A Simple Proof of Finite-Time Stabilizability of ... - Semantic Scholar
52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy
A simple proof of finite-time stabilizability of without drift systems by discontinuous feedback laws Chaker Jammazi Abstract— In this paper, we give a construction of discontinuous feedback laws stabilizing in finite-time all without drift systems. Our construction is based on “sampling control” defined by Clarke et al. [6] and on Bhat and Bernstein feedback laws for cascaded structures [3]. Moreover, our construction are extended to another family of systems as satellite with two controllers when stabilizability by pure discontinuous feedback laws fails.
I. INTRODUCTION In this paper, we present a new point of view for building stabilizing control laws for a class of nonlinear control systems which cannot be stabilized by regular state static feedback laws. As a benchmark example in this class is the Brockett’s integrator which given as follows x˙ 1 = u1 , x˙ 2 = u2 , x˙ 3 = x2 u1 .
x(t) ˙ = f (x(t)), x ∈ Rn x(t0 ) = x0 .
(1)
The state is given by x =: (x1 , x2 , x3 )0 ∈ R3 and the control is u := (u1 , u2 ) ∈ R2 . The system (1) models several physical situations. As shown in [4], the stabilization of (1) by regular state feedback laws is impossible; which make the stabilization of (1) by non-ordinary feedback law is challenging [13], [16], [19], [1], [7], [11], [24], [22], [9], [23]. In order to overcome the Brockett’s condition for many nonlinear controllable systems especially systems without drift, or systems with drift as some underactuated marine and aerospace systems such as the autonomous underwater vehicle, the ship, the rigid spacecraft, the airship, etc...; three approaches have been proposed: stabilization by discontinuous time-invariant feedback laws, time-varying stabilization and partial-finite time stabilization. In [13] we have shown that we can stabilize partially in finitetime the Brockett’s integrator -in terms of the state variables of a unicycle-like vehicle (reduced chained systems in R3 )by various approaches. The several stabilizing feedback laws are H¨older for the asymptotic partial stability, continuous and homogeneous for the finite-time partial stability and quasihomogeneous for finite-time partial stability by discontinuous feedbacks. This finite-time partial stability means that we can place in finite-time the unicycle in the equilibrium position, without taking into consideration its orientation. In [14] we have derived continuous or discontinuous feedback laws making all chained system finite-time stable with respect (n − 1) components while the last one remains This work was supported by LIM Laboratory of EPT. Chaker Jammazi is with Facult´e des Sciences de Bizerte, D´epartement de Math´ematiques, and LIM Laboratory of Ecole Polytechnique de Tunisie. Universit´e de Carthage, Tunisia. [email protected]
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constant for a large time. Moreover we have proved that last components is constant and cannot exceed a known values provided the initial condition is known in advance do not exceed this values. In this paper we progress in our construction and we give an explicit feedbacks that finite-time stabilize all nonholonomic chained systems. Our feedback laws are discontinuous in time or both discontinuous in state and time. The construction is based on part on the so called “sampling control” defined initially by Clarke et al. on the famous paper [6]. We will motivate our approach by the discussion on Brockett’s integrator, but before and for rigorous presentation of the problem, we introduce some definitions. We consider the Cauchy problem (2)
We consider the sampling schedule π = {ti }i≥0 of [0, +∞) we mean an infinite sequence 0 = t0 < t1 < t2 < ... with lim ti = ∞. We call i→∞
d(π) := supt≥0 (ti+1 − ti ). Next, we will recall the notion of generalized sampling solution initially introduced by Krasovskii and Subbotin, for more details, see [5]. The generalized sampling solutions are obtained as limits of solutions of a sequence of systems in which the control is piecewise constant. We consider the standard system of the form x˙ = f (x(t), k(t, x(t))), x(t0 ) = x0 ,
(3)
where k : R × Rn → Rm is, in general, a discontinuous function. Definition 1.1: [5], [6] An ε-trajectory associated to the Cauchy problem (3) and to the partition π = {t0 , t1 , ..., tN } of [t0 , T0 + a] with tN = t0 + a and dπ < ε, is a function obtained by iteratively solving the following integral equations ϕπ (t)
= + ϕπ (t) = +
x0 Rt
f (ϕπ (τ ), k(t0 , x0 )) dτ, t ∈ [t0 , t1 ], t0 ϕπ (ti ) Rt f (ϕπ (τ ), k(ti , ϕπ (ti ))) dτ, ti ∈ [ti , ti+1 ], ti (4) for all i = 1, ..., N − 1.
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Definition 1.2: [5], [6] A function ϕ : [t0 , t0 + a] → Rn is said to be a generalized sampling solution of x˙ = f (x), x(t0 ) = x0 if it is the uniform limit of a sequence of ε-trajectories as ε → 0. Now, we redefine the Krasovskii and Filippov solutions. We denote by co ¯ the convex closure and µ the usual Lebesgue measure in Rn , we have the following definitions. Definition 1.3: [5], [6] An absolutely continuous functions ϕ : [t0 , t0 + a] → Rn is said to be • a Krasovskii solution of (2) if it is a solution of the differential inclusion \ x˙ ∈ Kf (x) = cof ¯ (B(x, δ)) (5) δ>0
•
i.e. ϕ satisfies (5) for a.e. t ∈ [t0 , t0 + a], a Filippov solution of (2) if it is a solution of the differential inclusion \ \ x˙ ∈ Kf (x) = cof ¯ (B(x, δ) \ N ) (6)
holds for all (x1 (t0 ), x2 (t0 )) such that |x1 (t0 )| + |x2 (t0 )| < δ • there exists T = T (x(t0 )) > 0 called settling-time function, such that, x1 (t) = 0 for every t > t0 + T The control system x˙ = X(x, u), (7) X(0, x2 , 0) = 0, is p-partially stabilizable in finite- time by discontinuous feedback law, if there exists a discontinuous map u : Rn → Rm such for every x2 ∈ Rn−p , u(0, x2 ) = 0, and such that (0, 0) ∈ Rp × Rn−p is p-partially stable in finite-time for the closed loop system x˙ = X(x, u(x)). Of course, the case p = n corresponds to “complete” finitetime stability of the system (7). For simplicity, we denotes as in [18] by [z]r := sgn(z)|z|r , r > 0, z ∈ R where sgn is the sign function. II. C ONSTRUCTION OF THE STABILIZING FEEDBACK LAWS OF B ROCKETT ’ S INTEGRATOR
δ>0 µ(N )=0
i.e. ϕ satisfies (6) for a.e. t ∈ [t0 , t0 + a]. Definition 1.4: [6] Consider the control system x˙ = f (x, u), x ∈ Rn , u ∈ Rm where f is locally Lipschitz. Given a feedback k, a partition π, and an x0 ∈ Rn . For each i, i = 0, 1, 2, ..., recursively solve x(t) ˙ = f (x(t), k(x(ti ))), t ∈ [ti , ti +1] using as initial value x(ti ) the endpoint of the solution on the preceding interval (and starting with x(t0 ) = x0 ). The π− trajectory of x˙ = f (x, k(x)) starting from x0 is the function x(.) thus obtained. Now we recall the notion of finite-time partial stability. Of course, the meaning of the autonomous differential equation x˙ = X(x) with a piecewise continuous right-hand side is defined in the sense of Filippov (Def 1.3) and [10], [20], [21]). Let X : Rp × Rn−p ∼ = Rn → Rn , x = (x1 , x2 ) ∈ Rp × n−p n R 7→ X(x) ∈ R , be defined and discontinuous on a neighborhood of (0, 0) ∈ Rp × Rn−p . We assume that 1) X is measurable and locally bounded, 2) X(0, x2 ) = 0. We denote by x(., t0 , x0 ) a solution of x˙ = X(x) under the initial conditions x(t0 ) = x0 . The following definition is inspired from [21]. Definition 1.5 (p-finite-time partial stability): One says that (0, 0) ∈ Rp × Rn−p is p-partially stable in finite time for x˙ = X(x) if • the equilibrium (0, 0) is stable i.e. for each t0 ∈ R, ε > 0, there is δ(ε, t0 ) > 0 such that each solution x(., t0 , x0 ) of x˙ = X(x) with the initial data x0 s.t |x0 | < δ exists on [t0 , +∞) and satisfies the inequality |x(t, t0 , x0 )| < ε, ∀ t > t0 . •
In this section, we introduce the new concept called “partial planing and finite-time stability”. This means the following idea: consider system (1), over the time interval [0, t1 ] with t1 is sufficiently large, we take u1 = 1 i.e. the state x1 (t) follows the trajectory t + x1 (0) over the interval time [0, t1 ]. Then, the partial state (x2 , x3 ) of Brockett’s integrator becomes over [0, t1 ] x˙ 2 = u2 , x˙ 3 = x2 .
