README.md

MODELS

The Step Response Mode: FSRModel

The FSRModel is a C++ object creating the dynamics of a general linear step response model. The following equations and matrices are avaliable by calling the FSRModel functionality.

Output prediction equation

$$ Y(k+P) = \boldsymbol{\Theta} \Delta U(k+M)+ \boldsymbol{\Phi} \Delta \tilde{U}(k)+\Psi \tilde{U}(k-N)+B(k) $$

Theta-matrix definition:

$$ \boldsymbol{S} \triangleq\left[\begin{array}{cccc} s_1 & 0 & \cdots & 0 \\ s_2 & s_1 & 0 & \vdots \\ \vdots & \vdots & \ddots & 0 \\ s_M & s_{M-1} & \cdots & s_1 \\ s_{M+1} & s_M & \cdots & s_2 \\ \vdots & \vdots & \ddots & \vdots \\ s_P & s_{P-1} & \cdots & s_{P-M+1} \end{array}\right] $$

$$ \boldsymbol{\Theta} \triangleq\left[\begin{array}{cccc} \boldsymbol{S}_{11} & \boldsymbol{S}_{12} & \cdots & \boldsymbol{S}_{1 n_{MV}} \\ \boldsymbol{S}_{21} & \cdots & \cdots & \boldsymbol{S}_{2 n_{MV}} \\ \vdots & \vdots & \vdots & \vdots \\ \boldsymbol{S}_{n_{CV} 1} & \cdots & \cdots & \boldsymbol{S}_{n_{CV} n_{MV}} \end{array}\right]_{n_{CV} \cdot (P-W) \times M \cdot n_{MV}} $$

Phi-matrix definition: $$ \boldsymbol{\Phi}=\left[\begin{array}{cccc} \boldsymbol{\Phi_{1,1}} & \boldsymbol{\Phi_{1,2}} & \cdots & \boldsymbol{\Phi_{1, n_{M V}}} \ \boldsymbol{\Phi_{2,1}} & \boldsymbol{\Phi_{2,2}} & \cdots & \boldsymbol{\Phi_{2, n_{M V}}} \ \vdots & \vdots & \ddots & \vdots \ \boldsymbol{\Phi_{n_{C V}, 1}} & \boldsymbol{\Phi_{n_{C V}, 2}} & \cdots & \boldsymbol{\Phi_{n_{C V}, n_{M V}}} \end{array}\right]{n{C V}\left(P-W\right) \times \sum_{j=1}^{n_{M V}}\left(N-W-1\right)} $$

$$ \boldsymbol{\Phi_{i, j}} =\left[\begin{array}{ccccc} s_{W+1} & s_{W+2} & \ldots & s_{N-2} & s_{N-1} \\ s_{W+2} & s_{W+3} & \ldots & s_{N-1} & s_{N}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ s_{P+1} & s_{P+2} & \ldots & s_{N} & s_{N} \end{array}\right]_{\left(P-W\right) \times N-W-1} $$

Psi-matrix definition:

$$ \boldsymbol{\Psi} =\left[\begin{array}{cccc} \Psi_{1,1} & \Psi_{1,2} & \cdots & \Psi_{1, n_{M V}} \\ \Psi_{2,1} & \Psi_{2,2} & \cdots & \Psi_{2, n_{M V}} \\ \vdots & \vdots & \ddots & \vdots \\ \Psi_{n_{C V}, 1} & \Psi_{n_{C V}, 2} & \cdots & \Psi_{n_{C V}, n_{M V}} \end{array}\right]_{n_{C V}\left(P-W\right) \times n_{M V}} $$

$$ \Psi_{i, j} =\left[\begin{array}{c} s_{N} \\ s_{N} \\ \vdots \\ s_{N} \end{array}\right]_{\left(P-W\right) \times 1} $$

Simple first order model, siso_test

This is a module for generating customized step-response coefficients from a first order time delayed model.

$$ \frac{Y(s)}{U(s)}=\frac{K e^{-\theta s}}{\tau s+1}, $$

where $K$, $\theta$ and $\tau$ are respectively the gain, the time delay and the time constant of the system. $U(s)$ is modeled as a step input,

$$ U(s) = \frac{\Delta u}{s}, $$

where $\Delta u$ is the model input.