An intelligent control approach for defect-free friction stir welding

Abstract

An intelligent control approach is proposed as an alternative for the friction stir welding of an aluminum alloy. A validated empirical model is re-written from transfer functions to a set of ordinary differential equations, allowing to observe the force dynamics as a function of inputs of interest. A defect-free set-point is proposed for exploiting available labeled experimental data which defines operational boundaries and a region in which the probability of achieving defect-free welds with good mechanical properties is the highest. An intelligent controller in the fashion of a recurrent neural network is constructed. Computational experiments were carried out to verify the adequacy in disturbance rejection as well as to visualize the capabilities in achieving the proposed defect-free set-point by the controller. The intelligent approach is compared with a set of decoupled proportional-integral controllers and a linear model predictive control strategy. From this study, it is concluded that the intelligent controller shows superiority and good applicability for the studied problem.

Appendices

Appendix

Stability analysis

The existence of solutions of the proposed dynamic model in Eq. 1 is guaranteed because A is a matrix with constant coefficients. The functions d(t), v(t), and ω(t), and its first derivatives, are piece-wise continuous and bounded according to the behavior reported in Zhao et al. [42]. In fact, there is a fundamental matrix Φ(t) associated with the homogeneous system $\boldsymbol {u}^{\prime }(t) = A \boldsymbol {u}(t)$ such that an exponential matrix for the in-homogeneous system is e^At = Φ(t)Φ^− 1(0), and its solution is given by (refer to [31])

$$ \boldsymbol{u}(t)=e^{At}\boldsymbol{u}_{0}+{{\int}_{0}^{t}}e^{A(t-\xi)}\boldsymbol{G}(\xi) d\xi. $$

(5)

Based on this solution, the following stability results, in the sense of Liapunov [4], are obtained.

Theorem 1

For the regular linear system $\boldsymbol {u}^{\prime }(t)=A\boldsymbol {u}(t)$, the zero solution u(t)^∗≡0 is Lyapunov stable on t ≥ 0 if only if solution is bounded as $t\rightarrow \infty$.

Proof

Suppose that the zero solution u^∗(t) ≡0 is Lyapunov stable. By definition, there exists δ > 0 such that ∥u₀∥ < δ ⇒ ∥u(t)∥ < ε for all t ≥ 0. For the four linearly independent solutions {ϕ₁(t),ϕ₂,ϕ₃,ϕ₄} of $\boldsymbol {u}^{\prime }(t)=A\boldsymbol {u}(t)$, let us consider the fundamental matrix Φ(t) = [ϕ₁(t),ϕ₂,ϕ₃,ϕ₄] satisfying the initial condition $\displaystyle {{{\varPhi }}(0)=\frac {\delta }{2}I}$. Then, any solution for $\boldsymbol {u}^{\prime }(t)=A\boldsymbol {u}(t)$, with arbitrary initial conditions, can be written as u(t) = Φ(t)c where c is the column vector of components c_j (1 ≤ j ≤ 4) such that $\boldsymbol {c}={{\varPhi }}^{-1}(0)\boldsymbol {u}_{0}$. For any j = 1, 2, 3, 4, since $\Vert \boldsymbol {\phi }_{j}(0)\vert =\frac {\delta }{2}<\delta$, the Lyapunov stability implies that ∥ϕ_j(t)∥ < ε. Thus,

$$ \Vert\boldsymbol{u}(t)\Vert =\Vert {{\varPhi}}(t)\boldsymbol{c}\Vert= $$

$$ \biggl\Vert {\sum}_{j=1}^{4}c_{j}\boldsymbol{\phi}_{j}(t)\biggr\Vert\leq {\sum}_{j=1}^{2}|c_{j}|\Vert \boldsymbol{\phi}_{j}(t)\Vert\leq\varepsilon{\sum}_{j=1}^{4} |c_{j}|<\infty. $$

Conversely, suppose that every solution of $\boldsymbol {u}^{\prime }(t)=A\boldsymbol {u}(t)$ is bounded and let us consider Φ(t) be any fundamental matrix. Since u is bounded, then there exists M > 0 such that | | |Φ(t)| | | < M for all t ≥ 0 and any induced matrix norm | | |⋅| | |. Now, given any ε > 0, we choose $\displaystyle {\delta =\frac {\varepsilon }{M|\!|\!| {{\varPhi }}^{-1}(0)|\!|\!|}}>0$. As any solution of $\boldsymbol {u}^{\prime }(t)=A\boldsymbol {u}(t)$ has the form $\boldsymbol {u}(t)={{\varPhi }}(t){{\varPhi }}^{-1}(0)\boldsymbol {u}_{0}$, then for ∥u₀∥ < δ we have

$$ \Vert \boldsymbol{u}(t)\Vert\leq |\!|\!| {{\varPhi}}(t)|\!|\!| |\!|\!| {{\varPhi}}^{-1}(0)|\!|\!| \vert \boldsymbol{u}_{0}\Vert\\ \leq M|\!|\!| {{\varPhi}}^{-1}(0)|\!|\!|\delta=\varepsilon. $$

□

Using this theorem, the following stability results for the dynamic model introduced in Eq. 1 are obtained.

Theorem 2

All solutions of the dynamical model in Eq. 1 given by Eq. 5 have the same Lyapunov stability property as the zero solution of homogeneous linear system $\boldsymbol {u}^{\prime }(t)=A\boldsymbol {u}(t)$.

Proof

Let u^∗(t) be a solution of the dynamic model in Eq. 1, whose stability is to be determined. Let u(t) be another solution, a set v(t) := u(t) −u^∗(t). It follows that

$$ \begin{array}{@{}rcl@{}} \boldsymbol{v}^{\prime}(t)&=&A\boldsymbol{v}(t)\\ \boldsymbol{v}(0)&=&\boldsymbol{u}_{0}-\boldsymbol{u}^{\ast}_{0}. \end{array} $$

The Lyapunov stability of u^∗(t) implies that for all ε > 0, there exists δ > 0 such that $\Vert \boldsymbol {u}_{0}-\boldsymbol {u}^{\ast }_{0}\Vert <\delta$ ⇒ ∥u(t) −u^∗(t)∥ < ε for all t ≥ 0. In terms of v, this is equivalent to ∥v₀∥ < δ ⇒ ∥v(t)∥ < ε, which is the condition for the Lyapunov stability of the zero solution according to Theorem 1. □