CN104950682B - The underactuated system method of Constrained limitation - Google Patents

Abstract

A kind of underactuated system method of Constrained limitation, total mechanical energy to system has carried out labor, strengthen the coupled relation between rotation bead and translation vibration dolly by kinetic energy shaping on the basis of total mechanical energy, and ensure to calm rotation bead to balance point position using potential energy shaping, and ensure the slewing area of bead in control process, it is finally based on constructed energy function and proposes a kind of constrained gamma controller of tool.For can only ensure steady-state behaviour in conventional control, and it cannot be guaranteed that the defect of some of system transient modelling performance characteristic, the control method not only realizes the quick point stabilization to TORA systems, and ensure that rotation bead is rotated in default scope.Simulation comparison is carried out by with existing method, this method not only can guarantee that calms to stable equilibrium point position system, and can ensure that rotation bead is rotated in the range of constraint, be not in " circulation " phenomenon.

Description

The underactuated system method of Constrained limitation

Technical field

The invention belongs to Underactuated Mechanical Systems automatic control technology field, a kind of deficient drive of Constrained limitation is related generally to Dynamic system control method.

Background technology

Under-actuated systems refer to a class system of the free degree number more than control input dimension, because of its simple structure, flexibly Degree it is strong, lightweight many advantages, such as and be widely used in various places, so many Mechatronic Systems are designed to drive lacking, Also some systems, are also drive lacking due to being influenceed by non-integrable nonholonomic constraint, such as aerospace craft, movement Robot, submarine navigation device etc..Additionally, when the part actuator cisco unity malfunction of complete drive system, will also deteriorate to Under-actuated systems.However, due to the drive lacking characteristic of such system, causing that there is certain challenge to the control of such system again Property, thus the control problem of such system is controlled the extensive concern on boundary.Translation oscillator with rotation excitation (Translation oscillators with rotating actuator, TORA) (structure diagram is as shown in Figure 1) is made For a baseline system is applied to the design of gamma controller and realizes.The system is by a rotation bead and a translation Oscillator is constituted, wherein rotation bead can directly be controlled by external world's input, translation oscillator passes through a Hookean spring and fixation Metope connection, rotation that can only be by rotating bead controlled it indirectly.This drive lacking characteristic of TORA systems causes it The design of controller has certain difficulty.The system is proposed for research initially as the simplified model of dual spin spacecraft Covibration, was mainly used in the non-thread that inspection is carried out for drive lacking dynamic system as a non-linear baseline system later Property control design case.

Used as the simplified model and non-linear baseline system of dual spin spacecraft, TORA systems receive the wide of control field General concern.At present, many scholars have carried out substantial amounts of research to it both at home and abroad：Bupp etc. has built one for implementing first Tested with the TORA system experimental platforms of assessment gamma controller, and control performance to four kinds of gamma controllers Checking；Calani devises a kind of output feedback controller using certainty equivalent method, realizes to the quick of TORA systems Regulation control；Gao Bingtuan considers gravity factor, the TORA systems that rotation bead is moved in vertical plane is studied, profit With the method based on energy, a kind of state feedback controller is devised, achieve good point stabilization performance；Perhaps clear beautiful woman et al. The separation principle designed using High-gain observer and state feedback controller, it is proposed that a kind of nonlinear observer and be based on The control method that energy state feedback controller is combined, the angle displacement that need to only measure rotation bead is capable of achieving to TORA systems Control.Additionally, mainly having for the control method of TORA systems：Control, output feedback ontrol based on passivity, based on sight Survey the control methods such as the control of device.Verified by theory analysis and experiment simulation, the above method can obtain good control effect Really.

For different TORA systems, rotation bead can be rotated in horizontal plane or perpendicular.It is well known that State θ (t)=0 and state θ (the t)=2n π of bead are rotated, (n=1,2 ...) represent identical physical aspect.However, existing Equal distinguishable states θ (t)=0 of most control methods and state θ (t)=2n π, this characteristic of existing method causes Bad " circulation " behavior is likely to occur when controlling TORA systems, so as to cause energy waste, system can be caused not when serious Stabilization.

The content of the invention

The present invention will overcome current TORA system automatic control methods above shortcomings part, there is provided one kind has about The drive lacking TORA system control methods of beam.

