CN105182750A - Switching control method of linear/nonlinear active disturbance rejection control system - Google Patents

Abstract

The invention discloses a switching control method of a linear/nonlinear active disturbance rejection control system. The method comprises the step of establishing an active disturbance rejection control system, the step of performing linear/nonlinear switching on an extended state observer and the step of carrying out linear/nonlinear switching on a state error feedback control law. The method provided by the invention has the following advantages: the method has a higher anti-interference capability and higher control precision; the stability is irrelevant with an initial state; and a parameter setting method provided by the invention is integrated with the advantages of both a "bandwidth method" and an "experience method", influences of sampling step length, noise and the like are also taken into consideration, and the problems of deteriorating tracking performance and control performance which are caused by too large disturbance amplitude do not have to be considered.

Description

A kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method

Technical field

The invention belongs to technical field of automation, relate to a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method.

Background technology

Chinese Academy of Sciences Han Jingqing researcher recognizes that the control strategy that the modern control theory based on mathematical model provides is difficult to effectively be applied in working control engineering, to think deeply " control theory-model theory or kybernetics " for starting point, on the basis of self-examination classical control theory advantage, go on the road that is explored novel practical control technology resolutely.On the basis of priority invention Nonlinear Tracking Differentiator, non-linearity PID and extended state observer, formally automatic disturbance rejection controller was proposed in 1998.The birth of this achievement, has broken and has continued over half a century between Control Theory and Control Engineering and failed to obtain the fine discrepancy solved, and is expected to replace the PID control technology occupying dominant position at present in industry member.

Han Jingqing researcher advocates and uses nonlinear function to improve control performance, and therefore, initial Active Disturbance Rejection Control generally adopts nonlinear state Error Feedback control law and nonlinear extension state observer.But the introducing of nonlinear function, make Active Disturbance Rejection Control become difficulty in parameter tuning, stability analysis and performance evaluation, this is unfavorable for the promotion and application of Auto Disturbances Rejection Control Technique.Given this, professor Gao Zhiqiang proposes the linear active disturbance rejection controller of linearization, band broadening, not only parameter tuning simple, there is physical meaning, and stability analysis, control charac-teristic analyse by means of the classics/modern control theory of maturation, can greatly promote theoretical research and the engineer applied of Auto Disturbances Rejection Control Technique.At present, the theoretical research result that linear active disturbance rejection controls far exceedes non-linear Active Disturbance Rejection Control, and becomes the first-selection of engineer applied.

Active Disturbance Rejection Control marrow is estimation and the compensation of being carried out total disturbance by extended state observer, and the object introducing Nonlinear Mechanism promotes control performance further.Therefore, linear active disturbance rejection controls the actual needs that can meet most occasion, if but higher control accuracy, stronger interference rejection ability will be pursued, non-linear Active Disturbance Rejection Control is a kind of effectively selection.Also do not carry out switching the method controlled to linear/non-linear Active Disturbance Rejection Control system at present.

Summary of the invention

Technical matters to be solved by this invention is to provide the linear/non-linear Active Disturbance Rejection Control systematic evaluation control method that a kind of antijamming capability is strong, control accuracy is high, parameter tuning method is easy.

For solving the problems of the technologies described above adopted technical scheme be: a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method, comprises the steps:

(1) set up Active Disturbance Rejection Control system, it comprises controlled device and automatic disturbance rejection controller; Described automatic disturbance rejection controller comprises Nonlinear Tracking Differentiator, extended state observer and state error Feedback Control Laws;

Described Nonlinear Tracking Differentiator be input as r; The output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) with the output z of extended state observer _i(i=1,2 ..., n) do subtraction relatively after as the input e of described state error Feedback Control Laws _i; The output z of described state error Feedback Control Laws and described extended state observer _n+1do subtraction relatively after value as the first input signal of described extended state observer; The output z of described state error Feedback Control Laws and described extended state observer _n+1first do subtraction to compare, then after carrying out 1/b times of gain, as the input signal of described controlled device; The output y of described controlled device is as the second input signal of described extended state observer; N be greater than 1 positive integer;

(2) described extended state observer is carried out to the switching of linear/non-linear; The relation of sampling step length h, white noise amplitude and total disturbed value is as shown in table 1;

The corresponding relation of table 1 sampling step length h, white noise amplitude and total disturbed value

If the working time of Active Disturbance Rejection Control system is less than settling time; If or the tracing deviation e of described extended state observer is greater than 1; If or the total disturbed value corresponding with sampling step length h, white noise amplitude belongs to the scope shown in table 1; Then switch to linear extended state observer; Expression formula is

Otherwise, switch to nonlinear extension state observer; Expression formula is

In (formula 1) and (formula 2), y is the output of described controlled device, and u is the control inputs of controlled device, for the output of described extended state observer; for the gain of described linear state observer; for the gain of described nonlinear extension state observer; E is the tracing deviation of described extended state observer; Fal (e, α _i, δ) and be nonlinear function, wherein i=1,2 ..., n+1; Mathematic(al) representation is as follows:

