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Aerospace Systems

, Volume 2, Issue 2, pp 197–213 | Cite as

Nonlinear optimal control for autonomous hypersonic vehicles

  • G. RigatosEmail author
  • P. Wira
  • M. Abbaszadeh
  • K. Busawon
  • L. Dala
Original Paper
  • 19 Downloads

Abstract

The article proposes a nonlinear optimal (H-infinity) control method for a hypersonic aerial vehicle (HSV). The dynamic model of the hypersonic vehicle undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. This operating point consists of the present value of the system’s state vector and of the last value of the control inputs vector that was applied on the HSV. The linearization relies on Taylor series expansion and on the computation of the associated Jacobian matrices. For the approximately linearized model of the hypersonic aerial vehicle, an optimal (H-infinity) feedback controller was designed. To compute the controller’s feedback gains, an algebraic Riccati equation had to be repetitively solved at each iteration of the control algorithm. The global asymptotic stability of the control method is proven through Lyapunov analysis. The control scheme remains robust against model uncertainties an external perturbations.

Keywords

Autonomous hypersonic vehicle Nonlinear H-infinity control Approximate linearization Taylor series expansion Jacobian matrices Riccati equation Robust control Asymptotic stability 

1 Introduction

Control of hypersonic aerial vehicles (HSVs) has emerged as a significant research topic during the last years with possible use for both transportation and defense purposes [1, 2, 3, 4, 5]. Hypersonic aerial vehicles can develop speeds of about 5 to 6 Mach, which allows for shortening the duration of long-distance flights [6, 7, 8]. Consequently, HSVs can boost the operational capacity of airliners. On the other side, hypersonic aerial vehicles are more difficult to be withheld by aerial defense and protection systems, thus giving also a significant advantage in airforce operations [9, 10, 11, 12, 13]. The control of hypersonic aerial vehicles is a non-trivial problem due to the multi-variable and strongly nonlinear model of such aircrafts [14, 15, 16, 17, 18]. Besides, such systems are underactuated because of having more degrees of freedom than control inputs, and this imposes additional difficulty in achieving flight stabilization and control [19, 20, 21, 22]. Proof of stability is a main objective in the design of control methods for hypersonic vehicles during the last years. For instance, one can note model-based Lyapunov approaches for hypersonic vehicles control [14, 23, 24, 25, 26, 27]. Additionally, one can note model-free Lyapunov approaches for hypersonic vehicles control [28, 29, 30, 31, 32, 33].

In the present article, a nonlinear optimal (H-infinity) control method is developed for the dynamic model of hypersonic aerial vehicles [34, 35, 36, 37]. First, the dynamic model of the aerial vehicles undergoes approximate linearization around a temporary operating point which is recomputed at each time-step of the control method. This operating point consists of the present value of the system’s state vector and the last value of the control inputs vector that was applied on the HSV. The linearization relies on Taylor series expansion and on the computation of the Jacobian matrices of the associated state-space model [38, 39, 40]. For the approximately linearized model of the system, an H-infinity stabilizing feedback controller is designed. The modelling error which is due to the truncation of higher order terms in the Taylor series expansion is treated as a perturbation that is compensated by the robustness of the controller.

The proposed H-infinity controller achieves the solution of the nonlinear optimal control problem for the model of the hypersonic vehicle under model uncertainties and external perturbations. It actually represents the solution of a min–max differential game in which the controller tries to minimize a quadratic cost functional of the state vector’s error whereas the model uncertainty and perturbation terms try to maximize this cost functional. To compute the feedback gains of the controller an algebraic Riccati equation has to be solved at each time step of the control method [41, 42]. The stability properties of the control scheme are proven through Lyapunov analysis. First, it is demonstrated that the H-infinity tracking performance criterion is satisfied, which signifies elevated robustness against model uncertainties and external perturbations [34, 43]. Besides, under moderate conditions, the global asymptotic stability of the control loop has been demonstrated. Additionally, to implement state estimation-based control without measuring the entire state vector of the system, the H-infinity Kalman Filter has been used as a robust state estimator [34, 44].

The article’s nonlinear optimal control method for the model of the hypersonic aerial vehicles exhibits several advantages: (i) it avoids the complicated state variables transformations (diffeomorphisms) which are met in global linearization-based control schemes, (ii) the control inputs are applied directly on the initial nonlinear model of the hypersonic vehicle and not on a linearized equivalent model of it. Thus, inverse transformations and the related singularity problems are avoided, (iii) it retains the advantages of typical (linear) optimal control, that is fast and accurate tracking of the reference setpoints under moderate variations of the control inputs. Yet computationally simple, the proposed nonlinear optimal control approach for hypersonic vehicles is proven to be efficient and reliable.

The structure of the article is as follows: in Sect. 2, the dynamic model of the hypersonic aerial vehicle has been described. In Sect. 3, the differential flatness properties of the hypersonic vehicle have been explained. In Sect. 4, the approximately linearized model of the hypersonic vehicle has been given after applying Taylor series expansion and the computation of the associated Jacobian matrices. In Sect. 5, the H-infinity control problem for the model of the hypersonic vehicle has been analysed. In Sect. 6, the stability properties of the proposed H-infinity (optimal) control method have been proven through Lyapunov analysis. In Sect. 7, the H-infinity Kalman Filter has been used as a robust observer, so as to solve the state estimation-based control problem for the aerial vehicle. In Sect.  8, the performance of the control method has been tested through simulation experiments. Finally, in Sect. 9 concluding remarks have been stated.
Fig. 1

Diagram of the autonomous hypersonic vehicle

2 Dynamic model of the autonomous hypersonic aerial vehicle

2.1 Dynamics of the longitudinal motion of the hypersonic vehicle

The dynamic model of hypersonic aerial vehicles is a strongly nonlinear and multivariable one. The model of the longitudinal dynamics of the autonomous hypersonic vehicle shown in Fig. 1 is given by [1, 2, 3, 4, 5]:
$$\begin{aligned} {\dot{V}}= & {} {{T \cos (a)-D} \over m}-{{{\mu } \sin (\gamma )} \over r^2} \end{aligned}$$
(1)
$$\begin{aligned} {\dot{\gamma }}= & {} {{L+T \sin (a)} \over {mV}}-{{({\mu }-{V^2}r)\cos (\gamma )} \over {Vr^2}} \end{aligned}$$
(2)
$$\begin{aligned} {\dot{h}}= & {} V \sin (\gamma ) \end{aligned}$$
(3)
$$\begin{aligned} {\dot{a}}= & {} q{\dot{\gamma }} \end{aligned}$$
(4)
$$\begin{aligned} {\dot{q}}= & {} {M_{yy} \over I_{yy}} \end{aligned}$$
(5)
where
$$\begin{aligned}&L={1 \over 2}{\rho }{V^2}S{C_L}(a) \quad D={1 \over 2}{\rho }{V^2}S{C_L}(a) \nonumber \\&T={1 \over 2}{\rho }{V^2}S{C_T}(\beta ) \nonumber \\&M_{yy}={1 \over 2}{\rho }{V^2}S_{\bar{C}}[C_M(a)+C_M^{'}(q)+C_M^{'}(\delta _e,a)] \nonumber \\&r=h+R_E \end{aligned}$$
(6)
The state variables of the model are \(x=[V,\gamma ,h,a,q]^T\) where V is the velocity of the hypersonic aerial vehicle, \(\gamma \) is the flight path angle, h is the altitude, a is the angle of attack and q is the pitch rate. The control inputs of the model are \(\delta _e\) which is the elevator’s deflection and \(\beta \) which is the throttle’s angle.

Besides in the state-space model of Eqs. (1)–(5) L is a function describing lift effects, D is a function denoting drag effects, T is a function representing thrust effects, \(M_{yy}\) is a function providing the pitching moment and r is the function of the distance from earth’s center.

Other functions that appear in the previous state-space model are: \(C_L(a)\) providing the lift coefficient, \(C_D(a)\) providing the drag coefficient, \(C_T(\beta )\) standing for the thrust coefficient, \(C_M(a)\) which is the moment coefficient due to the angle of attack, \(C_M(a)\) which is the moment coefficient due to the pitch rate and \(C_M(\delta _e,a)\) which is the moment’s coefficient due to elevator’s deflection.

