Stable & Unstable Systems | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

A stable system satisfies the BIBO (bounded input for bounded output) condition. Here, bounded means finite in amplitude. For a stable system, output should be bounded or finite, for finite or bounded input, at every instant of time.

Some examples of bounded inputs are functions of sine, cosine, DC, signum and unit step.

Examples

a) y(t) = x(t)+10

Here, for a definite bounded input, we can get definite bounded output i.e. if we put x(t) = 2,y(t) = 12 which is bounded in nature. Therefore, the system is stable.

b) y(t) = sin[x(t)]

In the given expression, we know that sine functions have a definite boundary of values, which lies between -1 to +1. So, whatever values we will substitute at x(t), we will get the values within our boundary. Therefore, the system is stable.

Unstable systems do not satisfy the BIBO conditions. Therefore, for a bounded input, we cannot expect a bounded output in case of unstable systems.

Examples

a) y(t) = tx(t)

Here, for a finite input, we cannot expect a finite output. For example, if we will put x(t) = 2 ⇒ y(t) = 2t. This is not a finite value because we do not know the value of t. So, it can be ranged from anywhere. Therefore, this system is not stable. It is an unstable system.

Stable & Unstable Systems | Digital Signal Processing - Electronics and Communication Engineering (ECE)

We have discussed earlier, that the sine function has a definite range from -1 to +1; but here, it is present in the denominator. So, in worst case scenario, if we put t = 0 and sine function becomes zero, then the whole system will tend to infinity. Therefore, this type of system is not at all stable. Obviously, this is an unstable system.

The document Stable & Unstable Systems | Digital Signal Processing - Electronics and Communication Engineering (ECE) is a part of the Electronics and Communication Engineering (ECE) Course Digital Signal Processing.
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FAQs on Stable & Unstable Systems - Digital Signal Processing - Electronics and Communication Engineering (ECE)

1. What is stability in electrical systems?
Ans. Stability in electrical systems refers to the ability of the system to maintain a steady and predictable response under normal operating conditions. It ensures that the system's output remains within acceptable bounds and does not exhibit excessive oscillations or instability.
2. How can stability be determined in electrical systems?
Ans. Stability in electrical systems can be determined by analyzing the system's transfer function or frequency response. By examining the poles (zeros) of the transfer function, one can determine if the system is stable, marginally stable, or unstable. If all the poles lie in the left half of the complex plane, the system is stable.
3. What are the characteristics of stable systems in electrical engineering?
Ans. Stable systems in electrical engineering exhibit certain characteristics such as bounded input-output response, no sustained oscillations, and convergence to a steady-state condition. These systems do not exhibit exponential growth or decay in their response, ensuring reliable and predictable behavior.
4. What are the consequences of unstable systems in electrical engineering?
Ans. Unstable systems in electrical engineering can lead to various undesirable consequences, including excessive oscillations, unpredictable behavior, and even system failure. These systems may exhibit exponential growth or decay in their response, causing instability, overheating, or damage to components.
5. How can unstable systems be stabilized in electrical engineering?
Ans. Unstable systems in electrical engineering can be stabilized through various techniques, such as feedback control, adjusting system parameters, or implementing damping mechanisms. Feedback control techniques, such as proportional-integral-derivative (PID) controllers, can be used to stabilize the system by adjusting the input based on the system's output. Additionally, proper selection and tuning of system parameters can also help in stabilizing the system.
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