Nonlinear and Hybrid Control Systems

Introduction and Context

Classical linear control theory is powerful because linear models permit superposition, frequency-domain analysis, and systematic controller synthesis. However, most physical systems are nonlinear over at least part of their operating range. Actuator saturation, friction, backlash, dead zones, aerodynamic forces, switching converters, contact phenomena, biochemical reactions, and network-induced decisions all produce nonlinear behavior. In addition, modern control systems increasingly combine continuous-time dynamics with discrete logic. Such systems are called hybrid systems.

A nonlinear model is commonly written as

ẋ = f(x,u,t), y = h(x,u,t),

where x ∈ R^n is the state, u ∈ R^m is the input, y is the measured output, and f and h may be nonlinear functions. A linear model is a special case in which f and h are affine in x and u. Nonlinear systems may exhibit multiple equilibria, finite escape time, limit cycles, bifurcations, hysteresis, and dependence on initial conditions. Therefore, stability and performance must usually be analyzed locally or regionally rather than assumed globally.

Hybrid systems add discrete variables and mode-dependent dynamics. A general representation is

ẋ = f_q(x,u), q ∈ Q,

where q is a discrete mode, together with conditions for switching between modes. Hybrid behavior is central to robotics, automotive systems, aerospace guidance, power electronics, embedded control, and cyber-physical systems.

Learning Objectives

After completing this chapter, the reader should be able to:

1. Distinguish linear, nonlinear, switched, and hybrid system models.

2. Analyze planar nonlinear systems using phase portraits, nullclines, and local linearization.

3. Explain describing-function analysis and its assumptions.

4. Apply Lyapunov methods to assess stability and estimate regions of attraction.

5. Design and evaluate feedback-linearizing, sliding-mode, and other nonlinear controllers.

6. Model switching, hybrid, and event-triggered systems.

7. Recognize implementation limitations, safety issues, and robustness requirements.

1. Nonlinear System Behavior

1.1 Equilibria and trajectories

For the autonomous system

ẋ = f(x),

an equilibrium x_e satisfies f(x_e) = 0. If the initial state is x(0) = x_0, the resulting solution x(t) is a trajectory. Unlike linear systems, nonlinear systems can have several isolated equilibria or continuous sets of equilibria.

An equilibrium is stable in the sense of Lyapunov if, for every ε > 0, there exists δ > 0 such that ||x(0) − x_e|| < δ implies ||x(t) − x_e|| < ε for all t ≥ 0. It is asymptotically stable if it is stable and x(t) → x_e as t → ∞. It is exponentially stable if convergence is bounded by ||x(t) − x_e|| ≤ c exp(−λt)||x(0) − x_e|| for positive constants c and λ. Global stability means that the property holds for every initial condition in the state space; local stability applies only in a neighborhood.

1.2 Local linearization

Suppose f is continuously differentiable near an equilibrium x_e. With ξ = x − x_e, the first-order approximation is

ξ̇ ≈ Aξ, A = [∂f/∂x] evaluated at x_e.

If every eigenvalue of A has negative real part, the equilibrium is locally exponentially asymptotically stable. If at least one eigenvalue has positive real part, it is unstable. If eigenvalues lie on the imaginary axis, linearization is inconclusive and a nonlinear method, normally Lyapunov analysis or a higher-order argument, is required. This limitation is important: a stable linear approximation does not establish global nonlinear stability.

1.3 Phase-plane analysis

For a second-order autonomous system

ẋ_1 = f_1(x_1,x_2), ẋ_2 = f_2(x_1,x_2),

the phase plane is the x_1–x_2 plane. A phase portrait consists of trajectories, equilibrium points, and directional information. The nullcline f_1 = 0 identifies points where ẋ_1 = 0; f_2 = 0 identifies points where ẋ_2 = 0. Their intersections are equilibria.

At a point in the phase plane, the vector [f_1, f_2]^T gives the direction of motion. Closed trajectories may indicate periodic motion or a limit cycle. A stable limit cycle attracts nearby trajectories, whereas an unstable limit cycle repels them. Separatrices divide regions with qualitatively different behavior, such as convergence to different equilibria.

For the system

ẋ_1 = x_2,

ẋ_2 = −x_1 − x_2,

the system matrix is A = [[0,1], [−1,−1]]. Its characteristic polynomial is λ² + λ + 1, whose roots are (−1 ± j√3)/2. Both roots have negative real parts, so the origin is a locally—and here globally—exponentially stable focus. Trajectories spiral toward the origin. The conclusion follows from the global linear structure, not merely from a local phase-plane sketch.

