TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL FOR HYPERSONIC VEHICLES WITH INPUT SATURATION AND MISMATCHED UNCERTAINTIES

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1 International Journal of Innovative Computing, Information and Control ICIC International c 017 ISSN Volume 13, Number 6, December 017 pp TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL FOR HYPERSONIC VEHICLES WITH INPUT SATURATION AND MISMATCHED UNCERTAINTIES Jingguang Sun, Shenmin Song and Guanqun Wu Center for Control Theory and Guidance Technology Harbin Institute of Technology No. 9, West Dazhi Street, Harbin , P. R. China { sunjingguanghit; wgqwuguanqun }@163.com; Corresponding author: songshenmin@hit.edu.cn Received March 017; revised July 017 Abstract. This thesis investigates the tracking control problem of hypersonic vehicles subject to mismatched uncertainties and input saturation. Firstly, the feedback linearization for longitudinal model of hypersonic vehicles is reasonably decomposed into two subsystems that include velocity and altitude subsystem. Secondly, for these two subsystems, an anti-saturation robust dynamic surface controller is designed using the auxiliary system and dynamic surface control DSC) technique with compensating signals, respectively. The controllers can not only avoid the explosion of complexity in the backstepping design, but also remove the effect of the error caused by the first-order filter. Finally, Lyapunov theory is used to prove the stability of the designed controller strictly, and the numerical simulations of the longitudinal model of the hypersonic vehicles are carried out, which further demonstrate the robustness of the designed control scheme. Keywords: Hypersonic vehicles, Dynamic surface control, Tracking control, Input saturation, Mismatched uncertainty 1. Introduction. Hypersonic vehicles refer to the aircraft whose flight speed is greater than 5 Mach, which has fast speed, strong penetration ability, and therefore, it is of great military and economic value [1,]. In comparison with traditional vehicles, the unique integrated design of hypersonic vehicles results in the high nonlinearity and strong coupling among its airframe, propulsion system and structural dynamics [3,4]. From the above, we can see that the hypersonic vehicles have large flight envelop and complicated flight environment. Its aerodynamic characteristics can change violently, and its model uncertainty is strong. Therefore, the robust controller is designed for the hypersonic vehicles in presence of unknown factors, which is one of the key technologies to ensure the safe and effective flight of the hypersonic vehicles [5]. In recent years, domestic and foreign experts and scholars majoring in the field of control have embarked on extensive researches and explorations. Foreign scholars have made extensive researches based on linear and nonlinear methods. The linear design methods are shown in [6,7]. It is well known that due to the complex nonlinear characteristics of hypersonic vehicles, the controller design method based on linear mode cannot meet the control performance requirements of the system. In [6,7], the designed controllers within a small scope around a certain equilibrium limits do not have good robustness. In order to obtain better control performance, the nonlinear control method was applied to the design of the hypersonic vehicles controller [8-1]. In [8], the hypersonic vehicles with parametric uncertainties were chosen as control object. A nonlinear controller was designed based on the dynamic inverse method and stochastic robust control method, which can obtain good tracking performance. Based on the feedback linearization method, a sliding mode 067

2 068 J. SUN, S. SONG AND G. WU controller for hypersonic vehicles was designed in [9,10]. In [11], focused on hypersonic vehicle with TS disturbance modeling and disturbances, a robust tracking controller was designed using nonlinear disturbance observer. In [1], an anti-windup controller was proposed for air-breathing hypersonic vehicles, based on disturbance observer. However, it is more universal that mismatched disturbances the disturbances are not in the input channel) occur in hypersonic vehicles. The mismatched disturbances that resulted from the dramatic changes in flight environments may influence the states directly, for example, which may be some mismatched uncertainties between the feedback linearization model and the nonlinear system model in [13,14]. For nonlinear systems with mismatched uncertainties, the back-stepping method is widely used in aerospace field. In [15], an adaptive back-stepping controller was designed for hypersonic vehicles, which can solve parameter uncertainty and fault of actuator. In [16], hypersonic flight vehicles with mismatched disturbances were analyzed using disturbance observer. An exponential sliding mode controller was designed for hypersonic vehicle subject to the mismatched uncertainties based on back-stepping method in [17]. In [18,19], an adaptive neural network back-stepping controller was designed for hypersonic vehicles subject to unmodeled dynamics and input saturation. For the back-stepping method, when the order of the control system is high, the explosion of complexity is easy to appear. In order to solve this problem, focusing on nonlinear systems, an adaptive dynamic surface controller back-stepping controller was designed using dynamic surface control technique and fuzzy control theory, which guarantees the closed-loop system signal bounded in [0]. In [1], focusing on nonlinear systems, a robust back-stepping controller was designed, which compensates the error of the virtual derivative by using the compensation signal of the command filter. In [], concentrating on uncertain nonlinear systems, an adaptive dynamic surface controller was designed using command filter. Not only the explosion of complexity is avoided in the back-stepping, but also the effect of the errors is removed caused by command filter. In [3], based on the DSC and radial basis function neural network, an adaptive dynamic surface controller was designed for a class of nonlinear systems with unknown upper bound and all signals of the closed-loop system are bounded. During the process of actual design of control system, if the input constraint is not considered, the problem of actuator saturation may emerge in the practical actuation, which will lead to the weakening of the system control performance, or even the system instability. In [4], based on the command filter technology, an adaptive anti-saturation tracking controller was designed for uncertain MIMO nonlinear systems with input constraints. In [5], focusing on a class of nonlinear systems with input constraints, the input saturation function was processed by mean value theorem and a neural network dynamic surface controller was proposed combining DSC and neural network control theory. For hypersonic vehicle with actuator dynamics and disturbances, a robust attitude controller with use of predictive sliding mode control and nonlinear disturbance observer, was proposed. For a hypersonic reentry vehicle subject to actuator saturation, a linear controller was designed based on a nonlinear extended state observer. In [7], a dynamic surface tracking controller was proposed to deal with hypersonic vehicle with aerodynamic parameter uncertainty and input constraint, by combining nonlinear disturbance observer and dynamic surface control in [8]. In [9,30], focusing on the hypersonic vehicles with input constraint and uncertain aerodynamic parameters, an adaptive dynamic surface robust controller was designed, which can achieve the stable tracking control of velocity and altitude. However, the introduction of neural network in some way augmented the complexity of the controller design, making it difficult to be applied in the engineering. In [31], for the hypersonic vehicles subject to input saturation problem, based on high-order differentiator, an adaptive neural network controller was designed, which can guarantee

