CN104536295B - A kind of cantilever beam Robust Adaptive Control method - Google Patents

Abstract

The invention discloses a kind of cantilever beam Robust Adaptive Control method, design a preferable cantilever beam dynamic model, wherein comprising frequency signal abundant enough, it is used as system reference track, whole adaptive control system ensures reference locus on actual cantilever beam track following, a kind of preferable dynamic characteristic is reached, foozle and environmental disturbances is compensate for.Regard cantilever beam parameter itself as unknown systematic parameter, constitute a parameter error vector θ, one sliding formwork function of the design and derivative for making sliding formwork function is zero to draw equivalent controller, on this basis input signal is used as plus feedback term and robust, controller parameter θ adaptive law is designed based on Lyapunov methods, the stability of guarantee system, makes tracking error converge on zero, while all parameters converge on true value.

Description

A kind of cantilever beam Robust Adaptive Control method

Technical field

The present invention relates to a kind of cantilever beam Robust Adaptive Control method.

Background technology

Cantilever beam is that one end of fingerboard is the hold-down support for not producing axial direction, vertical displacement and rotation, and the other end is freely End (can be produced parallel to the power axially and perpendicular to axial direction).In engineering mechanics force analysis, than more typical simplified mould Type.In Practical Project analysis, most of Practical Project stressed member can be reduced to cantilever beam.

1985, whole piece piezoelectric membrane PVDF was affixed on as actuator on whole beam, using constant gain amplifier by Bailey etc. And normal amplitude (CAC) feedback has carried out experimental study to the vibration control of cantilever beam (CGC).1991, Lee etc. proposed piezoelectricity The design method of modal sensor and piezo modal driver, this method is affixed on the shape of the piezoelectric membrane on structure by adjustment Shape designs modal sensor and mode driver with the polarised direction of piezoelectric membrane is changed, but is intended to change the rank of controlled mode Number, then will change piezoelectric sensing layer and Piezoelectric Driving layer and shape and bonding method.In recent years, Sun Dongchang etc. is to control each rank mould Control electric energy needed for state is as small as possible as target, carries out the Study on Vibration Control of Piezoelectric Intelligent Beams；Wang Zongli etc. is proposed A kind of related LQR control methods of state.

But, such as modeling for piezoelectric cantilever is generally ignored due to pasting piezoelectric or embedment piezoresistive material Expect to bring the change in rigidity and quality to girder construction, and due to inevitable mismachining tolerance during the manufacturing and The influence of environment temperature, can cause the difference between original paper characteristic and design, cause cantilever beam to there is parameter uncertainty, it is difficult to The problems such as setting up accurate mathematical modeling.Along with the external disturbance effect in working environment be can not ignore so that cantilever beam Trajectory track control is difficult to, and robustness is relatively low.The nominal value parameter that traditional control method is based entirely on cantilever beam is set Meter, and ignore the effect of quadrature error and external disturbance, although system is still stable in most cases, but follows the trail of effect It is remote undesirable, it is this that there is very big use limitation for the controller that single environment is designed.Based on traditional control method Cantilever beam using it is upper also exist inconvenience and defect, it would be highly desirable to further improve.

The content of the invention

In view of the above-mentioned problems, the present invention provides a kind of cantilever beam Robust Adaptive Control method, added in control law Robust and feedback term, improve cantilever beam system to there is uncertain model, Parameter Perturbation and external disturbance power etc. various dry In the case of disturbing, the robustness of tracking performance and whole system to ideal trajectory, while being set based on Lyapunov Theory of Stability The adaptive law of meter ensure that the global stability of closed-loop system.

To realize above-mentioned technical purpose and the technique effect, the present invention is achieved through the following technical solutions：

A kind of cantilever beam Robust Adaptive Control method, it is characterised in that comprise the following steps：

S01：Set up the ideal kinetics model of cantilever beam：

x_m=Asin (wt),

In formula, A is amplitude of the cantilever beam on x coordinate direction of principal axis, and t is the time, and w is cantilever beam on x coordinate direction of principal axis Given vibration frequency, x_mFor preferable oscillation trajectory；

Then vector form is：

In formula, q_mFor x_m,k_mFor w²,For q_mSecond dervative；

S02：Set up cantilever beam dimensionless kinetics equation：

The non-dimension vector model of cantilever beam system is set up,

In formula, q is actual oscillation trajectory,Respectively q first derivative, second dervative, u is robust adaptive Control law, f is the external interference of cantilever beam, and C, K is the parameter of cantilever beam itself, and C is damping term, and K is frequency term；

S03：Sliding formwork function is set up, the derivative for making sliding formwork function against time is zero to obtain control law, and in control law Middle addition feedback term and robust；

S04：Robust adaptive rule is set up based on Lyapunov methods.

