Difference between revisions of "Problem 3 - Is the A matrix zero?"

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 $$z = (x,y,\theta)^{T}$$ and $$u=(v,\phi)^{T}$$. Use $$\phi =0$$ 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