A Separation Principle for a Class of Non

A Separation Principle for a Class of Non 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 of SISO systems which, unlike other techniques found in the literature, does not use quasi-linear high-gain observers and control input saturation to achieve separation between the state feedback and observer designs. Rather, we show that by using nonlinear high-gain observers working in state coordinates, together with a dynamic projection algorithm, the same kind of separation principle is achieved for a larger class of systems which are not uniformly completely observable. By working in state coordinates, this approach avoids using knowledge of the inverse of the observability mapping to estimate the state of the plant, which is otherwise needed when using high-gain observers to estimate the output time derivatives. Index Terms Nonlinear control, output feedback, separation principle, nonlinear observer.

I. I NTRODUCTION

T

HE area of nonlinear output feedback control has received much attention after the publication of the work [3], in which the authors developed a systematic strategy for the output feedback control of input-output linearizable systems with full relative degree, which employed two basic tools: an high-gain observer to estimate the derivatives of the outputs (and hence the system states in transformed coordinates), and control input saturation to isolate the peaking phenomenon of the observer from 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 papers found in the literature, (see, e.g., [3], [9], [12], [13], [7], [14]) the authors consider input-output feedback linearizable 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 with a chain of integrators at the input side. However, the results contained there are local. In [19], by putting together this idea with the approach found in [3], the authors were able to show how 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 control input 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 their input-output variable approach, which entails using the vector col (y, y, ˙ . . . , y (ny ) , u, u, ˙ . . . , u(nu ) ) as feedback, for some integers ny , nu , where y and u denote the system output and input, respectively. This in particular implies that, when dealing with systems which are not input-output feedback linearizable, such approach 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 a state-variable approach and achieves a separation principle for a class of non-UCO systems, specifically systems that are observable on an open region of the state space and input space, rather than everywhere. We impose a restriction on the topology of such an “observability region” assuming, among other things, that it contains a sufficiently large simply connected neighborhood of the origin. The main contributions are the development of a nonlinear observer working in state coordinates (which is proved to be equivalent to the standard high-gain observer in output coordinates), and a dynamic projection operating on the observer dynamics which eliminates the peaking phenomenon in the observer states, thus avoiding the need to use control input saturation. One of the benefits of a state-variable approach is 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 methodology yields closed-loop stability. In the particular case when the plant is globally stabilizable and UCO, this approach yields semiglobal stability, as in [19], provided a convexity requirement is satisfied. As in [21], [19], [1], our results rely on adding integrators at the input side of the plant and designing a stabilizing control law for the resulting augmented system. 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: [email protected]). 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: [email protected]). This work was supported by NASA Glenn Research Center, Grant NAG3-2084.

Published in IEEE 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 feedback control design for the augmented system and the observer design and not between the state feedback control design for the original system and the observer design. II. P ROBLEM F ORMULATION AND A SSUMPTIONS Consider the following dynamical system, x˙ = f (x, u)

(1)

y = h(x, u)

where x ∈ Rn , u, y ∈ R, f and h are known smooth functions, and f (0, 0) = 0. Our control objective is to construct a stabilizing controller for (1) without the availability of the system states x. In order to do so, we need an observability assumption. Define the observability mapping   y ´ ³ 4  .  ye =  ..  = H x, u, . . . , u(nu −1) y (n−1) 

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

  = 

h(x, u) ϕ1 (x, u, u(1) ) .. . ¡ ¢ ϕn−1 x, u, . . . , u(nu −1)

    

∂h (1) ∂h f (x, u) + u ∂x ∂u ∂ϕ1 ∂ϕ1 (1) ∂ϕ1 (2) ϕ2 (x, u, u(1) , u(2) ) = f (x, u) + u + u ∂x ∂u ∂u(1) .. . ϕ1 (x, u, u(1) ) =

ϕn−1 (x, . . .) =

nX u −2 ∂ϕn−2 (j+1) ∂ϕn−2 f (x, u) + u ∂x ∂u(j) j=0

where 0 ≤ nu ≤ n (nu = 0 indicates that there is no dependence on u). In the most general case, ϕi = ϕi (x, u, . . . , u(i) ), i = 1, 2, . . . , n − 1. In some cases, however, we may have that ϕi = ϕi (x, u) for all i = 1, . . . , r − 1 and some integer r > 1. This happens in particular when system (1) has a well-defined relative degree r. Here, we do not require the system to be input-output feedback linearizable, and hence to possess a well-defined relative degree. In the case of systems with well-defined relative degree, nu = 0 corresponds to having r ≥ n, while nu = n corresponds to having r = 0. Next, augment the system dynamics with nu integrators on the input side, which corresponds to using a compensator of order nu . System (1) can be rewritten as follows, (2) x˙ = f (x, z1 ), z˙1 = z2 , . . . , z˙nu = v Define the extended state variable X = col (x, z) ∈ Rn+nu , and the associated extended system X˙ = fe (X) + ge v y = he (X)

(3)

where fe (X) = col (f (x, z1 ), z2 , . . . , znu , 0), ge = col (0, . . . , 1), and he (X) = h(x, z1 ). Now, we are ready to state our first assumption. Assumption A1 (Observability): System (1) is observable over an open set O ⊂ Rn × Rnu containing the origin, i.e., the mapping F : O → Y (where Y = F(O)) defined by   · ¸ ye H(x, z)   Y = . . . = F(X) = (4) z z has a smooth inverse F −1 : Y → O,

F −1 (Y ) = F −1 (ye , z) =

· −1 ¸ H (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 set O to be Rn × 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, but rather be observable in some subset of Rn × 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 mapping H, viewed as a mapping acting on x parameterized by z, is assumed to be square (i.e., it maps spaces of equal dimension), thus implying that x can be expressed as a function of y, its n − 1 time derivatives and z, 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 of y, rather than just n − 1. In our setting, this is equivalent to assuming that F in A1, rather than being invertible, is just left-invertible. We are currently working 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 of x, i.e., there exists a smooth function u ¯(x) such that the origin is an asymptotically stable (or globally asymptotically stable) equilibrium point of x˙ = f (x, u ¯(x)). Remark 2: Assumption A2 implies that the origin of the extended system (3) is locally stabilizable (or stabilizable) by a function of X as well. A proof of the local stabilizability property for (3) may be found, e.g., in [17], while its global counterpart is a well known consequence of the integrator backstepping lemma (see, e.g., Theorem 9.2.3 in [5] or Corollary 2.10 in [10]). Therefore we conclude that for the extended system (3) there exists a smooth control v¯(X) such that its origin is asymptotically stable under closed-loop control. Let D be the domain of attraction of the origin of (3), and notice that, when A2 holds globally, D = Rn × Rnu . Remark 3: In [21] the authors consider affine systems and use a feedback linearizability 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). In this respect, our assumption A2 relaxes also the stabilizability assumption found in [19], while it is equivalent to Assumption 2 in [1].

