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*
Classical Feedback Control*
*with MATLAB*

**Boris J. Lurie and Paul J. Enright, Marcel Dekker, NY, 2000
**

Reprinted from Classical Feedback Control pp 94-129 by courtesy of Marcel Dekker, Inc. Copyright © 2000 Marcel Dekker, Inc. These materials may be copied for nonprofit use, so long as they are unaltered and accompanied by this header. If the materials are used for classroom instruction, mention is to be made of the source.

Chapter 4

SHAPING THE LOOP FREQUENCY RESPONSE

The problem of optimal loop shaping encompasses two fairly independent parts that can be solved sequentially (thus making the design structural):

The first part is *feedback bandwidth maximization*
which is solved by appropriately *shaping the feedback loop response at
higher frequencies* (in the region of crossover frequency and higher).

The second part is of *distribution of the available
feedback over the functional feedback band*.

The feedback bandwidth is limited by the sensor noise effect at the system output, the sensor noise effect at the actuator input, plant tolerances (including structural modes), and nonminimum phase lag (analog and digital) in the feedback loop. The optimal shape of the loop gain response at higher frequencies, subject to all these limitations except the first, includes a Bode step.

The Bode step is presented in detail as a loop shaping tool for maximizing the feedback bandwidth. The problem of optimal loop shaping is further described and the formulas are presented for calculation of the maximum available feedback over the specified bandwidth.

The above solution is then generalized by application of a Bode integral to reshaping the loop gain response over the functional bandwidth (i.e., for solving the second part of the loop shaping problem). It is shown that the feedback is larger and the disturbance rejection improved in Nyquist-stable systems.

Loop shaping is described for plants with flexible modes, for collocated and non-collocated control, and for the loops where the plant is unstable. The effect of resonance mode coupling on the loop shaping in MIMO systems is considered.

It is described how to shape the responses of parallel feedback channels to avoid nonminimum phase lag while providing good frequency selection between the channels.

When the book is used for an introductory control course, Sections 4.2.5 – 4.2.7, 4.3.3, 4.3.6, 4.3.7, 4.4, and 4.5 can be omitted.

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