Research on chaos control of permanent magnet synchronous motor based on the synthetical sliding mode control of inverse system decoupling

Abstract

This article focuses on realizing the chaos control of a permanent magnet synchronous motor by combining a pseudo-linear inverse system of the permanent magnet synchronous motor and synthetical sliding mode control. First, the permanent magnet synchronous motor dimensionless nonlinear mathematical model is established, and its chaos is analyzed by the Lyapunov exponent method. The permanent magnet synchronous motor parameter range when chaos appears is obtained. Then, the inverse system decoupling method is used to analyze the reversibility of the permanent magnet synchronous motor system, and the permanent magnet synchronous motor inverse system is obtained, which is compounded with the original system into a pseudo-linear inverse system that consists of two independent subsystems, including a first-order d-axis current system and a second-order rotational speed system, to decouple the permanent magnet synchronous motor system. Third, the first-order d-axis subsystem is controlled by sliding mode control with a hyperbolic tangent function as the switching function, and the second-order speed subsystem is controlled by super-twisting sliding mode control with a hyperbolic tangent function as the switching function, which is called the synthetical sliding mode control. The permanent magnet synchronous motor pseudo-linear inverse system is controlled by using the synthetical sliding mode to realize the chaos control of the permanent magnet synchronous motor. Finally, three kinds of permanent magnet synchronous motor chaos control systems are established in MATLAB/Simulink software, and the experimental tests are implemented. The results show that the proposed permanent magnet synchronous motor chaos control system has good performance, which can effectively eliminate chattering in sliding mode control and control chaos in the permanent magnet synchronous motor system.

Keywords

Permanent magnet synchronous motor chaos analysis chaos control synthetical sliding mode inverse system decoupling method

1. Introduction

Compared with conventional motors, the permanent magnet synchronous motor (PMSM) has many advantages, such as a high energy density, high torque inertia ratio, high reliability, and low torque ripple, which make it increasingly more widely used in the automotive, aerospace, precision processing industries, and other fields (Zhang et al., 2015). However, the PMSM is a multivariable and strongly coupled nonlinear system. When its parameters change in a certain range because of mechanical wear, modeling errors, or measurement errors, chaos will occur. Chaos is a very special nonlinear dynamic characteristic, which is sensitive to the initial conditions, and the motion trajectory is irregular and unpredictable (Huang et al., 2011). Therefore, the output of the PMSM will exhibit torque and speed fluctuations. Nevertheless, in most cases, chaos will adversely affect motor control performance, which is undesirable. Therefore, to improve the control performance of the motor, the chaos control of the motor has become a popular research topic.

Currently, the main methods that are successfully used to realize the chaos control of the PMSM are feedback control (Hu et al., 2015, 2016), adaptive control (Aguilar-Mejía et al., 2016; Wang et al., 2014), backstepping control (Liu et al., 2016; Singh et al., 2018), and sliding mode control (SMC) (Merzoug et al., 2012), but these methods have some shortcomings in controlling the PMSM chaos. When the control target is not an equilibrium point or an unstable periodic orbit, the effect of the feedback control will be affected. In addition, the time delay value of the feedback control is difficult to determine. Although the adaptive control has a better control effect when the system parameters change, the effect of the adaptive control will be greatly weakened when the external disturbance is large. The backstepping control has a better control effect on the system parameter uncertainty, particularly for uncertain systems that do not satisfy the switching conditions. However, the shortcoming is that the structure of the controller is more complex, which will produce many differential terms, result in “differential explosion”, and make the operation of the controller slow and the computational burden large. The SMC is an effective robust nonlinear control technology, which can ensure that the systems have good stability and robustness. However, because of the use of discontinuous high-speed switching functions in SMC, chattering will appear. The existence of chattering may initiate the system unmodeled high-order motion, which may cause the system high-frequency oscillation and may even destabilize the system in severe cases. In recent years, high-order SMC (Devdutt, 2019; Matraji et al., 2018) has been widely used, which has the advantages of producing a fast response, being insensitive to parameter changes and disturbances, not requiring online system identification, providing simple physical implementation, and eliminating chattering in traditional SMC.