III. N ONLINEAR O BSERVER : I TS N EED AND S TABILITY A NALYSIS Assumption A2 allows us to design a stabilizing state feedback control v = φ(x, z). In order to perform output feedback control x should be replaced by its estimate. Many researchers in the past adopted an input-output feedback linearizability assumption ([3], [9], [13], [7], [14]) and transformed the system into normal form π˙ i = πi+1 , 1≤i≤r−1 π˙ r = f¯(π, Π) + g¯(π, Π) u ˙ = Φ(π, Π), Π Π ∈ Rn−r

(6)

y = π1 .

In this framework the problem of output feedback control finds a very natural formulation, as the first r derivatives of y are equal to the states of the π-subsystem (i.e., the linear subsystem). The works [3], [9], [7] solve the output feedback control problem for systems with no zero dynamics (i.e., r = n), so that the first n − 1 derivatives of y provide the entire state of the system. In the presence of zero dynamics (Π-subsystem), the use of input-output feedback linearizing controllers for (6) forces the use of a minimum phase assumption (e.g, [13]) since the states of the Π-subsystem are made unobservable by such controllers and, hence, cannot be controlled by output feedback. For this reason the output feedback control of nonminimum phase systems has been regarded in the past as a particularly challenging problem. Researchers who have addressed this problem (e.g., [21], [19]) rely on the explicit knowledge of H−1 in (5), x = H−1 (ye , col (z1 , . . . , znu )), so that estimation of the first n − 1 derivatives of y (the vector ye ) provides an estimate of x, x ˆ = H−1 (ˆ ye , col (z1 , . . . , znu )), since the vector z, being the state of the controller, is known. Next, to estimate the derivatives of y, they employ an high-gain observer. Both the works [21] and [19] (the latter dealing with the larger class of stabilizable systems) rely on the knowledge of H−1 to prove closed-loop stability. In addition to this, the work in [1] proves that a separation principle holds for a quite general class of nonlinear systems which includes (1) provided that H−1 is explicitly known and that the system is uniformly completely observable. Sometimes however, even if it exists, H−1 cannot be explicitly calculated (see, e.g., the example in Section V) thus limiting the applicability of existing approaches. Hence, rather than estimating ye and using H−1 (·, ·) to get x, the approach adopted here is to estimate x directly using a nonlinear observer for system (1) and using the fact that the z-states are known.

4

The observer has the form1

x ˆ˙ = fˆ(ˆ x, z, y)

¸−1 ∂H(ˆ x, z) (7) E −1 L [y(t) − yˆ(t)] = f (ˆ x, z1 ) + ∂x ˆ yˆ(t) = h(ˆ x, z1 ) £ 2 ¤ where L is a n × 1 vector, E = diag ρ, ρ , . . . , ρn , and ρ ∈ (0, 1] is a fixed design constant. Notice that (7) does not require any knowledge of H−1 and has the advantage of operating in x-coordinates. The observability assumption A1 implies that the Jacobian of the mapping H with respect to x is invertible, and hence the inverse of ∂H(ˆ x, z)/∂ x ˆ in (7) is well-defined. In the work [2], the authors used an observer structurally identical to (7) for the more restrictive class of input-output feedback linearizable systems with full relative degree. Here, by modifying the definition of the mapping H and by introducing a dynamic projection, we considerably relax these conditions by just requiring the general observability assumption A1 to hold. Furthermore, we propose a different proof than the one found in [2] which clarifies the relationship among (7) and the high-gain observers commonly found in the literature. 4

·

Theorem 1 Consider system (2) and assume that A1 is satisfied for O = Rn+nu , the state X belongs to a positively invariant, ˆ Ω ˆ ⊃ Ω, which is positively invariant for (ˆ compact set Ω, supt≥0 v(t) < ∞, and that there exists a set Ω, x, z) and such that n ˆ {ye ∈ R | (ye , z) ∈ F(Ω)} is a compact set. Choose L = col (l1 , . . . , ln ) such that sn + l1 sn−1 + . . . + ln is Hurwitz. Under these conditions and using observer (7), the following two properties hold ˆ (i) Asymptotic stability of the estimation error: There exists ρ¯, 0 < ρ¯ ≤ 1, such that for all x ˆ(0) ∈ {ˆ x ∈ Rn | (ˆ x, z) ∈ Ω}, and all ρ ∈ (0, ρ¯), x ˆ(t) → x(t) as t → +∞. (ii) Arbitrarily fast rate of convergence: For each positive T, ², there exists ρ∗ , 0 < ρ∗ ≤ 1, such that for all ρ ∈ (0, ρ∗ ], kˆ x(t) − x(t)k ≤ ² ∀t ≥ T . In Section IV we show that, by applying to the vector field fˆ a suitable dynamic projection onto a fixed compact set, the ˆ is guaranteed. existence of the compact positively invariant set Ω Proof: Consider the smooth coordinate transformation ye = H(x, z) which maps (1) to y˙ e = Ac ye + Bc [α(ye , z) + β(ye , z)v] ,

(8)

where α(·, ·) and β(·, ·) are smooth functions and the pair (Ac , Bc ) is in Brunovsky canonical form. Similarly, it is not difficult to show that the coordinate transformation yˆe = H(ˆ x, z) maps the observer dynamics (7) to yˆ˙ e = Ac yˆe + Bc [α(ˆ ye , z) + β(ˆ ye , z)v] + E −1 L [ye,1 − yˆe,1 ].

(9)

Define the observer error in the new coordinates, y˜e = yˆe − ye . Then, the observer error dynamics are given by y˜˙ e = (Ac − E −1 L Cc )˜ ye + Bc [α(ˆ ye , z) + β(ˆ ye , z)v − α(ye , z) − β(ye , z)v] ,

(10)

where Cc = [1, 0, . . . , 0]. Note that, in ye coordinates, the observer (7) is almost identical to the standard high-gain observer used in, e.g., [19] with an important difference: while α and β in [19] depend on sat (ˆ ye ) (where sat(·) denotes the saturation function) and hence the nonlinear portion of the observer error dynamics is globally bounded when Y ∈ F(Ω), α and β in (10) do not contain saturation and thus one cannot guarantee global boundedness of the last term in (10). Since our observer operates in x coordinates, saturation cannot be induced in ye coordinates and therefore, without any further assumption, we ˆ addresses precisely this issue since cannot prove stability of the origin of (10). The assumption concerning the existence of Ω it guarantees that yˆe (t) = H(ˆ x(t), z(t)) is contained in a compact set for all t ≥ 0. Define the coordinate transformation ¸ · 1 1 4 (11) ν˜ = E 0 y˜e , E 0 = diag n−1 , n−2 , . . . , 1 . ρ ρ 1 Throughout this section we assume A1 to hold globally, since we are interested in the ideal convergence properties of the state estimates. In the next section we will show how to modify the observer equation in order to achieve the same convergence properties when A1 holds over the set O ⊂ Rn × Rnu .

