A Separation Principle for a Class ofNon Uniformly Completely Observable Systems

Manfredi Maggiore,Member, IEEE,and Kevin M. Passino,Senior Member, IEEE

Abstract

This paper introduces a new approach to output feedback stabilization ofSISO systems which, unlike other techniques foundin the literature, does not use quasi-linear high-gain observers and control input saturation to achieve separation between the statefeedback and observer designs. Rather, we show that by using nonlinear high-gain observers working in state coordinates, togetherwith a dynamic projection algorithm, the same kind of separation principle is achieved for a larger class of systems which arenot uniformly completely observable. By working in state coordinates, this approach avoids using knowledge of the inverse of theobservability mapping to estimate the state of the plant, which is otherwise needed when using high-gain observers to estimatethe output time derivatives.

Index Terms

Nonlinear control, output feedback, separation principle, nonlinear observer.

I. I NTRODUCTION

THE area of nonlinear output feedback control has received much attention after the publication of the work [3], in whichthe authors developed a systematic strategy for the output feedback control of input-output linearizable systems withfull

relative degree, which employed two basic tools: an high-gain observer to estimate the derivatives of the outputs (and hencethe system states in transformed coordinates), and controlinput saturation to isolate the peaking phenomenon of the observerfrom the system states. Essentially the same approach has later been applied in a number of papers by various researchers(see, e.g., [9], [12], [7], [13], [14], [1]) to solve different problems in output feedback control. In most of the papersfoundin the literature, (see, e.g., [3], [9], [12], [13], [7], [14]) the authors consider input-output feedback linearizable systems witheither full relative degree or minimum phase zero dynamics.The work in [21] showed that for nonminimum phase systemsthe problem can be solved by extending the system dynamics with a chain of integrators at the input side. However, the resultscontained there are local. In [19], by putting together thisidea with the approach found in [3], the authors were able to showhow to solve the output feedback stabilization problem for smoothly stabilizable and uniformly completely observable(UCO)systems. The work in [1] unifies these approaches to prove a separation principle for a rather general class of nonlinear systems.The recent work in [16] relaxes the uniformity requirement of the UCO assumption by assuming the existence of one controlinput for which the system is observable. On the other hand, however, [16] requires the observability property to be complete,i.e., to hold on the entire state space. Another feature of that work is the relaxation of the smooth stabilizability assumption,replaced by the notion of asymptotic controllability (which allows for possibly non-smooth stabilizers).

A common feature of the papers mentioned above is theirinput-output variable approach, which entails using the vectorcol(y, y, . . . , y(ny), u, u, . . . , u(nu)) as feedback, for some integersny, nu, wherey andu denote the system output and input,respectively. This in particular implies that, when dealing with systems which are not input-output feedback linearizable, suchapproach requires the explicit knowledge of the inverse of the observability mapping, which in some cases may not be available.

This paper develops a different methodology for output feedback stabilization which is based on astate-variable approachand achieves a separation principle for a class of non-UCO systems, specifically systems that are observable on an open regionof the state space and input space, rather than everywhere. We impose a restriction on the topology of such an “observabilityregion” assuming, among other things, that it contains a sufficiently large simply connected neighborhood of the origin. Themain contributions are the development of a nonlinear observer working in state coordinates (which is proved to be equivalentto the standard high-gain observer in output coordinates),and a dynamic projection operating on the observer dynamicswhicheliminatesthe peaking phenomenon in the observer states, thus avoiding the need to use control input saturation. One of thebenefits of astate-variable approachis that the knowledge of the inverse of the observability mapping is not needed.

It is proved that, provided the observable region satisfies suitable topological properties, the proposed methodologyyieldsclosed-loop stability. In the particular case when the plant is globally stabilizable and UCO, this approach yields semiglobalstability, as in [19], provided a convexity requirement is satisfied. As in [21], [19], [1], our results rely on adding integratorsat the input side of the plant and designing a stabilizing control law for the resultingaugmentedsystem. Thus, a drawback

M. Maggiore is with the Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Rd., Toronto, ON, M5S 3G4,Canada (e-mail: maggiore@control.utoronto.ca). Part of this research was performed when the author was at the Ohio State University.

K. M. Passino is with the Department of Electrical Engineering, The Ohio State University, 2015 Neil Avenue, Columbus, OH 43210 USA (e-mail:k.passino@osu.edu).

This work was supported by NASA Glenn Research Center, GrantNAG3-2084.

Published inIEEE Transactions on Automatic Control, vol. 48, no. 7, 2003, pp. 1122–1133

2

of our approach (as well as the approaches in [21], [19], [1])is that separation is only achieved between the state feedbackcontrol design for theaugmentedsystem and the observer design and not between the state feedback control design for theoriginal system and the observer design.

II. PROBLEM FORMULATION AND ASSUMPTIONS

Consider the following dynamical system,x = f(x, u)

y = h(x, u)(1)

where x ∈ Rn, u, y ∈ R, f and h are known smooth functions, andf(0, 0) = 0. Our control objective is to construct

a stabilizing controller for (1) without the availability of the system statesx. In order to do so, we need an observabilityassumption. Define the observability mapping

ye4=

y...

y(n−1)

= H

(

x, u, . . . , u(nu−1))

=

h(x, u)ϕ1(x, u, u(1))

...ϕn−1

(

x, u, . . . , u(nu−1))

(y(n−1) is then − 1-th derivative) where

ϕ1(x, u, u(1)) =∂h

∂xf(x, u) +

∂h

∂uu(1)

ϕ2(x, u, u(1), u(2)) =∂ϕ1

∂xf(x, u) +

∂ϕ1

∂uu(1) +

∂ϕ1

∂u(1)u(2)

...

ϕn−1 (x, . . .) =∂ϕn−2

∂xf(x, u) +

nu−2∑

j=0

∂ϕn−2

∂u(j)u(j+1)

where0 ≤ nu ≤ n (nu = 0 indicates that there is no dependence onu). In the most general case,ϕi = ϕi(x, u, . . . , u(i)),i = 1, 2, . . . , n− 1. In some cases, however, we may have thatϕi = ϕi(x, u) for all i = 1, . . . , r − 1 and some integerr > 1.This happens in particular when system (1) has a well-definedrelative degreer. Here, we do not require the system to beinput-output feedback linearizable, and hence to possess awell-defined relative degree. In the case of systems with well-definedrelative degree,nu = 0 corresponds to havingr ≥ n, while nu = n corresponds to havingr = 0. Next, augment the systemdynamics withnu integrators on the input side, which corresponds to using a compensator of ordernu. System (1) can berewritten as follows,

x = f(x, z1), z1 = z2, . . . , znu= v (2)

Define the extended state variableX = col(x, z) ∈ Rn+nu , and the associatedextended system

X = fe(X) + gev

y = he(X)(3)

wherefe(X) = col(f(x, z1), z2, . . . , znu, 0), ge = col(0, . . . , 1), andhe(X) = h(x, z1). Now, we are ready to state our first

assumption.Assumption A1 (Observability): System (1) is observable over an open setO ⊂ R

n × Rnu containing the origin, i.e., the

mappingF : O → Y (whereY = F(O)) defined by

Y =

ye

. . .z

= F(X) =

[

H(x, z)z

]

(4)

has a smooth inverseF−1 : Y → O,

F−1(Y ) = F−1(ye, z) =

[

H−1(ye, z)z

]

. (5)

3

Remark 1: In the existing literature, an assumption similar to A1 can be found in [21] and [19]. It is worth stressing, however,that in both works the authors assume the setO to beR

n × Rnu . When that is the case, the system is said to be ([20], [19])

uniformly completely observable(UCO). In many practical applications the system under consideration may not be UCO, butrather be observable in some subset ofR

n × Rnu only, thus preventing the use of most of the output feedback techniques

found in the literature, including the ones found in [21], [19], and [1]. On the other hand in A1 the mappingH, viewed as amapping acting onx parameterized byz, is assumed to be square (i.e., it maps spaces of equal dimension), thus implying thatx can be expressed as a function ofy, its n − 1 time derivatives andz, i.e., x = H−1(col(y, y, . . . , y(n−1)), z). In the works[19], [16], x is allowed to be a function depending on a possibly higher number of derivatives ofy, rather than justn − 1.In our setting, this is equivalent to assuming thatF in A1, rather than being invertible, is just left-invertible. We are currentlyworking on relaxing A1 and replace it by the weaker left-invertibility of F .

