Multi-objective control problems are characterized by complex design
specifications, and are naturally formulated as infinite-dimensional optimization
problems. In principle, the complexity of solving such problems is infinite.
The solvability of the problem defines when and in what sense this complexity can
be reduced, and it is characterized by:It turns out that duality theory is a fundamental tool in analyzing the
- Existence of computable exact solutions. (There is an underlying finite-dimensional structure)
- Existence of computable approximate solutions. (No finite-dimensional structure is apparent or exists)
- Accuracy of the approximations.
- Information provided by the approximate solutions about the structure of the optimal solution and optimal controller
solvability of infinite dimensional convex problems, and in providing generic computational
methods for such problems.
These methods translate into algorithms that are now the core of a
Computer Aided Control Design Methodology for linear systems that I have
implemented in a software package.
The methodology is quite powerful. It allows for direct incorporation of
time and frequency domain specifications in the design, including
typical specifications (overshoot, settling time, etc) on the response to
fixed inputs (step commands). It also provides the designer with information
about limitations and tradeoffs on the achievable performance, and delivers a
controller if the specifications can be met.The software has already been used in several applications at the Draper Laboratory, and in the design of high precision control of actuators for micro-surgery.