CN102900603B - Design method of wind turbine pitch controller based on finite-time non-fragile/cost-stable - Google Patents

Abstract

The invention provides a variable pitch controller design method based on a finite time non-crisp/guaranteed-cost stable wind turbine generator set. The method includes that a continuous time non-linear model of a wind turbine generator set variable pitch system is approximately indicated through a fuzzy T-S model; a dynamic fuzzy model is obtained through single-point fuzzification, product reasoning, defuzzification of the center of gravity according to the obtained fuzzy T-S model; and according to the obtained dynamic fuzzy model and finite time stable meaning, a state feedback controller of wind turbine generator set variable pitch is designed, and pitch angles of the wind turbine generator set, rotating speed of the wind turbine generator and output current of the wind turbine generator set are controlled through the obtained controller.

Description

Based on finite time non-crisp/protect the stable wind-powered electricity generation unit Variable-pitch Controller design method of cost

Technical field

The present invention relates to the control of wind-powered electricity generation unit feather, especially a kind of based on the stable controlling method of the non-crisp guarantor's cost of finite time.

background technique

Because wind energy is the randomness energy, when wind speed changes, the power of exporting on wind turbine shaft also changes thereupon.Therefore, how regulating the output power of wind energy conversion system is one of very important key technology for the wind-driven generator being incorporated into the power networks.At present, horizontal-shaft wind turbine power adjustments mode is mainly divided into two kinds, and fixed pitch stall-adjusted and feather power adjustments are two kinds.

The basic principle of fixed pitch stall power adjustments is: utilize the aerodynamic characteristic of blade itself,, in rated wind speed, the lift coefficient of blade is higher, the utilization factor C of wind energy _palso higher, and during wind speed overrate, blade enters stall conditions, just lift no longer increases, and wind speed round will no longer increase along with the increase of wind speed, thereby reaches the object of restriction wind energy conversion system output power.Put it briefly, stall power adjustments is to utilize the aerodynamic stalling power adjustments of blade, is again to utilize the aerodynamic stalling characteristic limitations pneumatic equipment blades made of blade to absorb wind energy, reaches and prevents that the output power of wind energy conversion system is excessive, thereby reach, maintains wind energy conversion system invariablenes turning speed.The shortcomings such as the advantage of this regulative mode is that variable propeller pitch adjusting mechanism is simple, and operational reliability is higher, but exists wind energy loss large, and the starting performance of wind energy conversion system is poor, and the pneumatic thrust that bears on blade is larger.

The basic principle of feather power adjustments mode is: when wind-force variation makes the wind speed round of wind energy conversion system depart from rated speed, in scheduled time, control by means of blade pitch adjusting color controls, change the propeller pitch angle of wind mill wind wheel blade, maintain the invariablenes turning speed of wind energy conversion system, thereby adjust the output power of wind energy conversion system.Common control algorithm has following several at present:

(1) the feather control technique based on Robust Control Algorithm, can realize at the maximal wind-energy capture having under modeling condition of uncertainty, the in the situation that of basic guarantee maximal wind-energy capture, can make the amplitude that on rotor shaft, torque changes reduce an order of magnitude.Robust control can also solve driftage problem, and the design that realizes fatigue loads controller in wind-energy changing system by the torque in control chain.

(2) the intelligent variable-pitch controller technology based on fuzzy algorithmic approach, can effectively adapt to nonlinear system, feather fuzzy control adopts change propeller pitch angle to change the method for aerodynamic torque, to regulate the power factor of wind mill wind wheel, and then controls the output power of wind energy conversion system.

(3) the wind-powered electricity generation unit feather based on Fuzzy RBF Neural Network is controlled, and adopts neuron network to realize FUZZY MAPPING process, according to input-output training data, automatically extracts control law, determines former piece and consequent parameter.This controller calculates based on real time data, can continue to optimize its Inter parameter and make system can overcome non-linear and time variation, has met dynamic characteristic and the steady-state behaviour of system.

summary of the invention

The present invention improves prior art, is intended to guarantee that wind-powered electricity generation unit variable-pitch control system " in short-term " is stable, and finite time is stable, and controller switches at different " in short-term ".

