I only know the "vanilla" use of a Kalman filter and I am currently trying to understand an article available here (the algorithm is presented in the 6 first pages) :
Adaptive Single Neuron Anti-Windup PID Controller Based on the Extended Kalman Filter Algorithm, open access
In which the plant is controlled by a PID with some anti wind-up scheme, but the trick is that the three PID gains (i.e $K_p$, $K_i$ and $K_d$ ) are dynamically estimated via an extended Kalman filter in which the state is $X = \begin{bmatrix}K_p \\ K_i \\K_d \end{bmatrix}$, what is considered to be the innovation is $e(k) = y_r(k)-y(k)$.
But then the matrix $H$ is defined as $H=\frac{\partial u}{\partial X}$ in the paper while, as far as I know, looking at the Kalman filtering formulas depicted in the paper (for instance the innovation defined above), it should be $H=\frac{\partial y}{\partial X}$.
Variable used :
- $u$ : the command sent to the plant
- $y$ : the measured output of the plant
- $y_r$ : the setpoint
- $K_p$, $K_i$ and $K_d$ : the three PID gains
To me, a Kalman filter estimate is an equilibrium point (that minimizes variance) between the state that best explains the measurement and the state you predicted and therefore, the absence of a model allowing to evaluate the measurement from the state (i.e $y_k = h(x_k)+noise$) should be a major deal breaker for the use of a Kalman filter. but I guess I'm wrong looking at the results of this paper.
Even if I forget this "mathematical" concern I also don't get how it proceeds to make the control error converge towards 0.