بسمه تعالي کنترل صنعتي 2 دکتر سید مجید اسما عیل زاده گروه کنترل دانشکده مهندسي برق دانشگاه علم و صنعت ایران پاییس 90

Transcription

1 بسمه تعالي کنترل صنعتي 2 دکتر سید مجید اسما عیل زاده گروه کنترل دانشکده مهندسي برق دانشگاه علم و صنعت ایران پاییس 90

2 Techniques of Model-Based Control By Coleman Brosilow, Babu Joseph Publisher : Prentice Hall PTR Pub Date : April 03, 2002 ISBN : X Pages : 704

3

4

5

6

7

8

9

10 Illustrative Example: Blending system Notation: w 1, w 2 and w are mass flow rates x 1, x 2 and x are mass fractions of component A

11 Assumptions: 1. w 1 is constant 2. x 2 = constant = 1 (stream 2 is pure A) 3. Perfect mixing in the tank Control Objective: Keep x at a desired value (or set point ) x sp, despite variations in x 1 (t). Flow rate w 2 can be adjusted for this purpose. Terminology: Controlled variable (or output variable ): x Manipulated variable (or input variable ): w 2 Disturbance variable (or load variable ): x 1

12 Design Question. What value of x x SP? w 2 is required to have Overall balance: Component A balance: 0 w w w (1-1) 1 2 w1 x1 w2 x2 wx 0 (1-2) (The overbars denote nominal steady-state design values.) x x SP At the design conditions,. Substitute Eq. 1-2, SPand x2 1, then solve Eq. 1-2 for w 2 : xsp x1 w2 w1 (1-3) 1 x SP x x

13 Equation 1-3 is the design equation for the blending system. If our assumptions are correct, then this value of w 2 will keep x at xsp. But what if conditions change? Control Question. Suppose that the inlet concentration x 1 changes with time. How can we ensure that x remains at or near the set point x SP? As a specific example, if x1 x1 and w2 w2, then x > x SP. Some Possible Control Strategies: Method 1. Measure x and adjust w 2. Intuitively, if x is too high, we should reduce w 2 ;

14 Manual control vs. automatic control Proportional feedback control law, w2 t w2 Kc x SP x t (1-4) 1. where K c is called the controller gain. 2. w 2 (t) and x(t) denote variables that change with time t. 3. The change in the flow rate, w 2 t w 2 is, proportional to the deviation from the set point, x SP x(t).

15

16 Method 2. Measure x 1 and adjust w 2. Thus, if x 1 is greater than x 1, we would decrease w 2 so that w w 2 2 ; One approach: Consider Eq. (1-3) and replace x1 and w2with x 1 (t) and w 2 (t) to get a control law: 1 xsp x t w2 t w1 (1-5) 1 x SP

17

18 Because Eq. (1-3) applies only at steady state, it is not clear how effective the control law in (1-5) will be for transient conditions. Method 3. Measure x 1 and x, adjust w 2. This approach is a combination of Methods 1 and 2. Method 4. Use a larger tank. If a larger tank is used, fluctuations in x 1 will tend to be damped out due to the larger capacitance of the tank contents. However, a larger tank means an increased capital cost.

19 1.2 Classification of Control Strategies Method Table. 1.1 Control Strategies for the Blending System Measured Variable Manipulated Variable Category 1 x w 2 FB a 2 x 1 w 2 FF 3 x 1 and x w 2 FF/FB Design change Feedback Control: Distinguishing feature: measure the controlled variable

20 Very oscillatory responses, or even instability It is important to make a distinction between negative feedback and positive feedback. Engineering Usage vs. Social Sciences Advantages: Corrective action is taken regardless of the source of the disturbance. Reduces sensitivity of the controlled variable to disturbances and changes in the process (shown later). Disadvantages: No corrective action occurs until after the disturbance has upset the process, that is, until after x differs from x sp.

21 Feedforward Control: Distinguishing feature: measure a disturbance variable Advantage: Correct for disturbance before it upsets the process. Disadvantage: Must be able to measure the disturbance. No corrective action for unmeasured disturbances.

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51 Process Modelling

52 Process Modelling

53 Development of Dynamic Models Illustrative Example: A Blending Process An unsteady-state mass balance for the blending system: rate of accumulation rate of rate of of mass in the tank mass in mass out (2-1)

54 or where w 1, w 2, and w are mass flow rates. d Vρ dt w w w 1 2 (2-2) The unsteady-state component balance is: d Vρx dt w x w x wx (2-3) The corresponding steady-state model was derived in Ch. 1 (cf. Eqs. 1-1 and 1-2). 0 w w w (2-4) w x w x wx (2-5)

