Stability Analysis and Chaos Control of Permanent-Magnet Synchronous Motor

Abstract

This paper investigates the dynamics of a permanent magnet synchronous motor (PMSM) and controls its chaotic speed behavior using the synergetic control technique (SCT). The model includes electrical dynamics in the dq frame and mechanical speed dynamics, with a scalar parameter $\gamma$ capturing cross-coupling effects. The equilibrium structure and local stability properties of the PMSM are analyzed. For zero input voltages and zero load torque, the system exhibits a pitchfork-type bifurcation in the electrical–mechanical equilibrium as $\gamma$ crosses a critical value. Explicit expressions are derived for all equilibria, and their stability is characterized using eigenvalue analysis and the Routh–Hurwitz criterion, and a secondary loss of stability via a Hopf-type mechanism is identified. The case of nonzero input voltages with zero load torque is also discussed. Numerical simulations confirm the analytical results and highlight the parameter regions that admit stable operation. Bifurcation diagrams show the different PMSM behaviors as the parameter $\gamma$ varies. For a certain interval of $\gamma$, the PMSM speed undergoes chaotic oscillations. The SCT is introduced to control the chaos. Macro variables are chosen to design the SCT. The derived SCT is implemented to eliminate the chaotic speed. The controller provides good performance in suppressing the chaos. The controller is tested under sudden reference speed change where the controller gets the new reference speed accurately. It is also evaluated under sudden and sinusoidal load torque variations.

1. Introduction

PMSM technology benefits significantly from advanced drive electronics and control techniques that enable precise torque and speed control over a broad range of speeds. Unlike induction motors which rely on induced flux, PMSMs have permanent magnets to create flux, which requires the variable frequency drive (VFD) controlling them to manage flux current differently for optimal performance. This allows for excellent efficiency, high power density, high torque-to-weight ratio, and precise dynamic response, making PMSMs very suitable as variable speed drives in many industrial and high-performance settings [1,2,3].

PMSMs are widely used in applications such as electric vehicles and hybrid vehicles [4,5], industrial robotics and CNC machines [6,7], HVAC systems (compressors, fans) and pumps [8,9], elevators and escalators [10], servo systems [11], and wind turbines [12].

However, the nonlinear nature of the differential equations governing the dynamics of these motors makes them susceptible to complex dynamic phenomena, most notably chaotic behavior. This undesirable behavior presents a significant challenge, leading to degraded motor performance, increased noise and vibration, and ultimately, system failure [13]. The bifurcation and chaos in nonlinear systems have been investigated in various scientific fields. Recently, in PMSM [14,15] induction motor drives [16], brushless DC motors [17], oil drilling systems actuated by induction motor [18], rotor stator system [19], 3D brushless DC motor [20], and DC motor [21] have been investigated.

Researchers have continued to deepen their understanding of the conditions that lead to disorder in PMSM systems, confirming that disorder typically arises when system parameters, such as source frequency, load torque, or resistor and inductor values, exceed certain critical limits [22].

Nonlinear systems and bifurcation theory provide a natural framework to analyze the qualitative behavior of such models [23,24,25]. In particular, the structure and stability of equilibrium points, as system parameters vary, can reveal critical thresholds beyond which conventional control strategies fail or require significant redesign.

The study of chaotic behavior relies primarily on the tools of nonlinear systems theory. Research studies focus on bifurcation diagrams, Lyapunov exponents, and Poincaré sections to identify chaotic operating regions [26]. For example, Huang et al. (2011) analyzed the fundamental dynamical properties of PMSMs, confirming their nonlinear nature and susceptibility to chaotic states [27].

More complex mathematical models of PMSMs, incorporating load torque effects or irregular air-gap conditions, have been developed, leading to the discovery of new chaotic patterns [28]. In this context, recent research (e.g., Fotsa et al., 2025) has highlighted the importance of studying motor dynamics under load torque, as this may result in unexpected chaotic behavior [29].

Controlling chaos in PMSMs represents one of the most active research directions, with proposed methods aiming to suppress chaotic behavior and restore the system to a desired stable or periodic state. These strategies can be categorized into several main groups, including adaptive control [30], single-state feedback control [31], inverse system control [32], fuzzy logic control [33], delay feedback control with analog circuit validation [34], adaptive terminal sliding mode control (ATSMC) [35], fixed-time adaptive control [36], sliding mode control (SMC) [37], time-limited control [38], adaptive backstepping-based tracking control [39], Lyapunov-based predictive control [40], neural network-based control [41], and genetic algorithm approaches [42].

Most of the aforementioned chaos-control methods for PMSMs are affected by parameter variations, and some, such as SMC, may introduce chattering due to discontinuous control action. In contrast, the synergetic control technique (SCT) offers a simpler design and is well suited for real-time implementation. Therefore, SCT is considered suitable for suppressing chaotic dynamics in PMSMs, as it is chattering-free and relatively independent of system parameter variations [43]. SCT has been successfully applied to control chaotic dynamics in power systems [44] and in indirect field-oriented induction motor drive systems [45].

In this paper, SCT is employed to eliminate chaotic dynamics due to its continuous control action and robustness against parameter variations. A simplified PMSM model is analyzed with a coupling parameter $\gamma$ representing cross-coupling effects in the dq frame. The analysis shows that:

• For small $\gamma$, the origin is the unique equilibrium and is locally asymptotically stable.
• When $\gamma$ exceeds a threshold, two additional nonzero equilibria appear, while the origin becomes unstable (pitchfork-type bifurcation).
• At a larger value of $\gamma$, the nonzero equilibria lose stability via a Hopf-type mechanism.

All results are obtained analytically and confirmed numerically. The SCT strategy is derived by proposing three macro variables and implemented numerically to suppress the chaotic oscillations in the PMSM dynamics. The SCT suppresses chaotic behavior by driving the system trajectories toward a stable manifold associated with a stable equilibrium point.

