Chaos control in networked permanent magnet synchronous motor using Lyapunov‐based model predictive subject to data loss

This study investigates the chaos control in the permanent magnet synchronous motor (PMSM) using a Lyapunov‐based model predictive approach subject to data loss. PMSM, as a rich nonlinear dynamic, can demonstrate chaotic behavior when its parameters are in a certain area. Thus, the performance of the system will degrade in this condition. Moreover, in some networked applications, especially in the modern automation industry, data can be lost at the sensor‐controller and controller‐actuator links. It will lead the system to illustrate the chaotic behavior. Thus, by assuming a PMSM under data loss, a Lyapunov‐based predictive model is applied to control the chaos in the PMSM in the presence of data loss. It shows that the PMSM system also be effective if the states of the system are not available or the system faces data loss. Sufficient conditions are discussed for the control of chaos in these conditions. Finally, the influence of the method on the control performance is evaluated via simulations on a nonlinear model for the PMSM.


INTRODUCTION
Nowadays the permanent magnet synchronous motor (PMSM) has been used in many applications such as robotic systems and automation. 1,2The dynamics and control analysis of the system plays a prominent role in modern technology since can exhibit rich dynamic behavior including regular, bifurcations and chaotic motions. 3,4Thus, researchers are willing to accept the chaotic behavior of the system.However, this chaotic behavior is not proper for many applications, and it is vital to control and eliminate the chaotic behavior in PMSM because it has a nonlinear and strongly coupled dynamic model. 5Recently, researchers have proposed many different control approaches for chaos control in the PMSM such as sliding mode control, 6 adaptive control, 7 and fractional control. 8However, all these approaches have their shortcoming and disadvantages.
With the growth of production scales and network technology, communication network technology has been implied in control systems.The networked control systems (NCSs) illustrate some advantages such as reducing the complexity TAHMASBI and easy installation, and have been applied in various fields. 9In recent years, modern industry such as modern automation systems demands more and more PMSMs that could be located in different separated areas, where the use of cable connection is not convenient for the installation and maintenance. 10However, one of the important points in the networked systems is when the states of the systems become unavailable.In this condition, the controller link would be disconnected and the system would be unstable.In other words, the use of NCS can create some problems such as data losses and delays in the feedback information, which may degrade the performance of system, and even cause instability. 11,12Due to these problems, researchers have focused on networked PMSM systems to propose various strategies for improving the performance of networked PMSM systems.A self-triggered protocol has been developed to monitor the states of the system and to improve the utilization of communication bandwidth. 13A novel periodic event-triggered sliding mode speed regulation is proposed for networked PMSM systems in Reference 14.The proposed method compensates the external disturbances and parameter uncertainties in the presence of communication network.In Reference 15, the performance of a networked PMSM is studied in the present of a stealthy attack under stochastic communication protocol.The study proposed a stealthy attack, and the effects of the attack on the performance of the networked PMSM are analyzed.For the speed control of networked PSMS, a novel robust control is proposed in Reference 16.A time-varying network delay is assumed on the transmission environment.By combination of the model predictive control and the sliding mode control, a controller is designed to guarantee the stability of the system.A suitable triggering mechanism and sliding mode control with network transmission are studied in References 17,18, respectively.In these studies, it is assumed that the communication is a wireless network.Also, a triggering based on model predictive and sliding mode control is discussed to improve the stability of PMSM system in the presence of network transmission.The Lyapunov function has been investigated in some applications to control PMSM.Due to stability requirements, there have been some applications of Lyapunov-based model predictive controller (LMPC) to system with uncertainty, 19,20 and to motor derive systems. 21,22In Reference 23, the proposed Lyapunov based function is suggested to minimize switching loss in PMSM.In Reference 24, a method for the nonlinear control of a PMSM has been investigated to overcome the problem of nonlinearity and feedback linearization technique.To overcome stability in both normal and faulty conditions, a LMPC and back-stepping method are applied to ensure stability. 25Unfortunately, till now, the control of chaotic behavior of a networked PMSM has not been studied in the presence of network transmission well.In fact, it is difficult to control the chaotic behavior of a PMSM in the presence of network problems such as delay, cyber-attacks, data loss, and so on.
Motivated by the above discussions, a LMPC is adopted to dealing with chaotic behavior of networked PMSM in the presence of data loss.The main idea of the LMPC is to formulate the controller problem in the predictive controller's optimization based on a Lyapunov-based controller.It allows not only to predict the future evolution of the system from current state along a given prediction horizon, also to use the robustness and stability properties of the Lyapunov-based controller.Thus, the LMPC framework is particularly appropriate for controlling system subject to data losses because the actuator can profit from the predicted evolution of the system, to update the input when feedback is lost, instead of setting the input to a fixed value.In other words, it guarantees the efficiency of the PMSM system in chaotic area subject to data loss in the communication link.
This study is organized as follows.In Section 2, the class of nonlinear systems in this study is considered.In Section 3, the chaotic phenomenon in PMSM is introduced.In Section 4, the nonlinear feedback control in PMSM for chaos control is introduced.In Section 5, the used LMPC scheme is introduced along with its stability conditions, and the study of chaos control in the presence of data loss in communication link is investigated.

