Dynamical analysis and robust control for dive plane of supercavitating vehicles

Applied Ocean Research 84 (2019) 259–267

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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor

Dynamical analysis and robust control for dive plane of supercavitating vehicles

T

⁎

Bui Duc Hong Phuca, Sam-Sang Youb, , Natwar Singh Rathorec, Hwan-Seong Kimd a

School of Intelligent Mechatronics Engineering, Sejong University, Seoul, Republic of Korea Division of Mechanical Engineering, Korea Maritime and Ocean University, Busan, Republic of Korea c Department of Electrical Engineering, G. H. Raisoni University, Amravati, India d Department of Logistics Engineering, Korea Maritime and Ocean University, Busan, Republic of Korea b

A R T I C LE I N FO

A B S T R A C T

Keywords: Supercavitation Underwater vehicles Dive plane H∞ control synthesis Modified PID control Loop shaping Planing force Disturbance Uncertainty Anti-windup

The high-speed supercavitating vehicle (HSSV) utilizes advanced technology that enables an underwater vehicle to reach its unprecedented high speed. The vertical motion control of the HSSV is challenging problem because of its complex dynamics with nonlinear planing force, parametric uncertainties, external disturbances, actuator saturation, and sensor noises. This paper deals with dynamical analysis and a robust H∞ loop-shaping synthesis with modified PID (proportional-integral-derivative) algorithm to control the dive plane maneuver of the HSSV. Typically, the control scheme has the low order structure and provides robustness against dynamic uncertainties, which can be implemented using the bilinear matrix inequality (BMI) optimization of an equivalent Schur formula. Simulation results show that the controlled vehicle system provides good performance and high robustness against uncertainties, ensuring no-overshoot and fast in time-domain responses. In addition, the control algorithm can decouple the dynamic interactions in the multi-input multi-output (MIMO) system, overcoming parametric uncertainty, planing force, and actuator saturation while minimizing the effect of the strong external disturbances and measurement noises.

1. Introduction

deflection angle, then it generates supercavitation technology providing lift force. The lift forces provided by the cavitator and control surface of the elevators and immersion planing force share the responsibility of equilibrium problem [2]. Depending on the shape of the cavitator and the size and immersion of the control surfaces, the vehicle body may be inherently stable or unstable [1]. The dynamic modelling of the HSSV is a very important step for its control mechanism design. Some models that have been proposed so far include the four-state rigid-body model used to study pitch-plane dynamics [2,3], the twelve-state rigid-body models [1], the vehicle models that reflect the effect of time delay [4,5], the numerical model incorporating structural elasticity [6], and the dynamical models that account for the non-cylindrical cavity effects [7]. The dive-plane model proposed by Dzielski and Kurdila [8] is the common for controller design purpose due to its simplification. In this model, the time-delay effects are not taken into account and the planing force is calculated based on the Logvinovich model [6]. Dzielski and Kurdila [8] also proposed a linear state-feedback scheme to control the HSSV but their basic controller could not robustly stabilize the underwater vehicle against uncertainty. Some advanced methods have been proposed for

Supercavitating vessel is a type of underwater vehicle that can reach extremely high speed by exploiting supercavitation technology. This emerging technology uses the proper design of a cavitator attached to the vehicle nose to create a large water vapor cavity that envelopes the vehicle body to eliminate skin friction drags. The technology could provide potential visions for underwater vehicle to travel through fluid at speed of sound underwater. However, the vehicle dynamics of supercavitating vehicles are significantly different from those of conventional underwater vehicles. The vehicle dynamics are highly coupled nonlinear with model uncertainties and external disturbances. In addition, the planing force, dynamic coupling, actuator saturation, and sensor noises are main factors that especially challenge underwater cruise control of the vehicle. The main actuators of an HSSV include the cavitator in the front and the fins in the aft part. The cavitator is intended to generate and maintain the cavity, together with the fins; it provides control mechanism for the vehicle direction and stability [1]. A typical dive plane configuration of the proposed HSSV is given in Fig. 1. At very high speeds, the cavitator contacts water at a certain

⁎

Corresponding author. E-mail address: [email protected] (S.-S. You).

https://doi.org/10.1016/j.apor.2019.01.022 Received 30 October 2018; Received in revised form 30 December 2018; Accepted 21 January 2019 0141-1187/ © 2019 Published by Elsevier Ltd.

