Analysis and design of complex-valued linear systems

This paper studies a class of complex-valued linear systems whose state evolution dependents on both the state vector and its conjugate. The complex-valued linear system comes from linear dynamical quantum control theory and is also encountered when a normal linear system is controlled by feedback containing both the state vector and its conjugate that can provide more design freedom. By introducing the concept of bimatrix and its properties, the considered system is transformed into an equivalent real-representation system and a non-equivalent complexlifting system, which are normal linear systems. Based on these two auxiliary systems and using the bimatrix as a fundamental tool, some analysis and design problems including solutions, controllability, observability, stability, pole assignment, stabilization, linear quadratic regulation (LQR), and state observer design are investigated. Criterion, conditions, and algorithms are provided in terms of the coefficients of the original system.

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