CN102799111A - Method for designing multi-balance point nonlinear system for dynamic stability control - Google Patents

Method for designing multi-balance point nonlinear system for dynamic stability control Download PDF

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CN102799111A
CN102799111A CN2012101679082A CN201210167908A CN102799111A CN 102799111 A CN102799111 A CN 102799111A CN 2012101679082 A CN2012101679082 A CN 2012101679082A CN 201210167908 A CN201210167908 A CN 201210167908A CN 102799111 A CN102799111 A CN 102799111A
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CN102799111B (en
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徐式蕴
汤涌
孙华东
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Beijing Institute of Technology BIT
China Electric Power Research Institute Co Ltd CEPRI
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China Electric Power Research Institute Co Ltd CEPRI
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Abstract

The invention provides a method for designing a multi-balance point nonlinear system for dynamic stability control. The method comprises the following steps of: 1) establishing a synchronous unit model, and determining a gradient-like definition; 2) judging the gradient-like performance of the synchronous unit model, and if the synchronous unit model is chaotic, executing the step 3); 3) performing controller design on the synchronous unit model in the step 1); and 4) performing dynamic stability analysis and control on a power system by using a controller in the step 3), so the dynamic stability of the whole power system is ensured. According to a dynamic stability control method which is employed in the invention, the linearization of the conventional like methods is eliminated, so accumulated errors which are generated in the method are greatly reduced, and a future practical engineering guiding function is realized.

Description

A kind of method for designing that is used for the multiple stable point NLS of dynamic stability control
Technical field
The present invention relates to field of power, be specifically related to a kind of method for designing that is used for the multiple stable point NLS of dynamic stability control.
Background technology
Formation along with the Da Qu interconnected network; It is more complicated that the dynamic security stable problem of electric system also becomes, and comprises the influencing each other etc. of voltage stability, ac and dc systems of operation control, the receiving-end system of dynamic perfromance, the Da Qu interconnected network of Da Qu interconnected electric power system.These problems are faced with new challenges the safe and stable operation of electric system, also power system safety and stability analysis and control have been proposed new requirement.Yet the method that up to the present, solves the control of electric system microvariations dynamic stability wide area is because the introducing of linearization technique has brought certain cumulative errors to design of Controller.In addition, existing big disturbance dynamic stability wide area control method depends on time-domain-simulation and simplification mostly, does not provide strict mathematical justification, has brought certain difficulty also for the design of controller.
Unit is the important component part of electric system synchronously, and whether dynamic stability synchronous and electric system has confidential relation for it.For synchronous unit, the class gradient property (being overall progressive stability) that characterizes the nonlinear kinetics equation of its dynamics is promptly corresponding to the synchronizing characteristics of unit synchronously.For guaranteeing the dynamic stability of electric system, for synchronous unit, that understands the chaotic oscillation that when can continue and how to avoid vibrating is extremely important.Its reason is: on the one hand, have under the situation of chaotic attractor the heavy losses that operation continuously can cause the destruction of synchronizing characteristics and cause valuable equipment in NLS; On the other hand, chaotic oscillation can cause the permanent harm of harmful transient state resonance.
In recent years, a lot of pertinent literatures were such as document one IEEE Transactions on Magnetics; 2003,39:2995-2997, Design of permanent magnets to avoid chaos in PM synchronous machines; Gao, Y.& Chau, K.T.; Document two IEEE Trans.on Circ.Sys.I., 1994,41:40-45; Strange attractors in brushless DC motors; Hemati N and document three IEEE Trans.Power System, 2004,19:1918-1924; Hysteresis and bifurcations in the classical model of generator; Subbarao D.and Singh K K has studied carefully the complex dynamic characteristics of synchrodyne group model in the document, can be described by a series of nonlinear differential equations.Wherein, document one has been studied the Hopf bifurcated and the chaos phenomenon of the synchronous unit of PM when not disposing the suitable dimension permanent magnet poles.The open loop dynamic perfromance of then having pointed out direct current generator in the document two is equivalent to the Lorenz system with chaotic characteristic.Exist a plurality of attractors that the possibility that dynamic system various control condition exists has been described in the dynamic system.Aspect the theory derivation, document four, Non-local methods for pendulum-like feedback systems (Teubner-Texte zur Mathematik Bd.132; B.G.; Teubner Stuttgart-Leipzig), 1992, Leonov G A; Reitmann V and Smirnova V B has provided the frequency domain criterion condition that guarantees the synchronous progressive stability of synchronous unit.Yet,, also need further it to be simplified because frequency domain condition has certain degree of difficulty in proof procedure.In addition, corresponding dynamic stability controller method for designing is not proposed yet in above-mentioned document.
