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COBEM 2021
26th International Congress of Mechanical Engineering
COMPUTATION OF MINIMUM-TIME SOLUTION OF A SEALING MACHINE THROUGH LINEAR PROGRAMMING WITH SENSITIVITY AND CLOSED-LOOP ROBUSTNESS ANALYSIS
Submission Author:
Yuri Mariano , SP
Co-Authors:
Yuri Mariano, José Jaime da Cruz, Fuad Kassab Junior
Presenter: Yuri Mariano
doi://10.26678/ABCM.COBEM2021.COB2021-1907
Abstract
Instances of minimum-time problems occur frequently in several engineering areas. A few relevant examples are path planning of industrial robots , of satellite maneuvers and of cranes. Generally a closed-loop system is built to track the planned path in order to provide robust performance. In this paper the problem of temperature control of a sealing machine used in automatic packing lines is considered where the input is the termal power applied to the jaw and the output is the jaw temperature. A second-order transfer function with real poles is adopted to relate the output to the input. The problem is to transfer in minimum-time the temperature of the jaw between two given initial and final values without overshoot and with null time-derivative of the jaw temperature at the final time. The power capacity of the heater must not be exceeded during all the time. After considering a stepwise constant approximation for the input power, it is shown that a sequence of Linear Programming problems can be used to solve the minimum-time problem. Then, the sensitivity of both the final temperature and of its time-derivative with respect to the parameters of the transfer function are evaluated through the integration of an appropriate system of linear differential equations. This is the second point adressed in the paper. In the planned path the resultant power to be delivered to the jaw is either zero (lower bound) or the power capacity of the heater (upper bound) since, as it is well known, the optimal control law in this case is bang-bang. Hence, when a closed-loop system is designed to track the planned path in the real world, due to robustness problems the resulting performance may be poor since these lower and upper bounds must still continue to be respected in closed-loop. Intuitively it should be expected that if a path is planned based on a lower value of the heater capacity, then the closed-loop system should exhibit a better performance since the upper bound becomes not tight. Nevertheless the price to be paid is a larger time to reach the desired final temperature. Using a SIMC-PID controller to close the loop, the Monte Carlo simulation is used in order to analyse these aspects when parameter uncertainties are present in the model. This is the third point addressed in the paper.
Keywords
linear programming, minimum time, Optimal Control, path planning, Sensitivity, robustness, bang-bang control
