UPSC ESE E&T Control Systems Syllabus

What does the Routh-Hurwitz criterion determine?

The number of roots of the characteristic equation in the right-half s-plane (number of sign changes in the first column) without solving for the roots, thus assessing stability.

In a Routh array, what does a sign change in the first column indicate?

Each sign change corresponds to one root in the right-half s-plane; for stability there must be no sign changes in the first column.

How do you handle a zero in the first column of a Routh array (with nonzero rest of row)?

Replace the zero with a small positive ε and continue; then examine signs as ε → 0.

What does a row of all zeros in the Routh array signify, and how is it handled?

It indicates symmetrically located roots (e.g., on the jω-axis). Form an auxiliary polynomial from the row above and differentiate it to replace the zero row.

What is the root locus?

The plot of the loci of the closed-loop poles in the s-plane as a system parameter (usually open-loop gain K) is varied from 0 to infinity.

From where do root locus branches start and where do they end?

Branches start at the open-loop poles (K = 0) and terminate at the open-loop zeros or at infinity (K = ∞).

How many root locus branches go to infinity, and along what asymptote angles?

Number going to infinity = P − Z (poles minus zeros). Asymptote angles = (2q+1)·180°/(P−Z), for q = 0,1,2,...

What is the formula for the centroid of root locus asymptotes?

σ = (Σ poles − Σ zeros) / (P − Z), located on the real axis.

What is the rule for which segments of the real axis are part of the root locus?

A real-axis point lies on the root locus if the total number of real poles and zeros to its right is odd.

What are the angle and magnitude conditions for a point to be on the root locus?

Angle condition: ∠G(s)H(s) = ±180°(2q+1). Magnitude condition: |G(s)H(s)| = 1, used to find the gain K at that point.

What is a Bode plot?

A pair of plots versus frequency (log scale): magnitude in decibels (20·log10|G(jω)|) and phase in degrees, used for frequency-response analysis.

What is the slope contribution of a simple pole and a simple zero on a Bode magnitude plot?

A simple pole contributes −20 dB/decade; a simple zero contributes +20 dB/decade (beyond their corner frequencies).

Define gain margin from a Bode plot.

Gain margin = the amount (in dB) by which the magnitude is below 0 dB at the phase crossover frequency (where phase = −180°). GM = −20log10|G(jω)| at that frequency.

Define phase margin from a Bode plot.

Phase margin = 180° + ∠G(jω) measured at the gain crossover frequency (where |G(jω)| = 1, i.e., 0 dB).

What is the gain crossover frequency and the phase crossover frequency?

Gain crossover frequency: where magnitude = 0 dB (|G|=1). Phase crossover frequency: where phase = −180°.

For stability via Bode plots, what must be true of gain margin and phase margin?

For a stable system both the gain margin and phase margin must be positive.

On a Bode magnitude plot, what is the slope at low frequency for a Type-N system?

The initial slope is −20N dB/decade, where N is the system type (number of poles at the origin).

What does the Nyquist plot represent?

A polar plot of the open-loop frequency response G(jω)H(jω) as ω varies from −∞ to +∞, in the complex plane.

State the Nyquist stability criterion.

N = P − Z, where N = number of clockwise encirclements of the point (−1+j0) by the G(jω)H(jω) plot, P = open-loop poles in the right-half plane, and Z = closed-loop poles in the right-half plane. For stability Z = 0.

In the Nyquist criterion, what is the critical point?

The point (−1 + j0) in the complex plane; encirclements of this point determine closed-loop stability.

For an open-loop stable system (P = 0), what does the Nyquist criterion require for closed-loop stability?

The Nyquist plot must not encircle the (−1+j0) point (N = 0).

What is the state-space (state-variable) representation of an LTI system?

ẋ = Ax + Bu (state equation) and y = Cx + Du (output equation), where x is the state vector, u input, y output, and A,B,C,D constant matrices.

In state-space, what do matrices A, B, C, and D represent?

A = system (state) matrix, B = input matrix, C = output matrix, D = direct (feedforward) transmission matrix.

How do you obtain the transfer function from a state-space model?

G(s) = C(sI − A)^(-1) B + D.

How are the eigenvalues of the system matrix A related to system poles?

The eigenvalues of A (roots of det(sI − A) = 0) are the poles of the system; they determine stability.

What is the state transition matrix φ(t)?

φ(t) = e^(At), the matrix that maps the initial state to the state at time t for the unforced system: x(t) = φ(t)x(0).

How can the state transition matrix be computed using Laplace transforms?

φ(t) = L^(-1){ (sI − A)^(-1) }.

State two key properties of the state transition matrix φ(t).

φ(0) = I (identity); φ(t1+t2) = φ(t1)φ(t2); and φ^(-1)(t) = φ(−t).

State the condition for complete controllability of an LTI system.

The controllability matrix Qc = [B AB A²B ... A^(n−1)B] must have full rank n (be nonsingular for single-input systems).

State the condition for complete observability of an LTI system.

The observability matrix Qo = [C; CA; CA²; ...; CA^(n−1)] must have full rank n.

What is the difference between controllability and observability?

Controllability concerns whether the input can drive the state to any desired value in finite time; observability concerns whether the state can be determined from the output over finite time. They are dual properties.