Abbreviations

AOA

angel-of-attack

BBT

bank to turn

DSC

dynamic surface control

HV

hypersonic vehicles

IPPDSC-NDO

improved prescribed performance dynamic surface control - nonlinear disturbance observer

NDO

nonlinear disturbance observer

PPC

prescribed performance control

PPF

prescribed performance function

PPSMC

prescribed performance sliding mode control

Nomenclature

$ {C}_{L\alpha }$

lift force aerodynamic coefficient related with angle-of-attack

$ {C}_{Y\beta }$

lateral force aerodynamic coefficient related with angle of sideslip

$ {C}_{D\alpha }$

drag force aerodynamic coefficient related with angle-of-attack

$ {C}_{m\alpha },{C}_{mq}$

pitch moment aerodynamic coefficient related with angle-of-attack and pitch angular velocity

$ {C}_{n\beta },{C}_{np},{C}_{nr}$

yaw moment aerodynamic coefficient related with angle of sideslip, roll angular velocity, and yaw angular velocity

$ {C}_{l\beta },{C}_{lp},{C}_{lr}$

roll moment aerodynamic coefficient related with angle of sideslip, roll angular velocity, and yaw angular velocity

$ {C}_{L{\delta }_{e}},{C}_{L{\delta }_{a}}$

lift force aerodynamic coefficient related with elevons

$ {C}_{Y{\delta }_{e}},{C}_{Y{\delta }_{a}},{C}_{Y{\delta }_{r}}$

lateral force aerodynamic coefficient related with elevons and rudder

$ {C}_{D{\delta }_{e}},{C}_{D{\delta }_{a}},{C}_{D{\delta }_{r}}$

drag force aerodynamic coefficient related with elevons and rudder

$ {C}_{m{\delta }_{e}},{C}_{m{\delta }_{a}}$

pitch moment aerodynamic coefficient related with elevons

$ {C}_{n{\delta }_{e}},{C}_{n{\delta }_{a}},{C}_{n{\delta }_{r}}$

yaw moment aerodynamic coefficient related with elevons and rudder

$ {C}_{l{\delta }_{e}},{C}_{l{\delta }_{a}},{C}_{l{\delta }_{r}}$

roll moment aerodynamic coefficient related with elevons and rudder

$ {C}_{{\mathrm{}}_{L{\delta }_{e}}}^{{\delta }_{e}},{C}_{{\mathrm{}}_{L{\delta }_{a}}}^{{\delta }_{a}}$

lift force aerodynamic coefficient derivatives related with elevons

$ {C}_{{\mathrm{}}_{Y{\delta }_{e}}}^{{\delta }_{e}},{C}_{{\mathrm{}}_{Y{\delta }_{a}}}^{{\delta }_{a}},{C}_{{\mathrm{}}_{Y{\delta }_{r}}}^{{\delta }_{r}}$

lateral force aerodynamic coefficient derivatives related with elevons and rudder

$ {C}_{{\mathrm{}}_{D{\delta }_{e}}}^{{\delta }_{e}},{C}_{{\mathrm{}}_{D{\delta }_{a}}}^{{\delta }_{a}},{C}_{{\mathrm{}}_{D{\delta }_{r}}}^{{\delta }_{r}}$

drag force aerodynamic coefficient derivatives related with elevons and rudder

$ {C}_{{\mathrm{}}_{m{\delta }_{e}}}^{{\delta }_{e}},{C}_{{\mathrm{}}_{m{\delta }_{a}}}^{{\delta }_{a}},{C}_{{\mathrm{}}_{m{\delta }_{r}}}^{{\delta }_{r}}$

pitch moment aerodynamic coefficient derivatives related with elevons and rudder

$ {C}_{{\mathrm{}}_{n{\delta }_{e}}}^{{\delta }_{e}},{C}_{{\mathrm{}}_{n{\delta }_{a}}}^{{\delta }_{a}},{C}_{{\mathrm{}}_{n{\delta }_{r}}}^{{\delta }_{r}}$

yaw moment aerodynamic coefficient derivatives related with elevons and rudder

$ {C}_{{\mathrm{}}_{l{\delta }_{e}}}^{{\delta }_{e}},{C}_{{\mathrm{}}_{l{\delta }_{a}}}^{{\delta }_{a}},{C}_{{\mathrm{}}_{l{\delta }_{r}}}^{{\delta }_{r}}$

roll moment aerodynamic coefficient derivatives related with elevons and rudder

2.0 Formulation

The tapered general-purpose winged-cone [Reference Keshmiri, Colgren, Mirmirani and Bai25] is taken as the research object. The overall shape of the winged-cone is a cone-shaped structure. Its front view and side view are shown in Fig. 1.

Figure 1. Front and side views of the winged-cone.

The front of the wing cone is equipped with canard wings, which can only be opened when flying at low speed (subsonic state). When entering hypersonic flight, the canard will be inserted into the belly of the wing cone to improve flight speed. When the winged-cone re-enters the atmosphere, the engine shuts down and the flight only depends on elevons and rudder control to glide in the atmosphere.

Before establishing the HV dynamical model, we need to make corresponding assumptions according to the flight characteristics of the HV to facilitate our subsequent modeling:

In the reentry attitude control problem of the HV, we tend to pay the most attention to two groups of states. They are attitude angle $ \theta ={\left(\alpha ,\beta ,\mu \right)}^{T}$ and attitude angular velocity $ \omega ={\left(q,r,p\right)}^{T}$ . The nonlinear dynamical model of the HV used in this paper can be written as follows [Reference Keshmiri, Colgren, Mirmirani and Bai25]:

(1) \begin{align} \dot{\alpha } & =q-\mathrm{tan}\,\beta \!\left(p\,\mathrm{cos}\,\alpha +r\,\mathrm{sin}\,\alpha \right)+\frac{-L+{m}_{0}g\,\mathrm{cos}\,\gamma \,\mathrm{cos}\,\mu }{{m}_{0}V\,\mathrm{cos}\,\beta }+{d}_{\alpha }\nonumber\\[3pt] \dot{\beta } & =p\,\mathrm{sin}\,\alpha -r\,\mathrm{cos}\,\alpha +\frac{Y+{m}_{0}g\,\mathrm{cos}\,\gamma \,\mathrm{sin}\,\mu }{{m}_{0}V} + {d}_{\beta }\nonumber\\[3pt] \dot{\mu }& =\frac{p\,\mathrm{cos}\,\alpha +r\,\mathrm{sin}\,\alpha }{\,\mathrm{cos}\,\beta }+\frac{L\!\left(\mathrm{tan}\,\beta +\mathrm{tan}\,\gamma \,\mathrm{sin}\,\mu \right)+Y\,\mathrm{tan}\,\gamma \,\mathrm{cos}\,\mu -{m}_{0}g\,\mathrm{cos}\,\gamma \,\mathrm{tan}\,\beta \,\mathrm{cos}\,\mu }{{m}_{0}V}+{d}_{\mu }\nonumber\\[3pt] \dot{q} & =\frac{{J}_{z}-{J}_{x}}{{J}_{y}}pr+\frac{m+{x}_{cg}\!\left(D\,\mathrm{sin}\,\alpha +L\,\mathrm{cos}\,\alpha \right)}{{J}_{y}}+{d}_{q}\nonumber\\[3pt] \dot{r}& =\frac{{J}_{x}-{J}_{y}}{{J}_{z}}pq+\frac{n-{x}_{cg}Y}{{J}_{z}}+{d}_{r}\nonumber\\[3pt] \dot{p} & =\frac{{J}_{y}-{J}_{z}}{{J}_{x}}qr+\frac{l}{{J}_{x}}+{d}_{p}\end{align}

where, $ \alpha $ , $ \beta $ , $ \mu $ respectively represent the angle-of-attack, angle of sideslip and angle of bank. $ q$ , $ r$ , $ p$ respectively represent the angular rates of pitch, yaw and roll. $ \gamma $ represents flight path angle. $ {J}_{x}$ , $ {J}_{y}$ , $ {J}_{z}$ are the rotational inertias of the body around three axes respectively. $ {m}_{0}$ and $ V$ are the mass and velocity of the HV respectively. $ {x}_{cg}$ represents the distance from the pressure centre to the mass center. $ L$ , $ Y$ , $ D$ are the lift, lateral and drag force of the HV respectively. $ m$ , $ n$ , $ l$ are the pitch, yaw and roll moment of the HV respectively. $ {d}_{\alpha }$ , $ {d}_{\beta }$ , $ {d}_{\mu }$ are the equivalent disturbance force of the HV model’s each channel respectively. $ {d}_{q}$ , $ {d}_{r}$ , $ {d}_{p}$ are the equivalent disturbance moment of the HV model’s each channel respectively. $ g={g}_{0}{\left(\dfrac{{R}_{e}}{{R}_{e}+H}\right)}^{2}$ , $ {g}_{0}$ is acceleration of gravity on the earth’s surface, $ {R}_{e}$ is Earth radius, $ H$ is the flight altitude.