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Fig. 1. Vehicle configuration with coordinate systems.

located at the center of gravity of the vehicle. In this study, the forward speed (V ) is assumed to be constant and defined along the longitudinal axis. The four states of the vehicle motion (heave and pitch) include the vertical position z, the vertical speed w, the pitch angle θ , and the pitch rate q. Two manipulated variables include the elevator deflection δe and the cavitator deflection δc . In this article, the controlled variables are vertical position z and pitch angle θ . The state-space representation in the body-fixed frame describing vertical dynamics is defined as follows:

the control of HSSVs, including the receding horizon control [9], the switch control [10], the adaptive control [11,12], the sliding mode control [13–19], the LPV control [15,20,21], the backstepping control [22,23], the robust control [21,24], the μ control [25], the model predictive control [26], the robust linear-time-invariant output-feedback control [27], and so on. Among the proposed control algorithms, robust controllers are most important to cruise the vehicle due to their ability to eliminate disturbances, noises, and uncertainties. However, those proposed controllers are often in high-order, which is hard to deploy into control hardware for real time implementation. For instance, the order of the proposed controller in [27] is up to 11, which is likely hard to be implemented due to its high order. Meanwhile, the other mentioned controllers are mostly susceptible to disturbances and uncertainties due to the lack of robustness. Some advanced approaches such as adaptive and sliding mode controllers have the ability to deal with the exogenous disturbances, inherent measurement noises and modelling uncertainties on certain levels. However, they cannot effectively cope with parameter perturbations and unmodelled dynamics in all frequencies. In this article, a robust H∞ control with modified PID scheme is proposed to robustly control the HSSV. This control algorithm provides the robustness with performance that can cope with external disturbances, measurement noises, and model uncertainties while preserving a low order control structure. Furthermore, an anti-windup compensator is also integrated into the control scheme to deal with actuator saturation. The saturation is the physical limits of deflection angles of control surfaces. It may occur when there is a sudden change in the vehicle direction. It is noted that saturation can cause high overshoots known as the windup phenomenon. The simulation result shows that the proposed controller can offer the maneuvering with fast response. It also shows the ability to attenuate the external disturbance and performs the robustness in case of high-amplitude planing forces and some fins saturation. Finally, because of its attractive features such as its low-order control scheme for real implementation and high robustness with performance against uncertainties, the proposed approach could provide a robust controller for new generations of uncertain dynamical system of HSSVs.

z˙ = w − Vθ θ˙ = q 7

17L 36

⎡ 9 ⎢ ⎢ 17L ⎣ 36

11R2

⎡0 V⎢ ⎢0 ⎣

7 9 ⎤ w ⎡ ⎤ 17L ⎥ ⎣ q ⎦ ⎥ 36 ⎦

60

+

−n ⎤ w˙ ⎡ ⎡ ⎤ = CV ⎢ mL − n ⎥ ⎣ q˙ ⎦ ⎢ ⎣m ⎦

⎥ 113L2 405

−n

⎡ + CV 2 ⎢ mL −n ⎢ ⎣m

−n m ⎤ w −n ⎥ ⎡ q ⎤+ ⎣ ⎦ mL ⎥ ⎦

1 mL ⎤ ⎡ δe ⎤

g 1 + ⎡ ⎤ + Fp ⎡ ⎤ ⎥ L 0 ⎥ δ ⎣ ⎦ 0 ⎥⎢ ⎣ ⎦ c ⎦⎣ ⎦

(1)

where L is the vehicle length, R the supercavitating body radius, V the forward speed, n the fin effectiveness ratio with respect to the cavitator, m the density ratio of the body to water, g the gravity acceleration, and Fp the planing force. C is further given by the following equation:

C = 0.5Cx

Rn2 , Cx = Cx 0 (1 + σ ) R2

(2)

where Cx0 is the cavitator lift force coefficient, σ the cavity number and Rn the cavitator radius. The planning force Fp in Eq. (1) is the nonlinear factor and can be calculated as follows: 2

