Some remarks on practical stabilization via CLF-based control under measurement noise

TL;DR

The work tackles practical stabilization of input-affine systems with measurement noise by combining a Lyapunov-based decay condition with a self-triggered sensing strategy. It derives state-dependent bounds on measurement error $\bar{\varepsilon}(\hat{x})$ ensuring a decay condition holds for all states in a neighborhood and introduces a triggering rule that guarantees convergence into and persistence within a target ball $\mathcal{B}_r$, while maintaining admissible controls. A hierarchical bound construction yields a core ball $\mathcal{B}_{r^*}$ and a lower bound on measurement error outside it, linking measurement precision, triggering, and stability. Three case studies with one- and two-input systems illustrate practical stabilization under noisy measurements, showing how per-state bounds can enable stable behavior with feasible sensing and sampling policies. The results offer a quantitative pathway to deploy CLF-based controllers in the presence of measurement imperfections and actuator limits, with potential extensions to delays and integration with optimization-based control.

Abstract

Practical stabilization of input-affine systems in the presence of measurement errors and input constraints is considered in this brief note. Assuming that a Lyapunov function and a stabilizing control exist for an input-affine system, the required measurement accuracy at each point of the state space is computed. This is done via the Lyapunov function-based decay condition, which describes along with the input constraints a set of admissible controls. Afterwards, the measurement time points are computed based on the system dynamics. It is shown that between these self-triggered measurement time points, the system evolves and converges into the so-called target ball, i.e. a vicinity of the origin, where it remains. Furthermore, it is shown that the approach ensures the existence of a control law, which is admissible for all possible states and it introduces a connection between measurement time points, measurement accuracy, target ball, and decay. The results of the approach are shown in three examples.

Figures (8)

• Figure 1: Visualization of $\varphi(u,\widehat{x};w) \leq 0$ (orange) and the $2^{2+1} = 8$ inequalities (blue) that are obtained with the bounds in \ref{['eqn:beta_min_beta_max_bounds']} as well as input constraints (violet). A control $u$ inside the white area is admissible, i.e. it ensures that $\varphi(u,x;w) \leq 0$ holds for all $x \in \mathcal{B}_{\varepsilon_i}(\widehat{x})$. This area shrinks with growing $\varepsilon$, where $\varepsilon_1$ of the left figure is smaller than $\varepsilon_2$ of the right figure.
• Figure 2: The effect of relaxing the decay condition. Upper pictures: For a required decay $w$ (blue dashed line) and a measured state $\widehat{x}$, $\beta_0(\widehat{x}; w)$ and $\beta_1(\widehat{x})$ are computed. The values $\beta_0(x; w)$ and $\beta_1(x)$ of the unknown state $x$ are located in the light gray shaded area, which is given via Lemma \ref{['lem:bounds-for-beta-Lip']}. The maximum measurement error at $\hat{x}$ is determined such that this gray shaded area remains in maximum two quadrants. If the decay is relaxed to $\widetilde{w}$, $\beta_0(x; \widetilde{w})$ shifts to the left along with the set where $\beta_0(x; \widetilde{w})$ and $\beta_1(x)$ are located. This relaxation of the decay might ensure that the maximum measurement error at $\widehat{x}$ can be enlarged. This case is presented on the right-hand side, where the shift to the left (caused by the relaxation of $w$) yields a buffer to the $\beta_1$ axis. It allows to enlarge the measurement error until the enlarged set touches the $\beta_1$ axis. Lower pictures: Due to the relaxation, the set of admissible controls enlarges.
• Figure 3: Visualization of \ref{['eqn:inequality-bar-eps']}: Based on the current measurement, $\widehat{x}^+(t_k)$ is obtained. All points starting in $\mathcal{B}_\varepsilon(\widehat{x}^+(t_k))$ remain in $\mathcal{B}_{\varepsilon + \overline F \delta_k}(\widehat{x}^+(t_k))$. After the measurement after $\delta_k$ section, the measurement is located in $\mathcal{B}_{2 \varepsilon + \overline F \delta_k}(\widehat{x}^+(t_k))$. Since the maximum measurement error is computed as $\overline \varepsilon(\widehat{x}^+(t_k))$, $\delta_k$ is chosen to satisfy this bound such that the set of admissible controls is nonempty after $\delta_k$ seconds and that the applied control is also admissible after a measurement.
• Figure 4: The procedure of the proof of Theorem \ref{['thm:max-meas-err']}: Once a measurement $\widehat{x}_k^+$ inside the core ball $\mathcal{B}_{r^\star}$ is obtained, an arbitrary control, e.g. $u = 0$, is applied as long as it is ensured that the next measurement $\widehat{x}_{k+1}^+$ is located in the target ball $\mathcal{B}_{\widetilde{r}}$. The set $\{ x \in \mathbb{R}^n: V(x) \leq V_r \}$ describes the level set of the CLF that the closed-loop trajectories do not leave since there exists a stabilizing control for all $\widehat{x} \in \mathcal{B}_\varepsilon(x_{k+1})$. Afterwards, the closed-loop trajectories evolve and remain in the target ball $\mathcal{B}_r$.
• Figure 5: Convergence of the closed-loop trajectories starting in $x_0$ and $\widehat{x}_0$ into the target ball $\mathcal{B}_r$ with $r = 1$. It can be seen that it remains in $\mathcal{B}_{r}$ after entering it once. The radius of the core ball is shown in cyan, while the radius of the target ball is added in magenta.

Theorems & Definitions (11)

• Lemma 1
• proof
• Theorem 1
• proof
• Lemma 2
• proof
• Remark 1
• Theorem 2
• proof
• Remark 2