05_B_Second_and_higher_order_systems_E2009

05_B_Second_and_higher_order_systems_E2009 - 28150...

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1 28150. Introduction to process control 5. Second and higher order systems Krist V. Gernaey 28 September 2009 28150 Learning objectives • At the end of this lesson you should be able to: – Explain the response of basic process transfer functions (first-order, second-order, integrating process) to standard inputs – Plot and interpret poles and zeros of a transfer function 28 September 2009 28150 Outline • Response of first-order processes (repetition) • Integrating processes • Response of second-order processes • Poles and zeros • Processes with time delays • Approximation of higher-order transfer functions 28 September 2009 28150 Outline Response of first-order processes (repetition) • Integrating processes • Response of second-order processes • Poles and zeros • Processes with time delays • Approximation of higher-order transfer functions 28 September 2009 28150 First-order process: step response • General transfer function: • Input: • Time domain response: Time 0 M ( ) ( ) 1 + = s K s U s Y τ ( ) s M s U S = ( ) ( ) 1 + = s s M K s Y τ ( ) ( ) τ / 1 t e M K t y - - = 28 September 2009 28150 First-order process: step response
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2 28 September 2009 28150 First-order process: ramp response • Input: Time domain response: – For t >> τ : ( ) 2 s a s U R = Time 0 Slope = a ( ) ( ) 2 3 2 1 2 1 1 s s s s s a K s Y α α τ α τ + + + = + = ( ) 2 2 1 s a K s a K s a K s Y + - + = τ τ τ ( ) ( ) t a K e a K t y t + - = - 1 / τ τ ( ) ( ) τ - = t a K t y 28 September 2009 28150 First-order process: ramp response 28 September 2009 28150 First order process: sinusoidal response • Input: Time domain response: – With: ( ) t A t U ω sin sin = Time 0 A P ( ) ( ) ( ) 2 2 1 ω τ ω + + = s s A K s Y ( ) + + + - + + = 2 2 2 2 2 2 2 1 1 ω ω ω τ ω τ τ ω τ ω s s s s A K s Y ( ) ( ) t t e A K t y t ω ω τ ω τ ω τ ω τ sin cos 1 / 2 2 + - + = - ( ) ( ) φ ω τ ω τ ω τ ω τ + + + + = - t A K e A K t y t sin 1 1 2 2 / 2 2 ( ) τ ω φ - = - 1 tan 28 September 2009 28150 First order process: sinusoidal response 28 September 2009 28150 Outline • Response of first-order processes (repetition) Integrating processes • Response of second-order processes • Poles and zeros • Processes with time delays • Approximation of higher-order transfer functions 28 September 2009 28150 Integrating processes: example • Assume a tank with a pump connected to the outflow line: • Steady-state at t = 0 • Switching to deviation variables: • Laplace transform: ( ) ( ) t q t q dt dh A i - = q i q Pump h h h q q i = = ( ) ( ) ( ) t q t q dt t h d A i - = ( ) ( ) ( ) s Q s Q s H A s i - =
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3 28 September 2009 28150 Integrating processes: example • After rearranging: • The two TFs in the system are integrating models : • Taking the integral of the differential equation on the previous slide: Integrating process ( ) ( ) ( ) [ ] s Q s Q A s s H i - = 1 ( ) ( ) A s s Q s H i = 1 ( ) ( ) A s s Q s H - = 1 ( ) ( ) [ ] * 0 * * * 1 dt t q t q A dh t i h h - = ( ) ( ) ( ) [ ] * 0 * * 1 dt t q t q A h t h t i - = - 28 September 2009 28150 Integrating processes • No steady-state gain
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