Time-delay root-locus GUI

Repetitive control analysis

Theory

This example shows how repetitive control analysis can be used for repetitive control method. Repetitive control method uses time-delay to enable a system to track periodic reference signals or to suppress periodic disturbances. It is based on Internal model principle, meaning that the controller must include the same poles as the reference signal’s transfer function for system to achieve zero steady-state error.
The reference signal can be represented as sum of sine and cosine functions with fundamental frequency \(\omega_\mathrm{0}\) and its harmonics. Signal with many higher harmonics requires controller of higher order. This can by bypassed by the introduction of time-delay and it's infinite poles.

Signal in form of: \[r(t) = a_\mathrm{0} + \sum_{k=1}^{K} a_\mathrm{k} \sin(k \omega_\mathrm{0} t) + \sum_{l=1}^{L} b_\mathrm{l} \cos(l \omega_\mathrm{0} t) \] requieres controller with poles at: \[ s_\mathrm{p,s} = \pm k \omega_\mathrm{0} i, \quad \forall k = 1,\dots,K \text{ such that } a_\mathrm{k} \neq 0, \] \[ s_\mathrm{p,c} = \pm l \omega_\mathrm{0} i, \quad \forall l = 1,\dots,L \text{ such that } b_\mathrm{l} \neq 0, \] \[ s_\mathrm{p,0} = 0 \quad \text{if } a_\mathrm{0} \neq 0\] This leads to controller in form of: \[ K(s) = \frac{K}{1 - e^{- \frac{2 \pi}{\omega_\mathrm{0}} s}} \]

tdrlocus example

Assuming simple first order system: \[ H_\mathrm{1}(s) = \frac{1}{s+3} \] and reference periodic signal with fundamental frequancy \( \omega_\mathrm{0} = 1 \) rad/s, the controller can be designed as: \[ K_\mathrm{1}(s) = \frac{4}{1 - e^{- 2 \pi s}} \] Such system can be called as:

reg = [-5, 2, 0, 20];
num = "1";
den = "s+3";
tdrlocus(reg, num, den);