Notes: Repetitive Control and Iterative Learning Control

POINT:

$e^{t_d\frac{d}{dx}}$ in Figure 1 is called shift operator , or translation operator ^[ref1] , or evolution operator ^[ref2] .

$x(t-t_d)=\sum _{n=0}^{\infty }\frac{x^{\left(n\right)}\left(t\right)}{n!}{(t-t_d-t)}^n=\sum _{n=0}^{\infty }\frac{{\displaystyle \frac{d^{\left(n\right)}}{dt}x\left(t\right)}}{n!}{(-t_d)}^n=e^{-t_d\frac{d}{dt}}x\left(t\right) \left(2\right)$

where $t_d>0$ , see [ref2] .

State space formulation of discrete time repetitive control ^{[LIT5] p8} :

Consider periodic discrete time disturbance $d(k-N)=d\left(k\right)$ , where $N$ is the period. We can formulate a disturbance generating exo-system ^{[ref3] p1} :

$\left\{\begin{array}{l}\left(\begin{array}{c}{x}_{d1}\\ x_{d2}\\ x_{d3}\\ ⋮\\ x_{dN}\end{array}\right)(k+1)=\left(\begin{array}{ccccc}0& 1& 0& 0& \cdots \\ 0& 0& 1& 0& \cdots \\ ⋮& ⋮& ⋮& ⋮& 0\\ 0& \cdots & 0& \cdots & 1\\ 1& 0& \cdots & 0& 0\end{array}\right)\left(\begin{array}{c}{x}_{d1}\\ x_{d2}\\ x_{d3}\\ ⋮\\ x_{dN}\end{array}\right)\left(k\right)\\ d\left(k\right)=x_{d1}\left(k\right)\end{array}\right. \left(4\right)$

This can be written as

$\left\{\begin{array}{l}{x}_d(k+1)=A_d{x}_d\left(k\right)\\ d\left(k\right)=C_d{x}_{d\left(k\right)}\end{array}\right. \left(5\right)$

If the plant to be controlled is:

$\left\{\begin{array}{l}x(k+1)=Ax\left(k\right)+B\left(u\right(k)+d(k\left)\right)\\ y\left(k\right)=Cx\left(k\right)\end{array}\right. \left(6\right)$

We proceed just like the disturbance-estimate feedback approach to IMP. A control strategy would be:

$u\left(k\right)=-Ke\left(k\right)-\hat{d}\left(k\right) \left(7\right)$

where $\hat{d}\left(k\right)$ is the estimate of $d\left(k\right)$ obtained using an observer:

$\left\{\begin{array}{l}\left(\begin{array}{c}\hat{x}\\ {\hat{x}}_d\end{array}\right)(k+1)=\left(\begin{array}{cc}A& B{C}_d\\ 0& A_d\end{array}\right)\left(\begin{array}{c}\hat{x}\\ {\hat{x}}_d\end{array}\right)\left(k\right)+\left(\begin{array}{c}B\\ 0\end{array}\right)u\left(k\right)-L\left(C\hat{e}\right(k)-y(k\left)\right)\\ \hat{d}\left(k\right)=\left[\begin{array}{cc}0& C_d\end{array}\right]{\hat{x}}_d\left(k\right)\end{array}\right. \left(8\right)$

where $L$ is the observer gain chosen such that all the eigenvalues of $\left(\begin{array}{cc}A& B{C}_d\\ 0& A_d\end{array}\right)-LC$ have absolute values less than 1. i.e. the eigenvalues should lie within the unit disk centered at the origin. This is the stability criterion for discrete time systems.
A difficulty in this approach is that $N$ the dimension of $x_d\left(k\right)$ is typically very large. The dimension, $N$ is the ratio of period of the periodic disturbance to the sampling time. This makes designing a stable observer difficult.
Moreover, $N$ is not always integer multiple of sampling time, how to deal with it?
# Kalman Filter # Observer Further reading at fractional order state space

Figure 2 State observer of repetitive system ^{ref4 p3}