# Discretizing stochastic dynamical systems using Lyapunov equations

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### Discretization of continuous-time models is hence fundamental.

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### Figure : The poles of the system

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### Solve (2) to find solution for (1).

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### Note: This is notfulfilled if the systemhas integrators!

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### How do we find Q if we have both integrators and strictly stable poles?

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### A 22 Q T12 + Q T12 A T11 = − V T12 − Q 22 A T12 , A 22 Q 22 + Q 22 A T22 = − V 22

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### 2. Find Q 12 by solving a Sylvester equation. 3. Find Q 11 by solving a Lyapunov equation.

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### 3. Find Q 11 by solving a Lyapunov equation.

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### 3. Then Q is given as Q = M 12 M T11 .

Van Loan, C.F. (1978).Computing integrals involving the matrix exponential.

IEEE Transactions on Automatic Control, 23(3), 395-404.

### Figure : The poles of the system

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### Figure : The poles of the system

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### Van Loan’s method performs better if the fastest pole is slow.

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