# Design of Asynchronous Motor Controller Based on Controlled Lagrangians Method

Authors: Yunbo Gao; Jianqiang Liu; Maoqing Li; Zhiyu Tian; Cuiran Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Asynchronous motor system has the characteristics of high order, strong coupling, and nonlinearity. From the dynamical model, it is the underactuated mechanical system, which means that the dimension of its input space is fewer than the degree of freedom. Following this perspective, the energy based nonlinear control technology-CL (controlled Lagrangians) method is used to solve the control problem in this paper. Based on the expected controlled energy and its derivative with respect to time, controlled Lagrangians and generalized force are constructed, and they produce the controlled equations. In order to ensure the complete matching between the controlled equation and the original equation, the gyroscopic forces containing the first-order term of velocity are innovatively introduced into the generalized force, and the matching conditions are obtained. By solving the matching conditions composed of some partial differential equations, the nonlinear smooth feedback control law can realize the global asymptotic stabilization of not only velocity but also position. Finally, the controlled energy is selected as the Lyapunov function, and the stability is proved according to the LaSalle invariant theorem. The effectiveness of the designed control law is demonstrated in the results of the simulation.


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## Body

## 1. Introduction

With the advantages of low price, simple structure, convenient maintenance, and reliable operation, the asynchronous motor has always been in a leading position in today’s social industrial production. Under the concept of advocating production environmental protection and low-carbon economy, the research on the control performance of the asynchronous motor has important theoretical significance and practical value [1].The asynchronous motor is a nonlinear system. However, the traditional linear control method cannot reveal its nonlinear nature. Therefore, the research on the control method of nonlinear theory is of great significance to improve the dynamic and static performance of the AC asynchronous motor. At present, the nonlinear control methods applied to the asynchronous motor mainly include feedback linearization control [2], backstepping control [3], sliding mode control [4], active disturbance rejection control [5, 6], and passive control theory [7, 8]. The control performance of the system has been significantly improved for application of the above method.Sun developed chopping control and energy-saving controller of a three-phase AC asynchronous motor [2]. Yu et al. designed the nonlinear adaptive controller of the asynchronous motor system by using the subsystem separation method and backstepping technology to ensure the stability of the system [3]. Lekhchine et al. designed a renewable energy storage electrical system for asynchronous motors. In this system, the motor is driven by sliding mode control, which can overcome the chattering phenomenon through the sliding surface based on fuzzy logic [4]. Li et al. proposed a second-order ADRC and AC excitation control system based on stator flux oriented control to control the active and reactive power of the variable-speed pumped storage unit [5, 6]. Wu et al. discussed the problem of asynchronous passive control of Markov jump systems and obtained three equivalent sufficient conditions to ensure the random passivity of hidden Markov jump systems by using matrix inequality technology. Based on the established conditions, an asynchronous controller is designed [7–9]. Yu et al. studied the tracking control of the underactuated dynamic system and proposed a six-step motion strategy of the pendulum driven vehicle rod system [10, 11]. By implementing feedback and other measures, the asymptotic stability of the control Hamiltonian system can be realized. The literature [12–15] reports some new results about the control of underactuated dynamic systems.In terms of mathematical equivalence, some studies analyze the controlled Lagrange (CL) method [16–19]. Compared with PBC, the CL method has a simpler mathematical form and clearer physical meaning, which is easy to understand. Usoro et al. described a Lagrangian method for solving nonlinear constrained optimization problems in set theory control problems. By introducing the matrix Lagrange multiplier, the problem is simplified to solve a set of nonlinear simultaneous matrix equations [16]. Műller et al. proved the possibility of maximizing the torque without exceeding the limit value of magnetic flux and stator current, which is independent of the number of revolutions of an asynchronous motor [17]. Lindgren et al. gave the exact slope distribution and other characteristic distributions of symmetric and asymmetric Lagrangian spatiotemporal waves at level crossings [18, 19]. In addition, some studies are extended to the general PBC method from the perspective of robust control and optimal control of general PCH systems [20–26]. These results have been proved to be expressed in a Lagrangian form.This paper applys CL method to analyze the asynchronous motor system from the perspective of an underdrive mechanical system. We will use the electromagnetic energy generated by the stator and rotor windings and the mechanical energy generated by the rotor to construct a controlled energy controller [12]. The controlled system maintains the Lagrangian mechanical structure in form, obtains the smooth nonlinear feedback control law, has a large convergence range, and helps to realize the CL method robust control and optimal control [14]. Compared with the port controlled dissipative Hamiltonian system, the nonlinear smooth feedback control law obtained in this paper can realize the global asymptotic stabilization of position and velocity at the same time.


