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

The gist This paper presents a novel mechanism to certify reach-stabilize behavior under parametric uncertainty for control-affine systems, ensuring safe set recovery and convergence to a goal point without requiring persistence of excitation

Problem Formulation

The problem considered is the reach-stabilize problem for a class of nonlinear control-affine systems with unknown parametric uncertainties, where the system states must remain in a safe set at all times, or enter a safe set in finite time and then remain in it, and converge to a goal point The state must reach the safe set in finite time, remain in the set thereafter, and converge to a goal point For known system models [4], [5], [6]. In practice, the model parameters are uncertain, and a controller certified under an assumed model need not retain either guarantee under the true parameters [7], [8] The safety guarantee of a standard CBF in terms of the forward invariance of a safe set only holds if the state starts inside the set

Parameter Estimation with Computable Error Bounds

The paper constructs an estimator that generates a parameter estimate and a computable, nonincreasing bound on the estimation error from a known initial error bound, without requiring persistence of excitation The uncertainty enters as a known regressor matrix multiplying an unknown constant parameter vector The state dynamics are written as x˙ = f(x) + g(x)u + φ(x)θ⋆, (1) where x ∈ D ⊆ R n is the measured state, u ∈ U ⊂ R m is the input, f: D → R n, g: D → R n×m, and φ: D → R n×p are known and locally Lipschitz The estimate is updated by the gradient law ˙ˆθ = ΓΦTf e, (9) where ˆθ ∈ R p and Γ > 0 is a scalar gain A bound on the estimation error is obtained by propagating an assumed bound on the initial error, and this bound is computable and nonincreasing

Control Design with Parameter Adaptation

The proposed method uses one online estimate of the parameter and one computable bound on its error jointly certify recovery of the safe set, forward invariance, and convergence to the goal point The construction has four steps. First, ˆθ is generated by a gradient update law driven by a filtered regressor Second, β is constructed from a known bound on the initial estimation error, and is tightened by an accumulated regressor when the regressor is excited over an interval, without requiring persistence of excitation Third, a margin proportional to β is added to the control barrier function condition and to the control Lyapunov function condition Fourth, the two modified conditions are enforced in a single quadratic program The margins absorb the estimation error, and thus the finite-time recovery and convergence guarantees of the known-parameter design hold for the unknown parameter whenever the quadratic program is feasible

Numerical Simulation Results

In numerical examples, with the initial state outside and inside the safe set, both baseline oracle control policy that uses the true parameter, both recover or remain in the safe set and converge to the goal point, while another baseline control policy that uses a fixed, incorrect, parameter does not remain in the safe set and does not converge to the goal point The adaptive controller uses ˆθ(t) together with δh(t) = Lφh(x)β(t) and δV (t) = LφV (x)β(t) as in (24) and (29), ensuring that the closed loop remains in the safe set from an initial state inside it, enters the safe set in finite time from an initial state outside it, and converges to the goal point The adaptive controller satisfies kuk∞ ≤ 3 for all three controllers

Conclusions

The paper presents a controller that solves the reach-stabilize problem for control-affine systems with an unknown constant parameter vector multiplying a known regressor matrix The closed loop remains in the safe set from an initial state inside it, enters the safe set in finite time from an initial state outside it, and converges to the goal point The recovery-time bound equals that of the known-parameter design and does not depend on the unknown parameter In all figures, blue denotes the oracle controller, red the naive controller, and yellow the adaptive controller The proposed control policy ensures that both recovery or remain in the safe set and converge to the goal point The bound on T in item 3 depends on x(0), γ, and ρ only, and thus does not depend on θ⋆ or β

References

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1. Safety certification for control-affine systems with unknown parameters: The system can be designed to satisfy reach-stabilize behavior, meaning it can enter C in finite time and then remain in it, and converge to a goal point, even when the system starts outside the safe set.

2. Robustness against parameter uncertainty: The controller design ensures that the certified finite-time recovery bound equals that of the known-parameter design and does not depend on the unknown parameter, meaning the uncertainty thus changes the control effort required to meet the bound, and does not change the bound.

3. Online, excitation-free estimation: The system incorporates an estimator that generates a parameter estimate and a computable error bound without requiring persistence of excitation, meaning that the guarantees hold when no such excitation occurs.

4. Guaranteed convergence to goal point: The control law is designed such that the closed-loop trajectories satisfy Stabilize: For all x(0) ∈ D, limt→∞ x(t) = xc, ensuring the state converges to a specific goal point exponentially.

5. Quadratic programming for control synthesis: The system employs a single quadratic program (QP) to enforce both modified Control Barrier Function (CBF) and Control Lyapunov Function (CLF) conditions simultaneously, allowing for the design of a feasible control input at each time t from the quadratic program.

This paper considers the reach-stabilize prob- lem for a class of nonlinear control-affine systems with unknown parametric uncertainties, where the system states must remain in a safe set at all times, or enter a safe set in finite time and then remain in it, and converge to a goal point. An estimator is designed that generates a parameter estimate and a computable, nonincreasing bound on the estimation error from a known initial error bound, without requiring persistence of excitation. The bound defines margins that are added to a control barrier and a control Lyapunov function condition, which are then enforced in a quadratic program for efficient control design. It is shown that, for the closed-loop system, the safe set is forward invariant when the system starts within the safe set and is finite-time reachable from outside the safe set with a recovery-time bound that does not depend on the unknown parameter, and that the goal point is exponentially stable. In two numerical examples, with the initial state outside and inside the safe set, the proposed control policy, a baseline oracle control policy that uses the true parameter, both recover or remain in the safe set and converge to the goal point, while another baseline control policy that uses a fixed, incorrect, parameter does not remain