System definition
In this study, it is assumed that the system is modeled by a nonlinear system with the following state space model: where x(t) ∈ R n is state variables vector, u(t) ∈ R n denotes the vector of input variables, f is locally Lipschitz, and w(t) ∈ R n denotes the vector of disturbance which is bounded.

Assumption
It is assumed that the system closed-loop system Equation ( 1) is stable for a given feedback control h(x(t)).It will be used as a feedback law in the design of the LMPC controller.

CHAOS IN PMSM
PMSM is one of high-efficient motor.Widely use of this kind of motor is in the motor derives servo systems and industry appliances such as robotics.The dynamic of a PMSM can be represented as follows 26 : where i d , i q are direct and quadrature axis stator currents, respectively, and  is angular frequency.This variables are selected as state variables of the dynamic system.u d and u q are stator voltages, J the polar moment of inertia,  the viscous damping coefficient, T L the external load torque, R 1 the stator winding resistance, L d and L q stator inductors,  r the permanent magnetic flux, and finally n p the number of the poles.The Equation ( 2) can get more simplified by the following variables transformation and assuming smooth air gap, where The parameters listed in Table 1 define the values used in Equation (3).The equilibrium points of the simplified system (Equation 3) can be obtained by putting the right hand side of it to  3) with these conditions, the equations to find the system's equilibrium points can be written as,

F I G U R E 2
The states of chaotic system.
The chaos phenomena of the system presented in Equation (3) has been studied in Reference 26.It can be illustrated that in the case of ũd = 0, ũq = 0, TL = 0,  = 5.46,  = 20, ( ĩd (0), ĩq (0), ω(0) ) = (20,0.01,−5) the system is chaotic.The chaotic attractor and system states for the condition are shown in Figures 1 and 2 respectively.As shown in Figure 2, the states of systems do not have a regular behavior and their evolution respect to time is chaotic.

CHAOS CONTROL IN PMSM
As mentioned, PMSM is an engineering system, and it is important to design a proper controller to eliminate the chaos phenomena in PMSM.In other words, ũd and ũq should be designed to control chaos in this system.In this manner, a nonlinear feedback control law is suggested in Reference 27 as follows, The state of chaotic closed-loop system.
where ĩ * d and ĩ * q are desired values or control objective.Also, K d and K q are controller gains which are positive values in this case.By substituting Equation (5) to Equation (3), the closed-loop system can be written as, According to Equation (4), the system presented in Equation (3) has three equilibrium points.In can be seen that one of the equilibrium points, named , is unstable.The objective is to stabilize the system in this equilibrium point by the proposed nonlinear feedback law in Equation (6).Moreover, for simplicity it is assumed that ĩ * d and ĩ * q are time invariant constant values, that means ), the closed loop system is simulated.The states of the closed-system are illustrated in Figure 3.As shown, the closed-loop system is stable and the states of the chaotic system converge to desired point.

