Help
Options
Didn't know it?
click below
click below
Knew it?
click below
click below
Don't Know
Remaining cards (0)
Know
retry
shuffle
restart
0:00
Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.
Normal Size Small Size show me how
Normal Size Small Size show me how
Nonlinear Control
| Question | Answer |
|---|---|
| Term | Definition |
| What is a nonlinear dynamical system? | A system whose state evolution is described by differential equations that are not linear in the state and/or input; superposition generally does not hold. |
| What is the standard state-space model for a nonlinear system? | ẋ = f(t, x, u), where x is the state vector, u the input, and f defines the state dynamics. |
| What is an output equation in state-space form? | y = h(t, x, u), where y contains the measured or otherwise relevant output variables. |
| What is an equilibrium point of an autonomous system ẋ = f(x)? | A point x* satisfying f(x*) = 0; if the system starts there, it remains there. |
| How do you find equilibria of a two-dimensional autonomous system? | Solve f₁(x₁,x₂) = 0 and f₂(x₁,x₂) = 0 simultaneously. |
| What is a phase portrait? | A qualitative picture of the family of system trajectories (orbits) in state space, often including equilibrium points and direction of motion. |
| What is an orbit in a phase portrait? | The curve traced by a solution in state space; time is suppressed, so the curve alone does not show how quickly it is traversed. |
| What does the vector field show for ẋ = f(x)? | At each state x, the vector f(x) gives the instantaneous direction and rate of motion. |
| Why are nonlinear systems often analyzed by linearization? | Near an equilibrium, a nonlinear system may be approximated by a linear system whose eigenvalues help predict local behavior. |
| How is a nonlinear system linearized near an equilibrium x*? | Set z = x − x* and approximate ż ≈ A z, where A = ∂f/∂x evaluated at x* (the Jacobian at the equilibrium). |
| What is the Jacobian matrix of f(x)? | The matrix of first partial derivatives: J(x) = [∂fᵢ/∂xⱼ]. It describes the local first-order variation of the vector field. |
| Why must the linearization be evaluated at an equilibrium? | The local stability test concerns motion near a stationary point; the relevant matrix is A = J(x*) with f(x*) = 0. |
| What does a negative real part of an eigenvalue imply for a linear mode? | That mode decays exponentially over time. |
| What does a positive real part of an eigenvalue imply for a linear mode? | That mode grows exponentially over time. |
| What is a stable node in a 2D linear system? | An equilibrium with two real negative eigenvalues; nearby trajectories approach it without spiral rotation. |
| What is an unstable node in a 2D linear system? | An equilibrium with two real positive eigenvalues; nearby trajectories move away without spiral rotation. |
| What is a stable focus (spiral sink)? | An equilibrium with complex-conjugate eigenvalues whose real part is negative; trajectories spiral inward. |
| What is an unstable focus (spiral source)? | An equilibrium with complex-conjugate eigenvalues whose real part is positive; trajectories spiral outward. |
| What is a saddle equilibrium? | An equilibrium with eigenvalues of opposite-sign real parts; it is unstable, with stable and unstable directions. |
| What is a center in a 2D linear system? | An equilibrium with nonzero purely imaginary eigenvalues; nearby linear trajectories are closed orbits and neither approach nor depart from the equilibrium. |
| How do you classify a 2×2 linear system when eigenvalues are real and have the same sign? | Both negative: stable node; both positive: unstable node. Repeated-eigenvalue cases may need additional inspection for the detailed portrait. |
| How do you classify a 2×2 system with real eigenvalues of opposite signs? | As a saddle; one mode decays and the other grows. |
| How do you classify complex eigenvalues λ = α ± jβ, β ≠ 0? | α < 0: stable focus; α > 0: unstable focus; α = 0: center for the linear system. |
| What is a hyperbolic equilibrium? | An equilibrium whose linearization has no eigenvalues on the imaginary axis (all eigenvalues have nonzero real parts). |
| Why is hyperbolicity important for nonlinear linearization? | For a hyperbolic equilibrium, the nonlinear system has the same local qualitative type as its linearization under the standard local linearization result. |
