Reach-Stabilize Control of Control-Affine Systems with Unknown Affine Parameters

I. INTRODUCTION

• Reach-stabilize differs from reach-avoid and stabilize-avoid by requiring safe-set entry from outside the set, invariance thereafter, and convergence to a goal point.In reach-avoid, reaching refers to a target; in reach-stabilize, it refers to the safe set, while stabilization refers to the goal.Source: §I · p. 1, Figure 1 · p. 1
• Unknown parameters can undermine guarantees certified under an assumed model, while disturbances or abrupt safe-set changes can leave a system outside the safe set.Existing recovery methods do not necessarily certify both remaining safe after recovery and convergence to a goal under unknown parameters.Source: §I · pp. 1–2
• The paper’s stated novelty is a single online estimate and error bound that certify recovery, invariance, and goal convergence without persistence of excitation.The cited closest work requires persistent excitation, does not address recovery from outside the safe set, and does not guarantee convergence to a goal point.Source: §I · p. 2
• The controller certifies safe-set invariance from inside the set, finite-time recovery from outside it, and convergence to a goal point under parametric uncertainty.Its recovery-time bound matches the known-parameter design and does not depend on the unknown parameter; a known initial estimation-error bound is required, but persistence of excitation is not.Source: §I · p. 2

II. PROBLEM FORMULATION

• The method adds estimation-error margins to CBF and CLF conditions evaluated at the estimate, then enforces both conditions in one quadratic program.The proposed dynamics have a known regressor matrix multiplying an unknown constant parameter vector; the estimate and computable error bound jointly certify the guarantees.Source: §I · p. 2, §II · p. 3

III. PARAMETER ESTIMATION WITH COMPUTABLE ERROR BOUNDS

• The estimator section constructs a computable, nonincreasing error bound from an assumed initial bound and tightens it using accumulated regressor information under interval excitation.This bound is intended to satisfy |˜θ(t)| ≤ β(t) for all t ≥ 0.Source: §III · p. 3

A. Gradient-based parameter estimator

• A filtered regressor equation removes dependence on the unmeasured state derivative, so its signals can be computed online from measured state and input.The filter is applied elementwise; the first-order choice uses R(s) = 1/(s + p_f), with p_f > 0.Source: A. Gradient-based parameter estimator · p. 3
• A gradient law updates the parameter estimate using the prediction error, which satisfies e = Φ_f ˜θ.The filtered signals are computable online, leaving the unknown parameter as the only unknown in the filtered regressor equation.Source: A. Gradient-based parameter estimator · p. 3

B. Computable bound on the estimation error

• βp(t) is nonnegative and nonincreasing, and bounds the parameter-estimation error for all t ≥ 0.The bound is well defined under the stated conditions.Source: B. Computable bound on the estimation… · p. 3
• βp is computable from online e and known Γ.The passage identifies a single-integrator computation.Source: B. Computable bound on the estimation… · p. 3
• The bound remains well defined at states where φ is unexcited because its computation does not require inverting Φf..Source: B. Computable bound on the estimation… · p. 3

C. Exact estimation error under interval excitation

• The initial-error bound has a conservatism floor fixed at initialization and not reduced by data; accumulated regressor signals are introduced to remove that floor.The passage contrasts this with concurrent learning and dynamic regressor extension and mixing as estimators that avoid persistence of excitation.Source: C. Exact estimation error under… · p. 4
• β is nonnegative and nonincreasing, and bounds the parameter-estimation error for all t ≥ 0.The bound combines βp with interval-derived bounds; the infimum in (20) is a running minimum computed by a monotone hold.Source: C. Exact estimation error under… · p. 4
• Interval excitation in every parameter direction makes Tt nonempty, rather than requiring persistent excitation.Φ(t) ≻ 0 exactly when no nonzero parameter direction is annihilated by Φf(τ) for almost every τ ∈ [0, t].Source: C. Exact estimation error under… · p. 4

IV. CONTROL DESIGN WITH PARAMETER

• The known-parameter CBF and CLF conditions cannot be evaluated because they contain the unknown parameter θ⋆.The design instead evaluates each condition at the estimate θ̂ and adds a margin bounding the contribution of estimation error through β.Source: §IV · p. 4
• The modified CBF condition certifies forward invariance of C and finite-time recovery to C with a recovery-time bound independent of θ⋆ and β.Source: §IV · p. 4
• The modified CLF condition yields exponential convergence of V, and both modified conditions are enforced in a single quadratic program.Source: §IV · p. 4

A. Modified CBF condition

• The modified CBF condition evaluates the unknown-parameter condition at θ̂ and adds a safety margin δh based on the estimation-error bound β.The stated choice is δh(t) = ∥Lφh(x)∥|β(t)|.Source: A. Modified CBF condition · p. 4
• If the condition holds along the trajectory, C is forward invariant for every initial state in C and is reached in finite time from outside.The theorem states this under its specified conditions and existence of a control signal satisfying the condition.Source: A. Modified CBF condition · p. 4
• After the trajectory reaches C, forward invariance holds from that time onward.Source: A. Modified CBF condition · p. 5