In (formula 3), α _i, δ is normal number respectively;

(3) described state error Feedback Control Laws is carried out to the switching of linear/non-linear;

If the working time of Active Disturbance Rejection Control system is less than settling time; If or described extended state observer tracing deviation e is greater than 1; If or the output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) export z with corresponding described extended state observer _ideviation be greater than 1, i.e. (v _i-z _i) > 1, then switch to linear state error feedback control to restrain; Expression formula is

Otherwise, switch to nonlinear state Error Feedback control law; Expression formula is

Wherein, u ₀₁for linear state error feedback control rule; u ₀₂for nonlinear state Error Feedback control law; v _i(i=1,2 ..., n) be the output of Nonlinear Tracking Differentiator, k ' _i, k _ibe respectively positive gain coefficient, α ' _ifor normal number;

(4) if described extended state observer and state error Feedback Control Laws are all linear, then described automatic disturbance rejection controller is linear active disturbance rejection controller, and described Active Disturbance Rejection Control system is linear active disturbance rejection control system; Otherwise described automatic disturbance rejection controller is non-linear automatic disturbance rejection controller, described Active Disturbance Rejection Control system is non-linear Active Disturbance Rejection Control system.

Described β ' _0i, k ' _iobtained by " Bandwidth Method ", i.e. β ' _0ifor by polynomial expression (s+ ω _o) ⁿ⁺¹s after launching ^n+1-ithe coefficient of item, wherein i=1,2 ..., n+1; β ' _0ifor ω _ofunction;

K ' _ifor by polynomial expression (s+ ω _c) ⁿs after launching ^i-1the coefficient of item, wherein i=1,2 ..., n; K ' _ifor ω _cfunction;

Wherein, ω _cfor linear active disturbance rejection controller bandwidth, ω _c> 0; ω _ofor the bandwidth of linear extended state observer, ω _o> 0.

Further, n=3; β ' ₀₁=3 ω ₀, β ' ₀₂=3 ω _o ², β ' ₀₃=ω ³ ₀, k ' ₁=ω ² _c, k ' ₂=2 ω _c.If described in order

$fal(e,{\mathrm{\α}}_i,\mathrm{\δ})=\frac{fal(e,{\mathrm{\α}}_i,\mathrm{\δ})}{e}e={\mathrm{\λ}}_{0i}\left(e\right)e,$ (formula 6)

Then obtained by (formula 2) and (formula 6):

$\left\{\begin{array}{c}e=z_1-y\\ {\stackrel{\·}{z}}_1=z_2-({\mathrm{\β}}_{01}\·{\mathrm{\λ}}_{01}(e\left)\right)e\\ {\stackrel{\·}{z}}_2=z_3-({\mathrm{\β}}_{02}\·{\mathrm{\λ}}_{02}(e\left)\right)e\\ .\\ .\\ .\\ {\stackrel{\·}{z}}_n=z_{n+1}-({\mathrm{\β}}_{0n}\·{\mathrm{\λ}}_{0n}(e\left)\right)e+b\·u\\ {\stackrel{\·}{z}}_{n+1}=-({\mathrm{\β}}_{0(n+1)}\·{\mathrm{\λ}}_{0(n+1)}(e\left)\right)e\end{array}\right.$

Formula (7)

Make e _i=v _i-z _i; Then to the fal (v in (formula 5) _i-z _i, α ' _i, δ) do as down conversion:

$fal(v_i-z_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})=fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})=\frac{fal(e_i.{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})}{e_i}{e}_i$

Order

$\frac{fal(e_i,{{\mathrm{\α}}^{\′}}_i,\mathrm{\δ})}{e_i}={\mathrm{\λ}}_i\left(e_i\right)$

Then

Fal (v _i-z _i, α ' _i, δ) and=λ _i(e _i) e _i(formula 8)

Obtained by (formula 5) and (formula 8):

$u_{02}=\underset{i=1}{\overset{n}{\Σ}}{k}_i{\mathrm{\λ}}_i\left(e_i\right)e_i$ (formula 9)

From (formula 6), (formula 7), (formula 8) and (formula 9), λ _0i(e) e, λ _i(e _i) e _ibe variable-gain linear function; (formula 7) is variable-gain extended state observer; (formula 9) is variable-gain linear state error feedback control rule; By λ _0ie () is abbreviated as λ _0i, λ _i(e _i) be abbreviated as λ _i; Then β _0i, β ' _0iwith λ _0iand k _i, k ' _iwith λ _ibetween pass be:

β _0i·(λ _0i) _min＜β′ _0i＜β _0i·(λ _0i) _max，k _i·(λ _i) _min＜k′ _i＜k _i·（λ _i） _max。

Further, 0.01≤δ≤0.1,0 < α _n+1< ... < α ₂< α ₁< 1,0.5 < α ' ₁< 1,1 < α ' ₂< 2.

Further, δ=0.01.

