Modified adaptive neural dynamic surface control for morphing aircraft with input and output constraints

Abstract

In this paper, a barrier Lyapunov function-based adaptive neural dynamic surface control approach is proposed for morphing aircraft subject to unknown parameters and input–output constraints. Based on the functional decomposition, the longitudinal dynamics can be divided into altitude and velocity subsystems. Minimal learning parameter (MLP) technique-based neural networks are used to estimate the model uncertainties; thus, the amount of online-updated parameters is largely reduced. To overcome the problem of ‘explosion of complexity’ in the back-stepping method, the first-order sliding mode differentiator (FOSD) is introduced to compute the derivative of virtual control laws. Combining MLP and FOSD technique, a composite adaptive neural control scheme is proposed by utilizing an auxiliary system to deal with the input saturation and a barrier Lyapunov function to counteract the output constraints. The highlight is that the proposed neural controller not only owns less online-updated neural parameters, but also has the ability of handling input–output constraints. The stability of the proposed control scheme is established using the Lyapunov theory. Simulation results show that the proposed controller can ensure good tracing performance of the morphing aircraft in the fixed configuration and morphing process.

Appendix

1.1 Proof of Theorem 1

Proof

Considering the following candidate Lyapunov function

$$\begin{aligned} L_V =\frac{1}{2}\log \left( {\frac{k_{b2}^2 }{k_{b2}^2 -z_V^2 }} \right) +\frac{1}{2\rho _{V1} }\tilde{{\varphi } }_V^2 +\frac{1}{2\rho _{V2} }\tilde{d}_V^2\nonumber \\ \end{aligned}$$

(58)

where $\rho _{V1}$ and $\rho _{V2}$ are positive design parameters, $\tilde{{\varphi } }_V ={\varphi } _V -\hat{{{\varphi } }}_V$ and $\tilde{d}_V =d_ VM -\hat{{d}}_V$.

Based on (25), (26) and (27), the time derivative of $L_V$ is given by

$$\begin{aligned} \dot{L}_V= & {} \frac{z_V \dot{z}_V }{k_{b2}^2 -z_V^2 }-\frac{1}{\rho _{V1} }\tilde{{\varphi } }_V \dot{\hat{{{\varphi } }}}_V -\frac{1}{\rho _{V2} }\tilde{d}_V \dot{\hat{{d}}}_V =\frac{z_V }{k_{b2}^2 -z_V^2 }\left[ {\begin{array}{l} -k_V z_V -\frac{z_V \hat{{{\varphi } }}_V \varPhi _V^T (\bar{{x}}_V ,T_f )\varPhi _V (\bar{{x}}_V ,T_f )}{4(k_{b2}^2 -z_V^2 )k_{22} }+d_V +W_V^{*T} \varPhi _V (\bar{{x}}_V ,T_f ) \\ -\hat{{d}}_V \tanh \left[ {\frac{z_V }{k_{b2}^2 -z_V^2 }\frac{1}{w_V }} \right] \\ \end{array}} \right] \nonumber \\&-\,\tilde{{\varphi } }_V \left[ {\frac{z_V^2 \varPhi _V^T (\bar{{x}}_V ,T_f )\varPhi _V (\bar{{x}}_V ,T_f )}{4(k_{b2}^2 -z_V^2 )^{2}k_{22} }-\frac{\sigma _{V1} }{\rho _{V1} }\hat{{{\varphi } }}_V } \right] -\,\tilde{d}_V \left[ {\frac{z_V }{k_{b2}^2 -z_V^2 }\tanh \left( {\frac{z_V }{k_{b2}^2 -z_V^2 }\frac{1}{w_V }} \right) -\sigma _{V2} \hat{{d}}_V } \right] \end{aligned}$$

(59)

Note that the following inequalities hold

$$\begin{aligned}&\frac{z_V }{k_{b2}^2 -z_V^2 }W_V^{*T} \varPhi _V^T (\bar{{x}}_V ,T_f )\nonumber \\&\quad \le \frac{{\varphi } _V z_V^2 \varPhi _V^T (\bar{{x}}_V ,T_f ) \varPhi _V (\bar{{x}}_V ,T_f )}{4(k_{b2}^2 -z_V^2 )^{2}k_{22} }+k_{22} \end{aligned}$$

(60)

$$\begin{aligned}&\left| {\frac{z_V }{k_{b2}^2 -z_V^2 }} \right| d_ VM -\frac{z_V }{k_{b2}^2 -z_V^2 }d_ VM \tanh \left( {\frac{z_V }{k_{b2}^2 -z_V^2 }\frac{1}{w_V }} \right) \nonumber \\&\quad \le \kappa _0 d_ VM w_V =\bar{{w}}_V \end{aligned}$$

(61)

where $k_{22} >0$.

