Performance-Aware Quantizer Synthesis for LQI Tracking Subject to Fixed Level Budgets

We study finite-level static measurement quantizers for discrete-time linear-quadratic-integral (LQI) tracking (servo) problems. In several practical cases, the hardware imposes a fixed communication-rate (quantization budget) constraint. In this setup, for the augmented LQI loop with quantized state and output measurements, we prove an ∞ –input-to-state stability (ISS) bound with respect to bounded reference inputs and bounded component-wise quantization errors. Furthermore, to ensure a well-posed performance criterion under practical stability, we introduce a discounted control-law distortion index measuring deviation from the nominal (unquantized) LQI command. We then derive an explicit upper bound that links this discounted distortion to a weighted sum-of-squared quantization errors, with constant sensitivity weights computed offline from the nominal closed-loop design. Under a component-wise scalar architecture, the resulting data-driven synthesis reduces to weighted 1D quantizer design problems and a finite level-allocation problem, both solved globally by dynamic programming. Simulations show that the proposed scheme outperforms uniform, logarithmic, and Lloyd-Max designs.