EA025476B1

EA025476B1 - Robust control system

Info

Publication number: EA025476B1
Application number: EA201400602A
Authority: EA
Inventors: Газанфар Арастун оглы Рустамов; Рустам Газанфар оглы Рустамов
Original assignee: Азербайджанский Технический Университет
Priority date: 2013-12-17
Filing date: 2013-12-17
Publication date: 2016-12-30
Also published as: EA201400602A1

Abstract

The invention relates to the field of automatic control and can be used for control of non-linear, non-stationary, a priori indefinite, one-dimensional objects having parameters and structure that vary over a wide range in unknown manner, in particular, for automatic control of mechanical and electromechanical facilities, airborne vehicles and production processes in various sectors of industry. The objective of the invention is higher operation speed, higher accuracy of tracking the reference path in operation with a priori indefinite dynamic objects, and broadening the area of the system application. The objective of the invention is attained by that in a control system comprising a comparison element, an aggregated variable forming unit, a repeated differentiation unit, a controller, a discrete integral I-observer, an adder, a power amplifier and a controlled object, the discrete integral I-observer is made as an analogue proportional-integral PI-observer with an adjustable amplification factor and with the possibility of unlimited amplification of the output signal, the input of the PI-observer being connected to the output of the aggregated variable forming unit, and the output being connected to the third input of the adder, wherein the PI-observer is defined by the following analytical expression:whereis evaluation of the object dynamics;is aggregated variable; k is regulation parameter; N is unlimitedly increasable amplification factor.

Description

The invention relates to the field of automatic control and can be used to control non-linear non-stationary a priori uncertain one-dimensional objects, the parameters and structure of which vary widely in an unknown way, in particular for automatic control of mechanical, electromechanical objects, aircraft, as well as technological processes in various industries industry. The objective of the invention is to increase the speed and accuracy of tracking the reference trajectory (tasks) when working with a priori uncertain dynamic objects and expanding the scope of the system. The objective of the invention is solved due to the fact that in the control system containing the comparison element, the block forming the aggregated variable, the unit of multiple differentiation, controller, discrete integral I-observer, adder, power amplifier, control object, discrete integral And-observer made in the form of analog proportional-integral PI-observer with adjustable gain with the possibility of unlimited amplification of the output signal, with the PI-observer input connected to output b localization of the aggregated variable, and the output is connected to the third input of the adder, while the PI-observer is determined / () = N · [ί + to the following analytical expression: ° where ^ b is the evaluation of the dynamics of the object;

025476 B1

- S Έ Έ ё ί ί ^ ^ М ^{7 1 2} - aggregate variable; k - setting parameter; Ν-gain with the possibility of unlimited increase.

The invention relates to the field of automatic control and can be used to control non-linear non-stationary a priori uncertain one-dimensional objects, the parameters and structure of which vary widely in unknown ways, in particular, to automate mechanical, electromechanical objects, aircraft, as well as technological processes in various industries industry.

The closest to the claimed technical essence is a robust control system with a dynamics observer, which is the prototype of the proposed [1]. The known system contains a comparison element, the output of which is simultaneously connected to the inputs of an aggregating variable shaping unit and a controller, the output of which is connected to the first input of the adder, the output of which is connected to the input of the power amplifier, the output of which is connected to the input of the object, the output of which is connected to the second negative input of the element comparison, the first positive input of which is connected to the master signal, which is also connected to the input of the block of multiple differentiation, the output of which is connected to the second The input to the adder, the third input of which is connected with the output of the discrete integrated AND observer, whose input is connected with the output of the aggregate variable generation unit.

The disadvantage of this system is that the speed (convergence) and accuracy of the evaluation of the dynamics of an object in a discrete integral I-observer without proportional (P) part is low. In addition, the integration step, which is a setting parameter, is rigidly embedded in the design of the observer and is not subject to operational reconfiguration, depending on the specific object. The absence of a setting parameter for a discrete I-observer narrows the scope of this system.

