a) D²(D²+2) y(t) = -3x(1) - Internal Stability: roots are either zero or have negative real parts, the system is internally stable. External Stability (BIBO Stability): integral can be shown to be convergent
b) D(D²+3D+2)y(t) = (D-3)x(t) -Internal Stability: All roots have negative real parts, indicating internal stability. External Stability (BIBO Stability):integral can be shown to be convergent
c) h(t) = t.e⁻ᵗ(t)-This system is not a linear time-invariant (LTI) system since it has a time-varying parameter in the exponential term.
d) h(t) = cos(t)-u(t)-This system is a linear time-varying (LTV) system due to the presence of the cosine term.
a) D²(D²+2) y(t) = -3x(1)
Internal Stability: In this case, the transfer function is given by: H(s) = -3 / (s^4 + 2s^2)
The characteristic equation is then: s^4 + 2s^2 = 0
Factoring the equation: s^2 (s^2 + 2) = 0
The roots of the characteristic equation are: s1 = 0, s2 = 0, s3 = -i√2, s4 = i√2
Since all roots are either zero or have negative real parts, the system is internally stable.
External Stability (BIBO Stability): To determine the external stability of the system, we need to apply the bounded-input, bounded-output (BIBO) test.
The BIBO test states that a system is BIBO stable if and only if its impulse response is absolutely integrable.
The impulse response of the system is obtained by taking the inverse Laplace transform of the transfer function.
In this case, the impulse response is given by: h(t) = -3/2 (t sin(√2t) - √2t cos(√2t)) u(t)
To check if the impulse response is absolutely integrable, we need to evaluate the following integral: ∫|h(t)| dt
Evaluating the integral: ∫|h(t)| dt = 3/2 ∫|t sin(√2t) - √2t cos(√2t)| dt
This integral can be shown to be convergent using various methods, such as the Cauchy-Schwarz inequality or direct integration techniques.
Therefore, the system is also externally stable (BIBO stable).
b) D(D²+3D+2)y(t) = (D-3)x(t)
Internal Stability:
The transfer function for this system is: H(s) = (D-3) / (s^3 + 3s^2 + 2s)
The characteristic equation is: s^3 + 3s^2 + 2s = 0
Factoring the equation: s (s^2 + 3s + 2) = 0
The roots of the characteristic equation are: s1 = 0, s2 = -1, s3 = -2
All roots have negative real parts, indicating internal stability.
External Stability (BIBO Stability): The impulse response for this system is:
h(t) = (t - e^{-3t}) u(t)
The integral of the absolute value of the impulse response is:
∫|h(t)| dt = ∫|t - e^{-3t}| dt
This integral can be shown to be convergent using various methods. Therefore, the system is also externally stable (BIBO stable).
c) h(t) = t.e⁻ᵗ(t)
This system is not a linear time-invariant (LTI) system since it has a time-varying parameter in the exponential term.
Therefore, the concepts of internal and external stability do not directly apply to this system.
d) h(t) = cos(t)-u(t)
This system is a linear time-varying (LTV) system due to the presence of the cosine term.
The analysis of stability for LTV systems is more complex and requires specific techniques tailored to the nature of the time-varying elements.