This invention address that proposing a kind of constrained stabilized control method of tool.To avoid the generation of " circulation " phenomenon, this Patent of invention proposes a kind of constrained underactuated system method first.For can only ensure stability in conventional control Can, and it cannot be guaranteed that the defect of some of system transient modelling performance characteristic, the control method is not only realized to the fast of TORA systems Fast point stabilization, and ensure that rotation bead is rotated in default scope.Specifically, it is exactly overall mechanical to system Labor can be carried out, has been strengthened by kinetic energy shaping on the basis of total mechanical energy between rotation bead and translation vibration dolly Coupled relation, and ensure to calm rotation bead to balance point position using potential energy shaping, and ensure bead in control process Slewing area, be finally based on constructed energy function and propose a kind of constrained gamma controller of tool.By with There is method to carry out simulation comparison and understand that this method not only can guarantee that calms to stable equilibrium point position system, and can protect Card rotation bead is rotated in the range of constraint, is not in " circulation " phenomenon.

The underactuated control method with Prescribed Properties that the present invention is provided, comprises the following steps：

1. drive lacking model and control targe are determined

For the under-actuated systems model in accompanying drawing 1, it is considered to rotate the drive lacking TORA that bead is rotated in perpendicular System, its dynamic characteristic can be described with equation below：

Wherein, q (t)=[x (t) θ (t)]^TThe state vector of expression system,Represent q (t) on the time respectively First order derivative and second derivative, the transposition of T representing matrixs or vector；U (t)=[0 τ (t)]^TThe control input of expression system Amount；T represents the time, and (t) behind variable represents that the variable is the function on time t；Most number variable has been omitted in formula (t) below；Companion matrix M (q),The expression of auxiliary vector G (q) is as follows：

G (q)=[kx mgr sin θs]^T

Wherein, M is the quality of translation vibration dolly；M is the quality for rotating bead, and the radius of gyration for rotating bead is r；k It is the stiffness factor of spring；Rotary inertia of the bead on its barycenter is designated as J；Acceleration of gravity is g；X (t) and θ (t) difference tables Show that dolly leaves the rotational angle in acceleration of gravity direction apart from the displacement of equalization point and bead edge counterclockwise,Represent bead Rotational angular velocity；τ (t) is to act on the input torque on rotation bead.Our control targe is in input torque τ (t) In the presence of, translation vibration dolly and rotation bead are calmed at stable equilibrium point simultaneously, and make θ (t) in the angle of regulation In the range of, that is, existUnder conditions of, makeIts Middle state [0 00 0]^TIt is poised state,The speed of dolly is represented, v is constrained for rotational angle.

2. system capacity analysis and equalization point determine

Total mechanical energy for system (1) can be expressed as：

Wherein E (t) represents the total mechanical energy of system.Derivation is carried out on the time to formula (2) both sides, and is entered using formula (1) Row can obtain the derivative of total mechanical energy after simplifying

Knowable to formula (2) and (3), for τ (t)=0 and θ ∈ (- π, π], system (1) is with two equalization points ( Unstable equilibrium point and an equalization point for stabilization), our target is the stable equilibrium adjusted system state amount to system At point, i.e. point:

3. control law is determined

Construction energy function, proposes a kind of non-linear bounded control method, realizes small to rotation bead and translation vibration The point stabilization of car, and ensure the anglec of rotation of bead in default scope, design control law is：

Wherein, k_p,k_d,k_v,k_θRepresent positive control gain；φ(t),It is auxiliary function, expression sees next section.

The theoretical foundation and derivation of the inventive method

1. system model and conversion

For convenience of analyzing, the kinetic model (1) of under-actuated systems is written as following form：

Above formula is arranged：

Wherein m (θ),The auxiliary function being expressed as follows respectively：

M (θ)=Mmr²+JM+Jm+m²r²sin²θ

2. controller design

For enhancing can drive rotation bead and can not be between driving oscillation dolly coupled relation, using rotating the dynamic of bead Mechanical characteristic such as formula (8), to following kinetic energy term is introduced in system total mechanical energy E (t), obtains following energy function：

Wherein k_vIt is positive control gain, to its both sides on time derivation, and using formula (3,4), after (6-8) is arranged It is available：

φ (t) is the auxiliary function that section is previously mentioned, and its expression formula is as follows：

To guarantee that rotation bead is calmed to stable equilibrium point position, in energy function V₀On the basis of (t), introduce as follows Potential-energy function, obtains following energy function

