The control and identification algorithm for devices with differential inductive
sensors. Sergey A. Kochetkov*. Pavel A. Shavrin**. Sergey A. Kiselyov***.
Togliatti ...
Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008
The control and identification algorithm for devices with differential inductive sensors Sergey A. Kochetkov*. Pavel A. Shavrin** Sergey A. Kiselyov*** Togliatti State University, Togliatti, 445667, Russia (Tel: +7 8482 247949; e-mail:
[email protected]*,
[email protected]**,
[email protected]***).
Abstract: The design algorithms of closed loop control system for surface profile measurement device are proposed. Control inputs are used to construct closed loop system, which oriented on identification of desired parameter that is estimated by high gain observer then. As a result, the designed nonlinear system has high selective properties with respect to registered value. The dynamic compensator approach is used to reduce the estimation error. 1. INTRODUCTION
2. PROBLEM STATEMENT
With evolution of scientific and technological progress and aggravation of competitive activity an increase of the requirements with respect to the quality of completed production takes place. Some of the main parameters, which characterize quality of completed production, are roughness, waviness. It is well known that modern contact devices for measuring roughness are based on open loop system. The traditional approach is based on tracing surface by the diamond needle of the sensor (Biryukov G.S. et al., 1987). The needle oscillation is transformed then with the help of sensitive transducer into variation of coil inductances. These inductances are connected into bridge scheme, one of the diagonal of which is supplied by the high frequency AC voltage. The primitive construction of the sensor is presented on the fig. 1. In the next step the signal from the sensor is amplified, detected, and filtered and so on similar to signal processing in the highway of the radio transmission. It is obviously that this approach has significant disadvantage. Parametric and additive disturbances can intervene to the output owing to linear character and the open loop of the processing. Moreover, each element of open loop system adds its own dynamics to the process of signal transmission. Other words, this approach is assigned for measuring values, which are varied slowly. Meanwhile, there is a need to design systems, which allow measuring surface profile with fast dynamic (Morrison E., 1996). Algorithms for design of close loop control system, which have no shortcomings listed up, are proposed in this paper. Basic algorithm consists in using control inputs to form close loop system, which is oriented on necessary parameter, which characterizes the surface profile. Then this parameter is identified by high-gain observer. The synthesized nonlinear system has high selective properties with respect to registered value. Simulation results represent the efficiency of proposed control and identification algorithm. The rate of necessary parameter identification is of order of transient time in the control object.
Let us consider the construction of the primary converter (fig. 1).
978-1-1234-7890-2/08/$20.00 © 2008 IFAC
1 – rotor, 2 – coils, 3 – mount point, 4 – diamond needle, 5 – controlled profile Fig. 1. The construction of the primary converter The differential equations of the sensor can be expressed in the following form (Zhuravlev Yu.N, 2003) ìj& = w ; ï ö M(t ) i22 i12 ï & mgc pk æç ÷+ ; ïw = J + 2 J ç 2 2 ÷ J0 ( d p j ) ( d + p j ) 0 0 0 0 è ø ïï í di1 pi1w i R (d + pj ) u1 (d 0 + pj ) = - 1 1 0 + ; ï k k ï dt d 0 + pj ï di i R (d - pj ) u 2 (d 0 - pj ) pi2w - 2 2 0 + , ï 2 =d 0 - pj k k îï dt
(1)
where m is mass of rotor, k is constant design coefficient, J 0 is inertia moment, d 0 is nominal air gap, M ( t ) is the reaction moment of N .
1809
10.3182/20080706-5-KR-1001.2801
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
It is necessary to note that for such sensor the next inequality is fulfilled
d 0 >> pj .
(2)
One can exclude from system (1) the mechanical equations and get ì ïj& = w ; ï ï di1 = -i1a 1 ( j ,w ) + b1 ( j )u1 ; í ï dt ï di2 ïî dt = -i2a 2 ( j ,w ) + b2 ( j )u 2 ,
(3)
a 1 ( j ,w ) = - pi1w / (d 0 + pj ) + R1 (d 0 + pj ) / k ; a 2 ( j ,w ) = pi2w / (d 0 - pj ) + R2 (d 0 - pj ) / k ; b1 ( j ) = (d 0 + pj ) / k ; b2 ( j ) = (d 0 - pj ) / k .
The substitution (6) to (3) leads to ì di1 = -a 1i1 + m b1b2i2 + b1g 1 ; ïï dt í ï di2 = -a i - m b b i + b g . 2 2 1 21 2 2 îï dt
(7)
Assumption 1. The variables j and w are assumed equal to sufficiently small values. Therefore the parameters a i , bi ( i = 1,2 ) are differed slightly from their nominal quantities
a i* = Rid 0 / k > 0 , bi* = b* = d 0 / k > 0 ( i = 1,2 ) .
where
(8)
One can neglect j и w and get linearized system in the form
The sensor is moved across controlled surface with constant velocity during measurement cycle. The diamond needle 4 touches with roughness and the position of rotor, determined by angle j , is changed. Therefore the parameter j is proportional to the height of controlled profile. Thus, the problem of identification for the value j can be established in view of the system (3) according to measurements of coil currents i1 , i2 . 3. SYNTHESIS OF THE CONTROL STRUCTURE It is well known that modern devices for measurement roughness are based on the principle of amplitude modulation of high frequency alternative voltage, which supplies one of diagonal of differential inductive sensor. This approach is used during many years and has good reputation. Therefore let us use oscillations as the basic mode for operation. So one can consider the system (3) and choose the control inputs u1 , u 2 in such a way, that limit cycle are excited in the following form i12 + i22 = C ,
where m - const , g 1 ,g 2 are new control inputs.
