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μ-Synthesis robust controller design for the supercavitating vehicle based on the BTT strategy Xin-hua Zhao n, Yao Sun, Guo-liang Zhao, Jia-li Fan College of Automation, Harbin Engineering University, Harbin 150001, China

art ic l e i nf o

a b s t r a c t

Article history: Received 26 December 2012 Accepted 22 June 2014 Available online 17 July 2014

In order to maintain the stability of the supercavitating vehicle in the turning process, a pair of elevator is used to achieve the functionality of the ailerons and the roll damping is provided by a pair of rudder. The turn centripetal force of the supercavitating vehicle is mainly supplied by a cavitator. In this paper, a control strategy based on BTT control for the supercavitating vehicle is proposed. The mathematical model of the BTT supercavitating vehicle is established, and the full state model is designed to include a pitch channel and a yaw–roll channel, and a μ-synthesis controller is designed respectively for the dual channels, and the simulation research of control effect is carried out. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Maneuver Stability BTT μ-Synthesis robust controller

1. Introduction The bank-to-turn (BTT) control technology is a turn control technology for some aircrafts and missiles whose aerodynamic conﬁgurations vary from the missiles which employ the conventional skid-to-turn control technology, and it is a polar coordinate control mode in McGehee (1979), McGehee and Emmert (1978), Humphrey and Sporing (1977) and Cunningham (1974). Scholars and researchers presented the turn mode and its control method as early as the beginning of the 1970s, whose purpose is to enhance the maneuvering ability, such as Froning Jr. (1981), Stallard (1985), Arrow (1985, 1982), Roddy et al. (1984) and Williams et al. (1987). The fundamental sign of the BTT control is to realize the maneuver ﬂight is on the premise that the sideslip angle is equal to zero or nearly zero. It is implemented by controlling the pitch angle and the roll angle of the vehicle, and in this way the maximum lift surface points at the maneuver direction. The BTT control possesses a lot of advantages, however, because the roll angle and the roll angle rate are no longer zero, and the missile dynamic cannot be divided into a longitude motion and a lateral motion and cannot be controlled simply. Therefore a new control method is required to solve the kinematic coupling and the inertial coupling and the aerodynamic coupling in the model. So far there have been a lot of BTT control methods. Initially, the controller design method is mainly the application of the classical single-input and single-output method of Arrow (1982) and Sheperd Christopher and Valavani (1988). In the late 1980s,

n

Corresponding author. Tel.: þ 86 15845038369. E-mail address: [email protected] (X.-h. Zhao).

http://dx.doi.org/10.1016/j.oceaneng.2014.06.035 0029-8018/& 2014 Elsevier Ltd. All rights reserved.

multi-input and multi-output method is introduced to the BTT control like Williams et al. (1987) and Williams and Friedland (1987). With the development of the robust control theory and the variable structure control theory, some scholars have applied it into the autopilot design of the BTT missile of Reichert (1989), Zhou et al. (1994) and Zhu and Yang (1997). In their work, coupling terms are considered as disturbances to design the dependent controller for each channel. In recent years, the nonlinear design method has also developed rapidly, some researchers applied the feedback linearization method and the backstepping design method into the BTT autopilot design to solve the problem of strong coupling nonlinear characteristics in the system in Adams and Banda (1993), Reiner et al. (1996) and Zhang et al. (2010). Some intelligent control methods such as the neural networks and the fuzzy control are attempted for the BTT controller design and ideal simulation results are obtained in McDowell et al. (1997), Kim et al. (1997), Lin and Wang (1998), McDowell and Irwin (1997), Huey and Bor (2002), Lin (2005), Duan et al. (2009) and Zheng et al. (2011). From the above results, we can see that the BTT strategy is mainly applied to missiles and aircrafts, and the research on the BTT control problem for the supercavitating vehicle has not attracted much attention (Wang et al., 2007). The supercavitating vehicle is a new type underwater vehicle which is enveloped by a cavity and the vehicle is traveling in the air such that the ﬂuid resistance is dramatically decreased, therefore its speed can achieve up to 100 m/s. The envelopment by the cavity makes the supercavitating vehicle possess complex and rich gas–liquid two phase dynamics characteristics, which make the control design more difﬁcult for the supercavitating vehicle. Recently research results mainly focus on the gesture control in dive-plane. A diveplane dynamic model and the feedback linearization controller

