The paper performs a comparison of the proposed Adaptive Neural Network Dynamic Surface Control (ANDSC) algorithm with other approaches which are Adaptive Neural Networks Backstepping Control (ANB) and Adaptive Neural Networks Sliding Mode Backstepping Control (ANSB).
Nghiên cứu khoa học công nghệ PERFORMANCE ASSESSMENT OF ADAPTIVE NEURAL NETWORK DYNAMIC SURFACE CONTROLLER WITH ADAPTIVE NEURAL NETWORK BACKSTEPPING CONTROLLER AND ADAPTIVE NEURAL NETWORK SLIDING MODE BACKSTEPPING CONTROLLER Hoang Thi Tu Uyen*, Pham Duc Tuan, Le Viet Anh, Dam Quang Truc, Phan Xuan Minh Abstract:In recent years, adaptive nonlinear control based on Backstepping, Sliding Mode Control, Dynamic Surface Control Techniques have been used to design controllers for surface ship to track desired trajectory The paper performs a comparison of the proposed Adaptive Neural Network Dynamic Surface Control (ANDSC) algorithm with other approaches which are Adaptive Neural Networks Backstepping Control (ANB) and Adaptive Neural Networks Sliding Mode Backstepping Control (ANSB) This comparison of these control methods is based on digital simulationsfor a surface ship The results of the simulations show that ANDSC and ANSB are given better control qualities than ANB Keywords:Sliding mode control, Dynamic surface control, Neural network, Backstepping, Surface ship INTRODUCTION The important of autopilot including unmanned surface vessels (USVs) and unmanned underwater vehicles (UUVs) has been demonstrated in portage, search and recovery, exploration, surveillance, monitoring marine environment, and military applications[1] Accurate trajectory tracking is crucial to the success of these vehicles and the development of the control in the maritime field However, there is much challenges in controlling marine vehicle The challenges might mainly stem from the fact that: 1) the working environment of marine vehicle are usually dynamic, complex and unstructured, which bring unpredictable perturbations to control system, for example, ocean currents, wave (wind generated) and wind, 2) the dynamic model of marine vehicle is high nonlinear, uncertain and time-varying which might be too complicated to be used for controller design Due to numerous important applications of marine vehicle, the identification and control problems of these vehicles have recently received a lot of attention A good number of novel model-based design techniques and interesting solution are presented for the motion control problems of marine vehicle [1] Typically, the design of model-based controller relies on system’s mathematical model and traditional model-based adaptive controller are useful when dealing with systems in which the dynamic are linear-in-theparameters, regressors are exactly known, uncertainties are parametric and time-invariant [2] However, the presence of modeling errors, in the form parametric, nonparametric uncertainties, unmodeled dynamics are almost inevitable in marine control applications which are characterized by large unknown perturbations from uncertain ocean environment including ocean currents, waves and wind The presence of nonparametric uncertainties, might excite high-frequency unmodeled dynamics, which could disrupt the function of adaptive controller and cause closed-loop system’s instability [3] So, how to handle such dynamical uncertainties is one of most important and challenging issues in the motion control of marine vehicles Tạp chí Nghiên cứu KH&CN quân sự, Số 52, 12 - 2017 35 Kỹ thuật điều khiển & Điện tử Owing to the universal approximation capabilities, learning and adaptation, parallel distributed structures of neural networks (NNs) [4], [5], it has been found that adaptive NN control is suitable for controlling complex, highly uncertain, nonlinear systems in industrial applications [6], [7], [8], as well as being used to resolve the challenge of navigating the maritime vehicle [9], [10], [11] Neural networks have been used with a variety of control methods for surface ships such as backstepping method, sliding mode control, dynamic surface control This paper performs a comparison and assessment of the adaptive backstepping control proposed in [12] with two methods proposed by the authors: the adaptive sliding mode backstepping [13] and the adaptive dynamic surface control [14] based on artificial neural network The purpose of the paper is to evaluate the quality of controllers when solving the problem of tracking the desired trajectory for surface ship within the conditions of dynamic model of ship containing uncertain components and to be affected by unknown environmental disturbances The comparisons are implemented by numerical simulations on the computer and are evaluated based on the criteria following: 1) The response of the ship to the various reference trajectories, 2) Tracking errors of the ship, 3) Oscillation of the control signals, 4) Transient time of the controllers These assessments assist navigational control system designers in selecting the control algorithms that meet the quality requirements for their control problem MARINE SURFACE SHIP DYNAMICS In this section, we consider multiple-input-multiple-output (MIMO) dynamics of a three degree-of-freedom (3DOF) surface ship with uncertainties The motion of the ship dynamics is described by the following ordinary differential equations [1],[15]: ̇ = ( )ʋ (1) ʋ̇ + (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ) = where = [ , , y] are the 3DOF position (x, y) and heading (y ) of ship in an earthfixed inertial frame; ʋ = [ , , ] are corresponding velocities in surge (u), sway (v) and yaw (r) in the boy-fixed frame; M is the ship inertia matrix; C(v) is the total Coriolis and centripetal acceleration matrix; D(v) is the damping matrix; J(η) is the 3DOF rotation matrix; g(η) is the vector of gravitational/buoyancy forces and moments; ∆( , ʋ)is the vector of model uncertainties or environmental disturbances which might be caused by unmodeled dynamics, sensor errors; ∈ is the vector of control inputs The coefficients in the rotation matrix J(η) are given by: 0 (ʋ) y y ( )= 0 (ʋ) y y ; (ʋ) = − (ʋ) − (ʋ) 0 (ʋ) 0 ( ) ∆ ( , ʋ) (ʋ) = (ʋ) (ʋ) ; ( ) = ( ) ∆( , ʋ) = ∆ ( , ʋ) (ʋ) (ʋ) ( ) ∆ ( , ʋ) where: ( ) ( ) = ; (ʋ), (ʋ), (ʋ), (ʋ), (ʋ), (ʋ), (ʋ), ( ), ( ), ( ), ∆ ( , ʋ), ∆ ( , ʋ), ∆ ( , ʋ) are unknown functions, and depended on velocity vector ʋ = [ , , ] UNCERTAINTY FUNCTIONS APROXIMATION BASED ON NEURAL NETWORKS The neural network is often used to approximate the uncertainty functions RBF neural network can be considered two-layer network In which the hidden layer consists of the 36 H T T Uyen, …, P X Minh, “Performance assessment of… backstepping controller.” Nghiên cứu khoa học công nghệ radial basis neural elements, the output layer is a linear neural and can be described in the following: ( )= ( )= ( ) (2) where ∈ W Ì is the input vector, W = [ , ,…, ] ∈ is the weight vector, > is the NN node number, S(Z) = [ ( ), … , ( )] is the regressor vector, with ( ) being a radial basis function The commonly used radial basis function is the Gaussian function: −( − ( )= ) ( − ) , = 1,2, … , (3) where =[ , , … , ] is the center of the receptive field and is the width of the receptive field The Gaussian function belongs to the class of localized radial basis functions in the sense that ( )® when ‖ đƠ s1 W1 s2 Z1 W2 Z2 s3 W3 Zn f(Z) Wn sn Input layer Hidden layer Output layer Figure 1.The structure of Gaussian RBF network It has been shown in [16], that for any continuous function f(Z): W ® , where W Ì is a compact set, and the NN approximation (2) (the node number N is sufficiently large), there exists an ideal constant weight vector ∗ such that for each ∗ > max |f(Z) − ∗ S(Z)| < ∗ (4) ∈W ( )= ∗ S(Z) + (Z), ∀ Z ∈ W (5) ∗ where| ( )| < Moreover, it was shown [17] that Gaussian RBF network ( ), with a finite (and also large enough) number of Gaussian nodes centered on a regular lattice on W , and with fixed variances, can uniformly approximate a smooth function f(Z): W R to any chosen tolerance ∗ according to (5) CONTROL ALGORITHMS 4.1 Adaptive Neural Networks Backstepping Controller In this section we present the controller design algorithm of ANB for surface ships The controller design steps are cited from the reference literature [12] Tạp chí Nghiên cứu KH&CN quân sự, Số 52, 12 - 2017 37 Kỹ thuật điều khiển & Điện tử Define the error variables: = h − h and = ʋ − , where h is the desired trajectory, is the virtual control law of the first subsystem The virtual controller