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16. Control Design Based on Optimisation

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 Thus far we have seen that design constraints arise from a number of different sources
  • structural plant properties, such as NMP zeros or unstable poles
  • disturbances, their frequency content, point of injection and measurability
  • architectural properties and the resulting algebraic laws of trade-off
  • integral constraints and the resulting integral laws of trade-off

The subtlety as well as complexity of the emergent trade-off web into which the designer needs to ease a solution, motivates interest in what is known as criterion based control design or optimal control theory: the aim here is to capture the control objective in a mathematical criterion and solve it for the controller that (depending on the formulation) maximizes or minimizes it.

Three questions arise:

  1. Is optimization of the criterion mathematically feasible?
  2. How good is the resulting controller?
  3. Can the constraint of the trade-off web be circumvented by optimization?

Question (1) has an affirmative answer for a number of criterion. In particular, quadratic formulations tend to favor tractability. Also, the affine parameterization of Chapter 15 is a key enabler, since it renders the sensitivity functions affine in the sought variable, Q.

The answer to question (2) has two facets: how good is the controller as measured by the criterion? Answer: it is optimal by construction; but how good is the resulting control loop performance as measured by the original performance specifications? Answer: as good, or as poor, as the criterion in use can capture the design intention and active trade-offs. A poorly formulated criterion will simply yield a controller which optimally implements the poor design. However when selected well, a design criterion can synthesize a controller which would have been difficult to conceive of by the techniques covered thus far; this is particularly true for multivariable systems covered in the next part of the book.

Question (3) is simply answered by no: all linear time invariant controllers, whether they were synthesized by trial and error, pole assignment or optimization, are subject to the same fundamental trade-off laws.

Summary
  • Optimization can often be used to assist with certain aspects of control system design.
     
  • The answer provided by an optimization strategy is only as good as the question that has been asked.
     
  • Optimization needs to be carefully employed keeping in mind the complex web of trade-offs involved in all control system design.
     
  • Quadratic optimization is a particularly simple strategy and leads to a closed form solution.
     
  • Quadratic optimization can be used for optimal Q synthesis.
     
  • We have also shown that quadratic optimization can be effectively used to formulate and solve robust control problems when the model uncertainty is specified in the form of a frequency domain probabilistic error.
     
  • Within this framework we have shown that the robust controller biases the nominal solution arising from optimal nominal design so as to create a gap for model uncertainty whilst attempting to minimize affecting the achieved performance.
     
  • This can be viewed as a formal way of achieving the bandwidth reduction which was discussed earlier as a mechanism for providing a robustness gap in control system design.