Control Systems on the FE Exam
Control systems appears on the FE Electrical exam (5-8%), the FE Mechanical exam within the measurements and controls section (5-8%), and the FE Chemical exam as process control (3-5%). The level of depth varies by discipline, but all require understanding of feedback systems, stability, and basic system response characteristics.
Transfer Functions
A transfer function relates the output of a linear time-invariant system to its input in the Laplace domain. It is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input, assuming zero initial conditions. Transfer functions are expressed as ratios of polynomials in the complex variable s.
The roots of the numerator polynomial are called zeros, and the roots of the denominator polynomial are called poles. The location of poles and zeros in the s-plane determines the system's stability and dynamic behavior.
Block Diagrams
Block diagrams represent the signal flow and mathematical operations in a control system. Key elements include summing junctions (add or subtract signals), blocks (multiply by a transfer function), and pickoff points (branch a signal). Know how to reduce complex block diagrams to a single transfer function using series, parallel, and feedback loop reduction rules.
For a standard negative feedback loop with forward path G(s) and feedback path H(s), the closed-loop transfer function equals G(s) divided by (1 plus G(s) times H(s)). This is the most commonly tested formula in control systems.
System Response
First-Order Systems
A first-order system has a single pole and is characterized by its time constant (tau). The step response rises exponentially toward its final value, reaching 63.2% of the final value at one time constant, 86.5% at two time constants, and 95% at three time constants. The settling time (within 2% of final value) is approximately 4 time constants.
Second-Order Systems
A second-order system is characterized by its natural frequency (omega n) and damping ratio (zeta). The damping ratio determines the nature of the response:
- Underdamped (0 less than zeta less than 1): Oscillatory response with overshoot
- Critically damped (zeta equals 1): Fastest non-oscillatory response
- Overdamped (zeta greater than 1): Slow non-oscillatory response
For underdamped systems, know how to calculate percent overshoot, peak time, settling time, and damped natural frequency from the natural frequency and damping ratio.
Stability Analysis
BIBO Stability
A system is bounded-input bounded-output (BIBO) stable if all poles of the closed-loop transfer function have negative real parts (lie in the left half of the s-plane). If any pole has a positive real part, the system is unstable. Poles on the imaginary axis represent marginally stable systems.
Routh-Hurwitz Criterion
The Routh-Hurwitz criterion determines stability without actually finding the poles. Construct the Routh array from the coefficients of the characteristic polynomial. The system is stable if and only if all entries in the first column of the Routh array are positive (for a positive leading coefficient). The number of sign changes in the first column equals the number of poles in the right half plane.
Root Locus
The root locus shows how the closed-loop poles move in the s-plane as a gain parameter varies from zero to infinity. Key rules for sketching the root locus include: the locus starts at the open-loop poles and ends at the open-loop zeros (or infinity), the number of branches equals the number of poles, the locus exists on the real axis to the left of an odd number of real-axis poles and zeros, and branches leave the real axis at breakaway points.
Frequency Response
Bode plots display the magnitude and phase of a transfer function as functions of frequency. The magnitude plot uses a logarithmic scale (decibels) and the frequency axis is also logarithmic. Know how to sketch Bode plots for common transfer function elements: gain, integrators, differentiators, first-order terms, and second-order terms.
Gain margin and phase margin are stability indicators read from the Bode plot. Gain margin is the amount of gain increase (in dB) needed to make the system unstable, measured at the phase crossover frequency. Phase margin is the additional phase lag needed to make the system unstable, measured at the gain crossover frequency. A stable system has positive gain and phase margins.
Exam Tips
Focus on the closed-loop transfer function formula, stability criteria, and second-order system response characteristics. These topics appear most frequently on the exam. Practice sketching root locus and Bode plots by hand, as the exam may ask you to identify the correct plot from given options.