Control Systems Theory Flashcards

A closed-loop control system has a characteristic equation given by s³ + 2s² + 4s + K = 0. For the system to remain stable, what is the valid range for the gain K?

Answer: 0 < K < 8

The Routh-Hurwitz stability criterion states that for a system to be stable, all the elements in the first column of the Routh array must have the same sign (and be non-zero). Constructing the Routh array for this characteristic equation yields the following first column elements: 1, 2, (8-K)/2, and K. For stability, all must be positive. This leads to the conditions K > 0 and (8-K)/2 > 0, which simplifies to K < 8. Therefore, the range for stability is 0 < K < 8.

In a Proportional-Integral-Derivative (PID) controller, what is the primary function of the integral (I) term?

Answer: To eliminate steady-state error for step inputs.

The integral term accumulates the error over time. This accumulated value drives the system to a point where the error becomes zero, thereby eliminating the steady-state error that might persist with only a proportional controller, especially for step inputs.

A unity feedback control system is subjected to a unit ramp input, r(t) = t. The system is observed to have a constant, finite steady-state error. Based on this observation, what is the 'type' of the system?

Answer: Type 1

The 'type' of a system is defined by the number of pure integrators (poles at s=0) in the open-loop transfer function. A Type 1 system has one integrator. This allows it to perfectly track a step input (zero steady-state error) and follow a ramp input with a constant, finite steady-state error. A Type 0 system would have infinite error for a ramp, and a Type 2 system would have zero error for a ramp.

Which of the following statements accurately describes a fundamental property of a root locus plot for a feedback control system?

Answer: The root locus branches start at the open-loop poles and terminate at the open-loop zeros or at infinity.

The root locus is a graphical method for examining how the closed-loop poles move in the s-plane as a parameter, usually gain K, is varied. A key rule for constructing the plot is that the branches of the locus originate from the open-loop poles (where K=0) and terminate at the open-loop zeros (as K approaches infinity). If there are more poles than zeros, the remaining branches go to infinity along asymptotes.

An engineer analyzes the Bode plot of a minimum-phase, open-loop transfer function G(s)H(s). The gain margin is found to be 10 dB and the phase margin is 45 degrees. What does this imply about the stability of the corresponding closed-loop system?

Answer: The closed-loop system is stable.

For a minimum-phase system, positive gain and phase margins are direct indicators of closed-loop stability. The gain margin (in dB) indicates how much the open-loop gain can be increased before the system becomes unstable. The phase margin indicates how much additional phase lag is required at the gain crossover frequency to make the system unstable. Since both are positive (10 dB and 45 degrees), the closed-loop system is stable.

Which of the following is a primary advantage of using the state-space representation of a system over the classical transfer function representation?

Answer: It provides a complete representation of the system's internal states, not just the input-output relationship.

The transfer function method describes a system based solely on its input-output relationship, treating the system as a "black box." In contrast, the state-space representation uses a set of first-order differential equations to model the internal state variables of the system. This provides a more complete model, revealing the internal behavior and dynamics, and is readily applicable to complex multiple-input, multiple-output (MIMO), non-linear, and time-varying systems.