Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay

Mechanical Systems and Signal Processing xxx (xxxx) xxx

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Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay Bo Li a,⇑, Xiaoting Rui b, Wei Tian a, Guangyu Cui a a b

College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao St., Nanjing 210016, China Institute of Launch Dynamics, Nanjing University of Science and Technology, 200 Xiaolingwei St., Nanjing 210094, China

a r t i c l e

i n f o

Article history: Received 12 June 2019 Received in revised form 14 September 2019 Accepted 28 October 2019 Available online xxxx Keywords: Actuator delay Dynamic recurrent neural network Motor-mechanism coupling system Multiple launch rocket system Orienting control

a b s t r a c t Development of multiple launch rocket system (MLRS) has been restricted for several decades due to the poor dispersion characteristics of rockets, which is caused by the orientation of the MLRS departing from that intended. Hence, it is vital to maintain the angles of MLRS at a desired value via a proper control strategy. In this paper, a new neural network predictive control is developed for orienting control of the MLRS with actuator delay. First, the dynamic model of motor-mechanism coupling system is established using Lagrange method and field-oriented control theory. Then, for cancelling the effects of nonlinearities and uncertainties, the concept of feedback linearization and a dynamic recurrent neural network are introduced. In addition, a modified Smith predictor is employed to maintain the desirable orienting performance in the occurrence of actuator delay. For the stability analysis, Lyapunov’s method is utilized to ensure uniform ultimate boundedness of the closed-loop system. The simulated and experimental results demonstrate the effectiveness of the proposed controller. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Multiple launch rocket system (MLRS) is a launching device that carries rockets and orients them to targets. In the last decades, MLRS has been intensively investigated [1–4] because of its potential applications in mobile military, aerospace engineering and civilian fields, e.g., anti-terrorist operations, anti-missile air defense and artificial rainfall. MLRS has two servomotors, namely, elevation and azimuth motors, which are individually connected to set of gear reducers and drive loads, to rotate about associated axes for providing desired angles. The MLRS is usually placed on a vehicle for high mobility. When the vehicle comes to a firing area, the MLRS directs the rockets to the desired orientation fast and accurately. Such process is typically called positioning. After the MLRS reaches the desired orientation, the rockets will be launched with a certain firing interval, which is called salvo firing. In the positioning process, very small deviations on the elevation and azimuth angles may cause significant rocket dispersions [5]. Therefore, the above-mentioned servomotors are used as actuators to obtain high positioning accuracy. Unfortunately, the elevation and azimuth angles are usually changed due to a jet force generated by firing a salvo of rockets, and thus deteriorating the dispersion characteristics of rockets. Consequently, control has become a promising technique to reduce the angular deviations. The MLRS is substantially an electromechanical system, including servo-motors and mechanisms. Inaccuracies of mechanical and electrical components and the nature of parametric variations and unmodeled dynamics mean that the MLRS model is nonlinear and uncertain. Therefore, the orienting control design of MLRS faces the difficulties of nonlinearities and uncertainties. ⇑ Corresponding author. E-mail address: [email protected] (B. Li). https://doi.org/10.1016/j.ymssp.2019.106489 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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In the last several decades, many methods have developed for MLRS control to improve the dispersions of rockets [6–15]. Nevertheless, few studies consider time delay in the actuator. This paper is motivated by this point. Actuator delay occurs frequently in control applications for industrial systems. The unavoidable time delay caused by the servomotors on MLRS can lead to poor control performance and potential instability. Recently, various methods are developed to compensate for the actuator delay; the predictor technique is the most widely used one. The predictive control approach originates from a technique known as the Smith predicator method [16], where the control design problem with delay can be converted to the one without delay. However, the Smith predictor can be only applied to a good match of plant [17]. To remove such constraint from the Smith predictor, many other predictor-based control approaches, e.g., Arstein model reduction [18], finite spectrum assignment [19], continuous pole placement [20] and partial differential equationsbased backstepping control [21] have been proposed to enhance the control applicability. In recent years, numerous predictive control techniques combined with some advanced methods have been reported to solve the time delay issues [22,23]. However, the above approaches are only suited for linear plants. For highly nonlinear systems, predictor-based control problem may become extremely difficult and sophisticated [24,25]. Compared to the time-delay linear systems, fewer outcomes are accessible to nonlinear systems. In [26], the Smith predictor-based control method is proposed for input delayed nonlinear systems. [27] presents a new adaptive fuzzy predictive sliding mode control for nonlinear systems with uncertain dynamics and unknown input delay. In [28], the control problem of strict-feedback nonlinear systems with time-varying input and output delays is considered, where the novelty lies in the use of the closed-loop dynamics in the predictor. In [29], the problem of feedback linearization for nonlinear systems with time-varying input and output delays is addressed, where a high-gain-observer is constructed as a predictor to solve the realization issue of future states. In [30], a predictor control for a class of multi-input non-linear systems with timeand state-dependent input delays is considered, and then a predictor-based control design is proposed in [31] for stabilizing discrete nonlinear systems with state-dependent input delays. In [32], the control algorithm based on the uncertainty and disturbance estimator is modified to extend its applicability to industrial processes with time delay. In [33], a new delay compensation algorithm for networked control systems was presented combining a feedback control law and the Smith predictor. Nonetheless, to the best of our knowledge, no attempt has been made towards controlling a time-delayed nonlinear MLRS with parametric and unstructured uncertainties, and external disturbances. Motivated by above analyses, a new neural network predictor-based approach is developed to cope with the orienting control problem for an uncertain nonlinear MLRS with actuator delay. The neural network predictor-based control has the following novelties: 1) Compared to the PID-type backstepping control [34], the proposed control has a better improvement in robustness to the variations of the actuator delay. 2) The traditional Smith predictor method is extended to MIMO uncertain nonlinear systems with actuator delay by using the DRNN and the feedback linearization technique. 3) The proposed control approach can be directly applied to other uncertain Euler-Lagrange systems with input delay such as robotics, manipulators, vehicle suspensions, etc. The main contributions of this paper are summarized as follows: 1) In comparison with our previous work [15] where a static neural network is chosen as uncertainty estimator, the present work introduces a DRNN which ensures a better model of the plant. 2) A new Smith predictor-based control approach integrated with a DRNN estimator is first proposed to control the orientation of a MIMO nonlinear uncertain MLRS with actuator delay. 3) An experiment is implemented to validate the effectiveness of the neural network-based control in a real-world MLRS. 2. Preliminaries and problem formulation 2.1. Preliminaries