3 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 069 that the closed-loop system is uniformly bounded. In [3], an adaptive robust controller was designed for the hypersonic vehicles with input saturation. In order to further solve the anti-saturation tracking control problem of hypersonic vehicles, this paper designs an anti-saturation robust dynamic surface controller on the basis of the command filtering and the dynamic surface control, which can eliminate the impact of unmatched uncertainties, and can be able to control the input constraint at the same time. Compared with the literature listed above, this paper has innovative aspects as the following. 1) Compared with [30], anti-saturation robust dynamic surface controller is designed using back-stepping method and command filter in the thesis, which adopts the robust compensation signals to eliminate the influence of mismatch. ) Compared with [9,3], the dynamic surface control technique is proposed with two kinds of first-order filters, which can remove the bounds of the derivative of the virtual control functions in this paper. 3) Compared with [8-10], this paper takes input constraint into consideration, which makes the designed controllers have practical significance. This paper is organized as follows. Firstly, the longitudinal input-output linearization model of hypersonic vehicles is established. Secondly, two anti-saturation robust dynamic surface back-stepping controllers are designed for velocity and altitude subsystem, respectively. Then, the corresponding stability of designed controller is analyzed by using the Lyapunov theory. Numerical simulations are given to verify the effectiveness of the designed tracking controllers in Section 4. In the end, the conclusion of the paper is presented. Nomenclature m Mass, kg) S Reference area, m ) c Mean aerodynamic chord, m) R E Earth radius, m) C α L First-order coefficient of α contribution to C L C D Drag coefficient C αi D ith-order coefficient of α contribution to C D C 0 D Constant term in C D µ Gravitational constant, N m/kg ) C T Thrust coefficient h Altitude, m) h d Reference command for altitude, m) V Velocity, m/s) V d Reference command for velocity, m/s) ρ Atmospheric density, kg/m 3 ) I yy Moment of inertia, kg m ) L Lift, N) D Drag, N) T Thrust, N) M Pitching, N m) C L Lift, coefficient β 0 and β 0 Fuel-to-air ratio contribution to C0 T β 1 Constant term in CT 0 C M,α Contribution to moment due to angle of attack CM,α αi ith-order coefficient of α contribution to C M,α CM,α 0 Constant term in C M,α C M,δe Contribution to moment due to elevator deflection c e Elevator coefficient in C M,δe C M,q Contribution to moment due to pitch rate CM,q αi ith-order coefficient of α contribution to C M,q C 0 M,q Constant term in C M,q

4 070 J. SUN, S. SONG AND G. WU. Preliminaries..1. Hypersonic vehicles non-linear mode. The rigid hypersonic vehicles model proposed by NASA Langley Research Center is below [10] V = T cos α D m ḣ = V sin θ α) α = L + T sin α mv θ = q µ sin θ α) r + q + g V V ) cos θ α) r 1) where q = M I yy L = 0.5ρV SC L D = 0.5ρV SC D T = 0.5ρV SC T M = 0.5ρV S cc M r = h + R E C L = C α Lα C D = CD α α + CDα α + CD 0 { C T = CT 0 β0 ϕ if ϕ < 1 ϕ) = β 1 + β 0ϕ if ϕ > 1 C M = C M α) + C M δ e ) + C M q) C M,α α) = CM,αα α + CM,αα α + CM,α 0 C M,δe δ e ) = c e δ e α) ) C M,q q) = c/v ) q CM,qα α + CM,qα α + CM,q 0 V, h, and q are velocity, altitude, angle of attack, pitch angle, pitch rate of the hypersonic aircrafts, T, D, L, M, δ e and ϕ mean thrust, drag, lift, pitching moment, the elevator deflection angle, the throttle setting, respectively. m, ρ, I yy, S, and R E denote mass of hypersonic vehicles, density of air, moment of inertia, gravitational constant, radius of the earth, c and c e are constants. Other related variables and parameters in this model are listed in the nomenclature. The second-order dynamical model of the engine can be described as ϕ = ζω n ϕ ω n ϕ + ω nϕ c ) where ϕ is the throttle setting, ϕ c is the demand of the control input, ω n is the undamped natural frequency of the engine dynamics, and ζ is the damping ratio... Input-output linearization model. In order to facilitate the design of hypersonic vehicles control system and controller, the linearization system model is as follows [10]. V 3) = f V + b 11 ϕ c + b 1 δ e + f V + b 11 ϕ c + b 1 δ e h 4) = f h + b 1 ϕ c + b δ e + f h + b 1 ϕ c + b δ e 3)

