Development of general multivariable run-by-run control methods with application to a sheet metal forming process

The problem of flexible-tool sheet metal forming and in-process shape control has been confounded by the lack of available measurements during the process. Run-by-run (rbr) control has been developed to achieve feedback control advantages in processes that do not lend themselves to in-process sensing and actuation. While the process cycle time sampling interval limits the performance, rbr can still achieve zero mean error and predictable variance on the output. This paper demonstrates a novel extension of typically single input–single output (SISO) rbr control to the control of discrete-die sheet metal forming which is a multivariable, regionally-coupled process; one where a single input affects a limited region of outputs and a single output is affected by multiple independent inputs. To this end, an easily calibrated, generic plant model is first reviewed. This model, along with others describing regional coupling, proves to be ill-conditioned for even moderately-sized processes. This renders the direct plant inversion used in SISO rbr control impractical. As a result, three controllers that do not rely on inverting the plant are considered: the linear-quadratic regulator (LQR), linear-quadratic Gaussian (LQG), and H-infinity. All three of these controllers are applicable to both square and non-square plants, and they are not limited to the class of plant explicitly considered in this paper. Next, the concept of robustness is introduced by establishing stability limits through structured singular values. All three controllers are experimentally tested on a discrete-die sheet metal forming press. In this stretch forming process, the traditionally monolithic die is divided into many small segments. These segments can then be rearranged to form a new die shape between sheet metal stretching cycles. The proper rearrangement of these inputs in response to output shape error is the goal of the three controllers. Experimental results are in close agreement with theoretical predictions, showing the validity of the modeling and control approaches.