The main idea, in this case, is to stabilize in finite-time the state (x2 , x3 ) ”faster“ than x1 . More precisely, if we denotes by T0 the settling time of (x2 , x3 ), of course we have T0 6 t1 . Over [0, t1 ], the system (8) is now presented as a double integrator when Bhat and Bernstein feedback law can be applied [2]. By taking u2 := −c1 [x2 ]α − c2 [x3 ]α/(2−α) , where c1 > 0, c2 > 0 and α ∈ (0, 1). Then u2 stabilizes in finite-time (x2 , x3 ). Thus, we get x2 (t) = x3 (t) = 0, ∀ t ∈ [T0 , t1 ]. Now, by choosing the feedback u1 of the form: 1, if 0 6 t 6 t1 , u1 (x(t)) := (9) −k [x1 ]α , if t > t1 , where k > 0, then u1 stabilizes x1 in finite-time. |t1 + x1 (0)|1−α |x1 (t1 )|1−α = t1 + the Let τ1 := t1 + k(1 − α) k(1 − α) settling time of x1 . Hence, we have τ1 > t1 . By an adequate choice of α ∈ (0, 1) and an adequate gain k and “good” t1 one can get x3 (τ1 ) = 0. |x2 (0)|1−α + τ1 . Clearly T¯ > τ1 . Now, we define T¯ := c1 (1 − α) Let Ω the set defined by Ω := {(x2 , x3 ) : x2 = x3 = 0, t > T¯},
The convergence of the state x1 lim |x1 (t, t0 , x0 )| = 0,
t→+∞
(8)
and we will show that this manifold is invariant when the state x1 is finite-time stable. 1302
We have x1 (t) = 0 for every t ≥ τ1 . Then we get by integration of the solution x3 that Z t x3 (t) = x3 (τ1 ) + u1 (x(s))x2 (s)ds = 0 + 0 = 0,
and u2 (x) = −0.5 [x2 ]1/2 − 1.5 [x3 ]4/3 . Clearly the simulations show the effectiveness of the proposed feedback laws and show that the state (x1 , x3 , x3 )0 is finite-time stable.
τ1
because, in closed loop, u1 (x(t)) = 0 for every t ≥ τ1 . In this case the dynamic of x2 becomes x˙ 2 = −c1 sgn(x2 )|x2 |α which is naturally finite-time stable with settling time |x2 (0)|1−α . Then the set Ω is invariant. c1 (1 − α) In the next, we present a numerical algorithm of the design of stabilizing feedback laws (since the settling time is not estimated for homogeneous systems). 1) choosing t1 > 0 large enough such that on [0, t1 ], u1 (x) = c > 0 for example, 2) taking the feedback given by u2 := −c1 [x2 ]α − c2 [x3 ]α/(2−α) , 3) we take the feedback given in (9) and by choosing “good constants” α ∈ (0, 1), an adequate gain k and “good” t1 one can get x3 (τ1 ) = 0 where τ1 := t1 + |x1 (t1 )|1−α > t1 . k(1 − α) In this case we have x1 (t) = x2 (t) = x3 (t) = 0 for |x2 (0)|1−α t > T¯ := + τ1 . c1 (1 − α) Instead of the above numerical algorithm, the next proposition gives a solution. Proposition 2.1: Let α and β be two constants in (0, 1). Let t1 > 0 be a large real number and three nonnegative constants a, c1 , c2 , c3 and c02 . Then, the following discontinuous feedback laws c1 , if 0 6 t 6 t1 , u1 (x(t)) := (10) −a [x1 ]α , if t > t1 ,
u2 (x(t)) :=
α −c2 [x2 ]α − c3 [x3 ] 2−α , if 0 6 t 6 t1 ,
−c02 [x2 ]β , if t > t1 ,
(11) system (1) is finite-time stable. Proof: Over the time interval [0, t1 ], the system described by (x2 , x3 ) is seen as a double integrator which is finite-time stable over [0, t1 ] by (11). Thus we get a settling time t0 6 t1 such that x2 (t) = x3 (t) = 0 for all t ∈ [t0 , t1 ]. For t > t1 , the feedback given in (10) stabilizes the state x1 in finite-time with settling time τ1 > t1 . Let remark that, under the feedback (11) we have x2 (t) = 0 for all t > t1 , now by writingRx3 (t) with integral representation t we get x3 (t) = x3 (t1 ) + t1 u1 (x(s))x2 (s)ds = 0 + 0 = 0 this shows also x3 (t) = 0 for every t > t1 . This completes the proof. Numerical Simulations To validate our result, we propose the following simulations, when the initial condition is x0 = (1.5, −2.5, 3.5)0 , and the feedback controllers are given by 1, if 0 6 t 6 20, u1 (x(t)) := −20 [x1 ]1/2 , if t > 20,
Trajectories of the state x1 , x2 and x3 .
