Across logarithmic frequency, which two quantities do Bode plots display?
APosition and velocity
BCurrent and voltage only
CMagnitude and phase
DPoles and zeros only
Answer: C. Magnitude and phase. Bode plots show frequency-response magnitude and phase versus logarithmic frequency.
Gain crossover occurs when open-loop magnitude reaches what level?
AZero radians per second
BOne, or zero dB
CMinus one decibel
DOne hundred decibels
Answer: B. One, or zero dB. Gain crossover is the frequency where open-loop magnitude is unity, or 0 dB.
The phase crossover occurs when open-loop phase reaches what value?
AMinus 180 degrees
BZero degrees
CPlus 90 degrees
DMinus 90 degrees
Answer: A. Minus 180 degrees. Phase crossover is the frequency at which open-loop phase reaches minus 180 degrees.
Phase margin is measured at which crossover frequency?
AGain crossover frequency
BNatural frequency only
CPhase crossover frequency
DSampling frequency
Answer: A. Gain crossover frequency. Phase margin is measured at gain crossover by comparing phase to minus 180 degrees.
Gain margin is measured at which crossover frequency?
AGain crossover frequency
BPhase crossover frequency
CResonant frequency only
DZero frequency only
Answer: B. Phase crossover frequency. Gain margin is assessed at phase crossover, where the phase is minus 180 degrees.
Nyquist stability analysis centers on which complex-plane point?
AOrigin of the plot
BPositive one on real axis
CMinus one on real axis
DPlus one on imaginary axis
Answer: C. Minus one on real axis. The Nyquist criterion examines encirclements of the critical point -1+j0.
Sign changes in a Routh array’s first column count which roots?
ALeft-half-plane zeros
BInput disturbances
CRight-half-plane roots
DClosed-loop zeros only
Answer: C. Right-half-plane roots. The number of first-column sign changes equals the number of right-half-plane roots.
Under what condition does a real-axis segment belong to root locus?
AEven number of poles only
BOdd pole-zero count to right
CNo zeros to its left
DGain is always negative
Answer: B. Odd pole-zero count to right. For positive gain, real-axis points with an odd number of real poles and zeros to their right lie on the root locus.
Root-locus branches begin at which locations when gain is zero?
AClosed-loop zeros
BImaginary-axis crossings
CThe origin always
DOpen-loop poles
Answer: D. Open-loop poles. At zero gain, closed-loop poles coincide with open-loop poles.
As gain approaches infinity, where do root-locus branches end?
AOpen-loop zeros or infinity
BOpen-loop poles only
CThe Nyquist point
DThe origin always
Answer: A. Open-loop zeros or infinity. As gain approaches infinity, branches terminate at open-loop zeros or go to infinity.
Closed-loop poles satisfy which characteristic equation?
ANumerator equals input
BFeedback equals reference
CSensor gain equals unity
D1 + KG(s)H(s) = 0
Answer: D. 1 + KG(s)H(s) = 0. For a standard negative-feedback loop, closed-loop poles are roots of the characteristic equation 1+KG(s)H(s)=0.
In y=Cx+Du, which matrix maps states to the output?
AOutput matrix C
BInput matrix B
CState matrix A
DFeedthrough matrix D
Answer: A. Output matrix C. In y=Cx+Du, matrix C maps state vector x to output y; D maps the input directly to the output.
How is state derivative expressed in standard state-space form?
Ay equals Ax plus Bu
Bu equals Cx plus D
Cxdot equals Ax plus Bu
Dx equals y plus time
Answer: C. xdot equals Ax plus Bu. The state equation is x-dot=Ax+Bu, with output y=Cx+Du.
The state vector contains which internal variables?
AOnly reference inputs
BController gains only
CSensor noise spectrum
DInternal system variables
Answer: D. Internal system variables. The state vector contains variables that summarize the system’s internal dynamic condition.
Controllability describes what ability?
AAbility to measure outputs
BAbility to drive states
CAbility to reject every noise
DAbility to set gain margin
Answer: B. Ability to drive states. A system is controllable if suitable inputs can move it between state configurations.
From output measurements over time, what property can be assessed?
AAbility to infer states
BAbility to drive states
CAbility to raise frequency
DAbility to cancel all poles
Answer: A. Ability to infer states. A system is observable if its internal states can be inferred from its outputs over time.
In state-space form, what role does matrix A play?
AInput locations only
BInternal state dynamics
COutput sensor scale only
DDirect feedthrough only
Answer: B. Internal state dynamics. Matrix A governs how the state evolves in the absence of external input.
For a minimal realization, eigenvalues of A correspond to what?
ASystem zeros only
BController setpoints
CSensor gains
DSystem poles
Answer: D. System poles. The eigenvalues of A are the natural modes and poles of a minimal state-space realization.
With loop gain L, which expression defines sensitivity S?
AL over one plus L
BOne plus L squared
CL minus one
DOne over one plus L
Answer: D. One over one plus L. For standard negative feedback, sensitivity is S=1/(1+L).
With loop gain L, which expression defines complementary sensitivity T?
AOne over one plus L
BOne minus L
CL²/(1+L)
DL over one plus L
Answer: D. L over one plus L. The complementary sensitivity is T=L/(1+L) for standard negative feedback.
Lead compensation provides which phase effect?
APositive phase lead
BNegative damping only
CA pure time delay
DA pole at origin
Answer: A. Positive phase lead. Lead compensation provides positive phase near its phase-boost region and can improve phase margin.
Lag compensation can improve which low-frequency performance?
AHigh-frequency noise gain
BActuator saturation limit
CLow-frequency tracking accuracy
DSampling resolution
Answer: C. Low-frequency tracking accuracy. Lag compensation can increase low-frequency loop gain and improve steady-state accuracy.
Feedforward action uses what to anticipate a plant disturbance?
AMeasured disturbance information
BOutput error only
CPole locations only
DRandom switching noise
Answer: A. Measured disturbance information. Feedforward measures a disturbance and applies a corrective action before its full effect appears at the output.
How can feedback limit the output effect of disturbances?
AAll sensor noise perfectly
BEvery actuator limit
CUnmeasured disturbances
DEvery plant nonlinearity
Answer: C. Unmeasured disturbances. Feedback can reduce the effect of disturbances by correcting the output error they create.
Under its stability conditions, what does the final-value theorem estimate?
AInitial output slope
BLong-term output value
CFirst overshoot peak
DGain crossover frequency
Answer: B. Long-term output value. Under its stability conditions, the final-value theorem estimates the steady-state value from a Laplace-domain response.