Summary of the invention
Deficiency to prior art; The present invention provides a kind of method for designing that is used for the multiple stable point NLS of dynamic stability control; Stability and chaotic Property Analysis from the synchrodyne group model of NLS; Utilize multiple stable point NLS time-frequency domain analytical approach to carry out design of Controller, resulting notional result can be avoided in the analysis of conventional electric power power system dynamic stability because the cumulative errors that linearization is introduced.
A kind of method for designing that is used for the multiple stable point NLS of dynamic stability control provided by the invention, its improvements are that said method comprises the steps:
1) sets up the synchrodyne group model, and confirm the definition of type gradient;
2) the class gradient property of judgement synchrodyne group model;
3) step 1) synchrodyne group model is carried out design of Controller;
4) controller with step 3) carries out power system dynamic stability analysis and control.
Wherein, the said synchrodyne group model of setting up of step 1):
dσ dt = η
Figure BDA00001685757200022
dy dt = Ay + Bf ( σ ) + α
Wherein, the transport function of linear segment is the matrix K (s) of m * m dimension:
K(s)=C T(A-sI) -1B
It is reduced to:
σ · = η
η · = ( - α 1 y 1 - α 2 y 2 ) sin σ + ( α 3 y 3 cos σ - α 4 sin σ cos σ )
y · 1 = α 5 - α 6 y 1 + α 7 y 2 + α 8 sin σ - - - ( 3 )
y · 2 = α 9 y 1 - α 10 y 2 + α 11 cos σ
y · 3 = - α 12 y 3 + α 13 sin σ
Wherein
A = - α 6 α 7 0 α 9 - α 10 0 0 0 - α 12 , B = α 8 0 α 11 0 0 - α 13
C = α 1 0 α 2 0 0 - α 3 , a = α 5 0 0 , f ( σ ) = cos σ - sin σ
Figure BDA00001685757200033
Wherein, being defined as of said type of gradient of step 1):
When model (1) t →+during ∞, it is separated
X ( t , X 0 ) = σ ( t , X 0 ) η ( t , X 0 ) y ( t , X 0 ) , X 0 = σ 0 η 0 y 0
Satisfy X (t, X 0) → c, then to be called be convergent to y;
If each of descriptive model (1) separated X (t, X 0) all restrain model (1) type of being gradient so.
Wherein, step 2) the class gradient property of said judgement synchrodyne group model comprises:
Suppose μ 1>=0, there is matrix P=P TMake assumed condition 1 in the lemma 2), 4), 5) and following time domain linear MATRIX INEQUALITIES condition
A T P + PA PB - A T C B T P - C T A - ( C T B + B T C ) < 0 - - - ( 5 )
Satisfy, then model (1) type of being gradient.
Wherein, step 3) is said carries out design of Controller to step 1) synchrodyne group model and comprises;
Choosing state feedback controller u=Ky, to carry out system calm, and system model is expressed as
d&sigma; dt = &eta;
Figure BDA00001685757200037
dy dt = Ay + Bf ( &sigma; ) + &alpha; + u
Wherein, u ∈ R mBe the control input, system transter becomes
K ~ ( s ) = C T ( A ~ - sI ) - 1 B
A=A+K wherein; K is a controller parameter.
Wherein, the method for designing of said controller parameter K comprises:
Suppose μ 1>=0, and the assumed condition in the lemma 2 4), 5) and following condition satisfy:
1. matrix
Figure BDA00001685757200041
is a diagonal matrix;
2. there is matrix W, Q=Q T>0 makes
QA T + AQ + W + W T B - QA T C - W T C B T - C T AQ - C T W - ( C T B + B T C ) < 0 - - - ( 8 )
Exist feedback controller u=Ky to make system's type of being gradient so, and controller parameter can be by K=WQ -1Draw.
