear combinations that minimize the difference between the dynamics of the original system and the reduced system. We describe this approach in detail for linear systems of ODEs. Numerical techniques and issues for carrying out the required minimization are presented. Examples of systems of linear ODEs and variable-coefficient linear PDEs are used to demonstrate the method. We show that the resulting approximate reduced system of ODEs gives good approximations to the original system. Finally, some directions for further work are outlined.

formula omitted] model reduction for uncertain switched linear discrete-time systems

To meet the challenge of handling dynamic systems in which continuous dynamics and discrete events coexist, we develop stability analysis of such systems governed by ordinary differential equations and stochastic differential equations with regime switching modulated by a Markov chain involving a small parameter. The small parameter is used to reflect either the in- herent two-time scales in the original system or the different rates of regime switching among a large number of states of the discrete events. The smaller the parameter is, the more rapid switching the system will experience. To reduce the complexity, one attempts to effectively replace the actual systems by a limit system with simpler structure. To ensure the validity of such a replacement in a longtime horizon, it is crucial that the original system is stable. The fast regime changes and the large state space of the Markov chain make the stability analysis difficult. Under suitable conditions, using the limit dynamic systems and the perturbed Lyapunov func- tion methods, we show that if the limit systems are stable, then so are the original systems. This justifies the replacement of a complicated original system by its limit from a longtime behavior point of view. Index Terms—Hybrid system, Lyapunov method, Markov chain, singular perturbation, stability, switching diffusion.