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So I have a state-space system, \begin{eqnarray} \dot{\mathbf{x}}&=&A\mathbf{x} + B\mathbf{u}\tag{1}\label{EqSS}\\ \mathbf{y}&=&C\mathbf{x}\\ \end{eqnarray} where, \begin{eqnarray} \mathbf{x}&=&\begin{bmatrix}x_1&x_2& \cdots&x_{11} & x_{12}\end{bmatrix}^\intercal\tag{2}\\ \mathbf{u}&=&\begin{bmatrix}u_1&u_2&u_3 & u_4\end{bmatrix}^\intercal\\ A&\in&\mathbb{R}^{12\times 12}\\ B&\in&\mathbb{R}^{12\times 4}\\ C&=&I\in\mathbb{R}^{12\times 12} \end{eqnarray} I am using the following control law for tracking commanded state values $\mathbf{x}^c\in\mathbb{R}^{12\times 1}$, \begin{equation}\label{EqUctrl} \mathbf{u}=-K\left(\mathbf{x}-\mathbf{x}^c\right)=K\left(\mathbf{x}^c-\mathbf{x}\right)\tag{3} \end{equation} where gains matrix $K\in\mathbb{R}^{4\times 12}$ is calculated using the $\texttt{lqr(A,B,Q,N)}$ function in $\texttt{Matlab}$.

It turns out tracking performance is quite good for all states but $x_{12}$ when using my selections for state weight and input weight matrices $Q\in\mathbb{R}^{12\times 12}$ and $R\in\mathbb{R}^{4\times 4}$. State $x_{12}$ tends to $x^c_{12}$ as simulation time goes by but is typically offset by say 10 %.

Questions

  1. How do I go about augmenting an integrator state $\dot{x}_{13}=x^c_{12}-x_{12}$ to \eqref{EqSS} and \eqref{EqUctrl}?

  2. My current plan for augmenting an integrator state $\dot{x}_{13}=x^c_{12}-x_{12}$ to \eqref{EqSS} and \eqref{EqUctrl} is to,

    • add a 13th row and column to system matrix $A$ according to, \begin{eqnarray} \text{Row 13: }&&\begin{bmatrix} a_{13,1} & a_{13,2} & \cdots & a_{13,11} & a_{13,12} & a_{13,13} \end{bmatrix}\phantom{\hspace{1mm}}= \begin{bmatrix} 0 & 0 & \cdots & 0 & -1 & 0 \end{bmatrix}\\ \text{Column 13: }&&\begin{bmatrix} a_{1,13} & a_{2,13} & \cdots & a_{11,13} & a_{12,13} & a_{13,13} \end{bmatrix}^\intercal= \begin{bmatrix} 0 & 0 & \cdots & 0 & \phantom{-}0 & 0 \end{bmatrix} \end{eqnarray}
    • add a 13th row to input signal matrix $B$ according to, \begin{equation} \text{Row 13: }\begin{bmatrix} b_{13,1} & b_{13,2} & b_{13,3} & b_{13,4} \end{bmatrix} \end{equation}
    • with the augmented matrices $A,B$ at hand calculate a new gain matrix $K$ for the control law \eqref{EqUctrl}.

    Do these steps seem appropriate to you guys? I am a bit puzzled on how to do this because of the material I have found on the Internet,
    Lec 6: State Feedback, Controllability, Integral Action (p.31-33)
    DC Motor Position: State-Space Methods for Controller Design

    What confuses me perhaps is the control law \eqref{EqUctrl} I am using which is a bit different from the often occuring $u=r-K\mathbf{x}$. Perhaps is $u=r-K\mathbf{x}$ applicable to single input systems only, but I guess it depends on how you would specify $r$?

  3. As for the control law I am using \eqref{EqUctrl} I found it here,
    MIT: Maneuvering and Control of Marine Vehicles
    StackExchange: How to improve reference tracking using state feedback?

    The control law performs well (apart from "that" state) and works nicely with the system description I have in terms of matrix sizes. Can anyone explain how you would expand or re-write $u=r-K\mathbf{x}$ to \eqref{EqUctrl}, derive \eqref{EqUctrl} or perhaps refer to a book or technical paper where this is done? I am more used to seeing $u=r-K\mathbf{x}$.

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  • $\begingroup$ I could be wrong, but AFAIK, LQR is really meant for driving the states towards zero and not towards a commanded value. According to the aircraft example in Matlab's help page for lqr(), they reduce tracking error by increasing the weights for the corresponding state. They appear to provide a different function lqgtrack for tracking and it mentions an integrator. $\endgroup$
    – AJN
    Feb 9 at 14:31
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    $\begingroup$ You’re definitely correct about the purpose of LQR driving states to zero. Perhaps one could view this problem in terms of driving the state vector $\tilde{\mathbf{x}}=\mathbf{x}-\mathbf{x}^c$ to zero? I’m going to investigate that I think. As for my idea on integration action it turned out my plan worked, the state tends to the desired value having introduced the augmented integrator state. $\endgroup$
    – wolfiesax
    Feb 9 at 21:56
  • $\begingroup$ If your implementation worked, you can post it as an answer here. It will help future readers. Self answers are allowed in SE. $\endgroup$
    – AJN
    Feb 10 at 4:10
  • $\begingroup$ Ah ok. I will try to compile an answer then. $\endgroup$
    – wolfiesax
    Feb 10 at 8:41

1 Answer 1

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I can confirm for the system at hand, augmenting an integrator state $\dot{x}_{13}=x^c_{12}-x_{12}$ can successfully be carried out by adding a $13^{\text{th}}$,

  1. row and column to system matrix $A$, \begin{eqnarray} \text{Row 13: }&&\begin{bmatrix} a_{13,1} & a_{13,2} & \cdots & a_{13,11} & a_{13,12} & a_{13,13} \end{bmatrix}\phantom{\hspace{1mm}}= \begin{bmatrix} 0 & 0 & \cdots & 0 & -1 & 0 \end{bmatrix}\\ \text{Column 13: }&&\begin{bmatrix} a_{1,13} & a_{2,13} & \cdots & a_{11,13} & a_{12,13} & a_{13,13} \end{bmatrix}^\intercal= \begin{bmatrix} 0 & 0 & \cdots & 0 & \phantom{-}0 & 0 \end{bmatrix} \end{eqnarray}
  2. row to input signal matrix $B$, \begin{equation} \text{Row 13: }\begin{bmatrix} b_{13,1} & b_{13,2} & b_{13,3} & b_{13,4} \end{bmatrix}=\begin{bmatrix} 0 & 0 & 0 & 0 \end{bmatrix} \end{equation}

Carrying out 1. and 2. means we have the augmented matrix sizes for the system and the controller, \begin{eqnarray} \mathbf{x}&=&\begin{bmatrix}x_1&x_2& \cdots&x_{11} & x_{12} & x_{13}\end{bmatrix}^\intercal\\ \mathbf{u}&=&\begin{bmatrix}u_1&u_2&u_3 & u_4\end{bmatrix}^\intercal\\ A&\in&\mathbb{R}^{13\times 13}\\ B&\in&\mathbb{R}^{13\times 4}\\ C&=&I\in\mathbb{R}^{13\times 13}\\ K&\in&\mathbb{R}^{4\times 13}\\ \end{eqnarray}

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