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Proportional integral derivative plus control for nonlinear discrete-time state-dependent parameter. . .

Figure 7. Onsite implementation of the SDP-PID+/LQ control for the bitumen tank, with the command
input shown alongside its ±5% tolerance. Abbreviations: LQ, linear quadratic; PID+, proportional integral
derivative plus; SDP, state-dependent parameter; TF, transfer function

Figure 8. The control process of the reeling/packing machine for Bitumen membrane sheets employs the
SDP-PID+ methodology, integrated in FPGAs, at INSUMAT company, Sadat City, Egypt. Abbreviations:
FPGA, Field programmable gate array; PID+, Proportional integral derivative plus; SDP, State-dependent
parameter.

Figure 9. Schematic of the reeling/packing process for bitumen membrane sheets at INSUMAT company,
Sadat City, Egypt. Abbreviation: AC, alternating current.

Using the model in Equation (43) and Table Equation (39). The time-variant transition ma-
1, the NMSS framework for nonlinear SDP-PID+ trix F k , time-invariant input vector g, and obser-
control is constructed, similar to the structure in vation vector h remain consistent with the state
T
vector x k = z k e k ∆e k u k−1 u k−2 .

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