Remark 1. Throughout this paper, the following annotations are defined as: For a vector x, jjxjj denotes the standard Euclidian norm. For a matrix A, jjAjjF denotes the Frobenius norm, and kmin ðAÞ, kmax ðAÞ denote the minimum and maximum eigenvalues of A, respectively. Lemma 1 ((Rayleigh-Ritz theorem)). Let A be a real, positive definite, symmetric matrix; hence, all the eigenvalues of A are positive and real. Then for an arbitrary vector x compatible with A, we have

kmin ðAÞjjxjj2  xT Ax  kmax ðAÞjjxjj2 :

ð1Þ

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Lemma 2 ((Young’s inequality)). For real numbers a, b, and k, the following inequality holds 2 2

2kab  a2 þ k b :

ð2Þ

2.2. Mechanism-motors modeling Fig. 1 depicts the schematic diagram of a MLRS, which is composed of vehicle, azimuth mechanism and elevation mechanism. / is the azimuth angle about oy axis; h is the elevation angle about o2 z2 axis. l1 and l2 are the lengths of the azimuth and elevation mechanisms, respectively; h2 is the height of the elevation mechanism; c1 and c2 are the centers of mass of the azimuth and elevation mechanisms, respectively. The masses of the azimuth and elevation mechanisms are m1 and m2 , respectively. The equation of motion of the MLRS can be obtained by using Lagrange’s approach as

€ þ V 0 ðq; qÞ _ þ d ¼ Tc; M 0 ðqÞq

ð3Þ


 J1 is J3

J q ¼ ½ h / T is the generalized state vector; M 0 ðqÞ ¼ o2zz the generalized mass matrix; J1 " # 2 _ _ _ ¼ J 2 / þ Be2h  Ge V 0 ðq; qÞ is the Coriolis, centrifugal, friction and gravity vector; d is the disturbance vector; _ _ _ 2J /h þ J h þ Ba /_ where

2

4

_ are T c ¼ ½T e T a T is the driving torque vector generated by the servomotors. In detail, the elements in M 0 ðqÞ and V 0 ðq; qÞ

J 1 ¼ J o2xz sinh þ Jo2yz cosh; J 2 ¼ ðJ o2xx  J o2yy Þsin2h þ 2J o2xy cos2h þ 0:5m2 l1 ðl2 sinh þ h2 coshÞ; 2

J 3 ¼ J 1yy þ J o2yy cos2 h þ J o2xx sin h þ J o2xy sin2h þ 0:25m2 l1 ðl1 þ 2h2 sinh  2l2 coshÞ; J 4 ¼ J o2xz cosh  J o2yz sinh; Ge ¼ 0:5m2 gðh2 sinh  l2 coshÞ; where Be and Ba are the damping coefficients of the elevation and azimuth axes, respectively; J o2i j (i = x, y, z, j = x, y, z) is the element of the inertia matrix J o2 of the elevation mechanism about point o2; J 1yy is moment of inertia of the azimuth mechanism about o1y1 axis. The 3-phase ac permanent magnet synchronous motor (PMSM), which is widely used in industrial applications, is chosen as the actuator in this work. The PMSM drive system can be reduced to a dc motor control system by utilizing the fieldoriented control theory [35]. Its control block is shown in Fig. 2. Further we have

T em ¼ 3np wf iq =2 ¼ K t iq ;

ð4Þ

Fig. 1. Schematic diagram of MLRS.