is chosen as: = − ( )( − ḣ ) (6) where is a positive design parameter (7) =ʋ− ̇ = ʋ̇ − ̇ = − (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ) − ̇ ̇ = − (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ) − ̇ (8) ( ) = (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ) Let: (9) ( ) is a function includes of all uncertain dynamics in (1) ( ) = [ ( ), ( ), ( )] ∈ where: (10) and Z = [ , ʋ ] ∈ The Gaussian RBF NNs are used to approximate uncertain function ( )can be expressed as: ( ) = ∗ ( ) + ( ) = 1,2,3 (11) ∗ ∗ ∗ ( )| where is ideal weight,| ≤ is NN approximation error with > Since ∗ are unknown, let be the estimate of ∗ The feedback control τ is chosen as: ( )+ (12) = − (h) − + ̇ where: ( )=[ ( ), ( ), ( )] (13) is used to approximate the unknown function F(Z) The updated adaptive laws are chosen as following [12]: ̇ = ̇ =− [ ( ) +s ( − (14) )], = 1,2,3 where s > 0, = 1,2,3 are the modification parameter, and are design parameters The stability of closed-system is proofed in [12] 4.2 Adaptive Neural Networks Sliding Mode Backstepping Controller The stabilization of dynamical systems with uncertainties has been studied in the recent years Most approaches are based upon Lyapunov and linearization methods to design a control Due to the inability of feedback linearization to handle uncertainties, much attention has been given recently to Lyapunov-based control design techniques, such as integrator backstepping (IB) and sliding mode control The backstepping approach presents a systematic method for designing a control to track a reference signal, by selecting an appropriate Lyapunov function by changing the coordinate The robust output tracking of nonlinear systems has been studied by many authors [18], [19] Sliding mode control (SMC) is a robust control method and backstepping is able considered to be a method of adaptive control The combination of these methods yields benefits from both approaches [18], [20] The design steps of ANSB controllers for surface ships to track the desired trajectory are detailed in the reference literature [13] Define the error variables: = h − h and =ʋ− where h is the desired trajectory, is the virtual control law of the first subsystem The virtual control is chosen as: 38 H T T Uyen, …, P X Minh, “Performance assessment of… backstepping controller.” Nghiên cứu khoa học công nghệ where = − ( )( − ḣ ) is a positivedesigned parameter =ʋ− ̇ = ʋ̇ − ̇ = − (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ) (15) − ̇ Let: ( )=− (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ) (16) ( )is function of uncertain dynamics, where ( ) = [ ( ), ( ), ( )] , Z = [ , ʋ ] The Gaussian RBF NNs are used to approximate uncertain function ( )can be expressed as (11) [ + ( )− [ + ∗ ( )+ ( )− (17) ̇ = ̇ ]= ̇ ] The sliding surface is defined in the following: (18) = + The feedback control τ is chosen as: ∗ ∗ ( ) (19) ( )− =− − ̇ + ̇ − ℎ( ) − + where , are arbitrarily small positive design parameters, h(s) is vector of sign function ( ); ( ); ( )] ; ( )=[ ( ), of sliding surface: ℎ( ) = [ ( ), ( )] is used to approximate the unknown function ( ) The updated adaptive laws are chosen as following: ̇ = ̇ = [ ( )s − s ], = 1,2,3 (20) 4.3 Adaptive Neural Networks Dynamic Surface Controller The backstepping technique is based on sequential design steps to determine Lyapunov's control function for strict-feedback nonlinear systems The disadvantage of the method is that the derivative of the virtual control signal must be determined after each step while these signals are a function that is dependent on state vector of the system thus the calculation becomes very difficult for high order nonlinear systems, the problem was called as “explosion of complexity” To overcome this disadvantage, Dynamic Surface Control technique [21], [22] was developed based on backstepping and MSS By using the first-order filter in each step of the backstepping method, it will overcome the downside of the traditional backstepping method The design steps of ANDSC controllers for surface ships to track the desired trajectory are detailed in the reference literature [14] Step 1: The first sliding surface is defined: = − (21) Choose a virtual control law for the first subsystem: =− + ( ) ̇ (22) is passed