Discrete-time PID-based model reference adaptive control

**Library:**Simscape / Electrical / Control / General Control

The Model Reference Adaptive Controller block implements discrete-time proportional-integral-derivative (PID) model reference adaptive control (MRAC). The three main components of an MRAC system are the reference model, the adjustment mechanism, and the controller.

The control equation is

${u}_{pid}(k)=\left[{K}_{p}+{K}_{i}\frac{{T}_{s}z}{z-1}+{K}_{d}\frac{z-1}{{T}_{s}z}\right]\text{e}(\text{k}),$

where:

*u*is the controller output._{pid}*K*is the proportional gain._{p}*K*is the integral gain._{i}*K*is the differential gain._{d}*T*is the sample time._{s}*e*is the error.

The reference model is the transfer function for the closed-loop system. This model captures the desired behavior of the closed-loop system. It is implemented as the discrete-time transfer function

${G}_{m}\left(z\right)=\frac{B(z)}{A(z)}.$

The adaptation mechanism adjusts the control action based on the error between the plant output and the reference model output as

$\theta =(y-{y}_{m}){y}_{m}\frac{-\gamma {T}_{s}z}{z-1},$

where:

*θ*is the adaptation parameter.*y*is the plant output.*y*is the reference model output._{m}*γ*is the learning rate.

Increasing the value of *γ* results in faster adaptation to plant
changes.

The adjusted control signal, *u*, is

$u(k)={u}_{pid}(k)\theta (k).$

[1] Butler, H. *Model-Reference Adaptive Control-From Theory to
Practice.* Upper Saddle River, NJ: Prentice Hall, 1992.