CN101776865A - Method for uniformly controlling mechanical system by utilizing differential inclusion - Google Patents

Abstract

Aiming at solving the shortcoming of the existing control technology for treating a differential inclusion system, the invention provides a control method directing at a feedback linear differential inclusion system. In the method, a Lyapunov equation is solved to construct a control Lyapunov function of a single-input differential inclusion system and design a continuous state feedback control law to ensure the global asymptotic stability of a closed-loop system. In order to achieve the purpose, the method which utilizes a differential inclusion system to uniformly control a mechanical system comprises the following specific steps: 1) firstly, identifying the model of the differential inclusion system; and 2), secondly, constructing a control Lyapunov function of a single-input differential inclusion system by a controller through solving the Lyapunov equation, designing the continuous state feedback control law to ensure the global asymptotic stability of the closed-loop system, carrying out D/A conversion, and then outputting to an actuator and acting on an controlled object until system nominal performance meeting engineering requirements is obtained.

Description

A kind of differential that adopts comprises the method that mechanical system is unified to control

Technical field

What the present invention relates to is a kind of method that is used for the Engineering Control technical field, specifically is that a kind of differential that adopts comprises the method that mechanical system is unified to control.

Background technology

Many engineering fields all need to comprise with differential the motion of the system of research, people such as Mazumder S.K. are in " IEEE Transactions on Power Electronics " (IEEE magazine) (the 16th the 2nd phase of volume of calendar year 2001, the 201-216 page or leaf) " Theoretical and experimental investigation of thefast-and slow-scale instabilities of a DC-DC converter " (the instable theoretical and experimental study of DC-DC transducer speed) delivered on, obtain because the DC-DC switch converters is in operation, its circuit topological structure is randomly changing in time, makes it become a typical sectionally smooth nonlinear kinetics system.And for example people such as RossI.M. was the collection of thesis of control academic conference " the 42nd U.S. electro-engineering association decision-making with " (2003, the 2210-2215 page or leaf) " the Unified computational framework for real-timeoptimal control " that delivers on (the unified calculation framework of optimum control in real time), proposing to act on behalf of one more the controlled dynamic system each agency dynamically, all available differential comprises describes its motion.People such as Johansson K.H. are by Unbehauen editor's " encyclopedia of life-support system " H.2004 year, on " Modeling ofhybrid systems " (the hybrid system model) delivered, the mechanical system that proposes to have gearing need comprise with differential describes its motion.In view of differential comprises the popularity of descriptive system, so entered since 21 century, this description is controlled the attention on boundary once more, and becomes one of focus of research.

People such as Goebel R. were in " IEEE Transactions on Automatic Control " (IEEE magazine) (2006 the 51st the 4th phases of volume in 2006, " Conjugate convexLyapunov functions for dual linear differential lnclusions " (dual linear differential comprises the conjugate convex cone Li Yapuluofu function of system) of delivering the 661-666 page or leaf), use the stability that convextiry analysis research linear differential comprises system and its antithesis.Hu is in " Automatica " (robotization magazine) (2007 the 43rd the 4th phases of volume, the 685-692 page or leaf) " the Nonlinear control design for linear differential inclusions via convexhull of quadratics " that delivers on (comprising gamma controller) by convex cone quadratic function design linear differential, utilize convex cone quadratic function structure nonlinear state FEEDBACK CONTROL rule, proposed the method for designing that the robust stabilizing linear differential comprises the gamma controller of system.Write by Isidori A., Springer Verlag publishing house, if nineteen ninety-five is published monograph " Nonlinear Control Systems " (nonlinear control system), proposes affine nonlinear system

For any x ∈ R ⁿ, rank r=n, and distribution G=span (g (x)) relatively is involutory, exists a differomorphism this nonlinear system to be changed into the system of feedback linearization so.This quasi-nonlinear system plays an important role in the control field.The differential of feedback linearization comprises the popularization that system is the feedback linearization system, because its structure is simpler, can be used for representing the system of many reality again, for example mechanical system, pneumatic system and flight control system etc. are studied and are had very important significance so this class differential is comprised system.But existing technology only comprises system to differential in theory to be analyzed, and perhaps analog differentiation comprises the model of system in the laboratory.Also there is the differential of on commercial production, attempting using to comprise system, can only controls single object by a controller but these differential comprise system.If be applied in the complicated control system, much more very the design of that controller will and be not easy to control, and not reach the effect of accurate control.Distance comprises system with differential and really is implemented in self industry and goes to exist great gap like this.Because no matter how outstanding method if can't combine with commercial production, just can't produce economic benefit, can not promote the development of society.

