# Controllability of a Class of Heterogeneous Networked Systems

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma**

**1**

**Lemma**

**2**

- (i)
- $(A\otimes B)(C\otimes D)=(AC\otimes BD)$.
- (ii)
- ${(A\otimes B)}^{-1}={A}^{-1}\otimes {B}^{-1}$ if A and B are invertible.
- (iii)
- $(A+B)\otimes C=A\otimes C+B\otimes C$.
- (iv)
- $A\otimes (B+C)=A\otimes B+A\otimes C$.
- (v)
- $A\otimes B=0$ if and only if $A=0$ or $B=0$.

**Lemma**

**3**

## 3. Model Formulation

## 4. Controllability Results in a General Network Topology

**Theorem**

**1**

- (i)
- The eigenspectrum of F is the union of eigenspectrum of ${A}_{i}+{\lambda}_{i}H$, where, $i=1,2,\dots ,N$. That is,$$\sigma \left(F\right)={\cup}_{i=1}^{N}\sigma ({A}_{i}+{\lambda}_{i}H)=\{{\mu}_{1}^{1},\dots ,{\mu}_{1}^{{q}_{1}},\dots ,{\mu}_{N}^{1},\dots ,{\mu}_{N}^{{q}_{N}}\}$$
- (ii)
- If J is a diagonal matrix, then ${e}_{i}T\otimes {\xi}_{ij}^{k},k=1,\dots ,{\gamma}_{ij}$ are the left eigenvectors of F corresponding to the eigenvalue ${\mu}_{i}^{j},j=1,\dots ,{q}_{i},i=1,\dots ,N$.
- (iii)
- If J contains a Jordan block of order $l\ge 2$ for some eigenvalue ${\lambda}_{{i}_{0}}$ of C with ${\xi}_{ij}^{k}H=0$ for all $i={i}_{0},{i}_{0}+1,\dots ,{i}_{0}+l-1,j=1,2,\dots ,{q}_{i},k=1,2,\dots ,{\gamma}_{ij}$, then ${e}_{i}T\otimes {\xi}_{ij}^{k},k=1,\dots ,{\gamma}_{ij}$ are the left eigenvectors of F corresponding to the eigenvalue ${\mu}_{i}^{j},i=1,2,\dots ,N,j=1,2,\dots ,{q}_{i}$.

**Theorem**

**2.**

- (i)
- ${e}_{i}TD\ne 0$ for all $i=1,\dots ,N$
- (ii)
- For a fixed i, each left eigenvector ξ of ${A}_{i}+{\lambda}_{i}H$, $\xi Bj\ne 0$ for some $j\in \{1,2,\cdots ,N\}$ with ${\left[{e}_{i}TD\right]}_{j}\ne 0$,
- (iii)
- If matrices ${A}_{{i}_{1}}+{\lambda}_{{i}_{1}}H,{A}_{{i}_{2}}+{\lambda}_{{i}_{2}}H,\dots ,{A}_{{i}_{p}}+{\lambda}_{{i}_{p}}H({\lambda}_{{i}_{k}}\in \sigma \left(C\right),k=1,\dots ,p,\phantom{\rule{4pt}{0ex}}where\phantom{\rule{4pt}{0ex}}p>1)$ have a common eigenvalue σ, then $({e}_{{i}_{1}}TD\otimes {\xi}_{{i}_{1}}^{1})B,\cdots ,({e}_{{i}_{1}}TD\otimes {\xi}_{{i}_{1}}^{{\gamma}_{{i}_{1}}})B,\dots ,({e}_{{i}_{p}}TD\otimes {\xi}_{{i}_{p}}^{1})B,\dots ,({e}_{{i}_{p}}TD\otimes {\xi}_{{i}_{p}}^{{\gamma}_{{i}_{p}}})B$ are linearly independent vectors, where ${\gamma}_{{i}_{k}}\ge 1$ is the geometric multiplicity of σ for ${A}_{{i}_{k}}+{\lambda}_{{i}_{k}}H$ and ${\xi}_{{i}_{k}}^{l}(l=1,\dots ,{\gamma}_{{i}_{k}})$ are the left eigenvectors of ${A}_{{i}_{k}}+{\lambda}_{{i}_{k}}H$ corresponding to $\sigma ,k=1,\dots ,p$.