Assumption A2 (Stabilizability): The origin of (1) is locally stabilizable (or stabilizable)by a static function ofx, i.e., thereexists a smooth functionu(x) such that the origin is an asymptotically stable (or globally asymptotically stable) equilibriumpoint of x = f(x, u(x)).

Remark 2: Assumption A2 implies that the origin of the extended system(3) is locally stabilizable (or stabilizable) bya function ofX as well. A proof of the local stabilizability property for (3) may be found, e.g., in [17], while its globalcounterpart is a well known consequence of the integrator backstepping lemma (see, e.g., Theorem 9.2.3 in [5] or Corollary2.10 in [10]). Therefore we conclude that for the extended system (3) there exists a smooth controlv(X) such that its origin isasymptotically stable under closed-loop control. LetD be the domain of attraction of the origin of (3), and notice that, whenA2 holds globally,D = R

n × Rnu .

Remark 3: In [21] the authors consider affine systems and use a feedbacklinearizability assumption in place of our A2.Here, we consider the more general class of non-affine systems for which the origin is locally stabilizable (stabilizable). Inthis respect, our assumption A2 relaxes also the stabilizability assumption found in [19], while it is equivalent to Assumption2 in [1].

III. N ONLINEAR OBSERVER: ITS NEED AND STABILITY ANALYSIS

Assumption A2 allows us to design a stabilizing state feedback control v = φ(x, z). In order to perform output feedbackcontrol x should be replaced by its estimate. Many researchers in the past adopted an input-output feedback linearizabilityassumption ([3], [9], [13], [7], [14]) and transformed the system into normal form

πi = πi+1, 1 ≤ i ≤ r − 1

πr = f(π,Π) + g(π,Π)u

Π = Φ(π,Π), Π ∈ Rn−r

y = π1.

(6)

In this framework the problem of output feedback control finds a very natural formulation, as the firstr derivatives ofy areequal to the states of theπ-subsystem (i.e., the linear subsystem). The works [3], [9], [7] solve the output feedback controlproblem for systems with no zero dynamics (i.e.,r = n), so that the firstn − 1 derivatives ofy provide the entire state ofthe system. In the presence of zero dynamics (Π-subsystem), the use of input-output feedback linearizingcontrollers for (6)forces the use of a minimum phase assumption (e.g, [13]) since the states of theΠ-subsystem are made unobservable by suchcontrollers and, hence, cannot be controlled by output feedback. For this reason the output feedback control of nonminimumphase systems has been regarded in the past as a particularlychallenging problem. Researchers who have addressed thisproblem (e.g., [21], [19]) rely on the explicit knowledge ofH−1 in (5), x = H−1 (ye, col(z1, . . . , znu

)), so that estimationof the firstn − 1 derivatives ofy (the vectorye) provides an estimate ofx, x = H−1 (ye, col(z1, . . . , znu

)), since the vectorz, being the state of the controller, is known. Next, to estimate the derivatives ofy, they employ an high-gain observer. Boththe works [21] and [19] (the latter dealing with the larger class of stabilizable systems) rely on the knowledge ofH−1 toprove closed-loop stability. In addition to this, the work in [1] proves that a separation principle holds for a quite general classof nonlinear systems which includes (1) provided thatH−1 is explicitly known and that the system is uniformly completelyobservable. Sometimes however, even if it exists,H−1 cannot be explicitly calculated (see, e.g., the example in Section V) thuslimiting the applicability of existing approaches. Hence,rather than estimatingye and usingH−1(·, ·) to getx, the approachadopted here is to estimatex directly using a nonlinear observer for system (1) and usingthe fact that thez-states are known.

4

The observer has the form1˙x = f(x, z, y)

4= f(x, z1) +

[

∂H(x, z)

∂x

]−1

E−1L [y(t) − y(t)]

y(t) = h(x, z1)

(7)

whereL is a n × 1 vector,E = diag[

ρ, ρ2, . . . , ρn]

, andρ ∈ (0, 1] is a fixed design constant.Notice that (7) does not require any knowledge ofH−1 and has the advantage of operating inx-coordinates. The observability

assumption A1 implies that the Jacobian of the mappingH with respect tox is invertible, and hence the inverse of∂H(x, z)/∂xin (7) is well-defined. In the work [2], the authors used an observer structurally identical to (7) for the more restrictive classof input-output feedback linearizable systems with full relative degree. Here, by modifying the definition of the mapping Hand by introducing a dynamic projection, we considerably relax these conditions by just requiring the general observabilityassumption A1 to hold. Furthermore, we propose a different proof than the one found in [2] which clarifies the relationshipamong (7) and the high-gain observers commonly found in the literature.

Theorem 1 Consider system (2) and assume that A1 is satisfied forO = Rn+nu , the stateX belongs to a positively invariant,

compact setΩ, supt≥0 v(t) < ∞, and that there exists a setΩ, Ω ⊃ Ω, which is positively invariant for(x, z) and such thatye ∈ R

n | (ye, z) ∈ F(Ω) is a compact set. ChooseL = col(l1, . . . , ln) such thatsn + l1sn−1 + . . . + ln is Hurwitz.

Under these conditions and using observer (7), the following two properties hold

(i) Asymptotic stability of the estimation error: There existsρ, 0 < ρ ≤ 1, such that for allx(0) ∈ x ∈ Rn | (x, z) ∈ Ω,

and all ρ ∈ (0, ρ), x(t) → x(t) as t → +∞.(ii) Arbitrarily fast rate of convergence: For each positiveT, ε, there existsρ∗, 0 < ρ∗ ≤ 1, such that for allρ ∈ (0, ρ∗],

‖x(t) − x(t)‖ ≤ ε ∀t ≥ T .

In Section IV we show that, by applying to the vector fieldf a suitable dynamic projection onto a fixed compact set, theexistence of the compact positively invariant setΩ is guaranteed.

Proof: Consider the smooth coordinate transformation

ye = H(x, z)

which maps (1) toye = Acye + Bc [α(ye, z) + β(ye, z)v] , (8)

whereα(·, ·) andβ(·, ·) are smooth functions and the pair(Ac, Bc) is in Brunovsky canonical form. Similarly, it is not difficultto show that the coordinate transformation

ye = H(x, z)

maps the observer dynamics (7) to

˙ye = Acye + Bc[α(ye, z) + β(ye, z)v] + E−1L [ye,1 − ye,1]. (9)

Define the observer error in the new coordinates,ye = ye − ye. Then, the observer error dynamics are given by

˙ye = (Ac − E−1LCc)ye

+ Bc [α(ye, z) + β(ye, z)v − α(ye, z) − β(ye, z)v] ,(10)

whereCc = [1, 0, . . . , 0]. Note that, inye coordinates, the observer (7) is almost identical to the standard high-gain observerused in, e.g., [19] with animportant difference:while α andβ in [19] depend on sat(ye) (where sat(·) denotes the saturationfunction) and hence the nonlinear portion of the observer error dynamics is globally bounded whenY ∈ F(Ω), α and β in(10) do not contain saturation and thus one cannot guaranteeglobal boundedness of the last term in (10). Since our observeroperates inx coordinates, saturationcannot be inducedin ye coordinates and therefore, without any further assumption, wecannot prove stability of the origin of (10). The assumptionconcerning the existence ofΩ addresses precisely this issue sinceit guarantees thatye(t) = H(x(t), z(t)) is contained in a compact set for allt ≥ 0.