Invoking $\tilde{{\varphi } }_V ={\varphi } _V -\hat{{{\varphi } }}_V$ and $\tilde{d}_V =d_ VM -\hat{{d}}_V$ yields

$$\begin{aligned} \frac{\sigma _{V1} }{\rho _{V1} }\tilde{{\varphi } }_V \hat{{{\varphi } }}_V= & {} \frac{\sigma _{V1} }{2\rho _{V1} }\left( {{\varphi } _V^2 -\tilde{{\varphi } }_V^2 -\hat{{{\varphi } }}_V^2 } \right) \nonumber \\\le & {} \frac{\sigma _{V1} }{2\rho _V }\left( {{\varphi } _V^2 -\tilde{{\varphi } }_V^2 } \right) , \nonumber \\ \sigma _{V2} \tilde{d}_V \hat{{d}}_V\le & {} \frac{\sigma _{V2} }{2}\left( {d_ VM ^2 -\tilde{d}_V^2 } \right) \end{aligned}$$

(62)

Consider that (60), (61), (62) and Lemma 3, (59) can be re-formulated as

$$\begin{aligned} \dot{L}_V\le & {} -\frac{k_V z_V^2 }{k_{b2}^2 -z_V^2 }-\frac{\sigma _{V1} }{2\rho _{V1} }\tilde{{\varphi } }_V^2 -\frac{\sigma _{V2} }{2}\tilde{d}_V^2 +\frac{\sigma _{V1} }{2\rho _{V1} }{\varphi } _V^2 \nonumber \\&\quad +\,\frac{\sigma _{V2} }{2}d_ VM ^2 +\bar{{w}}_V +k_{22} \nonumber \\\le & {} -k_V \log \frac{k_{b2}^2 }{k_{b2}^2 -z_V^2 }-\frac{\sigma _{V1} }{2\rho _{V1} }\tilde{{\varphi } }_V^2 \nonumber \\&\quad -\,\frac{\sigma _{V2} }{2}\tilde{d}_V^2 +C_{2V} \le -C_{1V} L_V +C_{2V} \end{aligned}$$

(63)

where $C_{1V} =\min \left\{ {2k_V ,\sigma _{V1} /\rho _{V1} ,\sigma _{V2} } \right\}$, $C_{V2} =\frac{\sigma _{V1} }{2\rho _{V1} }{\varphi } _V^2 +\frac{\sigma _{V2} }{2}d_ VM ^2 +\bar{{w}}_V +k_{22}$.

Multiplying (63) by $e^{C_{V1} t}$, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(L_V e^{C_{V1} t})\le C_{V2} e^{C_{V1} t} \end{aligned}$$

(64)

Integrating the above inequality, we obtain

$$\begin{aligned} L_V \le (L_V (0)-\frac{C_{V2} }{C_{V1} })e^{-C_{V1} t}+\frac{C_{V2} }{C_{V1} } \end{aligned}$$

(65)

Therefore, for $z_V$ we obtain

$$\begin{aligned} \frac{1}{2}\log \left( {\frac{k_{b2}^2 }{k_{b2}^2 -z_V^2 }} \right)\le & {} L_V (0)+\frac{C_{V2} }{C_{V1} }, \nonumber \\ \left| {z_V } \right|\le & {} k_{b2} \sqrt{(1-e^{-2(L_V (0)+C_{V2} /C_{V1} )})}<k_{b2}\nonumber \\ \end{aligned}$$

(66)

Similarly, we have $\left| {\tilde{{\varphi } }_V } \right| \le \sqrt{2\rho _{V1} (L_V (0)+C_{V2} /C_{V1} })$, $\left| {\tilde{d}_V } \right| \le \sqrt{2\rho _{V2} (L_V (0)+C_{V2} /C_{V1} })$.