The objective of the invention is to increase the speed and accuracy of tracking the reference trajectory (tasks) when working with a priori uncertain dynamic objects and expanding the scope of the system.

The objective of the invention is solved due to the fact that in a robust control system with an observer for a priori uncertain objects containing a comparison element, the output of which is simultaneously connected to the inputs of the aggregate variable generation unit and the controller, the output of which is connected to the first input of the adder, the output of which is connected to the amplifier input power, the output of which is connected to the input of the object, the output of which is connected to the second negative input of the comparison element, the first positive input of which is connected to the main signal, which is also connected to the input of the multiple differentiation unit, the output of which is connected to the second input of the adder, the third input of which is connected to the output of the discrete integrated I-observer, the input of which is connected to the output of the aggregating variable generation unit, the discrete I-observer is made in the form of analog proportional-integral PI-observer, with adjustable gain, with the possibility of unlimited amplification of the output signal, with the PIN of the watchman connected to the output of the bl the formation of an aggregated variable, and the output is connected to the third input of the adder, while the PI-observer is determined by the following analytical expression:

I / (0 = * [* + * / "*], o

where ΐ - assessment of the dynamics of the object; ί (ί) = с, е + с, е + ... + ¹² - aggregate variable;

k - setting parameter;

Ν-gain with the possibility of unlimited increase.

By performing a discrete I-observer in the form of an analog proportional-integral PI-observer, with adjustable gain, with the possibility of unlimited amplification of the output signal, it is possible to reduce the influence of all possible uncertainties to an arbitrarily small amount when controlling a priori undefined objects. As a result, with a sufficiently large value of the gain, the error of tracking the reference trajectory tends to zero, therefore, the accuracy and speed of the system, characterized by the time to establish the transient response, increase. In addition, the ability to adjust the gain of the PI observer for a specific object expands the scope of the system. All this shows that the new features listed above of the declared object are significant, which influence the solution of the problem of the invention, improving the speed and accuracy of tracking the reference trajectory (task) when working with a priori uncertain dynamic objects, i.e. these signs with achievable technical result are in a causal relationship.

FIG. 1 and 2 are block diagrams of the proposed robust control system and an analog proportional-integral PI observer.

The control system contains a comparison element 1, a unit for forming 2 an aggregated variable, a unit of multiple differentiation 3, a regulator 4, an analogue PI observer 5, a second adder 6, a power amplifier 7, an object 8.

- 1 025476 ί (ί) = c, e + c ₂ e + ... + e ^c

The observer contains amplifiers 9, 12, integrator 10, adder 11.

In this case, the following notation:

VA (1) - reference trajectory (task); at (1) - adjustable value; e = y, _y y - regulation error;

, (I-1)

- aggregate variable;

/ (!) - evaluation of the dynamics of an uncertain object; y ⁽ⁿ⁾ and (4) is the derivative of the ηth order of the master signal; and _{with the} signal control regulator;

and - the resulting amplified control signal.

The control system works as follows. The signal y (1) to regulate the output object 8 is supplied to the negative input of the second comparing element 1, on the first positive input of which is supplied _as a drive signal y (1), formed as a result of an error signal e (1) = va (1) = y ( 1), which is fed to the input of the block forming the aggregated variable 2, where the aggregated variable e, e + c, <is formed; + ... + <? '', which is fed to the input of the analog PI-observer 5, where the odds of / (/) = [s + k pe //] N is measured by the evaluation ° of the dynamics of an uncertain object (Fig. 2). Next, this signal enters the adder 6 and compensates for the a priori indeterminate dynamics G (1) of the real object, arriving through feedback. Error e (1) from the output of the comparison element 1 is also fed to the input of the PD controller 4, in which the control signal p · is generated

For η = 2, the settings are a ₀ = k-C1, and ₁ = k + c ₁ , where η is the order of the object. The parameters k, c _{1 of the} regulator are determined on the basis of the required establishment time of the transient response, which determines the system speed. For <\ = 2% permissible regulation error:

A =] - 1n (501 e (0) |) |, (1) where e (0) = y ₄ (0) -y (0) is the initial value of the error. With (1) the monotonicity of the transient response is ensured, i.e. the system has no overshoot, σ = 0%. The increase in speed is associated with a decrease in f. For small φ, the values of the parameters k and c ₁ increase. For example, when e (0) = 1, f = 1 s, the value of k = c1 "4; when e (0) = 1, f = 0.1 s., the value of k = c ₁ "40.