Technological scheme of the present invention is:

Based on the stable wind-powered electricity generation unit variable pitch control method of finite time, comprise the following steps:

The first step: for variable-pitch system of wind turbine generator, set up nonlinear model continuous time and by following T-S fuzzy model approximate representation:

Plant model rule i (i=1,2 ..., r)

If θ ₁(t) be N _i1, θ ₂(t) be N _i2θ ₃(t) be N _i3

So $\stackrel{\·}{x}\left(t\right)=A_{i}x\left(t\right)+B_{i}u\left(t\right)$

Wherein, θ ₁(t), θ ₂and θ (t) ₃(t) represent respectively wind speed, wind-driven generator rotating speed and output power; N _i1, N _i2and N _i3be respectively θ in i rule ₁(t), θ ₂and θ (t) ₃(t) corresponding linguistic variable; The vector of x (t) for being formed by propeller pitch angle, wind-driven generator rotating speed and wind-driven generator output current; U (t) represents the propeller pitch angle instruction input of expectation; (A _i, B _i) State Equation Coefficients corresponding to expression i bar plant model rule; R is control law number (value of the present invention is 9 or 16);

Second step: above-mentioned T-S fuzzy model is carried out to product reasoning, the processing of center of gravity defuzzification, obtain following dynamic fuzzy system:

$\stackrel{\·}{x}\left(t\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{h}_i\left(\mathrm{\θ}\left(t\right)\right)[A_{i}x\left(t\right)+B_{i}u\left(t\right)]$

Wherein, $h_i\left(\mathrm{\θ}\left(t\right)\right)=\frac{h_{\mathrm{il}}\left({\mathrm{\θ}}_1\left(t\right)\right)h_{i2}\left({\mathrm{\θ}}_2\left(t\right)\right)h_{i3}\left({\mathrm{\θ}}_3\left(t\right)\right)}{\underset{m=1}{\overset{r}{\mathrm{\Σ}}}{h}_{m1}\left({\mathrm{\θ}}_1\left(t\right)\right)h_{m2}\left({\mathrm{\θ}}_2\left(t\right)\right)h_{m3}\left({\mathrm{\θ}}_3\left(t\right)\right)}$ Represent that plant model meets the degree of i rule; h _i1(θ ₁(t)), h _i2(θ ₂) and h (t) _i3(θ ₃(t)) be respectively θ ₁(t), θ ₂and θ (t) ₃(t) membership function, works as θ ₁(t), θ ₂and θ (t) ₃(t), while being taken as concrete numerical value, its corresponding membership function value is respectively h _i1(θ ₁(t)), h _i2(θ ₂) and h (t) _i3(θ ₃(t));

The 3rd step: according to the stable connotation of finite time and above-mentioned plant model, the controller model that design is represented by following T-S fuzzy model, wherein, the corresponding controller model rule of each plant model rule:

Controller model rule j (j=1,2 ..., r)

If θ ₁(t) be N _j1, θ ₂(t) be N _j2θ ₃(t) be N _j3

U (t)=K so _jx (t)

Wherein, K _jfor gain matrix;

Above-mentioned controller model is carried out to product reasoning, center of gravity defuzzification, arranges and obtain following controller:

$u\left(t\right)=\underset{i=1}{\overset{r}{\mathrm{\Σ}}}{h}_j\left(\mathrm{\θ}\left(t\right)\right)K_{j}x\left(t\right)$

Wherein, N _jk(j=1,2 ..., r, k=1,2,3) with the first step in N _ik(i=1,2 ..., r, k=1,2,3) consistent, h _j(θ (t)) (j=1,2 ..., h r) and in second step _i(θ (t)) (i=1,2 ..., r) consistent;

The 4th step: the propeller pitch angle instruction input u (t) that utilizes the 3rd step to obtain, controls propeller pitch angle, wind-driven generator rotating speed and wind-driven generator output current.

Embodiment

[the feather Principles of Regulation of wind-powered electricity generation unit]

By power coefficient C _p=2P/ ρ v ³a knows, it is P=C that wind energy conversion system absorbs the output power that wind energy produces _pρ v ³a/2; Wind energy conversion system changes the energy of generation into mechanical energy and passes to load, mechanical energy representation:

P _m＝Tw (1)

In formula: P _m-mechanical energy; T-wind energy conversion system moment of torsion; ω-wind energy conversion system angular velocity, the torque T is here determined by load, can be obtained like this by formula (1):

ω＝ρπC _pR ²v ³/2T

When wind energy conversion system is under certain wind speed, for certain load, ρ, π, R are also constant, and rotating speed just depends on the size of power coefficient so, has ω ∝ C _p.The stressing conditions of blade while rotating with certain velocity-stabilization after starting according to foline characteristic theory analysis wind wheel, thus draw the relation at ideal situation downstream and each angle of blade:

I＝i+β

tgI＝v/ωr＝1/λ

In formula: I-inclination angle; The i-angle of attack; β-propeller pitch angle; λ-tip-speed ratio.