In addition, the motor dynamic and static performances are affected by the coupling characteristics of the stator current and rotating speed (Chen et al., 2018; Yong et al., 2018), which have seldom been considered in the research of the chaos control of the PMSM. Although the state feedback decoupling method is used to decouple the linear synchronous motor in the reference (Cai et al., 2018), the design of the chaos control controller based on the backstepping method is more complex. The inverse system is used to decouple the PMSM, and the pseudo-linear system, which is composed of an inverse system and the original system, is controlled by using the proportional-integral-derivative method (Zhao et al., 2012). The controller design is simple, but the control effect can still be improved. The PMSM is also decoupled by using the inverse system (Liu et al., 2017), and the pseudo-linear system is controlled by the traditional SMC. The control effect is improved, but there is chattering in the SMC system.

To overcome the shortcomings of the PMSM chaos control, a PMSM nonlinear mathematical model is established, and its chaos is analyzed by using the Lyapunov exponent and bifurcation diagram. The main contributions of the article are listed as follows: (1) aiming at the coupling characteristics of the PMSM stator current and rotating speed, the PMSM is decoupled into two independent pseudo-linear systems including a first-order d-axis current and a second-order rotating speed using the inverse system decoupling method, which makes the input and output of the PMSM system realize the linear decoupling and (2) the PMSM chaos control is realized by using the synthetical sliding mode method, which includes the SMC for the first-order d-axis current subsystem and the super-twisting sliding mode control (S-TSMC) for the second-order rotating speed subsystem. Simultaneously, the hyperbolic tangent function replaces the switching function to further eliminate chattering in the traditional SMC to improve the effect of the chaos control. Generally, the main novelty and superiority of the proposed control method are to decouple the stator current and rotating speed of the PMSM, implement high-order SMC in the PMSM speed loop, adopt the hyperbolic tangent function as the switching function to further eliminate the chattering in traditional SMC, and finally achieve a better chaos control of the PMSM.

The rest of the article is structured as follows. Section 2 includes the establishment of the PMSM nonlinear mathematical model and chaos analysis. The PMSM chaos control based on the inverse system decoupling and synthetical sliding mode is carried out in Section 3. In Sections 4 and 5, the numerical simulation and experimental analyses are, respectively, performed to verify the proposed chaos control algorithm, and the concluding remarks are presented in Section 6.

2. PMSM nonlinear mathematical model and chaos analysis

2.1. PMSM nonlinear mathematical model

In this article, the PMSM model with a uniform air gap (L_q = L_d) is the object of study. The PMSM mathematical model in the d–q coordinate system is expressed as follows

{\begin{cases} \frac{d i_{d}}{d t} = - \frac{R_{s}}{L_{d}} i_{d} + \frac{L_{q}}{L_{d}} ω i_{q} + \frac{u_{d}}{L_{d}} \\ \frac{d i_{q}}{d t} = - \frac{R_{s}}{L_{q}} i_{q} - \frac{L_{d}}{L_{q}} ω i_{d} - \frac{ψ ω}{L_{q}} + \frac{u_{q}}{L_{q}} \\ \frac{d ω}{d t} = \frac{n_{p}}{J} ψ i_{q} - \frac{B}{J} ω - \frac{T_{L}}{J} \end{cases}

(1)

In equation (1), $i_{d}$ and $i_{q}$ are the currents of the d- and q-axis, respectively; $R_{s}$ is the stator resistance; $L_{d} and L_{q}$ are the inductances of the d- and q-axis, respectively; $J$ is the rotational inertia of the rotor; $B$ is the rotational viscous coefficient of the motor; $ω$ is the motor angular speed; $ψ$ is the motor flux linkage; $n_{p}$ is the number of pole pairs; $u_{d}$ and $u_{q}$ are the voltage components of the d- and q-axis of the motor stator voltage, respectively.

To facilitate the PMSM chaos analysis, the affine and time scale transformations are performed as follows

{\begin{cases} x = Γ \hat{x} \\ t = τ \hat{t} \end{cases}

(2)

In equation (2), $x = {[\begin{matrix} i_{d} & i_{q} & ω \end{matrix}]}^{T}$ and $Γ = [\begin{matrix} b k \\ k \\ 1 / τ \end{matrix}]$ ; therefore, equation (1) can be transformed into the dimensionless state model in equation (3)

{\begin{cases} \frac{d {\hat{i}}_{d}}{d \hat{t}} = - b {\hat{i}}_{d} + {\hat{i}}_{q} \hat{ω} + {\hat{u}}_{d} \\ \frac{d {\hat{i}}_{q}}{d \hat{t}} = - {\hat{i}}_{q} - {\hat{i}}_{d} \hat{ω} + γ \hat{ω} + {\hat{u}}_{q} \\ \frac{d \hat{ω}}{d \hat{t}} = σ ({\hat{i}}_{q} - \hat{ω}) - {\hat{T}}_{L} \end{cases}