Further, n=3; α ₁=1, α ₂=0.5, α ₃=0.25, β ₀₁=3 ω _o, k ₂=2 ω _c, α ' ₁=0.75, α ' ₂=1.5; Wherein, ω _cfor linear active disturbance rejection controller bandwidth, ω _c> 0; ω _ofor the bandwidth of linear extended state observer, ω _o> 0.

The invention has the beneficial effects as follows: compared with independent linear or non-linear Active Disturbance Rejection Control system, the present invention has stronger antijamming capability, higher control accuracy; The present invention, in the starting stage, adopts linear active disturbance rejection control system, and its stability and original state have nothing to do; When disturbance is larger, switch to linear active disturbance rejection controller, the performance index such as classical frequency-domain analysis and stability margin still can be suitable for; Therefore, the present invention inherits the advantage of linear active disturbance rejection control system; Parameter tuning method of the present invention combines the advantage of " Bandwidth Method " and " empirical method ": be convenient to take into account and consider the impact such as sampling step length, noise, without the need to considering the excessive problem causing tracking performance, control performance to be deteriorated of perturbation amplitude.

Accompanying drawing explanation

Fig. 1 is Active Disturbance Rejection Control system architecture schematic diagram.

Fig. 2 is λ _0i(e) function characteristic.

Fig. 3 is tracking error curve figure under microvariations (M '=20).

Fig. 4 is tracking error curve figure under large disturbances (M '=200).

Fig. 5 is tracking accuracy analysis chart.

Fig. 6 is for being controlled quentity controlled variable analysis chart.

Embodiment

Below in conjunction with Fig. 1-Fig. 6 and embodiment 1 and embodiment 2, the invention will be further described.

Embodiment 1

(1) set up Active Disturbance Rejection Control system, it comprises controlled device and automatic disturbance rejection controller; Described automatic disturbance rejection controller comprises Nonlinear Tracking Differentiator, extended state observer and state error Feedback Control Laws;

Described Nonlinear Tracking Differentiator be input as r; The output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) with the output z of extended state observer _i(i=1,2 ..., n) do subtraction relatively after as the input e of described state error Feedback Control Laws _i; The output z of described state error Feedback Control Laws and described extended state observer _n+1do subtraction relatively after value as the first input signal of described extended state observer; The output z of described state error Feedback Control Laws and described extended state observer _n+1first do subtraction to compare, then after carrying out 1/b times of gain, as the input signal of described controlled device; The output y of described controlled device is as the second input signal of described extended state observer; N be greater than 1 positive integer;

For the interference rejection ability of each controller of more outstanding reaction, consider as lower integral tandem type controlled device:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x_2\\ {\stackrel{\&CenterDot;}{x}}_2=u\\ y=x_1+0.01n_0\left(t\right)\end{array}\right.$ (formula 10)

This system export pollute by the white noise of some strength, wherein n ₀t () is equally distributed white noise between ± 1.Given sampling step length h=0.001.

Based on particle swarm optimization algorithm, obtain the parameter optimization table on typical three rank extended state observer parameter (δ=0.01), as shown in table 2.

Table 2 parameter optimization table

Described extended state observer is carried out to the switching of linear/non-linear;

Settling time gets 1s, gets M=30 according to table 2.Determine linear/non-linear extended state observer switchover policy: if be less than 1s settling time the working time of Active Disturbance Rejection Control system; If or the tracing deviation e of described extended state observer is greater than 1; If or total disturbed value M > 30; Then switch to linear extended state observer; Expression formula is

Otherwise, switch to nonlinear extension state observer; Expression formula is

To described state error Feedback Control Laws u ₀carry out the switching of linear/non-linear;

If the working time of Active Disturbance Rejection Control system is less than 1s settling time; If or described extended state observer tracing deviation e is greater than 1; If or the output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) export z with corresponding described extended state observer _ideviation be greater than 1, i.e. (v _i-z _i) > 1, then switch to linear state error feedback control to restrain; Expression formula is

Otherwise, switch to nonlinear state Error Feedback control law; Expression formula is

Wherein, u ₀₁for linear state error feedback control rule; u ₀₂for nonlinear state Error Feedback control law; v _i(i=1,2 ..., n) be the output of Nonlinear Tracking Differentiator, k ' _i, k _ibe respectively positive gain coefficient, α ' _ifor normal number;

(4) if described extended state observer and state error Feedback Control Laws are all linear, then described automatic disturbance rejection controller is linear active disturbance rejection controller, and described Active Disturbance Rejection Control system is linear active disturbance rejection control system; Otherwise described automatic disturbance rejection controller is non-linear automatic disturbance rejection controller, described Active Disturbance Rejection Control system is non-linear Active Disturbance Rejection Control system.