Wherein, said type of gradient property is meant overall progressive stability.
Wherein, said lemma 2 is:
Make μ 1>=0, and following condition satisfies:
[1] matrix K (0) is a diagonal matrix;
[2] for all ω ∈ R, have Re [ 1 I&omega; K ( I&omega; ) ] > 0 ;
[3] lim &omega; &RightArrow; &infin; &omega; 2 Re [ 1 i&omega; K ( i&omega; ) ] > 0 ;
[4] function f (σ) has continuous second derivative, and at any interval (θ 1, θ 2) in all have f ' (σ) ≠ 0;
[5]
Figure BDA00001685757200045
The model of NLS (1) type of being gradient so.
With the prior art ratio, beneficial effect of the present invention is:
The present invention has considered the overall dynamic perfromance of synchrodyne group model under different system parameter and starting condition; Type of comprising gradient property and chaotic characteristic; And utilize the time-frequency domain method of multiple stable point NLS to carry out the synchronizing characteristics that feedback controller design guarantees synchronous unit, thereby guarantee the dynamic stability of whole electric system.Therefore linearization before the dynamic stability control method that the present invention utilized does not relate in the similar approach greatly reduces the cumulative errors that produces in this method.And the actual engineering to later has directive function.
The present invention utilizes the time-frequency domain disposal route of non-linear multiple stable point system, has provided synchrodyne category gradient sex determination theorem with the LMI form, and this criterion has utilized the MATLAB tool box to verify.
Description of drawings
Fig. 1 is σ provided by the invention (t), η (t), the time-domain-simulation figure of y (t).
Fig. 2 is σ provided by the invention (t), η (t), y (t), the time-domain-simulation figure of t ∈ [1000,1200].
Fig. 3 is y provided by the invention 1, y 2, y 3The phase space analogous diagram.
Fig. 4 is σ provided by the invention, y 1The phase plane analogous diagram.
Fig. 5 is η provided by the invention, y 1The phase plane analogous diagram.
Fig. 6 is σ provided by the invention (t), η (t), the time-domain-simulation figure of y (t).
Fig. 7 is η provided by the invention, y 2, y 3Phase space plot.
Fig. 8 is σ under the controller action provided by the invention (t), η (t), the time-domain-simulation figure of y (t).
Fig. 9 is a general flow chart of the present invention.
Embodiment
Do further to specify below in conjunction with the accompanying drawing specific embodiments of the invention.
The main thought of the present invention is to judge that system is whether stable earlier, if unstable, through CONTROLLER DESIGN it is controlled, and guarantees the stability of system.
The present invention at first utilizes KYP lemma that the frequency domain condition of class gradient property is transformed.KYP lemma is a basic conclusion of dynamic system analysis and the widespread use of FEEDBACK CONTROL field, and through the frequency domain inequality being converted into time domain linear MATRIX INEQUALITIES (LMI) of equal value, KYP lemma provides a kind of systematic analysis and comprehensive numerical analysis method.Based on resulting time domain linear MATRIX INEQUALITIES condition, the present invention further carries out design of Controller to closed-loop system and guarantees its type gradient property, and provides corresponding controller parametric solution method.
The method for designing process flow diagram of a kind of multiple stable point NLS that is used for dynamic stability control that concrete present embodiment provides is as shown in Figure 9, comprises the steps:
1) sets up the synchrodyne group model, and confirm the definition of type gradient;
Set up the synchrodyne group model:
d&sigma; dt = &eta;
Figure BDA00001685757200052
dy dt = Ay + Bf ( &sigma; ) + &alpha;
Wherein, the transport function of linear segment is the matrix K (s) of m * m dimension:
K(s)=C T(A-sI) -1B
It is reduced to:
&sigma; &CenterDot; = &eta;
&eta; &CenterDot; = ( - &alpha; 1 y 1 - &alpha; 2 y 2 ) sin &sigma; + ( &alpha; 3 y 3 cos &sigma; - &alpha; 4 sin &sigma; cos &sigma; )
y &CenterDot; 1 = &alpha; 5 - &alpha; 6 y 1 + &alpha; 7 y 2 + &alpha; 8 sin &sigma; - - - ( 3 )
y &CenterDot; 2 = &alpha; 9 y 1 - &alpha; 10 y 2 + &alpha; 11 cos &sigma;
y &CenterDot; 3 = - &alpha; 12 y 3 + &alpha; 13 sin &sigma;
Wherein
A = - &alpha; 6 &alpha; 7 0 &alpha; 9 - &alpha; 10 0 0 0 - &alpha; 12 , B = &alpha; 8 0 &alpha; 11 0 0 - &alpha; 13
C = &alpha; 1 0 &alpha; 2 0 0 - &alpha; 3 , a = &alpha; 5 0 0 , f ( &sigma; ) = cos &sigma; - sin &sigma;
Through given different parameter and initial value, the model of system (3) can demonstrate different dynamic behaviors.
