1. Introduction

In recent years, many countries are vigorously developing hypersonic weapons in near space, such as the United States (AHW, HTV-2, X-51 and X-43), India (HSTDV and RLV-TD), China (WU-14) and Russia (GLL-31). Because of its ultrahigh speed and non-fixed trajectory, the hypersonic weapon has become a great strategic threat to homeland air defense [1–5]. The hypersonic vehicle flies over 5 Mach in the near space covering distances of 20–100 km. Compared with the ballistic missile, the hypersonic weapon is usually designed in a lifting body to obtain stronger maneuverability. Traditional defense systems against cruise missiles in the atmosphere cannot reach the near space. Thereby, the near space hypersonic weapon is a threat to the current defense system.

There are mainly two kinds of hypersonic vehicles. One is the air-breathing cruise vehicle [6]. Its maneuverability is relatively weaker, thus its interception is relatively easier as its trajectory is predictable. The other is the gliding entry vehicle [7]. At the entry stage, its velocity is up to 25 Mach at maximum. In the entry phase, it is able to glide thousands of kilometers in the near space without any power. In the terminal phase, a dive attack is performed to the target on the ground [8]. Therefore, its trajectory is not predictable and its interception is a challenge. A lot of research on entry guidance techniques with no-fly zone constraints has been conducted for hypersonic weapons [9, 10]. However, there are few research works

on intercepting these vehicles [11]. Consequently, new technical challenges are raised to intercept these weapons [12].

pitch control. Simulations demonstrated that the FOPID controller using multiobjective bat-algorithm optimization had better performance than others [21]. Ref. [22] proposed a fractional order tension control law for deployment control of a space tether system, and its stability was proved. Ref. [23] realized a fractional order controller for position control of a one-DOF flight motion table. The flight motion table was used for simulating the rotational movement of flying vehicles. Experiments showed that tracking of a position profile using fractional order controller was feasible in real time. Ref. [24] presented a tuning method of a fractional order proportional derivative controller based on three points of the Bode magnitude diagram for vibration attenuation. An aluminum beam replicating an airplane

However, not much effort has been made to deal with the pursuit-evasion problem against target maneuver and guidance noise with the fractional order PID controller. Ye et al. presented a 3D extended PN guidance law for intercepting a maneuvering target based on fractional order PID control theory and demonstrated that the air-to-air missile had a smaller miss distance to a maneuvering target [25]. However, in their research, the velocity of the missile was twice as much as that of the target, and the noise impacting on the guidance state (such as line-of-sight rate) was not taken under consideration, which limits the proposed algorithm's practical engineering applications. For this reason, based on a nonlinear proportional and differential guidance law (NPDG) and fractional calculus technique, a fractional calculus guidance law (FCG) is proposed to intercept a hypersonic maneuverable target in this chapter. It is assumed that the velocity of the interceptor is same as that of the hypersonic target, which means the target can evade as fast as the interceptor, and the guidance noise of the line-of-sight rate is considered.

The rest of this chapter is organized as follows. Section 2 formulates the FCG and the system stability condition is given. Numerical experiments are carried out in

where q\_ is the line-of-sight (LOS) angular rate, aM(t) is the normal acceleration command of the interceptor, VR(t) is the approaching velocity between the inter-

For compensating the negative influence of the target maneuver, the LOS accel-

A nonlinear tracking differentiator is used to estimate q€. The state equation is

eration q€ is considered. A nonlinear proportional and differential guidance law

ceptor and the target, and KP is the proportional coefficient.

x\_ <sup>1</sup> ¼ x2,

x\_ <sup>2</sup> ¼ �K sgn

where KD is the differential coefficient.

8 < : aMðÞ¼ t KPVRð Þt q t \_ð Þ, (1)

aMðÞ¼ t KPVRð Þt q t \_ðÞþ KDVRð Þt q t €ð Þ, (2)

(3)

x<sup>1</sup> � q\_<sup>m</sup> þ j j x<sup>2</sup> x<sup>2</sup> 2K � �,

wing verified the proposed controller.

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

Section 3, and Section 4 concludes this work.

2. Guidance law design

2.1 Definition of the NPDG

The PNG is given by

(NPDG) is presented as

given by

107

The proportional navigation guidance law (PNG) for interception has a big disadvantage of the guidance command being behind the target maneuver [13]. Actually, PNG is a proportional controller belonging to the PID controller family. Since q€ embodies the target maneuver, to add a differential part KD q€ into the PNG is reasonable. Thus, the proportional and differential (PD) controller is utilized to formulate the guidance law in a hypersonic pursuit-evasion game.

