The stochastic stability and H∞‐fuzzy control of stochastic bifurcation of a doubly‐fed induction generator

Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

structure of a wind farm due to the operation mode, power outage, or a change in equivalent impedance on the PCC (point of common coupling) bus, and (2) according to the influence of stochastic processes (such as stochastic wind speed fluctuations, ambient noise in the transmission system and random fluctuations in electrical quantities).Therefore, studying the stochastic dynamic behavior of a DFIG model affected by environmental noise in space is necessary.Considering the complexity of wind farm operation modes, this paper focuses on the stability and nonlinear stochastic bifurcation behavior of a DFIG affected by the stochastic intensity in its space.
7][8][9] The Monte Carlo simulation (MCS) based numerical simulation method can statistically determine the probability distribution of system eigenvalues.In the context of small disturbance stability analysis, Shim et al. 6 investigated the eigenvalue distribution of stochastic small disturbance stability considering a high percentage of wind power connected to the grid using the MCS method.However, this numerical simulation method requires generating many scenarios with stochastic factors, making the calculation of eigenvalues complex and unable to accurately reflect the influence of stochastic intensity on the dynamic processes of the system.To address this limitation, Zhou et al. 7 proposed a method based on stochastic differential equations for the moment stability criterion, which investigates the impact of stochastic fluctuations in mechanical power on the stochastic stability of systems with wind power.Furthermore, instability in nonlinear dynamic systems can experience Hopf bifurcation, and a self-consistent calculation framework for stability has been proposed not only for random coupled networks but also for systems with a higher number of state variables. 8An incomplete theory has been proposed to address hidden bifurcations in dynamic systems. 9In the study of stochastic switched systems and time-delay networks, multiple Lyapunov functions and exponential stability have been proposed to analyze their global stability. 10,11esearch has indicated that noise intensity influences the dynamic characteristics of nonlinear systems, leading to increased complexity.3][14][15] However, stochastic bifurcation control methods for nonlinear dynamic models, particularly those involving Markov processes, are not yet fully established.Previous research has primarily focused on bifurcation control with random coefficients and stochastic control with the low-dimensional model. 16cently, a study examined a network control system with random network delay in the feedback channel, considering the instability caused by such delays.Stability criteria for closed-loop network predictive control systems were provided, offering valuable insights. 17,18Nonetheless, the specific mechanism of simplified random bifurcation in DFIGs has not been extensively investigated.Therefore, it is necessary to analyze the influence of noise intensity on the random space of the nonlinear model and study the trajectory of the DFIG system in the random space, considering the nonlinear term.This will enable an accurate characterization of the random stability and random bifurcation behavior of a DFIG under stochastic process intensity.
On the other hand, controlling nonlinear loads or noise interference may have an impact on wind power generation stability.In response, Srilakshmi et al. 19 suggest an intelligent hybrid controller that combines a proportional-integral controller (PI) with a fuzzy logic controller for wind power generation systems.Novel approaches to the stability control of nonlinear stochastic dynamic systems are offered by the fuzzy control strategy.To investigate bifurcation control in DFIG under stochastic processes, the Tkagi-Sugneo (T-S) fuzzy theory can be utilized for system reconstruction.This approach ensures that the reconstructed model is equivalent to the original nonlinear system. 20,21The refinement of the fuzzy set allows for a more accurate reconstruction of the original nonlinear system.Similarly, control systems can be developed using this approach.H  control determines the minimum amplitude frequency value of a system in the presence of external disturbances, and the inclusion of a controller strengthens its robustness. 22,23Therefore, by combining T-S fuzzy theory and H ∞ control, the instability issue in stochastic nonlinear systems can be effectively addressed.In cases where the state is challenging to measure or unmeasurable, it is advisable to prioritize the H ∞ output feedback control strategy.The main conclusions and innovative points of this article: (1) A research method is proposed to analyze the stochastic stability and stochastic bifurcation of DFIG.The local stability of DFIG is analyzed with the help of the quasi-Hamiltonian theory, and the global stability is studied with the help of the singular boundary theory.(2) It reveals the stochastic dynamic behavior of DFIG under the influence of noise intensity.The steady-state probability density of stochastic bifurcations (stochastic Hopf bifurcations and P-bifurcations) of DFIG subjected to random strength is analyzed with the help of the Fokker-Plank-Kolmogorov (FPK) equation.(3) The stochastic Hopf bifurcation can be avoided via a fuzzy output dynamic feedback strategy.The linear matrix inequality (LMI) method provides the asymptotic stability conditions for the closed-loop system with interference level, the dynamic output feedback controller is designed, and its gain matrix algorithm is provided, while the controller for the nonminimum phase system and the closed-loop system's small-signal stability are analyzed.
The remainder of this paper is organized as follows.In Section 2, a four-dimensional DFIG stochastic dynamic model is developed.In Section 3, the DFIG one-dimensional stochastic differential equation is established.In Section 4, the stochastic bifurcation behavior and the stability of the DFIG are analyzed.In Section 5, the design of stochastic bifurcation control based on H  -fuzzy dynamic output feedback is described.In Section 6, numerical simulation analyzes stochastic instability and bifurcation behavior, and provides instability parameters.Meanwhile, the stochastic bifurcation was controlled using a designed fuzzy controller.