(3)

where

b = (L_{q} / L_{d})

τ = (L_{q} / R_{s})

k = (B R_{s} / n_{p} ψ L_{q})

σ = (B L_{q} / J R_{s})

γ = - (ψ^{2} n_{p} / B R_{s})

{\hat{u}}_{d} = (u_{d} τ / L_{d} b k)

{\hat{u}}_{q} = (τ / L_{q} k) u_{q}

, and

{\hat{T}}_{L} = (τ^{2} / J) T_{L}

2.2. Chaos analysis

The affine and time scale transformations are linear transformations, so the original characteristics of the system will not change. For the convenience of the chaos analysis, the working condition of the sudden power failure of the PMSM under the no-load operation is considered, that is the system inputs ${\hat{u}}_{d}$ and ${\hat{u}}_{q}$ and ${\hat{T}}_{L}$ are 0.

The state variables can be set as $x = {[x_{1}, x_{2}, x_{3}]}^{T} = {[{\hat{i}}_{d}, {\hat{i}}_{q}, \hat{ω}]}^{T}$ ; then, equation (3) becomes

{\begin{cases} \frac{d x_{1}}{d t} = - b x_{1} + x_{2} x_{3} \\ \frac{d x_{2}}{d t} = - x_{1} - x_{1} x_{3} + γ x_{3} \\ \frac{d x_{3}}{d t} = σ (x_{2} - x_{3}) \end{cases}

(4)

According to equation (4), the PMSM system is a complex nonlinear system, and there is a coupling relationship between the current and the rotating speed. For a nonlinear system, chaos may occur when its parameters or external excitations change in a certain range. The Lyapunov exponential method is an important method to judge the chaos phenomenon, which represents the exponential convergence or divergence speed of the two adjacent trajectories of the system.

Equation (4) can be written as follows

\frac{d x}{d t} = f (x)

(5)

Then, the Lyapunov exponent of system (5) can be deduced as follows (Chen et al., 2019).

Assume that the initial conditions of two neighboring trajectories are $x (0) = x_{0}$ and $x (0) + δ x (0) = x_{0} + δ x_{0}$ , and $| δ x_{0} |$ is very small. After time t, the two trajectories are $x (t)$ and $x (t) + δ x (t)$ , respectively, and the following equations are obtained

\frac{d x (t)}{d t} = f (x (t))

(6)

\frac{d x (t)}{d t} + \frac{d δ x (t)}{d t} = f (x (t) + δ x (t))

(7)

By omitting the second-order terms from the above two equations, we obtain the linearized equations of system (5)

\frac{d δ x (t)}{d t} = D f (x (t) \cdot δ x (t))

(8)

where

D f (x (t))

is the derivative operator of the system of equations at

x (t)

Let the fundamental solution matrix of equation (8) be $φ (t, x)$ ; its corresponding eigenvalues are $μ_{1} (t), μ_{2} (t), and μ_{3} (t)$ , and $z_{j} (t)$ are the corresponding eigenvectors of $μ_{j} (t)$ . If the Lyapunov exponent of the system is $λ_{j}$ , which satisfies $λ_{1} \geq λ_{2} \geq λ_{3}$ , then $δ x_{0}$ can be expressed as

δ x_{0} = \sum_{j = 1}^{3} ς_{j} z_{j}

(9)

Similarly

\begin{array}{l} δ x (t) = φ (t, x) δ x_{0} \\ = \sum_{j = 1}^{3} ς_{j} φ (t, x) z_{j} = \sum_{j = 1}^{3} ς_{j} μ_{j} z_{j} \end{array}

(10)

By definition

ε_{j} (t) = λ_{k} - \frac{1}{t} | μ_{j} (t) |

(11)

According to the definition of the Lyapunov exponent, when the limit, $\lim_{t \to 0} ε_{j} (t) = 0$ , is applied to time in equation (11), then

μ_{j} = e^{λ_{j} t - ε_{j} t + i θ_{j}}

(12)

Substituting equation (12) into (10) gives

\begin{array}{l} δ x (t) = \sum_{j = 1}^{3} ς_{j} e^{λ_{j} t - ε_{j} t + i θ_{j}} z_{j} \\ = e^{λ_{1} t} (ς_{1} e^{- ε_{1} t + i θ_{1}} z_{1} + \sum_{j = 2}^{6} ς_{j} e^{λ_{j} t - ε_{j} t + i θ_{j} - λ_{1} t} z_{j}) \end{array}

