Large-scale Dynamical Systems Research Articles

$\mathcal{H}_2$-Optimal Model Reduction Using Projected Nonlinear Least Squares

In many applications throughout science and engineering, model reduction plays an important role replacing expensive large-scale linear dynamical systems by inexpensive reduced order models that capture key features of the original, full order model. One approach to model reduction finds reduced order models that are locally optimal approximations in the $\mathcal{H}_2$ norm, an approach taken by the Iterative Rational Krylov Algorithm (IRKA) among others. Here we introduce a new approach for $\mathcal{H}_2$-optimal model reduction using the projected nonlinear least squares framework previously introduced in [J. M. Hokanson, SIAM J. Sci. Comput. 39 (2017), pp. A3107--A3128]. At each iteration, we project the $\mathcal{H}_2$ optimization problem onto a finite-dimensional subspace yielding a weighted least squares rational approximation problem. Subsequent iterations append this subspace such that the least squares rational approximant asymptotically satisfies the first order necessary conditions of the original, $\mathcal{H}_2$ optimization problem. This enables us to build reduced order models with similar error in the $\mathcal{H}_2$ norm but using far fewer evaluations of the expensive, full order model compared to competing methods. Moreover, our new algorithm only requires access to the transfer function of the full order model, unlike IRKA which requires a state-space representation or TF-IRKA which requires both the transfer function and its derivative. Applying the projected nonlinear least squares framework to the $\mathcal{H}_2$-optimal model reduction problem open new avenues for related model reduction problems.

Minimal Driver Nodes for Structural Controllability of Large-Scale Dynamical Systems: Node Classification

This article considers the problem of minimal control inputs to affect the system states such that the resulting system is structurally controllable. This problem and the dual problem of minimal observability are claimed to have no polynomial-order exact solution and, therefore, are NP-hard. Here, adopting a graph-theoretic approach, this problem is solved for general nonlinear (and also structure-invariant) systems, and a P-order solution is proposed. In this direction, the dynamical system is modeled as a directed graph, called system digraph , and two types of graph components are introduced, which are tightly related to structural controllability. Two types of nodes, which are required to be affected (or driven) by an input, called driver nodes , are defined, and minimal number of these driver nodes are obtained. Polynomial-order complexity of the given algorithms to solve the problem ensures applicability of the solution for analysis of large-scale dynamical systems. The structural results in this article are significant as compared with the existing literature, which offer approximate and computationally less-efficient, e.g. Gramian-based, solutions for the problem, while this article provides an exact solution with the lower computational complexity and applicable for controllability analysis of nonlinear systems.

Trajectory Tracking Control Design for Large-Scale Linear Dynamical Systems With Applications to Soft Robotics

This article presents new results to control process modeled through linear large-scale systems. Numerical methods are widely used to model physical systems, and the finite-element method is one of the most common methods. However, for this method to be precise, it requires a precise spatial mesh of the process. Large-scale dynamical systems arise from this spatial discretization. We propose a methodology to design an observer-based output feedback controller. First, a model reduction step is used to get a system of acceptable dimension. Based on this low-order system, two linear matrix inequality problems provide us, respectively, with the observer and controller gains. In both the cases, model and reduction errors are taken into account in the computations. This provides robustness with respect to the reduction step and guarantees the stability of the original large-scale system. Finally, the proposed method is applied to a physical setup-a soft robotics platform-to show its feasibility.

Distributed Nonlinear Control Design Using Separable Control Contraction Metrics

This paper gives convex conditions for the synthesis of a distributed control system for large-scale networked nonlinear dynamic systems. It is shown that the technique of control contraction metrics can be extended to this problem by utilizing separable metric structures, resulting in controllers that only depend on information from local sensors and communications from immediate neighbors. The conditions given are pointwise linear matrix inequalities, and are necessary and sufficient for linear positive systems and certain monotone nonlinear systems. Distributed synthesis methods for systems on chordal graphs are also proposed based on semidefinite program decompositions. The results are illustrated on a problem of vehicle platooning with heterogeneous vehicles, and a network of nonlinear dynamic systems with over 1000 states that is not feedback linearizable and has an uncontrollable linearization.