Summary of the invention

The objective of the invention is to comprise the control technology deficiency of system at existing processing differential, a kind of control method that comprises system at the feedback linearization differential is proposed, by separating Li Yapuluofu (Lyapunov) equation, this class differential of the single input of structure comprises control Li Yapuluofu (Lyapunov) function of system, and design makes the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system.Control Li Yapuluofu (Lyapunov) function construction method that comprises system by single this class differential of input, obtain control Li Yapuluofu (Lyapunov) function that many these class differential of input comprise system, and design makes this class differential of many inputs comprise the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system.During the actual motion control system,, affact controlled device, meet the nominal performance of the system of engine request until acquisition by exporting actuator to after the D/A conversion.

For reaching described purpose, the present invention is a kind of to be adopted differential to comprise the method that mechanical system is unified to control to comprise following concrete steps:

1) at first picking out the model that differential comprises system is

Here co represents the convex closure of a set, x ∈ R ⁿ, u ∈ R ^mBe respectively state and input, F _i(x) ∈ R ^{L * 1}, G _i(x) ∈ R ^{L * m}All be continuous function, and F _i(0)=0, for any x ∈ R ⁿ, G _i(x) be the row full rank, A, B are following Bu Lunuo Paderewski (Brunovsky) canonical form

Here r ₁+ r ₂+ ... + r _k=n;

2) secondly by separating Li Yapuluofu (Lyapunov) equation, controller structure list input differential comprises control Li Yapuluofu (Lyapunov) function of system, and design can make the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system, by exporting actuator to after the D/A conversion, affact controlled device, meet the nominal performance of the system of engine request until acquisition;

3) import control Li Yapuluofu (Lyapunov) function construction method that differential comprises system by list, obtain control Li Yapuluofu (Lyapunov) function that many input differential comprise system, and design can make the differential of many inputs comprise the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system, by exporting actuator to after the D/A conversion, affact controlled device, meet the nominal performance of the system of engine request until acquisition.

Further, the model that comprises system for single this class differential of input also can specifically be expressed as

Here x ∈ R ⁿ, f _i(x), g _i(x) be continuous function, and f _i(0)=0, for i=1,2 ..., N, g _i(x) have same symbol, A, b are Bu Lunuo Paderewski (Brunovsky) canonical form;

1) by separating Li Yapuluofu (Lyapunov) equation, the step of controller structure control Li Yapuluofu (Lyapunov) function is:

A. appoint and get a positive number p ₂₂＞0 and Hull prestige thatch (Hurwitz) polynomial expression

λ ₁(β)＝λ ^n-1+β _1n-1λ ^n-2+…+β ₁₂λ+β ₁₁

β wherein _1j∈ R, j=1,2 ... n-1;

B. order

C. appoint and get a positive definite matrix Q ∈ R ^{(n-1) * (n-1)}, separate Li Yapuluofu (Lyapunov) equation Because C _{1 β}Be Hull prestige thatch (Hurwitz) matrix, by Li Yapuluofu (Lyapunov) theorem, then separating M is positive definite matrix;

D. calculate

Then

Be a positive definite matrix, and V (x)=x ^TPx is institute and asks control Li Yapuluofu (Lyapunov) function;

2)

For by 1) positive definite matrix of being constructed, h ∈ R ⁿBe last row of matrix P, Exist so and make the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system

Further, comprise system for this class differential of many inputs

3) matrix G _i(x), i=1,2 ..., N satisfies G _i(x)=G (x) D _i(x), D here _i(x)=diag[d _I1(x), d _I2(x) ..., d _Im(x)], d _Ij(x) be continuous function and d _Ij≠ 0, i=1,2 ..., N; J=1,2 ... m; To each j, j ∈ 1,2 ..., m}, function d _Ij(x), i=1,2 ... N sets up same symbol; G (x) is that row full rank l * m matrix and its each element are continuous function, chooses Λ (x)=diag[λ ₁(x), λ ₂(x) ... λ _m(x)], here

