Quaternion-Based Stabilization of Attitude ... - Semantic Scholar
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Quaternion-Based Stabilization of Attitude Dynamics Subject to Pointwise Delay in the Input Fr´ed´eric Mazenc,
Maruthi R. Akella
Abstract— The problem of stabilizing rigid-body attitude dynamics in the presence of pointwise time-delay for the input torque is considered. A quaternion-based linear state feedback controller is shown to achieve local stability in addition to the characterization of sufficient condition that depends only on the magnitude of the initial angular rates. More specifically, no restrictions are imposed on the body initial orientation which is a significant contrast with other results from recent literature that adopt three-dimensional representations for the attitude kinematics. Using the quaternion-based linear feedback structure, the closed-loop system is shown to never admit the possibility for finite-time escapes. While the actual magnitude of the time-delay can be unknown, an upper bound on the delay is assumed to be known. The proof relies on the construction of a functional which does not belong to the family of the strict Lyapunov-Krasovskii functionals, but shares important features with the functionals of this family. The stability conditions and results are illustrated through numerical simulations. Key Words: Delay, Attitude Stabilization, Strictification, Lyapunov-Krasovskii functionals.
I. I NTRODUCTION In many engineering applications, pointwise input delays arise from measurements and transport phenomena and may have a destabilizing effect on nonlinear controlled systems, even when they are arbitrarily small, as illustrated in [7]. Thus, the problem of designing for nonlinear systems stabilizing control laws in the presence of delays has been considered in many contributions, see in particular [5], [7], [9], [8], [2], [13] and the references therein. The problem of stabilizing attitude dynamics for a rigidbody with an unknown constant delay in feedback and with a known upper bound is very specific in the sense that the solutions belong to a particular manifold and the objective is not the stabilization of an equilibrium point but of some components of the state variables only. Assuming no delay in the feedback, the attitude control problem has been extensively studied [14] and continues to receive significant attention in recent literature. Specifically it has been established that “simple” proportional-derivative (PD) type controllers can indeed stabilize this system in the absence of delay [15] (and references therein). However, very few of these rigidbody attitude stabilization results carry over to the setting of time-delay systems. Recently, in [3], this problem has F. Mazenc is with EPI INRIA DISCO, L2S-CNRS-Sup´elec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France (e-mail: [email protected]). M. Akella is with The University of Texas at Austin, Aerospace Engineering and Engineering Mechanics, W.R. Woolrich Laboratories, E. 24th Street & Speedway, Austin, 78712, Texas, USA (email: [email protected]).
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been solved locally using the kinematically singular threedimensional Modified Rodrigues Parameter (MRP) representation for the attitude kinematics. In the present work, we revisit this attitude stabilization problem, which is obviously very significant with respect to a wide range of emerging applications, ranging from rigid aircraft and spacecraft systems to coordinated robot manipulators [18], [19], [17], [16]. In the presence of time-delay in feedback, we adopt, for the first time to our best knowledge, the globally non-singular and unit-norm constrained quaternion kinematics for the attitude parameterization. It should be emphasized that the approach taken in [3] does not carry over to this particular context given the fact that the kinematic differential equations governing the time-evolution of the quaternion vector do not lend themselves toward satisfying the Lipschitz-like condition that is enforced in [3]. In this work, we design quaternionbased linear feedback control that is locally stabilizing. The stability analysis is based on construction of a Lyapunov functional which relies on the ‘strictification’ technique (see for instance [6], [12] and the references therein for details on this notion), which consists in transforming a weak Lyapunov function (i.e. a function whose derivative along the trajectories of the studied system is nonpositive) into a strict one (i.e. a function whose derivative along the trajectories of the studied system is smaller than a negative definite function of the state variable). Using