In addition, to quantitatively demonstrate the effectiveness of the proposed approach, key performance indicators are explicitly reported in Section 7. These include a chaos suppression time of approximately 1.1 s following controller activation, fast speed tracking under multiple step changes (settling times below 1.1 s), and strong robustness against both sudden and sinusoidal load torque disturbances. These results confirm the dynamic performance and practical feasibility of the proposed SCT strategy.

The remainder of the paper is organized as follows. Section 2 presents the mathematical model of the PMSM. Section 3 provides the equilibrium and stability analysis. Section 4 presents numerical simulations, including bifurcation diagrams, Lyapunov exponents, Poincaré maps, and phase portraits. Section 5 derives the SCT controller. Section 6 introduces the extended state observer (ESO) for torque estimation. Section 7 discusses the control results, and Section 8 concludes the paper.

2. System State Equations

A three-phase PMSM model consists of three nonlinear differential equations [46]:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{i}_d}{dt}}& =(u_d-R_1{i}_d+\omega L_q{i}_q)/L_d\hfill \\ \hfill {\displaystyle \frac{d{i}_q}{dt}}& =(u_q-R_1{i}_q-\omega L_d{i}_d+\omega {\psi }_f)/L_q\hfill \\ \hfill {\displaystyle \frac{d\omega }{dt}}& =[n_p{\psi }_f{i}_q+n_p(L_d-L_q)i_d{i}_q-T_L-\beta \omega ]/J\hfill \end{array}$$

(1)

Here, $i_d$, $i_q$, $L_d$, $L_q$, $u_d$, and $u_q$ denote the direct–quadrature axis stator components of current, inductances, and voltages, respectively. The variables $\omega$, $R_1$, J, $T_L$, $\beta$, ${\psi }_f$, and $n_p$ represent the motor angular speed, stator winding resistance, moment of inertia, load torque, viscous damping coefficient, permanent magnet flux, and number of pole pairs, respectively.

By applying the transformation given in Equation (2) and the time-scaling transformation in Equation (3), the dimensionless model can be expressed as Equation (5):

$$x=S\tilde{x}$$

(2)

$$t=\gamma \tilde{t}$$

(3)

where

$$S=\left[\begin{array}{ccc}{s}_d& 0& 0\\ 0& s_q& 0\\ 0& 0& s_{\omega }\end{array}\right]=\left[\begin{array}{ccc}bk& 0& 0\\ 0& k& 0\\ 0& 0& \frac{1}{\tau }\end{array}\right]$$

(4)

with

$$k={\displaystyle \frac{\beta }{n_p\tau {\psi }_f}},b={\displaystyle \frac{L_q}{L_d}},\tau ={\displaystyle \frac{L_q}{R_1}}$$

After normalization, the dimensionless model becomes:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\widetilde{i_d}}{d\tilde{t}}}& =-\widetilde{i_d}+\tilde{\omega }\widetilde{i_q}+\widetilde{u_d}\hfill \\ \hfill {\displaystyle \frac{d\widetilde{i_q}}{d\tilde{t}}}& =-\widetilde{i_q}-\tilde{\omega }\widetilde{i_d}+\gamma \tilde{\omega }+\widetilde{u_q}\hfill \\ \hfill {\displaystyle \frac{d\tilde{\omega }}{d\tilde{t}}}& =\delta (\widetilde{i_q}-\tilde{\omega })+\epsilon \widetilde{i_d}\widetilde{i_q}-{\tilde{T}}_L\hfill \end{array}$$

(5)

The dimensionless parameters are defined as

$$\delta ={\displaystyle \frac{{\psi }_f}{k{L}_q}},\sigma ={\displaystyle \frac{\beta }{J\gamma }},ϵ={\displaystyle \frac{n_{p}b{\tau }^2{k}^2(L_d-L_q)}{J}},{\tilde{T}}_L={\displaystyle \frac{{\tau }^2}{J}}{T}_L,$$

and the scaled voltages are given by

$${\tilde{u}}_d={\displaystyle \frac{u_d}{R_{1}k}},{\tilde{u}}_q={\displaystyle \frac{u_q}{R_{1}k}}.$$

For a smooth-air-gap PMSM ($L_q=L_d$), the nonlinear normalized model can be written as:

$$\begin{array}{cc}\hfill {\dot{x}}_1& =-x_1+x_2{x}_3+{\tilde{u}}_d\hfill \\ \hfill {\dot{x}}_2& =-x_3{x}_1-x_2+\gamma x_3+{\tilde{u}}_q\hfill \\ \hfill {\dot{x}}_3& =\sigma (x_2-x_3)-{\tilde{T}}_L\hfill \end{array}$$

(6)

3. Dynamical Analysis

3.1. Equilibrium Equations

An equilibrium point $x^{*}=(x_1^{*},x_2^{*},x_3^{*})$ satisfies

$${\dot{x}}_1={\dot{x}}_2={\dot{x}}_3=0$$

Solving ${\dot{x}}_3=0$ gives

$$x_2^{*}=x_3^{*}+{\displaystyle \frac{{\tilde{T}}_L}{\sigma }}$$

Solving ${\dot{x}}_1=0$ yields

$$x_1^{*}=x_2^{*}{x}_3^{*}+{\tilde{u}}_d=x_3^{*}\left(x_3^{*}+{\displaystyle \frac{{\tilde{T}}_L}{\sigma }}\right)+{\tilde{u}}_d$$

Substituting these expressions into ${\dot{x}}_2=0$ gives

$$0=-x_3^{*}{x}_1^{*}-x_2^{*}+\gamma x_3^{*}+{\tilde{u}}_q$$

After simplification, the following cubic equation is obtained:

$${\left(x_3^{*}\right)}^3+{\displaystyle \frac{{\tilde{T}}_L}{\sigma }}{\left(x_3^{*}\right)}^2+({\tilde{u}}_d+1-\gamma )x_3^{*}+\left({\displaystyle \frac{{\tilde{T}}_L}{\sigma }}-{\tilde{u}}_q\right)=0$$

(7)

