Speaker: Babak Hassibi, Information Systems Laboratory, Stanford University, USA.

OH slides and book chapters:

Slides

Chapter 10: Input-Output Approach to H_2 and H_\infty Estimation

Chapter 11: Input-Output Approach to H_2 and H_\infty Control

Chapter 15: Optimal H_\infty solutions

Outline of Topics: 1. Input-output approach to control and estimation.
2. Input-output H-infinity full information control and estimation.
2.1. The optimal noncausal solution.
2.2. The optimal causal solution.
2.2.1. Laurent, Toeplitz and Hankel operators.
2.2.2. The Nehari problem.
2.2.3. Optimal solutions in the infinite-horizon case.
2.2.4. Optimal solutions in the finite-horizon case.
2.3. The suboptimal causal solution.
2.3.1. Canonical factorization.
3. Input-output measurement feedback control.
3.1. The suboptimal causal solution.
3.1.1. The separation principle.
3.2. The optimal noncausal solution.
3.3. The optimal causal solution.
4. State-space techniques and computational aspects.
4.1. Canonical factorization and algebraic Riccati equations.
4.2. The bounded-real lemma and linear matrix inequalities.
5. Applications and examples.
5.1. Tracking.
5.1.1. Full information tracking.
5.1.2. Measurement feedback tracking.
5.2. Equalization.
5.2.1. Linear equalization.
5.2.2. Decision-feedback equalization.
5.3. Filtering signals in additive noise.
5.4. Prediction.
5.4.1. H-infinity-optimality of H-2 predictors.
5.5. Adaptive signal processing and control.
5.5.1. Adaptive filtering.
5.5.2. Active noise cancellation.
5.6. Multirate signal processing.
5.6.1. Design of synthesis filters.
6. Worst-case controllability and estimability.
7. Open problems.

Synopsis: When H-infinity control was introduced in the early 1980's the first methods used to study and solve the problem were input-output operator-theoretic techniques. These methods quickly led to connections with, among others topics, analytic interpolation theory and J-spectral factorization, and required the computation of quantities such as the spectral radius of mixed Toeplitz-plus-Hankel operators. However, computational difficulties were encountered in performing such factorizations and determining such quantities, so that the input-output techniques were superceded by state-space techniques where the solutions could be obtained more directly using algebraic Riccati equations and linear matrix inequalities. Moreover, the simplicity of the state-space arguments, along with the formal similarities of the solutions to those of LQG control, the connections with game theory, etc., have made it the dominant approach to H-infinity control. This is is clearly reflected in many of the recent textbooks devoted to this subject.

The main message, nonetheless, of this mini course is that there is still much value in revisiting the input-output approach, especially with the hindsight obtained over the last fifteen years in H-infinity control. Although state-space techniques are extremely powerful for computing solutions (and other related quantities), and are growing even more so with the advent of efficient numerical techniques for solving complicated linear matrix inequalities, they do not always give much physical insight into the nature of the problem and what its fundamental limitations are. Such insight, however, can be obtained much more effectively using input-output techniques. The reason being that the input-output approach allows the solutions to be revealed and understood in their most transparent forms, and without the often obscuring fine details that occur in specific problems.

Therefore in this mini course we will present the basics of the input-output approach to H-infinity control, with an emphasis on obtaining closed-form solutions for optimal two-block and four-block H-infinity problems. Connections with state-space techniques, Riccati equations, and LMIs will also be reviewed. Using the solutions to the two-block and four-block problems, we study and explicitly find the H-infinity-optimal solutions for several important problems in tracking, equalization, prediction, adaptive signal processing and adaptive control, multirate signal processing, and others. An important feature of all these examples is that, since explicit closed-form solutions can be found, the fundamental limitations of the problems are exposed and much physical is insight obtained. Our studies also lead to the concepts of ``worst-case'' controllability (estimability), essentially the question of when a system (signal) is easy or difficult to control (estimate). Moreover, some open problems are also discussed.

We should also mention that many of our examples are from estimation and signal processing, since we believe that these are areas where the power and applicability of H-infinity theory has largely been unexplored.

Biographical sketch: Babak Hassibi was born in Tehran, Iran, in 1967. He received the B.S.~degree from the University of Tehran in 1989, and the M.S.~and Ph.D.~degrees from Stanford University in 1993 and 1996, respectively, all in electrical engineering.

From June 1992 to September 1992 he was with Ricoh California Research Center, Menlo Park, CA, and from August 1994 to December 1994 he was a short-term Research Fellow at the Indian Institute of Science, Bangalore India. Since September 1996 he has been a research associate at the Information Systems Laboratory, Stanford University, and starting November 1998 will be a Member of the Technical Staff at Bell Laboratories, Murray Hill, NJ. His research interests include robust estimation and control, equalization of communication channels, adaptive signal processing and neural networks, and linear algebra. He is the coauthor of the forthcoming books ``Indefinite Quadratic Estimation and Control: A Unified Approach to H$^2$ and H$^{\infty}$ Theories'' (New York: SIAM, 1998) and ``State Space Estimation'' (Englewood Cliffs, NJ: Prentice Hall, 1998).

Contact Urban Forssell for more info.

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