CN104503239B - A kind of controller design method of Lur ' e type differential inclusions - Google Patents
A kind of controller design method of Lur ' e type differential inclusions Download PDFInfo
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- CN104503239B CN104503239B CN201410780789.7A CN201410780789A CN104503239B CN 104503239 B CN104503239 B CN 104503239B CN 201410780789 A CN201410780789 A CN 201410780789A CN 104503239 B CN104503239 B CN 104503239B
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Abstract
The invention discloses a kind of controller design method of Lur ' e type differential inclusions, according to controller requirement to be designed, differential inclusions are built, reduced dimension observer is designed, the design of device is controlled based on reduced dimension observer.State of the feedback control based on observer of the present invention, can save the sensor in system, while the reliability of controller can also be improved；When constructing observer, Design of Reduced-Order Observers for Lipschitz technology is make use of, can both have been saved cost, in turn simplify the complexity of system, can be applied to the design such as wind generator system with less integrator and whole control system is simpler.
Description
Technical field
The present invention relates to a kind of design method of controller, and in particular to a kind of Lur ' e types based on reduced dimension observer are micro-
The design method of controller of the subpackage containing system.
Background technology
In control engineering field, particular system can be described using equation, and solves controller, to realize setting for system
Meter.For example, the rotor-support-foundation system of direct current generator can be built into by the metal cables of top rotary table, lower rotary table and the upper and lower rotating disk of connection
The system that line is constituted, because cable contains stiction, the system is actually a class Lur ' e type differential inclusions.
Lur ' e type systems are the special nonlinear systems of a class, from last century the fifties, and Lur ' e type systems are obtained
Sufficiently research, exists wherein foremost achievement is exactly former Soviet Union mathematician Lur ' e A.《Some Nonlinear Problem
in the Theory of Automatic Control》Conclusion in one book on superstability.With the reach of science and
That studies gos deep into, into after 21st century, and Lur ' e type differential inclusions start the extensive concern for obtaining controlling boundary, and
The characteristics of system is maximum is that non-linear partial contains set value function.Lur ' e type differential inclusions have extensive engineering
Application background and by numerous studies.In the research of the dynamic system with Coulomb frictional force, Pfeiffer F. and
Hajek M. exist《Philosophical Transactions: Physical Sciences and Engineering》（Wise man
Learn journal：Natural science and engineering science）（The 1651st phase of volume 338 in 1992, page 503-page 517）On " the Stick- that delivers
slip motions of turbine blade dampers”（The stick-slip of turbo blade damping）With Juloski A.
Et al.《Control Engineering Practice》（Control engineering practice）（Volume 12 the 10th phase, the 1241st in 2004
- page 1252 of page）In " the Data-based hybrid modelling of the component placement that deliver
process in pick-and-place machines”（Mixing based on data during pickup-mounting machine attachment is built
Mould）And Mihajlovic N. et al. exist《Nonlinear Dynamics》（Nonlinear kinetics）（2006 volume 46
3 phases, page 273-page 291）In " the Friction-induced limit cycling in flexible rotor that deliver
systems: An experimental drillstring set-up”（The limit caused by friction in flexible rotor system
Ring：One experimental drill string device）.At the same time, in the circuit system containing ideal crystal diode, linear complementary system
Deng being also widely used in field.
When controll plant is controllable, it is necessary to carry out measuring state variable using sensor to form feedback, still, due to being permitted
Many intermediatenesses are difficult or immeasurability, generally using the input quantity and output quantity of controll plant, by state observer come weight
Structure problem, controller is designed based on state observer.
Currently, the research for Lur ' e type differential inclusions is concentrated mainly on the design of observer, according to rendezvous value letter
Several differences, research method is also not quite similar.Such as Osorio M. and Moreno J. exist《45th IEEE decision-making
Collection of thesis with controlling meeting》（2006, page 5400-page 5405）On " the Dissipative design of that deliver
observers for multivalued nonlinear systems”（The dissipativeness observer of multivalue nonlinear system is set
Meter）In article, the method for being namely based on dissipativeness designs the observer of Lur ' e type differential inclusions.And Doris A. etc.
