Resilience of Linear Networks

Repository for the paper "Losing Control of your Network? Try Resilience Theory", which is available on ArXiv and submitted to the journal IEEE Transactions on Control of Network Systems. Here are all the codes and data necessary to reproduce the simulations of the paper. We study three different scenarios:

• a fully actuated 3-component network losing control over one of its actuators;
• an underactuated 3-component network losing control over one of its actuators;
• the IEEE 39-bus system losing control over one of its generator buses.

Fully actuated 3-component network

This is an academic example aimed at illustrating the resilience theory for fully actuated systems. The dynamics of the network are:

$$\dot \chi(t) = \begin{bmatrix} -1 & 0.3 \\ 0.3 & -1 \end{bmatrix} \chi(t) + \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} \hat{u}(t) + \begin{bmatrix} 0.3 \\ 0.3 \end{bmatrix} x_q(t),$$ $$\dot x_q(t) = -x_q(t) + u_q(t) + 2w_q(t) + \begin{array} 0.3 & 0.3 \end{array} \chi(t),$$

with

$$\chi(0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad x_q(0) = 0, \quad \hat{u}(t) = \begin{bmatrix} \hat{u}_1(t) \\ \hat{u}_1(t) \end{bmatrix} \in \hat{\mathcal{U}} = [-1, 1]^2,$$

$u_q(t) \in \mathcal{U}_q = [-1, 1]$ and $w_q(t) \in \mathcal{W}_q = [-1, 1]$. After a cyber-attack we lose control authority over the actuator producing $w_q$ Notice that control input $u_q$ cannot overcome all undesirable inputs $w_q$. Then, we want to bound how far away from the origin can $x_q$ be driven by $w_q$ despite our best effort with $u_q$. To do so, we first solve Lyapunov equations $A_q^\top P_q + P_q A_q = -Q_q$ and $(\hat{A}+\hat{D})^\top \hat{P} + \hat{P} (\hat{A}+\hat{D}) = -\hat{Q}$ with the function lyap on MATLAB:

$$Q_q = 1, \quad P_q = 0.5, \quad \hat{Q} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},\quad \text{and}\quad \hat{P} = \begin{bmatrix} 0.23 & 0.05 \\ 0.05 & 0.5 \end{bmatrix}.$$

Then, the resilient stabilizability conditions established in this paper are satisfied: $\gamma \gamma_q = 0.25 < \alpha \alpha_q = 0.7$ and $\gamma z_{max}^{P_q} = 0.5 < \alpha_q b_{min}^{\hat{P}} = 2$. As shown below, $\chi$ is indeed resiliently stabilizable in finite time by $\hat{B}\hat{u} = \frac{-\chi(t)}{ \| \chi(t) \| b_{min}^{\hat{P}}}$.

The numbers in parenthesis in the legend of the figures refer to equation numbers used in the paper. The state $x_q$ of the malfunctioning system can also be bounded even in the worst-case scenario as shown below.

These simulations are performed with test_full_actuation.m.

Underactuated 3-component network

We now modify $\hat{B}$ not to be full rank anymore, while maintinging controllability of the pair $(\hat{A} + \hat{D}, \hat{B})$. The updated network dynamics for $\chi$ are

$$\dot \chi(t) = \begin{bmatrix} -1 & 0.3 \\ 0.3 & -1 \end{bmatrix} \chi(t) + \begin{bmatrix} 2 \\ 0 \end{bmatrix} \hat{u}(t) + \begin{bmatrix} 0.3 \\ 0.3 \end{bmatrix} x_q(t),$$

and the dynamics of $x_q$ are left unchanged:

$$\dot x_q(t) = -x_q(t) + u_q(t) + 2w_q(t) + \begin{array} 0.3 & 0.3 \end{array} \chi(t),$$

with

$$\chi(0) = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad x_q(0) = 0, \quad \hat{u}(t) = \begin{bmatrix} \hat{u}_1(t) \\ \hat{u}_1(t) \end{bmatrix} \in \hat{\mathcal{U}} = [-1, 1]^2,$$

$u_q(t) \in \mathcal{U}_q = [-1, 1]$ and $w_q(t) \in \mathcal{W}_q = [-1, 1]$. As previously, we assume a loss of control authority over $w_q$ that control input $u_q$ is still unable to counteract. Because of the underactuation of dynamics $\chi$, we cannot use the same controler as above. Instead, we determine a gain matrix $K$ and positive definite matrices $\hat{P}$ and $\hat{Q}$ with MATLAB functions lqr and lyap:

$$K = \begin{bmatrix} 0.64 & 0.15 \end{bmatrix}, \quad \hat{P} = \begin{bmatrix} 0.22 & 0.04 \\ 0.04 & 0.5 \end{bmatrix},\quad \text{and} \quad \hat{Q} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$$

Then, the linear feedback control $\hat{u}(t) := -K\chi(t) \in [-1, 1]$ is admissible as shown below.

