½½ À ÈÌ Ê can be done using, for example, the MATLAB command ×ÕË Select ¯ N = U ¯ S , ¯ S ×ÕÖØѴ˵ . ¯ and note that N R n ×n and that NN T = U as desired. ¯ ¯ T ¯ ¯ ¯ T S U = U S U = , 5.4 ALGORITHMS PROVIDING GLOBAL GUARANTEES In this section constructive techniques for the selection of the dynamic anti-windup compensator matrices in (5.3) are given, with the goal of guaranteeing global guar- antees on the internal stability and input-output gain of the resulting closed-loop system. This section should be then understood as the dynamic counterpart of the previous Section 4.3 related to static DLAW. Similar to the static case, ex- ponential stability of the plant is a necessary condition for dynamic DLAW with global guarantees to be applicable, while to deal with non-exponentially stable plants the reader should refer to the regional algorithms reported in the subsequent Section 5.5. An appealing result that characterizes the dynamic DLAW techniques is that whenever the plant is exponentially stable, there always exists a dynamic anti-windup compensator of the same order as the plant that solves the global prob- lem. Therefore, dynamic anti-windup compensation constitutes a step forward as compared to static compensation techniques, where suitable feasibility conditions were required to hold for the algorithms to be applicable. 5.4.1 Global full-authority plant-order augmentation Full-authority linear dynamic anti-windup compensation synthesis amounts to se- lecting the matrices (A aw , B aw , C aw,1 , D aw,1 , C aw,2 , D aw,2 ) of the anti-windup com- pensator (5.3) corresponding to the block F in Figure 5.1 when the signal v acts both on the state and on the output equations of the unconstrained controller. An algorithm to construct a plant-order version of such a full-authority anti-windup compensation is offered in this section. Reduced-order design will be addressed in the following section. Algorithm 4 (Dynamic plant-order full-authority global DLAW) Applicability Exp Stab Marg Stab Marg Unst Exp Unst Architecture Lin/ NonL L Guarantee Global/ Regional G Dyn/ Static D Ext/ FullAu FA