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Fig. 2. Control block of PMSM drive system.

and

_ r; T em ¼ T L þ Bm xr þ J m x

ð5Þ

where np is number of pole pairs; wf is the permanent magnet flux; K t ¼ 3np wf =2 represents the torque coefficient; xr is the angular speed of rotor; Bm is the damping coefficient of rotor; J m is the inertia of rotor; and T L is the load torque. Fig. 3 shows a motor-gear transmission mechanism including a geared speed reducer with a gear ratio

n ¼ T=T L ¼ xr =x;

ð6Þ

where a and x are the angular displacement and velocity the azimuth or elevation mechanism, and T is the torque applied to the corresponding mechanism. Substituting (4) and (6) into (5) and considering the elevation and azimuth servomotors, one can obtain the dynamic equations of PMSM in a matrix form as

€Þ T c ¼ n ðK t iq  nBm q_  nJ m q

ð7Þ

where n ¼ diagðne ; na Þ, K t ¼ diagðK te ; K ta Þ, Bm ¼ diagðBme ; Bma Þ and J m ¼ diagðJ me ; J ma Þ represent the diagonal matrices composed of the elevation and azimuth servomotor parameters. Further, the mechanism-motors model can be formulated by combining (3) and (7) as

€ þ Vðq; qÞ _ þ dðtÞ ¼ LuðtÞ; MðqÞ q

ð8Þ

_ L ¼ nK t , and uðtÞ ¼ iq ðtÞ. _ ¼ V 0 ðq; qÞ _ þ n2 Bm q, with MðqÞ ¼ M 0 ðqÞ þ n2 J m , Vðq; qÞ Assumption 1. In this paper, due to the presence of the actuator delay in the actual MLRS, we assume that there exists a constant time delay s in the control input. Assumption 2. We assume that the inertia matrix MðqÞ is known and positive definite symmetric, and satisfies the following inequality

M1 I 22  MðqÞ  M 2 I 22 ;

ð9Þ

where M 1 and M 2 are positive constants, and I 22 denotes a 2-order identify matrix.

Fig. 3. Motor-gear transmission mechanism of MLRS.

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Assumption 3. For the unknown disturbance term dðtÞ, we assume that there exists a positive constant such that 

jjdðtÞjj  d; 8t 2 ½0; 1Þ. _ is unknown. Let x ¼ ½ qT Since uncertainties in the Coriolis and friction terms are unavoidable, therefore, Vðq; qÞ the MLRS dynamics equation (8) can be written in the state-space form as

x_ ¼ Ax þ B ½f ðxÞ þ M 1 ðxÞLuðtÞ  BM 1 ðxÞd;


where A ¼

022 A1

T

T q_  ,

ð10Þ


 I 22 022 ,B¼ , and f ðxÞ contains the unknown term in the plant, expressed as A2 I 22

f ðxÞ ¼ M 1 ðxÞ VðxÞ  ½ A 22

1

ð11Þ

A 2 x:

22

A 12R and A 2 2 R are selected such that A in (10) is stable. Then for an arbitrary symmetric and positive definite matrix Q , there exists a symmetric and positive definite matrix P satisfying the following Lyapunov equation

PA þ AT P ¼ Q :

ð12Þ

If f ðxÞ is well known and available, one can design the control law as

uðtÞ ¼ L1 MðxÞ½ua ðtÞ  f ðxÞ:

ð13Þ

Then the closed-loop system can be obtained by substituting (13) into (10) as

x_ ¼ Ax þ Bua ðtÞ  BM 1 ðxÞ d;

ð14Þ

with ua ðtÞ being an auxiliary control variable. Thus, the uncertainty term can be cancelled and the system is linearized. Further, some well-developed linear methods can be used to control such system for a good performance. In this section, the MLRS motor-mechanism model was formulated in order to facilitate the design of the proposed controller below. However, in practical MLRS, VðxÞ cannot be well known a priori. Hence, an alternative method is motivated to identify the unknown dynamics before designing the controller. In the next section, a DRNN estimator will be presented for this problem. 3. Estimation of un-delayed system DRNNs are such neural networks where two types of connections between neuronal units form a directed cycle. This distinguishes them from static feed-forward neural networks where the output of each unit is only connected to units in the next layer. The benefits resulting from the adoption of the DRNN in nonlinear mapping applications are found in [36]. Motivated by these works, we shall consider a three-layer network with a hidden layer composed of dynamical nodes to implement the estimation of the unknown uncertainty term (11) in MLRS-actuator dynamics. The dynamics of such network architecture shown in Fig. 4 can be depicted by T c Rðx ^Þ þ BM 1 ðxÞLu; ^_ ¼ Ax ^þBW x

ð15Þ ^

^

where x is the state vector of DRNN, i.e., the state estimate of the delay-free MLRS;A, W , BM 1 ðxÞL specify the connection ^

weights; and RðxÞ is a vector-valued function with Gaussian elements

^Þ ¼ exp ½jjx ^  c i jj2 = ð2bi Þ; R i ðx 2

ð16Þ

Fig. 4. Architecture of the DRNN.

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where ci and bi are the center vector and the width of the ith basis function, respectively; m is the number of the basis functions. It can be seen from Fig. 4 that fourth-layer structure with one input layer, two hidden layers and one output layer comprises the DRNN. In addition, there exist connections among nodes in different and same layers, which allow such network to exhibit dynamic temporal behavior. We assume that the unknown nonlinear function in (11) is continuous and can be completely described by a DRNN with a bounded formulation error g such that

^Þ þ g; f ðxÞ ¼ W T Rðx

ð17Þ



where W is the ideal weight that reaches the minimum estimation error. Thus the system (10) can be written as

^Þ þ g þ M 1 ðxÞLu  BM 1 ðxÞ d: x_ ¼ Ax þ B ½W T Rðx

ð18Þ

To compensate the estimation error g of the DRNN and the disturbance d of the system, a compensating term uc ðtÞ is introduced into (15). Then (15) can be written as

^ T Rðx ^Þ þ BM 1 ðxÞLu þ Buc : ^_ ¼ Ax ^ þ BW x

ð19Þ 



Assumption 4. The DRNN error is bounded, i.e., there exists a constant g such that jjgjj  g. 