2. Describing-Function Analysis

2.1 Purpose and concept

A describing function is an approximate frequency-domain representation of a static nonlinear element under sinusoidal excitation. It is especially useful for predicting possible limit cycles in systems containing saturation, relay nonlinearities, dead zones, or nonlinear gains.

Let the input to a memoryless nonlinearity be v(t) = A sin(ωt). If the output is approximated by its fundamental harmonic,

y(t) ≈ A|N(A)| sin(ωt + ∠N(A)),

then N(A) is the describing function. It generally depends on amplitude A and may be complex if the nonlinearity is asymmetric or has phase effects in the fundamental component.

For a symmetric nonlinearity with output y(θ), θ = ωt, the real describing function is

N(A) = (1/(πA)) ∫_0^{2π} y(A sin θ) sin θ dθ,

provided the output has odd symmetry and the fundamental cosine coefficient is zero. More generally, both sine and cosine Fourier coefficients must be calculated.

2.2 Relay example

For an ideal symmetric relay with output +M when v > 0 and −M when v < 0, the fundamental component of the square wave has amplitude 4M/π. Therefore,

N(A) = 4M/(πA).

This approximation neglects higher harmonics and assumes a sinusoidal steady-state input. It is not an exact replacement for the relay in transient or strongly nonlinear operation.

2.3 Harmonic-balance condition

For a feedback loop with linear transfer function G(s) and nonlinear element with describing function N(A), a predicted limit cycle satisfies approximately

1 + G(jω)N(A) = 0.

Equivalently, the Nyquist locus of G(jω) must intersect −1/N(A). The magnitude and phase conditions are

|G(jω)N(A)| = 1,

∠G(jω) + ∠N(A) = −π

modulo 2π. The result estimates amplitude A and frequency ω, but it does not prove that a limit cycle exists or that it is stable.

2.4 Limitations

Describing-function analysis is most credible when one harmonic dominates, the nonlinearity is memoryless or approximately so, and the loop is near sinusoidal steady state. It can fail for strongly distorted waveforms, multiple nonlinearities, subharmonic oscillations, unstable hidden oscillations, or systems with significant discontinuity-induced behavior. Numerical simulation and nonsmooth analysis should validate any design based on this method.

3. Lyapunov Stability Theory

3.1 Direct method

Consider ẋ = f(x), with equilibrium at the origin. A scalar function V(x) is a Lyapunov candidate if it is continuous, positive definite near the origin, and satisfies V(0) = 0. Its derivative along system trajectories is

V̇(x) = ∇V(x)^T f(x).

If V is positive definite and V̇ ≤ 0, the origin is stable. If V̇ is negative definite, the origin is asymptotically stable. If V is radially unbounded, meaning V(x) → ∞ as ||x|| → ∞, and V̇ is negative definite globally, global asymptotic stability follows under standard existence and uniqueness assumptions.

A positive semidefinite V̇ does not by itself establish asymptotic convergence. LaSalle’s invariance principle can be used: if trajectories remain in a compact positively invariant set and the largest invariant subset of {x : V̇ = 0} is the origin, then the origin is asymptotically stable.

3.2 Quadratic stability for linear systems

For ẋ = Ax, choose V = x^T P x, where P = P^T > 0. Then

V̇ = x^T(A^T P + PA)x.

If a positive definite Q can be selected such that

A^T P + PA = −Q,

then the origin is exponentially stable. For every Hurwitz matrix A and every Q > 0, this Lyapunov equation has a unique P > 0. This result is also a foundation for nonlinear and robust control analysis.

3.3 Worked example: nonlinear damping

Consider

ẋ_1 = x_2,

ẋ_2 = −x_1 − x_2^3.

Choose the energy-like function

V(x) = (1/2)x_1² + (1/2)x_2².

It is positive definite and radially unbounded. Its derivative is

V̇ = x_1ẋ_1 + x_2ẋ_2

= x_1x_2 + x_2(−x_1 − x_2³)

= −x_2⁴ ≤ 0.

The set where V̇ = 0 is x_2 = 0. On this set, ẋ_2 = −x_1, so the only invariant point is x_1 = 0. LaSalle’s principle therefore establishes global asymptotic stability of the origin. Notice that V̇ is not negative definite directly; the invariance argument is essential.

