I have a DC motor system with three states. The states are:
- $x_1$ = position (degrees),
- $x_2$ = velocity (degrees per second)
- $x_3$ = current (Amps).
My plant input $u(t)$ is measured in volts.
The dynamics are such that:
$$\begin{gather} \frac{dX}{dt} = AX + Bu(t) \\ Y = CX + Du(t) \end{gather}$$ And: $$\begin{align} A &= \left[\begin{matrix} 0 & 1 & 0 \\ 0 & -\dfrac{b_m}{J_m} & \dfrac{K_m}{J_m} \\ 0 & -\dfrac{K_m}{L} & -\dfrac{R}{L} \end{matrix}\right] \\ B &= \left[\begin{matrix} 0 \\ 0 \\ \dfrac{1}{L} \end{matrix}\right] \\ C &= \left[\begin{matrix} 1 & 0 & 0 \end{matrix}\right] \\ D &= [0] \end{align}$$
Where $J_m$ is the rotor inertia, $b_m$ is the viscous damping, $R$ is the armature resistance, $k_m$ is the motor constant, and $L$ is the inductance.
My objective is to control the position of my DC motor.
The rank of my observability matrix is equal to the number of states in my system, so the system is fully observable.
$$Q = \text{rank} \left(\left[\begin{matrix} C \\ CA \\ CA^2 \end{matrix}\right]\right)$$
I am measuring the position using an encoder and hope to employ a reduced order observer to do this.
Partitioning the A matrix into quadrants yields the measured and observed dynamics:
$$A = \left[\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}\right]$$
In my case: $$\begin{align} A11 &= [0] \\ A12 &= \left[\begin{matrix} 1 & 0 \end{matrix}\right] \\ A21 &= \left[\begin{matrix} 0 \\ 0 \end{matrix}\right] \\ A22 &= \left[\begin{matrix} -\dfrac{b_m}{J_m} & \dfrac{K_m}{J_m} \\ -\dfrac{K_m}{L} & -\dfrac{R}{L} \end{matrix}\right] \end{align}$$
Now the Observability of my reduced state observer partition is:
$$Q_{Obs} = \text{rank} \left(\left[\begin{matrix} A_{12} \\ A_{12}A_{22} \end{matrix}\right]\right)$$
The rank of this is 2 (full rank).
I've seen in other projects that the following condition must be satisfied:
$$\begin{align} Q_{Obs} &\approx Q-(\text{number of states}-\text{number of measured states}) \\ 2 &\approx 3-(3-1) \\ 2 &\approx 1 \end{align}$$
- How many states need to be measured in a reduced order observer?
- Why is it that my system is fully observable, but I cannot measure all the states of my system?
- If I measure both position and velocity it seems that $Q_{Obs}=1$. Making the equation:
$$Q_{Obs} \approx Q-(\text{number of states}-\text{number of measured states})$$
simplify to:
$$\begin{align} 1 &\approx 3-(3-2) \\ 1 &\approx 2 \end{align}$$
Does this mean the system is not observable without measuring all three states?
4. If I assume the mechanical time constant is much slower than the electrical time constant (poles are closer to 0 in the real axis):
Can I measure only position and estimate velocity while neglecting the electrical dynamics?
Will this provide a robust estimator when implemented?