1.2 Proof of Theorem 2

Proof

Choose the following candidate Lyapunov function

$$\begin{aligned} L=L_1 +L_2 +L_3 +L_4 \end{aligned}$$

(67)

where $L_1 =\frac{1}{2V}\log \left( {\frac{k_{b1}^2 }{k_{b1}^2 -z_1^2 }} \right) +\frac{1}{2\rho _{12} }\tilde{d}_1^2$, $L_2 =\frac{1}{2}z_2^2 +\frac{\tilde{{\varphi } }_2^2 }{2\rho _{21} }+\frac{\tilde{d}_2^2 }{2\rho _{22} }$, $L_3 =\frac{1}{2}z_3^2$, $L_4 =\frac{1}{2}z_4^2 +\frac{\tilde{{\varphi } }_4^2 }{2\rho _{41} }+\frac{\tilde{d}_4^2 }{2\rho _{42} }$, and $\tilde{d}_1 =d_{1M} -\hat{{d}}_1$, $\tilde{{\varphi } }_2 ={\varphi } _2 -\hat{{{\varphi } }}_2$, $\tilde{d}_2 =d_{2M} -\hat{{d}}_2$, $\tilde{{\varphi } }_4 ={\varphi } _4 -\hat{{{\varphi } }}_4$, $\tilde{d}_4 =d_{4M} -\hat{{d}}_4$.

According to (34), (35) and Lemma 2, the time derivative of $L_1$ is given by

$$\begin{aligned} \dot{L}_1= & {} \frac{1}{V}\frac{z_1 \dot{z}_1 }{k_{b1}^2 -z_1^2 }-\frac{\dot{V}}{2V^{2}}\log \frac{k_{b1}^2 }{k_{b1}^2 -z_1^2 }-\frac{1}{\rho _{12} }\tilde{d}_1 \dot{\hat{{d}}}_1 \nonumber \\= & {} \frac{z_1 }{k_{b1}^2 -z_1^2 }\left( z_2 -k_1 z_1 -\frac{z_1 }{2(k_{b1}^2 -z_1^2 )}\right. \nonumber \\&\left. -z_1 \hat{{d}}_1 \tanh \left( {\frac{z_1^2 }{k_{b1}^2 -z_1^2 }\frac{1}{w_1 }} \right) \right) \nonumber \\&+\,d_1 \log \frac{k_{b1}^2 }{k_{b1}^2 -z_1^2 }-\frac{1}{\rho _{12} }\tilde{d}_1 \dot{\hat{{d}}}_1 \nonumber \\\le & {} -\frac{k_1 z_1^2 }{k_{b1}^2 -z_1^2 }+\frac{z_1 z_2 }{k_{b1}^2 -z_1^2 }-\frac{z_1^2 }{2(k_{b1}^2 -z_1^2 )^{2}}\nonumber \\&+\,\left| {d_{1M} \frac{z_1^2 }{k_{b1}^2 -z_1^2 }} \right| -d_{1M} \frac{z_1^2 }{k_{b1}^2 -z_1^2 }\tanh \left( {\frac{z_1^2 }{k_{b1}^2 -z_1^2 }\frac{1}{w_1 }} \right) \nonumber \\&+\,\sigma _{12} \tilde{d}_1 \hat{{d}}_1 \end{aligned}$$

(68)

By invoking (40), (41), (42), the time derivative of $L_2$ is given by

$$\begin{aligned} \dot{L}_2= & {} z_2 \dot{z}_2 -\frac{1}{\rho _{21} }\tilde{{\varphi } }_2 \dot{\hat{{{\varphi } }}}_2 -\frac{1}{\rho _{22} }\tilde{d}_2 \dot{\hat{{d}}}_2 \nonumber \\\le & {} z_2 z_3 -k_2 z_2^2 +z_2 W_2^{*T} \varPhi _2 (x_2 ,x_{3f} )\nonumber \\&-\,\frac{1}{2}{\varphi } _2 z_2^2 \varPhi _2^T (x_2 ,x_{3f} )\varPhi _2 (x_2 ,x_{3f} ) \nonumber \\&+\,\left| {z_2 } \right| d_{2M} -z_2 d_{2M} \tanh \left( \frac{z_2 }{w_2 }\right) -z_2 \tau _1 \nonumber \\&+\,\frac{\sigma _{21} }{\rho _{21} }\tilde{{\varphi } }_2 \hat{{{\varphi } }}_2 +\sigma _{22} \tilde{d}_2 \hat{{d}}_2 \end{aligned}$$

