CN111022254A - Time-delay control method for maximum power point tracking of singularly perturbed wind power generation models - Google Patents

Abstract

The invention discloses a time-lag control method for tracking maximum power point of a singular perturbation wind power generation model, which comprises the steps of considering the condition that the wind speed is lower than the rated wind speed, collecting related data of a wind power generator system, establishing a nonlinear singular perturbation model aiming at a variable-speed variable-pitch type wind power generator, linearizing the wind power generation system model at a plurality of operating points aiming at the singular perturbation wind power generation model, then adopting a linear variable parameter model to approximate to the nonlinear model of the wind power generation system, adopting LPV technology and H_∞And time lag control is carried out, so that the aim of tracking the maximum power point of the wind power generation system is fulfilled. The mathematical model created by the method is closer to the original physical system, more accords with the mechanism characteristics of the wind power generation system, improves the modeling precision, reduces the error caused by modeling, and greatly reduces the conservatism and the calculation complexity of the controller.

Description

Time-delay control method for maximum power point tracking of singularly perturbed wind power generation models

technical field

The invention relates to the technical field of wind power generation, in particular to a time-delay control method for maximum power point tracking of a singular perturbation wind power generation model suitable for improving wind energy capture efficiency.

Background technique

The singular perturbation model is a very common dynamic system model. This type of model is the main tool used to describe the system behavior with multi-time scale dynamics, overcome the rigid ill-conditioned problems caused by multi-time scale, and obtain satisfactory control results.

Since the wind power generation system contains both a mechanical part (ie, a fan part) and an electromagnetic part (ie, a motor part). Compared with the dynamic change characteristics of the mechanical part, the change rate of the electromagnetic part is very fast, so the system has obvious dual time scale characteristics. The modeling of wind power generation system ignores the electromagnetic dynamic characteristics of the motor part. Obviously, this will inevitably lead to inaccurate modeling, and it is difficult to improve the control accuracy.

For the wind power generation system, in order to improve the wind energy capture efficiency in the range below the rated wind speed, the variable speed constant frequency wind turbine generally adopts the Maximum Power Point Tracking (MPPT) control strategy. MPPT adjusts the rotor speed to track a certain function of wind speed, so as to obtain greater wind energy capture efficiency. However, Zaiyu Chen et al. have demonstrated (Paper: Chen Z, Yin M, Zou Y, et al. Maximum WindEnergy Extraction for Variable Speed Wind Turbines With Slow Dynamic Behavior [J]. IEEE Transactions on Power Systems, 2016, PP (99 ): 1-2.), the fan tracks the slow dynamics in the wind speed (ie, the low-frequency fluctuation component in the wind speed), which can improve the capture efficiency of wind energy and reduce the mechanical load of the system. This conclusion provides a new idea and method to improve the efficiency of wind energy capture. According to this idea, the fast dynamics of wind speed (ie the high-frequency fluctuation components in wind speed) can be regarded as the external disturbance of the system. The H _∞ control method is an effective method to overcome the disturbance and improve the robustness of the system. In the current patent literature, there is no technical method to realize the maximum power point tracking of wind power generation system by using the H _∞ control method and the singular perturbation theory.

For the MPPT control problem, many scholars design the controller by linearizing the nonlinear system at a certain point. However, due to the real-time change of the natural wind speed, the method of linearizing the system and designing the controller in the case of a fixed wind speed is relatively conservative, and the use range of this controller is small. In addition, intelligent control algorithms, such as neural network control, genetic control algorithm, fuzzy control algorithm, etc., can be used. The intelligent control algorithm can better deal with the nonlinear characteristics of the system, but the calculation amount is large, the computer requirements are high, the calculation time is large, and it is easy to cause the accumulation of calculation errors in the computer, which ultimately makes the control effect of the system not good enough.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a time-delay control method for the maximum power point tracking of a singularly perturbed wind power generation model. It is close to the original physical system, more in line with the mechanism characteristics of the wind power generation system, improves the _modeling accuracy, and reduces the error caused by the modeling; The target of the maximum power point tracking of the power generation system.