According to equilibrium of forces relation, the moment of torsion of blade is:

T＝C _mρv ²AR/2

$W_r=\frac{v}{\sin I}$

$C_m=\frac{C_L(\sin I-\frac{1}{C_L/C_D}\cos I)}{{\sin }^{2}I}$

In formula: C _m-torque coefficient; The wind-exposuring area of A-wind wheel; R-wind wheel radius; W _rthe relative speed of wind of-blade.

For the wind energy conversion system turning round under certain rotating speed, as wind speed and direction one timing, W _rwith I be definite value.If increase the angle of attack (reducing propeller pitch angle), lift coefficient will increase, and ratio of lift coefficient to drag coefficient also will increase, and torque coefficient also can increase, and vice versa.So by changing wind energy conversion system propeller pitch angle β, just can change the rotating speed of wind energy conversion system, wind mill pitch-variable Principles of Regulation that Here it is.Normally using the rotating speed of wind speed and wind energy conversion system as the signal of blade pitch angle controller action.

[finite time is stable]

Progressive stable theory by Liapunov starts, and Theory of Stability is studied widely by people.In research process, General Definition a unlimited time interval,, when the time is tending towards infinite, parallel algorithm is stabilized in a field.And in actual applications, often the not consideration time is tending towards infinite stable case, and only consider the stable case within the scope of set time, introduce thus the stable concept of finite time, by the reduction of stability requirement, bring the dynamic performance of control system to promote.

Definition 1: for controlled device closed loop control system is called as that [0, T] interior finite time is stable to be referred to: have parameter (c ₁, c ₂, T, R _c) meet have $x^T\left(0\right)R_{C}x\left(0\right)\&le;c_1\&DoubleRightArrow;x^T\left(t\right)R_{C}x\left(t\right)\&le;c_2,$ 0 < c wherein ₁< c ₂, T ∈ R ₊and R _c> 0.

[variable pitch control method]

Utilize nonlinear model continuous time of T-S fuzzy model approximate representation variable-pitch system of wind turbine generator; According to the T-S fuzzy model obtaining, utilize single-point obfuscation, product reasoning, center of gravity defuzzification to obtain dynamic fuzzy system; According to the dynamic fuzzy system and the finite time that obtain, stablize connotation, design wind-powered electricity generation unit feather state feedback controller, and utilize the controller obtaining to control the propeller pitch angle of wind-powered electricity generation unit, wind-driven generator rotating speed and wind-powered electricity generation unit output current, concrete steps are as follows:

The first step: for variable-pitch system of wind turbine generator, set up nonlinear model continuous time and by following T-S fuzzy model approximate representation:

Plant model rule i (i=1,2 ..., r)

If θ ₁(t) be N _i1, θ ₂(t) be N _i2θ ₃(t) be N _i3

So $\stackrel{\&CenterDot;}{x}\left(t\right)=A_{i}x\left(t\right)+B_{i}u\left(t\right)$

Wherein, θ ₁(t), θ ₂and θ (t) ₃(t) represent respectively wind speed, wind-driven generator rotating speed and output power; N _i1, N _i2and N _i3be respectively θ in i rule ₁(t), θ ₂and θ (t) ₃(t) corresponding linguistic variable; The vector of x (t) for being formed by propeller pitch angle, wind-driven generator rotating speed and wind-driven generator output current; U (t) represents the propeller pitch angle instruction input of expectation; (A _i, B _i) State Equation Coefficients corresponding to expression i bar plant model rule; R is control law number (value of the present invention is 9 or 16);

Second step: above-mentioned T-S fuzzy model is carried out to product reasoning, the processing of center of gravity defuzzification, obtain following dynamic fuzzy system:

$\stackrel{\&CenterDot;}{x}\left(t\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{h}_i\left(\mathrm{\&theta;}\left(t\right)\right)[A_{i}x\left(t\right)+B_{i}u\left(t\right)]$