Define the coordinate transformation

ν = E ′ye, E ′ 4= diag

[

1

ρn−1,

1

ρn−2, . . . , 1

]

. (11)

1Throughout this section we assume A1 to hold globally, since we are interested in the ideal convergence properties of the state estimates. In the nextsection we will show how to modify the observer equation in order to achieve the same convergence properties when A1 holds over the setO ⊂ R

n ×Rnu .

5

In the new coordinates the observer error dynamics are givenby

˙ν =1

ρ(Ac − LCc)ν

+ Bc [α(ye, z) + β(ye, z)v − α(ye, z) − β(ye, z)v](12)

where, by assumption,Ac − LCc is Hurwitz. Let P be the solution to the Lyapunov equation

P (Ac − LCc) + (Ac − LCc)>P = −I, (13)

and consider the Lyapunov function candidateVo(ν) = ν>P ν. Calculate the time derivative ofVo along theν trajectories:

Vo = − ν>ν

ρ+ 2ν>PBc [α(ye, z) + β(ye, z)v

−α(ye, z) − β(ye, z)v] .

(14)

Since(x(t), z(t)) ∈ Ω for all t ≥ 0, we have that(ye(t), z(t)) ∈ F(Ω) and hence, for allt ≥ 0, ye(t) ∈ ye | (ye, z) ∈ F(Ω)which, by assumption, is a compact set. This, together with the fact that, for allt ≥ 0, X(t) ∈ Ω and supt≥0 v(t) < ∞,implies that there exists a fixed scalarγ > 0, independent ofρ, such that the time derivative ofV0 can be bounded as

Vo ≤ −‖ν‖2

ρ+ 2‖P‖γ‖ye‖‖ν‖ ≤ −‖ν‖2

ρ+ 2‖P‖γ‖ν‖2. (15)

Defining ρ = min1/(2‖P‖γ), 1, we conclude that, for allρ ∈ (0, ρ), the origin of (12) (and hence also the origin of (10))is asymptotically stable which, by the smoothness ofH, implies thatx(t) → x(t) ast → ∞. This concludes the proof of part(i).

As for part (ii), note thatλmin(E ′PE ′) ≥ λmin(E ′)2λmin(P ) = λmin(P ),

sinceλmin(E ′) = 1. Next,λmax(E ′PE ′) ≤ 1/(ρ2(n−1))λmax(P ),

sinceλmax(E ′) = 1/ρ(n−1). Therefore

λmin(P )‖ye‖2 ≤ y>e E ′PE ′ye ≤ 1

ρ2(n−1)λmax(P )‖ye‖2.

Defineε so that‖ye‖ ≤ ε implies that‖x−x‖ ≤ ε (the smoothness ofH−1 guarantees thatε is well-defined). By the inequality

above we have thatVo ≤ ε2λmin(P ) implies that‖ye‖ ≤ ε, andVo(0)4= Vo(ν(0)) ≤ (1/ρ2(n−1))λmax(P )‖ye(0)‖2. Moreover,

from (15)

Vo(t) ≤ −(

1

ρ− 2‖P‖γ

)

‖ν‖2

≤ − 1

λmax(P )

(

1

ρ− 2‖P‖γ

)

Vo(t).

Therefore, by the Comparison Lemma (see, e.g., [22]),Vo(t) satisfies the following inequality

Vo(t) ≤ Vo(0) exp

− 1

λmax(P )

(

1

ρ− 2‖P‖γ

)

t

≤ λmax(P )

ρ2(n−1)‖ye(0)‖2 exp−t/λmax(P )(ρ−1 − 2‖P‖γ)

(16)

which, for sufficiently smallρ, can be written as

Vo(t) ≤a1

ρ2nexp

−a2

ρt

, a1, a2 > 0.

An upper estimate of the timeT such that‖ye(t) − ye(t)‖ ≤ ε (and thus‖x(t) − x(t)‖ ≤ ε) for all t ≥ T , is calculated asfollows

a1

ρ2nexp

−a2

ρt

≤ ε2λmin(P ),

for all t ≥ T = ρ/a2 ln(

a1/(ε2ρ2nλmin(P )))

. Noticing thatT → 0 as ρ → 0, we conclude thatT can be made arbitrarilysmall by choosing a sufficiently smallρ∗, thus concluding the proof of part (ii).

6

Remark 4: Using inequality (16), we find the upper bound for the estimation error inye-coordinates

‖ye‖ ≤√

λmax(P )

λmin(P )ρ−(n−1)‖ye(0)‖

· exp−t/(2λmax(P ))(ρ−1 − 2‖P‖γ).(17)

Hence, during the initial transient,ye(t) may exhibit peaking, and the size of the peak grows larger asρ decreases andthe convergence rate is made faster. Refer to [18] for more details on the peaking phenomenon and to [3] for a study of itsimplications on output feedback control. The analysis in the latter paper shows that a way toisolatethe peaking of the observerestimates from the system states is to saturate the control input outside of the compact set of interest. The same idea hasthenbeen adopted in several other works in the output feedback control literature (see, e.g., [3], [9], [19], [12], [13], [7], [14], [1]).Rather than following this approach, in the next section we will present a new technique toeliminate(rather than just isolate)the peaking phenomenon which allows for the use of the weakerAssumption A1.

F

C

C

F G

G

O

F -1

CF -1( )

OF( )Ωc2

Ωc2( )

N

N

y

z

eN

N

ζ

z

'

-1

'

x

z

ye

C'

z-

z

ζ

z

Fig. 1. The mechanism behind the observer estimates projection.

IV. OUTPUT FEEDBACK STABILIZING CONTROL

Consider system (3), by using assumption A2 and Remark 2 we conclude that there exists a smooth stabilizing controlv = φ(x, z) = φ(X) which makes the origin of (3) an asymptotically stable equilibrium point with domain of attractionD. Bythe converse Lyapunov theorem found in [11], there exists a continuously differentiable functionV defined onD satisfying,for all X ∈ D,

α1(‖X‖) ≤ V (X) ≤ α2(‖X‖) (18)

limX → ∂D

α1(‖X‖) = ∞ (19)

∂V

∂X(fe(X) + ge v) ≤ −α3(‖X‖) (20)

whereαi, i = 1, 2, 3 are classK functions (see [8] for a definition), and∂D stands for the boundary of the setD. Given anyscalarc > 0, define

Ωc4= X ∈ R

n+nu |V ≤ c.Clearly, Ωc ⊂ D for all c > 0 and, from (19),Ωc becomes arbitrarily close toD as c → ∞. Next, the following assumptionis needed.Assumption A3 (Topology of O): Assume that there exists a constantc > 0 and a setC such that

F (Ωc) ⊂ C ⊂ Y (= F (O)) , (21)

whereC has the following properties(i) The boundary ofC, ∂C, is ann− 1 dimensional,C1 submanifold ofRn, i.e., there exists aC1 function g : C → R such

that ∂C = Y ∈ C | g(Y ) = 0, and(∂g/∂Y )> 6= 0 on ∂C.(ii) C z = ye ∈ R

n | (ye, z) ∈ C is convex for allz ∈ Rnu .

(iii) 0 is a regular value ofg(·, z) for each fixedz ∈ Rnu , i.e., for all ye ∈ C z, ∂g

∂ye(ye, z) 6= 0.

(iv)⋃

z∈RnuC z is compact.

7

C1

ye

z

C2 C3 C4

Fig. 2. The domainsC1, C2, C3 violate some of the requirements (i)-(iv) in A3, whileC4 does not.