(4)
ì di1 * * 2 * ïï dt = -i1a 1 + ( b ) mi2 + b g 1 ; í ï di2 = -i a * - ( b* )2 mi + b* g . 2 2 1 2 ïî dt
(9)
The characteristic equation of (9)
(
)
p 2 + a 1* + a *2 p + ( b* )4 m 2 = 0 .
(10)
Let us choose the parameter m so that roots of (10) are complex conjugate quantities, i.e.
(
D = a 1* + a *2
)
2
- 4( b* )4 m 2 >
(a
* 1
)
( )
+ a *2 / 2 / b*
2
= k 3 (R1 + R2 ) / 2d 03 .
(11)
It is necessary to note that differential inductive transformer is designed so that it’s inductive reactance greater than resistance for some frequency. Therefore we can choose the parameter m to guaranty (11) always. For example, one can accept m as
( )
2
where C is any constant.
m = 2pf nom / b*
Besides, if the equation (4) is fulfilled, we get that variables i1 , i2 are changed according to sine and cosine laws with am-
With the help of first term in (6) designed system obtain the oscillator property. Now we must choose second control components g 1 ,g 2 in such a way to guaranty that steady limit cycle are excited in (7).
plitude determined by constant C . And in this case i1 and i2 are orthogonal functions and can be used as reference signals for high gain identifier.
(
(5)
where b ( t ) is continuously differentiable, fulfils the equation (4).
)
where i = (i1
(13)
i2 )T .
It’s derivative according to (7) V& ( i ) = -a 1i12 - a 2 i22 + b1i1g 1 + b2 i2 g 2 .
Let us choose the control inputs in combined form ìu1 = mb2 i2 + g 1 ; í îu 2 = - mb1i1 + g 2 ,
(12)
Let us consider function V ( i ) V ( i ) = i12 + i22 - C / 2 ,
It is necessary to note that the non-stationary system di1 / dt = b ( t )i2 ; di2 / dt = - b ( t )i1 ,
= 2pf nom k 2 / d 02 .
Let us choose g 1 , g 2 so that next condition is fulfilled (6)
V& = -V ( i )s( i ) ,
1810
(14)
17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008
where s( i ) is any positive definite function, which is equal to zero in the origin of coordinates. The class of control inputs is determined by (14). We can choose g 1 ,g 2 in the following continuously differentiable form
(( ( (
)) ))
ìï g 1 = a 1Ci1 / b1 i12 + i22 ; í ïîg 2 = a 2Ci2 / b2 i12 + i22 ,
(15)
where C is constant which determine the amplitude of steady oscillations.
)
(
)
V& = -2 a 1i12 + a 2i22 V ( i ) / i12 + i22 . So we get that V& is negative and equal to zero in the origin of coordinates. In this case (4) is invariant set of the system (3) with control inputs (6), (15).
We get valuable control structure in the previous chapter. However it’s hard to realize (6) and (15) accurately in real devices because of unknown parameters a 1 , a 2 , b1 , b2 . Let us show that any limit cycle is excited in the system (3) with the help of control inputs in the form (6), (15) with some positive parameters aˆ 1 , aˆ 2 , bˆ1 , bˆ 2 instead of a 1 , a 2 , b1 , b2 . So according to reasoning we choose u1 , u2 as
( ( )) ( ( ))
(16)
Assumption 2. The variables a 1 , a 2 , b1 , b2 , aˆ 1 , aˆ 2 , bˆ1 , bˆ 2 are changed slowly in comparison with currents i1 , i2 . In other words we can say that a , a , b , b , aˆ , aˆ , bˆ , bˆ are quasi1
2
where I is constant determined by initial condition.
ì æ æ pj& ö÷ pj& ö÷ I sin 2 q - ç a 2 I cos 2 q + ï I& = -ça 1 + ç ÷ ç ˆ 2 kmb1b2 ø 2kmb2 bˆ 1 ÷ø ï è è ï 2 2 ï b1aˆ 1C sin q / bˆ 1 + b2aˆ 2 C cos q / bˆ 2 ; + ï mI b1bˆ 2 sin 2 q + b2 bˆ 1 cos 2 q ï í ææ pj& ö÷ æç pj& ö÷ ö÷ ï & çç ï Iq = -ç ç a 1 + 2 kmb bˆ ÷ + ç a 2 - 2kmb bˆ ÷ ÷ I sin q cos q + 1 2 ø è 2 1 øø èè ï ï æ b1aˆ 1 b2aˆ 2 C sin q cos q ï+ m b b bˆ bˆ I + ç + 1 2 1 2 2 2 ç bˆ ˆ ˆ ï m I b b sin q b b cos q bˆ 2 + 1 2 2 1 è 1 î
1
2
1
where
(17)
ì di* æ b1aˆ 1Ci1* pw ö÷ * ; i1 + m b1b2 bˆ 1bˆ 2 i*2 + ï 1 = -çç a 1 + 2kmb1bˆ 2 ÷ø bˆ 1 m b1bˆ 2 i1* 2 + b2 bˆ 1i*2 2 (18) ï dt è í * æ b2aˆ 2 Ci*2 ï di2 pw ö÷ * ç i2 - m b1b2 bˆ 1bˆ 2 i1* + . ï dt = -ç a 2 ÷ ˆ ˆ 2kmb2 b1 ø b2 m b1bˆ 2 i1* 2 + b2 bˆ1i*2 2 è î
(
)
(
)
According to assumption 1 the next inequality is right
)