this function, we construct a functional which is not Lyapunov-Krasovskii functional (see [4] for the notion of Lyapunov-Krasovskii functional) but shares important features with these functionals. The stability results presented here cannot be derived through the approaches of [10], [11] and [13] because those results apply only to systems that are globally asymptotically stable in the absence of delay, whereas the system we consider cannot be locally asymptotically stabilized (due to now well-understood topological obstructions, see [1]). We prove that finite-time escapes are impossible for the resulting closed-loop system in addition to deriving an estimate of the basin of attraction in terms of the magnitude of the initial angular rates. In accordance with the intuition, the size of these estimates increases with decreasing magnitude of the time-delay and goes to the infinity when the delay goes to zero. In sharp contrast with [3], given the fact that the present work adopts the quaternion representation for the attitude kinematics, our results do not impose any additional restrictions on the initial orientation of the body. The paper is organized as follows. The model is described in Section II and a stabilization result in the absence of delay is presented. The main result is established in Section III
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followed by an illustrative numerical example in Section IV. Concluding remarks in Section V end the paper. Preliminaries: The notation will be simplified whenever no confusion can arise from the context. Let n ∈ N be arbitrary. We let In denote the identity matrix in Rn×n , and | · | is the usual Euclidean norm of matrices and vectors. For any continuous function ϕ : [−τ, +∞) → Rn and all t ≥ 0, we define the function ϕt by ϕt (θ) = ϕ(t+θ) for all θ ∈ [−τ, 0]. Let C 1 denote the set of all continuously differentiable functions, and C 0 denote the set of all continuous functions, where the domains and ranges of the functions will be clear from the context. Given any constant τ > 0, we let C([−τ, 0], Rn ) denote the set of all continuous Rn -valued functions defined on a given interval [−τ, 0]. We often abbreviate this set as Cin , and we call it the set of initial functions. Let M ∈ Rn×n be a symmetric positive definite matrix. Then for all vectors µ ∈ Rn , ν ∈ Rn and all constant c > 0, the inequality 1 > ν Mν 4c is satisfied. Let v = (v1 , v2 , v3 ) ∈ R3 . Then v ∗ denotes the skew-symmetric matrix defined as 0 −v3 v2 0 −v1 . v ∗ = v3 −v2 v1 0 µ> M ν ≤ cµ> M µ +
The following will be used throughout our work [14]: when |s(0)| = 1, together with the fact that the timederivative of $(s) = s> s along the trajectories of (1) equals zero for all t ≥ 0, it follows that |s(t)| = 1 ,
|q(t)| ≤ 1 ,
(1)
with ω ∈ R3 , where u ∈ R3 is the input torque, τ ≥ 0 the time-delay, and T T s = p, q T ∈ R4 , E(s) = [−q, (pI3 − q ∗ )] , (2) where the moment-of-inertia matrix J is constant and symmetric, positive definite, i.e., J = J T > 0. Let us introduce the notation x = (s, ω) ∈ R7 . Throughout the paper, we consider initial conditions such that |s(θ)| = 1 for all θ ∈ [−τ, 0]. Let Jm be the smallest eigenvalue of J. Since J is symmetric positive-definite, Jm > 0. Note for later use that |J −1 | = 1/Jm . Let JM = |J|, the largest eigenvalue of J.
(4)
u(q, ω) = −aq − bω ,
(5)
which is s(t) ˙ = 21 E(s(t))ω(t) J ω(t) ˙ = −ω(t)∗ Jω(t) − aq(t) − bω(t) ,
(6)
are such that lim |ω(t)| = 0 ,
lim |q(t)| = 0
t→+∞
(7)
t→+∞
when |s(0)| = 1. Moreover, the derivative of the function V1 (x) = c1 a q > q + (p − 1)2 + c1 Q0 (ω) + q > Jω (8)
II. P ROBLEM STATEMENT AND PRELIMINARY RESULT
We consider the system s(t) ˙ = 21 E(s(t))ω(t) J ω(t) ˙ = −ω(t)∗ Jω(t) + u(t − τ ) ,
∀t ≥ 0 .
,
In this section, we state and prove the following result: Lemma 1: Let a and b be two positive real numbers. Then the solutions of (1) in closed-loop with the linear feedback
A. The model
|p(t)| ≤ 1
B. Stabilization of the model in the absence of delay
c1 = max In this section, we introduce the attitude kinematics based on the quaternion representation and Euler’s rotational dynamics. We first consider a classical controller that uses linear feedback for this problem in the delay-free case and present our ‘partial strictification’ approach. In Section III, we will show that this same control law still gives a local stability property to the system when it is affected by a sufficiently small pointwise delay.
(3)
which implies that
with
Note that for all v ∈ R3 , the inequality |v ∗ | ≤ |v| holds.
∀t ≥ 0
√ JM b JM + 1 2 + + √ , Jm b 2a 2a
(9)
and
1 > ω Jω 2 along the trajectories of (6) satisfies Q0 (ω) =
V˙ 1 (t) ≤
(10)
− a2 |q(t)|2 − |ω(t)|2 .