Indicative values for \(C_L(a)\), \(C_D(a)\), \(C_T(a)\), \(C_M(a)\), \(C_M^{'}(q)\), and \(C_M^{''}(\delta _e,a)\) are given in the following:
$$\begin{aligned}&C_L(a)=0.6203a \end{aligned}$$
(7)
$$\begin{aligned}&C_D(a)=0.6450{a^2}+0.0043379a+0.00372 \end{aligned}$$
(8)
$$\begin{aligned}&C_T(\beta )=0.02574{\beta } \ \ \text {if} \ \ \beta <1 \nonumber \\&C_T(\beta )=0.0224+0.00336{\beta } \ \ \text {if} \ \ \beta {\ge }1 \end{aligned}$$
(9)
$$\begin{aligned}&C_M(a)=0.035{a^2}+0.036617{a}+5.3261 \times 10^{-6} \end{aligned}$$
(10)
$$\begin{aligned}&C_M^{'}(V,a,q)={\ddot{c} \over {2V}}q(-6.796{a^2}+0.3015{a}-0.2289)\nonumber \\ \end{aligned}$$
(11)
$$\begin{aligned}&C_M^{''}(\delta _e,a)=C_M^{''}(\delta _e)+C_M^{''}(a)=0.0292{\delta _e}-0.0292{a}\nonumber \\ \end{aligned}$$
(12)
Regarding the parameters of the model these are defined as follows: \(\mu \) is the gravitational constant, \(\rho \) is the density of air, S is the reference area, \(\bar{c}\) is the mean aerodynamics cord and \(R_E\) is the radius of earth.
Using the previous state-space notation \(x=[x_1,x_2,x_3,x_4,x_5]^T=[V,\gamma ,h,a,q]^T\) and by defining the modified control inputs \(u_1=C_T(\beta )\) and \(u_2=C_M^{''}(\delta _e)\), one obtains the following state-space description about the dynamics of the hypersonic vehicle:
$$\begin{aligned} {\dot{x}}_1= & {} {1 \over m}{1 \over 2}{\rho }{x_1^2}S{u_1} \cos (x_4)-{1 \over m}{1 \over 2}{\rho }{x_1^2}S{C_D(x_4)}\nonumber \\&-{{{\mu } \sin (x_2)} \over {(x_3+R_E)^2}} \end{aligned}$$
(13)
$$\begin{aligned} {\dot{x}}_2= & {} {1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{C_L(x_4)}+{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{u_1}\sin (x_4)\nonumber \\&-{{(\mu -{x_1^2}r)\cos (x_2)} \over {{x_1}(x_3+R)^2}} \end{aligned}$$
(14)
$$\begin{aligned} {\dot{x}}_3= & {} {x_1}\cos (x_2) \end{aligned}$$
(15)
$$\begin{aligned} {\dot{x}}_4= & {} {x_5}-{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{C_L(x_4)}-{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{u_1} \sin (x_4)\nonumber \\&+{{(\mu -{x_1^2}r) \cos (x_2)} \over {{x_1}(x_3+R)^2}} \end{aligned}$$
(16)
$$\begin{aligned} {\dot{x}}_5= & {} {1 \over I_{yy}}{1 \over 2}{\rho }{x_1^2}{S_{\bar{c}}}[C_M(x_4)+u_2-0.0292{x_4}+{C_M^{'}}(x_1,x_4,x_5)]\nonumber \\ \end{aligned}$$
(17)
Next, the following functions are defined:
$$\begin{aligned} f_1= & {} -{1 \over m}{1 \over 2}{\rho }{x_1^2}S{C_D}(x_4)-{{{\mu } \sin (x_2)} \over {(x_3+R_E)^2}} \nonumber \\ g_{11}= & {} {1 \over m}{1 \over 2}{\rho }{x_1^2}S \cos (x_4) \quad g_{12}=0 \end{aligned}$$
(18)
$$\begin{aligned} f_2= & {} {1 \over {m{x_1}}}{1 \over 2}{\rho }{x_1^2}S{C_L(x_4)}-{{(\mu -{x_1^2}(x_3+R_E)) \cos (x_2)} \over {{x_1}(x_3+R_E)^2}} \nonumber \\ g_{21}= & {} {1 \over {m{x_1}}}{1 \over 2}{\rho }{x_1^2}S \sin (x_4) \quad g_{22}=0 \end{aligned}$$
(19)
$$\begin{aligned} f_3= & {} {x_1} \sin (x_2) \nonumber \\ g_{31}= & {} 0 g_{32}=0 \end{aligned}$$
(20)
$$\begin{aligned} f_4= & {} {x_5}-{1 \over {m{x_1}}}{1 \over 2}{\rho }{x_1^2}S{C_L(x_4)}+{{(\mu -{x_1^2}(x_3+R_E)) \cos (x_2)} \over {{x_1}(x_3+R_E)^2}} \nonumber \\ g_{41}= & {} -{1 \over {m{x_1}}}{1 \over 2}{\rho }{x_1^2}S \sin (x_4) \quad g_{42}=0 \end{aligned}$$
(21)
$$\begin{aligned} f_5= & {} {1 \over {I_{yy}}}{1 \over 2}{\rho }{x_1^2}{S_{\bar{c}}}[{C_M}(x_4)-0.0292{x_4}+{C_M^{'}}(x_1,x_4,x_5)] \nonumber \\ g_{51}= & {} 0 \quad g_{52}={1 \over {I_{yy}}}{1 \over 2}{\rho }{x_1^2}{S_{\bar{c}}} \end{aligned}$$
(22)
Consequently, the state-space description of the hypersonic vehicle is written in the affine-in-the-input form
$$\begin{aligned} \begin{pmatrix} {\dot{x}}_1 \\ {\dot{x}}_2 \\ {\dot{x}}_3 \\ {\dot{x}}_4 \\ {\dot{x}}_5 \end{pmatrix}= \begin{pmatrix} f_1(x) \\ f_2(x) \\ f_3(x) \\ f_4(x) \\ f_5(x) \end{pmatrix}+ \begin{pmatrix} g_{11}(x) &{} g_{12}(x) \\ g_{21}(x) &{} g_{22}(x) \\ g_{31}(x) &{} g_{32}(x) \\ g_{41}(x) &{} g_{42}(x) \\ g_{151}(x) &{} g_{52}(x) \end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \end{aligned}$$
(23)
By denoting \(x{\in }R^{5{\times }1}\), \(u{\in }R^{5{\times }2}\), \(f(x){\in }R^{5{\times }1}\) and \(G(x){\in }R^{5{\times }2}\), the previous state-space model can be also written in the concise form
$$\begin{aligned} {\dot{x}}=f(x)+G(x)u \end{aligned}$$
(24)

2.2 Dynamics of the HSV under a small angle of attack

Under a small angle of attack a one has that \(\cos (a){\simeq }1\) and \(\sin (a){\simeq }0\). Then the first equation of the previous state-space model of the HSV becomes
$$\begin{aligned} {\dot{x}}_1=-{1 \over m}{1 \over 2}{\rho }{x_1^2}S{C_D}(x_4)+{1 \over m}{1 \over 2}\rho {x_1^2}S{u_1} \end{aligned}$$
(25)
About the second state-space equation of the HSV one has
$$\begin{aligned} {\dot{x}}_2={1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{C_L}(x_4)-{{(\mu -{x_1^2}(x_3+R_E))\cos (x_2)} \over {{x_1}(x_3+R_E)^2}}\nonumber \\ \end{aligned}$$
(26)
About the third state-space equation of the HSV, one has
$$\begin{aligned} {\dot{x}}_3={x_1}\sin (x_2) \end{aligned}$$
(27)
About the fourth state-space equation of the HSV, one has
$$\begin{aligned} {\dot{x}}_4=x_5-{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{C_L}(x_4)+{{(\mu -{x_1^2}(x_3+R_E))\cos (x_2)} \over {{x_1}(x_3+R_E)^2}}\nonumber \\ \end{aligned}$$
(28)
Moreover, about the fifth state-space equation of the HSV, one obtains
$$\begin{aligned} {\dot{x}}_5= & {} {1 \over {I_{yy}}}{1 \over 2}{\rho }{x_1^2}{S_{\bar{c}}}[{C_M}(x_4)-0.0292{x_4}+{C_M^{'}}(x_1,x_4,x_5)]\nonumber \\&+{1 \over {I_{yy}}}{1 \over 2}{\rho }{x_1^2}{S_{\bar{c}}}{u_2} \end{aligned}$$
(29)

