Balanced Truncation Model Reduction Research Articles

Balanced truncation model reduction with a priori error bounds for LTI systems with nonzero initial value

In standard balanced truncation model order reduction, the initial condition is typically ignored in the reduction procedure and is assumed to be zero instead. However, such a reduced-order model may be a bad approximation to the full-order system, if the initial condition is not zero. In the literature there are several attempts for modified reduction methods at the price of having no error bound or only a posteriori error bounds which are often too expensive to evaluate. In this work we propose a new balancing procedure that is based on a shift transformation on the state. We first derive a joint projection reduced-order model in which the part of the system depending only on the input and the one depending only on the initial value are reduced at once and we prove an a priori error bound. With this result at hand, we derive a separate projection procedure in which the two parts are reduced separately. This gives the freedom to choose different reduction orders for the different subsystems. Moreover, we discuss how the reduced-order models can be constructed in practice. Since the error bounds are parameter-dependent we show how they can be optimized efficiently. We conclude this paper by comparing our results with the ones from the literature by a series of numerical experiments.

Control for stability improvement of high-speed train bogie with a balanced truncation reduced order model

The high-speed train bogie is a typical underactuated mechanical system. It has been difficult to design controls to increase the stability against hunting motion with limited sensor measurements and actuation hardware. This paper develops a control design for the stability improvement of the bogie by making use of the balanced truncation reduced order model of the bogie. When the train travels at speed higher than the critical speed, the bogie has unstable modes. The balanced truncation model reduction (BTMR) is usually applied to stable systems. In this paper, we first apply the real Schur decomposition to separate the stable and unstable modes of the bogie. The BTMR is then carried out for the stable subsystem. An augmented -balanced truncation algorithm is considered to deal with the dependence of the BTMR on initial conditions. The reduced order subsystem is then combined with the unstable subsystem. A linear quadratic regulator control together with a Luenberger observer for the state estimation is designed for the bogie with non-zero initial conditions. Simulation results show that the control designed with the balanced truncation reduced order model performs very well in terms of the stability improvement by significantly increasing the critical train speed.

Error bounds for port-Hamiltonian model and controller reduction based on system balancing

We study linear quadratic Gaussian (LQG) control design for linear port-Hamiltonian systems. To this end, we exploit the freedom in choosing the weighting matrices and propose a specific choice which leads to an LQG controller which is port-Hamiltonian and, thus, in particular stable and passive. Furthermore, we construct a reduced-order controller via balancing and subsequent truncation. This approach is closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. By exploiting the non-uniqueness of the Hamiltonian, we are able to determine an optimal pH representation of the full-order system in the sense that the error bound is minimized. In addition, we discuss consequences for pH-preserving balanced truncation model reduction which results in two different classical H∞-error bounds. Finally, we illustrate the theoretical findings by means of two numerical examples.

A dual reduction strategy for reduce-order modeling of periodic control system

Model order reduction (MOR) of periodic systems using the Krylov subspace methods received lots of interest in last few decades. In this paper, a structured Krylov subspace based model reduction for linear discrete-time periodic (LDTP) control system has been proposed using the corresponding lifted form. The proposed model reduction strategy first finds a reduced order Lyapunov equation via a projection where the projection subspace is computed by using the periodic Arnoldi method. Then, by applying the balanced truncation model reduction approach, it forms a reduced order model of the original system. In both cases, the periodic structure is ensured in the solutions and systems’ dynamics. This paper also presents numerical results to confirm the efficiency and accuracy of the proposed algorithm.

Robust Output Feedback Control Design for Inertia Emulation by Wind Turbine Generators

Wind generation has gained widespread use as a renewable energy source. Most wind turbines and other renewables connected to the grid through converters result in a reduction in the natural inertial response to grid frequency changes. The doubly-fed induction generator (DFIG) can be controlled to compensate for this reduction and, in fact, provide faster response than traditional synchronous machines. This paper proposes to design observer based output feedback linear quadratic regulator (LQR) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> control laws to realize the inertia emulation function and deliver fast frequency support. The aim is to track the reference speed by a diesel synchronous generator (DSG) in order to reach the desired inertia. The control signal is computed based on a reduced order model using the balanced truncation technique. A comparison with selective modal analysis (SMA) and balanced truncation model reduction techniques is presented. Comprehensive results show the effective emulation of synthetic inertia by implementing the control laws on a nonlinear three-phase diesel-wind system. The proposed technique is analyzed for different short circuit ratio (SCR) scenarios.