Assumption 5. The ideal weight W  is bounded by a positive constant W such that jjW  jj 



^



F

 W.

^

Define the state estimation error as x ¼ x  x and the weight error as W ¼ W   W and subtract (19) from (18), and we can obtain the state error equation  T   _ ^Þ þ g  uc   BM 1 ðxÞd: x ¼ A x þB ½W Rðx

ð20Þ

In order to stabilize the system given by governing equation (8) or (10) and the DRNN estimator (19), we present the following DRNN weight updated law ^_ T ^ W ¼ cR x PB  ck1 W;

ð21Þ

and the compensating term 

uc ¼ k2 x;

ð22Þ

where c, k1 and k2 are positive scalar design parameters. Select a Lyapunov function candidate as T





2

V 0 ¼ x P x =2 þ jj W jjF =ð2cÞ:

ð23Þ

Lemma 3. The time derivative of the Lyapunov function candidate can be upper bounded by

V_ 0  d0 V 0 þ e0 ; with d0 > 0 and

ð24Þ

e0 > 0.

Proof. Differentiating (23) with respect to time along (20) and (12), one can have  T   T  T  _ _ T  T  T T _ _ T  ^Þ þ tr½W W =c: V_ 0 ¼ ðx P x þ x P xÞ=2 þ tr½W W =c ¼ x Q x =2 þ x PBðg  uc þ M 1 dÞ þ x PB W Rðx  T

T

 T

ð25Þ

T

Utilizing x PB W R ¼ tr ðW R x PBÞ, and substituting (21) and (22) into (25) lead to  T

      ^ V_ 0 ¼ x Q x =2  k2 x PB x þx PBg þ x PBM 1 d þ k1 trðW WÞ: T

T

T

T

ð26Þ

Since  T

 T







2

^ ¼ tr½W ðW   W Þ  jj W jj jjW  jj  jj W jj ; trðW WÞ F F F

ð27Þ

and using Lemma 1 we have

V_ 0   ½kmin ðQ Þ=2 þ

 pffiffiffi  2  2k2 kmin ðP Þjj x jj þ q1 jj W jjF þ q2 jj x jj 



2

k1 jj W jjF ;

ð28Þ

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  pffiffiffi  where q1 ¼ k1 W , q2 ¼ 2kmax ðPÞ ð g þ d =M 1 Þ. There exist positive numbers j1 and j2 by using Lemma 2 such that the following inequalities hold:





2

2j1 q1 jj W jjF  q21 þ j21 jj W jjF ; 



2

2j2 q2 jj x jj  q22 þ j22 jj x jj :

ð29Þ ð30Þ

Then

V_ 0  ½kmin ðQ Þ=2 þ

2  pffiffiffi  2 2k2 kmin ðPÞ  j2 =2jj x jj  ðk1  j1 =2Þjj W jjF þ q21 =ð2j1 Þ þ q22 =ð2j2 Þ  d0 V 0 þ e0 ;

ð31Þ

where

e0 ¼ q21 =ð2j1 Þ þ q22 =ð2j2 Þ; and constants k1 ; k2 ; q1 ;

r1

ð32Þ

j1 ; j2 are selected to satisfy the following condition:

pffiffiffi ¼ kmin ðQ Þ=2 þ 2k2 kmin ðPÞ  j2 =2 > 0;

ð33Þ

r2 ¼ k1  j1 =2 > 0;

ð34Þ

d0 ¼ minfr1 =½kmax ðPÞ; cr2 g:

ð35Þ

Theorem 1. As the MLRS dynamics described by (8) or (18), and the DRNN estimator given by (19) with the weight-tuning algorithm provided by (21) and the proposed compensating term given by (22), given that the initial conditions are bounded, 



then the error signals x and W are uniformly ultimate bounded with specify bounds given by

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ½V 0 ð0Þ þ e0 =d0 =½kmin ðPÞ;  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jj W jjF  2c½V 0 ð0Þ þ e0 =d0 : 

jj x jj 

ð36Þ ð37Þ

Proof. Multiplying (24) by ed0 t yields

d ½VðtÞ ed0 t =dt 

e0 ed0 t :

ð38Þ

Integrating (38), we obtain

V 0 ðtÞ  ½V 0 ð0Þ  e0 =d0  ed0 t þ e0 =d0  V 0 ð0Þ þ e0 =d0 : 

2

T

ð39Þ



Since kmin ðPÞjj x jj  x P x, then 

2

kmin ðP Þjj x jj =2  V 0 ð0Þ þ e0 =d0 :