3.4 Control Lyapunov functions

For a control-affine system

ẋ = f(x) + g(x)u,

a control Lyapunov function (CLF) is a positive-definite function for which an admissible input can make V̇ negative. Since

V̇ = L_fV + L_gV u,

where L_fV = ∇V^T f and L_gV = ∇V^T g are Lie derivatives, a stabilizing control can be selected by enforcing L_fV + L_gV u < 0. Input constraints can be incorporated through quadratic programming, often together with control barrier functions for safety.

4. Nonlinear Controller Design

4.1 Feedback linearization

Feedback linearization cancels nonlinearities through a nonlinear coordinate transformation and input law. For the single-input system

ẋ = f(x) + g(x)u,

y = h(x),

if the output has relative degree r, differentiating y r times gives

y^(r) = a(x) + b(x)u,

with b(x) ≠ 0 in the operating region. Setting

u = [v − a(x)]/b(x)

produces y^(r) = v. A linear reference controller may then select v, for example

v = y_d^(r) − k_{r−1}e^(r−1) − ... − k_1ė − k_0e,

where e = y − y_d and the polynomial s^r + k_{r−1}s^(r−1) + ... + k_0 is Hurwitz.

The method requires an accurate model, measurable or estimable states, nonzero input gain b(x), and compatible relative degree. Internal dynamics, or zero dynamics, must be stable; otherwise input-output linearization may produce an unstable internal state. Exact cancellation is also sensitive to uncertainty and unmodeled dynamics.

4.2 Backstepping

Backstepping is a recursive design method for strict-feedback systems such as

ẋ_1 = f_1(x_1) + g_1(x_1)x_2,

ẋ_2 = f_2(x_1,x_2) + g_2(x_1,x_2)u.

The second state is first treated as a virtual control for stabilizing x_1. A Lyapunov function is augmented with the error between x_2 and its virtual-control command. The procedure is continued until the actual input is obtained. Backstepping avoids direct cancellation in some applications and can accommodate adaptive terms, but the resulting controllers may be algebraically complex and can suffer from actuator saturation.

4.3 Sliding-mode control

Sliding-mode control creates a discontinuous control action that drives trajectories to a designed sliding surface. For a tracking error e in a second-order system, a common surface is

s = ė + λe, λ > 0.

On s = 0, the reduced dynamics are ė + λe = 0, so tracking error converges exponentially. A simplified controller is

u = u_eq − K sign(s),

where u_eq maintains the nominal sliding condition, K > 0 is selected above a disturbance bound, and sign(s) is the sign function.

If the uncertain matched disturbance d satisfies |d| ≤ d_max and the remaining model terms are known sufficiently accurately, a gain satisfying K > d_max can establish a reaching condition such as s ṡ < 0, subject to actuator authority. In practice, ideal discontinuity causes chattering due to sampling, unmodeled dynamics, and actuator bandwidth. Replacing sign(s) with saturation(s/φ), where φ is a boundary-layer thickness, reduces chattering but introduces a tracking error near the surface. Higher-order sliding modes, such as super-twisting algorithms, can reduce chattering while preserving finite-time convergence under suitable regularity assumptions.

4.4 Gain scheduling and nonlinear model predictive control

Gain scheduling uses different linear controllers over operating regions. It is practical but does not automatically guarantee stability during transitions. Scheduling variables, interpolation, and region overlap must be designed carefully.

Nonlinear model predictive control (NMPC) repeatedly solves a finite-horizon optimization using a nonlinear model. It can handle constraints on states and inputs and is valuable for autonomous vehicles, chemical processes, and energy systems. Its disadvantages include computational demand, sensitivity to model quality, and the need for real-time feasibility. A terminal cost or terminal constraint is often used to support closed-loop stability.

5. Switching and Hybrid Systems

5.1 Definitions

A switched system has a finite set of continuous dynamics:

ẋ = f_{σ(t)}(x),

where σ(t) is a switching signal. A hybrid automaton supplements each mode q with an invariant set, flow map, guard condition, and reset map. During continuous evolution, ẋ = f_q(x) while x remains in the invariant. When a guard is reached, the mode may change and the state may reset according to x^+ = R_qr(x^-).

Examples include a thermostat, an anti-lock braking controller, a buck converter, and a robot changing between free motion and contact.

