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Anti-windup for sampled-data single-input-single-output linear

By Micheal White,2014-05-31 17:16
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Anti-windup for sampled-data single-input-single-output linear

April 2004

    Anti-windup

    for sampled-data single-input-single-output linear controllers

    Jaime Glaría

     Department of Electronic Engineering, Universidad Técnica Federico Santa María, Valparaiso, Chile

___________________________________________________________________________________________________________

Abstract

This note suggests a very simple way to insert anti-windup into sampled-data control, specially when controllers are single-input-

    single-output and linear.

     Key words Anti-windup, linear control, single-input-single-output control

    ___________________________________________________________________________________________________________

     1. INTRODUCTION

     Sampled-data linear controllers can be represented by linear

    recursive equations and easily handled by the z transform. In

    particular, a sampled-data P+I+D controller can be viewed as

    follows:

erykrk

    iieh/Tkk1ki

    deeT/h;;kkk1d

    uKeid;;kckkk

     Fig. 1: Sampled-data single-input-single-output linear controller EzRzYz;;;;;; 1 Y;;;;;;zIzzEzh/T 12. WINDUP ;;;;;; DzEzEzzT/h;;d UzKEzIzDz;;;;;;;;;;c Windup can appear when the controller output has to be

    constrained within extremes; mainly, as suggested by figure 2.

     EzRzYz;;;;;;

    112;;;;;;;;;; UzUzzEzAEzzBEzzC

    ?Thd? AK1c?Th~?i T?d BK12?ch~?

    T?d CK?ch~?

    erykrk uueAeBeCkk1kk1k2Fig. 2: Sampled-data single-input-single-output linear controller with constrained output In general, any sampled-data single-input-single-output linear controller can be viewed as: *( Windup is basically an issue arising from the fact that (uk can keep growing persistently in spite that u has hit one of the knmextreme values defined by the constraint, just because (1) is (1) uubeakkiikiiunstable; i.e., because the z transform for (1) yields poles 10iioutside or on the unit circle. (1) is equivalent to figure 1.

3. SUGGESTED ANTI-WINDUP

     Consider figure 3, where the controller output is constrained within the same extremes as before.

     Fig. 3: Sampled-data single-input-single-output linear controller with constrained output

     ** In figure 3, (u( can keep growing persistently if (e( grows kpersistently despite that u has hit one of the extreme values kdefined by the constraint; but it cannot keep growing persistently because (1) is unstable. Indeed, when u has hit the kextreme value u (either u or u) that value gradually extminmax*spreads through the delays in the loop of figure 3 and (u( k*converges to e+(b+b+…)u. Moreover, its is k12extstraightforward that, if such convergence takes place, the condition to get out of it is:

     nmn?? 1buea1bu??minmaxikiii101~?~?iii

     Figure 4 shows some simulation results for a P+I+D controller with h=1, K=1, T=1 and T=10; u=0 and u=255. Data is cicminmaxheld constant between sampling times.

     Fig. 4: Simulation results (top: e(t); center: u(t) according to

    figure 2; bottom: u(t) according to figure 3)

4. CONCLUSION

     A very simple way to insert anti-windup into sampled-data single-input-single-output linear controllers has been suggested.

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