for sampled-data single-input-single-output linear controllers
Department of Electronic Engineering, Universidad Técnica Federico Santa María, Valparaiso, Chile
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
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
Fig. 1: Sampled-data single-input-single-output linear controller ，Ez！Rz？Yz;；;；;； ？1 Y;；;；;；z！Iz；z？Ez；h/T ？12. WINDUP ;；;；;； Dz！Ez？Ez；z；T/h;；d Uz！K；Ez？Iz？Dz;；;；;；;；;；c Windup can appear when the controller output has to be
constrained within extremes; mainly, as suggested by figure 2.
：?Thd；? A！K；1？？c；?Th~?i T：?d B！K；1？2；；?ch~?
，e！r？ykrk u！u？e；A？e；B？e；Ckk？1kk？1k？2Fig. 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) u！u；b？e；a，？，？kkiikiiunstable; i.e., because the z transform for (1) yields poles ！1！0iioutside 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:
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)
A very simple way to insert anti-windup into sampled-data single-input-single-output linear controllers has been suggested.