From data and structure to models and controllers

Systems and control theory deals with analyzing dynamical systems and shaping their behavior by means of control. Dynamical systems are widespread, and control theory therefore has numerous applications ranging from the control of aircraft and spacecraft to chemical process control. During the last decades, a series of remarkable new control techniques have been developed. The majority of these techniques rely on mathematical models of the to-be-controlled system. However, the growing complexity of modern engineering systems complicates mathematical modeling. In this thesis, we therefore propose new methods to analyze and control dynamical systems without relying on a given system model. Models are thereby replaced by two other ingredients, namely measured data and system structure. In the first part of the thesis, we consider the problem of data-driven control. This problem involves the development of controllers for a dynamical system, purely on the basis of data. We consider both stabilizing controllers, and controllers that minimize a given cost function. Secondly, we focus on networked systems. A networked system is a collection of interconnected dynamical subsystems. For this type of systems, our aim is to reconstruct the interactions between subsystems on the basis of data. Finally, we consider the problem of assessing controllability of a dynamical system using its structure. We provide conditions under which this is possible for a general class of structured systems

H-∞ optimal actuator location

There is often freedom in choosing the location of actuators on systems governed by partial differential equations. The actuator locations should be selected in order to optimize the performance criterion of interest. The main focus of this thesis is to consider H-∞-performance with state-feedback. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output of a full-information plant. Optimal H-∞-disturbance attenuation as a function of actuator location is used as the cost function. It is shown that the corresponding actuator location problem is well-posed. In practice, approximations are used to determine the optimal actuator location. Conditions for the convergence of optimal performance and the corresponding actuator location to the exact performance and location are provided. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied. Systems of large model order arise in a number of situations; including approximation of partial differential equation models and power systems. The system descriptions are sparse when given in descriptor form but not when converted to standard first-order form. Numerical calculation of H-∞-attenuation involves iteratively solving large H-∞-algebraic Riccati equations (H-∞-AREs) given in the descriptor form. An iterative algorithm that preserves the sparsity of the system description to calculate the solutions of large H-∞-AREs is proposed. It is shown that the performance of our proposed algorithm is similar to a Schur method in many cases. However, on several examples, our algorithm is both faster and more accurate than other methods. The calculation of H-∞-optimal actuator locations is an additional layer of optimization over the calculation of optimal attenuation. An optimization algorithm to calculate H-∞-optimal actuator locations using a derivative-free method is proposed. The results are illustrated using several examples motivated by partial differential equation models that arise in control of vibration and diffusion

LQG Control and Sensing Co-Design

We investigate a Linear-Quadratic-Gaussian (LQG) control and sensing co-design problem, where one jointly designs sensing and control policies. We focus on the realistic case where the sensing design is selected among a finite set of available sensors, where each sensor is associated with a different cost (e.g., power consumption). We consider two dual problem instances: sensing-constrained LQG control, where one maximizes control performance subject to a sensor cost budget, and minimum-sensing LQG control, where one minimizes sensor cost subject to performance constraints. We prove no polynomial time algorithm guarantees across all problem instances a constant approximation factor from the optimal. Nonetheless, we present the first polynomial time algorithms with per-instance suboptimality guarantees. To this end, we leverage a separation principle, that partially decouples the design of sensing and control. Then, we frame LQG co-design as the optimization of approximately supermodular set functions; we develop novel algorithms to solve the problems; and we prove original results on the performance of the algorithms, and establish connections between their suboptimality and control-theoretic quantities. We conclude the paper by discussing two applications, namely, sensing-constrained formation control and resource-constrained robot navigation.Comment: Accepted to IEEE TAC. Includes contributions to submodular function optimization literature, and extends conference paper arXiv:1709.0882

Optimization methods for deadbeat control design: a state space approach

This thesis addresses the synthesis problem of state deadbeat regulator using state space techniques. Deadbeat control is a linear control strategy in discrete time systems and consists of driving the system from any arbitrary initial state to a desired final state infinite number of time steps. Having described the framework for development of the thesis which is in the form of a lower linear-fractional transformation (LFT), the conditions for internal stability based on the notion of coprime factorization over the set of proper and stable transfer matrices, namely RH, is discussed. This leads to the derivation of the class of all stabilizing linear controllers, which are parameterized affinely in terms of a stable but otherwise free parameter Q, usually known as the Q-parameterization. In this work, the classical Q- parameterization is generalized to deliver a parameterization for the family of deadbeat regulators. Time response characteristics of the deadbeat system are investigated. In particular, the deadbeat regulator design problem in which the system must satisfy time domain specifications and minimize a quadratic (LQG-type) performance criterion is examined. It is shown that the attained parameterization for deadbeat controllers leads to the formulation of the synthesis problem in a quadratic programming framework with Q regarded as the design variable. The equivalent formulation of this objective as a quadratic integral in the frequency domain provides the means for shaping the frequency response characteristics of the system. Using the LMI characterization of the standard H problem, a new scheme for shaping the system frequency response characteristics by minimizing the infinity norm of an appropriate closed-loop transfer function is introduced. As shown, the derived parameterization of deadbeat compensators simplifies considerably the formulation and solution of this problem. The last part of the work described in this thesis is devoted to addressing the synthesis problem of deadbeat regulators in a robust way, when the plant is subject to structured norm-bounded parametric uncertainties. A novel approach which is expressed as an LMI feasibility condition has been proposed and analysed