In the present embodiment, suppose
Figure BDA00001685757200069
It is synchrodyne group model (1) zero load.Wherein Δ is the cycle of nonlinear function
Figure BDA000016857572000610
.
Definition 1: when the t of system (1) →+during ∞, the separating of system (1):
X ( t , X 0 ) = &sigma; ( t , X 0 ) &eta; ( t , X 0 ) y ( t , X 0 ) , X 0 = &sigma; 0 &eta; 0 y 0
Satisfy X (t, X 0) → c, then to be called be convergent to y;
Definition 2: if each of synchrodyne group model (1) nonlinear equation of descriptive system separated X (t, X 0) all restrain model (1) type of being gradient so.
The class gradient property of present embodiment is meant overall progressive stability.
2) the class gradient property of judgement synchrodyne group model;
Lemma 1: (KYP lemma) given satisfies det (jwI-A) ≠ 0, the matrix A ∈ R of w ∈ R N * m, matrix B ∈ R N * mAnd M=M T∈ R (n+m) * (n+m), and (A, B) controlled.So following two conclusion equivalences:
( jwI - A ) - 1 B I * M ( jwI - A ) - 1 B I &le; 0 , &ForAll; w &Element; R
2 ° exist matrix P=P T∈ R N * nMake
M + A T P + PA PB B T P 0 &le; 0
Even when (A, when B) uncontrollable, corresponding inequality also is of equal value.
KYP lemma has provided a kind of relation of equivalence between 2 ° of 1 ° of frequency domain inequality and the time domain conditions, and it is one of basic conclusion in dynamical system analysis, FEEDBACK CONTROL and the signal Processing research field.Because the character of dynamical system can be described by a series of inequality conditions in the frequency domain; So through KYP lemma; Can this type frequency domain condition be converted into LMI condition of equal value, thereby utilize corresponding instrument to study the robustness and the design of Controller problem of system easily.
Lemma 2: make μ 1>=0, and following condition satisfies:
[1] matrix K (0) is a diagonal matrix;
[2] for all ω ∈ R, have Re [ 1 I&omega; K ( I&omega; ) ] > 0 ;
[3] lim &omega; &RightArrow; &infin; &omega; 2 Re [ 1 i&omega; K ( i&omega; ) ] > 0 ;
[4] function f (σ) has continuous second derivative, and at any interval (θ 1, θ 2) in all have f ' (σ) ≠ 0;
[5]
Figure BDA00001685757200075
NLS (1) type of being gradient so.
Theorem 1: suppose μ 1>=0, there is matrix P=P TMake assumed condition 1 in the lemma 2), 4), 5) and following LMI condition
A T P + PA PB - A T C B T P - C T A - ( C T B + B T C ) < 0 - - - ( 5 )
Satisfy, then model (1) type of being gradient.
Proof:
At first through the condition 2 of KYP lemma to lemma 2) carry out equivalence and simplify.Because
sK(s)=C TA(A-sI) -1-C TB
Can obtain
Re [ 1 i&omega; K ( i&omega; ) ] = Re [ 1 ( i&omega; ) 2 i &omega;K ( i&omega; ) ]
= Re [ - 1 &omega; 2 ( C T A ( A - i&omega;I ) - 1 - C T B ) ]
The condition 2 of lemma 2 so) is equivalent to for all ω ∈ R and has
Re[C TA(A-iωI) -1-C TB]<0 (6)
In addition, by condition 3) and the form of K (s) can know that the exponent number difference of the molecule of each element and denominator is at most 1 among the transport function K (s).Therefore, can be known by KYP lemma that inequality (6) is set up, and if only if exists P=P TMake inequality (5) set up.Theorem must be demonstrate,proved.
