Robust Control Lyapunov-Value Functions for Nonlinear Disturbed Systems

Abstract

Control Lyapunov Functions (CLFs) have been extensively used in the control community. A well-known drawback is the absence of a systematic way to construct CLFs for general nonlinear systems, and the problem can become more complex with input or state constraints. Our preliminary work on constructing Control Lyapunov Value Functions (CLVFs) using Hamilton-Jacobi (HJ) reachability analysis provides a method for finding a non-smooth CLF. In this paper, we extend our work on CLVFs to systems with bounded disturbance and define the Robust CLVF (R-CLVF). The R-CLVF naturally inherits all properties of the CLVF; i.e., it first identifies the "smallest robust control invariant set (SRCIS)" and stabilizes the system to it with a user-specified exponential rate. The region from which the exponential rate can be met is called the "region of exponential stabilizability (ROES)." We provide clearer definitions of the SRCIS and more rigorous proofs of several important theorems. Since the computation of the R-CLVF suffers from the "curse of dimensionality," we also provide two techniques (warmstart and system decomposition) that solve it, along with necessary proofs. Three numerical examples are provided, validating our definition of SRCIS, illustrating the trade-off between a faster decay rate and a smaller ROES, and demonstrating the efficiency of computation using warmstart and decomposition.

Figures (5)

• Figure 1: SRCIS corresponds to different loss functions for system \ref{['eqn:EX_IN2D']}. Top left to right: different loss functions, including the 2-norm, infinity norm, and weighted Q-norm ($Q = diag [0.2, 1]$). Middle left to right: R-CLVF ($\gamma = 0$) when $h(x) = ||x||_2$, $||x||_\infty$, $||x||_Q$, and $||x||_Q = \sqrt{x ^T Qx}$. Bottom left to right: the corresponding SRCIS and a trajectory starting inside the SRCIS. The robust control invariance is validated.
• Figure 2: Top: R-CLVF with $\gamma = 0.1$ (left) and $\gamma = 0.2$ (right). Bottom left: ROES, SRCIS, and the two optimal trajectories using R-CLVF-QP controller. The ROES and SRCIS for different $\gamma$ are all the same, while the optimal trajectories are different. To see this, first consider a point on the boundary of ROES, $[0.1,1]$, $d$ will make $x$ increase 0.1 to 1, while $u$ cannot decrease $y$. Since the distance is measured by $||x||_\infty$, we have $\ell (\xi(s;t, x, u^*(\cdot),\lambda[u^*])) = 1$, $\forall t < 0$. Using equation \ref{['eqn:finite_time_CLVF']} and \ref{['eqn:infinite_time_CLVF']}, the value will be infinite. However, for any $|y|<1$, the control can decrease $y$ to $0$, and for all $x$, it either goes to 0.5 or -0.5. Note both happen in a finite time horizon. Therefore, using equation \ref{['eqn:finite_time_CLVF']} and \ref{['eqn:infinite_time_CLVF']}, the value will be finite for all $\gamma \geq 0$. Bottom mid: value decay along the two optimal trajectories. All controllers were generated using R-CLVF-QP. With a 151-by-151 grid, the computation time for $\gamma = 0.1$ is 215.6s with warmstarting, and 289.7s w/o warmstarting, and 211.5s with warmstarting, and 258.4s w/o warmstarting for $\gamma=0.2$.
• Figure 3: Different SRCISs with different $h (x)$. Top left: SRCIS and optimal trajectory with $h (x) = ||x||_2$. Top right: SRCIS and optimal trajectory with $h (x) = ||x||_Q$, where $Q = diag[1,1,0]$. Bottom left: the value along the optimal trajectories. All controllers were generated using R-CLVF-QP. With a 51-by-51-by-53 grid, the computation time for $h(x) = ||x||_2$ is 264s with warmstarting, and 386.6s w/o warmstarting, and 143.4s with warmstarting, and 207.7s w/o warmstarting for $h(x) = ||x||_Q$.
• Figure 4: Comparison of R-CLVF with and without warmstarting for the Z-subsystem. The difference is almost negligible.
• Figure 5: Left: SRCIS of the reconstructed R-CLVF and the optimal trajectory. For Z-sys, with 101 grids for each state, time step = 0.1, convergence threshold = 0.0015, the computation time is 36.59s with warmstarting and 42.72s w/o warmstarting. For X(Y)-sys, with 17 grids for each state, time step = 0.1, convergence threshold = 0.02, the computation time is 828.27s with warmstarting and 887.79 w/o warmstarting. Right: the decay of the value along the optimal trajectory.

Theorems & Definitions (47)

• Definition 1
• Remark 1
• Definition 2
• Remark 2
• Definition 3
• Remark 3
• Remark 4
• Definition 4
• Proposition 1
• proof