Matlab Exercise

Chapter 6: オブザーバーと出力フィードバック

---

6.2 微分信号を利用した状態の復元

6.2.1 差分近似による速度の復元

6.2.2 入出力信号の時間微分を利用した状態変数の復元

$\left[\begin{array}{c}\mathit{\eta }\left(t\right)\\ \dot{\mathit{\eta }}\left(t\right)\\ \ddot{\mathit{\eta }}\left(t\right)\\ ⋮\\ {\mathit{\eta }}^{(n-1)}\left(t\right)\end{array}\right]=\left[\begin{array}{c}\overline{\mathit{C}}\\ \overline{\mathit{C}}\mathit{A}\\ \overline{\mathit{C}}{\mathit{A}}^2\\ ⋮\\ \overline{\mathit{C}}{\mathit{A}}^{n-1}\end{array}\right]\mathit{x}\left(t\right)+\left[\begin{array}{cccc}\mathbf{0}& \mathbf{0}& \cdots & \mathbf{0}\\ \overline{\mathit{C}}\mathit{B}& \mathbf{0}& \cdots & \mathbf{0}\\ \overline{\mathit{C}}\mathit{A}\mathit{B}& \overline{\mathit{C}}\mathit{B}& \cdots & \mathbf{0}\\ ⋮& ⋮& ⋮& ⋮\\ \overline{\mathit{C}}{\mathit{A}}^{n-2}\mathit{B}& \overline{\mathit{C}}{\mathit{A}}^{n-3}\mathit{B}& \cdots & \overline{\mathit{C}}\mathit{B}\end{array}\right]\left[\begin{array}{c}\mathit{u}\left(t\right)\\ \mathit{u}\left(t\right)\\ ⋮\\ {\mathit{u}}^{(n-2)}\left(t\right)\end{array}\right] (6.1)$

to be simplified,

$\mathit{\alpha }\left(t\right)={\mathit{V}}_o\mathit{x}\left(t\right)+\mathit{E}\mathit{\beta }\left(t\right) (6.2)$

where $\mathit{\eta }\left(t\right)=\overline{\mathit{C}}\mathit{x}\left(t\right)$ , ${\mathit{V}}_o={\left[\begin{array}{ccccc}\overline{\mathit{C}}& \overline{\mathit{C}}\mathit{A}& \overline{\mathit{C}}{\mathit{A}}^2& \cdots & \overline{\mathit{C}}{\mathit{A}}^{n-1}\end{array}\right]}^T$ is the observability matrix. NOTICE that $n$ is the dimension of $A$ . ^[ref]

Equation (6.1) is somehow similar to prediction equation ^{[ref]page 35, pp.8, eq1.12}

$\left[\begin{array}{c}\mathit{y}(k_i+1|k_i)\\ \mathit{y}(k_i+2|k_i)\\ \mathit{y}(k_i+3|k_i)\\ ⋮\\ \mathit{y}(k_i+N_p|k_i)\end{array}\right]=\left[\begin{array}{c}\mathit{C}\mathit{A}\\ \mathit{C}{\mathit{A}}^{\mathbf{3}}\\ \mathit{C}{\mathit{A}}^2\\ ⋮\\ \mathit{C}{\mathit{A}}^{N_p}\end{array}\right]\mathit{x}\left(k_i\right)+\left[\begin{array}{ccccc}\mathit{C}\mathit{B}& \mathbf{0}& \mathbf{0}& \cdots & \mathbf{0}\\ \mathit{C}\mathit{A}\mathit{B}& \mathit{C}\mathit{B}& \mathbf{0}& \cdots & \mathbf{0}\\ \mathit{C}{\mathit{A}}^{\mathbf{2}}\mathit{B}& \mathit{C}\mathit{A}\mathit{B}& \mathit{C}\mathit{B}& \cdots & \mathbf{0}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \mathit{C}{\mathit{A}}^{N_p-1}\mathit{B}& \mathit{C}{\mathit{A}}^{N_p-2}\mathit{B}& \mathit{C}{\mathit{A}}^{N_p-3}\mathit{B}& \cdots & \mathit{C}{\mathit{A}}^{N_p-N_c}\mathit{B}\end{array}\right]\left[\begin{array}{c}\Delta \mathit{u}\left(k_i\right)\\ \Delta \mathit{u}(k_i+1)\\ \Delta \mathit{u}(k_i+2)\\ ⋮\\ \Delta \mathit{u}(k_i+N_c-1)\end{array}\right] (6.3)$

To distinguish, 1. the dimensions of those two are:
Equation (6.1) :

$rn\times 1=rn\times n·n\times 1+rn\times (n-1)·p(n-1)\times 1$

where $n$ , $r$ and $p$ are dimensions of state variable, output and input respectively.

