Model Free Command Filtered Backstepping Control for Marine Power Systems

1College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, No. 29, Yudao Road, Nanjing 210016, China 2Institute of Information, JinLing Institute of Technology, No. 99, Hongjing Road, Nanjing 211169, China 3Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Institute of Automation, Jiangnan University, No. 1800, Lihu Road, Wuxi 214122, China


Introduction
The structure of modern marine power systems has been evermore complicated, especially the emergence of highperformance ship electric propulsion applications.With the development of modern marine power system, more reliable and stable requirements are needed for marine power systems.In recent years, researchers have found that chaotic oscillations occur in marine power system during the voyage or paroxysmal bursts.Chaotic oscillations could lead to the system instability, which poses a potential threat to the safe operation of the marine power grid [1][2][3].
At present, most of the power system chaos control methods mainly focus on land-based power systems, such as adaptive control, feedback control, and inverse system control [2,[4][5][6][7].Obviously, the marine power systems can be regarded as a special case of land-based power systems.As a result, a large number of control methods of land-based power systems can be extended to marine power systems.However, in the actual system, the accurate value of speed signal and the model parameters are difficult to obtain, which will make many model-based control algorithms difficult to be applied [8].
In control theory, backstepping is a technique developed in the 1990s for designing stabilizing controls for a special class of nonlinear dynamical systems [9], which has wider application range than output feedback [10,11].These systems are built from subsystems that radiate out from an irreducible subsystem stabilized by using some other methods.Thanks to this recursive structure, the designer can start the design process at the known-stable system and "back out" new controllers that progressively stabilize each outer subsystem.The process terminates when the final external control is reached.Therefore, this process is known as backstepping.So far, backstepping control has made a lot of achievements, such as adaptive backstepping control, adaptive sliding mode backstepping control, and dynamic surface control [12][13][14].
Recently, model free control has increasingly received more attention in solving complex and practical problems, such as active disturbance rejection control (ADRC) [15], model free adaptive control (MFAC) [16][17][18], and intelligent control [19].Based on the aforementioned works, this paper develops two model free adaptive backstepping controls with command-filtered compensation for marine power systems.
In order to suppress the chaotic marine power system oscillations, based on extended state observer (ESO) and adaptive neural network observer (NNO), the model free adaptive command-filtered backstepping chaos controller is designed.The rest of this paper is organized as follows.In Section 2, a brief description of two parallel nonlinear mathematical models is introduced.In Section 3, ESO and adaptive NNO are designed to estimate the velocity signal and the unknown dynamic model.Moreover, the uniform form of ESO and NNO is given.In Section 4, the model free command-filtered backstepping controller is proposed based on the uniform observer form.Simulation results are presented to show the effectiveness of the proposed control technique in Section 5.
Finally, some conclusions are made in Section 6.

Marine Power System Modeling and Problem Formulation
The basic structure of the power supply network for marine power system can be expressed as Figure 1, where  1 ∠ 1 and  2 ∠ 2 are emf of two generators in the system, respectively.  1 and   2 are synchronous reactances of two generators, respectively.  and   are the line resistance and reactance, respectively. and  denote the system loads.Because of the short-circuit in the marine power system, the line resistance is very small, which can often be neglected.
Considering the same case of generator parameters, let  =  1 −  2 and  =  1 −  2 become the relative power angle and relative power angle velocity of the two equivalent generators.Then two machines interconnected system can be described as follows: where  and  are equivalent inertia and damping, respectively.  is the input mechanical power of generator;   denotes the electromagnetic power of system output.  ⋅ Δ cos  represents the electromagnetic perturbation introduced to study chaotic motion for the marine power system under disturbance.  ⋅ Δ expresses the amplitude of disturbance, and  describes the frequency of disturbance.
Through the transformation  = √  /,  1 () = (), and  2 () = √/  ().Equation (1) can be written as where  = √  /,  =   /  ,  = Δ, and  = √  /.According to the transformation, we know that the system state variables  1 and  2 are obtained by the transformation of  and , which have the physical meanings of the power angle error and the power angle error relative velocity between the two generators.However, if the value √/  is imprecise, accurate state  2 () cannot be gained.In the following works, a novel model free control method is proposed under one condition; that is, only power angle () can be measured.
In order to ascertain subject for further elaboration, we define ȧ = /.In this paper, then (2) can be rewritten as Let  = [ 1 ,  2 ]  , and () = − sin  1 −  2 +  +  cos() sin  1 .In the above marine power system (3), when amplitude  and frequency  of disturbance satisfy certain conditions, the chaotic motion will occur.In the case of suppressing the chaotic motion, a controlled input  must be added to the equation of state (3); namely, If the parameters of model ( 3) cannot be obtained, () can be regarded as an unknown function, and the state  2 also cannot be measured.

