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Proportional integral derivative plus control for nonlinear discrete-time state-dependent parameter. . .
The time-invariant P matrix can be ob- enables efficient and intuitive control of these sys-
tained by freezing F k at a specific operating tems, which are characterized by discrete-time
point u k−2 = u k−3 = 0. The weights R = 1 TFs of any order and sampling time delays.
and Q = diag {1 1 1 1 1} were selected to ob- The selected demonstrators represent the first
tain the time-invariant P matrix using algebraic industrial implementations of the SDP-PID+ con-
Riccatti equation as defined in Equation (29). trol methodology, showcasing its applicability
The time-variant SDP-PID+ gain vector, k + is across systems with varying response times, 240
k
then computed using Equation (30). The control and 0.5 s, respectively, providing a comprehensive
law in Equation (24) is employed to control the evaluation of their performance.
drum’s reeling/packing machine speed, as shown In all cases, the systems were modeled using
in Figure 12. the quasi-linear SDP model structure, where pa-
rameters depend on other system variables. This
model structure allowed for designing an SDP-
PID+ control law to leverage linear system design
strategies such as suboptimal LQ optimization or
pole placement. Importantly, the control gains
were made state-dependent, enhancing adaptabil-
ity and performance. Notably, the time-invariant
numerator parameter in models in Equations (38)
and (43) satisfies the two controllability condi-
tions discussed in Section (3.2).
The proposed control strategy markedly im-
proved temperature regulation for the industrial
bitumen tank system, maintaining it consistently
within the ±5% permissible range while avoid-
ing overshoot. This enhanced performance min-
imized the risk of vapor self-ignition inside the
tank, ensuring greater operational safety and im-
proved quality of the bitumen mixture.
Figure 12. Onsite application of FPGAs/SDP-PID+ Finally, implementing the FPGAs/SDP-
control for the industrial reeling/packing machine.
PID+ control for the industrial reeling/packing
Abbreviations: FPGA: Field programmable gate ar-
ray; PID+: Proportional integral derivative plus; machine demonstrated robust closed-loop perfor-
SDP: State-dependent parameter; VDC: Volts direct mance. The system exhibited no overshoot, no
current. severe oscillations, and effective noise reduction,
attributed to the advanced real-time processing
capabilities of FPGA technology.
The onsite implementation demonstrates that
The results from these diverse demonstrators
the FPGAs/SDP-PID+ control ensures accept-
confirm the effectiveness of the extended SDP-
able performance with significant noise suppres-
PID+ control in optimizing design time, enhanc-
sion and negligible differences between practical
ing practicality, and improving adaptability and
and SDP-TF model in Equation (40) responses.
robustness. Its ability to accommodate systems
No overshoot or severe oscillations were observed,
with varying dynamics while ensuring superior
confirming robust control for the reeling/packing
performance and operational reliability highlights
process. It is worth noting that the feasibility its potential as a versatile control strategy for in-
of FPGA implementation for real-scale industrial dustrial applications. Future work may extend
systems, including processing power and memory this approach to more complex multivariable sys-
constraints, is detailed in Shaban et al., 15 where tems and optimize its implementation for im-
all relevant hardware features are analyzed.
proved computational efficiency.
6. Conclusion Acknowledgments

This paper presented the expansion and prac- The author gratefully acknowledges the support
tical implementation of SDP-PID+ control sys- of the Deanship of Graduate Studies and Scien-
tems across two distinct demonstrators: an indus- tific Research, Jazan University, Saudi Arabia.
trial bitumen tank system and an industrial reel-
Funding
ing/packing machine used in the bitumen mem-
brane sheet production line. This enhancement None.
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