(13)

When $ς_{1}$ is nonzero, we take the logarithm and obtain

λ_{1} = \lim_{t \to + \infty} \frac{1}{t} \ln \frac{| δ x (t) |}{| δ x_{0} |}

(14)

From (14), when the system operation time t is large, then $| δ x (t) | = e^{λ_{1} t} | δ x_{0} |$ ; when $λ_{1} > 0$ , $δ x (t)$ increases according to the exponential $e^{λ_{1} t}$ , that is when the initial states of the two trajectories of the system are similar to each other, the distance between the two trajectories increases exponentially.

To make the deduction process generic, assume that there are $k$ initial states $δ x_{1} (0), δ x_{2} (0), \dots, δ x_{k} (0)$ ; therefore, we have

{\begin{matrix} | δ x_{1} (t) | = e^{λ_{1} t} | δ x_{1} (0) | \\ | δ x_{2} (t) | = e^{λ_{2} t} | δ x_{2} (0) | \\ ⋮ \\ | δ x_{k} (t) | = e^{λ_{k} t} | δ x_{k} (0) | \end{matrix}}

(15)

Then, the volume of the system after time t of the operation is expressed as

\begin{array}{l} | V_{k} (t) | = | δ x_{1} (t) | \dots | δ x_{k} (t) | \\ \approx e^{(λ_{1} + \dots λ_{k}) t} | V_{k (0)} | \end{array}

(16)

Consequently

\lim_{t \to + \infty} \frac{1}{t} \frac{| V_{k} (t) |}{| V_{k} (0) |} = λ_{1} + λ_{2} + \dots + λ_{k}

(17)

Let $λ^{k} = λ_{1} + λ_{2} + \dots + λ_{k}$ , which describes the growth rate of the k-dimensional volume, and $| V_{k} (0) | = | δ x_{1} (0) | \dots | δ x_{k} (0) |$ .

For the dissipation systems, equation (18) holds

λ^{k} = λ_{1} + λ_{2} + \dots + λ_{k} < 0

(18)

At least one Lyapunov exponent of the chaos system is positive, that is it is stretched in one or more directions, but the volume of the entire system is contracted. Therefore, to judge whether a nonlinear system will enter a chaotic state, we must only determine whether the maximum Lyapunov exponent is positive or negative. When the maximum Lyapunov exponent is less than zero, the distance between the neighboring trajectories will converge with time, and chaos will not occur. On the contrary, chaos will occur.

In this article, the Lyapunov exponent method is used to analyze the chaos of system (4), and the chaotic attractor and bifurcation diagram are used to verify the analysis results.

As observed in Figure 1(a), when the system parameter γ is greater than 11.6, the corresponding maximum Lyapunov exponential value begins to be greater than zero. Therefore, chaos will occur in the system. Figure 1(b) shows that when the system parameter γ is greater than 11.6, state x₃ begins to discontinuously change with the change in γ, that is it will start to abruptly change, which also indicates that chaos appears in the system. The chaotic attractor is one of the important characteristics of chaos. From Figure 1(c), we observe that the structure of the chaotic attractor does not continuously change with system parameter γ. During the process of parameter change, the entire structure of the chaotic attractor abruptly changes, which indicates that the system state irregularly and nonperiodically changes with time, that is chaos appears. This simulation further proves that chaos appears in the PMSM system when the parameter changes.

3. PMSM chaos control based on the inverse system decoupling and synthetical sliding mode

To realize the chaos control of the PMSM, a method including the inverse system decoupling and synthetical sliding mode is proposed in this article and shown in Figure 2. First, the reversibility of the PMSM mathematical model is analyzed, and a pseudo-linear inverse system including a first-order d-axis current subsystem and a second-order speed subsystem is established. Then, the first-order d-axis current subsystem is controlled by the SMC, and the second-order speed subsystem is controlled by the S-TSMC. Similarly, to improve the PMSM control performance, the switching function in the SMC is replaced by the hyperbolic tangent function.

3.1. Inverse system decoupling of the PMSM

The PMSM mathematical model clearly shows that the stator current and rotating speed have coupling characteristics, which will adversely affect the motor dynamic and static characteristics, and they need to be decoupled. In this article, the PMSM is decoupled by using the inverse system method.