Rc − R ⎛ ⎞ ⎤ 1 + h′ Fp = −πρRc2 V 2 ⎡ ⎢1 − h′R + R c − R ⎥ 1 + 2h′ α ⎝ ⎠⎦ ⎣ ⎜

⎟

(3)

where ρ is the fluid density, h’ the immersion depth, α the angle of attack. In addition, h’ and α are noncontinuous functions that relate to the vertical speed as expressed in the following equations:

2. Vehicle model and dynamical behavior

h′ = 2.1. Mathematical model Aiming at designing the robust controller for HSSVs, this paper focuses on the dynamical model proposed by Dzielski and Kurdila [8] which has four states to describe the dive-plane dynamics. This model is well defined and is suitable for controller design purpose. As illustrated in Fig. 1, the earth-fixed frame is denoted as (OE − XE , ZE ) and the body-fixed frame is (Ob − x , z ). The origin Ob of the body frame is

⎧0

if

⎨ L |w| − ⎩ RV w − R˙ c V ⎨ w + R˙ c

⎧ α=

⎩

V

Rc − R R

Rc − R R

>

L |w| RV

otherwise

if

w V

(4)

>0

otherwise

(5)

In Eq. (3), Rc is the cavity radius and its contraction rate R˙ c can be calculated using the following equations: 260

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Table 1 Model parameters of the vehicle system. Parameter

Value

Cx0 g L m n Rn R V σ

[0.47,1] 9.81 1.18 2 0.5 0.0191 0.0508 [63-112] [0.0198, 0.0368]

m/s2 m

m m m/s

1+σ K2 σ

(6)

4.5σ ⎞ 40/17 1 − ⎛1 − K1 1 + σ⎠ ⎝

(7)

R c = Rn 0.82

K2 =

Unit

L

K1 = Rn

(

1.92 σ

)

Fig. 3. Phase portrait diagram of the vehicle model.

−1

−3

(8)

For dynamical analysis and control synthesis, the related vehicle parameters are described in Table 1.

system stability. Fig. 3 is the phase portrait of the two basic states, the vertical speed w, and the pitching rate q. The phase plane trajectories show that the equilibrium point is unstable focus. Any initial values of the state variables will lead the system states away from them to infinity. Through the nonlinear analyses, it strongly indicates that the system needs an effective controller to keep the vehicle stable while producing precise vehicle maneuvering.

2.2. Dynamical behavior

3. Loop-shaping framework with modified PID synthesis

In this model, the planing force is the nonlinear factor affecting to the system dynamics. The planing force is often modelled as too simple and therefore inaccurate to describe the dynamical behavior. In general, this force depends on the vertical velocity and is a monotonically increasing function of the immersion h’ and attack angle α . Fig. 2 shows the dynamic relation between the normalized planing force and vertical speed w. It can be observed that the planing force is noncontinuous and only occurs if the magnitude of the vertical velocity is greater than wo, where the vehicle aft end pierces the supercavity. This force has a tendency of pushing the vehicle aft end toward the center of the supercavity bubble. When alternatively occurring in the two opposite directions, the force will cause a tail-slap phenomenon, which will eventually lead to instability unless a proper active controller is applied. The nonlinear behavior of the HSSV is then analyzed to describe the

3.1. Modified PID structure

−1.176 R˙ c =

(0.82 ) V (1 − − 3) K ( 1+σ σ 2

4.5σ 1+σ

)K

23/17 1

1.92 σ

(9)

The conventional PID scheme is well-defined and widely used but lack of robustness against uncertainties. In this paper, the following structure is introduced into H∞ loop shaping framework to gain robustness,

KPIDij (s ) = k Pij + k Iij

1 s + k Dij s ts + 1

(10)

where KPIDij (s ) is the ijth element of the transfer function matrix KPID (s ) , k Pij the proportional gain of the ijth element, k Iij the integral gain of the ijth element, k Dij the derivative gain of the ijth element, and t is the derivative action time constant. Then, Eq. (10) can be rewritten as

KPIDij (s ) = ≜

(k Iij t −

k Dij t

) s + k Iij

ts 2 + s

K Dij s + K Iij ts 2 + s

+ k Pij +

k Dij t

+ K Pij

(11)

k Dij

k Dij

where K Pij = k Pij + t , K Iij = k Iij , K Dij = k Iij t − t The current structure of the PID controller for the HSSV with two inputs and two outputs can be written in the follow form: KD s + KI

