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# More lecture notes - Learning objectives 28150 Introduction...

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1 28150. Introduction to process control 7. Dynamics and stability of closed-loop systems Krist V. Gernaey 19 October 2009 28150 Learning objectives • At the end of this lesson you should be able to: – Derive closed-loop transfer functions based on a given block diagram – Investigate the stability of a closed-loop system with a feedback loop based on the roots of the characteristic equation / the Routh stability criterion – Predict the shape of the output of the closed loop system based on the position of the poles in the complex plane (root locus) 19 October 2009 28150 Outline • Block diagrams • Closed-loop transfer functions • Closed-loop responses • Stability of closed-loop control systems • Root locus diagrams 19 October 2009 28150 Outline Block diagrams • Closed-loop transfer functions • Closed-loop responses • Stability of closed-loop control systems • Root locus diagrams 19 October 2009 28150 Blending tank example ( ) ( ) x V w x x V w dt dx - + - = 1 2 1 1 ρ ρ x 1 is the disturbance; For constant feed flow, and for x 1 < x << 1 After switching to deviation variables: 1 w w x w w x w dt dx V - + = 2 1 ρ x w w x dt x d w V - + = 2 1 1 ρ 19 October 2009 28150 Blending tank example K w = 1 τ ρ = w V Substitution: Taking Laplace transforms of both sides: The following transfer functions can be defined: Linear systems! x w K x dt x d - + = 2 1 τ ( ) ( ) ( ) ( ) s W K s X s X s 2 1 1 + = + τ ( ) ( ) ( ) s W s K s X s s X 2 1 1 1 1 + + + = τ τ ( ) ( ) ( ) 1 1 1 1 + = = s s G s X s X τ ( ) ( ) ( ) 1 2 2 + = = s K s G s W s X τ

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2 19 October 2009 28150 Process TF: Blending tank example G 2 (s) G 1 (s) X’ 1 (s) + + W’ 2 (s) X’(s) X’ u (s) X’ d (s) G p G d D + + U Y Y u Y d Block diagram: general notation ! ( ) ( ) ( ) ( ) ( ) s W s G s X s G s X 2 2 1 1 + = 19 October 2009 28150 Blending tank with composition control Measurement: G m Controller: G c Signal Transducer: K IP Control valve: G v 19 October 2009 28150 Sensor and transmitter • Sensor: provides a measured value for a process variable (e.g. T, pressure, concentration, …) • Transmitter: Converts the sensor output to a signal appropriate for input to a controller (e.g. 4 – 20 mA) • Sensor-transmitter combination can be approximated by a TF: – Dynamics can be neglected when τ >> τ m ( ) ( ) m m m m G s K s X s X = + = 1 τ ( ) ( ) m m K s X s X = 19 October 2009 28150 Controller • For example a PI controller ( ) ( ) ( ) s G s K s E s U c I C = + = τ 1 1 19 October 2009 28150 Current-to-pressure (I/P) transducer • Transducer can be assumed linear, with negligible dynamics (time constant is very small compared to process time constant): ( ) ( ) IP t K s U s U = 19 October 2009 28150 Manipulated variable • Actuators/manipulated variables are not ideal • The dynamics of a control valve can be represented as a first-order TF: ( ) ( ) v v v t G s K s U s W = + = 1 2 τ
3 19 October 2009 28150 Block diagram of blending tank with composition control G 2 (s) G 1 (s) X’ 1 (s) + + W’ 2 (s) X’ u (s) X’ d (s) G v (s) U’ t (s) K IP U’(s) G c (s) E(s) G

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