Strictly positive real systems based on reduced-order observers

Abstract

We study the extension of the class of linear timeinvariant open-loop systems that may be transformed into SPR systems introducing a reduced-order observer. It is shown that for open-loop stable systems a cascaded observer achieves the result. For open-loop unstable systems observer-based feedback is required to succeed. In general, any stabilizable and observable system may be transformed into an SPR system defining a new output based on the observer state. This overcomes the old conditions of minimum phase and relative degree one for the case of keeping the original output. The result is illustrated with some examples.

Model Matching and Passivation of MIMO Linear Systems via Dynamic Output Feedback and Feedforward

“…Since the design of the compensator Σ c as in 7is such that the matrix transfer function between v(s) andŷ(s) is given by W d (s), it follows that the selection of the matrices in (17) ensures stability of the interconnected system together with the property that…”

“…By inspecting the structure of the internal transfer matrices of the interconnected system given in (17), the following statement is straightforward, hence its proof is omitted. Proposition 4: Suppose that the assumptions of Proposition 1 hold.…”

“…Therefore, a passivating technique consisting of static feedback/feedforward actions is proposed in [41], while [10], based on the characterization of SPR systems in terms of the Kalman-Yakubovich-Popov lemma [9], [33], envisions the use of observers, thus leading to a dynamic, rather than static, construction and yielding an LMI-based approach for passivation of MIMO systems. The design introduced in [17], instead, is based on reduced-order observers for the original linear system. A suitably designed input-output transformation matrix is proposed in [43]: this allows to guarantee positive passivity indices.…”