People exists《IEEE Transactions on Control Systems Technology》（IEEE control system technology proceedings）
（The 6th phase of volume 16 in 2008, page 1323-page 1332）On " the Observer designs for that deliver
experimental non-smooth and discontinuous systems”（The observation of the discontinuous experimental system of Non-smooth surface
Device is designed）And Brogliato B. and Heemels W. exist《IEEE Transactions on Automatic
Control》（IEEE automatically controls proceedings）（The 8th phase of volume 54 in 2009, page 1996-page 2001）On deliver
“Observer design for Lur’e systems with multivalued mappings:a passivity
approach”（The Design of Observer of Lur ' the e type systems of multivalued mappings：A kind of passivity-based method）, both utilize positive reality
Property method design Lur ' e type differential inclusions observer.
However, in actual applications, it is necessary to realize the design of controller.How to make whole control system more simple, be
The problem of this area needs to solve.
The content of the invention
The goal of the invention of the present invention is to provide a kind of controller of Lur ' the e type differential inclusions based on reduced dimension observer
Design method, to simplify the complexity of system.
To achieve the above object of the invention, the technical solution adopted by the present invention is：A kind of control of Lur ' e type differential inclusions
Device design method processed, it is characterised in that comprise the following steps：
(1) according to system to be designed, quantity of state, input quantity and the output quantity of system is provided, is by system constructing
In formula,, it is the quantity of state of system, n is the dimension of system mode, and R is set of real numbers,, it is system
Input quantity, m is the dimension of system input,, it is the output quantity of system, q is the dimension of system output；
It is set value function, r is gathers the dimension of value function, and t represents the time；It isOutput, A, G, B, H, C are constant value
Matrix, and A dimension is that n × n, G dimension are that n × r, B dimension are that n × m, H dimension are that r × n, C dimension are q × n；
(2) reduced dimension observer is designed：
1）Find out the matrix for meeting following equalitiesP：
,
In formula,,,,,A、C、G、H、n、q、r
It is identical with step (1)；
2）Do block form as follows：
,,,,,,,
In formula,x 1∈Rq,,,,,
,；
3）Take；
4）The reduced dimension observer form for constructing Lur ' e type differential inclusions is as follows：
(3) design of device is controlled based on reduced dimension observer：
1）By solution such as lower inequality, feedback oscillator is designedK 2 Value：
In formula,,,It is r × r diagonal matrix, and=
diag（λ 1, λ 2 ... λ r）, λi>0, i=1,2 ... r；
2）Done and fed back using observer, the final form for providing controller is：
,。
Because above-mentioned technical proposal is used, the present invention has following advantages compared with prior art：
1. because designed feedback control is the state based on observer, therefore the sensor in system can be saved,
The reliability of controller can also be improved simultaneously；
2. construct observer when, make use of Design of Reduced-Order Observers for Lipschitz technology, can with less integrator and entirely
Control system is simpler, has both saved cost, in turn simplify the complexity of system.
Brief description of the drawings
Fig. 1 is the construction Simplified analysis figure of the rotor-support-foundation system in embodiment；
Fig. 2, Fig. 3, Fig. 4 are the state x that closed-loop system after feedback control is carried out in embodiment1、x2、x3Response curve；
Fig. 5 and Fig. 6 are the error re2 and re3 between original system and reduced dimension observer system.
Wherein：1st, direct current generator；2nd, top rotary table；3rd, lower rotary table；4th, metal cable.
Embodiment
Below in conjunction with the accompanying drawings and embodiment the invention will be further described：
Embodiment one：The present embodiment is directed to the rotor-support-foundation system of direct current generator, and its controller is designed based on reduced dimension observer.
The specific configuration of rotor-support-foundation system is shown in that provided in Fig. 1, figure is Simplified analysis figure, the driving top rotary table 2 of direct current generator 1,
Lower rotary table 3 is driven through metal cable 4.Wherein,WithThe angular speed of top rotary table 2 and lower rotary table 3 is represented respectively,
WithThe torque produced by the frictional force of top rotary table 2 and lower rotary table 3 is represented respectively,uIt is the voltage of direct current generator.Rotor
The operation mechanism of system is that top rotary table 2 and lower rotary table 3 are connected to drive after electricity on metal cable 4, direct current generator 1
Rotating disk 2 is first rotated, in the presence of metal cable 4, and lower rotary table 3 also can be rotated and then.Gained knowledge according to power, by
Contain stiction in metal cable, therefore the system is actually a class Lur ' e type differential inclusions.If by direct current generator
Voltage regard the input of system as, then the control problem formulation of the system is, the input of design system causes turning for upper lower rotary table
Speed reaches unanimously.
