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Candidate: Zhong Fang

Date: September 2, 2025

Time: 2:30pm

Location: EIT 3145

Supervisors: Dr. Michael Fisher

All are welcome!

Abstract:

The design of controllers with mixed H2/Hinf cost functions remains a challenging problem in control theory, with pervasive applications across diverse engineering fields. The main difficulties arise from nonconvexity and infinite dimensionality of the associated optimization problem for the design. Recently, several new approaches were developed to tackle nonconvexity by reparameterizing the variables to transform the optimization into a convex but infinite-dimensional formulation incorporating additional affine constraints in the design problem. For state feedback design, system level synthesis is focused since input output parameterization is primarily intended for output feedback. To make the problem tractable, and to address limitations of historical approximation methods, a new Galerkin-type method for finite-dimensional approximations of transfer functions in Hardy space with a selection of simple poles was recently developed.

However, prior applications of this simple pole approximation resulted in a design problem that required an additional approximation of a finite time horizon to compute H2 and Hinf norms for the closed-loop response. This finite horizon resulted in increased suboptimality, degraded performance, and increased problem size and memory requirements. To address these limitations, this thesis presents a novel control design framework that combines the frequency domain convex reparameterization affine constraints with a state space formulation of the H2 and Hinf norms using linear matrix inequality. This state space formulation eliminates the need for a finite time horizon approximation, and results in a convex and tractable semidefinite program for the control design. Suboptimality bounds are provided for the method which guarantee convergence to the global optimum of the infinite dimensional problem as the number of poles approaches infinity with a convergence rate that depends on the geometry of the pole selection.

The recently developed convex reparameterization methods have been challenging to adapt to continuous time control design in practice, because they typically rely on finite dimensional approximations for tractability that lead to numerical ill-conditioning or even closed-loop instability. In this work, the hybrid state space and frequency domain control design method is adapted to develop the first practical and tractable continuous time control design based on these convex reparameterizations that does not suffer from ill-conditioning and that guarantees closed-loop stability for stabilizable plants. Approximation error bounds are established for the first time for the simple pole approximation in continuous time. These bound the error based on the geometry of the pole selection, and show that this error goes to zero as the number of poles approaches infinity. These bounds are particularly challenging to obtain compared to the discrete time case due to the noncompactness of the domain of integration for computing the H2 and Hinf norms in continuous time. These approximation error bounds are then used to develop suboptimality guarantees of an analogous nature to those in discrete time. This is the first time that suboptimality bounds with zero asymptotic error have been developed for a control design method using these recent convex reparameterization approaches in continuous time. Again, the noncompactness represents a major challenge that must be overcome to establish these results.

There exist several recently developed convex reparameterizations for output feedback control design in discrete time (including system level synthesis and input output parameterization). However, all of these methods currently lack rigorous suboptimality guarantees that establish convergence to the solution of the infinite dimensional problem as the approximation dimension approaches infinity. This is largely due to the additional complexity introduced by the output feedback case compared to state feedback. This work develops novel output feedback control design methods using four different convex reparameterization approaches, each with different benefits and trade-offs, that result in convex and tractable control design formulations. Moreover, a single unified approximation theory is developed that simultaneously establishes suboptimality bounds for all four methods that recovers analogous results to the state feedback setting. In particular, they show a convergence rate to the global optimum that depends on the geometry of the pole selection in a similar fashion to the state feedback case.

The novel methods are applied to design controllers for power converter interfaced devices to provide frequency and voltage regulation to the power grid. Practical case studies demonstrate the ability of the methods to match desired dynamic behavior for these services. We consider multi-controller scenarios involving distributed energy resources where multiple power converters must coordinate to provide grid services while respecting physical and engineering constraints, including state, input, and output limits for each device, enabling coordinated control of dynamic virtual power plants. Case studies involving power converter voltage and frequency regulation, as well as multi-controller coordination in the IEEE 9-bus system, demonstrate superior performance.