On Fundamental Limitations of Dynamic Feedback Control in Regular Large-Scale Networks

In this paper, we study fundamental performance limitations of distributed feedback control in large-scale networked dynamical systems. Specifically, we address the question of whether dynamic feedback controllers perform better than static (memoryless) ones when subject to locality constraints. We consider distributed linear consensus and vehicular formation control problems modeled over toric lattice networks. For the resulting spatially invariant systems, we study the large-scale asymptotics (in network size) of global performance metrics that quantify the level of network coherence. With static feedback from relative state measurements, such metrics are known to scale unfavorably in lattices of low spatial dimensions, preventing, for example, a one-dimensional string of vehicles to move like a rigid object. We show that the same limitations in general apply also to dynamic feedback control that is locally of first order. This means that the addition of one local state to the controller gives a similar asymptotic performance to the memoryless case. This holds unless the controller can access noiseless measurements of its local state with respect to an absolute reference frame, in which case the addition of controller memory may fundamentally improve performance. In simulations of platoons with 20–200 vehicles, we show that the performance limitations we derive manifest as unwanted accordionlike motions. Similar behaviors are to be expected in any network that is embeddable in a low-dimensional toric lattice, and the same fundamental limitations would apply. To derive our results, we present a general technical framework for the analysis of stability and performance of spatially invariant systems in the limit of large networks.

Condensed matter physics, hybrid energy and entropy principles, and the hybrid first and second laws of thermodynamics

In this paper, we develop an energy-based, large-scale hybrid dynamical system model to present a generic framework for hybrid thermodynamics involving hybrid energy and entropy conservation and nonconservation principles. Specifically, using a hybrid compartmental dynamical system energy flow model we develop a hybrid state-space dynamical system formalism for addressing critical phenomena and discontinuous phase transitions in thermodynamics and provide hybrid extensions to the first and second laws of thermodynamics. In addition, using Lyapunov stability theory for impulsive differential equations, we show that our hybrid large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies. Moreover, we show that the steady-state distribution of the hybrid large-scale system energies is uniformly distributed over all of the subsystems, leading to system energy equipartitioning corresponding to a maximum entropy equilibrium state.

A comprehensive scheme for reduction of nonlinear dynamical systems

Model order reduction (MOR) also known as dimension reduction is a computational tool to obtain cost-effective lower order approximations of large scale dynamical systems. This paper presents a detailed yet simplified MOR approach using nonlinear moment matching (NLMM) in conjuncture with the Discrete Empirical Interpolation Method (DEIM). NLMM avoids the expensive simulation of the underlying nonlinear Sylvester partial differential equation by reducing it to a system of nonlinear algebraic equations using proper step-by-step simplifications. This reduces the offline computational cost of generating the orthonormal projection basis substantially. This is followed by the DEIM algorithm, resulting in comprehensive savings in computational resources. The proposed algorithms are tested on two benchmark problems and the results so obtained are compared with proper orthogonal decomposition for different test inputs.

A Tangential Block Lanczos Method for Model Reduction of Large-Scale First and Second Order Dynamical Systems

In this paper, we present a new approach for model reduction of large scale first and second order dynamical systems with multiple inputs and multiple outputs. This approach is based on the projection of the initial problem onto tangential subspaces to produce a simpler reduced-order model that approximates well the behaviour of the original model. We present an algorithm named: adaptive block tangential Lanczos-type algorithm. We give some algebraic properties and present some numerical experiences to show the effectiveness of the proposed algorithms.

Structure-preserving model reduction for dynamical systems with a first integral

Since the expense of the numerical integration of large scale dynamical systems is often computationally prohibitive, model reduction methods, which approximate such systems by simpler and much lower order ones, are often employed to reduce the computational effort. In this paper, for dynamical systems with a first integral, new structure-preserving model reduction approaches are presented that yield reduced-order systems while preserving the first integral. We apply energy-preserving integrators to the reduced-order systems and show some numerical experiments that demonstrate the favourable behaviour of the proposed approaches.