Motion of the unicycle in the plane (x2 , x3 ). A. Discontinuous feedback laws with respect to time and states The following proposition shows that we can stabilize the system (1) by discontinuous state feedback laws with respect to time and states. The proof of the result is identical to the first case using discontinuous Orlov’s feedback laws for double integrator [21]. Proposition 2.2: Let α and β be two constants in (0, 1), Let t1 > 0 large and three nonnegative constants a, c1 , c2 , c3 and c02 such that c3 > c2 . Then under the following discontinuous feedback laws c1 > 0 if 0 6 t 6 t0 , u1 (x(t)) := −a sgn(x1 ), if t > t0 , −c2 sgn(x2 ) − c3 sgn(x3 ) if 0 6 t 6 t0 , u2 (x(t)) := −c02 sgn(x2 ), if t > t0 , stabilize in finite-time the system (1). Proof: The proof is identical to the proof of Proposition 2.1. The simulation result shows the effectiveness of our feedback laws. In this simulation the initial condition is x0 = (1.5, −2.5, 3.5)0 and the feedback laws are 1 if 0 6 t 6 19, u1 (x(t)) := −100 sgn(x1 ), if t > 19, and u2 (x) = −5 sgn(x2 ) − 20 sgn(x3 ).
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system (13) is globally finite-time stable through continuous state feedback. The construction of the finite-time stabilizing feedback laws of (13) is given by various method. The recent one is given by Moulay in [18] which generalize the result given by Bhat and Bernstein in [3, Theorem 8.1, pp123]. This feedbacks is given by [18, Proposition 11, pp. 1384] by the following: Let k := (k1 , ..., kn−1 ) ∈ Rn−1 such that sn−1 + kn−1 sn−2 + 1 , 1] ... + k2 s + k1 is Hurwitz. Then there exists ε ∈ [1 − n−2 and εi for 1 6 i 6 (n − 1) such that for all α ∈ (1 − ε, 1), ki0 ∈ (ki − εi , ki + εi ). Then (13) is globally finitetime stable under the continuous feedback control
Trajectories of the state x1 , x2 and x3 .
0 [xn−1 ]αn−1 , u ¯2 (x(t)) = −k10 [x1 ]α1 − . . . − kn−1
where α1 , ..., αn−1 satisfy αi αi+1 αi−1 = , i = 2, . . . , (n − 1), 2αi+1 − αi
(14)
(15)
with αn = 1 and αn−1 = α.
Motion of the unicycle in the plane (x2 , x3 ).
The black part in the simulation is explained by the so called chattering phenomenon which is due to the presence of sign function in these feedbacks. III. G ENERAL CHAINED SYSTEMS Recall that chained systems are described by the set of equations x˙ 1 x˙ 2
x˙ n−2 x˙ n−1 x˙ n
= = . . . = = =
x2 u1 x3 u1
In this case, and just after the time t1 , we turn to stabilize the state x1 by taking 1 if 0 6 t 6 t1 , u1 (x(t)) := −β [xn ]α , β > 0, if t > t1 . |xn (t1 )|1−α the settling time of β(1 − α) xn , then we have the following proposition. |xn−1 (0)|1−α Proposition 3.1: Let T0 := τ1 + 0 = τ1 + kn−1 (1 − αn−1 ) |xn−1 (0)|1−αn−1 , then the manifold defined by 0 kn−1 (1 − α) If we choosing τ1 := t1 +
Ω := {xi , 1 6 i 6 (n − 1) : xi (t) = 0, ∀ t > T0 } (12) xn−1 u1 u2 u1 ,
where the state is x := (x1 , x2 , ..., xn ) ∈ Rn and u := (u1 , u2 ) ∈ R2 is the control. The main contribution in this section is the building of discontinuous feedback laws making the system (12) in closed loop finite-time stable. The main idea for this construction is to planing the state xn over a “large” interval time [0, t1 ] and to stabilize in finite-time the partial state (x1 , x2 , ..., , xn−1 )0 ∈ Rn−1 “faster” than xn . At this stage, we put the state xn in finite-time by the action of u1 . Similarly to first case n = 3, over a time interval [0, t1 ] we take the path xn (t) = t + xn (0) i.e. u1 = 1. In this case, the partial-chained system becomes in cascaded integrators form with length (n − 1) : x˙ 1 = x2 , x˙ 2 = x3 , ..., x˙ n−1 = u2 .
(13)
Since (13) is controllable, because the system (13) satisfies Kalman’s condition. Then by using [3, Theorem 8.1, pp 123],
is invariant under the semi-flow of the system (13) in closed loop. Proof: Since for all t > T0 the control law u1 (x) vanishes because T0 is bigger than the settling time of xn , then for all 1 6 j 6 n − 2, we have Rt xj (t) = xj (T0 ) + T0 xj−1 (s)u1 (x(s))ds Rt = 0 + T0 0, = 0. Now we turn to see xn−1 . We have after the settling time T0 , the state xn−1 satisfies the equation 0 x˙ n−1 = −kn−1 sgn(xn−1 )|xn−1 |αn−1 ,
which is naturally finite-time stable with settling time |xn−1 (0)|1−α , then the manifold Ω is an invariant set. 0 kn−1 (1 − α) Proposition 3.2: The following discontinuous state feedback laws c > 0, if 0 6 t 6 t1 , u1 (x(t)) := (16) −a sgn(xn )|xn |α , if t > t1 ,
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Clearly, the dynamic of ψ is equivalent to the following u2 (x(t)) :=
¯2 (x(t)), if 0 6 t 6 t1 , u
ψ (3) (t) = u2 ,
(20)
0 −kn−1 sgn(xn−1 )|xn−1 |α , if t > t1 , (17) stabilize in finite-time chained system (12). Proof: The proof is similar to the proof of Proposition 2.1.. The following Theorem extends Proposition3.2 to all multichained systems [19]. Theorem 3.3: Let m be a positive integer, let n1 , ...nm be m X m nonnegative integers such that n = 1 + m + nj . The
j=1
following m-chain single-generator chained form: z˙1,0 = v1 , z˙1,1 = v0 z1,0 , .. .
z˙2,0 = v2 , z˙2,1 = v0 z2,0 , .. .
...
by using feedbacks cited in [3, Theorem 8.1, pp123], for triple integrator, and improved recently in [18, Proposition 11, pp. 1384], we can put in finite-time the states ω2 , ω3 and ψ in zero. This feedback can be given by 11 11 3 3 (21) u ¯2 (x(t)) = −[ψ]11/38 − [ω3 ] 29 − [ω2 ] 20 . 2 2 In this case, there exists T0 6 t1 such that for all t ∈ [T0 , t1 ], ω2 (t) = ω3 (t) = ψ(t) = 0. Now we turn to study the variables ω1 , φ and θ. Clearly after T0 , the dynamic of these states becomes ω˙ 1 = u1 φ˙ = ω1 (22) z˙m,0 = vm , ˙ θ = 0, z˙m,1 = v0 zm,0 , .. .