In order to achieve the above object, with reference to FIG. 1 , the present invention proposes a time-delay control method for maximum power point tracking of a singularly perturbed wind power generation model. The time-delay control method includes:

S1: Considering the situation that the wind speed is lower than the rated wind speed, collect the relevant data of the wind turbine system, and establish a nonlinear singular perturbation model for the wind turbine of variable speed and pitch type;

S2: Select 5 operating points θ _j , j=1,2,...,5, so that the set with the operating points as vertices forms a convex hull Θ, Θ=Co{θ ₁ , θ ₂ , θ ₃ , θ ₄ , θ ₅ }, for any point θ in the convex hull Θ, θ ∈ Θ, there is a set of non-negative numbers α _j ≥ 0, j = 1, 2,..., 5, such that:

且

and

S3: By linearizing the nonlinear singular perturbation model at multiple operating points θ _j , five linear time-invariant singular perturbation models are obtained;

S4: For five linear time-invariant singular perturbation models, combined with the given γ and each operating point θ _j , by solving the matrix inequality, the H _∞ robust time-delay controller K _j (θ _j ) is designed, such that The closed-loop linear time-invariant singular perturbation model is robust and stable, and γ is the index requirement for the infinite norm of the system transfer function, that is, the infinite norm of the system transfer function is required ||G(s)|| _∞ <γ;

S5: At time t _k , θ(t _k ) is obtained by measurement, and the weight coefficient α _j is calculated so that it satisfies:

S6: At time t _k , the controller for the original system is designed as:

S7: Calculate the control input u(t _k ) according to the following formula: u(t _k )=K(θ(t _k ))X(th), where K(θ(t _k )) is the controller gain, X( th) is the state variable of the singular perturbation model, t is the time, h is the time delay, and the calculation control input u(t _k ) is applied to the original nonlinear wind power generation system;

S8: Set t _k =t _k+1 , and repeat steps S5-S8 to control the wind power generation system in real time.

Compared with the existing technical solutions of the present invention, the significant beneficial effects are:

(1) Fully considering the dual time scale characteristics of the wind power generation system, the singular perturbation method is adopted to model the electromagnetic part and the mechanical part in a unified manner, which improves the modeling accuracy.

(2) Linearize the wind power system model at multiple operating points, and then use the Linear Parameter Varying (LPV) model to approximate the nonlinear model of the wind power system, which can greatly reduce the conservatism of the controller, and this method Simple and effective, the computational complexity is limited; at the same time, since the essence of LPV technology is between several operating points and inside the convex hull surrounded by the operating points, the method of weighted summation is used to achieve flexible switching, which effectively avoids switching control. The jitter problem caused.

(3) H _∞ robust time-delay controller is designed to effectively improve the wind energy capture efficiency of wind turbines and make full use of wind energy.

It is to be understood that all combinations of the foregoing concepts, as well as additional concepts described in greater detail below, are considered to be part of the inventive subject matter of the present disclosure to the extent that such concepts are not contradictory. Additionally, all combinations of the claimed subject matter are considered to be part of the inventive subject matter of this disclosure.

The foregoing and other aspects, embodiments and features of the present teachings can be more fully understood from the following description when taken in conjunction with the accompanying drawings. Other additional aspects of the invention, such as features and/or benefits of the exemplary embodiments, will be apparent from the description below, or learned by practice of specific embodiments in accordance with the teachings of this invention.

Description of drawings

The drawings are not intended to be drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures may be represented by the same reference numeral. For clarity, not every component is labeled in every figure. Embodiments of various aspects of the present invention will now be described by way of example and with reference to the accompanying drawings, wherein:

FIG. 1 is a flow chart of the MPPT control method for a wind power system based on the singular perturbation modeling theory and the robust time-delay control method of the present invention.

FIG. 2 is a schematic diagram showing the comparison of the tracking effect of the rotor speed under the control of the robust time-delay control method of the present invention and the optimal torque method.

3 is a schematic diagram showing the comparison of the tracking effect of the rotor speed under the control of the robust time-delay control method of the present invention and the optimal torque method.

Detailed ways

In order to better understand the technical content of the present invention, specific embodiments are given and described below in conjunction with the accompanying drawings.

