An Improved Internal Model Principle Based Multivariable Nonlinear Control Method with Multiclass Nonharmonic Disturbances and Its Application to Speed Control of a Motor Drive System

and Applied Analysis 3 Remark 3. Generally speaking, there exist numerous nonlinear systems satisfying Assumption 2, such as harmonic functions and limit cycles of nonlinear dynamic system. The well-known Van der Pol oscillator [20] is selected as the exosystem in the paper and can be expressed as below:


Introduction
Due to its importance, the problem of disturbance rejection generated by the exosystem has been paid great attention in the world over the past few decades [1][2][3].Especially, this problem in nonlinear system has been concerned in a recent period of time [4][5][6].In terms of internal model principle (IMP), one of the key technologies of rejection of exogenous disturbance is to construct reasonable internal model equations, which is able to duplicate the characteristics of the external disturbance.
In addition, many industrial applications, such as servo system, wind power system, photovoltaic system, and other modern systems, which rely on power electronic equipments to work, [7][8][9], are frequently influenced by the external disturbances generated by exosystem [10,11].In the circumstance of the existence of external disturbances, the conventional PID controller with constant parameters is difficult to satisfy the requirement of high precision control for these systems [12,13].Therefore, rejection of external disturbances by means of new ways [14,15] is an important work to improve the working performance of these industrial systems.From the state-of-the-art technologies in disturbance rejection, the researchers frequently assume the following: (1) the external disturbances are frequently sinusoidal; that is to say, the exosystem generally satisfies the condition of neutral stability such as references [16,17] discussing the suppression of sinusoidal interference with known and unknown frequency, respectively; (2) currently, the focus of the research on disturbance rejection is gradually converting from linear systems to nonlinear systems; nevertheless, the disturbance rejections with semiglobal stability [18] and singleinput and single-output (SISO) [19] systems are concerned. 2 Abstract and Applied Analysis Therefore, the disturbance rejection problem with nonharmonic interferences and multivariable nonlinear systems has not received adequate attention in current studies.However, the nonharmonic disturbances produced by nonlinear exosystem can induce noise and reduce accuracy to these industrial systems [20].In addition, just a few articles on nonharmonic interference suppression aim at one class of disturbance [21].Therefore, the research of multiclass disturbances rejection in multivariable nonlinear system can extend the application of present control theory to a generalized range [22].Hence, in [23], a global multivariable control algorithm is proposed to reject multiclass external nonharmonic disturbances.
In view of the issues described above, the purpose of the paper is to propose an improved internal model principle (IMP) based multivariable nonlinear control algorithm with multiclass nonharmonic disturbances and utilize the proposed method to control a motor drive system to run stably and suppress multiclass external nonharmonic disturbances.As written in [23], the critical points of the problem in disturbance rejection are to model the nonlinear exosystem and present a corresponding control method to suppress the disturbance.Therefore, the essential distinction with the work in [23], this paper addresses the problem of multivariable disturbances rejection with other multiclass exosystems, additionally, which are provided by matrix form.For the difference of external exosystems, the entirely different internal model and a matching control method should be proposed to model and reject the external disturbances in terms of the intrinsic characteristics of exosystem.In practical terms, the construction of internal model replicating with the exogenous interference in the paper is on the basis of selfcontained concepts of expanded steady state generator and internal model proposed in [24].Similarly, the problem of nonharmonic disturbances rejection in the paper is different from the work in [22], which aims to reject a constant disturbance.
The innovative works in the paper are the following: (1) the previous work in references [20,21] is extended to a generalized field with multivariable inputs of nonlinear systems and multiclass nonlinear exosystems; (2) an improved IMP based multivariable nonlinear control algorithm with another multiclass nonharmonic disturbance is presented on the basis of certain characteristic conditions satisfied by exosystem and state information of closed-loop system; (3) a new type of nonlinear multivariable internal models for another multiclass nonlinear nonharmonic disturbance is constructed in terms of the expanded steady state generator and internal model.
The structures of the paper are arranged as follows.It begins in Section 1 with an analysis of the state of the art of disturbance rejection.The formulation of control problem in the paper is given in Section 2. The nonlinear internal models of multiclass external disturbances are designed in Section 3. The nonlinear multivariable state feedback controller is devised in Section 4, and the global convergence of the presented controller is theoretically proved.The implementation of the proposed algorithm to a two-input motor drive system is performed in Section 5 and the conclusions of the research are obtained in the last section.

