Problem 3 - Is the A matrix zero?

From MurrayWiki

Q: Is the A matrix zero in problem 3?

A: No it is not.

The problem dynamics are given as $\dot{z} = f(z,u)$ where $z = (x, y,θ) T$ and $u = (v,φ) T$ . Use $φ = 0$ for simplicity.

The estimator can be built directly from the system dynamics: $\dot{\hat{z}} = f(\hat{z},u) + L(y - C \hat{z})$ . As Kalman showed, the optimal observer gain is $L = P (t) C T R - 1 (t)$ , where $P (t)$ comes from a solution of the Riccati equation

$\dot{P} = AP + PA^{T} - P C^{T} R^{-1}CP + FQF^{T}$

Note, there are no random disturbances affecting the system dynamics, so F=0 in these equations. The entries of matrix $A (t)$ come from linearization of the observer: $A = \left.\frac{\partial f}{\partial \hat{z}}\right|_{\hat{z}_{ref},u}$ . If you utilize the A matrix with the initial state of the observer with the Matlab lqe or kalman command, you will receive an error. This is because these Matlab functions assume $\dot{P} = 0$ . This will not work in this problem due to its nonlinearity.

You will need to numerically integrate the Riccati equation to get $P (t)$ . Additionally, over time the observer state will also change, and the linearization A at time zero will not be valid for all time. Thus, your numerical integration will need to update the A matrix for each step of your iteration.

Suggestion: Utilize the same numerical integration call to integrate the original dynamics $\dot{z}$ , the estimator dynamics $\dot{\hat{z}}$ , and the Riccati Equation $\dot{P}$ simultaneously. This will allow you to simultaneously update the A matrix, P matrix, and L matrix.

--John

Problem 3 - Is the A matrix zero?

From MurrayWiki

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