Abstract
In this paper, a pole-independent, single-input, multi-output explicit linear MPC controller is proposed to stabilize the fourth-order cart–inverted-pendulum system around the desired equilibrium points. To circumvent an obvious stability problem, a generalized prediction model is proposed that yields an MPC controller with four tuning parameters. The first two parameters, namely the horizon time and the relative cart–pendulum weight factor, are automatically adjusted to ensure a priori prescribed system gain margin and fast pendulum response while the remaining two parameters, namely the pendulum and cart velocity weight factors, are maintained as free tuning parameters. The comparison of the proposed method with some optimal control methods in the absence of disturbance input shows an obvious advantage in the average peak efficiency in favor of the proposed SIMO MPC controller at the price of slightly reduced speed efficiency. Additionally, none of the compared controllers can achieve a system gain margin greater than 1.63, while the proposed one can go beyond that limit at the price of additional degradation in the speed efficiency.
Highlights
The cart–inverted pendulum (CIP) system that belongs to the class of fast singleinput, multiple-output (SIMO), under-actuated systems and satisfies a set of complicated characteristics, such as fourth-order highly nonlinear dynamics, open-loop instability, state coupling, and non-minimum-phase (NMP) behavior, provides many challenging problems to standard and modern control techniques [1]
NMP, and under-actuated systems, such as the CIP system, the above design issues appear to be more challenging when dealing with the design of SIMO model predictive control (MPC) controllers, especially if the stabilization requirements are to obtain (i) prescribed system gain margin, (ii) short CIP settling time with insignificant overshoot and undershoot, and (iii) reduced control effort
The controller ensures a priori a prescribed system gain margin and a two-time scale structure, allowing the response of the pendulum to be fast in comparison to the cart response
Summary
The cart–inverted pendulum (CIP) system that belongs to the class of fast singleinput, multiple-output (SIMO), under-actuated systems and satisfies a set of complicated characteristics, such as fourth-order highly nonlinear dynamics, open-loop instability, state coupling, and non-minimum-phase (NMP) behavior, provides many challenging problems to standard and modern control techniques [1]. The comparison, conducted in [25], between the MPC and LQR has shown that the MPC method is more suitable for the trajectory tracking task and smoothing in the control input, while the LQR is more convenient for fixed-value control and disturbance rejection, but it may generate adverse and rapid changes in the control signal For both approaches, MPC and non-MPC, the presence of real NMP zeros in the cart part of the fourth-order linearized CIP transfer function limits the robustness performance and prevents the achievement of monotonic cart step responses. On the other hand, addressing the SFC gains tuning problem to achieve monotonic step responses with pole-independent tuning methods can be performed for all-pole systems using the well-known coefficient diagram method (CDM) [27,28,31] In this method, controllers are designed via the assignment of the so-called characteristic ratios and generalized time constant (GTC), which may have a strong physical relationship with the damping (i.e., overshoot) and speed of response of the closed-loop system, respectively.
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