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Automatic Control System

Automatic Control System. Modelling. Modelling dynamical systems. Engineers use models which are based upon mathematical relationships between two variables . We can define the mathematical equations : Measuring the responses of the built process (b lack model )

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Automatic Control System

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  1. Automatic ControlSystem Modelling

  2. Modelling dynamical systems • Engineers use models which are based upon mathematical relationships between two variables. • We can define the mathematical equations: • Measuring the responses of the built process (black model) • Not interested in the number and the real value of the time constants of the process, only it is enough that the response is similar sufficient accuracy • Using the basic physical principles (greymodel). • In order to simplification of mathematical model the small effects are neglected and idealised relationships are assumed. • Developing a new technology or a new construction nowadays it’s very helpful applying computer aided simulation technique. • This technique is very cost effective, because one can create a model from the physical principles without building of process.

  3. Grey box model

  4. Modelling mechanical systems Newton’s law one-dimensional translation and rotational systems. F: force [N]; FD: absorber’s force; Fs: spring’s force M: moment about centre of mass of body [Nm] I: the body’s moment of inertia [kgm2] s, : displacement [m; rad] v, ω: velocity[m/sec; rad/sec] a, : acceleration [m/sec2; rad/sec2] C: friction constant[Nsec/m] k: spring constant[N/m] velocity acceleration • Assign variables and sufficient to describe an arbitrary position of the object. • Draw a free-body diagram of each components and indicate all forces acting. • Apply Newton’s law in translation and rotational form. • Combine the equations to eliminate internal forces.

  5. y r m2 t t k2 C m1 k1 Modelling mechanical systems Road surface The car’s wheel vertical motion is assumed one-dimensional and the mass hasn’t got extension.The sock absorber is represented by a dashpot symbol with friction constant C. The force from the spring acts on both masses in proportional their relative displacement with spring constant k. The equilibrium positions of the two mass are offset from the spring’s unstretched positions becauseof the force of gravity. Nowadays the mathematical programs allow one to create a better model. The system can be approximated by the simplified system shown in left. x t

  6. y r m2 t t k2 C m1 k1 Modeling mechanical systems x t The positive displacements or velocity of mass, and so the positive force are signed by up arrows.

  7. Simulating the system by MATLAB

  8. Difference equations A block input is energised by x(t), and the response is y(t). Because of the response needs any time or in simulation the calculation of the response also needs any time, and so y(iT0) can be calculated from previous value than x(iT0). The sampled block has a delay time (T0). Example: y7 y6 y5 y4 y3 y2 y1 y0 0 T0 2T0 3T0 4T0 5T0 6T0 7T0

  9. Using difference equation

  10. m2 k2 C m1 k1 Using Laplace transform to model mechanical systems

  11. Block modeling mechanical system

  12. Block modeling mechanical systemBlock reduction

  13. Modeling electromechanical systems DC motor’s voltages. The electromotive force based on: DC motor’s tongues. B: magnet field [T: Tesla]; Ta armature torque Ia: armature current Tf friction torque UE: electromotive force [V] TL load torque UL: voltage on inductance [V] C: friction constant [Nsec/m] UR: voltage on resistance [V]

  14. M Modeling DC motor  angular displacement Ta armature torque Tf friction torque TL load torque Ia armature current Ra armature resistance La armature inductance armature inertia Generated electromotive force (emf) against the applied armature voltage Ke electromotive force constant Ka motor torque constant C rotational friction constant

  15. Simulating the systemby MATLAB

  16. M Example: Block modeling DC motor

  17. In time domain using difference equations

  18. M In operator frequency domain Assuming than Udc is constant andthe system is steady-state when onechange the value of Udc Examination of dynamic behaviourcan be used the Laplace transform. G(s)

  19. M Block modeling DC motor

  20. Simpler block model of DC motor

  21. Models of electronic circuit Kirchhoff’s current law: The algebraic sum of current leaving a junction or node equals the algebraic sum of the current entering that node. Kirchhoff’s current law: The algebraic sum of all voltages taken around a closed path in a circuit is zero. Resistor Capacitor Inductor

  22. Models of electronic circuit I3 All resistance equal R and all capacitanceequal C. Point “A” is nearly ground and such as a summing junction for the currents. “B” is a take off point for U2. The OpAmp amplitude gain is Au(s) I2 I1 B A U1 U2 U1 U1 U2 I1 I1 I2 U2 I2 I3 U2 I3

  23. Modeling heat flow Thermal conductivity: Heat energy flow. A: cross-sectional areal: length of the heat-flow path k: thermal conductivityconstant Temperature as a function of heat-energy flow: Specific heat: q: heat energy flow J/secR: thermal resistance °C/JT: temperature °C C: thermal capacity J/°C

  24. Heat flow models T0 q2 R0 m: the mass of the substance cv: specific heatconstant q1 room T1 R1 The net heat-energy flow into a substance: The heat energy flows through substances (across the room’s wall): The heat can also flow when a warmer mass flow into a cooler mass or vice versa:

  25. Heat flow models The total heat-energy flow: It’s non-linear, except T1=T2.

  26. water steam Modelling a heat exchanger The time delay between the measurementand the exit flow of the water: As: area of the steam inlet valve, Aw: area of the water inlet Ks: flow coefficient of the inlet valve, Kw: flow coefficient of the water inlet cvs: specific heat of steam, cvw: specific heat of water Tsi: temperature of inflow steam, Twi: temperature of inflow water Ts: temperature of outflow steam, Ts: temperature of outflow water Cs=mscvs thermal capacity of the steam, Cw=mwcvw thermal capacity of the water R: thermal resistance (average over the entire exchanger)

  27. Simulating heat exchanger

  28. Laplace form heat exchanger The equation is nonlinear because the state variable Ts is multiplied by the the controlinput As. The equation can be linearized at the working point Ts0, and so Tsi-Ts0=Tsnearly constant. To measure all temperature from Twi, it’s eliminated Twi=0.

  29. Block model of heat exchanger

  30. Black box model Modeling by reaction curve

  31. Feedback control plant GW(s) controller GA(s) GP1(s) GP2(s) GC(s) A/M GT(s) Process field When auto / manual switch is manual position (open), then GC(s)=1 W GW(s) R0+r U0+u GC(s) GA(s) GP(s) A/M YM0+yM GT(s)

  32. Modelled the process from reaction curveby dead-timeproportional first order transfer function HPT1

  33. The error of the model The principle of the less squares: The better model the less sum of value of the squares. The error of the model:

  34. A better model of process from reaction curveby dead-time second order transfer function HPT2 It needs computer! The beginning parameters:

  35. Modelled the process from reaction curveby “n” order transfer function PTn

  36. Modelled the process from reaction curveby dead-time integral first order transfer function HIT1

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