Formulation of Control Problem
The multivariable nonlinear system with multiclass external nonharmonic disturbances studied in the paper is described in an affine form shown as below [23]: where x ∈ R  indicate the state vectors, u  ∈ R describe the control inputs, f(x) and g  (x) denote the known smooth vector fields, k  (w) represent the external nonlinear disturbances, w ∈ R  are the exogenous vectors generated by the nonlinear exosystem and shown as follows [24]: where matrices  1 and   ∈    ×  ,  = 2, . . ., , and   (w) :    →  are sufficient smooth functions, and   (0) = 0.
For the stability problems of multivariable input system, the key of the study is to convert multivariable nonlinear system into multiple single-input systems [25].
Assumption 1.For the multivariable nonlinear disturbancefree system shown below: there exists control law of sate feedback   (x), which can make the system asymptotically converge to the origin.In consequence, a Lyapunov function V(x) exists and makes the following inequalities hold [20,23]: where , , and  0 belong to class  ∞ functions.
Assumption 2. The flows of the vector field for nonlinear exosystem (2) are bounded.
Remark 3. Generally speaking, there exist numerous nonlinear systems satisfying Assumption 2, such as harmonic functions and limit cycles of nonlinear dynamic system.The well-known Van der Pol oscillator [20] is selected as the exosystem in the paper and can be expressed as below: where  and  are constants greater than zero.Under such circumstances, the limit cycle generated by Van der Pol oscillator is stable.Consequently, ( 6) is written in the form of ( 2), and we can obtain where Assumption 4.There exist a constant   and a set of real numbers  0 ,  1 , . . .,  (  −1) satisfying the following equation: where k  (w)/w) 1 w,  = 2, 3, . . .,   , and  is an operator of Lee derivative.
Apparently, if k  (w) denotes a linear expression of w, Assumption 4 will hold automatically.

Assumption 5. There exists a smooth function h
where G  is a nonzero constant vector defined in R   .
The questions resolved in the paper can be depicted as below.Definition 6.For an arbitrarily given compact subset D w ∈ R   , a state feedback controller u  can always be found to guarantee the solution of original multivariable closed-loop system (1) existence and boundness under arbitrary initial conditions for all w(0) ∈ D w , when  ≥ 0, and even lim  → ∞ x() = C, where C is a constant vector representing the reference value.
In terms of linear observer theory, a Hurwitz matrix F  is chosen to make matrix pair (F  , G  ) be controllable for nonzero constant vector   defined in (9).Due to the observability of matrix pair (Φ  ,   ) and the fact that F  and Φ  have nonintersecting frequency spectrum, hence Sylvester equation   Φ  −    =     has a unique nonsingular solution   .
Let   =    −1  , the nonlinear exosystem can immerse into the system shown as below: In consequence, multiclass nonlinear internal models can be designed as follows: Define an auxiliary error e  as and with derivative of ( 16) along with (1), (14), and (15), we can get

Design of Multivariable Nonlinear State Feedback Controller
On the basis of the established multiclass nonlinear internal models (15) and Assumption 1, the multivariable nonlinear state feedback controller can be designed as follows: Construct a Lyapunov function  as below: where  is a positive real constant.With derivative of function  along with originally nonlinear system (1) and auxiliary error (17), we can obtain ( Assume that there exists a compact set (w) making (w) be a symmetric matrix of negative definiteness for all w ∈ (w).Hence, there exists a positive real number  satisfying the following inequality for all w ∈ (w):

