Control Systems
What is Control Systems?
A system that manages, commands, directs, or regulates the behavior of other systems or devices.
Key formula / rule: General Transfer Function
Key points
- Model physical systems using transfer functions and state-space equations.
- Analyze the time-domain response of first and second-order systems.
- Determine system stability using Routh-Hurwitz criterion, Root Locus, Bode plots, and Nyquist criterion.
- Design lead, lag, and PID controllers to meet performance specifications.
Common exam trap
Incorrectly applying Routh-Hurwitz criterion (e.g., sign changes in the first column).
Definitions
- Term
Control System
- Meaning
A system that manages, commands, directs, or regulates the behavior of other systems or devices.
- Term
Open-Loop System
- Meaning
A control system in which the control action is independent of the output.
- Term
Closed-Loop System (Feedback System)
- Meaning
A control system in which the control action is dependent on the output, using feedback to compare the output with the desired input.
- Term
Transfer Function
- Meaning
The ratio of the Laplace transform of the output to the Laplace transform of the input of a linear, time-invariant system, assuming all initial conditions are zero.
- Term
Stability
- Meaning
The property of a system to return to its equilibrium state after a disturbance, or to remain bounded for a bounded input.
- Term
Steady-State Error
- Meaning
The difference between the desired output and the actual output of a system as time approaches ∞.
- Term
Root Locus
- Meaning
A graphical method that shows the locations of the closed-loop poles as a system parameter (usually gain K) is varied from zero to ∞.
- Term
Bode Plot
- Meaning
A pair of plots (magnitude and phase) showing the frequency response of a system.
- Term
Nyquist Plot
- Meaning
A polar plot of the open-loop transfer function G(jω)H(jω) as ω varies from 0 to ∞, used for stability analysis.
- Term
Gain Margin (GM)
- Meaning
The amount of gain that can be added to the system before it becomes unstable.
- Term
Phase Margin (PM)
- Meaning
The amount of phase lag that can be added to the system before it becomes unstable.
Learning objectives
Model physical systems using transfer functions and state-space equations.
Analyze the time-domain response of first and second-order systems.
Determine system stability using Routh-Hurwitz criterion, Root Locus, Bode plots, and Nyquist criterion.
Design lead, lag, and PID controllers to meet performance specifications.
Understand the concepts of gain margin and phase margin.
Formulae
- Name
General Transfer Function
- Note
Ratio of Laplace transform of output to Laplace transform of input, assuming zero initial conditions.
- Expression
G(s) = Y(s) / U(s)
- Name
Closed-loop Characteristic Equation
- Note
Roots of this equation are the closed-loop poles, determining system stability.
- Expression
1 + G(s)H(s) = 0
- Name
Position Error Constant (Kp)
- Note
For Type 0 system, ess = 1 / (1 + Kp) for step input.
- Expression
Kp = lim(s→0) G(s)
- Name
Velocity Error Constant (Kv)
- Note
For Type 1 system, ess = 1 / Kv for ramp input.
- Expression
Kv = lim(s→0) sG(s)
- Name
Acceleration Error Constant (Ka)
- Note
For Type 2 system, ess = 1 / Ka for parabolic input.
- Expression
Ka = lim(s→0) s2 G(s)
- Name
Damped Frequency
- Note
Actual frequency of oscillation for underdamped systems.
- Expression
ωd = ωn * √(1 - ζ2)
- Name
Peak Time (Tp)
- Note
Time to reach the first peak overshoot.
- Expression
Tp = π / ωd
- Name
Maximum Overshoot (Mp)
- Note
Maximum percentage overshoot from the final value.
- Expression
Mp = exp(-ζπ / √(1 - ζ2)) * 100%
- Name
Settling Time (Ts, 2% criterion)
- Note
Time for the response to settle within 2% of the final value.
- Expression
Ts = 4 / (ζωn)
- Name
PID Controller Transfer Function
- Note
Proportional, Integral, and Derivative gains.
- Expression
Gc(s) = Kp + Ki/s + Kd*s
Prerequisites
Basic knowledge of differential equations and Laplace transforms.
Understanding of complex numbers and functions.
Linear algebra (for state-space analysis).
Circuit theory fundamentals.
Common mistakes
Incorrectly applying Routh-Hurwitz criterion (e.g., sign changes in the first column).
Errors in block diagram reduction or signal flow graph calculations.
Misinterpreting stability conditions from Bode, Nyquist, or Root Locus plots.
Forgetting to check for non-minimum phase systems when using Bode plots.
Not considering the effect of zeros on the Root Locus.
Confusing steady-state error for different types of systems (Type 0, 1, 2).
Keywords
Control System
Feedback
Open-loop
Closed-loop
Transfer Function
State-Space
Time Domain
Frequency Domain
Stability
Routh-Hurwitz
Root Locus
Bode Plot
Nyquist Plot
PID Controller
Lead Compensator
Lag Compensator
Transient Response
Steady-State Error
Gain Margin
Phase Margin
Poles
Zeros
Practice preview
For a system with transfer function G(s) = K / (s(s+a)), what is the condition for the system to be marginally stable?…
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The root locus of a system starts from the open-loop poles and terminates at the open-loop zeros or at infinity. Which of the following statements about root locus properties is INCORRECT?…
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A second-order system has a damping ratio (ζ) of 0.5 and a natural frequency (ωn) of 10 rad/s. What is the peak overshoot of the system's step response?…
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