Stability of switched seesaw systems with application to the stabilization of underactuated vehicles

Transcription

1 Stability of switched seesaw systems with application to the stabilization of underactuated vehicles A. Pedro Aguiar, João P. Hespanha, and António M. Pascoal Abstract This paper addresses the stabilization of a class of nonlinear systems in the presence of disturbances, using hybrid control. To this effect we introduce two new classes of switched systems and provide conditions under which they are input-to-state practically stable ISpS. These results lead to a control design methodology called switched seesaw control design that allows for the development of nonlinear control laws yielding input-to-state stability, using switching. The range of applicability and the efficacy of the methodology proposed are illustrated via two non-trivial design examples. Namely, stabilization of the extended nonholonomic double integrator ENDI and stabilization of an underactuated autonomous underwater vehicle AUV in the presence of input disturbances and measurement noise. I. INTRODUCTION There has been increasing interest in hybrid control in recent years, in part due to its potential to overcome the basic limitations to nonlinear system stabilization introduced by Brockett s celebrated result in the area of nonholonomic systems control [1]. Hybrid controllers that combine timedriven with event-driven dynamics have been developed by a number of authors and their design is by now firmly rooted in a solid theoretical background [], [3]. Inspired by the progress in the area, the first part of this paper offers a new design methodology for the stabilization of nonlinear systems in the presence of external disturbances by resorting to hybrid control. To this effect, two classes of switched systems are introduced: unstable/stable switched systems and switched seesaw systems. The first have the property of alternating between an unstable and a stable mode during consecutive periods of time. The latter can be viewed as the interconnection of two unstable/stable systems such that when one is stable the other is unstable, and viceversa. Conditions are given under which these systems are input-to-state practically stable ISpS. The results are then used to develop a control design framework called switched seesaw control design that allows for the solution of robust control problems using switching. To illustrate the scope of the new design methodology proposed, the second part of the paper solves the challenging problems of stabilizing the so-called extended nonholonomic double integrator ENDI and an underactuated autonomous Research supported by NSF grant ECS-4798, ARO grant DAAD , Portuguese FCT POSI programme under framework QCA III and by project MAYA-Sub of the AdI. Part of this research was done while the first author was with the Center for Control Engineering and Computation, University of California, Santa Barbara, CA A. Pedro Aguiar and António M. Pascoal are with the ISR/IST Institute for Systems and Robotics, Instituto Superior Técnico, Torre Norte 8, Av. Rovisco Pais, Lisboa, Portugal. {pedro,antonio}@isr.ist.utl.pt João P. Hespanha is with Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA , USA. hespanha@ece.ucsb.edu underwater vehicle AUV in the presence of input disturbances and measurement noise. These examples were motivated by the problem of point stabilization, that is, the problem of steering an autonomous vehicle to a point with a desired orientation. For underactuated vehicles, i.e., systems with fewer actuators than degrees-of-freedom, pointstabilization is particularly challenging because most of the vehicles exhibit second-order acceleration nonholonomic constraints. As pointed out in a famous paper of Brockett [1], nonholonomic systems cannot be stabilized by continuously differentiable time invariant static state feedback control laws. To overcome this basic