Parameter tuning:

Linear active disturbance rejection controller parameter is adjusted according to " Bandwidth Method ", makes ω _o=30, ω _c=10, so β ' ₀₁=3 ω ₀, ${{\mathrm{\&beta;}}^{\&prime;}}_{02}=3{\mathrm{\&omega;}}_o^2,{{\mathrm{\&beta;}}^{\&prime;}}_{03}={\mathrm{\&omega;}}_0^3,{k^{\&prime;}}_1={\mathrm{\&omega;}}_c^2,$ k′ ₂＝2ω _c。

Non-linear automatic disturbance rejection controller parameter based on experience value and experimental formula determine, that is, δ=0.01, α ₁=1, α ₂=0.5, α ₃=0.25, β ₀₁=3 ω _o,

After determining automatic disturbance rejection controller, t ∈ [5 when the system is operated, 6], get the square wave disturbance of amplitude M '=20, contrast the interference rejection ability that independent linear active disturbance rejection controls (LADRC), independent non-linear Active Disturbance Rejection Control (NLADRC) and the present invention-linear/non-linear Active Disturbance Rejection Control (SADRC), under microvariations, (M '=20) tracking error change curve as shown in Figure 3.As shown in Figure 3, under microvariations (M '=20), the interference rejection ability of the present invention and independent non-linear Active Disturbance Rejection Control is better than independent linear active disturbance rejection and controls; Independent non-linear Active Disturbance Rejection Control is substantially identical with interference rejection ability of the present invention, only in some difference of starting stage; Starting stage is that stability and initial value have nothing to do by the benefit that linear active disturbance rejection controls, and is favourable for system stability.

Keep the parameter constant of automatic disturbance rejection controller, when the system is operated, t ∈ [5,6], gets the square wave disturbance of amplitude M '=200, and under large disturbances, (M '=200) tracking error curve as shown in Figure 4.As shown in Figure 4, under large disturbances (M '=200), independent linear active disturbance rejection controls to be better than independent non-linear Active Disturbance Rejection Control with interference rejection ability of the present invention; Independent linear active disturbance rejection controls substantially identical with interference rejection ability of the present invention.In summary, interference rejection ability of the present invention will be better than the interference rejection ability of the control of independent linear active disturbance rejection and independent non-linear Active Disturbance Rejection Control on the whole respectively.

Embodiment 2

For more clearly reacting each controller tracking accuracy, consider controlled device

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=x2\\ {\stackrel{\&CenterDot;}{x}}_2=20sin\left(\mathrm{\&omega;}t\right)+15sign\left(\sin \left(\mathrm{\&omega;}t\right)\right)+u\\ y=x_1+0.01n_0\left(t\right)\end{array}\right.$

Switch controller design and setting parameter consistent with embodiment 1.

Contrast the control of independent linear active disturbance rejection, independent non-linear Active Disturbance Rejection Control and tracking accuracy of the present invention, tracking error change curve as shown in Figure 5.As shown in Figure 5, independent non-linear Active Disturbance Rejection Control and tracking accuracy of the present invention are better than the tracking accuracy that independent linear active disturbance rejection controls, substantially identical with the tracking accuracy that independent linear active disturbance rejection controls in starting stage the present invention.

Usual control quantitative analysis is usually uncared-for problem.But in engineering practice, if controlled quentity controlled variable significantly high frequency flutter, can adverse influence be brought for topworks.For avoiding the flutter of controlled quentity controlled variable high frequency, nonlinear Feedback Control rule power is unsuitable too small.Fig. 6 is that independent linear active disturbance rejection controls, the controlled quentity controlled variable of independent non-linear Active Disturbance Rejection Control and three kinds of control algolithms of the present invention, as can be seen here, when considering noise, three kinds of controlled quentity controlled variables are substantially suitable, to noise, there is substantially identical enlarge-effect, and all there is not high frequency flutter.

Described in summary, interference free performance of the present invention is better than the control of independent linear active disturbance rejection and independent non-linear Active Disturbance Rejection Control on the whole; Tracking accuracy of the present invention is substantially identical with independent non-linear Active Disturbance Rejection Control, and tracking accuracy of the present invention is better than independent linear active disturbance rejection and controls; In the presence of noise, controlled quentity controlled variable of the present invention not significantly high frequency flutter, controls substantially suitable with independent linear active disturbance rejection; Most importantly parameter tuning of the present invention and stability analysis, makes the application of non-linear Active Disturbance Rejection Control become more simple.

Active Disturbance Rejection Control marrow is estimation and the compensation of being carried out total disturbance by extended state observer, and the object introducing Nonlinear Mechanism promotes control performance further.Therefore, linear active disturbance rejection controls the actual needs that can meet most occasion, if but higher control accuracy, stronger interference rejection ability will be pursued, non-linear Active Disturbance Rejection Control is a kind of effectively selection.For reducing non-linear Active Disturbance Rejection Control threshold further, playing the superiority of Nonlinear Mechanism better, the invention provides a kind of linear/non-linear active disturbance rejection method for handover control.

In automatic disturbance rejection controller, α _i, δ is two undetermined constants: work as α _iduring < 1, this function has " big error, little gain; Little error, large gain " characteristic; δ represents between linear zone, and object avoids zero crossings high-gain to cause flutter.