The aerodynamic forces and aerodynamic moments of the HV are expressed as follows :

(2) \begin{align} L & = {C}_{L}\bar{q}S \nonumber\\ m & ={C}_{m}\bar{q}cS\nonumber\\ Y & ={C}_{Y}\bar{q}S\\ n & ={C}_{n}\bar{q}bS\nonumber\\ D & ={C}_{D}\bar{q}S\nonumber\\ l & ={C}_{l}\bar{q}bS\nonumber \end{align}

where, $ {C}_{i}\left(i = L,Y,D,m,n,l\right)$ represents the relevant force aerodynamic and moment aerodynamic coefficient. $ \bar{q}$ , $ c$ , $ b$ , $ S$ are the dynamic pressure, reference aerodynamic chord, wing span and reference area of the HV respectively. Besides, $ \bar{q}=\frac{1}{2}\rho {V}^{2}$ , $ \rho ={\rho }_{0}{e}^{-\frac{H}{{H}_{0}}}$ , where $ \rho $ is atmospheric density, $ {\rho }_{0}$ is atmospheric density on the earth’s surface, $ {H}_{0}$ is equivalent atmospheric scale height.

The aerodynamic coefficients’ composition are as follows [Reference Keshmiri, Colgren, Mirmirani and Bai25]:

(3) \begin{align} {C}_{L} & = {C}_{L\alpha }+{C}_{L{\delta }_{e}}+{C}_{L{\delta }_{a}}\nonumber\\ {C}_{Y} & ={C}_{Y\beta }+{C}_{Y{\delta }_{e}}+{C}_{Y{\delta }_{a}}+{C}_{Y{\delta }_{r}}\nonumber\\ {C}_{D} & ={C}_{D\alpha }+{C}_{D{\delta }_{e}}+{C}_{D{\delta }_{a}}+{C}_{D{\delta }_{r}}\nonumber\\ {C}_{m} & ={C}_{m\alpha }+{C}_{m{\delta }_{e}}+{C}_{m{\delta }_{a}}+{C}_{m{\delta }_{r}}+{C}_{mq}\frac{qc}{2V}\\[4pt] {C}_{n} & ={C}_{n\beta }\beta +{C}_{n{\delta }_{e}}+{C}_{n{\delta }_{a}}+{C}_{n{\delta }_{r}}+{C}_{np}\frac{pb}{2V}+{C}_{nr}\frac{rb}{2V}\nonumber\\[4pt] {C}_{l} & ={C}_{l\beta }\beta +{C}_{l{\delta }_{e}}+{C}_{l{\delta }_{a}}+{C}_{l{\delta }_{r}}+{C}_{lp}\frac{pb}{2V}+{C}_{lr}\frac{rb}{2V}\nonumber \end{align}

where, $ {\delta }_{e}$ , $ {\delta }_{a}$ , $ {\delta }_{r}$ represent the left elevon angle, the right elevon angle and the rudder angle respectively.

The final actual control are the deflection angles of the HV. The control input is defined as $ U={\left({\delta }_{e},{\delta }_{a},{\delta }_{r}\right)}^{T}$ . In order to facilitate the subsequent controller design, Equation (1) is further sorted and rewritten into an affine nonlinear form (without disturbances):

(4) \begin{align} \dot{\theta } & ={B}_{\delta }{g}_{s\delta }U+{f}_{s}\!\left(\theta \right)+{g}_{s}\omega \nonumber\\ \dot{\omega } & ={J}^{-1}{g}_{f\delta }U+{f}_{f}\!\left(\omega \right) \end{align}

where, $ {J}^{-1}=diag\{{{J}_{y}}^{-1},{{J}_{z}}^{-1},{{J}_{x}}^{-1}\}$ . $ {B}_{\delta }$ is the equivalent control force efficiency matrix. $ {g}_{s}$ represents the system state matrix of angle loop. The specific expression of matrix $ {B}_{\delta }$ and $ {g}_{s}$ are shown in Equations (5 and 6):

(5) \begin{align} {B}_{\delta } & =\frac{1}{{m}_{0}V}\left(\begin{array}{c@{\quad}c@{\quad}c}\dfrac{1}{\,\mathrm{cos}\,\beta } & 0 & 0\\[6pt] 0 & 1 & 0\\[3pt] 0 & 0 & 1\\[3pt] \end{array}\right)\end{align}

(6) \begin{align} {g_s} = \left( {\begin{array}{c@{\quad}c@{\quad}c}1 & { - {\rm{sin}}\,\alpha \,{\rm{tan}}\,\beta } & { - {\rm{cos}}\,\alpha \,{\rm{tan}}\,\beta }\\[3pt]0 & { - {\rm{cos}}\,\alpha } & {\,{\rm{sin}}\,\alpha }\\[3pt]0 & {\,{\rm{sin}}\,\alpha \,{\rm{sec}}\,\beta } & {\,{\rm{cos}}\,\alpha \,{\rm{sec}}\,\beta }\end{array}} \right)\\[-10pt]\nonumber\end{align}

where, $ {f}_{\mathrm{s}}\!\left(\omega \right)={\left(\begin{array}{l@{\quad}l@{\quad}l}{f}_{\alpha } & {f}_{\beta } & {f}_{\mu } \end{array}\right)}^{T}$ is a vector function of attitude angle. $ {f}_{f}\left(\omega \right)={\left(\begin{array}{l@{\quad}l@{\quad}l}{f}_{q} & {f}_{r} & {f}_{p} \end{array}\right)}^{T}$ is a vector function of attitude angular velocity. The specific expression of vector function $ {f}_{\mathrm{s}}\!\left(\omega \right)$ and $ {f}_{f}\!\left(\omega \right)$ are shown in Equations (7 and 8):

(7) \begin{align}{f_\alpha } & = \frac{{ - {C_{L\alpha }}\bar{q}S + {m_0}g\,{\rm{cos}}\,\gamma \,{\rm{cos}}\,\mu }}{{{m_0}V\,{\rm{cos}}\,\beta }}\nonumber\\{f_\beta } & = \frac{{ - {C_{Y\beta }}\bar{q}S + {m_0}g\,{\rm{cos}}\,\gamma \,{\rm{sin}}\,\mu }}{{{m_0}V}}\\{f_\mu } & = \frac{{{C_{Y\beta }}\bar{q}S\,{\rm{cos}}\,\mu + {C_{L\alpha }}\bar{q}S\,{\rm{sin}}\,\mu }}{{{m_0}V}}\,{\rm{tan}}\,\gamma + \frac{{{C_{L\alpha }}\bar{q}S - {m_0}g\,{\rm{cos}}\,\gamma \,{\rm{cos}}\,\mu }}{{{m_0}V}}\,{\rm{tan}}\,\beta \nonumber\end{align}

