An Optimal Solution to Infinite Horizon Nonholonomic and Discounted Nonlinear Control Problems

Abstract

This paper considers the infinite horizon optimal control problem for nonlinear systems. Under the condition of nonlinear controllability of the system to any terminal set containing the origin and forward invariance of the terminal set, we establish a regularized solution approach consisting of a ``finite free final time" optimal transfer problem to the terminal set, which renders the set globally asymptotically stable. Further, we show that the approximations converge to the optimal infinite horizon cost as the size of the terminal set decreases to zero. We also perform the analysis for the discounted problem and show that the terminal set is asymptotically stable only for a subset of the state space and not globally. The theory is empirically evaluated on various nonholonomic robotic systems to show that the cost of our approximate problem converges and the transfer time into the terminal set is dependent on the initial state of the system, necessitating the free final time formulation. We also do comparisons of our free-final time approach with nonlinear MPC.

Figures (4)

• Figure 1: An illustration of the discounted cost problem. We will show that given any $\Omega^0$, there exists a $\beta<1$, s.t., any $x_0 \in \Omega^0$ may be controlled into the terminal set $\Omega_M$.
• Figure 2: Fish Model simulated in MuJoCo in their initial and final states.
• Figure 3: Results for the car-like robot (a)-(d), and the fish model (e)-(h) with two different initial conditions - labeled Case 1 and 2. The parameters of the simulation are shown in Table \ref{['tab:experiments']}. It is observed that different initial conditions correspond to different transfer times in both cases.
• Figure 4: Comparison of different MPC policies on the car-like robot system. It can be observed that the optimal horizon for MPC depends on the initial state, and choosing a small horizon leads to suboptimal performance.

Theorems & Definitions (22)

• Corollary 1
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 2
• Theorem 1