limitation, a variety of approaches have been proposed in the literature [], [4] [15]. From a practical point of view, the above problem has been the subject of much debate within the ground robotics community. However, it was only recently that the problem of point stabilization of underactuated AUVs received special consideration in the literature [16] [19]. Point stabilization of AUVs poses considerable challenges to control system designers because the dynamics of these vehicles are complicated due to the presence of complex, uncertain hydrodynamic terms. These, in turn, require that model uncertainty be taken into account explicitly at the control design level. Furthermore, the models of the vehicles are normally secondorder nonholonomic and include a drift vector field that is not in the span of the input vector fields, thus precluding the use of input transformations to bring them to driftless form. In this paper, the problem of stabilizing an underactuated AUV in the horizontal plane is solved by resorting to the seesaw control design technique referred to before. For reasons of clarity, the technique is first applied to stabilize, in the presence of input disturbances and measurement noise, the so-called extended nonholonomic double integrator ENDI, which captures the kinematic and dynamic equations of a wheeled robot. The methodology adopted is then extended to deal with an AUV by showing that its dynamics can be cast in a form similar to but more complex than that of the ENDI. One of the key contributions of the paper is the fact that the solution proposed for point stabilization of an AUV addresses explicitly the existence of external disturbances and measurement errors. In a general setting this topic has only been partially addressed in the literature and in many aspects it still remains an open problem. Noteworthy exceptions are e.g., [9], [14], [15]. Due to space limitations, all the proofs are omitted. These can be found in []. Notation and definitions: denotes the standard Euclidean norm of a vector in R n and u I is the essential supremum norm of a signal u : I R n on an interval I [,. Let a b := max{a, b} and denote by M W the set of measurable, essentially bounded signals w :[t, W, where W R m. A function γ :[, [, is of class K γ K

2 if it is continuous, strictly increasing, and γ = and of class K if in addition it is unbounded. A function β : [, R [, is of class 1 KL if it is continuous, for each fixed t R the function β,t is of class K, and for each fixed r the function βr, t decreases with respect to t and βr, t as t. We denote the identity function from R to R by id, and the composition of two functions γ i : R R; i =1, in this order by γ γ 1.The acronym w.r.t. stands for with respect to. II. DWELL-TIME SWITCHING THEOREMS AND HYBRID CONTROL This section introduces and analyzes stability related results for two classes of systems that will be henceforth called unstable/stable and seesaw switched systems. A. Unstable/stable switched system Consider the switched system ẋ = f σ x, w, xt =x 1 where x X R n is the state, w M W is a disturbance, and σ :[t, {1, } is a piecewise constant switching signal that is continuous from the right and evolves according { to 1, t [t k 1,t k,kodd σt =, t [t k 1,t k,keven In, {t k } := {t 1,t,t 3,...} is a sequence of strictly increasing infinity switching times in [t, and t is the initial time. We assume that both f i ; i = 1, are locally Lipschitz w.r.t. x, w and that the solutions of 1 lie in X and are defined for all t t. Let ω : R n [t, be a continuous nonnegative real function called a measuring function. For a given switching signal σ, system 1 is said to be input-to-state practically stable ISpS on X w.r.t. ω if there exist functions β KL, γ K, and a nonnegative constant c such that for every initial condition xt and every input w M W such that the solution xt of 1 lies entirely in X, xt satisfies ωxt βωxt,t t γ w