4) to A _k, b _k, k=1,2 ..., l utilizes 2 (this section the has been quoted claim 2) method that the single input of structure differential comprises the control Li Yapuluofu (Lyapunov) of system function that proposes, and obtains

So

And V (x)=x ^TPx is that this class is imported control Li Yapuluofu (Lyapunov) function that differential comprises system more;

5) for satisfying 1) this class import differential more and comprise system, continuous state FEEDBACK CONTROL rule

Wherein

Make the closed-loop system asymptotically stable in the large.

Owing to adopted described technical scheme, a complete set of regulating and controlling process of the present invention can be finished on industrial computer.Compare with traditional method for designing, the advantage that a kind of differential that the present invention provides comprises the system controller method for designing is: at complex mechanical system, the step of control Li Yapuluofu (Lyapunov) construction of function and the general analytic formula of CONTROL LAW DESIGN are proposed, by exporting actuator to after the D/A conversion, affact a plurality of controlled devices, nominal performance until the system that obtains to meet engine request, easy and simple to handle directly perceived.Break through a controller and can only control the traditional control method of single object.Making differential comprise system can really be applied in the commercial production activity.

What set forth below in conjunction with accompanying drawing is the good control effect that a embodiment that the present invention provides shows, it may be noted that, the present invention is not only limited to following embodiment, the controller design method that provides, be applicable to that the linearizing differential of various different available feedback comprises the control system of description, but the widespread use robotic arm, the unified control of complex mechanical systems such as pneumatic system and flight control system.

Description of drawings

Fig. 1 is a kind of closed loop controlling structure synoptic diagram that adopts differential to comprise the method that mechanical system is unified to control of the present invention.

Fig. 2 adopts differential to comprise the method that mechanical system is unified to control and is applied to synoptic diagram to the control of one group of simple joint robotic arm for the present invention is a kind of.

Fig. 3 the present invention is a kind of to be adopted differential to comprise the method that mechanical system is unified to control to be applied to state path to one group of simple joint robotic arm closed-loop system under the control law effect.

Fig. 4 adopts differential to comprise the method that mechanical system is unified to control and is applied to the unified control law path of controlling of one group of simple joint robotic arm for the present invention is a kind of.

Embodiment

As Fig. 1, shown in be that a kind of method that adopts differential to comprise mechanical system is unified to control of the present invention is unified control at one group by the simple joint robotic arm, concrete implementation step:

1. at first write by Isidori A., Springer Verlag publishing house, nineteen ninety-five is published monograph " NonlinearControl Systems " (nonlinear control system), and one group of simple joint robotic arm is formed the model that differential comprises system and is as can be known

X=[x wherein ₁x ₂x ₃x ₄] ^TBe state, x ₁Be the angular displacement of robotic arm, control input u is the torque of driving shaft, and

Here J _I1, J _I2Represent inertia, F _iRepresent the viscous friction coefficient, K _iRepresent the elastic constant of spring, N is the transmission gear ratio, m _iWith d _iRepresent the position of the quality and the center of gravity of robotic arm.

2. by separating Li Yapuluofu (Lyapunov) equation, controller 6 structure control Li Yapuluofu (Lyapunov) functions

Appoint and get a positive number p ₂₂=1 and Hull prestige thatch (Hurwitz) polynomial expression

λ ₁(β)＝λ ³+3λ ²+3λ+1

Order

Appoint and get a positive definite matrix Separate Li Yapuluofu (Lyapunov) equation

Then

Thus

Be a positive definite matrix, and V (x)=x ^TPx is institute and asks control Li Yapuluofu (Lyapunov) function.