Example 1

1, the present invention refers to a time-delay control method for maximum power point tracking of a singularly perturbed wind power generation model, which specifically includes the following steps:

Step 1: Considering the situation that the wind speed is lower than the rated wind speed, determine the system parameters (gearbox reduction ratio, gearbox efficiency, fan inertia moment, motor inertia moment, electromagnetic torque of generator, rigidity coefficient of high-speed drive shaft, high-speed drive shaft The value of damping coefficient, stator resistance, stator inductance of d-axis and q-axis components, number of pole pairs, and magnetic flux), a singular perturbation model for variable-speed and variable-pitch wind turbines is established as follows:

p为极对数个数，φ_m是磁通量。Among them, ω _r (t) is the rotor speed, i is the gearbox reduction ratio, η is the gearbox efficiency, J _r is the inertia moment of the fan, Tr is the aerodynamic torque, ω _g ( _t ) is the motor speed, J _g is the motor inertia moment, T _H (t) is the torque of the high-speed shaft, T _g (t) is the electromagnetic torque of the generator, K _g is the rigidity coefficient of the high-speed drive shaft, B _g is the damping coefficient of the high-speed drive shaft, ε =0.01 is the singular perturbation parameter, i _d (t), L _d , _ud (t) and i _q (t), L _q , u _q (t) are the stator current and inductance of the d-axis and q-axis components, respectively and voltage, R _s is the resistance of the stator,

p is the number of pole pairs, and φ _m is the magnetic flux.

其中ρ是空气密度，V(t)为风速，R为风机平面半径，功率系数C_Q(λ)是由叶尖速比λ(t)的二次多项式逼近的：C_Q(λ)＝C_Qmax-k_Q(λ(t)-λ_Qmax)²，C_Qmax是最大力矩系数，λ_Qmax表示对应最大力矩系数的叶尖速比，k_Q为逼近系数。The dynamic characteristics of the aerodynamic torque T _r are described as

where ρ is the air density, V(t) is the wind speed, R is the plane radius of the fan, and the power coefficient C _Q (λ) is approximated by a quadratic polynomial of the tip speed ratio λ(t): C _Q (λ)=C _Qmax -k _Q (λ(t)-λ _Qmax ) ² , C _Qmax is the maximum torque coefficient, λ _Qmax represents the tip speed ratio corresponding to the maximum torque coefficient, and k _Q is the approximation coefficient.

发电机的电磁转矩T_g(t)为T_g(t)＝pφ_mi_q(t)。Definition of tip speed ratio λ(t):

The electromagnetic torque T _g (t) of the generator is T _g (t)=pφm _{i q} (t).

使得以操作点为顶点的集合构成一个凸包Θ，即Θ＝Co{θ₁,θ₂,θ₃,θ₄,θ₅}。那么凸包内任何一个点可以由操作点θ_j的线性组合表示出来，即任意θ∈Θ，都存在一组非负数α_j≥0,j＝1,2,…,5，使得Step 2: Choose 5 Operating Points Appropriately

Let the set of operating points as vertices form a convex hull Θ, that is, Θ=Co{θ ₁ , θ ₂ , θ ₃ , θ ₄ , θ ₅ }. Then any point in the convex hull can be represented by a linear combination of operating points θ _j , that is, for any θ∈Θ, there is a set of non-negative numbers α _j ≥ 0, j = 1, 2,..., 5, such that

处，计算对应的发电机的电磁转矩

和空气动态力矩

从而可以由(3)式计算获得操作点θ_j对应的

令

δV(t)＝V(t)-V^j，

将非线性奇异摄动模型线性化，可得：Step 3: At the operating point

, calculate the electromagnetic torque of the corresponding generator

and aerodynamic torque

Therefore, the corresponding operation point θ _j can be obtained by formula (3).

make

δV(t)=V(t)-V ^j ,

Linearizing the nonlinear singular perturbation model, we get:

where δV(t) is regarded as a disturbance, and the coefficient matrix is as follows:

in,

The LPV singular perturbation model can then be written as:

那么mark

So

B(θ_j)＝[B₁(θ_j) B₂]。in

B(θ _j )=[B ₁ (θ _j ) B ₂ ].

又因为B_gq(θ_j)、B_qg(θ_j)、B_gd(θ_j)、B_dg(θ_j)、B_r(θ_j)、K_rv(θ_j)是θ_j的仿射函数，所以对于任意θ∈Θ，都存在一组正数α_j＞0,j＝1,2,…5使得known from step 2

And because B _gq (θ _j ), B _qg (θ _j ), B _gd (θ _j ), B _dg (θ _j ), B _r (θ _j ), K _rv (θ _j ) are affine functions of θ _j , so for any θ∈Θ, there is a set of positive numbers α _j > 0, j = 1, 2, ... 5 such that

Therefore, at any operating point θ∈Θ, a linearly variable parameter singular perturbation model can be obtained:

代入系统(8)，可得Step 4: Design H _∞ robust time-delay controller for given γ and operating point θ _j , j = 1, 2, ... 5

Substitute into system (8), we can get

in,

To simplify the notation, the coefficient matrix A(θ _j ) is abbreviated as A in the matrix inequality condition and proof process.