X.-h. Zhao et al. / Ocean Engineering 88 (2014) 280–288

were proposed in Dzielski and Kurdila (2003), and a steady supercavity and an analytical ﬁn force expression were exploited. A full-state dynamic model with the cavitator and the ﬁn force data generated by numerical calculate was developed and a linear quadratic regulator (LQR) was designed for a straight and level ﬂight in Kirschner et al. (2002). The advanced control method was also designed for the supercavitating vehicle, like the switch control in Lin et al. (2004, 2008), the sliding mode control in Lv et al. (2010, 2011), Fan et al. (2010) and Zhao et al. (2011), the LPV control in Balint et al. (2010) and Mao and Wang (2009), the backstepping control in Han et al. (2010) and Lin et al. (2006), the robust control in Anukul (2005) and Mao and Wang (2009) and so on. A linear control scheme and a switching control scheme were designed to stabilize the vehicle at a desired equilibrium point even when the initial motions are out of the dead zone in Lin et al. (2004, 2008). In Lin et al. (2006), controllers that provided absolute stability were designed by modeling the planing force as sector-bounded uncertainties, and a backstepping controller was designed for the nominal system. The sliding mode control and the quasi linear-parameter-varying control were proposed for the dive-plane dynamics model of the supercavitating vehicle, and a saturation compensator was designed to compensate the physical limits of deﬂection angles of control surfaces of the cavitator and ﬁns in Mao and Wang (2009). The pitch-plane dynamics of the supercavitating vehicle as a time-delay quasi-linear-parametervarying system was reformulated and delay-dependent H1 controllers were developed in Mao and Wang (2011). All the above research are carried out by assuming that the cavity state is stable. Therefore, stable cavity is the precondition for the supercavitating control. Furthermore, to realize the stable turning travel of the vehicle, and the most enveloped by the cavity, that is, the cavity stability is necessary condition. The traditional skid-to-turn (STT) of ordinary underwater vehicle is not beneﬁcial to the stability of the cavity, and here the BTT strategy is proposed to implement turning. In this paper, the cavitator has only one degree of freedom, and rotates around the axis that is parallel with the lateral axis of the vehicle. Through the supercavitating vehicle heeling, centripetal force of turn is provided by horizontal component of forces on the cavitator. The input of the elevator is differential instruction, and a pair of elevators is used to achieve the function of ailerons, and elevators will produce a roll torque around the longitudinal axis. While a pair of rudder provides the roll damping for the supercavitating vehicle. For the most BTT missiles, their dynamic is designed to include a pitch–yaw channel and a roll channel to design the controller respectively. While for the supercavitating vehicle, in order to be fully enveloped by the cavity, the BTT strategy is used. Due to the existence of the roll angular rate, there are the kinematics coupling and the inertial coupling between the yaw channel and the pitch channel. The coupling effect is not very strong for the constraint of the attack angle and the sideslip angle in the maneuver process. Due to the axis deformation of the cavity on the aft of the supercavitating vehicle, it produces the roll torque when operating the rudder, which leads to some coupling effect that exists between the yaw channel and the roll channel. According to the above analysis, the dynamic state of the supercavitating vehicle is designed to include a pitch channel and a yaw–roll channel to design the BTT controller. There is much model uncertainty between the nominal model established and the actual model of the pitch channel and the yaw–roll channel of the supercavitating vehicle. Therefore, the designed controller should be robustly stable and it is robust to the disturbance. For the system described by the parameter uncertainty and the multi-point independent bounded uncertainty, specially, the designed system requires both robust stability and robust performance, in generally, only μ-synthesis method can be

281

used to design the robust performance of the system. Therefore, the μ-synthesis method is exploited to design the BTT controller for the supercavitating vehicle. In this paper, a μ-synthesis controller for the BTT supercavitating vehicle is presented. In Section 2, the mathematical model suited for the BTT control of the supercavitating vehicle is established. Robust controllers design based on the BTT strategy, includes μsynthesis controllers for the pitch channel and the yaw–roll channel, are given in Section 3. Section 4 is the simulation research. Finally, in Section 5 conclusions are drawn.