Balanced Truncation Model Reduction for Symmetric Second Order Systems---A Passivity-Based Approach

We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally, we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.

Delay-dependent supplementary damping controller of TCSC for interconnected power system with time-delays and actuator saturation

In this paper, a delay-dependent state feedback based supplementary damping controller of Thyristor Controlled Series Capacitor (TCSC) is designed to enhance the damping of inter-area oscillations of an interconnected power system. Although wide-area feedback signals are advantageous for damping enhancement of inter-area modes, but its usage as feedback signals in designing of a damping controller together with physical limitation of the actuator introduces time-delay and actuator saturation in the feedback loop. This results in degradation of the power system performance and even cause instability. The paper focuses on designing a damping controller considering the aforesaid time-delay and actuator saturation effects. Residue approach and Schur balanced truncation model reduction methods are used to obtain input-output control signals and reduced order power system model. The nonlinear effects of the actuator saturation are consider in the closed-loop system by using generalized sector condition. Based on Lyapunov–Krasovskii functional approach, a delay-dependent stability and stabilization conditions via linear matrix inequalities (LMIs) formulation are derived to guarantee the asymptotic stability of the closed-loop system. Then the problem of designing a state feedback control law which maximizes the region of attraction is formulated and solved as an optimization problem with LMI constraints. Simulation studies in the two-area four-machine power system demonstrate the effectiveness of the proposed controller for damping enhancement of the inter-area oscillations and compensating the effect of time-delays and actuator saturation.

Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Numerical study of polynomial feedback laws for a bilinear control problem

An infinite-dimensional bilinear optimal control problem with infinite-time horizon is considered. The associated value function can be expanded in a Taylor series around the equilibrium, the Taylor series involving multilinear forms which are uniquely characterized by generalized Lyapunov equations. A numerical method for solving these equations is proposed. It is based on a generalization of the balanced truncation model reduction method and some techniques of tensor calculus, in order to attenuate the curse of dimensionality. Polynomial feedback laws are derived from the Taylor expansion and are numerically investigated for a control problem of the Fokker-Planck equation. Their efficiency is demonstrated for initial values which are sufficiently close to the equilibrium.

Model reduction of constrained mechanical systems in M-M.E.S.S.

We discuss balanced truncation model reduction of constrained mechanical systems. The incorporation of the algebraic constraints leads to linear differential-algebraic control systems of index 2 or 3. Then for the square-root method in balanced truncation, the solution of projected Lyapunov equations is required. In this paper we discuss how to avoid the explicit construction of the projectors and the formulation of a projection-avoiding balanced truncation method which is suitable for efficient numerical computations. We have implemented this method in the Matlab package M-M.E.S.S. and present some numerical results for illustration.

Balanced truncation model reduction of nonstationary systems interconnected over arbitrary graphs

This paper deals with the balanced truncation of discrete-time, linear time-varying, heterogeneous subsystems interconnected over finite arbitrary directed graphs. The information transfer between the subsystems is subject to a communication latency of one time-step. The method guarantees the preservation of the interconnection structure and further allows for its simplification. In addition to truncating temporal states associated with the subsystems, the method allows for the order reduction of spatial states associated with the interconnections between the subsystems and even the removal of whole interconnections. Upper bounds on the ℓ2-induced norm of the resulting error system are derived. The method is illustrated through an example.

The design of delay-dependent wide-area DOFC with prescribed degree of stability α for damping inter-area low-frequency oscillations in power system

In this paper, the delay-dependent wide-area dynamic output feedback controller (DOFC) with prescribed degree of stability is proposed for interconnected power system to damp inter-area low-frequency oscillations. Here, the prescribed degree of stability α is used to maintain all the poles on the left of s=−α in the s-plane. Firstly, residue approach is adopted to select input-output control signals and the schur balanced truncation model reduction method is utilized to obtain the reduced power system model. Secondly, based on Lyapunov stability theory and transformation operation in complex plane, the sufficient condition of asymptotic stability for closed-loop power system with prescribed degree of stability α is derived. Then, a novel method based on linear matrix inequalities (LMIs) is presented to obtain the parameters of DOFC and calculate delay margin of the closed-loop system considering the prescribed degree of stability α. Finally, case studies are carried out on the two-area four-machine system, which is controlled by classical wide-area power system stabilizer (WAPSS) in reported reference and our proposed DOFC respectively. The effectiveness and advantages of the proposed method are verified by the simulation results under different operating conditions.