Analysis of Structural Properties of Complex and Networked Systems

Over the past decades, science and society have been experiencing systems that tend to be increasingly sophisticated and interconnected. Although it would be challenging to understand and control complex systems fully, the analysis and control of such systems can be partially realized only after applying some reasonable simplifications. In particular, for the analysis of certain control properties, such as controllability, a complex system can be simplified to a linear structured system capturing an essential part of the structural information in that system, such as the existence or absence of relations between components of the system. This thesis has studied the effect of the interconnection structure of complex systems on their control properties following a structural analysis approach. More explicitly, we have analyzed strong structural properties of complex systems. The main contributions have been split into two parts:1. We have introduced a new framework for linear structured systems in which the relations between the components of the systems are allowed to be unknown. This kind of systems has been formalized in terms of pattern matrices whose entries are either fixed zero, arbitrary nonzero, or arbitrary. We have dealt with strong structural controllability and the solvability of the FDI problem of this kind of linear structured systems.2. We have introduced a novel framework for linear structured systems in which a priori given entries in the system matrices are restricted to take arbitrary but identical values. Several sufficient algebraic and graph theoretic conditions were established under which these systems are strongly structurally controllable.Finally, in the outlook subsection, we have suggested some future research problems concerning the analysis of strong structural properties of complex systems

Control and Estimation Oriented Model Order Reduction for Linear and Nonlinear Systems

Optimization based controls are advantageous in meeting stringent performance requirements and accommodating constraints. Although computers are becoming more powerful, solving optimization problems in real-time remains an obstacle because of associated computational complexity. Research efforts to address real-time optimization with limited computational power have intensified over the last decade, and one direction that has shown some success is model order reduction. This dissertation contains a collection of results relating to open- and closed-loop reduction techniques for large scale unconstrained linear descriptor systems, constrained linear systems, and nonlinear systems. For unconstrained linear descriptor systems, this dissertation develops novel gramian and Riccati solution approximation techniques. The gramian approximation is used for an open-loop reduction technique following that of balanced truncation proposed by (Moore, 1981) for ordinary linear systems and (Stykel, 2004) for linear descriptor systems. The Riccati solution is used to generalize the Linear Quadratic Gaussian balanced truncation (LQGBT) of (Verriest, 1981) and (Jonckheere and Silverman, 1983). These are applied to an electric machine model to reduce the number of states from $>$100000 to 8 while improving accuracy over the state-of-the-art modal truncation of (Zhou, 2015) for the purpose of condition monitoring. Furthermore, a link between unconstrained model predictive control (MPC) with a terminal penalty and LQG of a linear system is noted, suggesting an LQGBT reduced model as a natural model for reduced MPC design. The efficacy of such a reduced controller is demonstrated by the real-time control of a diesel airpath. Model reduction generally introduces modeling errors, and controlling a constrained plant subject to modeling errors falls squarely into robust control. A standard assumption of robust control is that inputs/states/outputs are constrained by convex sets, and these sets are ``tightened'' for robust constraint satisfaction. However, robust control is often overly conservative, and resulting control strategies cannot take advantage of the true admissible sets. A new reduction problem is proposed that considers the reduced order model accuracy and constraint conservativeness. A constant tube methodology for reduced order constrained MPC is presented, and the proposed reduced order model is found to decrease the constraint conservativeness of the reduced order MPC law compared to reduced order models obtained by gramian and LQG reductions. For nonlinear systems, a reformulation of the empirical gramians of (Lall et al., 1999) and (Hahn et al., 2003) into simpler, yet more general forms is provided. The modified definitions are used in the balanced truncation of a nonlinear diesel airpath model, and the reduced order model is used to design a reduced MPC law for tracking control. Further exploiting the link between the gramian and Riccati solution for linear systems, the new empirical gramian formulation is extended to obtain empirical Riccati covariance matrices used for closed-loop model order reduction of a nonlinear system. Balanced truncation using the empirical Riccati covariance matrices is demonstrated to result in a closer-to-optimal nonlinear compensator than the previous balanced truncation techniques discussed in the dissertation.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/140839/1/riboch_1.pd

Regelungstheorie

The workshop “Regelungstheorie” (control theory) covered a broad variety of topics that were either concerned with fundamental mathematical aspects of control or with its strong impact in various fields of engineering