## Background

Feedback error learning (FEL) is a useful tool for controlling some uncertain systems. For example, to control a robot arm catching some loads with uncertain weight, conventional control method (like a feedback control) may have poor performance. FEL, however, could achieve better result by online adjustment. FEL contains a fixed feedback block for closed-loop stabilization, and a tunable feedforward block for response tracking. The tracking performance is mainly influenced by the tuning law for the feedforward.

In this research, we concern an FEL scheme under some strictly positive real (SPR) condition. However, the SPR condition is strict, and sometimes it is difficult to attain the condition. The non-SPRness can be caused by two problems: a certain filter denominator is not suitable (denominator problem), and the plant has relative degree two (relative degree problem). Our research target is to propose solutions for these two problems.

## Achievements

We proposed two methods for two non-SPR problems respectively. For the denominator problem, we established a systematic approach using Kalman-Yakubovich-Popov lemma to calculate the denominator polynomial. This approach is mathematically proved. Compared with trial and error, this proposal offers a reliable way to achieve the SPR condition.

The performance of this approach is verified by simulations. It could attain the SPR condition more easily, which leads to a better response performance.

For the relative degree problem, we introduce a virtual plant with relative degree one. Then the real plant is divided into the virtual plant and a filter. We then design an extended tuning law to control the virtual plant indirectly.

The extended tuning law is verified to be effective. The conventional method cannot even ensure stability, while the proposed method achieves a rather better result.