For known ε>0, h>0, if there is a 5×5-dimensional matrix P _ε , such that E _ε P _ε >0 is established, 2×2-dimensional matrix Q, 2×2-dimensional matrix R ₁ >0, 2×2 The dimensional matrix R ₂ >0 and the 5×5 dimensional matrix H make matrix inequality (13) true:

in

其中

为

的广义逆矩阵。Then the closed-loop linear time-invariant singular perturbation model (12) is robust and stable, and the gain coefficient of the controller is

in

for

The generalized inverse matrix of .

Next, in order to verify the reasonable validity of the proposed control scheme, a robust stability proof is carried out.

(1) First, it is proved that the closed-loop system is asymptotically stable in the case of zero disturbance.

定义Lyapunov函数(为简化符号，标记X_t＝X(t))：Assuming δV(t)=0, let

Define the Lyapunov function (notation X _t =X(t) for simplified notation):

in,

沿着系统(12)对时间t求导，可得on the Lyapunov function

Taking the derivative of time t along the system (12), we get

A known

Integrating both sides, it can be derived that

F

(21) Substitute into (17) to get

Because for any α, β∈R ⁿ and any symmetric positive definite n×n-dimensional matrix H, we have

-2α ^T β≤α ^T H ^-1 α+β ^T Hβ

Then for any 2×2-dimensional symmetric positive definite matrix R ₁ and 2×2-dimensional symmetric positive definite matrix R ₂ , there are

A known

So

Substitute into (22) to get

Eliminate the underlined part in the above formula, you can get

To further sort out the underlined part, remember

Then, it can be sorted out

According to Schur's lemma, it can be known from the matrix inequality (13) that

由此可得，当扰动δV(t)＝0时，系统(12)的平衡点是渐近稳定的。Combining inequality (25), it is easy to know

Therefore, when the disturbance δV(t)=0, the equilibrium point of the system (12) is asymptotically stable.

(2) Next, it is proved that when δV(t)≠0, the system is robust under the condition of matrix inequality (13).

The same Lyapunov function as (16) is used

沿系统(26)对时间t求导for functions

Derivative with respect to time t along the system (26)

利用Schur补引理，由条件(13)可知then there is,

Using Schur's complement lemma, it can be seen from condition (13) that

The above inequality is equivalent to

The above inequality can be further transformed into

Substitute into equation (27)

also known,

so,

Based on the proof of asymptotic stability, it can be known that X _t (∞)=0. Now assume that X _t (0)=0, integrate both sides of inequality (29), we can get:

which is

即||C(sE_ε-A(θ))^-1B₁(θ)||_∞＜γ。证毕It is evident from this

That is, ||C(sE _ε -A(θ)) ^-1 B ₁ (θ)|| _∞ <γ. certificated

其中

为

的广义逆矩阵。By solving the matrix inequality (13), the gain coefficient of the controller can be obtained as

in

for

The generalized inverse matrix of .

Step 5: For the LPV singular perturbation model, at time t _k , the parameter θ(t _k )=[ω _r V ω _g i _d i _q ] is measured and obtained, and the weight coefficient α _j is calculated so that it satisfies:

Step 6: At time t _k , calculate the controller gain of the original wind power generation system:

Step 7: Calculate the control input u(t _k )=K(θ _k )X(th), and apply u(t _k ) to the original nonlinear wind power generation system;

Step 8: At time t _k+1 , repeat steps 5-8.

Embodiment 2

This example uses a CART3 blade wind turbine built by the U.S. Department of Energy's Renewable Energy Laboratory (NREL) as the research object. The parameters of the wind turbine are shown in Table 1.

