Automatica 43 (2007) 1997–2008www.elsevier.com/locate/automatica

Switched seesaw control for the stabilization of underactuated vehicles�

A. Pedro Aguiara,∗, João P. Hespanhab, António M. Pascoala

aISR/IST - Institute for Systems and Robotics, Instituto Superior Técnico, Torre Norte 8, Av. Rovisco Pais, 1049-001 Lisbon, PortugalbCenter for Control Engineering and Computation, University of California, Santa Barbara, CA 93106-9560, USA

Received 11 October 2005; received in revised form 9 October 2006; accepted 28 March 2007Available online 21 August 2007

Abstract

This paper addresses the stabilization of a class of nonlinear systems in the presence of disturbances, using switching controllers. Tothis effect we introduce two new classes of switched systems and provide conditions under which they are input-to-state practically stable(ISpS). By exploiting these results, a methodology for control systems design—called switched seesaw control—is obtained that allows forthe development of nonlinear control laws yielding input-to-state stability. The range of applicability and the efficacy of the methodologyproposed are illustrated via two nontrivial design examples. Namely, stabilization of the extended nonholonomic double integrator (ENDI) andstabilization of an underactuated autonomous underwater vehicle (AUV) in the presence of input disturbances and measurement noise.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Switched systems; Hybrid control; Stabilization; Nonholonomic systems; Autonomous underwater vehicles

1. Introduction

There has been increasing interest in hybrid control in recentyears, in part due to its potential to overcome the basic limita-tions to nonlinear system stabilization introduced by Brockett’scelebrated result in the area of nonholonomic systems control(Brockett, 1983). Hybrid controllers that combine time-drivenwith event-driven dynamics have been developed by a num-ber of authors and their design is by now firmly rooted in asolid theoretical background. See for example Kolmanovskyand McClamroch (1996), Tomlin, Pappas, and Sastry (1998),

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor HenkNijmeijer under the direction of Editor Hassan Khalil. Research supportedin part by project MAYA-Sub of the AdI, project GREX/CEC-IST (ContractNo. 035223) of the Commission of the European Communities, and by theFCT-ISR/IST plurianual funding through the POS-C Program that includesFEDER funds. Part of this research was done while the first author was withthe Center for Control Engineering and Computation, University of California,Santa Barbara, CA, USA.

∗ Corresponding author. Tel.: +351 21 841 8056; fax:+351 21 841 8291.E-mail addresses: pedro@isr.ist.utl.pt (A. Pedro Aguiar),

hespanha@ece.ucsb.edu (J.P. Hespanha), antonio@isr.ist.utl.pt(A.M. Pascoal).

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2007.03.023

Morse (1995), Hespanha (1996), and Liberzon (2003) and thereferences therein.

Inspired by progress in the area, the first part of this paper of-fers a new design methodology for the stabilization of nonlinearsystems in the presence of external disturbances by resorting tohybrid control. To this effect, two classes of switched systemsare introduced: unstable/stable switched systems and switchedseesaw systems. The first, as their name indicates, have theproperty of alternating between an unstable and a stable modeduring consecutive periods of time. The latter can be viewedas the interconnection of two unstable/stable systems such thatwhen one is stable the other is unstable, and vice versa. Condi-tions are given under which the interconnection is input-to-statepractically stable (ISpS). The results are then used to developa control design framework called switched seesaw control de-sign that allows for the solution of robust (in an appropriatelydefined sense) control problems using switching.

To illustrate the scope of the new design methodology pro-posed, the second part of the paper solves the challenging prob-lems of stabilizing the so-called extended nonholonomic doubleintegrator (ENDI) (Aguiar & Pascoal, 2000) and an underac-tuated autonomous underwater vehicle (AUV) in the presenceof input disturbances and measurement noise. These exampleswere motivated by the problem of point stabilization, that is,

1998 A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008

the problem of steering an autonomous vehicle to a pointwith a desired orientation. The complexity of the point stabi-lization problem is highly dependent on the configuration ofthe vehicle under consideration. For underactuated vehicles,i.e., systems with fewer actuators than degrees-of-freedom,point-stabilization is particularly challenging because most ofthe vehicles exhibit second-order (acceleration) nonholonomicconstraints. As pointed out by Brockett (1983), nonholonomicsystems cannot be stabilized by continuously differentiable (oreven simply continuous) time invariant static state feedbackcontrol laws. To overcome this basic limitation, a variety ofapproaches have been proposed in the literature. Among theproposed solutions are continuous smooth or almost smoothtime-varying (periodic) controllers (Dixon, Jiang, & Dawson,2000; Godhavn & Egeland, 1997; M’Closkey & Murray, 1997;Morin & Samson, 2000, 2003; Samson, 1995; Tell, Murray, &Walsh, 1995), discontinuous or piecewise time-invariant smoothcontrol laws (Aguiar & Pascoal, 2001; Aicardi, Casalino,Bicchi, & Balestrino, 1995; Astolfi, 1998; Bloch & Drakunov,1994; Canudas-de-Wit & SZrdalen, 1992), and hybrid con-trollers (Aguiar & Pascoal, 2000, 2002; Bloch, Reyhanoglu,& McClamroch, 1992; Hespanha, 1996; Lizárraga, Aneke, &Nijmeijer, 2004; Prieur & Astolfi, 2003).

From a practical point of view, the above problem has beenthe subject of much debate within the ground robotics commu-nity. However, it was only recently that the problem of pointstabilization of underactuated AUVs received special consid-eration in the literature (Do, Jiang, Pan, & Nijmeijer, 2004;Leonard, 1995; Pettersen & Egeland, 1999; Pettersen & Fossen,2000). Point stabilization of AUVs poses considerable chal-lenges to control system designers because the dynamics ofthese vehicles are complicated due to the presence of complex,uncertain hydrodynamic terms.

One of the key contributions of the paper is the fact that thesolution proposed for point stabilization of an AUV addressesexplicitly the existence of external disturbances and mea-surement errors. In a general setting this topic has only beenpartially addressed in the literature and in many aspects itstill remains an open problem. Noteworthy exceptions aree.g., Morin and Samson (2003), where smooth time-varyingfeedback control laws for practical stabilization of driftlessnonlinear systems subjected to known or measured additiveperturbations are derived by using the transverse function ap-proach; Prieur and Astolfi (2003), where a hybrid control lawis proposed for stabilization of nonholonomic chained systemsthat yields global exponential stability and global robustnessagainst a class of small measurements errors; and Lizárragaet al. (2004) that addresses the point stabilization for the ex-tended chained form in the presence of additive disturbances.

The paper is organized as follows. In Section 2 we intro-duce and analyze the stability of two new classes of switchedsystems: unstable/stable and seesaw switched systems. The re-sults obtained are then used to derive a switched seesaw controldesign methodology that allows for the development of a newclass of nonlinear control laws yielding input-to-state stability.In Section 3 we illustrate the applicability and the efficacy ofthe theoretical results derived in the previous section via two

non-trivial design examples. Concluding remarks are given inSection 4.

Notation and definitions: | · | denotes the standard Euclideannorm of a vector in Rn and ‖u‖I is the (essential) supremumnorm of a signal u : [0, ∞) → Rn on an interval I ⊂ [0, ∞).Let a ⊕ b := max{a, b} and denote by MW the set of mea-surable, essentially bounded signals w : [t0, ∞) → W, whereW ⊂ Rm. A function � : [0, ∞) → [0, ∞) is of class K(� ∈ K) if it is continuous, strictly increasing, and �(0) = 0and of class K∞ if in addition it is unbounded. A function� : [0, ∞)× R → [0, ∞) is of class1 KL if it is continuous,for each fixed t ∈ R the function �(·, t) is of class K, and foreach fixed r �0 the function �(r, t) decreases with respect tot and �(r, t) → 0 as t → ∞. A class KL function �(r, t) iscalled exponential if �(r, t)� �re−�t , � > 0, � > 0. We denotethe identity function from R to R by id, and the compositionof two functions �i : R → R; i = 1, 2 in this order by �2 ◦ �1.The acronym w.r.t. stands for “with respect to”.