w [t,t] c. 3 for all t t. When X = R n, W = R m, ωx = x and c =, ISpS is equivalent to the by now classical definition of input-to-state stability ISS [1]. With respect to 1, assume that the following conditions hold: 1. Unstability σ = 1. For ẋ = f 1 x, w, there exist functions β 1 KL, γ1 w K, and a nonnegative constant c 1 such that for every initial condition xt and every input w M W for which the solution xt of 1 lies entirely in X, xt satisfies 3 ωxt β 1 ωxt γ1 w w [t,t] c 1, t t 4 1 Our definition of KL functions is slightly different from the standard one because the domain of the second argument has been extended from [, to R. This will allow us to consider the case βr, t which may grow unbounded as t. On a first reading, one can consider that X = R n. In this case, the reference to the set X is omitted. We will need the more general setting when we consider applications to the stabilization of underactuated vehicles. 3 Another alternative is to consider that xt satisfies ωxt β1 x ωxt, t t β1 w w [t,t], t t β1 c c1, t t with β1 x,βw 1,βc 1 KL. There is no loss of generality in considering 4, because one can always take β 1 r, t =β1 xr, t βw 1 r, t β1 c r, t with the advantage of introducing a less complicated notation. However, this may lead to more conservative estimates. t t. Notice how the term t t in the second argument of β 1 captures the unstable characteristics of the system.. Stability σ =. System ẋ = f x, w is ISpS on X w.r.t. ω, that is, for every initial condition xt and every input w M W such that the solution xt of system 1 lies entirely in X, xt satisfies ωxt β ωxt,t t γ w w [t,t] c, 5 t t, where β KL, γ w K, c. If conditions 1 above are met, we call 1 an unstable/stable switched system on X w.r.t. ω. The definition of a stable/unstable switched is done in the obvious manner. The following result provides conditions under which an unstable/stable switched system is ISpS. Lemma 1: Consider an unstable/stable switched system on X w.r.t. ω. Let t i ; i N be a sequence of strictly increasing switching times {t i } such that the differences between consecutive instants of times i := t i t i 1 satisfy β β1 r, k+1, k+ id αr, r r 6 for k =,, 4,..., and for some choice of class K function α and r. Then, system 1 is ISpS at t = t k, that is, xt satisfies the ISpS condition 3 at t = t k. Similarly, if β 1 β r, k, k+1 id αr, r r 7 for k =, 4, 6,..., and for some choice of class K function α and r, then system 1 is ISpS at t = t k+1.if either 6 or 7 hold and the differences between consecutive switching times i are uniformly bounded, then system 1 is ISpS. Remark 1: If c 1 = c = r =, ωx = x, X = R n, and W = R m and all the conditions of Lemma 1 above are met, then system 1 is ISS. Remark : If 4 5 hold with exponential class KL functions, i.e., β i r, t = ˆβ i re λit, i =1,, and α can be taken as αr =ˆαr; ˆα>, with id α K, then 6 7 become independent of r. In particular, 6 7 degenerate into k+ λ 1 k ln ˆβ 1 ˆβ ; k =,, 4,... λ λ 1 ˆα k+1 λ λ 1 k + 1 λ 1 ln 1 ˆα ˆβ 1 ˆβ ; k =, 4, 6,... B. Switched seesaw system This section introduces the key concept of switched seesaw system. To this effect, consider the switched system 1 where the signal σ :[t, {1, } is a piecewise constant switching signal, continuous from the right, and evolves according to. Given two measuring functions ω su, ω us and a set X R n we call 1 a switched seesaw system on X w.r.t. ω su,ω us if it is a stable/unstable system w.r.t. ω su when ω us x and w are regarded as inputs, and an unstable/stable w.r.t. ω us when ω su x and w are regarded as inputs, see Table I. More precisely, the following conditions must hold:

3 TABLE I TEMPORAL REPRESENTATION OF THE SWITCHED SEESAW SYSTEM σ 1 1 ω su ω us Stable Unstable C1. For ẋ = f 1 x, w, that is, σ =1, there exist β 11,β 1 KL, γ ωus 11,γωsu 1,γw 11,γ1 w K, c 11,c 1 such that for every solution x X ω su xt β 11 ω su xt,t t γ ωus 11 ω usx [t,t] γ11 w w [t,t] c 11, 8 ω us xt β 1 ωus xt γ ωsu 1 ω sux [t,t] γ1 w w [t,t] c 1, t t. 9 C. For ẋ = f x, w, that is, σ =, there exist β 1,β KL, γ ωus 1,γωsu,γw 1,γ w K, c 1,c such that for every solution x X ω su xt β 1 ωsu xt γ ωus 1 ω usx [t,t] γ1 w w [t,t] c 1, t t, 1 ω us xt β ω us xt,t t γ ωsu ω sux [t,t] γ w w [t,t] c. 11 The following theorem gives conditions under which a switched seesaw system is ISpS. Theorem 1: Let τ 1, τ be two positive constants called dwell times, {t k }; k N a sequence of strictly increasing switching times, and k = t k t k 1 a sequence of intervals satisfying 1 = 3 = 5 = = τ 1, = 4 = 6 = = τ. Assume there exist α i K ; i =1, such that β 1 β11 r, τ 1, τ id α1 r, r r, 1 β β1 r, τ 1,τ id α r, r r, 13 for some r and γ ωsu γ ωus 1 r <r, r >ˆr, 14 γ ωus 1 γ ωsu r <r, r >ˆr, 15 for some ˆr, where γ ωus 1 r :=α1 1 ρ 1 1 β 1 γ ω us 11 r γωus 1 r, τ, γ ωsu r :=α 1 ρ 1 [ id α γ ωsu 1 r γωsu r] and ρ i K ; i =1, are arbitrary functions such that id ρ i K. Then, the seesaw switched system 1 is ISpS on X w.r.t. to ω su ω us. Remark 3: If the KL functions β ij are exponential, that is, if β ij r, t = ˆβ ij re λijt, with ˆβ ij > 1, and α i can be taken as α i r = ˆα i r;ˆα i >, with id α i K, then conditions 1 13 imply the necessary condition λ 1 λ 1 < 1. λ 11 λ The above expression sets an upper bound on the ratio of λ 1 λ 1 product of the rates of explosion versus λ 11 λ product of the rates of implosion. Remark 4: If ˆr =, 14 and 15 are equivalent. If ˆr >, the same is also true but possibly with a larger value of ˆr. C. Seesaw control systems design Equipped with the mathematical results derived, this section proposes a new methodology for the design of stabilizing feedback control laws for nonlinear systems of the form ẋ = fx, u, w, 16 where x X R n is the state, u U R m is the control input, and w M W, W R nw is a disturbance signal. The first step consist of finding two measuring functions ω su x, ω us x such that 16 is measure-to-state practically stable MSpS w.r.t. ω su x ω us x, that is, there exist a class KL function β, a class K function γ, and a nonnegative constant c such that xt β xt,t t γ ω su x ω us x [t,t] w [t,t] c. 17 The choice of ω su x, ω us x is strongly motivated by the physics of the problem at hand, as the examples in Section 3 reveal. In fact, given the original state x, ω su x and ω us x will in general be functions of disjoint, yet complementary partitions of x. The next step involves the design of two feedback laws α 1 x, α x, such that 16 together with the switching controller u = α σ x, σ {1, } is a switched seesaw system w.r.t. ω su ω us. It is now easy to show that if σt is chosen such that the conditions of Theorem 1 hold and if the MSpS condition applies, then the closed-loop system ẋ = fx, α σ x, w, is ISpS w.r.t. ωx = x. III. STABILIZATION OF UNDERACTUATED VEHICLES A. The Extended nonholonomic double integrator In [1], the equations of the nonholonomic integrator system were introduced. The nonholonomic integrator displays all basic properties of nonholonomic systems and is often quoted in the literature as a benchmark for control system design. Under suitable state and control transformations, it captures the kinematics of a wheeled robot. However, to tackle the realistic case where both the kinematics and dynamics of a wheeled robot must be taken into account, the nonholonomic integrator model must be extended. In [11], it is shown that the dynamic equations of motion of a mobile robot of the unicycle type can be transformed into the system ẍ 1 = u 1, ẍ = u, ẋ 3 = x 1 ẋ x ẋ 1, 18 where x := x 1,x,x 3, ẋ 1, ẋ R 5 is the state vector and u := u 1,u R is a two-dimensional control vector. System 18 will be referred to as the extended nonholonomic double integrator ENDI. The ENDI falls into the class of control affine nonlinear systems with drift and cannot be stabilizable via a time-invariant continuously differentiable feedback law cf., e.g., [].