**Proof.**

**(Necessary part)**Fix i. Let $\xi $ be an arbitrary left eigenvector of ${A}_{i}+{\lambda}_{i}H$. From Theorem 1, we find that ${e}_{i}T\otimes \xi $ is a left eigenvector of F. By Lemma 3, for the networked system (3) to be controllable, we must have

**(Sufficiency part)**To prove the converse part, we will show that if the networked system is uncontrollable, at least one condition in Theorem 1 does not hold. Suppose that the networked system (3) is not controllable. Then by Lemma 3, there exists a left eigenpair $(\tilde{\mu},\tilde{v})$ of F, such that $\tilde{v}G=0$.

**Example**

**1.**

- (i)
- as $TD=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 0& 1\end{array}\right]$, ${e}_{i}TD\ne 0$ for all $i=1,2,3$.
- (ii)
- for ${A}_{1}+{\lambda}_{1}H={A}_{1}=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]$, the only left eigenvector is ${\xi}_{11}^{1}=\left[\begin{array}{ccc}0& 0& 1\end{array}\right]$. We have ${\left[{e}_{1}TD\right]}_{1}\ne 0$ and ${\xi}_{11}^{1}{B}_{1}\ne 0$.For the matrix ${A}_{2}+{\lambda}_{2}H={A}_{2}=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& 1\\ 1& 1& 2\end{array}\right]$ the left eigenvectors are, respectively, ${\xi}_{21}^{1}=\left[\begin{array}{ccc}0.44062& 0.828911& 1\end{array}\right],$${\xi}_{22}^{1}=\left[\begin{array}{ccc}-0.72031-0.784805i& -0.914456+1.47641i& 1\end{array}\right],$ and ${\xi}_{23}^{1}=\left[\begin{array}{ccc}-0.72031+0.784805i& -0.914456-1.47641i& 1\end{array}\right]$. We have ${\left[{e}_{2}TD\right]}_{3}\ne 0$ and ${\xi}_{21}^{1}{B}_{3},{\xi}_{22}^{1}{B}_{3},{\xi}_{23}^{1}{B}_{3}\ne 0$.For the matrix ${A}_{3}+{\lambda}_{3}H={A}_{3}+H=\left[\begin{array}{ccc}2& 3& 1\\ 1& 2& 2\\ 1& 1& 2\end{array}\right]$, the left eigenvectors are, respectively,${\xi}_{31}^{1}=\left[\begin{array}{ccc}0.720551& 1.09001& 1\end{array}\right],$${\xi}_{32}^{1}=\left[\begin{array}{ccc}-0.0875483-0.34424i& -0.681369+0.450503i& 1\end{array}\right],$and ${\xi}_{33}^{1}=\left[\begin{array}{ccc}-0.0875483+0.34424i& -0.681369-0.450503i& 1\end{array}\right]$. We have ${\left[{e}_{3}TD\right]}_{3}\ne 0$ and ${\xi}_{31}^{1}{B}_{3},{\xi}_{32}^{1}{B}_{3},{\xi}_{33}^{1}{B}_{3}\ne 0$.
- (iii)
- as the matrices ${A}_{1},{A}_{2}$ and ${A}_{3}+H$ do not have any common eigenvalues, third condition of Theorem 2 is satisfied.

**Example**

**2.**

- (i)
- as $TD=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& \frac{1}{2}& 0\end{array}\right]$, ${e}_{i}TD\ne 0$ for all $i=1,2,3$.
- (ii)
- for ${A}_{1}+H=\left[\begin{array}{ccc}1& 1& 0\\ 0& 2& -1\\ 0& 1& 0\end{array}\right]$, the only left eigenvector is ${\xi}_{11}^{1}=\left[\begin{array}{ccc}0& -1& 1\end{array}\right]$. We have ${\left[{e}_{1}TD\right]}_{1}\ne 0$ and ${\xi}_{11}^{1}{B}_{1}\ne 0$.For ${A}_{2}+H=\left[\begin{array}{ccc}1& 2& 1\\ 2& 2& 0\\ 0& 3& 0\end{array}\right]$, the left eigenvectors are ${\xi}_{21}^{1}=\left[\begin{array}{ccc}3.90547& 5.67363& 1\end{array}\right],$${\xi}_{22}^{1}=\left[\begin{array}{ccc}-0.452737+1.15383i& -0.336813-1.0993i& 1\end{array}\right],$ and ${\xi}_{23}^{1}=\left[\begin{array}{ccc}-0.452737-1.15383i& -0.336813+1.0993i& 1\end{array}\right]$. We have ${\left[{e}_{2}TD\right]}_{2}\ne 0$ and ${\xi}_{21}^{1}{B}_{2},{\xi}_{22}^{1}{B}_{2},{\xi}_{23}^{1}{B}_{2}\ne 0$.and for the matrix ${A}_{3}-H=\left[\begin{array}{ccc}-1& 0& 1\\ 2& 0& -2\\ 0& 1& -2\end{array}\right]$, the left eigenvectors are ${\xi}_{31}^{1}=\left[\begin{array}{ccc}2& 1& 0\end{array}\right],$${\xi}_{32}^{1}=\left[\begin{array}{ccc}-0.25+0.661438i& -0.375-0.330719i& 1\end{array}\right],$ and ${\xi}_{33}^{1}=\left[\begin{array}{ccc}-0.25-0.661438i& -0.375+0.330719i& 1\end{array}\right]$. We have ${\left[{e}_{3}TD\right]}_{2}\ne 0$ and ${\xi}_{31}^{1}{B}_{2},{\xi}_{32}^{1}{B}_{2},{\xi}_{33}^{1}{B}_{2}\ne 0$.
- (iii)
- as the matrices ${A}_{1}+H,{A}_{2}+H$ and ${A}_{3}-H$ do not have any common eigenvalues, third condition of Theorem 2 is satisfied.