2. Dwell-time switching theorems and hybrid control

This section introduces and analyzes stability related resultsfor two classes of systems that will be henceforth called unsta-ble/stable and seesaw switched systems. The results obtainedare key to the derivation of a new hybrid control methodol-ogy for nonlinear system stabilization in the presence of dis-turbances.

2.1. Unstable/stable switched system

Consider the switched system

x = f�(x, w), x(t0) = x0, (1)

where x ∈ X ⊂ Rn is the state, w ∈ MW is a disturbance, and� : [t0, ∞) → {1, 2} is a piecewise constant switching signalthat is continuous from the right and evolves according to

�(t) ={

1, t ∈ [tk−1, tk), k odd,

2, t ∈ [tk−1, tk), k even.(2)

In (2), {tk} := {t1, t2, t3, . . .} is a sequence of strictly increasinginfinite switching times in [t0, ∞) and t0 is the initial time. Weassume that both fi; i = 1, 2 are locally Lipschitz w.r.t. (x, w)

and that the solutions of (1) lie in X and are defined for allt � t0.

Let � : Rn → [0, ∞) be a continuous nonnegative realfunction called a measuring function. For a given switchingsignal �, system (1) is said to be ISpS2 on X w.r.t. � if thereexist functions � ∈ KL, �w ∈ K, and a nonnegative constant

1 Our definition of KL functions is slightly different from the standardone because the domain of the second argument has been extended from[0, ∞) to R. This will allow us to consider the case �(r, −t) which maygrow unbounded as t → ∞.

2 On a first reading, one can consider that X = Rn. In this case, thereference to the set X is omitted. However, we will need the more generalsetting when we consider applications to the stabilization of underactuatedvehicles.

A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008 1999

c such that for every initial condition x(t0) and every inputw ∈ MW such that the solution x(t) of (1) lies entirely in X,x(t) satisfies

�(x(t))��(�(x(t0)), t − t0) ⊕ �w(‖w‖[t0,t]) ⊕ c (3)

for all t � t0. When X = Rn, W = Rm, �(x) = |x| and c = 0,ISpS is equivalent to the by now classical definition of input-to-state stability (ISS) (Sontag, 1989).

With respect to (1), assume the following conditions hold:

1. Instability (� = 1). For x = f1(x, w), there exist functions�1 ∈ KL, �w

1 ∈ K, and a nonnegative constant c1 suchthat for every initial condition x(t0) and every input w ∈MW for which the solution x(t) of (1) lies entirely in X,x(t) satisfies3

�(x(t))��1(�(x(t0)) ⊕ �w1 (‖w‖[t0,t])

⊕ c1, −(t − t0)), t � t0. (4)

Notice how the negative term −(t − t0) in the second ar-gument of �1 captures the unstable characteristics of thesystem when � = 1.

2. Stability (�=2). System x =f2(x, w) is ISpS on X w.r.t. �,that is, for every initial condition x(t0) and every input w ∈MW such that the solution x(t) of system (1) lies entirelyin X, x(t) satisfies

�(x(t))��2(�(x(t0)), t − t0) ⊕ �w2 (‖w‖[t0,t])

⊕ c2, t � t0, (5)

where �2 ∈ KL, �w2 ∈ K, c2 �0.

If conditions 1–2 above are met, we call (1)–(2) an unsta-ble/stable switched system on X w.r.t. �. The definition of astable/unstable switched is done in the obvious manner.

The following result provides conditions under which an un-stable/stable switched system is ISpS.

Lemma 1. Consider an unstable/stable switched system onX w.r.t. �. Let ti; i ∈ N be a sequence of strictly increasingswitching times {ti} such that the differences between consecu-tive instants of times �i := ti − ti−1 satisfy

�2(�1(r, −�k+1), �k+2)�(id − �)(r) ∀r �r0 (6)

for k = 0, 2, 4, . . . , and for some class K∞ function �(·) andr0 �0. Then, system (1)–(2) is ISpS at t=tk , that is, x(t) satisfiesthe ISpS condition (3) at t = tk . Similarly, if

�1(�2(r, �k), −�k+1)�(id − �)(r) ∀r �r0 (7)

3 Another alternative is to consider that x(t) satisfies

�(x(t))��x1 (�(x(t0)), −(t − t0)) ⊕ �w

1 (‖w‖[t0,t], −(t − t0))

⊕�c1(c1, −(t − t0))

with �x1 , �w

1 , �c1 ∈ KL. There is no loss of generality in considering (4),

because one can always take �1(r, −t) = �x1 (r, −t) ⊕ �w

1 (r, −t) ⊕ �c1(r, −t)

with the advantage of introducing a less complicated notation. However, thismay lead to more conservative estimates.

for k = 2, 4, 6, . . . , and for some class K∞ function �(·) andr0 �0, then system (1)–(2) is ISpS at t = tk+1. If either (6) or (7)holds and the piecewise continuous function that captures thedifferences between consecutive switching times � : [t0, ∞) →[0, ∞) defined by �(t) := �i; t ∈ [ti−1, ti), i ∈ N is bounded,then system (1)–(2) is ISpS.

Remark 2. If c1=c2=r0=0, �(x)=|x|, X=Rn, and W=Rm

and all the conditions of Lemma 1 are met, then system (1)–(2)is ISS.

Remark 3. If (4)–(5) hold with exponential class KL func-tions, i.e., �i (r, t)� �i re−�i t , i = 1, 2, and � can be taken as�(r) = �r; � ∈ (0, 1), then inequalities (6)–(7) become inde-pendent of r. In particular, (6) and (7) above degenerate into

�k+2 � �1

�2�k+1 + 1

�2ln

�1�2

1 − �; k = 0, 2, 4, . . .

and

�k+1 � �2

�1�k + 1

�1ln

1 − �

�1�2

; k = 2, 4, 6, . . . ,

respectively. Notice how the first condition sets lower boundson the periods of time over which the switching system (1) isrequired to be stable. Similarly, the second condition enforcesupper bounds on the periods of time over which the switchingsystem may be unstable.

Remark 4. The results above can be extended to sta-ble/unstable switched systems in the obvious manner.

Proof of Lemma 1. Select some switching time tk such that�(t) = 1 for all t ∈ [tk, tk+1) (unstable mode). From (4), weconclude that

�(x(t−k+1))��1(�(x(tk)) ⊕ �w1 (‖w‖[tk,tk+1)) ⊕ c1, −�k+1),

where x(t−k+1) denotes the limit from the left. Using (5) andthe continuity of x(t) it follows from the inequality �2(a ⊕b, c)��2(b, c) ⊕ �2(a, c) that

�(x(tk+2))��2(�1(�(x(tk)), −�k+1), �k+2)

⊕ �2(�1(�w1 (‖w‖[tk,tk+1)), −�k+1), �k+2)

⊕ �2(�1(c1, −�k+1), �k+2)

⊕ �w2 (‖w‖[tk+1,tk+2)) ⊕ c2.

