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Homework 2: Interacting Tank-in-Series System

Chapter 2. Examples of Dynamic Mathematical Models. Homework 2: Interacting Tank-in-Series System. 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:.

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Homework 2: Interacting Tank-in-Series System

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  1. Chapter 2 Examples of Dynamic Mathematical Models Homework 2: Interacting Tank-in-Series System

  2. 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

  3. 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

  4. 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 equatios can be written compactly as:

  5. 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 equatios can be written compactly as:

  6. 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

  7. 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

  8. 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

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

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

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

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

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

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

  15. 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).

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

  17. 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:

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

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

  20. 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

  21. 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

  22. Chapter 2 Linearization Homework 3: Interacting Tank-in-Series System 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 digit of your Student ID. For example: Kartika 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

  23. Chapter 2 Linearization Homework 3: Triangular-Prism-Shaped Tank 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 2 digits of your Student ID. For example: Bernard Andrew  qi2= 0.3 liter/s, Sugianto qi2= 1.0 liter/s. • Submit the mdl-file and the screenshots of the Matlab-Simulink file + scope. NEW qi2 qi1 hmax h a qo v

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