It is preferred that, step S03 specifically includes following steps：

Defining sliding formwork function s is：

In formula, e=q-q_mFor tracking error,For e first derivative, λ is sliding formwork parameter；

The derivative of sliding formwork function against time is：

I.e.：

Definition：θ^*=[C, K]^TFor parameter error vector,

Then,

MakeTo obtain equivalent control u_εq：

u_εq=Y θ^*- Q-f,

Addition feedback term and robust set up Robust Adaptive Control rule and are：

U=Y θ-Q+u_s1+u_s2=Y θ-Q-k_sS- η tanh (s),

In formula, u_s1=-k_sS is feedback term, u_s2=-η tanh (s) are robust, k_sFor feedback factor, η is robust coefficient, θ is parameter error vector θ^*Estimate.

It is preferred that, step S04 specifically includes following steps：

Liapunov function is：

Make v derivativeObtain robust adaptive rule：

In formula, γ, P are symmetric positive definite matrix,S is sliding-mode surface.

The beneficial effects of the invention are as follows：

First, the dynamic characteristic of cantilever beam is a kind of idealized model, compensate for foozle and and environmental disturbances；

2nd, feedback term is added in control algolithm, cantilever beam oscillation trajectory tracking velocity is substantially increased and parameter is estimated Speed is counted, while reducing concussion amplitude；

3rd, robust is added in control algolithm, the parameter uncertainty of environmental disturbances and cantilever beam in itself is counteracted, changes It has been apt to the robustness and dynamic characteristic of system；

4th, the adaptive algorithm set up based on Lyapunov methods ensure that the Globally asymptotic of whole closed-loop system Property；

5th, control of this method to cantilever beam need not be set up on the basis of object Accurate Model, save modeling Expense.

Brief description of the drawings

Fig. 1 is the simplified model schematic diagram of cantilever beam system of the present invention；

Fig. 2 is the schematic diagram of the present invention；

Fig. 3 is the track following effect curve figure of cantilever beam in the specific embodiment of the invention；

Fig. 4 is the estimation response curve of cantilever beam parameter C in the specific embodiment of the invention；

Fig. 5 is the estimation response curve of cantilever beam parameter K in the specific embodiment of the invention.

Embodiment

Technical solution of the present invention is described in further detail with specific embodiment below in conjunction with the accompanying drawings, so that ability The technical staff in domain can be better understood from the present invention and can be practiced, but illustrated embodiment is not as the limit to the present invention It is fixed.

The simplified model schematic diagram of cantilever beam system is as shown in figure 1, a kind of cantilever beam Robust Adaptive Control method, is utilized The control method of robust adaptive, adds feedback term and robust in control law, applied in the controller of cantilever beam, to outstanding Arm beam is controlled.Specifically include following steps：

S01：Set up the ideal kinetics model of cantilever beam：

x_m=Asin (wt),

In formula, A is amplitude of the cantilever beam on x coordinate direction of principal axis, and t is the time, and w is cantilever beam on x coordinate direction of principal axis Given vibration frequency, x_mFor preferable oscillation trajectory；

Then vector form is：

In formula, q_mFor x_m,k_mFor w²,For q_mSecond dervative；

Ideal kinetics model is reference model.

S02：Set up cantilever beam dimensionless kinetics equation：

The non-dimension vector model of cantilever beam system is set up,

In formula, q is actual oscillation trajectory,Respectively q first derivative, second dervative, u is robust adaptive Control law, f is the external interference of cantilever beam, and C, K is the parameter of cantilever beam itself, and C is damping term, and K is frequency term；

S03：Sliding formwork function is set up, the derivative for making sliding formwork function against time is zero to obtain control law, and in control law Middle addition feedback term and robust；

S04：Robust adaptive rule is set up based on Lyapunov methods.

It is preferred that, step S03 specifically includes following steps：

Defining sliding formwork function s is：

In formula, behalf sliding-mode surface, e=q-q_mFor tracking error,For e first derivative, λ is sliding-mode surface parameter；

The derivative of sliding formwork function against time is：

Arranged and be for the form with parameter error vector：

Definition：Be known to a parameter 1 × 2 matrix, θ^*=[C, K]^TFor parameter error vector,It is a known constant,

Then,

MakeTo obtain equivalent control robust adaptive rule u_εq, to makeThen

Addition feedback term and robust set up Robust Adaptive Control rule and are：

U=Y θ-Q+u_s1+u_s2=Y θ-Q-k_sS- η tanh (s),

In formula, u_s1=-k_sS is feedback term, u_s2=-η tanh (s) are robust, k_sFor feedback factor, η is robust coefficient, θ is parameter error vector θ^*Estimate.