4 1 Seesaw control design: We now solve the problem of practical stabilization of the ENDI system 18 subject to input disturbances v M V, V := {v R : v v} and measurement noise n M N, N := {n R 5 : n ɛ}, where v and ɛ are finite but otherwise arbitrary. To this effect, the dynamics of 18 are first extended to ẍ 1 = u 1 + v 1, ẍ = u + v, ẋ 3 = x 1 ẋ x ẋ 1, 19 y = x + n where y R 5 is the vector of state measurements corrupted by noise n. Following the procedure described in Section II-C we first introduce the measuring functions ω su := z, z := ẋ 3 + λ 1 x 3,λ 1 > 1 ω us := x 1 +ẋ 1 + x +ẋ. and the feedback laws [ α 1 x := k ẋ 1 k ẋ k 3 x 1 z ] [ ],α x := kẋ 1 k 1x 1 κ k ẋ k 1x 3 where κ, k 1,k,k 3 >. To provide some insight into the choice of the relevant functions note that ω su and ω us can be viewed as positive semi-definite Lyapunov functions of z and x 1, ẋ 1,x, ẋ, respectively, the time-derivatives of which are given by ω su =z [ x 1 u + v + λ 1 ẋ x u 1 + v 1 + λ 1 ẋ 1 ] 4 ω us =ẋ 1 x 1 + u 1 + v 1 +ẋ x + u + v 5 In the absence of input disturbances and measurement noise, it is straightforward to conclude that with the control law u = α 1 x, the measuring function ω su satisfies ω su = k 3 ω su as long as x 1. This in turn implies that ω su converges exponentially fast to zero during the intervals of time in which u = α 1 x is applied. In a similar vein, consider the evolution of ω us under the influence of the control law u = α x. Simple computations show that ẍ 1 = k ẋ 1 k 1 x 1 κ, ẍ = k ẋ k 1 x and therefore ω us converges exponentially fast to κ during the intervals of time in which u = α x is applied. We now proceed with the seesaw control design as explained in Section II-C Proposition 1: The ENDI system together with the measuring functions ω su x and ω us x is measure-to-state stable 4 MSS w.r.t. ω su x ω us x. Proposition : The ENDI system 19 in closed-loop with u = α σ y 6 defined in 3 verifies conditions C1 and C of a switched seesaw system w.r.t. ω su ω us on X {x R 5 : x 1 δ} for some δ>. Proof: [Outline] We start by showing that C1 is observed when σ =1. In the presence of input disturbances and measurement noise, the control input u = α 1 x+n is given by u 1 = k ẋ 1 +n 4, u = k ẋ +n 5 k3 x 1+n 1 z+n z, n z := x 1 n 5 + n 1 ẋ + n 1 n 5 x n 4 n ẋ 1 n n 4 + k n 3 which, from the fact that n ɛ, satisfies n z 4 A system is MSS w.r.t. ω sux ω usx if it is MSpS w.r.t. ω sux ω usx and c = see 17. 4ɛ ω us +ɛ + λɛ. For x 1 >δ>ɛ, a bound for ω su is determined as ω su λ 11 ω su + λ 11 [ where λ 11 = k 3 1+ ɛ δ 11 ω us + λ 11 3 c 11, θ 1,θ > ] ɛk θ v, 3 ˆγωus k3 θ 1 1 ɛ δ k ɛ δ θ 1 ɛ + ɛk + 1 θ v, and λ11 3 c 11 = k3 1 ɛ δ Therefore, λ 11 3 ˆγωus 11 = 1 θ 1 ɛ ɛ + λ. ω su t 3ω su t e λ11t t ˆγ ωus 11 ω us [t,t] c 11 7 Notice the absence of the term γ11 w due to the fact that the disturbances and noise are assumed to be bounded and their bounds are known in advance 5. We now establish a bound for ω us. Computing its time-derivative yields ω us λ 1 ω us + λ 1ˆγ ωsu 1 ω su + λ 1 γ1 v +λ v 1 c 1, where λ 1 =+ k3 δ ɛ +4ɛ + θ3, θ 3 >, λ 1ˆγ ωsu 1 = k3 λ 1 γ1r v = r, and λ 1 c 1 = 4kɛ + k3ɛ δ ɛ Therefore, ω us satisfies δ ɛ, ɛ+k θ 3. ω us t 4 ω us t ˆγ ωsu 1 ω su [t,t] γ v 1 v [t,t] c 1 e λ 1t t. 8 Similarly, we check that condition C is satisfied when σ =. In this case, the control input u = α x + n is given by u 1 = k ẋ 1 + n 4 k 1 x 1 + n 1 κ, u = k ẋ +n 5 k 1 x +n. Substituting the above equations into 4 yields ω su λ 1 ω su + λ 1ˆγ ωus 1 ω us, θ 4,θ 5 > where λ 1 = k+k1+k1κ + v θ 4 θ 5 and λ 1ˆγ ωus 1 = θ 4 [k + k 1 ɛ + k 1 κ]+θ 5 v. Therefore, ω su ω su t ˆγ ωus 1 ω us [t,t] e λ 1t t. 9 To compute a bound for ω us t, we first observe that ω