Observer-Based Model Identification and State Estimation
In this section, there are two methods proposed to estimate the () and the state  2 of (4).One is the ESO method; the other is the neural network observer method.

Extended State Observer Design.
It is assumed that only power angle  =  can be measured for marine power system (3).Hence in this paper, the third-order ESO is designed, which is used to estimate the state  2 and the unknown function ().Define the unknown function () as an extended state  3 .Let  3 = () and ẋ 3 = , where () is an unknown function.Assume that |()| < , and then system ( 4) is equivalent to For purpose of estimating the state  2 and the unknown function (), the following third-order ESO [13,17] is designed: where and   > 0,  = 1, 2, 3, are parameters of observer (6).And the nonlinear function fal(⋅) is defined as Let  be the sampling period of control.In general,  is selected as  = 5 ∼ 10.Until now, there is no reliable and theoretical analysis method available for the third-order ESO.Fortunately, based on [20], if the suitable parameters of observer (6) are selected, the following results can be gained:

Adaptive Neural Network
Observer Design.The model ( 4) can be represented as where Radical basis function (RBF) neural network is usually used to model nonlinear functions for its good capabilities in function approximation [21].It is a well-known result that for  restricted to a compact set of  and for some sufficiently large numbers of hidden layer neurons, there exist weights and thresholds so that any continuous function on the compact set  can be represented by a recurrent neural network.The functions () are approximated as follows by using neural network system with x, their input being estimated: where x is the estimation of .Ŵ ∈  ×3 is the estimation weight matrix of the RBF neural network.Therein,  is the number of nodes of the implicit layer.And Φ(⋅) = [(⋅), . . .,   (⋅)]  acts as an activation function vector, which is usually assumed to be a Gaussian function as follows:

𝑗
) ,  = 1, . . ., , where ]  ∈  3×1 and   represent the center vector and the width vector of the basis function, respectively.The approximating property of the nonlinear models depends on the center vector, width vector of Gaussian function, and the number of implicit layers .The original functions () in ( 11) can be expressed as where  is the functional reconstruction error of the neural network.In general, even given the best-possible weight values, the given nonlinear function is not exactly approximated and the functional reconstruction error is still remaining. * is the optimal parameter vector required for analytical purpose satisfying with bounded as ‖ * ‖ ≤ .
With the neural network approximation, the dynamic equation of a NNO which estimates the states in ( 4) is given as follows: Mathematical Problems in Engineering where  = [ 1 ,  2 ]  is the observer gain vector; x = [x 1 , x2 ]  , and  0 will be designed later.Define the state and the output estimated errors as x =  − x and yield the error dynamics from ( 4) and ( 15): where  = −  , W =  * − Ŵ, and  =  *  [Φ()−Φ( x)].In general, the neural network basis function is bounded.This implies that every element of Φ() − Φ( x) is bounded; that is, ‖Φ() − Φ( x)‖ ≤ Φ  with Φ  constant.
In order to construct the vector  0 , the algebraic Ricatilike equation   Γ + Γ + Γ 2 ≤ − 1 for  1 > 0 is considered [22].Using the positive definite matrix Γ, the vector  0 is chosen as  0 = Γ −1 .It is shown below that this choice will guarantee the stability of the observer.
Theorem 1. Considering the observer system (15), the update law for the parameters of the neural network system is where Υ = Υ  > 0 and  > 0.Then, the state estimation errors x1 , x2 and parameter estimation errors are uniformly ultimately bounded.
Proof.The proof is similar as [22].

