Unit 3 - Practice Quiz

A step signal represents a sudden change in the input, which is zero before a certain time and constant afterwards. A unit step signal has an amplitude .

The Laplace transform of the function is . For a unit ramp function, , so its Laplace transform is .

For a first-order system, the response to a unit step input is . At , the value is , which is approximately 0.632 or 63.2% of the final value.

A smaller time constant means the system reaches its final steady-state value more quickly. Therefore, a smaller corresponds to a faster system response.

A damping ratio of means there is no damping in the system. The response will be a pure sinusoidal oscillation that continues indefinitely.

When the damping ratio is greater than one, the system is overdamped. Its response to a step input is slow and sluggish, approaching the final value without any oscillations.

Peak Time () is specifically the time it takes for the system's response to a step input to reach its first (and highest) peak.

Maximum overshoot is the maximum peak value of the response curve measured from the final steady-state value. It is often expressed as a percentage and indicates the relative stability of the system.

Settling time indicates how long it takes for the transient oscillations to decay. It is defined as the time after which the response remains within a certain percentage (usually 2% or 5%) of the final steady-state value.

Steady-state error is the error that remains after all the transient behavior has disappeared. Mathematically, it is , where r(t) is the input and c(t) is the output.

The 'type' of a system is the number of pure integrators (poles at the origin) in the open-loop transfer function. A Type 1 system has one integrator, which allows it to eliminate steady-state error for a step input.

The position error coefficient () is specifically associated with the step input. The steady-state error for a step input is given by .

The static velocity error coefficient () is used to find the steady-state error for a ramp input. It is calculated by taking the limit of as approaches zero.

The location of poles determines the system's stability. For a system to be stable, all its closed-loop poles must have negative real parts, placing them in the left-half of the complex s-plane.

This is the fundamental definition of BIBO stability. It means that if the input signal does not go to infinity, the output signal will also not go to infinity.

Absolute stability is a binary (yes/no) concept. It simply determines if the system's output will remain bounded or grow indefinitely over time.

Relative stability is a quantitative measure of how 'good' the stability is. It is often described by metrics like gain margin and phase margin, which indicate the safety margin before a system becomes unstable.

The Routh-Hurwitz criterion is an algebraic method that allows us to check for system stability by analyzing the coefficients of the characteristic equation, thereby identifying the presence of any unstable poles (in the right-half plane).

A system is stable if and only if there are no sign changes in the first column of the Routh array. The number of sign changes directly corresponds to the number of poles in the right-half of the s-plane.

The natural frequency () is the oscillation frequency without damping. When damping is present (), the oscillations occur at a lower frequency called the damped natural frequency, .

To determine stability, we construct the Routh array. The first column elements are . For stability, all elements in the first column must have the same sign. This leads to two conditions: , and . Combining these gives the range .

Comparing the transfer function with the standard form , we find rad/s, and . The percentage overshoot is calculated using the formula: .

The system is Type 1. The steady-state error for a ramp input is given by , where is the velocity error constant. . Therefore, the steady-state error is .

The step response of a first-order system is . To find the time to reach 95% of the final value (which is 1), we set . So, , which gives . Taking the natural logarithm of both sides gives . Thus, .

The standard poles for an underdamped second-order system are . The distance from the origin to a pole in the s-plane is equal to the natural frequency . Using the Pythagorean theorem, rad/s.

The acceleration error constant is defined as . For the given transfer function, .

An auxiliary equation is formed when an entire row in the Routh array becomes zero. This indicates the presence of roots that are symmetrically located about the origin, such as roots on the imaginary axis. This leads to marginal stability for that specific value of K that causes the row of zeros.

The settling time for a 2% tolerance band is approximated by the formula . Given and rad/s, the settling time is seconds.

Relative stability is determined by how far the dominant poles are from the imaginary axis in the left-half s-plane. The real part of the poles determines the rate of decay of the transient response. System A's dominant poles have a real part of -1, while System B's have a real part of -2. Since |-2| > |-1|, System B's poles are farther from the imaginary axis, indicating a faster decay and thus greater relative stability.

For a Type 2 system, the acceleration error constant is a finite non-zero value. The steady-state error for a parabolic input is given by . Since is finite and non-zero, the steady-state error will also be a finite non-zero constant.

A system is BIBO stable if its impulse response is absolutely integrable, i.e., . Here, is non-zero only for . The integral is . Since the integral is finite, the system is stable. Note that this is a non-causal system.

The time to reach the first peak, or peak time (), is given by the formula . Given seconds, we have . Solving for the damped natural frequency, we get rad/s.

The Routh-Hurwitz stability criterion states that the number of roots of the characteristic equation in the right-half s-plane is equal to the number of sign changes in the first column of the Routh array. The sequence of signs is +, +, -, +, +. There is one sign change from +3 to -4, and a second sign change from -4 to +1. Therefore, there are 2 poles in the right-half s-plane.

The type of a system is defined as the number of pure integrators in the open-loop transfer function, which corresponds to the number of poles at the origin (). In the given , the denominator has a term , which indicates three poles at the origin. Therefore, the system is a Type 3 system.

The response is for a step input of magnitude 5, since the final value is 5. The standard form for a unit step input is . Here, , so the time constant . The system transfer function is of the form . The DC gain is , which is the final value for a unit step input. Since the response to a step of magnitude 5 is 5, the DC gain is . Wait, let's recheck. The question is for a step input, but not specified unit. Let's assume a unit step. The final value is 5, so DC gain is 5. The TF is . Let's re-verify: Laplace of step input is . Output . . So . Inverse Laplace is . This matches. The correct TF is .

We know , so we can find from . The rise time is given by , where . By substituting the calculated value of into the rise time equation, we can solve for and subsequently find the damping ratio . The other options are incorrect relationships.

A differentiator has a transfer function . A unit ramp signal is , and its Laplace transform is . The output in the Laplace domain is . The inverse Laplace transform of is the unit step signal, .

Even though the RHP pole at is cancelled by a zero, the system is still internally unstable. The mode corresponding to the pole at () exists within the system and will grow unbounded if excited, even if it is not visible at the output for all inputs. Therefore, stability is determined by the un-cancelled poles, and the presence of any RHP pole makes the system internally unstable.

The steady-state error is determined by the highest power of in the input, as it will dominate. The input is a composite of step, ramp, and parabolic signals. The system is Type 2. A Type 2 system has zero error for step and ramp inputs. For the parabolic component (, which corresponds to a standard parabolic input of ), the error is where A=2. The acceleration constant is . Thus, the error is .

Percentage overshoot (%OS) is given by . As increases (in the underdamped range 0 to 1), the exponent becomes more negative, so %OS strictly decreases. Settling time is . The term is the real part of the pole. Increasing while keeping constant moves the pole along a circular path towards the real axis. The real part, , might not strictly increase or decrease without more information about . However, typically settling time decreases as damping increases toward critical damping. But if the question implies only changes, we have to consider its effect on . Let's assume the question implies is constant. Then as increases, increases, and decreases. A better phrasing for the correct option would reflect this common scenario. Re-evaluating the options. Option B is the most plausible intended answer in a typical exam context, where increasing damping is associated with faster settling. Let's select a better option. The term dictates the settling time. Increasing for a fixed will decrease the settling time. So, %OS decreases and decreases.