Output feedback control of nonlinear systems with uncertain ISS / iISS supply rates and noises ∗

Abstract. This paper considers the problem of global output feedback control for a class of nonlinear systems with inverse dynamics. The main contribution of paper is that: For the inverse dynamics with uncertain ISS/iISS supply rates, and the systems being disturbed by L noises, we construct a reduced-order observer-based output feedback controller, which drives the output of system to zero and maintain other closed-loop signals bounded. Finally, a simulation example shows the effectiveness of the control scheme.


Introduction
Since the notion of input-to-state stability (ISS) was first introduced in [1], it has been recognized as a central concept in nonlinear control systems.[2][3][4][5] and the references therein investigated many kinds of properties of ISS.[6][7][8][9] and the references therein considered controller design and stability analysis for various classes of nonlinear systems with ISS (or ISpS) inverse dynamics.Subsequently, another important concept, integral input-to-state stability (iISS), was firstly presented in [10], and several characterizations on iISS were investigated in [11], in which iISS is proved to be strictly weaker than ISS.In [12], the authors analyzed nonlinear cascades in which the driven subsystem is iISS, and characterized the admissible iISS-gains for stability.Recently, [13][14][15][16] gave several Lyapunov-based small-gain theorems covering iISS systems.
So far, in addition to the above literatures, there are many other results on the design and analysis of controller for nonlinear systems with ISS/iISS inverse dynamics.For example, Arcak et al. in [12] applied the admissible iISS-gains for stability of cascade systems to develop a new observer-based backstepping design.Jiang et al. in [17] firstly presented a unifying framework for the robust global regulation via output feedback for nonlinear systems with iISS inverse dynamics.Recently, [18] further studied output feedback regulation for a class of nonlinear systems with iISS inverse dynamics, in which the observer gain is governed by a Riccati differential equation, and Xu and Huang in [19] considered the output regulation problem for output feedback systems with relative degree one and iISS inverse dynamics.In [20], the authors considered reduced-order observer-based output feedback regulation for a class of nonlinear systems with iISS inverse dynamics.Recently, Yu et al. in [21,22] extended the notion and some properties of iISS to stochastic nonlinear systems.
However, almost of the above papers only consider the ISS/iISS inverse dynamics with known ISS/iISS supply rates.When the inverse dynamics with uncertain ISS/iISS supply rates, how to design a feedback controller for nonlinear systems seems to be an interesting work.
The main contribution of paper is that: For the inverse dynamics with uncertain ISS/iISS supply rates, and the systems being disturbed by L 2 noises, we construct a reduced-order observer-based output feedback controller, which drives the output of system to zero and maintain other closed-loop signals bounded.
The remainder of paper is organized as follows.Section 2 is problem statements.Section 3 gives the design of output feedback controller.Section 4 is the main results.A simulation example is given in Section 5. Section 6 concludes the paper.

Notations
R + stands for the set of all nonnegative real numbers, R n is the n-dimensional Euclidean space, |x| is the usual Euclidean norm of a vector x.K denotes the set of all functions γ : R + → R + , which are continuous, strictly increasing and γ(0) = 0; K ∞ is the set of all functions which are of class K and unbounded, KL denotes the set of all functions β(s, t) : R + × R + → R + , which are of class K for each fixed t, and decrease to zero as t → ∞ for each fixed s. σ 1 (s) = O(σ 2 (s)) as s → 0+ means that σ 1 (s) c 1 σ 2 (s) for some constant c 1 > 0 and all s in a small neighborhood of zero, and σ 1 (s) = O(σ 2 (s)) as s → ∞ means that σ 1 (s) c 2 σ 2 (s) for some constant c 2 > 0 and all large enough s.L 2 (R + ; R) is the family of all functions l : R
The control objective is to design an output feedback controller for system (1) based on a reduced-order observer.Such controller drives the output of systems to zero asymptotically and maintains other closed-loop signals bounded.
The main results of paper are based on the following assumptions.
Assumption 1.For η-system of (1), there is a positive definite function where α 0 , α 0 , γ 0 are class K ∞ functions, π 0 is a positive-definite continuous function, and p 0 is an uncertain positive constant.
Since p 0 in (2) is unknown, the inverse dynamics have uncertain ISS/iISS supply rates.
Remark 2. Assumption 3 shows that f i includes not only the output, but also the unmeasured state variables.Moreover, f i (x i ) can be any smooth function with respect to measurable variable x 1 , and be Lipschitz function with respect to the unmeasurable variables x 2 , . . ., x i with the Lipschitz constant satisfying LMI (3).Assumption 3.For each 1 i n, there exist unknown positive constants p i1 , p i2 , and known positive-definite smooth functions φ i1 , φ i2 such that Remark 3. Assumption 3 is an usual condition in output feedback control of nonlinear systems (e.g., see [17,18,20,23]).Assumption 4 shows that system (1) is disturbed by L 2 noises.

Output feedback controller design
This section gives the design procedure of global output feedback controller by using the method of adaptive backstepping.
Remark 5.In [20], there is a mistake in the choice of observer gain.Here we correct it and give a LMI algorithm of it.

Adaptive controller design
Now, we give the adaptive controller design procedure by using the backstepping method.
Step 1. Begin with the y-subsystem of (11) and consider ξ 2 as the virtual dynamic control input.We define the 1st dynamic virtual control input where Γ , c are two positive parameters and ψ 1 is a smooth positive design function, introduce a new intermediate variable www.mii.lt/NA Step 2. Denoting V 1 = (1/2)y 2 , viewing ξ 2 as the virtual control input, and considering the Lyapunov function , with the use of ( 11)-( 13), one has By Young's inequality, one leads to where 2 is a small design parameter to be determined in Appendix.We define an unknown constant θ such that θ (2 2 p * + p * 2 )/(2 2 ), and set Φ 1 (t, ē2 , η, y) = y(p * ē2 + 14) and ( 15), some simple manipulations lead to and the variable z 2 satisfies , Step i (3 i n).At step i, one can obtain the similar property to (16).Such a result is presented by the following lemma, for notational coherence, denote u = ξ n+1 .
Lemma 1.For each i = 3, . . ., n, there exist smooth functions α i , Proof.With the aid of the completion of squares, it follows directly from the definitions of Φ n−1 , G and D(t).

A simulation example
Consider the following nonlinear system with inverse dynamics and noises: where By (8), the reduced-order observer is given by According to Section 3, the dynamic output feedback control law can be designed as where

Conclusions
This paper considers global output feedback control for a class of nonlinear systems with inverse dynamics and L 2 noise.For the inverse dynamics with uncertain supply rates, the reduced-order observer based output feedback controller is constructed, which drives the output of system to zero asymptotically and maintains other closed-loop signals bounded.

Appendix. The proof of Lemma 1
Assuming that Vi−1 satisfies the similar properties to (17), noticing that , there holds Vi −cκψ 1 (y)y 2