(8) \begin{align}{f}_{q} & =\frac{{J}_{z}-{J}_{x}}{{J}_{y}}pr+\frac{\bar{q}cS\!\left({C}_{m\alpha }+{C}_{mq}\dfrac{qc}{2V}\right)+{x}_{cg}\bar{q}S\!\left({C}_{D\alpha }\,\mathrm{sin}\,\alpha +{C}_{L\alpha }\,\mathrm{cos}\,\alpha \right)}{{J}_{y}}\nonumber\\[5pt] {f}_{r} & =\frac{{J}_{x}-{J}_{y}}{{J}_{z}}pq+\frac{\bar{q}bS\!\left({C}_{n\beta }+{C}_{np}\dfrac{pb}{2V}+{C}_{nr}\dfrac{rb}{2V}\right)-{x}_{cg}\bar{q}S\ {C}_{Y\beta }}{{J}_{z}}\\[5pt] {f}_{p} & =\frac{{J}_{y}-{J}_{z}}{{J}_{x}}qr+\frac{\bar{q}bS\!\left({C}_{l\beta }+{C}_{lp}\dfrac{pb}{2V}+{C}_{lr}\dfrac{rb}{2V}\right)}{{J}_{x}}\nonumber \end{align}

where, $ {g}_{s\delta }$ is the control matrix generated by deflection angle to attitude angle. $ {g}_{f\delta }$ is the control matrix generated by deflection angle to attitude angular velocity. The specific expressions of matrix $ {g}_{s\delta }$ and $ {g}_{f\delta }$ are shown in Equations (9 and 10):

(9) \begin{equation} {g}_{s\delta }=\bar{q}s\!\left(\begin{array}{l@{\quad}l@{\quad}l}{g}_{\alpha {\delta }_{e}} & {g}_{\alpha {\delta }_{a}} & {g}_{\alpha {\delta }_{r}}\\[4pt] {g}_{\beta {\delta }_{e}} & {g}_{\beta {\delta }_{a}} & {g}_{\beta {\delta }_{r}}\\[4pt] {g}_{\mu {\delta }_{e}} & {g}_{\mu {\delta }_{a}} & {g}_{\mu {\delta }_{r}} \end{array}\right)\end{equation}

(10) \begin{equation}{g}_{f\delta }=\bar{q}s\!\left(\begin{array}{l@{\quad}l@{\quad}l}{g}_{q{\delta }_{e}} & {g}_{q{\delta }_{a}} & {g}_{q{\delta }_{r}}\\[4pt] {g}_{r{\delta }_{e}} & {g}_{r{\delta }_{a}} & {g}_{r{\delta }_{r}}\\[4pt] {g}_{p{\delta }_{e}} & {g}_{p{\delta }_{a}} & {g}_{p{\delta }_{r}} \end{array}\right)\end{equation}

Equation (11) is the specific expression of variables in Equations (9 and 10):

(11) \begin{align} {g}_{\alpha {\delta }_{e}} & =-{C}_{{\mathrm{}}_{L{\delta }_{e}}}^{{\delta }_{e}};\,{g}_{\beta {\delta }_{e}}={C}_{{\mathrm{}}_{Y{\delta }_{e}}}^{{\delta }_{e}};\,{g}_{\mu {\delta }_{e}}={C}_{{\mathrm{}}_{Y{\delta }_{e}}}^{{\delta }_{e}}\,\mathrm{tan}\,\gamma \,\mathrm{cos}\,\mu +{C}_{{\mathrm{}}_{L{\delta }_{e}}}^{{\delta }_{e}}\left(\mathrm{tan}\,\gamma \,\mathrm{sin}\,\mu +\mathrm{tan}\,\beta \right);\,\nonumber\\[3pt] {g}_{\alpha {\delta }_{a}} & =-{C}_{{\mathrm{}}_{L{\delta }_{a}}}^{{\delta }_{a}};\,{g}_{\beta {\delta }_{a}}={C}_{{\mathrm{}}_{Y{\delta }_{a}}}^{{\delta }_{a}};\,{g}_{\mu {\delta }_{a}}={C}_{{\mathrm{}}_{Y{\delta }_{a}}}^{{\delta }_{a}}\,\mathrm{tan}\,\gamma \,\mathrm{cos}\,\mu +{C}_{{\mathrm{}}_{L{\delta }_{a}}}^{{\delta }_{a}}\left(\mathrm{tan}\,\gamma \,\mathrm{sin}\,\mu +\mathrm{tan}\,\beta \right);\,\nonumber\\[3pt] {g}_{\alpha {\delta }_{r}} & =0;\,{g}_{\beta {\delta }_{r}}={C}_{{\mathrm{}}_{Y{\delta }_{r}}}^{{\delta }_{r}};\,{g}_{\mu {\delta }_{r}}={C}_{{\mathrm{}}_{Y{\delta }_{r}}}^{{\delta }_{r}}\,\mathrm{tan}\,\gamma \,\mathrm{cos}\,\mu ;\,\nonumber\\[3pt] {g}_{q{\delta }_{e}} & =c{C}_{{\mathrm{}}_{m{\delta }_{e}}}^{{\delta }_{e}}+{x}_{cg}\!\left({C}_{{\mathrm{}}_{D{\delta }_{e}}}^{{\delta }_{e}}\,\mathrm{sin}\,\alpha +{C}_{{\mathrm{}}_{L{\delta }_{e}}}^{{\delta }_{e}}\,\mathrm{cos}\,\alpha \right);\,{g}_{r{\delta }_{e}}=b{C}_{{\mathrm{}}_{n{\delta }_{e}}}^{{\delta }_{e}}-{x}_{cg}{C}_{{\mathrm{}}_{Y{\delta }_{e}}}^{{\delta }_{e}};\,{g}_{p{\delta }_{e}}=b{C}_{{\mathrm{}}_{l{\delta }_{e}}}^{{\delta }_{e}};\,\\[3pt] {g}_{q{\delta }_{a}} & =c{C}_{{\mathrm{}}_{m{\delta }_{a}}}^{{\delta }_{a}}+{x}_{cg}\!\left({C}_{{\mathrm{}}_{D{\delta }_{a}}}^{{\delta }_{a}}\,\mathrm{sin}\,\alpha +{C}_{{\mathrm{}}_{L{\delta }_{a}}}^{{\delta }_{a}}\,\mathrm{cos}\,\alpha \right);\,{g}_{r{\delta }_{a}}=b{C}_{{\mathrm{}}_{n{\delta }_{a}}}^{{\delta }_{a}}-{x}_{cg}{C}_{{\mathrm{}}_{Y{\delta }_{a}}}^{{\delta }_{a}};\,{g}_{p{\delta }_{a}}=b{C}_{{\mathrm{}}_{l{\delta }_{a}}}^{{\delta }_{a}};\,\nonumber\\[3pt] {g}_{q{\delta }_{r}} & =c{C}_{{\mathrm{}}_{m{\delta }_{r}}}^{{\delta }_{r}}+{x}_{cg}{C}_{{\mathrm{}}_{D{\delta }_{r}}}^{{\delta }_{r}}\,\mathrm{sin}\,\alpha ;\,{g}_{r{\delta }_{r}}=b{C}_{{\mathrm{}}_{n{\delta }_{r}}}^{{\delta }_{r}}-{x}_{cg}{C}_{{\mathrm{}}_{Y{\delta }_{r}}}^{{\delta }_{r}};\,{g}_{p{\delta }_{r}}=b{C}_{{\mathrm{}}_{l{\delta }_{r}}}^{{\delta }_{r}};\,\nonumber \end{align}

The $ {B}_{\delta }{g}_{s\delta }U$ and $ {f}_{s}\!\left(\theta \right)$ in the Equation (4) are small quantities, and consider the existence of disturbances and unmodeled dynamics. In order to facilitate the subsequent controller design, Equation (4) is further rewritten. These small quantities and unmodeled dynamics are treated as internal disturbances of the model, and the final affine nonlinear model can be written as follows :

(12) \begin{align}\dot{\theta } & ={g}_{s}\omega +{\mathrm{\Delta }}_{s}\nonumber\\ \dot{\omega } & ={g}_{f}U+{f}_{f}\!\left(\omega \right)+{\mathrm{\Delta }}_{f} \end{align}

where, $ {g}_{f}={J}^{-1}{g}_{f\delta }$ . $ {\mathrm{\Delta }}_{s}={B}_{\delta }{g}_{s\delta }U+{f}_{s}\!\left(\theta \right)+{D}_{1}$ . $ {\mathrm{\Delta }}_{f}={J}^{-1}{M}_{\delta }+{D}_{2}$ . $ {D}_{1}$ and $ {D}_{2}$ are the sum of external disturbances and other internal disturbances such as system uncertainty and unmodeled dynamics, $ {D}_{1},{D}_{2}\in {R}^{3\times 1}$ .