Structure preserving iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

In this paper, we develop structure preserving iterative schemes to solve the periodic discrete-time projected Lyapunov equations associated to analysis and design of discrete-time descriptor systems exploiting the reflexive generalized inverses of the periodic matrices associated with these systems. In particular, we extend the Smith method to solve the large-scale projected periodic discrete-time algebraic Lyapunov equations in lifted form. A low-rank version of this method is also presented, avoiding the explicit lifted formulation and working directly with the periodic matrix coefficients. Moreover, we consider an application of the Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Solving Parameter-Dependent Lyapunov Equations Using the Reduced Basis Method with Application to Parametric Model Order Reduction

Our aim is to numerically solve parameter-dependent Lyapunov equations using the reduced basis method. Such equations arise in parametric model order reduction. We restrict ourselves to the systems that affinely depend on the parameter, as our main strategy is the min-$\theta$ approach. In those cases, we derive various a posteriori error estimates. Based on these estimates, a greedy algorithm for constructing reduced bases is formulated. Thanks to the derived results, a novel so-called parametric balanced truncation model reduction method is developed. Numerical examples are presented.

Efficient Techniques for Solving the Periodic Projected Lyapunov Equations and Model Reduction of Periodic Systems

We have presented the efficient techniques for the solutions of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form. These types of problems arise in model reduction and state feedback problems of periodic descriptor systems. Two most popular techniques to solve such Lyapunov equations iteratively are the low-rank alternating direction implicit (LR-ADI) method and the low-rank Smith method. The main contribution of this paper is to update the LR-ADI method by exploiting the ideas of the adaptive shift parameters computation and the efficient handling of complex shift parameters. These approaches efficiently reduce the computational cost with respect to time and memory. We also apply these iterative Lyapunov solvers in balanced truncation model reduction of periodic discrete-time descriptor systems. We illustrate numerical results to show the performance and accuracy of the proposed methods.

Balanced truncation model reduction for linear time-varying systems

ABSTRACTA practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high-order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort in treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.

Low-rank iterative methods for periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systems

We discuss the numerical solution of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.

Active Integrated Devices PDN Balanced Truncation Reduction Method

For EMC simulation, the vector fitting model is transformed into time domain state equation model. Then the system is balanced. Then the reduced model can be obtained by removing the states corresponding to the small HSV (Hankel Singular Values). The order of the reduced model is determined by the singular curvature spectrum. Finally, balanced truncation model reduction method is used on a sample chip PDN (Power Distribution Network) and compared with the performance of the AWE (Asymptotic Waveform Evaluation) method. Simulation results show that the proposed method can operate in a wide frequency range and has smaller error and faster speed.

Reduced-Order &lt;i&gt;H&lt;sub&gt;∞&lt;/sub&gt; &lt;/i&gt;Controller Design Using the &lt;i&gt;μ&lt;/i&gt;-Synthesis Applied to Lateral Control of Highly Flexible Aircraft

A lateral flight reduced-order controller for a flexible aircraft is presented. It is based on the reduced-order model of the flexible aircraft and theμ-analysis and synthesis toolbox. The first step deals with the order reduction of the full-model by using the balanced-truncation model reduction method. Then the reduced-orderHcontroller is achieved by using theμ-analysis and synthesis theory and applied to the reduced-order model and full-order model of the flexible aircraft. Finally, numerical results are presented and discussed. The simulation results show the efficiency of the reduced-order controller based on the reduced-order model since the close-loop system of the full-order model meets a set of realistic specifications in the frequency and time domains.

Goal-Oriented Inference: Approach, Linear Theory, and Application to Advection Diffusion

Inference of model parameters is one step in an engineering process often ending in predictions that support decision in the form of design or control. Incorporation of end goals into the inference process leads to more efficient goal-oriented algorithms that automatically target the most relevant parameters for prediction. In the linear setting the control-theoretic concepts underlying balanced truncation model reduction can be exploited in inference through a dimensionally optimal subspace regularizer. The inference-for-prediction method exactly replicates the prediction results of either truncated singular value decomposition, Tikhonov-regularized, or Gaussian statistical inverse problem formulations independent of data; it sacrifices accuracy in parameter estimate for online efficiency. The new method leads to low-dimensional parameterization of the inverse problem enabling solution on smartphones or laptops in the field.