3 Differential flatness properties of the hypersonic vehicle

It is proven that the dynamic model of the hypersonic aerial vehicle is a differentially flat one [34, 42]. The flat outputs of the system are defined as
$$\begin{aligned}{}[y_1,y_2]^T=[V,h]^T \end{aligned}$$
(30)
Under the assumption of a small angle of attack it holds: \(\cos (a){\simeq }1\) and \(\sin (a){\simeq }0\). Consequently, from the previous states-space model of Eqs. (18)–(22)
$$\begin{aligned} f_1= & {} -{1 \over m}{1 \over 2}{\rho }{x_1^2}S{C_D}(x_4) \nonumber \\ g_{11}= & {} {1 \over m}{1 \over 2}{\rho }{x_1^2}S \quad g_{12}=0 \end{aligned}$$
(31)
$$\begin{aligned} f_2= & {} {1 \over {m{x_1}}}{1 \over 2}{\rho }{x_1^2}S{C_L(x_4)}-{{(\mu -{x_1^2}(x_3+R_E))\cos (x_2)} \over {{x_1}(x_3+R_E)^2}} \nonumber \\ g_{21}= & {} 0 \quad g_{22}=0 \end{aligned}$$
(32)
$$\begin{aligned} f_3= & {} {x_1}\sin (x_2) \nonumber \\ g_{31}= & {} 0 \quad g_{32}=0 \end{aligned}$$
(33)
$$\begin{aligned} f_4= & {} {x_5}-{1 \over {m{x_1}}}{1 \over 2}{\rho }{x_1^2}S{C_L(x_4)}+{{(\mu -{x_1^2}(x_3+R_E))\cos (x_2)} \over {{x_1}(x_3+R_E)^2}} \nonumber \\ g_{41}= & {} 0 \quad g_{42}=0 \end{aligned}$$
(34)
$$\begin{aligned} f_5= & {} {1 \over {I_{yy}}}{1 \over 2}{\rho }{x_1^2}{S_{\bar{c}}}[{C_M}(x_4)-0.0292{x_4}+{C_M^{'}}(x_1,x_4,x_5)] \nonumber \\ g_{51}= & {} 0 \quad g_{52}={1 \over {I_{yy}}}{1 \over 2}{\rho }{x_1^2}{S_{\bar{c}}} \end{aligned}$$
(35)
Consequently, the state-space model of the hypersonic vehicle, under a small angle of attack, becomes
$$\begin{aligned} {\dot{x}}_1= & {} f_1(x)+g_{11}(x)u_1 \end{aligned}$$
(36)
$$\begin{aligned} {\dot{x}}_2= & {} f_2(x) \end{aligned}$$
(37)
$$\begin{aligned} {\dot{x}}_3= & {} f_3(x) \end{aligned}$$
(38)
$$\begin{aligned} {\dot{x}}_4= & {} f_4(x) \end{aligned}$$
(39)
$$\begin{aligned} {\dot{x}}_5= & {} f_5(x)+g_{52}(x)u_2 \end{aligned}$$
(40)
Using the previously defined flat outputs of the system \(y_1\) and \(y_2\), it holds that
$$\begin{aligned} x_1=y_1&x_2=y_2 \end{aligned}$$
(41)
From Eq. (38), one has
$$\begin{aligned} \begin{array}{c} {\dot{x}}_3={x_1}\sin (x_2){\Rightarrow }{x_2}=\sin ^{-1}({{\dot{x}}_3 \over x_1})\\ {\Rightarrow }{x_2}=\sin ^{-1}({{\dot{y}}_2 \over y_1}){\Rightarrow }x_2=h_2(y,{\dot{y}}) \end{array} \end{aligned}$$
(42)
which signifies that state variable \(x_2\) is a differential function of the system’s flat outputs. From Eq. (37), one has
$$\begin{aligned} {\dot{x}}_2= & {} {1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{C_L}(x_4)-{(\mu -{x_1^2}(x_3+R_E)) \over {{x_1}(x_3+R_E)^2}}{\Rightarrow }\nonumber \\ {C_L}(x_4)= & {} {{{\dot{x}}_2+{(\mu -{x_1^2}(x_3+R_E)) \over {{x_1}(x_3+R_E)^2}}} \over {{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S}} \end{aligned}$$
(43)
and using that \(C_L(x_4)=0.6203{x_4}\) one obtains
$$\begin{aligned} x_4= & {} {1 \over 0.6203}{{{\dot{x}}_2+{(\mu -{x_1^2}(x_3+R_E)) \over {{x_1}(x_3+R_E)^2}}} \over {{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S}}\nonumber \\\Rightarrow & {} x_4=h_4(y,{\dot{y}}) \end{aligned}$$
(44)
which signifies that state variable \(x_4\) is a differential function of the system’s flat outputs. From Eq. (39), one has
$$\begin{aligned} {\dot{x}}_4= & {} {x_5}-{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{C_L(x_4)}+{{(\mu -{x_1^2}({x_3+R_E}))} \over {x_1({x_3+R_E})^2}}{\Rightarrow }\nonumber \\ x_5= & {} \dot{x_4}+{1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{C_L(x_4)}-{{(\mu -{x_1^2}({x_3+R_E}))} \over {x_1({x_3+R_E})^2}}{\Rightarrow } \nonumber \\ x_5= & {} h_5(y,{\dot{y}}) \end{aligned}$$
(45)
which signifies that state variable \(x_5\) is a differential function of the system’s flat outputs. From Eq. (36), one has
$$\begin{aligned} u_1={{{\dot{x}}_1-f_1(x)} \over {g_{11}(x)}} \end{aligned}$$
(46)
Since it has been proven that all state variables of the system are differential functions of its flat outputs, the previous relation signifies that control input \(u_1\) is also a differential function of the flat outputs of the hypersonic vehicle’s model. Moreover, from Eq. (40) one has
$$\begin{aligned} u_2={{{\dot{x}}_5-f_5(x)} \over {g_{52}(x)}} \end{aligned}$$
(47)
Since it has been proven that all state variables of the system are differential functions of its flat outputs, the previous relation signifies that control input \(u_2\) is also a differential function of the flat outputs of the hypersonic vehicle’s model.

As a result of the above, all state variables and the control inputs of the hypersonic aerial vehicle can be written as differential functions of its flat outputs and, consequently, it is proven that this system is a differentially flat one. The differential flatness property of the system allows for solving the problem of reference trajectories (setpoints) selection for the HSV’s state variables. Actually, by choosing reference values for the flat outputs of the system \(y_1=V\) (velocity of the HSV) and \(y_2=h\) (height of the USV) and using the previous relations one can also compute the reference trajectories for the rest of the system’s state variables that is \(x_2=\gamma \), \(x_4=a\) and \(x_5=q\).