Let z be a real root of the above cubic equation. Then the equilibrium point is given by

$$\begin{array}{|c|}\hline \begin{array}{cc}\hfill x_3^{*}& =z\hfill \\ \hfill x_2^{*}& =z+{\displaystyle \frac{{\tilde{T}}_l}{\sigma }}\hfill \\ \hfill x_1^{*}& =z\left(z+{\displaystyle \frac{{\tilde{T}}_L}{\sigma }}\right)+{\tilde{u}}_d\hfill \end{array}\\ \hline\end{array}$$

(8)

3.2. Equilibrium Analysis for Special Case

Special case: ${\tilde{T}}_L={\tilde{u}}_d={\tilde{u}}_q=0$, so that Equation (6) reduces to

$$\begin{array}{cc}\hfill {\dot{x}}_1& =-x_1+x_2{x}_3\hfill \\ \hfill {\dot{x}}_2& =-x_3{x}_1-x_2+\gamma x_3\hfill \\ \hfill {\dot{x}}_3& =\sigma (x_2-x_3)\hfill \end{array}$$

(9)

Thus, using Equations (7) and (8):

• If $\gamma \le 1$, the origin is the unique equilibrium point:

  $$(0,0,0).$$
• If $\gamma >1$, three equilibrium points exist:

  $$(0,0,0),(\gamma -1,\pm \sqrt{\gamma -1},\pm \sqrt{\gamma -1}).$$

Figure 1a shows the branches of $x_3^{★}\left(\gamma \right)$: the trivial branch $x_3^{★}=0$ exists for all $\gamma$, while the two nonzero branches $\pm \sqrt{\gamma -1}$ exist only for $\gamma >1$. This structure corresponds to a pitchfork-type bifurcation [25].

3.2.1. Linearization and Local Stability

The Jacobian of Equation (9) at a point x is

$$J\left(x\right)=\left[\begin{array}{ccc}-1& x_3& x_2\\ -x_3& -1& -x_1+\gamma \\ 0& \sigma & -\sigma \end{array}\right].$$

(10)

3.2.2. Stability of the Origin

At the origin, the Jacobian becomes

$$J_0\left(\gamma \right)=\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& \gamma \\ 0& \sigma & -\sigma \end{array}\right].$$

(11)

Its characteristic polynomial is

$$\begin{array}{cc}\hfill det(\lambda I-J_0)& =(\lambda +1)\left({\lambda }^2+11\lambda +\sigma (1-\gamma )\right).\hfill \end{array}$$

The quadratic factor has the form

$${\lambda }^2+a_1\lambda +a_0,a_1=11,a_0=\sigma (1-\gamma ).$$

For Equation (9):

• For $0\le \gamma <1$, $a_1>0$, $a_0>0$, so both roots are in the open left half-plane by the Routh–Hurwitz criterion [24] (all eigenvalues of $J_0$ satisfy $\Re \lambda <0$). Together with the eigenvalue $\lambda =-1$, this implies asymptotic stability of the origin.
• For $\gamma =1$, $a_0=0$ and one eigenvalue is zero. Hence, the origin is non-hyperbolic and a bifurcation occurs.
• For $\gamma >1$, $a_0<0$ and the product of the two roots is negative, so one is positive. Therefore, the origin becomes unstable (a saddle point).

3.2.3. Stability of the Nonzero Equilibria

For $\gamma >1$, consider the equilibrium point

$$x^{★}=(\gamma -1,a,a),a=\sqrt{\gamma -1}$$

(the other nonzero equilibrium is symmetric with $a\mapsto -a$ and has the same eigenvalues). Substituting into Equation (10) gives

$$J_{\pm }\left(\gamma \right)=\left[\begin{array}{ccc}-1& a& a\\ -a& -1& 1\\ 0& \sigma & -\sigma \end{array}\right].$$

(12)

Its characteristic polynomial is

$${\lambda }^3+(\sigma +2){\lambda }^2+(\sigma +a^2+1)\lambda +\sigma (2a^2+1)=0.$$

(13)

The two nonzero equilibria of Equation (9) are:

$$(\gamma -1,\pm \sqrt{\gamma -1},\pm \sqrt{\gamma -1})$$

The characteristic polynomial can be written in the standard form

$${\lambda }^3+a_1{\lambda }^2+a_2\lambda +a_3,$$

For $\sigma =10$, this becomes

$$a_1=12,a_2=10+\gamma ,a_3=20(\gamma -1)$$

The third-order Routh–Hurwitz conditions [23,24] require

$$a_1>0,a_2>0,a_3>0,a_1{a}_2>a_3.$$

For $\gamma >1$, it is clear that $a_1>0$, $a_2>0$, and $a_3>0$. The remaining condition yields

$$12(10+\gamma )>20(\gamma -1)⟺\gamma <17.5.$$

• For $1<\gamma <17.5$, all eigenvalues have negative real parts; therefore, the nonzero equilibria are asymptotically stable.
• For $\gamma >17.5$, the inequality is reversed, and at least one eigenvalue has a positive real part, implying instability.

Figure 1 illustrates the equilibrium structure and local stability of the normalized PMSM model under the special case ${\tilde{T}}_L={\tilde{u}}_d={\tilde{u}}_q=0$. The bifurcation diagram in Figure 1a represents the evolution of equilibrium points in the dq-frame state space as the coupling parameter $\gamma$ varies. For $\gamma$ < 1, the origin is the unique asymptotically stable equilibrium. At $\gamma$ = 1, an eigenvalue crosses zero (Figure 1b), leading to loss of hyperbolicity and the emergence of two symmetric nonzero equilibria for $\gamma$ = 1, consistent with a pitchfork-type bifurcation. The eigenvalue trajectories in Figure 1b corroborate the Routh–Hurwitz stability analysis presented above. Figure 1 therefore provides a dynamical stability map of the reduced PMSM system, rather than an electromagnetic field representation.

To further characterize the loss of stability at $\gamma =17.5$, observe that the Routh–Hurwitz condition $a_1{a}_2=a_3$ defines the critical boundary where the real part of a complex-conjugate pair of eigenvalues becomes zero. Since the coefficients $a_1$, $a_2$, and $a_3$ remain positive at the critical value, and the instability occurs through violation of the inequality $a_1{a}_2>a_3$, the transition is associated with a pair of eigenvalues crossing the imaginary axis.

