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Quiz: Linearization and Nonlinear Effects

Test your understanding of linearization techniques and common nonlinear phenomena in control systems.

1. What is the purpose of linearization in control systems analysis?

  1. To make any system stable
  2. To approximate nonlinear behavior with a linear model valid near an operating point
  3. To eliminate all system dynamics
  4. To increase system gain
Show Answer

The correct answer is B. Linearization approximates a nonlinear system's behavior around a specific operating point using a linear model. This allows engineers to apply powerful LTI analysis tools (transfer functions, Bode plots, root locus) to systems that are fundamentally nonlinear.

Concept Tested: Linearization

2. An equilibrium point of a system is a state where:

  1. The system oscillates with constant amplitude
  2. All derivatives equal zero and the system remains stationary if undisturbed
  3. The output equals the input
  4. The transfer function equals unity
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The correct answer is B. At an equilibrium point, all time derivatives are zero ($\dot{x} = 0$), meaning the system stays at that state indefinitely unless disturbed. Linearization is performed around these equilibrium points to analyze small deviations.

Concept Tested: Equilibrium Point

3. Taylor series expansion is used in linearization to:

  1. Find the roots of a polynomial
  2. Approximate a nonlinear function with linear terms near an operating point
  3. Calculate the Laplace transform
  4. Determine system stability
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The correct answer is B. Taylor series expansion approximates a nonlinear function as a sum of terms. Linearization keeps only the first-order (linear) terms, giving $f(x) \approx f(x_0) + \frac{df}{dx}|_{x_0}(x - x_0)$. Higher-order terms are neglected for small deviations.

Concept Tested: Taylor Series Expansion

4. Saturation is a nonlinear effect that occurs when:

  1. The system oscillates too rapidly
  2. An actuator reaches its physical output limit and cannot increase further
  3. The damping ratio exceeds unity
  4. The system has too many poles
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The correct answer is B. Saturation occurs when an actuator (motor, amplifier, valve) hits its maximum output and cannot respond proportionally to further increases in input. The relationship becomes flat instead of linear, violating the homogeneity property.

Concept Tested: Saturation

5. Small signal analysis assumes:

  1. The system has no input
  2. Deviations from the operating point are small enough that linear approximations are valid
  3. All signals are sinusoidal
  4. The system is unstable
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The correct answer is B. Small signal analysis examines small perturbations around an operating point where the linearized model is valid. The key assumption is that deviations are small enough that higher-order nonlinear terms can be neglected.

Concept Tested: Small Signal Analysis

6. A dead zone is a nonlinear effect where:

  1. The system output oscillates uncontrollably
  2. The output remains zero until the input exceeds a threshold
  3. The system becomes unstable
  4. Energy is dissipated as heat
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The correct answer is B. A dead zone (or dead band) is a region where small inputs produce no output—the system appears "dead." Common examples include static friction that must be overcome before motion begins, or valve stiction requiring minimum pressure to open.

Concept Tested: Dead Zone

7. Backlash is commonly found in:

  1. Electrical resistors
  2. Gear trains and mechanical linkages
  3. Capacitors
  4. Ideal springs
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The correct answer is B. Backlash is the "play" or clearance between mating gear teeth or mechanical components. When the input reverses direction, there's a small range where the output doesn't change because the gears must take up the slack before making contact again.

Concept Tested: Backlash

8. Hysteresis causes the system output to depend on:

  1. Only the current input value
  2. The current input and the history of past inputs
  3. Only the input frequency
  4. The number of system poles
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The correct answer is B. Hysteresis means the output depends on both the current input and the direction of approach—the system "remembers" its history. A thermostat with hysteresis turns on at a different temperature than it turns off, creating a hysteresis loop.

Concept Tested: Hysteresis

9. For the pendulum equation $mL^2\ddot{\theta} + mgL\sin\theta = 0$, linearization around $\theta = 0$ replaces $\sin\theta$ with:

  1. $\theta^2$
  2. $\cos\theta$
  3. $\theta$
  4. $1$
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The correct answer is C. For small angles, $\sin\theta \approx \theta$ (in radians). This is the first term of the Taylor series: $\sin\theta = \theta - \theta^3/6 + ...$. This linearization is accurate to within 1% for angles less than about 14°.

Concept Tested: Linearization, Taylor Series Expansion

10. Which statement about nonlinear systems is TRUE?

  1. All physical systems are perfectly linear
  2. The superposition principle applies to nonlinear systems
  3. Most real physical systems are inherently nonlinear but often behave approximately linear near an operating point
  4. Nonlinear systems cannot be analyzed mathematically
Show Answer

The correct answer is C. Almost every real physical system is fundamentally nonlinear—springs stiffen, amplifiers saturate, friction is complex. However, many systems behave approximately linear over a limited operating range, which justifies using LTI analysis tools for practical design.

Concept Tested: Nonlinear System, Linearization