4 Approximate linearization for the dynamic model of the hypersonic vehicle

The previously analysed dynamic model of the hypersonic vehicle undergoes approximate linearization around the temporary operating point \((x^{*},u^{*})\), where \(x^{*}\) is the present value of the system’s state vector and \(u^{*}\) is the last value of the control inputs vector that was applied to it. The concise model of the system was
$$\begin{aligned} {\dot{x}}=f(x)+G(x)u \end{aligned}$$
(48)
The linearization relies on Taylor-series expansion and on the computation of the associate Jacobian matrices. The equivalent linearized model is obtained through this procedure
$$\begin{aligned} {\dot{x}}=Ax+Bu+{\tilde{d}} \end{aligned}$$
(49)
where \({\tilde{d}}\) is the disturbance vector which is due to approximate linearization and the truncation of higher-order terms in the Taylor series expansion. This term can also comprise external perturbations. Besides, matrices A and B are given by
$$\begin{aligned}&A={\nabla _x}[f(x)+G(x)u]\mid _{(x^{*},u^{*})}{\Rightarrow } \nonumber \\&A=[{\nabla _x}[f(x)]\mid _{(x^{*},u^{*})}+{\nabla _x}[g_1(x)]{u_1}\mid _{(x^{*},u^{*})}\nonumber \\&\qquad +{\nabla _x}[g_2(x)]{u_2}\mid _{(x^{*},u^{*})} \end{aligned}$$
(50)
$$\begin{aligned}&B={\nabla _u}[f(x)+G(x)u]\mid _{(x^{*},u^{*})}{\Rightarrow }B=G(x)\mid _{(x^{*},u^{*})}\nonumber \\ \end{aligned}$$
(51)
where \(g_i(x) \ i=1,2\) denotes the ith column of the control inputs gain matrix G(x). About the computation of the Jacobian matrix \({\nabla _x}[f(x)]\mid _{(x^{*},u^{*})}\), one has:
First row of the Jacobian matrix \({\nabla _x}[f(x)]\mid _{(x^{*},u^{*})}\):
$$\begin{aligned} {{{\partial }{f_1}} \over {{\partial }{x_1}}}= & {} -{1 \over m}{\rho }{{x_1}}S{C_B(x_4)} \\ {{{\partial }{f_1}} \over {{\partial }{x_2}}}= & {} -{{{\mu }\cos (x_2)} \over {(x_3+R_E)^2}} \\ {{\partial }{f_1} \over {{\partial }{x_3}}}= & {} {{2{\mu }\sin (x_2)} \over {({x_3+R_E})^3}} \\ {{\partial }{f_1} \over {{\partial }{x_4}}}= & {} -{1 \over m}{1 \over 2}\rho {x_1^2}S{{{\partial }C_D(x_4)} \over {{\partial }{x_4}}}\\ {{{\partial }{f_1}} \over {{\partial }{x_5}}}= & {} 0. \end{aligned}$$
where \({{{\partial }{C_D}(x_4)} \over {{\partial }{x_4}}}=0.6450{\cdot }2{\cdot }{x_4}+0.0043378\)
Second row of the Jacobian matrix \({\nabla _x}[f(x)]\mid _{(x^{*},u^{*})}\):
$$\begin{aligned} {{{\partial }{f_2}} \over {{\partial }{x_1}}}= & {} -{1 \over m}{1 \over 2}S{C_L(x_4)}-{{[(-2{x_1}{x_3+R_E})(x_1{x_3+R_E}^2)\cos (x_2)-(\mu -{x_1^2}(x_3+R_E))(x_3+R_E)^2]} \over {(x_1(x_3+R_E)^2)^2}}, \\ {{{\partial }{f_2}} \over {{\partial }{x_2}}}= & {} {{(\mu -{x_1^2}(x_3+R_E))\sin (x_2)} \over {{x_1}{(x_3+R_E)}^2}}, \\ {{{\partial }{f_2}} \over {{\partial }{x_3}}}= & {} {{[{x_1^2}\cos (x_2)(x_1(x_3+R_E)^2)+(\mu -{x_1^2}(x_3+R_E))\cos (x_2){x_1}2(x_3+R_E)]} \over {({x_1}{(x_3+R_E)}^2)^2}} \\ {{{\partial }{f_2}} \over {{\partial }{x_4}}}= & {} {1 \over {m{x_1}}}{1 \over 2}\rho {x_1^2}S{{\partial {C_L(x_4)}} \over {{\partial }{x_4}}} \\ {{{\partial }{f_2}} \over {{\partial }{x_5}}}= & {} 0 \end{aligned}$$
where \({{{\partial }{C_L}(x_4)} \over {{\partial }{x_4}}}=0.6203\).

Third row of the Jacobian matrix \({\nabla _x}[f(x)]\mid _{(x^{*},u^{*})}\): \({{{\partial }{f_3}} \over {{\partial }{x_1}}}=\sin (x_2)\), \({{{\partial }{f_3}} \over {{\partial }{x_2}}}={x_1}\cos (x_2)\), \({{{\partial }{f_3}} \over {{\partial }{x_3}}}=0\), \({{{\partial }{f_2}} \over {{\partial }{x_4}}}=0\) and \({{{\partial }{f_3}} \over {{\partial }{x_5}}}=0\).

Fourth row of the Jacobian matrix \({\nabla _x}[f(x)]\mid _{(x^{*},u^{*})}\)
$$\begin{aligned} {{{\partial }{f_4}} \over {{\partial }{x_1}}}= & {} -{1 \over m}{1 \over 2}S{C_L(x_4)} +{{[(-2{x_1}{x_3+R_E})(x_1({x_3+R_E})^2)\cos (x_2)-(\mu -{x_1^2}(x_3+R_E))(x_3+R_E)^2]} \over {(x_1((x_3+R_E))^2)^2}},\\ {{{\partial }{f_4}} \over {{\partial }{x_2}}}= & {} -{{(\mu -{x_1^2}(x_3+R_E))\sin (x_2)} \over {{x_1}{(x_3+R_E)}^2}}, \\ {{{\partial }{f_4}} \over {{\partial }{x_3}}}= & {} -{{[{x_1^2}\cos (x_2)(x_1(x_3+R_E)^2)+(\mu -{x_1^2}(x_3+R_E))\cos (x_2){x_1}2(x_3+R_E)]} \over {({x_1}{(x_3+R_E)}^2)^2}} \\ {{{\partial }{f_4}} \over {{\partial }{x_4}}}= & {} -{1 \over {m{x_2}}}{1 \over 2}\rho {x_1^2}S{{\partial {C_L(x_4)}} \over {{\partial }{x_4}}} \\ {{{\partial }{f_4}} \over {{\partial }{x_5}}}= & {} 0 \end{aligned}$$
Fifth row of the Jacobian matrix \({\nabla _x}[f(x)]\mid _{(x^{*},u^{*})}\)
$$\begin{aligned} {{{\partial }{f_5}} \over {{\partial }{x_1}}}= & {} {1 \over I_{yy}}\rho {x_1}S_{\bar{C}}[{C_M}(x_4)-0.0292{x_4}+C_M^{'}(x_1,x_4,x_5)] \\&\quad +{1 \over I_{yy}}\rho {x_1}S_{\bar{C}}{{{\partial }C_{M}^{'}(x_1,x_4,x_5)} \over {{\partial }x_1}} \end{aligned}$$
where \({{{\partial }C_{M}^{'}(x_1,x_4,x_5)} \over {{\partial }x_1}}=-{\ddot{c} \over {2{x_1^2}}}{x_5}(-6.796{x_4^2+0.3015{x_4}-}\)0.2284).
$$\begin{aligned}&{{{\partial }{f_5}} \over {{\partial }{x_2}}}=0 \hbox { and } {{{\partial }{f_5}} \over {{\partial }{x_3}}}=0. \\&{{{\partial }{f_5}} \over {{\partial }{x_4}}}={1 \over I_{yy}}{1 \over 2}{\rho }{x_1^2}S_{\bar{C}}\left[ {{{\partial }{C_M}(x_4)} \over {{\partial }{x_4}}}-0.0292\right. \\&\quad \left. +{{{\partial }{C_M^{'}}(x_1,x_4,x_5)} \over {{\partial }{x_4}}}\right] \end{aligned}$$
where \({{{\partial }{C_M}(x_4)} \over {{\partial }{x_4}}}=-2{\cdot }0.035{x_4}+0.036617\) and \({{{\partial }{C_M^{'}}(x_1,x_4,x_5)} \over {{\partial }{x_4}}}={\ddot{c} \over {2{x_1}}}{x_5}(-2{\cdot }6.796{x_4}+0.3015)\)
$$\begin{aligned} {{{\partial }{f_5}} \over {{\partial }{x_5}}}={1 \over I_{yy}}\rho {x_1}S_{\bar{C}}{{{\partial }{C_M^{'}}(x_1,x_4,x_5)} \over {{\partial }{x_5}}} \end{aligned}$$
where \({{{\partial }{C_M^{'}}(x_1,x_4,x_5)} \over {{\partial }{x_5}}}={{\ddot{c} \over {2{x_1}}}(-6.796{x_4^2}+0.3015{x_4}-0.2289)}\)

About the computation of the Jacobian matrix \({\nabla _x}[g_1(x)]\mid _{(x^{*},u^{*})}\), one has:

1st row of the Jacobian matrix \({{\nabla }_x}g_1(x)\): \({{{\partial }g_{11}} \over {{\partial }{x_1}}}={1 \over m}{\rho }{x_1}S\cos (x_4)\), \({{{\partial }g_{11}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{11}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{11}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{11}} \over {{\partial }{x_5}}}=0\).