Wherein k_pIt is positive control gain, to its both sides on time derivation, and can be obtained using formula (10)：

To ensure the rotational angle of rotation bead in default scope, in energy function V₁On the basis of (t), introduce as follows Potential function：

Wherein, k_θIt is positive control gain, v is the default rotational angle constraint for rotating bead.To formula (14) both sides on when Between derivation, and using formula (13) can obtain：

Wherein,The auxiliary function being previously mentioned as above is saved, its expression formula is as follows：

According to energy function V (t) and its derivativePropose following gamma controller：

3. stability analysis

TORA systems are put forward for this paper, under control law (5) effect, closed-loop system can be with asymptotic convergence to stable equilibrium Point, in whole process, it is ensured that rotate the rotational angle of bead in default scope, i.e.,

And

To prove the theorem, proof is carried out in two steps：(1) ensure the rotational angle of rotation bead in default scope； (2) convergence of system mode is ensured.

To the proof of the first step, it is Lyapunov candidate functions to choose stored-energy function defined in formula (14), and to formula (14) both sides carry out derivation on the time, and carried gamma controller (5) is updated in result expression and can be obtained

Obviously, V (t) ∈ L_∞, so closed-loop system is stable under Lyapunov meanings.With reference to Lyapunov candidate's letters The structure of number (14) can further deduce the signal whole bounded of closed-loop system, i.e.,

To inquire into 1/ (v²-θ²) boundedness, a point the following two kinds situation is analyzed：

Situation 1.Convolution (20) conclusion understands

So just like drawing a conclusion：

| the θ (t) | of situation 2. → 0.Now have：

Can be obtained with reference to above two situation：

Association type (20) conclusion and controller (5) expression formula understand：

τ(t)∈L_∞ (25)

To rotate the scope of bead rotational angle during analysis and Control, it is assumed herein that

|θ(t)|→v (26)

Then have

Contradicted with formula (24) conclusion.Therefore, the hypothesis is invalid.So just like drawing a conclusion in control process；

It is the convergence of analysis system state to the proof of second step, is defined as follows invariant set S：

Understood in invariant set S by formula (19)

Therefore can obtain in invariant set S

Wherein, α is constant.To further determine that in invariant setValue, a point the following two kinds situation is analyzed；

Situation 1. is in invariant set SThen x (t) is constant, thenThen convolution (6), (30) and (31) x (t)=0 is understood.Controller (5) is substituted into formula (6) and is arranged：

Therefore, for θ ∈, (- v, v), equation (32) has unique solution θ (t)=0.

Situation 2. is in invariant set SIt is nonidentical in zero, i.e., in S more at least causeAnd becauseThusContinuously, so in the presence of oneNeighborhoodSo that in neighborhoodInIn neighborhoodIn analyzed as follows, by controller (5) substitute into (6) and convolution (30), (31) carry out arrange abbreviation can obtain：

Know that θ (t) is constant in S by formula (31), understand that x (t) is also constant with reference to above formula, soWith hypothesis In neighborhoodIt is middle to there is certain pointContradiction, therefore assume invalid.

The above analysis understands：In whole control process, rotate bead rotational angle θ (t) remain at it is pre- If in the range of；Meanwhile, stable equilibrium point is only included in invariant set S

Then, it can be seen from LaSalle principles of invariance, system mode in the presence of gamma controller (5) at any time Between elapse asymptotic convergence at stable equilibrium point, you can to realize expected control targe.

It is an advantage of the invention that：For drive lacking model, existing method distinguishable states θ (t)=0 and state θ (t)=2n π, this characteristic causes to be likely to occur bad " circulation " behavior when TORA systems are controlled, and the present invention proposes one kind and carries The TORA model control methods of constraints.Compared to existing control method, this method can guarantee that system can realize quick calm control System, moreover it is possible to ensure to rotate rotation bead control in set scope.It is right simultaneously in the presence of the designed controllers of this paper Can be suppressed in different types of external interference and quickly eliminated, robustness is good, has a good application prospect.

Brief description of the drawings

Fig. 1 is TORA system construction drawings.

Fig. 2 is Simulation results figure of the present invention.

Fig. 3 is original methods and resultses figure.

Fig. 4 is robust performance test result figure.

Specific embodiment

The present invention is further illustrated below in conjunction with the accompanying drawings.