us = χ 1 + χ, where χ 1 := x 1, ẋ 1, χ := x, ẋ, and χ 1, χ satisfy χ i = Aχ i + Bd i, i =1, with A := [ ] 1 k 1 k, B =[, 1], d 1 = k n 4 k 1 n 1 + κ + v 1, and d = k n 5 k 1 n + v.letλ>be an arbitrary constant such that A+ λ I is Hurwitz. Further, let P>satisfy A + λ I P + P A + λ I + BB. Define V i := χ i P 1 χ i and compute V i to obtain V i = χ ip 1 A + A P 1 χ i +χ P 1 Bd i λ θ 6 V i, V i d i, θ 6,λ θ 6 From the above, it follows that V i t V i t e λ θ6t t di θ 6, and therefore ω us ˆβ ω us t e λt t γ v v [t,t] c, where ˆβ =3 λmaxp λ minp, λ =λ θ 6, γ v r =6r, and c =6 [ k + k 1 ɛ + k 1 κ +k + k 1 ɛ ]. 5 It is possible to avoid this at the cost of introducing the K function γ11 v r and making γωsu 1 r a quadratic function.

5 It is now easy to conclude that if the switched seesaw controller 6 is applied to the ENDI system and a suitable selection of the dwell times τ 1, τ is made such that conditions 1 15 hold, then the resulting closed-loop system is ISpS as long as x 1 t δ. It remains to state conditions under which x 1 is indeed bounded away from. Proposition 3: Consider the closed-loop system 19, 3 and 6. Let S := {x R 5 : x 1 t κ, ẋ 1 t µ}, forsomeµ>. Then, under a suitable choice of the controller gains, for every initial condition xt S, the resulting solution x lies in X {x R 5 : x 1 δ; δ>}. From Propositions 1 3 and Theorem 1 we finally conclude Theorem : Consider the ENDI system 19 subject to input disturbances and measurement noise, together with the switching control law 3, 6. Assume the conditions of Theorem 1 hold and let the initial conditions of the closed-loop system be in S, defined in Proposition 3. Then, the switching controller stabilizes the state around a neighborhood of the origin, that is, it achieves ISpS of the closed-loop system on X w.r.t. ωx = x. Remark 5: It is always possible to make sure that x starts in S because x 1, ẋ 1 can initially be brought as close as required to κ, by applying u = α x during a finite amount of time before the normal switching takes over. In fact, from 3 it is clear that with u = α x, x 1, ẋ 1 converges to κ,. Fig. 1. x 1 x x 3 ω 1 ω Time evolution of the state variables x 1 t, x t, andx 3 t σ Fig.. Time evolution of the measuring functions ω sut, ω ust, and the switching signal σt. Simulation results: Figures 1 show the simulations results. The measurement noise is a zero mean uniform random noise with amplitude.1, and the input disturbances is v 1 =.1sint and v =.1sint + π/.with the dwell-time constants set to τ 1 = 1. s and τ = 5. s, the assumptions of Theorem were verified to hold. Notice how the state variables converge to a small neighborhood of the origin. Fig. shows clearly, during the first switching intervals, how the behavior of ω su and ω us capture the successive stable/unstable and the unstable/stable cycles, respectively. B. The underactuated autonomous underwater vehicle Based on the results derived, this section addresses the problem of stabilizing an underactuated AUV in the horizontal plane to a point, with a desired orientation. The AUV has no side thruster, and its control inputs are the thruster surge force τ u and the thruster yaw torque τ r.theauv 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 feedback law that asymptotically stabilizes the closed-loop system to an equilibrium point []. 