5

In the new coordinates the observer error dynamics are given by 1 ν˜˙ = (Ac − LCc )˜ ν ρ + Bc [α(ˆ ye , z) + β(ˆ 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 candidate Vo (˜ ν ) = ν˜> P ν˜. Calculate the time derivative of Vo along the ν˜ trajectories: ν˜> ν˜ + 2˜ ν > P Bc [α(ˆ ye , z) + β(ˆ ye , z)v V˙ o = − ρ −α(ye , z) − β(ye , z)v] .

(14)

ˆ for all t ≥ 0, we have that (ye (t), z(t)) ∈ F(Ω) ˆ and hence, for all t ≥ 0, ye (t) ∈ {ye | (ye , z) ∈ F(Ω)} ˆ Since (ˆ x(t), z(t)) ∈ Ω which, by assumption, is a compact set. This, together with the fact that, for all t ≥ 0, X(t) ∈ Ω and supt≥0 v(t) < ∞, implies that there exists a fixed scalar γ > 0, independent of ρ, such that the time derivative of V0 can be bounded as 2

2

k˜ νk k˜ νk + 2kP kγk˜ ye kk˜ νk ≤ − + 2kP kγk˜ ν k2 . V˙ o ≤ − ρ ρ

(15)

Defining ρ¯ = min{1/(2kP kγ), 1}, we conclude that, for all ρ ∈ (0, ρ¯), the origin of (12) (and hence also the origin of (10)) is asymptotically stable which, by the smoothness of H, implies that x ˆ(t) → x(t) as t → ∞. This concludes the proof of part (i). As for part (ii), note that λmin (E 0 P E 0 ) ≥ λmin (E 0 )2 λmin (P ) = λmin (P ), since λmin (E 0 ) = 1. Next,

λmax (E 0 P E 0 ) ≤ 1/(ρ2(n−1) )λmax (P ),

since λmax (E 0 ) = 1/ρ(n−1) . Therefore λmin (P )k˜ ye k2 ≤ y˜e> E 0 P E 0 y˜e ≤

1 λmax (P )k˜ ye k2 . ρ2(n−1)

Define ²¯ so that k˜ ye k ≤ ²¯ implies that kˆ x −xk ≤ ² (the smoothness of H−1 guarantees that ²¯ is well-defined). By the inequality 4 2 above we have that Vo ≤ ²¯ λmin (P ) implies that k˜ ye k ≤ ²¯, and Vo (0) = Vo (˜ ν (0)) ≤ (1/ρ2(n−1) )λmax (P )k˜ ye (0)k2 . Moreover, from (15) ¶ µ 1 − 2kP kγ k˜ ν k2 V˙ o (t) ≤ − ρ µ ¶ 1 1 ≤− − 2kP kγ Vo (t). λmax (P ) ρ Therefore, by the Comparison Lemma (see, e.g., [22]), Vo (t) satisfies the following inequality µ ½ ¶ ¾ 1 1 Vo (t) ≤ Vo (0) exp − − 2kP kγ t λmax (P ) ρ λmax (P ) ye (0)k2 exp{−t/λmax (P )(ρ−1 − 2kP kγ)} ≤ 2(n−1) k˜ ρ

(16)

which, for sufficiently small ρ, can be written as ½ ¾ a2 a1 Vo (t) ≤ 2n exp − t , ρ ρ

a1 , a2 > 0.

An upper estimate of the time T such that kˆ ye (t) − ye (t)k ≤ ²¯ (and thus kˆ x(t) − x(t)k ≤ ²) for all t ≥ T , is calculated as follows ½ ¾ a1 a2 exp − t ≤ ²¯2 λmin (P ), ρ2n ρ ¡ ¢ for all t ≥ T = ρ/a2 ln a1 /(¯ ²2 ρ2n λmin (P )) . Noticing that T → 0 as ρ → 0, we conclude that T can be made arbitrarily small by choosing a sufficiently small ρ∗ , thus concluding the proof of part (ii).

6

Remark 4: Using inequality (16), we find the upper bound for the estimation error in ye -coordinates s λmax (P ) −(n−1) k˜ ye k ≤ ρ k˜ ye (0)k λmin (P ) −1

· exp{−t/(2λmax (P ))(ρ

(17)

− 2kP kγ)}.

Hence, during the initial transient, y˜e (t) may exhibit peaking, and the size of the peak grows larger as ρ decreases and the convergence rate is made faster. Refer to [18] for more details on the peaking phenomenon and to [3] for a study of its implications on output feedback control. The analysis in the latter paper shows that a way to isolate the peaking of the observer estimates from the system states is to saturate the control input outside of the compact set of interest. The same idea has then been 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 to eliminate (rather than just isolate) the peaking phenomenon which allows for the use of the weaker Assumption A1.

Nz'

Nz F ( Ω c2)

F

Nζ'

Nye G

Ω c2

C' F -1

z

O

C

G -1

C

F -1 ( C )

z

x Fig. 1.

F(O)

z-

z ye

ζ

The mechanism behind the observer estimates projection.

IV. O UTPUT F EEDBACK S TABILIZING C ONTROL Consider system (3), by using assumption A2 and Remark 2 we conclude that there exists a smooth stabilizing control v = φ(x, z) = φ(X) which makes the origin of (3) an asymptotically stable equilibrium point with domain of attraction D. By the converse Lyapunov theorem found in [11], there exists a continuously differentiable function V defined on D satisfying, for all X ∈ D, α1 (kXk) ≤ V (X) ≤ α2 (kXk) (18) lim α1 (kXk) = ∞ (19) X → ∂D ∂V (fe (X) + ge v) ≤ −α3 (kXk) (20) ∂X where αi , i = 1, 2, 3 are class K functions (see [8] for a definition), and ∂D stands for the boundary of the set D. Given any scalar c > 0, define 4 Ωc = {X ∈ Rn+nu | V ≤ c}. Clearly, Ωc ⊂ D for all c > 0 and, from (19), Ωc becomes arbitrarily close to D as c → ∞. Next, the following assumption is needed. Assumption A3 (Topology of O): Assume that there exists a constant c¯ > 0 and a set C such that F (Ωc¯) ⊂ C ⊂ Y (= F (O)) ,

(21)

where C has the following properties (i) The boundary of C, ∂C, is an n − 1 dimensional, C 1 submanifold of Rn , i.e., there exists a C 1 function g : C → R such that ∂C = {Y ∈ C | g(Y ) = 0}, and (∂g/∂Y )> 6= 0 on ∂C. (ii) C z¯ = {ye ∈ Rn | (ye , z¯) ∈ C} is convex for all z¯ ∈ Rnu . ∂g (iii) 0 is a regular value of g(·, z¯) for each fixed z¯ ∈ Rnu , i.e., for all ye ∈ C z¯, (y , z¯) 6= 0. ∂ye e S z¯ (iv) z¯∈Rnu C is compact.