## 2. Mathematical Model

For the convenience of writing, we will indicate the independent variables of the functions and matrices that appear, which will be omitted when they appear below.l,m,n∈N, and Ni=1,⋯,i, Nn means a collection consisting of the first n quantity of natural numbers. Let ze represents the function of the vector e=e1,⋯,e5T, Ykj corresponds to the element at the kth row and the jth column of the function matrix vector Ye, xi denotes the ith element of function vector Xe: R5⟶R5, where i∈N5, I means the five-order identity matrix. Besides, some notations as given below:(1)z,i=∂ze∂ei,Y,i=Y11,i⋮Yl1,i⋯⋯Y1m,i⋮Ylm,i∂ez=z,1⋮z,n,∂eTX=x1,1⋮xm1⋯⋯x1,n⋮xm,n.The generalized coordinate variables of the AC asynchronous motor isq=q1,…,q5T, where q1,q2,q3,q4 are the components of stator inductance charge and rotor inductance charge on d−q axis, and q5 is the angular position of motor rotor. u=u1,…,u4,0T is the original control input, where u1,u2,u3,u4 are the elements of the stator voltage and rotor voltage on the d−q axis. In addition, u=Oqv, where the input coupling matrix O=I01,I02,I03,I04, and v∈R4.In view of the mathematic model of a three-phase asynchronous motor, the following assumptions are expressed as follows [9]:(1)
The spatial harmonic and the spatial difference between three-phase windings are ignored. Meanwhile, it is assumed that the generated magnetomotive force is distributed sinusoidally along the circumference of the air gap.(2)
Magnetic circuit saturation and core loss are ignored. At the same time, it is assumed that the inductance parameters of every phase winding, not only self-inductance but also mutual one, are constant.(3)
It is not considered of the influence of frequency and temperature changes on the variation of winding resistance.According to the above assumptions for the AC asynchronous motor, its mathematical model in thed−q coordinate system can be obtained by(2)L110L13000L110L130L130L33000L130L3300000J/npq¨1q¨2q¨3q¨4q¨5+0L11q˙50L13q˙5−npψsq−L11q˙50−L13q˙50npψsd0L13q˙50L33q˙50−L13q˙50−L33q˙500npψsq−npψsd000q˙1q˙2q˙3q˙4q˙5+R1−ω1L110−ω1L130ω1L11R1ω1L13000−ω1L13R2−ω1L330ω1L130ω1L33R2000000q˙1q˙2q˙3q˙4q˙5+0000TL=u1u2u3u40.In (2), ψsd=L11q˙1+L13q˙3, ψsq=L11q˙2+L13q˙4, L11, and L33 denote the equivalent self-inductance of the stator and rotor phase windings, respectively. And L13 is the equivalent mutual inductance of the stator and rotor phase windings. The load torque is denoted by TL, and TL=T1+TL', where T1 includes no-load torque and the external one, and TL′=Hq5 denotes the torsional torque generated when the motor and the mechanical load are connected with a relatively long shaft, where H is the deformation coefficient. The equation (2) is abbreviated as(3)Mq¨+Cq,q˙q˙+C0q˙+∂qEp=u,where qT,q˙T=qdt,0T is the equilibrium point, and its input must satisfy O¯u=0, where O¯∈O⊥T and O⊥TO=0, so O¯=0,0,0,0,1.