Equation (3) can be written in the form of the state equation as follows

{\begin{cases} \frac{d x}{d t} = f (x, u) \\ y = Cx \end{cases}

(19)

where

x

is the state vector and

x = {[\begin{matrix} {\hat{i}}_{d} & {\hat{i}}_{q} & \hat{ω} \end{matrix}]}^{T}

u

is the input vector and

u = {[\begin{matrix} {\hat{u}}_{d} & {\hat{u}}_{q} \end{matrix}]}^{T}

, and

y

is the output vector and

y = {[\begin{matrix} {\hat{i}}_{d} & \hat{ω} \end{matrix}]}^{T}

Taking the derivative of the output ${\hat{i}}_{d}$

\frac{d {\hat{i}}_{d}}{d t} = - b {\hat{i}}_{d} + {\hat{i}}_{q} \hat{ω} + {\hat{u}}_{d}

(20)

Because the first-order derivative of output ${\hat{i}}_{d}$ contains input ${\hat{u}}_{d}$ , therefore, stop taking the derivative. Let $Y_{1} = d {\hat{i}}_{d} / d t$ , then the Jacobian matrix of $Y_{1}$ to $u$ is

\frac{\partial Y_{1}}{\partial u} = [\begin{matrix} 0 & 1 \end{matrix}]

(21)

Thus, $α_{1} = rank (\partial Y_{1} / \partial u) = 1$ .

Taking the derivative of output $\hat{ω}$ , we obtain

\frac{d \hat{ω}}{d t} = σ {\hat{i}}_{q} - σ \hat{ω} - {\hat{T}}_{L}

(22)

From equation (22), its first-order derivative does not include input ${\hat{u}}_{q}$ , so we continue to take its second-order derivative as follows

\begin{array}{l} \frac{d^{2} \hat{ω}}{d t^{2}} = σ [- (1 + σ) {\hat{i}}_{q} - {\hat{i}}_{d} \hat{ω} + (γ + σ) \hat{ω}] \\ = + σ {\hat{u}}_{q} + σ {\hat{T}}_{L} - \frac{d {\hat{T}}_{L}}{d t} \end{array}

(23)

Because the second-order derivative of output $\hat{ω}$ contains input ${\hat{u}}_{q}$ , therefore, we stop taking the derivative. And let $Y_{2} = {[\begin{matrix} d \hat{ω} / d t & d^{2} \hat{ω} / d t^{2} \end{matrix}]}^{T}$ , then the Jacobian matrix of $Y_{2}$ to ${\hat{u}}_{q}$ is

\frac{\partial Y_{2}}{\partial u} = [\begin{matrix} 1 & 0 \\ 0 & σ \end{matrix}]

(24)

Because $σ$ is not zero, therefore, $α_{2} = rank (\partial Y_{2} / \partial u) = 2$ .

In summary, the relative order of the PMSM dimensionless system is $α = {α_{1}, α_{2}} = {1,2}$ , that is $\sum_{i = 1}^{2} α_{i} = 3$ , so the system has an inverse system.

From equations (20) and (23), the output can be expressed as follows

{\begin{cases} {\hat{u}}_{d} = b {\hat{i}}_{d} - {\hat{i}}_{q} \hat{ω} + {\hat{i}}_{d}^{'} \\ {\hat{u}}_{q} = {\hat{ω}}^{″} - [- (1 + σ) {\hat{i}}_{q} - {\hat{i}}_{d} \hat{ω} + (γ + σ) \hat{ω}] - {\hat{T}}_{L} + \frac{1}{σ} \frac{d {\hat{T}}_{L}}{d t} \end{cases}

(25)

Equation (25) can be written as

\hat{u} = h_{2}^{- 1} [x, {\hat{i}}_{d}^{'}, {\hat{ω}}^{″}]

(26)

By substituting equation (26) into (19), using variable substitution, and replacing the variables $u, y, and x$ with $w, v, and z$ , we obtain the inverse system of equation (3) as follows

{\begin{cases} \frac{d z}{d t} = f {z, h_{2}^{- 1} [z, v_{1}^{'}, v_{2}^{″}]} \\ w = h_{2}^{- 1} [z, v_{1}^{'}, v_{2}^{″}] \end{cases}

(27)

By connecting the original system expressed by equation (3) with its inverse system in equation (27), a physically implementable pseudo-linear system can be constructed, as shown in Figure 3. Therefore, the control problem of the original system is transformed into the control problem of the pseudo-linear system.