z˙1,n1 = v0 z1,n1 −1 , z˙2,n2 = v0 z2,n2 −1 , z˙n = v0 ,
z˙m,nm = v0 zm,nm −1 , third equation show that θ remains constant for a large time. The states ω1 and φ can be controlled by is finite-time stabilizable under discontinuous controllers (the 0, if 0 ≤ t 6 t1 , last component of the state being zn ). u1 (x(t)) := Proof: The considered system is a collection of mα chain of integrators, then the construction of discontinuous −c1 [ω1 ]α − c2 [φ] (2−α) , if t > t1 , stabilizing feedback law of the multi-chained system of Now by choosing the settling time T is t , we show that 0 1 Murray and Sastry follows from Proposition 3.2. the set Ω defined by IV. A PPLICATION TO PARTIAL ATTITUDE CONTROL OF
Ω := {(ω2 , ω3 , ψ) : ω2 (t) = ω3 (t) = ψ(t) = 0, ∀ t > t1 },
SATELLITE
The problem of attitude control of satellite with two controllers is well known. Let us recall that this system is not stabilizable in usual sense by regular state feedback laws. A great effort has been done to overcome this problem, see [8], [17], [12], [15], and the references therein. In this section we shall solve this question by using finite-time discontinuous controllers. This system is presented in the following configuration ω˙ 1 = u1 ω˙ 2 = u2 ω˙ 3 = ω1 ω2 (18) φ˙ = ω1 + (ω2 sinφ + ω3 cosφ) tgθ ˙ θ = ω2 cosφ − ω3 sinφ ψ˙ = ω2 sinφ + ω3 cosφ . cosθ The state is given by x := (ωi , φ, θ, ψ)0 ∈ R6 , i = 1, 2, 3 and u = (u1 , u2 ) ∈ R2 the control. Let be t1 > 0 a large time. Over the time interval [0, t1 ] we consider for the partial state (ω1 , φ, θ)0 the following path: ω1 (t) = c1 > 0, φ(t) = 0 and θ(t) = c3 ∈ (0, π/2). The objective is to stabilize in finite-time the state (ω2 , ω3 , ψ)0 faster than (ω1 , φ, θ)0 . Over [0, t1 ], the dynamic of (ω2 , ω3 , ψ)0 becomes ω˙ 2 = u2 ω˙ 3 = ω2 (19) ω3 ψ˙ = . cos c3
is invariant. Indeed: for every t > t1 we have φ(t) = ω2 (t) = 0, then we get Z t ω3 (t) = ω3 (t1 ) + ω1 (s) ω2 (s) ds = 0 + 0 = 0. t1
˙ Therefore we get ψ(t) = 0 for every t > t1 , this implies that ψ(t) = ψ(t1 ) = 0. In the end, for t > t1 , the state ω2 satisfies ω˙ 2 = −1.5sgn(ω2 )|ω2 |11/20 which is finite-time stable. Finally, ˙ = 0 for t > t1 which the state θ satisfies the equation θ(t) implies that θ(t) remains constant for a large time. Thus we have proved the following proposition. Proposition 4.1: Under the following discontinuous feedback laws c > 0, if 0 ≤ t 6 t1 , u1 (x(t)) := (23) α −c1 [ω1 ]α − c2 [φ] (2−α) , if t > t1 , and u2 (x(t)) :=
¯2 (x(t)), if 0 ≤ t 6 t1 , u
(24)
11
− 23 [ω2 ] 20 , if t > t1 ,
the system (18) is five-partially finite-time stable. More precisely, the partial state (ω1 , ω2 , ω3 , φ, ψ)0 is finite-time stable and θ(t) is constant for a large time t. Numerical Simulations The following simulations show the effectiveness of our feedbacks. The initial condition is
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x0 = (.5, 0.7, 1.8, 0, 0.5, 1.9)0 , and in the feedbacks (2324): t1 = 30, c = 0 and c1 = c2 = 2. Clearly the partial state (ω1 , ω2 , ω3 , φ, ψ)0 is finite-time stable and θ(t) ' −3 for t > 40.
Trajectories of the state ω1 , ω2 and ω3 .
Trajectories of the state φ, θ and φ. Remark 4.2: • The finite-time partial stabilization of the satellite is sufficient, since (18) is an example of affine control systems when Coron’s et al. [10] fails for the stabilizability by pure discontinuous feedback laws. • The above proposition shows that with the action of the proposed feedbacks we can place in finite-time the satellite in the equilibrium position without taking into consideration his orientation θ with respect to second axis. V. C ONCLUSION In this paper, the problem of finite-time stabilizability of chained and multi-chained system is solved. Our consideration gives a new alternative to overcome the obstruction of Brockett’s condition for stabilizability by regular feedback laws. The established stabilizing feedbacks are discontinuous both in time and states. Moreover, according to definition given for “sampling control” in [6], the solutions are as well as defined and have meaning for the two classes of constructed feedback laws. As defined, this idea combined the planning motion and finite-time stability theory and can be applied to solve the partial orientation of satellite with two controllers, and shows the effectiveness of our approach. R EFERENCES
[2] B HAT, S. P., AND B ERNSTEIN , D. S. Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Trans. Automatic Control 43, 5 (1998), 678–682. [3] B HAT, S. P., AND B ERNSTEIN , D. S. Geometric homogeneity with applications to finite-time stability. Math Control Signals Systems 17 (2005), 101–127. [4] B ROCKETT, R. W. Asymptotic stability and feedback stabilization. Differential geometric control theory, Progress in Math. 27 (1983), 181–191. [5] C ERAGIOLI , F. M. Discontinuous Ordinary Differential Equations and Stabilization. Tesi di dottorato di ricerca in matematica, Consorzio delle universit‘a di Cagliari, Firenze, Modena, Perugia e Siena, 1999. [6] C LARKE , F. H., ABD E. D S ONTAG , Y. S. L., AND S UBBOTION , A. I. Asymptotic controllability implies feedback stabilization. IEEE Trans. Automat. Control 42 (1997), 1394–1407. [7] C ORON , J.-M., AND D ’A NDR E´ A N OVEL , B. Smooth stabilizing timevarying control laws for a class of nonlinear systems. Applications to mobile robots. In IFAC Nonlinear Control Systems Design (Bordeaux, France, 1992), M. Fliess, Ed. 1992, pp. 413–418. [8] C ORON , J.-M., AND K ERA¨I , E. Y. Explicit feedbacks stabilizing the attitude of a rigid spacecraft with two torques. Automatica 32 (1996), 669–677. [9] C ORON , J.-M., AND P OMET, J.-B. A remark on the design of timevarying stabilizing feedback laws for controllable systems without drift. In IFAC Nonlinear Control Systems Design (Bordeaux, France, 1992), M. Fliess, Ed. 1992, pp. 397–401. [10] C ORON , J.-M., AND ROSIER , L. A relation between continuous timevarying and discontinuous feedback stabilization. J. Math. Systems Estimation and Control 4 (1994), 67–84. [11] DE W IT, C. C., AND S ØRDALEN , O. J. Exponential stabilization of mobile robots with nonholonomic constraints. IEEE Trans. Aut. Contr. 37, 11 (November 1992), 1791–1797. [12] JAMMAZI , C. Backstepping and Partial Asymptotic Stabilization. Applications to Partial Attitude Control. Inter. J. of Control, Automation, and Systems 6, 6 (2008), 859–872. [13] JAMMAZI , C. A discussion on the H¨older and finite-time partial stabilizability of Brockett’s Integrator. ESAIM: Control, Optimisation and Calculus of Variations 18, 2 (2012), 360–382. [14] JAMMAZI , C. Continuous and discontinuous homogeneous feedbacks finite-time partially stabilizing controllable multi-chained systems. Submitted to SIAM Control and Optimization (2013). [15] JAMMAZI , C., AND Z AGHDOUDI , M. On the rational stabiliy of autonomous dynamical systems. Applications to chained systems. Appl. Math. Comput. 