Model Reduction of Discrete-Time Index-3 Second-Order Form Systems for Limited Frequency Intervals ^†

Abstract

A model order reduction framework for limited frequency interval response optimization of reduced order models (ROMs) for index-3 second-order form of systems (SOSs) is presented in this paper. Firstly, an index-3 SOS system is transformed into index-0 and the corresponding generalized first-order form. In order to emphasize ROM response over the required frequency interval, frequency-limited gramians and corresponding generalized Lyapunov equations are presented and balancing of gramians is obtained by solving Lyapunov equations to obtain the required ROM exhibiting a good response in the intended frequency interval. The developments are tested on multiple systems and the superiority of the proposed extension over existing methods is certified. Propositions can be utilized for frequency-limited applications for index-3 SOSs.

1. Introduction

In simulations, the physical model of a system is written in the form of a mathematical model. Real-time applications most of the time involve linear time-invariant (LTI) systems. In many modeling processes, the mathematical model takes a generalized form such as descriptor, second order, index-2, index-3, etc. These index-3 forms appear in fields such as microelectronics, circuit simulations, aerospace, signal processing, etc. [1,2,3,4]. In most cases in a physical system, there are many disparate devices, and these devices form the components of the system. Modeling such systems becomes difficult, and mathematical models can be composed of several large and sparse matrices. A large-scale system is often difficult to implement due to high order because it requires more computational power, storage capacity, and cost. To reduce the complexity of the system, a reduced model is computed that reflects the behavior of the original model with a lower order model and is easy to implement [5]. The most prominent techniques for model order reduction (MOR) are balanced truncation (B and transfer function rational interpolation using the iterative rational Krylov algorithm (IRKA). Stykel [6], introduced the balanced truncation of second-order systems. There is a drawback of this method in that it does not give an error bound, and stability is not guaranteed. Whereas, a balanced truncation-based model gives the global error bound by preserving stability. The only congestion of balanced truncation is the solution of two Lyapunov equations for the computation of gramians. Balance truncation techniques are well designed for standard state-space systems and generalized systems in [7]. Standard system algorithms cannot be applied directly for descriptor systems due to the lower accuracy of reduced-order models. Such techniques for MOR of first-order systems utilizing balance truncation are explained in [7]. In such cases, certain projections must be applied for the decoupling of differential and algebraic equations. However, the application of projections arises computational issues and robustness if the number of rows of a projector is increased [7]. The design of both IRKA and BT are presented in generalized form for large-scale descriptor systems [8]. In some recent works, some methods have been proposed for MOR of descriptor systems that do not require the computation of projectors separately. It becomes easier to define a projector separately for the differential part of the system from given data. This method eliminates the algebraic equation. Then, leaving behind the technicalities of the projector, we can directly apply the DAE systems to the projected system. All the above procedures have been designed for the interpolation of projection and balancing for MOR of the first-order index-3 descriptor system in [8]. As far as real-time analysis is concerned, many applications such as filter design, controller design, and signal reconstruction of system design require ROM optimization in limited frequency intervals for data analysis. A finite interval model order reduction of linear time-invariant standard state-space systems is described in [5]. A stability preservation MOR technique using frequency-limited gramians for a discrete-time system is presented in [8]. This research article discusses the MOR of the second-order index-3 system. In a generalized system in [2], $E$ matrix often becomes singular, which halts the reduction process. To overcome singularity, iterative numerical techniques are utilized to make $E$ non-singular. The number of iterations required to make $E$ nonsingular represent the index of the system. Multiple MOR/BT techniques have been proposed in the literature for index 0, 1, 2, and 3 systems [7,8,9]. A discrete time index-2 descriptor system is designed in [10]. DAEs compute gramians for discrete-time indices. A new method of solving DAEs is developed which is called index aware model order reduction. Partitioning the discrete-time second-order system gramians into position and velocity components preserving the structure of a system is discussed in [11]. The proposed technique first converts the index-3 system to the index-0 system and then reduces the order of the system. The wide range of limited frequency interval applications for discrete-time index-3 s order systems (SOSs) for which the MOR technique does not exist in literature is presented in the paper. Results are validated according to the practice followed in the literature by comparing the infinite interval response with the limited interval response. The results show that our approximation is correct. The paper is organized as follows: Section 1 presents the introduction, Section 2 covers the proposed technique, and Section 3 is based on numerical results.