**Corollary**

**1.**

- (i)
- ${e}_{i}TD\ne 0$ for all $i=1,\dots ,N$;
- (ii)
- For a fixed i, $({A}_{i}+{\lambda}_{i}H,{B}_{j})$ is controllable for some $j\in \{1,2,\cdots ,N\}$ with ${\left[{e}_{i}TD\right]}_{j}\ne 0$;
- (iii)
- If matrices ${A}_{{i}_{1}}+{\lambda}_{{i}_{1}}H,{A}_{{i}_{2}}+{\lambda}_{{i}_{2}}H,\dots ,{A}_{{i}_{p}}+{\lambda}_{{i}_{p}}H({\lambda}_{{i}_{k}}\in \sigma \left(C\right),k=1,\dots ,p,\phantom{\rule{4pt}{0ex}}where\phantom{\rule{4pt}{0ex}}p>1)$ have a common eigenvalue σ, then $({e}_{{i}_{1}}TD\otimes {\xi}_{{i}_{1}}^{1})B,\cdots ,({e}_{{i}_{1}}TD\otimes {\xi}_{{i}_{1}}^{{\gamma}_{{i}_{1}}})B,\dots ,({e}_{{i}_{p}}TD\otimes {\xi}_{{i}_{p}}^{1})B,\dots ,({e}_{{i}_{p}}TD\otimes {\xi}_{{i}_{p}}^{{\gamma}_{{i}_{p}}})B$ are linearly independent vectors, where ${\gamma}_{{i}_{k}}\ge 1$ is the geometric multiplicity of σ for ${A}_{{i}_{k}}+{\lambda}_{{i}_{k}}H$ and ${\xi}_{{i}_{k}}^{l}(l=1,\dots ,{\gamma}_{{i}_{k}})$ are the left eigenvectors of ${A}_{{i}_{k}}+{\lambda}_{{i}_{k}}H$ corresponding to $\sigma ,k=1,\dots ,p$.

**Example**

**3.**

- (i)
- as $TD=\left[\begin{array}{ccc}0& 1& 0\\ \frac{\sqrt{3}}{2}& 0& 0\\ -\frac{\sqrt{3}}{2}& 0& 0\end{array}\right]$, ${e}_{i}TD\ne 0$ for all $i=1,2,3$.
- (ii)
- We have ${\left[{e}_{1}TD\right]}_{2},{\left[{e}_{2}TD\right]}_{1},{\left[{e}_{3}TD\right]}_{1}\ne 0$. Here $({A}_{1},{B}_{2}),({A}_{2}+H,{B}_{1})$ and $({A}_{1}-H,{B}_{2})$ are controllable.
- (iii)
- Here, ${A}_{1}$ and ${A}_{3}-H$ has a common eigenvalue, $\sigma =1$. The corresponding left eigenvectors are, respectively, $\xi =\left[\begin{array}{ccc}-2& 0& 1\end{array}\right]$ and $\nu =\left[\begin{array}{ccc}1& 0& 0\end{array}\right]$. Clearly, $({e}_{1}TD\otimes \xi )B\ne 0$ and $({e}_{3}TD\otimes \nu )B\ne 0$.

