CN109901633A - Linear feedback gain scheduling control method based on complex mode - Google Patents
Linear feedback gain scheduling control method based on complex mode Download PDFInfo
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Abstract
The invention provides a linear feedback gain scheduling control method based on a complex mode. The method comprises the following steps: s1, establishing a dynamic model of the controlled object, and acquiring the complex modal characteristics of the controlled object; s2, performing modal reduction on the dynamic model by using a modal truncation method, and designing a linear feedback controller in a complex space based on a linear Riccati equation; and S3, considering the saturation condition of the controller, designing a gain scheduling controller, and utilizing the efficiency of the controller to the maximum extent under the condition of ensuring the stability of a closed loop. The invention solves the problem of controller design when the controlled object adopts complex modal representation.
Description
Technical field
The present invention relates to Study on Vibration Control fields, specifically design a kind of linear feedback gain scheduling control based on complex mode
Method processed.
Background technique
Vibration control problem is one of multiple branch of engineering common problems, especially large-size pliable structure.By
To internal or external load excitation when, such system generates serious vibration problem, influences the system accuracy, service life, tired
The performances such as labor and safety.Therefore, designing effective controller is one of the major issue that engineer faces.Nearly tens
Nian Lai, for the vibration control of flexible structure, scholar has carried out numerous studies and has proposed various control theory, such as directly speed
Degree control, strain feedback control, optimum control, sliding formwork control (SMC), Independent modal system control and robust control etc..
However, overwhelming majority control theories are directed to the system design for being under the jurisdiction of the real space at present, i.e., state space and its it is
The system that matrix number belongs to the real space.And for such as rotating containing Structural Damping Systems or for considering the system of gyroscopic effect
The oscillation crosswise of shafting system, high-speed flexible push and pull system etc. are belonged to multiple using its state space after modal coordinate characterization system
Space.It is few for the Study on Vibration Control of complex mode system at present.
Summary of the invention
It is an object of the invention to propose a kind of linear feedback gain scheduling control method based on complex mode, solve
Controlled device using complex mode characterize when controller design the problem of.
A kind of linear feedback gain scheduling control method based on complex mode of the present invention comprising the steps of:
S1 establishes the kinetic model of controlled device, obtains the complex mode feature of controlled device；
S2 carries out modal reduction method to kinetic model using modal truncation, based on the multiple sky of linear Riccati equation design
Between under linear feedback controller；
S3 considers the case where controller is saturated, designing gain scheduling controller, it was demonstrated that the closed loop containing gaing scheduling control
The stability of system.
Compared with the prior art, the present invention has the following advantages:
The present invention is directed to the system indicated in complex space, proposes and devises a kind of linear feedback gain scheduling control
Controller theoretical extension to complex space can be used for the system using complex mode characterization by device.Basic thought is divided in complex space
The complex mode feature for analysing controlled device designs linear feedback controller based on complex mode, and considers the case where controller is saturated, root
The state of system will be divided into the ellipsoid collection of multiple nestings according to domain of attraction, design a kind of gaing scheduling control, to reach maximum
Utilize the purpose of controller efficiency.
Detailed description of the invention
Fig. 1 flow chart of the method for the present invention
Fig. 2 nest set schematic diagram
Specific embodiment
The specific embodiments are described below with reference to the accompanying drawings.
The present invention mainly has three steps, as shown in Figure 1, being below described in detail as follows detailed process:
S1: establishing the kinetic model of controlled device, and analysis obtains the complex mode feature of controlled device under complex space；
Sub-step S11: the kinetic model of controlled device is established；
The mathematical model of controlled device is established based on Hamiton's principle, governing equation is as follows
In formula, δ (N × 1, N degree of freedom in system number) is degree of freedom in system；M (N × N), K (N × N) be respectively quality and just
Matrix is spent, is symmetrical matrix；C (N × N) is the sum of damping matrix and gyroscopic matrix.The damping matrix when ignoring, Matrix C are anti-
Symmetrical matrix.P (N × 1) indicates that gyro force vector, F (N × 1) indicate external applied load vector.