1 Vehicle Modeling: In the horizontal plane, the kinematic equations of motion of the vehicle can be written as ẋ = u cos ψ v sin ψ, ẏ = u sin ψ + v cos ψ, ψ = r, where, following standard notation, u surge speed and v sway speed are the body fixed frame components of the vehicle s velocity, x and y are the cartesian coordinates of its center of mass, ψ defines its orientation, and r is the vehicle s angular speed. Neglecting the motions in heave, roll, and pitch the simplified dynamic equations of motion for surge, sway and heading yield [] m u u m v vr + d u u = τ u, 3 m v v + m u ur + d v v =, 31 m r ṙ m uv uv + d r r = τ r, 3 where the positive constants m u = m X u, m v = m Y v, m r = I z Nṙ, and m uv = m u m v capture the effect of mass and hydrodynamic added mass terms, and d u = X u X u u u, d v = Y v Y v v v, and d r = N r N r r r capture hydrodynamic damping effects. Coordinate Transformation: Consider the global diffeomorphism given by the state and control coordinate transformation [] x 1 = ψ x = x cos ψ + y sin ψ x 3 = x sin ψ y cos ψ + ψ x cos ψ + y sin ψ u 1 = 1 τ r + muv uv dr r m r m r m r u = mv vr du u + 1 x 1x x 3 τ u u 1 + vr r z m u m u m u that yields ẍ 1 = u 1, ẍ = u, ẋ 3 = x 1 ẋ x ẋ 1 +v, 33 x 1 x x 3 m v v + m u ẋ +ẋ 1 ẋ 1 + d v v =. 34 Throughout the paper, q := colx, v, x := x 1,x,x 3, ẋ 1, ẋ and u = u 1,u denote the state vector and the input vector of 33 34, respectively.

6 3 Seesaw control design: We now design a switching feedback control law for system so as to stabilize in an ISpS sense the state q around a small neighborhood of the origin. A comparison of with the ENDI system 18 shows the presence of an extra state variable v that is not in the span of the input vector field but enters as an input perturbation in the x 3 dynamics. We also note that since dv m v >, 34 is ISS when x is regarded as input. Motivated by these observations, we select for measuring functions ω su, ω us the ones given in 1. Using Proposition 1 and the fact that v satisfies vt ˆβ v vt e λvt t γ v ω su [t,t] ω us [t,t] for some ˆβ v,λ v >, and γ v r Kwe conclude that system is MSS w.r.t. ω su ω us. Before we define the feedback laws α 1, α we compute the time-derivatives of ω su and ω us to obtain ω su =z [ x 1 u + v + k ẋ x u 1 + v 1 + k ẋ 1 +ν ] ω us =ẋ 1 x 1 + u 1 + v 1 +ẋ x + u + v, where ν := v + k v satisfies the linear bound ν ˆγ v v + ˆγ ωus ω us +ˆγ z z provided that x 1 κ, ẋ 1 µ, fora given µ>. Comparing ω su, ω us with 4 5 and using?? together with the previous results for the ENDI case, it is straightforward to conclude that if α 1, α are selected as in 3, then the following result holds: Theorem 3: For every initial condition in S := {q R 6 : x 1, ẋ 1 µ },forsomeµ >, and selecting σ such that the assumptions of Theorem 1 hold, system subject to input disturbances and measurement noise in closed-loop with the seesaw controller u = αq + n is ISpS on X w.r.t. ωq = q. 4 Simulation results: Simulations were done using a dynamic model of the Sirene AUV []. Figure 3 shows the simulation results for a sample initial condition given by x, y,ψ,u, v, r = 4 m, 4 m, π/4,,,. The amplitudes of the noise signals were set to.5 m,.5 m, 5π/18,.1,.1,.1. There is also a small input disturbance: v 1 =1sint, v =1sint + π/. The dwell-time constants were set to τ 1 =15s and τ =s. IV. CONCLUSIONS A new class of switched systems was introduced and mathematical tools were developed to analyze their stability and disturbance/noise attenuation properties. A so-called seesaw control design methodology was also proposed. To illustrate the potential of this control design methodology, applications were made to the stabilization of the ENDI and to the dynamic model of an underactuated AUV in the presence of input disturbances and measurement noise. Future research will address the generalization of the seesaw control design methodology to tackle a wider range of control problems. REFERENCES [1] R. W. Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory, R. W. Brockett, R. S. 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