55

56 General Modeling Principles The model equations are at best an approximation to the real process. Adage: All models are wrong, but some are useful. Modeling inherently involves a compromise between model accuracy and complexity on one hand, and the cost and effort required to develop the model, on the other hand. Process modeling is both an art and a science. Creativity is required to make simplifying assumptions that result in an appropriate model. Dynamic models of processes consist of ordinary differential equations (ODE) and/or partial differential equations (PDE), plus related algebraic equations. 56

57 Table 2.1. A Systematic Approach for Developing Dynamic Models 1. State the modeling objectives and the end use of the model. They determine the required levels of model detail and model accuracy. 2. Draw a schematic diagram of the process and label all process variables. 3. List all of the assumptions that are involved in developing the model. Try for parsimony; the model should be no more complicated than necessary to meet the modeling objectives. 4. Determine whether spatial variations of process variables are important. If so, a partial differential equation model will be required. 5. Write appropriate conservation equations (mass, component, energy, and so forth). 57

58 Table 2.1. (continued) 6. Introduce equilibrium relations and other algebraic equations (from thermodynamics, transport phenomena, chemical kinetics, equipment geometry, etc.). 7. Perform a degrees of freedom analysis (Section 2.3) to ensure that the model equations can be solved. 8. Simplify the model. It is often possible to arrange the equations so that the dependent variables (outputs) appear on the left side and the independent variables (inputs) appear on the right side. This model form is convenient for computer simulation and subsequent analysis. 9. Classify inputs as disturbance variables or as manipulated variables. 58

59 Modeling Approaches Physical/chemical (fundamental, global) Model structure by theoretical analysis Material/energy balances Heat, mass, and momentum transfer Thermodynamics, chemical kinetics Physical property relationships Model complexity must be determined (assumptions) Can be computationally expensive (not real-time) May be expensive/time-consuming to obtain Good for extrapolation, scale-up Does not require experimental data to obtain (data required for validation and fitting)

60 Black box (empirical) Large number of unknown parameters Can be obtained quickly (e.g., linear regression) Model structure is subjective Dangerous to extrapolate Semi-empirical Compromise of first two approaches Model structure may be simpler Typically 2 to 10 physical parameters estimated (nonlinear regression) Good versatility, can be extrapolated Can be run in real-time

61 linear regression y c 2 0 c1x c2x nonlinear regression y K 1 e t / number of parameters affects accuracy of model, but confidence limits on the parameters fitted must be evaluated objective function for data fitting minimize sum of squares of errors between data points and model predictions (use optimization code to fit parameters) nonlinear models such as neural nets are becoming popular (automatic modeling)

62 Conservation Laws Theoretical models of chemical processes are based on conservation laws. Conservation of Mass rate of mass rate of mass rate of mass accumulation in out (2-6) Conservation of Component i rate of component i rate of component i accumulation in rate of component i rate of component i out produced 62 (2-7)

63 Conservation of Energy The general law of energy conservation is also called the First Law of Thermodynamics. It can be expressed as: rate of energy rate of energy in rate of energy out accumulation by convection by convection net rate of heat addition to the system from the surroundings net rate of work performed on the system (2-8) by the surroundings The total energy of a thermodynamic system, U tot, is the sum of its internal energy, kinetic energy, and potential energy: Utot Uint UKE UPE (2-9) 63

64 For the processes and examples considered, it is appropriate to make two assumptions: 1. Changes in potential energy and kinetic energy can be neglected because they are small in comparison with changes in internal energy. 2. The net rate of work can be neglected because it is small compared to the rates of heat transfer and convection. For these reasonable assumptions, the energy balance in Eq. 2-8 can be written as du int wh Q dt (2-10) U int H w Q the internal energy of the system enthalpy per unit mass mass flow rate rate of heat transfer to the system 64 denotes the difference between outlet and inlet conditions of the flowing streams; therefore -Δ wh = rate of enthalpy of the inlet stream(s) - the enthalpy of the outlet stream(s)

65 The analogous equation for molar quantities is, du int wh Q dt (2-11) where H is the enthalpy per mole and w is the molar flow rate. In order to derive dynamic models of processes from the general energy balances in Eqs and 2-11, expressions for U int and Ĥ or H are required, which can be derived from thermodynamics.

66 The Blending Process Revisited For constant, Eqs. 2-2 and 2-3 become: dv w1 w2 w (2-12) dt d Vx dt w1 x1 w2 x2 wx (2-13)

67 Equation 2-13 can be simplified by expanding the accumulation term using the chain rule for differentiation of a product: d Vx dx dv V x dt dt dt Substitution of (2-14) into (2-13) gives: (2-14) dx dv V x w1 x1 w2 x2 wx (2-15) dt dt Substitution of the mass balance in (2-12) for dv/ dt in (2-15) gives: dx V x w1 w2 w w1 x1 w2 x2 wx (2-16) dt After canceling common terms and rearranging (2-12) and (2-16), a more convenient model form is obtained: dv 1 w1 w2 w (2-17) dt dx w1 w2 1 2 dt V V x x x x (2-18)

68

69

70

71

72

73