| Deterministic mathematical model of the DFIG
Doubly-fed asynchronous motors in wind power systems are gradually becoming mainstream generators, which, unlike direct-drive generators, enable energy exchange between the stator and rotor sides and the grid.This paper simplifies the dynamic links of rotor-side and stator-side control links, as shown in Figure 1.
Equation (1) was chosen to describe the mathematical model of the doubly-fed asynchronous power generation system. 24c x c u where , x 1 is the stator q-axis magnetic chain, x 2 is the stator d-axis magnetic chain, x 3 is the speed difference, x 4 is the output of indirect magnetic field control, and the physical meaning of the remaining parameters can be found in Yang et al. 24 and Bazanella et al., 25 the equilibrium point of the deterministic DFIG is given by Equation (2): (2) where s x u = / e 4 2 .To facilitate the solution of the coefficients of the central manifold expansion at the equilibrium point, it is necessary to translate the original equilibrium point to the origin E. Let ; then, the stability of the study Equation ( 1) at the equilibrium point x* i is the same as that at the equilibrium point E(0, 0, 0, 0).

| DFIG nonlinear stochastic dynamic model
Considering that the original determination of the DFIG model on each state space will be disturbed by noise and that the random noise space is complex, 25 therefore, white noise is selected as a random factor to simulate the process of environmental noise.
From Equation (1), the perturbation of each state variable of the deterministic DFIG on the random space by multiplicative noise can be written as Equation (3).
At the noise intensity parameter σ = 0, Equation (3) reduces to the deterministic model in Equation (1).Let l x = *    26 that the stability of a doubly-fed asynchronous wind turbine is closely related to its internal parameters, which include the tuning parameter k of the system and the parameters of the speed control system k p , k i , and T w .However, the parameters of the speed regulation system are generally constant and controllable, and the rotor-side variables play a dominant role in the stability of the DFIG system.Therefore, considering the speed dynamic response characteristics of the DFIG system, the tuning coefficient k is selected as an important factor affecting the stability of the system.Let If ε k ( ) = 0 in Equation ( 5), then the deterministic DFIG model undergoes Hopf bifurcation, which can then be divided into two subspaces, that is, C U V =  , where C is the total space of U and V. U is the two-dimensional subspace formed by the eigenvector p corresponding to the eigenvalue λ 2,3 and V is the two-dimensional subspace formed by the eigenvector q corresponding to the eigenvalue λ c = − 1,4 1 , which results in D λ ( ) = 0. Let Q be the matrix formed by the eigenvectors P and q such that X QY = , where X 4 , so that Equation (4) can be organized into a general expression as shown in Equation (6).
where A 1 represents the eigenmatrix of the U subspace and g u v δ ( , , ) represents the nonlinear function of the U subspace.Similarly, A 2 represents the eigenmatrix of the V subspace, and h u v δ ( , , ) represents a nonlinear function of the V subspace.When the deterministic DFIG model is in the neighborhood where Hopf bifurcation occurs, it is known from nonlinear canonical type theory that Equation (6) can be written as Equation (7).
To reduce the error in the calculation results, four valid digits are retained and the nonlinear part of the stochastic DFIG model is Considering the small noise intensity, the central flow equation of the DFIG system can be written by the central manifold shape theorem as y h y y σ y h y y σ = ( , , ), = ( , , ).
