TRANSTATION

I. Internal Model Principle and Repetitive Control [link]

1. Disturbance and reference Signal model

If reference signal or disturbance signal $f (t)$ satisfy differential equation as:

$\frac{d^{(n)}}{d t^{n}} f (t) + γ_{n - 1} \frac{d^{(n - 1)}}{d t^{n - 1}} f (t) + \dots + γ_{1} \frac{d}{d t} f (t) + γ_{0} f (t) = 0 (1.1)$

Then, taking Laplace Transform :

$\underset{Γ (s)}{\underset{⏟}{[s^{n} + γ_{n - 1} s^{n - 1} + \dots + γ_{0}]} F (s) = l (0, s)} (1.2)$

where $l (0, s)$ is a polynomial in $s$ series because of initial conditions $f (0), \dot{f} (0), \dots, f^{(n)} (0)$ .
We call $Γ (s)$ disturbance generating polynomial .

-> Examples:

$f (t)$	$F (s)$	$Γ (s)$
$α_{0}$ (constant)	$\frac{α_{0}}{s}$	$s$
$e^{a t}$	$\frac{1}{s - a}$	$s - a$
$α_{0} + α_{1} e^{a t}$	$\frac{α_{0}}{s} + \frac{α_{1}}{s - a}$	$s (s - a)$
$\sin (ω t)$	$\frac{ω}{s^{2} + ω^{2}}$	$s^{2} + ω^{2}$

Table 1. Laplace transform and Generating polynomial

2. Internal Model Principle

For a fundamental review of IMP (Internal Model Principle), see [exercise]

Figure 2.1 feedback control system [ref]

If disturbance $d (s)$ and reference $y^{r e f} (s)$ as is shown in Figure 2.1 both have $Γ (s)$ as the generating polynomial , then using a controller of the form:

$C_{c l} (s) : = \frac{N_{C} (s)}{D_{C} (s)} = \frac{N_{C} (s)}{Γ (s) {\tilde{D}}_{C} (s)} (2.1)$

in the standard one degree-of-freedom control architecture can asymptotically reject the effect of the disturbance and cause the output to track the reference.

Let the plant model $P_{c l} (s)$ in Figure 2.1 be $\frac{N_{P} (s)}{D_{P} (s)}$ , assume $Γ (s)$ is not a factor of $N_{P} (s)$ .

(a)

(b)

(c)

Figure 2.2 Diagram of (a) sensitivity ; (b) input sensitivity ; (c) complementary sensitivity .

The sensitivity function $T_{d o} (s)$ , a.k.a. the transfer function between an output disturbance and output as is shown in Figure 2.2(a) , the input sensitivity function $T_{d i} (s)$ , a.k.a. the transfer function between an intput disturbance and output as is shown in Figure 2.2(b) , the complementary sensitivity function $T (s)$ , a.k.a. the transfer function between the reference input and the output as is shown in Figure 2.2(c) , are obtained as:

$T_{d o} (s) : = \frac{y (s)}{d_{o} (s)} = \frac{Γ (s) {\tilde{D}}_{C} (s) D_{P} (s)}{Γ (s) {\tilde{D}}_{C} (s) D_{P} (s) + N_{C} (s) N_{P} (s)} (2.2 a)$

$T_{d i} (s) : = \frac{y (s)}{d_{i} (s)} = \frac{Γ (s) {\tilde{D}}_{C} (s) N_{P} (s)}{Γ (s) {\tilde{D}}_{C} (s) D_{P} (s) + N_{C} (s) N_{P} (s)} (2.2 b)$

$T (s) : = \frac{y (s)}{y^{r e f} (s)} = \frac{N_{C} (s) N_{P} (s)}{Γ (s) {\tilde{D}}_{C} (s) D_{P} (s) + N_{C} (s) N_{P} (s)} (2.2 c)$

Suppose that ${\tilde{D}}_{C} (s)$ and $N_{C} (s)$ have been designed to have all the roots of closed loop characteristic equation

