Matlab Exercise

Figure 5.1 track control using feedforwad

Figure 5.2

Theorem. Internal Model Principle

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$\begin{array}{c}e\left(s\right)=y^{ref}\left(s\right)-y\left(s\right) \\ =y^{ref}\left(s\right)-P_{cl}\left(s\right)\left(d\right(s)+v(s\left)\right) \\ =y^{ref}\left(s\right)-P_{cl}\left(s\right)\left(d\right(s)+C_{cl}(s\left)e\right(s\left)\right)\\ \Rightarrow e\left(s\right)=\frac{y^{ref}\left(s\right)-P_{cl}\left(s\right)d\left(s\right)}{1+P_{cl}\left(s\right)C_{cl}\left(s\right)} \end{array} (5.24)$

Reference and disturbance are step signal:

$y^{ref}\left(t\right)={\alpha }_{1}f\left(t\right), d\left(t\right)={\alpha }_2\left(t\right)f\left(t\right)\Rightarrow y^{ref}\left(s\right)=\frac{{\alpha }_1}{s}, d\left(s\right)=\frac{{\alpha }_2}{s} (5.25)$

Decompose $P_{cl}\left(s\right)$ and $C_{cl}\left(s\right)$ in Num and Den:

$P_{cl}\left(s\right)=\frac{N_P\left(s\right)}{D_P\left(s\right)}, C_{cl}\left(s\right)=\frac{N_C\left(s\right)}{D_C\left(s\right)} (5.26)$

$e\left(s\right)=\frac{D_C\left(s\right)D_P\left(s\right){\alpha }_1-D_C\left(s\right)N_P\left(s\right){\alpha }_2}{s\left(D_P\right(s\left)D_C\right(s)+N_P(s\left)N_C\right(s\left)\right)}:=\frac{D_C\left(s\right)D_P\left(s\right){\alpha }_1-D_C\left(s\right)N_P\left(s\right){\alpha }_2}{s{\Delta }_{cl}\left(s\right)} (5.27)$

$s=p_1, \dots , p_N$ are the solutions of ${\Delta }_{cl}\left(s\right)=0$ , if all of the roots $p_1, \dots , p_N$ are zero, the feedback control system in Figure 5.3 is Internal stable .

If there are no cancellations of Num and Den in Equation (5.27), $e\left(s\right)=ℒ\left[e\right(t\left)\right]$ is:

$e\left(s\right)=\frac{h}{s}+\frac{k_1}{s-p_1}+\cdots +\frac{k_N}{s-p_N} (5.28)$

$e\left(t\right)={ℒ}^{-1}\left[e\right(s\left)\right]=h+k_1{e}^{p_{1}t}+\cdots +k_N{e}^{p_{N}t}\Rightarrow e_{\infty }=h\ne 0 (5.29)$

There is stationary error in Equation (5.29). Accordingly, modify construction of controller transfer function $C_{cl}\left(s\right)$ as:

$C_{cl}\left(s\right)=\frac{1}{s}\times \frac{N_C\left(s\right)}{D_C\left(s\right)} (5.30)$

$\begin{array}{c}e\left(s\right)=\frac{{\tilde{D}}_C\left(s\right)\left({\alpha }_1{D}_P\right(s)-{\alpha }_2{N}_P(s\left)\right)}{s{\Delta }_{cl}\left(s\right)} =\frac{{\tilde{k}}_1}{s-p_1}+\cdots +\frac{{\tilde{k}}_N}{s-p_N} \\ \Rightarrow e\left(t\right)={ℒ}^{-1}\left[e\right(s\left)\right]={\tilde{k}}_1{e}^{p_{1}t}+\cdots +{\tilde{k}}_N{e}^{p_{N}t}\Rightarrow e_{\infty }=0\end{array} (5.31)$

Internal model principle [ref] :

Condition : reference and disturbance are step signal
Target : With step target $y^{ref}\left(t\right)$ , input disturbance $d\left(t\right)$ , steady-state deviation is zero ( $e_{\infty }=0$ )
Necessary and sufficient condition :

1. Feedback control system in figure 5.3 is internal stable.
2. Transfer function ${\mathcal{P}}_{cl}\left(s\right)$ has no zeros at origin $s=0$ .
3. Transfer function ${\mathcal{C}}_{cl}\left(s\right)$ has at least one integrator $1/s$ .

Further [ref] [note] :

Condition : reference and disturbance are normal signals other than step ones.
Using generating polynomial instead of integrator (actually it's a differentiator) in the denominator of ${\mathcal{C}}_{cl}\left(s\right)$ , see [note] .

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5.2.2 状態フィードバック形式の積分型コントローラの設計

$K_{cl}:v\left(s\right)=C_{cl}\left(s\right)e\left(s\right), C_{cl}\left(s\right)=g\frac{1}{s}\Rightarrow K_{cl}:v\left(t\right)=g{\int }_0^{t}e\left(t\right)dt (5.32)$

For system as (p2):

In a view of cart #1 state, a state feedback for cart #1 and I-PD controller for cart #2 is designed as:

$\mathcal{K}:u\left(t\right)=-k_{P1}{z}_1\left(t\right)-k_{D1}{\dot{z}}_1\left(t\right)-k_{P2}{z}_2\left(t\right)+k_{I2}{\int }_0^{t}e\left(t\right)dt-k_{D2}{\dot{z}}_2\left(t\right) (5.33)$

which can be rewritten in 状態フィードバック形式( state variable feedback ) ^[ref] の積分型コントローラ as:

$\mathcal{H}:u\left(t\right)=\left[\begin{array}{cccc}-k_{P1}& -k_{D1}& -k_{P2}& -k_{D2}\end{array}\right]\left[\begin{array}{c}{z}_1\left(t\right)\\ {\dot{z}}_1\left(t\right)\\ z_2\left(t\right)\\ {\dot{z}}_2\left(t\right)\end{array}\right]+k_{I2}{\int }_0^{t}e\left(t\right)dt (5.34)$

State feedback style integrator controller:

$\mathcal{K}:u\left(t\right)=\mathit{k}\mathit{x}\left(t\right)+g\omega \left(t\right), \omega \left(t\right):={\int }_0^{t}e\left(t\right)dt (5.35)$

Consequently, does PD controller equal to feedback controller?

As [ref] said, "Proportional+Derivative (e.g. from PID) is a very similar to State Variable feedback for a second order system. The difference is in how you obtain the state variables." "PD says to use a derivative to obtain velocity in a 2nd order system."
In my opinion, PD controller is a output feedback, in many systems, say, position-velocity, voltage-current, etc. states can be obtained via derivation of output. Other system may use state observer.

Further reading:

Servo Motors and Industrial Control Theory.pdf chap3
Double Loop Control Design.pdf

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