Solution to Homework 2

2. Chapter 2 Examples of Dynamic Mathematical Models Solution to Homework 2

3. Chapter 2 General Process Models State Equations • A suitable model for a large class of continuous theoretical processes is a set of ordinary differential equations of the form: t : Time variable x1,...,xn : State variables u1,...,um :Manipulated variables r1,...,rs :Disturbance, nonmanipulable variables f1,...,fn :Functions

4. Chapter 2 General Process Models Output Equations • A model of process measurement can be written as a set of algebraic equations: t : Time variable x1,...,xn : State variables u1,...,um :Manipulated variables rm1,...,rmt :Disturbance, nonmanipulable variables at output y1,...,yr : Measurable output variables g1,...,gr :Functions

5. Chapter 2 General Process Models State Equations in Vector Form • If the vectors of state variables x, manipulated variables u, disturbance variables r, and vectors of functions f are defined as: • Then the set of state equations can be written compactly as:

6. Chapter 2 General Process Models Output Equations in Vector Form • If the vectors of output variables y, disturbance variables rm, and vectors of functions g are defined as: • Then the set of algebraic output equations can be written compactly as:

7. Chapter 2 General Process Models Heat Exchanger in State Space Form Tl q V ρ T cp T q Tj If , then State Space Equations

8. Chapter 2 General Process Models Double-Pipe Heat Exchanger in State Space Form • Processes with distributed parameters are usually approximated by a series of well-mixed lumped parameter processes. • This is also the case for the heat exchanger, as shown in the next figure, which is divided into n well-mixed heat exchangers. • The space variable is divided into n equal lengths within the interval [0, L]. • After rearrangement, the mathematical model of the heat exchanger is of the form: • After rearrangement, the mathematical model of the heat exchanger is of the form: where and

9. Chapter 2 General Process Models Double-Pipe Heat Exchanger in State Space Form • We introduce the state parameters

10. Chapter 2 General Process Models Difference Quotient • The derivation with respect to space, δT/δτ, will now be approximated by using a difference quotient. • The difference quotient itself is the equation that can be used to approximately calculate the slope of a function at a certain point. • There are three formations of difference quotient: • Forward Difference • Backward Difference • Central Difference

11. Chapter 2 General Process Models Double-Pipe Heat Exchanger in State Space Form • Replacing δT/δτ with its corresponding difference will result a model that consists of a set of ordinary differential equations only. • The equation set can then be re-written in state space form. State Space Equations

12. Chapter 2 Linearization Linearization • Linearization is a procedure to replace a nonlinear original model with its linear approximation. • Linearization is done around a constant operating point. • It is assumed that the process variables change only very little and their deviations from steady state remain small. Operating point Linearization Taylor series expansion Nonlinear Model Linear Model

13. Chapter 2 Linearization Linearization • The approximation model will be in the form of state space equations • An operating point x0(t) is chosen, and the input u0(t) is required to maintain this operating point. • In steady state, there will be no state change at the operating point, or x0(t) = 0

14. Chapter 2 Linearization Taylor Expansion Series • Scalar Case A point near x0 Only the linear terms are used for the linearization

15. Chapter 2 Linearization Taylor Expansion Series • Vector Case where

16. Chapter 2 Linearization Taylor Expansion Series n: Number of states m : Number of inputs

17. Chapter 2 Linearization Taylor Expansion Series • Performing the same procedure for the output equations,

18. Chapter 2 Linearization Taylor Expansion Series r: Number of outputs

19. Chapter 2 Linearization Taylor Expansion Series Nonlinear Model Linear Model

20. Chapter 2 Linearization Single Tank System qi • The model of the system is already derived as: V h qo v1 • The relationship betweenh and h in the above equation is nonlinear. • An operating point for the linearization is chosen, (h0,qi,0).

21. Chapter 2 Linearization Single Tank System • The linearization around (h0,qi,0) for the state equation can be calculated as:

22. Chapter 2 Linearization Single Tank System • The linearization for the ouput equation is: • Note that the input of the linearized model is now Δqi. • To obtain the actual value of state and output, the following equation must be enacted:

23. Chapter 2 Linearization Single Tank System • The Matlab-Simulink model of the linearized system is shown below. All parameters take the previous values.

24. Chapter 2 Linearization Single Tank System • The simulation results : Original model : Linearized model

25. Chapter 2 Linearization Single Tank System • If the input qi deviates from the operating point, the linearized model will deliver inaccurate output. : Original model : Linearized model

26. Chapter 2 Linearization Single Tank System • If the input qi deviates from the operating point, the linearized model will deliver inaccurate output. : Original model : Linearized model

27. Chapter 2 Linearization Homework 3 Linearize the the interacting tank-in-series system for the operating point resulted by the parameter values as given in Homework 2. • For qi, use the last two digits of your Student ID. For example: 08  qi= 8 liters/s. • Submit the mdl-file and the screenshots of the Matlab-Simulink file + scope. qi h1 h2 q1 a1 a2 v2 v1

28. Chapter 2 Linearization Homework 3A Linearize the the triangular-prism-shaped tank for the operating point resulted by the parameter values as given in Homework 2 (New). • For qi2, use the last two digits of your Student ID divided by 10. For example: 03  qi2= 0.3 liter/s, 17  qi2= 1.7 liter/s. • Submit the mdl-file (softcopy) and the screenshots of the Matlab-Simulink file + scope (hardcopy). qi2 qi1 hmax h a qo v