2nd row of the Jacobian matrix \({{\nabla }_x}g_1(x)\): \({{{\partial }g_{21}} \over {{\partial }{x_1}}}=-{1 \over {m{x_1^2}}}{1 \over 2}{\rho }S\cos (x_4)\), \({{{\partial }g_{21}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{21}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{21}} \over {{\partial }{x_4}}}={1 \over {m}}{1 \over 2}{\rho }{x_1}S\cos (x_4)\), and \({{{\partial }g_{11}} \over {{\partial }{x_5}}}=0\).

3rd row of the Jacobian matrix \({{\nabla }_x}g_1(x)\): \({{{\partial }g_{31}} \over {{\partial }{x_1}}}=0\), \({{{\partial }g_{32}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{33}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{34}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{35}} \over {{\partial }{x_5}}}=0\).

4th row of the Jacobian matrix \({{\nabla }_x}g_1(x)\): \({{{\partial }g_{41}} \over {{\partial }{x_1}}}=-{1 \over m}{1 \over 2}{\rho }S\sin (x_4)\), \({{{\partial }g_{42}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{43}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{44}} \over {{\partial }{x_4}}}=-{1 \over {m{x_1}}}{1\over 2}{\rho }{x_1^2}{S}\cos (x_4)\), and \({{{\partial }g_{45}} \over {{\partial }{x_5}}}=0\).

5th row of the Jacobian matrix \({{\nabla }_x}g_1(x)\): \({{{\partial }g_{51}} \over {{\partial }{x_1}}}=0\), \({{{\partial }g_{52}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{53}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{54}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{55}} \over {{\partial }{x_5}}}=0\).

About the computation of the Jacobian matrix \({\nabla _x}[g_2(x)]\mid _{(x^{*},u^{*})}\) one has:

1st row of the Jacobian matrix \({{\nabla }_x}g_2(x)\): \({{{\partial }g_{12}} \over {{\partial }{x_1}}}=0\), \({{{\partial }g_{12}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{12}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{12}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{12}} \over {{\partial }{x_5}}}=0\).

2nd row of the Jacobian matrix \({{\nabla }_x}g_2(x)\): \({{{\partial }g_{22}} \over {{\partial }{x_1}}}=0\), \({{{\partial }g_{22}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{22}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{22}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{22}} \over {{\partial }{x_5}}}=0\).

3rd row of the Jacobian matrix \({{\nabla }_x}g_2(x)\): \({{{\partial }g_{32}} \over {{\partial }{x_1}}}=0\), \({{{\partial }g_{32}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{32}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{32}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{32}} \over {{\partial }{x_5}}}=0\).

4th row of the Jacobian matrix \({{\nabla }_x}g_2(x)\): \({{{\partial }g_{42}} \over {{\partial }{x_1}}}=0\), \({{{\partial }g_{42}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{42}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{42}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{42}} \over {{\partial }{x_5}}}=0\).

5th row of the Jacobian matrix \({{\nabla }_x}g_2(x)\): \({{{\partial }g_{52}} \over {{\partial }{x_1}}}={1 \over I_{yy}}{\rho }{x_1}S_{\bar{C}}\), \({{{\partial }g_{52}} \over {{\partial }{x_2}}}=0\), \({{{\partial }g_{52}} \over {{\partial }{x_3}}}=0\), \({{{\partial }g_{52}} \over {{\partial }{x_4}}}=0\), and \({{{\partial }g_{52}} \over {{\partial }{x_5}}}=0\).

5 The nonlinear H-infinity control

5.1 Tracking error dynamics

Next, a nonlinear optimal (H-infinity) controller will be developed for the hypersonic aerial vehicle. The initial nonlinear model of the hypersonic aerial vehicle is in the form
$$\begin{aligned} {\dot{x}}=f(x,u) \ \ x{\in }R^n, \ u{\in }R^m \end{aligned}$$
(52)
Linearization is performed at each iteration of the control algorithm around its present operating point \({(x^{*},u^{*})}=(x(t),u(t-T_s))\). The linearized equivalent of the hypersonic aerial vehicle is described by
$$\begin{aligned} {\dot{x}}=Ax+Bu+L{\tilde{d}} \ \ x{\in }R^n, \ u{\in }R^m, \ {\tilde{d}}{\in }R^q \end{aligned}$$
(53)
Thus, after linearization round its current operating point, the model of the hypersonic aerial vehicle with flexible joints is written as
$$\begin{aligned} {\dot{x}}=Ax+Bu+d_1 \end{aligned}$$
(54)
Parameter \(d_1\) stands for the linearization error in the model of the hypersonic aerial vehicle appearing in Eq. (54). The reference setpoints for the system are denoted by \(\mathbf{{x_d}}=[x_1^{d},\cdots ,x_4^{d}]\). Tracking of this trajectory is achieved after applying the control input \(u^{*}\). At every time instant, the control input \(u^{*}\) is assumed to differ from the control input u appearing in Eq. (54) by an amount equal to \({\Delta }u\), that is \(u^{*}=u+{\Delta }u\). One can write
$$\begin{aligned} {\dot{x}}_d=Ax_d+Bu^{*}+d_2 \end{aligned}$$
(55)
The dynamics of the controlled system described in Eq. (55) can be also written as
$$\begin{aligned} {\dot{x}}=Ax+Bu+Bu^{*}-Bu^{*}+d_1 \end{aligned}$$
(56)
and by denoting \(d_3=-Bu^{*}+d_1\) as an aggregate disturbance term one obtains
$$\begin{aligned} {\dot{x}}=Ax+Bu+Bu^{*}+d_3 \end{aligned}$$
(57)
By subtracting Eq. (55) from Eq. (57), one has
$$\begin{aligned} {\dot{x}}-{\dot{x}}_d=A(x-x_d)+Bu+d_3-d_2 \end{aligned}$$
(58)
By denoting the tracking error as \(e=x-x_d\) and the aggregate disturbance term as \({\tilde{d}}=d_3-d_2\), the tracking error dynamics becomes
$$\begin{aligned} {\dot{e}}=Ae+Bu+{\tilde{d}} \end{aligned}$$
(59)
Fig. 2

Diagram of the control scheme for the hypersonic aerial vehicle

The above linearized form of the model of the hypersonic aerial vehicle can be efficiently controlled after applying an H-infinity feedback control scheme.

5.2 Min–max control and disturbance rejection

The initial nonlinear model of the hypersonic aerial vehicle is in the form
$$\begin{aligned} {\dot{x}}=f(x,u) \ \ x{\in }R^n, \ u{\in }R^m \end{aligned}$$
(60)
Linearization of the model of the hypersonic aerial vehicle is performed at each iteration of the control algorithm round its present operating point \({(x^{*},u^{*})}=(x(t),u(t-T_s))\). The linearized equivalent of the system is described by
$$\begin{aligned} {\dot{x}}=Ax+Bu+L{\tilde{d}} \ \ x{\in }R^n, \ u{\in }R^m, \ {\tilde{d}}{\in }R^q \end{aligned}$$
(61)
where matrices A and B are obtained from the computation of the previously defined Jacobians and vector \({\tilde{d}}\) denotes disturbance terms due to linearization errors. The problem of disturbance rejection for the linearized model that is described by
$$\begin{aligned} {\dot{x}}= & {} Ax+Bu+L{\tilde{d}} \nonumber \\ y= & {} Cx \end{aligned}$$
(62)
where \(x{\in }R^n\), \(u{\in }R^m\), \({\tilde{d}}{\in }R^q\) and \(y{\in }R^p\), cannot be handled efficiently if the classical LQR control scheme is applied. This is because of the existence of the perturbation term \({\tilde{d}}\). The disturbance term \({\tilde{d}}\) can represent (i) modeling (parametric) uncertainty and external perturbation terms that affect the hypersonic aerial vehicle, (ii) noise terms of any distribution.
In the \(H_{\infty }\) control approach, a feedback control scheme is designed for trajectory tracking by the system’s state vector and simultaneous disturbance rejection, considering that the disturbance affects the system in the worst possible manner. The effects that disturbances have on the hypersonic aerial vehicle are incorporated in the following quadratic cost function:
$$\begin{aligned} J(t)= & {} {1 \over 2}{\int _0^T}[{y^T}(t)y(t)+r{u^T}(t)u(t)\nonumber \\&-{\rho ^2}{{\tilde{d}}^T}(t){\tilde{d}}(t)]\mathrm{d}t, \ \ r,{\rho }>0 \end{aligned}$$
(63)
Equation (63) denotes a min–max differential game taking place between disturbance and control inputs. Actually, the control inputs try to minimize this cost function while the disturbance inputs try to maximize it. Then, the optimal feedback control law is given by (Fig. 2)
$$\begin{aligned} u(t)=-Kx(t) \end{aligned}$$
(64)
with \(K={1 \over r}{B^T}P\), while P is a positive semi-definite symmetric matrix which is obtained from the solution of the Riccati equation of the form
$$\begin{aligned} {A^T}P+PA+Q-P\left( {1 \over r}B{B^T}-{1 \over {2\rho ^2}}L{L^T}\right) P=0 \end{aligned}$$
(65)
The transients of the control algorithm are determined by matrix Q and also by gains r and \(\rho \). The latter gain is the H-infinity attenuation coefficient and its minimum value that allows solution of Eq. (65) is the one that provides maximum robustness to the control algorithm for the hypersonic aerial vehicle.