It is preferred that, step S04 specifically includes following steps：

Liapunov (Lyapunov) function is：

V is the scalar function of a positive definite, P=P^T, γ=γ^TFor symmetric positive definite matrix, s is sliding-mode surface, The derivative of Lyapunov function against time is：

To ensureNeed to meet η ＞ f, make v derivativeObtain robust adaptive rule：

Cantilever beam Robust Adaptive Control of the present invention based on addition robust and feedback term in control law Method is to be applied to robust adaptive method in the control of cantilever beam, designs a preferable cantilever beam dynamic model, wherein Comprising frequency signal abundant enough, as system reference track, whole adaptive control system ensures actual cantilever beam track Reference locus in tracking, reaches a kind of preferable dynamic characteristic, compensate for foozle and environmental disturbances.Cantilever beam is joined in itself Number regards unknown systematic parameter as, constitutes a parameter error vector θ, designs a sliding formwork function and makes the derivative of sliding formwork function It is zero to draw equivalent controller, on this basis plus feedback term and robust as input signal, based on Lyapunov methods Design controller parameter θ adaptive law, it is ensured that the stability of system, tracking error is converged on zero, while all parameters are received Hold back in true value.

Compared with prior art, the beneficial effects of the invention are as follows：

First, the dynamic characteristic of cantilever beam is a kind of idealized model, compensate for foozle and and environmental disturbances；

2nd, feedback term is added in control algolithm, cantilever beam oscillation trajectory tracking velocity is substantially increased and parameter is estimated Speed is counted, while reducing concussion amplitude；

3rd, robust is added in control algolithm, the parameter uncertainty of environmental disturbances and cantilever beam in itself is counteracted, changes It has been apt to the robustness and dynamic characteristic of system；

4th, the adaptive algorithm set up based on Lyapunov methods ensure that the Globally asymptotic of whole closed-loop system Property；

5th, control of this method to cantilever beam need not be set up on the basis of object Accurate Model, save modeling Expense.

By taking the cantilever beam system shown in Fig. 1 as an example, its principle is as shown in Fig. 2 piezoelectric intelligent cantilever beam is in piezoelectric actuator Effect under mode motion equation be：

In formula, ξ_iIt is the damping of the i-th stage structure；w_iIt is the i-th rank intrinsic frequency；B_i=K_a[φ'_i(x₂)-φ'_i(x₁)], φ '_i (x₂)、φ'_i(x₁) be mass normalisation orthogonal modes matrix, K_aFor piezoelectric coupling coefficient, U_aFor input voltage, actual conditions In be considered as external disturbance, our existing mode motion equation (1) formulas by cantilever beam rewrite following non-dimension vector form：

In formula, C, K ∈ R^i*iFor systematic parameter, wherein C is damping term, and K is frequency term, and f is disturbance term, u for input to Amount, wherein u=B_iU_a。

Track following error：

E=q-q_m (3)

Setting up sliding formwork function is：

λ=λ in formula^T＞ 0, is sliding-mode surface parameter, is typically taken as the diagonal matrix that element is all positive.Derivation is carried out to s to bring into (2) formula is obtained：

Formula (5) is organized into the form with parameter error vector, this is also the conventional change of one kind of Self Adaptive Control analysis Change method.

Definition：θ^*=[C, K]^T,So, formula (6) can be write as：

In formula, Y is known to a parameter 1 × 2 matrix, and Q is a known constant, θ^*It is one and includes 2 not Know 2 × 1 parameter error vector of systematic parameter.MakeTo obtain equivalent control u_eq：

u_εq=Y θ^*-Q-f (8)

So design Robust Adaptive Control rule is：

U=Y θ-Q+u_s1+u_s2=Y θ-Q-k_ss-ηtanh(s) (9)

In formula, u_s1=-k_sS is feedback term, u_s2=-η tanh (s) are robust (tanh is hyperbolic tangent function), and θ is ginseng Number error vector θ^*Estimate.

Formula (9) is brought into formula (7) to obtain：

In formula,

Liapunov (Lyapunov) function is used below, and from the stability of the overall situation is ensured, design controller is joined Several adaptive laws.

In view of the derivative formula (10) of sliding formwork function, design liapunov function is：

In formula, P=P^T, γ=γ^TFor symmetric positive definite matrix, at the same this function contain sliding formwork function and parameter Estimation to Amount.The adaptive law designed based on this liapunov function is it is ensured that they all change according to the rule of regulation.

The derivative of Lyapunov function against time is：

To makeSimplest method is exactly to meet η ＞ f, while making last be zero.Require：

Therefore adaptive law is：

Have after selecting above-mentioned adaptive law：

Meet the requirements.

It is proposed by the present invention based on adding the outstanding of robust and feedback term in control law in order to more intuitively show Arm beam Robust Adaptive Control method, now carries out computer to this control program using perceptive construction on mathematics/SIMULINK and imitates True experiment.