3. CONTROLLER DESIGN 6,

Exist and make the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system

Wherein

As shown in Figure 2, during emulation experiment, the parameter of constructing two simple joint robotic arm models by controller 6 is respectively

J ₁₁＝0.7，J ₁₂＝0.5，m ₁＝0.5，K ₁＝8，d ₁＝0.3，N＝10，F ₁₁＝0.07，F ₁₂＝0.04，g＝9.8；

J ₂₁＝0.9，J ₂₂＝0.8，m ₂＝0.7，K ₂＝9，d ₂＝0.4，N＝10，F ₂₁＝0.08，F ₂₂＝0.07，g＝9.8。The steering order of controller 6 is transported to respectively in two motors 5, and motor 5 drives driving shaft 4, drives knob spring 2 under the cooperation of wheel box 3, drives the movable part 1 of mechanical arm.And two mechanical arms can be made different actions simultaneously.Use a controller 6 just a plurality of members in one group of mechanical system to be controlled.

As Fig. 3, the differential that is depicted as two simple joint robotic arms compositions with different model parameters comprises the different conditions path of system's closed-loop system under the control law effect, Figure 4 shows that the differential of two simple joint robotic arms compositions with different model parameters comprises the control corresponding rule path of system's closed-loop system under the control law effect.The control method that adopts the present invention to provide as can be seen from Figure utilizes a control law can unify the mechanical system that control is made up of two different model parameter simple joint robotic arms well.Break through a controller and can only control the traditional control method of single object.

Claims (2)

1. one kind is adopted differential to comprise the method that mechanical system is unified to control, and it is characterized in that comprising following concrete steps:

1) at first picking out the model that differential comprises system is

$\stackrel{\·}{x}\∈\mathrm{co}\{\mathrm{Ax}+B[F_i\left(x\right)+G_i\left(x\right)u\left]\right\},$ i＝1，2，…N，

Here co represents the convex closure of a set, x ∈ R ⁿ, u ∈ R ^mBe respectively state and input, F _i(x) ∈ R ^{L * 1}, G _i(x) ∈ R ^{L * m}All be continuous function, and F _i(0)=0, for any x ∈ R ⁿ, G _i(x) be the row full rank, A, B are following Bu Lunuo Paderewski (Brunovsky) canonical form

$b_k={\left[\begin{array}{c}0\\ \·\\ \·\\ \·\\ 0\\ 1\end{array}\right]}_{r_k\×1}$

Here r ₁+ r ₂+ ... + r _k=n;

2) secondly by separating Li Yapuluofu (Lyapunov) equation, the single input of controller (6) structure differential comprises control Li Yapuluofu (Lyapunov) function of system, and design can make the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system, by exporting actuator to after the D/A conversion, affact controlled device, meet the nominal performance of the system of engine request until acquisition;

3) import control Li Yapuluofu (Lyapunov) function construction method that differential comprises system by list, controller (6) obtains control Li Yapuluofu (Lyapunov) function that many input differential comprise system, and design can make the differential of many inputs comprise the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system, by exporting actuator to after the D/A conversion, affact controlled device, meet the nominal performance of the system of engine request until acquisition.

2. a kind of differential that adopts according to claim 1 comprises the method that mechanical system is unified to control, and it is characterized in that described step 2) in, the model that comprises system for single this class differential of input also can specifically be expressed as

$\stackrel{\·}{x}\∈\mathrm{co}\{\mathrm{Ax}+b[f_i\left(x\right)+g_i\left(x\right)u\left]\right\},$ i＝1，2，…N，

Here x ∈ R ⁿ, f _i(x), g _i(x) be continuous function, and f _i(0)=0, for i=1,2 ..., N, g _i(x) have same symbol, A, b are Bu Lunuo Paderewski (Brunovsky) canonical form;

1) by separating Li Yapuluofu (Lyapunov) equation, the step of controller (6) structure control Li Yapuluofu (Lyapunov) function is:

A. appoint and get a positive number p ₂₂＞0 and Hull prestige thatch (Hurwitz) polynomial expression

λ ₁(β)＝λ ^n-1+β _1n-1λ ^n-2+…+β ₁₂λ+β ₁₁

β wherein _1j∈ R, j=1,2 ... n-1;

B. order $P_{12}^T=p_{22}[{\mathrm{\β}}_{11},\·\·\·,{\mathrm{\β}}_{1n-1}];$

C. appoint and get a positive definite matrix Q ∈ R ^{(n-1) * (n-1)}, separate Li Yapuluofu (Lyapunov) equation ${\mathrm{MC}}_{1\mathrm{\β}}+C_{1\mathrm{\β}}^{T}M=-Q,$ Because C _{1 β}Be Hull prestige thatch (Hurwitz) matrix, by Li Yapuluofu (Lyapunov) theorem, then separating M is positive definite matrix;

