Solution_problem_4.15 - one of the state variables as shown...

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4.15 – 1 Solution to problem 4.15 From example 4.4, the system equations are: 1 1 1 1 1 1 1 1 2 1 2 2 2 2 1 2 2 2 ' 1 1 1 ' ' , ' ' ' 1 1 1 ' ' , ' ' i dh q h q h dt A A R R dh h h q h dt A R A R R = - = = - = In this system, the modeled variables are 1 ' h and 2 ' h (liquid heights in tanks 1 and 2 in deviation form). These variables give information about the state of the system at any time. In a state-space representation the state variables are grouped in a vector x . Note that the vector x could contain variables with different magnitudes, for example the state of a reactor could be determined by the concentration of component A and the temperature in the reactor. The state-space representation is a convenient way to organize the information in our model. It separates the variables into state variables ( x ), inputs ( u ), disturbances ( d ) and outputs ( y ). In general, a state-space representation has the form: = + + = x Ax Bu Ed y Cx d where d dt = x x d . The outputs are the variables in which we are interested. They are usually the measured variables. In this case, we are interested in the flow exiting tank 2. This flow rate is related to
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Unformatted text preview: one of the state variables as shown by the equation 2 2 2 1 ' ' q h R = . As explained above, the state variables are: 1 2 ' ' h h = x While the input is ' i q = u and the output is 2 ' q = y . We can write the state-space model as follows: 1 1 1 1 1 2 2 2 1 2 2 1 ' 1 ' ' ' ' 1 1 i dh A R h dt A q h dh A R A R dt - = + - The output can be related to the state variables through the equation: 1 2 2 2 ' 1 ' ' h q h R = x d x u B x A C y 4.15 – 2 Now we just have to identify the matrices A , B , E and C : 1 1 2 1 2 2 1 1 1 A R A R A R - = - A 1 1 A = B 2 1 R = C = E...
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This note was uploaded on 11/20/2009 for the course CHME DTU-abroad taught by Professor Rafiqulgani during the Spring '09 term at Rensselaer Polytechnic Institute.

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Solution_problem_4.15 - one of the state variables as shown...

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