through a first-order filter with time constant and output of the filter is : ̇ + = (23) The filter error: = − (24) Step 2: Tạp chí Nghiên cứu KH&CN quân sự, Số 52, 12 - 2017 39 Kỹ thuật điều khiển & Điện tử The second sliding surface is defined: =ʋ− (25) (26) [ − ( (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ)) − ̇ ] ( ) = − (ʋ)ʋ + (ʋ)ʋ + ( ) + ∆( , ʋ) + ̇ Let: (27) is a function of the uncertain dynamics, where F( ) = [ ( ), ( ), ( )] , Z = [ , ʋ, ̇ ] The Gaussian RBF NNs are used to approximate uncertain function F( )can be expressed as (11) The feedback control τ is chosen as: ̇ = =− − − ( ) [− − − ( ) + ( )] (30) − ̇ (31) (28) ( )=[ ( ), ( ), ( )] is used to approximate the unknown where: function F( )(27)(27).The updated adaptive laws are chosen as follows: ̇ = ̇ = [ ( ) − s ], = 1,2,3 (29) ̇ = ̇ = ̇ − ̇ =− Let B( , ʋ) = − ̇ , with assumption that B( , ʋ) is bounded function B( , ʋ) ≤ NUMERICAL SIMULATIONS To demonstrate the effectiveness of the proposed control design, we perform numerical simulation on system (1) under the following choices of plant parameters [15], [23], [12]: ( ) = −24.6612 − 1.0948 , ( ) = −25.8 , ( ) = 0.7225 + 1.3274| | + ( ) = −0.1079 + 5.8664| | , ( ) = 0.8612 + 36.2823| | + 8.05| | , ( ) = −0.1052 − 5.0437| | − 0.13| | , ( ) = 1.09 − 0.845| | + 3.450| | , ( ) = ( ) = ( ) = The known ship inertia matrix 0.08| | + 0.75| | , parameters [15] are given by = [25.800 0; 25.6612 1.0948; 1.0948 2.7600] We assume there exist model uncertainties Δ( , ), in the ship dynamics system (1), which are described by Δ( , ) = [1,0.01| | + 0.05, −0.1| | + sin( )] 5.1 Reference trajectory is step function with disturbance In the simulation, we choose the reference trajectory h are step functions Figure Tracking errors 40 H T T Uyen, …, P X Minh, “Performance assessment of… backstepping controller.” Nghiên cứu khoa học công nghệ Figure Tracking trajectories 5.2 Reference trajectory is since function with disturbance In the simulation, we choose the reference trajectory h = [sin( ) , cos( ) , sin ( )] The simulation results show: 1) All of the controllers track reference trajectory 2) The tracking error of the controllers proposed by the authors (ANSB and ANDSC) is smaller than ANB (Figure 2, Figure 7), especially with the reference trajectory is step function, ANDSC tracks faster, quality of tracking is better (Figure 3) 3) Figure Control Inputs Figure 5.Tracking trajectories Tạp chí Nghiên cứu KH&CN quân sự, Số 52, 12 - 2017 41 Kỹ thuật điều khiển & Điện tử Figure Control inputs Figure Tracking errors The control signals of both ANSB and ANDSC controllers oscillate more than the ANB because these two controllers are associated with the sliding mode control (Figure 6) 4) The transition time of the two ANSB and ANDSC controllers is 10 seconds, then these control signals stabilize, similar to the control signal of ANB CONCLUSIONS The simulation results show that: ANDSC and ANSB have better tracking qualities than the ANB proposed in the literature [12] The control signals of the both controllers initially oscillate, but only exist for short periods of time (10 seconds), then stabilize like the control signals of ANB Acknowledgment: This work was supported by PhD training project 911 of Ministry of Education and Training REFERENCES [1] T I Fossen, "Review of Marine Control Systems: Guidance, Navigation, and Control of Ships, Rigs and Underwater Vehicles", vol 28, no 2002 42 H T T Uyen, …, P X Minh, “Performance assessment of… backstepping controller.” Nghiên cứu khoa học công nghệ [2] K S Narendra and A M Annaswamy, "Stable Adaptive Systems" Dover Publication, 2012 [3] P A Ioannou and J Sun, “Robust Adaptive Control,”N/a, vol N/A, no TFRT-1035, p 825, 1996 [4] B M Wilamowski, “Neural network architectures and learning algorithms - 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The stability of closed-system is proofed in [12] 4.2 Adaptive Neural Networks Sliding Mode Backstepping Controller The stabilization of dynamical systems with uncertainties has been studied in... dynamic surface control This paper performs a comparison and assessment of the adaptive backstepping control proposed in [12] with two methods proposed by the authors: the adaptive sliding mode backstepping