Consideration of plant behaviour in optimal servo-compensator design

ABSTRACT

Where the most prevalent optimal servo-compensator formulations penalise the behaviour of an error system, this paper considers the problem of additionally penalising the actual states and inputs of the plant. Doing so has the advantage of enabling the penalty function to better resemble an economic cost. This is especially true of problems where control effort needs to be sensibly allocated across weakly redundant inputs or where one wishes to use penalties to soft-constrain certain states or inputs. It is shown that, although the resulting cost function grows unbounded as its horizon approaches infinity, it is possible to formulate an equivalent optimisation problem with a bounded cost. The resulting optimisation problem is similar to those in earlier studies but has an additional ‘correction term’ in the cost function, and a set of equality constraints that arise when there are redundant inputs. A numerical approach to solve the resulting optimisation problem is presented, followed by simulations on a micro–macro positioner that illustrate the benefits of the proposed servo-compensator design approach.

Keywords:

• Optimal control
• servo-compensator
• input redundancy

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Davison and Ferguson (1981) actually assume var(x(0)) = I_N; however, the extension of their formulation to allow for arbitrary var(x(0)) is trivial.

2. Since the nature of the optimisation surfaces has not been thoroughly investigated, it is not possible to state that local optimality implies global optimality.

3. Care must be taken to not over-penalise the actual plant states and inputs. Doing so can yield optimal controllers with arbitrarily slow convergence.

4. To be precise, there are m nontrivial equations in the (·)^⋆ = (·)° case; however, due to Assumption 2.2, m = M.