Through theoretical analysis and a large amount of simulation study, the respective feature of LESO (linear extended state observer), NLESO (nonlinear extension state observer) is summarized as follows:

LESO parameter tuning is convenient, and theoretical analysis is simple, and changes with perturbation amplitude hardly to disturbance tracking performance; Therefore, LESO is popular in actual applications;

NLESO advantage is: parameter efficiency is higher, and tracking accuracy is higher, and response speed is very fast; Shortcoming is: tracking performance is relevant with perturbation amplitude, by noise limit, limited (for significantly disturbance to the tracking power of significantly disturbance, often need the tracking performance that very large parameter just can reach comparatively satisfied, but also significantly can amplify noise, this is the contradiction that cannot be in harmonious proportion simultaneously), simultaneously, parameter tuning is more complicated, and theoretical analysis is more difficult; Therefore, when not giving full play to non-linear effectiveness, its result of use is often not as LESO.

In view of LESO, NLESO feature separately, a kind of naturally idea adopts LESO when perturbation amplitude is larger, switches to NLESO, so just can play LESO, NLESO advantage separately when amplitude is less.

Corresponding states Error Feedback control law, as α ' _iduring < 1, be non-smooth feedback, its efficiency is good more than smooth feedback, and error attenuated is fast, and antijamming capability is strong.But in actual applications, parameter tuning relative complex, stability analysis is also comparatively difficult, and easily causes controlled quentity controlled variable flutter during small-signal, so, still in the majority to use Linear Control to restrain.The switching condition of linear/non-linear control law inherits the switching condition of LESO/NLESO, also sum functions fal (e, α _i, δ) and characteristic is relevant, both in order to ensure stability, is also to improve control performance.

This switch controller overall thought is: in the control starting stage, utilize linear active disturbance rejection controller roughly to follow the tracks of reference input, then switch to non-linear automatic disturbance rejection controller, to improve tracking accuracy and interference rejection ability; When disturbance is larger, or output state evaluated error is larger, or input signal and each rank differential signal thereof depart from corresponding extended state observer output state when estimating far away, are ensure system stability and control performance, should switch to linear active disturbance rejection controller; Certainly, if controlled device condition of work is not harsh, non-linear automatic disturbance rejection controller can be adopted separately.

In theory, this switch controller has the advantage of linear/non-linear automatic disturbance rejection controller concurrently, but wants real practical implementation, also needs parameter tuning and method for analyzing stability easily.Parameter tuning:

The parameter tuning of this switch controller neutral line automatic disturbance rejection controller is easily resolved by " Bandwidth Method " or its modified, and difficult point is still the parameter tuning of non-linear automatic disturbance rejection controller.At present, in non-linear automatic disturbance rejection controller parameter tuning, a large amount of work is had.

Non-linear automatic disturbance rejection controller parameter tuning method mainly can be divided into " empirical method ", artificial intelligence method and additive method.The relation arranged with Fibonacci numbers based on the experimental formula adopting the power form of step-length to represent and parameter that in " empirical method ", the foremost Han Jingqing of being researcher proposes, facilitates and reference for parameter tuning provides.These two kinds of methods are all relatively simple, and obtain based on certain performance index simulation optimization, and by emulating the summary of experience obtained in a large number.But must consider bandwidth, noise, disturbance amplitude in the actual middle controller parameter of engineering and adopt the factors such as step-length and compromise, therefore, the parameter obtained based on ideal situation (not considering noise) is still subject to certain restriction in practical application.Parameter optimization method based on artificial intelligence etc. occupies half of the country, dynamic parameter such as based on neural network is adjusted, based on the parameter tuning of immune bifurcation Particle Swarm, based on the parameter tuning of chaotic particle swarm optimization, based on the Parameter Self-learning algorithm in conjunction with continuous action intensified learning device framework; Etc..Although these optimization methods can ensure good effect in theory, practice is got up more loaded down with trivial details, and the restrictions such as less consideration bandwidth, noise and employing step-length, not there is generality, not easily by engineering reality is accepted.In addition, the parameter tuning method etc. based on time scale is also had.

According to the inherent law of non-linear automatic disturbance rejection controller parameter, and combine " Bandwidth Method " and propose a kind of parameter tuning method being applicable to engineering debug.

Get special case α _i=0.25, δ=0.05, λ _0ie () function characteristic as shown in Figure 2.

As shown in Figure 2, λ _0ie () function is a constant value in linear interval δ, that is, be greater than δ, increase along with error e and reduce, i.e. " big error, little gain; Little error, large gain ", for method for handover control, when | during e|>1, | λ _0i| <1, gain is less, therefore using | e|>1 is suitable as switching condition.Therefore can think that non-linear Active Disturbance Rejection Control is the linear active disturbance rejection control of a variable element.The non-linear ESO stability of such complexity and the stability of switched system prove, can solve comparatively simply by this conversion.