(11)
Moreover
√ √ a 2 a JM − Jm + √ |q|2 + |ω|2 , ∀ x ∈ R7 . Jm Jm 2JM (12) Remark 1: The system (6) admits two equilibria: (p = 1, q = 0, ω = 0) and (p = −1, q = 0, ω = 0). Therefore it does not admit a globally asymptotically stable equilibrium point. Moreover, neither (p = 1, q = 0, ω = 0) nor (p = −1, q = 0, ω = 0) is a locally exponentially stable equilibrium of (6). This can be readily checked by analyzing the linear approximation of (6) around these equilibria. Proof. The function V1 defined in (8) satisfies V1 (x) = c1 a q > q + (p − 1)2 + c21 ω > Jω + q > Jω ≥ Jc1Ma q > Jq + c21 ω > Jω + q > Jω . (13) Since (c1 /2 − 1/Jm ) > 0, triangle inequality implies that
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V1 (x) ≥
V1 (x) ≥
=
c1 a > c1 > JM q Jq + 2 ω Jω − 2c −1 4 q > Jq − c21 1 Jm
h
c1 a JM
−
1 2c1 − J4m
i
>
−
q Jq +
1 Jm
ω > Jω
1 > Jm ω Jω
.
(14)
From (9), we deduce that (12) is satisfied. Next, let us introduce a function V0 (x) = a q > q + (p − 1)2 + Q0 (ω) .
(15)
Then (3) implies that, along the trajectories of (1) we consider, V0 satisfies V0 (x(t)) = 2a [1 − p(t)] + Q0 (ω(t)) .
V˙ 0 (t) = −2ap(t) ˙ + ω(t)> J ω(t) ˙ = −b|ω(t)|2 .
(17)
where we used ω(t)> ω(t)∗ = 0. Therefore V0 is nonincreasing along the trajectories of the considered system. Moreover, V0 is nonnegative. However, stability properties cannot be easily derived from these facts. Indeed, one cannot apply the Lyapunov theorem because the right-hand-side of (17) is not a negative definite function of x(t). However, we manage to prove (7) by partially ‘strictifying’ V0 , i.e. by considering V1 whose derivative contains extra nonpositive terms which enable us to prove (7). More importantly, V1 will be used in the next section as an important building block to establish a stability result in the presence of a time-delay in the input u. To ease the analysis of V˙ 1 , consider the function: N (x) = q > Jω .
(18)
Its derivative along the trajectories of (6) satisfies: N˙ (t) = −q(t)> ω(t)∗ Jω(t) − a|q(t)|2 bq(t)> ω(t) + 12 [q ∗ (t)ω(t) + p(t)ω(t)]> Jω(t) = −a|q(t)|2 − bq(t)> ω(t) + 21 p(t)ω(t)> Jω(t) − 12 q(t)> ω ∗ (t)Jω(t) ,
N˙ (t) ≤ −a|q(t)| − bq(t) ω(t) + JM |ω(t)|
2
Moreover, (21) and (17) imply that h 2 i V˙ 1 (t) ≤ − a2 |q(t)|2 + b2a + JM − c1 b |ω(t)|2 .
(20)
(22)
From (9), the inequality (11) is satisfied. The inequality (22) implies that ω is bounded. Since s is bounded, we deduce that ω and q are uniformly continuous. Then Barbalat’s lemma allows us to conclude that the equalities (7) hold. III. M AIN RESULT In this section we establish the main result of the paper. Let us introduce the constant:
and let us state and prove the following result:
2 Jm −8b2 τ 2 2 c b2 τ 2 32JM 2
3
(23)
,
2
2 2 with ξ1 (τ ) = Jm c2 τ / 2 Jm − 8b2 τ , 2 32ac21 + 72 b2 τ 2 ≤ Jm
(25)
and 2 2 2 Jm a + 4b4 c2 τ 2 ≤ Jm − 8b2 τ 2 /4 ,
(26)
with c1 defined in (9) and c2 defined in (23). Then the solutions of (1) in closed-loop with the feedback u(t) = −aq(t − τ ) − bω(t − τ )
(27)
with initial conditions in Cin satisfying |s(0)| = 1 and |ω(t)| ≤ Ω for all t ∈ [−τ, 0] are such that lim |ω(t)| = 0 ,
lim |q(t)| = 0 .
