# On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

^{*}

## Abstract

**:**

## 1. Introduction

#### Main Contribution

## 2. Spectral Expansions of Gramians of Linear Systems

#### 2.1. Eigenmode Decompositions of the Dynamics of a Linear System

**Proposition**

**1.**

**Proof.**

**Proposition**

**2**

**Proof.**

#### 2.2. Modal Observability and Controllability of a Linear System

**Definition**

**1.**

**observable**in the linear system (1) at the moment ${t}_{0}$, when ${y}_{i}(t,{t}_{0},{x}_{0},u=0)\equiv 0$ at $t\ge {t}_{0}$ if, and only if, ${x}_{i}\left({t}_{0}\right)=0$.

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Definition**

**2.**

**is controllable**, if for each event $({t}_{0},{x}_{0}={x}_{i}\left({t}_{0}\right))$, there is a control $u\left(t\right)$, which brings the system to the zero state in a finite time.

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Corollary**

**1.**

#### 2.3. Spectral Decompositions of Gramians of a Linear System

**Theorem**

**1**

**Proposition**

**7**

**Proof.**

**Proposition**

**8.**

**Proposition**

**9**

**Proposition**

**10.**

## 3. Spectral Decompositions of Gramians of a Bilinear Control System

#### 3.1. Partitioning the Solution into Generalized Modes of the Matrix A

**Definition**

**3.**

**The generalized mode**of the bilinear system (21) corresponding to the eigenvalue ${\lambda}_{i}$ of the matrix A is the sum of the mode ${x}_{i}^{\left(1\right)}\left(t\right)$ of the linear part of the system and non-linear corrections generated by this mode, obtained in the course of solving the recursive system (22), i.e.,