CHAOS CONTROL OF PMSM UNDER DATA LOSS
In this section, a LMPC is introduced.It is adopted for the presented closed-loop PMSM model (Equation 6) subject to data loss. 28For this purpose, it is assumed that the PMSM is in a network setting.Figure 4 illustrates the schematic F I G U R E 4 Networked control system subject to data loss.
of NCS for PMSM.In the real application, transmission channels can be lost whether at the sensor-controller link or controller-actuator links.To deal with the condition, the proposed model based nonlinear model predictive control is useful.By applying the approach, data at each sampling time can be predicted.However, it is vital that there is no data loss in ultimately bounded.Also, it can be proved that, 28 the closed-loop system will be stable under data loss if the maximum time in which the loop is open, is shorter than a given constant that depends on the parameters of the system.
In the rest of the present section, data loss model and LMPC algorithm are investigated.Finally, this approach is applied in PMSM.

Data loss model and closed-loop system under sample
To model data loss, an extra variable s(t k ) is introduced where t k = t 0 + kΔ with t 0 is initial time, k = 0, 1, … ., and Δ is sampling time.When s(t k ) = 1 the full state vector is available for the controller and when s(t k ) = 0, the full state is not available.It means that, the system operates in open loop.In this regards, the LMPC uses the model of system to predict the trajectory of system for a given u(t) with t ∈ [t k , t k+N ] where N is the prediction horizon.
Definition 1.The sampled trajectory of the system in Equation (1) associated with a feedback law h(x) and sampling time Δ starting at x(t 0 ), is demonstrated by x(t), and it is obtained by solving the dynamic of system recursively, where t k = t 0 + kΔ, k = 0, 1, 2, … , and x(t 0 ) = x(t 0 ).

LMPC definition and algorithm
The adopted LMPC subject to data loss can be written as a finite horizon constrained optimal control as follows, 28 min Subject to: where S(Δ) is a piece-wise constant function with sampling period Δ, x(t) denotes the predicted sampled states of a nonlinear system for the input computed by Equation ( 8), x(t) is sampled states under the Lyapunov-based controller u = h(x(t)), and Q c , R c are weight matrices that define the cost effort for state and input vector respectively.As shown in the Equation (8b), it is assumed that w(t) = 0.It means that the nominal closed-loop system (system (1) with w(t) ≡ 0 for all t) has an asymptotically stable equilibrium at origin x = 0 for a given feedback control.This assumption is equivalent to the existence of a control Lyapunov function for the system ̇x = f (x, u, 0).Also, it shows that the optimization problem does not depend on the uncertainty and assures that the states of the nonlinear system would be predicted in there bounded uncertainty.The optimization problem is solved at each sampling time to prove that the LMPC inherits the stability properties of the Lyapunov-based controller.When data losses are taken to account, in order to prove that the LMPC inherits the same properties of the Lyapunov-based controller, the constraint must hold along the whole prediction horizon.In fact, when data is lost, the optimal input trajectory evaluated guarantees that the predicted decrease the of the Lyapunov function using the nominal model is at least equal to the one obtained applying the Lyapunov-based controller.
The procedure of the LMPC algorithm is summarized in Algorithm 1.According to stability theorem presented in Reference 28, if no data losses are present, the system converge to a neighborhood of the origin, and it is ultimately bounded.Moreover, it is stated that in the presence of data loss, the closed-loop system state will be bonded if the maximum lost data is smaller than sampling time plus prediction horizon that means T data loss ≤ NΔ, where N is prediction horizon, and Δ is sampling time.In other words, if the system meets the condition for each with V(x) ≤ .The details of the theories are explained in Reference 28, and readers can refer to the reference for more information.