219 (2013), 10158–10171. [16] M’C LOSKEY, R. T. Exponential stabilization of driftless nonlinear control systems. Ph.D Thesis, Departement of Mechanical Engineering California Institute of Technology, 1994. [17] M ORIN , P., AND S AMSON , C. Time-varying exponential stabilization of a rigid spacecraft with two control torques. IEEE Trans. Autom. Cont. 42 (1997), 528–534. [18] M OULAY, E. Stability and stabilization of homogeneous systems depending on a parameter. IEEE Transactions on Automatic Control 54, 6 (2009), 1342–1385. [19] M URRAY, R., L I , Z., AND S ASRTY, S. S. A Mathematical Introduction to Robotic Manipulation, CRC ed. Library of Congress Cataloging-in-Publication Data, 1993. [20] O RLOV, Y. Finite time stability and robust control synthesis of uncertain switched systems. SIAM J. Control Optim. 43, 4 (2005), 1253–1273. [21] O RLOV, Y. Discontinuous systems - Lyapunov Analysis and Robust Synthesis under Uncertainty Conditions, first ed. Communications and Control Engineering. Springer-Verlag, London, 2009. [22] P OMET, J. B. Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift. Syst. and control lett. 18 (1992), 147–158. [23] S AMSON , C. Velocity and torque feedback control of a nonholonomic cart. In Proceeding of International Workshop on Nonlinear and Adaptive Control, vol. 162. Springer-Verlag, November 1991, ch. Advanced Robot Control, pp. 125–151. [24] S AMSON , C. Control of chained systems: Application to path following and time-varying point-stabilization of mobile robots. IEEE Trans. Autom. Cont. 40, 1 (1995), 64–77.
[1] A STOLFI , A. Asymptotic stabilization of nonholonomic systems with discontinuous control. Ph. D Thesis, Swiss Federal Institute of Technology Zurich, 1996.
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A simple proof of finite-time stabilizability of without drift systems by discontinuous feedback laws Chaker Jammazi Abstract— In this paper, we give a construction of discontinuous feedback laws stabilizing in finite-time all without drift systems. Our construction is based on “sampling control” defined by Clarke et al. [6] and on Bhat and Bernstein feedback laws for cascaded structures [3]. Moreover, our construction are extended to another family of systems as satellite with two controllers when stabilizability by pure discontinuous feedback laws fails.
I. INTRODUCTION In this paper, we present a new point of view for building stabilizing control laws for a class of nonlinear control systems which cannot be stabilized by regular state static feedback laws. As a benchmark example in this class is the Brockett’s integrator which given as follows x˙ 1 = u1 , x˙ 2 = u2 , x˙ 3 = x2 u1 .
x(t) ˙ = f (x(t)), x ∈ Rn x(t0 ) = x0 .
(1)
The state is given by x =: (x1 , x2 , x3 )0 ∈ R3 and the control is u := (u1 , u2 ) ∈ R2 . The system (1) models several physical situations. As shown in [4], the stabilization of (1) by regular state feedback laws is impossible; which make the stabilization of (1) by non-ordinary feedback law is challenging [13], [16], [19], [1], [7], [11], [24], [22], [9], [23]. In order to overcome the Brockett’s condition for many nonlinear controllable systems especially systems without drift, or systems with drift as some underactuated marine and aerospace systems such as the autonomous underwater vehicle, the ship, the rigid spacecraft, the airship, etc...; three approaches have been proposed: stabilization by discontinuous time-invariant feedback laws, time-varying stabilization and partial-finite time stabilization. In [13] we have shown that we can stabilize partially in finitetime the Brockett’s integrator -in terms of the state variables of a unicycle-like vehicle (reduced chained systems in R3 )by various approaches. The several stabilizing feedback laws are H¨older for the asymptotic partial stability, continuous and homogeneous for the finite-time partial stability and quasihomogeneous for finite-time partial stability by discontinuous feedbacks. This finite-time partial stability means that we can place in finite-time the unicycle in the equilibrium position, without taking into consideration its orientation. In [14] we have derived continuous or discontinuous feedback laws making all chained system finite-time stable with respect (n − 1) components while the last one remains This work was supported by LIM Laboratory of EPT. Chaker Jammazi is with Facult´e des Sciences de Bizerte, D´epartement de Math´ematiques, and LIM Laboratory of Ecole Polytechnique de Tunisie. Universit´e de Carthage, Tunisia. [email protected]
978-1-4673-5716-6/13/$31.00 ©2013 IEEE
constant for a large time. Moreover we have proved that last components is constant and cannot exceed a known values provided the initial condition is known in advance do not exceed this values. In this paper we progress in our construction and we give an explicit feedbacks that finite-time stabilize all nonholonomic chained systems. Our feedback laws are discontinuous in time or both discontinuous in state and time. The construction is based on part on the so called “sampling control” defined initially by Clarke et al. on the famous paper [6]. We will motivate our approach by the discussion on Brockett’s integrator, but before and for rigorous presentation of the problem, we introduce some definitions. We consider the Cauchy problem (2)
We consider the sampling schedule π = {ti }i≥0 of [0, +∞) we mean an infinite sequence 0 = t0 < t1 < t2 < ... with lim ti = ∞. We call i→∞
d(π) := supt≥0 (ti+1 − ti ). Next, we will recall the notion of generalized sampling solution initially introduced by Krasovskii and Subbotin, for more details, see [5]. The generalized sampling solutions are obtained as limits of solutions of a sequence of systems in which the control is piecewise constant. We consider the standard system of the form x˙ = f (x(t), k(t, x(t))), x(t0 ) = x0 ,
(3)
where k : R × Rn → Rm is, in general, a discontinuous function. Definition 1.1: [5], [6] An ε-trajectory associated to the Cauchy problem (3) and to the partition π = {t0 , t1 , ..., tN } of [t0 , T0 + a] with tN = t0 + a and dπ < ε, is a function obtained by iteratively solving the following integral equations ϕπ (t)
= + ϕπ (t) = +
x0 Rt
f (ϕπ (τ ), k(t0 , x0 )) dτ, t ∈ [t0 , t1 ], t0 ϕπ (ti ) Rt f (ϕπ (τ ), k(ti , ϕπ (ti ))) dτ, ti ∈ [ti , ti+1 ], ti (4) for all i = 1, ..., N − 1.