7

C1

C3

C2

C4

z ye

Fig. 2.

The domains C1 , C2 , C3 violate some of the requirements (i)-(iv) in A3, while C4 does not.

Remark 5: See Figure 1 for a pictorial representation of condition (21). This assumption requires in primis that there exists a set C in Y coordinates which contains the image under F of a level set Ωc¯ of the Lyapunov function V and is contained in the image under F of the observable set O. This guarantees in particular that Ωc¯ ⊂ O, and thus, when the state feedback controller is employed, the phase curves leaving from Ωc¯ never exit the observable region O. Furthermore, it is required that C possesses some basic topological properties: It is asked that the boundary of C be continuously differentiable (part (i)), every slice of C obtained by holding z constant at z¯, C z¯, is convex (part (ii)), the normal vector to each slice C z¯ (which is given by [∂g/∂ye (ye , z¯)]> ) does not vanish anywhere on the slice (part (iii)) and, finally, it is asked S that the set C is compact in the ye direction (part (iv)). Part (iv) can be replaced by the slightly weaker requirement that z¯∈Ωz C z¯ is compact (Ωz is defined in the next section), with minor changes in the analysis to follow. To further clarify the topology of the domains under consideration, consider the sets C1 to C4 in Figure 2, corresponding to the case ye ∈ R2 , z ∈ R. While they all satisfy part (i), C1 does not satisfy part (ii) since its slices along z are not convex. C2 satisfies part (ii) but violates requirement (iii) because the normal vector to one of the slices has no components in the ye direction. C3 satisfies parts (i)-(iii) but violates (iv) since the area of its slices grows unbounded as z → ∞. C4 satisfies all the requirements above. Note that, when the plant is UCO (and thus O = Rn ×Rnu ) and F(Rn+nu ) = Rn+nu , A3 is always satisfied by a sufficiently large set C and any c¯ > 0. In order to see that, pick any c¯ and choose C to be any cylinder {Y ∈ Rn+nu | kye k ≤ M }, where M > 0, containing F(Ωc¯). The existence of C is guaranteed by the fact that the set F(Ωc¯) is bounded. More generally, the 4 same holds when F(Rn+nu ) is not all of Rn+nu and Y z¯ = F(Rn , z¯) is convex for all z¯ ∈ Rnu . Finally, notice that when the plant in not UCO but O = X × Rnu , where X is an open set which is not all of Rn , and the origin is globally stabilizable (i.e., A2 holds globally), one can choose C = D × Rnu , where D ⊂ Rn is any convex compact set with smooth boundary contained in X and containing the point H(0, 0) (i.e., the origin in ye coordinates). The scalar c¯ is then the largest number such that F(Ωc¯) ⊂ C (notice, however, that c¯ does not need to be known for design purposes). Consequently, in the particular case when nu = 0 (and hence the control input does not affect the mapping H) and A2 holds globally, one can choose C = D, where D 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 approach generally adopted in several papers is to saturate the control input to prevent it from growing above a given threshold. This technique, however, does not eliminate the peak in the observer estimate and, hence, cannot be used to control general systems like the ones satisfying assumption A1, since even when the system state lies in the observable region O ⊂ Rn × Rnu , the observer estimates may enter the unobservable domain where (7) is not well-defined. It appears that in order to deal with systems that are not completely observable, one has to eliminate the peaking from the observer by guaranteeing its estimates to 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 within a desired convex set is gradient projection. This idea cannot be directly applied to our problem, mainly because fˆ is not proportional to the gradient of the observer Lyapunov function and, thus, the projection cannot be guaranteed to preserve the convergence properties of the estimate. Inspired by this idea, however, we propose a way to modify the vector field fˆ which confines x ˆ to within a prespecified compact set while preserving its convergence properties. Recall the coordinate transformation defined in (4) and let yˆeP = H(ˆ xP , z), y˜eP = yˆeP − ye Yˆ P = F(ˆ xP , z) = col (ˆ y P , z), e

(22)

8

where x ˆP is the state of the projected observer defined as2 ) · ¸−1(  ΓNye (Yˆ P )LGˆ g ∂H ∂H   LFˆ H − − z˙   ∂x ∂z ˆP Nye (Yˆ P )> ΓNye (Yˆ P ) P ˙x ˆ =  if LGˆ g ≥ 0 and Yˆ P ∈ ∂C    ˆ P f (ˆ x , z, y) otherwise

(23)

where

¸> ¸> · ∂g P ∂g P P ˆ , z) , z) (ˆ y (ˆ y , N ( Y ) = z e ∂z e ∂ yˆeP represent the ye and z components of the normal vector N (Yˆ P ) to the boundary of C at Yˆ P , i.e., N (Yˆ P ) = col (Nye (Yˆ P ), Nz (Yˆ P )) (the function g is defined in A3). Further, LFˆ H and LGˆ g are the Lie derivatives of yˆeP and g(ˆ yeP , z) along the vector fields ˆ ˆ ˆ F = col (f , z2 , . . . , znu , v) and G = LFˆ F, respectively, i.e., Nye (Yˆ P ) =

·

∂H ∂H ˆ P f (ˆ x , z, y) + z˙ ∂z ∂x ˆP ∂g ∂g LGˆ g = P LFˆ H + ˙ z˙ = Nye (Yˆ P )> LFˆ H + Nz (Yˆ P )> z, ∂z ∂ yˆe LFˆ H =

and Γ = (SE 0 )−1 (SE 0 )−> , where S = S > denotes the matrix square root of P (defined in (13)). The dynamic projection (23) is well-defined since A3, part (iii), guarantees that Nye does not vanish (see also Remark 5). The following lemma shows that (23) guarantees boundedness and preserves convergence of x ˆP . Lemma 1 If A3 holds and (23) is used: (i) Positive invariance of F −1 (C): if (ˆ xP (0), z(0)) ∈ F −1 (C), then (ˆ 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 property holds 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 x ˆP . Proof: We begin by introducing another coordinate transformation, ζ = SE 0 ye , (similarly, let ζˆP = SE 0 yˆeP , ζ˜P = 4 letting G = diag [SE 0 , Inªu ×nu ], and letting C 0 be the image of the set C under the linear map G, i.e., C 0 = −1 0 0 n+nu | G col (ζ, z) ∈ C . Let Nζ (ζ, z), Nz (ζ, z) be the ζ and z components of the normal vector to the boundary (ζ, z) ∈ R of C 0 . The reader may refer to Figure 1 for a pictorial representation of the sets under consideration. In order to prove part (i) of the Lemma, we have to show that the projection (23) renders C a positively invariant set for Yˆ P . The coordinate transformation (22) maps (23) to · ¸ © ª ∂H ˙ P ∂H P ˙yˆPe = d H(ˆ x ˆ + x , z) = z˙ (24) dt ∂z ∂x ˆP  Nye (Yˆ P )LGˆ g    L H − Γ ˆ  F Nye (Yˆ P )> ΓNye (Yˆ P ) (25) =  if LGˆ g ≥ 0 and Yˆ P ∈ ∂C    LFˆ H otherwise