Sub-step S12: the complex mode feature of controlled device is extracted；
System equation is expressed as state space form
In formula
The conjugated system of formula (2) is
SubscriptTExpression turns order,HIndicate that conjugation turns order.
Above system and the Free Vibration Equations of its conjugated system are represented by
Asterisk indicates the conjugation of scalar, vector or matrix.The solution of above-mentioned two system is represented by
In formula, Φ is left eigenvector, and Ψ is right feature vector, and above formula is substituted into formula (5), can be obtained
Matrix AδIt is asymmetric, therefore eigenvalue λ and feature vector are represented by complex conjugate pair
In formula
Matrix can piecemeal be
Multiple left eigenvector and multiple right feature vector meet following orthogonality condition
arAnd brFor scalar, meet following formula
Definition modal matrix Φ is transition matrix
Element in vector x is real number, therefore modal coordinate can be divided into the subvector of two conjugation.
Formula (13) are substituted into formula (2), and consider formula (11), system equation is
In formula
Step S2: modal reduction method is carried out to kinetic model using modal truncation, is designed based on linear Riccati equation
Linear feedback controller under complex space；
Sub-step S21: modal reduction method is carried out to kinetic model using modal truncation, is write as the form of state space；
Under normal circumstances, system is mainly influenced by lower mode, therefore only retains preceding 2n rank mode Φc, and ignore outer
Load.System representation is
In formula
Sub-step S22: the linear feedback controller under complex space is designed based on linear Riccati equation；
For positive definite matrix Q, existence anduniquess steady-state solution P (2n × 2n) makes
PHA+AHP-PBBHP+Q=0 (18)
State feedback controller with following form
U=-BHPx (19)
Meet u*=u, and stablize system (16).
Step S3: the case where considering controller saturation, designing gain scheduling controller, in the case where guaranteeing closed-loop stabilization
Maximally utilise controller efficiency.
Sub-step S31: defining a series of domains of attraction, designs following gaing scheduling control；
In the case where making moving vector u saturation, nonlinear system is represented by
Function sat (u) indicates actuator saturation, and dimension is m × 1, component sat (uj) such as give a definition
In formula, uj maxThe extreme value inputted for j-th, function sign () indicate sign function.
Low gain control law is represented by
uL=FL(ε)x (22)
In formula
FL(ε) :=- BHPε,ε∈(0,1] (23)
In formula, PεFor unique steady-state solution of following equation
PHA+AHP-PBBHP+Qε=0 (24)
For any ε > 0, domain of attraction may be defined as ellipsoid collection
ε(Pε){x∈2n:xHPεx≤c} (25)
Wherein, defining c > 0 is
Consider set
ε={ ε0,ε1,...,εN}εi∈+ and εi<εi+1(i=0,1 ..., N) (27)
In formula,For positive integer.
As shown in Fig. 2, corresponding ellipsoid collection is
Based on the ellipsoid collection of above-mentioned nesting, defining nested controller is
Feedback oscillator with time-varying
And
tiIndicate that system mode reaches i-th of ellipsoid collection ε (Pi) boundary at the time of, λmin() indicates positive definite matrix characteristic value
The minimum value of real part.
Sub-step S32: the stability of the closed-loop system containing gaing scheduling control is proved；
Closed-loop system containing gaing scheduling control shown in formula (29) is represented by
Choose Li Yapu promise husband's function
V(Pi, t) and=x (t)HPix(t) (33)
V(Pi, t) be for the derivative of time
Therefore for any t ∈ [ti,ti+1)
For any time t ∈ [ti,ti+1), such as lower inequality can be obtained
Work as t=ti, factor alphaiIt is zero, system mode meets
I.e.
Introduce intermediate variable
Choose intermediate variable Li Yapu promise husband's function be
According to formula (36), can obtain
Therefore, intermediate variable is located in domain of attraction, and system is not up to saturated at this time, i.e.,
And
Therefore the controller input that formula (29) indicates meets constraint condition.