## 3. Design of Asynchronous Motor Controller Based on CL Method

### 3.1. Construction of Controlled Energy and Generalized Force

According to (3), the controlled kinetic energy of the controlled system is E¯kq,q˙=1/2q˙TM¯q˙. The controlled kinetic energy matrix satisfies M¯:R5⟶R5×5,M¯=M¯T and M¯≠0.Take the controlled potential energy asE¯p:R5⟶R, and generalized force u¯∈R5, then controlled Lagrangian function L¯q,q˙ and controlled energy E¯q,q˙ of the system are as follows:(4)L¯=E¯k−E¯p,E¯=E¯k+E¯p.Sometimes, the controlled energy has physical meaning, such as controlled kinetic energy or controlled potential energy, and maybe it has only a mathematical meaning which is sufficient and necessary. According toL¯ and u¯, we obtain the controlled equations of the system as follows:(5)u¯=M¯q¨+∂qTM¯q˙q˙−12∂qq˙TM¯q˙+∂qE¯p.Using (4) and (5), we get(6)E¯˙=q˙Tddt∂q˙L¯−∂qL¯=q˙Tu¯.The generalized force of the system consists of two parts, namely gyroscopic forcesGq,q˙q˙ and dissipation force −Dqq˙, where gyroscopic forces matrix is G=−GT=∑i∈N5IoiGio=∑i,j∈N5Ioiq˙Tgioq+G0.Furthermore,gijkq represents the element at the kth row and jth column of function matrix gio, and for i,j,k∈N5, gijk is the kth component function of the element Gij of the gyroscopic forces matrix.Since the gyroscopic forces matrix is an anti-symmetric one, there isgijk=−gjik. Similarly, the constant elements of the gyroscopic forces matrix G also satisfy Gij0=−Gji0.Remark 1.
Since there isC0q˙ term in the original system, constant term G0 is introduced into the matrix G to construct the gyroscopic forces consisting of the first term of the velocity, which matches C0q˙ of the original system. Maybe these forces do not exist in real world, and only mathematical meaning is necessary for them. As we know, this introduction is for the first time.
When generalized forceu¯=G−Dq˙, we obtain from (5) that(7)E¯˙=q˙TG−Dq˙≤0,which indicates that the energy of the closed system is decreasing. Multiplying MM¯=Νq −N(q) at the two sides of (4) at the same time, we obtain(8)Nu¯=Mq¨+N∂qTM¯q˙q˙−12∂qq˙TM¯q˙+∂qE¯p.
According to (2) and (7), the original control input u and the control input u¯ of the controlled system are obtained. The relationship between them can be given as(9)u=C+C0q˙−N∂qE¯p+∂qEp+Nu¯−N∂qTM¯q˙q˙−12∂qq˙TM¯q˙.
According to (9)–(14) in [14] and (8) in this article, we obtain(10)u=C+C0q˙+∂qEp−N∂qE¯p−∑i∈N512NIoiq˙TTiq˙+Nu¯−G¯q˙.
The deduction of (9) is tedious, so we borrow the similar process in literature [12] for abbreviation.