From Equation ( 12), the DFIG stochastic model reduces to a deterministic model when the noise intensity is equal to 0. Starting with Equation ( 12), let ; after polar transformation, A t θ t ( ), ( ) are nonlinear differential equations with respect to time.Then, Equation ( 13) can be written as  , CHEN ET AL.
| 621 We multiply both sides of Equation (13a) by θ cos and both sides of Equation (13b) by θ sin to obtain Equation (14).
The differential equation in DFIG polar coordinates can be obtained by solving Equations ( 12) and ( 14) jointly, as shown in Equation (15).
Since the model can be considered a proposed Hamiltonian system in the Hopf bifurcation neighborhood of the DFIG, the solution orbit of the model at this point can be called the canonical orbit.Since actual Gaussian white noise is often a smooth process with a very short correlation time, the Stratonovich stochastic differential equation is more exact when analyzing the response of Gaussian white noise to a real dynamic system. 27The general form of its stochastic differential is shown in Equation (16). where The general form of the stochastic differential equation in the sense of Stratonovich is 28 where the second of these terms is the Wong-Zakai correction term, which, if σ X t X ( , )/ = 0   , is consistent with the expression of the Ito ˆstochastic differential equation.Conversely, the Ito ˆstochastic differential equation must account for the correction term and can be written as From Equation (18), it is not easy to find the exact solution of the Ito stochastic differential equation for the DFIG model.However, according to R. Khasminskii's theorem, when the noise intensity σ is small, A t θ t { ( ), ( )} weakly converges to a two-dimensional Markov diffusion process.According to the stochastic averaging method, retaining the slow variables of the model A t ( ), combining Equations ( 16) and (18) yields that the DFIG's Ito ŝtochastic differential equation can be written as Equation ( 18) by using the theory of such equation.
where ( ) is the drift coefficient and ( )

STOCHASTIC STABILITY AND BIFURCATION BEHAVIOR OF DFIG
In nonlinear dynamic models, stochastic bifurcations can be classified as kinetic bifurcations and phenomenological bifurcations. 29The kinetic bifurcation is also called a D-bifurcation and the phenomenological bifurcation is a P-bifurcation.

| Stochastic D-bifurcation based on the Lyapunov exponent
Since the DFIG model has coupled nonlinear factors, it is more difficult to analyze the model directly with the help of SDE theory.However, as shown in Equation (19), in the neighborhood of the Hopf bifurcation point of DFIG, it can be regarded as a proposed Hamiltonian system. 30,31herefore, in the neighborhood of the equilibrium point, the linearized exact solution of the equation of A(t) can be obtained by applying the theory of the linear stochastic differential equation, as shown in Equation (20). where ; with the help of the proposed Hamiltonian theory, it is known that the Lyapunov exponent of the DFIG can be expressed as , the linearized DFIG stochastic model is in a stable state with probability 1.

| Stochastic P-bifurcation analysis of DFIG
From Namachivaya theory, it is known that the essential information about the steady-state behavior of a nonlinear stochastic dynamical system is the extreme value of the invariant measure.When the noise intensity is sufficiently small, the p A ( ) extrema converges to determine the steady-state line of the system.Since the system response A t ( ) weakly converges to a onedimensional Markov process, it is known from Oseledec traversal theory that p A ( ) can be considered a measure of the access time of the sample trajectory in the neighborhood of A t ( ).The FPK equation can be obtained by the DFIG's Ito ˆslow variable A t ( ) as in Equation (22).
where the drift coefficient is