Applying (6) it is straightforward to obtain

�(x(tk+2)) − �(x(tk))� − �(�(x(tk)))

⊕ �w1 (‖w‖[tk,tk+2)) ⊕ c1,

where �w1 (r) := (id − �) ◦ �w

1 (r) ⊕ �w2 (r) and c1 := (id −

�)(c1 ⊕ �w1 (r0) ⊕ r0) ⊕ c2. It can now be shown that (3) is

satisfied at t = tk; k = 0, 2, 4, . . . by using the same argumentsas in Jiang and Wang (2001, Lemma 3.5) and by viewing (witha slight abuse of terminology) �(·) as a discrete-time ISS-Lyapunov function. Estimates for �w and c in (3) can be derivedby assuming without loss of generality that id − � ∈ K (cf.

2000 A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008

Jiang & Wang, 2001, Lemma B.1) and by choosing any ∈K∞ such that id − is of class K. Then, (3) holds with�w(r) := �−1 ◦ −1 ◦ �w

1 (r) and c := �−1 ◦ −1(c1).To prove (3) at t = tk; k = 2, 4, . . . using (7) instead of (6),

select a switching time tk−1 such that �(t) = 2 for all t ∈[tk−1, tk) (stable mode). From (5), a bound on x(tk) can bewritten as

�(x(t−k ))��2(�(x(tk−1)), �k) ⊕ �w2 (‖w‖[tk−1,tk)) ⊕ c2.

Using the continuity of x(t) and (4) yields

�(x(tk+1))��1(�2(�(x(tk−1)), �k), −�k+1)

⊕ �1(�w2 (‖w‖[tk−1,tk)) ⊕ c2

⊕ �w1 (‖w‖[tk,tk+1)) ⊕ c1, −�k+1).

From (7) it follows that

�(x(tk+1)) − �(x(tk−1))� − �(�(x(tk−1)))

⊕ �w2 (‖w‖[tk−1,tk+1)) ⊕ c2,

where �w2 (r) := �1(�

w2 (r)⊕ �w

1 (r), −�k+1), and c2 := �1(c2 ⊕c1, −�k+1)⊕(id−�)(r0). Again, using the arguments advancedin Jiang and Wang (2001, Lemma 3.5) we conclude that (3)applies, possibly with different estimates for �w and c. Theproof that system (1)–(2) is ISpS (at all times t) if either (6)or (7) hold and �(t) is bounded is straightforward and followsfrom simple algebra. �

2.2. Switched seesaw system

This section introduces the concept of switched seesaw sys-tem. To this effect, consider the switched system (1)–(2). Giventwo measuring functions �su, �us and a set X ⊂ Rn we call(1) a switched seesaw system on X w.r.t. (�su, �us) if the fol-lowing conditions hold:

C1. For x=f1(x, w), that is, �=1, there exist �11, �12 ∈ KL,��us

11 , ��su

12 , �w11, �

w12 ∈ K, c11, c12 �0 such that for every

solution x(·) ∈ X

�su(x(t))��11(�su(x(t0)), t − t0)

⊕ ��us

11 (‖�us(x)‖[t0,t]) ⊕ �w11(‖w‖[t0,t])

⊕ c11, (8)

�us(x(t))��12(�us(x(t0)) ⊕ ��su

12 (‖�su(x)‖[t0,t])⊕ �w

12(‖w‖[t0,t]) ⊕ c12, −(t − t0)). (9)

C2. For x=f2(x, w), that is, �=2, there exist �21, �22 ∈ KL,��us

21 , ��su

22 , �w21, �

w22 ∈ K, c21, c22 �0 such that for every

solution x(·) ∈ X

�su(x(t))��21(�su(x(t0)) ⊕ ��us

21 (‖�us(x)‖[t0,t])⊕ �w

21(‖w‖[t0,t]) ⊕ c21, −(t − t0)), (10)

�us(x(t))��22(�us(x(t0)), t − t0)

⊕ ��su

22 (‖�su(x)‖[t0,t]) ⊕ �w22(‖w‖[t0,t])

⊕ c22. (11)

In view of the above, the switched seesaw system can beinterpreted as a stable/unstable system w.r.t. �su when �us(x)

Table 1Temporal representation of the switched seesaw system

�1 �2 �3 �4 · · ·� 1 2 1 2 · · ·�su ↘ ↗ ↘ ↗ · · ·�us ↗ ↘ ↗ ↘ · · ·↘, Stable; ↗, Unstable.

and w are regarded as inputs, and an unstable/stable w.r.t. �us

when �su(x) and w are regarded as inputs, see Table 1.The following theorem gives conditions under which a

switched seesaw system is ISpS.

Theorem 5. Consider the switched seesaw system on Xw.r.t. (�su, �us). Let min

1 , max1 , min

2 , max2 be positive con-

stants, called dwell time bounds, such that max1 �min

1 > 0,max

2 �min2 > 0, {tk}, k ∈ N a sequence of strictly increasing

switching times, and �k = tk − tk−1 a sequence of intervalssatisfying

�i ∈ [min1 , max

1 ], �i+1 ∈ [min2 , max

2 ], i = 1, 3, 5, . . . .

Assume there exist �i ∈ K∞; i = 1, 2 such that

�21(�11(r, min1 ), −max

2 )�(id − �1)(r) ∀r �r0, (12)

�22(�12(r, −max1 ), min

2 )�(id − �2)(r) ∀r �r0 (13)

for some r0 �0 and

��su

2 ◦ ��us

1 (r) < r ∀r > r0, (14)

��us

1 ◦ ��su

2 (r) < r ∀r > r0 (15)

for some r0 �0, where

��us

1 (r) := �−11 ◦ −1

1 ◦ �21(��us

11 (r) ⊕ ��us

21 (r), −max2 ), (16)

��su

2 (r) := �−12 ◦ −1

2 ◦ [(id − �2) ◦ ��su

12 (r) ⊕ ��su

22 (r)], (17)

and i ∈ K∞; i = 1, 2 are arbitrary functions such that id −i ∈ K. Then, the seesaw switched system (1) is ISpS on Xw.r.t. to �su ⊕ �us .

Remark 6. If the KL functions �ij are exponential, that is,

if �ij (r, t)� �ij re−�ij t , with �ij > 1, and the �i can be taken as�i (r) = �i r; � ∈ (0, 1), then inequalities (12)–(13) with 1 :=min

1 =max1 , 2 := min

2 =max2 are equivalent to the linear matrix

inequality (LMI)

��b, (18)

where � = [ �11−�12

−�21�22

], = (1, 2)′, and b = (b1, b2)

′ is apositive vector, i.e., b1, b2 > 0. This LMI together with the factthat > 0 imply the necessary condition

�12

�11

�21

�22< 1.

The above expression sets an upper bound on the ratio of �12�21(product of the rates of explosion) versus �11�22 (product ofthe rates of implosion).