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Definition 1.2: [5], [6] A function ϕ : [t0 , t0 + a] → Rn is said to be a generalized sampling solution of x˙ = f (x), x(t0 ) = x0 if it is the uniform limit of a sequence of ε-trajectories as ε → 0. Now, we redefine the Krasovskii and Filippov solutions. We denote by co ¯ the convex closure and µ the usual Lebesgue measure in Rn , we have the following definitions. Definition 1.3: [5], [6] An absolutely continuous functions ϕ : [t0 , t0 + a] → Rn is said to be • a Krasovskii solution of (2) if it is a solution of the differential inclusion \ x˙ ∈ Kf (x) = cof ¯ (B(x, δ)) (5) δ>0
•
i.e. ϕ satisfies (5) for a.e. t ∈ [t0 , t0 + a], a Filippov solution of (2) if it is a solution of the differential inclusion \ \ x˙ ∈ Kf (x) = cof ¯ (B(x, δ) \ N ) (6)
holds for all (x1 (t0 ), x2 (t0 )) such that |x1 (t0 )| + |x2 (t0 )| < δ • there exists T = T (x(t0 )) > 0 called settling-time function, such that, x1 (t) = 0 for every t > t0 + T The control system x˙ = X(x, u), (7) X(0, x2 , 0) = 0, is p-partially stabilizable in finite- time by discontinuous feedback law, if there exists a discontinuous map u : Rn → Rm such for every x2 ∈ Rn−p , u(0, x2 ) = 0, and such that (0, 0) ∈ Rp × Rn−p is p-partially stable in finite-time for the closed loop system x˙ = X(x, u(x)). Of course, the case p = n corresponds to “complete” finitetime stability of the system (7). For simplicity, we denotes as in [18] by [z]r := sgn(z)|z|r , r > 0, z ∈ R where sgn is the sign function. II. C ONSTRUCTION OF THE STABILIZING FEEDBACK LAWS OF B ROCKETT ’ S INTEGRATOR
δ>0 µ(N )=0
i.e. ϕ satisfies (6) for a.e. t ∈ [t0 , t0 + a]. Definition 1.4: [6] Consider the control system x˙ = f (x, u), x ∈ Rn , u ∈ Rm where f is locally Lipschitz. Given a feedback k, a partition π, and an x0 ∈ Rn . For each i, i = 0, 1, 2, ..., recursively solve x(t) ˙ = f (x(t), k(x(ti ))), t ∈ [ti , ti +1] using as initial value x(ti ) the endpoint of the solution on the preceding interval (and starting with x(t0 ) = x0 ). The π− trajectory of x˙ = f (x, k(x)) starting from x0 is the function x(.) thus obtained. Now we recall the notion of finite-time partial stability. Of course, the meaning of the autonomous differential equation x˙ = X(x) with a piecewise continuous right-hand side is defined in the sense of Filippov (Def 1.3) and [10], [20], [21]). Let X : Rp × Rn−p ∼ = Rn → Rn , x = (x1 , x2 ) ∈ Rp × n−p n R 7→ X(x) ∈ R , be defined and discontinuous on a neighborhood of (0, 0) ∈ Rp × Rn−p . We assume that 1) X is measurable and locally bounded, 2) X(0, x2 ) = 0. We denote by x(., t0 , x0 ) a solution of x˙ = X(x) under the initial conditions x(t0 ) = x0 . The following definition is inspired from [21]. Definition 1.5 (p-finite-time partial stability): One says that (0, 0) ∈ Rp × Rn−p is p-partially stable in finite time for x˙ = X(x) if • the equilibrium (0, 0) is stable i.e. for each t0 ∈ R, ε > 0, there is δ(ε, t0 ) > 0 such that each solution x(., t0 , x0 ) of x˙ = X(x) with the initial data x0 s.t |x0 | < δ exists on [t0 , +∞) and satisfies the inequality |x(t, t0 , x0 )| < ε, ∀ t > t0 . •
In this section, we introduce the new concept called “partial planing and finite-time stability”. This means the following idea: consider system (1), over the time interval [0, t1 ] with t1 is sufficiently large, we take u1 = 1 i.e. the state x1 (t) follows the trajectory t + x1 (0) over the interval time [0, t1 ]. Then, the partial state (x2 , x3 ) of Brockett’s integrator becomes over [0, t1 ] x˙ 2 = u2 , x˙ 3 = x2 .
The main idea, in this case, is to stabilize in finite-time the state (x2 , x3 ) ”faster“ than x1 . More precisely, if we denotes by T0 the settling time of (x2 , x3 ), of course we have T0 6 t1 . Over [0, t1 ], the system (8) is now presented as a double integrator when Bhat and Bernstein feedback law can be applied [2]. By taking u2 := −c1 [x2 ]α − c2 [x3 ]α/(2−α) , where c1 > 0, c2 > 0 and α ∈ (0, 1). Then u2 stabilizes in finite-time (x2 , x3 ). Thus, we get x2 (t) = x3 (t) = 0, ∀ t ∈ [T0 , t1 ]. Now, by choosing the feedback u1 of the form: 1, if 0 6 t 6 t1 , u1 (x(t)) := (9) −k [x1 ]α , if t > t1 , where k > 0, then u1 stabilizes x1 in finite-time. |t1 + x1 (0)|1−α |x1 (t1 )|1−α = t1 + the Let τ1 := t1 + k(1 − α) k(1 − α) settling time of x1 . Hence, we have τ1 > t1 . By an adequate choice of α ∈ (0, 1) and an adequate gain k and “good” t1 one can get x3 (τ1 ) = 0. |x2 (0)|1−α + τ1 . Clearly T¯ > τ1 . Now, we define T¯ := c1 (1 − α) Let Ω the set defined by Ω := {(x2 , x3 ) : x2 = x3 = 0, t > T¯},
The convergence of the state x1 lim |x1 (t, t0 , x0 )| = 0,
t→+∞
(8)
and we will show that this manifold is invariant when the state x1 is finite-time stable. 1302
We have x1 (t) = 0 for every t ≥ τ1 . Then we get by integration of the solution x3 that Z t x3 (t) = x3 (τ1 ) + u1 (x(s))x2 (s)ds = 0 + 0 = 0,
and u2 (x) = −0.5 [x2 ]1/2 − 1.5 [x3 ]4/3 . Clearly the simulations show the effectiveness of the proposed feedback laws and show that the state (x1 , x3 , x3 )0 is finite-time stable.