ð40Þ

Furthermore, we can obtain and from. j This section focuses on the design of the DRNN estimator to compensate for uncertainties and unknown dynamics in the MLRS model regardless of the time delay. Further, to improve control accuracy, a prediction control law will be designed in the presence of actuator delay next section. 4. Predictive control design for time-delay compensation In this section, a predictor is proposed to compensate for the actuator delay for improving orienting performances based on the results of Section III. According to Section III, the unknown term f in can (10) be identified by

^f ¼ W ^ T Rðx ^Þ:

ð41Þ

Considering the actuator delay and to compensate for the nonlinear part of the system, the control input (13) is rewritten as

^ T Rðx ^Þ: uðtÞ ¼ L1 MðxÞ ½ua ðt  sÞ  W

ð42Þ

Substituting (42) into (18), we can obtain a linearized model plus a disturbance

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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B. Li et al. / Mechanical Systems and Signal Processing xxx (xxxx) xxx  T

^ðtÞ  BM 1 ½xðtÞ dðtÞ þ B gðtÞ: _ xðtÞ ¼ AxðtÞ þ Bua ðt  sÞ þ B W ðtÞR ½x

ð43Þ

Subsequently, we will find a control uðtÞ by predictive control approach such that the system state xðtÞ track accurately a desired trajectory, denoted by T xd ðtÞ ¼ ½ qTd ðtÞ q_ Td ðtÞ  ;

ð44Þ

in the presence of the time delay s. According to Theorem 1, assume that the nonlinear unknown term and the disturbance term is compensated for completely. Thus the MLRS dynamics (43) can be written as

_ xðtÞ ¼ AxðtÞ þ Bua ðt  sÞ:

ð45Þ

Based on Watanabe’s Smith predictor [37], the block diagram of the proposed DRNN predictive control structure is shown in Fig. 5. Now let the predictor be [37]

x_ p ðtÞ ¼ Axp ðtÞ þ B ua ðtÞ; yp ðtÞ ¼ eA s xp ðtÞ  xp ðt  sÞ:

ð46Þ

where xp ðtÞ and yp ðtÞ are the predictor state and output, respectively. Define an error vector as

e1 ðtÞ ¼ xd ðtÞ  xðtÞ  yp ðtÞ:

ð47Þ

h i eðtÞ ¼ N e1 ðtÞ ¼ N I 22 e1 ðtÞ;

ð48Þ

Let



with N 2 R22 > 0 being a diagonal matrix selected by designers. Differentiating eðtÞ and using (45), lead to

    _ eðtÞ ¼ N x_ d ðtÞ  AxðtÞ  eA s Axp ðtÞ  N eA s Bua ðtÞ  Axp ðt  sÞ :

ð49Þ

Now the following control is designed

   N eA s Bua ðtÞ ¼ K d eðtÞ þ NAxp ðt  sÞ þ N½x_ d ðtÞ  AxðtÞ  eA s Axp ðtÞ ;

ð50Þ

where K d > 0 is a diagonal gain matrix. Substituting (50) into (49), we can obtain

_ þ K d eðtÞ ¼ 02 : eðtÞ

ð51Þ

Define a Lyapunov function candidate as

VðtÞ ¼ V 0 ðtÞ þ V 0 ðt  sÞ þ eT ðtÞ eðtÞ=2:

ð52Þ

Fig. 5. DRNN predictive control structure for MLRS with actuator delay.

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Lemma 4. The time derivative of the Lyapunov function candidate (52) can be upper bounded by

V_  dV þ e; with d > 0 and

ð53Þ

e > 0.

Proof. Differentiating (52) with respect to time results in

_ _ VðtÞ ¼ V_ 0 ðtÞ þ V_ 0 ðt  sÞ þ eT ðtÞeðtÞ:

ð54Þ

Arranging (51) and substituting it into (54), we obtain

_ VðtÞ ¼ V_ 0 ðtÞ þ V_ 0 ðt  sÞ  eT ðtÞK d eðtÞ  d0 V 0 ðtÞ þ e0  d0 V 0 ðt  sÞ þ e0  kmin ðK d ÞjjeðtÞjj2  dVðtÞ þ e; where d ¼ min½d0 ; 2kmin ðK d Þ > 0,

ð55Þ

e ¼ 2e0 > 0. j

Theorem 2. Consider the MLRS dynamics described by (8) with input delay, the DRNN weight update law (21) and the 





compensating term (22). Let the control law be provided (42) by and (50). Then the error signals eðtÞ, xðtÞ, xðt  sÞ, W ðtÞ, and 

W ðt  sÞ are uniformly ultimate bounded.

Proof. The proof of Theorem 2 is similar to that of Theorem 1; therefore, it is omitted here for space reasons. j 5. Numerical simulations In this section, a simulation is carried out to demonstrate the control performance of the proposed controller for the MLRS with actuator delay. Our previous NN control [15], Dokumaci’s control [12], and Sharma’s control [34] are implemented for comparative study. The control objectives in this paper are to (1) attain the desired angles for the elevation and azimuth mechanisms in positioning process with desired speeds and accuracies, and (2) keep the angles on or reach to the desired orientation during salvo firing in the presence of actuator delay. The main parameters of the MLRS model and PMSMs are tabulated in Tables 1 and 2, respectively. The desired position was selected as qd ¼ ½0:345 0T rad. The actuator delay is assumed to be s ¼ 0:02 s:The control cycle time is 1 ms. In the salvo firing, assume that 40 kNm and 10 kNm disturbances for each rocket are applied to the elevation and azimuth mechanisms, respectively, as pulse during 300 ms to simulate a salvo of eighteen rockets with 1 s firing interval. Clearly, the external disturbances are shown in Fig. 6(a) and (b). Remark 2. It should be noted that the simulation conditions (e.g., disturbance, firing interval, control cycle time, delayed time and desired position) for all the control methods are the same.