Numerically, for $\gamma \ge 17.5$, the system exhibits sustained oscillatory behavior, as observed in the bifurcation diagrams and time-series simulations (Section 4). This combination of (i) the analytical stability boundary determined via the Routh–Hurwitz conditions and (ii) the numerical emergence of persistent oscillations is consistent with a Hopf-type bifurcation mechanism.

A full center-manifold reduction and normal-form derivation of the Hopf bifurcation conditions is beyond the scope of the present study. Nevertheless, the analytical eigenvalue crossing condition, together with the observed oscillatory attractor, provides strong evidence supporting the Hopf-type interpretation.

3.2.4. Summary

• For $\gamma <17.5$, the nonzero equilibria are asymptotically stable.
• At $\gamma \approx 17.5$, the eigenvalues cross the imaginary axis, leading to critical slowing down.
• For $\gamma >17.5$, the nonlinear PMSM model exhibits a stable oscillatory attractor, consistent with a supercritical Hopf-type bifurcation.

These results demonstrate that the bifurcation parameter $\gamma$ fundamentally alters the stability of the PMSM equilibrium, a finding consistent with well-known nonlinear motor dynamics [23,24].

3.3. Equilibrium and Stability for Constant q–Axis Excitation

In this section, the PMSM model in Equation (6) is revisited under the following operating condition:

$${\tilde{u}}_d=0,{\tilde{u}}_q=10,T_L=0,$$

(14)

Then, Equation (6) reduces to

$$\begin{array}{cc}\hfill {\dot{x}}_1& =-x_1+x_2{x}_3\hfill \\ \hfill {\dot{x}}_2& =-x_3{x}_1-x_2+\gamma x_3+{\tilde{u}}_q,\hfill \\ \hfill {\dot{x}}_3& =\sigma (x_2-x_3),\hfill \end{array}$$

(15)

which corresponds to a constant q-axis excitation and zero d-axis voltage and load torque. Such operating modes are common in torque-producing PMSM.

Equilibria and their stability are analyzed for $\gamma \in [0,20]$. The equilibrium $x^{★}=(x_1^{★},x_2^{★},x_3^{★})$ of Equation (15) can be satisfied by making:

$$\dot{x_1}=\dot{x_2}=\dot{x_3}=0.$$

From ${\dot{x}}_3=0$,

$$x_2^{★}=x_3^{★}.$$

(16)

From ${\dot{x}}_1=0$,

$$x_1^{★}=x_2^{★}{x}_3^{★}.$$

(17)

Define

$$z:=x_3^{★}.$$

Then from Equations (16) and (17)

$$x_2^{★}=z,x_1^{★}=z^2.$$

Substituting into ${\dot{x}}_2=0$ yields

$$0=-x_3^{★}{x}_1^{★}-x_2^{★}+\gamma x_3^{★}+\sigma =-z·z^2-z+\gamma z+\sigma ,$$

that is,

$$-z^3+(\gamma -1)z+\sigma =0.$$

(18)

or equivalently,

$$z^3-(\gamma -1)z-\sigma =0.$$

(19)

For $\sigma =10$, this equation can be written in the depressed cubic form $z^3+pz+q=0$ with

$$p=-(\gamma -1),q=-10.$$

Its discriminant is

$$\Delta =-\left(4p^3+27q^2\right)=4{(\gamma -1)}^3-2700.$$

According to classical cubic theory [25]:

• $\Delta <0$ implies one real root and two complex-conjugate roots;
• $\Delta =0$ implies multiple (coincident) roots;
• $\Delta >0$ implies three distinct real roots.

Solving $\Delta =0$ gives

$$4{(\gamma -1)}^3-2700=0\Rightarrow {\gamma }_{\mathrm{m}}=1+{675}^{1/3}\approx 9.77.$$

Hence, there is always at least one real root, and for $\gamma >{\gamma }_{\mathrm{m}}$ there are three distinct real roots. The corresponding equilibria follow from $x_2^{★}=x_3^{★}=z$ and $x_1^{★}=z^2$.

• For $\gamma \in [0,20]$, the “main” physically relevant equilibrium branch corresponds to the positive root $z=z\left(\gamma \right)$ of Equation (19), which increases monotonically from

  $$z\left(0\right)=2\mathrm{to}z\left(20\right)\approx 4.60.$$
• For $\gamma >{\gamma }_{\mathrm{m}}$, two additional equilibria with $z<0$ appear, as expected from the discriminant analysis.

3.3.1. Linearization and Characteristic Polynomial

The Jacobian matrix of system (15), for $\sigma =10$, is

$$J\left(x\right)=\left[\begin{array}{ccc}-1& x_3& x_2\\ -x_3& -1& -x_1+\gamma \\ 0& 10& -10\end{array}\right].$$

(20)

At an equilibrium $(z^2,z,z)$, the Jacobian becomes

$$J^{★}(z,\gamma )=\left[\begin{array}{ccc}-1& z& z\\ -z& -1& -z^2+\gamma \\ 0& 10& -10\end{array}\right].$$

(21)

A straightforward computation yields the characteristic polynomial

$${\lambda }^3+a_1{\lambda }^2+a_2\lambda +a_3=0,$$

(22)

with

$$a_1=12,a_2=-10\gamma +11z^2+21,a_3=-10\gamma +30z^2+10.$$

(23)

Using the equilibrium relation (19), we can eliminate $\gamma$ in favor of z. From

$$z^3-(\gamma -1)z-10=0\Rightarrow \gamma =1+{\displaystyle \frac{z^3-10}{z}},$$

Substituting into the expressions for $a_2$ and $a_3$ gives

$$\begin{array}{cc}\hfill a_1& =12,\hfill \\ \hfill a_2& =z^2+11+{\displaystyle \frac{100}{z}},\hfill \\ \hfill a_3& =20{\displaystyle \frac{z^3+5}{z}}.\hfill \end{array}$$

(24)