A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008 2001

Proof of Theorem 5. We start by computing the evolution of�us(x(tk)); k=2, 4, 6, . . . . Condition (13) and Lemma 1 yield

�us(x(tk))� �2(�us(x(t0)), tk − t0) ⊕ ��su

2 (‖�su(x)‖[t0,tk])⊕ �w

2 (‖w‖[t0,tk]) ⊕ c2, (19)

where ��su

2 is defined in (17), c2 := �−12 ◦ −1

2 ((id − �2)(c12 ⊕��su

12 (r0) ⊕ �w12(r0) ⊕ r0) ⊕ c22), and �2 and �w

2 are KL andK functions, respectively, the form of which is not relevant. Ina similar manner, consider the evolution of �su(x(tk)). Condi-tion (12) and a straightforward reformulation of Lemma 1 forstable/unstable switched systems yield

�su(x(tk))� �1(�su(x(t0)), tk − t0) ⊕ ��us

1 (‖�us(x)‖[t0,tk])⊕ �w

1 (‖w‖[t0,tk]) ⊕ c1, (20)

where ��us

2 is defined in (16), c1 := �−11 ◦ −1

1 (�21(c11 ⊕c21, max

2 ) ⊕ (id − �2)(r0)), and �2 and �w2 are KL and K

functions, respectively. Notice in (19) and (20) the existenceof a cross-coupling term from �su(·) to �us(·). A straight-forward application of the small-gain theorem (Jiang, Teel, &Praly, 1994; Jiang & Wang, 2001) implies that (1) is ISpS w.r.t.�su ⊕ �us at t = tk , k = 2, 4, . . . if (14) is satisfied. The proofthat (1) is ISpS w.r.t. (�su, �us) for all t � t0 follows from thefact that �(t) defined in Lemma 1 is uniformly bounded. �

2.3. Seesaw control systems design

Equipped with the mathematical results derived, this sectionproposes a new methodology for the design of stabilizing feed-back control laws for nonlinear systems of the form

x = f (x, u, w), (21)

where x ∈ X ⊂ Rn is the state, u ∈ U ⊂ Rm is the controlinput, and w ∈ MW, W ⊂ Rnw is a disturbance signal. Inparticular, we seek to derive a switching control law for u thatwill render the resulting closed-loop system ISpS w.r.t. �(x)=|x|.

The first step consist of finding two measuring functions�su(x), �us(x) that satisfy the following detectability property:if ‖�su(x)⊕�us(x)‖ and ‖w‖ converge to zero as t → ∞, then|x(t)| tends also to zero as t → ∞. More precisely, (21) associ-ated with the outputs �su(x) and �us(x), must be input–output-to-state stable (IOSS) (Sontag & Wang, 1997) with respect tow, that is, there must exist a class KL function � and classK function � such that4

|x(t)|��(|x(t0)|, t − t0)

⊕ �(‖�su(x) ⊕ �us(x)‖[t0,t] ⊕ ‖w‖[t0,t]). (22)

The choices of �su(x) and �us(x) are strongly motivatedby the physics of the problem at hand, as the examples in

4 In fact, since we will only prove ISpS of the closed-loop system, itis sufficient that (21) be input–output-to-state practically stable (IOSpS) withrespect to w, that is, that there exist a class KL function �, a class K

function �, and a nonnegative constant c such that |x(t)|��(|x(t0)|, t − t0)⊕�(‖�su(x) ⊕ �us(x)‖[t0,t] ⊕ ‖w‖[t0,t]) ⊕ c.

Section 3 reveal. It will be later seen that in general �su(x)

and �us(x) are not functions of the whole state x, but rather ofdisjoint, yet complementary collections of the elements of x.

The next step involves the design of two feedback laws �1(x),�2(x), such that (21) together with the switching controller

u = ��(x), � ∈ {1, 2}becomes a switched seesaw system w.r.t. �su ⊕ �us . It is theneasy to show that if �(t) is chosen such that the conditions ofTheorem 5 hold and if the IOSS condition applies, then theclosed-loop system

x = f (x, ��(x), w)

is ISpS w.r.t. �(x) = |x|.The methodology proposed brings together tools from

Lyapunov-stability and switching system analysis. Its rationalecan be explained in simple terms, if one compares the strat-egy proposed against that of “classical” nonlinear controllerdesign by resorting to a single Lyapunov function, a task thatcan be impossible or extremely difficult at best. Instead, theinitial stabilization problem is somehow broken up into twoseparate, simpler problems. This is done by viewing the twomeasuring functions �su(x), �us(x) as candidate Lyapunovfunctions defined over two different collections of the elementsof the original state. With a proper choice of these functions,the task of finding the Lyapunov-based control laws �1(x),�2(x) becomes simple, as the examples in Section 3 will show.Switching between the control laws is the final ingredient thatwill yield overall ISpS of the complete system w.r.t. �(x)=|x|.

3. Stabilization of underactuated vehicles

The present section focuses on control system design andprovides the insight that goes into the choice of the switchingcontrol laws referred to before. This is done by addressing thenontrivial problem of underactuated underwater vehicle stabi-lization in the presence of disturbances and measurement noise.For the sake of clarity, the illustrative example proceeds in twosteps. First, the techniques are applied to the stabilization of so-called extended nonholonomic double integrator (ENDI), whichcaptures the kinematic and dynamic equations of a wheeledrobot. The methodology adopted is then extended to deal withan underwater vehicle by showing that its dynamics can be castin a form similar to (but more complex than) that of the ENDI.

3.1. The extended nonholonomic double integrator

The nonholonomic integrator introduced in Brockett (1983)captures (under suitable state and control transformations) thekinematics of a wheeled robot, displays all basic propertiesof nonholonomic systems, and is often quoted in the litera-ture as a benchmark for control system design, e.g., Bloch andDrakunov (1994), Hespanha (1996) and Astolfi (1998). How-ever, to tackle the realistic case where both the kinematics anddynamics of a wheeled robot must be taken into account, thenonholonomic integrator model must be extended. In Aguiar

2002 A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008

and Pascoal (2000), it is shown that the dynamic equations ofmotion of a mobile robot of the unicycle type can be trans-formed into the system

x1 = u1, x2 = u2, x3 = x1x2 − x2x1, (23)

where x := (x1, x2, x3, x1, x2)′ ∈ R5 is the state vector and

u := (u1, u2)′ ∈ R2 is a two-dimensional control vector. Sys-

tem (23) will be referred to as the ENDI.The ENDI falls into the class of control affine nonlinear sys-

tems with drift and cannot be stabilizable via a time-invariantcontinuously differentiable feedback law (cf., e.g., Aguiar,2002).

3.1.1. Seesaw control designWe now solve the problem of practical stabilization of the

ENDI system (23) subject to input disturbances v ∈ MV, V :={v ∈ R2 : ‖v‖[0,∞) � v} and measurement noise n ∈ MN,N := {n ∈ R5 : ‖n‖� n}, where v and n are finite but oth-erwise arbitrary. To this effect, the dynamics of (23) are firstextended to

x1 = u1 + v1, x2 = u2 + v2, x3 = x1x2 − x2x1, (24)

y = x + n, (25)

where y ∈ R5 is the vector of state measurements corrupted bynoise n. Following the procedure described in Section 2.3 wefirst introduce the measuring functions

�su := z2, z := x3 + �1x3, �1 > 0, (26)

�us := x21 + x2

1 + x22 + x2

2 (27)

and the feedback laws

�1(x) :=[ −k2x1

−k2x2 − k3

x1z

],

�2(x) :=[−k2x1 − k1(x1 − �)

−k2x2 − k1x2

], (28)

where �, k1, k2, k3 > 0. Notice that in order for the first con-trol law to be well defined, x1 must be bounded away from0. This justifies the need to required that all trajectories lie insome specific set X ⊂ {x ∈ R5 : |x1|� } for some > 0,as explained later. To provide some insight into (26)–(28), ob-serve that �su and �us can be viewed as positive semi-definiteLyapunov functions of z and (x1, x1, x2, x2)

′, respectively, thetime-derivatives of which are given by

�su = 2z[x1(u2 + v2 + k2x2) − x2(u1 + v1 + k2x1)], (29)

�us = 2x1(x1 + u1 + v1) + 2x2(x2 + u2 + v2). (30)

In the absence of input disturbances and measurement noise,it is straightforward to conclude that with the control law u =�1(x), the measuring function �su satisfies �su = −2k3�su

as long as x1 �= 0. This in turn implies that �su convergesexponentially fast to zero during the intervals of time in whichu = �1(x) is applied. In a similar vein, consider the evolution

of �us under the influence of the control law u=�2(x). Simplecomputations show that

x1 = −k2x1 − k1(x1 − �), x2 = −k2x2 − k1x2

and therefore �us converges exponentially fast to �2 during theintervals of time in which u = �2(x) is applied.