Application to Speed Control of a Two-Input Nonlinear System
Speed control of a permanent magnet synchronous motor drive system is widely utilized in various industrial fields.Hence, consider a two-input variable motor drive system shown as below [25], which is not able to tackle with a singleinput method of previous disturbance rejection: where   and   represent the current in -axis and -axis, respectively,   and   indicate the voltage in -axis and axis, respectively,   and   denote the resistance of stator and flux linkage of permanent magnet,   is the speed,  and   denote number of pole pairs and moment of inertia of rotor, respectively, and   is an increasing load.With conversion of (26) into an affine form as (1), we can get where and control inputs For the sake of understandability, the external disturbances inputs, V 1 and V 2 , which denote different nonlinear nonharmonic disturbance input signals, are both produced by Van der Pol oscillator shown in (6), and let  =  = 1.Hence, Evidently,  21 =  11 and  22 =  12 .It is important to note that  11 ,  21 ,  12 , and  22 are not elements in matrix  1 or  2 , which are all independent matrices defined in (2).For the sake of convenience, let  11 =  21 =  1 ,  12 =  22 =  2 , and  12 (w) =  22 (w) =  2 1 .The bounded limit cycle will be generated by the oscillator; hence, Assumption 2 holds.
Let  1 ,  2 , and  3 all be certain positive constants; the control law of disturbance-free system for (27) can be obtained by means of backstepping control and can be shown as follows: where  ref is the reference speed.It can be verified that disturbance-free system (3) can be stabilized by (x).Owing to its unimportance for the study of the paper, the derivation process of the controller for the disturbance-free system is omitted.Let By means of some calculations and simplifications, we can get To be more specific, the relevant parameters of the motor drive system are depicted as follows: the rated torque of motor   = 5.0 N⋅m, number of rotor pole pairs  = 4, flux linkage of permanent magnet   = 0.18 Wb, resistance of stator   = 1.95 Ω, inductances of stator in -axis and -axis   =   = 0.0115 H, moment of inertia of rotor   = 0.008 kg⋅m 2 , and damping coefficient of motor   = 0.01 N/rad/s.
In addition, we assume ]  ; by application of (30) and (31) and selection of  1 = 8000,  2 = 40, and  3 = 8000, we can get Hence, Assumption 1 is satisfied.Let Hence, Assumption 5 holds.Assume that V 1 and V 2 denote different nonlinear disturbance input signals and act on the currents of -axis and axis, respectively.Choose V 1 =  1 and V 2 =  2 .In other words, V 1 and V 2 represent that the system are immersed into two differently external disturbances.
In terms of (35), let Then, there exist matrices Φ 1 = [ According to (36), let In that way, there exist matrices Φ Choose Φ 12 =  2 ; after some calculations it can be gotten that Select Φ 22 = [ −1 0 −1 0 ]; after some calculations we can get Consequently, Assumption 7 is satisfied.Assume τ1 (w) =  1  1 (w), where  1 ∈   1 × 1 is a nonsingular matrix, and by derivation of τ1 (w) along with (2), we can get where  1 (w) = Φ 1 +  1 (w),  1 (w) = Φ 12  12 (w).Suppose τ2 (w) =  2  2 (w), where  2 ∈   2 × 2 is a nonsingular matrix and by derivation of τ2 (w) along with (2), we can get where  2 (w) = Φ 2 +  2 (w),  2 (w) = Φ 22  22 (w).Choose It has already been verified that all assumed conditions required for systems (27) and ( 6) are satisfied.Following the design procedure given in Sections 3 and 4, the corresponding nonlinear internal models and state feedback controller are able to be designed as below: In order to evaluate the validity of the proposed method, the numerical simulations are performed in Matlab software with the sampling interval Δ = 0.001 s and initial conditions (0) = [0.1 0 1.0] , η(0) = [0 0 0 0], and w(0) = [1 −1].In addition, the reference values of -axis current   and speed   are set to zero and 600 r/min, respectively, and -axis current   tracks the linear variation value of load   .Ultimately, the simulation results are depicted in Figures 1, 2, and 3. Figure 1 describes the multiclass nonlinear nonharmonic disturbances V 1 and V 2 , in which we can see that the exogenous disturbances immersing into the system are bounded and different, and they can originate from different exosystems so long as the exosystems can be written in the form of (2). Figure 2 demonstrates the control inputs   and   , which matches the external disturbances and reference values; and their combined actions make the different disturbances be rejected and ensure the system outputs to track the reference values.Figure 3 indicates the state outputs of the system, in which we can see that the -axis current   and speed   are able to track their own reference values in high precision.In other words, the simulation results signify that the proposed algorithm is able to reject the multiclass external disturbance effectively and ensures that the closed-loop system rapidly converge to the reference values.In consequence, the designed internal models can replicate with the different exogenous disturbances and the proposed controller has an excellent control performance.

Conclusions
In the paper, an IMP based multivariable nonlinear control algorithm with multiclass nonharmonic disturbances is proposed to suppress the differently external nonharmonic disturbances and control the closed-loop system to track the reference values.The major conclusions of the research are summarized as follows: (1) in light of the multiclass nonharmonic disturbances, a class of new multiclass nonlinear internal models can be constructed on the basis of the definitions of expanded steady state generator and IMP, and