0 P SE © y˜e ),

yeP , z) and Nz0 (ζˆP , z), recall from A3 that the boundary of C is expressed In order to relate Nζ0 (ζˆP , z) and Nz0 (ζˆP , z) to Nye (ˆ n+nu as the set ∂C = {Y ∈ R | g(Y ) = 0} and hence the boundary of C 0 is the set ∂C 0 = {ζ ∈ Rn | g((SE 0 )−1 ζ, z) = 0}. From this definition we find the expression of Nζ0 and Nz0 as Nζ0 (ζˆP , z) = (SE 0 )−> [∂g(Yˆ P )/∂yeP ]> = (SE 0 )−> Nye (Yˆ P ) Nz0 (ζˆP , z) = Nz (Yˆ P ). The expression of the projection (23) in ζ coordinates is found by noting that ˙ ζˆP = SE 0 yˆ˙Pe ¡ ¢  N Ny>e LFˆ H + Nz> z˙  0 0 −> ye   SE LFˆ H − (SE ) Ny>e ΓNye = P ˆ  if LGˆ g ≥ 0 and Y ∈ ∂C   SE 0 LFˆ H otherwise

(26)

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

9

and then substituting Nζ0 = (SE 0 )−> Nye , Nz0 = Nz , and LFˆ H = (SE 0 )−1 (LFˆ SE 0 H), to find that ³ ´   Nζ0 Nζ0 > (LFˆ SE 0 H) + Nz0 > z˙    (LFˆ SE 0 H) − ˙ Nζ0 > Nζ0 ζˆP =  0 > 0 0 > P 0   if Nζ (LFˆ SE H) + Nz z˙ ≥ 0 and (ζˆ , z) ∈ ∂C  0 (LFˆ SE H) otherwise

(27)

Next, we show that the boundary of the domain¡ C 0 is positively ¢ invariant with respect to (27). In order to do that, consider the continuously differentiable function VC 0 = g (SE 0 )−1 ζ, z and calculate its time derivative along the vector field of (27) when (ζˆP , z) ∈ ∂C 0 , ˙ V˙ C 0 = Nζ0 (ζˆP , z)> ζˆP + Nz0 (ζˆP , z)z˙ ³ ´ Nζ0 > Nζ0 Nζ0 > (LFˆ SE 0 H) + Nz0 > z˙ = Nζ0 > (LFˆ SE 0 H) − Nζ0 > Nζ0 + Nz0 z˙ = 0

Since C 0 is positively invariant, so is C and, therefore, F −1 (C). The proof of part (ii) is based on the knowledge of a Lyapunov function for the observer in ν˜ coordinates (see (11)). Letting ˜ Now let V P = ζ˜P > ζ˜P be a Lyapunov function ζ˜ = S ν˜, we have that, in new coordinates, Vo = ν˜> P ν˜ = (˜ ν > S)(S ν˜) = ζ˜> ζ. o 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 0 . From (27), when (ζˆP , z) is in the interior of C 0 , or (ζˆP , z) is on the boundary of C 0 and Nζ0 > (LFˆ SE 0 H) + Nz0 > z˙ < 0 (i.e., the unprojected update is pointed to the interior of C 0 ), we have that V˙ oP = V˙ o < 0. Let us now consider all remaining cases, i.e., (ζˆP , z) ∈ ∂C and Nζ0 > (LFˆ SE 0 H) + Nz0 > z˙ ≥ 0, h ˙ ˙ V˙ oP = 2ζ˜P ζ˜P = 2ζ˜P > ζ˜P = 2ζ˜P > (LFˆ SE 0 H) − ζ˙ i (28) −p(ζˆP , (LFˆ SE 0 H), z, z)N ˙ ζ0 (ζˆP , z) = V˙ o (ζ˜P ) − 2pζ˜P > Nζ0

where p(ζˆP , (LFˆ SE 0 H), z, z) ˙ =

Nζ0 > (LFˆ SE 0 H) + Nz0 > z˙ Nζ0 > Nζ0

is nonnegative since, by assumption, Nζ0 > (LFˆ SE 0 H) + Nz0 > z˙ ≥ 0. Using the fact that (ζ, z) ∈ C 0 and that (ζˆP , z) lies on the boundary of C 0 , we have that the difference vector (ζˆP − ζ, 0) points outside of C 0 or, equivalently, ζˆP − ζ points outside of the slice C 0z = {ζ ∈ Rn | (ζ, z) ∈ C 0 }. Using the definition of C 0 , we have that the set C 0z is the image of the convex compact set C z , defined in A3, under the linear map SE 0 and is therefore compact and convex as well. Combining these two facts we have that ζ˜P > Nζ0 ≥ 0, thus proving that V˙ oP ≤ V˙ o < 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 a ˆ for (ˆ ˆ = F −1 (C). To see that, note that positively invariant set Ω x, z) is satisfied with Ω ´ ³[ z C × Rnu C⊂ n z∈R

and thus

u

ˆ = {ye ∈ Rn | (ye , z) ∈ C} {ye ∈ Rn | (ye , z) ∈ F(Ω)} [ Cz ⊂ n z∈R

u

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 idea introduced in [15]. In this case, condition (21) in A3 should be 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 ratio Nye LGˆ g/Nye > ΓNye in (23) should be multiplied by g, and a slight modification of Lemma 1 would show that, by applying ¯ (rather than the set C) is made positively invariant. the smooth projection to fˆ, the set F −1 (C) Remark 8: From the proof of Lemma 1 we conclude that (23) confines (ˆ xP , z) within the set F −1 (C) which is, in general, −1 unknown since we do not know H , and is generally not convex (see Figure 1). It is interesting to note that applying a standard gradient projection for x ˆ over an arbitrary convex domain does not necessarily preserve the convergence result (ii) in Theorem 1.