${\tilde{Δ}}_{c l} (s) : = Γ (s) {\tilde{D}}_{C} (s) D_{P} (s) + N_{C} (s) N_{P} (s) (2.3)$

is negative in real parts. The response of the system to the output disturbance $d_{o} (s)$ with generating polynomial $Γ (s)$ is:

$y (s) = T_{o} (s) d_{o} (s) = T_{o} (s) \frac{l_{o} (0, s)}{Γ (s)} = \frac{{\tilde{D}}_{C} (s) D_{P} (s)}{{\tilde{Δ}}_{c l} (s)} l_{o} (0, s) (2.4)$

Similarily, the response of the system to the input disturbance $d_{i} (s)$ and the response of the system to the reference input may be derived as also. Notice

$y (s) = T (s) y^{r e f} (s) = T (s) \frac{l^{r e f} (0, s)}{Γ (s)} = \frac{N_{C} (s) N_{P} (s)}{Γ (s) {\tilde{Δ}}_{c l} (s)} l^{r e f} (0, s) (2.5)$

may has non-negative roots in real parts, still

$e (s) = (1 - T (s)) y^{r e f} (s) = (1 - T (s)) \frac{l^{r e f} (0, s)}{Γ (s)} = T_{o} (s) \frac{l^{r e f} (0, s)}{Γ (s)} = \frac{{\tilde{D}}_{C} (s) D_{P} (s)}{{\tilde{Δ}}_{c l} (s)} l^{r e f} (0, s) (2.6)$

has all roots' real parts in negative domain. Then $e (t \to \infty) = 0 \Rightarrow y (t) \to r (t)$ .

Figure 2.3 Control input time delayed

Transfer function:

$T (s) = \frac{y (s)}{y^{r e f} (s)} = \frac{e^{- t_{d} s} P_{c l} (s) C_{c l} (s)}{1 + e^{- t_{d} s} P_{c l} (s) C_{c l} (s)} = \frac{e^{- t_{d} s} N_{P} (s) N_{C} (s)}{D_{P} (s) D_{C} (s) + e^{- t_{d} s} N_{P} (s) N_{C} (s)} (2.7)$

Error function:

$e (s) = y^{r e f} (s) - y (s) = \frac{1}{1 + e^{- t_{d} s} P_{c l} (s) C_{c l} (s)} y^{r e f} (s) = \frac{D_{P} (s) D_{C} (s)}{D_{P} (s) D_{C} (s) + e^{- t_{d} s} N_{P} (s) N_{C} (s)} y^{r e f} (s) (2.8)$

Notes: Repetitive Control and Iterative Learning Control

I. Internal Model Principle and Repetitive Control [link]

1. Disturbance and reference Signal model

Generating polynomial:

If reference signal or disturbance signal $f (t)$ satisfy differential equation as:

Then, taking Laplace Transform :

where $l (0, s)$ is a polynomial in $s$ series because of initial conditions $f (0), \dot{f} (0), \dots, f^{(n)} (0)$ .
We call $Γ (s)$ disturbance generating polynomial .

-> Examples:

Table 1. Laplace transform and Generating polynomial

2. Internal Model Principle

For a fundamental review of IMP (Internal Model Principle), see [exercise]

Internal Model Principle:

Figure 2.1 feedback control system [ref]

If disturbance $d (s)$ and reference $y^{r e f} (s)$ as is shown in Figure 2.1 both have $Γ (s)$ as the generating polynomial , then using a controller of the form:

in the standard one degree-of-freedom control architecture can asymptotically reject the effect of the disturbance and cause the output to track the reference.

-> Derivation:

Let the plant model $P_{c l} (s)$ in Figure 2.1 be $\frac{N_{P} (s)}{D_{P} (s)}$ , assume $Γ (s)$ is not a factor of $N_{P} (s)$ .

(a)

(b)

(c)

Figure 2.2 Diagram of (a) sensitivity ; (b) input sensitivity ; (c) complementary sensitivity .

