Plant identification

In control systems, a plant is usually modeled by an ordinary differential equation

$$\dot{X} = f_c(X, U)$$

where $X \quad C^1:\mathbb{R} -> \mathbb{R}^n$ is the state of the system and $U \quad C^0:\mathbb{R} -> \mathbb{R}^m$ is its input

When working with digital systems, it is common practice to discretize the above continuous-time model by considering the input $U$ to be constant on the time interval $[t, t+dt]$ and use a difference equation like

$$X_{k+1} = f_d(X_k, U_k)$$

As a universal function approximator, an Artificial Neural Networks (ANN) can be trained to approximate the plant’s dynamics.

Fig1. - Input-Output Plant Identification.

Input/Output identification

NARMA model

$$y_{k+d} = h\left( y_k, y_{k-1} \dots y_{k-n}, u_k, u_{k-1} \dots u_{k-m} \right)$$

Control-Affine model

$$y_{k+d} = h\left( y_k, y_{k-1} \dots y_{k-n}, u_{k-1} \dots u_{k-m} \right) + g\left( y_k, y_{k-1} \dots y_{k-n}, u_{k-1} \dots u_{k-m} \right).u_k$$

Examples

• First Order LTI

• Second Order LTI

• Electric DC motor

• Robot Arm

• Mobile Inverted Pendulum