Remark 5: See Figure 1 for a pictorial representation of condition (21). This assumption requiresin primis that there existsa setC in Y coordinates which contains the image underF of a level setΩc of the Lyapunov functionV and is containedin the image underF of the observable setO. This guarantees in particular thatΩc ⊂ O, and thus, when the state feedbackcontroller is employed, the phase curves leaving fromΩc never exit the observable regionO. Furthermore, it is required thatCpossesses some basic topological properties: It is asked that the boundary ofC be continuously differentiable (part (i)), everyslice of C obtained by holdingz constant atz, C z, is convex (part (ii)), the normal vector to each sliceC z (which is given by[∂g/∂ye(ye, z)]>) does not vanish anywhere on the slice (part (iii)) and, finally, it is asked that the setC is compact in theye

direction (part (iv)). Part (iv) can be replaced by the slightly weaker requirement that⋃

z∈ΩzC z is compact (Ωz is defined inthe next section), with minor changes in the analysis to follow.

To further clarify the topology of the domains under consideration, consider the setsC1 to C4 in Figure 2, corresponding tothe caseye ∈ R

2, z ∈ R. While they all satisfy part (i),C1 does not satisfy part (ii) since its slices alongz are not convex.C2 satisfies part (ii) but violates requirement (iii) because the normal vector to one of the slices has no components in theye

direction.C3 satisfies parts (i)-(iii) but violates (iv) since the area ofits slices grows unbounded asz → ∞. C4 satisfies all therequirements above.

Note that, when the plant is UCO (and thusO = Rn×R

nu ) andF(Rn+nu) = Rn+nu , A3 is always satisfied by a sufficiently

large setC and anyc > 0. In order to see that, pickany c and chooseC to be any cylinderY ∈ Rn+nu | ‖ye‖ ≤ M, where

M > 0, containingF(Ωc). The existence ofC is guaranteed by the fact that the setF(Ωc) is bounded. More generally, the

same holds whenF(Rn+nu) is not all of Rn+nu andY z 4

= F(Rn, z) is convex for allz ∈ Rnu .

Finally, notice that when the plant innot UCO butO = X × Rnu , whereX is an open set which is not all ofRn, and the

origin is globally stabilizable (i.e., A2 holds globally),one can chooseC = D × Rnu , whereD ⊂ R

n is any convex compactset with smooth boundary contained inX and containing the pointH(0, 0) (i.e., the origin inye coordinates). The scalarcis then the largest number such thatF(Ωc) ⊂ C (notice, however, thatc does not need to be known for design purposes).Consequently, in the particular case whennu = 0 (and hence the control input does not affect the mappingH) and A2 holdsglobally, one can chooseC = D, whereD is defined above.

A. Observer Estimates Projection

As we already pointed out in Remark 4, in order to isolate the peaking phenomenon from the system states, the approachgenerally adopted in several papers is to saturate the control input to prevent it from growing above a given threshold. Thistechnique, however, does not eliminate the peak in the observer estimate and, hence, cannot be used to control general systemslike the ones satisfying assumption A1, since even when the system state lies in the observable regionO ⊂ R

n × Rnu , the

observer estimates may enter the unobservable domain where(7) is not well-defined. It appears that in order to deal withsystems that are not completely observable, one has to eliminate the peaking from the observer by guaranteeing its estimatesto be confined in a prespecified observable compact set.

A common procedure used in the adaptive control literature (see [4]) to confine vectors of parameter estimates withina desired convex set is gradient projection. This idea cannot be directly applied to our problem, mainly becausef is notproportional to the gradient of the observer Lyapunov function and, thus, the projection cannot be guaranteed to preserve theconvergence properties of the estimate. Inspired by this idea, however, we propose a way to modify the vector fieldf whichconfinesx to within a prespecified compact set while preserving its convergence properties.

Recall the coordinate transformation defined in (4) and let

yPe = H(xP , z), yP

e = yPe − ye

Y P = F(xP , z) = col(yPe , z),

(22)

8

wherexP is the state of theprojectedobserver defined as2

˙xP =

[

∂H∂xP

]−1

LFH− ΓNye

(Y P )LG

g

Nye(Y P )>ΓNye

(Y P )− ∂H

∂zz

if LG

g ≥ 0 and Y P ∈ ∂Cf(xP , z, y) otherwise

(23)

where

Nye(Y P ) =

[

∂g

∂yPe

(yPe , z)

]>

, Nz(YP ) =

[

∂g

∂z(yP

e , z)

]>

represent theye andz components of the normal vectorN(Y P ) to the boundary ofC at Y P , i.e.,N(Y P ) = col(Nye(Y P ), Nz(Y

P ))(the functiong is defined in A3). Further,L

FH andL

Gg are the Lie derivatives ofyP

e andg(yPe , z) along the vector fields

F = col(f , z2, . . . , znu, v) and G = L

FF , respectively, i.e.,

LFH =

∂H∂xP

f(xP , z, y) +∂H∂z

z

LG

g =∂g

∂yPe

LFH +

∂g

∂zz = Nye

(Y P )>LFH + Nz(Y

P )>z,

andΓ = (SE ′)−1(SE ′)−>, whereS = S> denotes the matrix square root ofP (defined in (13)).The dynamic projection (23) is well-defined since A3, part (iii), guarantees thatNye

does not vanish (see also Remark 5).The following lemma shows that (23) guarantees boundednessand preserves convergence ofxP .

Lemma 1 If A3 holds and (23) is used:(i) Positive invariance ofF−1(C): if (xP (0), z(0)) ∈ F−1(C), then(xP (t), z(t)) ∈ F−1(C) for all t ≥ 0.If, in addition, (x(t), z(t)) ∈ Ωc for all t ≥ 0 and the assumptions of Theorem 1 are satisfied, then the following propertyholds for the integral curves of the projected observer dynamics (7), (23)(ii) Preservation of original convergence characteristics:properties (i) and (ii) established by Theorem 1 remain valid for xP .

Proof: We begin by introducing another coordinate transformation, ζ = SE ′ye, (similarly, let ζP = SE ′yPe , ζP =

SE ′yPe ), letting G = diag [SE ′, Inu×nu

], and lettingC′ be the image of the setC under the linear mapG, i.e., C′ 4=

(ζ, z) ∈ Rn+nu | G−1col(ζ, z) ∈ C

. Let N ′ζ(ζ, z), N ′

z(ζ, z) be theζ andz components of the normal vector to the boundaryof C′. The reader may refer to Figure 1 for a pictorial representation of the sets under consideration. In order to prove part (i) ofthe Lemma, we have to show that the projection (23) rendersC a positively invariant set forY P . The coordinate transformation(22) maps (23) to

˙yeP =

d

dt

H(xP , z)

=

[

∂H∂xP

˙xP +∂H∂z

z

]

(24)

=

LFH− Γ

Nye(Y P )L

Gg

Nye(Y P )>ΓNye

(Y P )

if LG

g ≥ 0 and Y P ∈ ∂CL

FH otherwise

(25)

In order to relateN ′ζ(ζ

P , z) andN ′z(ζ

P , z) to Nye(yP

e , z) andN ′z(ζ

P , z), recall from A3 that the boundary ofC is expressedas the set∂C = Y ∈ R

n+nu | g(Y ) = 0 and hence the boundary ofC′ is the set∂C′ = ζ ∈ Rn | g((SE ′)−1ζ, z) = 0.

From this definition we find the expression ofN ′ζ andN ′

z as

N ′ζ(ζ

P , z) = (SE ′)−>[∂g(Y P )/∂yPe ]> = (SE ′)−>Nye

(Y P )

N ′z(ζ

P , z) = Nz(YP ).

The expression of the projection (23) inζ coordinates is found by noting that

˙ζP = SE ′ ˙ye

P

=

SE ′LFH− (SE ′)−>

Nye

(

N>ye

LFH + N>

z z)

N>ye

ΓNye

if LG

g ≥ 0 and Y P ∈ ∂CSE ′L

FH otherwise

(26)

2The projection defined in (23) is discontinuous in the variable ye, therefore raising the issue of the existence and uniqueness of its solutions. We refer thereader to Remark 7, were this issue is addressed and a solutionis proposed.