3.3.2. Stability of the Positive Equilibrium Branch

We focus on the positive-root branch $z=z\left(\gamma \right)>0$ of Equation (19), which exists for all $\gamma \in [0,20]$ and satisfies $z\left(\gamma \right)\in [2,4.7)$. For a cubic ${\lambda }^3+a_1{\lambda }^2+a_2\lambda +a_3$, the Routh–Hurwitz conditions for all roots to lie in the open left half-plane are [23,24]:

$$a_1>0,a_2>0,a_3>0,a_1{a}_2>a_3.$$

(25)

Along the positive branch (with $\gamma \in [0,20]$), $z>0$. From Equation (24),

$$a_1=12>0.$$

For $a_2$,

$$a_2=z^2+11+{\displaystyle \frac{100}{z}}>0\mathrm{for}\mathrm{all}z>0.$$

For $a_3$,

$$a_3=20{\displaystyle \frac{z^3+5}{z}}.$$

Along the equilibrium branch, we have $z\ge 2$ (since $z\left(0\right)=2$ and $z\left(\gamma \right)$ increases with $\gamma$). Hence, $z^3+5>0$ and $z>0$, which implies that $a_3>0$.

Finally,

$$a_1{a}_2-a_3=12\left(z^2+11+\frac{100}{z}\right)-20\frac{z^3+5}{z}={\displaystyle \frac{-8z^3+132z+1100}{z}}.$$

Define

$$h\left(z\right):=-8z^3+132z+1100.$$

Then

$$a_1{a}_2-a_3={\displaystyle \frac{h\left(z\right)}{z}}.$$

For $z>0$, the sign of $a_1{a}_2-a_3$ coincides with the sign of $h\left(z\right)$.

Differentiating,

$$h^{\prime }\left(z\right)=-24z^2+132,$$

which vanishes at $z=\sqrt{132/24}=\sqrt{5.5}\approx 2.345$. Thus, $h\left(z\right)$ has a unique maximum at $z\approx 2.345$ and is strictly decreasing for $z>2.345$.

Moreover,

$$h\left(2\right)=-8\left(8\right)+132\left(2\right)+1100=1300>0,$$

and

$$h\left(5\right)=-8\left(125\right)+132\left(5\right)+1100=760>0.$$

Since the positive equilibrium branch satisfies $z\left(\gamma \right)\in [2,4.7)$ for $\gamma \in [0,20]$, it follows that $h\left(z\right)>0$ on this interval.

Hence,

$$a_1{a}_2-a_3>0\mathrm{for}\mathrm{all}\gamma \in [0,20].$$

All Routh–Hurwitz conditions are therefore satisfied for the positive equilibrium branch over $\gamma \in [0,20]$, which proves that all eigenvalues of $J^{*}(z\left(\gamma \right),\gamma )$ have negative real parts. Local asymptotic stability follows from standard linearization theory [24].

Figure 2 illustrates the equilibrium bifurcation diagram and corresponding eigenvalue trajectories for the PMSM model under constant q-axis excitation.

3.3.3. Summary

For $\gamma >{\gamma }_m\approx 9.77$, the cubic Equation (19) admits two additional real roots with $z<0$, and hence two additional equilibria $(z^2,z,z)$. Direct evaluation of the Jacobian matrix in Equation (21) at these points shows that at least one eigenvalue has a positive real part (for example, for $\gamma =10$, the negative roots $z=-2$ and $z\approx -1.45$ each yield one eigenvalue with positive real part). Therefore, these additional equilibria are unstable, and the positive equilibrium branch $x^{*}\left(\gamma \right)$ is the unique locally asymptotically stable equilibrium for $\gamma \in [0,20]$.

3.4. Equilibria and Their Stability for ${\tilde{T}}_L=1$, ${\tilde{u}}_d=5$, ${\tilde{u}}_q=10$

3.4.1. The Equilibria

For the PMSM system (6) with fixed parameters

$${\tilde{T}}_L=1,{\tilde{u}}_d=5,{\tilde{u}}_q=10,\sigma =10,$$

an equilibrium $x^{*}=(x_1^{*},x_2^{*},x_3^{*})$ satisfies ${\dot{x}}_1={\dot{x}}_2={\dot{x}}_3=0$.

From Equation (8), we obtain

$$x_2^{*}=z+{\displaystyle \frac{{\tilde{T}}_L}{\sigma }}=z+0.1.$$

(26)

$$x_1^{*}=z\ast (z+0.1)+5.$$

(27)

and

$$z^3+0.1z^2+(6-\gamma )z-9.9=0.$$

(28)

For each fixed $\gamma$, the real roots z of the above cubic equation determine the equilibrium values

$$x_3^{*}=z,x_2^{*}=z+0.1,x_1^{*}=z(z+0.1)+5.$$

(29)

3.4.2. Linearization and Local Stability

The Jacobian matrix of the vector field at a state $x=(x_1,x_2,x_3)$ is given in Equation (10). The linear stability of an equilibrium point $x^{*}$ is determined by the eigenvalues of $J(x^{*},\gamma )$.

Numerical evaluation of the roots of the cubic equation (Equation (28)) and the eigenvalues of $J(x^{*},\gamma )$ for $\gamma \in [0,20]$ shows the following behavior:

• For $0\le \gamma \lesssim 14.542$, the cubic equation has a single real root $z\left(\gamma \right)>0$. The corresponding equilibrium

  $$x^{*}\left(\gamma \right)=(x_1^{*}\left(\gamma \right),x_2^{*}\left(\gamma \right),x_3^{*}\left(\gamma \right))$$

  has all eigenvalues of $J(x^{*}\left(\gamma \right),\gamma )$ with negative real parts, and is therefore asymptotically stable.
• For $\gamma \gtrsim 14.542$, the cubic equation has three distinct real roots:

  $$z_{+}\left(\gamma \right)>0,z_m\left(\gamma \right)<0,z_{-}\left(\gamma \right)<0.$$
  The equilibrium associated with the positive root $z_{+}\left(\gamma \right)$ remains asymptotically stable (all eigenvalues of J have negative real parts).
  In contrast, the two equilibria corresponding to the negative roots $z_m\left(\gamma \right)$ and $z_{-}\left(\gamma \right)$ each have one eigenvalue with a positive real part and are therefore unstable (saddle-type equilibria).

