Understanding Notations of Matrix Calculus in Controller Tuning Article

Question

I have been working my way through the paper "Iterative feedback tuning: theory and applications" (Hjalmarsson, Gevers et al IEEE Control Systems Magazine , vol. 18, no. 4, pp. 26-41, Aug. 1998) (link to PDF) which seems to be fairly widely cited.

I'm struggling a bit with the notation and I’d appreciate a little confirmation as to whether I’m on the right track.

The paper involves minimising a quadratic criterion function $J(\boldsymbol\rho)$ for a closed-loop system with a (known) controller parameterised by a vector $\boldsymbol\rho$, through an iterative solution of:

$ \frac{\partial J}{\partial\boldsymbol\rho}(\boldsymbol\rho)=\tfrac1N\Bigg(\sum_{t=1}^N\tilde y_t(\boldsymbol\rho)\,\frac{\partial\tilde y_t}{\partial\boldsymbol\rho}(\boldsymbol\rho)\ +\ \lambda\sum_{t=1}^Nu_t(\boldsymbol\rho)\,\frac{\partial u_t}{\partial\boldsymbol\rho}(\boldsymbol\rho)\Bigg)=0 $$\phantom{tab}$(eq 11 of ref.)

where $\tilde y_t(\boldsymbol\rho)$ and $u_t(\boldsymbol\rho)$ are the discrete time values of the error signal and controller output for some parameters $\boldsymbol\rho$.

Later the paper derives approximations of the partial derivatives at the current iteration, such as:

$ \frac{\partial\tilde y}{\partial\boldsymbol\rho}(\boldsymbol\rho_i) \approx \frac1{C_r(\boldsymbol\rho_i)}\Bigg(\bigg(\frac{\partial C_r}{\partial\boldsymbol\rho}(\boldsymbol\rho_i)-\frac{\partial C_y}{\partial\boldsymbol\rho}(\boldsymbol\rho_i)\bigg)\,y_3(\boldsymbol\rho_i)\,+\,\frac{\partial C_y}{\partial\boldsymbol\rho}(\boldsymbol\rho_i)\,y_2(\boldsymbol\rho_i)\Bigg) $$\phantom{tab}$(eq 19 of ref.)

where $y_2(\boldsymbol\rho_i)$ and $y_3(\boldsymbol\rho_i)$ are discrete time signals measured at $\boldsymbol\rho=\boldsymbol\rho_i$, and $C_r(\boldsymbol\rho_i)$, $C_y(\boldsymbol\rho_i)$ are the known transfer functions of the controller.

My current interpretation of these expressions is as follows:

1. Discrete time signals such as $y_2(\boldsymbol\rho_i)$ can be viewed as ordered sets of sample values.
2. Multiplying a signal by a function such as $C_r(\boldsymbol\rho_i)$ corresponds to a mapping of the ordered set (e.g. via a difference equation).
3. A term such as $\frac{\partial C_r}{\partial\boldsymbol\rho}(\boldsymbol\rho_i)$ is a vector of functions composed of the partial derivatives of $C_r$ with respect to the components of $\boldsymbol\rho$, evaluated at $\boldsymbol\rho=\boldsymbol\rho_i$. That is: $ \frac{\partial C_r}{\partial\boldsymbol\rho}(\boldsymbol\rho_i)= \begin{bmatrix} \tfrac{\partial C_r}{\partial\rho_1}(\boldsymbol\rho_i)\\ \tfrac{\partial C_r}{\partial\rho_2}(\boldsymbol\rho_i)\\ \vdots\\ \tfrac{\partial C_r}{\partial\rho_{n_\rho}}(\boldsymbol\rho_i)\\ \end{bmatrix} $
4. Multiplying a signal by a partial derivative, as in $\frac{\partial C_y}{\partial\boldsymbol\rho}(\boldsymbol\rho_i)\,y_2(\boldsymbol\rho_i)$, results in a vector of ordered sets (signals) due to applying each of the partial derivative functions to the signal. That is: $\frac{\partial C_r}{\partial\boldsymbol\rho}(\boldsymbol\rho_i)\,y(\boldsymbol\rho_i)= \begin{bmatrix} \Big\{\tfrac{\partial C_r}{\partial\rho_1}(\boldsymbol\rho_i)\,y(\boldsymbol\rho_i,t)\Big\}_{1\,\le\,t\,\le\,N}\\ \Big\{\tfrac{\partial C_r}{\partial\rho_2}(\boldsymbol\rho_i)\,y(\boldsymbol\rho_i,t)\Big\}_{1\,\le\,t\,\le\,N}\\ \vdots\\ \Big\{\tfrac{\partial C_r}{\partial\rho_{n_\rho}}(\boldsymbol\rho_i)\,y(\boldsymbol\rho_i,t)\Big\}_{1\,\le\,t\,\le\,N}\\ \end{bmatrix}$
5. In the expression for the criterion, quantities such as the one above are summed over $t$ to produce a vector of real values.

Does this look right?

Answer 1

I think you're mostly right.

I am not sure what you mean in (4).
You may look at $C, {y}_{i}$ as multivariate functions.
Their input is a vector and output is a scalar.

A simple rule for matrix calculus is the gradient dimension must match the input dimensions.
Hence all gradients with respect to $\boldsymbol{\rho}$ are vectors.

Now, if $\boldsymbol{y}$ is a vector matrix, namely the input is a vector and the output is a vector its "Gradient" is actually the Jacobian where each row is the gradient of the specific multi variate function.

I guess this is what you meant in the matrix in (4).

The minimization happens where each gradient is zero.

A great resource for Matrix Calculus is given by: MIT - 18.S096 - Matrix Calculus for Machine Learning and Beyond.