Wherein, $h_i\left(\mathrm{\&theta;}\left(t\right)\right)=\frac{h_{\mathrm{il}}\left({\mathrm{\&theta;}}_1\left(t\right)\right)h_{i2}\left({\mathrm{\&theta;}}_2\left(t\right)\right)h_{i3}\left({\mathrm{\&theta;}}_3\left(t\right)\right)}{\underset{m=1}{\overset{r}{\mathrm{\&Sigma;}}}{h}_{m1}\left({\mathrm{\&theta;}}_1\left(t\right)\right)h_{m2}\left({\mathrm{\&theta;}}_2\left(t\right)\right)h_{m3}\left({\mathrm{\&theta;}}_3\left(t\right)\right)}$ Represent that plant model meets the degree of i rule; h _i1(θ ₁(t)), h _i2(θ ₂) and h (t) _i3(θ ₃(t)) be respectively θ ₁(t), θ ₂and θ (t) ₃(t) membership function, works as θ ₁(t), θ ₂and θ (t) ₃(t), while being taken as concrete numerical value, its corresponding membership function value is respectively h _i1(θ ₁(t)), h _i2(θ ₂) and h (t) _i3(θ ₃(t));

The 3rd step: according to the stable connotation of finite time and above-mentioned plant model, the controller model that design is represented by following T-S fuzzy model, wherein, the corresponding controller model rule of each plant model rule:

Controller model rule j (j=1,2 ..., r)

If θ ₁(t) be N _j1, θ ₂(t) be N _j2θ ₃(t) be N _j3

U (t)=K so _jx (t)

Wherein, K _jfor gain matrix;

Above-mentioned controller model is carried out to product reasoning, center of gravity defuzzification, arranges and obtain following controller:

$u\left(t\right)=\underset{i=1}{\overset{r}{\mathrm{\&Sigma;}}}{h}_j\left(\mathrm{\&theta;}\left(t\right)\right)K_{j}x\left(t\right)$

Wherein, N _jk(j=1,2 ..., r, k=1,2,3) with the first step in N _ik(i=1,2 ..., r, k=1,2,3) consistent, h _j(θ (t)) (j=1,2 ..., h r) and in second step _i(θ (t)) (i=1,2 ..., r) consistent;

The 4th step: the propeller pitch angle instruction input u (t) that utilizes the 3rd step to obtain, controls propeller pitch angle, wind-driven generator rotating speed and wind-driven generator output current.

[control parameter designing]

According to above definition 1, if there is scalar ce>=0, symmetric positive definite matrix Q ∈ R ^{n * n}and matrix W _j(j=1,2 ..., r) meet certain relation, controller parameter is taken as stable at [0, T] interior finite time to meet control system, described relation is:

$\left\{\begin{array}{c}{A}_i\tilde{Q}+\tilde{Q}{A}_i^T+B_i{W}_j+W_j^T{B}_i^T-\mathrm{\&alpha;}\tilde{Q}<0(1\&le;i,j\&le;r)\\ \frac{c_1}{{\mathrm{\&lambda;}}_{\min }\left(Q\right)}<\frac{c_2{e}^{-\mathrm{\&alpha;T}}}{{\mathrm{\&lambda;}}_{\max }\left(Q\right)}\end{array}\right.$

Wherein parameter (c ₁, c ₂, T, R _c) meet have and, 0 < c ₁< c ₂, T ∈ R ₊and R _c> 0, R _cexpression state gain matrix, c ₁represent the x that original state x (0) is corresponding ^t(0) R _cx (0) the value upper limit, c ₂be illustrated in the time (0, T] x that internal state x (t) is corresponding ^t(t) R _cx (t) the value upper limit, λ _min(Q) minimal eigenvalue of representing matrix Q, λ _max(Q) eigenvalue of maximum of representing matrix Q.

Above relation can utilize the LMI toolbox of Matlab to solve.

Be noted that the mode that the controlling method of the embodiment of the present invention can add essential general hardware platform by software realizes.Understanding based on such, the part that the technological scheme of the embodiment of the present invention contributes to prior art in essence in other words can embody with the form of software product, this computer software product is stored in a storage medium, comprises that some instructions are in order to carry out the method described in each embodiment of the present invention.Here alleged storage medium, as: ROM/RAM, disk, CD etc.In sum, these are only preferred embodiment of the present invention, be not intended to limit protection scope of the present invention.Within the spirit and principles in the present invention all, any modification of doing, be equal to replacement, improvement etc., within all should being included in protection scope of the present invention.