### 3.2. Determination of the Matching Conditions

If the controlled equation (5) matches the original (3), the control input determined by (9) is true; that is, u5=0 is true for any point q,q˙.In the same form as gyroscopic forces matrixG, matrix G⌢q,q˙ is given as follows:(11)G⌢q,q˙=Gq,q˙−G¯q,q˙,where G⌢ gives out functions g⌢jikq and g⌢ij similar to gjikq and gij. Substituting u¯=G−Dq˙ and (11) into (10), we get(12)u=−∑i∈N512NIoiq˙TTiq˙+C+C0q˙+NG^−Dq˙−N∂qE¯p+∂qEp.LetN¯=O¯N, multiplying the line vector O¯ from the left to (12) and taking the obtained left side zero constantly, we acquire the matching conditions as follows (13)–(16):(13)0=∑i∈N5N¯iTjji+∑i≠j2N¯igjii,where j∈N5 and each j represents an equation, and(14)0=∑i∈N5N¯iTfji+gjif+gfij+hnpL13.In equation (14), each pair of (j, f) corresponds to an equation. Otherwise, j,f∈N5 and j > f, When f + j ≠ 5, h = 0; then h = (−1)f.(15)0=∑i∈N5N¯iDij+∑i≠jN¯ig⌢ji,where j∈N5, and each j corresponds to an equation.(16)0=∑i∈N5N¯iE¯p,i−Ep,5.Multiplying (13) by N¯12,⋯,N¯52, and multiplying (14) by 2N¯1N¯2, 2N¯1N¯3, 2N¯1N¯4, 2N¯1N¯5, 2N¯2N¯3, 2N¯2N¯4, 2N¯2N¯5, 2N¯3N¯4, 2N¯3N¯5, and 2N¯4N¯5 , then the sum to obtain a equation has nothing to do with gyroscopic forces:(17)2npL13N¯2N¯3−N¯1N¯4=∑i∈N5O¯N¯iW,i−1O¯T.Remark 2.
Equation (17) is obtained without gyroscopic forces terms due to anti-symmetric property of gyroscopic forces matrix, which indicates that the quadratic form of the anti-symmetric matrix is equal to zero.
LetW−1=Kq, then (17) can be concisely expressed as(18)2npL13N¯2N¯3−N¯1N¯4=∑i∈N5O¯N¯iK,iO¯T.
For the controlled kinetic energy matrix, its regular condition isK≠0. According to literature [12], Wq=M−1M¯M−1 is known, so N=KM−1 can be obtained from the definition of matrixes N,W, and K. If N¯1,⋯,N¯5 are zero, then K=0 is available. Therefore, N¯1,⋯,N¯5 cannot be zero at the same time. In order to facilitate the subsequent calculations, suppose N¯5≠0. Multiplying (15) by N¯1,⋯,N¯5 and summing them, we get(19)N¯DN¯T=0.
In summary, the combination of (16), (18), and (19) and K≠0 is the condition under which the controlled system matches the original one.Remark 3.
Equations (16) and (18) are partial differential equations, and equations (13)–(15) related to gyroscopic forces terms are linear algebraic equations. They are cascaded. PDEs are resolved at first to decrease difficulty, and then there are only linear algebraic equations which could be solved explicitly with introducing the previous solution.
Except algebraic equations, there are only two PDEs contained in the matching condition for the underactuation degree one system. They could be solved with involving enough independent variables according to some examples applied in CL methods.


### 3.3. Determination of the Matching Controller

For convenience, some notations can be expressed as follows:(20)O⌢=OTO−1OT,N⌢=O⌢N.MultiplyingO⌢ from the left to (12), the matching control of the system is(21)v=−12N⌢∑i∈N5Ioiq˙TTiq˙+O⌢C+C0q˙+N⌢G⌢−Dq˙−N⌢∂qE¯p+O⌢∂qEp.