Remark 3. Because of the limitation of input saturation in actuator, the control current uðtÞ is confined to uðtÞ 2 ½80; 80 A. Remark 4. For the unknown nonlinear function, the DRNN estimator is online, and the computation time is 0.001 s.

Table 1 Partial physical parameters for the MLRS model. Parameters

Description

Values

m2 Be m1 Ba l1 l2 h2 J1yy J o2

Mass of the elevation mechanism Viscous damping coefficient of elevation axis Mass of the azimuth mechanism Viscous damping coefficient of azimuth axis Length of the azimuth mechanism Length of the elevation mechanism Height of the elevation mechanism Moment of inertia of the azimuth mechanism about o1y1 axis Inertia matrix of the elevation mechanism about point o2

2968.62 kg 9134 Nms/rad 1383.96 kg 9244 Nms/rad 1.70 m 1.88 m 1.50 m 390.54 kgm2 2 2722:41 2504:59 4 2504:59 5369:65 43:70 37:57

3 43:70 37:57 5 kgm2 7005:36

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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B. Li et al. / Mechanical Systems and Signal Processing xxx (xxxx) xxx Table 2 Physical parameters for the PMSMs. Parameters

Description

Values

ne na K te K ta Bme

Gear ratio of the elevation mechanism Gear ratio of the azimuth mechanism Torque coefficient of the elevation motor Torque coefficient of the azimuth motor Damping coefficient of the elevation motor

2257 2153 1:5N  m=A 1:5N  m=A 5:78  103 N  m  s=rad

Bma

Damping coefficient of the azimuth motor

5:78  103 N  m  s=rad

Jme

Inertia of the elevation motor

1:03  103 kg  m2

Jma

Inertia of the azimuth motor

1:03  103 kg  m2

Fig. 6. Disturbance torques applied to (a) elevation mechanism and (b) azimuth mechanism.

5.1. Dokumaci’s control In this part, Dokumaci’s control method is utilized to achieve good performance of orienting control for MLRS. The controller parameters are selected as kp ¼ diag ð178:74; 153:63Þ, ki ¼ diag ð0:1; 0:1Þ, and kd ¼ diag ð22:97; 19:52Þ. Please refer to [12] for the detailed controller and its parameters. We obtain the results of h, / and the associated control efforts, which are shown in Fig. 7. From Fig. 7(a) and (b), it can be seen that Dokumaci’s control has an acceptable performance for both positioning and salvo firing. The maximum overshoots of h and / are both less than 10% in positioning process. The adjustment time is about 0.6 s and the steady-state error is approximately zero. When the MLRS is firing rockets, it can be found from Fig. 7(c) and (d) that the external disturbances change h and / by approximately 7.3  103 rad and 4.7  104 rad, respectively. Nonetheless, the angles reach the desired values in a short time under the action of controller. The repositioning time is less than the firing interval 1 s, which means that the controller can keep the MLRS on target during salvo firing. 5.2. Sharma’s control To achieve good performance of orienting control, Sharma’s control parameters are chosen as a11 = 3.19, a12 = 18.60, a21 = 34.01, a22 = 18.58, ka1 = 7.20  105, ka2 = 7.00  105, where the second subscript denotes the elevation and azimuth channels respectively. The controller structure and its parameters are referred to [34] for more details. The simulated results are shown in Fig. 8. We can find from Fig. 8(a) and (b) that the maximum overshoots of h and / for Sharma’s control are both less than 10%. The settling time for h is 0.9 s, and for / 1.12 s and the steady-state error is approximately zero. Due to the effect of the disturbances, it can be observed from Fig. 8(c) and (d) that h and / have changed by 4.4  103 rad and 3.6  104 rad, respectively, in the salvo firing. However, the repositioning error converges to a neighborhood around zero quickly under the control action. 5.3. DRNN predictor-based control As for the DRNN predictor-based control, the center vector ci and the width bi of the used Gaussian functions can be determined according to the value of the desired state x. The other parameters, e.g., the matrix Q and the controller parameters were optimized using genetic algorithm, taking the minimum error between the desired state and the actual state as objective function. They are given as k1 = 0.10, k2 = 9.95, m = 12, A1 = diag (1, 1), A2 = diag (2, 2), c = 9.99, Kd = diag (31.87, 

31.62), Q = diag (990, 990, 990, 990), N= diag (7.96, 7.91). The initial conditions of the states for the MLRS, the predictor, the ^

DRNN and of the weights W are all zeros. The simulated results of the proposed controller are shown in Fig. 9. It is clear from Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

B. Li et al. / Mechanical Systems and Signal Processing xxx (xxxx) xxx

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Fig. 7. Simulated results of Dokumaci’s control with time delay s = 0.02 s. (a), (b) Elevation and azimuth angles; (c), (d) Angular errors for elevation and azimuth mechanisms; (e), (f) Control efforts.

