Solution_to_problem_4 17

# Solution_to_problem_4 17 - 4.17 – 1 Solution to problem...

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Unformatted text preview: 4.17 – 1 Solution to problem 4.17 (Corrected formulation!) System of 2 coupled differential equations: 1 max f X D X S K S dt dX S = ⋅- ⋅ + ⋅ = μ ( ) 2 / max f S S D X S K S Y dt dS f S S X =- ⋅ + ⋅ + ⋅- = μ Input: D States: X, S Disturbance: S f Remember that the following applies for linearisation: Assume a non-linear model: ) , ( u y f dt dy = Linearization is possible using Taylor series expansion, and truncating after the first order terms: ) ( ) ( ) , ( ) , ( , , u u u f y y y f u y f u y f u y u y- ∂ ∂ +- ∂ ∂ + ≅ And if we substitute with deviation variables, the linearised model is: u u f y y f dt y d s s ′ ∂ ∂ + ′ ∂ ∂ = ′ For this problem, we apply linearisation of non-linear terms around steady-state, using the values at steady-state ( ) , , , D X S S f ) to calculate the numerical values of the different partial derivatives. max 1 11 =- + ⋅ = ∂ ∂ = D S K S X f a S μ ( ) 1125 ....
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Solution_to_problem_4 17 - 4.17 – 1 Solution to problem...

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