In classical control theory, known increase system open loop gain can reduce steady-state error.The parameter efficiency height of nonlinear extension state observer and Nonlinear control law and non-linear λ _0i(e), λ _i(e _i) the function characteristic with " little error, large gain " is closely-related.Can by β _0iλ _0i(e), k _iλ _i(e _i) be interpreted as linear active disturbance rejection controller parameter.Therefore, in conjunction with linear automatic disturbance rejection controller " Bandwidth Method ", following parameter tuning principle is proposed:

1) linear active disturbance rejection of adjusting controling parameters.Consider sampling step length, noise etc., observer and controller bandwidth of adjusting respectively, is designated as: ω _o, ω _c, correspondingly, parameter is designated as: β ' _0i, k ' _i;

2) cardinal rule of non-linear automatic disturbance rejection controller parameter choose: β _0i(λ _0i) _min< β ' _0i< β _0i(λ _0i) _max, k _i(λ _i) _min< k ' _i< k _i(λ _i) _max.This principle is consider system performance, also needs for guarantee system stability simultaneously.

3) to adjust extended state observer parameter.According to Fig. 2, between linear zone, δ should not get excessive, and the advantage of excessive non-linear gain has been lost; But can not be too small, the too small system that easily causes is unstable; Generally get 0.01 < δ < 0.1, δ=0.01 to be relatively suitable for.Power general satisfaction 0 < α _n+1< ... < α ₂< α ₁< 1, usually get empirical value α ₁=1, α ₂=0.5, α ₃=0.25, α ₄=0.125.In fact the three above ESO's in rank is less, and High order Plant is by multiple ESO series connection or utilize depression of order automatic disturbance rejection controller to control according to Relative order concept.Therefore, the key parameter that remaining needs are adjusted in switching controls, for the situation that disturbance amplitude is larger, LESO is adopted to estimate and compensate; And NLESO is only used for estimating disturbance that is less, amplitude known (artificially arranging), therefore, the key factor considered is needed when sampling step length, noise etc. are only optimum configurations.Reference Han Jingqing researcher obtains some conventional parameter optimization values based on the thinking of simulation optimization.Based on particle swarm optimization algorithm, obtain typical three rank ESO parameter (δ=0.01) as shown in table 2, and matching obtains experimental formula.As can be seen here, the optimum configurations of nonlinear extension state observer is by tabling look-up or utilizing experimental formula just can solve well.This parameter list can be improved further as required.

By matching, above-mentioned publicity table is roughly satisfied

β ₀₁＝3ω _o， ${\mathrm{\&beta;}}_{02}=3{\mathrm{\&omega;}}_o^2/5,{\mathrm{\&beta;}}_{03}={\mathrm{\&omega;}}_o^3/9.$

Note: sampling step length is larger, the disturbance amplitude that NLESO can follow the tracks of is less; In addition, suitably β is increased ₀₃although optimality criterion can be better, easily cause the hyperharmonic of total disturbance estimated value to vibrate, and cause the vibration of controlled quentity controlled variable further, therefore, get proper.

4) Nonlinear control law parameter tuning.Although power α ' _iobtain less, error attenuated speed is faster, and antijamming capability is stronger, but too small power α ' _ithe high frequency flutter of controlled quentity controlled variable can be caused, often harmful effect is brought for actual topworks.For second nonlinear control law, get α ' ₁=0.75, α ' ₂=1.5, error attenuated speed and controlled quentity controlled variable are all more gratifying.For k _i, still can set according to " Bandwidth Method ", i.e. k _i=k ' _i.Also can nearby adjust slightly.

In sum, the present invention combines the advantage of " Bandwidth Method " and " empirical method ": be convenient to take into account consider and adopt the impact such as step-length, noise; The parameter of non-linear automatic disturbance rejection controller is by tabling look-up or utilizing experimental formula just can solve, without the need to considering the excessive problem causing tracking performance, control performance to be deteriorated of perturbation amplitude.Like this, non-linear automatic disturbance rejection controller parameter tuning becomes simple and easy.

The above embodiment is only the preferred embodiments of the present invention, and and non-invention possible embodiments exhaustive.For persons skilled in the art, to any apparent change done by it under the prerequisite not deviating from the principle of the invention and spirit, all should be contemplated as falling with within claims of the present invention.

Claims (7)

1. a linear/non-linear Active Disturbance Rejection Control systematic evaluation control method, is characterized in that comprising the steps:

(1) set up Active Disturbance Rejection Control system, it comprises controlled device and automatic disturbance rejection controller; Described automatic disturbance rejection controller comprises Nonlinear Tracking Differentiator, extended state observer and state error Feedback Control Laws;

Described Nonlinear Tracking Differentiator be input as r; The output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) with the output z of extended state observer _i(i=1,2 ..., n) do subtraction relatively after as the input e of described state error Feedback Control Laws _i; The output z of described state error Feedback Control Laws and described extended state observer _n+1do subtraction relatively after value as the first input signal of described extended state observer; The output z of described state error Feedback Control Laws and described extended state observer _n+1first do subtraction to compare, then after carrying out 1/b times of gain, as the input signal of described controlled device; The output y of described controlled device is as the second input signal of described extended state observer; N be greater than 1 positive integer;

