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.
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
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
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.
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 )
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
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
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 3..5..5..5 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
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.8.6.4. 3 -.8.6.4. B 3 B 3 R K B - K R( K ) ( K )( K R) α α
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 - -.7 E E.8.9.9 Qi Qo..4.6.8 K
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..4.6.8 5 E
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 - - -3 - - K,3,,3 3 φ 3 φ u I,3,,3 φ, w