As illustrated in Figure 3, the positive equilibrium branch remains asymptotically stable over the considered range of $\gamma$, while additional negative branches are unstable.

3.4.3. Summary

In summary, for the parameter set $({\tilde{T}}_L,{\tilde{u}}_d,{\tilde{u}}_q)=(1,5,10)$ and $\gamma \in [0,20]$:

• the positive $x_3^{*}$ equilibrium branch is asymptotically stable over the entire interval;
• the additional equilibria that appear for $\gamma \gtrsim 14.542$ and satisfy $x_3^{*}<0$ are unstable.

4. Numerical Results of the PMSM Dynamics

This section presents numerical simulations of the nonlinear PMSM model. The PMSM parameters are [47] $L_d=1.0\times {10}^{-3}$ H, $L_q=1.0\times {10}^{-3}$ H, $R_1=0.24\mathrm{\Omega }$, $J=4.7\times {10}^{-5}{\mathrm{kg}·\mathrm{m}}^2$, $\beta =0.01619$, ${\psi }_f=0.06784$ Wb, and $n_p=4$.

To visualize the analytical results, the PMSM dynamics are analyzed at $\sigma =10$, while $\gamma$ varies in the range $[0,30]$ with a resolution of $0.01$.

Two bifurcation diagrams are shown in Figure 4, obtained in two directions. The first is achieved by increasing $\gamma$ from 0 to 30 (blue curve), and the second by decreasing $\gamma$ from 30 to 0 (red curve).

For each direction, the initial values of the state variables are set to $(1,1,1)$ for the first run, while the initial values for the subsequent run are taken as the final values of the previous run. The integration was performed using a fourth-order Runge–Kutta method with a sampling interval of $1\times {10}^{-3}$ s over a 100 s time horizon. From Figure 4, the system trajectories converge to the zero equilibrium for $0\le \gamma <1$ in both directions. In the forward direction (blue curve), the trajectories converge to the corresponding stable nonzero equilibrium for $1<\gamma <14.8$, while the speed exhibits chaotic oscillations for $\gamma \ge 14.8$. In the backward direction (red curve), the trajectories converge to the nonzero stable equilibrium for $1<\gamma <17.5$, while the PMSM speed exhibits chaotic behavior for $\gamma \ge 17.5$. These numerical results are consistent with the mathematical analysis.

Quantitative identification of chaos can be achieved using indicators such as Lyapunov exponents, which measure the exponential divergence of nearby trajectories, and Poincaré maps, which reveal the geometric structure of attractors.

Figure 5 shows the Lyapunov exponents of the PMSM for $\gamma \in [0,30]$ with $\sigma =10$ in the backward direction, using initial conditions $(1,1,1)$. It is observed that all exponents are negative for $0\le \gamma <14.8$, indicating a stable fixed-point response, while the largest exponent becomes positive for $\gamma \in [14.8,30]$, which indicates chaotic behavior. The Lyapunov exponent plot confirms the system behavior observed previously in the bifurcation diagram (Figure 4).

Figure 6 displays the Poincaré map of $x_2$ for $\sigma =10$, $\gamma =16$, ${\tilde{u}}_d=0$, ${\tilde{u}}_q=0$, and $T_L=0$. The Poincaré section consists of multiple scattered points, which indicates chaotic behavior.

The PMSM exhibits coexistence of attractors for $\gamma \in [14.8,17.5]$, as illustrated in Figure 4. The phase portraits of these attractors with $\sigma =10$ and $\gamma =16$ are shown in Figure 7. Figure 7a represents the chaotic regime for initial conditions $(1,1,1)$, where the trajectories exhibit chaotic oscillations away from the unstable equilibria (red markers). In contrast, Figure 7b, with initial conditions $(14,3.8,3.8)$, shows trajectories converging to the stable equilibrium, indicating asymptotic stability.

Figure 8 presents a two-dimensional bifurcation diagram for $\sigma \in [9,12]$ and $\gamma \in [0,20]$.

The results show that the PMSM exhibits chaotic speed oscillations for $\gamma >15$ within $\sigma \in [9,12]$. The chaotic behavior becomes more pronounced as $\gamma$ increases.

5. Synergetic Control Technique

The synergetic control technique (SCT) implies that the time required to reach a steady state generally depends on the initial conditions of the system. It aims to force the trajectories to follow a specific desired path, called the synergetic manifold. SCT is known for avoiding the “chattering” issue present in some other control techniques, such as SMC.

The steps for designing the SCT begin with selecting a macro-variable ($\psi$) according to the system variables to ensure that the system trajectories converge to a stable manifold, after which they slide along it toward a stable equilibrium point at the origin.

The manifold equation is described as follows:

$$T\dot{\psi }+\varphi \left(\psi \right)=0$$

(30)

where T is a time constant designed to determine the convergence time toward the manifold, and $\psi$ is a selected function that must satisfy the following three conditions:

• it must be invertible and differentiable;
• $\varphi \left(0\right)=0$;
• $\varphi \left(\psi \right)\psi >0$.

In the following SCT design procedure, the summation of term $\varphi \left(\psi \right)$ in Equation (30) will be determined by:

$$\varphi \left(\psi \right)=\psi$$

(31)

The mathematical model of the PMSM under SCT control can be expressed as follows:

$$\dot{\mathit{x}}=\mathit{f}\left(\mathit{x}\right)+\mathit{g}\mathit{u}(\mathit{x},t)$$

(32)

where $\mathit{x}\in {\mathbb{R}}^n$ is the vector of the state variables, $\mathit{g}={[g_1,g_2,0]}^T$, $\mathit{u}(\mathit{x},t)=[u_d(\mathit{x},t),u_q(\mathit{x},t),0]$ is the SCT control input, and $\mathit{f}\left(\mathit{x}\right)$ is defined as

$$\mathit{f}\left(\mathit{x}\right)=(f_1,f_2,f_3)=\left[\begin{array}{c}-x_1+x_2{x}_3\\ -x_3{x}_1-x_2+\gamma x_3\\ \sigma (x_2-x_3)+{\tilde{T}}_L\end{array}\right]$$

The controller is implemented in the PMSM model described in Equation (6) to ensure stabilization of its equilibrium point under a wide range of parameter variations, thereby suppressing unwanted chaotic behavior.