3.0 Controller design

3.1 Prescribed performance control structure

The design method of the prescribed performance controller is as follows. First, the PPF is designed to constrain the state, and then the corresponding control strategy is proposed for the constrained state. The control performance of the system can be guaranteed. In order to obtain the better comprehensive performance of angle tracking control, the PPC structure is adopted in the outer loop. The error transformation function in the outer loop is used to transform the actual control error into virtual error. Then a dynamic surface controller is designed for the virtual error to ensure the boundedness of the transformed system error. Finally, design the relevant dynamic surface controller in the inner loop.

In this section, the method of unconstrained translation of state under homeomorphic mapping is adopted. It can transform the constrained state control problem into a new unconstrained state control problem. The whole system control structure is shown in Fig. 2.

Figure 2. Control system structure block diagram.

3.1.1 General PPF and error transformation form

The controlled state is generally taken as error $ e\!\left(t\right)$ in the prescribed performance tracking control. In most current researches related to the PPC method [Reference Zhang, Ma, Sun and Li26–Reference Bu28], the PPF is usually used:

(13) \begin{align}\rho \!\left(t\right)=k\!\left({\rho }_{0}-{\rho }_{\mathrm{\infty }}\right){e}^{-\lambda t}+{\rho }_{\mathrm{\infty }}\end{align}

where, $ {\rho }_{0}$ represents the initial error limit, $ {\rho }_{\mathrm{\infty }}$ represents the steady-state error limit, $ \lambda $ represents the error convergence rate limit, which can be designed manually. $ \rho \!\left(t\right)$ satisfies $ \mathrm{l}\mathrm{i}{\mathrm{m}}_{t\to \mathrm{\infty }}\rho \!\left(t\right)={\rho }_{\mathrm{\infty }}$ with $ {\rho }_{0}\gt {\rho }_{\mathrm{\infty }}\gt 0$ . $ k$ is the scaling coefficient, and the default value is 1. Then, $ e\!\left(t\right)$ satisfies:

(14) \begin{align}-L\rho \!\left(t\right)\le {e}_{i}\!\left(t\right)\le U\rho \!\left(t\right)\end{align}

Generally, the unconstrained error transformation function is chosen as follows:

(15) \begin{align}{\varepsilon }_{i}\!\left(t\right) & =\frac{1}{2}\mathrm{l}\mathrm{n}\!\left(\frac{LU+U{x}_{i}\!\left(t\right)}{LU-L{x}_{i}\!\left(t\right)}\right)\nonumber\\ {x}_{i}\!\left(t\right) & =\frac{{e}_{i}\!\left(t\right)}{\rho \!\left(t\right)} \end{align}

where, $ {\varepsilon }_{i}\!\left(t\right)\in R\!\left(i=\mathrm{1,2},3\right)$ is the unconstrained virtual error after conversion. $ {e}_{i}\!\left(t\right)\in R\!\left(i=\mathrm{1,2},3\right)$ is the actual error.

In view of the Equation (15), taking the derivative of $ \varepsilon \!\left(t\right)={\left[{\varepsilon }_{1}\left(t\right),{\varepsilon }_{2}\left(t\right),{\varepsilon }_{3}\left(t\right)\right]}^{T}$ with respect to time, we can obtain

(16) \begin{align} \dot{\varepsilon }\!\left(t\right)={T}_{d}\dot{e}\!\left(t\right)+{T}_{c}\end{align}

(17) \begin{align} {T}_{di}=\frac{\dfrac{1}{{x}_{i}\!\left(t\right)+L}-\dfrac{1}{{x}_{i}\!\left(t\right)-U}}{2\rho \!\left(t\right)}\left(i=\mathrm{1,2},3\right)\end{align}

(18) \begin{align} {T}_{ci}=-{T}_{di}\frac{{e}_{i}\!\left(t\right)\dot{\rho }\!\left(t\right)}{\rho \!\left(t\right)}\left(i=\mathrm{1,2},3\right)\end{align}

where, $ {T}_{d}=diag\!\left({T}_{d1},{T}_{d2},{T}_{d3}\right)$ , $ {T}_{c}={\left[{T}_{c1},{T}_{c2},{T}_{c3}\right]}^{T}$ . $ L$ and $ U$ represent the upper and lower error overshoot constraint coefficients, with the range of $ \left(\mathrm{0,1}\right)$ . An example of PPF curve form is shown in Fig. 3 ( $ L=U=1$ ).

Figure 3. PPF curve.

3.1.2 PPF with adaptive scaling strategy

Due to the disturbances of the system and drastic changes of reference command, the tracking error may suddenly become large and violate the constraint. In Equation (15), the unconstrained virtual state will result in a computational singularity, even a system divergence caused by the error violation. Therefore, this section proposes a PPC method with an adaptive scaling strategy. The scheme provides a PPF that can adjust adaptively if the tracking error violation occurs. It can ensure that the tracking error no longer violates the constraint through adaptive expansion.

In addition, the PPF structural parameters $ {\rho }_{0}$ is set according to the initial error size, $ l$ , $ {\psi }_{1}$ and $ {\psi }_{2}$ are the constant parameters we can design. It can make the PPF structural parameters to be set more flexible.

Besides, a periodic scaling scheme is adopted. It can avoid the buffeting effect of the control response caused by frequent PPF scaling. According to the above scheme, the adaptive scaling PPF can be realised.

We summarise the implementation of adaptive scaling strategy in Algorithm 1.

Algorithm 1 The implementation of adaptive scaling strategy.

Substituting the coefficients $ k$ , $ {\rho }_{0}$ and $ {\rho }_{\mathrm{\infty }}$ of Algorithm 1 into the Equations (13–18), then we can get the new $ \varepsilon $ , $ {T}_{d}$ and $ {T}_{c}$ . Therefore, according to the PPC principle, as long as we design the controller to ensure that the virtual error $ \varepsilon $ is bounded, the tracking error of the original system will satisfy the constraint performance $ -L\rho \!\left(t\right)\le {e}_{i}\!\left(t\right)\le U\rho \!\left(t\right)$ we designed.

Remark 1. The Algorithm 1 includes the PPF of expanding and resuming. If the signal of the tracking error violation occurs, the PPF is adaptively expanded according to the angle error and angle error change rate, where $ h$ is the sensitivity coefficient. $ \vartheta $ represents the error trend angle of the scaling point, while $ tan\!\left(\vartheta \right)$ corresponds to the error derivative of the scaling point. By adjusting the appropriate $ \vartheta $ , we can obtain two types of scaling. We hope that the expanded PPF can reasonably cover the error changes. Expansion methods based on error and error change rate are more predictive and reasonable. Too frequent or excessive expansion is detrimental to the tracking performance of the system. Compared to arbitrary expansion methods, this scheme can ensure the use of reasonable expansion times and degrees. Based on the angle error and angle error change rate, we can design two expansion schemes with different degrees of conservatism. It will guarantee as proper times and degrees of expansion as possible.

Remark 2. The expansion in algorithm1 has high priority. As long as the tracking error violation occurs at any time, the expansion in the Algorithm 1 will be triggered and the expansion coefficient will be locked. Once triggered, the scaling coefficient will be locked and maintained for time $ T$ . At the end of the time $ T$ , the algorithm can recover to normal PPF ( $ k=1$ ) immediately after judging no violation. If the tracking error still violates the constraint after expanding, it can still expand according to the previous expansion until no violation, and so on. The setting of $ T$ ensures the buffer effect of PPF’s expansion after the violation.

