Robustness is a defining feature of successful complex systems. In this talk I will describe a fundamental limit on the robustness of complex systems achievable through feedback control. Based on a generic formulation of the control problem, we define /fragility/ in terms of the sensitivity of an output to a disturbance and derive an integral constraint (lower-bound) on the net fragility of a system. Put simply, reducing the sensitivity to disturbances at one range of frequencies by feedback control will necessarily amplify disturbances at other frequencies. For the special case of linear feedback systems, this result reduces to Bode's integral formula. We illustrate the implications of this robustness tradeoff for biological systems, and identify feedforward control and buffering as strategies for ameliorating the tradeoff. Finally, the theory provides an alternate interpretation of Shannon's channel capacity theorem for nonlinear control systems.
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