Robust control Flashcards
(25 cards)
Q: What is a common Lyapunov function
A: A single positive definite Vx whose derivative is negative definite for every model in the uncertainty set
Q: Robust stability certificate for a polytopic set of systems
A: Find P > 0 with AiᵀP + PAi < 0 for all vertices Ai so every convex combination is stable
Q: Meaning of LMI
A: Linear Matrix Inequality a constraint of the form F0 + ∑θiFi ⪯ 0 that is convex in the decision variables
Q: S-procedure purpose
A: Converts an implication between two quadratic forms into a single matrix inequality using a nonnegative scalar multiplier
Q: Sector bound l u for nonlinearity Δ
A: Condition lq ≤ Δq ≤ uq rewritten as quadratic constraint (p − lq)(p − uq) ≤ 0 with p = Δq
Q: Closed loop state matrix with state feedback u = Kx
A: Acl = A + BK
Q: Bounded real lemma goal
A: Gives an LMI that guarantees L2 gain from w to z is below γ
Q: Storage function inequality for L2 gain
A: V̇x ≤ γ² wᵀw − zᵀz ensures ∥z∥L2 ≤ γ∥w∥L2
Q: Change of variables for controller synthesis
A: Set S = P⁻¹ and Y = KS to make bilinear terms linear in S and Y
Q: Recovery of controller after LMI solution
A: K = YS⁻¹
Q: Condition for exponential stability with rate α using P
A: P > 0 and AᵀP + PA ⪯ −αP
Q: Radially unbounded function definition
A: Vx → ∞ as ∥x∥ → ∞
Q: Global asymptotic stability via Lyapunov
A: Need V positive definite radially unbounded and V̇ negative definite for all x≠0
Q: Role of congruence transformation in LMIs
A: Multiplies an inequality by Tᵀ and T to keep definiteness while reparameterising variables
Q: Definition of polytopic uncertainty
A: System matrices vary inside the convex hull of a finite set of vertex matrices
Q: Why robust analysis may be conservative
A: Single Lyapunov function may not exist even if every individual model is stable so test can fail though system is actually robust
Q: Switched system uniform stability condition
A: Existence of one P with AσᵀP + PAσ < 0 for every mode σ guarantees stability under arbitrary switching
Q: Meaning of H∞ performance bound γ
A: Worst case L2 gain from disturbance to performance output is less than γ
Q: LQR objective J
A: Integral from zero to infinity of xᵀQx + uᵀRu dt
Q: Bilinearity issue in design LMIs
A: Terms like BKP are nonlinear in both K and P so require variable change to get convexity
Q: Observability Gramian role in H2 norm
A: M solves AᵀM + MA + CᵀC = 0 and H2 norm squared equals trace MGGᵀ
Q: Interval matrix stability test complexity
A: Deciding if every matrix in an interval family is Hurwitz is NP hard
Q: Purpose of slack variable Z in LQR LMI
A: Linearises trace S⁻¹GGᵀ objective via Schur complement making optimisation convex
Q: Linear time varying uncertainty handled by common P
A: If P satisfies inequalities for all vertices then inequality holds for any time varying convex combination A t