9

and then substitutingN ′ζ = (SE ′)−>Nye

, N ′z = Nz, andL

FH = (SE ′)−1(L

FSE ′H), to find that

˙ζP =

(LFSE ′H) −

N ′ζ

(

N ′ζ>(L

FSE ′H) + N ′

z>z

)

N ′ζ>N ′

ζ

if N ′ζ>(L

FSE ′H) + N ′

z>z ≥ 0 and (ζP , z) ∈ ∂C′

(LFSE ′H) otherwise

(27)

Next, we show that the boundary of the domainC′ is positively invariant with respect to (27). In order to do that, considerthe continuously differentiable functionVC′ = g

(

(SE ′)−1ζ, z)

and calculate its time derivative along the vector field of (27)when (ζP , z) ∈ ∂C′,

VC′ = N ′ζ(ζ

P , z)>˙ζP + N ′

z(ζP , z)z

= N ′ζ>(L

FSE ′H)−

N ′ζ>N ′

ζ

(

N ′ζ>(L

FSE ′H) + N ′

z>z

)

N ′ζ>N ′

ζ

+ N ′z z = 0

SinceC′ is positively invariant, so isC and, therefore,F−1(C).The proof of part (ii) is based on the knowledge of a Lyapunov function for the observer inν coordinates (see (11)). Letting

ζ = Sν, we have that, in new coordinates,Vo = ν>P ν = (ν>S)(Sν) = ζ>ζ. Now let V Po = ζP

>ζP be a Lyapunov function

candidate for the projected observer error dynamics in transformed coordinates and recall that, by assumption,F(Ωc) ⊂ C,and thus(x, z) ∈ Ωc. The latter fact implies that(ye, z) ∈ C or, what is the same,(ζ, z) ∈ C′. From (27), when(ζP , z) isin the interior ofC′, or (ζP , z) is on the boundary ofC′ and N ′

ζ>(L

FSE ′H) + N ′

z>z < 0 (i.e., theunprojectedupdate is

pointed to the interior ofC′), we have thatV Po = Vo < 0. Let us now consider all remaining cases, i.e.,(ζP , z) ∈ ∂C and

N ′ζ>(L

FSE ′H) + N ′

z>z ≥ 0,

V Po = 2ζP ˙

ζP = 2ζP > ˙ζP = 2ζP >

[

(LFSE ′H) − ζ

−p(ζP , (LF

SE ′H), z, z)N ′ζ(ζ

P , z)]

= Vo(ζP ) − 2pζP >N ′

ζ

(28)

where

p(ζP , (LFSE ′H), z, z) =

N ′ζ>(L

FSE ′H) + N ′

z>z

N ′ζ>N ′

ζ

is nonnegative since, by assumption,N ′

ζ>(L

FSE ′H) + N ′

z>z ≥ 0.

Using the fact that(ζ, z) ∈ C′ and that(ζP , z) lies on the boundary ofC′, we have that the difference vector(ζP − ζ, 0) pointsoutside ofC′ or, equivalently,ζP − ζ points outside of the sliceC′z = ζ ∈ R

n | (ζ, z) ∈ C′. Using the definition ofC′, wehave that the setC′z is the image of the convex compact setCz, defined in A3, under the linear mapSE ′ and is thereforecompact and convex as well. Combining these two facts we havethat ζP >N ′

ζ ≥ 0, thus proving thatV Po ≤ Vo < 0, which

concludes the proof of part (ii).

Remark 6: Part (i) of Lemma 1 implies that, with dynamic projection, the requirement in Theorem 1 that there exists apositively invariant setΩ for (x, z) is satisfied withΩ = F−1(C). To see that, note that

C ⊂(

⋃

z∈Rnu

Cz)

× Rnu

and thusye ∈ R

n | (ye, z) ∈ F(Ω) = ye ∈ Rn | (ye, z) ∈ C

⊂⋃

z∈Rnu

Cz

which, by part (iv) in A3, is a compact set.

Remark 7: In order to avoid the discontinuity in the right hand side of (23) one can employ the smooth projection ideaintroduced in [15]. In this case, condition (21) in A3 shouldbe replaced by the following

F(Ωc) ⊂ C ⊂ C ⊂ Y (29)

10

where C and C satisfy properties (i)-(iv) in A3 and∂C = Y ∈ C | g(Y ) = 0, ∂C = Y ∈ C | g(Y ) = 1. The ratioNye

LG

g/Nye

>ΓNyein (23) should be multiplied byg, and a slight modification of Lemma 1 would show that, by applying

the smooth projection tof , the setF−1(C) (rather than the setC) is made positively invariant.

Remark 8: From the proof of Lemma 1 we conclude that (23) confines(xP , z) within the setF−1(C) which is, in general,unknown since we do not knowH−1, and is generallynot convex (see Figure 1). It is interesting to note that applying astandard gradient projection forx over an arbitrary convex domain does not necessarily preserve the convergence result (ii) inTheorem 1.

B. Closed-Loop Stability

To perform output feedback control we replace the state feedback law v = φ(x, z) with v = φ(xP , z) which, by thesmoothness ofφ and the fact thatxP is guaranteed to belong toH−1(C), is bounded provided thatz is confined to within acompact set. In the following we will show thatv makes the origin of (3) asymptotically stable and thatΩc is contained in itsregion of attraction, for all0 < c < c 3. The proof is divided in three steps:

1) (Lemma 2).Invariance ofΩc and uniform ultimate boundedness:Using the arbitrarily fast rate of convergence of theobserver (see part (ii) in Theorem 1), we show that any phase curve leaving fromΩc cannot exit the setΩc andconverges in finite time to an arbitrarily small neighborhood of the origin. Here, Lemma 1 plays an important role, inthat it guarantees that the peaking phenomenon is eliminated and thus it does not affectv, allowing us to use the sameidea found in [3] to prove stability.

2) (Lemma 3).Asymptotic stability of the origin:By using Lemma 2 and the exponential stability of the observer estimate,we prove that the origin of the closed-loop system is asymptotically stable.

3) (Theorem 2).Closed-loop stability:Finally, by putting together the results of Lemma 2 and 3, we conclude the closed-loopstability proof.

The arguments of the proofs are relatively standard (see, e.g., [1], [19]) and so are sketched.

Lemma 2 For all c ∈ (0, c), ε > 0 and µ > 1 such that

dε = α2 α−13 (µA γ ε) < c,

there exists a numberρ ∈ (0, 1) such that, for allρ ∈ (0, ρ], for all (X(0), xP (0)) ∈ A, where

A =

(X, xP ) ∈ R2n+nu |X ∈ Ωc, (x

P , z) ∈ F−1(C)

,

the phase curvesX(t) of the closed-loop system remain confined inΩc, the setΩdε⊂ Ωc is positively invariant, and is reached

in finite time.

Sketch of the proof:Since (xP (0), z(0)) ∈ F−1(C), by Lemma 1, part(i),(xP (t), z(t)) ∈ F−1(C) for all t ≥ 0. LetΩz

c = z ∈ Rnu | (x, z) ∈ Ωc and notice that

X ∈ Ωc and (xP , z) ∈ F−1(C)

=⇒ xP ∈(

⋃

z∈Ωz

c

Cz

)

× Ωzc ,

which is a compact set independent ofρ. Hence, there exists a bounded positive real numberD independent ofρ such that,for all X ∈ Ωc and all (xP , z) ∈ F−1(C),

‖fe(X) + geφ(xP , z)‖ ≤ D.