**Theorem**

**3**

- (i)
- ${e}_{i}TD\ne 0$ for all $i=1,\dots ,N$;
- (ii)
- $({A}_{i}+{\lambda}_{i}H,B)$ is controllable, for $i=1,2,\cdots ,N$;
- (iii)
- If matrices ${A}_{{i}_{1}}+{\lambda}_{{i}_{1}}H,{A}_{{i}_{2}}+{\lambda}_{{i}_{2}}H,\dots ,{A}_{{i}_{p}}+{\lambda}_{{i}_{p}}H({\lambda}_{{i}_{k}}\in \sigma \left(C\right),k=1,\dots ,p,\phantom{\rule{4pt}{0ex}}where\phantom{\rule{4pt}{0ex}}p>1)$ have a common eigenvalue σ, then$({e}_{{i}_{1}}TD)\otimes ({\xi}_{{i}_{1}}^{1}B),\cdots ,({e}_{{i}_{1}}TD)\otimes ({\xi}_{{i}_{1}}^{{\gamma}_{{i}_{1}}}B),\dots ,({e}_{{i}_{p}}TD)\otimes ({\xi}_{{i}_{p}}^{1}B),\dots ,({e}_{{i}_{p}}TD)\otimes ({\xi}_{{i}_{p}}^{{\gamma}_{{i}_{p}}}B)$are linearly independent vectors, where ${\gamma}_{{i}_{k}}\ge 1$ is the geometric multiplicity of σ for ${A}_{{i}_{k}}+{\lambda}_{{i}_{k}}H$ and ${\xi}_{{i}_{k}}^{l}(l=1,\dots ,{\gamma}_{{i}_{k}})$ are the left eigenvectors of ${A}_{{i}_{k}}+{\lambda}_{{i}_{k}}H$ corresponding to $\sigma ,k=1,\dots ,p$.

#### Controllability Results in a Special Network Topology

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

- (i)
- Every node have external control input.
- (ii)
- $\left({A}_{i}+{c}_{ii}{H}_{i},{B}_{i}\right)$ is controllable for all $i=1,2,\dots N$.

**Proof.**

**Example**

**4.**

- (i)
- From Figure 4, it is clear that all the nodes have external control input.
- (ii)
- $({A}_{1},{B}_{1}),({A}_{2},{B}_{2})$ and $({A}_{3}+{H}_{3},{B}_{3})$ are controllable.