11 + K P11 ⎡ 112 ts + s KPID (s ) = ⎢ K s + K ⎢ D212 I21 + K P21 ⎣ ts + s

K D12 s + K I12 ts 2 + s K D22 s + K I22 ts 2 + s

+ K P12 ⎤ ⎥ + K P22 ⎥ ⎦

(12)

Let τ = 1/ t , then a partial fraction expansion of KPID (s ) is derived as

KPID (s ) = DK +

BK 1 B + K2 s s+τ

(13)

where the control gains are given by

k P11 + τk D11 k P12 + τk D12 ⎤ DK = KP = ⎡ ⎢ k P21 + τk D21 k P22 + τk D22 ⎥ ⎦ ⎣

Fig. 2. Nonlinear characteristics of the planing force. 261

(14)

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γ ≥ T⎡d ⎤

z ⎤ ⎢ r ⎥ →⎡ ⎣u ⎦ ⎣n ⎦ ∞

=

(I − W2 GKPID )−1W2 ⎡ Gs ⎢ GKPID F − T r ⎢ ⎢ −1 −1 W −1 K PID F 1 ⎢ (I − KPID W2 G ) W1 ⎢K G ⎣ PID s

(I − W2 GKPID )−1W2 ⎤ ⎥ GKPID ⎥ ⎥ KPID ⎥ ⎥ ⎦

∞

Fig. 4. Loop shaping setup for tuning PID parameters.

k I k I12 ⎤ BK 1 = KI = ⎡ 11 ⎢ k I21 k I22 ⎥ ⎦ ⎣

BK 2 = KD −

− τ 2k D11 − τ 2k D12 ⎤ KI =⎡ 2 2 ⎢ ⎥ τ ⎣− τ k D21 − τ k D22 ⎦

(I − Gˆs KPID )−1Gˆs ⎡ Gˆs W1 ⎢ KPID F − T r ⎢ = ⎢ −1 −1 ˆ − I K G W1−1 KPID F ( PID s ) W1 ⎢ ⎢ K Gˆ W ⎣ PID s 1

(15)

BK 1 ⎤ A B BK 2 ⎥ = ⎡ K K ⎤ CK DK ⎦ ⎢ ⎥ ⎥ ⎣ DK ⎦

∞

(20) (16)

where Gˆs = W2 G1, Gs = W2 GW1 = Ms−1 Ns , Ms and Ns are nominator and denominator normalized coprime factors realized in Fig. 6. There are some types of uncertainty model to integrate uncertainties in the robust control synthesis. In this study, the coprime factor uncertainty model is exploited due to its ability to cover a variety of uncertainty types at different frequencies without prior uncertainty information. The obtained controller will robustly stabilize the perturbed plant for a very general uncertainty. The robust stabilization problem is to stabilize the set of perturbed plants:

Finally, from partial fraction expansion in Eq. (13), a minimal statespace realization of KPID (s ) can be obtained as follows:

⎡0 0 KPID (s ) = ⎢ 0 −τI ⎢ ⎣I I

(I − Gˆs KPID )−1Gˆs KPID ⎤ ⎥ ⎥ ⎥ KPID ⎥ ⎥ ⎦

(17)

3.2. PID control in the H∞ loop shaping framework

Gp (s ) = (Ms + ΔM )−1 (Ns + ΔN ),

H∞ loop shaping procedure proposed by McFarlane and Glover [28] is an efficient method to design robust controllers. In this framework, the designers can shape the open-loop plant G with the pre-compensator W1 and post-compensator W2 as shown in Fig. 4 to get the shaped loop Gs with desired loop gains at specified frequencies. Typically, the loop gains have to be large at low frequencies for good disturbance rejection at both the input and output of the plant, and small at high frequencies for noise rejection. In addition, the desired shapes should approximately -20 dB/decade roll-off around the crossover frequency to achieve the desired robust stability, gain and phase margins, overshoot, and damping. Following the method of Genc [29], the controller K∞ is constructed by

where ΔM and ΔN denote bounded uncertainty functions in the nominator and denominator coprime factors. The inverse of γ is the so-called robust stability margin ε, and this is the indicator of the achieved robust stability of the shaped loop.