6 Lyapunov stability analysis

Through Lyapunov stability analysis, it will be shown that the proposed nonlinear control scheme assures \(H_{\infty }\) tracking performance for hypersonic aerial vehicle and that under moderate conditions about the disturbance terms, asymptotic convergence to the reference setpoints is achieved. The tracking error dynamics for the hypersonic aerial vehicle is written in the form
$$\begin{aligned} {\dot{e}}=Ae+Bu+L{\tilde{d}} \end{aligned}$$
(66)
where in the hypersonic aerial vehicle’s case \(L=I{\in }R^{5{\times }5}\) with I being the identity matrix. Variable \({\tilde{d}}\) denotes model uncertainties and external disturbances of the motor’s model. The following Lyapunov equation is considered:
$$\begin{aligned} V={1 \over 2}{e^T}Pe \end{aligned}$$
(67)
where \(e=x-x_d\) is the tracking error. By differentiating with respect to time, one obtains
$$\begin{aligned} {\dot{V}}= & {} {1 \over 2}{{\dot{e}}^T}Pe+{1 \over 2}eP{\dot{e}}{\Rightarrow }\nonumber \\ {\dot{V}}= & {} {1 \over 2}{[Ae+Bu+L{\tilde{d}}]^T}P+{1 \over 2}{e^T}P[Ae+Bu+L{\tilde{d}}]{\Rightarrow }\nonumber \\ \end{aligned}$$
(68)
$$\begin{aligned} {\dot{V}}= & {} {1 \over 2}[{e^T}{A^T}+{u^T}{B^T}+{{\tilde{d}}^T}{L^T}]Pe\nonumber \\&+{1 \over 2}{e^T}P[Ae+Bu+L{\tilde{d}}]{\Rightarrow } \end{aligned}$$
(69)
$$\begin{aligned} {\dot{V}}= & {} {1 \over 2}{e^T}{A^T}Pe+{1 \over 2}{u^T}{B^T}Pe+{1 \over 2}{{\tilde{d}}^T}{L^T}Pe \nonumber \\&+{1 \over 2}{e^T}PAe+{1 \over 2}{e^T}PBu+{1 \over 2}{e^T}PL{\tilde{d}} \end{aligned}$$
(70)
The previous equation is rewritten as
$$\begin{aligned} {\dot{V}}= & {} {1 \over 2}{e^T}({A^T}P+PA)e+\left( {1 \over 2}{u^T}{B^T}Pe+{1 \over 2}{e^T}PBu\right) \nonumber \\&+\left( {1 \over 2}{{\tilde{d}}^T}{L^T}Pe+{1 \over 2}{e^T}PL{\tilde{d}}\right) \end{aligned}$$
(71)

Assumption

For given positive definite matrix Q and coefficients r and \(\rho \), there exists a positive definite matrix P, which is the solution of the following matrix equation:
$$\begin{aligned} {A^T}P+PA=-Q+P\left( {2 \over r}B{B^T}-{1 \over \rho ^2}L{L^T}\right) P \end{aligned}$$
(72)
Moreover, the following feedback control law is applied to the system:
$$\begin{aligned} u=-{1 \over r}{B^T}Pe \end{aligned}$$
(73)
By substituting Eqs. (72) and (73), one obtains
$$\begin{aligned} {\dot{V}}= & {} {1 \over 2}{e^T}[-Q+P\left( {2 \over r}B{B^T}-{1 \over {\rho ^2}}L{L^T}\right) P]e \nonumber \\&+{e^T}PB\left( -{1 \over r}{B^T}Pe\right) +{e^T}PL{\tilde{d}}{\Rightarrow } \end{aligned}$$
(74)
$$\begin{aligned} {\dot{V}}= & {} -{1 \over 2}{e^T}Qe+\left( {1 \over r}{e^T}PB{B^T}Pe-{1 \over {2\rho ^2}}{e^T}PL{L^T}Pe\right. \nonumber \\&\left. -{1 \over r}{e^T}PB{B^T}Pe\right) +{e^T}PL{\tilde{d}} \end{aligned}$$
(75)
which after intermediate operations gives
$$\begin{aligned} {\dot{V}}=-{1 \over 2}{e^T}Qe-{1 \over {2\rho ^2}}{e^T}PL{L^T}Pe+{e^T}PL{\tilde{d}} \end{aligned}$$
(76)
or, equivalently
$$\begin{aligned} {\dot{V}}= & {} -{1 \over 2}{e^T}Qe-{1 \over {2\rho ^2}}{e^T}PL{L^T}Pe \nonumber \\&+{1 \over 2}{e^T}PL{\tilde{d}}+{1 \over 2}{{\tilde{d}}^T}{L^T}Pe \end{aligned}$$
(77)

Lemma

The following inequality holds:
$$\begin{aligned} {1 \over 2}{e^T}PL{\tilde{d}}+{1 \over 2}{\tilde{d}}{L^T}Pe-{1 \over {2\rho ^2}}{e^T}PL{L^T}Pe{\le }{1 \over 2}{\rho ^2}{{\tilde{d}}^T}{\tilde{d}}\nonumber \\ \end{aligned}$$
(78)

Proof

The binomial \(({\rho }{\alpha }-{1 \over \rho }b)^2\) is considered. Expanding the left part of the above inequality, one gets
$$\begin{aligned}&{\rho ^2}{a^2}+{1 \over {\rho ^2}}{b^2}-2ab \ge 0 \Rightarrow {1 \over 2}{\rho ^2}{a^2}+{1 \over {2\rho ^2}}{b^2}-ab \ge 0 \nonumber \\&\quad \Rightarrow ab-{1 \over {2\rho ^2}}{b^2} \le {1 \over 2}{\rho ^2}{a^2} \Rightarrow {1 \over 2}ab \nonumber \\&\qquad +{1 \over 2}ab-{1 \over {2\rho ^2}}{b^2} \le {1 \over 2}{\rho ^2}{a^2} \end{aligned}$$
(79)
The following substitutions are carried out: \(a={\tilde{d}}\) and \(b={e^T}{P}L\) and the previous relation becomes
$$\begin{aligned} {1 \over 2}{{\tilde{d}}^T}{L^T}Pe+{1 \over 2}{e^T}PL{\tilde{d}}-{1 \over {2\rho ^2}}{e^T}PL{L^T}Pe{\le }{1 \over 2}{\rho ^2}{\tilde{d}}^T{\tilde{d}}\nonumber \\ \end{aligned}$$
(80)
Equation (80) is substituted in Eq. (77) and the inequality is enforced, thus giving
$$\begin{aligned} {\dot{V}}{\le }-{1 \over 2}{e^T}Qe+{1 \over 2}{\rho ^2}{{\tilde{d}}^T}{\tilde{d}} \end{aligned}$$
(81)
Equation (81) shows that the \(H_{\infty }\) tracking performance criterion is satisfied. The integration of \({\dot{V}}\) from 0 to T gives
$$\begin{aligned}&{\int _0^T}{\dot{V}}(t)\mathrm{d}t{\le }-{1 \over 2}{\int _0^T}{||e||_Q^2}{\mathrm{d}t}+{1 \over 2}{\rho ^2}{\int _0^T}{||{\tilde{d}}||^2}{\mathrm{d}t}\nonumber \\&\quad {\Rightarrow }2V(T)+{\int _0^T}{||e||_Q^2}{\mathrm{d}t}{\le }2V(0)+{\rho ^2}{\int _0^T}{||{\tilde{d}}||^2}\mathrm{d}t\nonumber \\ \end{aligned}$$
(82)
Moreover, if there exists a positive constant \(M_d>0\) such that
$$\begin{aligned} \int _0^{\infty }{||{\tilde{d}}||^2}\mathrm{d}t \le M_d \end{aligned}$$
(83)
then one gets
$$\begin{aligned} {\int _0^{\infty }}{||e||_Q^2}\mathrm{d}tt \le 2V(0)+{\rho ^2}{M_d} \end{aligned}$$
(84)
Thus, the integral \({\int _0^{\infty }}{||e||_Q^2}\mathrm{d}tt\) is bounded. Moreover, V(T) is bounded and from the definition of the Lyapunov function V in Eq. (67) it becomes clear that e(t) will be also bounded since \(e(t) \ \in \ \Omega _e=\{e|{e^T}Pe{\le }2V(0)+{\rho ^2}{M_d}\}\). \(\square \)