## 5. Conclusions and Future Scope of Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LTI | Linear Time-Invariant |

MIMO | Multi Input Multi Output |

## References

- Strogatz, S.H. Exploring complex networks. Nature
**2001**, 410, 268–276. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wang, X.F.; Chen, G. Complex networks: Small-world, scale-free and beyond. IEEE Circuits Syst. Mag.
**2003**, 3, 6–20. [Google Scholar] [CrossRef][Green Version] - Xiang, L.; Chen, F.; Ren, W.; Chen, G. Advances in network controllability. IEEE Circuits Syst. Mag.
**2019**, 19, 8–32. [Google Scholar] [CrossRef] - Kalman, R.E. On the general theory of control systems. IRE Trans. Autom. Control
**1959**, 4, 110. [Google Scholar] [CrossRef] - Lin, C.T. Structural controllability. IEEE Trans. Autom. Control
**1974**, 19, 201–208. [Google Scholar] - Hautus, M.L. Controllability and observability conditions of linear autonomous systems. Ned. Akad. Wet.
**1969**, 72, 443–448. [Google Scholar] - Glover, K.; Silverman, L. Characterization of structural controllability. IEEE Trans. Autom. Control
**1976**, 21, 534–537. [Google Scholar] [CrossRef] - Mayeda, H. On structural controllability theorem. IEEE Trans. Autom. Control
**1981**, 26, 795–798. [Google Scholar] [CrossRef] - Tarokh, M. Measures for controllability, observability and fixed modes. IEEE Trans. Autom. Control
**1992**, 37, 1268–1273. [Google Scholar] [CrossRef] - Tanner, H.G. On the controllability of nearest neighbor interconnections. In Proceedings of the 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No. 04CH37601), Nassau, Bahamas, 14–17 December 2004; Volume 3, pp. 2467–2472. [Google Scholar]
- Rahmani, A.; Mesbahi, M. Pulling the strings on agreement: Anchoring, controllability, and graph automorphisms. In Proceedings of the 2007 American Control Conference, New York, NY, USA, 9–11 July 2007; pp. 2738–2743. [Google Scholar]
- Liu, X.; Lin, H.; Chen, B.M. Graph-theoretic characterisations of structural controllability for multi-agent system with switching topology. Int. J. Control
**2013**, 86, 222–231. [Google Scholar] [CrossRef] - Yazıcıoğlu, A.Y.; Abbas, W.; Egerstedt, M. Graph distances and controllability of networks. IEEE Trans. Autom. Control
**2016**, 61, 4125–4130. [Google Scholar] [CrossRef][Green Version] - Farhangi, H. The path of the smart grid. IEEE Power Energy Mag.
**2009**, 8, 18–28. [Google Scholar] [CrossRef] - Wuchty, S. Controllability in protein interaction networks. Proc. Natl. Acad. Sci. USA
**2014**, 111, 7156–7160. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gu, S.; Pasqualetti, F.; Cieslak, M.; Telesford, Q.K.; Alfred, B.Y.; Kahn, A.E.; Medaglia, J.D.; Vettel, J.M.; Miller, M.B.; Grafton, S.T.; et al. Controllability of structural brain networks. IEEE Nat. Commun.
**2015**, 6, 1–10. [Google Scholar] [CrossRef][Green Version] - Bassett, D.S.; Sporns, O. Network neuroscience. Nat. Neurosci.
**2017**, 20, 353–364. [Google Scholar] [CrossRef][Green Version] - Zhou, T. On the controllability and observability of networked dynamic systems. Automatica
**2015**, 52, 63–75. [Google Scholar] [CrossRef][Green Version] - Wang, L.; Chen, G.; Wang, X.; Tang, W.K. Controllability of networked MIMO systems. Automatica
**2016**, 69, 405–409. [Google Scholar] [CrossRef][Green Version] - Wang, L.; Wang, X.; Chen, G. Controllability of networked higher-dimensional systems with one-dimensional communication. In Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences; Royal Society: London, UK, 2017; Volume 375, p. 20160215. [Google Scholar]
- Hao, Y.; Duan, Z.; Chen, G. Further on the controllability of networked MIMO LTI systems. Int. J. Robust Nonlinear Control
**2018**, 28, 1778–1788. [Google Scholar] [CrossRef] - Wang, P.; Xiang, L.; Chen, F. Controllability of heterogeneous networked MIMO systems. In Proceedings of the 2017 International Workshop on Complex Systems and Networks (IWCSN), Doha, Qatar, 8–10 December 2017; pp. 45–49. [Google Scholar]
- Xiang, L.; Wang, P.; Chen, F.; Chen, G. Controllability of directed networked MIMO systems with heterogeneous dynamics. IEEE Trans. Control Netw. Syst.
**2019**, 7, 807–817. [Google Scholar] [CrossRef] - Ajayakumar, A.; George, R.K. A Note on Controllability of Directed Networked System with Heterogeneous Dynamics. IEEE Trans. Control Netw. Syst.
**2022**, 2022, 1–4. [Google Scholar] [CrossRef] - Kong, Z.; Cao, L.; Wang, L.; Guo, G. Controllability of Heterogeneous Networked Systems with Non-identical Inner-coupling Matrices. IEEE Trans. Control Netw. Syst.
**2022**, 9, 867–878. [Google Scholar] [CrossRef] - Ajayakumar, A.; George, R.K. Controllability of networked systems with heterogeneous dynamics. Math. Control Signals Syst.
**2023**, 1–20. [Google Scholar] [CrossRef] - Horn, R.A.; Johnson, C.R. Topics in Matrix Analysis; Cambridge University Press: Cambridge, UK, 1994. [Google Scholar]
- Horn, R.A.; Johnson, C.R. Matrix Analysis; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Rugh, W.J. Linear System Theory; Prentice-Hall, Inc.: Kalamazoo, MI, USA, 1996. [Google Scholar]

**Figure 4.**Take ${c}_{12}={c}_{13}={c}_{23}={c}_{33}=1,$ otherwise ${c}_{ij}=0$ and ${d}_{1}={d}_{2}={d}_{3}=1$.

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**MDPI and ACS Style**

Ajayakumar, A.; George, R.K.
Controllability of a Class of Heterogeneous Networked Systems. *Foundations* **2023**, *3*, 167-180.
https://doi.org/10.3390/foundations3020015

**AMA Style**

Ajayakumar A, George RK.
Controllability of a Class of Heterogeneous Networked Systems. *Foundations*. 2023; 3(2):167-180.
https://doi.org/10.3390/foundations3020015

**Chicago/Turabian Style**

Ajayakumar, Abhijith, and Raju K. George.
2023. "Controllability of a Class of Heterogeneous Networked Systems" *Foundations* 3, no. 2: 167-180.
https://doi.org/10.3390/foundations3020015