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.

Background and Objective: New proposals to improve the regulation of hypnosis in anaesthesia based on the development of advanced control structures emerge continuously. However, a fair study to analyse the real benefits of these structures compared to simpler clinically validated PID-based solutions has not been presented so far. The main objective of this work is to analyse the performance limitations associated with using a filtered PID controller, as compared to a high-order controller, represented through a Youla parameter.Methods: The comparison consists of a two-steps methodology. First, two robust optimal filtered PID controllers, considering the effect of the inter-patient variability, are synthesised. A set of 47 validated paediatric pharmacological models, identified from clinical data, is used to this end. This model set provides representative inter-patient variability Second, individualised filtered PID and Youla controllers are synthesised for each model in the set. For fairness of comparison, the same performance objective is optimised for all designs, and the same robustness constraints are considered. Controller synthesis is performed utilising convex optimisation and gradient-based methods relying on algebraic differentiation. The worst-case performance over the patient model set is used for the comparison.

ControlSystems.jl enables the powerful features of the Julia language to be leveraged for control design and analysis. The toolbox provides types for state-space, transfer-function, and timedelay models, together with algorithms for design and analysis. Julia's mathematically-oriented syntax is convenient for implementing control algorithms, and its just-in-time compilation gives performance on par with C. The multiple-dispatch paradigm makes it easy to combine the algorithms with powerful tools from Julia's ecosystem, such as automatic differentiation, arbitrary-precision arithmetic, GPU arrays, and probability distributions. We demonstrate how these features allow complex problems to be solved with little effort. I. INTRODUCTIONThe Julia programming language [1] has, over the last couple of years, revolutionized technical computing. It is a high-level language with mathematically-oriented syntax and semantics, but still achieves execution speeds comparable to C by relying on just-in-time compilation. Julia is free, open source and due to its suitability for technical computing, already has a rich ecosystem with high-quality packages for applied mathematics. Julia's use of multiple-dispatch and duck typing simplifies code reuse and composability across packages, making it possible to achieve complex functionality with little effort. The power of the Julia language has been demonstrated in numerous applications [2].ControlSystems.jl [3] provides a large set of algorithms for control design and analysis, enabling the control community to leverage the power of the Julia language and its ecosystem. For example, ControlSystems.jl makes it easy to tune controller parameters using automatic differentiation and optimize performance subject to uncertainty by propagating probability distributions.The ControlSystems.jl toolbox supports common functionality such as creating linear time-invariant (LTI) systems using either a state-space representation or as transfer functions with either polynomials or zeropole-gain representations. Systems with delays are supported, with the time-response capabilities provided

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