Find the transfer function model for the given state space model. [RH]++ C Y = [136][*]
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- What is the general solution of the following inhomogeneous state space equation?Find the generating functions a(x) and b(x) with system of recurrent relationships:The Hamiltonian operator of a system is H=-(d2f/dx2) +x2 . Show that Nx exp (-x2/2) is an eigenfunction of H and determine the eigenvalue. Also evaluate N by normalization of the function.
- Classify the following system of first-order partial differential equations (k=const.):Find the general solution of the linear system (1) when A is then x n diagonal matrix A = diag[Aj, A21... ,An]- What condition onthe eigenvalues Al, ... , An will guarantee that limt_,,. x(t) = 0 for allsolutions x(t) of (1)?At any given time, a subatomic particle can be in one of two states, and it moves randomly from one state to another when it is excited. If it is in state 1 on one observation, then it is 3 times as likely to be in state 1 as state 2 on the next observation. Likewise, if it is in the state 2 on one observation, then it is 3 times as likely to be in state 2 as state 1 on the next observation. 1. Find the transition matrix for this Markov chain. 2. Researchers estimate that the particle is currently 4 times as like to be in state 1 as state 2. Find the probability vector representing this estimation. 3. Based on the estimation, what is the probability that the particle will be in state 2 two weeks from now? 4. What is the probability that the particle will be in state 1 three weeks from now?
- If the eigenvalues from a linearized system of non-linear differential equations indicate a fixed point at the origin is an unstable spiral, yet the non-linearized system shows that a limit cycle exists at the circle r=1, do we still classify the origin as an unstable spiral even though it approaches the limit cycle? In other words, does an unstable spiral have to approach infinity as time increases? Or can we say an unstable spiral approaches a limit cycle? I'm wondering how do we classify the stability of the origin when a limit cycle exists? Hope this makes sense.This problem is exercise 5 from Robert L. Devaney's An Introduction to Chaotic Dynamical SystemsThe function f (x, y) = 2xy certainly has a saddle point and not a minimum at (0, 0). What symmetric matrix S produces this f? What are its eigenvalues?
- Note: The following matrix represents the Jacobian at the equilibrium point .li ba] Use Jury Conditions to find the conditions on the parameter b so that the positive equilibrium is locally asymptotica13.Assume that the system described by the equation mu″ + γu′ + ku = 0 is either critically damped or overdamped. Show that the mass can pass through the equilibrium position at most once, regardless of the initial conditions. Hint: Determine all possible values of t for which u = 0.A second order system with a single input and a single output is the state equation given below It is represented by. According to this; a) Find the Φ(t) state transition matrix by the Laplace Transform method. b) If the input sign is the unit digit, find the recess and forced solution for the state variable x2(t). c) Obtain the transfer function of the system H(s)=Y(s)/E(s).