GATE EE 2015 Set 2 — Question 62
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Control Systems → Laplace & Z Transforms → Difference Equations
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Question
The following discrete-time equations result from the numerical integration of the differential equations of an un-damped simple harmonic oscillator with state variables and . The integration time step is .For this discrete-time system, which one of the following statements is TRUE?
Correct answer
(A) The system is not stable for h 0
Solution
The discrete-time state equations can be rewritten as:
In matrix form: Let . The eigenvalues of are found by solving :
The magnitude of the eigenvalues is .
For a discrete-time system to be stable, all eigenvalues must lie within or on the unit circle, i.e., .
Here, .
Since is the integration time step and , is always positive. Therefore, the system is never stable for any . Statement (A) is true.
In matrix form: Let . The eigenvalues of are found by solving :
The magnitude of the eigenvalues is .
For a discrete-time system to be stable, all eigenvalues must lie within or on the unit circle, i.e., .
Here, .
Since is the integration time step and , is always positive. Therefore, the system is never stable for any . Statement (A) is true.
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