B. Closed-Loop Stability To perform output feedback control we replace the state feedback law v = φ(x, z) with vˆ = φ(ˆ xP , z) which, by the P −1 smoothness of φ and the fact that x ˆ is guaranteed to belong to H (C), is bounded provided that z is confined to within a compact set. In the following we will show that vˆ makes the origin of (3) asymptotically stable and that Ωc is contained in its region of attraction, for all 0 < c < c¯ 3 . The proof is divided in three steps: 1) (Lemma 2). Invariance of Ωc¯ and uniform ultimate boundedness: Using the arbitrarily fast rate of convergence of the observer (see part (ii) in Theorem 1), we show that any phase curve leaving from Ωc cannot exit the set Ωc¯ and converges in finite time to an arbitrarily small neighborhood of the origin. Here, Lemma 1 plays an important role, in that it guarantees that the peaking phenomenon is eliminated and thus it does not affect vˆ, allowing us to use the same idea 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-loop stability 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¯), ² > 0 and µ > 1 such that

d² = α2 ◦ α3−1 (µ A γ¯ ²) < c¯,

there exists a number ρ¯ ∈ (0, 1) such that, for all ρ ∈ (0, ρ¯], for all (X(0), x ˆP (0)) ∈ A, where ª © xP , z) ∈ F −1 (C) , A = (X, x ˆP ) ∈ R2n+nu | X ∈ Ωc , (ˆ

the phase curves X(t) of the closed-loop system remain confined in Ωc¯, the set Ωd² ⊂ Ωc is positively invariant, and is reached in finite time. Ωzc¯

Sketch of the proof: Since (ˆ xP (0), z(0)) ∈ F −1 (C), by Lemma 1, part(i), (ˆ xP (t), z(t)) ∈ F −1 (C) for all t ≥ 0. Let nu = {z ∈ R | (x, z) ∈ Ωc¯} and notice that X ∈ Ωc¯ and (ˆ xP , z) ∈ F −1 (C) P

=⇒ x ˆ ∈

µ[

z∈Ωcz¯

C

z

¶

× Ωzc¯ ,

which is a compact set independent of ρ. Hence, there exists a bounded positive real number D independent of ρ such that, for all X ∈ Ωc¯ and all (ˆ xP , z) ∈ F −1 (C), kfe (X) + ge φ(ˆ xP , z)k ≤ D. Therefore, for all t such that X(t) ∈ Ωc¯, kX(t) − X(0)k ≤ Dt, proving that the exit time from the set Ωc¯ has a positive X(0) lower bound, T1 , independent of ρ. Further, letting d = dist(Ωc¯, Ωc ) = inf X1 ∈Ωc¯,X2 ∈Ωc kX1 − X2 k, it is readily seen that X(0)

4

inf X(0)∈Ωc (T1 ) = d/D = T1 . We have thus shown that for any phase curve X(t) leaving from the set Ωc there exists a uniform lower bound T1 to the exit time from Ωc¯ which is independent of ρ. Choose T0 ∈ (0, T1 ), let A = supX∈Ωc¯ k∂V /∂Xk, and γ¯ be the Lipschtiz constant of φ over the compact set ¶ µ[ z z −1 C × Ωc¯ . F z z∈Ωc¯

3 Recall

that c¯ is a positive constant satisfying A3 and hence its size is constrained by the topology of the observable set O.

11

By A3, X(t) ∈ Ωc ⊂ Ωc¯ implies X(t) ∈ F −1 (C) and thus from Theorem 1, part (ii), and Lemma 1 there exists ρ¯ ∈ (0, 1) such that, for all ρ ∈ (0, ρ¯], kx(t) − x ˆP (t)k ≤ ² for all t ∈ [T0 , T1 ). Now using the function V defined in (18)-(20), a standard Lyapunov analysis shows that, for all t ∈ [T0 , T1 ), V ≥ d²

=⇒

V˙ ≤ −(µ − 1)A¯ γ ²,

which, since d² < c¯, implies that Ωc¯ is positively invariant (T1 = ∞), and that the phase curves X(t) enter and stay in the set Ω² in finite time. Remark 9: The use of the projection for the observer estimate plays a crucial role in the proof of Lemma 2. As ρ is made smaller, the observer peak may grow larger, thus generating a large control input, which in turn might drive the system states X outside of Ωc in shorter time. The dynamic projection makes sure that the exit time T1 is independent of ρ, thus allowing one to choose ² independently of T1 . Lemma 3 Under the assumptions of Lemma 2, there exists a positive scalar ²∗ such that for all ² ∈ (0, ²∗ ] the origin (X, x − x ˆP ) = (0, 0) is asymptotically stable. Sketch of the proof: Let x ˜P = x − x ˆP . By (17) and Lemma 1, the origin of the ye − yˆeP dynamics is exponentially stable. P −1 −1 P Recalling that x ˜ = H (ye , z) − H (ˆ ye , z), from the smoothness of H we conclude that the origin of the x ˜P dynamics is 0 P exponentially stable as well. By the converse Lyapunov theorem there exists a Lyapunov function Vo (˜ x ), a positive number r, and positive constants k1 , k2 , k3 such that, for all x ˜P ∈ Br = {˜ xP ∈ Rn | k˜ xP k ≤ r}, k1 k˜ xP k2 ≤ Vo0 ≤ k2 k˜ xP k2 xP k2 . V˙ o0 ≤ −k3 k˜

Choose ²∗ ∈ (0, r) such that d²∗ = α2 ◦ α3−1 (µ A γ¯ ²∗ ) < c¯. For any ² ∈ (0, ²∗ ], choose ρ¯ as in Lemma 2 and pick any ρ ∈ (0, ρ¯]. Then, the following Lyapunov function √ q 2 c¯2 γ¯ A xP ), λ> Vc (X, x ˜P ) = V (X) + λ Vo0 (˜ c¯3 readily allows one to conclude that the origin (X, x ˜P ) = (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), with control law vˆ = φ(ˆ xP , z), there exists a scalar ρ∗ ∈ (0, 1) such that, for all ρ ∈ (0, ρ∗ ], the set A is contained in the region of attraction of the origin (X, x ˆP ) = (0, 0). Sketch of the proof: By Lemma 3, there exists ²∗ > 0 such that, for all ² ∈ (0, ²∗ ], the origin (X, x ˜P ) = (0, 0) is P asymptotically stable. Let ∆² be its domain of attraction. For any (X(0), x ˆ (0)) ∈ A, use Lemma 2 to find ρ¯ ∈ (0, 1) so that for all ρ ∈ (0, ρ¯] the phase curves of the closed-loop system enter ∆² in finite time. Remark 10: Theorem 2 proves regional stability of the closed-loop system, since given an observability domain O, and provided A3 is satisfied, the control law vˆ, together with (7) and (23), make the compact set A an estimate of the domain of attraction for the origin of the closed-loop system. The difference between Theorem 2 and a local stability result lies in the fact that here the domain of attraction for x is at least as large as Ωc and not restricted to be a small unknown neighborhood of the origin. Further, the size of Ωc is independent of ρ and thus the domain of attraction does not shrink as the rate of convergence of the observer is made faster (that is, when ρ → 0). The next corollary gives conditions for the recovery of the domain of attraction of the state feedback controller by output feedback. Corollary 1 Assume that A1 is satisfied on O = Rn+nu , Y z¯ = F(Rn , z¯) is convex for all z¯ ∈ Rnu , and A2 holds. Then, given any set D0 ⊂ int(D), there exists a scalar ρ∗ ∈ (0, 1) such that, for all ρ ∈ (0, ρ∗ ], the set D0 is contained in the region of attraction of the origin X = 0. Moreover, if D = Rn+nu , the origin X = 0 is semiglobally asymptotically stable. Proof: By property (19) of the Lyapunov function V (X), given any set D0 ⊂ int(D), there exists c > 0 such that D0 ⊂ Ωc . By Remark 5, if O = Rn+nu , A3 is satisfied by any c¯ ∈ (0, ∞) and a sufficiently large C. Choose c¯ > c. By Theorem 2 we conclude that there exists ρ∗ ∈ (0, 1) such that, for all ρ ∈ (0, ρ∗ ], theª origin of the closed-loop system is asymptotically © stable and the set A = (X, x ˆP ) ∈ R2n+nu | X ∈ Ωc , (ˆ xP , z) ∈ F −1 (C) is contained in its domain of attraction. Thus, in 0 particular, D is contained in the domain of attraction of X = 0. If D = Rn+nu then D0 can be chosen to be any compact set and thus the origin X = 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 achieved between the observer design and the state feedback control design for the augmented system (3). In order to avoid increased