Fig. 9(a) and (b) that when the DRNN predictive control is applied, the maximum overshoot of h and / are less than 10% in the positioning process. The adjustment time is less than 1.5 s, and the steady-state error is approximately zero. In the shooting process, it is clear from Fig. 9(c) and (d) that h and / have changed by 1.5  103 rad and 3.5  104 rad, respectively, due to the effect of the disturbances. But under the controller action, both angles return to the desired angle before the next rocket launch. From Fig. 9(e) and (f), the DRNN estimates the MLRS state well and the estimation accuracy is high. Synthesizing the above analyses, the performance of the proposed control is acceptable. To be more specific, the proposed control can achieve comparable performance in positioning and salvo firing. 5.4. NN control in [15] In this section, the performance of the NN control in [15] is simulated for comparison. The controller and its parameters can be found in [15]. It should be pointed out that the NN control in [15] does not consider the effect of time delay. The associated simulated results are shown in Fig. 10. Obviously, owing to the existence of the time delay, this control method has Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Fig. 8. Simulated results of Sharma’s control with time delay s = 0.02 s. (a), (b) Elevation and azimuth angles; (c), (d) Angular errors for elevation and azimuth mechanisms; (e), (f) Control efforts.

poor transient and steady responses. The angular amplitude of the MLRS significantly increases and the angles at each shot seriously deviate from the expected values. It can be seen that when there is a time delay, the NN control performance without considering the time delay is drastically deteriorated and presents a diverging trend. 5.5. Comparisons of different controllers for different delays To sum up, the above-mentioned controllers have a good performance on orienting control of the MLRS with actuator delay s ¼ 0:02 by adjusting control parameters except for the NN controller. However, the time delay is unknown in actual applications. Hence, it is necessary to investigate the effect of different s on control performance for the Dokumaci, Sharma and DRNN predictive control methods. We have tested different values of s for the performances of the three controllers as 0.02, 0.06 and 0.1, respectively. The corresponding results are shown in Fig. 11. It can be seen from Fig. 11, as the time delay s increases, the control performance of the Dokumaci and Sharma controllers gradually deteriorates and even becomes unstable. When s achieves 0.06, for Dokumaci’s control, the responses of h and / Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Fig. 9. Simulated results of the proposed control with time delay s = 0.02 s. (a), (b) Desired and actual values for elevation and azimuth angles; (c), (d) Angular errors for elevation and azimuth mechanisms; (e), (f) Theoretical and estimated values for elevation and azimuth angles; (g), (h) Control efforts.

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Fig. 10. Simulated results of NN control in [15] with time delay s = 0.02 s. (a), (b) Elevation and azimuth angles; (c), (d) Control efforts.

oscillate significantly in positioning process, and the orienting performance deteriorates in salvo firing. Similar conclusion can be obtained as that in Fig. 11(e) and (f), for Sharma’s control, when s reaches 0.1. On the contrary, the proposed control keeps an excellent performance in spite of the increase on s, as shown in Fig. 11. The aforementioned fact indicates that the proposed control in this paper is robust to the variations of s. The above analyses demonstrate that the proposed controller is feasible and robust in the presence of actuator delay. 5.6. Control performance with time-varying delay It can be seen from the above analyses that the proposed control is effective with the constant actuator delay. In this subsection, we assessed the control performance of the proposed control method with time-varying time delay, which alters randomly with time. Particularly, a uniformly distributed random signal within [0.02, 0.06] s is selected as the delay, which is shown in Fig. 12. The simulated results are shown in Fig. 13. Obviously, the control performance with varying-time delay in the positioning process is similar to that with constant time delay from Fig. 13(a) and (b). From Fig. 13(c), the maximum error of the elevation angle with time-varying delay is 6  103 rad, which is 75% larger than that with constant time delay. It is observed from Fig. 13(d) that the maximum error of the azimuth angle with time-varying delay is 4.4  104 rad, which is 20.5% larger than that with constant time delay. The azimuth error of varying-time delay returns to zero before the next rocket launch, which is similar to that of constant time delay. Although the elevation error of varying-time delay at the beginning is large and the steady-error does not go back to zero before the next rocket launch, it gradually decreases and returns back to zero under the action of the proposed control. In addition, from Fig. 13(e) and (f), the DRNN estimator still has a good estimation accuracy despite of the time-varying actuator delay. It can be concluded that the performance of the proposed control approach remains satisfactory in the case of time-varying actuator delay.

6. Experimental verification 6.1. Experimental setup The time delay in the actual control system is about 0.035 s. In order to experimentally validate the proposed control strategy, a salvo of eighteen rockets is fired with 1 s interval. The schematic and photograph of the experimental setup are shown in Fig. 14. The experimental setup consists of an 18-tube MLRS, upper computer, embedded controller (MS320F2812 DSP, TI Corporation), servo drivers, PMSMs, speed reducers, fiber optical gyroscope (type: SDI-TX70M, sampling frequency: 200 Hz, Beijing Seven Dimension Information Corporation). The fiber optic gyroscope mounted on the elevation mechanism senses the attitude signal and transmits it to the embedded controller through the signal cable. Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Fig. 11. Simulated results of elevation and azimuth angles for different controllers with (a), (b) time delay s = 0.02 s; (c), (d) time delay s = 0.06 s; (e), (f) time delay s = 0.1 s.