With this controller the resilience condition $\gamma \gamma_q = 0.24 < \alpha \alpha_q = 0.98$ holds. Thus, network state $\chi$ can be resiliently bounded as illustrated below.

The state $x_q$ of the malfunctioning system can also be bounded even in the worst-case scenario as shown below.

These simulations are performed with test_underactuation.m.

IEEE 39-bus system

We now study the resilience of the IEEE 39-bus system. This network is composed of 29 load buses numbered 1 to 29 on the figure below, and 10 generator buses numbered 30 to 39. The picture of the IEEE 39-bus system is taken from [1].

We obtain the linearized network equation from [2]. After the loss of control authority over generator bus 39, we split the network state between $x_q = \big( \delta_{39}, \dot \delta_{39} \big)$ and $\chi$ for the remaining states. The malfunctioning dynamics are

$$\begin{bmatrix} \dot \delta_{39}(t) \\ \dot \omega_{39}(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -18.6 & -11.2 \end{bmatrix} \begin{bmatrix} \delta_{39}(t) \\ \omega_{39}(t) \end{bmatrix} + \begin{bmatrix} 0 \\ 0.22 \end{bmatrix} w_q(t) + D_{q,\_} \chi(t).$$

We choose initial states $\chi(0) = \mathbf{1}$, $\delta_{39}(0) = 0$ rad and $\omega_{39}(0) = 0$ Hz. Since pair $(\hat{A} + \hat{D}, \hat{B})$ is controllable, we can find a stabilizing gain matrix $K$ for the network dynamics. However, the resilient stability condition $\gamma \gamma_q < \alpha \alpha_q$ is not satisfied. Indeed, $\gamma \gamma_q = 6.3 \times 10^4$, while $\alpha \alpha_q = 5.7 \times 10^{-3}$. This magnitude difference leads to the exponential divergence of bounds (15) and (16), as seen below.

Bound (7) remains a reasonable bound for malfunctioning state $x_q$ over a much longer time horizon as illustrated below.

The choice of $K$ ensures admissibility of controller $\hat{u} = -K\chi$ by guaranteeing $\underset{i, t}{\max} |K\chi_i(t)| \leq 1$ as shown below.

Despite having $\gamma \gamma_q\gg \alpha \alpha_q$, the coupling does not destabilize states $x_q$ and $\chi$, which are both bounded, as shown on this last figure.

These simulations are performed with main_IEEE_net.m.

File Structure

• main_IEEE_net.m runs the simulation of the IEEE 39-bus network and compute all discussed bounds on the states $\chi$ and $x_q$.
• test_full_actuation.m runs the simulation of the fully actuated 3-component network.
• test_underactuation.m runs the simulation of the underactuated 3-component network.
• Pnorm.m calculates the $P$-norm of a vector $x$ as $\|x\|_P = \sqrt{x^\top P x}$, where $P$ is a positive definite matrix.

Citation

@article{bouvier2023networks,  
  title = {Losing Control of your Network? Try Resilience Theory},   
  author = {Jean-Baptiste Bouvier and Sai Pushpak Nandanoori and Melkior Ornik},    
  journal = {},    
  year = {2023},   
  volume = {},
  pages = {},
  doi = {}
}

Contributors

• Jean-Baptiste Bouvier
• Sai Pushpak Nandanoori
• Melkior Ornik

References

[1] T. Athay, R. Podmore, and S. Virmani, “A practical method for the direct analysis of transient stability,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-98, no. 2, pp. 573 – 584, 1979.

[2] S. P. Nandanoori, S. Kundu, J. Lian, U. Vaidya, D. Vrabie, and K. Kalsi, “Sparse control synthesis for uncertain responsive loads with stochastic stability guarantees,” IEEE Transactions on Power Systems, vol. 37, no. 1, pp. 167 – 178, 2022.