5 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 071 where ϕ c and δ e are control inputs, f V, f h, b 11, b 1, b 1, b are bounded items created by parameter uncertainly and external disturbance, and the definitions of f V, f h, b 11, b 1, b 1 and b can be seen in [10]. Let x V 3 = f V + b 11 δ e + b 1 ϕ c x h4 = f h + b 1 δ e + b ϕ c u V = b 11 ϕ c + b 1 δ z u h = b 1 ϕ c + b δ 4) z x V = [ ] T [ ] T x V 1 x V x V 3 = V V V x h = [ ] T [ x h1 x h x h3 x h4 = h ḣ ḧ... h ] T By applying 4), the system model 3) can be transformed into velocity and altitude subsystem which have triangular form with mismatched uncertainties ẋ V 1 = x V + x V 1 ẋ V = x V 3 + x V ẋ V 3 = f V + u V + x V 3 5) ẋ h1 = x h + x h1 ẋ h = x h3 + x h ẋ h3 = x h4 + x h3 ẋ V 3 = f h + u h + x h4 6) where x V 1, x V, x V 3, x h1, x h, x h3 and x h4 are lumped mismatched uncertainties. Consider the following forms of input saturation u i max, u ic u i max u i = u ic, u i min < u ic < u i max i = V, h) 7) u i min, u ic u i min where u i is desired control input, u i min is the minimum values of u i, u i max is the maximum values of u i. Control object: In the thesis, two anti-saturation robust dynamic surface controllers are designed for the system mode 5)-7), so that the velocity V and the altitude h could track the reference signals V d and h d in finite-time subject to mismatched uncertainties and input saturation, respectively. Meanwhile, angle of attack α, pitch angle θ and pitch rate q are kept within a certain range..3. Related lemmas and assumptions. To facilitate the control system design in this subsection, the following assumption is introduced firstly. Lemma.1. [1]: The compensating signals ξ i for i =,..., n are defined as ξ 1 = c 1 ξ 1 ξ 1 + ξ + x,c a 1 ) ξ i = c i ξ i ξ i 1 + ξ i+1 + x i+1,c a i ) ξ n = c n ξ n ξ n 1 8) where a i and x i+1,c are first-order filter input and output. We can get that ξ i is bounded. Assumption.1. The disturbance x V i i = 1,, 3) is the unknown bounded uncertainty that has bounded derivative. That is to say, there is a positive constant σ V i i = 1,, 3) satisfying the inequality as the following x V i σ V i i = 1,, 3) 9)

6 07 J. SUN, S. SONG AND G. WU Assumption.. The disturbance x hj j = 1,, 3, 4) is the unknown bounded uncertainty that has bounded derivative. That is to say, there is a positive constant σ hj j = 1,, 3, 4) satisfying the inequality as the following x hj σ hj j = 1,, 3, 4) 10) 3. Main Results. A robust DSC back-stepping control scheme is design to cope with mismatched uncertainties and input saturation of the system, which uses dynamic surface control with signal compensation and auxiliary system. The proposed approach not only avoids the explosion of complexity in the back-stepping method, but also removes the effect of the error caused by the first-order filter, which guarantees the fast convergence of tracking error and the system robustness for uncertainties Robust DSC back-stepping controller design for velocity subsystem. For system 5), an anti-saturation robust dynamic surface controller is designed. The specific process is as follows. Step 1: Define the tracking error variable z V 1 as where x V d is reference velocity. Computing the first order derivative of 11), we obtain Define the virtual control functions x V as z V 1 = x V 1 x V d 11) ż V 1 = ẋ V 1 ẋ V d = x V ẋ V d + x V 1 1) x V = k V 1 z V 1 + ẋ V d + ω V 1 13) where k V 1 > 0 is a positive constant. The virtual controller 13) consists of two parts. 1) The nominal intermediate input k V 1 x V 1 + ẋ V d. ) ω V 1 is the robust compensation input, which can eliminate the effect of uncertainties. Substituting 13) into 1) yields ż V 1 = k V 1 z V 1 + ω V 1 + x V 1 14) The compensation signal ω V 1 is constructed by a low-pass filter ω V 1 = F V 1 s) x V 1 15) where F V 1 s) = λ V 1 /s + λ V 1 is a low-pass filter. When the bandwidth λ V 1 of the filter is chosen large enough, the signal ω V 1 approximately equals x V 1, which can eliminate the effects of mismatched uncertainties. We define x V d satisfying as τ V ẋ V d + x V d = x V, x V d 0) = x V 0) 16) where x V d and x V are first-order filter input and output respectively, τ V is a positive constant. To eliminate the effect of the error x V d x V, the compensating signal is defined as ξ V 1 = k V 1 ξ V 1 + ξ V + x V d x V ) 17) The compensated tracking error signal ν V 1 = z V 1 ξ V 1 and the compensation error signal ρ V 1 = ω V 1 + x V 1. Consider Lyapunov function as V V 1 = 1 ν V ρ V 1 18)

