Let's say I have a linear time invariant dynamic system with following state space model
$$ \begin{eqnarray} \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x} &=& \mathbf{A}\cdot\mathbf{x} + \mathbf{B}\cdot\mathbf{u}\newline \mathbf{y} &=& \mathbf{C}\cdot\mathbf{x} \end{eqnarray} $$
$$ \mathbf{A}= \begin{bmatrix} \frac{-RS\cdot (LL + LM)^2 - RR\cdot LM^2}{LL\cdot LM\cdot (LL + LM)}&0&\frac{RR}{LL\cdot(LL + LM)}&0\newline 0&\frac{-RS\cdot (LL + LM)^2 - RR\cdot LM^2}{LL\cdot LM\cdot (LL + LM)}&0&\frac{RR}{LL\cdot(LL + LM)}\newline \frac{LM\cdot RR}{LL + LM}&0&\frac{-RR}{LL + LM}&0\newline 0&\frac{LM\cdot RR}{LL + LM}&0&\frac{-RR}{LL + LM} \end{bmatrix} $$
$$ \mathbf{B}= \begin{bmatrix} \frac{LL + LM}{LL\cdot LM}&0\newline 0&\frac{LL + LM}{LL\cdot LM}\newline 0&0\newline 0&0\newline \end{bmatrix} $$
$$ \mathbf{C}= \begin{bmatrix} 1&0&0&0 \newline 0&1&0&0 \end{bmatrix} $$
and I have a Luenberger observer for that system in following form
$$ \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathbf{x}} = \mathbf{A}_n\cdot\hat{\mathbf{x}} + \mathbf{B}_n\cdot\mathbf{u} + \mathbf{L}\cdot\left(\mathbf{y} - \hat{\mathbf{y}}\right) = \mathbf{A}_n\cdot\hat{\mathbf{x}} + \mathbf{B}_n\cdot\mathbf{u} + \mathbf{L}\cdot\left(\mathbf{C}\cdot\mathbf{x} - \mathbf{C}_n\cdot\hat{\mathbf{x}}\right) $$
$$ \mathbf{A}_n= \begin{bmatrix} \frac{-RS_n\cdot (LL_n + LM_n)^2 - RR_n\cdot LM_n^2}{LL_n\cdot LM_n\cdot (LL_n + LM_n)}& 0&\frac{RR_n}{LL_n\cdot(LL_n + LM_n)}&0\newline 0&\frac{-RS_n\cdot (LL_n + LM_n)^2 - RR_n\cdot LM_n^2}{LL_n\cdot LM_n\cdot (LL_n + LM_n)}& 0&\frac{RR_n}{LL_n\cdot(LL_n + LM_n)}\newline \frac{LM_n\cdot RR_n}{LL_n + LM_n}&0&\frac{-RR_n}{LL_n + LM_n}&0\newline 0&\frac{LM_n\cdot RR_n}{LL_n + LM_n}&0&\frac{-RR_n}{LL_n + LM_n} \end{bmatrix} $$
$$ \mathbf{B}_n= \begin{bmatrix} \frac{LL_n + LM_n}{LL_n\cdot LM_n}&0\newline 0&\frac{LL_n + LM_n}{LL_n\cdot LM_n}\newline 0&0\newline 0&0 \end{bmatrix} $$
$$\mathbf{C}_n = \mathbf{C}$$
$$ \mathbf{L}= \begin{bmatrix} l_1&0\newline 0&l_1\newline l_3&0\newline 0&l_3 \end{bmatrix} $$
$$ \begin{eqnarray} l_1 &=& (K - 1)\cdot\left(\frac{RSn\cdot(LLn + LMn)^2 + RRn\cdot LMn^2}{LLn\cdot LMn\cdot (LLn + LMn)} + \frac{RRn}{LLn + LMn}\right)\newline l_3 &=& (K^2 - 1)\cdot\left(\frac{RSn\cdot(LLn + LMn)^2 + RRn\cdot LMn^2}{LMn\cdot(LLn + LMn)} - \frac{LMn\cdot RRn}{LLn + LMn}\right) - (K - 1)\cdot\left(\frac{RSn\cdot(LLn + LMn)^2 + RRn\cdot LMn^2}{LMn\cdot(LLn + LMn)} + \frac{LLn\cdot RRn}{LLn + LMn}\right) \end{eqnarray} $$
I have attempted to describe in mathematical form the practical situation where there is an uncertainty in the system parameters. In other words the paramaters used by the observer are different from the actual parameters of the system
$$ \begin{eqnarray} \mathbf{A} &=& \mathbf{A}_n - \Delta\mathbf{A}, \\ \mathbf{B} &=& \mathbf{B}_n - \Delta\mathbf{B}, \\ \mathbf{C} &=& \mathbf{C}_n - \Delta\mathbf{C} \\ \end{eqnarray} $$
I would like to analyze the dynamics of the error estimate and steady state value of the error estimate in respect to uncertainties in individual parameters of the system. For that purposes I have derived below given equation describing the dynamics of the error estimate
$$ \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathbf{x}} = \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathbf{x}} - \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x} = \left(\mathbf{A}_n - \mathbf{L}\cdot\mathbf{C}_n\right)\cdot\tilde{\mathbf{x}} + \left(\Delta\mathbf{A} - \mathbf{L}\cdot\Delta\mathbf{C}\right)\cdot\mathbf{x} + \Delta\mathbf{B}\cdot\mathbf{u} $$
