I am designing a PI controller for an electrical system consisting of several parallel-connected branches, where each branch current will be controlled by a converter.
\begin{equation}\label{eq:current_dynamics} \begin{split} \frac{d}{dt}I_k=\frac{1}{\left(L_{line}-L\right)}&\left[-R_{line}I_k+\frac{3\sqrt{3}}{\pi}A_fV_{m_k}\right.\\ &+\frac{3\sqrt{3}}{\pi}A_dV_{m_k}\cos(\alpha_k)-V_{DC_y}^{*}\\ &\left.-\sum_{j\neq k}\left(R_{line}I_j+L_{line}\frac{d}{dt}I_j\right)\right]\quad k\in[1,N] \end{split} \end{equation}
Where $I_k$ and $I_j$ are my currents (states), $V_{m_k}$ are voltage sources that I do not control (their value may change regardless of my will) and $\alpha_k$ are my thyristor firing angles, the only variables that I can to use to control my system. The voltage source $V_{DC_y}^{*}$ is much more of a fixed input to the system, so can be considered as constant. All other parameters such as $L$, $R_{line}$ and $L_{line}$ are constants.
I have here some questions:
- My first attempt was defining:
\begin{equation} \begin{split} \mathbf{x}&=\left[I_1\quad I_2\quad\cdots\quad I_N\right]^{T} \in\mathbb{R}^{N}\\ \mathbf{d}&=\left[V_{m_1}\quad V_{m_2}\quad\cdots\quad V_{m_N}\quad V_{DC_y}^{*}\right]^{T}\in\mathbb{R}^{N+1}\\ \mathbf{u}&=\left[\alpha_1\quad \alpha_2\quad\cdots\quad \alpha_N\right]^{T}\in\mathbb{R}^{N} \end{split} \end{equation}
Then, I have coupling not only by other other states, but also by their derivatives. Also, I have a nonlinearity where I have $V_{m_k}\cos(\alpha_k)$. So what I did was to define:
\begin{equation} \mathbf{h}(\mathbf{u},\mathbf{d})=\frac{1}{\left(L_{line}-L\right)}\begin{bmatrix}\frac{3\sqrt{3}}{\pi}A_fV_{m_1}+\frac{3\sqrt{3}}{\pi}A_dV_{m_1}\cos(\alpha_1)-V_{DC_y}^{*}\\ \frac{3\sqrt{3}}{\pi}A_fV_{m_2}+\frac{3\sqrt{3}}{\pi}A_dV_{m_2}\cos(\alpha_2)-V_{DC_y}^{*}\\ \vdots\\ \frac{3\sqrt{3}}{\pi}A_fV_{m_N}+\frac{3\sqrt{3}}{\pi}A_dV_{m_N}\cos(\alpha_N)-V_{DC_y}^{*}\end{bmatrix} \end{equation}
And rewrite the equation as:
\begin{equation} \begin{split} \frac{d}{dt}x_k+\frac{L_{line}}{\left(L_{line}-L\right)}\sum_{j\neq k}\frac{d}{dt}x_j=\frac{1}{\left(L_{line}-L\right)}&\left[-R_{line}x_k-\sum_{j\neq k}R_{line}x_j\right]+h_k \end{split} \end{equation}
From which I get the following nonlinear model:
\begin{equation} \mathbf{E}\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{h}(\mathbf{u},\mathbf{d}) \end{equation}
If I go through withe linearization, I can obtain the following:
\begin{equation} \delta\dot{\mathbf{x}}=\mathbf{E}^{-1}\mathbf{A}\delta\mathbf{x}+\mathbf{E}^{-1}\mathbf{B}\delta\mathbf{u}+\mathbf{E}^{-1}\mathbf{G}\delta\mathbf{d} \end{equation}
How do I design a PI controller from this? Should I treat $\mathbf{u}$ as a disturbance vector? What would be the procedure?
- The second attempt was defining:
\begin{equation} \begin{split} \mathbf{x}&=\left[I_1\quad I_2\quad\cdots\quad I_N\right]^{T} \in\mathbb{R}^{N}\\ \mathbf{u}&=\left[V_{DC_y}^{*} \quad V_{m_1}\quad V_{m_2}\quad\cdots\quad V_{m_N}\quad \alpha_1\quad \alpha_2\quad\cdots\quad \alpha_N \right]^{T}\in\mathbb{R}^{2N+1} \end{split} \end{equation}
which basically means that I put all the system inputs, whether I can manipulate them or not, into vector $\mathbf{u}$.
Following the same procedure gives me in the end:
\begin{equation} \delta\dot{\mathbf{x}}=\mathbf{E}^{-1}\mathbf{A}\delta\mathbf{x}+\mathbf{E}^{-1}\mathbf{B}\delta\mathbf{u} \end{equation}
Which is the classical state-space representation, but now I have an input vector $\mathbf{u}$ that comprises both inputs that I can and cannot manipulate.
How should I deal with this? I want a PI control that generates the $\alpha$ firing angles such that the currents $I_k$ follow $I_k^{*}$ reference values.
Thank you!