We now proceed with the seesaw control design as explainedin Section 2.3. For clarity of exposition all the proofs relatedto this example are at the end of this section. Following theprocedure described in Section 2.3, the first step is to show thatthe measuring functions satisfy the IOSS detectability property.

Proposition 7. The ENDI system together with the measuringfunctions �su(x) and �us(x) of (26) and (27), respectively, asoutputs, is IOSS.

The next step consists in showing that the closed-loop systemdescribed by the ENDI system and the control law

u = ��(x + n) (31)

defined in (28) verifies the seesaw conditions C1 and C2.

Proposition 8. Consider the ENDI system subject to input dis-turbances and measurement noise and the control law (31). Forevery > n�0, there are control gains such that the closed-loop system

x = f (x, ��(x + n), w)

with w = (v, n) verifies the seesaw conditions C1 and C2 w.r.t.�su ⊕ �us on X ⊂ {x ∈ R5 : |x1|� }.

It is now easy to conclude that if the switched seesaw con-troller (31) is applied to the ENDI system and a suitable se-lection of the dwell times 1, 2 is made such that conditions(12)–(15) hold, then the resulting closed-loop system is ISpS aslong as |x1(t)|� . It remains to state conditions under which|x1| is indeed bounded away from zero.

Proposition 9. Consider the closed-loop system that consistsof (24)–(25) and the feedback control law (28), (31). Given any > n�0, there exists � > 0 such that under a suitable choiceof the controller gains, for every initial condition x(t0) ∈ S0 :={x ∈ R5 : |(x1 − �, x1)|��}, the resulting solution x(·) lies inX ⊂ {x ∈ R5 : |x1|� }.

From Propositions 7–9 and Theorem 5 we finally conclude

Theorem 10. Consider the ENDI system subject to input dis-turbances and measurement noise, together with the switchingcontrol law (28), (31). Assume the conditions of Theorem 5 holdand let the initial conditions of the closed-loop system be in S0,defined in Proposition 9. Then, the switching controller stabi-lizes the state around a neighborhood of the origin, that is, itachieves ISpS of the closed-loop system on X w.r.t. �(x)=|x|.

Remark 11. It is always possible to make sure that x startsin S0 by applying u = �2(y) during a finite amount of time

A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008 2003

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

x1

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

x2

0 10 20 30 40 50 60 70 80 90 100

−10

−5

0

5

x3

time [s]

time [s]

time [s]

Fig. 1. Time evolution of state variables x1(t), x2(t), and x3(t).

before the normal switching takes over. In fact, from (24), (28)it is clear that with u=�2(y), (x1, x1) reaches S0 in finite time.From the particular evolution of (x1, x1), �us , �su during thistime interval, Proposition 7, and Theorem 10, we can also con-clude that with the procedure adopted the resulting switchingcontroller achieves ISpS of the closed-loop system on Rn w.r.t.�(x) = |x|.

Remark 12. We have eschewed the general problem of deriv-ing a procedure to choose the dwell time bounds (if they exist)that satisfy the conditions of Theorem 5. For the example con-sidered, however, this turns out to be simple because all theKL functions are exponential and the K functions are linear(see also Remark 6).

3.1.2. Simulation resultsNumerical simulations were done to illustrate the perfor-

mance of the switching controller proposed, when applied tothe ENDI. Figs. 1 and 2 show the time evolution of state vari-ables x1, x2, x3 and signals �su, �us, �, respectively, in thepresence of measurement noise and input disturbances. In thesimulations, the measurement noise is a zero mean uniform ran-dom noise with amplitude 0.1, and the input disturbances arev1=0.1 sin(t) and v2=0.1 sin(t+�/2), in the appropriate units.With the dwell-time constants set to 1 =min

1 =max1 =1.0 s and

2 = min2 = max

2 = 5.0 s, the assumptions of Theorem 10 wereverified to hold. Notice how the state variables converge to a

small neighborhood of the origin. Fig. 2 shows clearly, duringthe first switching intervals, how the behavior of �su and �us

capture the successive “stable/unstable” and “unstable/stable”cycles, respectively.

3.1.3. Proofs

Proof of Proposition 7. Let x=(x1, x1, x2, x2, x3)′. From (26)

using the variation of constant formula and taking norms, x3can be bounded as

|x3(t)|�e−k2(t−t0)|x3(t0)| + 1

k2

√‖�su(x)‖[t0,t).

Therefore,

|x(t)| =√

�us(x(t)) + x23 (t)�

√�us(x(t)) + |x3(t)|

��(|x(t0)|, t − t0) ⊕ �(‖�su(x)‖[t0,t] ⊕ ‖�us(x)‖[t0,t])

with �(r, t) := 2e−k2t r and �(r) := 2(1/k2 + 1)√

r thus satis-fying (22). �

Proof of Proposition 8. We start by showing that C1 is ob-served when � = 1. In the presence of measurement noise, thecontrol input u = �1(x + n) is given by

u1 = −k2(x1 + n4),

u2 = −k2(x2 + n5) − k3

x1 + n1(z + nz),

2004 A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008

0 5 10 15 20 25 30 35 40 45 50

0

100

200

300

wsu

wsu

time [s]

0 10 20 30 40 50 60 70 80 90 100

0

100

200

300

time [s]

0 5 10 15 20 25 30 35 40 45 50

0

1

2

3

time [s]

�

Fig. 2. Time evolution of measuring functions �su(t), �us(t), and the switching signal �(t).

where nz = x1n5 + n1x2 + n1n5 − x2n4 − n2x1 − n2n4 +k2n3 which, from the fact that ‖n‖[0,∞) � n, satisfies|nz|�4n

√�us + 2n2 + �n. For |x1| > > n, a bound for �su

is determined by computing the time-derivative of �su as

�su = − 2k3

1 + n1

x1

z2 + 2zx1

[−k2n5 − k3

x1 + n1nz + v2

]

− 2zx2[−k2n4 + v1]� − �11�su + �11

3��us

11 �us

+ �11

3c11, �1, �2 > 0,

where

�11 = 2

⎡⎢⎣ k3

1 + n

− k3

1 − n

�1

2− nk2 − �2v

⎤⎥⎦ ,

�11

3��us

11 = 2k3

1 − n

16

�1n2 + nk2 + 1

�2v and

�11

3c11 = 2

k3

1 − n

1

�1n2(2n + �)2.

Therefore,5

�su(t)�3�su(t0)e−�11(t−t0) ⊕ ��us

11 ‖�us‖[t0,t] ⊕ c11. (32)

Notice the absence of the term �w11 due to the fact that the

disturbances and noise are assumed to be bounded and theirbounds are known in advance.6 We now establish a bound for�us . Computing its time-derivative yields7

�us = 2(x1x1 + x2x2) − 2k2(x21 + x2

2 ) + 2x1(−k2n4 + v1)

+ 2x2(−k2n5 + v2) − 2x2k3

x1 + n1(z + nz)

��12�us + �12��su

12 �su + �12�v12(|v|) + �12c12,

where �12 = 2 + k3/( − n)+ 4n+ �3/2, �12��su

12 = k3/( − n),�12�v

12(r) = 2r2, and �12c12 = 4k22 n2 + [k3n

2/( − n)](2n +k2)

2/(2�3). Therefore, �us satisfies

�us(t)�4(�us(t0) ⊕ ��su

12 ‖�su‖[t0,t]⊕ �v

12(‖v‖[t0,t]) ⊕ c12)e�12(t−t0). (33)

5 We exploit the fact that for every class K function � and arbitrarypositive numbers r1, r2, . . . , rk we have �(r1+· · ·+rk)��(kr1)+· · ·+�(krn).