| What is the main limitation of linearization? | It predicts local behavior near the equilibrium, not necessarily global behavior far from it. |
| Can linearization conclusively classify a nonlinear equilibrium when an eigenvalue has zero real part? | Generally no. The linearization is inconclusive in non-hyperbolic cases; nonlinear analysis may be required. |
| Why is a center structurally fragile? | Its eigenvalues lie on the imaginary axis; arbitrarily small perturbations can move them into the left or right half-plane, producing a stable or unstable focus. |
| What does structural stability mean in this context? | The qualitative phase portrait persists under sufficiently small perturbations of the system parameters. |
| Which common 2D equilibrium types are structurally stable under small perturbations? | Hyperbolic types such as nodes, foci, and saddles; a center is generally not structurally stable. |
| What is the stable manifold of a saddle? | The set of states whose trajectories approach the saddle as time increases. |
| What is the unstable manifold of a saddle? | The set of states whose trajectories depart from the saddle in forward time (equivalently, approach it in backward time). |
| What are separatrices in a phase portrait? | Special trajectories, often stable manifolds of saddles, that divide regions with different qualitative trajectory behavior. |
| What is a limit cycle? | An isolated periodic orbit of a dynamical system; nearby trajectories do not form a continuous family of periodic orbits. |
| What condition defines a periodic solution with period T? | x(t + T) = x(t) for all t in its domain, for some T > 0. |
| Why are limit cycles a specifically important nonlinear phenomenon? | A nonlinear system can sustain an isolated oscillation, often robustly, unlike the non-robust purely imaginary eigenvalue condition required for oscillations in an unforced linear system. |
| What is finite escape time? | A solution becomes unbounded in a finite amount of time, rather than only tending to infinity as time tends to infinity. |
| What are multiple isolated equilibria? | Two or more distinct equilibrium points in the same system; their presence is a common nonlinear phenomenon. |
| What is a key difference between linear and nonlinear systems regarding equilibria? | A linear system has equilibria determined by Ax = 0 (and may have a whole subspace of equilibria); nonlinear systems can have multiple isolated equilibria. |
| What is the principle of superposition? | For a linear system, sums of solutions (with corresponding input combinations where applicable) remain solutions; nonlinear systems generally do not satisfy this principle. |
| What does continuous dependence of eigenvalues on matrix entries imply for stability? | Eigenvalues strictly in either open half-plane remain there under sufficiently small perturbations, but eigenvalues on the imaginary axis can move to either side. |
| What is the undamped pendulum model in state variables? | ẋ₁ = x₂ and ẋ₂ = −sin(x₁), where x₁ is angular position and x₂ is angular velocity. |
| Where are the equilibria of the undamped pendulum model ẋ₁ = x₂, ẋ₂ = −sin(x₁)? | At (x₁,x₂) = (kπ, 0) for any integer k. |
| What changes when friction is added to the pendulum model? | A damping term proportional to angular velocity is added, e.g. ẋ₂ = −sin(x₁) − c x₂ with c > 0; energy is dissipated. |
| Why can a nonlinear system have behavior that a local linear model misses? | Nonlinearities can create global effects such as multiple equilibria, limit cycles, finite escape time, and complex or chaotic behavior. |
| What is a harmonic oscillator in the 2D linear-system setting? | A system with purely imaginary eigenvalues and oscillatory solutions; for example, ẍ + ω²x = 0. |
| What is the qualitative effect of a stable eigenmode? | Its contribution to the state decays exponentially as time increases. |
| What is the qualitative effect of an unstable eigenmode? | Its contribution grows exponentially and typically causes nearby trajectories to move away from the equilibrium. |
| Why can the same nonlinear system show different behaviors in different regions of state space? | Its dynamics depend on the state, so different regions can contain different equilibria, attraction basins, or oscillatory behaviors. |
| What is the main purpose of nonlinear control techniques? | To analyze and design control laws for systems whose dynamics cannot be adequately handled by linear methods alone. |
Created by:
Filotì