Chattering Phenomena in Time-Optimal Control for High-Order Chain-of-Integrator Systems with Full State Constraints (Extended Version)

TL;DR

This work tackles the longstanding open problem of time-optimal control for high-order chain-of-integrator systems with full state constraints and establishes a rigorous framework for chattering. It proves that chattering exists for 4th-order problems with velocity constraints and a unique chattering mode, while nonexistence holds for lower orders; for the critical case, it derives an explicit attenuating chattering sequence with attenuation factor $\alpha^*\approx0.1660687$ and computes associated time scales. The authors also connect the finite-time chattering to an infinite-horizon reformulation, provide closed-form optimal trajectories, and demonstrate practical realizability via finite-frequency interpolation, rectifying industry misconceptions about the time-optimality of conventional S-shaped profiles. The results enable strictly time-optimal snap-limited trajectories under full state constraints and offer switching-surface geometry and costate dynamics that underpin future numerical methods and tooling for ultra-precision motion control.

Abstract

Time-optimal control for high-order chain-of-integrator systems with full state constraints remains an open and challenging problem within the discipline of optimal control. The behavior of optimal control in high-order problems lacks precise characterization, and even the existence of the chattering phenomenon, i.e., the control switches for infinitely many times over a finite period, remains unknown and overlooked. This paper establishes a theoretical framework for chattering phenomena in the considered problem, providing novel findings on the uniqueness of state constraints inducing chattering, the upper bound of switching times in an unconstrained arc during chattering, and the convergence of states and costates to the chattering limit point. For the first time, this paper proves the existence of the chattering phenomenon in the considered problem. The chattering optimal control for 4th-order problems with velocity constraints is precisely solved, providing an approach to plan time-optimal snap-limited trajectories. Other cases of order $n\leq4$ are proved not to allow chattering. The conclusions rectify a longstanding misconception in the industry concerning the time-optimality of S-shaped trajectories with minimal switching times.

Figures (8)

• Figure 1: (a-d) Strictly optimal trajectories for position-to-position problems of order $n=1,2,3,4$, respectively. (e) A suboptimal trajectory planned by the MIM method in our previous work wang2024time. (f) Augmented switching laws (ASL, see Definition \ref{['def:AugmentedSwitchingLaw']}) of trajectories in (d-e). $M_0=1$, $M_1=1$, $M_2=1.5$, $M_3=4$, $M_4=15$. For an $n$th-order problem, ${\boldsymbol{x}}_0=-M_n{\boldsymbol{e}}_n$, ${\boldsymbol{x}}_{\mathrm{f}}=M_n{\boldsymbol{e}}_n$. In (a-e), $\bar{u}=\frac{u}{M_0}$. $\forall1\leq k\leq4$, $\bar{x}_k=\frac{x_k}{M_k}$, and $\bar{\lambda}_k=\frac{\lambda_k}{\left\|\lambda_k\right\|_{\infty}}$. (d3-d4) show the enlargements of (d1-d2) during the chattering period. The abscissa is in logarithmic scale with respect to time, i.e., $-\log_{10}\left(t_{\infty}-t\right)$, where $t_\infty\approx6.0732$ is the first chattering limit time. $\forall k=1,2,3$, $\hat{x}_k\left(t\right)=\frac{x_k^*\left(t\right)\left(t_\infty-t\right)^{-k}}{\left\|x_k^*\left(t\right)\left(t_\infty-t\right)^{-k}\right\|_{\infty}}$, and $\hat{\lambda}_k\left(t\right)=\frac{\lambda_k\left(t\right)\left(t_\infty-t\right)^{k-4}}{\left\|\lambda_k\left(t\right)\left(t_\infty-t\right)^{k-4}\right\|_{\infty}}$.
• Figure 2: A 3rd-order optimal trajectory planned by wang2024time, whose ASL is $S=\overline{01}\underline{0}\left(\underline{3},2\right)\underline{0}\overline{0}\underline{0}$. In this example, $\lambda_0>0$, ${\boldsymbol{x}}_0=\left(0.9,-0.715,0.1288\right)$, ${{\boldsymbol{x}}_\mathrm{f}}=\left(0,0,-0.15\right)$, and ${\boldsymbol{M}}=\left(1,1,1.5,0.15\right)$. (a) The state vector. (b) The costate vector. (c) The trajectory ${\boldsymbol{x}}={\boldsymbol{x}}\left(t\right)$. (d) The flow chat of $S$.
• Figure 3: Chattering arcs in problem \ref{['eq:optimalproblem']}. (a) Multiple active constraints. (b) Constrained arcs. (c) Unconstrained arcs connected at the constraint's boundary. (a) and (b) are not allowed, while (c) exists in chattering periods.
• Figure 4: Optimal solution of problem \ref{['eq:optimalproblem_n4s3_equivalent']}. (a) The optimal trajectory ${\boldsymbol{y}}^*\left(\tau\right)$ and the optimal control $v^*\left(\tau\right)$. (b) The optimal costate vector $\frac{1}{p_0}{\boldsymbol{p}}^*\left(\tau\right)$. (c-d) Enlargement of (a-b) during the chattering period. The abscissa is in logarithmic scale with respect to time, i.e., $-\log_{10}\left(\tau_{\infty}^*-\tau\right)$. $\forall k=1,2,3$, $\hat{y}_k^*\left(\tau\right)=y_k^*\left(\tau\right)\left(1-\frac{\tau}{\tau_{\infty}^*}\right)^{-k}$, and $\hat{p}_k^*\left(\tau\right)=p_k^*\left(\tau\right)\left(1-\frac{\tau}{\tau_{\infty}^*}\right)^{k-4}$.
• Figure 5: Loss function $\widehat{J}\left(\alpha\right)$ when choosing different chattering attenuation rate $\alpha$ in problem \ref{['eq:optimalproblem_n4s3_equivalent']}. (a) and (b) are in different scales.

Theorems & Definitions (59)

• Proposition 1: Junction condition in problem \ref{['eq:optimalproblem']}
• Remark
• Proposition 2: System dynamics of problem \ref{['eq:optimalproblem']}
• proof
• Lemma 1: Optimal Control's Behavior of Problem \ref{['eq:optimalproblem']} wang2024time
• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Remark