7 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 073 The time derivative of V V 1 is given as V 1 = ν V 1 ν V 1 + ρ V 1 ρ V 1 = ν V 1 k V 1 z V 1 + ω V 1 + x V 1 + z V + x V d x V + k V 1 ξ V 1 ξ V x V d + x V ) + ρ V 1 ρ V 1 = ν V 1 k V 1 z V 1 + ρ V 1 + z V + k V 1 ξ V 1 ξ V ) + ρ V 1 λ V 1 ρ V 1 + x V 1 ) 19) = k V 1 νv 1 + ν V 1 z V ξ V ) + ν V 1 ρ V 1 λ V 1 ρ V 1 + x V 1 ρ V 1 By using Young s inequality ν V 1 ρ V 1 1 v V ρ V 1, xv 1 ρ V 1 1 V ρ V 1 0) Substituting 0) into 19) yields V 1 = k V 1 νv 1 + ν V 1 z V ξ V ) + 1 v V ρ V 1 λ V 1 ρ V V ρ V 1 = k V 1 1 ) νv 1 + ν V 1 z V ξ V ) λ V 1 1) ρ V δ V 1 1) Step : Define the tracking error variable The time derivative of ) can be written as Define the virtual control functions x V 3 as z V = x V x V d ) ż V = ẋ V ẋ V d = x V 3 + x V ẋ V d 3) x V 3 = k V z V z V 1 + ẋ V d + ω V 4) where k V > 0 is a positive constant. The compensation signal ω V is constructed by a low-pass filter ω V = F V s) x V 5) where F V s) = λ V /s + λ V is a low-pass filter. When the bandwidth λ V of filter is chosen large enough, the signal ω V approximately equals x V, which can eliminate the effects of mismatched uncertainties. x V 3d is defined as τ V 3 ẋ V 3d + x V 3d = x V 3, x V 3d 0) = x V 3 0) 6) where x V 3d and x V 3 are the input and output of the first-order filter, respectively, τ V 3 is a positive constant. To eliminate the effect of the error x V 3d x V 3, the compensating signals are defined as ξ V = k V ξ V ξ V 1 + ξ V 3 + x V 3d x V 3 ) 7) We define the compensated tracking error signal ν V = z V ξ V and the compensation error signal ρ V = ω V + x V. Choose the Lyapunov function as V V = V V ν V + 1 ρ V 8)

8 074 J. SUN, S. SONG AND G. WU Computing the first order derivative of V V yields V = V 1 + ν V ν V + ρ V ρ V = V 1 + ν V z V 3 k V z V z V 1 + k V ξ V + ξ V 1 ξ V 3 + ρ V ) + ρ V ρ V = k V 1 1 ) νv 1 λ V 1 1) ρ V 1 + ν V 1 ν V + ν V k V ν V + z V 3 ξ V 3 + ρ V ν V 1 ) λ V ρ V + V ρ V + 1 δ V 1 = k V 1 1 ) νv 1 λ V 1 1) ρ V 1 + ν V k V ν V + z V 3 ξ V 3 ) λ V ρ V 9) + ν V ρ V + V ρ V + 1 δ V 1 By using Young s inequality ν V ρ V 1 v V + 1 ρ V, V ρ V 1 V + 1 ρ V 30) Substituting 30) into 9), we have V V k V 1 1 ) ν V 1 λ V 1 1) ρ V 1 k V 1 ) νv λ V 1) ρ V + ν V z V 3 ξ V 3 ) + 1 δ V δ V k V i 1 ) k V i νv i λ V i 1) ρ V i 31) + ν V z V 3 ξ V 3 ) + 1 δ V δ V Step 3: Define the tracking error variable Computing the first order derivative of z V 3, we obtain z V 3 = x V 3 x V 3d 3) ż V 3 = f V + u V + x V 3 ẋ V 3d 33) To handle input saturation, the auxiliary system 34) is introduced k V η η V 1 η V = η V ν V 3 u V u V ) η V u V k V γ sig γ ν V 3 ), η V σ V 0 η V < σ V 34) where sig γ ν V 3 ) = ν V 3 γ sign ν V 3 ). k V η, k V γ and γ are positive parameters and η V is the state variable of the auxiliary system. σ V is positive constant, u V = u V u V c. Remark 3.1. From the hypersonic vehicles background, it can be seen that the input saturation may be symmetrical or asymmetric. The auxiliary system 34) can handle symmetrical and asymmetric input saturation. The compensated tracking error signal ν V 3 = z V 3 ξ V 3, and compensation error signal ρ V 3 = ω V 3 + x V 3. The compensating signal ξ V 3 is defined as ξ V 3 = k V 3 ξ V 3 ξ V 35) where k V 3 is a positive constant.

