Control Systems Flashcards

The characteristic equation of a closed-loop system is 1 + G(s)H(s) = 0. The roots of this equation are:

Answer: Closed-loop poles

The roots of the characteristic equation 1 + G(s)H(s) = 0 are the closed-loop poles of the system.

Which criterion is used to determine the ABSOLUTE stability of a linear time-invariant system without finding the roots of the characteristic polynomial?

Answer: Routh-Hurwitz criterion

The Routh-Hurwitz criterion uses an array of coefficients to determine the number of RHP roots without solving the polynomial.

For a unity negative feedback system, if the open-loop transfer function is G(s) = K/[s(s+2)], the system is critically damped when K equals:

Answer: 1

Closed-loop characteristic equation is s² + 2s + K = 0; critical damping requires ζ = 1, so (2/(2√K)) = 1, giving K = 1.

The phase margin of a system is defined as 180° plus the phase angle of the open-loop transfer function at:

Answer: Gain crossover frequency

Phase margin = 180° + ∠G(jωgc)H(jωgc), where ωgc is the gain crossover frequency where |GH| = 1 (0 dB).

State transition matrix Φ(t) = e^(At) satisfies which property?

Answer: Φ(t1 + t2) = Φ(t1) · Φ(t2)

The state transition matrix satisfies the semigroup property: Φ(t1 + t2) = Φ(t1)·Φ(t2), analogous to the scalar exponential.

In the Nyquist stability criterion, the number of clockwise encirclements N of the (-1, j0) point is related to Z (RHP closed-loop poles) and P (RHP open-loop poles) by:

Answer: N = Z - P

Nyquist criterion states N = Z - P; for a stable system with no open-loop RHP poles, Z = 0 requires N = 0.

A system is said to be observable if:

Answer: The initial state can be determined from the output over a finite time interval

Observability means the initial state x(0) can be uniquely determined from the output y(t) over a finite time interval [0, T].