Predictive Control of Unmeasured Outputs

*† This paper describes a method to control an output or outputs that are not measured as in the case of controlling the output at a location where a sensor cannot be placed in actual operation. The same method can be used to produce a back-up controller when a sensor suddenly fails. Both the stabilization and tracking cases are treated. The design is data-based in that the controller is directly derived from a set of input-output data instead of from a known model of the system, and is based on model predictive control theory. When unknown periodic disturbances are present, or if they are artificially injected in the inputoutput data, the resulting controller will also have built-in capability to reject such disturbances. I. Introduction YPICAL control algorithms for multiple-input multiple-output (MIMO) systems assume knowledge of all the outputs being controlled, as well as the inputs at all actuators. In some situations, measurements of one or more of those outputs become unavailable during operation of the controller, for example by failure of a sensor. When it is known that a particular output is no longer available during operation, the controller could be re-designed to ignore that output completely. However, such a design would give up the opportunity to control that output. A perfectly reasonable option is to involve an extra identification step to establish a relationship between the unmeasured and measured outputs so that the unmeasured outputs could be estimated. What we are aiming for in this paper is a controller that, once designed, does not require measurement of at least one of multiple controlled outputs, and that, during design, does not involve an explicit derivation of the relationship between the outputs that are accessible and those that may become inaccessible during operation. In other words we look for a solution where one or more missing outputs are considered in the design from the beginning, that the controller has an implicit observer for the unmeasured outputs built in. Another application of the design method is to control the output of a system at a location where a sensor is available only during the design stage, but that sensor cannot be installed during the operation of the controller. For example, in the problem of interior noise control inside the cabin of an aircraft, the ideal sensor would be located exactly at the passenger’s head area, but such a sensor cannot be installed in practice, and nearby sensors must be used instead. We choose to derive such a controller within the framework of predictive control theory. 1-6 In the predictive control approach the controller is designed to minimize a receding-horizon cost function. Mathematically the challenge here is that the cost function is specified in terms of the outputs that include both the measured and unmeasured outputs, yet the controller to minimize such a cost function does not involve the outputs that later become unmeasured. A related situation was treated in Ref. 7, where the objective was to make a measured output track an unmeasured target. Another objective of the design is that the controller is capable of rejecting unknown periodic disturbances, both while stabilizing and while tracking the system outputs, including output(s) at the inaccessible location(s). The controller design algorithm requires no a priori knowledge of the system or disturbance characteristics. It turns out the data-based predictive control approach can be used to address this problem. 8-12 Only open-loop input-output data—corrupted by a similar, but not necessarily identical, unknown periodic disturbance—is required for controller design. In other words, the controller gains are to be designed from a set of input-output data instead of from a known model of the system. The output data could be disturbance-corrupted, in which case the resultant controller will have built-in capability to reject similar disturbances. In this paper we impose an additional requirement that