Table 1 Wind turbine parameters

name symbol Numerical value name symbol Numerical value Optimum tip speed ratio λ_opt 5.8 Fan radius R 20m Optimal wind energy utilization coefficient C_Pmax 0.467 Air density ρ 0.98Kg/m³ Gearbox factor n 1 Fan inertia J_r 3.88Kg·m² Motor inertia J_g 0.22Kg m² number of pole pairs P 3 gear ratio i 43.165 magnetic flux φ_m 0.4382wb Motor damping coefficient B_g 0.3Kg m²/s Motor stiffness coefficient K_g 75Nm/rad Stator d-axis inductance L_d 41.56mH Stator q-axis inductance L_q 41.56mH Stator damping R_s 3.3Ω

Using the parameter values in Table 1, the nonlinear singular perturbation model can be established as follows:

Then, the above wind power generation system is controlled by the control method proposed by the present invention, and compared with the optimal torque method, a comparison chart of the tracking effect of the rotational speed of the wind rotor can be obtained. It can be seen from FIG. 1 that the robust time-delay control method proposed by the present invention based on the singular perturbation method has a more accurate tracking effect. Figure 2 is a comparison chart of tracking errors. The rotor speed tracking error obtained by the robust time-delay control method is smaller than that of the optimal torque method, which further verifies the effectiveness and superiority of the method proposed in the present invention.

The robust time-delay control method and optimal torque control are used to control the wind turbine. The simulation time is 500 seconds, and the average wind energy capture efficiency of the two methods is calculated. The results are shown in Table 2. It can be clearly seen from Table 2 that compared with the optimal torque control method, the robust time-delay control method can achieve higher wind energy capture efficiency and has advantages.

Table 2 Comparison of control effects

method Average wind energy capture efficiency Robust Delay Control 0.4496 optimal torque control 0.4455

Aspects of the invention are described in this disclosure with reference to the accompanying drawings, in which a number of illustrative embodiments are shown. Embodiments of the present disclosure are not necessarily defined to include all aspects of the invention. It should be understood that the various concepts and embodiments described above, as well as those described in greater detail below, can be implemented in any of a variety of ways, as the concepts and embodiments disclosed herein do not limited to any implementation. Additionally, some aspects of the present disclosure may be used alone or in any suitable combination with other aspects of the present disclosure.

Although the present invention has been disclosed above with preferred embodiments, it is not intended to limit the present invention. Those skilled in the art to which the present invention pertains can make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, the protection scope of the present invention should be determined according to the claims.

Claims (6)

1. A singular perturbation wind power generation model maximum power point tracking time lag control method is characterized by comprising the following steps:

s1: the method comprises the steps of taking the situation that the wind speed is lower than the rated wind speed into consideration, collecting relevant data of a wind driven generator system, and establishing a nonlinear singular perturbation model for a variable-speed variable-pitch type wind driven generator;

s2: select 5 operating points θ_jJ is 1,2, …,5, so that a set of vertices with the operation point constitutes a convex hull Θ, and Θ is Co { θ {₁,θ₂,θ₃,θ₄,θ₅There is a set of nonnegative numbers α for any point theta, theta e theta, within the convex hull theta_j0, j ≧ 1,2, …,5, such that:

and is

S3: by at a plurality of operating points theta_jLinearizing the nonlinear singular shooting model to obtain 5 linear time invariant singular shooting models;

s4: combining a given gamma and each operating point theta for 5 linear time invariant singular perturbation models_jH is obtained by solving the matrix inequality through design_∞Robust time lag controller K_j(θ_j) The closed-loop linear time invariant singular perturbation model is robust and stable, and gamma is an index requirement on infinite norm of a system transfer function;

s5: at t_kAt the moment, θ (t) is measured_k) And calculating a weight coefficient α_jSuch that it satisfies:

s6: at t_kAt the moment, the controller of the original system is designed as follows:

s7: calculating a control input u (t) according to the following formula_k)：u(t_k)＝K(θ(t_k) X (t-h), where K (θ (t)_k) X (t-h) is the state variable of the singular perturbation model, t is time, h is time lag; will calculate the control input u (t)_k) The method is applied to the original nonlinear wind power generation system;

s8: let t_k＝t_k+1And repeating the steps S5-S8 to control the wind power generation system in real time.

2. The singularity perturbation wind power generation model maximum power point tracking time lag control method according to claim 1, wherein the related data of the wind power generator system comprise a gearbox reduction ratio, gearbox efficiency, fan inertia moment, motor inertia moment, electromagnetic torque of a generator, a rigidity coefficient of a high-speed transmission shaft, a damping coefficient of the high-speed transmission shaft, resistance of a stator, stator inductance of d-axis and q-axis components, the number of pole pairs and magnetic flux.