In this article, a synthetical SMC is applied to control the pseudo-linear system. The purpose of the design of the synthetical SMC system is to suppress the chaos phenomenon of the PMSM and make the motor have better dynamic performance. Therefore, the sliding mode controllers are designed for the d-axis current subsystem and rotating speed subsystem to realize the PMSM chaos control based on the inverse system decoupling and synthetical sliding mode.

3.2. Sliding mode controller design of the d-axis current subsystem

To control the d-axis current subsystem, $s_{i} = {\hat{i}}_{d} - {\hat{i}}_{d}^{*}$ is selected as the sliding mode variable, where ${\hat{i}}_{d}^{*}$ is the expected d-axis current and is set as 0. Take the derivative of $s_{i}$

\frac{d s_{i}}{d t} = - b {\hat{i}}_{d} + {\hat{i}}_{q} \hat{ω} + {\hat{u}}_{d}

(28)

To satisfy the sliding condition, the control law can be designed as follows

{\hat{u}}_{d} = - {\hat{i}}_{q} \hat{ω} - α_{1} \tanh (s_{1})

(29)

where

α_{1} > 0

and the switching function is replaced by the hyperbolic tangent function to effectively reduce the chattering and improve the control accuracy.

Under the control law shown in equation (29), the Lyapunov function expressed as equation (30) is established to prove that the system state can converge to the sliding surface in finite time

V = \frac{1}{2} s_{1}^{2}

(30)

Taking the derivative of equation (3) and combining it with equation (29), we obtain

\begin{array}{l} \frac{d V}{d t} = s_{1} \frac{d s_{1}}{d t} \\ = {\hat{i}}_{d} (- b {\hat{i}}_{d} + {\hat{i}}_{q} \hat{ω} + {\hat{u}}_{d}) \\ = - b {\hat{i}}_{d}^{2} + {\hat{i}}_{d} {\hat{i}}_{q} \hat{ω} + {\hat{i}}_{d} {\hat{u}}_{d} \\ = - b {\hat{i}}_{d}^{2} + {\hat{i}}_{d} {\hat{i}}_{q} \hat{ω} + {\hat{i}}_{d} (- {\hat{i}}_{q} \hat{ω} - α_{1} \tanh (s_{1})) \\ = - b {\hat{i}}_{d}^{2} - α_{1} {\hat{i}}_{d} \tanh (s_{1}) \\ \leq - b {\hat{i}}_{d}^{2} - α_{1} | {\hat{i}}_{d} | < 0 \end{array}

(31)

From equation (31), when $s_{i} \neq 0$ , then $d V / d t < 0$ , and when $s_{i} = 0$ , then $d V / d t = 0$ , that is $d V / d t \leq 0$ , which implies that the sliding condition is satisfied, and the d-axis current can be made to track the expected values, which satisfies the design objective of the d-axis current subsystem controller.

3.3. Super-twisting sliding mode controller design of the rotating speed subsystem

When controlling the rotating speed subsystem, $s_{2} = \hat{ω} - {\hat{ω}}^{*}$ is selected as the sliding variable, and ${\hat{ω}}^{*}$ is the expected rotating speed. When we take the derivative of $s_{2}$ , the control input ${\hat{u}}_{q}$ only appears in its second-order derivative, so its relative order is 2. To eliminate the chattering in the traditional SMC and accurately track the expected rotating speed values, a PMSM rotating speed controller is designed based on the S-TSMC.

The PMSM rotating speed controller can be designed as follows

{\begin{cases} {\hat{u}}_{q} = - k_{1} {| s_{2} |}^{1 / 2} sgn (s_{2}) + {\hat{u}}_{q 1} \\ \frac{d {\hat{u}}_{q 1}}{d t} = - k_{3} sgn (s_{2}) \end{cases}

(32)

To make all trajectories of the system converge to zero in finite time, which is proven by constructing the Lyapunov function as shown in equation (33)

V = 2 k_{2} | s_{2} | + \frac{1}{2} {\hat{u}}_{q 1}^{2} + \frac{1}{2} {(k_{1} {| s_{2} |}^{1 / 2} sgn (s_{2}) - {\hat{u}}_{q 1}^{2})}^{2}

(33)

The Lyapunov function is rewritten in the quadratic form as follows

V = ς^{T} Pς

(34)

where

ς^{T} = [\begin{matrix} {| s_{2} |}^{1 / 2} & sgn (s_{2}) & {\hat{u}}_{q 1} \end{matrix}]

and

P = 1 / 2 [\begin{matrix} 4 k_{2} + k_{1}^{2} & - k_{1} \\ - k_{1} & 2 \end{matrix}]