(69)

Considering (47) and (48), the time derivative of $L_3$ is obtained as

$$\begin{aligned} \dot{L}_3= & {} z_3 \dot{z}_3 =z_3 \left( {z_4 -z_2 -k_3 z_3 -\tau _2 } \right) \nonumber \\\le & {} -\left( k_3 -\frac{1}{2k_{11} }\right) z_3^2 -z_2 z_3 +z_3 z_4 +\frac{k_{11} }{2}\bar{{\tau }}_2^2\nonumber \\ \end{aligned}$$

(70)

Using (53), (54) and (55) results in the time derivative of $L_4$

$$\begin{aligned} \dot{L}_4= & {} z_4 \dot{z}_4 -\frac{1}{\rho _{41} }\tilde{{\varphi } }_4 \dot{\hat{{{\varphi } }}}_4 -\frac{1}{\rho _{42} }\tilde{d}_4 \dot{\hat{{d}}}_4 \nonumber \\= & {} z_4 \left( -k_4 z_4 -z_3 +W_4^{*T} \varPhi _4 (\bar{{x}},u_f )\right. \nonumber \\&\qquad -\,\frac{1}{2}z_4 \hat{{{\varphi } }}_4 \varPhi _4^T (\bar{{x}},u_f )\varPhi _4 (\bar{{x}},u_f )+d_4\nonumber \\&\qquad \left. -\,\hat{{d}}_4 \tanh \left( \frac{z_4 }{w_4 }\right) -\tau _3 \right) \nonumber \\&-\,\frac{1}{\rho _{41} }\tilde{{\varphi } }_4 \dot{\hat{{{\varphi } }}}_4 -\frac{1}{\rho _{42} }\tilde{d}_4 \dot{\hat{{d}}}_4 \nonumber \\\le & {} -k_4 z_4^2 -z_3 z_4 +z_4 W_4^{*T} \varPhi _4 (\bar{{x}},u_f )\nonumber \\&-\,\frac{1}{2}z_4^2 {\varphi } _4 \varPhi _4^T (\bar{{x}},u_f )\varPhi _4 (\bar{{x}},u_f )+\left| {z_4 } \right| d_{4M} \nonumber \\&-\,z_4 d_{4M} \tanh \left( \frac{z_4 }{w_4 }\right) -z_4 \tau _3 +\frac{1}{\rho _{41} }\tilde{{\varphi } }_4 \hat{{{\varphi } }}_4 +\frac{1}{\rho _{42} }\tilde{d}_4 \hat{{d}}_4\nonumber \\ \end{aligned}$$

(71)

Consider the following facts

$$\begin{aligned}&\sigma _{12} \tilde{d}_1 \hat{{d}}_1 \le \frac{\sigma _{12} }{2}\left( {d_{1M}^2 -\tilde{d}_1^2 } \right) , \quad \frac{z_1 z_2 }{k_{b1}^2 -z_1^2 }\nonumber \\&\quad \le \frac{z_1^2 }{2(k_{b1}^2 -z_1^2 )^{2}}+\frac{1}{2}z_2^2 , \quad -z_2 \tau _1 \le \frac{1}{2}z_2^2 +\frac{1}{2}\bar{{\tau }}_1^2\nonumber \\ \end{aligned}$$