5.2 Stability under switching

Each mode can be stable while arbitrary switching is unstable. Switching may repeatedly excite energy that no individual mode generates. A common sufficient condition is a common Lyapunov function V satisfying

∇V(x)^T f_q(x) ≤ −W(x)

for every mode q, with W positive definite. Then arbitrary switching preserves asymptotic stability. If no common Lyapunov function exists, dwell-time or average-dwell-time constraints can limit switching frequency and recover stability.

Hybrid systems may also exhibit Zeno behavior: infinitely many discrete transitions occur in finite time. A physically meaningful model must specify how Zeno executions are interpreted or prevented. Hysteresis, minimum dwell time, and actuator dynamics are practical remedies.

5.3 Event-triggered control

In periodic digital control, measurements and control updates occur at fixed sampling times. Event-triggered control updates only when a condition indicates that the measurement error has become significant. Let x(t_k) be the latest transmitted state and e_s(t) = x(t_k) − x(t) be the sampling error. A typical trigger is

||e_s(t)|| ≥ σ||x(t)||,

where 0 < σ < 1, although practical conditions may include absolute thresholds and noise margins.

Event-triggering can reduce communication, computation, and energy consumption. Its analysis must address stability, minimum inter-event time, measurement noise, packet loss, quantization, and denial-of-service risks. A positive lower bound on inter-event times is desirable to exclude Zeno behavior. Dynamic triggering introduces an auxiliary variable whose dynamics regulate transmission decisions and can improve the trade-off between performance and communication.

6. Illustrative Case: Nonlinear Position Control of a DC Motor

The following is an illustrative engineering case. Consider a motor position θ with dynamics

Jθ̈ + bθ̇ + τ_f(θ̇) = K_t i,

L i̇ + Ri = v − K_e θ̇,

where J is inertia, b is viscous friction, τ_f is nonlinear friction, K_t is torque constant, L and R are armature inductance and resistance, K_e is back-emf constant, i is current, and v is applied voltage. Suppose Coulomb friction is approximated by τ_f(θ̇) = T_c sign(θ̇), with a smooth approximation used in simulation.

Background and problem

A linear controller designed around zero velocity performs adequately for small motions but produces overshoot and oscillatory behavior during large position commands. Voltage saturation limits acceleration, while friction creates a dead-zone-like effect near zero velocity.

Analysis

The state is x = [θ, θ̇, i]^T. The model is nonlinear because of sign(θ̇) and may be nonsmooth at zero velocity. The actuator constraint is |v| ≤ V_max. A local linear model cannot predict all transitions through friction reversal and saturation. A phase-plane projection using position and velocity reveals slow motion near the origin and faster trajectories away from it.

Approach

A practical design combines several tools:

1. Use an inner current loop to make electromagnetic torque approximately track a commanded torque.

2. Use a nonlinear outer controller with a Lyapunov-based error function.

3. Add saturation-aware anti-windup or a reference governor.

4. Use a smoothed sliding surface for robust rejection of bounded load torque.

5. Verify the complete switched implementation, including current-loop limits and mode changes.

For position error e = θ − θ_d and velocity error ė = θ̇ − θ̇_d, select s = ė + λe. The commanded torque contains a nominal compensation term and a robust term −K sat(s/φ). The gain K must account for the worst credible load and model error, while φ is selected according to sensor resolution and actuator bandwidth. A supervisory mode can reduce the command near the target to prevent chatter and excessive current.

Outcome and validation

The controller should not be accepted solely because a nominal simulation converges. Validation should include parameter variation, friction uncertainty, voltage saturation, sensor noise, sampling delay, quantization, load disturbances, and emergency-stop transitions. Key measures include settling time, maximum overshoot, RMS current, energy use, steady-state error, and minimum safety margin to mechanical limits.

Learning points

The case demonstrates that nonlinear design is inseparable from implementation constraints. Robustness may be improved by sliding-mode action, but chattering and actuator limits must be managed. Hybrid supervisory logic can improve safety, but its transitions require explicit analysis. A Lyapunov argument for an ideal continuous model is not automatically a proof for the sampled, saturated, quantized product.