Equation (6.3) :

$N_p(n+r)\times 1=N_p(n+r)\times n·(n+r)\times 1+N_p(n+r)\times p(N_c-1)·p(N_c-1)\times 1$

where $N_p$ and $N_c$ are prediction horizon and control horizon respectively.

2. the receding equations are:

Equation (6.1) :

$\dot{\mathit{x}}\left(t\right)=\mathit{A}\mathit{x}\left(t\right)+\mathit{B}\mathit{u}\left(t\right) (6.4)$

Equation (6.3) :

$\left[\begin{array}{c}\Delta {\mathit{x}}_m(k+1)\\ \mathit{y}(k+1)\end{array}\right]=\left[\begin{array}{cc}{\mathit{A}}_m& \mathbf{0}\\ {\mathit{C}}_m& \mathit{I}\end{array}\right]\left[\begin{array}{c}\Delta {\mathit{x}}_m\left(k\right)\\ \mathit{y}\left(k\right)\end{array}\right]+\left[\begin{array}{c}{\mathit{B}}_m\\ {\mathit{C}}_m{\mathit{B}}_m\end{array}\right]\Delta \mathit{u}\left(k\right) (6.5)$

状態変数が下より復元できる

$\mathit{x}\left(t\right)=V_o^{-1}\left(\mathit{\alpha }\right(t)-\mathit{E}\mathit{\beta }(t\left)\right) (6.6)$

しかし、微分の定義より、

${\omega }_i\left(t\right)={\dot{\theta }}_i\left(t\right)=\underset{\Delta t\to 0}{\lim }\frac{{\theta }_i\left(t\right)-{\theta }_i(t-\Delta t)}{t-(t-\Delta t)}=\underset{\Delta t\to 0}{\lim }\frac{{\theta }_i\left(t\right)-{\theta }_i(t-\Delta t)}{\Delta t} (6.7)$

6.3 同一次元オブザーバーによる状態推定

6.3.1 同一次元オブザーバーの構成

結論

State estimator in equation (6.6) has a fatal shortcome that it cannot deal with noise, comparing to observer in equation (6.8)

$\dot{\hat{\mathit{x}}}\left(t\right)=\mathit{A}\hat{\mathit{x}}\left(t\right)+\mathit{B}\mathit{u}\left(t\right)-\mathit{L}\left(\mathit{\eta }\right(t)-\hat{\mathit{\eta }}(t\left)\right) (6.8)$

Q: 1. HOW TO WORK OUT $\hat{\mathit{x}}\left(t\right)$ ?
2. HOW TO SELECT OBSERVER GAIN $L$ ?
3. HOW TO DESIGN EMBEDDED CODE?

A:1. It costs time iterating to converge.

6.4 同一次元オブザーバーを利用した出力フィードバック制御

A:2.

$\left[\begin{array}{c}\dot{\mathit{x}}\left(t\right)\\ \dot{\hat{\mathit{x}}}\left(t\right)\end{array}\right]=\left[\begin{array}{c}\mathit{A}\mathit{x}\left(t\right)+\mathit{B}\left({\mathit{C}}_k\hat{\mathit{x}}\right(t)+\mathit{H}{\mathit{y}}^{ref}(t)\\ {\mathit{A}}_k\hat{\mathit{x}}\left(t\right)+{\mathit{B}}_k\overline{\mathit{C}}\mathit{x}\left(t\right)+\mathit{B}\mathit{H}{\mathit{y}}^{ref}\left(t\right)\end{array}\right]=\left[\begin{array}{cc}\mathit{A}& \mathit{B}{\mathit{C}}_k\\ {\mathit{B}}_k\overline{\mathit{C}}& {\mathit{A}}_k\end{array}\right]\left[\begin{array}{c}\mathit{x}\left(t\right)\\ \hat{\mathit{x}}\left(t\right)\end{array}\right]+\left[\begin{array}{c}\mathit{B}\mathit{H}\\ \mathit{B}\mathit{H}\end{array}\right]{\mathit{y}}^{ref}\left(t\right) (6.9)$

where ${\mathit{A}}_k=\mathit{A}+\mathit{B}\mathit{K}+\mathit{L}\overline{\mathit{C}}, {\mathit{B}}_k=-\mathit{L}, {\mathit{C}}_k=\mathit{K}$ .