3. The singular perturbation wind power generation model maximum power point tracking time-lag control method according to claim 1, wherein in step S1, the nonlinear singular perturbation model is:

wherein, ω is_r(t) rotor speed, i represents gearbox reduction ratio, η represents gearbox efficiency, J_rIs the fan moment of inertia, T_rIs an air dynamic moment, omega_g(t) motor speed, J_gIs the moment of inertia of the motor, T_H(T) is the high-speed shaft torque, T_g(t) is the electromagnetic torque of the generator, K_gIs the rigidity coefficient of high-speed transmission shaft, B_gIs the damping coefficient of high-speed drive shaft, and is a singular perturbation parameter, i ═ 0.01_d(t)、L_d、u_d(t) and i_q(t)、L_q、u_q(t) stator currents, inductances and voltages, respectively d-and q-axis components, R_sIs the resistance of the stator and is,

p is the number of pole pairs, phi_mIs the magnetic flux; air dynamic moment T_rIs described as

Where ρ is the air density, V (t) is the wind speed, R is the fan plane radius, the power coefficient C_Q(λ) is approximated by a quadratic polynomial of the tip speed ratio λ (t): c_Q(λ)＝C_Qmax-k_Q(λ(t)-λ_Qmax)²，C_QmaxIs the maximum moment coefficient, λ_QmaxRepresenting tip speed ratio, k, corresponding to the maximum moment coefficient_QIs an approximation coefficient; the tip speed ratio λ (t) is defined as:

electromagnetic torque T of generator_g(T) is T_g(t)＝pφ_mi_q(t)。

4. The singular perturbation wind power generation model maximum power point tracking time-lag control method according to claim 3, wherein in step S2, the 5 operation points θ_jExpressed as:

5. the singular perturbation wind power generation model maximum power point tracking time-lag control method according to claim 4, wherein in step S3, the passing is at a plurality of operating points θ_jThe method for linearizing the nonlinear singular shooting model to obtain 5 linear time invariant singular shooting models comprises the following steps:

s31: at the operating point

Calculating the electromagnetic torque of the corresponding generator

And air dynamic moment

Thereby calculating the operation point theta_jCorresponding to

S32: order to

δV(t)＝V(t)-V^j，

Linearizing the nonlinear singular shooting model to obtain:

where δ v (t) is taken as a perturbation, the coefficient matrix is as follows:

in the formula, B_gq(θ_j)、B_qg(θ_j)、B_gd(θ_j)、B_dg(θ_j)、B_r(θ_j)、K_rv(θ_j) Is theta_jThe affine function of (a) is,

s33: obtaining a linear variable parameter singular perturbation model:

s331: marking

Then

Wherein

B(θ_j)＝[B₁(θ_j) B₂]；

S332, setting that for any theta epsilon theta, a group of positive numbers α exists_j> 0, j ═ 1,2, … 5 such that:

then at any operation point theta epsilon theta, the linear variable parameter singular perturbation model is transformed into:

6. the singular perturbation wind power generation model maximum power point tracking time-lag control method according to claim 5, wherein in step S4, said combining given γ and each operating point θ for 5 linear time-invariant singular perturbation models_jH is obtained by solving the matrix inequality through design_∞Robust time lag controller K_j(θ_j) The process for making the closed-loop linear time-invariant singular perturbation model robust and stable comprises the following steps:

s41: for a given gamma and operating point theta_jJ-1, 2, … 5, design H_∞The robust time lag controller is as follows:

substituting the linear variable parameter into the singular perturbation model to obtain

Wherein,

s42: setting matrix inequality conditions and coefficient matrix A (theta) in the proving process_j) Abbreviated as a;

for known ε > 0, h > 0, if a 5 × 5 dimensional matrix P exists_εSo that E_εP_εGreater than 0 true, 2 x 2 dimensional matrix Q,2 x 2 dimensional matrix R₁Greater than 0,2 x 2 dimensional matrix R₂The following matrix inequality is established when the dimension H of the matrix is more than 0 and 5 multiplied by 5, the closed loop linear time invariant singular perturbation model is determined to be robust and stable, and the gain coefficient of the controller is

Wherein

Is composed of

Generalized inverse matrix of (1):

wherein:

P₁＝P₁ ^T＞0，

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