Nonlinear adaptive tracking control of non-minimum phase hypersonic flight vehicles with unknown input nonlinearity

Abstract

A nonlinear adaptive controller design method for output tracking of hypersonic flight vehicles (HFVs) subject to input nonlinearity is proposed in this paper. The parameters of the input nonlinearity are assumed to be unknown. This problem is challenging because of the complex nonlinearity of HFVs and the existence of the unknown input nonlinearity. The nonlinear model of HFVs is linearized, and the internal dynamics of this model are constructed. Since the internal dynamics of HFVs are non-minimum, an ideal internal dynamics (IID)-based control strategy is utilized and a states tracking model is constructed based on the computed IID. Then a nonlinear adaptive control strategy is designed to guarantee the bounded of output tracking error in the existence of unknown input nonlinearity. Comparison simulation results are provided to verify the available of the proposed method.

Appendix

1.1 Appendix A

Proof of Theorem 1.

Choose a Lyapunov function for (13) as

$$\begin{aligned} V(t)=\frac{1}{2}e^\mathrm{T}Pe+\frac{1}{2q_{1}}\tilde{\rho }_{0}^{2}+\frac{1}{2q_{2}} \tilde{\rho }_{1}^{2} \end{aligned}$$

(20)

where $\tilde{\rho }_{0}=\rho _{0}-\hat{\rho }_{0}$, $\tilde{\rho }_{1}=\rho _{1}-\hat{\rho }_{1}$, $q_{1}$ and $q_{2}$ are given scalars. Notice that $\dot{\tilde{\rho }}_{0}=-\dot{\hat{\rho }}_{0}$, and $\dot{\tilde{\rho }} _{1}(t)=-\dot{\hat{\rho }}_{1}$, taking the time derivative, we have

$$\begin{aligned} \dot{V}= & {} \frac{1}{2}\dot{e}^\mathrm{T}Pe+\,\frac{1}{2}e^\mathrm{T}P\dot{e}-\,\frac{1}{q_{1}} \tilde{\rho }_{0}\dot{\hat{\rho }}_{0}-\,\frac{1}{q_{2}}\tilde{\rho }_{1}\dot{ \hat{\rho }}_{1} \\= & {} \frac{1}{2}\left\{ \left( A+B(I+k(t))K\right) e(t)\right. \\&\left. +\,B\left[ -k(t)F(x)+\,\left( I+k(t)\right) F(x^{d})\right. \right. \\&\left. \left. +\,k(t)y^{d(r)}+\left( I+k(t)\right) u_{s}+\,d(t)\right] \right\} ^\mathrm{T}Pe\\&+\,\frac{1}{2}e^\mathrm{T}P\left\{ \left( A+B(I+k(t))K\right) e(t)\right. \\&\left. +\,B\left[ -k(t)F(x)+\,\left( I+k(t)\right) F(x^{d})\right. \right. \\&\left. \left. +\,k(t)y^{d(r)}+\,\left( I+k(t)\right) u_{s}+\,d(t)\right] \right\} \\&-\,\frac{1}{q_{1}}\tilde{\rho }_{0}\dot{\hat{\rho }}_{0}-\,\frac{1}{q_{2}}\tilde{ \rho }_{1}\dot{\hat{\rho }}_{1} \\= & {} \frac{1}{2}e^\mathrm{T}\left[ P\left( A+B(I+k(t))K\right) +\left( A+B(I+k(t))K\right) ^\mathrm{T}P\right] e \\&+\,e^\mathrm{T}PB\left[ -k(t)F(x)+\,\left( I+k(t)\right) F(x^{d})\right. \\&\left. +\,k(t)y^{d(r)}+\left( I+k(t)\right) u_{s}+d(t)\right] \\&-\,\frac{1}{q_{1}}\tilde{\rho }_{0}\dot{\hat{\rho }}_{0}-\,\frac{1}{q_{2}}\tilde{ \rho }_{1}\dot{\hat{\rho }}_{1} \\= & {} e^\mathrm{T}\left[ P\left( A+B(I+k(t))K\right) +\,\left( A+B(I+k(t))K\right) ^\mathrm{T}P \right] e \\&+\,e^\mathrm{T}PB\left( k(t)y^{d(r)}+F(x^{d})\right) \\&+\,e^\mathrm{T}PBd(t)+\,e^\mathrm{T}PBk(t)\left[ -\,F(x)+F(x^{d})\right] \\&+\,e^\mathrm{T}PB\left( I+k(t)\right) u_{s}-\,\frac{1}{q_{1}}\tilde{\rho }_{0}\dot{ \hat{\rho }}_{0}-\frac{1}{q_{2}}\tilde{\rho }_{1}\dot{\hat{\rho }}_{1} \end{aligned}$$