According to the above and with the use of Barbalat’s Lemma one obtains lim\(_{t \rightarrow \infty }{e(t)}=0\).

The outline of the global stability proof is that at each iteration of the control algorithm the state vector of the hypersonic aerial vehicle converges towards the temporary equilibrium and the temporary equilibrium in turn converges towards the reference trajectory [34]. Thus, the control scheme exhibits global asymptotic stability properties and not local stability. Assume the ith iteration of the control algorithm and the ith time interval about which a positive definite symmetric matrix P is obtained from the solution of the Riccati equation appearing in Eq. (72). By following the stages of the stability proof one arrives at Eq. (81) which shows that the H-infinity tracking performance criterion holds. By selecting the attenuation coefficient \(\rho \) to be sufficiently small and in particular to satisfy \(\rho ^2<||e||^2_Q / ||{\tilde{d}}||^2\), one has that the first derivative of the Lyapunov function is upper bounded by 0. Therefore, for the ith time interval it is proven that the Lyapunov function defined in Eq. (67) is a decreasing one. This signifies that between the beginning and the end of the ith time interval, there will be a drop of the value of the Lyapunov function and since matrix P is a positive definite one, the only way for this to happen is the Euclidean norm of the state vector error e to be decreasing. This means that compared to the beginning of each time interval, the distance of the state vector error from 0 at the end of the time interval has diminished. Consequently, as the iterations of the control algorithm advance the tracking error will approach zero and this is a global asymptotic stability condition.

Remark 1

The linearization of the nonlinear state-space model of the hypersonic vehicle relies on first order Taylor series expansion, which means that only the first order of the model’s Taylor series is retained. The linearization process is repeated at each sampling period of the control algorithm. The linearization point is a time-varying one and is also updated at each sampling instant. It consists of the present value of the state vector of the hypersonic vehicle and of the last (most recent) value of the control inputs vector that was applied on it. The modelling error, which is due to the truncation of higher-order terms in the Taylor series expansion, is considered to be a perturbation that is asymptotically compensated by the robustness of the H-infinity control algorithm. Actually, as proven in the Lyapunov stability analysis, the proposed control method achieves global asymptotic stability for the control loop of the hypersonic vehicle. The disturbances vector \({\tilde{d}}\) which appears in the linearized dynamics of the hypersonic vehicle that is given in Eq. (49) may represent not only the modelling error due to truncation of higher-order terms in the Taylor series expansion, but it may also accumulate the effects of external perturbation, as well as of sensors’ noise of any distribution. Yet conceptually simple, the proposed nonlinear optimal control scheme is a reliable and computationally efficient solution to the control problem of hypersonic aerial vehicles.

7 Robust state estimation with the use of the \(H_{\infty }\) Kalman Filter

The control loop can be implemented with the use of information provided by a small number of sensors and by processing only a small number of state variables. To reconstruct the missing information about the state vector of the hypersonic aerial vehicle, it is proposed to use a filtering scheme and based on it to apply state estimation-based control [42, 44]. The recursion of the \(H_{\infty }\) Kalman Filter, for the model of the autonomous aerial vehicle, can be formulated in terms of a measurement update and a time updatepart

Measurement update:
$$\begin{aligned} D(k)= & {} [I-{\theta }W(k)P^{-}(k)+{C^T}(k)R(k)^{-1}C(k)P^{-}(k)]^{-1}\nonumber \\ K(k)= & {} P^{-}(k)D(k){C^T}(k)R(k)^{-1}\nonumber \\ {\hat{x}}(k)= & {} {\hat{x}}^{-}(k)+K(k)[y(k)-C{\hat{x}}^{-}(k)] \end{aligned}$$
(85)
Time update:
$$\begin{aligned} {\hat{x}}^{-}(k+1)= & {} A(k)x(k)+B(k)u(k) \nonumber \\ P^{-}(k+1)= & {} A(k)P^{-}(k)D(k)A^T(k)+Q(k) \end{aligned}$$
(86)
Fig. 3

Tracking of setpoint 1 for the HSV state variables a convergence of state variables \(x_1=V\) and \(x_2=\gamma \) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines), b convergence of state variables \(x_3=h\) and \(x_5=q\) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines)

Fig. 4

Tracking of setpoint 2 for the HSV state variables a convergence of state variables \(x_1=V\) and \(x_2=\gamma \) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines), b convergence of state variables \(x_3=h\) and \(x_5=q\) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines)

Fig. 5

Variation of the control inputs \(u_1=\delta _e\) and \(u_2=\beta \) of the HSV a when tracking of setpoint 1, b when tracking setpoint 2

Fig. 6

Tracking of setpoint 3 for the HSV state variables a convergence of state variables \(x_1=V\) and \(x_2=\gamma \) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines). b convergence of state variables \(x_3=h\) and \(x_5=q\) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines)

Fig. 7

Tracking of setpoint 4 for the HSV state variables a convergence of state variables \(x_1=V\) and \(x_2=\gamma \) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines), b convergence of state variables \(x_3=h\) and \(x_5=q\) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines)

Fig. 8

Variation of the control inputs \(u_1=\delta _e\) and \(u_2=\beta \) of the HSV a when tracking of setpoint 3, b when tracking setpoint 4

Fig. 9

Tracking of setpoint 5 for the HSV state variables. a Convergence of state variables \(x_1=V\) and \(x_2=\gamma \) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines), b convergence of state variables \(x_3=h\) and \(x_5=q\) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines)

Fig. 10

Tracking of setpoint 6 for the HSV state variables. a Convergence of state variables \(x_1=V\) and \(x_2=\gamma \) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines). b Convergence of state variables \(x_3=h\) and \(x_5=q\) (blue lines) to the reference setpoints (red lines) and their estimated values (green lines)

Fig. 11

Variation of the control inputs \(u_1=\delta _e\) and \(u_2=\beta \) of the HSV a when tracking of setpoint 5, b when tracking setpoint 6

where it is assumed that parameter \(\theta \) is sufficiently small to assure that the covariance matrix \({P^{-}(k)}^{-1}-{\theta }W(k)+C^T(k)R(k)^{-1}C(k)\) will be positive definite. When \(\theta =0\) the \(H_{\infty }\) Kalman Filter becomes equivalent to the standard Kalman Filter. One can measure only a part of the state vector of the hypersonic aerial vehicle, such as state variables \(x_1=V\) and \(x_3=h\), and can estimate through filtering the rest of the state vector elements. Moreover, the proposed Kalman filtering method can be used for sensor fusion purposes.