| Steady-state probability densities for Ito ˆstochastic differential equations
As shown in Equation ( 21), it describes the flow of the transfer probability density p x t x t ( , | , ) 0 0 of the diffusion process when b ( ) = 0 ij  , which describes the deterministic variation of the process, and when a ( ) = 0 i  , which describes the pure diffusive motion.Otherwise, it describes the deterministic variation of the diffusion process with diffusive motion.When τ p t , / 0      tends toward the steady state, the FPK equation for the steady state of Equation ( 23) can be written as Equation ( 23) is an elliptic linear variable coefficient partial differential equation, often referred to as the smooth FPK equation, which has a solution in the form of a smooth probability density p x ( ).To determine the solution of the FPK equation, in addition to the drift coefficient a ( ) ij  and the diffusion coefficient b ( ) ij  , the initial conditions need to be known.The initial conditions are as follows: Equation ( 24) indicates that at t t = 0 the system converges with probability 1 to the initial state x 0 , and the FPK equation can be written in operator form as In Equation ( 25), L* is the accompanying operator of the elliptic operator L.
From Equation ( 19), the steady-state probability density of A t ( ) is determined by the Fokker-Planck equation.The drift coefficient and A t ( ) are related.Therefore, the Wong-Zakai correction term must be considered for the Ito stochastic differential equation.Therefore, the steadystate probability density of the DFIG is Then, the steady-state probability density of the DFIG p A ( ) where st reaches a large value at A A = *, the sample trajectory stays longer in the neighborhood of A*; that is, the system is probabilistically stable at A*.

| Global stochastic stability
In the one-dimensional Ito ˆstochastic differential equation, the probabilistic asymptotic stability of the system and the existence of a smooth probability density are determined entirely by the nature of the stochastic process on the boundary.When there exists a point x s such that the diffusion coefficient σ x ( ) = 0 s is unbounded from the drift coefficient, it is called the first class and second singularity boundary.
From Equation (18), by singular boundary theory, 32,33 we know that A = 0 is the first type of singular boundary.When A  , M A ( )  , and therefore A   is the second type of singular boundary.The global stability of the stochastic DFIG can be determined by calculating the diffusion index, drift index, and eigenscale on the boundary, and thus the global stability can be classified into two cases.
From the DFIG Ito ˆstochastic differential Equation ( 18), m 0 2  ; therefore, only two cases, A A l , then the boundary A = 0 is the natural attraction boundary.
, then the boundary A = 0 is a natural exclusion boundary. .

Ⅱ. Scenario 2 (A
, then boundary A = + is a natural exclusion boundary.
, then boundary A = + is the entry boundary.
In summary, the global stability of the DFIG stochastic model is shown in Table 1.
From Table 1, it can be seen that the one-dimensional time-dispersive process A(t) is defined on the interval A A ( , ) , the left boundary is the natural attraction boundary, while the right boundary is the entry boundary, at which time the orbit of A(t) converges asymptotically to the natural attraction boundary.That is, A(t) has global stability at A = 0.
From the above conclusion, it is clear that the DFIG random model converges asymptotically from the entry boundary A = + to the natural attraction boundary A = 0 for all solution curves of the model at m m m 2 < (1 + ) 1 2 3 and m > −2

3
. Therefore, the DFIG random model can be considered to have global stability at the equilibrium point. 21,34le i : if z 1 (t) is M, i1 ⋯ , and z q (t) is M iq , then

| T-S fuzzy reconstruction of nonlinear systems
where the weights of the system under rule i are generally determined by the following equation, which is normalized to yield where q is a nonlinear term, r is the number of fuzzy rules, and finally, the fuzzy reconstruction model of the nonlinear model is Equation (33): Similarly, from this rule, the rule that outputs the H  control system can be described as follows: Rule i : if z 1 (t) is M, i1 … , and z q (t) is M iq , then .
Then the DFIG based on H  fuzzy output dynamic feedback can be described as where Global stochastic stability of the doubly-fed induction generator.