By Schur complement, we can get

$$\begin{aligned}&P\left( A+B(I+k(t))K\right) +\,\left( A+B(I+k(t))K\right) ^\mathrm{T}P \\&\quad =P\left( A+BK\right) +\,\left( A+BK\right) ^\mathrm{T}P+\,PBk(t)K\\&\qquad +\,\left( k(t)K\right) ^\mathrm{T}B^\mathrm{T}P \\&\quad \le P\left( A+BK\right) +\,\left( A+BK\right) ^\mathrm{T}P\\&\qquad +\,\varepsilon \hbox {PBB}^\mathrm{T}P\\&\qquad +\,\varepsilon ^{-1}\left( k(t)K\right) ^\mathrm{T}k(t)K \end{aligned}$$

where $\varepsilon >0$ is a scalar. From Assumption 5, $\left\| k(t)\right\| \le \mu$, then

$$\begin{aligned}&P\left( A+BK\right) +\,\left( A+BK\right) ^\mathrm{T}P\\&\qquad +\,\varepsilon PBB^\mathrm{T}P+\varepsilon ^{-1}\left( k(t)K\right) ^\mathrm{T}k(t)K \\&\quad \le PA+\,PBK+\,A^\mathrm{T}+\,K^\mathrm{T}B^{T}P+\,\varepsilon PBB^\mathrm{T}P\\&\qquad +\,\varepsilon ^{-1}\mu ^{2}K^\mathrm{T}K \end{aligned}$$

Defining $X=P^{-1}$, $Y=KX$, and multiplying $P^{-1}$ to both left and right side of the above equation

$$\begin{aligned} AX+\,BY+\,XA^\mathrm{T}+\,Y^{T}B^\mathrm{T}+\,\epsilon BB^\mathrm{T}+\,\epsilon ^{-1}\mu ^{2}Y^\mathrm{T}Y \end{aligned}$$

From (15),

$$\begin{aligned} AX+\,BY+\,XA^\mathrm{T}+\,Y^{T}B^\mathrm{T}+\,\epsilon BB^\mathrm{T}+\,\epsilon ^{-1}\mu ^{2}Y^\mathrm{T}Y<0 \end{aligned}$$

then

$$\begin{aligned}&e^\mathrm{T}[ P\left( A+B(I+k(t))K\right) \\&\quad +\left( A+B(I+k(t))K\right) ^\mathrm{T}P ] e<0 \end{aligned}$$

Since $\left\| k(t)\right\| ,\left\| y^{d(r)}\right\|$, $\left\| F(x^{d})\right\|$ and $\left\| d(t)\right\|$ are all bounded, $\left\| k(t)y^{d(r)}+F(x^{d})+d(t)\right\|$ is bounded. Assume that $\sup \left\| k(t)y^{d(r)}+F(x^{d})+d(t)\right\| =\rho _{0}$. Then

$$\begin{aligned}&\left\| e^\mathrm{T}PB\right\| \left\| \left( k(t)y^{d(r)}+F(x^{d})\right) +d(t)\right\| \\&\quad \le \left\| e^\mathrm{T}PB\right\| \rho _{0} \end{aligned}$$

From Assumption 4, F(x) is Lipschitz, and $F(x^{d})$ is bounded, so $\left\| -F(x)+F(x^{d})\right\| \le L\left\| x-x^{d}\right\|$. Assume that $\sup \left\| k(t)L\right\| =\rho _{1}$, then

$$\begin{aligned}&\left\| e^\mathrm{T}PB\right\| \left\| k(t)\left[ -F(x)+F(x^{d})\right] \right\| \\&\quad \le \left\| e^\mathrm{T}PB\right\| L\left\| k(t)(x-x^{d})\right\| \le \rho _{1}\left\| e^\mathrm{T}PB\right\| \left\| e\right\| \end{aligned}$$