2. Mathematical model of the supercavitating vehicle for the BTT control According to the Newton's law and acknowledgment of hydrodynamics, the 6DOF dynamic model of the supercavitating vehicle is established by assuming that the mass of the supercavitating vehicle is constant, and the additional mass is neglected. The dynamic model of the supercavitating vehicle is in the following form (Anukul, 2005): 2 3 u_ þ qw vr 6 7 m4 v_ þ ur pw 5 ¼ F ð1Þ _ w þ pv uq _ b

where m is the mass of the vehicle, u; v; and w represent the velocity components of the vehicle in body-ﬁxed frame, F is the total forces acted on the vehicle. F consists of the components as follows: F ¼ F R1 þ F R2 þ F E1 þ F E2 þ F c þ F planing þ F g þF pr

ð2Þ

where F R1 ; F R2 ; F E1 ; F E2 ; F c ; F planing ; F g ; and F pr are hydrodynamic forces on rudders, elevators and the cavitator, and the planing force, the gravitational force and the thrust, respectively. Especially, the planing force is caused due to the interaction between the cavity and the aft vehicle, which has relationship with the cavity shape. It depends on the cavity radius and its contraction rate, as well as other parameters. In updated literatures like Mao and Wang (2009, 2011), Balint et al. (2010) and Zhao et al. (2011), they take into account the memory effect (time-delay) which is the function of the travel speed and the vehicle length and is usually micro-second. In this paper, the BTT model and the control strategy is mainly discussed, the memory effect is included in the unmodeled dynamic uncertainty. In the further work, the supercavitating vehicle model with memory effect for the BTT model and the control strategy should be conducted deeply. According to the deﬁnition of the body-ﬁxed frame of the supercavitating vehicle, ox and oy are both on the symmetry plane of the vehicle, therefore, I xz ¼ I zy ¼ 0. ox is coincided with the principal axis of inertial, so I xy ¼ 0. According to the Newton's law for rotation, we have 2 32 3 _ _ _ Ix 0 0 p_ b1 b2 b3 6 7 60 I 07 q r ¼ M ð3Þ y 4 54 q_ 5 þ p 0 0 Iz r_ Ix p Iy q Iz r where M is the total moments acted on the vehicle, included M ¼ M R1 þM R2 þ M E1 þ M E2 þM c þ M planing

ð4Þ

where M R1 ; M R2 ; M E1 ; M E2 ; M c ; and M planing are the moments produced by hydrodynamics on rudders, elevators, the cavitator and the planing force. Let u ¼ vx ; v ¼ vy ; w ¼ vz ; p ¼ wx ; q ¼ wy ; r ¼ wz and for the attack angle and the sideslip angle are very small, we have

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X.-h. Zhao et al. / Ocean Engineering 88 (2014) 280–288

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v2x þ v2z ¼

α ¼ arctan ðvz =vx Þ ðvz =vx Þ, and β ¼ arctan vy = ðvy =vx Þ, then the kinematic model is rewritten as 8 φ_ ¼ wx þ ðwy sin φ þ wz cos φÞ tan θ > < : θ_ ¼ wy cos φ wz sin φ > : _ ϕ ¼ secθðw sin φ þ w cos φÞ y

ð5Þ

Fy ; mg

nz ¼

Fz mg

ð6Þ

Then the observable full state dynamic model (Yufang Zhang et al., 2011; Yichao Zhang et al., 2011) of the supercavitating vehicle is

α_ ¼

z

mg

α

wy

y

δ cz cz

ð9Þ

δe e

z

where ϕ is roll angle, θ is pitch angle and φ is the yaw angle. Normal overload is ny ¼

channel for the supercavitating vehicle is 8 Zδ Zw Z g cos θ cos φ Zα > þ 1 þ vx my wy þ β wx þ vx cmz δcz þ vxδme δe > vx > α_ ¼ vx mα þ > < L L Ix _ y ¼ LI α α þ Iwy wy þ δI cz δcz þ LIδe δe þ Iz w I y wx wz y y y y > > > > : n ¼ 1 ðZ α þ Z w þ Z δ þZ δ Þ Mathematical model of the yaw–roll channel is 8 Y Y Y wz Y wx > wz wx α vgx cos θ sin φ mvδrx δr mvβx β mv β_ ¼ 1 mv wx > > x x > > > > N N