Claims (1)

1. A wind turbine pitch control method based on finite time stability, comprising the following steps: Step 1: For the wind turbine pitch system, establish a continuous-time nonlinear model And it is approximated by the following fuzzy TS model: Controlled object model rule i, i=1, 2, ..., r If θ ₁ (t) is N _i1 , θ ₂ (t) is N _i2 , θ ₃ (t) is N _i3 So $\stackrel{\&Center\; Dot;}{x}\left(t\right)=A_{i}x\left(t\right)+B_{i}u\left(t\right)$ Among them, θ ₁ (t), θ ₂ (t) and θ ₃ (t) represent wind speed, wind turbine speed and output power respectively; N _i1 , N _i2 and N _i3 are θ ₁ (t ), the linguistic variables corresponding to θ ₂ (t) and θ ₃ (t); x(t) is a vector composed of pitch angle, wind turbine speed and wind turbine output current; u(t) represents the desired propeller The distance angle command input; (A _i , B _i ) represents the coefficient of the state equation corresponding to the i-th controlled object model rule; r is the number of control rules, and its value is 9 or 16; Step 2: Perform product reasoning and defuzzification of the center of gravity on the above fuzzy T-S model, and obtain the controlled object model represented by the following dynamic fuzzy model: $\stackrel{<span\; class="notranslate">\; \&CenterDot;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; r</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$ in, $h_i\left(\mathrm{\&theta;}\left(t\right)\right)=\frac{h_{i1}\left({\mathrm{\&theta;}}_1\left(t\right)\right)h_{i2}\left({\mathrm{\&theta;}}_2\left(t\right)\right)h_{i3}\left({\mathrm{\&theta;}}_3\left(t\right)\right)}{\underset{m=1}{\overset{r}{\mathrm{\&Sigma;}}}{h}_{m1}\left({\mathrm{\&theta;}}_1\left(t\right)\right)h_{m2}\left({\mathrm{\&theta;}}_2\left(t\right)\right)h_{m3}\left({\mathrm{\&theta;}}_3\left(t\right)\right)}$ Indicates the extent to which the controlled object model conforms to the i-th rule; h _i1 (θ ₁ (t)), h _i2 (θ ₂ (t)) and h _i3 (θ ₃ (t)) are θ _i (t), The membership function of θ ₂ (t) and θ ₃ (t); The third step: according to the implication of finite time stability and the controlled object model, design a controller model represented by the following fuzzy T-S model, wherein each controlled object model rule corresponds to a controller model rule: Controller model rule j, j = 1, 2, ..., r If θ ₁ (t) is N _j1 , θ ₂ (t) is N _j2 , θ ₃ (t) is N _j3 Then u(t)=K _j x(t) Among them, K _j is the gain matrix, that is, the control coefficient; The product reasoning and defuzzification of the center of gravity are carried out on the above controller model, and the following controller is obtained after sorting out: $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; r</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{\mathrm{<span\; class="notranslate">\; h</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>$ Among them, N _jk , j=1, 2,..., r; k=1, 2, 3 and N _ik in the first step, i=1, 2,..., r; k=1, 2 , 3 are consistent, h _j (θ(t)), j=1, 2, ..., r is the same as h _i (θ(t)), i = 1, 2, ..., r in the second step consistent; Step 4: Use the pitch angle command input u(t) obtained in the third step to control the pitch angle, the speed of the wind turbine and the output current of the wind turbine, where, When the scalar α≥0, the symmetric positive definite matrix Q∈R ^nxn , and the matrix W _j , j=1, 2,..., r satisfy a certain relationship, the control coefficient K _j is taken as That is to say, the control system is stable within [0, T] in a finite time, and the relational expression is: $\left\{\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; Q</span>}}<span\; class="notranslate">\; +</span>\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; Q</span>}}{\mathrm{<span\; class="notranslate">\; A</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; W</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;</span>}\stackrel{<span\; class="notranslate">\; ~</span>}{\mathrm{<span\; class="notranslate">\; Q</span>}}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; r</span>}\\ \frac{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; min</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; <</span>\frac{{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; e</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&alpha;T</span>}}}{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; max</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; Q</span>}<span\; class="notranslate">\; )</span>}& \end{array}\right.$ in, Parameters c ₁ , c ₂ , T, R _C satisfy have And, 0<c ₁ <c ₂ , T∈R ₊ and R _C >0, R _C represents the state gain matrix, c ₁ represents the x ^T (0)R _C x(0) corresponding to the initial state x(0) The upper limit of the value, c ₂ means the upper limit of the value of x ^T (t) R _C x (t) corresponding to the state x (t) in time (0, T], λ min (Q) means the minimum eigenvalue of the matrix Q, λ max ( Q) represents the largest eigenvalue of matrix Q.