9 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 075 The control law u V c is defined as u V c = k V 3 z V 3 f V z V + ẋ V 3d + ω V 3 + k V 4 η V v V 3µ V v V 3 ) ψv + v V 3 36) k V ψ ψ V µ V v V 3 ) ψ V ψ V = ψ V + v V 3, v V 3 ψ ν 37) 0, v V 3 < ψ ν where µ V v V 3 ) = 0.5k V 4 v V 3. k V 3, k V 4, k V ψ and ψ ν are positive constants. The compensation signal ω V 3 is constructed by a low-pass filter where F V 3 s) = λ V 3 /s + λ V 3 is a low-pass filter. ω V 3 = F V 3 s) x V 3 38) Theorem 3.1. Considering system 5) with Assumption.1, the state of closed-loop system is regulated under the anti-saturation robust dynamic surface controllers 34)-37). The following conclusions can be obtained. i) The variable v V i i = 1,, 3), ρ V i i = 1,, 3), η V and ψ V are uniformly ultimately bounded respectively; ii) The velocity tracking error converges to any small neighborhood. Proof: Choose Lyapunov function as Computing the derivative of V V 3, we have V V 3 = V V + 1 ν V ρ V η V + 1 ψ V 39) V V 3 = V V + ν V 3 ν V 3 + ρ V 3 ρ V 3 + η V η V + ψ V ψ V 40) According to 34)-37), we can obtain V V 3 = V V + ν V 3 ω V 3 + x V 3 k V 3 z V 3 z V ξ V 3 ) + k V 4 ν V 3 η V + ν V 3 u V + ρ V 3 ρ V 3 + η V η V + ψ V ψ V v V 3 µ V v V 3 ) ψv + v V 3 = k V i 1 ) νv i λ V i 1) ρ V i + 1 δ V δ V + ν V 3 k V 3 z V 3 + k V 3 ξ V 3 + ρ V 3 ) + ρ V 3 ρ V 3 + k V 4 ν V 3 η V + ν V 3 u V + η V η V + ψ V ψ V v V 3 µ V v V 3 ) ψv + v V 3 = k V i 1 ) νv i λ V i 1) ρ V i + 1 δ V δ V 41) k V 3 ν V 3 + ν V 3 ρ V 3 λ V 3 ρ V 3 + ρ V 3 xv 3 + k V 4 ν V 3 η V + ν V 3 u V + η V η V + ψ V ψ V v V 3 µ V v V 3 ) ψ V + v V 3 By using Young s inequality ν V 3 ρ V 3 1 v V ρ V 3, V 3 ρ V 3 1 V ρ V 3 4)

10 076 J. SUN, S. SONG AND G. WU Substituting 4) into 41), we have V V 3 k V i 1 ) νv i λ V i 1) ρ V i k V 3 1 ) νv 3 λ V 3 1) ρ V 3 + k V 4 ν V 3 η V + ν V 3 u V + η V η V + ψ V + 1 δ V δ V + 1 δ V 3 v V 3 µ V v V 3 ) ψv + v V 3 = k V i 1 ) νv i λ V i 1) ρ V i + k V 4 ν V 3 η V + ν V 3 u V + η V η V + ψ V ψ V + 1 δ V δ V + 1 δ V 3 v V 3 µ V v V 3 ) ψ V + v V 3 Applying 34) and 37) into 43) can be written as V V 3 = k V i 1 ) νv i λ V i 1) ρ V i k V η ηv k V ψ ψv + ν V 3 u V + k V 4 ν V 3 η V ν V 3 u V 1 u V η V u V v V 3 µ V v V 3 ) ψv + v V 3 µ V v 3 ) ψv ψv + v V 3 ψ V 43) 44) + 1 δ V δ V + 1 δ V 3 k V γ ν V 3 sig γ ν V 3 ) As v V 3 µ V v V 3 ) ψv + v V 3 = µ V v V 3 ) + µ V v V 3 ) ψv ψv + v V 3 µ V v V 3 ) = 0.5kV 4 v V 3, ν V 3 u V ν V 3 u V 0 45) k V 4 ν V 3 η V η V u V 1 k V 4 ν V 3 + η V + 1 u V According to 45), then 43) can be further simplified as V V 3 + k V i 1 ) νv i λ V i 1) ρ V i k V η 1) ηv k V ψ ψv 1 δ V i k V γ ν V 3 sig γ ν V 3 ) k V i 1 ) νv i λ V i 1) ρ V i k V η 1) ηv k V ψ ψ V + ε 1 V 3 + C 1 1 δ V i 46) where ε 1 = min 1 i 3 { kv i 1 ), λv i 1), k V η 1), k V ψ }, C1 = 3 1 δ V i, k V η > 1. If V V 3 = P 1, then V V 3 εp 1 +C 1. If ε 1 > C 1 P 1, then V V 3 = P 1 and V V 3 0. Therefore, V V 3 P 1 is an invariant set. In other words, if V V 3 0) P 1, then V V 3 P 1.

