System-Decomposition Based Multi-level Control Approach

Abstract

A system-decomposition based multi-level control method is developed for complex forging processes with uncertainty in this chapter. The key idea in this proposed method is to decompose the system complexity into a group of simple sub-systems and the control task is shared by a group of simple sub-controllers. Under this framework, a sequence control strategy is developed to help these sub-controllers to handle the coupling between sub-systems, which can ensure the desirable global control performance for the complex forging process.

Appendices

Appendix A

Derivation of the sliding mode surface (10.19), we have

$$ \begin{aligned} & \dot{s}_{1} = C_{1} \dot{e}(t) + C_{2} \ddot{e}(t) \\ & \quad = C_{1} (\dot{r} - \dot{X}_{1} ) + C_{2} (\ddot{r} - \ddot{X}_{1} ) \\ \end{aligned} $$

(A.1)

According to the variable structure theory [5, 18], if

$$ s_{1}^{T} \dot{s}_{1} < 0 $$

(A.2)

Then the system can reach the sliding surface and thus is stable and can obtain the satisfactory tracking performance.

From the sub-system 1 in (10.13) and the sliding mode surface (10.19), we have

$$ \begin{aligned} s_{1}^{T} \dot{s}_{1} & = s_{1}^{T} \{ C_{1} (\dot{r} - \dot{X}_{1} ) + C_{2} (\ddot{r} - \ddot{X}_{1} )\} \\ & = s_{1}^{T} \{ C_{1} \dot{r} - C_{1} X_{2} + C_{2} \ddot{r} - C_{2} \dot{X}_{2} \} \\ \end{aligned} $$

(A.3)

From (10.13) and (10.21), we have

$$ \dot{X}_{2} = \varepsilon - C_{2}^{ - 1} C_{1} X_{2} + C_{2}^{ - 1} C_{1} \dot{r} + \ddot{r} + s_{1} + \beta $$

(A.4)

Inserting (A.4) into (A.3), we have

$$ \begin{aligned} s_{1}^{T} \dot{s}_{1} & = s_{1}^{T} C_{2} ( - s_{1} - \beta - \varepsilon ) \\ & = - s_{1}^{T} C_{2} s_{1} - \beta_{1} C_{2,(1,1)} s_{1,1} \text{sgn} (s_{1,1} ) \cdots \beta_{n} C_{2,(n,n)} \text{sgn} (s_{n,n} ) - s_{1}^{T} C_{2} \varepsilon \\ & = - s_{2}^{T} C_{2} s_{1} - \beta_{1} C_{2,(1,1)} \left| {s_{1,1} } \right| - \cdots - \beta_{n} C_{2,(n,n)} \left| {s_{n,n} } \right| - (\varepsilon_{1} C_{2,(1,1)} s_{1,1} + \cdots + \varepsilon_{n} C_{2,(n,n)} s_{n,n} ) \\ \end{aligned} $$

(A.5)

Since C₂ is diagonal and positive matrix, its every element is positive. Thus, we have

$$ \begin{aligned} s_{1}^{T} \dot{s}_{1} & \le - s_{1}^{T} C_{2} s_{1} - \beta_{1} C_{2,(1,1)} \left| {s_{1,1} } \right| - \cdots - \beta_{n} C_{2,(n,n)} \left| {s_{n,n} } \right| + (\left| {\varepsilon_{1} } \right|C_{2,(1,1)} \left| {s_{1,1} } \right| + \cdots + \left| {\varepsilon_{n} } \right|C_{2,(n,n)} \left| {s_{n,n} } \right|) \\ & = - s_{2}^{T} C_{2} s_{1} + (\left| {\varepsilon_{1} } \right| - \beta_{1} )C_{2,(1,1)} \left| {s_{1,1} } \right| + \cdots + (\left| {\varepsilon_{n} } \right| - \beta_{n} )C_{2,(n,n)} \left| {s_{n,n} } \right| \\ \end{aligned} $$

(A.6)

Since $\left[ {\begin{array}{*{20}c} {\beta_{1} } & \cdots & {\beta_{n} } \\ \end{array} } \right]^{T}$ is the bound of the modeling error $\varepsilon = \left[ {\begin{array}{*{20}c} {\varepsilon_{1} } & \cdots & {\varepsilon_{n} } \\ \end{array} } \right]^{T}$ and $\left| {\varepsilon_{i} } \right| \le \beta_{i}$, we can obtain

$$ \left| {\varepsilon_{i} } \right| - \beta_{i} \prec 0\quad (i = 1, \cdots ,n) $$

(A.7)

According to (A.6) and (A.7), the following inequality is satisfied

$$ s_{1}^{T} \dot{s}_{1} \le 0 $$

(A.8)

Thus, the sub system 1 is stable and the sliding mode s₁ = 0 will be reached in a finite time. In the sliding mode s₁ = 0, the tracking error e(t) will converge to zero through choosing the suitable parameters C₁ and C₂, which means that the working plate position X ₁(t) can track the reference signal r(t). Thus, the sub system 1 can obtain the satisfactory tracking performance.

Appendix B

From (10.13) and (10.22), we have

$$ \dot{P}_{i} = \dot{P}_{i,r} + \frac{{\beta_{t} }}{{V_{i} }}\{ \alpha \,\text{sgn} (s_{2} ) - (\Delta g_{i} - \Delta \tilde{g}_{i} )\} $$

(B.1)

Derivation of the sliding mode surface s ₂ and consideration of (B.1), we have

$$ \begin{aligned} \dot{s}_{2} & = \dot{P}_{i,r} - \dot{P}_{i} \\ & = \frac{{\beta_{t} }}{{V_{i} }}\{ \alpha \,\text{sgn} (s_{2} ) - (\Delta g_{i} - \Delta \tilde{g}_{i} )\} \\ \end{aligned} $$

(B.2)

Choosing a Lyapunov function as $V = 0.5S_{2}^{2}$ and considering Eq. (B.2) and the coefficient $\beta_{t} /V_{i} > 0$, we have

$$ \begin{aligned} \dot{V} & = s_{2} \dot{s}_{2} \\ & = \frac{{\beta_{t} }}{{V_{i} }}\{ - s_{2} a\,\text{sgn} (s_{2} ) - s_{2} (\Delta g_{i} - \Delta \tilde{g}_{i} )\} \\ & = \frac{{\beta_{t} }}{{V_{i} }}\{ - \left| {s_{2} } \right|a + s_{2} (\Delta g_{i} - \Delta \tilde{g}_{i} )\} \\ & \le \frac{{\beta_{t} }}{{V_{i} }}\{ - \left| {s_{2} } \right|a + \left| {s_{2} } \right|(\left| {\Delta g_{i} - \Delta \tilde{g}_{i} } \right|)\} \\ & = \frac{{\beta_{t} }}{{V_{i} }}\{ - \left| {s_{2} } \right|(a - \left| {\Delta g_{i} - \Delta \tilde{g}_{i} } \right|)\} \\ \end{aligned} $$

(B.3)

Since α is the bound of the modeling error $\Delta g_{i} - \Delta \tilde{g}_{i}$ and

$$ \left| {\Delta g_{i} - \Delta \tilde{g}_{i} } \right| \prec a $$

(B.4)

From (B.3) and (B.4), we have

$$ \dot{V} < 0 $$

(B.5)

Thus, the sub system 2 is stable and can obtain the satisfactory tracking performance.