With step 2) synchronous unit is judged, be chaos when judging the synchrodyne group model, then carry out step 3);
3) step 1) synchrodyne group model is carried out design of Controller;
When the model (1) of system has chaos or periodic oscillation when separating, i.e. during electrical network generation dynamic buckling, can on the basis of theorem 1, provide controller design method, thus the class gradient property (overall progressive stability) of the system of assurance.The present invention chooses state feedback controller u=Ky, and to carry out system calm.At this moment, system model can be expressed as
d&sigma; dt = &eta;
Figure BDA00001685757200084
dy dt = Ay + Bf ( &sigma; ) + &alpha; + u
Wherein, u ∈ R mBe the control input, system transter becomes
K ~ ( s ) = C T ( A ~ - sI ) - 1 B
Wherein
Figure BDA00001685757200087
K is a controller parameter.
The method for designing of controller parameter K comprises:
Theorem 2: suppose μ 1>=0, and the assumed condition in the lemma 2 4), 5) and following condition satisfy:
2. matrix
Figure BDA00001685757200088
is a diagonal matrix;
2. there is matrix W, Q=Q T>0 makes
QA T + AQ + W + W T B - QA T C - W T C B T - C T AQ - C T W - ( C T B + B T C ) < 0 - - - ( 8 )
Exist feedback controller u=Ky to make system's type of being gradient so, and controller parameter can be by K=WQ -1Draw, in other words, Q is through finding the solution the matrix that (8) formula draws, if (8) formula is separated, then have Q, and controller K can obtaining through Q.
4) controller with step 3) carries out power system dynamic stability analysis and control, guarantees power system stability.
Concrete, present embodiment is through given different parameter and initial value, and model (3) can demonstrate different dynamic behaviors, at first the different dynamic behavior of synchronous unit is carried out simulation analysis.In following emulation, unified initial value X (0)=[1.2,0.4,1.0,0.5,1.05] of getting T
Class gradient property:
The selective system parameter is following
α 1=1.2137,α 2=2.1106,α 3=7.8436,α 4=1.4764
α 5=1.0003,α 6=6.1205,α 7=0.0752,α 8=200.04
(10)
α 9=0.9037,α 10=20.0377,α 11=5.4428,α 12=1.1826
α 13=2.2242
The time-domain-simulation figure of model (3) with above-mentioned parameter (10) is as shown in Figure 1, and system is stable.
Chaotic characteristic:
Get parameter
α 1=1.1106,α 2=2.0871,α 3=-9.3000,α 4=1.0280
α 5=1.0003,α 6=6.1205,α 7=2.0352,α 8=2.0242
(11)
α 9=1.9137,α 10=2.0377,α 11=5.4428,α 12=4.1826
α 13=2.2242
At this moment, have time-domain-simulation curve such as Fig. 2 of the model (3) of parameter (11)-shown in Figure 5, this system is a chaos.
Have the Lyapunov index of parameter (11) drag (3) through calculating, obtain
λ 1=0.796,λ 2=0.000,λ 3=2.536
λ 4=4.472,λ 5=-19.986
Wherein there is positive Lyapunov index λ 1=0.796 has explained the chaotic characteristic of system.Owing to have under the situation of chaotic attractor the heavy losses that operation continuously can cause the destruction of synchronizing characteristics and cause valuable equipment in non-linear electric system, thereby the dynamic stability of electrical network caused serious threat.In addition, chaotic oscillation can cause the permanent harm of harmful transient state resonance, therefore for synchronous unit, how to avoid being extremely important of chaotic oscillation.In other words, for guaranteeing the dynamic stability of electric system, need the suitable controller of design that synchronous unit is controlled.Provide on the basis that system's initial value carries out simulation result above-mentioned, the following step will confirm the validity according to the dynamic stability controller that the present invention designed with simulation result.