K∞ = W1−1 KPID

− ΔM ] ||∞ ≤ ε

(21)

3.3. Control synthesis and optimization From the formulations described above, the optimization problem in Eq. (20) will be solved using a bilinear matrix inequality (BMI) based on state-space realizations of the component matrices. At first, a minimal state-space realization of the transfer matrix T in Eq. (20) is obtained as

T⎡d ⎤

z ⎤ ⎢ r ⎥ →⎡ ⎣u ⎦ ⎣n ⎦

(18)

Hence, the final controller K has a desired PID structure, since

K = W1 K∞ W2 = KPID W2

||[ΔN

⎡ Ai ⎢0 ⎢0 ⎢ ⎢0 =⎢ 0 ⎢ ⎢0 ⎢ Ci ⎢ ⎣ 0

(19)

The initial simulation shows that the controller in the above configuration cannot meet both time-domain constraints and robust gain margin for the current system, because the current system is highly unstable while the controller has only one degree. Therefore, a twodegree of freedom (2DOF) control strategy is proposed including a feedback component as mentioned in Eq (19), and a set-point filter. As illustrated in Fig. 5, the set-point filter F is to improve settling times while decreases overshoots and the dynamic interactions between system channels. The filter guarantees that the performance of the feedback system matches as closely as possible that of the reference model Tr. It is interesting to note that in this new form, W1 and W1−1 only affect the closed loop indirectly though KPID since KPID is formed with respect to the shaped plant Gs that includes W1 and W2. The optimization problem is to minimize the H∞ norm of the transfer function matrix T from inputs disturbance d, reference r, and noise n to the regulated output z and control effort u by designing the controller KPID and the filter F to obtain a desired value γ, in which

AT BT ⎤ =⎡ C ⎢ ⎣ T DT ⎥ ⎦

Bi DK C 0 Bi CK Bi DK CF 0 0 Bi DK DF Bi DK ⎤ A1 0 0 0 0 B1 0 0 ⎥ BK DF BK ⎥ 0 AK BK CF BK C 0 0 ⎥ BF 0 0 ⎥ AF 0 0 0 0 BTr 0 0 ⎥ ATr 0 0 0 0 ⎥ BC1 BCK BDK CF 0 A + BDK C BD1 BDK DF BDK ⎥ Di DK C 0 Di CK Di DK CF 0 0 Di DK DF Di DK ⎥ ⎥ 0 −DTr 0 ⎦ C 0 0 0 −CTr (22)

where the relevant matrices are given in the following state-space forms:

262

A B⎤ Gˆs (s ) = W2 G = ⎡ ⎣C D⎦

(23)

A B W1 (s ) = ⎡ 1 1 ⎤ C ⎢ ⎦ ⎣ 1 D1⎥

(24)

Ai Bi ⎤ W1−1 (s ) = ⎡ ⎢ Ci Di ⎥ ⎦ ⎣

(25)

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Fig. 5. 2DOF control structure of H∞ formulation. T T ⎛ PAT + AT P PBT ⎞ + ⎛ CT ⎞ γ −1I C D < 0 ( T T) ⎜ ⎟ ⎜ T⎟ T BT P − γI ⎠ ⎝ DT ⎠ ⎝

(30)

Then the optimization problem in Eq. (20) can be written in the equivalent form of Schur complement formula as min γ (P , AK , BK , CK , DK , AF , BF , CF , DF ) such that T T ⎛ PAT + AT P PBT CT ⎞ T ⎜ BT P − γI DTT ⎟ < 0, ⎜ C DT − γI ⎟⎠ T ⎝

(26)

A B Tr (s ) = ⎡ Tr Tr ⎤ ⎢ CTr DTr ⎥ ⎦ ⎣

(27)

⎧|| T (s ) ||∞ ≤ γ ⎨ ⎩|| P (s ) ||∞ ≤ β

According to the bounded real lemma, for any γ > 0, the norm condition || T ||∞ < γ in Eq. (20) holds if and only if there exists a symmetric positive definite matrix X such that T T T ⎛ AT X + XAT + CT CT XBT + CT DT ⎞ < 0 ⎜ ⎟ T T T BT X + DT CT DT DT − γ 2I ⎠ ⎝

