� Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski � Interpretations for Standard Optimization Setup 1 This is the second lecture on standard feedback optimization setup. It describes a variety of ways to come up with H2 and H-Infinity performance measures, provides hints for reducing non-standard objectives to the standard format. 2.1 Systems and signals background This section provides some minimal background in systems and signals needed for under- standing this lecture. 2.1.1 Signals and systems in continuous time It is convenient to think of continuous time (CT) signals as real vector-valued functions of time t ≤ [0, →), integrable over any bounded interval. From this viewpoint, f 1 (t) = t −1/2 (defined, for the sake of mathematical accuracy, as zero at t = 0) and f 2 (t)= e t 2 are signals, while f 3 (t)= t −1 (where f 3 (0) = 0) and f 4 (t)= � (t) (Dirac delta) are not. The set of all signals with values in R k will be denoted by L k . A continuous time system S with a k-dimensional input and m-dimensional output is simply a map S : L k ∞�L m (usually multi-valued, so that one input f ≤L k corresponds c A. Megretski, 2004 1 Version of February 9, 2004