t→+∞
t→+∞
(28)
Remark 2: The conditions imposed in Theorem 1 seem algebraically complicated, however they can be easily checked. J2 Moreover, by observing that (25) implies that 8b2 τ 2 ≤ 9m we deduce the simpler sufficient conditions: i h 2 τ τ (e − 1) + JM Ω c1 4a + e2 JM Ω2 + (a+bΩ) 2J m h 2 2 i 9c2 τ 3 1 2 2 2 + a Ω + J 2 JM Ω + a + bΩ b 16 m
is satisfied. From the triangle inequality, it follows that 2 a b N˙ (t) ≤ − |q(t)|2 + + JM |ω(t)|2 . (21) 2 2a
c2 = c21 + 2/a
ω ∗ = ω > (q ∗ )> . Since |s(t)| = 1 for all t ≥ 0, it follows that the inequality >
m
(16)
The derivative of V0 along the trajectories of (1) in closedloop with (5) satisfies
2
Theorem 1: Let a and b be two positive real numbers. Let τ > 0 and Ω > 0 be two constants such that the following inequalities hold: i h τ 2 τ + JM Ω c1 4a + e2 JM Ω2 + (e −1)(a+bΩ) 2Jm h i 2 (24) + a2 Ω2 + J12 JM Ω2 + a + bΩ b2 ξ1 (τ )
> = − 2 |q(t)| − |ω(t)| + [c1 ω(t) + q(t) ]r(xt ) . (32) Next, we define a functional: Rt Rt 2 V2 (xt ) = V1 (x(t)) + γ1 t−τ ` |q(r)| ˙ drd` Rt Rt (33) 2 +γ2 t−τ ` |ω(r)| ˙ drd` , where γ1 > 0 and γ2 > 0 are constants to be selected later. Then elementary calculations give V˙ 2 (t) ≤ − a2 |q(t)|2 − |ω(t)|2 + |c1 ω(t) + q(t)||r(xt )| Rt 2 +τ γ1 |q(t)| ˙ 2 + τ γ2 |ω(t)| ˙ − γ1 t−τ |q(l)| ˙ 2 dl Rt 2 −γ2 t−τ |ω(l)| ˙ dl. (34) From (30), we deduce that |q(t)| ˙ 2 ≤ |ω(t)|2 ≤ |ω(t)∗ Jω(t) + aq(t) + bω(t) − r(xt )|2 2 |ω(t)|4 + |r(xt )|2 + a2 |q(t)|2 + b2 |ω(t)|2 . ≤ 4 JM (35) Combining (34) and (35), we obtain 2 2 |ω(t)| ˙ Jm
V˙ 2 (t) ≤ − a2 |q(t)|2 − |ω(t)|2 + |c1 ω(t) + q(t)||r(xt )| + τ γ1 |ω(t)|2 2 +τ ς2 JM |ω(t)|4 + |r(xt )|2 + a2 |q(t)|2 + b2 |ω(t)|2 Rt Rt 2 −γ1 t−τ |q(l)| ˙ 2 dl − γ2 t−τ |ω(l)| ˙ dl 2 2 + ς1 τ + τ ς2 JM |ω(t)| − 1 |ω(t)|2 +c1 |ω(t)||r(xt )| + |q(t)||r(xt )| + τ ς2 |r(xt )|2 Rt Rt 2 −γ1 t−τ |q(l)| ˙ 2 dl − γ2 t−τ |ω(l)| ˙ dl , 2 2 2 with ς1 = γ1 Jm + 4γ2 b2 /Jm , ς2 = 4γ2 /Jm . From the triangle inequality, we deduce that V˙ 2 (t) ≤ τ ς2 a − 12 a|q(t)|2 4τ γ J 2 + ς1 τ + J22 M |ω(t)|2 − 1 |ω(t)|2 m
+ + −
c21 1 a 2 2 2 2 |ω(t)| + 2 |r(xt )| + 4 |q(t)| 4τ γ2 1 2 2 2 |r(xt )| a |r(xt )| + Jm Rt Rt 2 2 γ1 t−τ |q(l)| ˙ dl − γ2 t−τ |ω(l)| ˙ dl
(36) .