In this implementation step, choose initial value X (0)=[2.2,4,1.0,1.5,2.5] again THas systematic parameter so
α 1=1.1106,α 2=2.1271,α 3=9.0000,α 4=1.0280
α 5=1.0003,α 6=6.1205,α 7=2.0352,α 8=200.0
α 9=-1.9122,α 10=0.0377,α 11=5.4428,α 12=20.0,α 13=2.2242
The dynamics of synchrodyne group model (3) describe by Fig. 6, Fig. 7.
Be not difficult to find out that from Fig. 6, Fig. 7 synchrodyne group model this moment (3) has the typical nonlinear chaotic characteristic, corresponding, by the electric system generation dynamic buckling of system (3) formation.Consider the closed-loop system characteristic under controller u=Ky effect below.Obviously, the condition 4 in the lemma 2 this moment), 5) all satisfy.Through finding the solution the LMI condition (8) in the theorem 2, can obtain following state feedback controller:
K = 0.9916 - 1.3074 - 0.2350 3.7681 - 4.9681 - 0.8930 - 0.1598 - 0.0408 16.0344 - - - ( 9 )
Thereby can know the class gradient property of the system that guaranteed by theorem 2.
Validity for checking gained notional result; Provided the system stability simulation curve of system under controller (9) effect among Fig. 8; Graphic result verifies again that also synchrodyne group model (3) has reached overall progressive stable, thereby has guaranteed the dynamic stability of electric system.
Need to prove that each alphabetical physical significance of the present invention is following:
σ, η, y corresponding to the physical parameter in the synchronous unit, are time dependent vectors, wherein vectorial y=[y 1, y 2, y 3] T
A, B, C are the known parameters matrixes in the synchrodyne group model, are constant matricess;
C is an arbitrary constant;
(η is about the cycle nonlinear function of σ in the synchrodyne group model σ) for f (σ),
Figure BDA00001685757200102
g;
α is the known parameters in the general synchrodyne group model;
U is the control input variable in the synchronous unit;
X 0Be σ 0, η 0, y 0The initial value column vector of forming, and have: X ( t , X 0 ) = &sigma; ( t , X 0 ) &eta; ( t , X 0 ) y ( t , X 0 ) , X 0 = &sigma; 0 &eta; 0 y 0 ;
σ 0, η 0, y 0It is respectively the initial value of σ, η, y;
Figure BDA00001685757200104
is respectively σ, η, y derivative in time;
α 113Be equation coefficient;
Figure BDA00001685757200105
is the transport function after the conversion;
Figure BDA00001685757200111
is the matrix after the conversion,
Figure BDA00001685757200112
Syst.Contr Lett.1996 is come from KYP lemma provided by the invention, 28:7-10.On the Kalman-Yakubovich-Popov lemma, Rantzer A.
Lemma 2 provided by the invention comes from 1992.; Non-local methods for endulum-like feedback systems (Teubner-Texte zur Mathematik Bd.132; B.G.; Teubner Stuttgart-Leipzig), Leonov G A, Reitmann V and Smirnova VB.
Should be noted that at last: above embodiment is only in order to technical scheme of the present invention to be described but not to its restriction; Although the present invention has been carried out detailed explanation with reference to the foregoing description; Under the those of ordinary skill in field be to be understood that: still can specific embodiments of the invention make amendment or be equal to replacement; And do not break away from any modification of spirit and scope of the invention or be equal to replacement, it all should be encompassed in the middle of the claim scope of the present invention.

Claims (8)

1. a method for designing that is used for the multiple stable point NLS of dynamic stability control is characterized in that said method comprises the steps:
1) sets up the synchrodyne group model, and confirm the definition of type gradient;
2) the class gradient property of judgement synchrodyne group model;
3) step 1) synchrodyne group model is carried out design of Controller;
4) controller with step 3) carries out power system dynamic stability analysis and control.
