Immersion and invariance-based adaptive wing rock control with nonlinear terminal manifold

Abstract

This paper develops a new adaptive control system based on the immersion and invariance (I&I) theory for the wing rock motion control using state variable feedback. An I&I-based adaptive control system is designed for the regulation of the roll angle trajectory to a nonlinear terminal manifold. The trajectory evolving on this manifold converges to the origin in a finite time. The control system includes a control module and a parameter estimator which can be tuned independently for shaping the responses. A special feature of this adaptive law is that once the estimated parameters coincide with the actual values, then subsequently these parameter estimates remain frozen, and adaptation stops. Furthermore the trajectories of the closed-loop system eventually evolve on a manifold in an extended state space. Simulation results are presented which show that the adaptive controller suppresses the wing rock motion of delta wing at different angles of attack, despite uncertainties.

Appendix

1.1 1. Computation of $\beta _1$ for $\phi _\mathrm{r}\ne 0$

For $\phi _\mathrm{r} \ne 0$, (29) gives

$$\begin{aligned} \beta _1= \int _0^ {\dot{\phi }} \gamma _1 [\phi ,\xi , \xi |\xi |, \phi ^3, \phi ^2 \xi ]^\mathrm{T} \lambda \nu |\xi -\dot{\phi }_\mathrm{r}|^{\nu -1} \text {d} \xi \nonumber \\ \end{aligned}$$

(60)

Note $\dot{\phi }$ has been replaced by variable $\xi$ within the integrand. Integral of each term in (60) is obtained now.

$$\begin{aligned}&\int _0^{\dot{\phi }} \phi |\xi -\dot{\phi }_\mathrm{r}|^{\nu -1} \hbox {d}\xi \nonumber \\&\quad =\phi \left[ \frac{ |\dot{\phi }-\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(\dot{\phi }-\dot{\phi }_\mathrm{r}) - \frac{ |\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(-\dot{\phi }_\mathrm{r}) \right] \doteq \phi B_1\nonumber \\&\int _0^{\dot{\phi }} \xi |\xi -\dot{\phi }_\mathrm{r}|^{\nu -1} \hbox {d}\xi =\frac{ \dot{\phi }|\dot{\phi }-\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(\dot{\phi }-\dot{\phi }_\mathrm{r}) \nonumber \\&\quad \quad -\,\int _0^{\dot{\phi }} \frac{ |\xi -\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(\xi -\dot{\phi }_\mathrm{r}) \hbox {d}\xi \nonumber \\&\quad = \frac{ \dot{\phi }|\dot{\phi }-\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(\dot{\phi }-\dot{\phi }_\mathrm{r})\nonumber \\&\quad \quad -\,\left[ \frac{ |\dot{\phi }-\dot{\phi }_\mathrm{r}|^{\nu +1}}{\nu (\nu +1)} - \frac{|\dot{\phi }_\mathrm{r}|^{\nu +1}}{\nu (\nu +1)} \right] \doteq B_2 \end{aligned}$$

(61)

$$\begin{aligned}&\int _0^{\dot{\phi }} \xi |\xi ||\xi -\dot{\phi }_\mathrm{r}|^{\nu -1}d \xi =\int _0^{\dot{\phi }}\xi ^2\hbox {sgn}(\xi ) |\xi -\dot{\phi }_\mathrm{r}| ^{\nu -1} \hbox {d}\xi \nonumber \\&\quad =\frac{ \dot{\phi }^2 \hbox {sgn}(\dot{\phi })|\dot{\phi }-\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(\dot{\phi }-\dot{\phi }_\mathrm{r}) \nonumber \\&\quad \quad -\,\int _0^{\dot{\phi }} \frac{ 2\xi \hbox {sgn}(\xi ) |\xi -\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(\xi -\dot{\phi }_\mathrm{r}) \hbox {d}\xi \nonumber \\&\quad =\frac{ \dot{\phi }^2 \hbox {sgn}(\dot{\phi })|\dot{\phi }-\dot{\phi }_\mathrm{r}|^\nu }{\nu } \hbox {sgn}(\dot{\phi }-\dot{\phi }_\mathrm{r}) \nonumber \\&\quad \quad -\,\frac{2 \hbox {sgn}(\dot{\phi })}{\nu } \left[ \frac{ \dot{\phi }|\dot{\phi }-\dot{\phi }_\mathrm{r}|^{\nu +1}}{\nu (\nu +1)} - \int _0^{\dot{\phi }} \frac{ |\xi -\dot{\phi }_\mathrm{r}|^{\nu +1}}{\nu (\nu +1)} \hbox {d}\xi \right] \nonumber \\&\quad \doteq B_3 \end{aligned}$$

(62)

Using equations for $B_i$, one obtains $\beta _1$ of the form

$$\begin{aligned} \beta _1=\gamma _1 \lambda \nu \left[ \begin{array}{cc} \phi B_1 \\ B_2 \\ B_3 \\ \phi ^3 B_1 \\ \phi ^2 B_2 \end{array} \right] \in R^5 \end{aligned}$$

(63)

The partial derivatives of $\beta _1$ with respect to $\hat{\Theta }$ and $\phi$ required to compute the update law for $\hat{\Theta }$ are

$$\begin{aligned}&\frac{\partial \beta _1}{\partial \hat{\Theta }}=0_{5 \times 6}\nonumber \\&\frac{ \partial \beta _1}{\partial \phi } =\gamma _1 \lambda \nu \left[ \begin{array}{cc} B_1 \\ 0 \\ 0 \\ 3\phi ^2 B_1 \\ 2\phi B_2 \end{array} \right] \in R^5 \end{aligned}$$

(64)

This completes the derivation of $\beta _1$.