Remark 3. PPF structure parameter $ {\rho }_{0}$ is set to $ {\rho }_{0}=l\cdot \left|{e}_{0}\right|+{\psi }_{1}$ . In this way, it is associated with the initial error and become more flexible. By adding a set of smaller constants $ {\psi }_{1}$ and $ {\psi }_{2}$ as constraints, it can avoid PPF being too tight and causing frequent state violation.

3.2 Disturbance observer

Considering the uncertainties and disturbances in Equation (12), the controller own robustness cannot effectively resist the disturbance. So a nonlinear disturbance observer (NDO) is adopted to observe and compensate the disturbance [Reference Bu, Wu, Chen and Bai29], making the system more robust.

Lemma 1. [Reference Bu, Wu, Chen and Bai29] Consider the following system:

(19) \begin{align}{\dot{x}}_{1} & ={x}_{2}\nonumber\\ {\dot{x}}_{2} & =F\!\left({x}_{1},{x}_{2}\right) \end{align}

where $ {x}_{1},{x}_{2}\in {R}^{1\times 1}$ . If system (19) satisfies the conditions of $ {x}_{1}\!\left(t\right)\to 0,{x}_{2}\!\left(t\right)\to 0\!\left(t\to \infty \right)$ , then for any arbitrary continuous bounded function $ v\!\left(t\right)$ , $ T\gt 0$ and $ Q\gt 0$ , the solution of the following system:

(20) \begin{align} {\dot{x}}_{1} & ={x}_{2}\nonumber\\ {\dot{x}}_{2} & ={Q}^{2}F\!\left({x}_{1}-v\!\left(t\right),\frac{{x}_{2}}{Q}\right) \end{align}

satisfies:

(21) \begin{align}\underset{Q\to \infty }{\mathop{\lim }}{\int }_{0}^{T} \left|{x}_{1}-v\!\left(t\right)\right|dt=0\end{align}

Lemma 1 indicates that if $ Q$ is selected to be sufficient large, then $ {x}_{1}$ averagely converges to the input signal $ v\!\left(t\right)$ and $ {x}_{2}$ weakly converges to the generalised differentiation of $ \dot{v}\!\left(t\right)$ .

Theorem 1. Consider the following system:

(22) \begin{align}{\dot{x}}_{1} & ={x}_{2}\nonumber\\ {\dot{x}}_{2} & ={Q}^{2}\left(-{h}_{1}\,\mathrm{tan}\mathrm{h}\!\left({x}_{1}\right)-{h}_{2}\,\mathrm{tanh}\!\left(\frac{{x}_{2}}{Q}\right)\right)\end{align}

where, $ {x}_{1},{x}_{2},{h}_{1},{h}_{2},Q\in {R}^{1\times 1}$ . Define the candidate Lyapunov function as follows:

(23) \begin{align}{V}_{1}={\int }_{0}^{{x}_{1}}\mathrm{ }{Q}^{2}{h}_{1}\,\mathrm{tanh}\!\left(\xi \right)d\xi +\frac{1}{2}{x}_{2}^{2}\end{align}

The derivative of Equation (23) can be obtained,

(24) \begin{align} {\dot{V}}_{1} & ={h}_{1}{Q}^{2}\,\mathrm{tanh}\!\left({x}_{1}\right){x}_{2}+{x}_{2}{\dot{x}}_{2}\nonumber\\ & =-{h}_{2}{Q}^{2}\,\mathrm{tanh}\!\left(\frac{{x}_{2}}{Q}\right){x}_{2}\\ & \le 0\nonumber \end{align}

Therefore, when $ {h}_{1},{h}_{2}\gt 0$ , the system is asymptotically stable at $ \left(\mathrm{0,0}\right)$ . The $ F\!\left({x}_{1},{x}_{2}\right)$ we selected satisfies the Lemma 1.

Now, consider the system as follows :

(25) \begin{align} \dot{z}=f\!\left(z\right)+g\!\left(z\right)u+d\end{align}

where, $ z\in {R}^{n\times 1}$ is the state, $ f\!\left(z\right)\in {R}^{n\times 1}$ and $ g\!\left(z\right)\in {R}^{n\times n}$ are continuous functions, $ u\in {R}^{n\times 1}$ is the control quantity, $ d\in {R}^{n\times 1}$ is the sum of unmodeled dynamics and disturbances.

Theorem 2. [Reference Bu, Wu, Chen and Bai29] Consider the following system:

(26) \begin{align} \dot{\hat{z}} & =f\!\left(z\right)+g\!\left(z\right)u+\hat{d}\nonumber\\ \dot{\hat{d}} & ={Q}^{2}\left(-{h}_{1}\,\mathrm{tanh}\!\left(\hat{z}-z\right)-{h}_{2}\,\mathrm{tanh}\!\left(\frac{\hat{d}}{Q}\right)\right) \end{align}

where, $ \hat{z}$ and $ \hat{d}$ are the estimates of $ z$ and $ d$ . $ {h}_{1},{h}_{2},Q\in {R}^{1\times 1}$ represent adjustable observer parameters. If $ T\gt 0$ and $ Q\gt 0$ , the

(27) \begin{align}\underset{Q\to \infty }{\mathop{\lim }}{\int }_{0}^{T}\mathrm{ }\left|\hat{z}-z\right|dt=0\end{align}

holds true, it means that $ \hat{z}$ approaches $ z$ . Furthermore, based on Equations (25 and 26), we can obtain that $ \hat{d}\to d$ .

According to Theorem 1 and Lemma 1, it can be seen that the value of $ \left|\dot{\hat{d}}\right|=\left|{Q}^{2}F\!\left(\hat{z}-z,\frac{\hat{d}}{Q}\right)\right|$ may be infinitely large when $ Q\to \infty $ . It denotes that $ \hat{d}$ varies much faster than $ f\!\left(z\right)+g\!\left(z\right)u$ . Then we have

(28) \begin{align}\underset{Q\to \infty }{\mathop{\lim }}\frac{d\!\left(f\!\left(z\right)+g\!\left(z\right)u+\hat{d}\right)}{dt}\approx \dot{\hat{d}}\end{align}

(29) \begin{align}\underset{Q\to \infty }{\mathop{\lim }}\frac{f\!\left(z\right)+g\!\left(z\right)u+\hat{d}}{Q}\approx \frac{\hat{d}}{Q}\end{align}

Then, replacing $ {x}_{2}$ in Equation (20) by $ f\!\left(z\right)+g\!\left(z\right)u+\hat{d}$ , the Equations (26 and 27) always hold true.

Assumption 4. [Reference Hao, Zhang, Cao and Tang30] The $ d$ and $ \dot{d}$ are bounded. Based on this assumption, the system disturbance could be observed. Thus all of the continuous and bounded system disturbances satisfy this assumption. The stability analysis of disturbance estimation is feasible.

Remark 4. [Reference Bu, Wu, Zhang, Ma and Huang31] Although only when $ Q\to \infty $ one can get $ \hat{d}\to d$ in theory. Simulation studies show that an effective estimation performance can be also obtained using the proposed NDO (26) by choosing a finite $ Q$ . $ \hat{d}\to d$ can still converge to a small convergence domain.

Remark 5. The parameter analysis of the NDO is as follows: $ Q$ , $ {h}_{1}$ and $ {h}_{2}$ are related to the convergence speed and convergence accuracy. Increasing $ Q$ or $ {h}_{1}$ will improve the convergence speed and convergence accuracy, but an excessively large $ Q$ or $ {h}_{1}$ will increase the high-frequency oscillation during convergence process; $ {h}_{2}$ has the opposite effect. Increasing $ {h}_{2}$ is beneficial for suppressing high frequency oscillation, but excessive $ {h}_{2}$ can slow down convergence speed and reduce accuracy. The parameter tuning process is as follows: First, select a large $ Q$ as the dominant factor. Second, select a small $ {h}_{1}$ to continue to fine-tune the system. Last, select a small $ {h}_{2}$ to suppress the high frequency oscillation during convergence process. We provide the effectiveness verification of NDO in section 4.