Iz

Iz

Iz

Iz

M > > _ x ¼ MI β β þ MIwz wz þ MIwx wx þ MI δr δr þ I δd δd þ Iy I Iz wy wz > w > x x x x x x > > > :φ _ ¼ wx þ wy sin φ þ wz cos φ tan θ

:

ð10Þ

g cos θ cos φ Fz þ wy þ vx m þ vx

β wx Fy g _ β ¼ wz αwx vx cos θ sin φ mv x

3. Robust controller designs for the supercavitating vehicle based on the BTT strategy

Iy Þ _ z ¼ MI z þ ðIx w I z wx wy z Ix _ y ¼ MI y þ Iz w I y wz wx y

ð7Þ

_ x ¼ MI x þ Iy I Iz wy wz w x x

φ_ ¼ wx þ ðwy sin φ þ wz cos φÞ tan θ ny ¼ F y =mg nz ¼ F z =mg

Ideally, only the cavitator and four ﬁns produce hydrodynamic forces acting on the supercavitating vehicle. However, because of the deformation of the cavity and maneuvering of the vehicle, the planing force generates on the aft of the vehicle. The planing force is related with attack angle and sideslip angle and could be controlled by restricting the two angles. Ignoring a few minor factors and coupling terms, hydrodynamics and moments in Eq. (7) could be expressed as follows: 8 F y ¼ Y β β þ Y wz w z þ Y w x w x þ Y δr δ r > > > > > F z ¼ Z α α þ Z w y w y þ Z δ cz δ c z þ Z δ e δ e > > < M x ¼ M β β þ M wz wz þ M wx wx þM δr δr þM δd δd ð8Þ > > > M ¼ L α þL w þ L δ þ L δ y α w y c e > δ δ y z c e z > > > : M z ¼ N β β þ N w wx þ N w wz þ N δ δr x

z

r

where Y β is the partial derivative of the lateral force with respect to the sideslip angle, and the other Y n has the same deﬁnition. Z α is the partial derivative of the lift with respect to the attack angle, the other Z n has the same deﬁnition. M δd is the partial derivative of the roll torque with respect to differential steering angle. Lα is the partial derivative of the pitch torque with respect to the attack angle. Nβ is the partial derivative of the yaw torque with respect to the sideslip angle. The other Ln , M n , and N n have the same deﬁnition. Noted that, M δr δr is the roll torque which is produced by different wetted areas of upper and lower rudders caused by axis deformation of the cavity as operating the rudders. From Eq. (7), the coupling effect between the channels is caused by roll angle rate wx , where ððI x I y Þ=I z Þwx wy and ðI z I x =I y Þwz wx and ðI y I z =I x Þwy wz are inertial coupling terms, βwx and αwx are kinematic coupling terms. Due to the presence of coupling terms, the model is multi-variable nonlinear model. These coupling terms are considered as disturbance for the supercavitating vehicle because requirements of the attack angle α and the sideslip angle are not too much in the movement process. Therefore, the dynamic model is divided into the pitch channel and the yaw–roll channel to design the robust controller independently. The kinematic and inertial coupling between the channels is treated as interference. The mathematical model of the pitch

3.1. Requirement of the supercavitating vehicle based on the BTT strategy According to the requirement of the BTT missile and the particular characteristics of the supercavitating vehicle, the main anticipant performance criterions of the control system are as follows (Yufang Zhang et al., 2011; Yichao Zhang et al., 2011): (1) The pitch channel and the yaw–roll channel have perfect command tracking performance and dynamic characteristics. (2) The sideslip angle is β r 5 3 . This requirement does not only reduce the coupling effect between the pitch channel and the yaw–roll channel, but also decrease or suppress the lateral planing force on the aft section of the supercavitating vehicle in the turning process. (3) The hull of the supercavitating vehicle is enveloped by the cavity and the operating force is totally produced by control surfaces (the cavitator, elevators, and rudders). Thus the planing force is not expected under the condition of conﬁguration rudders. Furthermore, the value of the planing force is related with the attack angle of the supercavitating vehicle directly, so the requirement of the attack angle at mass center in the maneuver process is jαj r 5 3 . (4) Deﬂection angle of the cavitator is δcz r20 3 , and its deﬂec tion angle rate is δ_ cz r25 3 =s. Deﬂection angles of elevators and rudders are δe r 30 3 , δr r 30 3 , δd r 10 3 , and its deﬂection angle rate of elevators and rudders is δ_ e;r;d r25 3 =s.