Within the PMSM drive structure, three macro-variables (${\psi }_d$, ${\psi }_q$, and ${\psi }_{\omega }$) are selected to achieve the control objectives.

5.1. Direct-Axis Path

The d-axis part macro variable can be chosen as the sum of d-axis current error and its integral, as follows:

$${\psi }_d=x_1-x_{1ref}+k_1\int (x_1-x_{1ref})dt$$

(33)

Impose the synergetic dynamics

$$T_d{\dot{\psi }}_d+{\psi }_d=0.$$

(34)

From the system model in Equations (32)–(34), solving for the control input $u_d(x,t)$ yields

$$u_d(\mathit{x},t)=-{\displaystyle \frac{1}{g_1}}\left[f_1-{\dot{x}}_{1ref}+k_1(x_1-x_{1ref})+{\displaystyle \frac{1}{T_d}}{\psi }_d\right].$$

(35)

5.2. Speed Path

This path consists of two loops: an outer synergetic speed loop and an inner synergetic q-axis loop.

5.2.1. Outer Synergetic Speed Loop

For the outer speed loop, the macro-variable is chosen as

$${\psi }_{\omega }=x_3-x_{3ref}.$$

(36)

Impose

$$T_{\omega }{\dot{\psi }}_{\omega }+{\psi }_{\omega }=0,$$

(37)

From the system model in Equations (32), (36), and (37), solving for the virtual control $x_{2ref}$ gives

$$\sigma (x_{2ref}-x_3)+ϵx_{2ref}{x}_1+{\tilde{T}}_L={\dot{x}}_{3ref}-{\displaystyle \frac{1}{T_{\omega }}}(x_3-x_{3ref}).$$

(38)

Hence,

$$x_{2ref}={\displaystyle \frac{{\dot{x}}_{3ref}-\frac{1}{T_{\omega }}(x_3-x_{3ref})+\sigma x_3-{\tilde{T}}_L}{\sigma +ϵx_1}}.$$

(39)

5.2.2. Inner Synergetic q-Axis Loop

In the inner loop, $x_2$ is forced to track $x_{2ref}$ by defining

$${\psi }_q=x_2-x_{2ref}+k_2\int \left(x_2-x_{2ref}\right).$$

(40)

Impose

$$T_q{\dot{\psi }}_q+{\psi }_q=0,$$

(41)

From the system model in Equations (32), (40), and (41), solving for the control input $u_q(\mathit{x},t)$ yields

$$u_q(\mathit{x},t)=-{\displaystyle \frac{1}{g_2}}\left[f_2-{\dot{x}}_{2ref}+k_2\left(x_3-x_{3ref}\right)+{\displaystyle \frac{1}{T_q}}{\psi }_q\right]$$

(42)

Here, $T_d$, $T_{\omega }$, $T_q$, $k_1$, $k_2$, $g_1$, and $g_2$ are positive design constants. The selection of macro variables ${\psi }_d$, ${\psi }_{\omega }$, and ${\psi }_q$ is motivated by the cascade control structure of the PMSM drive. The d-axis macro-variable ${\psi }_d$ (Equation (33)) incorporates integral action to eliminate steady-state current error and ensure convergence of the d-axis current. The speed macro-variable ${\psi }_{\omega }$ (Equation (36)) defines the outer-loop speed regulation objective and generates the virtual q-axis reference through Equation (39). The inner-loop macro-variable ${\psi }_q$ (Equation (40)) ensures convergence of the actual q-axis current to the virtual reference $x_{2ref}$.

The imposed manifold dynamics (Equations (34) and (37)) are first-order stable differential equations with positive time constants $T_d$, $T_{\omega }$, and $T_q$. Under standard synergetic control assumptions, these dynamics guarantee exponential convergence of the macro-variables to zero. Consequently, the tracking errors converge asymptotically, and the closed-loop trajectories are driven toward the desired stable manifold, resulting in suppression of chaotic oscillations and stable speed regulation.

The SCT controller represented by Equations (35), (39), and (42) is illustrated in Figure 9.

6. Torque Observer

The extended state observer (ESO) is adopted for this purpose. ESO was first proposed by Han [48,49]. It has been used with IPMSMs for position and speed estimation [50] and as a speed observer for PMSMs [51].

The mechanical subsystem $\left(\sigma (x_2-x_3)+{\tilde{T}}_L\right)$ in Equation (6) can be rewritten as follows:

$${\dot{x}}_3=a\left(x\right)+\widetilde{T_L},$$

(43)

where

$$a\left(x\right)=\sigma (x_2-x_3),$$

(44)

The ESO observer dynamics is defined as:

$$\begin{array}{cc}\hfill {\dot{\widehat{x}}}_3& =a\left(x\right)+{\widehat{\tilde{T}}}_L+l_1(x_3-{\widehat{x}}_3),\hfill \\ \hfill {\dot{\widehat{\tilde{T}}}}_L& =l_2(x_3-{\widehat{x}}_3),\hfill \end{array}$$

(45)

where ${\widehat{x}}_3$ and ${\widehat{\tilde{T}}}_L$ are the estimated speed and load torque, respectively, and $l_1,l_2>0$ are observer gains.