The above description is only a preferred embodiment of the present invention, is not intended to limit the scope of the invention, all at this
Under the inventive concept of invention, using equivalent structure transformation made by description of the invention and accompanying drawing content, or directly/use indirectly
It is included in other related technical areas in scope of patent protection of the invention.
Claims (4)
1. a kind of linear feedback gain scheduling control method based on complex mode, which is characterized in that comprise the steps of:
S1 establishes the kinetic model of controlled device, obtains the complex mode feature of controlled device；
S2 carries out modal reduction method to kinetic model using modal truncation, based under linear Riccati equation design complex space
Linear feedback controller；
S3 considers the case where controller is saturated, designing gain scheduling controller, it was demonstrated that the closed-loop system containing gaing scheduling control
Stability.
2. the linear feedback gain scheduling control method based on complex mode as described in claim 1, which is characterized in that described
Step S1 the following steps are included:
S11 establishes the kinetic model of controlled device are as follows:
In formula, δ (N × 1, N are degree of freedom in system number) is degree of freedom in system；M (N × N), K (N × N) is respectively quality and rigidity
Matrix is symmetrical matrix；C (N × N) is the sum of damping matrix and gyroscopic matrix；P (N × 1) indicates gyro force vector, F (N × 1)
Indicate external applied load vector；
S12 extracts the complex mode feature of controlled device,
System equation is expressed as state space form
In formula
State transformation is carried out by complex eigenvector Φ, system equation is
In formula, Π is complex matrix, and λ is characterized value vector, and
。
3. the linear feedback gain scheduling control method based on complex mode as described in claim 1, which is characterized in that described
Step S2 the following steps are included:
S21 carries out modal reduction method to kinetic model using modal truncation, is write as the form of state space
In formula
U is control amount；
S22, the linear feedback controller under complex space are as follows:
In formula, u is control amount, and Q is positive definite matrix, and P is steady-state solution.
4. the linear feedback gain scheduling control method based on complex mode as described in claim 1, which is characterized in that described
Step S3 the following steps are included:
S31, designing gain scheduling controller are as follows:
In formula,For nest set, αiFor gain coefficient；
S32, it was demonstrated that the stability of closed-loop system: liapunov function V (Pi, t) derivative of time is met
In formula, ηiFor positive real number, tiIndicate that system mode reaches i-th of ellipsoid collectionAt the time of boundary.
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| CN108170967A (en) * | 2018-01-05 | 2018-06-15 | 哈尔滨工程大学 | A kind of elastic beam structure PVDF modal sensor generation methods |
| CN108158581A (en) * | 2017-12-14 | 2018-06-15 | 燕山大学 | A kind of multi-channel nerve signal analysis method based on phase pattern complexity |
| CN109060292A (en) * | 2018-06-01 | 2018-12-21 | 华南理工大学 | Consider the wind shake implementation method of double coupled systems of radio frequency plasma CVD test |
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Patent Citations (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| US7028912B1 (en) * | 2003-09-03 | 2006-04-18 | Howard Rosen | Programmable thermostat incorporating a display screen selectively presenting system modes that includes a simple mode |
| CN102129520A (en) * | 2011-03-16 | 2011-07-20 | 哈尔滨工程大学 | Self-adaptive time domain identification method of dynamic load |
| CN108158581A (en) * | 2017-12-14 | 2018-06-15 | 燕山大学 | A kind of multi-channel nerve signal analysis method based on phase pattern complexity |
| CN108170967A (en) * | 2018-01-05 | 2018-06-15 | 哈尔滨工程大学 | A kind of elastic beam structure PVDF modal sensor generation methods |
| CN109060292A (en) * | 2018-06-01 | 2018-12-21 | 华南理工大学 | Consider the wind shake implementation method of double coupled systems of radio frequency plasma CVD test |
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