## 3.1. Construction of Controlled Energy and Generalized Force

According to (3), the controlled kinetic energy of the controlled system is E¯kq,q˙=1/2q˙TM¯q˙. The controlled kinetic energy matrix satisfies M¯:R5⟶R5×5,M¯=M¯T and M¯≠0.Take the controlled potential energy asE¯p:R5⟶R, and generalized force u¯∈R5, then controlled Lagrangian function L¯q,q˙ and controlled energy E¯q,q˙ of the system are as follows:(4)L¯=E¯k−E¯p,E¯=E¯k+E¯p.Sometimes, the controlled energy has physical meaning, such as controlled kinetic energy or controlled potential energy, and maybe it has only a mathematical meaning which is sufficient and necessary. According toL¯ and u¯, we obtain the controlled equations of the system as follows:(5)u¯=M¯q¨+∂qTM¯q˙q˙−12∂qq˙TM¯q˙+∂qE¯p.Using (4) and (5), we get(6)E¯˙=q˙Tddt∂q˙L¯−∂qL¯=q˙Tu¯.The generalized force of the system consists of two parts, namely gyroscopic forcesGq,q˙q˙ and dissipation force −Dqq˙, where gyroscopic forces matrix is G=−GT=∑i∈N5IoiGio=∑i,j∈N5Ioiq˙Tgioq+G0.Furthermore,gijkq represents the element at the kth row and jth column of function matrix gio, and for i,j,k∈N5, gijk is the kth component function of the element Gij of the gyroscopic forces matrix.Since the gyroscopic forces matrix is an anti-symmetric one, there isgijk=−gjik. Similarly, the constant elements of the gyroscopic forces matrix G also satisfy Gij0=−Gji0.Remark 1.
Since there isC0q˙ term in the original system, constant term G0 is introduced into the matrix G to construct the gyroscopic forces consisting of the first term of the velocity, which matches C0q˙ of the original system. Maybe these forces do not exist in real world, and only mathematical meaning is necessary for them. As we know, this introduction is for the first time.
When generalized forceu¯=G−Dq˙, we obtain from (5) that(7)E¯˙=q˙TG−Dq˙≤0,which indicates that the energy of the closed system is decreasing. Multiplying MM¯=Νq −N(q) at the two sides of (4) at the same time, we obtain(8)Nu¯=Mq¨+N∂qTM¯q˙q˙−12∂qq˙TM¯q˙+∂qE¯p.
According to (2) and (7), the original control input u and the control input u¯ of the controlled system are obtained. The relationship between them can be given as(9)u=C+C0q˙−N∂qE¯p+∂qEp+Nu¯−N∂qTM¯q˙q˙−12∂qq˙TM¯q˙.
According to (9)–(14) in [14] and (8) in this article, we obtain(10)u=C+C0q˙+∂qEp−N∂qE¯p−∑i∈N512NIoiq˙TTiq˙+Nu¯−G¯q˙.
The deduction of (9) is tedious, so we borrow the similar process in literature [12] for abbreviation.


## 3.2. Determination of the Matching Conditions

If the controlled equation (5) matches the original (3), the control input determined by (9) is true; that is, u5=0 is true for any point q,q˙.In the same form as gyroscopic forces matrixG, matrix G⌢q,q˙ is given as follows:(11)G⌢q,q˙=Gq,q˙−G¯q,q˙,where G⌢ gives out functions g⌢jikq and g⌢ij similar to gjikq and gij. Substituting u¯=G−Dq˙ and (11) into (10), we get(12)u=−∑i∈N512NIoiq˙TTiq˙+C+C0q˙+NG^−Dq˙−N∂qE¯p+∂qEp.LetN¯=O¯N, multiplying the line vector O¯ from the left to (12) and taking the obtained left side zero constantly, we acquire the matching conditions as follows (13)–(16):(13)0=∑i∈N5N¯iTjji+∑i≠j2N¯igjii,where j∈N5 and each j represents an equation, and(14)0=∑i∈N5N¯iTfji+gjif+gfij+hnpL13.In equation (14), each pair of (j, f) corresponds to an equation. Otherwise, j,f∈N5 and j > f, When f + j ≠ 5, h = 0; then h = (−1)f.(15)0=∑i∈N5N¯iDij+∑i≠jN¯ig⌢ji,where j∈N5, and each j corresponds to an equation.(16)0=∑i∈N5N¯iE¯p,i−Ep,5.Multiplying (13) by N¯12,⋯,N¯52, and multiplying (14) by 2N¯1N¯2, 2N¯1N¯3, 2N¯1N¯4, 2N¯1N¯5, 2N¯2N¯3, 2N¯2N¯4, 2N¯2N¯5, 2N¯3N¯4, 2N¯3N¯5, and 2N¯4N¯5 , then the sum to obtain a equation has nothing to do with gyroscopic forces:(17)2npL13N¯2N¯3−N¯1N¯4=∑i∈N5O¯N¯iW,i−1O¯T.Remark 2.
Equation (17) is obtained without gyroscopic forces terms due to anti-symmetric property of gyroscopic forces matrix, which indicates that the quadratic form of the anti-symmetric matrix is equal to zero.
LetW−1=Kq, then (17) can be concisely expressed as(18)2npL13N¯2N¯3−N¯1N¯4=∑i∈N5O¯N¯iK,iO¯T.
For the controlled kinetic energy matrix, its regular condition isK≠0. According to literature [12], Wq=M−1M¯M−1 is known, so N=KM−1 can be obtained from the definition of matrixes N,W, and K. If N¯1,⋯,N¯5 are zero, then K=0 is available. Therefore, N¯1,⋯,N¯5 cannot be zero at the same time. In order to facilitate the subsequent calculations, suppose N¯5≠0. Multiplying (15) by N¯1,⋯,N¯5 and summing them, we get(19)N¯DN¯T=0.
In summary, the combination of (16), (18), and (19) and K≠0 is the condition under which the controlled system matches the original one.Remark 3.
Equations (16) and (18) are partial differential equations, and equations (13)–(15) related to gyroscopic forces terms are linear algebraic equations. They are cascaded. PDEs are resolved at first to decrease difficulty, and then there are only linear algebraic equations which could be solved explicitly with introducing the previous solution.
Except algebraic equations, there are only two PDEs contained in the matching condition for the underactuation degree one system. They could be solved with involving enough independent variables according to some examples applied in CL methods.