2. the method for designing of multiple stable point NLS as claimed in claim 1 is characterized in that, the said synchrodyne group model of setting up of step 1):
d&sigma; dt = &eta;
Figure FDA00001685757100012
dy dt = Ay + Bf ( &sigma; ) + &alpha;
Wherein, the transport function of linear segment is the matrix K (s) of m * m dimension:
K(s)=C T(A-sI) -1B
It is reduced to:
&sigma; &CenterDot; = &eta;
&eta; &CenterDot; = ( - &alpha; 1 y 1 - &alpha; 2 y 2 ) sin &sigma; + ( &alpha; 3 y 3 cos &sigma; - &alpha; 4 sin &sigma; cos &sigma; )
y &CenterDot; 1 = &alpha; 5 - &alpha; 6 y 1 + &alpha; 7 y 2 + &alpha; 8 sin &sigma; - - - ( 3 )
y &CenterDot; 2 = &alpha; 9 y 1 - &alpha; 10 y 2 + &alpha; 11 cos &sigma;
y &CenterDot; 3 = - &alpha; 12 y 3 + &alpha; 13 sin &sigma;
Wherein
A = - &alpha; 6 &alpha; 7 0 &alpha; 9 - &alpha; 10 0 0 0 - &alpha; 12 , B = &alpha; 8 0 &alpha; 11 0 0 - &alpha; 13
C = &alpha; 1 0 &alpha; 2 0 0 - &alpha; 3 , a = &alpha; 5 0 0 , f ( &sigma; ) = cos &sigma; - sin &sigma;
Figure FDA000016857571000111
3. the method for designing of multiple stable point NLS as claimed in claim 1 is characterized in that, being defined as of the said type gradient of step 1):
When model (1) t →+during ∞, it is separated
X ( t , X 0 ) = &sigma; ( t , X 0 ) &eta; ( t , X 0 ) y ( t , X 0 ) , X 0 = &sigma; 0 &eta; 0 y 0
Satisfy X (t, X 0) → c, then to be called be convergent to y;
If each of descriptive model (1) separated X (t, X 0) all restrain model (1) type of being gradient so.
4. the method for designing of multiple stable point NLS as claimed in claim 1 is characterized in that step 2) the class gradient property of said judgement synchrodyne group model comprises:
Suppose μ 1>=0, there is matrix P=P TMake assumed condition 1 in the lemma 2), 4), 5) and following time domain linear MATRIX INEQUALITIES condition
A T P + PA PB - A T C B T P - C T A - ( C T B + B T C ) < 0 - - - ( 5 )
Satisfy, then model (1) type of being gradient.
5. the method for designing of multiple stable point NLS as claimed in claim 1 is characterized in that, step 3) is said carries out design of Controller to step 1) synchrodyne group model and comprise;
Choosing state feedback controller u=Ky, to carry out system calm, and system model is expressed as
d&sigma; dt = &eta;
Figure FDA00001685757100024
dy dt = Ay + Bf ( &sigma; ) + &alpha; + u
Wherein, u ∈ R mBe the control input, system transter becomes
K ~ ( s ) = C T ( A ~ - sI ) - 1 B
Wherein
Figure FDA00001685757100027
K is a controller parameter.
6. the method for designing of multiple stable point NLS as claimed in claim 5 is characterized in that, the method for designing of said controller parameter K comprises:
Suppose μ 1>=0, and the assumed condition in the lemma 2 4), 5) and following condition satisfy:
1. matrix
Figure FDA00001685757100028
is a diagonal matrix;
2. there is matrix W, Q=Q T>0 makes
QA T + AQ + W + W T B - QA T C - W T C B T - C T AQ - C T W - ( C T B + B T C ) < 0 - - - ( 8 )
Exist feedback controller u=Ky to make system's type of being gradient so, and controller parameter can be by K=WQ -1Draw.
7. like the method for designing of claim 1,3,4 or 6 arbitrary described multiple stable point NLSs, it is characterized in that said type of gradient property is meant overall progressive stability.
8. the method for designing of multiple stable point NLS as claimed in claim 5 is characterized in that, said lemma 2 is:
Make μ 1>=0, and following condition satisfies:
[1] matrix K (0) is a diagonal matrix;
[2] for all ω ∈ R, have Re [ 1 I&omega; K ( I&omega; ) ] > 0 ;
[3] lim &omega; &RightArrow; &infin; &omega; 2 Re [ 1 i&omega; K ( i&omega; ) ] > 0 ;
[4] function f (σ) has continuous second derivative, and at any interval (θ 1, θ 2) in all have f ' (σ) ≠ 0;
[5]
Figure FDA00001685757100034
The model of NLS (1) type of being gradient so.
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