B. Modified CLF condition

• The unknown parameter prevents direct enforcement of the known-parameter CLF condition, so the modified condition evaluates it at the estimate and adds margin δV.The margin bounds the estimation error’s contribution to ˙V; choosing δV(t) = ∥LφV(x)∥|β(t)| ensures convergence.Source: B. Modified CLF condition · p. 5
• With the modified CLF condition, V converges to zero exponentially and the state converges to the goal point from every initial state in D.The result assumes the modified condition holds for all t ≥ 0 along the solution.Source: B. Modified CLF condition · p. 5
• For a known parameter, the CLF inequality implies V(x(t)) ≤ e^-λtV(x(0)), which yields convergence to xc.This result follows from ˙V(x(t)) ≤ −λV(x(t)) and the CLF lower bound on distance to xc.Source: B. Modified CLF condition · p. 5

C. Control design using quadratic programming

• Feasibility is a hypothesis of the closed-loop guarantee, and the QP is not guaranteed to be feasible.The modified CBF and CLF conditions are hard constraints.Source: C. Control design using quadratic… · pp. 5–6
• Bounding U can cause infeasibility because recovery terms and estimation-error margins are largest at initialization; enlarging U enlarges the set of states where the QP is feasible.With U = R^m, infeasibility can still occur when a relevant control direction vanishes or the CBF and CLF constraints are incompatible.Source: C. Control design using quadratic… · p. 6
• When the QP is feasible throughout D and its solution is continuous, V converges exponentially to zero, the safe set is forward invariant, and states outside it enter within the stated bound.For x(0) outside C, the bound is |h(x(0))|^(1−ρ)/(γ(1−ρ)).Source: C. Control design using quadratic… · p. 6

• The simulation applies the proposed controller to a constructed unstable system with state-dependent f, g, and φ.Source: §V · p. 6
• The adaptive controller is compared with an oracle using θ⋆ and a naive controller using an incorrect constant parameter.Source: §V · p. 6
• An initial condition outside C tests finite-time recovery, while an initial condition inside C tests forward invariance.Source: §V · p. 6

A. Setup

• The example uses C = {x : h(x) ≥ 0} with h(x) = r² − ∥x∥², r = 1.5, goal xc = (0, 1), and U = {u : ∥u∥∞≤3}.The CLF is V(x) = ∥x − xc∥².Source: A. Setup · p. 6
• Choosing xc with φ(xc) ≠ 0 keeps the regressor nonzero at steady state, allowing the estimator bound β to contract with data.If xc = 0, the trajectory settles where φ(x) = 0 and β stops contracting.Source: A. Setup · p. 6
• The example uses θ⋆ = 2, ˆθ(0) = 0, and β0 = 2 = |θ⋆−ˆθ(0)|; figures color the oracle blue, naive red, and adaptive yellow.Source: A. Setup · p. 6
• The adaptive controller uses ˆθ(t) and margins δh and δV, while the oracle uses θ⋆ without margins and the naive controller uses the initial estimate without margins.The QP is feasible when g(x) retains rank two on the visited region, making the CBF and CLF constraints generically compatible half-planes.Source: A. Setup · p. 6

B. Finite-Time Recovery

• The outside-safe-set test starts at x(0) = (1.8, −0.6), where h(x(0)) = −1.35; Figures 2 and 3 show diagnostics and phase-plane trajectories.Source: B. Finite-Time Recovery · p. 6, Figure 2 · p. 7, Figure 3 · p. 7
• The oracle and adaptive controllers enter C below the recovery-time bound, remain in C, and drive V toward zero; the naive controller leaves C and V plateaus near 3.2.The naive controller crosses h = 0 transiently and does not remain in C.Source: Table I · p. 6, B. Finite-Time Recovery · p. 6
• For the adaptive controller, β is nonincreasing and the accumulated-regressor term reduces it below βp; all three controllers satisfy ∥u∥∞≤3.Source: B. Finite-Time Recovery · p. 6

C. Forward Invariance

• For x(0) = (0.5, −1.2), Fig. 4 tracks h, log-scale V, β(t), |˜θ|, and control input u.The initial state is inside the safe set, with h(x(0)) = 0.560.Source: Figure 4 · p. 7, C. Forward Invariance · p. 7
• At 4.54 s, the naive controller leaves C, while the oracle and adaptive controllers remain in C and drive V toward zero over the horizon.The result is for x(0) = (0.5, −1.2), which lies inside the safe set.Source: Table II · p. 7, C. Forward Invariance · p. 7

VI. CONCLUSIONS

• The proposed controller combines an online nonincreasing estimation-error bound with CBF and CLF margins enforced in one QP, guaranteeing safety, finite-time recovery, and goal convergence.The recovery-time bound equals that of the known-parameter design and does not depend on the unknown parameter.Source: §VI · p. 7