6 To simplify the control algorithm, we use explicitly in advancethe fact that the disturbances and noise are bounded by ‖v‖[0,∞) � v and‖n‖[0,∞) � n, respectively. It is possible to avoid this at the cost of introduc-ing the K function �v

11(r) and making ��su12 (r) a quadratic function.

7 We have used the fact that |x2||nz|�4n�us + n2(2n + k2)2/2�3 +�us�3/2, �3 > 0.

A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008 2005

From (32) and (33), we can now conclude that condition C1holds by identifying w in C1 with the input disturbance v.

Similarly, we check that condition C2 is satisfied when �=2.In this case, the control input u = �2(x + n) is given by

u1 = −k2(x1 + n4) − k1(x1 + n1 − �),

u2 = −k2(x2 + n5) − k1(x2 + n2).

Substituting the above equations into (29) yields

�su = 2z[x1(−k2n5 − k1n2 + v2)

− x2(−k2n4 − k1n1 + k1� + v1)]��21�su + �21�

�us

21 �us, �4, �5 > 0,

where �21 = (2(k2 + k1) + k1�)/�4 + 2v/�5 and �21��us

21 =�4[(k2 + k1)n + k1�] + �5v. Therefore,

�su �2(�su(t0) ⊕ ��us

21 ‖�us‖[t0,t])e�21(t−t0). (34)

To compute a bound for �us(t), we first observe that

�us = |�1|2 + |�2|2, (35)

where �1 := [x1, x1]′, �2 := [x2, x2]′, and �1, �2 satisfy

�i = A�i + Bdi, i = 1, 2 (36)

with A := [ 0−k1

1−k2

], B =[0, 1]′, d1 =−k2n4 −k1n1 +�+v1,and d2 =−k2n5 − k1n2 +v2. Let � > 0 be an arbitrary constantsuch that (A + (�/2)I ) is Hurwitz. Further, let P > 0 satisfy

(A + �

2I

)P + P

(A + �

2I

)′+ BB ′ �0. (37)

Define Vi := �′iP

−1�i and compute Vi to obtain

Vi = �′i (P

−1A + A′P −1)�i + 2�′P −1Bdi

� − (� − �6)Vi, Vi �|di |2�6

, �6 ∈ (0, �).

From the above, it follows that Vi(t)�Vi(t0)e−(�−�6)(t−t0) ⊕|di |2/�6, and therefore

�us � �22�us(t0)e−�22(t−t0) ⊕ �v

22(‖v‖[t0,t]) ⊕ c22,

where �22 = 3�max(P )/�min(P ), �22 = (� − �6), �v22(r) = 6r2,

and c22 = 6[((k2 + k1)n + k1�)2 + (k2 + k1)2n2]. �

Proof of Proposition 9. Let � := (�1, �2)′ := (x1 − �, x1)

′.For � = i; i ∈ 1, 2, consider the dynamics

� = Ai� + Bdi , (38)

where A1 := [ 00

1−k2

], A2 := [ 0−k1

1−k2

], B = [0, 1]′,|d1|�k2n+|v|, and |d2|�(k2+k1)n+|v|. Let ��1

, ��2> 0 be ar-

bitrary constants such that (A1−(��1/2)I ), and (A2+(��2

/2)I )

are Hurwitz. Further let Pi > 0; i = 1, 2 satisfy

(A1 − ��1

2I

)P1 + P1

(A1 − ��1

2I

)′+ BB ′ �0,

(A2 + ��2

2I

)P2 + P2

(A2 + ��2

2I

)′+ BB ′ �0,

and define Vi = �′P −1i �. A reasoning similar to the one used

in the last part of the proof of Proposition 8 shows that

V1(t)�(

V1(t0) + |d1|2��1

)e��1

(t−t0),

V2(t)�V2(t0)e−(��2

−�7)(t−t0) ⊕ |d2|2�7

, �7 ∈ (0, ��2).

Therefore, for � = 1,

|�(t)|�2

[�1/2

max(P1)

�1/2min(P1)

|�(t0)| ⊕ �1/2max(P1)‖d1‖[t0,t]

]e(��1

/2)(t−t0)

and for � = 2

|�(t)|� �1/2max(P2)

�1/2min(P2)

|�(t0)|e−[(��2−�7)/2](t−t0)

⊕ �1/2max(P2)

‖d2‖[t0,t]�7

.

Consider now the switched system with state � and externalinput d = (d1, d2)

′ satisfying the two inequalities above. Sincethe unstable mode ��1

can be made arbitrarily close to zero,simple but lengthy computations show that there is always achoice of controller gains k2 and k3 such that the conditions ofLemma 1 are met. Therefore, the switched system (38) is ISpS.In particular, �(t) satisfies |�(t)|��(|�(t0)|, 0)⊕ �(‖v‖[t0,t))⊕c��(�, 0) ⊕ �(v) ⊕ c, for some � ∈ KL, � ∈ K, and c�0.Therefore, choosing � large enough, with � > �(�, 0)⊕ �(v)⊕c ⊕ , it follows that |x1(t)| > for all t � t0. �

3.2. The underactuated AUV

This section addresses the problem of stabilizing an under-actuated AUV in the horizontal plane to a point, with a desiredorientation. The AUV has no side thruster, and its control in-puts are the thruster surge force u and the thruster yaw torquer . The AUV model is a second-order nonholonomic system,falls into the class of control affine nonlinear systems with drift,and there is no time-invariant continuously differentiable feed-back law that asymptotically stabilizes the closed-loop systemto an equilibrium point (Aguiar, 2002; Aguiar & Pascoal, 2001,2002).

2006 A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008

3.2.1. Vehicle modelingIn the horizontal plane, the kinematic equations of motion

of the vehicle can be written as

x = u cos � − v sin �, y = u sin � + v cos �, � = r,

where, following standard notation, u (surge speed) and v (swayspeed) are the body fixed frame components of the vehicle’svelocity, x and y are the cartesian coordinates of its center ofmass, � defines its orientation, and r is the vehicle’s angularspeed. Neglecting the motions in heave, roll, and pitch thesimplified dynamic equations of motion in the horizontal planefor surge, sway and heading yield (Aguiar, 2002)

muu − mvvr + duu = u, (39)

mv v + muur + dvv = 0, (40)

mr r − muvuv + drr = r , (41)

where the positive constants mu = m − Xu, mv = m − Yv ,mr = Iz − Nr , and muv = mu − mv capture the effect of massand hydrodynamic added mass terms, and du=−Xu−X|u|u|u|,dv = −Yv − Y|v|v|v|, and dr = −Nr − N|r|r |r| capture hydro-dynamic damping effects. The symbols u and r denote theexternal force in surge and the external torque about the z-axisof the vehicle, respectively. Since there is no thruster capable ofimparting a direct thrust on sway, the vehicle is underactuated.

3.2.2. Coordinate transformationConsider the global diffeomorphism given by the state and

control coordinate transformation (Aguiar, 2002)

x1 = �,

x2 = x cos � + y sin �,

x3 = −2(x sin � − y cos �) + �(x cos � + y sin �),

u1 = 1

mr

r + muv

mr

uv − dr

mr

r,

u2 = mv

mu

vr − du

mu

u + 1

mu

u

− u1x1x2 − x3

2+ vr − r2z2

that yields

x1 = u1, x2 = u2, x3 = x1x2 − x2x1 + 2v, (42)

and transforms the second-order constraint (40) for the swayvelocity into

mv v + mu

(x2 + x1

x1x2 − x3

2

)x1 + dvv = 0. (43)

Throughout the paper, q := col(x, v), x := (x1, x2, x3, x1, x2)′

and u = (u1, u2)′ denote the state vector and the input vector

of (42)–(43), respectively.