Fig. 12. Time-varying actuator delay.

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Fig. 13. Simulated results of the proposed control with time-varying delay. (a), (b) Desired and actual values for elevation and azimuth angles; (c), (d) Angular errors for elevation and azimuth mechanisms; (e), (f) Theoretical and estimated values for elevation and azimuth angles; (g),(h) Control efforts.

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Elevation mechanism Elevation PMSM

Elevation speed reducer

Elevation servo driver Upper computer

Embedded controller Azimuth servo driver Azimuth PMSM

Azimuth speed reducer Azimuth mechanism

Fiber optical gyroscope

Vehicle

(a)

(b) Fig. 14. Experimental setup. (a) Schematic diagram; and (b) Photograph.

Fig. 15. Experimental results of DRNN control without considering time delay. (a) and (b) Angular errors for elevation and azimuth mechanisms.

6.2. Experimental results To prove the validity of the DRNN predictor-based controller, the DRNN controller without considering time delay is compared for the response of the MLRS. The DRNN controller without considering time delay means that the predictor is ignored in the proposed control strategy. The time histories for the angular errors of the elevation and azimuth mechanisms for the two controllers are shown in Figs. 15 and 16. From Figs. 15 and 16, it is clear that the angular errors for the elevation and

Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489

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Fig. 16. Experimental results of the proposed control. (a) and (b) Angular errors for elevation and azimuth mechanisms.

Table 3 Comparison of the two orienting control system. Control method

MAE for Dh (rad)

MAE for D/ (rad)

RMS for Dh (rad)

RMS for D/ (rad)

DRNN predictive DRNN un-delayed

2.34  103 6.25  103

4.48  104 2.07  103

1.22  104 3.36  104

1.82  105 1.40  104

azimuth mechanisms of the DRNN predictive control drop significantly compared with that of the DRNN un-delayed control, which indicates that the proposed control is effective in reducing orientation deviation. The angular error responses of the orienting control system with different control methods are summarized in Table 3. It can be observed from Table 3 that the proposed control method can reduce the maximum absolute error (MAE) and the root mean square (RMS) of the angular errors relative to the controller without delay compensation, by 62.6% and 63.7% for the elevation mechanism, 78.4% and 87.0% for the azimuth mechanism, respectively. The above analyses indicate that a high orienting performance is achieved by the DRNN predictive control, which also validates the proposed control method. 7. Conclusion In this paper, a neural network predictor-based control was developed for an uncertain multiple launch rocket system (MLRS) with actuator delay. The nonlinear MLRS is approximately linearized by using the dynamic recurrent neural network and the feedback linearization method. Then the modified Smith predictor is used to deal with the time delay for the linearized system. The simulated results showed that the proposed control have a good orienting performance for the MLRS. Comparisons of different controllers with different time delays demonstrate that the proposed control has a better robustness regarding the variations of time delay. Finally, an experiment of the salvo firing of MLRS was implemented and the experimental results further proved the validity of the proposed control. However, some aspects still need to be improved; for example, how to deal with control problem for the nonlinear uncertain MLRS with time-varying actuator delay is an open and interesting topic. Acknowledgement The authors would like to thank the support by the Natural Science Foundation of Jiangsu Province, Grant No. BK20190417; and National Natural Science Foundation of China, Grant No. 51875287. References [1] A.E. Gamble, P.N. Jenkins, Low cost guidance for the Multiple Launch Rocket System (MLRS) artillery rocket, IEEE Aero. El. Sys. Mag. 16 (2001) 33–39. [2] T. Jitpraphai, M. Costello, Dispersion reduction of a direct fire rocket using lateral pulse jets, J. Spacecraft Rockets 38 (2001) 929–936. [3] B. Pavkovic, M. Pavic, D. Cuk, Enhancing the precision of artillery rockets using pulsejet control systems with active damping, Sci. Tech. Rev. 62 (2012) 10–19. [4] W. Tang, X. Rui, G. Wang, Z. Song, L. Gu, Dynamics design for multiple launch rocket system using transfer matrix method for multibody system, P. I. Mech. Eng. G-J. Aer. 230 (2016). [5] F. Scheurpflug, A. Kallenbach, F. Cremaschi, Sounding rocket dispersion reduction impact by second stage pointing control, J. Spacecraft Rockets 49 (2015) 1159–1162. [6] J.E. Cochran, D.E. Christensen, Launcher/rocket dynamics and passive control, Proceedings of the 7th AIAA Atmospheric Flight Mechanics Conference, 1981. [7] J. Hu, D.W. Ma, Y.J. Guo, W.X. Zhuang, F. Yang, Optimal PID Position Controller of Multi-Rocket Launcher Using Improved Elman Network, Proceedings of the 8th World Congress on Intelligent Control and Automation Jinan, China, pp. 2424-2429.

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Please cite this article as: B. Li, X. Rui, W. Tian et al., Neural-network-predictor-based control for an uncertain multiple launch rocket system with actuator delay, Mechanical Systems and Signal Processing, https://doi.org/10.1016/j.ymssp.2019.106489