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A complex number z = a + j b has real part a, imaginary part b, modulus |z| = sqrt(a squared + b squared) and argument (phase) theta = arctan(b/a). Polar form: z = |z| exp(j theta) (Euler's formula exp(j theta) = cos theta + j sin theta). Multiplication multiplies moduli and adds phases; the conjugate has the imaginary part negated. Roots of real polynomials are real or occur in conjugate pairs.

Matrix operations: sum, product (not commutative), transpose, determinant, inverse (exists if the determinant is not zero), eigenvalues (roots of det(A - lambda I) = 0) and eigenvectors. A polynomial of degree n has n roots. A rational function is a ratio of polynomials N(s)/D(s): zeros are roots of the numerator, poles are roots of the denominator. Partial fraction expansion writes it as a sum of simple terms, which is used to invert Laplace transforms.

1. Write the polar form of 3 + 4j. 2. What are poles and zeros? 3. How are eigenvalues defined?

1. Modulus 5, argument arctan(4/3) about 53 degrees: 5 exp(j 53 degrees). 2. Poles are the roots of the denominator and zeros the roots of the numerator of a rational function. 3. As the roots of det(A - lambda I) = 0.

Signals: unit step, ramp, impulse (Dirac delta), exponential and sinusoid. The Laplace transform F(s) = integral from 0 to infinity of f(t) exp(-s t) dt, with s a complex variable. Transforms: impulse 1; step 1/s; ramp 1/s squared; exp(-a t) gives 1/(s + a); sin(w t) gives w/(s squared + w squared); cos(w t) gives s/(s squared + w squared). Properties: linearity; derivative (the transform of df/dt is s F(s) - f(0)); integral (division by s); time shift; final value theorem (limit of f(t) as t tends to infinity is limit of s F(s) as s tends to 0, if the poles are stable); initial value theorem.

To find f(t) from F(s) = N(s)/D(s): factor the denominator, expand in partial fractions, and invert each term using the table. Distinct real poles give real exponentials; complex conjugate poles give damped sinusoids; repeated poles give terms multiplied by t.

1. What is the Laplace transform of a unit step? 2. State the final value theorem. 3. How does a complex conjugate pair of poles appear in time?

1. 1/s. 2. If all poles of sF(s) are in the left half plane, f(infinity) = limit of s F(s) as s tends to 0. 3. As a damped sinusoid.

State-space model: dx/dt = A x + B u, y = C x + D u, where x is the state vector, u the input and y the output. Transfer function G(s) = Y(s)/U(s) with zero initial conditions: G(s) = C (s I - A)^-1 B + D. The poles of G(s) are the eigenvalues of A. System response: the response to an impulse (the impulse response) is the inverse Laplace transform of G(s); the response to any input is the convolution of the input with the impulse response (in the Laplace domain Y(s) = G(s) U(s)).

First-order system G(s) = K/(tau s + 1): step response K (1 - exp(-t/tau)), reaches 63 percent of the final value at t = tau, settles (2 percent) at about 4 tau. Second-order system G(s) = K omega_n squared/(s squared + 2 zeta omega_n s + omega_n squared): natural frequency omega_n and damping ratio zeta. For zeta below 1 the response oscillates, with percent overshoot exp(-pi zeta/sqrt(1 - zeta squared)); zeta = 1 is critical damping; above 1 the response is overdamped. Rise time, settling time and overshoot are the usual performance specifications.

Asymptotic stability: all poles have negative real part (left half of the s plane). Poles on the imaginary axis (simple) give marginal stability (sustained oscillation); any pole in the right half plane gives instability (exponential growth). For a polynomial, the Routh criterion tests the signs of the roots without computing them; a necessary condition is that all coefficients of the characteristic polynomial have the same sign.

1. What is the transfer function of a first-order system and its time constant? 2. Define damping ratio effects on the step response. 3. What is the stability condition on the poles? 4. What relation do the poles of G(s) have with matrix A?

1. G(s) = K/(tau s + 1) with time constant tau. 2. Below 1: oscillatory with overshoot; equal to 1: critical damping; above 1: slow, no overshoot. 3. All poles must have negative real part. 4. They are the eigenvalues of A.

For a stable system with a sinusoidal input of frequency w, the steady-state output is a sinusoid at the same frequency with amplitude multiplied by |G(jw)| and phase shifted by the argument of G(jw). The Bode diagram plots 20 log10 |G(jw)| in decibels and the phase versus log frequency. Each first-order factor 1/(1 + s/w_c) contributes a corner at w_c: magnitude slope changes by -20 dB per decade and phase goes from 0 to -90 degrees (-45 degrees at the corner). Bandwidth is the frequency at which the gain drops by 3 dB. The Nyquist diagram plots G(jw) in the complex plane and is used to assess closed-loop stability.

Elements: blocks (transfer functions), summing junctions and pick-off points. Series: transfer functions multiply. Parallel: they add. Negative feedback loop with forward path G and feedback H: closed-loop transfer function G/(1 + G H); the loop gain is L = G H. Feedback reduces sensitivity to parameter changes and disturbances but can cause instability.

Controller: u = Kp e + Ki integral of e + Kd de/dt, where e is the error between reference and output. Proportional action reduces error but leaves steady-state error for a step; integral action eliminates it (type 1 systems track a step with zero error) but can reduce stability; derivative action anticipates and adds damping but amplifies noise. Steady-state error depends on system type (number of integrators in the loop). Stability margins (gain margin and phase margin, read from Bode or Nyquist plots) quantify robustness: a phase margin of about 45-60 degrees is a typical target. Biomedical examples: insulin pumps, ventilator control and temperature regulation.

1. What does a Bode magnitude slope of -20 dB/decade indicate? 2. Write the closed-loop transfer function of a negative feedback system. 3. What is the effect of integral action? 4. Define phase margin.

1. One first-order pole (above its corner frequency). 2. G/(1 + G H). 3. It removes steady-state error for constant references, but may reduce stability. 4. The additional phase lag at the gain crossover frequency that would bring the loop to instability (180 degrees minus the phase there).