3.2.3. Seesaw control designWe now design a switching feedback control law for system

(42)–(43) so as to stabilize (in an ISpS sense) the state q arounda small neighborhood of the origin. We omit many of the details,because the methodology adopted for control system designfollows closely that adopted for the ENDI. A comparison of(42)–(43) with the ENDI system (23) shows the presence of anextra state variable v that is not in the span of the input vectorfield but enters as an input perturbation in the x3 dynamics. Wealso note that since dv/mv > 0, (43) is ISS when x is regarded asinput. Motivated by these observations, we select for measuringfunctions �su(·), �us(·) the ones given in (26)–(27). UsingProposition 7 and the fact that v satisfies

|v(t)|� �v|v(t0)|e−�v(t−t0) ⊕ �v(‖�su‖[t0,t] ⊕ ‖�us‖[t0,t])

for some �v, �v > 0, and �v(r) ∈ K we conclude that system(42)–(43) with �su and �us as outputs is IOSS.

Before we define the feedback laws �1(·), �2(·) we computethe time-derivatives of �su and �us to obtain

�su = 2z[x1(u2 + v2 + k2x2) − x2(u1 + v1 + k2x1 + 2�)]�us = 2x1(x1 + u1 + v1) + 2x2(x2 + u2 + v2),

where � := v + k2v satisfies the linear bound

|�|� �v|v| + ��us|�us | + �z|z|, (44)

for some positive constants �v , ��us, �z, and assuming that

‖(x1 −�, x1)‖[0,∞) ��, for a given � > 0. Comparing �su, �us

with (29)–(30) and using (44) together with the previous resultsfor the ENDI case, it is straightforward to conclude that if �1(·),�2(·) are selected as in (28), then (42)–(43) (with input dis-turbances v ∈ MV) in closed-loop with the seesaw controlleru=��(q +n), n ∈ MN, N := {n ∈ R6 : |n|� n} is a switchedseesaw system on X ⊂ {q ∈ R6 : |x1|� , |(x1 − �, x1)|��}.The existence of such a set X can be proved using the samearguments as in Proposition 9. These results are summarizedin the following theorem.

Theorem 13. Consider system (42)–(43) subject to input dis-turbances and measurement noise, and select � such that the as-sumptions of Theorem 5 hold. Then, there exists �0 > 0 such thatfor every initial condition in S0 := {q ∈ R6 : |(x1−�, x1)|��0}the closed-loop system with the seesaw controller u= �(q +n)

is ISpS on X w.r.t. �(q) = |q|.Remark 14. As in the ENDI case (see Remark 11), it is alwayspossible to make sure that q starts in S0 by applying duringa finite amount of time the control law u = �2(q) before thenormal switching takes over.

3.2.4. Simulation resultsSimulations were done using a dynamic model of the Sirene

AUV (Aguiar, 2002). Fig. 3 shows the simulation resultsfor a sample initial condition given by (x, y, �, u, v, r) =(−4 m, −4 m, �/4, 0, 0, 0). Zero mean uniform random noisewas introduced in every sensed signal: the x and y positions,the orientation angle �, the linear velocities u, v, and the an-gular velocity r. The amplitudes of the noise signals were set

A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008 2007

0 20 40 60 80 100 120 140 160 180 200

−4

−2

0

2

x [m

]

time [s]

0 20 40 60 80 100 120 140 160 180 200

−4

−2

0

2

y [m

]

time [s]

0 20 40 60 80 100 120 140 160 180 200

0

0.2

0.4

0.6

0.8

time [s]

ψ [rad

]

Fig. 3. Time evolution of the position x, y and orientation �.

to (0.5 m, 0.5 m, 5�/180, 0.1, 0.1, 0.1). There is also a smallinput disturbance: v1 = 10 sin(t), v2 = 10 sin(t + �/2). Thedwell-time constants were set to 1 = min

1 = max1 = 15 s and

2 = min2 = max

2 = 20 s. Clearly, the vehicle converges to asmall neighborhood of the target position while the headingangle is attracted to a neighborhood around zero.

4. Conclusions

A new class of switched systems was introduced andmathematical tools were developed to analyze their stabilityand disturbance/noise attenuation properties. A so-called see-saw control design methodology was also proposed that yieldsISS of these systems using switching. Applications were madeto the stabilization of the ENDI and to the dynamic model ofan underactuated AUV in the presence of input disturbancesand measurement noise.

Seesaw controllers explore switching between two modes,each one driving a different sub-component of the closed-loopstate to the origin. Recent work in Hespanha, Liberzon, andTeel (2005) suggests that instead of switching between differentmodes of operation, one could use the continuous flow to drivea subset of the state to the origin and instantaneous jumps todrive a complementary subset of the state to the origin. Theseideas were suggested by one of the anonymous reviewers andprovide a promising direction for future work.

References

Aguiar, A. P. (2002). Nonlinear motion control of nonholonomic andunderactuated systems. Ph.D. Thesis. Instituto Superior Técnico, TechnicalUniversity of Lisbon, Lisbon, Portugal.

Aguiar, A. P., & Pascoal, A. M. (2000). Stabilization of the extendednonholonomic double integrator via logic-based hybrid control. InProceedings of the 6th international IFAC symposium on robot control,Vienna, Austria.

Aguiar, A. P. & Pascoal, A. M. (2001). Regulation of a nonholonomicautonomous underwater vehicle with parametric modeling uncertaintyusing Lyapunov functions. In Proceedings of the 40th conference ondecision and control, Orlando, FL, USA.

Aguiar, A. P., & Pascoal, A. M. (2002). Global stabilization of anunderactuated autonomous underwater vehicle via logic-based switching.In Proceedings of the 41st conference on decision and control, Las Vegas,NV, USA.

Aicardi, M., Casalino, G., Bicchi, A., & Balestrino, A. (1995). Closed loopsteering of unicycle-like vehicles via Lyapunov techniques. IEEE Robotics& Automation Magazine, 2(1), 27–35.

Astolfi, A. (1998). Discontinuous control of the Brockett integrator. EuropeanJournal of Control, 4(1), 49–63.

Bloch, A., & Drakunov, S. (1994). Stabilization of a nonholonomic systemvia sliding modes. In Proceedings of the 33rd conference on decision andcontrol, Orlando, FL, USA.

Bloch, A. M., Reyhanoglu, M., & McClamroch, N. H. (1992). Control andstabilization of nonholonomic dynamic systems. IEEE Transactions onAutomatic Control, 37(11), 1746–1757.

Brockett, R. W. (1983). Asymptotic stability and feedback stabilization. In:R. W. Brockett, R. S. Millman, & H. J. Sussman (Eds.), Differentialgeometric control theory (pp. 181–191). Boston, USA: Birkhäuser.

2008 A. Pedro Aguiar et al. / Automatica 43 (2007) 1997–2008

Canudas-de-Wit, C., & SZrdalen, O. J. (1992). Exponential stabilizationof mobile robots with nonholonomic constraints. IEEE Transactions onAutomatic Control, 37(11), 1791–1797.

Dixon, W., Jiang, Z., & Dawson, D. (2000). Global exponential setpointcontrol of wheeled mobile robots: A Lyapunov approach. Automatica, 36,1741–1746.

Do, K. D., Jiang, Z. P., Pan, J., & Nijmeijer, H. (2004). A global output-feedback controller for stabilization and tracking of underactuated ODIN:A spherical underwater vehicle. Automatica, 40, 117–124.

