Understanding an adaptative single neuron PID controller

Question

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.

Answer 1

The paper algebra is correct, it is $H$ in terms of $u$ and not $y$.

This paper is as best, misleading, and has some serious mistakes, for the following reasons:

1. The EKF is applied over the neuron only, and not over the system under control. The key win of EKF is to reduce variance over known systems, as an extension of KF for nonlinear models. In here, EKF is used just for training, not for filtering and or estimation. This is acceptable, provided everybody understand this.

This reason is the key for questioning why you would wish to use EKF for training a controller.

2. A controller is not stochastic. Does not have $w_k$ ($Q$) and $v_k$ ($R$) noises. Lets assume that noise comes from the uncertainty of not knowing $w_k$. At best, we would explain $Q$. But $R$ has no reason at all to have a value.

3. The additional term $\eta$, as adaptative, has nothing to do with EKF. It is a custom modification, such as RLS which is another adaptive algorithm.

4. The results?. Anybody will defend their own results. Call the creators of the compared methods, and they will train their own methods to get the best. Not a big deal on that.

There are more remarks, but this is enough for making the point. There is no reason at all to prove why this application would converge, or be optimal or robust in any sense.

Proposed EKF

Standard EKF

Edit

This Paper EKF is applied on the Controller (!?) with input output $e$ and $u$, not on the Plant with input output $u$ and $y$ (which should be the way). For the Paper EKF, $u$ is an output (!?), so $H={\partial h \over \partial x}$ is a derivative over the nonlinear component of the neuron, since $z=h(x)$ (in the Standard EKF) $\to u=h(\omega)$ (in the paper "way of thinking").

Besides, the $\eta$ factor obfuscates even more the Paper EKF. The term instead of $\eta K (y-y_r)$ should be $K (u-h(\omega))$. Again, why there is an stochastic uncertainty assumption (i.e. a noise in $u=h(\omega)+v$) between $u$ and $h(\omega)$ is an out of comprehension mistake from the authors and editors in wherever that paper got published.

In doing this, the Paper EKF in a huge naive way assumes the plant (inverse) response, instead of a proper dynamical model $G^{-1}(y-y_r)$, is simply a variant scalar $\eta$. Too disappointing.