7. Challenges, Governance, and Best Practice

Nonlinear control design faces several recurring difficulties:

• Model uncertainty: Exact cancellation can amplify errors. Robust or adaptive methods should be used only with stated assumptions.
• State availability: Many methods require velocity, acceleration, or internal states. Observers and differentiators introduce noise and delay.
• Discontinuities: Relays, friction, impacts, and switching require nonsmooth models and careful numerical simulation.
• Constraints: Input saturation can invalidate stability proofs and cause integrator windup.
• Computational limits: NMPC and online optimization must meet worst-case execution deadlines.
• Verification: Hybrid mode combinations create a large state-space and logic-space verification problem.

Safety-critical applications should maintain traceable requirements, conservative fallback modes, fault detection, independent monitoring, and validated software. Standards depend on the industry, but functional-safety principles generally require hazard analysis, verification evidence, controlled changes, and fail-safe behavior. Cybersecurity is also relevant because event-triggered and networked controllers expose communication channels to spoofing, replay, and denial-of-service attacks.

Artificial intelligence can assist system identification, anomaly detection, and policy optimization, but learned components should not replace stability and safety analysis without evidence. Safe reinforcement learning commonly combines a learned policy with a certified safety filter, control barrier function, backup controller, or runtime monitor. Dataset shift, adversarial inputs, and opaque decisions remain significant concerns.

Best practice is to proceed in layers: derive a physically meaningful model; identify equilibria and constraints; establish local or regional stability; design the controller; analyze hybrid transitions; simulate adverse cases; test on hardware-in-the-loop; and finally validate experimentally with safety interlocks. Assumptions, uncertainty bounds, sampling rates, and failure responses should be documented.

Contemporary Developments

Current research includes nonlinear control of autonomous robots, distributed hybrid systems, event-triggered networked control, learning-enhanced model predictive control, barrier-function-based safety, and finite-time or fixed-time stabilization. Fixed-time methods seek convergence within a bound independent of initial condition, while prescribed-time methods target a designer-selected deadline; both require careful treatment of input magnitude and model uncertainty. Digital-twin platforms increasingly combine high-fidelity simulation with online monitoring, but model validation and data governance remain essential.

Chapter Summary

Nonlinear systems cannot generally be understood through superposition or a single transfer function. Equilibria, phase portraits, nullclines, local linearization, and Lyapunov methods provide complementary tools for understanding their behavior. Describing functions offer an approximate harmonic-balance method for predicting oscillations, but their assumptions and limitations must be recognized.

Feedback linearization transforms suitable nonlinear systems into linear input-output dynamics, provided that relative degree, input gain, and internal stability conditions are satisfied. Backstepping provides a recursive Lyapunov-based alternative for strict-feedback structures. Sliding-mode control offers robustness against matched uncertainties but introduces chattering and implementation challenges. NMPC and gain scheduling extend nonlinear control to constrained and varying operating conditions.

Switched and hybrid systems combine continuous evolution with discrete decisions. Stability may require a common Lyapunov function, dwell-time restrictions, or explicit analysis of guards and resets. Event-triggered control can reduce communication and computation but must address noise, delays, packet loss, and Zeno behavior. In professional practice, mathematical guarantees must be reconciled with saturation, sampling, quantization, safety, cybersecurity, and verification requirements.

Key Terms

Nonlinear system: A system whose governing equations are not linear in states, inputs, or outputs.

Equilibrium: A state at which the system derivative is zero for a specified input.

Phase portrait: A graphical representation of trajectories in a state plane.

Nullcline: A locus where one state derivative is zero.

Limit cycle: An isolated periodic trajectory of a dynamical system.

Local linearization: First-order approximation about an equilibrium.

Describing function: An approximate amplitude-dependent frequency representation of a nonlinearity.

Harmonic balance: An approximate method equating fundamental harmonic terms in a feedback loop.

Lyapunov function: A scalar function used to assess stability through its value and derivative.

LaSalle invariance principle: A result that establishes convergence using the invariant subset of V̇ = 0.

Control Lyapunov function: A function for which an input can enforce stabilizing decrease.

Feedback linearization: Nonlinear state or input transformation that produces linear input-output behavior.

Relative degree: Number of output differentiations before the input explicitly appears.

Zero dynamics: Internal dynamics remaining when the output is constrained to zero.

Sliding surface: A designed manifold to which sliding-mode trajectories are driven.

Chattering: High-frequency oscillation caused by discontinuous control implementation.

Switched system: A system whose dynamics change according to a switching signal.

Hybrid automaton: A formal model combining continuous flows, discrete modes, guards, and resets.