11 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 077 Integrate 46) and we can obtain 0 V V 3 t) C 1 ε 1 + V V 3 0) C 1 ε 1 ) exp ε 1 t), t 0 47) Based on the above 47), the bound of V 3 t) is C 1 / ε 1 ). For any t 0, 0 V V 3 t) C 1 / ε 1 ). Therefore, v V i i = 1,, 3), ρ V i i = 1,, 3), η V and ψ V are uniformly ultimately bounded. Therefore, conclusion i) is proved According to 47), one can conclude that the ν V i = z V i ξ V i i = 1,, 3) is satisfied following C 1 ν V i V V 3 0) C ) 1 exp ε 1 t) 48) ε 1 ε 1 Then lim ν V i C 1 /ε 1 49) t { From 49), we can conclude that ν V i converges to compact set R vi = ν V i νv i } C1 /ε 1. That is to say, when ε 1 is chosen large enough, ν V i converges to any small neighborhood. From Lemma.1, we can get that ξ V i is bounded. According to ν V i = z V i ξ V i, further analysis, the velocity tracking error z V i also converges to any small neighborhood. Therefore, conclusion ii) is proved. Theorem 3.1 is proved. Remark 3.. In controller 36), we introduce k V 4 η V to handle the input saturation of velocity subsystem, mainly for the following two conditions. When η V σ V > 0, there is an input saturation in control system. a) When u V c u V max, k V 4 η V can guarantee that u V c can reduce to u V c = u V max. b) When u V c u V min, k V 4 η V can guarantee that u V c can increase to u V c = u V min. Thus, u V c = u V max or u V c = u V min. When η V < σ V, η V = 0, there is no input saturation in control system, that is to say u V = 0, the k V 4 η V can guarantee that u V c satisfies u V min < u V c < u V max. Thus, u V = u V c. 3.. Adaptive integral terminal sliding mode controller design. For System 6), refer to the method of velocity subsystem design. Specific process is as follows. Define the tracking error of the altitude as the following z 1h = x h1 x hd z h = x h x hd z 3h = x h3 x h3d z 4h = x h4 x h4d 50) where x hd is altitude reference signal, x hjd j =, 3, 4) the output of the first-order filter with virtual control functions x hj j =, 3, 4) as the input. Define the virtual control functions as where k h1, k h and k h3 are positive constants. x h = k h1 z h1 + ẋ hd + ω h1 x h3 = k h z h z h1 + ẋ hd + ω h x h4 = k h3 z h3 z h + ẋ h3d + ω h3 51)

12 078 J. SUN, S. SONG AND G. WU Define the first-order filter as τ ẋ hd + x hd = x h, x hd 0) = x h 0) τ 3 ẋ h3d + x h3d = x h3, x h3d 0) = x h3 0) τ 4 ẋ h4d + x h4d = x h4, x h4d 0) = x h4 0) 5) The compensation signal ω hj j = 1,, 3, 4) is constructed by a low-pass filter as ω hj = F hj s) x hj j = 1,, 3, 4) 53) where F hj s) = λ hj / s + λ hj ) is a low-pass filter. To eliminate the effect of the errors x hjd x hj j =, 3, 4) caused by the first-order filter, we will design the compensating signals. The compensating signals are defined as ξ h1 = k h1 ξ h1 + ξ h + x hd x h ) ξ h = k h ξ h ξ h1 + ξ h3 + x h3d x h3 ) 54) ξ h3 = k h3 ξ h3 ξ h + ξ h4 + x h4d x h4 ) ξ 4 = k h4 ξ h4 ξ h3 where k h4 is a positive constant. To deal with input saturation problem, the auxiliary system 55) is built up as the following k hη η h 1 η h = η h ν h4 u h u h ) η h u h k hr sig r ν h4 ), η h σ h 0, η h < σ h 55) where sig r ν h4 ) = ν h4 r sign ν h4 ). k hη, k hr and r are positive parameters and η h is the state variable of the auxiliary system. σ h is a positive constant, u h = u h u hc. Then, the controller u hc is designed as u hc = k h4 z h4 f h z h3 + ẋ h4d + ω 4 + k h5 η h v h4µ h v h4 ) ψ h + v h4 56) k hψ ψ h µ h v h4 ) ψ h ψ h = ψ h + v h4, v h4 ψ hν 57) 0, v h4 < ψ hν where µ h v h4 ) = 0.5k h5 v h4, k h4, k h5, k hψ and ψ hν are positive constants. Theorem 3.. Considering system 6) with Assumption., the state of closed-loop system is regulated under the anti-saturation robust dynamic surface controllers 55)-57). We can draw the following conclusions. i) The variable v hj j = 1,, 3, 4), η h, ψ h are uniformly ultimately bounded; ii) The altitude tracking error converges to any small neighborhood. Proof: The compensated tracking error signal ν hj = z hj ξ hj j = 1,, 3, 4) and the compensation error signal ρ hj = ω hj + x hj j = 1,, 3, 4). Consider the Lyapunov function as V h4 = 1 ν hj + 1 ρ hj + 1 η h + 1 ψ h 58)