I have written a Scilab code for solving the ODEs of the system ($\mathbf{x}(t)$) along with the ODEs of the error estimate ($\tilde{\mathbf{x}}(t)$) for analysis of the error estimate dynamics. I have run the simulation with increasing value of the observer gain $K$ and 10% uncertainty in the system parameter $RR$ i.e. $RR = 1.1\cdot RR_n = 1.1\cdot 7.548\cdot 10^{-3}$. The values of the rest parameters were $RS=RS_n=7.4\cdot 10^{-3}, LM=LM_n=4.265\cdot 10^{-3}, LL=LL_n=0.231\cdot 10^{-3}$. The system input was $$ \mathbf{u}= \begin{bmatrix} 20.0\cdot\cos(2\pi\cdot 0.1\cdot t)\newline 20.0\cdot\sin(2\pi\cdot 0.1\cdot t) \end{bmatrix} $$
where $t\in\left<0, 20\right>\,s$
The simulation provided following outcomes
- $\tilde{\mathbf{x}}(t)$ with $K=1.5$
- $\tilde{\mathbf{x}}(t)$ with $K=150.0$
Based on the graphs it seems to me that $\tilde{x}_1$ and $\tilde{x}_2$ tend to decay to zero more rapidly with increasing value of gain $K$ (I expected this behavior because increasing value of $K$ results in "faster" poles (eigenvalues) of the $\mathbf{A}_n-\mathbf{L}\cdot\mathbf{C}_n$) but the $\tilde{x}_3$ and $\tilde{x}_4$ elements of the error estimate vector are more or less constant ($\sim 0.3$) independently from the value of the gain $K$ (I haven't expected this behavior). Can anybody help me to understand why the error estimates $\tilde{x}_3$ and $\tilde{x}_4$ don't decay with the increasing value of the observer gain? It is maybe related to my other question.
Notes regarding the structure of the $\mathbf{L}$ matrix:
The dynamic system for which the observer is being designed is the three phase induction motor. It's system matrix $\mathbf{A}$ has several elements dependent on the mechanical speed. I have decided to use the well known methodology for the $\mathbf{L}$ matrix design based on requirement that the observer poles are $K\times$ faster than the system poles. This requirement results in speed dependency of several elements of the $\mathbf{L}$ matrix. In respect to the fact that I have been analyzing error estimates in case the mechanical speed is zero I have the speed dependent elements of the system matrix $\mathbf{A}$ and the observer matrix $\mathbf{L}$ with zero values.
In my opinion the reason why the $\tilde{x}_3$ and $\tilde{x}_4$ error estimates don't decay with increasing value of $K$ in case there are uncertainties in parameters values is given by the equation describing the error estimate dynamics
$$ \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathbf{x}} = \left(\mathbf{A}_n - \mathbf{L}\cdot\mathbf{C}_n\right)\cdot\tilde{\mathbf{x}} + \left(\Delta\mathbf{A} - \mathbf{L}\cdot\Delta\mathbf{C}\right)\cdot\mathbf{x} + \Delta\mathbf{B}\cdot\mathbf{u} $$
In my situation where the only one source of uncertainty in parameters values is the value of $RR = 1.1\cdot RR_n$ following conditions hold $\Delta\mathbf{B}=\mathbf{0}$, $\Delta\mathbf{C}=\mathbf{0}$. It means that there will exist nonzero steady state value of the $\tilde{\mathbf{x}}_e = -\left(\mathbf{A}_n - \mathbf{L}\cdot\mathbf{C}_n\right)^{-1}\cdot\Delta\mathbf{A}\cdot\mathbf{x}$.
There is still one questionmark regarding the explanation above. I have originally thought that the state observers are in general a tool for overcomme just the problems with uncertainties in parameters values. According to the equation describing the dynamics of the error estimate it seems to me that it isn't truth which seems to me weird.