## 3.3. Determination of the Matching Controller

For convenience, some notations can be expressed as follows:(20)O⌢=OTO−1OT,N⌢=O⌢N.MultiplyingO⌢ from the left to (12), the matching control of the system is(21)v=−12N⌢∑i∈N5Ioiq˙TTiq˙+O⌢C+C0q˙+N⌢G⌢−Dq˙−N⌢∂qE¯p+O⌢∂qEp.


## 4. Matching Conditions Solution

For the asynchronous motors model, there is no control inputu5=0 at the fifth degree of freedom except the ath degree of freedom, where a∈N4, at the same time, the following notations are defined as(22)O⌢=OT,N¯=N50,OaoN⌢=NaoIntroduce the function vectorΓT=−N5a/N55 from the definition of matrixes N, W, and K, we get K50=−N55ΦqT, where ΦT=Φ1,⋯,Φ5=ΓTM, and thus the element in the fifth row of matrix K is K5a=K55Φa/Φ5. To ensure K>0, we choose Kaa=K55kaΦa2/Φ52+ka+n, where ka and ka+n are constants. The system matching conditions expressed by Γ, K55, and E¯p are(23)∑i∈N5ΓiK55,i=2npL13Γ2Γ3−Γ1Γ4K55ΓTM05,(24)∑i∈N5ΓiE¯p,i=ΓTM05K55Ep,5,(25)ΓTDΓ=0.AssumeΓ1=0,⋯,Γ4=0,Γ5=−1, and find a special solution to (23) as follows:(26)K55=k5.In the above (23), k5 is a constant.TheK matrix of the system is(27)K=k5k600000k700000k800000k9000001.It can be seen from theK matrix that its determinant is K=k55k6k7k8k9, so a sufficient condition for K>0 is(28)k5,k6,k7,k8,k9>0.From (24), the positive definite solution to the controlled potential energy E¯p and the satisfied conditions are(29)E¯p=∑i∈N5qi−ai2;k5=JH2np,a5=−JT12npk5,where a1,⋯,a5 are constants. After some calculations, the Hessian matrix of E¯p is(30)∂2E¯p∂q∂qT=2000002000002000002000002.It is clear that the Hessian matrix is positive definite. And fromE¯p,1a1,⋯,E¯p,5a5=0, we know that a1,⋯,a5 is the minimum point of the controlled potential energy.Remark 4.
The controlled energy consists of controlled kinetic energy and potential energy. On the one hand, the controlled kinetic energy is in quadratic form of velocity and achieves positive definiteness with supplied condition. On the other hand, the controlled potential energy is constructed in the form to be positive definite conveniently.
According to (30), the dissipation matrix is chosen in diagonal form as follows:(31)D=d1d2d3d40,where d1,⋯,d4 can be any value greater than zero.