12

fˆ(ˆ x, z, y) = ·

·

ˆ2 ª ©x (1 + x ˆ1 ) exp (ˆ x1 )2 + z1 − 1

¸

0ª £ ¤ + 2(ˆ x2 − 1) exp (ˆ x1 )2 2(ˆ x1 )2 + 2ˆ x1 + 1 ©

¸ −1 2(ˆ x© 2 − 1) ª ¤ E −1 L[y − (ˆ x2 − 1)2 ] 2 (1 + x ˆ1 ) exp (ˆ x1 )2 + z1 − 1

(35)

£

complexity of the controller, it would be more desirable to achieve separation between the observer design and the state feedback control design for the original system (1). Remark 12: As mentioned in Remark 5, if the plant is UCO (and hence O = Rn × Rnu ) and Y z¯ = F(Rn , z¯) is a convex set for all z¯ ∈ Rnu , assumption A3 is automatically satisfied by a sufficiently large cylindrical set C. Even in this case, if H−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 in ye coordinates. Clearly, the only instance when dynamic projection can be replaced by saturation of the observer estimates is when the observability mapping is the identity.

V. E XAMPLE Consider the following input-output linearizable system: x˙ 1 = x2 x˙ 2 = (1 + x1 ) exp(x21 ) + u − 1

(30)

2

y = (x2 − 1) . 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 multiplying u vanishes when x2 = 1, and hence system (30) does not have a well-defined relative degree everywhere. Since u appears in y, ˙ we have that nu = 1, therefore we add one integrator at the input side, z˙1 = v, The mapping F is given by

u = z1 .

  ¸ · y H(x, z1 )   Y = y˙ = F(x, z1 ) = z1 z1   (x2 − 1)2 = 2(x2 − 1)[(1 + x1 ) exp(x21 ) + (z1 − 1)] . z1

The first equation in (32) is invertible for all x2 < 1, and its inverse is given by x2 = 1 − equation in (32) and isolating the term in x1 , we get √ y˙ + 2 y(z1 − 1) . (1 + x1 ) exp(x21 ) = √ −2 y

(31)

(32)

√ y. Substituting x2 into the second

(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 domain O = {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 nonexistence 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 4 x3 = (1 + x1 ) exp(x21 ) + z1 − 1 and rewriting the system in new coordinates xe = col (x1 , x2 , x3 ): x˙ 1 = x2 x˙ 2 = x3 x˙ 3 = −x2 exp(x21 )(2x21

x2 exp(x21 )(2x21

(34) + 2x1 + 1) + v.

Choose v = + 2x1 + 1) − Kxe , where K = [1, 3, 3], so that the closed-loop system becomes x˙ e = (Ac − Bc K) xe with poles placed at −1. Then, the origin xe = 0 is a globally asymptotically equilibrium point of (34), and

13

Assumption A2 is satisfied with D = R3 . Let P¯ be the solution of the Lyapunov equation associated to Ac − Bc K, so that a ª 4 © 3 ¯ Lyapunov function for system (34) is V = x> e P xe , and any set Ωc = xe ∈ R | V (xe ) ≤ c , with c > 0, is contained in the region of attraction for the origin. Next, we seek to find a set C 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 cylinder D × R, where D is any compact convex set in the upper half plane {R+ − 0} × R containing the point H(0, 0) = (1, 0). For the sake of simplicity choose D to be the disk of radius ω < 1 centered at (1, 0), so that C = {Y ∈ R3 | (Y1 − 1)2 + Y22 < ω 2 }, and C ⊂ Y, as depicted in Figure 3. We stress that our choice is quite conservative and is made exclusively for the sake of illustration. Once C has been Unobservable boundary

C

0

1

ω

F (O )

Fig. 3.

The projection domain C.

chosen, the control design is complete and the output feedback controller is given by © P 2ª z˙1 = − x ˆP x1 ) [2(ˆ xP )2 + 2ˆ xP ˆP xP 2 exp (ˆ 1 + 1] − x 1 − 3ˆ 2 £ © P1 2 ª ¤ P − 3 (1 + x ˆ1 ) exp (ˆ x1 ) + z1 − 1 ,

(36)

where x ˆP (t) is the solution of (23) with fˆ(ˆ x, z, y) defined in (35) and · ¸ ¸ · ∂H 0 1 P P ye , z) = yˆe − ,Nz (ˆ yeP , z) = 0, , Nye (ˆ = P 0 2(ˆ x − 1) ∂z 2 · ¸ 2 (ˆ xP P 2 − 1) (37) yˆe = 2 2(ˆ xP ˆP xP 2 − 1)[(1 + x 1 ) exp((ˆ 1 ) ) + (z1 − 1)] ∂H ˆ P ∂H LFˆ H = f (ˆ x , z, y) + z. ˙ P ∂z ∂x ˆ Using controller (36), Theorem 2 guarantees that the origin xe = 0 of the closed-loop system is asymptotically stable and it provides an estimate of its domain of attraction. Specifically, given any positive scalar c < c¯, there exists ρ∗ > 0 such that Ωc

14

is contained in the domain of attraction for all 0 < ρ < ρ∗ . In what follows we will find the set Ωc¯ satisfying A3. Recalling that, in xe -coordinates, Ωc¯ is expressed as {xe ∈ R2 | xe > P¯ xe }, we have that xe ∈ Ωc¯ implies |xi | ≤ (¯ c/λmin (P¯ )), i = 1, 2, 3, 4 3 and hence Ωc¯ ⊂ Ξ = {xe ∈ R | |xi | ≤ (¯ c/λmin (P¯ )), i = 1, 2, 3}. Now let c¯ < λmin (P¯ ) and note that, for all xe ∈ ye , µ ¶2 ¶2 µ c¯ c¯ 2 ≤ |x2 − 1| ≤ 1 + 1− λmin (P¯ ) λ (P¯ ) µ ¶min c¯ c¯ |2(x2 − 1)x3 | ≤ 2 1 + . λmin (P¯ ) λmin (P¯ ) Next, we seek to find c¯ such that F(Ωc¯) ⊂ C. Using the inequalities above we have that if c¯ = min{c∗ , λmin (P¯ )}, where c∗ is the largest scalar satisfying #2 "µ ¶2 c∗ −1 1+ λmin (P¯ ) (38) µ ¶2 µ ¶2 c∗ c¯2 +4 1+ < ω2 , λmin (P¯ ) λmin (P¯ )