Godhavn, J. M., & Egeland, O. (1997). A Lyapunov approach to exponentialstabilization of nonholonomic systems in power form. IEEE Transactionson Automatic Control, 42(7), 1028–1032.

Hespanha, J. P. (1996). Stabilization of nonholonomic integrators via logic-based switching. In Proceedings of the 13th World congress of internationalfederation of automatic control (Vol. E, pp. 467–472). S. Francisco, CA,USA.

Hespanha, J. P., Liberzon, D., & Teel, A. (2005). On input-to-state stabilityof impulsive systems. In Proceedings of the 44th conference on decisionand control, Seville, Spain.

Jiang, Z. P., Teel, A., & Praly, L. (1994). Small-gain theorem for ISS systemsand applications. Mathematics of Control, Signals, and Systems, 7, 95–120.

Jiang, Z.-P., & Wang, Y. (2001). Input-to-state stability for discrete-timenonlinear systems. Automatica, 37, 857–869.

Kolmanovsky, I., & McClamroch, N. H. (1996). Hybrid feedback lawsfor a class of cascade nonlinear control systems. IEEE Transactions onAutomatic Control, 41(9), 1271–1282.

Leonard, N. E. (1995). Control synthesis and adaptation for an underactuatedautonomous underwater vehicle. IEEE Journal of Oceanic Engineering,20(3), 211–220.

Liberzon, D. (2003). Switching in systems and control. Systems and control:Foundations and applications. Boston, MA: Birkhäuser.

Lizárraga, D. A., Aneke, N. P. I., & Nijmeijer, H. (2004). Robust pointstabilization of underactuated mechanical systems via the extended chainedform. SIAM Journal of Control Optimization, 42(6), 2172–2199.

M’Closkey, R. T., & Murray, R. M. (1997). Exponential stabilization ofdriftless nonlinear control systems using homogeneous feedback. IEEETransactions on Automatic Control, 42(5), 614–628.

Morin, P., & Samson, C. (2000). Control of nonlinear chained systems:From the Routh–Hurwitz stability criterion to time-varying exponentialstabilizers. IEEE Transactions on Automatic Control, 45(1), 141–146.

Morin, P., & Samson, C. (2003). Practical stabilization of driftless systemson lie groups: The transverse function approach. IEEE Transactions onAutomatic Control, 48(9), 1496–1508.

Morse, A. S. (1995). Control using logic-based switching. In A. Isidori (Ed.),Trends in control (pp. 69–114). New York, USA: Springer.

Pettersen, K. Y., & Egeland, O. (1999). Time-varying exponential stabilizationof the position and attitude of an underactuated autonomous underwatervehicle. IEEE Transactions on Automatic Control, 44, 112–115.

Pettersen, K. Y., & Fossen, T. I. (2000). Underactuated dynamic positioningof a ship—experimental results. IEEE Transactions on Control SystemsTechnology, 8(5), 856–863.

Prieur, C., & Astolfi, A. (2003). Robust stabilization of chained systems viahybrid control. IEEE Transactions on Automatic Control, 48(10), 1768–1772.

Samson, C. (1995). Control of chained systems: Application to path followingand time-varying point-stabilization of mobile robots. IEEE Transactionson Automatic Control, 40(1), 64–77.

Sontag, E. (1989). Smooth stabilization implies coprime factorization. IEEETransactions on Automatic Control, 34(4), 435–443.

Sontag, E., & Wang, Y. (1997). Output-to-state stability and detectability ofnonlinear systems. Systems & Control Letters, 29, 279–290.

Tell, A. R., Murray, R. M., & Walsh, G. C. (1995). Nonholonomic controlsystems: From steering to stabilization with sinusoids. InternationalJournal of Control, 62, 849–870.

Tomlin, C., Pappas, G. L., & Sastry, S. (1998). Conflict resolution for air trafficmanagement: A study in multiagent hybrid systems. IEEE Transactionson Automatic Control, 43(4), 509–521.

A. Pedro Aguiar received the Licenciatura,M.S. and Ph.D. in electrical and computer en-gineering from the Instituto Superior Técnico,Technical University of Lisbon, Portugal in1994, 1998 and 2002, respectively. From 2002to 2005, he was a post-doctoral researcher atthe Center for Control, Dynamical-Systems,and Computation at the University of Califor-nia, Santa Barbara. Currently, Dr. Aguiar holdsan Invited Assistant Professor position withthe Department of Electrical and ComputerEngineering, Instituto Superior Técnico, and a

Senior Researcher position with the Institute for Systems and Robotics,Instituto Superior Técnico (ISR/IST).His research interests include modeling, control, navigation, and guidance ofautonomous vehicles; nonlinear control; switched and hybrid systems; track-ing, path-following; performance limitations; nonlinear observers; the inte-gration of machine vision with feedback control; and coordinated/cooperativecontrol of multiple autonomous robotic vehicles.Further information related to Dr. Aguiar’s research can be found athttp://users.isr.ist.utl.pt/∼pedro.

João P. Hespanha received the Licenciatura inelectrical and computer engineering from theInstituto Superior Técnico, Lisbon, Portugal in1991 and the Ph.D. degree in electrical engi-neering and applied science from Yale Univer-sity, New Haven, Connecticut in 1998, respec-tively. He currently holds a Professor positionwith the Department of Electrical and ComputerEngineering, the University of California, SantaBarbara. From 1999 to 2001, he was an As-sistant Professor at the University of SouthernCalifornia, Los Angeles.

His research interests include hybrid and switched systems; the modeling andcontrol of communication networks; distributed control over communicationnetworks (also known as networked control systems); the use of vision infeedback control; stochastic modeling in biology; and the control of hapticdevices. He is the author of numerous technical papers and the PI and co-PIin several federally funded projects.Dr. Hespanha is the recipient of the Yale University’s Henry Prentiss BectonGraduate Prize for exceptional achievement in research in Engineering andApplied Science, a National Science Foundation CAREER Award, the 2005best paper award at the 2nd International Conference on Intelligent Sensingand Information Processing, the 2005 Automatica Theory/Methodology bestpaper prize, and the 2006 George S. Axelby Outstanding Paper Award. Since2003, he has been an Associate Editor of the IEEE Transactions on AutomaticControl.

António M. Pascoal received his Ph.D. inControl Science from the University of Min-nesota, Minneapolis, MN, USA in 1987. From1987 to 1988, he was a Research Scientist withIntegrated Systems Incorporated, Santa Clara,California were he participated in the devel-opment of advanced robotic systems for theUS Air Force and Army. He has held visitingpositions with the Department of ElectricalEngineering University of Michigan, USA, theDepartment Aeronautics and Astronautics, theNaval Postgraduate School (NPS), Monterey,

California, USA, and the National Institute of Oceanography, Goa, India. Hewas the coordinator of two EC funded projects (UBC and SOUV) that led tothe development of the first European civilian autonomous underwater vehi-cle (AUV) named MARIUS. He has been active in the design, development,and operation of autonomous underwater and surface vehicles for scientificapplications in the scope of several projects funded by the Portuguese Gov-ernment, the NSF/NPS (USA), the EU (projects DESIBEL, EXOCET, andASIMOV) and the Government of India/Agency for Innovation of Portugal(MAYA project). He is an Associate Professor of the Instituto Superior Téc-nico (IST, Lisbon, Portugal) and a Coordinator of the Dynamical Systems andOcean Robotics Lab of ISR (Institute for Systems and Robotics) of IST. Hisareas of expertise include Dynamical Systems Theory, Navigation, Guidance,and Control of Autonomous Vehicles, and Coordinated Motion Control ofMarine Robots. He has an interest in history of science and in the applicationof robotics to underwater archaeology.