Signals and Control Systems Flashcards

The characteristic equation of a closed-loop control system is given by s³ + 6s² + 11s + K = 0. For the system to be stable, which of the following conditions must be met for the gain K?

Answer: 0 < K < 66

According to the Routh-Hurwitz stability criterion, for a third-order system with characteristic equation as³ + bs² + cs + d = 0, all coefficients must be positive, and the condition bc > ad must be satisfied for stability. In this case, a=1, b=6, c=11, and d=K. The conditions are K > 0 and (6)(11) > (1)(K), which simplifies to 66 > K. Combining these, the condition for stability is 0 < K < 66.

An engineer is analyzing a system's frequency response using a Bode magnitude plot. In a specific frequency range, the plot shows a consistent downward slope of -40 dB/decade. What does this slope most likely indicate about the system's transfer function in that range?

Answer: Two poles or a double pole

In a Bode magnitude plot, each pole in the transfer function contributes a slope of -20 dB/decade after its corner frequency. A slope of -40 dB/decade indicates the combined effect of two poles, which could be two distinct poles whose corner frequencies have been passed, or a double pole at the same frequency.

A control systems engineer observes that a process under proportional-integral (PI) control exhibits a persistent steady-state error after a step change in the setpoint. Which of the following is the most likely cause?

Answer: The integral time (Ti) is too large, resulting in weak integral action.

The primary role of the integral (I) term in a PID controller is to eliminate steady-state error by accumulating the error over time. A very large integral time (Ti) corresponds to a very small integral gain (Ki = 1/Ti), making the integral action too weak to effectively eliminate the offset. Integral windup typically occurs from large, prolonged errors causing saturation, not a persistent small error. Proportional gain affects the speed and stability but doesn't guarantee zero steady-state error on its own. Derivative action is not primarily used for steady-state error.

In the context of a system's transfer function in the s-plane, what is the primary significance of a pole's location?

Answer: It dictates the system's stability and transient response characteristics.

The location of the poles (roots of the denominator of the transfer function) in the complex s-plane is fundamental to the system's behavior. Poles in the left-half plane result in a stable system where the transient response decays over time. Poles in the right-half plane cause an unstable response that grows without bound. Poles on the imaginary axis lead to marginal stability or sustained oscillations.

A continuous-time signal is band-limited to a maximum frequency of 4 kHz. According to the Nyquist-Shannon sampling theorem, what is the minimum sampling rate required to avoid aliasing and allow for perfect reconstruction of the signal?

Answer: 8 kHz

The Nyquist-Shannon sampling theorem states that to perfectly reconstruct a continuous-time signal from its samples, the sampling frequency (fs) must be at least twice the maximum frequency (f_max) present in the signal. This minimum rate is known as the Nyquist rate. Here, f_max = 4 kHz, so the minimum sampling rate is 2 * 4 kHz = 8 kHz.

Which of the following input-output relationships, y(t), describes a system that is linear but NOT time-invariant?

Answer: y(t) = x(t) * cos(2πt)

The system y(t) = x(t) * cos(2πt) is linear because it satisfies the superposition principle: a*x1(t)*cos(2πt) + b*x2(t)*cos(2πt) = (a*x1(t) + b*x2(t))*cos(2πt). However, it is not time-invariant because a shift in the input, x(t-T), yields x(t-T)*cos(2πt), which is not the same as a shift in the output, y(t-T) = x(t-T)*cos(2π(t-T)). The cosine term is dependent on absolute time 't'.