Benchmark problems for robust control design

Journal of Guidance, Control, and Dynamics, 1995

Design of fuel/time optimal control of the benchmark two-mass/spring system is addressed in the frequency domain. The optimal control profile is represented as the output of a time-delay filter, where the amplitude of the time-delayed signals are constrainted to satisfy the control bounds. The time delays of the filter are determined by solving a parameter optimization problem that minimizes a weighted fuel/time cost function subject to the constraint that the tune-delay filter cancel all the poles of the system and the control profile satisfies the rigid-body boundary conditions. It is shown that three control structures exist: a three-switch profile corresponding to the time optimal control problem that changes to a six-switch profile corresponding to a cost function that includes a small weight on the fuel. As the weight on the fuel increases beyond a critical value, the control profile changes to a two-switch profile. The value of the critical weight that represents the transition of the control profile from a six-switch to a two-switch control profile is determined.

Automatica, 1993

This bibliography includes a compilation of selected books and journal articles written on the subject of Robust Control during the period 1987-1991. Papers are further divided into theory and applications. The theoretical subject matter of each article is identified by appropriate acronyms and application papers are separated into specific areas. * The theoretical subject matter of each paper is identified by appropriate acronyms (see the list below) written at the end of each citation. Many of the acronyms correspond to standard usage in the literature, for example [LQG] Verlag, Berlin. Reinschke, K. J. (1988). Multivariable control, a graph theory approach.

IFAC-PapersOnLine

The paper discusses performance measures used dominantly in the robust and optimal control design. By simple examples of the first order time delayed (FOTD) system control it illustrates that the usually prescribed levels of the maximal and complementary sensitivity functions indeed define situations with interesting loop properties, but may not be universally applied to the robust and optimal design of systems with an uncertain feedback variable in a broader range. For this purpose the shape related measures based on deviations from monotonicity yield results matching the technological requirements of practice in a much more appropriate way. It is also shown that in an optimal nominal controller design the monotonicity based performance measures nearly coincide with the multiple real dominant pole (MRDP) method. For the loop optimization using a broader spectrum of different performance measures, the performance portrait method may be recommended. It avoids the problems of convergence to the absolute optimum and, once generated, the performance portrait may repeatedly be used with a limited effort for a broader spectrum of different cost function specifications.

IUTAM Symposium on Vibration Control of Nonlinear …, 2005

European Journal of Control, 2011

This paper will review recent progress on developing a unified theory for complex networks from biological systems and physics to engineering and technology. Insights into what the potential universal laws, architecture, and organizational principles are can be drawn from three converging research themes: growing attention to complexity and robustness in systems biology, layering and organization in network technology, and new mathematical frameworks for the study of complex networks. We will illustrate how tools in robust control theory and optimization can be integrated towards such unified theory by focusing on their applications in biology, physics, network design, and electric grid.

2014

The results and opinions expressed in this paper are not necessarily those of the Market Participants Group on Reference Rate Reform, a committee chaired by Duffie that was established by the Financial Stability Board and that published its final report in 2014, nor do they necessarily reflect the views of the National Bureau of Economic Research. At least one co-author has disclosed a financial relationship of potential relevance for this research. Further information is available online at http://www.nber.org/papers/w20540.ack NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies official NBER publications.

Smoothing, Filtering and Prediction: Estimating the Past, Present and Future Second Edition, 2019

The previously-discussed optimum predictor, filter and smoother solutions assume that the model parameters are correct, the noise processes are white and their associated covariances are known precisely. These solutions are optimal in a mean-square-error sense, that is they provide the best average performance. If the above assumptions are correct, then the filter’s mean-square-error equals the trace of design error covariance. The underlying modelling and noise assumptions are a often convenient fiction. They do, however, serve to allow estimated performance to be weighed against implementation complexity. In general, robustness means “the persistence of a system’s characteristic behaviour under perturbations or conditions of uncertainty” [1]. In an estimation context, robust solutions refer to those that accommodate uncertainties in problem specifications. They are also known as worst-case or peak error designs. The standard predictor, filter and smoother structures are retained but a larger design error covariance is used to account for the presence of modelling error. Designs that cater for worst cases are likely to exhibit poor average performance. Suppose that a bridge designed for average loading conditions returns an acceptable cost benefit. Then a robust design that is focussed on accommodating infrequent peak loads is likely to provide worse cost performance. Similarly, a worst-case shoe design that accommodates rarely occurring large feet would provide poor fitting performance on average. That is, robust designs tend to be conservative. In practice, a trade-off may be desired between optimum and robust designs. The material canvassed herein is based on the H∞ filtering results from robust control. The robust control literature is vast, see [2] – [33] and the references therein. As suggested above, the H∞ solutions of interest here involve observers having gains that are obtained by solving Riccati equations. This Riccati equation solution approach relies on the Bounded Real Lemma – see the pioneering work by Vaidyanathan [2] and Petersen [3]. The Bounded Real Lemma is implicit with game theory [9] – [19]. Indeed, the continuous-time solutions presented in this section originate from the game theoretic approach of Doyle, Glover, Khargonekar, Francis Limebeer, Anderson, Khargonekar, Green, Theodore and Shaked, see [4], [13], [15], [21]. The discussed discrete-time versions stem from the results of Limebeer, Green, Walker, Yaesh, Shaked, Xie, de Souza and Wang, see [5], [11], [18], [19], [21]. In the parlance of game theory: “a statistician is trying to best estimate a linear combination of the states of a system that is driven by nature; nature is trying to cause the statistician’s estimate to be as erroneous as possible, while trying to minimise the energy it invests in driving the system” [19]. Pertinent state-space H∞ predictors, filters and smoothers are described in [4] – [19]. Some prediction, filtering and smoothing results are summarised in [13] and methods for accommodating model uncertainty are described in [14], [18], [19]. The aforementioned methods for handling model uncertainty can result in conservative designs (that depart far from optimality). This has prompted the use of linear matrix inequality solvers in [20], [23] to search for optimal solutions to model uncertainty problems. It is explained in [15], [19], [21] that a saddle-point strategy for the games leads to robust estimators, and the resulting robust smoothing, filtering and prediction solutions are summarised below. While the solution structures remain unchanged, designers need to tweak the scalar within the underlying Riccati equations. This chapter has two main parts. Section 9.2 describes robust continuous-time solutions and the discrete-time counterparts are presented in Section 9.3. The previously discussed techniques each rely on a trick. The optimum filters and smoothers arise by completing the square. In maximum-likelihood estimation, a function is differentiated with respect to an unknown parameter and then set to zero. The trick behind the described robust estimation techniques is the Bounded Real Lemma, which opens the discussions.

Automatica, 1994

Robust fixed-gain controllers were sattsfactordy designed for stress levels 1 and 2 of the benchmark process using H® loop-shaping and a numerical approach to the selection of weights At stress level 3, it was found that the performance of a fixed-gain controller was Improved usmg a simple adaptive gain scheme.

Mathematical Programming, 2006

In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set. The extension of Robust Optimization methodology developed in this paper opens up new possibilities to solve efficiently multi-stage finite-horizon uncertain optimization problems, in particular, to analyze and to synthesize linear controllers for discrete time dynamical systems.

1996