By introducing fractional calculus to PID control, the fractional order PID control has become an emerging field since the 1990s [14]. Fractional calculus is a generalization of the classical integer order calculus. There are mainly three fractional calculus definitions, including Riemann-Liouville (RL) definition, Grünwald-Letnikov (GL) definition and Caputo definition. Since the Gamma function and precise solution of fractional order equations are developed, fractional calculus has appeared in the control field [15, 16]. Like integer order PID controllers, the fractional order PID controller can also be classified into PI<sup>λ</sup> , PD<sup>μ</sup> and PI<sup>λ</sup> D<sup>μ</sup> (λ and μ represent fractional orders). Compared to integer order PD controller concerned in this chapter, the fractional order PD<sup>μ</sup> controller has the following advantages. First, a fractional order controller has greater control flexibility. There are proportional and differential fractional order μ in the PD<sup>μ</sup> controller. The selection of fractional order makes it more flexible than the integer order PD controllers. Secondly, fractional order makes the controller more robust. Fractional order controller is insensitive to the parameter uncertainties of the controller and controlled plant. Even if the system parameters change a lot, a fractional order controller can still work well.

The memory function and stability characteristic make the fractional order PID controller widely applicable in the field of aircraft guidance and control [15, 16], such as pitch loop control of a vertical takeoff and landing unmanned aerial vehicle (UAV) [17], roll control of a small fixed-wing UAV [18], perturbed UAV roll control [19], hypersonic vehicle attitude control [20], aircraft pitch control [21], deployment control of a space tether system [22], position control of a one-DOF flight motion table [23], and vibration attenuation to airplane wings [24]. The viscosity of the atmosphere interacting with air vehicles has given the aircrafts the similar aerodynamics to the fractional order systems, thus the fractional order PID control theory is appropriate to design aircraft guidance and control systems.

Han et al. designed a fractional order strategy to control the pitch loop of a vertical takeoff and landing UAV. Simulations verified that the proposed controller was superior to an integer order PI controller based on the modified Ziegler-Nichols tuning rule and a general integer order PID controller in robustness and disturbance rejection [17]. Luo et al. developed a fractional order PI<sup>λ</sup> controller to control the roll channel of a small fixed-wing UAV. From both simulation and real flight experiments, the fractional order controller outperformed the modified Ziegler-Nichols PI and the integer order PID controllers [18]. Seyedtabaii applied a fractional order PID controller to the roll control of a small UAV in dealing with system uncertainties, where the aerodynamic parameters are often approximated roughly [19]. Song et al. proposed a nonlinear fractional order proportion integral derivative (NFOPI<sup>λ</sup> Dμ ) active disturbance rejection control strategy for hypersonic vehicle flight control. The proposed method was composed of a tracking-differentiator, an NFOPI<sup>λ</sup> D<sup>μ</sup> controller and an extended state observer. Simulations showed that the proposed method made the hypersonic vehicle nonlinear model track-desired commands quickly and accurately, and it has robustness against disturbances [20]. Kumar et al. developed the fractional order PID (FOPID) and integer order PID controllers using multi-objective optimization based on the bat algorithm and differential evolution technique. The proposed controllers were applied to the aircraft

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

on intercepting these vehicles [11]. Consequently, new technical challenges are

formulate the guidance law in a hypersonic pursuit-evasion game.

change a lot, a fractional order controller can still work well.

The proportional navigation guidance law (PNG) for interception has a big disadvantage of the guidance command being behind the target maneuver [13]. Actually, PNG is a proportional controller belonging to the PID controller family. Since q€ embodies the target maneuver, to add a differential part KD q€ into the PNG is reasonable. Thus, the proportional and differential (PD) controller is utilized to

By introducing fractional calculus to PID control, the fractional order PID control has become an emerging field since the 1990s [14]. Fractional calculus is a generalization of the classical integer order calculus. There are mainly three fractional calculus definitions, including Riemann-Liouville (RL) definition, Grünwald-Letnikov (GL) definition and Caputo definition. Since the Gamma function and precise solution of fractional order equations are developed, fractional calculus has appeared in the control field [15, 16]. Like integer order PID controllers, the fractional order PID

, PD<sup>μ</sup> and PI<sup>λ</sup>

The memory function and stability characteristic make the fractional order PID controller widely applicable in the field of aircraft guidance and control [15, 16], such as pitch loop control of a vertical takeoff and landing unmanned aerial vehicle (UAV) [17], roll control of a small fixed-wing UAV [18], perturbed UAV roll control [19], hypersonic vehicle attitude control [20], aircraft pitch control [21], deployment control of a space tether system [22], position control of a one-DOF flight motion table [23], and vibration attenuation to airplane wings [24]. The viscosity of the atmosphere interacting with air vehicles has given the aircrafts the similar aerodynamics to the fractional order systems, thus the fractional order PID control theory is appropriate to design aircraft guidance and control systems. Han et al. designed a fractional order strategy to control the pitch loop of a vertical takeoff and landing UAV. Simulations verified that the proposed controller was superior to an integer order PI controller based on the modified Ziegler-Nichols tuning rule and a general integer order PID controller in robustness and disturbance rejection [17]. Luo et al. developed a fractional order PI<sup>λ</sup> controller to control the roll channel of a small fixed-wing UAV. From both simulation and real flight experiments, the fractional order controller outperformed the modified Ziegler-Nichols PI and the integer order PID controllers [18]. Seyedtabaii applied a fractional order PID controller to the roll control of a small UAV in dealing with system uncertainties, where the aerodynamic parameters are often approximated roughly [19]. Song et al. proposed a nonlinear fractional order proportion integral derivative