Conditions
The first type of singular boundary

Stability Left border
Right border

of fuzzy outputs based on the LMI method
Theorem For any given constant γ > 0, the dynamic output feedback of a fuzzy system with H  interference suppression level γ is asymptotically stable if there exist symmetric positive definite matrices P i and Q i such that the following matrix inequality holds, as shown in Equation (36).
where N N , oi ci are the matrices formed by any set of basis vectors as column vectors in subspaces and ( ) , respectively, and the proof procedure is given in Li et al. 35 and Modares et al. 36

| The design of the H ∞ output controller
Algorithm 1.Output H  -fuzzy design algorithm.
Step 1: Solve P i and Q i in the above theorem using LMI, Step 2: Solve to satisfy −1 and use P i and M i to construct G i , Step 3: Substitute the obtained i i  and solve for the H  controller gain matrix K i with the help of LMI.where 6 | ANALYSIS OF CALCULATION CASES

| Global stochastic stability analysis of the DFIG model
From Section 4.3, when considering the effect of noise intensity on the model, the stochastic process can be transformed into a random quantity and then analyzed.When the noise intensity is σ = 0.5 and the tuning factor is 3 and m > −2

3
, the DFIG stochastic model is globally stable, as shown in Figure 2.
Taking the intensity parameter σ = 0.9 and the tuning factor k = 0.8, the DFIG stochastic dynamic model becomes globally unstable, as shown in Figure 3.By comparison with the deterministic model in Equation ( 1), it can be seen that the addition of noise leads to the occurrence of Hopf bifurcation in advance.

| MCS verification
It can be seen from Equation ( 18) that its stochastic differential equation is simulated for 1000 times by using MCS.If the random strength is determined and all sample trajectories of the system converge to the equilibrium point, then it is demonstrated that the stochastic system is stochastically stable.
When σ A = 0.5, (0) = 0.1, the A(t) values of the 1000 sample trajectories all converge to 0, as shown in Figure 4. Therefore, A = 0 is the global stable equilibrium point of the DFIG stochastic model.

| P-bifurcation analysis of the DFIG stochastic dynamic model
From Equation (26), it can be seen that the analytical solution of A* is difficult to find, but its numerical solution can be found for a given interval.Now, consider (A) (B) 3 ).
the stochastic P-bifurcation of the DFIG stochastic dynamic model caused by the variation of random coefficients.
6.3.1 | V 1 and V 2 are unchanged, and V 3 is changed When given the initial value A (0, 5] ∈ , the locations and probability densities of the DFIG stochastic Hopf bifurcations for the m 3 = 0.05, 0.06, 0.07, and 0.08 variations are discussed, as shown in Table 2 and Figure 5.

| V 3 and V 2 are unchanged, and V 1 is changed
From the destabilization condition, when considering unchanged V 3 and V 2 and a 3 is satisfied, the DFIG stochastic model will be bifurcated by the stochastic Hopf bifurcation phenomenon after destabilization.The location and probability densities of stochastic Hopf bifurcation occurring in DFIG are shown in Table 3 and Figure 6.
From Figure 5, it can be seen that when the internal parameters of the DFIG satisfy 2m 1 > (1 + m 2 )m 3 , the CHEN ET AL.
| 627 system is in an unstable state and reaches a large value at A*, indicating that the trajectory of the DFIG stochastic model is round-trip on the interval of the invariant measure.

| Steady-state probability density when the DFIG stochastic model is stable
From the conclusion of 4, it is clear that when 2m 1 < (1 + m 2 )m 3 , all orbits of the DFIG stochastic model converge asymptotically converge to the natural attraction boundary, and thus the smooth probability density of the diffusion process is a delta function on the natural attraction boundary (A = 0), as shown in Figure 6A, which shows invariant m 1 and m 2 and varying m 3 , and Figure 6B shows invariant m 2 and m 3 and varying m 1 .
From Figure 6, when 2m 1 < (1 + m 2 )m 3 , the DFIG system is in a steady state.Its steady-state probability density is p (0) = st , indicating that all trajectories of the DFIG model converge to the equilibrium point.

