A sharp weighted Wirtinger inequality by Ricciardi T.

By Ricciardi T.

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Extra info for A sharp weighted Wirtinger inequality

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Um . Let the process state variables xi change in the neighbourhood of the steady-state xis under the influence of the manipulated variables ui . Then it is possible to approximate the process nonlinearities. 3) We suppose that the solution of these equations is known for some ujs , j = 1, . . , m. The function fi (•) is approximated by the Taylor series expansion truncated to only first order terms as . 4) s s (∂fi /∂xl ) , l = 1, . . , n and ∂fi /∂uj , j = 1, . . , m denote partial derivatives for xl = xls and uj = ujs , respectively.

5 Solution of Linear Differential Equations by Laplace Transform Techniques Linear differential equations are solved by the means of the Laplace transform very simply with the following procedure: 1. Take Laplace transform of the differential equation, 2. solve the resulting algebraic equation, 3. find the inverse of the transformed output variable. 1: Solution of the 1st order ODE with zero initial condition Consider the heat exchanger shown in Fig. 3. The state equation is of the form 1 Z1 Z2 dϑ(t) = − ϑ(t) + ϑp (t) + ϑv (t) dt T1 T1 T1 The exchanger is in a steady-state if dϑ(t)/dt = 0.

Um (t), rm1 (t), . . , rmt (t)) = g2 (t, x1 (t), . . , xn (t), u1 (t), . . , um (t), rm1 (t), . . , rmt (t)) .. yr (t) = gr (t, x1 (t), . . , xn (t), u1 (t), . . , um (t), rm1 (t), . . 2) where y1 , . . , yr - measurable process output variables, rm1 , . . , rmt - disturbance variables, g1 , . . , gr - functions. 3) x =  ...  , u =  ...  , r =  ...  , f =  ... 4) If the vectors of output variables y, disturbance variables r m , and vectors of functions g are defined as       y1 rm1 g1       y =  ...

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