## 5. Matching Control Law and Stability Analysis

Based on the above analysis,N=KM−1 is known, so the matrix N of the system is(32)N=β1k6L330−k6L13000k7L330−k7L130−k8L130k8L11000−k9L130k9L1100000β2.In equation (32), β1 =  k5/L11L33−L132 and β2 = npL11L33−L132/J, because Ti=MK,i−1M, where i∈N5, so we get(33)Ti=0,i∈N5.From (13)–(15) and from (30)–(32), we have the gyroscopic forces component function g⌢ijk as follows:(34)g⌢ijk=γijk,g⌢ij=γij,i,j,k∈N5.In equation (33), one part of the value of gyroscopic forces component function γijk needs to satisfy γ152=−γ251,γ153=−γ351,γ254=−γ452,γ354=−γ453 and be arbitrary values, the other part needs to be taken according to the following (35), and the rest can be arbitrary values.(35)γ151=0,γ154=2JL13/k5,γ155=0,γ252=0,γ253=−2JL13/k5,γ255=0,γ352=JL13/k5,γ353=0,γ355=0,γ451=−JL13/k5γ454=0,γ455=0,γ15=0,γ25=0,γ35=0,γ45=0.To keep the control law simple, take the value ofγ125,γ153,γ254,γ354,γ12, γ13,γ14,γ23,γ24,γ34 to be one and the rest to be zero.Remark 5.
A large part of variables,g⌢ijk and g⌢ij in equation (34), take the value zero for convenience to get the simpler control law. The different evaluation for them could still assure stability of the system and maybe affect the convergence rate of the system. Further, there is a room to search control law for better performance in some respects.
Substituting (20), (27)–(35) into (21), the obtained matching control law of the motor system is described as(36)u1=R1q˙1+q˙5−ω1L11q˙2+L13q˙4−npψsqq˙5+k5k6L11L33−L132L33q˙2+q˙3+q˙4+q˙3q˙5+q˙2q˙5+L13q˙1+q˙2−q˙4−q˙4q˙5+L13Jq˙5∗2L33q˙4−L13q˙2/k5+L13d3q˙3−L33d1q˙1−2L33q1−a1−L13q3−a3,(37)u2=R1q˙2+ω1−q˙5L11q˙1+L13q˙3+npψsdq˙5+k5k7L11L33−L132L33q˙3+q˙4−q˙1−q˙1q˙5+q˙4q˙5+L13q˙1+q˙2+q˙3+L13Jq˙5k5∗L13q˙1−2L33q˙3+L13d4q˙4−L33d2q˙2−2L33q2−a2−L13q4−a4,(38)u3=q˙5−ω1L13q˙2+L33q˙4+R2q˙3+k5k8L11L33−L132L11q˙4−q˙1−q˙2+q˙4q˙5−L13q˙2+q˙3+q˙2q˙5+q˙3q˙5+L13Jq˙5L11q˙2−2L13q˙4/k5+L13d1q˙1−L11d3q˙3−2L11q3−a3−L13q1−a1(39)u4=ω1−q˙5L13q˙1+L33q˙3+R2q˙4+k5k9L11L33−L132L13q˙1−q˙3−q˙4+q˙1q˙5−q˙4q˙5−L11q˙1+q˙2+q˙3+L13Jq˙52L13q˙3−L11q˙1/k5+L13d2q˙2−L11d4q˙4−2L11q4−a4−L13q2−a2.
In summary, the conclusion is listed as follows:.Proposition 1.
For AC asynchronous motor systems, if the parameters of controller meet the following conditions:(40)k5,⋯,k9>0,d1,⋯,d4>0,then the smooth feedback control law expressed by equations (35)–(38) can stabilize the motor globally asymptotically at qdT,0T. It is the desired equilibrium point at which the controlled potential energy achieves the minimum.Proof.
Let Lyapunov candidate functionV=E¯. If the system controller parameters are selected according to (39), then the function is positive definite. From equation (6), there is V˙≤0. Therefore, the control law given by equations (35)–(38) enables the induction motor to achieve global asymptotic stabilization at qdT,0T.
For the asymptotic stability, it can be proved that there is no trajectory of isolated points other than equilibrium points in the set ofV˙=0.
Assuming that there is such a trajectory in the set, it can be obtained as(41)q˙i≡0,i∈N4.
There is a certain pointq0 on this trajectory, and at the same time, (39) also holds in a certain area δ0 of this point q0. Differentiating and integrating (40) along the trajectory, we get(42)q¨i≡0,qi=αi,i∈N4,where α1,⋯,α4 are constant.
Substituting (35)–(38), (40), and (41) into the first four equations of (1), we get(43)2k5k6L11L33−L132L33α1−a1−L13α3−a3=02k5k7L11L33−L132L33α2−a2−L13α4−a4=02k5k8L11L33−L132L11α3−a3−L13α1−a1=02k5k9L11L33−L132L11α4−a4−L13α2−a2=0.
It can be seen from observation that (43) is true if and only if α1=a1,⋯,α4=a4, and vice versa. Therefore, the assumption that there are isolated points belonging to the point group of V˙=0 does not hold.
From (4) and (41), we obtain ∂qE¯p=0. This shows that the trajectory in the set V˙=0 can only be the equilibrium point, and the Hessian matrix of E¯p is positive definite, so point qd is the sole extreme point of E¯pq. According to the LaSalle principle, the system can achieve global asymptotic stabilization by the proposed control law.