(31)

where P is a symmetric positive matrix called the Lyapunov function. This is a BMI optimization problem. The local optimal solution is found through alternately minimizing the optimal cost γ with respect to P with controller parameters fixed and vice versa [29]. Finally, an antiwindup compensator is also introduced in the control scheme to deal with actuator saturation. The control framework with the compensator J is constructed as in Fig. 7. Then the idea can be justified by the multi-objective H∞ control problem [30]:

Fig. 6. Closed-loop control system with coprime plant perturbation.

AF BF ⎤ F (s ) = ⎡ ⎢ ⎦ ⎣ CF DF ⎥

P>0

(32)

where P(s) is the equal robust plant including the perturbed plant with controller KPID and the anti-windup compensator J; β is the second optimal cost emphasizing to meet system’s performances. Two optimization problems are solved simultaneously to obtain the controller and the compensator in the same process that guarantees the robustness with performance for the controlled system.

(28)

In the sequence of solving the inequality, Eq. (28) is rearranged as T T ⎛ XAT + AT X XBT ⎞ + ⎛ CT ⎞ I C D < 0 T) ⎜ ⎟ ⎜ T⎟ ( T T 2 BT X − γ I ⎠ ⎝ DT ⎠ ⎝

1

4. Simulation results and discussions In order to simulate the dive motions of the HSSV, the manipulated variables are chosen as the fin deflection δe and the cavitator deflection δc . The controlled variables are the vertical position z and the pitch angle θ . In the practical perspective of the HSSV maneuver, the

(29)

Aiming at realization of the Schur complement formula, let P = γ– X and multiply Eq. (29) by γ–1 to yield

Fig. 7. Anti-windup control structure. 263

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transient responses must be fast enough with a short rise and a fast settling without substantial overshoots. Special care must be taken to ensure that the high robustness with stability should be maintained even if actuator saturation, disturbance, and planing force occur simultaneously. At such high speed, a slight overshoot, slow response or disturbance can make serious problems such as collision or target missing. Besides, the active controller has to minimize the dynamic coupling between inputs and outputs of the MIMO system for more precise control. Considering all design requirements stated in [24], the strict design criteria are described as follows:

• The stability margin is expected to be in ε > 0.25. • The effects of external disturbance and measurement noises are reduced by at least 50%. • The transient responses for step-type inputs have to be in settling • •

times less than 0.5 s, overshoots less than 5%, and zero steady state errors. The interactions between channels are less than 20% at rising time. No overshoots are guaranteed in case of actuator saturations.

Practical implementation shows that ε > 0.25 is acceptable for stability margin, which means that the compensated system will be both satisfying robust stability and robust performance simultaneously. In this simulation, the robust stability margin is achieved a satisfied value at 0.63. The state space representations of the reference system Tr and compensators W1 and W2 that enable the system designer to achieve the desired loop shaping are chosen as follows:

0 ⎤ ⎡4 ⎡−32 −16 0 0 0 ⎥ −16 0 0 , BTr = ⎢ ATr = ⎢ 0 −28 −15.25⎥ ⎢0 ⎢ 0 ⎢ ⎢ 0 16 0 ⎥ ⎦ ⎣ 0 ⎣0 0 4 0 0 0 0 ⎤, DTr = ⎡ ⎤ CTr = ⎡ ⎣0 0⎦ ⎣ 0 0 0 3.06 ⎦

Fig. 8. Frequency responses of the nominal and shaped model.

