Complex-coefficient systems in control

Abstract

Complex-valued dynamics can be used for modeling rotationally invariant two-input two-output systems and bandpass systems when they are considered in the baseband. In a few instances, control design has been done in the complex domain, which facilitated analysis and synthesis. While previous work has been application specific, we will discuss more generally how complex valued dynamics arise, basic properties of these systems, revisit some classic control theoretic results in the complex setting, and discuss two novel examples of control design in the complex domain-accelerator cavity field control and feedback linearization of RF amplifiers.

Complex-Valued Multifrequency Admittance Model of Three-Phase VSCs in Unbalanced Grids

“…where the real vectors in the αβand dq-frames represented by [v α , v β ] T and [v d , v q ] T are mapped as two base vectors in the complex space, i.e., [v, v * ] T and [v dq , v * dq ] T , and " * " denotes the complex conjugate operator. Therefore, the model derived based on complex vectors in the complex space is called a complex-valued model [8], [18]- [20].…”

“…The off-diagonal elements are zero since the LTI system does not contribute to any frequency couplings. The form of (20) applies to the transfer functions of the CC, the time delay, and the converter plants, which are represented by G i (s), G d (s), Y o (s), and Y p (s), respectively [18].…”

“…In contrast to the real-vector representations, the multifrequency model based on the complex vectors is recently reported in [5] and [17]- [19], which is thus called the complex-valued model [20]. With complex vectors, the threephase balanced voltage or current can be represented by a single variable, which helps reduce the model order.…”

Elementary Derivation of the Nyquist Criterion for Fractional-Order Feedback Systems

“…where subscript a stands for asymmetric. The consideration of asymmetric sectors may turn out useful, e.g., in root-clustering problems [20] for complex polynomials [27], [28]. Remark 3.3: Removing Assumption 3.1 is not conceptually difficult but implies an increase of notational complexity at the expense of clarity and the resort to artifices similar to those adopted in the classic Mikhailov stability test to deal with the critical cases (see, e.g., [29]).…”

Complex-Based Controller for a Three-Phase Inverter With an <italic>LCL</italic> Filter Connected to Unbalanced Grids

“…The main advantage is the reduction of the order of the system that facilitates the analysis and the synthesis of the controllers [34]. Recently, some frequencydomain results were revisited in [35]. Additionally, filters based on complex coefficients have been proposed for PLLs and synchronization techniques [36].…”