Because $V$ is continuous, it is nondifferentiable at $s_{2} = 0$ . When $k_{2} > 0$ , $V$ is positive definite and radial unbounded, then

λ_{\min} {P} {‖ ς ‖}_{2}^{2} \leq V \leq λ_{\max} {P} {‖ ς ‖}_{2}^{2}

(35)

where

{‖ ς ‖}_{2}^{2}

is the Euclidean norm of

ς

, and its value is

{‖ ς ‖}_{2}^{2} = | s_{2} | + {\hat{u}}_{q 1}^{2}

Taking the derivative of equation (34) yields

\frac{d V}{d t} = - \frac{1}{{| s_{2} |}^{1 / 2}} ς^{T} Qς \leq - \frac{1}{{| s_{2} |}^{1 / 2}} λ_{\min} {Q} {‖ ς ‖}_{2}^{2}

(36)

where

Q = k_{1} / 2 [\begin{matrix} 2 k_{2} + k_{1}^{2} & - k_{1} \\ - k_{1} & 1 \end{matrix}]

If $k_{1}, k_{2} > 0$ , the values of matrix $Q$ are greater than 0, so $(d V / d t)$ is negative definite.

Combining equations (35) and (36), we obtain the following equations

{| s_{2} |}^{1 / 2} \leq {‖ ς ‖}_{2} \leq \frac{V^{1 / 2}}{λ_{\min}^{1 / 2} {P}}

(37)

\frac{d V}{d t} \leq - γ V^{1 / 2}

(38)

where

γ = (λ_{\min}^{1 / 2} {P} λ_{\min} {Q} / λ_{\min} {P})

The following differential equations are solved

{\begin{cases} \frac{d υ}{d t} = - γ υ^{1 / 2} \\ υ (0) = υ_{0} > 0 \end{cases}

(39)

The solution is

υ (t) = {(υ_{0}^{1 / 2} - \frac{γ}{2} t)}^{2}

(40)

According to the comparison principle, when $V \leq υ_{0}$ , then $V (t) \leq υ (t)$ . From equation (40), after time $T = ({2 V^{1 / 2} |}_{s_{2} = s_{20}} / γ)$ , the system trajectory can reach sliding surface $s_{2}$ . $s_{20}$ is the initial error between the actual trajectory and the desired trajectory. Therefore, the rotating speed S-TSMC controller is stable and can make the system motion achieve the expected motion state in finite time.

Similarly, the switching function in equation (32) is replaced with the hyperbolic tangent function to further improve the control effect of the S-TSMC.

4. Simulation analysis

To verify the proposed chaos control algorithm based on the inverse system decoupling and synthetical sliding mode, the PMSM system model shown in equation (1) is established by using MATLAB/Simulink software. Three kinds of chaos controllers are designed: SMC of the noninverse system decoupling, synthetical sliding mode I with inverse system decoupling (the switching function is used to switch), and synthetical sliding mode II with inverse system decoupling (the hyperbolic tangent function is used to switch). A comparative analysis of the simulation is performed to verify the validity of the proposed control algorithm.

The initial values of the state in the simulation are ${[\begin{matrix} {\hat{i}}_{d} & {\hat{i}}_{q} & \hat{ω} \end{matrix}]}^{T} = {[\begin{matrix} 0.03 & 0.03 & 0.03 \end{matrix}]}^{T}$ , and parameters $σ$ , $γ$ , and $b$ are 5.46, 17, and 1, respectively. When these parameter values are adopted, chaos will appear in the system. The no-load operation of the PMSM is considered as the simulation working condition. The chaos controller begins working at the 5th s. Figures 5–9 show the simulation results.

The expected rotating speed of the motor is set as shown in Figure 4, which is a square wave signal, the period is 5 s, the duty cycle is 50%, the maximum rotating speed is 500 r/min, and the minimum value is 0 r/min. It is used to simulate the sudden change in the motor rotating speed from one stable operation state to another. During the simulation, the PMSM system chaos control is not performed in the first 5 s, so the motor is in a chaotic state. The chaos control is applied to the PMSM system after the 5th s. As shown in Figure 5, the PMSM chaos can be suppressed by the three kinds of chaos control methods, so that the PMSM can remove the chaos. However, it can be seen that synthetical sliding mode II with inverse system decoupling has the shortest trajectory, which indicates that the overshoot is the smallest. Therefore, Figure 5 shows that synthetical sliding mode II with inverse system decoupling is obviously better than the other two kinds of chaos control methods.