2. Proposed Technique

2.1. Discrete-Time Index-3 Second-Order System

A model of the discrete-time linear time-invariant system can be written as:

$$\begin{gathered} M\eta \left[k+2\right]+D\eta \left[k+1\right]+K\eta \left[k\right]+G^T\mu \left[k\right]=B^{\prime }u\left[k\right] \\ 0=G\eta \left[k\right] \\ y\left[k\right]=C_1\eta \left[k\right] \end{gathered}$$

(1)

System of Equation (1) is a second-order descriptor system.

Where $Mϵ{ℛ}^{n\times n},$ $Dϵ{ℛ}^{n\times n},Kϵ{ℛ}^{n\times n}$, $Bϵ{ℛ}^{n\times m}$, $C_1ϵ{ℛ}^{p\times n}$, $x\left[k\right]ϵ{ℛ}^n$, $\mu \left[k\right]ϵ{ℛ}^m$, $y\left[k\right]ϵ{ℛ}^p$, n is the order of the system, m is the number of inputs, and $p$ is the number of outputs. $M,D,K$ in the equation represent mass, stiffness, and damping of the system. Moreover, matrix $M$ is an invertible matrix [2]. The system of Equation (1) is a second-order index 3 descriptor system.

Using the mathematical operations, the second-order index-3 is converted into-0 form as given below:

$$\begin{gathered} \Theta M{\Theta }^T\eta \left[k+2\right]+\Theta D{\Theta }^T\eta \left[k+1\right]+\Theta K{\Theta }^T\eta \left[k\right]=\Theta B_{2}u\left[k\right] \\ y\left[k\right]=C{\Theta }^T\eta \left[k\right] \end{gathered}$$

(2)

$$\begin{gathered} {\Theta }_r^{T}M{\Theta }_r\ddot{\check{\eta }}\left[k\right]=-{\Theta }_r^{T}D{\Theta }_r\dot{\check{\eta }}\left[k\right]-=-{\Theta }_r^{T}K{\Theta }_r\check{\eta }\left[k\right]+{\Theta }_r^T{B}^{\prime }u\left[k\right] \\ y\left[k\right]=C_1{\Theta }_r\check{\eta }\left[k\right] \end{gathered}$$

(3)

This is a system of (3) with projection ${\Theta }_r$; the projection removes the redundant equations. The three dynamical systems of (1), (2), and (3) are different realizations of the same transfer function. Moreover, their finite spectrum will be the same.

The system will be converted into the generalized first-order form and written as a generalized system in first-order form

$$\begin{gathered} Ex\left[k+1\right]=A x\left[k\right]+Bu\left[k\right] \\ y\left[k\right]=C x\left[k\right] \end{gathered}$$

(4)

2.2. Limited Frequency Interval Gramians

Frequency limited gramians [12] of Equation (5) are as

$$W_{c\delta }=\frac{1}{2\pi }{{\displaystyle \int }}_{\delta }{\left(e^{j\omega }E-A\right)}^{-1}B{B}^T{\left(e^{-j\omega }E-A\right)}^{-T}d\omega , W_{o\delta }=\frac{1}{2\pi }{{\displaystyle \int }}_{\delta }{\left(e^{j\omega }E-A\right)}^{-T}{C}^{T}C{\left(e^{-j\omega }E-A\right)}^{-1}d\omega$$

(5)

where $\delta$ = [−${\omega }_{2,}-{\omega }_1$] ${\displaystyle \cup }$ [${\omega }_1,{\omega }_2$] should be chosen in such a way that the limited frequency interval gramians are ensured to be real, symmetric, and positive definite.