Controller design is comprised the following steps that：
1. by the method for System Discrimination, the mathematical modeling for providing rotor-support-foundation system is as follows：
In formula,
,,
,,,
It is used as state, x1It is the differential seat angle of top rotary table and lower rotary table, x2、x3It is upper turn respectively
The angular speed of disk and the angular speed of lower rotary table.
2. corresponding observer is gone out according to reduced dimension observer Theoretical Design.
By solving equation：
,
,
Obtain, take, other are according to matrix
Piecemeal calculated, then.The numeric form of the reduced dimension observer of rotor-support-foundation system is：
3. design controller.
By solving inequality：
So as to obtain controller gain, controlled with this
Device processed, i.e.,, that is, obtain。
As shown in Figure 2, Figure 3, Figure 4, respectively be carry out feedback control after closed-loop system state x1、x2、x3Response it is bent
Line.Fig. 5 and Fig. 6 are then the error re2 and re3 between original system and reduced dimension observer system.It can clearly see from figure, close
The state x of loop system1、x2、x3Asymptotic convergence is in 0 in 1 second, and error re2 and re3 then in 0.6 second asymptotic convergence in 0.
I.e. controller design is that effective, all state finally all converges to equalization point, the final Asymptotic Stability of original system.By this reality
The method for applying example, can be such that the rotor velocity of top rotary table and lower rotary table reaches unanimously, and now the moment of torsion on metal cable is 0,
The mechanical stress of metal cable when therefore can reduce system work.
In systems in practice, such as wind-powered electricity generation field, often there is gear-box between wind turbine and generator, be used as wind turbine
Group fault rate highest part, its torsion problem is a key reason.The method of the present invention can also be applied to the field, be
Greatly improve Wind turbines service life and provide possible.
Claims (1)
1. a kind of controller design method of the rotor-support-foundation system of direct current generator, based on Design of Reduced-Order Observers for Lipschitz, it is characterised in that straight
The rotor-support-foundation system of stream motor is reduced to DC motor Driver top rotary table, and lower rotary table is driven through metal cable,WithRepresent respectively
The angular speed of top rotary table and lower rotary table,WithProduced by the frictional force for representing top rotary table and lower rotary table respectively
Torque, the input of design system causes the rotating speed of upper lower rotary table to reach that unanimously design method comprises the following steps：
(1) by the method for System Discrimination, the mathematical modeling for providing rotor-support-foundation system is as follows：
In formula,, it is the quantity of state of system, dimension n=3, x of system mode1It is top rotary table and lower turn
The differential seat angle of disk, x2、x3It is the angular speed of top rotary table and the angular speed of lower rotary table respectively；uIt is the input quantity of system, represents direct current
The voltage of machine,, m is the dimension of system input,, it is the output quantity of system, q is the dimension of system output；
It is set value function, r is gathers the dimension of value function, and t represents the time；It is's
Output, A, G, B, H, C are constant value matrixes, and A dimension be n × n, G dimension be n × r, B dimension be n × m, H dimension
It is that r × n, C dimension are q × n；
(2) reduced dimension observer is designed：
1）Find out the matrix for meeting following equalitiesP：
,
In formula,,,,,A、C、G、H, n, q, r and step
Suddenly (1) is identical；
2）Do block form as follows：
,,,,,,,
In formula,x 1∈Rq,,,,, ,；
3）Take；
4）The reduced dimension observer form for constructing rotor-support-foundation system is as follows：
(3) design of device is controlled based on reduced dimension observer：
1）By solution such as lower inequality, feedback oscillator is designedK 2 Value：
In formula,,,It is r × r diagonal matrix, and=diag（λ
1, λ 2 ... λ r）, λi>0, i=1,2 ... r；
2）Done and fed back using observer, the final form for providing controller is：
,。
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