Figures 6 and 7 shows that in the first 5 s, the d-axis and q-axis currents of the PMSM greatly fluctuate and are in a chaotic state. When the chaos control is applied to the PMSM system after the 5th s, although the three kinds of chaos control methods can make the PMSM break away from the chaos, the current fluctuation and overshoot of the SMC of noninverse system decoupling are the largest in the process of reaching the stable operation state, and after reaching the stable state, the current value still fluctuates slightly near the stable value. Nevertheless, among the three kinds of chaos control methods, synthetical SMC II with inverse system decoupling has the smallest overshoot. When the motor current reaches a stable state, the current value also has the slightest fluctuations. The results show that synthetical SMC II with inverse system decoupling has a better control effect than the other two kinds of chaos control methods.

Figures 8 and 9 shows that in the first five seconds, the PMSM rotating speed fluctuates, and it cannot track the expected rotating speed. Its operation is in a chaotic state, which is not conducive to the application of the PMSM. When the chaos control is applied to the PMSM system after the 5th s, all three kinds of chaos control methods can make the PMSM rotating speed track the expected rotating speed. However, the synthetical SMC II with inverse system decoupling has the smallest overshoot and the slightest fluctuation after reaching a stable value, which indicates that synthetical SMC II with inverse system decoupling has an obvious chaotic suppression effect compared with that of the other two kinds of chaos control methods and can effectively separate the PMSM from the chaotic state.

5. Experimental analysis

To further verify the practical effect of the chaos control algorithm proposed in this article, the motor test platform shown in Figure 10 is built. The motor test platform includes a motor-driven test box, an upper monitor, a PMSM, and a loading motor. The control circuit and driving circuit are integrated in the motor driving experiment box. The main control chip TMS320F28335 DSP provides the SVPWM signal, and the driving circuit drives the PMSM to realize the driving control of the PMSM.

Similar to the simulation, three kinds of chaos controllers are designed in the experiment: SMC of noninverse system decoupling, synthetical sliding mode I with inverse system decoupling, and synthetical sliding mode II with inverse system decoupling. The expected rotating speed of the motor is shown in Figure 4. The experimental conditions are consistent with the simulation, and the experimental results are shown in Figures 11–14.

As seen from Figures 11 and 12, when the chaos control is applied to the PMSM system after the 5th s, although the three kinds of chaos control methods can effectively control the PMSM, the current fluctuation and overshoot of the SMC of noninverse system decoupling are the largest in the process of reaching the stable operation state, and after reaching the stable state, the current value still slightly fluctuates near the stable value. Synthetic SMC II with inverse system decoupling has the smallest overshoot. When the motor current reaches the stable state, the current value also has the slightest fluctuations. Therefore, the experimental results are consistent with the simulation results, which show that synthetical SMC II with inverse system decoupling has a better control effect than the other two kinds of chaos control methods. The results of Figures 13–14 are also consistent with the simulation results, which also show that synthetical SMC II with inverse system decoupling has an obvious control effect and can effectively control the PMSM compared with the other two kinds of chaos control.

6. Conclusion

Aiming at the chaos phenomenon in the operation of the PMSM, first, the PMSM nonlinear model is established, and the chaos analysis is performed using the Lyapunov exponent method. The analysis reveals that when the motor parameters are in a certain range, chaos will occur in the operation of the motor and be manifested by the sharp fluctuation of the torque and speed. To control the chaos of the PMSM, the reversibility of the PMSM model is analyzed, and a pseudo-linear inverse system, which includes a first-order d-axis current subsystem and a second-order rotating speed subsystem, is established to decouple the PMSM. Then, synthetical SMC II with inverse system decoupling is proposed to control the chaos of the PMSM, where the d-axis current is controlled by the SMC, and the rotating speed is controlled by the S-TSMC, both of which use the hyperbolic tangent function to switch. Finally, the simulation analysis and experiment analysis are performed, which indicate that synthetical SMC II with inverse system decoupling has a good control effect on the chaos of the PMSM compared with the another two control methods.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This project was supported by the National Natural Science Foundation of China (Grant No. 51605003 and 51575001), the Anhui Science and Technology Project (Grant No. 1604a0902158), and the introduction of talents science research foundation of Anhui Polytechnic University (Grant No. 2016YQQ002). In addition, I am very grateful to my wife and children for their selfless support in my work and life.

ORCID iD

Rongyun Zhang

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