According to the above analysis, combing the nonlinear model (12), we can write the disturbance observer model in the form of (30–31) :

(30) \begin{align} \dot{\hat{\theta }} & ={g}_{s}\omega +{\hat{\mathrm{\Delta }}}_{s}\nonumber\\[3pt] {\dot{\hat{\mathrm{\Delta }}}}_{s} & ={{R}_{s}}^{2}\left(-{h}_{s1}\,\mathrm{tanh}\!\left(\hat{\theta }-\theta \right)-{h}_{s2}\,\mathrm{tanh}\!\left(\frac{{\hat{\mathrm{\Delta }}}_{s}}{{R}_{s}}\right)\right) \end{align}

(31) \begin{align} \dot{\hat{\omega }} & ={g}_{f}U+{f}_{f}\!\left(\omega \right)+{\hat{\mathrm{\Delta }}}_{f}\nonumber\\[3pt] {\dot{\hat{\mathrm{\Delta }}}}_{f} & ={{R}_{f}}^{2}\left(-{h}_{f1}\,\mathrm{tanh}\!\left(\hat{\omega }-\omega \right)-{h}_{f2}\,\mathrm{tanh}\!\left(\frac{{\hat{\mathrm{\Delta }}}_{f}}{{R}_{f}}\right)\right) \end{align}

where, $ \hat{\theta }$ , $ \hat{\omega }$ , $ {\hat{\mathrm{\Delta }}}_{s}$ , $ {\hat{\mathrm{\Delta }}}_{f}$ are the estimated values of $ \theta $ , $ \omega $ , $ {\mathrm{\Delta }}_{s}$ , $ {\mathrm{\Delta }}_{f}$ . In addition, we define $ \tilde{\theta}=\hat{\theta }-\theta $ , $ \tilde{\omega}=\hat{\omega }-\omega $ , $ \tilde{\Delta}_{s}={\hat{\mathrm{\Delta }}}_{s}-{\mathrm{\Delta }}_{s}$ , $ \tilde{\Delta}_{f}={\hat{\mathrm{\Delta }}}_{f}-{\mathrm{\Delta }}_{f}$ as the corresponding estimation errors of NDO. By designing and adjusting the appropriate parameter $ {R}_{s}$ , $ {h}_{s1}$ , $ {h}_{s2}$ , $ {R}_{f}$ , $ {h}_{f1}$ , $ {h}_{f2}$ , we can ensure that the estimation errors satisfy $ \|\tilde{\theta}\|\lt \bar{\theta}$ , $ \|\tilde{\omega}\|\lt \bar{\omega}$ , $ \|\tilde{\Delta}_{s}\|\lt \bar{\Delta}_{s}$ , $ \|\tilde{\Delta}_{f}\|\lt \bar{\Delta}_{f}$ . $ \bar{\theta}$ , $ \bar{\omega}$ , $ \bar{\Delta}_{s}$ and $ \bar{\Delta}_{f}$ are their upper bounds.

3.3 Design of prescribed performance dynamic surface controller

Step 1: Define the first dynamic surface $ {S}_{1}=\varepsilon $ and error $ {e}_{1}=\theta -{\theta }_{d}$ . $ {\theta }_{d}\in {R}^{3\times 1}$ is the given command angle. Combined with the Equations (16–18), we can get the derivative of $ {S}_{1}$ as follows:

(32) \begin{align} {\dot{e}}_{1} & ={g}_{s}\omega +{\mathrm{\Delta }}_{s}-{\dot{\theta }}_{d}\nonumber\\[3pt] {\dot{S}}_{1} & ={T}_{d}{\dot{e}}_{1}+{T}_{c} \end{align}

According to method of dynamic surface controller, the virtual control law $ {\omega }_{1}$ is proposed as:

(33) \begin{align} {\omega }_{1}={{g}_{s}}^{-1}\left({\dot{\theta }}_{d}-{\hat{\mathrm{\Delta }}}_{s}-{{T}_{d}}^{-1}{T}_{c}-{C}_{1}{S}_{1}\right)\end{align}

where, $ {C}_{1}=diag\!\left({C}_{11},{C}_{12},{C}_{13}\right)$ . $ {C}_{11},{C}_{12},{C}_{13}$ are positive constants.

Next, in order to avoid the complexity, let $ {\omega }_{1}$ pass through a first-order filter with time constant $ \tau $ , given by:

(34) \begin{align} \tau {\dot{\omega }}_{d}+{\omega }_{d}={\omega }_{1},{\omega }_{d}\!\left(0\right)={\omega }_{1}\!\left(0\right)\end{align}

Within this setting, $ {\dot{\omega }}_{d}$ can be calculated by $ \dfrac{({\omega }_{1}-{\omega }_{d})}{\tau }$ without direct differential operation of $ {\omega }_{d}$ . Further, the error of the low filter is defined as:

(35) \begin{align} \sigma ={\omega }_{d}-{\omega }_{1}\end{align}

Step 2: Define the second dynamic surface $ {S}_{2}=\omega -{\omega }_{d}$ . Taking the derivative of $ {S}_{2}$ with respect to time yields:

(36) \begin{align} {\dot{S}}_{2}=\dot{\omega }-{\dot{\omega }}_{d}\end{align}

Consider the candidate Lyapunov function as :

(37) \begin{align} {V}_{2}=\frac{1}{2}{{S}_{1}}^{T}{S}_{1}+\frac{1}{2}{{S}_{2}}^{T}{S}_{2}+\frac{1}{2}{\sigma }^{T}\sigma \end{align}

The derivative of $ {V}_{2}$ can be written as :

(38) \begin{align}{\dot{V}}_{2} & ={{S}_{1}}^{T}{\dot{S}}_{1}+{{S}_{2}}^{T}{\dot{S}}_{2}+{\sigma }^{T}\dot{\sigma }\nonumber\\[4pt] & ={{S}_{1}}^{T}\left({T}_{d}{\dot{e}}_{1}+{T}_{c}\right)+{{S}_{2}}^{T}\left({g}_{f}U+{f}_{f}\left(\omega \right)+{\mathrm{\Delta }}_{f}-{\dot{\omega }}_{d}\right)+{\sigma }^{T}\dot{\sigma }\nonumber\\[4pt] & ={{S}_{1}}^{T}\left({T}_{d}\left({g}_{s}\sigma +{g}_{s}{S}_{2}+{g}_{s}{\omega }_{1}+{\mathrm{\Delta }}_{s}-{\dot{\theta }}_{d}\right)+{T}_{c}\right)\nonumber\\[4pt] & \quad +{{S}_{2}}^{T}\left({g}_{f}U+{f}_{f}\left(\omega \right)+{\mathrm{\Delta }}_{f}-{\dot{\omega }}_{d}\right)+{\sigma }^{T}\dot{\sigma }\nonumber\\[4pt] & ={{S}_{1}}^{T}\left({T}_{d}\left(-\tilde{\Delta}_{s}+{g}_{s}{S}_{2}+{g}_{s}\sigma -{C}_{1}{S}_{1}\right)\right)\\[4pt] & \quad +{{S}_{2}}^{T}\left({g}_{f}U+{f}_{f}\left(\omega \right)+{\mathrm{\Delta }}_{f}-{\dot{\omega }}_{d}\right)+{\sigma }^{T}\dot{\sigma }\nonumber\\[4pt] & =-{\left({T}_{d}{S}_{1}\right)}^{T}\tilde{\Delta}_{s}+{{S}_{1}}^{T}\left({T}_{d}{g}_{s}{S}_{2}\right)+{{S}_{1}}^{T}\left({T}_{d}{g}_{s}\sigma \right)\nonumber\\[4pt] & \quad -{{S}_{1}}^{T}\left({T}_{d}{C}_{1}\right){S}_{1}+{{S}_{2}}^{T}\left({g}_{f}U+{f}_{f}\left(\omega \right)+{\mathrm{\Delta }}_{f}-{\dot{\omega }}_{d}\right)+{\sigma }^{T}\dot{\sigma }\nonumber\end{align}

To stabilise $ {V}_{2}$ , the control law $ U$ is proposed as:

(39) \begin{align} U={{g}_{f}}^{-1}\left(-{f}_{f}-{\hat{\mathrm{\Delta }}}_{f}+{\dot{\omega }}_{d}-{T}_{d}{g}_{s}{S}_{1}-{C}_{2}{S}_{2}-\frac{{{S}_{1}}^{T}\left({T}_{d}{g}_{s}\sigma \right){S}_{2}}{{\|{S}_{2}\|}^{2}}-{T}_{d}{{g}_{s}}^{T}{S}_{1}\right)\end{align}

where, $ {C}_{2}=diag\!\left({C}_{21},{C}_{22},{C}_{23}\right)$ . $ {C}_{21},{C}_{22},{C}_{23}$ are positive constants.

