For a nonlinear system ẋ₁ = x₂, ẋ₂ = −sin(x₁) − x₂ + u (a damped pendulum with control torque), you propose Lyapunov function V = ½x₁² + ½x₂². To make dV/dt negative, what control law u = u(x₁, x₂) should you choose?
Au = 0 (no control needed, the system is already stable from the damping term)
Bu = x₁ + x₂ (add linear feedback)
Cu = x₁ + 2x₂ (cancel the nonlinearity and add extra damping)
Du = sin(x₁) + 2x₂ (nonlinear feedback that exactly cancels the sine term and adds damping)
Computing dV/dt = x₁·ẋ₁ + x₂·ẋ₂ = x₁x₂ + x₂(−sin(x₁) − x₂ + u). To make dV/dt < 0, we need −x₂² + x₂(sin(x₁) + u) < 0 for all x₂. Choosing u = −sin(x₁) − 2x₂ gives dV/dt = −x₂² − x₂² = −2x₂² ≤ 0, with equality only at x₂ = 0. By the second derivative test (LaSalle invariance principle), all trajectories converge to x₂ = 0, and then from ẋ₁ = 0 and the system dynamics, x₁ → 0. This nonlinear feedback law is derived purely from the Lyapunov function, with no need to linearize or compute eigenvalues.
Question 2 Multiple Choice
Feedback linearization transforms a nonlinear system into an equivalent linear system by choosing u as a nonlinear function of state and a linear control input v: ẍ = f(x, ẋ) + g(x, ẋ)u → you set u = [f(x, ẋ) + v]/g(x, ẋ). What is the main practical limitation?
AFeedback linearization only works for systems that are already linear
BThe nonlinear control law requires precise knowledge of f and g (the system model) and often involves dividing by g, which amplifies errors when g is small; measurement noise is amplified through differentiation
CFeedback linearization is computationally too expensive for real-time control
DThe linear system resulting from feedback linearization is always unstable
Feedback linearization is model-dependent: if your model of f or g is incorrect, the linearization is inexact and the closed-loop system may not be linear or stable. Since the control law often includes terms like u = [−f + v]/g, a small error in estimating g becomes a large error in u. Additionally, if the control law requires derivatives of measured signals (e.g., ∂f/∂ẋ multiplied by estimated acceleration), measurement noise is differentiated and amplified. In practice, feedback linearization works best for systems where the model is accurate and measurement noise is low. Many applications blend feedback linearization (in the low-frequency control law) with robustness techniques (high-frequency correction) to mitigate model uncertainty.
Question 3 True / False
Backstepping is a recursive control design method that stabilizes a nonlinear system by stabilizing lower-order subsystems one at a time. If a subsystem has a control-affine structure ż = f(z) + g(z)·ζ (where ζ is a 'virtual control'), backstepping treats ζ as a design variable. What does it mean to design ζ as a function of z?
TTrue
FFalse
Answer: True
In backstepping, you first design a stabilizing 'virtual control' law ζ = α(z) such that dV₁/dt = ∂V₁/∂z·f(z) + ∂V₁/∂z·g(z)·α(z) < 0. This ensures the z-subsystem is stable when ζ = α(z). Then, you augment the Lyapunov function to include a term for ζ − α(z) (the error between the actual and desired virtual control), and design the real control input u to stabilize this augmented system. The recursion continues, one state at a time. This is powerful for cascade-like systems where you can stabilize the innermost subsystem and build outward.
Question 4 True / False
In passivity-based control, a system is passive if its energy is non-increasing: the work done by external inputs is always ≥ the change in stored energy. How does this structure enable simple stabilizing controllers?
TTrue
FFalse
Answer: True
A passive system with a stable energy minimum (e.g., a mechanical system with friction or damping) is inherently stable at the energy-minimizing equilibrium. A passive controller (one that dissipates energy) connected to a passive system remains stable by the passivity theorem — no destabilizing feedback loops can arise. This is why a simple proportional feedback u = −K·(q − q_ref) (proportional control about a desired position) stabilizes a passive mechanical system: the controller dissipates energy proportional to position error, and the system's natural damping does the rest. Passivity-based design is particularly powerful for mechanical and electrical systems where the Lagrangian structure is known.
Question 5 Short Answer
Explain why a Lyapunov function is more than just a mathematical tool and represents a fundamentally different design philosophy from linear frequency-domain control (Bode plots, Nyquist criterion).
Think about your answer, then reveal below.
Model answer: Frequency-domain control (Bode, Nyquist) treats the plant as a black box: you measure its frequency response or transfer function and design compensators to shape magnitude and phase at each frequency, ensuring stability margins and bandwidth. This approach is powerful for linear systems but breaks down for nonlinear systems (no frequency response, no transfer function). Lyapunov design is model-based and time-domain: it uses the state equations directly and designs feedback to drive an energy-like function to decrease. The design provides provable guarantees for the *nonlinear* system, not a linearization. More fundamentally, a Lyapunov function is a way of saying 'I know what system behavior I want to achieve (drive V toward a minimum), and I can compute control actions that steer toward it.' It's intuitive (like a ball rolling down a landscape toward the lowest point) but requires deeper system understanding — you must propose the right Lyapunov function, and for complex systems this can be as much art as science.
Modern control design increasingly blends both philosophies: use frequency-domain techniques for the nominal model (ensuring robustness), then use Lyapunov methods to extend the design to handle nonlinearities and larger disturbances. Some systems are purely nonlinear (soft robotics, contact dynamics) where frequency-domain tools don't apply; others are mildly nonlinear and benefit from linear design with nonlinear corrections. The Lyapunov perspective is essential for systems where equilibrium changes (e.g., a robot balancing while walking) or disturbances are large enough that linearization is invalid.