τ1
because, in closed loop, u1 (x(t)) = 0 for every t ≥ τ1 . In this case the dynamic of x2 becomes x˙ 2 = −c1 sgn(x2 )|x2 |α which is naturally finite-time stable with settling time |x2 (0)|1−α . Then the set Ω is invariant. c1 (1 − α) In the next, we present a numerical algorithm of the design of stabilizing feedback laws (since the settling time is not estimated for homogeneous systems). 1) choosing t1 > 0 large enough such that on [0, t1 ], u1 (x) = c > 0 for example, 2) taking the feedback given by u2 := −c1 [x2 ]α − c2 [x3 ]α/(2−α) , 3) we take the feedback given in (9) and by choosing “good constants” α ∈ (0, 1), an adequate gain k and “good” t1 one can get x3 (τ1 ) = 0 where τ1 := t1 + |x1 (t1 )|1−α > t1 . k(1 − α) In this case we have x1 (t) = x2 (t) = x3 (t) = 0 for |x2 (0)|1−α t > T¯ := + τ1 . c1 (1 − α) Instead of the above numerical algorithm, the next proposition gives a solution. Proposition 2.1: Let α and β be two constants in (0, 1). Let t1 > 0 be a large real number and three nonnegative constants a, c1 , c2 , c3 and c02 . Then, the following discontinuous feedback laws c1 , if 0 6 t 6 t1 , u1 (x(t)) := (10) −a [x1 ]α , if t > t1 ,
u2 (x(t)) :=
α −c2 [x2 ]α − c3 [x3 ] 2−α , if 0 6 t 6 t1 ,
−c02 [x2 ]β , if t > t1 ,
(11) system (1) is finite-time stable. Proof: Over the time interval [0, t1 ], the system described by (x2 , x3 ) is seen as a double integrator which is finite-time stable over [0, t1 ] by (11). Thus we get a settling time t0 6 t1 such that x2 (t) = x3 (t) = 0 for all t ∈ [t0 , t1 ]. For t > t1 , the feedback given in (10) stabilizes the state x1 in finite-time with settling time τ1 > t1 . Let remark that, under the feedback (11) we have x2 (t) = 0 for all t > t1 , now by writingRx3 (t) with integral representation t we get x3 (t) = x3 (t1 ) + t1 u1 (x(s))x2 (s)ds = 0 + 0 = 0 this shows also x3 (t) = 0 for every t > t1 . This completes the proof. Numerical Simulations To validate our result, we propose the following simulations, when the initial condition is x0 = (1.5, −2.5, 3.5)0 , and the feedback controllers are given by 1, if 0 6 t 6 20, u1 (x(t)) := −20 [x1 ]1/2 , if t > 20,
Trajectories of the state x1 , x2 and x3 .
Motion of the unicycle in the plane (x2 , x3 ). A. Discontinuous feedback laws with respect to time and states The following proposition shows that we can stabilize the system (1) by discontinuous state feedback laws with respect to time and states. The proof of the result is identical to the first case using discontinuous Orlov’s feedback laws for double integrator [21]. Proposition 2.2: Let α and β be two constants in (0, 1), Let t1 > 0 large and three nonnegative constants a, c1 , c2 , c3 and c02 such that c3 > c2 . Then under the following discontinuous feedback laws c1 > 0 if 0 6 t 6 t0 , u1 (x(t)) := −a sgn(x1 ), if t > t0 , −c2 sgn(x2 ) − c3 sgn(x3 ) if 0 6 t 6 t0 , u2 (x(t)) := −c02 sgn(x2 ), if t > t0 , stabilize in finite-time the system (1). Proof: The proof is identical to the proof of Proposition 2.1. The simulation result shows the effectiveness of our feedback laws. In this simulation the initial condition is x0 = (1.5, −2.5, 3.5)0 and the feedback laws are 1 if 0 6 t 6 19, u1 (x(t)) := −100 sgn(x1 ), if t > 19, and u2 (x) = −5 sgn(x2 ) − 20 sgn(x3 ).
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system (13) is globally finite-time stable through continuous state feedback. The construction of the finite-time stabilizing feedback laws of (13) is given by various method. The recent one is given by Moulay in [18] which generalize the result given by Bhat and Bernstein in [3, Theorem 8.1, pp123]. This feedbacks is given by [18, Proposition 11, pp. 1384] by the following: Let k := (k1 , ..., kn−1 ) ∈ Rn−1 such that sn−1 + kn−1 sn−2 + 1 , 1] ... + k2 s + k1 is Hurwitz. Then there exists ε ∈ [1 − n−2 and εi for 1 6 i 6 (n − 1) such that for all α ∈ (1 − ε, 1), ki0 ∈ (ki − εi , ki + εi ). Then (13) is globally finitetime stable under the continuous feedback control
Trajectories of the state x1 , x2 and x3 .
0 [xn−1 ]αn−1 , u ¯2 (x(t)) = −k10 [x1 ]α1 − . . . − kn−1
where α1 , ..., αn−1 satisfy αi αi+1 αi−1 = , i = 2, . . . , (n − 1), 2αi+1 − αi
(14)
(15)
with αn = 1 and αn−1 = α.
Motion of the unicycle in the plane (x2 , x3 ).
The black part in the simulation is explained by the so called chattering phenomenon which is due to the presence of sign function in these feedbacks. III. G ENERAL CHAINED SYSTEMS Recall that chained systems are described by the set of equations x˙ 1 x˙ 2
x˙ n−2 x˙ n−1 x˙ n
= = . . . = = =
x2 u1 x3 u1
In this case, and just after the time t1 , we turn to stabilize the state x1 by taking 1 if 0 6 t 6 t1 , u1 (x(t)) := −β [xn ]α , β > 0, if t > t1 . |xn (t1 )|1−α the settling time of β(1 − α) xn , then we have the following proposition. |xn−1 (0)|1−α Proposition 3.1: Let T0 := τ1 + 0 = τ1 + kn−1 (1 − αn−1 ) |xn−1 (0)|1−αn−1 , then the manifold defined by 0 kn−1 (1 − α) If we choosing τ1 := t1 +
Ω := {xi , 1 6 i 6 (n − 1) : xi (t) = 0, ∀ t > T0 } (12) xn−1 u1 u2 u1 ,
where the state is x := (x1 , x2 , ..., xn ) ∈ Rn and u := (u1 , u2 ) ∈ R2 is the control. The main contribution in this section is the building of discontinuous feedback laws making the system (12) in closed loop finite-time stable. The main idea for this construction is to planing the state xn over a “large” interval time [0, t1 ] and to stabilize in finite-time the partial state (x1 , x2 , ..., , xn−1 )0 ∈ Rn−1 “faster” than xn . At this stage, we put the state xn in finite-time by the action of u1 . Similarly to first case n = 3, over a time interval [0, t1 ] we take the path xn (t) = t + xn (0) i.e. u1 = 1. In this case, the partial-chained system becomes in cascaded integrators form with length (n − 1) : x˙ 1 = x2 , x˙ 2 = x3 , ..., x˙ n−1 = u2 .