Event-triggered control: Control in which updates occur when a state-dependent condition is satisfied.

Zeno behavior: Infinitely many discrete events occurring in finite time.

Review Questions

Short Answer Questions

1. Define local asymptotic stability and distinguish it from global asymptotic stability.

2. What information is provided by the nullclines of a planar nonlinear system?

3. State the main assumptions behind describing-function analysis.

4. Why is a positive semidefinite Lyapunov derivative not always sufficient for asymptotic stability?

5. What is the purpose of a sliding surface?

Long Answer Questions

1. Explain phase-plane analysis, including equilibria, nullclines, trajectories, limit cycles, and separatrices.

2. Derive the feedback-linearizing control law for a system with output relative degree r and discuss zero-dynamics stability.

3. Compare feedback linearization, backstepping, sliding-mode control, and nonlinear model predictive control.

4. Explain how a common Lyapunov function can establish stability under arbitrary switching.

5. Discuss the practical and theoretical challenges of event-triggered control in a networked cyber-physical system.

Analytical / Application Questions

1. For ẋ_1 = x_2 and ẋ_2 = −x_1 − x_2^3, use a Lyapunov function and LaSalle’s principle to prove global asymptotic stability.

2. Design a sliding surface for a second-order position-tracking system and identify the assumptions needed for a reaching-condition proof.

3. Propose a hybrid control architecture for an autonomous mobile robot that switches between cruise, obstacle-avoidance, docking, and emergency-stop modes. Specify states, guards, resets, safety constraints, and a validation strategy.

Multiple Choice Questions with Answers

Question 1

For a continuously differentiable nonlinear system, what does a Jacobian with all eigenvalues having negative real parts establish about an equilibrium?

A. Global finite-time stability

B. Local exponential asymptotic stability

C. Instability under all inputs

D. Existence of a limit cycle

Correct Answer: B. Local exponential asymptotic stability

Question 2

For an ideal symmetric relay with output levels +M and −M, what is its describing function for sinusoidal input amplitude A?

A. M/(πA)

B. 2M/(πA)

C. 4M/(πA)

D. πM/(4A)

Correct Answer: C. 4M/(πA)

Question 3

If a Lyapunov function is positive definite and its derivative is negative definite in a neighborhood, the equilibrium is generally

A. Asymptotically stable

B. Only marginally stable

C. Necessarily unstable

D. A periodic orbit

Correct Answer: A. Asymptotically stable

Question 4

Which condition is required for the basic feedback-linearization law u = [v − a(x)]/b(x) to be well defined?

A. b(x) must be zero

B. b(x) must be nonzero in the operating region

C. The system must have no internal dynamics

D. The output must be discontinuous

Correct Answer: B. b(x) must be nonzero in the operating region

Question 5

On the sliding surface s = ė + λe = 0 with λ > 0, the tracking-error dynamics are

A. ė − λe = 0

B. ë + λe = 0

C. ė + λe = 0

D. ë + λė = 0

Correct Answer: C. ė + λe = 0

Question 6

A principal practical disadvantage of ideal sign-based sliding-mode control is

A. Lack of robustness to matched disturbances

B. Chattering caused by high-frequency switching

C. Requirement for a linear plant

D. Inability to use feedback

Correct Answer: B. Chattering caused by high-frequency switching

Question 7

A common Lyapunov function for a switched system is useful because it can establish stability

A. Only for one selected mode

B. Under arbitrary switching among the modes

C. Only when all modes are unstable

D. Only when no state is measured

Correct Answer: B. Under arbitrary switching among the modes

Question 8

What distinguishes a hybrid automaton from a purely continuous nonlinear model?

A. It has no state variables

B. It combines continuous flows with discrete modes and transition conditions

C. It must be linear in every mode

D. It cannot contain resets

Correct Answer: B. It combines continuous flows with discrete modes and transition conditions

Question 9

In event-triggered control, a transmission is commonly initiated when

A. The sampling error reaches a prescribed condition

B. Time is always exactly one second

C. The plant becomes linear

D. The controller is turned off permanently

Correct Answer: A. The sampling error reaches a prescribed condition

Question 10

Which phenomenon consists of infinitely many discrete transitions in finite time?

A. Harmonic balance

B. Zero dynamics

C. Zeno behavior

D. Gain scheduling

Correct Answer: C. Zeno behavior