The stationary point of observer-based dynamic LQR is characterized by a pair of symmetric discrete-time Sylvester equations, and the usual separated LQR-plus-minimum-trace-observer design is not optimal.
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On the Optimization Landscape of Observer-based Dynamic Linear Quadratic Control
The stationary point of observer-based dynamic LQR is characterized by a pair of symmetric discrete-time Sylvester equations, and the usual separated LQR-plus-minimum-trace-observer design is not optimal.