Random disturbances in machines and production processes

Transcription

1 VŠB echnical Uniersit of Ostraa Facult of mechanical Engineering Department of Control Sstems and Instrumentation Random disturbances in machines and production processes Prof. Ing. Jiří ůma, CSc.

2 Outline Introduction properties of random ariables Losses as a consequence of the random disturbance influence on production processes Loss reduction can be reached b Homogenization of input raw materials Controlling the production processes Homogenization heaps, continuous drum mixers and bins Control sstems

3 Statistical properties of random errors and disturbances ime histor Autocorrelation function Power spectral densit R ( τ) δ( τ) S( ω) time -τ τ -ω ω α τ R( τ) e α S ω ω α ( ) tim e -τ τ -ω ω lag frequenc Ealuation the ariance of the linear dnamic sstem output x ( jω) x ( jω) Sxx( ω) dω π π j c x ( z) ( z ) S ( z) x xx z dz

4 Losses as a consequence of the random disturbance influence on the production process ) Reduction the percentage of wastes 3) Loss reduction as a consequence of nonlinearit Example: controlled ariable ) Reduction the margin of materials controlled ariable controlled ariable Loss is proportional to the ariance of the fuel-air ratio.

5 Reduction ariation of production processes Homogenisation heaps, natural non-homogeneit x(m) otal mass... M Number of laers N Model (moing aerage) Homogenisation principles N * ( m) x ( m i M N ) N i N * x N N k (m) ransfer function for ariances ( km N ) ρ ( km N ) N log x x(m) M/N M/N 3M/N 4M/N m x * (m) M/N M/N 3M/N 4M/N m N ρ ( m) exp( α m ) Ealuation * ρ log( N ) ( m)... Homogenisation efficienc Limit efficienc f ( α M )

6 Reduction ariation of production processes wo-component homogenisation heap Input flow of different ores X a X b Component alternation: Modelled b the Marko chain p aa a a b a a b a. Rich ore b. Poor ore a p ab p ba b p bb m Homogenisation efficienc log x Quotient q has to be minimal q p ba pa p p Optimal input flow alternation: p p p, p a b laers ba ab b q bb Number of batches: 5

7 Reduction ariation of production processes Continuous mixing drums and bins x(t) I (s) k I (s) (t) Parameter identification Impulse response ~ probabilit densit function g () t k k ( k )! t k exp k t, t x(t), (t).. Some component content Ideal mixer Y () () s s I X () s s.. ime constant (retain time interal) Real mixer k Y () () s I s X () s ( s k) k Peclet number Mixing efficienc in the standard deiation ratio k,8 x,6 k 5,4, k Pe L D k α, k

8 Random disturbance and measurement error in automatic control sstems w Disturbance R S (ω) ω S δ Frequenc spectrum of the random process (pink noise) and step function S(ω) ω α ω Measurement error S δδ (ω) ω -3α -α -α α α 3α ω Controller snthesis is focused on minimizing the effect of the disturbance and measurement error ariance on the controlled ariable ariance

9 Analsis of an effect resulting from controlling the static sstem with a transport dela b the PS-controller w - e PS Controller output: u Assessing the relationship between the disturbance and measurement error standard deiations: B ( E)( KE) 3 ( E( K R ) KE )( K )( K R) δ Let : E exp α ransfer function relating the disturbance ariance to the controlled ariable ariance : B δ u k Ke ( ) k R k i e i K R B 3 B 3 R K B - K R( K ) ( K )( K R) α α

10 Analsis of an effect resulting from controlling the integration sstem with a transport dela onl b using the feedback P-controller ŽH - K R..ř. Q i Q o - P Qo Let : Qo Qi Qo K E exp( α ), K K ransfer function relating the disturbance ariance to the controlled and actuating ariable ariance : ( E K )( K ) E( K ) ( K )( K )( E E K ) ( K )( E K ) KE( K ) ( K )( K )( E E K ) R E E Qi Qo K

11 Analsis of an effect resulting from controlling the integration sstem with a transport dela b the feedback and forward P-controllers ŽH - K R..ř. δ Q i - Q o Qo P A Let : Qo ( ) E exp α δ K K Parameter assessing the relationship between the standard deiation of disturbance and measurement error: A Qo ransfer function relating the disturbance ariance to the controlled ariable ariance : K( E K ) KE( K ) ( K )( K )( E E K ) A K R - - A 5 5 K opt E

12 Analsis of an effect resulting from controlling the first order sstem b a PI-controller w - e PI ransfer functions for the controller and plant: () R s K S () s I s s Controller tuning b emploing the inerse dnamic : I Let : ransfer function relating the disturbance ariance to the controlled ariable and actuating ariable ariance : α I u K( K α αi K ) ( K α )( α ( K ) K ) α K ( K α )( α ( K ) K ) α K K w ϕ α I K,3,,3 3 φ 3 φ u I,3,,3 φ, w