(2) described extended state observer is carried out to the switching of linear/non-linear; The relation of sampling step length h, white noise amplitude and total disturbed value is as shown in table 1;

The corresponding relation of table 1 sampling step length h, white noise amplitude and total disturbed value

If the working time of Active Disturbance Rejection Control system is less than settling time; If or the tracing deviation e of described extended state observer is greater than 1; If or the total disturbed value corresponding with sampling step length h, white noise amplitude belongs to the scope shown in table 1; Then switch to linear extended state observer; Expression formula is

$\left(\begin{array}{c}e=z_1-y\\ {\stackrel{\&CenterDot;}{z}}_1=z_2-{{\mathrm{\&beta;}}^{\&prime;}}_{01}\&CenterDot;e\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-{{\mathrm{\&beta;}}^{\&prime;}}_{02}\&CenterDot;e\\ .\\ .\\ .\\ {\stackrel{\&CenterDot;}{z}}_n=z_{n+1}-{{\mathrm{\&beta;}}^{\&prime;}}_{0n}\&CenterDot;e+b\&CenterDot;u\\ {\stackrel{\&CenterDot;}{z}}_{n+1}=-{{\mathrm{\&beta;}}^{\&prime;}}_{0\left(n+1\right)}\&CenterDot;e\end{array}\right.;$ (formula 1)

Otherwise, switch to nonlinear extension state observer; Expression formula is

$\left(\begin{array}{c}e=z_1-y\\ {\stackrel{\&CenterDot;}{z}}_1=z_2-{\mathrm{\&beta;}}_{01}\&CenterDot;fal(e,{\mathrm{\&alpha;}}_1,\mathrm{\&delta;})\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-{\mathrm{\&beta;}}_{02}\&CenterDot;fal(e,{\mathrm{\&alpha;}}_2,\mathrm{\&delta;})\\ .\\ .\\ .\\ {\stackrel{\&CenterDot;}{z}}_n=z_{n+1}-{\mathrm{\&beta;}}_{0n}\&CenterDot;fal(e,{\mathrm{\&alpha;}}_n,\mathrm{\&delta;})b\&CenterDot;u\\ {\stackrel{\&CenterDot;}{z}}_{n+1}=-{\mathrm{\&beta;}}_{0\left(n+1\right)}\&CenterDot;fal(e,{\mathrm{\&alpha;}}_{n+1},\mathrm{\&delta;})\end{array}\right.;$ (formula 2)

In (formula 1) and (formula 2), y is the output of described controlled device, and u is the control inputs of controlled device, z _l(i=1,2 ..., n+1) be the output of described extended state observer; β ' _oi(i=1,2 ..., n+1) and be the gain of described linear state observer; β _oi(i=1,2 ..., n+1) and be the gain of described nonlinear extension state observer; E is the tracing deviation of described extended state observer; Fal (e, α _i, δ) and be nonlinear function, wherein i=1,2 ..., n+1; Mathematic(al) representation is as follows:

$\mathrm{fal}(e,{\mathrm{\&alpha;}}_i,\mathrm{\&delta;})=\left\{\begin{array}{cc}e/{\mathrm{\&delta;}}^{1-{\mathrm{\&alpha;}}_i}& \left|e\right|\&le;\mathrm{\&delta;}\\ {\left|e\right|}^{{\mathrm{\&alpha;}}_i}\mathrm{sgn}\left(e\right)& \left|e\right|>\mathrm{\&delta;}\end{array}\right.$ (formula 3)

In (formula 3), α _i, δ is normal number respectively; Sgn () represents sign function;

(3) described state error Feedback Control Laws is carried out to the switching of linear/non-linear;

If the working time of Active Disturbance Rejection Control system is less than settling time; If or described extended state observer tracing deviation e is greater than 1; If or the output v of described Nonlinear Tracking Differentiator _i(i=1,2 ..., n) export z with corresponding described extended state observer _ideviation be greater than 1, i.e. (v _i-z _i) > 1, then switch to linear state error feedback control to restrain; Expression formula is

$u_{01}=\underset{i=1}{\overset{n}{\&Sigma;}}{k^{\&prime;}}_i(v_i-z_i);$ (formula 4)

Otherwise, switch to nonlinear state Error Feedback control law; Expression formula is

$u_{02}=\underset{i=1}{\overset{n}{\&Sigma;}}{k}_{i}fal(v_i-z_i,{{\mathrm{\&alpha;}}^{\&prime;}}_i,\mathrm{\&delta;});$ (formula 5)

Wherein, u ₀₁for linear state error feedback control rule; u ₀₂for nonlinear state Error Feedback control law; v _i(i=1,2 ..., n) be the output of Nonlinear Tracking Differentiator, k ' _i, k _ibe respectively positive gain coefficient, α ' _ifor normal number;

(4) if described extended state observer and state error Feedback Control Laws are all linear, then described automatic disturbance rejection controller is linear active disturbance rejection controller, and described Active Disturbance Rejection Control system is linear active disturbance rejection control system; Otherwise described automatic disturbance rejection controller is non-linear automatic disturbance rejection controller, described Active Disturbance Rejection Control system is non-linear Active Disturbance Rejection Control system.

