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Trajectory Tracking with Output Constraints on a Gantry Crane

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Fig. 1: Trajectory tracking with contraints. The load must travel through the parcours without touching the columns.

In this project, we apply Iterative Learning Control (ILC) to trajectory tracking on a lab-scaled gantry crane. However, in constrast to previous work, we assume that the load is only allowed to move in the close proximity of the reference trajectory. Since these output constraints lead to disrupted trials, the pass length in this ILC system is not constant. Therefore, we present new methods for the class of ILC systems with variable trial duration and apply these methods to the given application. Both simulation and experimental results are provided which demonstrate that both maximum pass length and small tracking error can be achieved in very few iterations even in the presence of tight constraints.

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We consider the special case of ILC application systems in which the initial input, i.e., the input that is applied in the first trial, drives the system to violate constraints before the trial has completed and consequently, that trial is aborted before it is finished. In that case, the incomplete output information must nonetheless be used to improve the input trajectory such that, in the second trial, the constraint is not violated, or at least violated at a later (cycle) time.

As an application example, we consider trajectory tracking with output constraints on a lab-scale gantry crane, as depicted in the figure above.

To simplify the dynamics that the ILC must face, we design an observer and a linear-quadratic regulator for the cart position. This inner feedback loop is cascaded with an iterative learning controllers, as depicted in the following.

Fig. 2: Cascaded control structure. The velocities of the crane drive are set by two state feedback loops, one of which is driven by an outer ILC loop.

To use model-based ILC algorithms, we use a simple step test and standard system identification methods to approximate the input-output dynamics of the dashed box in the figure above by a linear time-discrete transfer function. The Markov parameters of that transfer function are used to determine a lifted-systems representation of the dynamics.

We design two different approaches: a standard diagonal learning gain matrix and an inversion-based design. The Figure below shows results for the diagonal approach. The trials are disrupted as soon as the load leaves the band (break points). After four iterations, the maximum pass length is achieved.

Fig. 3: Experimental results for iterative learning control with diagonal learning gain.

The inversion-based algorithm appears to be more precise than the algorithm based on a diagonal learning gain matrix. In both cases the maximum pass length is achieved after four iterations and monotonic convergence is observed even for very tight constraints. The proposed approach shows sufficiently low sensitivity with respect to constant and iteration invariant disturbances.

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