A first-order system is represented by the timedomain differential equation x = x + u. A

Chapter 0, Problem P11.1

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A first-order system is represented by the timedomain differential equation x = x + u. A feedback controller is to be designed such that u(t) = -kx, and the desired equilibrium condition is x(t) = 0 as t > oo. The performance integral is defined as J = x 2 dt, and the initial value of the state variable is x(0) = Obtain the value of k in order to make J a minimum. Is this k physically realizable? Select a practical value for the gain k and evaluate the performance index with that gain. Is the system stable without the feedback due to u(t)l

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