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Examples Modeling
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1 Single Input-Single Output

Linear control theory, as taught to undergraduate students, is primarily concerned with Single Input-Single Output (SISO) systems. Many real world systems are linear and while technically Multiple Input-Multiple Output (MIMO) the coupled axes are so weakly coupled that the coupling can be neglected. As a result the system can be approximated as SISO. An example of this the hovering helicopter where at hover the pitch attitude and horizontal speed can be decoupled because they should both be nearly zero minimizing any coupling between them.

SISO systems are typically less complex than MIMO systems. Usually, it is also easier to make order of magnitude or trending predictions "on the fly" or "back of the envelope". MIMO systems have too many interactions for most of us to trace through them quickly, thoroughly, and effectively in our heads.

Frequency domain techniques for analysis and controller design dominate SISO control system theory. Bode, Nyquist, Nichols, and root locus are the usual tools for SISO system analysis. Controllers can be designed through the polynomial design, root locus design methods to name just 2 of the more popular. Often SISO controllers will be PI, PID, or Lead-Lag.

What follows are some design goals to keep in mind.

2 Big Picture

Design the pre-filter (W) and controller (K) on the basis of a nominal model LaTeX:  P_0 for the plant LaTeX:  P such that the feedback system exhibits the following properties:

  1. Stability: if the system is perturbed then the system will return to equilibrium
  2. Small tracking error
  • Good low frequency command following
  • Good low frequency disturbance attenuation
  • Good high frequency noise attenuation

The stated goals must be achieved in the presence of the following sources of uncertainty

  • LaTeX: P_0 \ne P
  • LaTeX: H is not known exactly
  • LaTeX: d_i and LaTeX: d_o are not known exactly
  • LaTeX: n is not known exactly