Conditions for Complete Decentralization of the Linear Quadratic Regulator

Abstract

An unconstrained optimal control policy is completely decentralized if computing actuation for each subsystem only requires information directly available to its own subcontroller. Parameters that admit a completely decentralized optimal controller have been characterized in a variety of systems, but attempts to physically explain the phenomenon have been limited. As a step toward a general characterization of complete decentralization, this paper presents conditions for complete decentralization of Linear Quadratic Regulators for several simple cases and physically interprets these conditions with illustrative examples. These simple cases are then leveraged to characterize complete decentralization of more complex systems.

Figures (5)

• Figure 1: Visualization of distributed (top) and decentralized (bottom) control policies for three subsystems with $N_1=\{x_1,x_2\}, N_2=\{x_3\}, N_3=\{x_4,x_5\}$. In the distributed policy, the subcontrollers $Z_i$ are shown relaying state information.
• Figure 2: Heat plot of closed loop $H_2$ norm with varying choices of $Q$ and $R$. The point in red marks the values of $q0/q2$ and $\gamma_0/\gamma_2$ that result in decentralization.
• Figure 3: Heat plot of closed loop $H_2$ norm with varying choices of $Q$ and ${a_2}/{a_0}$. The curve of decentralization is shown in black.
• Figure 4: Diagram for physical example of $2\times 2$ system with bass as the predators and shrimp as the prey.
• Figure 5: Heat transfer across a wall