A Separation Principle for a Class of Non Uniformly Completely Observable Systems

This paper introduces a new approach to output feedback stabilization of SISO systems which, unlike other techniques found in the literature, does not use quasi-linear high-gain observers and con trol input saturation to achieve separation between the state feedback and observer designs. Rather, we show that by using nonlin ear high-gain observers working in state coordinates, together with a dynamic projection algorithm, the same kind of separation principle is ac hieved for a larger class of systems which are not uniformly completely observable . By working in state coordinates, this approach avoids using knowledge o f the inverse of the observability mapping to estimate the state of the plant, which is otherwise neede d when using high-gain observers to estimate the output time derivatives. Index Terms Nonlinear control, output feedback, separation principle, nonlinear ob server. I. I NTRODUCTION THE area of nonlinear output feedback control has received mu ch attention after the publication of the work [3], in which the authors developed a systematic strategy for the output f eedback control of input-output linearizable systems with full relative degree, which employed two basic tools: an high-ga in observer to estimate the derivatives of the outputs (and h ence the system states in transformed coordinates), and control i put saturation to isolate the peaking phenomenon of the ob s rver from the system states. Essentially the same approach has la ter been applied in a number of papers by various researchers (see, e.g., [9], [12], [7], [13], [14], [1]) to solve differe nt problems in output feedback control. In most of the papers found in the literature, (see, e.g., [3], [9], [12], [13], [7], [14 ]) the authors consider input-output feedback linearizabl e systems with either full relative degree or minimum phase zero dynamics. The work in [21] showed that for nonminimum phase systems the problem can be solved by extending the system dynamics wi th a chain of integrators at the input side. However, the resu lts contained there are local. In [19], by putting together this idea with the approach found in [3], the authors were able to s how how to solve the output feedback stabilization problem for s moothly stabilizable and uniformly completely observable (UCO) systems. The work in [1] unifies these approaches to prove a se paration principle for a rather general class of nonlinear s ystems. The recent work in [16] relaxes the uniformity requirement o f he UCO assumption by assuming the existence of one control input for which the system is observable. On the other hand, h owever, [16] requires the observability property to be comp lete, i.e., to hold on the entire state space. Another feature of th at work is the relaxation of the smooth stabilizability assu mption, replaced by the notion of asymptotic controllability (whicallows for possibly non-smooth stabilizers). A common feature of the papers mentioned above is their input-output variable approach , which entails using the vector col(y, ẏ, . . . , yy, u, u̇, . . . , uu) as feedback, for some integers ny, nu, wherey andu denote the system output and input, respectively. This in particular implies that, when dealin g with systems which are not input-output feedback lineariz able, such approach requires the explicit knowledge of the inverse of t he observability mapping, which in some cases may not be avai lable. This paper develops a different methodology for output feed back stabilization which is based on a state-variable approach and achieves a separation principle for a class of non-UCO sy stems, specifically systems that are observable on an open re gion of the state space and input space, rather than everywhere. W impose a restriction on the topology of such an “observabil ity region” assuming, among other things, that it contains a suf ficiently large simply connected neighborhood of the origin . The main contributions are the development of a nonlinear obser ver working in state coordinates (which is proved to be equiv alent to the standard high-gain observer in output coordinates), and a dynamic projection operating on the observer dynamics which eliminatesthe peaking phenomenon in the observer states, thus avoidin g the need to use control input saturation. One of the benefits of astate-variable approachis that the knowledge of the inverse of the observability map ping is not needed. It is proved that, provided the observable region satisfies s uitable topological properties, the proposed methodology yields closed-loop stability. In the particular case when the plan t is globally stabilizable and UCO, this approach yields sem iglobal stability, as in [19], provided a convexity requirement is s ati fied. As in [21], [19], [1], our results rely on adding int egrators at the input side of the plant and designing a stabilizing con trol law for the resultingaugmentedsystem. Thus, a drawback M. Maggiore is with the Department of Electrical and Computer E ngineering, University of Toronto, 10 King’s College Rd., T oronto, ON, M5S 3G4, Canada (e-mail: maggiore@control.utoronto.ca). Part of thi s research was performed when the author was at the Ohio State U niv rsity. K. M. Passino is with the Department of Electrical Engineerin g, The Ohio State University, 2015 Neil Avenue, Columbus, OH 4 3210 USA (e-mail: k.passino@osu.edu). This work was supported by NASA Glenn Research Center, Grant NAG3-2084. Published inIEEE Transactions on Automatic Control , vol. 48, no. 7, 2003, pp. 1122–1133