3.2. The

μ-synthesis controller for the pitch channel

H1 synthesis control and μ synthesis control are mostly widely used in the robust control theory. The H1 synthesis control could handle the nominal performance and the robust stability. In some situation, robust performance criterion could be transformed into the H1 synthesis problem, whereas it is too conservative to use the H1 synthesis control to settle the robust performance problem of the system, and the μ synthesis control could settle this problem in a better way. The μ synthesis control is also called structured singular value method. The concept of the structured singular value was ﬁrst presented by Doyle in 1982 which attracted much attention by many scholars of control theory as soon as it is proposed. Especially, after further study and develop by Doyle himself and

X.-h. Zhao et al. / Ocean Engineering 88 (2014) 280–288

Packard and so on, the structured singular value theory is established and perfected gradually, which becomes an important constituent in modern robust control theory. In the control system design, robust performance of the system is expected to be designed, which denotes the performance of the controlled system impacted by the uncertainty. The H1 synthesis control does not distinguish between real and complex forms of the uncertainty. The H1 norm fundamentally analyzes with a full Δ matrix, so that all inputs are related to all outputs. This is not the special case when we are dealing with the parametric uncertainty. Being too conservative it has drawbacks. Mainly during design the system performance is sacriﬁced to get better robust stability. As a result the operational envelope is limited in order to avoid conditions where the dynamics might change from the model. A better way to analyze the robust stability of a closed loop system is by using the μ synthesis control. The μ synthesis control could indicate the effect of every parameter on the system performance accurately in the control plant. The block uncertainty could reﬂect independent action of the limitary norm uncertainty of some subsystems in the original system. Therefore, the structured singular value could act as nonconservative tool of the robust stability and the robust performance of the controlled system with uncertainty. Hence, it is suitable to use the μ synthesis control method as analysis tool of the robust stability and the robust performance for the system which is described by parameter uncertainty and bounded norm uncertainty of the multi-point independent, especially for the system whose robust stability and robust performance are both needed to design simultaneity. For the pitch channel, the state vector is picked as xf ¼ h iT h iT α wy , the control vector is picked as u ¼ δcz δe , the h i output vector is picked as y ¼ α wy nz , and neglecting the coupling term, then the state space equation is (

x_ f ¼ Af xf þBf u

ð11Þ

y ¼ C f xf þDf u The input matrix is 2

Zα

6 vx m Af ¼ 4 L α Iy

Zw

1 þ vx my Lw y Iy

3 7 5;

2

Z δc

3

z

6 vx m B f ¼ 4 Lδ c z

Iy

Z δe vx m 7 Lδ e Iy

5

Fig. 1. The μ-synthesis scheme of the pitch channel.

283

The output matrix is 2 3 2 1 0 0 6 7 6 0 1 6 7 6 Cf ¼ 4 ; Df ¼ 4 0 Z δc Z wy 5 Z α

mg

z

mg

mg

0

3

7 0 7 5

Z δe mg

There is much model uncertainty between the actual dynamics model and the nominal model. For the pitch dynamic of the supercavitating vehicle based on the BTT strategy, the uncertainty and the disturbance of the system mainly include following aspects: (1) Coefﬁcients in the nominal model Eq. (11) are obtained by the partial derivative of the hydrodynamic force with respect to the corresponding variable, in which the high-order terms and coupling terms are neglected. On the other side, the memory effect (time-delay) of the acting force is neglected which should be taken into account. So there is unmodeled dynamic uncertainty between the nominal model and the actual model. (2) The mass of the supercavitating vehicle and the travel speed are impossible to remain unchanged as the vehicle is running. Therefore, there are errors between the actual model and the nominal model. (3) The disturbances in the system mainly consist of the cavity deformation, the planing force on the aft section, the sensor measurement noise and the actuator deﬂection etc.