With reference to existing literature, the parameter for choosing cantilever beam is：C=0.18, K=56.4.Ideal trajectory is described as：Q=sin (t).Cantilever beam is zero original state.Consider that external disturbance act as the noise resonated with ideal trajectory, external disturbance takes f= randn(1).In l-G simulation test, two parameters to be set of adaptive law take γ=diag { 10,10 }, P=1000, feedback Term coefficient k_s=1000, robust term coefficient η=100.Simulated program is run, the simulation result curve of invention specific embodiment is obtained As shown in accompanying drawing 3,4,5.

Wherein, accompanying drawing 3 illustrates the track following effect curve of the cantilever beam under control method proposed by the present invention.From Accompanying drawing can be seen that the output that control system enables to cantilever beam, is not knowing cantilever beam parameter and structure and is existing outer In the case of boundary's interference effect, given ideal trajectory, whole closed-loop system asymptotically stability can be promptly tracked, and follow the trail of Error very little, has reached satisfied effect.

Attached Figure 4 and 5 illustrate cantilever beam parameter C and K parameter estimation response curve, as a result show that they can be converged to respectively From true value, and regulating time is shorter.

Can be seen that control method proposed by the present invention from above analogous diagram has well to the track following of cantilever beam Control effect, substantially increases the tracking performance and robustness of cantilever beam system, the high-precision control to cantilever beam oscillation trajectory There is provided theoretical foundation and Math.

The content not being described in detail in description of the invention belongs to technological know-how known to professional and technical personnel in the field.

The preferred embodiments of the present invention are these are only, are not intended to limit the scope of the invention, it is every to utilize this hair The equivalent structure that bright specification and accompanying drawing content are made either equivalent flow conversion or to be directly or indirectly used in other related Technical field, be included within the scope of the present invention.

Claims (2)

1. a kind of cantilever beam Robust Adaptive Control method, it is characterised in that comprise the following steps：

S01：Set up the ideal kinetics model of cantilever beam：

x_m=Asin (wt),

In formula, A is amplitude of the cantilever beam on x coordinate direction of principal axis, and t is the time, and w is that cantilever beam gives on x coordinate direction of principal axis Vibration frequency, x_mFor preferable oscillation trajectory；

Then vector form is：

${\stackrel{\·\·}{q}}_m+k_m{q}_m=0,$

In formula, q_mFor x_m,k_mFor w²,For q_mSecond dervative；

S02：Set up the dimensionless kinetics equation of cantilever beam：

The non-dimension vector model of cantilever beam system is set up,

$\stackrel{\·\·}{q}+C\stackrel{\·}{q}+Kq=u+f,$

In formula, q is actual oscillation trajectory,Respectively q first derivative, second dervative, u is Robust Adaptive Control Rule, f is the external interference of cantilever beam, and C, K is the parameter of cantilever beam itself, and C is damping term, and K is frequency term；

S03：Set up sliding formwork function, the derivative for making sliding formwork function against time is zero to obtain control law, and in control law plus Enter feedback term and robust；

S04：Robust adaptive rule is set up based on Lyapunov methods；

Step S03 specifically includes following steps：

Defining sliding formwork function s is：

$s=\stackrel{\·}{e}+\mathrm{\λ}e,$

In formula, e=q-q_mFor tracking error,For e first derivative, λ is sliding-mode surface parameter；

The derivative of sliding formwork function against time is：

$\stackrel{\·}{s}=u+f-C\stackrel{\·}{q}-Kq+\mathrm{\λ}(\stackrel{\·}{q}-{\stackrel{\·}{q}}_m)+k_m{q}_m,$

I.e.：

Definition：θ^*=[C, K]^TFor parameter error vector,

Then,

MakeTo obtain equivalent control u_εq：

u_εq=Y θ^*- Q-f,

Addition feedback term and robust set up Robust Adaptive Control rule and are：

U=Y θ-Q+u_s1+u_s2=Y θ-Q-k_sS- η tanh (s),

In formula, u_s1=-k_sS is feedback term, u_s2=-η tanh (s) are robust, k_sFor feedback factor, η is robust coefficient, and θ is Parameter error vector θ^*Estimate.

2. a kind of cantilever beam Robust Adaptive Control method according to claim 1, it is characterised in that step S04 is specific Comprise the following steps：

Liapunov function is：

$v=\frac{1}{2}{s}^{T}Ps+\frac{1}{2}{\widetilde{\mathrm{\θ}}}^T{\mathrm{\γ}}^{-1}\widetilde{\mathrm{\θ}},$

Make v derivativeObtain robust adaptive rule：

In formula, γ, P are symmetric positive definite matrix,S is sliding-mode surface.