If POST is used to submit the form, enter the value = '". $ _ POST [' test ']."' In the value property of the required input where test is the name of the desired input. If GET is used, then $ _GET ['test']
update: If you use selects, then in the place where they are generated, you need to add an option tag
if ('Значение данного option'==$_POST['test']){ echo ' selected ';}
An example for your code. (Provided that you do not use a generator for this form, a template engine or ajax)
<select class="btn dropdown-toggle selectpicker btnfull btn-warning" name="filter_высота" size="1"> <option <?php if ($_POST['filter_высота']==40){ echo ' selected '; } ?> value="40">10.50</option> <option>....</option> ... </select>