Considering the uncertainty and disturbances discussed above, robust stability and robust performance are demanded for the controller. For the system with the parameter uncertainty and the multi-point independent bounded uncertainty, in particular, when the designed system requires both robust stability and robust performance, the μ-synthesis method is suitable to be adopted. Here, the μ-synthesis control is exploited to design the BTT controller for the supercavitating vehicle. The structure of the μ-synthesis robust controller of the pitch channel for the supercavitating vehicle is shown in Fig. 1. Where, GBTTP represents a nominal model of the pitch channel of the supercavitating vehicle, K μ is the robust controller of which the input is the deﬂection angle command δcz C of the cavitator in the pitch direction and the deﬂection angle command δeC of elevators, the output of which is the attack angle α, the pitch angle rate wy and the actual deﬂection angle δcz of the cavitator, and the actual deﬂection angle δe of elevators and the lateral load nz . The actual model of the pitch channel is an addition of the input multiplicative uncertainty and the nominal model. ACS and AES represent the servo model of the cavitator and elevators, the output of which is the actual deﬂection angle δcz , δe and the deﬂection angle rate δ_ cz and δ_ e . W ideal denotes the ideal instruction tracking model, and W noise is the weighting function of noise, and W act is the weighting function of the deﬂection angle and the deﬂection angle rate of the servo, and W perf is the weighting function of the performance index, and W unc is the weighting function of the uncertainty (Simone, 1999). W unc weights the unmodeled dynamic and the parameter uncertainty between the actual model and the nominal model. The parameter uncertainty is from lift coefﬁcient and drag coefﬁcient of the cavitator and elevators. The unmodeled dynamic is usually acting at high frequency band, and the parameter uncertainty is expressed as low frequency perturbance. Therefore W unc should have larger gain at high frequency to suppress vibration of the control input, and its amplitude is larger than maximum singular value of model error caused by the parameter uncertainty. Without increasing the orders of the system, it is a priority to select a ﬁrst order function.

284

X.-h. Zhao et al. / Ocean Engineering 88 (2014) 280–288

Then W unc is " sþ5 # 0 3s þ 140 W unc ¼ 25 0 2ssþþ175

ð12Þ

The Bode diagram is shown in Fig. 2. W perf has a large amplitude at low frequency to guarantee tracking ability of the low frequency command and suppressing the low frequency disturbance. Usually, there is no punish function, then W perf is " # s þ 10 0 1:80:1s þ1 W perf ¼ ð13Þ s þ 10 0 1:80:1s þ1 The measurement noise input weighting function W noise reﬂects the inﬂuence caused by the measurement error. The weighting function is chosen as a high-pass ﬁlter form. In order to decrease the orders of the controller, W noise is usually chosen as a constant matrix here 2 3 0:1 0 0 0 0 6 0 0:1 0 0 0 7 6 7 6 7 0 0:1 0 0 7 W noise ¼ 6 ð14Þ 6 0 7 6 7 0 0 0:1 0 5 4 0 0 0 0 0 0:1 In order to reﬂect the restrain of the deﬂection angle and the rate of the actuators, the input weighting function W act is chosen as a constant. 2 3 57:3 0 0 0 20 6 7 6 0 57:3 0 0 7 20 6 7 W act ¼ 6 ð15Þ 7 6 0 0 57:3 0 7 25 4 5 0 0 0 57:3 25

the controller is too high to be realized in engineering, using balanced truncated model reduction method, reduced-order model of K μ is calculated. 3.3. The

μ-synthesis controller for the yaw–roll channel

The state-space model of the yaw–roll channel for the supercavitating vehicle is ( x_ l ¼ Al xl þBl u ð17Þ yl ¼ C l xl þ Dl u h xl ¼ β iT wz φ .

where h β wx

wx

The state matrix is 2 Yβ Y Y wx mv 1 mvwzx x 6 mvx 6 Mβ M wz M wx 6 Ix Ix Al ¼ 6 I x 6 Nβ N wz N wx 4 I I I 2

1

60 6 Cl ¼ 6 40 0

z

z

z

0

1

0

0

0

1

0

0 0

1 0

0

3

iT

φ ,

wz

h u ¼ δr

3

2

7 7 0 7 7 7 0 5 0

6 6 6 Bl ¼ 6 6 4

vgx

i

δd ,

Y

mvδrx

0

M δr Ix N δr Iz

Mδ d Ix

0

0

yl ¼

3

7 7 7 7; 7 0 5

07 7 7: 05 1

The design method and design steps of the μ-synthesis controller for the yaw–roll channel is similar with the pitch channel. Input commands of the yaw–roll channel are the roll angle command φc and the angle of sideslip command βcom . The structure diagram of the μ-synthesis controller of the yaw–roll channel is analogous to Fig. 1, and the corresponding weighting function is chosen as follows:

According to the technology requirements of the BTT control of the supercavitating vehicle, the ideal response model of overload command of the pitch direction and the attack angle is selected as 2 3 25 0 s2 þ 7s þ 25 5 ð16Þ W ideal ¼ 4 25 0 s2 þ 7s þ 25

W unc of the yaw–roll channel is still chosen as " sþ5 # 0 3s þ 140 W unc ¼ 25 0 2ssþþ175

Under the premise of knowing the pitch dynamic model of the supercavitating vehicle, together with weighting functions selected above, the structure shown in Fig. 1 could be transformed into a standard μ synthesis problem. The μ-synthesis function in MATLAB is exploited to obtain the controllerK μ . Since the order of

The ideal instruction tracking model for the sideslip angle command and the roll angle command is 2 3 16 0 s2 þ 5:6s þ 16 4 5 ð19Þ W ideal ¼ 6 0 sþ6

10

W perf is

5

W perf ¼

Relative error on nominal supercavitating model as a function of frequency

" 0:4s þ 5

0

s þ 0:1

# ð20Þ

s þ 10 0:05s þ 20

0

ð18Þ

Magnitude (dB)

0

To reﬂect the restrictions of the deﬂection angle and the rate of the actuator, like the pitch channel, the weighting function of the actuator is

-5 -10

2

-15

6 6 0 6 W act ¼ 6 6 0 4 0

-20 -25 -30 -1 10

57:3 20

10

0

10

1

10

2

10

3

10

0

0

57:3 20

0

0

57:3 25

0

0

0

3

7 0 7 7 7 0 7 5

ð21Þ

57:3 25

4

Frequency (rad/sec) Fig. 2. Frequency characteristics of weighting functions to the model uncertainty.

A 6-inputs, 34-states and 2-outputs μ-synthesis controller K μL can be obtained through MATLAB robust control toolbox. Similarly,

X.-h. Zhao et al. / Ocean Engineering 88 (2014) 280–288

as the order of the controller is too high to be realized in projects. Reducing order process is conducted to get the ﬁnal controller.

6 5

az/(m/s2)

4. Simulation research In order to test the effectiveness and robustness to disturbances of the μ-controller designed for the two channels, simulations are performed in Matlab/Simulink environment. All the simulations are performed by considering the full set of the vehicle conﬁguration parameter variables. Same system parameters as in Seong (2007) and Dzielski and Kurdila (2003) are used in the nominal simulations in Table 1. Putting all parameters into the nominal model Eq. (11) of the pitch channel, we get 1:18 1:03 0:79 0:39 Af ¼ ; Bf ¼ ; 1:75 1:73 207:31 58:31 2 3 1 0 0 0 21:1 T 6 1 7 ð22Þ Cf ¼ 4 0 5; Df ¼ 0 0 5:9 0:12 0:1

4 3 2 1 0

0

1

2

3

4

5

t/s Fig. 4. Pitch overload response curve. 15 10 5 0

0

1

2

3

4

5

0

1

2

3

4

5

30

The servo model of the simulation is selected as Acs ¼ Aes ¼

285

20

15 s þ 15

10

ð23Þ

0

The amplitude of overload command in the pitch direction is 5 g and the attack angle command is 01. Simulation results are shown in Figs. 3–5. We can conclude that the robust controller is effective and can satisfy the technical indexes of the autopilot based on the BTT strategy of the supercavitating vehicle proposed above. In order to verify the robustness of the controller, the uncertainty of the hydrodynamics coefﬁcient and 750% disturbance of the parameters are added to the simulation, and the overload response curve is shown in Fig. 6. It can be found that the controlled system possesses stronger robustness.

Fig. 5. Control input of the pitch channel.

Table 1 Simulation parameters. Notation

Description

Value and unit

g m ne Rn R L XG Xf Zf V σ Cx0

Gravity acceleration Vehicle mass Efﬁciency of ﬁns Cavitator radius Vehicle radius Vehicle length Vehicle mass center location Fin's root location Fin span length Vehicle speed Cavitation number Drag coefﬁcient of cavitator

9.81 m/s2 150 kg 0.5 0.04 m 0.1 m 4m 2.31 m 3.87 m 0.2 m 85 m/s 0.03 0.82

Fig. 6. Response curves of the pitch overload command with system uncertainty. 0.6

β /deg

0.4

0.2

0

-0.2

0

1

2

3

4

5

t/s 2.5

Fig. 7. Response curve of the sideslip angle.