Therefore, for allt such thatX(t) ∈ Ωc, ‖X(t) − X(0)‖ ≤ Dt, proving that the exit time from the setΩc has a positivelower bound,TX(0)

1 , independent ofρ. Further, lettingd = dist(Ωc,Ωc) = infX1∈Ωc,X2∈Ωc‖X1 − X2‖, it is readily seen that

infX(0)∈Ωc(T

X(0)1 ) = d/D

4= T1. We have thus shown that for any phase curveX(t) leaving from the setΩc there exists a

uniform lower boundT1 to the exit time fromΩc which is independent ofρ. ChooseT0 ∈ (0, T1), let A = supX∈Ωc‖∂V/∂X‖,

and γ be the Lipschtiz constant ofφ over the compact set

F−1

(

⋃

z∈Ωz

c

Cz × Ωzc

)

.

3Recall thatc is a positive constant satisfying A3 and hence its size is constrained by the topology of the observable setO.

11

By A3, X(t) ∈ Ωc ⊂ Ωc implies X(t) ∈ F−1(C) and thus from Theorem 1, part (ii), and Lemma 1 there existsρ ∈ (0, 1)such that, for allρ ∈ (0, ρ], ‖x(t)− xP (t)‖ ≤ ε for all t ∈ [T0, T1). Now using the functionV defined in (18)-(20), a standardLyapunov analysis shows that, for allt ∈ [T0, T1),

V ≥ dε =⇒ V ≤ −(µ − 1)Aγε,

which, sincedε < c, implies thatΩc is positively invariant (T1 = ∞), and that the phase curvesX(t) enter and stay in the setΩε in finite time.

Remark 9: The use of the projection for the observer estimate plays a crucial role in the proof of Lemma 2. Asρ is madesmaller, the observer peak may grow larger, thus generatinga large control input, which in turn might drive the system statesX outside ofΩc in shorter time. The dynamic projection makes sure that the exit time T1 is independent ofρ, thus allowingone to chooseε independently ofT1.

Lemma 3 Under the assumptions of Lemma 2, there exists a positive scalar ε∗ such that for all ε ∈ (0, ε∗] the origin(X,x − xP ) = (0, 0) is asymptotically stable.

Sketch of the proof:Let xP = x− xP . By (17) and Lemma 1, the origin of theye − yPe dynamics is exponentially stable.

Recalling thatxP = H−1(ye, z)−H−1(yPe , z), from the smoothness ofH we conclude that the origin of thexP dynamics is

exponentially stable as well. By the converse Lyapunov theorem there exists a Lyapunov functionV ′o(xP ), a positive number

r, and positive constantsk1, k2, k3 such that, for allxP ∈ Br = xP ∈ Rn | ‖xP ‖ ≤ r,

k1‖xP ‖2 ≤ V ′o ≤ k2‖xP ‖2

V ′o ≤ −k3‖xP ‖2.

Chooseε∗ ∈ (0, r) such thatdε∗ = α2 α−13 (µA γ ε∗) < c. For any ε ∈ (0, ε∗], chooseρ as in Lemma 2 and pick any

ρ ∈ (0, ρ]. Then, the following Lyapunov function

Vc(X, xP ) = V (X) + λ√

V ′o(xP ), λ >

2√

c2γA

c3

readily allows one to conclude that the origin(X, xP ) = (0, 0) is asymptotically stable.We are now ready to state the following closed-loop stability theorem.

Theorem 2 Suppose that assumptions A1, A2, and A3 are satisfied. Then, for the closed-loop system (3), (7), (23), withcontrol law v = φ(xP , z), there exists a scalarρ∗ ∈ (0, 1) such that, for allρ ∈ (0, ρ∗], the setA is contained in the regionof attraction of the origin(X, xP ) = (0, 0).

Sketch of the proof:By Lemma 3, there existsε∗ > 0 such that, for allε ∈ (0, ε∗], the origin (X, xP ) = (0, 0) isasymptotically stable. Let∆ε be its domain of attraction. For any(X(0), xP (0)) ∈ A, use Lemma 2 to findρ ∈ (0, 1) so thatfor all ρ ∈ (0, ρ] the phase curves of the closed-loop system enter∆ε in finite time.

Remark 10: Theorem 2 proves regional stability of the closed-loop system, since given an observability domainO, andprovided A3 is satisfied, the control lawv, together with (7) and (23), make the compact setA an estimate of the domain ofattraction for the origin of the closed-loop system. The difference between Theorem 2 and a local stability result lies in thefact that here the domain of attraction forx is at least as large asΩc and not restricted to be a smallunknownneighborhoodof the origin. Further, the size ofΩc is independent ofρ and thus the domain of attractiondoes notshrink as the rate ofconvergence of the observer is made faster (that is, whenρ → 0). The next corollary gives conditions for the recovery of thedomain of attraction of the state feedback controller by output feedback.

Corollary 1 Assume that A1 is satisfied onO = Rn+nu , Y z = F(Rn, z) is convex for allz ∈ R

nu , and A2 holds. Then,givenany setD′ ⊂ int(D), there exists a scalarρ∗ ∈ (0, 1) such that, for allρ ∈ (0, ρ∗], the setD′ is contained in the regionof attraction of the originX = 0. Moreover, ifD = R

n+nu , the origin X = 0 is semiglobally asymptotically stable.

Proof: By property (19) of the Lyapunov functionV (X), given any setD′ ⊂ int(D), there existsc > 0 such thatD′ ⊂ Ωc.By Remark 5, ifO = R

n+nu , A3 is satisfied byany c ∈ (0,∞) and a sufficiently largeC. Choosec > c. By Theorem 2we conclude that there existsρ∗ ∈ (0, 1) such that, for allρ ∈ (0, ρ∗], the origin of the closed-loop system is asymptoticallystable and the setA =

(X, xP ) ∈ R2n+nu |X ∈ Ωc, (x

P , z) ∈ F−1(C)

is contained in its domain of attraction. Thus, inparticular,D′ is contained in the domain of attraction ofX = 0. If D = R

n+nu thenD′ can be chosen to be any compact setand thus the originX = 0 is semiglobally asymptotically stable.

Remark 11: A drawback of the result in Theorem 2, shared by the works in [21], [19], [1], is that separation is achievedbetween the observer design and the state feedback control design for theaugmentedsystem (3). In order to avoid increased

12

f(x, z, y) =

[

x2

(1 + x1) exp

(x1)2

+ z1 − 1

]

+

[

0 2(x2 − 1)2(x2 − 1) exp

(x1)2 [

2(x1)2 + 2x1 + 1

]

2[

(1 + x1) exp

(x1)2

+ z1 − 1]

]−1

E−1L[y − (x2 − 1)2]

(35)

complexity of the controller, it would be more desirable to achieve separation between the observer design and the statefeedback control design for theoriginal system (1).

Remark 12: As mentioned in Remark 5, if the plant is UCO (and henceO = Rn × R

nu ) andY z = F(Rn, z) is a convexset for all z ∈ R

nu , assumption A3 is automatically satisfied by a sufficiently large cylindrical setC. Even in this case, ifH−1

is not explicitly known and one wants to directly estimate the state of the plant, one should employ the dynamic projection(23) since the standard saturation used, e.g., in [3], [19] can only be applied to a high-gain observer inye coordinates. Clearly,the only instance when dynamic projection can be replaced bysaturation of the observer estimates is when the observabilitymapping is the identity.

V. EXAMPLE

Consider the following input-output linearizable system:

x1 = x2

x2 = (1 + x1) exp(x21) + u − 1

y = (x2 − 1)2.

(30)

The control input appears in the first derivative of the output:

y = 2(x2 − 1)(1 + x1) exp(x21) + 2(x2 − 1)(u − 1).

Notice, however, that the coefficient multiplyingu vanishes whenx2 = 1, and hence system (30) does not have a well-definedrelative degree everywhere. Sinceu appears iny, we have thatnu = 1, therefore we add one integrator at the input side,

z1 = v, u = z1. (31)

The mappingF is given by

Y =

yyz1

= F(x, z1) =

[

H(x, z1)z1

]

=

(x2 − 1)2

2(x2 − 1)[(1 + x1) exp(x21) + (z1 − 1)]

z1

.