which can be simplified as

$$\begin{gathered} {\dot{\mathit{x}}}_{cl}\left(t\right)={\mathit{A}}_{cl}{\mathit{x}}_{cl}\left(t\right)+{\mathit{B}}_{cl}{\mathit{y}}^{ref}\left(t\right) (6.10) \\ {\mathit{A}}_{cl}:=\left[\begin{array}{cc}\mathit{A}& \mathit{B}{\mathit{C}}_k\\ {\mathit{B}}_k\overline{\mathit{C}}& {\mathit{A}}_k\end{array}\right]=\left[\begin{array}{cc}\mathit{A}& \mathit{B}\mathit{K}\\ -\mathit{L}\overline{\mathit{C}}& \mathit{A}+\mathit{B}\mathit{K}+\mathit{L}\overline{\mathit{C}}\end{array}\right], \\ {\mathit{B}}_{cl}=\left[\begin{array}{c}\mathit{B}\mathit{H}\\ \mathit{B}\mathit{H}\end{array}\right] \end{gathered}$$

分離定理：状態フィードバックゲイン $K$ とオブザーバーゲイン $L$ を独立に設計すること。

That, Ackermann's formula are utilized twice: 1. work out $K$ and reference gain $H$ , 2. work out $L$

Q4: WHAT IS $H$ FOR?
Q5: CAN Ackermann's formula USED IN MOMI SYSTEMS?

Ackermann's formula in pole placement: ^{[ref1]page46, pp.82 [ref2]}

$\left\{\begin{array}{l}\mathit{K}=-\mathit{e}{\mathit{V}}_c^{-1}{\mathit{\Delta }}_{\mathbf{A}}\\ \mathit{L}=-{\mathit{\Delta }}_{\mathbf{A}}{\mathit{V}}_o^{-1}{\mathit{e}}^T\end{array}\right. (6.11)$

where ${\mathit{V}}_c=\left[\begin{array}{cccc}\mathit{B}& \mathit{A}\mathit{B}& \cdots & {\mathit{A}}^{n-1}\mathit{B}\end{array}\right], {\mathit{V}}_o={\left[\begin{array}{cccc}\overline{\mathit{C}}& \overline{\mathit{C}}\mathit{A}& \cdots & \overline{\mathit{C}}{\mathit{A}}^{n-1}\end{array}\right]}^T$ are controllability matrix and observability matrix respectively. $\mathit{e}=\left[\begin{array}{cccc}0& \cdots & 0& 1\end{array}\right]$ , ${\mathit{\Delta }}_{\mathbf{A}}:={\mathit{A}}^n+{\delta }_{n-1}{\mathit{A}}^{n-1}+\cdots +{\delta }_1\mathit{A}+{\delta }_0\mathit{I}$ , where ${\delta }_0, {\delta }_1, \dots , {\delta }_{n-1}$ can be derived according to [ref]^{page70, pp.131例6.5} .

A4: The method here using $H$ is tracking control mentioned in [ref]^{page53, pp.96, chap.5} , which cannot deal with disturbance as is stated in [ref]^{page54, pp.98, sec.5.1.3} .

A5: Ackermann's formula should NEVER be used in MOMI systems, see [ref]^{page46, pp.83, sec.4.3.4} . Conclusively, when MI, $K$ は無数に存在する、 when MO, $L$ は無数に存在する。 check [ref]^{page47, pp.85} for pole placement of MIMO systems.

A3: For embedded coding, we should work out $K$ , $H$ and $L$ ahead of main loop as a setup step. Then fulfill 2 source files with functionalities of 同一次元オブザーバー and 出力フィードバック形式のコントローラ as is shown in figure 6.2 . The code of main loop can refer to [ref] .