) active disturbance rejection control strategy for hypersonic vehicle

D<sup>μ</sup> controller and an extended state observer. Simulations showed that the

flight control. The proposed method was composed of a tracking-differentiator, an

proposed method made the hypersonic vehicle nonlinear model track-desired commands quickly and accurately, and it has robustness against disturbances [20]. Kumar et al. developed the fractional order PID (FOPID) and integer order PID controllers using multi-objective optimization based on the bat algorithm and differential evolution technique. The proposed controllers were applied to the aircraft

orders). Compared to integer order PD controller concerned in this chapter, the fractional order PD<sup>μ</sup> controller has the following advantages. First, a fractional order controller has greater control flexibility. There are proportional and differential fractional order μ in the PD<sup>μ</sup> controller. The selection of fractional order makes it more flexible than the integer order PD controllers. Secondly, fractional order makes the controller more robust. Fractional order controller is insensitive to the parameter uncertainties of the controller and controlled plant. Even if the system parameters

D<sup>μ</sup> (λ and μ represent fractional

raised to intercept these weapons [12].

Military Engineering

controller can also be classified into PI<sup>λ</sup>

(NFOPI<sup>λ</sup>

NFOPI<sup>λ</sup>

106

Dμ

pitch control. Simulations demonstrated that the FOPID controller using multiobjective bat-algorithm optimization had better performance than others [21]. Ref. [22] proposed a fractional order tension control law for deployment control of a space tether system, and its stability was proved. Ref. [23] realized a fractional order controller for position control of a one-DOF flight motion table. The flight motion table was used for simulating the rotational movement of flying vehicles. Experiments showed that tracking of a position profile using fractional order controller was feasible in real time. Ref. [24] presented a tuning method of a fractional order proportional derivative controller based on three points of the Bode magnitude diagram for vibration attenuation. An aluminum beam replicating an airplane wing verified the proposed controller.

However, not much effort has been made to deal with the pursuit-evasion problem against target maneuver and guidance noise with the fractional order PID controller. Ye et al. presented a 3D extended PN guidance law for intercepting a maneuvering target based on fractional order PID control theory and demonstrated that the air-to-air missile had a smaller miss distance to a maneuvering target [25]. However, in their research, the velocity of the missile was twice as much as that of the target, and the noise impacting on the guidance state (such as line-of-sight rate) was not taken under consideration, which limits the proposed algorithm's practical engineering applications. For this reason, based on a nonlinear proportional and differential guidance law (NPDG) and fractional calculus technique, a fractional calculus guidance law (FCG) is proposed to intercept a hypersonic maneuverable target in this chapter. It is assumed that the velocity of the interceptor is same as that of the hypersonic target, which means the target can evade as fast as the interceptor, and the guidance noise of the line-of-sight rate is considered.

The rest of this chapter is organized as follows. Section 2 formulates the FCG and the system stability condition is given. Numerical experiments are carried out in Section 3, and Section 4 concludes this work.

### 2. Guidance law design

#### 2.1 Definition of the NPDG

The PNG is given by

$$a\_M(t) = K\_P V\_R(t) \dot{q}(t),\tag{1}$$

where q\_ is the line-of-sight (LOS) angular rate, aM(t) is the normal acceleration command of the interceptor, VR(t) is the approaching velocity between the interceptor and the target, and KP is the proportional coefficient.

For compensating the negative influence of the target maneuver, the LOS acceleration q€ is considered. A nonlinear proportional and differential guidance law (NPDG) is presented as

$$a\_M(t) = K\_P V\_R(t)\dot{q}(t) + K\_D V\_R(t)\ddot{q}(t),\tag{2}$$

where KD is the differential coefficient.

A nonlinear tracking differentiator is used to estimate q€. The state equation is given by

$$\begin{cases} \dot{\varkappa}\_1 = \varkappa\_2\\ \dot{\varkappa}\_2 = -K \operatorname{sgn} \left( \frac{\varkappa\_1 - \dot{q}\_m + |\varkappa\_2|\varkappa\_2}{2K} \right), \end{cases} \tag{3}$$

where K is the estimation coefficient, q\_ <sup>m</sup>ð Þt is the LOS rate measured by the seeker, q\_mð Þt and q€mð Þt are estimated by x<sup>1</sup> and x2, namely x<sup>1</sup> ¼ q ^\_mð Þ<sup>t</sup> and x<sup>2</sup> ¼ q ^€mð Þ<sup>t</sup> . It is not easy to determine the value of <sup>K</sup>. If <sup>K</sup> is larger, the estimation will be more precise and the phase lag will be less, but the estimation will be noisier. Therefore, a fractional calculus guidance law is presented.