The underactuated control method with Prescribed Properties that the present invention is provided, implementation method is as follows：

1. drive lacking model and control targe are determined

For the under-actuated systems model in accompanying drawing 1, it is considered to rotate the drive lacking TORA that bead is rotated in perpendicular System, its dynamic characteristic can be described with equation below：

Wherein, q (t)=[x (t) θ (t)]^TThe state vector of expression system；Represent q (t) on the time respectively First order derivative and second derivative, the transposition of T representing matrixs or vector；U (t)=[0 τ (t)]^TThe control input of expression system Amount；T represents the time, and (t) behind variable represents that the variable is the function on time t；M(q),It is companion matrix, G Q () is auxiliary vector, the expression of the model is as follows：

Above formula is arranged：

Wherein m (θ),The auxiliary function being expressed as follows respectively：

M (θ)=Mmr²+JM+Jm+m²r²sin²θ

M is the quality of translation vibration dolly；M is the quality for rotating bead, and the radius of gyration for rotating bead is r；K is spring Stiffness factor；Rotary inertia of the bead on its barycenter is designated as J；Acceleration of gravity is g；X (t) and θ (t) represent dolly respectively Prolong the rotational angle for leaving acceleration of gravity direction counterclockwise apart from the displacement of equalization point and bead,Represent the rotation of bead Angular speed；τ (t) is to act on the input torque on rotation bead.Our control targe is the effect in input torque τ (t) Under, will translation vibration dolly and rotation bead simultaneously calm at stable equilibrium point, and make θ (t) regulation angular range in, ExistUnder conditions of, makeWherein state [0 0 0 0]^TIt is poised state.

2. system capacity analysis and equalization point determine

Total mechanical energy for system (1) can be expressed as：

Wherein E (t) represents the total mechanical energy of system.Derivation is carried out on the time to formula (2) both sides, and is entered using formula (1) Row can obtain the derivative of total mechanical energy after simplifying

For τ (t)=0 and θ ∈ (- π, π], system has two equalization points (unstable equilibrium point and a stabilizations Equalization point), our target is to adjust to the stable equilibrium point of system system state amount, i.e. point:

3. control law is determined

Construction energy function, proposes a kind of non-linear bounded control method, realizes small to rotation bead and translation vibration The point stabilization of car, and ensure the anglec of rotation of bead in default scope.Its control law is：

Wherein, k_p,k_d,k_v,k_θRepresent positive control gain.

4. it is determined that after control method, proposing that the present invention solves the scheme that its technical problem is used：

Emulation experiment checking is carried out in Matlab/Simulink environment.In emulation in systematic parameter selection Bupp et al. Experiment porch parameter, it is specific as follows；

M=1.3608kg, m=0.096kg, k=186.3N/m, r=0.0592m, J=0.0002175kgm²

Emulation experiment checking carries out point two parts, test respectively institute's extracting method of the present invention Control with constraints performance with Robustness, and carry out restraint performance comparative analysis with the control method designed by Gao Bingtuan.The present invention proposes the gain of controller Choose as follows；

k_p=0.0145, k_d=0.0024, k_v=0.06, k_θ=0.0001

Constrained performance test.The original state of system is：

The anglec of rotation constraint of bead is chosen for：

V=20 °

Accompanying drawing 2,3 sets forth the simulation result of control method designed by the present invention and third high puies forward control method Control effect.By contrast, two methods are capable of achieving the quick point stabilization to TORA systems, and catch time is basic It is identical.But in whole control process, former control method cannot ensure to rotate the rotating range of bead, and the present invention proposes control Strategy can ensure to rotate rotation bead control in set scope.

Robustness is tested.Further to test the robustness of this paper institutes extracting method, artificially to dolly in control process Add different types of interference.Specifically, the original state of system is set to；

And the sinusoidal wave interference that frequency is 4 π was added between the 8th second to the 8.1st second, added between the 15th second to the 21st second Enter random disturbances, its amplitude is 1.Meanwhile, the anglec of rotation constraint of bead is chosen for

Accompanying drawing 4 gives robustness test result, it can be seen that in the presence of the designed controller of invention, Can be suppressed for different types of external interference and quickly eliminated, and ensure rotation bead in default slewing area Rotate.Simulation result shows that the inventive method has good robustness.