0⎤ 0⎥ 4⎥ 0⎥ ⎦ (33)

12.11 −18.34 ⎤ 10.74 11.9 ⎤ kP = ⎡ , kI = ⎡ 3.24 ⎦ ⎣ 1.04 ⎣−0.67 −1.81⎦ −42.7 82.2 ⎤ kD = ⎡ , τ = 4.05 ⎣ −1 −13.69 ⎦

(36)

−10 0 ⎤ 4 0⎤ AF = ⎡ , BF = ⎡ ⎣ 0 −10 ⎦ ⎣0 4⎦ 2.5 0 ⎤ 0 0⎤ CF = ⎡ , DF = ⎡ ⎣0 0⎦ ⎣ 0 2.5⎦

(37)

−13.18 7.3 ⎤ −1.15 −0.52 ⎤ AJ = ⎡ , BJ = ⎡ ⎣−13.68 −2.6 ⎦ ⎣−2.69 −3.88 ⎦ 9.56 −9.14 ⎤ 0.08 1.96 ⎤ CJ = ⎡ , DJ = ⎡ ⎣ 3.25 1.39 ⎦ ⎣ 0.58 −1.29 ⎦

(38)

0 ⎤ 32 0 ⎤ −105 A1 = ⎡ , B =⎡ −105⎦ 1 ⎣ 0 32 ⎦ ⎣ 0 32.8 0 ⎤ 0 0⎤ C1 = ⎡ , D1 = ⎡ ⎣0 0⎦ ⎣ 0 30.6 ⎦

(34)

0 ⎤ 128 0 ⎤ −600 , B =⎡ A2 = ⎡ −600 ⎦ 2 ⎣ 0 128⎦ ⎣ 0 0 ⎤ 15 0 ⎤ −65.5 C2 = ⎡ , D2 = ⎡ −65.5⎦ ⎣ 0 ⎣ 0 15⎦

(35)

Fig. 9 shows the time-domain responses of the controlled system to a step input in each channel. It can be seen that both controlled variables z and θ realize fast transient responses without overshoots, and the settling times ensuring less than 0.5 s. They are almost identical to the ideal reference. Note that it is very important for the HSSV to have such responses, which guarantee its dynamic behavior being precise with

The frequency response of the shaped plant relating to the original gain is illustrated in Fig. 8. For the singular value specifications on the design purposes, it can be observed that the shaped plant has high gain at low frequency and low gain at high frequency as desired. More precisely, the system designer should push up the smallest singular value at low frequencies for performance bound ensuring disturbance attenuation and reference tracking, and make sure that the largest singular value is small at high frequencies for robustness bound guaranteeing noise rejection and control activity reduction. Typically, the shaped plant has a slope (-20 dB/decade) around crossover (when 0 dB) focusing on stability margins, and a larger roll-off at higher frequencies. The desired slope at low frequencies depends on the nature of disturbances or reference signal. Based on the shaped loop gain and reference system Tr, a first-order filter F, a third-order H∞ loop shaping PID controller, and a secondorder anti-windup compensator are synthesized to control the HSSV. After some trials and errors with the pre-compensator W1, the control parameters are tuned by the shaped plant satisfying above mentioned design requirements. The achieved value of stability margin (ε ) is 0.63, which indicates good stability gain. Then 63% of coprime factor uncertainty is allowed around the cross-over frequency range. Controller parameters are given as follows:

Fig. 9. Transient responses due to step-type input of the dive plane variables. 264

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Fig. 10. Dynamic coupling effects between dive plane variables: (a) δe to θ and (b) δc to z .

Fig. 11. Time-domain responses due to parameter variations: (a) vertical position and (b) pitch angle.

fast responses to properly react to control commands and the surrounding environment changes. In fact, a slight overshoot also leads to undesired and catastrophic collisions at that high speed. Thus, the transient responses including such as rise time, overshoot, and settling time are so crucial to the safe operations of underwater vehicle. The transient response results presented in this paper is in superior tracking performance compared with the results of the adaptive control reported in [11] where the settling time is about 0.9 s, the results of the sliding mode control in [15] where the settling time is about 0.9 s with 10% overshoots, the result of the LPV controller in [21] where the settling time is more than 1 s, and the results in [31] where the constraints are also taken into account and the settling time is about 2 s. For the MIMO dynamical system, the interactions between channels are always challenge issues for system designers. In this study, the modified PID control structure has an ability to deal with the complex HSSV by the decoupling, illustrated in Fig. 10(a). The dynamic interactions from δe to θ and from δc to z are effectively minimized. There exist only some slight couplings in the rising time. In consequence, the interactions are perfectly decoupled in the steady-state. This feature could improve the stability and make the controller more applicable. The corresponding control signals are also shown in Fig. 10(b) which poses no actuator saturation. The robustness of the controlled system has been evaluated via high stability margin of ε = 0.63. Furthermore, this feature is also visualized by its ability to cope with parameter uncertainties. In this simulation, the effectiveness of the controller has been demonstrated via 30 wide random set of varying parameters given in Table 1. As illustrated in Fig. 11, the time-domain responses are still almost identical even if the