Application in the networked-PMSM
Consider the PMSM system presented in Equation (3).As mentioned, the system has three equilibrium points (two locally asymptotically stable and one unstable) which can be obtained through Equation ( 4).The nonlinear feedback control objective is to stabilize the closed-loop system at ( ĩ * d , ĩ * q , ω * ) = (3, 5, 5) as an unstable equilibrium point (Section 4).To apply the introduced LMPC controller when there is data loss on actuator communication link, it can be shown that the system presented in Equation (3) belongs to the following class of nonlinear system, It can be assumed that there is a bounded disturbance to analyze the system in the real application.So, we could rewrite the Equation (9) in the following form: By considering the control Lyapunov function V(x) = x T Px for the system in Equation ( 6) with It can be illustrate that the closed-loop system by introduced nonlinear control law and defined Lyapunov function is asymptotically stable.The system is implemented by using the introduced LMPC controller with sample time Δ = 0.05 s , and a prediction horizon N = 5.The cost functions in Equation ( 8) are defined by the weight matrices Q c = P and R c = 10 −6 .Due to stability condition in the presence of data loss, the closed-loop system would be stable for T data loss ≤ NΔ, otherwise the system would be unstable.The model has been simulated in MATLAB using fmincon and a Runge-Kutta solver.The fmincon command is according to, By comparing Equation (12) with Equation ( 8), it can be seen that, The noise of the system is modeled by using randn command in the MATLAB.We assumed that the system in t ∈ [0, 5] s is open loop, and for t ∈ [5, 10] s the system is closed-loop.Moreover, it is assumed that sampling time is Δ = 0.05 s and N = 5.Thus, for t ∈ [7,7.2],there is a data loss with T data loss ≤ NΔ, and for t ∈ [8, 9] the data loss is implemented with T data loss > NΔ.
The Figure 5 illustrates the states of the system.As shown, when the system is open loop (t ∈ [0, 5]), the system is chaotic and when the system is closed-loop the states converge to desired values.As mentioned, it is assumed that for t ∈ [7,7.2]there is data loss, however, the system is stable and LMPC estimates the control input properly since ΔN = 0.25 s < 1 which it satisfies the condition of stability.However, for t ∈ [8, 9], the system is unstable and states diverge from desired values since 1 s > ΔN, and the condition of stability is not satisfied.
Figure 6 shows the control inputs for the system subject to data loss.As seen, the control inputs are zero when the system operates in open loop.When the system is in the closed-loop condition and there is data loss, the control inputs would be obtained using the LMPC presented in Equation (8).The obtained controls inputs are the family of piece-wise constant function, and for every prediction horizon, there is a control signal.According to Figure 6, the obtained control commands, when the stability condition not satisfied, are not proper.The fact can be also shown in the trajectory of the system in Figure 7 for states ĩd and ĩq with the initial condition, ( ĩd (0), ĩq (0), ω(0) ) = (20,0.01,−5).As shown, the system is chaotic when it is in the open loop condition.When it is closed by a nonlinear feedback controller, the system would converge to unstable equilibrium point.If the condition of stability is satisfied, the system would remain around of the equilibrium point.However, if it is not satisfied, the trajectory will diverge from equilibrium point and system is unstable.In this condition, by removing the data loss, the system would be stable and trajectory would converge to the  equilibrium points.It should be noted that stability region is calculated for the equilibrium point, where   is V ≤ 1.441, that is,  = 1.441.
Figure 8, denotes the fmincon parameters related to simulation of the LMPC.According the Equation ( 12), for the fmincon function, the conditions C eqi ≅ 0 (i = 1, 2, 3) and C ≤ 0 should be satisfied.These conditions show in Figure 8.As seen, when the stability condition in the presence of data loss is satisfied, we have C eqi ≅ 0 (i = 1, 2, 3) and C ≤ 0, otherwise C eqi ≠ 0 (i = 1, 2, 3) and C ≰ 0. It means that the implemented optimization problem works accurately according to Equation (12).

) TA B L E 1
Parameters of a PMSM.Parameters Definition ĩd Transformed direct axis stator current ĩq Transformed quadrature axis stator current ω Transformed angle speed of motor ũd Transformed direct axis stator voltage ũq Transformed the quadrature axis stator voltage TL Transformed external load torque  System parameters  System parameters F I G U R E 1 Strange attractor in permanent magnet synchronous motor.

F I G U R E 5
States of the system subject to data loss.

F I G U R E 6
Control commands.

F I G U R E 7
Trajectory of system.