Control of Deposited Zinc in Hot Dip Gal

8.7.4 Controller Design

RST controllers (with integrator behavior) have been designed for the various operation regions using tracking and regulation with independent objectives (see Chapter 3, Section 3.4) since the model always has stable zeros (the sampling period is tied to the steel-strip speed). An important requirement is the achievement of a delay margin of 2 T_s.

The polynomial P(q^-1) defining the closed loop poles will have a maximum degree given by




degP(q^1) (n_A n_B d n_HS-1
In this application one gets (n_HS 1)

Practical Aspects of Digital Control 369
n_A n_B d n_HS 19and therefore one can specify nine poles. The real poles of the model vary from 0.15 to 0.4 over the entire range of operating points. Two poles have been assigned to 0.2 and 0.3 (respectively 0.4 depending on the operating region and the values of the identified pole of the model) and the other seven auxiliary poles have been assigned at 0.1¹¹. These auxiliary poles introduce an attenuation at high frequencies (outside the band pass of the system) and improve the robustness of the system (in particular, the delay margin is increased) without affecting the regulation performance at low frequencies.
Figure 8.49a gives the Nyquist plot for the case where only two poles are specified for P(q^-1) (0.2 and 0.3) and Figure 8.49b allows one to see the effect of the additional auxiliary poles (seven poles at 0.1). One observes an important increase of the delay margin from 10.9 s to 25.7 s (T_s = 12s) since the Nyquist plot does no longer intersect the unit circle at high frequencies. The introduction of the auxiliary poles also produces a significant reduction of the magnitude of the input sensitivity function at high frequencies and, therefore, a reduction of the stress of the actuator in this frequency region. The other robustness margins have satisfactory values with or without the auxiliary poles (∆M > - 6dB ∆G > 2 ∆> 60 deg).
Simulation tests have shown that the closed loop system tolerates delay variations of ± T_s (one sampling period corresponds to about 15% variations of the delay) and ±50% variations of other model parameters (b₁ and a₁).
Procedures for taking into account the quantization, the saturation and the bumpless transfer from open loop to closed loop (see Sections 8.1.3 through 8.1.5) have been implemented.
Since the speed of the steel strip is measured, it was also possible to make an “open loop compensation” of this disturbance. This has a beneficial effect upon the regulation transient; however, it is the integrator included in the controller which cancels the effect of this disturbance in steady state.