13 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 079 Computing the first order derivative of 58), we obtain V h4 = ν h1 ν h1 + ν h ν h + ν h3 ν h3 + ν h4 ν h4 + = k hj 1 ) νhj ρ hj ρ hj + η h η h + ψ h ψ h λ hj 1) ρ hj + + ν h4 ν h3 k h4 z h4 + k h5 η h z h3 ξ 3 ) + ν h4 u h λ h4 ρ h4 + ν h4 ρ h4 + x h4 ρ h4 + η h η h + ψ h ψ h v h4 µ h v h4 ) ψ h + v h4 σ hj 59) By using Young s inequality Substituting 60) into 59), we have ν h4 ρ h4 1 v h4 + 1 ρ h4, h4 ρ h4 1 h4 + 1 ρ h4 60) V h4 k hj 1 ) νhj λ hj 1) ρ hj + σhj k h4 1 ) νh4 λ h4 1) ρ h4 + k h5 ν h4 η h + ν h4 u h + η h η h + ψ h ψ h + 1 σ h4 v h4 µ h ν h4 ) ψh + ν h4 = k hj 1 ) νhj λ hj 1) ρ hj + σhj + k h5 ν h4 η h + ν h4 u h + η h η h + ψ h ψ h v h4 µ h ν h4 ) ψ h + ν h4 61) Substituting 55) and 57) into 61), we obtain V h4 = k hj 1 ) νhj λ hj 1) ρ hj k hη ηh k hψ ψh + k h5 ν h4 η h + ν h4 u h ν h4 u h 1 u h η h u h v h4 µ h ν h4 ) ψh + ν h4 µ h ν h4 ) ψh ψh + ν h4 + σhj k hr ν h4 sig r ν h4 ) 6) As v h4 µ h ν h4 ) ψ h + ν h4 = µ h v h4 ) + µ h ν h4 ) ψ h ψ h + ν h4 µ h v h4 ) = 0.5k h5 v h4, ν h4 u h ν h4 u h 0 63) k h5 ν h4 η h η h u h 1 k h5 v h4 + η h + 1 u h

14 080 J. SUN, S. SONG AND G. WU according to 63), then 6) can be further simplified as V h4 k hj 1 ) νhj λ hj 1) ρ hj k hη 1) ηh k hψ ψh + σhj k hr ν h4 sig r ν h4 ) k hj 1 ) νhj λ hj 1) ρ hj k hη 1) ηh k hψ ψh + σ hj 64) ε V h4 + C { ) } where ε = min 1 j 4 khj 1, λv j 1), k V η 1), k V ψ, C = 4 σhi, k hη > 1. Similar to Theorem 3.1, one can conclude that the ν hj = z hj ξ hj j = 1,, 3, 4) is satisfied following C ν hj V h4 0) C ) exp ε t) 65) ε ε Then lim ν hj C /ε 66) t { From 66), we can conclude that ν hj converges to compact set R hj = ν hj νhj } C /ε, that is to say, when ε is chosen large enough, ν hj converges to any small neighborhood. From Lemma.1, we can get that the ξ hj is bounded. According to ν hj = z hj ξ hj, further analysis, the velocity tracking error z hj also converges to any small neighborhood. Therefore, conclusion ii) is proved Therefore, Theorem 3. is proved. Remark 3.3. In controller 56), we introduce k h5 η h to handle the input saturation of velocity subsystem, mainly for the following two conditions. When η h σ h > 0, there is input saturation in control system. a) When u hc u h max, k h5 η h can guarantee that u hc can reduce to u hc = u h max. b) When u hc < u h max, k h5 η h can guarantee that u hc can increase to u hc = u h min. Thus, u hc = u h max or u hc = u h min. When η h < σ h, η h = 0, there is no input saturation in control system, that is to say u h = 0. k h5 η h can guarantee that u hc satisfies u h min < u hc < u h max. Thus u h = u hc. 4. Numerical Examples. In order to verify the effectiveness of the two robust backstepping controllers, we take the nonlinear longitudinal motion 1) of the hypersonic vehicles in this chapter as the simulation object. The parameters of the hypersonic vehicles and flight environment of [10] are shown in Table 1 and the values of the aerodynamic coefficients are shown in Table. Based on the basic parameters of the above hypersonic vehicles, we can figure out a group of equilibrium points as the initial condition, and the initial value of simulation is set as x0) = [ ] T. In the simulation process, the external disturbances d 1 t) = sin0.t), d t) = 0.01 sin0.t). The parameter