| Stochastic P-bifurcation of DFIGs
From Equation ( 33), the steady-state probability density of the DFIG varies with the sizes of m 1 and m 3 .To verify that the stochastic Hopf bifurcation of the DFIG is related to the random parameters, the joint probability density of the DFIG random model is shown below.From Equation ( 12), the reduced-order system of the DFIG has the same nonlinear characteristics as the original system in the neighborhood of the equilibrium point.When the reduced-order system is verified with the original system, the DFIG reduced-order system undergoes a stochastic Hopf bifurcation in the neighborhood of the bifurcation value when m 1 = 0.042, m 2 = − 0.0082, and m 3 = 0.05, as shown in Figure 9D.
As shown in Figure 7, m 1 = 0.042, m 2 = −0.0082,and m 3 = 0.09 satisfy 2m 1 < (1 + m 2 )m 3 , and the DFIG stochastic model is in a steady state with a single-peaked joint probability density and a steady-state probability density at t   and p A ( = 0) = st , indicating that all tracks of the stochastic model converge to the origin.Similarly, when m 1 = 0.042, m 2 = −0.0082,and m 3 = 0.05, 2m 1 > (1 + m 2 )m 3 is satisfied, and then the joint probability density has a crater shape, that is, a stochastic Hopf bifurcation occurs.

| The connection between Dbifurcation and P-bifurcation in the DFIG
As shown in Figure 3, with the increase in random factors, it can be seen from the comparison with the deterministic model bifurcation in Equation ( 5) that the Hopf bifurcation of a random DFIG will form a limit cycle in advance.
From the summary of 4.1, it can be seen that the Dbifurcation point of the DFIG is at 16m 1 − (m 2 + 8) m 3 = 0, and at this point, the maximum Lyapunov exponent of DIFG is 0. From Equation ( 27), it can be seen that at this point, the DFIG did not undergo a Pbifurcation, but with an increase in random intensity, the DFIG experienced a P-bifurcation, as shown in Tables 2  and 3 and Figure 5. Therefore, there is no direct connection between D-bifurcation and P-bifurcation in the DFIG, as shown in Figure 8.
In summary, when the shape of the joint probability density changes from a single peak to a crater, stochastic P-bifurcation occurs in the DFIG system.

| DFIG fuzzy reconstruction
From Equations ( 3) and ( 31), we know that  where From the time-domain simulation results of the DFIG, it can be assumed that x k k i (− , ), = 1, 2, 3, 4 i i i

∈
. The fuzzy design of the DFIG using the triangular affiliation function is given as From Section 4.1, the fuzzy rules of the DFIG can be described as follows: Rule The affiliation function can be described as Equation (41): Therefore, from Equation (32), the weights of the fuzzy reconstruction are Then, from Equation (33), the DFIG reconstruction system based on T-S fuzzy is where  The same approach for the design of the H  controller is Equation (34), then 2 fuzzy rules can be designed as Equation ( 44) .
Detailed parameters can be found in Appendix A.3.Then the system with the addition of the controller can be described as Equation (45).
From Section 4, we can see that when the strength parameter σ > 0.9 and k = 0.8, a stochastic Hopf bifurcation occurs in the DFIG system.Therefore, in this section of control, we select σ = 1 and k = 1, and the system will inevitably experience Hopf bifurcation with a stronger degree than in Figure 7C.Taking the parameters into the constructed fuzzy system, we can see from Figure 9A that all state response curves are asymptotically stable, and Figure 9B reflects the maximum probability of the equilibrium point at the convergence point of the state CHEN ET AL.
| curve.Therefore, the addition of H ∞ -fuzzy control can suppress instability and bifurcation caused by noise intensity.

