On Robustness in the Gap Metric and Coprime Factor Uncertainty for LTV Systems

In this paper, we study the problem of robust stabilization for linear time-varying (LTV) systems subject to time-varying normalized coprime factor uncertainty. Operator theoretic results which generalize similar results known to hold for linear time-invariant (infinite-dimensional) systems are developed. In particular, we compute an upper bound for the maximal achievable stability margin under TV normalized coprime factor uncertainty in terms of the norm of an operator with a time-varying Hankel structure. We point to a necessary and sufficient condition which guarantees compactness of the TV Hankel operator, and in which case singular values and vectors can be used to compute the time-varying stability margin and TV controller. A connection between robust stabilization for LTV systems and an Operator Corona Theorem is also pointed out.Comment: 20 page

Optimal Disturbance Rejection and Robustness for Infinite Dimensional LTV Systems

In this paper, we consider the optimal disturbance rejection problem for possibly infinite dimensional linear time-varying (LTV) systems using a framework based on operator algebras of classes of bounded linear operators. This approach does not assume any state space representation and views LTV systems as causal operators. After reducing the problem to a shortest distance minimization in a space of bounded linear operators, duality theory is applied to show existence of optimal solutions, which satisfy a "time-varying" allpass or flatness condition. Under mild assumptions the optimal TV controller is shown to be essentially unique. Next, the concept of M-ideals of operators is used to show that the computation of time-varying (TV) controllers reduces to a search over compact TV Youla parameters. This involves the norm of a TV compact Hankel operator defined on the space of causal trace-class 2 operators and its maximal vectors. Moreover, an operator identity to compute the optimal TV Youla parameter is provided. These results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed Hankel-Toeplitz. The final outcome of the approach developed here is that it leads to two tractable finite dimensional convex optimizations producing estimates to the optimum within desired tolerances, and a method to compute optimal time-varying controllers.Comment: 30 pages, 1 figur

Sets and Constraints in the Analysis Of Uncertain Systems

This thesis is concerned with the analysis of dynamical systems in the presence of model uncertainty. The approach of robust control theory has been to describe uncertainty in terms of a structured set of models, and has proven successful for questions, like stability, which call for a worst-case evaluation over this set. In this respect, a first contribution of this thesis is to provide robust stability tests for the situation of combined time varying, time invariant and parametric uncertainties. The worst-case setting has not been so attractive for questions of disturbance rejection, since the resulting performance criteria (e.g., ℋ∞,) treat the disturbance as an adversary and ignore important spectral structure, usually better characterized by the theory of stochastic processes. The main contribution of this thesis is to show that the set-based methodology can indeed be extended to the modeling of white noise, by employing standard statistical tests in order to identify a typical set, and performing subsequent analysis in a worst-case setting. Particularly attractive sets are those described by quadratic signal constraints, which have proven to be very powerful for the characterization of unmodeled dynamics. The combination of white noise and unmodeled dynamics constitutes the Robust ℋ2 performance problem, which is rooted in the origins of robust control theory. By extending the scope of the quadratic constraint methodology we obtain a solution to this problem in terms of a convex condition for robustness analysis, which for the first time places it on an equal footing with the ℋ∞ performance measure. A separate contribution of this thesis is the development of a framework for analysis of uncertain systems in implicit form, in terms of equations rather than input-output maps. This formulation is motivated from first principles modeling, and provides an extension of the standard input-output robustness theory. In particular, we obtain in this way a standard form for robustness analysis problems with constraints, which also provides a common setting for robustness analysis and questions of model validation and system identification

Recent Advances in Robust Control

Robust control has been a topic of active research in the last three decades culminating in H_2/H_\infty and \mu design methods followed by research on parametric robustness, initially motivated by Kharitonov's theorem, the extension to non-linear time delay systems, and other more recent methods. The two volumes of Recent Advances in Robust Control give a selective overview of recent theoretical developments and present selected application examples. The volumes comprise 39 contributions covering various theoretical aspects as well as different application areas. The first volume covers selected problems in the theory of robust control and its application to robotic and electromechanical systems. The second volume is dedicated to special topics in robust control and problem specific solutions. Recent Advances in Robust Control will be a valuable reference for those interested in the recent theoretical advances and for researchers working in the broad field of robotics and mechatronics

Resource-aware motion control:feedforward, learning, and feedback

Controllers with new sampling schemes improve motion systems’ performanc