Then for the Lyapunov function,

$$\begin{aligned} \dot{V}= & {} e^\mathrm{T}[ P\left( A+B(I+k(t))K\right) \\&+\,\left( A+B(I+k(t))K\right) ^\mathrm{T}P] e \\&+\,e^\mathrm{T}PB\left( k(t)y^{d(r)}+F(x^{d})\right) \\&+\,e^\mathrm{T}PBd(t)+e^\mathrm{T}PBk(t)\left[ -F(x)+F(x^{d})\right] \\&-\,e^\mathrm{T}PB\left( 1+k(t)\right) u_{s}-\frac{1}{q_{1}}\tilde{\rho }_{0}\dot{ \hat{\rho }}_{0}-\frac{1}{q_{2}}\tilde{\rho }_{1}\dot{\hat{\rho }}_{1} \\< & {} \left\| e^\mathrm{T}PB\right\| \rho _{0}+\left\| e^\mathrm{T}PB\right\| \rho _{1}\left\| e\right\| \\&+\,e^\mathrm{T}PB\left( I+k(t)\right) u_{s}\\&-\,\frac{1}{q_{1}}\tilde{\rho }_{0}\dot{\hat{\rho }}_{0}-\frac{1}{q_{2}}\tilde{\rho }_{1} \dot{\hat{\rho }}_{1} \end{aligned}$$

From (16)

$$\begin{aligned} u_{s}= & {} -\,\hat{\rho }_{0}\frac{B^\mathrm{T}X^{-1}e}{\left\| e^\mathrm{T}X^{-1}B\right\| +\xi _{1}}-\hat{\rho }_{1}\frac{B^\mathrm{T}X^{-1}e}{ \left\| e\right\| \left\| e^\mathrm{T}X^{-1}B\right\| +\xi _{2}} \\= & {} -\,\hat{\rho }_{0}\frac{B^\mathrm{T}Pe}{\left\| e^\mathrm{T}PB\right\| +\xi _{1}}- \hat{\rho }_{1}\frac{B^\mathrm{T}Pe}{\left\| e\right\| \left\| e^\mathrm{T}PB\right\| +\xi _{2}} \end{aligned}$$

and $\hat{\rho }_{0}>0$, $\hat{\rho }_{1}>0$, so

$$\begin{aligned} -\,k(t)\hat{\rho }_{0}\frac{e^\mathrm{T}PBB^{T}Pe}{\left\| e^\mathrm{T}PB\right\| +\xi _{1}}<0\\ -\,k(t)\hat{\rho }_{1}\frac{e^\mathrm{T}PBB^{T}Pe}{\left\| e^\mathrm{T}PB\right\| +\xi _{2}}<0 \end{aligned}$$

then

$$\begin{aligned} \dot{V}< & {} \left\| e^\mathrm{T}PB\right\| \rho _{0}+\left\| e^\mathrm{T}PB\right\| \rho _{1}\left\| e\right\| \\&-\hat{\rho }_{0}\frac{ e^\mathrm{T}PBB^{T}Pe}{\left\| e^\mathrm{T}PB\right\| +\xi _{1}}\\&-\,\hat{\rho }_{1}\frac{e^\mathrm{T}PBB^{T}Pe}{\left\| e\right\| \left\| e^\mathrm{T}PB\right\| +\xi _{2}} \\&-\,\frac{1}{q_{1}}\tilde{\rho }_{0}(t)\dot{\hat{\rho }}_{0}(t)-\frac{1}{q_{2}} \tilde{\rho }_{1}(t)\dot{\hat{\rho }}_{1}(t) \\\le & {} \left\| e^\mathrm{T}PB\right\| \left( \rho _{0}-\hat{\rho }_{0}\right) +\,\left\| e^\mathrm{T}PB\right\| \left\| e\right\| \left( \rho _{1}-\hat{ \rho }_{1}\right) \\&-\,\frac{1}{q_{1}}\left( \rho _{0}-\hat{\rho }_{0}\right) \dot{ \hat{\rho }}_{0}-\frac{1}{q_{2}}\left( \rho _{1}-\hat{\rho }_{1}\right) \dot{ \hat{\rho }}_{1} \end{aligned}$$

Since the adaptive law of $\hat{\rho }_{0}$ and $\hat{\rho }_{1}$ is

$$\begin{aligned} \dot{\hat{\rho }}_{0}=q_{1}\left\| e^\mathrm{T}PB\right\| ,\text { }\dot{\hat{ \rho }}_{1}=q_{2}\left\| e\right\| \left\| e^\mathrm{T}PB\right\| \end{aligned}$$

By plugging the adaptive law (17) into the right hand of the above equation, we obtain

$$\begin{aligned} \dot{V}<0 \end{aligned}$$

The proof is completed.$\square$