Dynamic Aggregation Modeling of Grid-Connected Inverters Using Hamilton's-Action-Based Coherent Equivalence

The high penetration of renewable energy generation (REG) systems influences the dynamics of power systems. Accurate transient simulations and dynamic models are indispensable for the research on the dynamic behavior of REG systems. A large-scale REG system consisting of numerous grid-connected inverters consumes a large amount of computation time, so a dynamic aggregated model is needed for efficient simulation. The aggregated model established by coherent equivalence fits the detailed model well, and thus it is widely used in the aggregation of large-scale dynamic systems. Presently, the key challenge of the application of the coherent-equivalence method is to identify inverters that have similar dynamic characteristics. The Hamilton principle expressed by Hamilton's action has a simple form and indicates the comprehensive dynamic characteristics of inverters. Thus, this paper proposes a coherent-equivalence method of inverters based on Hamilton's action. The proposed method is applied to a wind farm of 16 permanent magnet synchronous generator-based wind turbines for demonstration. Compared with the existing equivalence method without considering coherency, the proposed method shows advantages in effectiveness and accuracy.

Noise-induced limitations to the scalability of distributed integral control

We study performance limitations of distributed feedback control in large-scale networked dynamical systems. Specifically, we address the question of how the performance of distributed integral control is affected by measurement noise. We consider second-order consensus-like problems modeled over a toric lattice network, and study asymptotic scalings (in network size) of H2 performance metrics that quantify the variance of nodal state fluctuations. While previous studies have shown that distributed integral control fundamentally improves these performance scalings compared to distributed proportional feedback control, our results show that an explicit inclusion of measurement noise leads to the opposite conclusion. The noise’s impact on performance is shown to decrease with an increased inter-nodal alignment of the local integral states. However, even though the controller can be tuned for acceptable performance for any given network size, performance will degrade as the network grows, limiting the scalability of any such controller tuning. In particular, the requirement for inter-nodal alignment increases with network size. We show that this may in practice imply that very large and sparse networks will require any integral control to be centralized, rather than distributed. In this case, the best-achievable performance scaling, which is shown to be that of proportional feedback control, is retrieved.

Approximation of large-scale dynamical systems for Bench-mark Collection

Approximation of large-scale dynamical systems for Bench-mark Collection

Manifold learning for parameter reduction

Large scale dynamical systems (e.g. many nonlinear coupled differential equations) can often be summarized in terms of only a few state variables (a few equations), a trait that reduces complexity and facilitates exploration of behavioral aspects of otherwise intractable models. High model dimensionality and complexity makes symbolic, pen–and–paper model reduction tedious and impractical, a difficulty addressed by recently developed frameworks that computerize reduction. Symbolic work has the benefit, however, of identifying both reduced state variables and parameter combinations that matter most (effective parameters, “inputs”); whereas current computational reduction schemes leave the parameter reduction aspect mostly unaddressed. As the interest in mapping out and optimizing complex input–output relations keeps growing, it becomes clear that combating the curse of dimensionality also requires efficient schemes for input space exploration and reduction. Here, we explore systematic, data-driven parameter reduction by means of effective parameter identification, starting from current nonlinear manifold-learning techniques enabling state space reduction. Our approach aspires to extend the data-driven determination of effective state variables with the data-driven discovery of effective model parameters, and thus to accelerate the exploration of high-dimensional parameter spaces associated with complex models.

A tangential method for the balanced truncation in model reduction

In this paper, we present a new approach for large-scale Lyapunov matrix equations, where we present two algorithms named: Adaptive Block Tangential Lanczos-type and Arnoldi-type algorithms (ABTL and ABTA). This approach is based on the projection of the initial problem onto tangential Krylov subspaces to produce a low-rank approximate solution of large Lyapunov equations. These approximations are used in model reduction of large-scale dynamical systems with multiple inputs and multiple outputs (MIMO). We give some algebraic properties and present some numerical experiences to show the effectiveness of the proposed algorithms.

Reduced order models for dynamic molecular weight distribution in polymerization processes

Key polymer properties are substantially related to the polymer molecular weight distribution (MWD). Modeling polymerization processes based on MWD with kinetic-based approach involves the mass balances for all chain types and for all possible chain lengths. These balances exhibit a large-scale dynamic system which is computationally expensive to obtain numerical solutions. Moreover, these models are hard to apply for dynamic optimization. In this work, a reduced order model of dynamic MWD is proposed with a novel idea to reformulate and decouple the population balances based on nullspace projection method. The reformulation first separates the fast equilibrium of living chains from the accumulative dead chains, and then decomposes the coupling in the living chain lengths by further derivation. Dynamic simulation and optimization results demonstrate that the reduced model with manageable scale benefits computational efficiency, which provides a perspective for on-line applications.