parameter perturbations exist. It indicates that the controlled system is still stable with some higher level of uncertainties. The controller robustness also includes the ability to reject external disturbance. There are many disturbance sources in underwater environments. In this simulation, wave disturbances are introduced to describe some effects on the outputs. Fig. 12(a) shows that high-amplitude disturbances can only cause some slight variations on the system outputs. In fact, a parametric calculation gives a result that 80% of disturbance has been rejected. This performance is obtained from the result of the high-slope shaped loop at the low-frequency range in Fig. 8. Fig. 12(b) illustrates the corresponding planing force and vertical speed. In general, when the planing force occurs, the system dynamics will be dramatically changed. However, in this simulation, the robust controller can successfully overcome large external disturbances and planing forces. The actuator saturation often occurs in sudden-direction changes, since the deflection angles of the fins and the cavitator are often small. Under a disturbance of 0.5 m in the vertical position at 7th second, there exists a saturation in the deflection of the fins δe , which is supposed to be limited at 0.52 rad. Usually, the integrator part in some conventional controllers will accumulate the error during the saturation and may cause a windup afterward. In this study, the designed anti-windup compensator will cope with the effect of the saturation to stabilize the control signal. It can be observed from Fig. 13 that the vehicle vertical position is still stabilized and can well track a 2 m reference signal under disturbances with fins deflection saturation, without any windup. Another inherent problem in any controlled system is the influence of sensor noises. Measurement noises can cause some negative effects 265

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Fig. 13. System responses with anti-windup: (a) vertical position and (b) control signals with actuator saturations.

Fig. 12. Time-domain responses due to external disturbances: (a) dive plane responses and (b) planing force and vertical speed.

on the controlled systems, especially dynamical system like the HSSV. In this simulation, a white noise with 10Khz frequency is added into the feedback outputs. Fig. 14 illustrates that the controlled variables are fluctuating under the effects of the noises, but showing very mall magnitudes comparing with noise levels. A calculation points out that almost 80% of the sensor noises have been eliminated. Finally, the simulation results confirm both robustness and performance of the designed control algorithm against uncertainty and disturbance.

5. Conclusions This paper presents the dive plane model, dynamical analysis and robust control synthesis of supercavitating underwater vehicles. The HSSV modelling includes many components of the system dynamics such as cavity, cavitator, body, and fins. First, a simplified model of vertical plane dynamics has been presented for vehicle control synthesis. The maneuvering control of supercavitating vehicles is extremely challenging due to its high speed under a variety of effecting factors. Particularly, the control system is vulnerable to its sensitivity to measurement errors and lack of robustness and even unstable in the presence of strong disturbances and uncertainties. Next, this paper introduced the H∞ robust control with a modified PID algorithm to robustly control the MIMO HSSV. The control synthesis consists of a modified PID structure incorporated in the loop-shaping to get desired loop shape, a first order filter, and an anti-windup compensator. The multi-objective control problems are solved using BMI optimization of an equivalent Schur formula. The robustness issue is proven by the

Fig. 14. System responses due to noise inputs.

ability of the proposed controller to cope with parametric uncertainties, external disturbances, actuator saturations, and sensor noises. Furthermore, the dynamic interactions are highly decoupled which allows the PID structured controller to be well applied for the MIMO HSSV. The effectiveness and efficiency of control algorithms are validated through numerical simulations under uncertainty and

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disturbance. The numerical simulation shows that the controlled vehicle system has good transient responses without overshoots, guaranteeing high robustness. More precisely, the robust controller can deal with up to 63% of uncertainties in the vehicle model and can successfully eliminate up to 80% of exogenous disturbances and sensor noises. Finally, the integrated robust control system provides high robustness with performance while maintaining the simple control structure through its low-order PID algorithm which will make the active controller easy for real implementation.

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