The output of the controller 4 is connected to the first input of the adder 6, to the second input of which, from the output of the multiple differentiation unit 3, a signal y ⁽ⁿ⁾ a (1) is generated, which is formed on the basis of the incoming driving signal uTcf). With piecewise constant setting signals, this block is not used. The adder 6 the summation of input signals and forming the resulting management ^and ~ ^and? ⁺ Ud ⁺ / (?)> Which, after amplification, K = 1 / b in power amplifier 7 in the form u = K and _c is applied to control object 8. Here b is the known gain of the object.

The observer’s tuning parameter N is associated with the dominant uncertainty of the real object and is determined experimentally for a specific object.

The observer works as follows. From the output of block 2, the signal 8 (D) after amplification in the amplifier 9 is fed to the input of the integrator 10.

I

I / = k ^ L

The output signal of the integrator about is fed to the adder 11, to the second input of which is fed ί

signal 8. After summation, the resulting signal o after amplification in the amplifier / (/) = TC + k p *] by multiplying by a sufficiently large number N in the form ° is fed to the adder 6 to compensate for the dynamics G (1) of the real object coming through feedback. As a result, the dynamics Γ (Φ) of an indefinite object is weakened N times: ε = Γ (ΐ) / Ν, where Ж 100 = 500.

In the limit there is a ratio:

Nt {D>)} = / (/).

Given this relationship, with N g, a system with a priori undefined object is converted into a deterministic system, described by a linear equation of the first order:

ίΖϊ / ί / ί + & ϊ (ί) = 0, (0) = ίθ. (2)

Thus, as a result of evaluating and suppressing by the PI observer the dynamics of G (F) a priori undefined object, the original undefined system is converted to a deterministic system (2), which does not include the details of the a priori undefined object. This feature allows

- 2 025476 set the desired quality indicators of the real system (establishment time I, and the type of the reference trajectory y (1)) for the model (2) and with a sufficiently large value of the gain N, significantly increase the speed and accuracy of tracking the reference trajectory y <(1 ) (see Appendix 1, Fig. 1 (a)). In addition, the ability to reconfigure the gain N of the observer expands the scope of the system.

This device can be implemented industrially on standard elements of analog and digital technology.

Computer simulation results

In general, a priori undefined object is described by a differential equation of the form:

Y ^(l) = DRLO) + b, y = {y, y, ..., y ⁽ⁿ ~ ' ^} } ^t = {χλ, χ1, ..., χ „) ^τ eE - <e [0, oo) (3) where 'νν> · ι /> - is the state vector available for measurement or assessment;

y - controlled output; and - scalar control;

Г (·) is a bounded unknown non-linear non-stationary function (dynamics of an indefinite object);

ν (ΐ) is an unknown perturbation;

B = cong> () is the known gain of the object.

The component G (yy (0) of model (3) is a source of uncertainty, but it may also contain known components. In any case, G (y ^ (4)) is taken as a potential source of parasitic dynamics and is suppressed, that is, its value decreases in N times, where N is a sufficiently large number and is determined experimentally.

In the case of the robust system under consideration, the control object is an asymmetric nonlinear generator, described by the Van der Pol equation [2]:

„2 у = -р (у - 1) у - у + и + т + 005 (20 /).

We assume that the object has interval uncertainty, i.e. has coefficients varying in an unknown way in a given interval.

Object order η = 2 ^ = ·

The external perturbation ν (ΐ) enters linearly into the equation. The nominal values of the parameters are μ = 10, then = 2. We assume that with the operation of the system, the nonlinearity coefficient μ and the constant component m of the disturbance change by ± 50% of the nominal: -5 <μ <+15; -1 <t <+3.