Suppose that ${\tilde{D}}_{C} (s)$ and $N_{C} (s)$ have been designed to have all the roots of closed loop characteristic equation

is negative in real parts. The response of the system to the output disturbance $d_{o} (s)$ with generating polynomial $Γ (s)$ is:

Similarily, the response of the system to the input disturbance $d_{i} (s)$ and the response of the system to the reference input may be derived as also. Notice

may has non-negative roots in real parts, still

has all roots' real parts in negative domain. Then $e (t \to \infty) = 0 \Rightarrow y (t) \to r (t)$ .

-> Exercise: Input signal is time-delayed ( $t_{d} \in ℝ > 0$ )

Figure 2.3 Control input time delayed

Transfer function:

Error function:

CONTENTS

I. Internal Model Principle and Repetitive Control

ELSEWHERE

Notes: Repetitive Control and Iterative Learning Control

I. Internal Model Principle and Repetitive Control [link]

1. Disturbance and reference Signal model

Generating polynomial:

If reference signal or disturbance signal f(t) satisfy differential equation as:

Then, taking Laplace Transform :

where l(0,s) is a polynomial in s series because of initial conditions f(0), f˙(0), …, f(n)(0) . We call Γ(s) disturbance generating polynomial .

-> Examples:

Table 1. Laplace transform and Generating polynomial

2. Internal Model Principle

For a fundamental review of IMP (Internal Model Principle), see [exercise]

Internal Model Principle:

Figure 2.1 feedback control system [ref]

If disturbance d(s) and reference yref(s) as is shown in Figure 2.1 both have Γ(s) as the generating polynomial , then using a controller of the form:

in the standard one degree-of-freedom control architecture can asymptotically reject the effect of the disturbance and cause the output to track the reference.

-> Derivation:

Let the plant model Pcl(s) in Figure 2.1 be NP(s)DP(s) , assume Γ(s) is not a factor of NP(s) .

(a)

(b)

(c)

Figure 2.2 Diagram of (a) sensitivity ; (b) input sensitivity ; (c) complementary sensitivity .

Suppose that D~C(s) and NC(s) have been designed to have all the roots of closed loop characteristic equation

is negative in real parts. The response of the system to the output disturbance do(s) with generating polynomial Γ(s) is:

Similarily, the response of the system to the input disturbance di(s) and the response of the system to the reference input may be derived as also. Notice

may has non-negative roots in real parts, still

has all roots' real parts in negative domain. Then e(t→∞)=0⇒y(t)→r(t) .

-> Exercise: Input signal is time-delayed ( td∈ℝ>0 )

Figure 2.3 Control input time delayed

Transfer function:

Error function:

CONTENTS

I. Internal Model Principle and Repetitive Control

ELSEWHERE

If reference signal or disturbance signal $f (t)$ satisfy differential equation as:

where $l (0, s)$ is a polynomial in $s$ series because of initial conditions $f (0), \dot{f} (0), \dots, f^{(n)} (0)$ .
We call $Γ (s)$ disturbance generating polynomial .

If disturbance $d (s)$ and reference $y^{r e f} (s)$ as is shown in Figure 2.1 both have $Γ (s)$ as the generating polynomial , then using a controller of the form:

Let the plant model $P_{c l} (s)$ in Figure 2.1 be $\frac{N_{P} (s)}{D_{P} (s)}$ , assume $Γ (s)$ is not a factor of $N_{P} (s)$ .

Suppose that ${\tilde{D}}_{C} (s)$ and $N_{C} (s)$ have been designed to have all the roots of closed loop characteristic equation

is negative in real parts. The response of the system to the output disturbance $d_{o} (s)$ with generating polynomial $Γ (s)$ is:

Similarily, the response of the system to the input disturbance $d_{i} (s)$ and the response of the system to the reference input may be derived as also. Notice

has all roots' real parts in negative domain. Then $e (t \to \infty) = 0 \Rightarrow y (t) \to r (t)$ .

-> Exercise: Input signal is time-delayed ( $t_{d} \in ℝ > 0$ )