Content described in this specification embodiment is only enumerating to the way of realization of inventive concept, protection of the invention Scope is not construed as being only limitted to the concrete form that embodiment is stated, protection scope of the present invention is also and in art technology Personnel according to present inventive concept it is conceivable that equivalent technologies mean.

Claims (1)

1. a kind of underactuated system method of Constrained limitation, comprises the following steps：

Step 1. determines drive lacking model and control targe

Dynamic characteristic to under-actuated systems model is described with equation below：

$M\left(q\right)\stackrel{\·\·}{q}+V(q,\stackrel{\·}{q})\stackrel{\·}{q}+G\left(q\right)=u---\left(1\right)$

Wherein, q (t)=[x (t) θ (t)]^TThe state vector of expression system,Q (t) on the time one is represented respectively The transposition of subderivative and second derivative, T representing matrixs or vector；U (t)=[0 τ (t)]^TThe control input amount of expression system；t Expression time, (t) behind variable represents that the variable is the function on time t；Omitted in formula behind most number variable (t)；Companion matrix M (q),The expression of auxiliary vector G (q) is as follows：

$M\left(q\right)=\left(\begin{array}{cc}M+m& mrcos\mathrm{\θ}\\ mr\cos \mathrm{\θ}& {\mathrm{mr}}^2+J\end{array}\right)$

$V(q,\stackrel{\·}{q})=\left[\begin{array}{cc}0& -mr\stackrel{\·}{\mathrm{\θ}}sin\mathrm{\θ}\\ 0& 0\end{array}\right]$

G (q)=[kx mgr sin θs]^T

Wherein, M is the quality of translation vibration dolly；M is the quality for rotating bead, and the radius of gyration for rotating bead is r；K is bullet The stiffness factor of spring；Rotary inertia of the bead on its barycenter is designated as J；Acceleration of gravity is g；X (t) and θ (t) represent small respectively Car leaves the rotational angle in acceleration of gravity direction apart from the displacement of equalization point and bead edge counterclockwise,Represent turning for bead Dynamic angular speed；τ (t) is to act on the input torque on rotation bead；Control targe is in the presence of input torque τ (t), to incite somebody to action Translation vibration dolly and rotation bead are calmed at stable equilibrium point simultaneously, and make θ (t) in the angular range of regulation, that is, exist | θ (t) | ＜ v,Under conditions of, makeWherein state [0 00 0]^TIt is poised state,The speed of dolly is represented, v is constrained for rotational angle；

Step 2. system capacity is analyzed and equalization point determines

Total mechanical energy for system (1) can be expressed as：

$E=\frac{1}{2}{\stackrel{\·}{q}}^{T}M\left(q\right)\stackrel{\·}{q}+\frac{1}{2}{\mathrm{kx}}^2+mgr(1-cos\mathrm{\θ})---\left(2\right)$

Wherein E (t) represents the total mechanical energy of system；Derivation is carried out on the time to formula (2) both sides, and letter is carried out using formula (1) The derivative of total mechanical energy can be obtained after change

$\stackrel{\·}{E}={\stackrel{\·}{q}}^{T}M\left(q\right)\stackrel{\·\·}{q}+\frac{1}{2}{\stackrel{\·}{q}}^T\stackrel{\·}{M}\left(q\right)\stackrel{\·}{q}+{\stackrel{\·}{q}}^{T}G\left(q\right)=\stackrel{\·}{\mathrm{\θ}}\mathrm{\τ}---\left(3\right)$

Knowable to formula (2) and (3), for τ (t)=0 and θ ∈ (- π, π], system (1) with two equalization points, i.e., one not Stable equilibrium point and an equalization point for stabilization, target is to adjust to the stable equilibrium point of system system state amount, i.e. point:

${\left[\begin{array}{cccc}x& \stackrel{\·}{x}& \mathrm{\θ}& \stackrel{\·}{\mathrm{\θ}}\end{array}\right]}^T={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^T---\left(4\right)$

Step 3. determines control law

Construction energy function, proposes a kind of non-linear bounded control method, realizes the town to rotation bead and translation vibration dolly Fixed control, and ensure the anglec of rotation of bead in default scope, design control law is：

Wherein, k_p,k_d,k_v,k_θRepresent positive control gain；φ(t),It is auxiliary function, their expression is respectively：

$\mathrm{\φ}\left(t\right)=\frac{kxmrcos\mathrm{\θ}}{M+m}-mgrsin\mathrm{\θ}---\left(11\right)$