President UniversityErwin SitompulSMI 3/1 Dr.-Ing. Erwin Sitompul President University Lecture 3 System Modeling and Identification

President UniversityErwin SitompulSMI 3/2 Homework 2: Interacting Tank-in-Series System Chapter 2Examples of Dynamic Mathematical Models

President UniversityErwin SitompulSMI 3/3 A suitable model for a large class of continuous theoretical processes is a set of ordinary differential equations of the form: State Equations Chapter 2General Process Models t:Time variable x 1,...,x n :State variables u 1,...,u m :Manipulated variables r 1,...,r s :Disturbance, nonmanipulable variables f 1,...,f n :Functions

President UniversityErwin SitompulSMI 3/4 A model of process measurement can be written as a set of algebraic equations: Output Equations Chapter 2General Process Models t:Time variable x 1,...,x n :State variables u 1,...,u m :Manipulated variables r m1,...,r mt :Disturbance, nonmanipulable variables at output y 1,...,y r :Measurable output variables g 1,...,g r :Functions

President UniversityErwin SitompulSMI 3/5 State Equations in Vector Form Chapter 2General Process Models 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:

President UniversityErwin SitompulSMI 3/6 Output Equations in Vector Form Chapter 2General Process Models If the vectors of output variables y, disturbance variables r m, and vectors of functions g are defined as: Then the set of algebraic output equatios can be written compactly as:

President UniversityErwin SitompulSMI 3/7 Chapter 2 Heat Exchanger in State Space Form TjTj q TlTl q T V ρ T cpcp If, then State Space Equations General Process Models

President UniversityErwin SitompulSMI 3/8 Chapter 2Linearization 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 Nonlinear Model Linear Model Taylor series expansion

President UniversityErwin SitompulSMI 3/9 Chapter 2Linearization The approximation model will be in the form of state space equations An operating point x 0 (t) is chosen, and the input u 0 (t) is required to maintain this operating point. In steady state, there will be no state change at the operating point, or x 0 (t) = 0

President UniversityErwin SitompulSMI 3/10 Chapter 2Linearization Taylor Expansion Series Scalar Case A point near x 0 Only the linear terms are used for the linearization

President UniversityErwin SitompulSMI 3/11 Chapter 2Linearization Taylor Expansion Series Vector Case where

President UniversityErwin SitompulSMI 3/12 Chapter 2Linearization Taylor Expansion Series n:Number of states m:Number of inputs

President UniversityErwin SitompulSMI 3/13 Chapter 2Linearization Taylor Expansion Series Performing the same procedure for the output equations,

President UniversityErwin SitompulSMI 3/14 Chapter 2Linearization Taylor Expansion Series r:Number of outputs

President UniversityErwin SitompulSMI 3/15 Chapter 2Linearization Taylor Expansion Series Nonlinear ModelLinear Model

President UniversityErwin SitompulSMI 3/16 Chapter 2Linearization Single Tank System v1v1 qiqi qo qo V h The model of the system is already derived as: The relationship between h and h in the above equation is nonlinear. An operating point for the linearization is chosen, (h 0,q i,0 ).

President UniversityErwin SitompulSMI 3/17 Chapter 2Linearization Single Tank System The linearization around (h 0,q i,0 ) for the state equation can be calculated as:

President UniversityErwin SitompulSMI 3/18 Chapter 2Linearization Single Tank System The linearization for the ouput equation is: Note that the input of the linearized model is now Δq i. To obtain the actual value of state and output, the following equation must be enacted:

President UniversityErwin SitompulSMI 3/19 Chapter 2Linearization Single Tank System The Matlab-Simulink model of the linearized system is shown below. All parameters take the previous values.

President UniversityErwin SitompulSMI 3/20 Chapter 2Linearization Single Tank System The simulation results : Original model : Linearized model

President UniversityErwin SitompulSMI 3/21 Chapter 2Linearization Single Tank System If the input q i deviates from the operating point, the linearized model will deliver inaccurate output. : Original model : Linearized model

President UniversityErwin SitompulSMI 3/22 Chapter 2Linearization Single Tank System If the input q i deviates from the operating point, the linearized model will deliver inaccurate output. : Original model : Linearized model

President UniversityErwin SitompulSMI 3/23 Chapter 2Linearization Homework 3: Interacting Tank-in-Series System v1v1 qiqi h1h1 h2h2 v2v2 q1 q1 a1 a1 a2 a2 Linearize the the interacting tank-in-series system for the operating point resulted by the parameter values as given in Homework 2.  For q i, use the last digit of your Student ID. For example: Kartika  q i = 8 liters/s.  Submit the mdl-file and the screenshots of the Matlab- Simulink file + scope.

President UniversityErwin SitompulSMI 3/24 Chapter 2Linearization 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 q i2, use the last 2 digits of your Student ID. For example: Bernard Andrew  q i2 = 0.3 liter/s, Sugianto  q i2 = 1.0 liter/s.  Submit the mdl-file and the screenshots of the Matlab- Simulink file + scope. NEW v q i1 qo qo a q i2 h max h