GRADS – Gradient search optimization

Block SymbolLicensing group: ADVANCED

Function Description
The GRADS block performs one-dimensional optimization of the $f (x, v)$ function by gradient method, where $x \in <xmin, xmax>$ is the optimized variable and $v$ is an arbitrary vector variable. It is assumed that the value of the function $f (x, v)$ for given x at time $k$ is enumerated and fed to the f input at time $k + n * T_{S}$ , where $T_{S}$ is the execution period of the GRADS block. This means that the individual optimization iterations have a period of $n * T_{S}$ . The length of step of the gradient method is given by

$\begin{array}{rcl} g r a d & = & (f_{i} - f_{i - 1}) * {(d x)}_{i - 1} \\ {(d x)}_{i} & = & - gamma * g r a d, \end{array}$

where $i$ stands for $i$ -th iteration. The step size is restricted to lie within the interval $⟨ dmin, dmax ⟩$ . The value of the optimized variable for the next iteration is given by

x i + 1 = x i + (d x) i <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow> <msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo class="MathClass-bin">+</mo><mn>1</mn></mrow></msub> <mo class="MathClass-rel">=</mo> <msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub> <mo class="MathClass-bin">+</mo> <msub><mrow><mrow><mo class="MathClass-open">(</mo><mrow><mi>d</mi><mi>x</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mrow><mi>i</mi></mrow></msub> </mrow></math>

Inputs

f	Value of the optimized $f (.)$ for given variable x	Double (F64)
x0	Optimization starting point	Double (F64)
START	Starting signal (rising edge)	Bool
BRK	Termination signal	Bool

Outputs

x	Current value of the optimized variable	Double (F64)
xopt	Resulting optimal value of the x variable	Double (F64)
fopt	Resulting optimal value of the function $f (x, v)$	Double (F64)
BSY	Indicator of running optimization	Bool
iter	Number of current iteration	Long (I32)
E	Error flag	Bool
iE	Error code	Long (I32)
	1 .... $x \notin < xmin, xmax >$ 2 .... $x = xmin$ or $x = xmax$

Parameters

xmin	Lower limit for the x variable	Double (F64)
xmax	Upper limit for the x variable $⊙$ 10.0	Double (F64)
gamma	Coefficient for determining the step size of the gradient optimization method $⊙$ 0.3	Double (F64)
d0	Initial step size $⊙$ 0.05	Double (F64)
dmin	Minimum step size $⊙$ 0.01	Double (F64)
dmax	Maximum step size $⊙$ 1.0	Double (F64)
n	Iteration period (in sampling periods $T_{S}$ ) $⊙$ 100	Long (I32)
itermax	Maximum number of iterations $⊙$ 20	Long (I32)

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