2. a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method according to claim 1, is characterized in that: described β ' _0i, k ' _iobtained by " Bandwidth Method ", i.e. β ' _0ifor by polynomial expression (s+ ω _o) ⁿ⁺¹s after launching ^n+1-ithe coefficient of item, wherein i=1,2 ..., n+1; β ' _0ifor ω _ofunction;

K ' _ifor by polynomial expression (s+ ω _c) ⁿs after launching ^i-1the coefficient of item, wherein i=1,2 ..., n; K ' _ifor ω _cfunction;

Wherein, ω _cfor linear active disturbance rejection controller bandwidth, ω _c> 0; ω _ofor the bandwidth of linear extended state observer, ω _o> 0.

3. a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method according to claim 2, is characterized in that: n=3; β ' ₀₁=3 ω ₀, k ' ₂=2 ω _c.

4. a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method according to claim 1, is characterized in that: if described in order

$fal(e,{\mathrm{\&alpha;}}_i,\mathrm{\&delta;})=\frac{fal(e,{\mathrm{\&alpha;}}_i,\mathrm{\&delta;})}{e}e={\mathrm{\&lambda;}}_{0i}\left(e\right)e,$ (formula 6)

Then obtained by (formula 2) and (formula 6):

$\left\{\begin{array}{c}e=z_1-y\\ {\stackrel{\&CenterDot;}{z}}_1=z_2-({\mathrm{\&beta;}}_{01}\&CenterDot;{\mathrm{\&lambda;}}_{01}(e\left)\right)e\\ {\stackrel{\&CenterDot;}{z}}_2=z_3-({\mathrm{\&beta;}}_{02}\&CenterDot;{\mathrm{\&lambda;}}_{02}(e\left)\right)e\\ .\\ .\\ .\\ {\stackrel{\&CenterDot;}{z}}_n=z_{n+1}-({\mathrm{\&beta;}}_{0n}\&CenterDot;{\mathrm{\&lambda;}}_{0n}(e\left)\right)e+b\&CenterDot;u\\ {\stackrel{\&CenterDot;}{z}}_{n+1}=-({\mathrm{\&beta;}}_{0(n+1)}\&CenterDot;{\mathrm{\&lambda;}}_{0(n+1)}(e\left)\right)e\end{array}\right.$

Formula (7)

Make e _i=v _i-z _i; Then to the fal (v in (formula 5) _i-z _i, α ' _i, δ) do as down conversion:

$fal(v_i-z_i,{{\mathrm{\&alpha;}}^{\&prime;}}_i\mathrm{\&delta;})=fal(e_i\&CenterDot;{{\mathrm{\&alpha;}}^{\&prime;}}_i,\mathrm{\&delta;})=\frac{fal(e_i,{{\mathrm{\&alpha;}}^{\&prime;}}_i,\mathrm{\&delta;})}{e_i}ei$

Order

$\frac{fal(e_i,{{\mathrm{\&alpha;}}^{\&prime;}}_i,\mathrm{\&delta;})}{e_i}={\mathrm{\&lambda;}}_i\left(e_i\right)$

Then

Fal (v _i-z _i, α ' _i, δ) and=λ _i(e _i) e _i(formula 8)

Obtained by (formula 5) and (formula 8):

$u_{02}=\underset{i=1}{\overset{n}{\&Sigma;}}{k}_i{\mathrm{\&lambda;}}_i\left(e_i\right)e_i$ (formula 9)

From (formula 6), (formula 7), (formula 8) and (formula 9), λ _0i(e) e, λ _i(e _i) e _ibe variable-gain linear function; (formula 7) is variable-gain extended state observer; (formula 9) is variable-gain linear state error feedback control rule; By λ _0ie () is abbreviated as λ _0i, λ _i(e _i) be abbreviated as λ _i;

Then β _0i, β ' _0iwith λ _0iand k _i, k ' _iwith λ _ibetween pass be:

β _0i·(λ _0i) _min＜β′ _0i＜β _0i·(λ _0i) _max，k _i·(λ _i) _min＜k′ _i＜k _i·(λ _i) _max。

5. a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method according to claim 4, is characterized in that: 0.01≤δ≤0.1,0 < α _n+1< ... < α ₂< α ₁< 1,0.5 < α ' ₁< 1,1 < α ' ₂< 2.

6. a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method according to claim 5, is characterized in that: δ=0.01.

7. a kind of linear/non-linear Active Disturbance Rejection Control systematic evaluation control method according to claim 6, is characterized in that: n=3; α ₁=1, α ₂=0.5, α ₃=0.25, β ₀₁=3 ω _o, k ₂=2 ω _c, α ' ₁=0.75, α ' ₂=1.5; Wherein, ω _cfor linear active disturbance rejection controller bandwidth, ω _c> 0; ω _ofor the bandwidth of linear extended state observer, ω _o> 0.