Recursive Analytic Solution of Nonlinear Optimal Regulators

Abstract

The paper develops an optimal regulator for a general class of multi-input affine nonlinear systems minimizing a nonlinear cost functional with infinite horizon. The cost functional is general enough to enforce saturation limits on the control input if desired. An efficient algorithm utilizing tensor algebra is employed to compute the tensor coefficients of the Taylor series expansion of the value function (i.e., optimal cost-to-go). The tensor coefficients are found by solving a set of nonlinear matrix equations recursively generalizing the well-known linear quadratic solution. The resulting solution generates the optimal controller as a nonlinear function of the state vector up to a prescribed truncation order. Moreover, a complete convergence of the computed solution together with an estimation of its applicability domain are provided to further guide the user. The algorithm's computational complexity is shown to grow only polynomially with respect to the series order. Finally, several nonlinear examples including some with input saturation are presented to demonstrate the efficacy of the algorithm to generate high order Taylor series solution of the optimal controller.

Safety Embedded Control of Nonlinear Systems via Barrier States

“…As remarked earlier in the paper, the embedded system (7) preserves the continuous differentiability and stabilizability properties of (1) and thus satisfies the analyticity and stabilizability assumptions in [12]- [14] hence guaranteeing the existence and uniqueness of the value function V * (x) and the corresponding optimal controller u * saf e (x) described in (15). Furthermore, the origin of the resulting closed loop system is asymptotically stable by Lyapunov stability theory [13], [15] and by Theorem 3, u * saf e (x) is also safe which completes the proof.…”

“…Various techniques have been proposed in the literature to approximate the solution to the HJB equation for the infinite horizon optimal control problem or approximate the associated optimal control [12]- [14], [16]- [21]. In the next section, we utilize these efforts to produce a power series of the value function and it's gradient which we then utilize to produce the optimal safe control (15) for the optimal control problem (13). Specifically, we mainly utilize the recursive analytic solution proposed in [13], which is a generalization of the nonlinear quadratic regulator (NLQR) in [12].…”

“…In the next section, we utilize these efforts to produce a power series of the value function and it's gradient which we then utilize to produce the optimal safe control (15) for the optimal control problem (13). Specifically, we mainly utilize the recursive analytic solution proposed in [13], which is a generalization of the nonlinear quadratic regulator (NLQR) in [12].…”

“…where z ci = (z i + c i ), c i = 1 hi(0) = 1 3 and γ i = 5 for i = 1, 2. To generate the optimal safe controller, it is chosen to minimize the functional (13), with weighting matrices…”