**Proposition**

**11.**

**Proof.**

#### 3.2. Spectral Decompositions of Gramians

**Theorem**

**2**

**Theorem**

**3.**

**Proposition**

**12.**

**Proof.**

**Definition**

**4.**

**Controllability sub-Gramians and pairwise sub-Gramians**of the bilinear system (21) are, respectively, the matrices

**Proof.**

**Property**

**2.**

**Proof.**

**Property**

**3**

**Property**

**4.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Property**

**5.**

**Definition**

**5.**

**Observability sub-Gramians and pairwise sub-Gramians**of the bilinear system (21) are, respectively, the matrices

## 4. Iterative Algorithms for Computing Gramians and Sub-Gramians

#### 4.1. Algorithm for the Element-Wise Computation of Gramian in the Eigenvector Basis

#### 4.2. Novel Criterion for the Existence of Gramians

**Theorem**

**4.**

**Proof.**

**Example**

**1.**

#### 4.3. Iterative Algorithm for Computing Sub-Gramians

**Example**

**2.**

**Example**

**3.**

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Häger, U.; Rehtanz, C.; Voropai, N.I. Monitoring, Control and Protection of Interconnected Power Systems; Springer: New York, NY, USA, 2014; 391p. [Google Scholar]
- D’Alessandro, P.; Isidori, A.; Ruberti, A. Realization and structure theory of bilinear dynamic systems. SIAM J. Control.
**1974**, 12, 517–535. [Google Scholar] [CrossRef] - Pupkov, K.A.; Kapalin, V.I.; Yushchenko, A.S. Functional Series in the Theory of Nonlinear Systems; Nauka: Moscow, Russia, 1976; 448p. (In Russian) [Google Scholar]
- Flagg, G.M.; Gugercin, S. Multipoint Volterra Series Interpolation and H2 Optimal Model Reduction of Bilinear Systems. SIAM J. Matrix Anal. Appl.
**2015**, 36, 549–579. [Google Scholar] [CrossRef][Green Version] - Al-Baiyat, S.; Farag, A.S.; Bettayeb, M. Transient approximation of a bilinear two-area interconnected power system. Electr. Power Syst. Res.
**1993**, 26, 11–19. [Google Scholar] [CrossRef] - Zhang, L.; Lam, J. On the H
_{2}model reduction of bilinear systems. Automatica**2002**, 38, 205–216. [Google Scholar] [CrossRef] - Antoulas, A.C. Approximation of Large-Scale Dynamical Systems: Advances in Design and Control; SIAM: Philadephia, PA, USA, 2005; 479p. [Google Scholar]
- Benner, P.; Breiten, T. Interpolation-based H2-model reduction of bilinear control systems. SIAM J. Matrix Anal. Appl.
**2012**, 33, 859–881. [Google Scholar] [CrossRef][Green Version] - Benner, P.; Cao, X.; Schilders, W. A bilinear H2 model order reduction approach to bilinear parameter-varying systems. Adv. Comput. Math.
**2019**, 45, 2241–2271. [Google Scholar] [CrossRef][Green Version] - Gibbard, M.J.; Pourbeik, P.; Vowles, D.J. Small-Signal Stability, Control and Dynamic Performance of Power Systems; University of Adelaide Press: Adelaide, Australia, 2015. [Google Scholar]
- Jang, G.; Vittal, V.; Kliemann, W. Effect of nonlinear modal interaction on control performance: Use of normal forms technique in control design, Part 1: General theory and procedure. IEEE Trans. Power Syst.
**1998**, 13, 401–407. [Google Scholar] [CrossRef] - Pariz, N.; Shanechi, H.M.; Vaahedi, E. Explaining and validating stressed power systems behavior using modal series. IEEE Trans. Power Syst.
**2003**, 18, 778–785. [Google Scholar] [CrossRef] - Arroyo, J.; Betancourt, R.; Messina, A.R.; Barocio, E.D. Development of bilinear power system representations for small-signal stability analysis. Electr. Power Syst. Res.
**2007**, 77, 1239–1248. [Google Scholar] [CrossRef] - Ugwuanyi, N.S.; Kestelyn, X.; Thomas, O.; Marinescu, B.; Messina, A.R. New Fast Track to Nonlinear Modal Analysis of Power System Using Normal Form. IEEE Trans. Power Syst.
**2020**, 35, 3247–3257. [Google Scholar] [CrossRef][Green Version] - Hamzi, B.; Abed, E.H. Local modal participation analysis of nonlinear systems using Poincaré linearization. Nonlinear Dyn.
**2020**, 99, 803–811. [Google Scholar] [CrossRef][Green Version] - Benner, P.; Damm, T. Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems. SIAM J. Control. Optim.
**2011**, 49, 686–711. [Google Scholar] [CrossRef][Green Version] - Yadykin, I.B.; Iskakov, A.B.; Akhmetzyanov, A.V. Stability analysis of large-scale dynamical systems by sub-Gramian approach. Int. J. Robust. Nonlin. Control
**2014**, 24, 1361–1379. [Google Scholar] [CrossRef] - Yadykin, I.B.; Iskakov, A.B. Spectral Decompositions for the Solutions of Sylvester, Lyapunov, and Krein Equations. Dokl. Math.
**2017**, 95, 103–107. [Google Scholar] [CrossRef] - Yadykin, I.B.; Iskakov, A.B. Spectral decompositions for the solutions of Lyapunov equations for bilinear dynamical systems. Dokl. Math.
**2019**, 100, 501–504. [Google Scholar] [CrossRef] - Iskakov, A.B.; Yadykin, I.B. Analysis of a bilinear model of an electric power system using spectral decompositions of Lyapunov functions. IFAC-PapersOnLine
**2020**, 53, 13514–13519. [Google Scholar] [CrossRef] - Garofalo, F.; Iannelli, L.; Vasca, F. Participation Factors and their Connections to Residues and Relative Gain Array. IFAC Proc. Vol.
**2002**, 35, 125–130. [Google Scholar] [CrossRef][Green Version] - Bruni, C.; Dipillo, G.; Koch, G. On the mathematical models of bilinear systems. Ric. Di Autom.
**1971**, 2, 11–26. [Google Scholar] - Siu, T.; Schetzen, M. Convergence of Volterra series representation and BIBO stability of bilinear systems. Int. J. Syst. Sci.
**1991**, 22, 2679–2684. [Google Scholar] [CrossRef] - Yadykin, I.; Galyaev, A. On the methods for calculation of Gramians and their use in analysis of linear dynamic systems. Autom. Remote Control
**2013**, 74, 207–224. [Google Scholar] [CrossRef] - Khlebnikov, M.V. Quadratic Stabilization of Bilinear Control Systems. Autom. Remote Control
**2016**, 77, 980–991. [Google Scholar] [CrossRef] - Vadivel, R.; Hammachukiattikul, P.; Gunasekaran, N.; Saravanakumar, R.; Dutta, H. Strict dissipativity synchronization for delayed static neural networks: An event-triggered scheme. Chaos Solitons Fractals
**2021**, 150, 111212. [Google Scholar] [CrossRef] - Gunasekaran, N.; Thoiyab, N.M.; Zhu, Q.; Cao, J.; Muruganantham, P. New Global Asymptotic Robust Stability of Dynamical Delayed Neural Networks via Intervalized Interconnection Matrices. IEEE Trans. Cybern.
**2021**. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The Frobenius norm of sub-Gramians ${\tilde{P}}_{i}$ for generalized eigenmodes as a function of the weighting coefficient $\alpha $ in the test experiment in [20].

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Iskakov, A.; Yadykin, I. On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems. *Mathematics* **2021**, *9*, 3288.
https://doi.org/10.3390/math9243288

**AMA Style**

Iskakov A, Yadykin I. On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems. *Mathematics*. 2021; 9(24):3288.
https://doi.org/10.3390/math9243288

**Chicago/Turabian Style**

Iskakov, Alexey, and Igor Yadykin. 2021. "On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems" *Mathematics* 9, no. 24: 3288.
https://doi.org/10.3390/math9243288