Combining the disturbance observer form designed by Equations (30 and 31) and substituting the Equations (33) and (39) into Equation (38), we can get :

(40) \begin{align}{\dot{V}}_{2} & =-{\left({T}_{d}{S}_{1}\right)}^{T}\tilde{\Delta}_{s}+{{S}_{1}}^{T}\left({T}_{d}{g}_{s}\sigma \right)-{{S}_{1}}^{T}\left({T}_{d}{C}_{1}\right){S}_{1}\nonumber\\[4pt]& -{{S}_{2}}^{T}\tilde{\Delta}_{f}-{{S}_{2}}^{T}{C}_{2}{S}_{2}+{\sigma }^{T}\dot{\sigma }\end{align}

According to the well-known Young’s inequality [Reference Hu, Shi and Shao32], we can obtain the following inequality:

(41) \begin{align} -{\left({T}_{d}{S}_{1}\right)}^{T}\tilde{\Delta}_{s}\le \frac{{\chi }_{1}}{2}{\left({T}_{d}{S}_{1}\right)}^{T}\left({T}_{d}{S}_{1}\right)+\frac{1}{2{\chi }_{1}}\tilde{\Delta}^{T}_{s}\tilde{\Delta}_{s}\end{align}

(42) \begin{align} -{{S}_{2}}^{T}\tilde{\Delta}_{f}\le \frac{{\chi }_{2}}{2}{{S}_{2}}^{T}{S}_{2}+\frac{1}{2{\chi }_{2}}\tilde{\Delta}^{T}_{f}\tilde{\Delta}_{f}\end{align}

where $ {\chi }_{1}$ and $ {\chi }_{2}$ is a positive tuning parameter. Then, we have

(43) \begin{align}{\dot{V}}_{2} & \le \frac{{\chi }_{1}}{2}{\left({T}_{d}{S}_{1}\right)}^{T}\left({T}_{d}{S}_{1}\right)+\frac{1}{2{\chi }_{1}}\tilde{\Delta}^{T}_{s}\tilde{\Delta}_{s}+\frac{{\chi }_{1}}{2}{{S}_{2}}^{T}{S}_{2}\nonumber\\[5pt]& +\frac{1}{2{\chi }_{1}}\tilde{\Delta}^{T}_{f}\tilde{\Delta}_{f}-{{S}_{1}}^{T}\left({T}_{d}{C}_{1}\right){S}_{1}-{{S}_{2}}^{T}{C}_{2}{S}_{2}+{\sigma }^{T}\dot{\sigma }\end{align}

Integrating the Equations (34 and 35), we have

(44) \begin{align}{\sigma }^{T}\dot{\sigma } & ={\sigma }^{T}\left(-\frac{\sigma }{\tau }-{\dot{\omega }}_{1}\right)\nonumber\\[3pt]& =-\frac{1}{\tau }{\sigma }^{T}\sigma -{\sigma }^{T}{\dot{\omega }}_{1}\end{align}

Lemma 2. [Reference Hou, Liang, Duan and Bai33] The virtual control law $ {\omega }_{1}$ is bounded and satisfies: $ \|{\dot{\omega }}_{1}\|\le \eta $ , $ \eta $ is a nonnegative continuous function.

Therefore, according to the Lemma 2 and Young’s inequality, we can obtain the following inequality:

(45) \begin{align}{\sigma }^{T}\dot{\sigma } & =-\frac{1}{\tau }{\sigma }^{T}\sigma -{\sigma }^{T}{\dot{\omega }}_{1}\nonumber\\& \le -\frac{1}{\tau }{\sigma }^{T}\sigma +\frac{{\chi }_{3}}{2}{\sigma }^{T}\sigma +\frac{1}{2{\chi }_{3}}{{\dot{\omega }}_{1}}^{T}{\dot{\omega }}_{1}\\ & \le \left(\frac{{\chi }_{3}}{2}-\frac{1}{\tau }\right){\sigma }^{T}\sigma +\frac{{\eta }^{2}}{2{\chi }_{3}}\nonumber\end{align}

where $ {\chi }_{3}$ is a positive tuning parameter.

Integrating the Equations (43) and (45), $ {\dot{V}}_{2}$ satisfies the following inequality :

(46) \begin{align}{\dot{V}}_{2} & \le \frac{{\chi }_{1}}{2}{\left({T}_{d}{S}_{1}\right)}^{T}\left({T}_{d}{S}_{1}\right)+\frac{1}{2{\chi }_{1}}\tilde{\Delta}^{T}_{s}\tilde{\Delta}_{s}-{{S}_{1}}^{T}\left({T}_{d}{C}_{1}\right){S}_{1}+\frac{{\chi }_{2}}{2}{{S}_{2}}^{T}{S}_{2}\nonumber\\& \quad +\frac{1}{2{\chi }_{2}}\tilde{\Delta}^{T}_{f}\tilde{\Delta}_{f}-{{S}_{2}}^{T}{C}_{2}{S}_{2}+\left(\frac{{\chi }_{3}}{2}-\frac{1}{\tau }\right){\sigma }^{T}\sigma +\frac{{\eta }^{2}}{2{\chi }_{3}}\nonumber\\& =-{{S}_{1}}^{T}\left[{T}_{d}\left({C}_{1}-\frac{{\chi }_{1}}{2}{T}_{d}\right)\right]{S}_{1}+\frac{1}{2{\chi }_{1}}{\|\tilde{\Delta}_{s}\|}^{2}-{S}_{2}^{T}\left({C}_{2}-\frac{{\chi }_{2}}{2}\cdot {I}_{3\times 3}\right){S}_{2}\\& \quad +\frac{1}{2{\chi }_{2}}{\|\tilde{\Delta}_{f}\|}^{2}-\left(\frac{1}{\tau }-\frac{{\chi }_{3}}{2}\right){\sigma }^{T}\sigma +\frac{{\eta }^{2}}{2{\chi }_{3}}\nonumber\end{align}

Let $ {M}_{1}=\frac{1}{2{\chi }_{1}}{\|\tilde{\Delta}_{s}\|}^{2}$ , $ {M}_{2}=\frac{1}{2{\chi }_{2}}{\|\tilde{\Delta}_{f}\|}^{2}$ , $ {M}_{3}=\dfrac{{\eta }^{2}}{2{\chi }_{3}}$ . They are satisfied:

(47) \begin{align}& {M}_{1}\lt \frac{1}{2{\chi }_{1}}\bar{\Delta}^{2}_{s}\nonumber\\ & {M}_{2}\lt \frac{1}{2{\chi }_{2}}\bar{\Delta}^{2}_{f} \end{align}

Consider $ \mathrm{\Omega }=\left\{\left({S}_{1},{S}_{2},\sigma ,\tilde{\Delta}_{s},\tilde{\Delta}_{f}\right):{V}_{2}\le p\right\}$ is a compact set. Thereby, $ \eta $ has a maximum value $ \bar{\eta}$ on $ \mathrm{\Omega }$ . let $ M=\dfrac{1}{2{\chi }_{1}}\bar{\Delta}^{2}_{s}+\dfrac{1}{2{\chi }_{2}}\bar{\Delta}^{2}_{f}+\dfrac{{\bar{\eta}}^{2}}{2{\chi }_{3}}$ , which leads to