## 6. Simulation Result Analysis

The simulation parameters of the system are selected asL11 = 0.45 H, L13 = 0.42 H, L33 = 0.45 H, J = 0.2 kg m2H, ω1=10 rad/s, T1 = 8Nm, H = 8Nm, np = 8, and R1 = R2 = 0.97 Ω. The parameters of the controller that satisfy the positive definite of the controlled energy are k5 = 0.1, k6 = 4, k7 = 6, k8 = 0.5, and k9 = 0.4.Whend1 = 7, d2 = 7, d3 = 5, and d4 = 5, the simulation results are as follows:The desired equilibrium point of the system isqdt,0T , where qdT=1,1,1,1,−0.5. It can be seen from Figures 1–3 that when the AC asynchronous motor is disturbed by some uncertain factors, the control target of the system deviates from the expected balance point. Under the control input of Figure 4, the control target of the system can return to the desired equilibrium point as soon as possible. In this process, the change of electromagnetic torque, original, and controlled energy are showed in Figures 5 and 6.Figure 1
Stator inductive charge and current.Figure 2
Charge and current of rotor.Figure 3
Rotor angular displacement and angular velocity.Figure 4
Control input.Figure 5
Electromagnetic torque.Figure 6
Original energy and controlled energy.Remark 6.
Parametersk5, k6, k7, k8, and k9 in controller are connected with the controlled kinetic energy, and their evaluation should ensure controlled kinetic energy positive definite. Furthermore, they will affect primarily the amplitude of inputs.Remark 7.
Parametersd1, d2, d3, and d4 are related only to the dissipated forces, and thus the changing of them could be able to improve the convergence rate.
Compared with other control methods [8, 9], the nonlinear smooth feedback control law obtained in this article has a larger convergence range for its global asymptotic stabilization.


## 7. Conclusion

The control of an asynchronous motor is studied in this paper. By applying the controlled Lagrange function method to high-order, strongly coupled, and time-varying nonlinear systems, the controlled equation matching the original equation of the system is derived by using the expected controlled energy and its time derivative. Because the primary term of velocity exists in the original equation, the gyroscopic forces of the generalized primary term of velocity is innovatively introduced into the controlled equation, and the condition of complete matching between the original equation and the controlled equation can be obtained. By solving the matching condition composed of some partial differential equations, the specific matching control law of the system is obtained, and the global asymptotic stabilization of not only velocity but also position can be realized at the desired equilibrium point at the same time. Finally, the controlled energy of the system is chosen as the Lyapunov function for its property, which facilitates the proof of stability.In the research process, we can see that the CL method analyzes the asynchronous motor system from the perspective of underactuated mechanical system, so that the controlled system maintains the Lagrangian mechanical structure in form, and the nonlinear smooth feedback control law can be obtained, which has a large convergence range and is helpful to realize robust control and optimal control. On the basis of the work in this paper, the nonlinear CL method will be improved by the introducing observer and be intended to solve tracking problem in following work.



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