D. calculate $P_{n-1}=M+P_{12}{p}_{22}^{-1}{P}_{12}^T,$ Then $P=\left[\begin{array}{cc}{P}_{n-1}& P_{12}\\ P_{12}^T& p_{22}\end{array}\right]$ Be a positive definite matrix, and V (x)=x ^TPx is institute and asks control Li Yapuluofu (Lyapunov) function;

2) $P=\left[\begin{array}{cc}{P}_{n-1}& P_{12}\\ P_{12}^T& p_{22}\end{array}\right]$ For by 1) positive definite matrix of being constructed, h ∈ R ⁿBe last row of matrix P, $\mathrm{\ζ}\left(x\right)=x^T\mathrm{PAx}+\underset{i}{\max }{x}^T{\mathrm{hf}}_i\left(x\right),$ $\mathrm{\η}\left(x\right)=x^{T}h\underset{i}{\min }{g}_i\left(x\right),$ Exist so and make the globally asymptotically stable continuous state FEEDBACK CONTROL rule of closed-loop system

$u=\left\{\begin{array}{cc}-\frac{\mathrm{\ζ}\left(x\right)+\sqrt{{\left(\mathrm{\ζ}\left(x\right)\right)}^2+{\left(\mathrm{\η}\left(x\right)\right)}^4}}{\mathrm{\η}\left(x\right)},& x^{T}h\≠0,\\ 0,& x^{T}h=0.\end{array}\right.$

Further, comprise system for this class differential of many inputs

3) matrix G _i(x), i=1,2 ..., N satisfies G _i(x)=G (x) D _i(x), D here _i(x)=diag[d _I1(x), d _I2(x) ..., d _Im(x)], d _Ij(x) be continuous function and d _Ij≠ 0, i=1,2 ..., N; J=1,2 ... m; To each j, j ∈ 1,2 ..., m}, function d _Ij(x), i=1,2 ... N sets up same symbol; G (x) is that row full rank l * m matrix and its each element are continuous function, chooses Λ (x)=diag[λ ₁(x), λ ₂(x) ... λ _m(x)], here

${\mathrm{\&lambda;}}_j\left(x\right)=\left\{\begin{array}{cc}\underset{i}{\min d_{\mathrm{ij}}\left(x\right),}& d_{\mathrm{ij}}\left(x\right)>0\\ \underset{i}{\max d_{\mathrm{ij}}\left(x\right),}& d_{\mathrm{ij}}\left(x\right)<0\end{array}\right.$

4) to A _k, b _k, k=1,2 ..., l utilizes the method that the single input of structure differential comprises the control Li Yapuluofu (Lyapunov) of system function that proposes, and obtains $P_k=\left[\begin{array}{cc}{P}_{k-1}& P_{k2}\\ P_{k2}^T& p_{k3}\end{array}\right],$ So

And V (x)=x ^TPx

Be that this class is imported control Li Yapuluofu (Lyapunov) function that differential comprises system more;

5) for satisfying 1) this class import differential more and comprise system, continuous state FEEDBACK CONTROL rule

$u=\left\{\begin{array}{cc}-\mathrm{\&eta;}\left(x\right)\frac{\mathrm{\&zeta;}\left(x\right)+\sqrt{{\left(\mathrm{\&zeta;}\left(x\right)\right)}^2+{\left({\mathrm{\&eta;}}^T\left(x\right)\mathrm{\&eta;}\left(x\right)\right)}^2}}{{\mathrm{\&eta;}}^T\left(x\right)\mathrm{\&eta;}\left(x\right)},& x^T\mathrm{PB}\&NotEqual;0\\ 0,& x^T\mathrm{PB}=0\end{array}\right.$

Wherein $\mathrm{\&zeta;}\left(x\right)=x^T\mathrm{PAx}+\underset{i}{\max }{x}^T{\mathrm{PBF}}_i\left(x\right),$ η (x)=Λ (x) G ^T(x) B ^TPx makes the closed-loop system asymptotically stable in the large.

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