(32)

The first equation in (32) is invertible for allx2 < 1, and its inverse is given byx2 = 1−√y. Substitutingx2 into the second

equation in (32) and isolating the term inx1, we get

(1 + x1) exp(x21) =

y + 2√

y(z1 − 1)

−2√

y. (33)

Since(1 + x1) exp(x21) is a strictly increasing function, it follows that (33) is invertible for all x1 ∈ R, however, an analytical

solution to this equation cannot be found. In conclusion, Assumption A1 is satisfied on the domainO = x ∈ R2 |x2 < 1×R,

but an explicit inverse(x, z1) = F−1(ye, z1) is not known. The fact that system (30) is not UCO, together with the non-existence of an explicit inverse to (32), prevents the application of the output feedback control approaches in [3], [21], [9],[19], [12], [13], [7], [14], [1].

To find a stabilizing state feedback controller, note that the extended system (30), (31) can be feedback linearized by letting

x3 = (1 + x1) exp(x21) + z1 − 1 and rewriting the system in new coordinatesxe

4= col(x1, x2, x3):

x1 = x2

x2 = x3

x3 = x2 exp(x21)(2x

21 + 2x1 + 1) + v.

(34)

Choosev = −x2 exp(x21)(2x

21 + 2x1 + 1) − Kxe, where K = [1, 3, 3], so that the closed-loop system becomesxe =

(Ac − BcK)xe with poles placed at−1. Then, the originxe = 0 is a globally asymptotically equilibrium point of (34), and

13

Assumption A2 is satisfied withD = R3. Let P be the solution of the Lyapunov equation associated toAc − BcK, so that a

Lyapunov function for system (34) isV = x>e P xe, and any setΩc

4=

xe ∈ R3 |V (xe) ≤ c

, with c > 0, is contained in theregion of attraction for the origin.

Next, we seek to find a setC satisfying Assumption A3. To this end, notice that

Y = F(O) = F(

x ∈ R2, z1 ∈ R |x2 < 1

)

= R+ − 0 × R × R.

Next recall from Remark 5 that, since A2 holds globally,C can be chosen to be the cylinderD×R, whereD is any compactconvex set in the upper half planeR+ − 0 × R containing the pointH(0, 0) = (1, 0). For the sake of simplicity chooseDto be the disk of radiusω < 1 centered at(1, 0), so thatC = Y ∈ R

3 | (Y1 − 1)2 + Y 22 < ω2, andC ⊂ Y, as depicted in

Figure 3. We stress that our choice is quite conservative andis made exclusively for the sake of illustration. OnceC has been

0

F (O )

Unobservable boundary

1 ω

C

Fig. 3. The projection domainC.

chosen, the control design is complete and the output feedback controller is given by

z1 = − xP2 exp

(xP1 )2

[2(xP1 )2 + 2xP

1 + 1] − xP1 − 3xP

2

− 3[

(1 + xP1 ) exp

(xP1 )2

+ z1 − 1]

,(36)

wherexP (t) is the solution of (23) withf(x, z, y) defined in (35) and

∂H∂z

=

[

02(xP

2 − 1)

]

, Nye(yP

e , z) = yPe −

[

10

]

,Nz(yPe , z) = 0,

yPe =

[

(xP2 − 1)2

2(xP2 − 1)[(1 + xP

1 ) exp((xP1 )2) + (z1 − 1)]

]

LFH =

∂H∂xP

f(xP , z, y) +∂H∂z

z.

(37)

Using controller (36), Theorem 2 guarantees that the originxe = 0 of the closed-loop system is asymptotically stable and itprovides an estimate of its domain of attraction. Specifically, given any positive scalarc < c, there existsρ∗ > 0 such thatΩc

14

is contained in the domain of attraction for all0 < ρ < ρ∗. In what follows we will find the setΩc satisfying A3. Recallingthat, inxe-coordinates,Ωc is expressed asxe ∈ R

2 |xe>P xe, we have thatxe ∈ Ωc implies |xi| ≤ (c/λmin(P )), i = 1, 2, 3,

and henceΩc ⊂ Ξ4= xe ∈ R

3 | |xi| ≤ (c/λmin(P )), i = 1, 2, 3. Now let c < λmin(P ) and note that, for allxe ∈ ye,(

1 − c

λmin(P )

)2

≤ |x2 − 1|2 ≤(

1 +c

λmin(P )

)2

|2(x2 − 1)x3| ≤ 2

(

1 +c

λmin(P )

)

c

λmin(P ).

Next, we seek to findc such thatF(Ωc) ⊂ C. Using the inequalities above we have that ifc = minc∗, λmin(P ), wherec∗

is the largest scalar satisfying[

(

1 +c∗

λmin(P )

)2

− 1

]2

+ 4

(

1 +c∗

λmin(P )

)2 (

c2

λmin(P )

)2

< ω2,

(38)

(note thatc∗ > 0 satisfying the inequality above always exists) thenF(Ωc) ⊂ F(Ξ) ⊂ C, and hencec satisfies A3.For our simulations we chooseω = 0.9 and, from (38), we getc = 0.06. The initial condition of the extended system is

set tox1(0) = 0.05, x2(0) = 0.07, x3(0) = 0.1 (or z1(0) = 0.0474), which is contained insideΩc so that Theorem 2 can beapplied. Finally, we choose the observer gainL to be col(4, 4), so that its associated polynomial is Hurwitz with both polesplaced at−2. We present four different situations to illustrate four features of our output feedback controller:

1) Arbitrary fast rate of convergence of the observer. Figure 4 shows the evolution of the integral curveX(t), as wellas the control inputv, for ρ = 0.2 andρ = 0.05. The convergence in the latter case is faster, as predicted by Theorem1 (see Remark 4).

2) Observer estimate projection. Figure 5 shows the evolution ofx and v for ρ = 10−3 with and without projection.The dynamic projection successfully eliminates the peak inthe observer states, thus yielding a bounded control input,aspredicted by the result of Lemma 1. Figure 6 shows that the phase curveY (t) = F(X(t)) is contained within the setCfor all t ≥ 0, confirming the result of Lemma 1. In particular, Figure 6 shows the operation of the projection when thephase curve of the observer (inY coordinates) hits the boundary ofC: it forces Y P (t) to “slide” along the boundaryof C and preserves its convergence characteristics. This is equivalent, in thex domain, to saying that the phase curve(xP (t), z1(t)) slides along the boundary ofF−1(C) and xP (t) converges tox(t).

3) Observer estimate projection and closed-loop stability. In Figure 7 a phase plane plot forx(t) is shown with andwithout observer projection whenρ = 10−4. The small value ofρ generates a significant peak which, if projection isnot employed, drives the phase curve of the output feedback system away form that of the state feedback system and,in general, may drive the system to instability (see Remark 9). On the other hand, using dynamic projection, the phasecurve of the output feedback system is almost indistinguishable from the phase curve of the state feedback system.

4) Trajectory recovery. The evolution of the phase curveX(t) for decreasing values ofρ, in Figure 8, shows that thephase curve of the output feedback system approaches the phase curve of the state feedback one asρ → 0 (see Remark11).

ACKNOWLEDGMENTS

The authors would like to thank Mireille Broucke for fruitful discussions concerning the dynamic projection.

REFERENCES

[1] A. Atassi and H. Khalil, “A separation principle for the stabilization of a class of nonlinear systems,”IEEE Transactions on Automatic Control, vol. 44,no. 9, pp. 1672–1687, September 1999.

[2] G. Ciccarella, M. Dalla Mora, and A. Germani, “A Luenberger-like observer for nonlinear systems,”International Journal of Control, vol. 57, no. 3,pp. 537–556, 1993.

[3] F. Esfandiari and H. Khalil, “Output feedback stabilization of fully linearizable systems,”International Journal of Control, vol. 56, no. 5, pp. 1007–1037,1992.

