Gradient Methods for Large-Scale and Distributed Linear Quadratic Control

This thesis considers methods for synthesis of linear quadratic controllers for large-scale, interconnected systems. Conventional methods that solve the linear quadratic control problem are only applicable to systems with moderate size, due to the rapid increase in both computational time and memory requirements as the system size increases. The methods presented in this thesis show a much slower increase in these requirements when faced with system matrices with a sparse structure. Hence, they are useful for control design for systems of large order, since they usually have sparse systems matrices. An equally important feature of the methods is that the controllers are restricted to have a distributed nature, meaning that they respect a potential interconnection structure of the system. The controllers considered in the thesis have the same structure as the centralized LQG solution, that is, they are consisting of a state predictor and feedback from the estimated states. Strategies for determining the feedback matrix and predictor matrix separately, are suggested. The strategies use gradient directions of the cost function to iteratively approach a locally optimal solution in either problem. A scheme to determine bounds on the degree of suboptimality of the partial solution in every iteration, is presented. It is also shown that these bounds can be combined to give a bound on the degree of suboptimality of the full output feedback controller. Another method that treats the synthesis of the feedback matrix and predictor matrix simultaneously is also presented. The functionality of the developed methods is illustrated by an application, where the methods are used to compute controllers for a large deformable mirror, found in a telescope to compensate for atmospheric disturbances. The model of the mirror is obtained by discretizing a partial differential equation. This gives a linear, sparse representation of the mirror with a very large state space, which is suitable for the methods presented in the thesis. The performance of the controllers is evaluated using performance measures from the adaptive optics community

Static output-feedback stabilization of discrete-time Markovian jump linear systems: a system augmentation approach

This paper studies the static output-feedback (SOF) stabilization problem for discrete-time Markovian jump systems from a novel perspective. The closed-loop system is represented in a system augmentation form, in which input and gain-output matrices are separated. By virtue of the system augmentation, a novel necessary and sufficient condition for the existence of desired controllers is established in terms of a set of nonlinear matrix inequalities, which possess a monotonic structure for a linearized computation, and a convergent iteration algorithm is given to solve such inequalities. In addition, a special property of the feasible solutions enables one to further improve the solvability via a simple D-K type optimization on the initial values. An extension to mode-independent SOF stabilization is provided as well. Compared with some existing approaches to SOF synthesis, the proposed one has several advantages that make it specific for Markovian jump systems. The effectiveness and merit of the theoretical results are shown through some numerical example

Event-triggered Consensus Control of Heterogeneous Multi-agent Systems: Model- and Data-based Analysis

This article deals with model- and data-based consensus control of heterogenous leader-following multi-agent systems (MASs) under an event-triggering transmission scheme. A dynamic periodic transmission protocol is developed to significantly alleviate the transmission frequency and computational burden, where the followers can interact locally with each other approaching the dynamics of the leader. Capitalizing on a discrete-time looped-functional, a model-based consensus condition for the closed-loop MASs is derived in form of linear matrix inequalities (LMIs), as well as a design method for obtaining the distributed controllers and event-triggering parameters. Upon collecting noise-corrupted state-input measurements during open-loop operation, a data-driven leader-following MAS representation is presented, and employed to solve the data-driven consensus control problem without requiring any knowledge of the agents' models. This result is then extended to the case of guaranteeing an $\mathcal{H}_{\infty}$ performance. A simulation example is finally given to corroborate the efficacy of the proposed distributed event-triggering scheme in cutting off data transmissions and the data-driven design method.Comment: 13 pages, 6 figures. This draft was firstly submitted to IEEE Open Journal of Control Systems on April 30, 2022, but rejected on June 19, 2022. Later, on July 23, 2022, this paper was submitted to the journal SCIENCE CHINA information scienc

Learning Stable Koopman Models for Identification and Control of Dynamical Systems

Learning models of dynamical systems from data is a widely-studied problem in control theory and machine learning. One recent approach for modelling nonlinear systems considers the class of Koopman models, which embeds the nonlinear dynamics in a higher-dimensional linear subspace. Learning a Koopman embedding would allow for the analysis and control of nonlinear systems using tools from linear systems theory. Many recent methods have been proposed for data-driven learning of such Koopman embeddings, but most of these methods do not consider the stability of the Koopman model. Stability is an important and desirable property for models of dynamical systems. Unstable models tend to be non-robust to input perturbations and can produce unbounded outputs, which are both undesirable when the model is used for prediction and control. In addition, recent work has shown that stability guarantees may act as a regularizer for model fitting. As such, a natural direction would be to construct Koopman models with inherent stability guarantees. Two new classes of Koopman models are proposed that bridge the gap between Koopman-based methods and learning stable nonlinear models. The first model class is guaranteed to be stable, while the second is guaranteed to be stabilizable with an explicit stabilizing controller that renders the model stable in closed-loop. Furthermore, these models are unconstrained in their parameter sets, thereby enabling efficient optimization via gradient-based methods. Theoretical connections between the stability of Koopman models and forms of nonlinear stability such as contraction are established. To demonstrate the effect of the stability guarantees, the stable Koopman model is applied to a system identification problem, while the stabilizable model is applied to an imitation learning problem. Experimental results show empirically that the proposed models achieve better performance over prior methods without stability guarantees

Discrete-time infinity control problem with measurement feedback

The paper is concerned with the discrete-time H(sub infinity) control problem with measurement feedback. The authors extend previous results by having weaker assumptions on the system parameters. The authors also show explicitly the structure of H(sub infinity) controllers. Finally, they show that it is in certain cases possible, without loss of performance, to reduce the dynamical order of the controllers