When μ <0, the object is unstable, i.e. there is a structural change.

The reference trajectory is defined _as a harmonic function y (1) = 51P (21). The initial condition of the reference path and object, respectively at ₄ (0) = (0.0) and y ^m (0) = (3.0) ^t. We determine the initial value of the error e (0) = y, | (0) -y (0) = (- 3.0) ' ^g ^ C] (0) = -3.

Let it be required to ensure the establishment time 1, = 1 s. with δ, = 2%. From (1) we determine the parameters k = 5 with ₁ = 5. Regulator settings a ₀ = k · c ₁ = 25, and! = K + s ₁ = 10.

Aggregated variable

The regulator equation and _p = α _χ έ + α ^ = 10e + 25e.

The observer equation:

I / (/) = 200 [5 + 5 pL].

about

FIG. 3 (a) - (d) shows the characteristics of the system while simultaneously changing the parameters within the above limits. Characteristics are obtained at nominal settings with ₁ = 5, k = 5, and ₁ = 10, and ₀ = 25 and N = 200 for 14 discrete combinations of parameters: μ = [-5 -2 0 2 6 10 15 -5 - 2 0 2 6 10 15], m ₀ = [- 1 -0.5 -0.2 0 12 3-1 -0.5 -0.2 0123].

As can be seen from FIG. 3 (a), despite the change in the parameters of the object in a wide range, the deviation of the 14 transient characteristics of (4) from the reference trajectory is ν <ι (1) = 5ίη (2ΐ) for 1> 1, = 1 s. visually imperceptible. This confirms the high accuracy of tracking the reference trajectory y <| (1). In other words, the system has a high degree of robustness.

FIG. 3 (b) and (c), respectively, the families of regulation errors e (1) and control signals and (1) are shown. Circles represent families enlarged 20 times.

FIG. The 3rd (d) shows the actual dynamics Γ (y, ν) = - (Γ (y, ΐ) + ν (ΐ)) and its assessment to Al) when used as part of the control signal y ¹² 'l (1).

As can be seen, with N = 200, the estimation accuracy is very high.

FIG. 4 shows the simulation scheme of the proposed robust control system on the package.

- 3 025476

MayaBshiNpk.

Configuration parameters - Dash: H \ e4-51: Mer 51 / e: 0.001: 8011: o4e2.

Literature

1. Rustamov GA, Gardashov S.G., Rustamov R.G. Stabilization of nonlinear objects based on the Lyapunov function method with nonlinearity and disturbance estimation. Automation and computing. 2010, No. 1, pp.65-73 (prototype).

2. VNeN N. еe1G-Ogdash8aNop ίη (Not Wap 4g Ro1 depegUg.Te Ishuegvyu oG SNeGPI, 2009.

Claims

A robust control system with an observer for a priori indefinite objects, containing a comparison element, the output of which is simultaneously connected to the inputs of the aggregate variable formation unit and the controller, the output of which is connected to the first input of the adder, the output of which is connected to the input of the power amplifier, the output of which is connected to the input of the object, the output of which is connected to the second negative input of the comparison element, the first positive input of which is connected to the master signal, which is also connected to the input multiple differentiation, the output of which is connected to the second input of the adder, the third input of which is connected to the output of the discrete integral I-observer, the input of which is connected to the output of the aggregate variable generating unit, characterized in that the discrete I-observer is made in the form of an analog proportional-integral PI -observer, with adjustable gain, with the possibility of unlimited amplification of the output signal, and the input of the PI observer is connected to the output of the aggregation forming unit variable, and the output is connected to the third input of the adder, while the PI observer is determined by the following analytical expression:

ί / (1) = Ν- [а + к [нН], о

where is the assessment of the dynamics of the object;

- aggregated variable:

k is the setting parameter;

N - gain, with the possibility of unlimited increase.

FIG. one

FIG. 2

- 4,025476

FIG. 3 (a-6). Robust control system simulation scheme while changing parameters

FIG. 4. System characteristics while changing parameters