Control Systems
Open Loop Systems
No feedback, examples, advantages and limitations.
Closed Loop Systems
Feedback path, error signal, self-correcting behavior.
Transfer Function
G(s) = Output/Input, rational polynomial in s.
Block Diagram Basics
Summing point, take-off point, forward and feedback path.
Block Diagram Reduction
Series, parallel, feedback rules, cascade simplification.
Signal Flow Graph Construction
Nodes, branches, gains, SFG from block diagram.
Mason Gain Formula
Forward paths, loops, non-touching loops, delta calculation.
Test Input Signals
Impulse, step, ramp, parabolic standard test inputs.
First Order System Response
Time constant tau, rise time, settling time.
Second Order System Parameters
Damping ratio zeta, natural frequency wn, categories.
Second Order Underdamped Response
Oscillatory, overshoot, peak time, settling time.
Time Domain Specifications
Rise time, peak time, settling time, percentage overshoot.
System Type Number
Type 0, 1, 2 systems based on poles at origin.
Steady State Error
Error constants Kp Kv Ka, error for different inputs.
Steady State Error Improvement
Increasing type number, integral control effect.
BIBO Stability
Bounded input bounded output, pole location requirement.
Routh-Hurwitz Criterion
Routh array construction, necessary and sufficient conditions.
Routh Array Special Case Zero Element
Epsilon method, stability determination.
Routh Array Special Case Zero Row
Auxiliary polynomial, imaginary axis poles.
Root Locus Concept
Locus of closed-loop poles as gain K varies.
Root Locus Construction Rules
Number of branches, symmetry, asymptotes, angles.
Root Locus Breakaway Points
Departure from real axis, dK/ds = 0 condition.
Bode Plot Construction
Magnitude in dB and phase plots, asymptotic approximation.
Bode Plot of First Order System
Single pole corner frequency, -20dB/dec slope.
Bode Plot of Second Order System
Resonant peak, damping effect on magnitude.
Gain Margin
dB above 0 at phase crossover frequency.
Phase Margin
Degrees above -180 at gain crossover frequency.
Gain and Phase Margin from Bode
Reading margins from Bode plot, stability.
Polar Plot
G(jw) in complex plane, magnitude and angle locus.
Nyquist Plot Construction
Contour mapping, semicircular s-plane contour.
Nyquist Stability Criterion
N = P - Z, encirclements of -1 point.
Lag Compensator Design
Improving steady state error without affecting transient.
PID Controller
Combined P+I+D action, tuning methods.
Ziegler-Nichols Tuning
Step response and ultimate gain methods for PID.
State Space Representation
Dot-x = Ax + Bu, y = Cx + Du matrices.
State Space from Transfer Function
Controllable and observable canonical forms.
Transfer Function from State Space
H(s) = C(sI-A)^(-1)B + D derivation.
State Transition Matrix
e^(At), matrix exponential, homogeneous solution.
Controllability
Controllability matrix rank test, Mc = [B AB A²B...].
Observability
Observability matrix rank test, Mo = [C CA CA²...].
Sampled Data Systems
Ideal sampler, zero order hold, pulse transfer function.
Z-Domain Analysis
Z-transform in control, mapping from s to z domain.
Stability in Z-Domain
Unit circle criterion, Jury stability test.
Second Order Overdamped Response
No oscillation, sum of exponentials, slow response.
Control System Components
Plant, controller, sensor, actuator, comparator.
Second Order Critically Damped Response
Fastest non-oscillatory, repeated real poles.
Dominant Poles
Poles closest to imaginary axis, approximation criteria.
Root Locus Angle of Departure
Angle condition at complex poles and zeros.
Complementary Root Locus
Negative feedback, K from 0 to -infinity.
Frequency Response Concept
Magnitude and phase as function of frequency.
Nichols Chart
M and N circles, gain-phase plane, closed loop from open loop.
All Pass and Minimum Phase Systems
Minimum phase, non-minimum phase, all-pass factor.
Lead-Lag Compensator
Combined lead and lag, simultaneous improvement.
P Controller
Proportional control, gain adjustment, offset error.
PI Controller
Proportional-integral, zero steady state error.
PD Controller
Proportional-derivative, anticipatory action, damping.
State Variables
Minimum set of variables to describe system state.
State Feedback Control
u = -Kx + r, pole placement by state feedback.
Observer Design
State estimation, Luenberger observer, observability.
Diagonalization
Eigenvalue decomposition, decoupled state equations.
Digital PID Controller
Discretization methods, difference equation implementation.
Bilinear Transformation w-domain
Tustin approximation, frequency warping.
Root Locus and Stability
Gain margin from root locus, critical gain.
Lead Compensator Design
Phase lead network, improving transient response.