These gramians are the solution to discrete-time algebraic generalized Lyapunov equations:

$$E{W}_{c\delta }{A}^T+A{W}_{c\delta }{E}^T=-EHB{B}^T-B{B}^T{H}^{*}{E}^T$$

(6)

$$E{W}_{o\delta }{A}^T+A{W}_{o\delta }{E}^T=-E^T{H}^{*}{C}^{T}C-C^{T}CHE$$

(7)

where H* is the conjugate of H transpose and E, A, B, and C possess second-order structure, here H is $H=\frac{-\left({\omega }_2-{\omega }_1\right)}{4\pi }I+E{H}_1$ and $H_1=\frac{1}{2\pi }{{\displaystyle \int }}_{\delta }{\left(E-A{e}^{-j\omega }\right)}^{-1}d\omega$.

2.3. Balance Truncation for Frequency Limited Second-Order System

The relations in (5) are used to obtain frequency-limited gramians. Singular value decomposition is given by: $L_{\delta }^T \Theta E \Theta R_{\delta }=\left[\begin{array}{cc}{U}_{\delta 1}& U_{\delta 2}\end{array}\right]\left[\begin{array}{cc}{\mathsf{\Sigma }}_{\delta 1}& 0\\ 0& {\mathsf{\Sigma }}_{\delta 2}\end{array}\right]\left[\begin{array}{c}{V}_{\delta 1}^T\\ V_{\delta 2}^T\end{array}\right].$ Thus, right- and left-hand side transformation matrices are: $\widetilde{R_{\delta }}=R_{\delta } V_{\delta 1}{\mathsf{\Sigma }}_{\delta 1}{}^{\left(-\frac{1}{2}\right)},$ $\widetilde{L_{\delta }}=L_{\delta } U_{\delta 1}{\mathsf{\Sigma }}_{\delta 1}{}^{\left(-\frac{1}{2}\right)}$.

Applying this transformation, a reduced index -3 system of second order can be computed at a limited frequency interval [13].

$$E_{r\delta }={\widetilde{L_{\delta }}}^{T}E \widetilde{R_{\delta },}{A}_{r\delta }={\widetilde{L_{\delta }}}^{T}A \widetilde{R_{\delta }},B_{r\delta }= {\widetilde{L_{\delta }}}^{T}B,C_{r\delta }=C \widetilde{R_{\delta }}$$

3. Numerical Results with Example

The proposed technique is applied to a single input single output (SISO) given in the Appendix A, and the results are discussed below. A 12th-order system that is reduced to the 5th order for the infinite interval. The response of the original system and reduced system is shown in Figure 1. As we can see, the ROM response is not catching the original system response in the intervals $\delta =\left[0.3,0.5\right],\left[0,0.2\right],\left[0.6,1\right]$. Therefore, we apply the proposed technique for a limited interval to obtain performance emphasis over these regions (one at a time), and responses are shown in Figure 2. The topmost plot of Figure 2 shows the response of the original and ROM obtained for the finite and infinite intervals. Note that the proposed finite response of ROM approximates the original system near to perfection in the emphasized intervals $\delta =\left[0.3,0.5\right] and \left[0,0.2\right].$ Similarly, for interval $\delta =\left[0.6,1\right]$ the proposed technique matches to the original system response. Thus, the proposed technique is validated from the responses of the system.

4. Conclusions

Research into a discrete-time index-3 s order system reduction technique using frequency-limited gramians is proposed. It has been shown that a discrete-time index-3 system can be converted into an index-0 generalized system. The Lyapunov equations are solved over a limited frequency interval. The gramians are balanced by their corresponding HSV balancing. The proposed technique is very useful for discrete-time index-3 system design over a limited frequency interval. The proposed technique is widely useful for finite frequency interval applications of index-3 SOSs such as filter design, signal reconstruction, and controller design.