8 Simulation tests

The performance of the proposed nonlinear optimal (H-infinity) control for the hypersonic aerial vehicle has been tested through simulation experiments, which were obtained with the use of the small-angle-of-attack HSV state-space model. To implement the control method, the dynamic model of the hypersonic aerial vehicle was subject to consecutive linearizations through Taylor series expansion that were taking place at each iteration of the control algorithm. Besides, to compute the feedback control gains, the algebraic Riccati equation of Eq. (72) had to be repetitively solved at each time step of the control scheme. As shown in Figs. 3, 4, 5, 6, 7, 8, 9, 10 and 11, the proposed control approach achieved minimization of the tracking error for all state variables while the control inputs varied within the allowed ranges.

The transient performance of the control algorithm is determined by the selection of specific parameters in the Riccati equation of Eq. (72), such as weight matrix Q, control gain r and the attenuation coefficient \(\rho \). Actually, \(\rho \) is the parameter that determines also the robustness of the control scheme. The smallest value of \(\rho \) for which an admissible solution of the Riccati equation of Eq. (72) can be obtained is the one that provides the control scheme with maximum robustness. To implement feedback control without the need to measure the entire state vector of the hypersonic aerial vehicle, but after reading only specific state variables out of it, the H-infinity Kalman Filter has been used as a robust state estimator. In the provided simulation diagrams, the real values of the state variables of the hypersonic vehicle are printed in blue, the estimated values are plotted in green and the reference setpoints are depicted in red.

Remark 2

Viewed against other nonlinear control methods for the same problem, the article’s nonlinear optimal control approach exhibits specific advantages: (i) compared to global linearization control schemes (such as Lie algebra-based control and differential flatness theory-based control) the nonlinear optimal control method does not need complicated transformation of the state variables of the controlled system (diffeomorphisms). The control inputs are applied directly on the initial nonlinear model of the hypersonic vehicles and thus inverse transformations and the related singularity problems are avoided, (ii) compared to popular approaches for optimal control which are met in industry, for instance model predictive control (MPC) and nonlinear model predictive control (NMPC), the article’s nonlinear optimal control method is of assured convergence and stability. Actually, MPC is a linear control scheme and its application to the highly nonlinear model of the hypersonic vehicle will cause the loss of stability of the control loop. Besides, the convergence of the iterative search for an optimum that is performed by NMPC is dependent on initialization and on parameter values’ selection; therefore, stability issues for this method may also arise. (iii) unlike sliding-mode control the article’s nonlinear optimal control method avoids the intuitive definition of variables such as the sliding surface. It is noted that there is no generic procedure for selecting the sliding surface unless the system is found in the input-output linearized form (that is the canonical form). For other types of state-space description of the controlled system, the selection of the sliding surface follows an ad hoc procedure, (iv) unlike backstepping control, the article’s nonlinear optimal control method does not require the controlled system to be found in specific state-space form (for instance the triangular form or the backstepping integral form) (v) unlike PID control, the article’s nonlinear optimal control method is of proven global stability, and retains its stability properties at any change of operating points. Besides, the new control method avoids the empirical selection of controller gains and assures satisfactory performance at changes of the operating conditions, (vi) unlike multiple local models-based control, the article’s nonlinear optimal control method requires linearization around only one operating point and does not involve the ad hoc definition of equilibria. Besides it needs to solve only one Riccati equation, thus avoiding the computational effort related with the solution of multiple Riccati equations or linear matrix inequalities.

Table 1

RMSE of the hypersonic vehicle’s state variables

\(\text {Setpoint}\)

\(\text {RMSE} \ x_1\)

\(\text {RMSE} \ x_2\)

\(\text {RMSE} \ x_3\)

\(\text {RMSE} \ x_4\)

1

\(2.90 \times 10^{-3}\)

\(1.80 \times 10^{-3}\)

\(0.50 \times 10^{-3}\)

\(0.90 \times 10^{-3}\)

2

\(3.00 \times 10^{-3}\)

\(1.60 \times 10^{-3}\)

\(1.90 \times 10^{-3}\)

\(0.90 \times 10^{-3}\)

3

\(1.80 \times 10^{-3}\)

\(1.10 \times 10^{-3}\)

\(1.20 \times 10^{-3}\)

\(0.90\times 10^{-3}\)

4

\(1.30\times 10^{-3}\)

\(2.20\times 10^{-3}\)

\(2.40\times 10^{-3}\)

\(0.90\times 10^{-3}\)

5

\(1.20\times 10^{-3}\)

\(5.60\times 10^{-3}\)

\(2.20\times 10^{-3}\)

\(0.90\times 10^{-3}\)

6

\(0.30\times 10^{-3}\)

\(0.50\times 10^{-3}\)

\(1.20\times 10^{-3}\)

\(0.90\times 10^{-3}\)

Table 2

RMSE of the hypersonic vehicle under disturbances

\({\Delta }g_{11} \%\)

\(\text {RMSE} \ x_1\)

\(\text {RMSE} \ x_2\)

\(\text {RMSE} \ x_3\)

\(\text {RMSE} \ x_4\)

0

\(1.80 10^{-3}\)

\(1.10\times 10^{-3}\)

\(1.20 \times 10^{-3}\)

\(0.90\times 10^{-3}\)

10

\(1.80\times 10^{-3}\)

\(1.10\times 10^{-3}\)

\(1.20 \times 10^{-3}\)

\(2.90\times 10^{-3}\)

20

\(1.80\times 10^{-3}\)

\(1.10\times 10^{-3}\)

\(1.20 \times 10^{-3}\)

\(4.30\times 10^{-3}\)

30

\(1.80\times 10^{-3}\)

\(1.10\times 10^{-3}\)

\(1.20 \times 10^{-3}\)

\(4.20\times 10^{-3}\)

40

\(1.80\times 10^{-3}\)

\(1.10\times 10^{-3}\)

\(1.20 \times 10^{-3}\)

\(3.10\times 10^{-3}\)

50

\(1.80\times 10^{-3}\)

\(1.10\times 10^{-3}\)

\(1.20\times 10^{-3}\)

\(1.60\times 10^{-3}\)

60

\(1.80\times 10^{-3}\)

\(1.10\times 10^{-3}\)

\(1.20\times 10^{-3}\)

\(0.60\times 10^{-3}\)

Indicative results about the tracking accuracy of the state variables of the hypersonic aerial vehicle under the nonlinear optimal control method are given in Table 1. Moreover, results about the robustness of the control method under parametric changes, for instance a change of \({\Delta }{g_{12}}\) of control input gain \({g_{12}}\) from its nominal value, are given in Table 2.

9 Conclusions

The article has proposed a nonlinear optimal control approach for the model of hypersonic aerial vehicles. For the implementation of this method, the associated dynamic model had to undergo approximate linearization with the use of Taylor series expansion around a time-varying operating point that was recomputed at each time step of the control algorithm. The linearization relied on the computation of the Jacobian matrices of the system’s state-space model. For the linearized state-space description of the system, an algebraic Riccati equation had to be solved at each time step of the control algorithm. This allowed to compute the controller’s feedback gains.

It was shown that the H-infinity controller stands for the solution of the optimal control problem for the hypersonic aerial vehicle, under modelling uncertainty and external perturbations. The global stability properties of the control scheme were proven through Lyapunov analysis. First, by proving that the H-infinity tracking performance criterion is satisfied it was shown that the control loop has elevated robustness against model uncertainties and external perturbations. Moreover, under moderate conditions it was shown that the control loop is globally asymptotically stable. Finally, to implement state estimation-based control without the need to measure the entire state vector of the hypersonic vehicle, the H-infinity Kalman Filter has been used as a robust state estimator.

Notes

Acknowledgements

Funding was provided by Unit of Industrial Automation/Industrial Systems Institute (Grant no. Ref 5352/Nonlinear Control and Filtering).

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Copyright information

© Shanghai Jiao Tong University 2019

Authors and Affiliations

  • G. Rigatos
    • 1
    Email author
  • P. Wira
    • 2
  • M. Abbaszadeh
    • 3
  • K. Busawon
    • 4
  • L. Dala
    • 5
  1. 1.Unit of Industrial Automation Industrial Systems InstituteRion PatrasGreece
  2. 2.IRIMAS Université d’ Haute AlsaceMulhouseFrance
  3. 3.GE Global Research General ElectricNiskayunaUSA
  4. 4.Nonlinear Control Group University of NorthumbriaNewcastleUK
  5. 5.Department of Mechanical EngineeringUniversity of NorthumbriaNewcastleUK

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