(13)
Since (13) is controllable, because the system (13) satisfies Kalman’s condition. Then by using [3, Theorem 8.1, pp 123],
is invariant under the semi-flow of the system (13) in closed loop. Proof: Since for all t > T0 the control law u1 (x) vanishes because T0 is bigger than the settling time of xn , then for all 1 6 j 6 n − 2, we have Rt xj (t) = xj (T0 ) + T0 xj−1 (s)u1 (x(s))ds Rt = 0 + T0 0, = 0. Now we turn to see xn−1 . We have after the settling time T0 , the state xn−1 satisfies the equation 0 x˙ n−1 = −kn−1 sgn(xn−1 )|xn−1 |αn−1 ,
which is naturally finite-time stable with settling time |xn−1 (0)|1−α , then the manifold Ω is an invariant set. 0 kn−1 (1 − α) Proposition 3.2: The following discontinuous state feedback laws c > 0, if 0 6 t 6 t1 , u1 (x(t)) := (16) −a sgn(xn )|xn |α , if t > t1 ,
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Clearly, the dynamic of ψ is equivalent to the following u2 (x(t)) :=
¯2 (x(t)), if 0 6 t 6 t1 , u
ψ (3) (t) = u2 ,
(20)
0 −kn−1 sgn(xn−1 )|xn−1 |α , if t > t1 , (17) stabilize in finite-time chained system (12). Proof: The proof is similar to the proof of Proposition 2.1.. The following Theorem extends Proposition3.2 to all multichained systems [19]. Theorem 3.3: Let m be a positive integer, let n1 , ...nm be m X m nonnegative integers such that n = 1 + m + nj . The
j=1
following m-chain single-generator chained form: z˙1,0 = v1 , z˙1,1 = v0 z1,0 , .. .
z˙2,0 = v2 , z˙2,1 = v0 z2,0 , .. .
...
by using feedbacks cited in [3, Theorem 8.1, pp123], for triple integrator, and improved recently in [18, Proposition 11, pp. 1384], we can put in finite-time the states ω2 , ω3 and ψ in zero. This feedback can be given by 11 11 3 3 (21) u ¯2 (x(t)) = −[ψ]11/38 − [ω3 ] 29 − [ω2 ] 20 . 2 2 In this case, there exists T0 6 t1 such that for all t ∈ [T0 , t1 ], ω2 (t) = ω3 (t) = ψ(t) = 0. Now we turn to study the variables ω1 , φ and θ. Clearly after T0 , the dynamic of these states becomes ω˙ 1 = u1 φ˙ = ω1 (22) z˙m,0 = vm , ˙ θ = 0, z˙m,1 = v0 zm,0 , .. .
z˙1,n1 = v0 z1,n1 −1 , z˙2,n2 = v0 z2,n2 −1 , z˙n = v0 ,
z˙m,nm = v0 zm,nm −1 , third equation show that θ remains constant for a large time. The states ω1 and φ can be controlled by is finite-time stabilizable under discontinuous controllers (the 0, if 0 ≤ t 6 t1 , last component of the state being zn ). u1 (x(t)) := Proof: The considered system is a collection of mα chain of integrators, then the construction of discontinuous −c1 [ω1 ]α − c2 [φ] (2−α) , if t > t1 , stabilizing feedback law of the multi-chained system of Now by choosing the settling time T is t , we show that 0 1 Murray and Sastry follows from Proposition 3.2. the set Ω defined by IV. A PPLICATION TO PARTIAL ATTITUDE CONTROL OF
Ω := {(ω2 , ω3 , ψ) : ω2 (t) = ω3 (t) = ψ(t) = 0, ∀ t > t1 },
SATELLITE
The problem of attitude control of satellite with two controllers is well known. Let us recall that this system is not stabilizable in usual sense by regular state feedback laws. A great effort has been done to overcome this problem, see [8], [17], [12], [15], and the references therein. In this section we shall solve this question by using finite-time discontinuous controllers. This system is presented in the following configuration ω˙ 1 = u1 ω˙ 2 = u2 ω˙ 3 = ω1 ω2 (18) φ˙ = ω1 + (ω2 sinφ + ω3 cosφ) tgθ ˙ θ = ω2 cosφ − ω3 sinφ ψ˙ = ω2 sinφ + ω3 cosφ . cosθ The state is given by x := (ωi , φ, θ, ψ)0 ∈ R6 , i = 1, 2, 3 and u = (u1 , u2 ) ∈ R2 the control. Let be t1 > 0 a large time. Over the time interval [0, t1 ] we consider for the partial state (ω1 , φ, θ)0 the following path: ω1 (t) = c1 > 0, φ(t) = 0 and θ(t) = c3 ∈ (0, π/2). The objective is to stabilize in finite-time the state (ω2 , ω3 , ψ)0 faster than (ω1 , φ, θ)0 . Over [0, t1 ], the dynamic of (ω2 , ω3 , ψ)0 becomes ω˙ 2 = u2 ω˙ 3 = ω2 (19) ω3 ψ˙ = . cos c3
is invariant. Indeed: for every t > t1 we have φ(t) = ω2 (t) = 0, then we get Z t ω3 (t) = ω3 (t1 ) + ω1 (s) ω2 (s) ds = 0 + 0 = 0. t1
˙ Therefore we get ψ(t) = 0 for every t > t1 , this implies that ψ(t) = ψ(t1 ) = 0. In the end, for t > t1 , the state ω2 satisfies ω˙ 2 = −1.5sgn(ω2 )|ω2 |11/20 which is finite-time stable. Finally, ˙ = 0 for t > t1 which the state θ satisfies the equation θ(t) implies that θ(t) remains constant for a large time. Thus we have proved the following proposition. Proposition 4.1: Under the following discontinuous feedback laws c > 0, if 0 ≤ t 6 t1 , u1 (x(t)) := (23) α −c1 [ω1 ]α − c2 [φ] (2−α) , if t > t1 , and u2 (x(t)) :=
¯2 (x(t)), if 0 ≤ t 6 t1 , u
(24)
11
− 23 [ω2 ] 20 , if t > t1 ,
the system (18) is five-partially finite-time stable. More precisely, the partial state (ω1 , ω2 , ω3 , φ, ψ)0 is finite-time stable and θ(t) is constant for a large time t. Numerical Simulations The following simulations show the effectiveness of our feedbacks. The initial condition is
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x0 = (.5, 0.7, 1.8, 0, 0.5, 1.9)0 , and in the feedbacks (2324): t1 = 30, c = 0 and c1 = c2 = 2. Clearly the partial state (ω1 , ω2 , ω3 , φ, ψ)0 is finite-time stable and θ(t) ' −3 for t > 40.
Trajectories of the state ω1 , ω2 and ω3 .
Trajectories of the state φ, θ and φ. Remark 4.2: • The finite-time partial stabilization of the satellite is sufficient, since (18) is an example of affine control systems when Coron’s et al. [10] fails for the stabilizability by pure discontinuous feedback laws. • The above proposition shows that with the action of the proposed feedbacks we can place in finite-time the satellite in the equilibrium position without taking into consideration his orientation θ with respect to second axis. V. C ONCLUSION In this paper, the problem of finite-time stabilizability of chained and multi-chained system is solved. Our consideration gives a new alternative to overcome the obstruction of Brockett’s condition for stabilizability by regular feedback laws. The established stabilizing feedbacks are discontinuous both in time and states. Moreover, according to definition given for “sampling control” in [6], the solutions are as well as defined and have meaning for the two classes of constructed feedback laws. As defined, this idea combined the planning motion and finite-time stability theory and can be applied to solve the partial orientation of satellite with two controllers, and shows the effectiveness of our approach. R EFERENCES
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