2

All parameters are put into Eq. (17), and then the mathematic model of the yaw–roll channel is

1.5 1

2

0.5 0

0

1

2

3

4

5

0:39 6 1:89 6 Al ¼ 6 4 58:31 0

Fig. 3. Attack angle response curve.

0

1:05

6:01

1:5

0:14

0:86

1

0

0:11

3

0 7 7 7; 0 5 0

2

0:4 6 127:8 6 Bl ¼ 6 4 58:3 0

0

3

2555:2 7 7 7: 0 5 0 ð24Þ

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X.-h. Zhao et al. / Ocean Engineering 88 (2014) 280–288

Let the autopilot track a step signal whose amplitude is 57.31, and the sideslip angle command is selected as 01. The simulation results are shown in Figs. 7–11, which proves that the controller has good tracking performance.

8

r/deg/s

6

Fig. 12. The roll angle response curves with system uncertainty.

4 2 0 -2

0

1

2

3

4

5

t/s Fig. 8. Response curve of the yaw angle rate.

60 50

φ /deg

40 30

Fig. 13. The sideslip angle response with system uncertainty. 20 10 0

0

1

2

3

4

5

4

5

t/s Fig. 9. Response curve of the roll angle.

p/(deg/s)

60

40

20

0

0

1

2

3

To verify the robustness of the yaw–roll channel controller, 750% variation in the hydrodynamic coefﬁcient of the nominal values is included in the simulation. The roll angle command is a step signal with 57.31 amplitude and the sideslip angle command is 01. The simulation results are shown in Figs. 12 and 13. We can observe that the controller also has better robustness. Finally, the robust controller designed for two channels and Eq. (7) are used to perform the 6DOF simulation of the BTT supercavitating vehicle. Both commands of the attack angle and the sideslip angle are 01. The overload command in the pitch is selected as 5 g and the roll angle command is 57.31. Using the ﬁrst roll and then the pitch strategy, the roll angle command is added ﬁrst, 1 s later, the pitch overload command is added. The simulation curve is shown in Fig. 14 which presents that the controller designed is effective.

t/s

5. Conclusions

Fig. 10. Response curve of the roll angle rate.

δ r/deg

1

0

-1

0

1

2

3

4

5

3

4

5

t/s

δ d/deg

0.2

0

-0.2

0

1

2

t/s Fig. 11. Input signal of the yaw–roll channel.

To ensure the stability of the cavity and the supercavitating vehicle in the process of turning, the BTT strategy is proposed in this paper. Using modeling theory of rigid dynamics, full state observable dynamics model is established for the BTT supercavitating vehicle. The kinematics coupling terms and the inertial coupling terms are considered as interference accounting for coupling circumstance of each channel. The supercavitating vehicle dynamic is designed to include a pitch channel and a yaw–roll channel for which the μ-synthesis robust controller is designed respectively. For the purpose of testing the robustness of the system, simulation is accomplished for the robust controller of the pitch channel and the yaw–roll channel. In the end, the ﬁrst roll and then pitch tactics is exploited. The 6DOF nonlinear simulation is carried out based on the BTT control supercavitating vehicle. The 6DOF nonlinear simulation veriﬁes that the designed controller can satisfy the system's control requirement excellently with a high tracking precision.

4

1

2

0.5

β/deg

α/deg

X.-h. Zhao et al. / Ocean Engineering 88 (2014) 280–288

0 -2

0

1

2

3

4

0 -0.5

5

287

0

1

2

t/s

4

5

100

p/(deg/s)

q/(deg/s)

500

0

-500

3

t/s

0

1

2

3

4

50

0

5

0

1

2

3

t/s

4

5

t/s

10

r/deg/s

az /(10m/s2)

10

0

-10

0

1

2

3

4

0 -10

5

0

1

2

3

t/s

4

5

t/s

φ /deg

100

50

0

0

1

2

3

4

5

t/s Fig. 14. 6DOF simulation curves of the supercavitating vehicle.

Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grant 50909027 and the Post Doctor Foundation of Heilongjiang Province under grant 01106948.

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