(48) \begin{align}{\dot{V}}_{2} & \le -{{S}_{1}}^{T}\left[{T}_{d}\left({C}_{1}-\frac{{\chi }_{1}}{2}{T}_{d}\right)\right]{S}_{1}+{M}_{1}-{S}_{2}^{T}\left({C}_{2}-\frac{{\chi }_{2}}{2}\cdot {I}_{3\times 3}\right){S}_{2}+{M}_{2}-\left(\frac{1}{\tau }-\frac{{\chi }_{3}}{2}\right){\sigma }^{T}\sigma +{M}_{3}\nonumber\\[5pt]& \le -{{S}_{1}}^{T}\left[{T}_{d}\left({C}_{1}-\frac{{\chi }_{1}}{2}{T}_{d}\right)\right]{S}_{1}-{S}_{2}^{T}\left({C}_{2}-\frac{{\chi }_{2}}{2}\cdot {I}_{3\times 3}\right){S}_{2}-\left(\frac{1}{\tau }-\frac{{\chi }_{3}}{2}\right){\sigma }^{T}\sigma +M\end{align}

Lemma 3. [Reference Hu, Shi and Shao32] $ {\zeta }_{min}{X}^{T}X\le {X}^{T}AX\le {\zeta }_{max}{X}^{T}X$ , $ A\in {R}^{n\times n}$ is a real symmetric matrix. $ {\zeta }_{min}$ and $ {\zeta }_{max}$ are the minimum and maximum eigenvalues of matrix $ A$ . $ X\in {R}^{n\times 1}$ is the arbitrary matrix.

According to the Lemma 3, by choosing appropriate parameters $ {c}_{1}$ , $ {c}_{2}$ , $ \tau $ , $ {\chi }_{1}$ , $ {\chi }_{2}$ , $ {\chi }_{3}$ and define $ A=\left({T}_{d}{C}_{1}-\dfrac{{\chi }_{1}}{2}{{T}_{d}}^{T}{T}_{d}\right)$ , $ B=\left({C}_{2}-\dfrac{{\chi }_{2}}{2}\cdot {I}_{3\times 3}\right)$ , $ C=\left(\dfrac{1}{\tau }-\dfrac{{\chi }_{3}}{2}\right)$ as real symmetric matrixs. Then $ {\dot{V}}_{2}$ is satisfied:

(49) \begin{align}{\dot{V}}_{2} & \le -{\zeta }_{1}{{S}_{1}}^{T}{S}_{1}-{\zeta }_{2}{{S}_{2}}^{T}{S}_{2}-{\zeta }_{3}{\sigma }^{T}\sigma +M\nonumber\\& \le -\zeta {{S}_{1}}^{T}{S}_{1}-\zeta {{S}_{2}}^{T}{S}_{2}-\zeta {\sigma }^{T}\sigma +M\\ & \le -2\zeta {V}_{2}+M\nonumber\end{align}

where, $ {\zeta }_{1}$ , $ {\zeta }_{2}$ , $ {\zeta }_{3}$ are the positive minimum eigenvalues of the real symmetric matrixs $ A$ , $ B$ , $ C$ . $ \zeta =\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\zeta }_{1},{\zeta }_{2},{\zeta }_{3}\right\}$ .

By comparison principle, it is easy from Equation (49) to obtain that

(50) \begin{align}{V}_{2}\le \frac{M}{2\zeta }+\left({V}_{2}\left(0\right)-\frac{M}{2\zeta }\right){e}^{-2\zeta t}\end{align}

According to (50), if $ 2\zeta \ge \dfrac{M}{p}$ , then we have $ {\dot{V}}_{2}\le 0$ and $ {V}_{2}\le p$ is an invariant set. Therefore, if $ {V}_{2}\left(0\right)\le p$ , then $ {V}_{2}\left(t\right)\le p$ , for all $ t\gt 0$ . In addition, $ {S}_{1}$ , $ {S}_{2}$ , $ \sigma $ are all semi-globally uniformly ultimately bounded. By choosing proper $ \zeta $ , and the convergence bound can be set as small as possible. Furthermore, the unconstrained error is bounded. According to the previous conclusion, $ {e}_{1}$ can also meet the prescribed performance requirements.

4.0 Numerical simulations

This section will use the control laws in Equations (33 and 34) and (39), disturbance estimations Equations (30 and 31) for simulation verification and analysis. The simulation verification of the command tracking with continuous time-varying external disturbances and other uncertain parameters are implemented.

The parameters of HV model in this paper are shown in Table 1.

Table 1. HV and environment parameters

The parameters of controller and disturbance observer in this paper are shown in Table 2.

Table 2. Controller and disturbance observer parameters

The parameters related to the adaptive scaling strategy are shown in Table 3.

Table 3. Relevant parameters of adaptive scaling strategy

The HV uncertainty model and the time-varying external disturbances (apply within 20s–30s) are shown in Table 4.

Table 4. Uncertainty and disturbances parameters

We take the attitude control problem of HV longitudinal channel as an example. We focus on the angle-of-attack simulation results. The 6-degree of freedom simulations are implemented. The command angle of sideslip and command angle of bank are set to fixed $ 0\left(rad\right)$ . In the simulation, we set the initial angle $ \theta ={\left[\mathrm{0.14,0},0\right]}^{T}\left(rad\right)$ , initial angle velocity $ \omega ={\left[\mathrm{0,0},0\right]}^{T}\left(rad/s\right)$ . Besides, considering the limit of the elevons, we set $ \left|\delta \right|\le 2{0}^{\circ }$ .

Due to the inability to provide explicit forms of system disturbances in the model of HV. Therefore, it is difficult to provide a comparison chart between the disturbance terms and NDO observation results. So the observation effect of NDO on system disturbances is verified here through the simple system shown as Equation (25). The specific form is shown in Equation (51). The disturbance estimation method is shown as Equation (26). The selected NDO parameters are also shown as $ Q=60$ , $ {h}_{1}=1$ , $ {h}_{2}=0.8$ .

(51) \begin{align} \dot{z} & =f\!\left(z\right)+g\!\left(z\right)U+d\nonumber\\ {z}_{d} & =\,\mathrm{sin}\!\left(t\right)\nonumber\\ U & ={g}^{-1}\left(z\right)\left[-10\!\left(z-{z}_{d}\right)-f\!\left(z\right)-\hat{d}+{\dot{z}}_{d}\right]\\ f\!\left(z\right) & ={z}^{2}+z+2\,\mathrm{sin}\left(z\right)+1\nonumber\\ g\!\left(z\right) & =10\nonumber\end{align}

where, the given test disturbance is set to $ 0.1sin\!\left(0.5t\right)+0.1sin\left(t\right)+0.1sin\left(1.5t\right)+0.1sin\left(2t\right)$ . Figure 4(a–d) show the relevant test results of NDO.

Figure 4. Simulation results of the NDO.

Comparing with the dynamic surface control method under observer compensation (DSC-NDO, the green line in Figs. 5(b) and 6(b)) and the prescribed performance sliding mode control method (PPSMC, the red line in Figs. 5(b) and 6(b)) to verify the advantages of the proposed method (IPPDSC-NDO, the blue line in Figs. 5(b) and 6(b), the black line in Fig. 7). To prove the effectiveness of adaptive scaling strategy, the method in this paper is also compared with the scheme without applying the adaptive scaling strategy (PPDSC-NDO, the red line in Fig. 7).

Figure 5. Simulation results with uncertainties and small external disturbances.

Figure 6. Simulation results with uncertainties and large external disturbances.

Figure 7. Simulation results with and without adaptive scaling strategy under uncertainties and large external disturbances.

5.0 Conclusion

This paper proposes an improved prescribed performance dynamic surface control method with an adaptive scaling strategy. This method achieves good tracking control of the HV attitude command angle. Compared with the traditional PPF, the PPF designed in this paper can scale adaptively according to the relevant state and avoid the tracking error violation. In addition, a NDO is introduced to enhance the robustness of the system. Thus, based on the PPC framework and NDO, it improves both robustness and dynamic performance of the system. Finally, the numerical simulation results verify the effectiveness of the control method we design.