15 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 081 Table 1. The parameters of hypersonic aircrafts and flight environment Physical quantity Symbol Value Mass kg) m Moment of inertia kg m ) I yy Reference area m ) S 335. Mean aerodynamic chord m) c Earth radius m) R E Gravitational constant N m/kg ) µ Altitude m) h 3358 Velocity m/s) V Atmospheric density kg/m 3 ) ρ Table. The values of the aerodynamic coefficients Coefficient Value Coefficient Value Coefficient Value CL α CM,α α β CD α CM,α β CD α CM CM,α α CD CM,q α CM,q β CM,q α c e 1897 uncertainties of the model are selected m = m m), I yy = I I yy ) S = S S), c = c c) c e = c e0 1 + c e0 ), ρ = ρ ρ) m 0.05, I yy 0.05, S 0.05 c 0.05, ρ 0.05, c e 0.05 where m 0, I 0, S 0, c 0, c e0, ρ 0 are normal values respectively, the parameter uncertainties of the model are set as m = 0.05, I yy = 0.05, c = 0.05, c e = 0.05, ρ = 0.05, S = Simulation analysis of the robust dynamic surface controller considering the control saturation. To verify the validity of the control strategy of the hypersonic vehicles designed in this paper, we have added a comparison with second-order terminal sliding mode control TSMTC) [33]. The simulations select two kinds of reference signals as tracking signals. The control gains and parameters are shown in Table 3. Situation 1: The reference velocity of the hypersonic vehicles is V d = m/s, that is, V = 100 m/s, and the reference altitude is h d = 3508 m, that is, h = 1500 m. Situation : The reference velocity of the hypersonic vehicles is V d = m/s, that is, V = t) m/s, and the reference altitude is h d = 3508 m, that is, h = t ) m. 1) For the situation 1, the simulation results are shown in Figures 1-4. Figure 1 and Figure show the tracking curves of velocity and altitude of hypersonic vehicles, respectively. From Figures 1 and, it can be seen that even if there are mismatched uncertainties and input saturation, the velocity error and altitude error can converge rapidly, satisfying the requirement of control accuracy. Figure 3 illustrates the 67)

16 08 J. SUN, S. SONG AND G. WU Table 3. The set of the parameters of controllers Parameter Value Parameter Value Parameter Value Parameter Value k V 1 15 ψ V 0.01 k V η 3.5 λ V 0.1 k V 5 τ V 0.01 k V γ 1.5 γ 0.75 k V 3 1 τ V k V ψ 0.8 λ V k V λ V σ V 0.01 τ V k h1 15 τ h 0.01 k hη.5 λ h3 0.1 k h 8 τ h k hψ 0.65 λ h4 0.1 k h3 3. τ h σ h 0.01 k hr 3.5 k h4 16 λ h1 0.1 ψ h 0.01 r 0.8 k h5 0.6 λ h 0.1 Figure 1. The tracking curves of velocity curves of control input, suggesting that the amplitude of the control force accords with the control constraint. As the simulation results show, the control input signals ϕ c and δ e are continuous and smooth which make the proposed control laws possible for the actual flight implementation. The curves of the attack angle, the pitch angle and the pitch rate depicted in Figure 4, are quite smooth and within their rational bounds. ) For the situation, its control parameters and gains and reference signal are the same as those of the situation 1 and the simulation results are shown in Figures 5-8. From Figures 5-8, when the reference signals are time-varying signals, for the longitudinal model of the hypersonic vehicle, both the velocity and altitude can track their reference signals under proposed control strategy. So this part will focus on the analysis of the differences between the two situations. From Figures 5 and 6, it can be obtained that the proposed method and TSMTC can achieve the stable tracking of the velocity V and altitude h of the reference command. The curves of control inputs under the proposed method and TSMTC are shown in Figure 7. It can be obtained that the curve of the control input ϕ c and δ e under the proposed method are smooth and can converge to

17 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 083 Figure. The tracking curves of altitude Figure 3. The curves of the control inputs ϕ c, δ e a small neighbourhood of zero in tracking process with a smaller control gain compared with TSMTC. From Figure 8, it can be known that the curves of the angle of attack α, the pitch angle θ, and the pitch rate q approach their steady-state values in short time. In summary, simulation results show that the robust DSC control scheme possesses a good adjustment capacity and a strong robustness for the longitudinal model of the hypersonic vehicle with mismatched uncertainties and input saturation, which can guarantee the stability of the system tracking performance.

18 084 J. SUN, S. SONG AND G. WU Figure 4. The curves of the α, θ and q Figure 5. The tracking curves of velocity 5. Conclusions. This paper studies the saturated tracking performance of hypersonic vehicles on the basis of the DSC with signal compensation and auxiliary system. The conclusions are as the following. 1) The non linear control system model of hypersonic vehicles is simplified by the input and output linearization, and reasonably decomposed into subsystems that include velocity subsystem and altitude subsystem. ) A robust back-stepping control scheme is designed using the DSC with signal compensation method and the auxiliary system, which can eliminate the effect of mismatched uncertainties equivalents thoroughly and handle the impact of input saturation.

19 TRACKING CONTROL VIA ROBUST DYNAMIC SURFACE CONTROL 085 Figure 6. The tracking curves of altitude Figure 7. The curves of the control inputs ϕ c, δ e 3) The simulation results demonstrate the robustness of controllers for mismatched uncertainties and good tracking performance of the desired reference signals. 4) In the future, the problem of elasticity and non-minimum phase should be considered in the design hypersonic control system. Based on the existing control algorithm, the different advanced control methods should be combined with each other to improve the performance of the control algorithm and solve the control problem.

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