(note that c∗ > 0 satisfying the inequality above always exists) then F(Ωc¯) ⊂ F(Ξ) ⊂ C, and hence c¯ satisfies A3. For our simulations we choose ω = 0.9 and, from (38), we get c¯ = 0.06. The initial condition of the extended system is set to x1 (0) = 0.05, x2 (0) = 0.07, x3 (0) = 0.1 (or z1 (0) = 0.0474), which is contained inside Ωc¯ so that Theorem 2 can be applied. Finally, we choose the observer gain L to be col (4, 4), so that its associated polynomial is Hurwitz with both poles placed 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 curve X(t), as well as the control input v, for ρ = 0.2 and ρ = 0.05. The convergence in the latter case is faster, as predicted by Theorem 1 (see Remark 4). 2) Observer estimate projection. Figure 5 shows the evolution of x ˆ and v for ρ = 10−3 with and without projection. The dynamic projection successfully eliminates the peak in the observer states, thus yielding a bounded control input, as predicted by the result of Lemma 1. Figure 6 shows that the phase curve Y (t) = F(X(t)) is contained within the set C for all t ≥ 0, confirming the result of Lemma 1. In particular, Figure 6 shows the operation of the projection when the phase curve of the observer (in Y coordinates) hits the boundary of C: it forces Yˆ P (t) to “slide” along the boundary of C and preserves its convergence characteristics. This is equivalent, in the x domain, to saying that the phase curve (ˆ xP (t), z1 (t)) slides along the boundary of F −1 (C) and x ˆP (t) converges to x(t). 3) Observer estimate projection and closed-loop stability. In Figure 7 a phase plane plot for x(t) is shown with and without observer projection when ρ = 10−4 . The small value of ρ generates a significant peak which, if projection is not 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 phase curve 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 curve X(t) for decreasing values of ρ, in Figure 8, shows that the phase curve of the output feedback system approaches the phase curve of the state feedback one as ρ → 0 (see Remark 11). ACKNOWLEDGMENTS The authors would like to thank Mireille Broucke for fruitful discussions concerning the dynamic projection. R EFERENCES [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 problem of 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. Krsti´c, I. Kanellakopoulos, and P. Kokotovi´c, Nonlinear and Adaptive Control Design. NY: John Wiley & Sons, Inc., 1995.

15

x and its estimate

x and its estimate

1

2

0.25

0.1 0.08

0.2

0.06 0.15

0.04

0.1

0.02 0

0.05 0

-0.02 0

2

4

6

8

10

-0.04

0

2

4

t

6

8

10

8

10

8

10

8

10

t

Integrator state z

Control input

1

0.1

0.5

0.05 0

0 -0.05

-0.5 -0.1 -0.15

-1

-0.2 -0.25

0

2

4

6

8

10

-1.5

0

2

4

6

t

ρ = 0.2. x1 and its estimate

x2 and its estimate

0.35

0.08

0.3

0.06

0.25

0.04

0.2 0.02 0.15 0

0.1 0.05

-0.02

0

-0.04

0

2

4

6

8

10

0

2

4

t

6 t

Integrator state z1

Control input

0.05

0.5

0

0 -0.5

-0.05 -1 -0.1 -1.5 -0.15 -0.2

-2 0

2

4

6

8

10

-2.5

0

2

4

6

t

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 stabilization of minimum-phase input-output linearizable systems via partial state and output feedback,” IEEE Transactions on Automatic Control , vol. 40, no. 6, pp. 1029–1041, 1995. [13] N. Mahmoud and H. Khalil, “Asymptotic regulation of minimum phase 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 imply semiglobal 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. Kokotovi´c, “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. Tornamb`e, “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

2

without projection with projection

1.5

xˆ

1 0.5 0

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.4 0.3

xˆ

0.2 0.1 0 0

-100

u -200 -300

Fig. 5.

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

C

2

Y3

0

-2 0 0.2 2

0.4 1.5

0.6 1

0.8

Y1

0.5

1 0

1.2 -0.5

1.4 1.8

-1.5 2

Fig. 6.

Y2

-1

1.6 -2

Phase curve of the observer with dynamic projection in the transformed domain Y = F (X).

PLACE PHOTO HERE

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

17

0.1

state feedback output feedback without projection output feedback with projection 0.08

0.06

0.04

x

2

0.02

0

-0.02

-0.04

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x1

Fig. 7.

Phase curve x(t) with and without projection, ρ = 10−3 . state feedback output feedback: ρ=0.2 output feedback: ρ=0.01 output feedback: ρ=0.001

0.1 0 0.1

z -0.1 1

0.08

-0.2 0.06 0.04

0 0.02

0.02

0.04 0.06

0 0.08

-0.02

0.1

x 0.12

-0.04

2

x

1

Fig. 8.

Phase curve X(t) of the closed-loop system for decreasing values of ρ.

Kevin M. Passino Kevin M. Passino (S’79, M’90, SM’96) received his Ph.D. in Electrical Engineering from the University of Notre Dame in 1989. He has worked on control systems research at Magnavox Electronic Systems Co. and McDonnell Aircraft Co. He spent a year at Notre Dame as a Visiting Assistant Professor and is currently Professor of Electrical Engineering at The Ohio State University. Also, he is the Director of the OSU Collaborative Center of Control Science that is funded by AFRL/VA and AFOSR. PLACE He has served as the Vice President of Technical Activities of the IEEE Control Systems Society (CSS); was an elected member of PHOTO the IEEE Control Systems Society Board of Governors; was Chair for the IEEE CSS Technical Committee on Intelligent Control; HERE 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 Editor for a special track of papers on Intelligent Control for IEEE Expert Magazine in 1996; and was on the Editorial Board of the Int. Journal for Engineering Applications of Artificial Intelligence. He was a Program Chairman for the 8th IEEE Int. Symp. on Intelligent Control, 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 ”Fuzzy Control,” Addison Wesley Longman Pub., 1998; co-author (with K.L. Burgess) of the book ”Stability Analysis of Discrete Event Systems,” John Wiley and Sons, 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 Control Systems Software Development”, John Wiley and Sons, NY, 2001; co-author (with J.T. Spooner, M. Maggiore, R. Ordonez) of the book ”Stable Adaptive Control and Estimation for Nonlinear Systems: Neural and Fuzzy Approximator Techniques,” John Wiley and Sons, NY, 2002.