Contractor-Expander and Universal Inverse Optimal Positive Nonlinear Control

Abstract

For general control-affine nonlinear systems in the positive orthant, and with positive controls, we show how strict CLFs can be utilized for inverse optimal stabilization. Conventional ``LgV'' inverse optimal feedback laws, for systems with unconstrained states and controls, assume sign-unconstrained inputs and input penalties that are class-K in the input magnitude, hence symmetric about zero. Such techniques do not extend to positive-state-and-control systems. Major customizations are needed, and introduced in this paper, for positive systems where highly asymmetric (or unconventionally symmetric) costs not only on the state but also on control are necessary. For the predator-prey positive-state positive-input benchmark system, with a strict CLF built in our previous paper, we prototype two inverse optimal methodological frameworks that employ particular ``contractor and expander functions.'' One framework (A) employs a triple consisting of a CLF, a stabilizing feedback, and an expander, whereas the other framework (B) employs a pair of a CLF and a contractor function. Both frameworks yield inverse optimal stabilizer constructions, on positive orthants of arbitrary dimensions. Framework B demands more design effort than framework A but is free of conditions that may fail to hold in general. Biological interpretation for the predator-prey model illuminates that such inverse optimal control constructions are bio-ecologically meaningful. In addition to general frameworks, we present one fully explicit design: a Sontag-like universal formula for inverse optimal stabilization of positive-orthant systems by positive feedback.

Figures (7)

• Figure 1: The "expander" nonlinearity $\Sigma(s)$, defined by \ref{['eq-def-Sigma']}, \ref{['eq-def-Theta']}, for $\Psi=\Omega$ defined in \ref{['eq-Volterra-Lyapunov']}, and used in the controller \ref{['eq-optimal-U*']}. The identity function $s$ in thin green gives the nominal backstepping feedback $U_0=Y^2/X$.
• Figure 2: The functions that govern the feedback laws \ref{['eq-u02']} and \ref{['eq-u*2']}. Compared to the blue curve, for the nominal control, the red curve, representing the optimal control, decays twice as fast near $z=0$, a tell-tale sign of optimality (the gain margin of 1/2), and at a rate that gets even faster --- $\ln(z)$ faster --- as $z$ grows.
• Figure 3: The "contractor" nonlinearities $\Theta_i$ defined in \ref{['eq-Theta0i']}, with their growth trends quantified in \ref{['eq-Theta0i+']}.
• Figure 4: The control cost functions $\Psi_i$, obtained from the respective $\Theta_i$, using \ref{['eq-Theta-to-Psi']}. Both $\Psi_1$ and $\Psi_3$ have barrier behavior at $s=0$, but $\Psi_2$ does not. Of the three, $\Psi_1$ and $\Psi_1$ have the least aggressive growth for large $s$ --- linear.
• Figure 5: With initial prey slightly depleted and predator dominant, the expander $\Sigma$ makes the action of optimal control $U^*=Y \Sigma(Y/X)$ stronger than the dominant control $U_0=Y^2/X$ and reduces the excursion of the trajectory on the way to the equilibrium $X=Y=U=1$. Perhaps counterintuitively, the optimal controller, which harvests the predator more aggressively, prevents a deeper depletion of the prey than the nominal controller. While the control cost $r(X,Y) \Omega(X,Y)$ is higher for the optimal control initially, the state cost $q(X,Y)$ is lower for the duration of the transient, resulting in the overall cost $J^*$ that is lower than $J$ achieved under the nominal control, as predicted by the theory, and this reduction happens to be considerable: 20%.

Theorems & Definitions (9)

• Theorem 1
• Lemma 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Theorem 5
• Corollary 2
• Corollary 3