(72)

$$\begin{aligned}&d_{1M} \left| {\frac{z_1^2 }{k_{b1}^2 -z_1^2 }} \right| -d_{1M} \frac{z_1^2 }{k_{b1}^2 -z_1^2 }\tanh \left( {\frac{z_1^2 }{k_{b1}^2 -z_1^2 }\frac{1}{w_1 }} \right) \nonumber \\&\quad \le \kappa _0 d_{1M} w_1 =\bar{{w}}_1 ,\nonumber \\&\quad \frac{\sigma _{21} }{\rho _{21} }\tilde{{\varphi } }_2 \hat{{{\varphi } }}_2 \le \frac{\sigma _{21} }{2\rho _{21} }{\varphi } _2^2 -\frac{\sigma _{21} }{2\rho _{21} }\tilde{{\varphi } }_2^2 \end{aligned}$$

(73)

$$\begin{aligned}&\left| {z_2 } \right| d_{2M} -z_2 d_{2M} \tanh \left( {\frac{z_2 }{w_2 }} \right) \le \kappa _0 d_{2M} w_2 =\bar{{w}}_2 ,\nonumber \\&\quad \sigma _{22} \tilde{d}_2 \hat{{d}}_2 \le \frac{1}{2}\sigma _{22} d_{2M}^2 -\frac{1}{2}\sigma _{22} \tilde{d}_2^2 \end{aligned}$$

(74)

$$\begin{aligned}&z_2 W_2^{*T} \varPhi (\bar{{x}},u_f )\le \frac{1}{2}z_2^2 {\varphi } _2 \varPhi _2^T (\bar{{x}},u_f )\varPhi _2 (\bar{{x}},u_f )+\frac{1}{2},\nonumber \\&\quad \frac{\sigma _{41} }{\rho _{41} }\tilde{{\varphi } }_4 \hat{{{\varphi } }}_4 \le \frac{\sigma _{41} }{2\rho _{41} }\left( {{\varphi } _4^2 -\tilde{{\varphi } }_4^2 } \right) \end{aligned}$$

(75)

$$\begin{aligned}&-\,z_4 \tau _3 \le \frac{1}{2}\left( {z_4^2 +\bar{{\tau }}_3^2 } \right) ,\nonumber \\&\qquad \left| {z_4 } \right| d_{4M} -z_4 d_{4M} \tanh \left( {\frac{z_4 }{w_4 }} \right) \le \kappa _0 d_{4M} w_4 =\bar{{w}}_4 \nonumber \\\end{aligned}$$

(76)

$$\begin{aligned}&\sigma _{42} \tilde{d}_4 \hat{{d}}_4 \le \frac{1}{2}\sigma _{42} \left( {d_{4M}^2 -\tilde{d}_4^2 } \right) ,\nonumber \\&\quad z_4 W_4^{*T} \varPhi (\bar{{x}},u_f )\le \frac{1}{2}z_4^2 {\varphi } _4 \varPhi _4^T (\bar{{x}},u_f )\varPhi _4 (\bar{{x}},u_f )+\frac{1}{2}\nonumber \\ \end{aligned}$$

(77)

The derivative of L is obtained as

$$\begin{aligned} \dot{L}= & {} \dot{L}_1 +\dot{L}_2 +\dot{L}_3 +\dot{L}_4 \nonumber \\\le & {} -k_1 \log \frac{k_{b1}^2 }{k_{b1}^2 -z_1^2 }-\left( {k_2 -1} \right) z_2^2 -\left( {k_3 -\frac{1}{2k_{11} }} \right) z_3^2 \nonumber \\&-\,(k_4 -0.5)z_4^2 -\frac{\sigma _{12} }{2}\tilde{d}_1^2 -\frac{\sigma _{21} }{2\rho _{21} }\tilde{{\varphi } }_2^2 \nonumber \\&-\,\frac{\sigma _{22} }{2}\tilde{d}_2^2 -\frac{\sigma _{41} }{2\rho _{41} }\tilde{{\varphi } }_4^2 -\frac{\sigma _{42} }{2}\tilde{d}_4^2 +\frac{\sigma _{12} }{2}d_{1M}^2 +\frac{\sigma _{22} }{2}d_{2M}^2\nonumber \\&+\,\frac{\sigma _{42} }{2}d_{4M}^2 +\frac{\sigma _{21} }{2\rho _{21} }{\varphi } _2^2 \nonumber \\&+\,\frac{\sigma _{41} }{2\rho _{41} }{\varphi } _4^2 +\bar{{w}}_1 +\bar{{w}}_2 +\bar{{w}}_4 +\frac{1}{2}\bar{{\tau }}_1^2 +\frac{k_{11} }{2}\bar{{\tau }}_2^2 \nonumber \\&+\frac{1}{2}\bar{{\tau }}_3^2 +1\;\le -C_1 L+C_2 \end{aligned}$$