| Small-signal stability analysis of a DFIG with fuzzy reconstruction
The reconstruction findings show that we created a fuzzy set based on the response of the nonlinear stochastic system.
As can be seen from Figure 10A, the system of the controller surrounds (−1, j0) one lap clockwise, so , N k is the number of laps that surrounds (−1, j0), and P k is the pole of the open-loop transfer function, so the output of the designed controller has a time-varying characteristic, and as can be seen from Figure 10B, it is possible to assume that when GM = − ω   is added, and at this time, it can be utilized to determine the stabilization, but it loses its significance of the GM.Similarly, from Figure 10C, it can be seen that the closed-loop system Z , so the closed-loop system after adding H  is stable, and from Figure 10D, it can be seen that the closed-loop system's Bode plot to judge the destabilization has been invalidated.
The maximum response amplitude in the entire frequency domain, or the point at which the amplitudefrequency response reaches its maximum, is referred to as the system's modulus infinity.Minimizing modulus infinity entails limiting the maximum amplitude of the amplitudefrequency response in the entire frequency domain to a value greater than 1.Therefore, by combining the smallsignal stability analysis with the large-signal reconstruction model, we can conclude that the closed-loop system after adding the controller is consistent with the time-domain simulation results under small perturbations.

| CONCLUSION
In this paper, the stochastic stability and stochastic bifurcation control strategy of DFIG under the influence of stochastic intensity in random space are investigated by establishing a dynamic model of DFIG, and the main conclusions are as follows: (1) Stochastic Hopf bifurcation occurs in the stochastic DFIG model when the intensity parameter σ > 0.9 is used.The steady-state probability density p A ( ) st is investigated with the help of FPK equations, Thus, the position and probability density of occurrence of its stochastic Hopf bifurcation are shown, demonstrating that noise intensity can generate a stochastic bifurcation in the DFIG to destabilize it.
(2) A fuzzy output dynamic feedback strategy based on H ∞ is able to eliminate the stochastic Hopf bifurcation.The asymptotic stability conditions for the closed-loop system with H  interference level γ 0 < < 1 are given by the LMI method, the dynamic output feedback controller is designed and its gain matrix algorithm is given.u srd and u srq are the d-and q-axis components of the voltage, respectively, i rsd and i rsq are the d-and q-axis components of the current, respectively.r sr is the stator and rotor resistances.ω n is the synchronous speed, ω r is the rotor speed, and ω s is the relative speed of the rotor's d-q coordinate system.ψ srd is the d-axis components of the stator and rotor magnetic chain.and ψ ψ , sq rq are the q-axis components of the stator and rotor magnetic chains, respectively.L m , L s , and L r are the equivalent mutual inductances between the stator and rotor in the coaxial phase, and the equivalent self-inductances between the stator and rotor in the two phases, respectively.J g is the generator inertia, D g is the torque damping factor, p n is the number of pole pairs of the asynchronous motor, and T w is the mechanical torque.c ˆ1 is the estimated time constant of the rotor, c 1 is the real rotor time regular of the doubly-fed asynchronous motor, k p and k i are the PI parameters of the speed controller, respectively, and δ, δ ref are the rotor angle and the rotor reference angle, respectively.the tuning parameter is defined as k = c ˆ1/c 1 .The physical meaning of the remaining variables can be found in Bazanella et al. 25 and Reginatto and Bazanella.Step 5: If ϕ ϕ t = 0, / = 0   , get the center manifold of the system, marked as W c , Step 6: Taylor expansion of W c at the equilibrium point, mapping it onto an unstable manifold, thus completing the reduction of order.  .

1 1 ;
then, the characteristic equation of the DFIG model after it becomes dimensionless is Equation (4).

F I G U R E 9 F
State response curve and stochastic bifurcation control.I G U R E 10 Nyquist curve and Bode diagram of the doubly-fed induction generator H  controller and response system.(A) The Nyquist plot of control, (B) the Bode plot, and (C, D) the Nyquist and Bode plots of the response system, respectively.
This paper linearly combines it under this set, which means we utilize the linearized model to approximate the nonlinear stochastic model of the DFIG.From Equation (44), the small-signal frequency domain model of the H  controller designed for random bifurcation is as follows Equation (46).For small-signal stability of the minimum phase, we use the Nyquist criterion and Bode diagram for joint determination, as shown in Figure10.