[4] P. Ioannou and J. Sun,Stable and Robust Adaptive Control. Englewood Cliffs, NJ: Prentice-Hall, 1995.[5] A. Isidori, Nonlinear Control Systems, 3rd ed. London: Springer-Verlag, 1995.[6] A. Isidori, A. R. Teel, and L. Praly, “A note on the problemof semiglobal practical stabilization of uncertain nonlinear systems via dynamic output

feedback,”Systems & Control Letters, vol. 39, pp. 165–179, 2000.[7] M. Jankovic, “Adaptive output feedback control of non-linear feedback linearizable systems,”International Journal of Adaptive Control and Signal

Processing, vol. 10, pp. 1–18, 1996.[8] H. Khalil, Nonlinear Systems, 2nd ed. NJ: Prentice-Hall, 1996.[9] H. Khalil and F. Esfandiari, “Semiglobal stabilization of a class of nonlinear systems using output feedback,”IEEE Transactions on Automatic Control

, vol. 38, no. 9, pp. 1412–1415, 1993.[10] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. NY: John Wiley & Sons, Inc., 1995.

15

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

x1 and its estimate

t0 2 4 6 8 10

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

x2 and its estimate

t

0 2 4 6 8 10-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Integrator state z1

t0 2 4 6 8 10

-1.5

-1

-0.5

0

0.5Control input

ρ = 0.2.

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x1 and its estimate

t0 2 4 6 8 10

-0.04

-0.02

0

0.02

0.04

0.06

0.08

x2 and its estimate

t

0 2 4 6 8 10-0.2

-0.15

-0.1

-0.05

0

0.05

Integrator state z1

t0 2 4 6 8 10

-2.5

-2

-1.5

-1

-0.5

0

0.5Control input

ρ = 0.05.

Fig. 4. Integral curve of the closed-loop system under output feedback.

[11] J. Kurzweil, “On the inversion of Ljapunov’s second theorem on stability of motion,”American Mathematical Society Translations,Series 2, vol. 24,pp. 19–77, 1956.

[12] Z. Lin and A. Saberi, “Robust semi-global stabilizationof minimum-phase input-output linearizable systems via partial state and output feedback,”IEEETransactions on Automatic Control, vol. 40, no. 6, pp. 1029–1041, 1995.

[13] N. Mahmoud and H. Khalil, “Asymptotic regulation of minimumphase nonlinear systems using output feedback,”IEEE Trans. on Automatic Control,vol. 41, no. 10, pp. 1402–1412, 1996.

[14] ——, “Robust control for a nonlinear servomechanism problem,” International Journal of Control, vol. 66, no. 6, pp. 779–802, 1997.[15] J. Pomet and L. Praly, “Adaptive nonlinear regulation: Estimation from the Lyapunov equation,”IEEE Transactions on Automatic Control, vol. 37,

no. 6, pp. 729–740, 1992.[16] H. Shim and A. R. Teel, “Asymptotic controllability implysemiglobal practical asymptotic stabilizability by sampled-data output feedback,” preprint.[17] E. D. Sontag, “Remarks on stabilization and input to state stability,” in Proceedings of the IEEE Conference on Decision and Control, Tampa, FL,

December 1989, pp. 1376–1378.[18] J. H. Sussmann and P. V. Kokotovic, “The peaking phenomenon and the global stabilization of nonlinear systems,”IEEE Transactions on Automatic

Control , vol. 36, no. 4, pp. 424–440, 1991.[19] A. Teel and L. Praly, “Global stabilizability and observability imply semi-global stabilizability by output feedback,” Systems & Control Letters, vol. 22,

pp. 313–325, 1994.[20] ——, “Tools for the semiglobal stabilization by partial state and output feedback,”SIAM J. Control Optim., vol. 33, no. 5, pp. 1443–1488, 1995.[21] A. Tornambe, “Output feedback stabilization of a class of non-minimum phase nonlinear systems,”Systems & Control Letters, vol. 19, pp. 193–204,

1992.[22] T. Yoshizawa,Stability Theory by Lyapunov’s Second Method. The Mathematical Society of Japan, Tokyo, 1966.

16

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.5

1

1.5

2

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020

0.1

0.2

0.3

0.4

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02-300

-200

-100

0

without projectionwith projection

x

ˆ

x

ˆ

u

Fig. 5. Observer states during the initial peaking phase with and without projection,ρ = 10−3.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-2

0

2

CCCC

Y1

Y2

Y3

Fig. 6. Phase curve of the observer with dynamic projection inthe transformed domainY = F(X).

PLACEPHOTOHERE

Manfredi Maggiore Manfredi Maggiore (M’00) received the “Laurea” degree in Electronic Engineering in 1996 from the Universityof Genoa, Italy, and the PhD degree in Electrical Engineering from the Ohio State University, USA, in 2000. He is currentlyAssistant Professor in the Edward S. Rogers Sr. Department ofElectrical and Computer Engineering, University of Toronto, Canada.His research interests include various aspects of nonlinear control, among them output feedback control, output tracking, and nonlinearestimation.

17

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

x1

x2

state feedbackoutput feedback without projectionoutput feedback with projection

Fig. 7. Phase curvex(t) with and without projection,ρ = 10−3.

0

0.02

0.04

0.06

0.08

0.1

0.12 -0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

-0.2

-0.1

0

0.1

x2

x1

z1

state feedbackoutput feedback: ρ=0.2output feedback: ρ=0.01output feedback: ρ=0.001

Fig. 8. Phase curveX(t) of the closed-loop system for decreasing values ofρ.

PLACEPHOTOHERE

Kevin M. Passino Kevin M. Passino (S’79, M’90, SM’96) received his Ph.D. in Electrical Engineering from the University of NotreDame in 1989. He has worked on control systems research at Magnavox Electronic Systems Co. and McDonnell Aircraft Co. Hespent a year at Notre Dame as a Visiting Assistant Professor and is currently Professor of Electrical Engineering at The Ohio StateUniversity. Also, he is the Director of the OSU Collaborative Center of Control Science that is funded by AFRL/VA and AFOSR.He has served as the Vice President of Technical Activities of the IEEE Control Systems Society (CSS); was an elected member ofthe IEEE Control Systems Society Board of Governors; was Chair for the IEEE CSS Technical Committee on Intelligent Control;has been an Associate Editor for the IEEE Transactions on Automatic Control and also for the IEEE Transactions on Fuzzy Systems;served as the Guest Editor for the 1993 IEEE Control Systems Magazine Special Issue on Intelligent Control; and a Guest Editorfor a special track of papers on Intelligent Control for IEEEExpert Magazine in 1996; and was on the Editorial Board of theInt.Journal for Engineering Applications of Artificial Intelligence. He was a Program Chairman for the 8th IEEE Int. Symp. on IntelligentControl, 1993 and was the General Chair for the 11th IEEE Int.Symp. on Intelligent Control. He was the Program Chair of the 2001

IEEE Conf. on Decision and Control. He is currently a Distinguished Lecturer for the IEEE Control Systems Society. He is co-editor (with P.J. Antsaklis)of the book ”An Introduction to Intelligent and Autonomous Control,” Kluwer Academic Press, 1993; co-author (with S. Yurkovich) of the book ”FuzzyControl,” Addison Wesley Longman Pub., 1998; co-author (with K.L. Burgess) of the book ”Stability Analysis of Discrete Event Systems,” John Wiley andSons, 1998; co-author (with V. Gazi, M.L. Moore, W. Shackleford, F. Proctor, and J.S. Albus) of the book ”The RCS Handbook: Tools for Real Time ControlSystems Software Development”, John Wiley and Sons, NY, 2001;co-author (with J.T. Spooner, M. Maggiore, R. Ordonez) of the book ”Stable AdaptiveControl and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques,” John Wiley and Sons, NY, 2002.