6.1. Observer Stability

Define the estimation errors:

$$e_1=x_3-{\widehat{x}}_3,e_2={\tilde{T}}_L-{\widehat{\tilde{T}}}_L$$

(46)

The error dynamics become:

$$\begin{array}{cc}\hfill {\dot{e}}_1& =-l_1{e}_1+e_2\hfill \\ \hfill {\dot{e}}_2& =-l_2{e}_1\hfill \end{array}$$

(47)

This yields the second-order characteristic polynomial:

$$s^2+l_{1}s+l_2=0$$

(48)

Equation (48) is asymptotically stable for

$$l_1>0,l_2>0.$$

6.2. Observer Gains Selection

A commonly used choice is critical damping [51]:

$$l_1=2{\omega }_o,l_2={\omega }_o^2$$

(49)

where ${\omega }_o$ is the ESO bandwidth. The closed-loop speed error dynamics can be obtained from Equations (36) and (37) as

$${\dot{e}}_{\omega }=-{\displaystyle \frac{1}{T_{\omega }}}{e}_{\omega }$$

(50)

where $e_{\omega }=x_3-x_{3ref}$

Thus, the corresponding pole is

$$s=-{\displaystyle \frac{1}{T_{\omega }}}$$

(51)

Typically:

$${\omega }_o=(3–8){\omega }_{\mathrm{bw}}$$

where ${\omega }_{\mathrm{bw}}$ is the speed-loop bandwidth. Since the speed-loop is first-order, the pole magnitude corresponding to the $-3$ dB bandwidth is

$${\omega }_{bw}={\displaystyle \frac{1}{T_{\omega }}}rad/s$$

7. Results and Discussion

The drive system consists of a PMSM controlled by SCT, as illustrated in Figure 9. The controller outputs ($u_q(x,t)$ and $u_d(x,t)$) are applied to the PMSM drive at $t=87$ s, as shown in Figure 10. The controller parameters are $T_d=0.05$, $T_q=0.05$, $T_{\omega }=0.01$, $k_1=1$, $k_2=40$, $g_1=0.3$, and $g_2=0.5$. These values were selected to ensure fast manifold convergence while maintaining smooth control action (thus avoiding chattering) and stable tracking under torque disturbances. From Figure 10a, the controller stabilizes the chaotic behavior within approximately 1.1 s. The d-axis and q-axis control signals are illustrated in Figure 10b and Figure 10c, respectively.

The controller performance is tested under sudden changes in the reference speed, as shown in Figure 11. First, the reference speed is varied from 0.6 to 1.3, where the controller requires approximately 0.9 s to reach the new reference value. The second speed step is from 1.3 to 1.8, where the PMSM reaches the final speed within approximately 0.75 s. The third step is from 1.8 to 1.3, which requires about 1.05 s to track the new reference. Finally, approximately 1.1 s is required when the reference speed is changed from 1.3 to 0.6. These results indicate that the SCT controller effectively tracks the new command under successive reference speed changes.

To evaluate the SCT performance under load torque variations, two scenarios are considered: a sudden change and a sinusoidal variation, as presented in Figure 12a. From the time response, the SCT controller maintains the motor speed close to the reference value when the load torque changes suddenly from 0 to 1.5. In this case, the q-axis current increases from 0.6 to 0.75 within approximately 0.14 s, as shown in Figure 12a. For the sinusoidal load torque, the motor speed remains near the reference value (0.6), as illustrated in Figure 12b. These results demonstrate that the SCT controller preserves the PMSM speed at the desired reference under load disturbances.

For clarity, the main quantitative performance indicators are summarized as follows. The SCT suppresses chaotic oscillations within approximately 1.1 s after controller activation (Figure 10a). During step reference transitions (Figure 11), the speed tracking response time ranges between 0.75 s and 1.1 s, depending on the step magnitude. Under sudden load torque variation (0 to 1.5), the speed deviation remains small and recovers within approximately 0.14 s. Under sinusoidal torque disturbances, only minor speed fluctuations are observed (Figure 12). The control signals shown in Figure 10b,c confirm smooth control action without high-frequency chattering.

For contextual comparison, the SCT is qualitatively compared with widely used chaos-control strategies such as sliding mode control (SMC) and fuzzy logic control. While SMC offers robustness, it may introduce chattering due to discontinuous control action, which can excite high-frequency oscillations in motor drives [52]. Fuzzy approaches reduce chattering but often require heuristic tuning and may lack explicit stability guarantees [53,54].

In contrast, the proposed SCT employs continuous manifold-based dynamics (Equations (34), (37), and (41)), resulting in smooth control signals (Figure 10b,c) and rapid chaos suppression (approximately 1.1 s; Figure 10a), with stable tracking under speed and torque variations (Figure 11 and Figure 12). A comprehensive benchmark comparison under identical tuning conditions is left for future work.

8. Conclusions

This study investigated the nonlinear dynamics and chaos control of a permanent magnet synchronous motor (PMSM) using a normalized electromechanical model with coupling parameter $\gamma$. The analytical results revealed a structural transition in the equilibrium configuration: a pitchfork-type bifurcation occurs at $\gamma =1$, where the origin loses stability and two symmetric nonzero equilibria emerge. For the special case with zero input voltages and load torque, further analysis using the Routh–Hurwitz conditions showed that the nonzero equilibria remain asymptotically stable for $1<\gamma <17.5$, with loss of stability occurring near $\gamma =17.5$, consistent with a Hopf-type mechanism.

Numerical simulations confirmed the analytical predictions and demonstrated coexistence of attractors and chaotic oscillations for larger values of $\gamma$. The bifurcation diagrams and phase portraits illustrated the transition from stable equilibrium behavior to oscillatory and chaotic regimes.

To suppress chaotic oscillations, a synergetic control technique (SCT) was designed based on manifold dynamics and implemented in the PMSM model. The proposed controller achieved chaos suppression within approximately 1.1 s after activation (Figure 10), ensured reliable speed tracking under step reference changes (Figure 11), and maintained robustness under sudden and sinusoidal load torque disturbances (Figure 12). The continuous control action avoided chattering and provided smooth control signals.

Overall, the combined analytical and numerical results demonstrate that the proposed SCT effectively stabilizes chaotic PMSM behavior while preserving fast dynamic response and disturbance robustness. Future work may include extended quantitative chaos diagnostics and comprehensive benchmark comparisons with alternative nonlinear control strategies.