1.2 2. Computation of $\beta _2$ for $\phi _\mathrm{r}=0$

Now the computation of $\beta _2$ is considered. As noted earlier, closed-loop responses are satisfactory for the choice of $\phi _\mathrm{r}=0$, in the sequel computation of $\beta _2$ is given by setting $\phi _\mathrm{r}=0$ for simplicity. Equation (29) gives

$$\begin{aligned} \beta _2= -\int _0^{\dot{\phi }} \gamma _2 w \lambda \nu |\dot{\phi }|^{\nu -1} \hbox {d}\dot{\phi }\end{aligned}$$

(65)

with

$$\begin{aligned} w= & {} -\varPsi _1^\mathrm{T}(\hat{\theta }_1 +\beta _1)-k_1 s-k_2 |s|^\mu \hbox {sgn}(s)\nonumber \\&-\,(\lambda \nu )^{-1}|\dot{\phi }|^{2-\nu } \hbox {sgn}(\dot{\phi })\nonumber \\ s= & {} \phi +\lambda |\dot{\phi }|^{\nu } \hbox {sgn}(\dot{\phi }) \end{aligned}$$

(66)

Now we can get $\beta _2$ by collecting all.

$$\begin{aligned}&\beta _2=\gamma _2 \lambda \nu \biggl [ \int _0^{\dot{\phi }} (\psi _1^\mathrm{T} |\dot{\phi }|^{\nu -1}\hbox {d}\dot{\phi }) \hat{\theta }_1 \nonumber \\&\quad \quad +\, \int _0^{\dot{\phi }} \psi _1^\mathrm{T} \beta _1|\dot{\phi }|^{\nu -1}\hbox {d}\dot{\phi }+ k_1\int _0^{\dot{\phi }} s |\dot{\phi }|^{\nu -1}\hbox {d}\dot{\phi }\nonumber \\&\quad \quad +\, k_2 \int _0^{\dot{\phi }} |s|^{\mu } \hbox {sgn}(s) |\dot{\phi }|^{\nu -1}\hbox {d}\dot{\phi }+ \frac{1}{\lambda \nu } \int _0^{\dot{\phi }}{ |\dot{\phi }|\hbox {sgn}(\dot{\phi }) \hbox {d}\dot{\phi }}\biggr ]\nonumber \\&\quad \doteq A_1+A_2+A_3+A_4+A_5 \end{aligned}$$

(67)

Computing $A_i$, $i=1,\ldots , 5$, gives

$$\begin{aligned}&A_1=\frac{\gamma _2\beta _1^\mathrm{T}\hat{\theta }_1}{\gamma _1} \end{aligned}$$

(68)

$$\begin{aligned}&A_2=\gamma _1\gamma _2 (\lambda \nu )^2\left[ \frac{\phi ^2|\dot{\phi }|^{2\nu }}{2\nu ^2} +\frac{|\dot{\phi }|^{2\nu +2}}{(2\nu +2)(\nu +1)}\right. \nonumber \\&\left. \quad \quad + \frac{|\dot{\phi }|^{2\nu +4}}{(\nu +2)(2\nu +4)}+\frac{\phi ^6 |\dot{\phi }|^{2\nu }}{2\nu ^2}+\frac{\phi ^4 |\dot{\phi }|^{2\nu +2}}{(2\nu +2)(\nu +1)}\right] \nonumber \\\end{aligned}$$

(69)

$$\begin{aligned}&A_3=\gamma _2\lambda \nu k_1\left[ \frac{\phi |\dot{\phi }|^{\nu }\hbox {sgn}(\dot{\phi })}{\nu }+\frac{\lambda |\dot{\phi }|^{2\nu }}{2 \nu }\right] \end{aligned}$$

(70)

$$\begin{aligned}&A_4= \frac{\gamma _2 k_2}{\mu +1}[|\phi +\lambda |\dot{\phi }|^{\nu }\hbox {sgn}(\dot{\phi })|^{\mu +1}-|\phi |^{\mu +1}] \nonumber \\&\quad \quad = \frac{\gamma _2 \lambda \nu k_2}{\mu +1}[|s|^{\mu +1}-|\phi |^{\mu +1}] \end{aligned}$$

(71)

$$\begin{aligned}&A_5=\gamma _2\frac{|\dot{\phi }|^2}{2} \end{aligned}$$

(72)

Now $\beta _2$ is obtained by adding the expressions for $A_i$.

$$\begin{aligned} \beta _2=A_1+A_2+A_3+A_4+A_5 \end{aligned}$$

(73)

This completes computation of $\beta _2$.