(78)

where $C_1 :=\left( 2k_1 ,2k_2 -2,2k_3 -\frac{1}{k_{11} },2k_4 -1,\right.$ $\left. \sigma _{12} ,\frac{\sigma _{21} }{\rho _{21} },\sigma _{22} ,\frac{\sigma _{41} }{\rho _{41} },\sigma _{42} \right)$,

$$\begin{aligned} C_2 :=\left( {\begin{array}{l} 0.5\sigma _{12} d_{1M}^2 +0.5\sigma _{22} d_{2M}^2 +0.5\sigma _{42} d_{4M}^2 +0.5\sigma _{21} /\rho _{21} {\varphi } _2^2 +0.5\sigma _{41} /\rho _{41} {\varphi } _4^2 +\bar{{w}}_1 +\bar{{w}}_2 +\bar{{w}}_4 \\ +0.5\bar{{\tau }}_1^2 +0.5k_{11} \bar{{\tau }}_2^2 +0.5\bar{{\tau }}_3^2 +1 \\ \end{array}} \right) . \end{aligned}$$

To ensure $C_1 >0$, the corresponding design parameters $k_{i=1,2,3,4}$, $\sigma _{ij,i=1,2,4,j=1,2}$ and $\rho _{ij,i=1,2,3,j=1,2}$ should be chosen such that $k_1 >0$, $k_2 -1>0$, $k_3 -0.5/k_{11} >0$, $k_4 -0.5>0,\sigma _{ij,i=1,2,3,j=1,2} >0$ and $\rho _{ij,i=1,2,4,j=1,} >0$. Multiplying (78) by $e^{C_1 t}$, we can get $\frac{\mathrm{d}}{\mathrm{d}t}(Le^{C_1 t})\le C_2 e^{C_1 t}$. Integrating inequality $\frac{\mathrm{d}}{\mathrm{d}t}(Le^{C_1 t})\le C_2 e^{C_1 t}$, we obtain

$$\begin{aligned} L\le (L(0)-\frac{C_2 }{C_1 })e^{-C_1 t}+\frac{C_2 }{C_1 } \end{aligned}$$

(79)

Therefore, we have

$$\begin{aligned} \frac{1}{2}\log \left( {\frac{k_{b1}^2 }{k_{b1}^2 -z_1^2 }} \right)\le & {} L(0)+\frac{C_2 }{C_1 }, \nonumber \\ \left| {z_1 } \right|\le & {} k_{b1} \sqrt{(1-e^{-2(L(0)+C_2 /C_1 )})}<k_{b1}\nonumber \\ \end{aligned}$$

(80)

Similarly, we obtain

$$\begin{aligned} \left| {z_2 } \right|\le & {} \sqrt{2(L(0)+C_2 /C_1 }), \\ \left| {z_3 } \right|\le & {} \sqrt{2(L(0)+C_2 /C_1 }), \\ \left| {z_4 } \right|\le & {} \sqrt{2(L(0)+C_2 /C_1 }), \\ \left| {\tilde{d}_1 } \right|\le & {} \sqrt{2\rho _{12} (L(0)+C_2 /C_1 }), \\ \left| {\tilde{{\varphi } }_2 } \right|\le & {} \sqrt{2\rho _{21} (L(0)+C_2 /C_1 }), \\ \left| {\tilde{d}_2 } \right|\le & {} \sqrt{2\rho _{22} (L(0)+C_2 /C_1 }), \\ \left| {\tilde{{\varphi } }_4 } \right|\le & {} \sqrt{2\rho _{41} (L(0)+C_2 /C_1 }),\\ \left| {\tilde{d}_4 } \right|\le & {} \sqrt{2\rho _{42} (L(0)+C_2 /C_1 }). \end{aligned}$$