An adaptive block tangential method for multi-input multi-output dynamical systems

Abstract In this paper, we present a new approach for model order reduction in large-scale dynamical systems, with multiple inputs and multiple outputs (MIMO). This approach will be named: Adaptive Block Tangential Arnoldi Algorithm (ABTAA) and is based on interpolation via block tangential Krylov subspaces requiring the selection of shifts and tangent directions via an adaptive procedure. We give some algebraic properties and present some numerical examples to show the effectiveness of the proposed method.

Dominant Pole Based Approximation for Discrete Time System

This paper presents an effective procedure for model order reduction of discrete time control system. The exact model derived from complex dynamic systems proves to be very complicated for analysis, control and design. This necessity brings about using a tool known as model order reduction technique or model simplification. A novel mixed method has been implemented in this paper for reducing the order of the large scale dynamic discrete system. Dominant pole based pole clustering method has been used to derive the coefficients of denominator polynomial while Padé approximation has been applied to obtain the coefficients of numerator polynomial of the reduced order model. The proposed method is quite simple and able to generate a stable reduced order model from high order stable discrete systems. The dominancy of poles has been decided by values of the ratio of residue to its pole. The pole is considered dominant which have larger ratio value. An illustrative example has been considered to show the various reduction steps. The result obtained confirms the effectiveness of the approach.

Iterative methods for solving large sparse Lyapunov equations and application to model reduction of index 1 differential-algebraic-equations

To implement the balancing based model reduction of large-scale dynamical systems we need to compute the low-rank (controllability and observability) Gramian factors by solving Lyapunov equations. In recent time, Rational Krylov Subspace Method (RKSM) is considered as one of the efficient methods for solving the Lyapunov equations of large-scale sparse dynamical systems. The method is well established for solving the Lyapunov equations of the standard or generalized state space systems. In this paper, we develop algorithms for solving the Lyapunov equations for large-sparse structured descriptor system of index-1. The resulting algorithm is applied for the balancing based model reduction of large sparse power system model. Numerical results are presented to show the efficiency and capability of the proposed algorithm.

Unstable modes in projection-based reduced-order models: How many can there be, and what do they tell you?

Projection methods provide an appealing way to construct reduced-order models of large-scale linear dynamical systems: they are intuitively motivated and fairly easy to compute. Unfortunately, the resulting reduced models need not inherit the stability of the original system. How many unstable modes can these reduced models have? This note investigates this question, using theory originally motivated by iterative methods for linear algebraic systems and eigenvalue problems, and illustrating the theory with a number of small examples. From these results follow rigorous upper bounds on the number of unstable modes in reduced models generated via orthogonal projection, for both continuous- and discrete-time systems. Can anything be learned from the unstable modes in reduced-order models? Several examples illustrate how such instability can helpfully signal transient growth in the original system.

Generalized Discrete-Time Equivalent Model for Dynamic Simulation of Regional Power Area

The introduction of an equivalent model for a regional power area in a large-scale power system with complex loads is essential for reducing the computation burden in real-time dynamic analyses. In this paper, we propose a generalized discrete-time equivalent model (GDEM) with specific relations among model parameters for simulating the physical characteristics of a regional power area. The GDEM facilitates the interconnection of equivalent models representing regional power areas and improving the simulation accuracy and speed in large-scale dynamic power systems. This paper first investigates the inherent relations among GDEM parameters in the discrete-time models of synchronous generators and composite loads so as to guide the GDEM parameter estimation in regional power areas. This paper then develops relations among GDEM parameters for a regional power area. Numerical experiments are conducted by simulating ground faults in the China Electric Power Research Institute system and the accuracy of the proposed GDEM is verified by analyzing the dynamic simulation results. In addition, this paper has applied the GDEM to study the regional power area of Central China, which validates the use of GDEM in practical power system analyses.