Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

and added to the objective function:
yk =argmin
y∈S {f (y)+µkB (y)} k =0, 1,... (4.8)
The subscript k indicates the iteration number, and the weight of the barrier function
relative to the objective decreases at each iteration allowing the search to approach the
constraint boundary. The search direction can be determined through any standard
gradient search procedure such as steepest descent, Newton’s method or conjugate
gradient (See Appendix A). However, the step-size must be properly selected to
ensure that all iterates lie within the feasible region, S. An interior point method is
attractive for the problem of hardware tuning since all iterates are feasible.
The barrier method is implemented in MATLAB as outlined in Figure 4-8. The
initial iterate, x0, termination condition tolerances and barrier parameters, ɛ and µ0,
are inputs to the algorithm. The iteration loop begins by initializing the algorithm
as shown. Then the constraint equations, their derivatives and the barrier function
(Equation 4.6) are calculated analytically at the current iterate. The tuning param-
eters are set to the current iterate on the hardware and a test is run. This data is
saved as f (yk), and the objective cost at this iterate is evaluated. Next, the gradients
of the objective are needed to generate a new search direction:
∂J (yk)
∂y
= ∂f (yk) ∂B (yk)
+ µk
∂y ∂y
∇J (yk) = ∇f (yk)+µk∇B (yk) (4.9)
The performance gradients, ∇f (yk), are determined with Equation 4.3, requiring
an additional hardware setup and test, while the barrier gradients are calculated
analytically:
∇B (yk) =−
r�
j=1
�
− 1
g (yk)
�
∂g (yk)
∂y
(4.10)
The objective gradients are then used to find the new search direction. The MATLAB
implementation developed for this thesis allows either a steepest descent or conjugate
gradient directions. Once the search direction is obtained the step-size is calculated
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