Command-Filtered Backstepping Controller Design
It can be seen above that ( 19) is a strict-feedback form, so the controller can be designed via the backstepping idea.The differential expansion and control saturation problems exist in the traditional backstepping control.Therefore, Farrell et al. have introduced a constrained command filter into the adaptive backstepping control system [23], and the command filter is used to eliminate the impact of derivative of the "virtual controls" and the control saturation.Commandfiltered backstepping control is different from backstepping control, such as the design procedure.The block diagram of the proposed control algorithm is described as Figure 2.
Define the tracking error variables  1 and  2 which are introduced as follows: where   1 and x 2 are the filtered-command of x1 and x2 , respectively.From (19) and (20), it can be seen that Consider the candidate Lyapunov function The time derivative of  1 with respect to time is given by The virtual controller (i.e., outer-loop controller) can be designed as where  1 is a positive definite constant to be designed.Substituting ( 24) into ( 23), we have V 1 ≤ 0. Pass x 2 through a filter [24], which is shown in Figure 3.
And the state-space model of constrained command filter can be described as where and   is the output of the constrained command filter.Therein,  and   represent the damping and the bandwidth of filter, respectively.
Redefine tracking error  1 =  1 − , and the filter error compensation is designed as We choose the Lyapunov function Then, taking the time derivative of  2 yields If the global control law is extracted as the time derivative of Lyapunov function  2 is described as where  2 is also a positive definite constant.Equation (31) means that  1 and  2 are uniformly ultimately bounded.Further, combined with the results of Section 3, we know all error signals are bounded.

Simulation Results
Simulations are performed in the MATLAB/SIMULINK environment.From the numerical analysis of the marine power system's chaotic motion, the results can be obtained that when the amplitude  = 1.3, the chaos will appear in the marine power system under  = 0.4,  = 0.2 and disturbance frequency  = 0.8.It can be obtained that the motion state of the marine power system is in Figures 4-5.From Figure 4, it can be seen that the system power angle and the angular velocity of the phase diagram of movement are ergodic, which shows that the system appears in chaos. of commandfilter are chosen as  =  = 0.1 and   = 20.The parameters of ESO are designed as  1 =  2 = 0.9,  1 = 100,  2 = 1000,  1 = 10,  2 = 100, and  3 = 1000.The initial state is  0 = [0.1,0].In the following simulation, the control signal  is added to the marine power system when the chaotic motion occurs after 100 seconds.Figures 6 and  7 show the curves of power angle and the angular velocity of marine power system with ESO method.And the phase diagram is expressed in Figure 8. Figure 9 shows the Actual () function and its estimation f().
It can be seen from Figures 6 and 7, before 100 seconds, that the power angle and the relative power angle velocity  are in chaotic state.While the designed controller is added after 100 seconds, the system is quickly stabilized, which indicates that the proposed ESO-based control algorithm has a very reliable and stable ability for the marine power system's chaotic motion. 0 = [0, 1]  , Υ = diag[5 × 10 4 ], and  = 0.001.The initial states of the marine power system and NNO are  0 = x0 = [0.1,0].In the following simulation, the control signal  is also added to the marine power system when the chaotic motion occurs after 100 seconds.Figures 10 and 11 show the curves of power angle and the angular velocity of marine power system under NNO method.And the phase diagram is shown in Figure 12. Figure 13 shows the actual () function and the estimation f().

Simulation 2 (NNO-Based Model Free Adaptive
It can be seen from Figures 10 and 11, after 100 seconds, that the designed controller is added to the chaotic marine power system, and the system is quickly stabilized, which indicates that the proposed NNO-based control algorithm also has a very reliable and stable ability for the marine power system's chaotic motion.

Conclusion
Based on observer techniques, two novel model free adaptive command-filtered backstepping control methods are proposed for marine power systems.In the developed two model free adaptive command-filtered backstepping controls, there are three main problems solved.They are the following: (1) Velocity signal does not need to be known.The proposed control algorithms can achieve the closed-loop stability without speed sensor.(2) The proposed control methods do not need dynamic mathematical model of marine power systems.
(3) The proposed two controls can eliminate the impact of derivative signal and control saturation.In addition, the stability analysis is given for closed-loop control system.Simulation results show that the proposed method not only guarantees the closed-loop stability of the controlled marine power system, but also well identifies the velocity state and the unknown dynamic model.Universities (JUSRP11562), and the Natural Science Foundation of Jiangsu Higher Education Institution (14KJB120013).

Figure 1 :
Figure 1: Block diagram of marine power systems.

Figure 2 :
Figure 2: Proposed two model free adaptive command-filtered backstepping controls block diagram.

Figure 3 :
Figure 3: Structure of constrained command filters.

Figure 8 :Figure 9 :
Figure 8: Phase diagram of power angle  and relative power angle velocity  (Simulation 1).

Figure 12 :Figure 13 :
Figure 12: Phase diagram of power angle  and relative power angle velocity  (Simulation 2).