Analysis and design of robust disturbance observers

Abstract

Robustness has been one of the most defining features of control systems since the classical control period. In the early days, the robustness of the control system was expressed using concepts like phase margin and gain margin, adapted from telecommunications engineering, and this terminology was faithfully used during the period when the significant achievements of modern control theory were demonstrated. However, by the end of the 70s, two separate developments marked the beginning of the golden age of robust control theory. The first of the developments that heralded this new era is Kharitonov's theorem, which established a new field of research for examining the stability of systems with parametric uncertainty. The other is John Doyle's demonstration that even in a single-input, single-output system, the LQG regulator does not have any guaranteed robustness margin, unlike the LQ regulator. While the first formed the basis of the research field known as the parametric approach, the other was one of the precursors of the $H_{\infty}$ theory. Since then, robust control has been seen as an independent sub-branch of control theory. Both approaches reached their peak with both theoretical and practical applications throughout the 1980s and 1990s. On the other hand, it has been shown that more robust closed-loop systems can be developed by changing the structure of the controller. One of the prominent methods is the approach known in the literature as the disturbance-observer (DOB). This approach, which enables the prediction and cancellation of disturbances and uncertainties that impact the system at its input, has been widely implemented, particularly in practical applications. On the other hand, the theoretical limits of the method, its analysis under uncertainty, and its design with newly developed robust control methods have lagged behind practical applications. Although theoretical studies have been carried out especially with the $H_{\infty}$ approach since the 2000s, DOB design and analysis under parametric uncertainties have not attracted the attention of researchers sufficiently. The main purpose of this thesis is to develop new approaches for both the analysis and design of disturbance observers under parametric uncertainties. In the analysis of systems with parametric uncertainty, how the uncertainties are modeled is the factor that directly affects the analysis method. In Kharitonov's paradigm, the parametric uncertainty bounding set is usually expressed as a box, which corresponds to the $l_{\infty}$ representation of the parameter box. However, the $l_{2}$ analog of the same representation is also possible. In fact, this representation is more suitable for the situation where the mathematical model is obtained by linear or nonlinear regression methods under system identification approach. Based on this, in the first part of the thesis, the answer to the question of "How much uncertainty can be tolerated with the DOB structure?", has been sought. Although approaches in the frequency domain produce effective results for DOB analysis, new challenges arise when the problem is expressed in the state space. Two approaches have come to the fore for examining parametric uncertainties in the state space. The first of these is to move the problem to the frequency domain where there are theorems and mathematical tools mature enough to examine parametric uncertainties. However, when this method is utilized, even the simplest interval system matrices show themselves as a affine-linear or more complex polynomial when expressed as a polynomial. Therefore, design in state space was seen as a "hard nut to crack" problem, in Yedevalli's words, and pushed control theorists to different research directions. The other method is to consider the problem directly in the state space. Although similar difficulties exist in this approach, when designing directly in the state space, the use of proven state space methods is also possible. Although new solutions are proposed, especially under the concept of quadratic stability, the nature of the problem condemns control theorists to use conservative approaches. In addition, a suitable Lyapunov function has not yet been proposed in the case where the design regions used to limit the parametric uncertainties are disjoint. The second contribution put forward within the scope of the thesis is the guardian-map approach, which offers less conservative disturbance observer design. Thanks to the method, robustness criteria can be assigned for each nominal eigenvalue separately and the disturbance observer is designed to meet this criterion. In this way, the inherent trade-off between robustness of the disturbance observer and the disturbance observer bandwidth is decided according to whether the closed-loop system satisfies the previously determined eigenvalue spread criterion. Advantages of considering the problem in state space include the possibility to use LMI tools and the incorporation of useful methods such as eigenstructure assignment into the solution of the problem. Many control problems can be expressed in LMI form, and these LMIs can be formulated as appropriate convex optimization problems. The LMI framework is particularly useful for expressing parametric uncertainties and constraining eigenvalue spread. However, when the dominant methods in the literature are examined, the design regions defined by the LMI approach are not defined separately for each eigenvalue, but a combined LMI design region is defined for all eigenvalues. This situation complicates the eigenvalue assignment problem and does not allow defining different robustness criteria between the eigenvalues in the non-dominant region, which is less important for the design, and the dominant region eigenvalues, which determine the behavior of the system. In addition, when the eigenstructure assignment methods are considered, the methods for minimizing the sensitivity of the system dominate the literature, instead of expressing the parametric uncertainties directly. Although robust eigenstructure assignment methods based on $H_{\infty}$-based approaches have been proposed, eigenstructure assignment methods have not been sufficiently studied in direct parametric uncertainty system design. In the eigenstructure assignment methods, since the eigenvalues are assigned strictly at the beginning of the design, the vector space to which the eigenvectors can be assigned in the rest of the design is also limited. In order to overcome this, although methods such as regional assignment, partial eigenvalue assignment and loose eigenstructure assignment are suggested in the literature, suppressing the effect of parametric uncertainties has not been the primary design criterion in these approaches. In order to fill these gaps in the literature, a new design method has been proposed, and in this approach, the robustness of the system to parametric uncertainties has been made the primary criterion of the design, and a novel disturbance observer design method has been proposed by using eigenstructure assignment and LMI approaches together for this purpose. The approach does not require any heuristic algorithms or global optimization methods, as well as allowing the solution of the robust root clustering problem for disjoint design regions. As a result, the method inevitably suffers from conservatism. However, the design reduces the problem of finding robust eigenvectors to finding the appropriate one among a finite number of eigenvectors. As a conclusion, within the scope of this thesis, a method is proposed to examine the robustness of the disturbance observer under parametric uncertainties, and two new design methods are proposed to limit the eigenvalue spread in the state space within the disjoint design regions determined for each nominal eigenvalue. By using the obtained results, a disturbance observer in the state space is designed for systems with parametric uncertainty and the results are shared.