By grouping the terms, we obtain V˙ 2 (t) ≤ ς2 τ a − 14 a|q(t)|2 2 2 + ς1 τ + ς2 JM τ |ω(t)| − 12 |ω(t)|2 +
−
c21 2
1 a
+ ς2 τ |r(xt )|2 Rt Rt 2 ˙ dl . γ1 t−τ |q(l)| ˙ 2 dl − γ2 t−τ |ω(l)| +
Now, observe, that, for all t ≥ τ , the equality holds: Rt Rt r(xt ) = a t−τ q(l)dl ˙ + b t−τ ω(l)dl ˙ .
We deduce that V˙ 2 (t) ≤ ς2 aτ − 14 a|q(t)|2 2 + ς1 τ + ς2 τ JM |ω(t)|2 − 12 |ω(t)|2 Rt Rt 2 +ξ2 (τ ) a2 t−τ |q(l)| ˙ 2 dl + b2 t−τ |ω(l)| ˙ dl Rt R t 2 2 −γ1 t−τ |q(l)| ˙ dl − γ2 t−τ |ω(l)| ˙ dl ,
2 with ξ2 (τ ) = c2 τ + 8τ 2 γ2 /Jm , where c2 is the constant defined in (23) and the choice
γ1 = (c2 + 2ς2 τ ) a2 τ
(38)
From Cauchy-Schwarz’s inequality, we have for all t ≥ τ , Rt Rt 2 |r(xt )|2 ≤ 2a2 τ t−τ |q(l)| ˙ 2 dl + 2b2 τ t−τ |ω(l)| ˙ dl . (39)
(41)
gives 1 2 V˙ 2 (t) ≤ ς aτ − a|q(t)| 2 4 2 2 2 + ξ2 (τ )a + ς2 b τ + ς2 τ JM |ω(t)|2 − 21 |ω(t)|2 R R t t 2 2 + c22 + ς2 τ 2b2 τ t−τ |ω(l)| ˙ dl − γ2 t−τ |ω(l)| ˙ dl . (42) By grouping the terms, we obtain V˙ 2 (t) ≤ ς2 τ a − 41 a|q(t)|2 1 2 2 2 (43) h+ ξ3 (τ )τ+ ς22 τ2JM |ω(t)| i R− 2 |ω(t)| t 2 + c2 b2 τ + 8bJ 2τ − 1 γ2 t−τ |ω(l)| ˙ dl , m
where ξ3 (τ ) = (c2 + 2τ ς2 ) a2 τ + ς2 b2 . Since (25) implies 2 2 > 0, we can choose − 8b2 τ 2 /Jm that Jm 2 2 γ2 = c2 b2 Jm τ / Jm − 8b2 τ 2 . (44) This choice yields 2 2 1 2b τ V˙ 2 (t) ≤ J4ac a|q(t)|2 2 −8b2 τ 2 − 4 m (45) + ξ4 (τ )τ 2 + ξ5 (τ )τ 2 |ω(t)|2 − 21 |ω(t)|2 , 4 2 2 τ 2b with ξ4 (τ ) = 1 + J 28b c2 a2 + J 2 4c 2 2 2 τ 2 , ξ5 (τ ) = −8b τ −8b m m 2 2 2 2 2 4JM c2 b / Jm − 8b τ . The inequality (25) implies that 2 ξ4 (τ )τ 2 = 4ac2 b2 τ 2 / Jm − 8b2 τ 2 ≤ 1/8 . (46) The inequality (26) implies that 2 2 2 c2 τ 2 Jm a + 4b4 / Jm − 8b2 τ 2 ≤ 1/4 .
(47)
It follows that a 1 2 2 2 ˙ V2 (t) ≤ − |q(t)| + ξ6 (τ )τ |ω(t)| − |ω(t)|2 (48) 8 4 2 2 − 8b2 τ 2 . From (12) and with ξ6 (τ ) = 4JM c2 b2 / Jm the definition of V2 , we deduce that |ω(t)|2
(37)
(40)
≤ V1 (x(t)) ≤ V2 (xt ) .
(49)
It follows that, for all t ≥ τ , the inequality a 1 4J 2 c2 b2 V˙ 2 (t) ≤ − |q(t)|2 + − + 2 M 2 2 τ 2 V2 (xt ) |ω(t)|2 8 4 Jm − 8b τ (50) is satisfied. Step 2. Let K be a real number such that 2 2 0 < K < Jm − 8b2 τ 2 / 16JM c2 b2 τ 2 (51) and assume that V2 (xτ ) < K. Then let us show that V2 (xt ) < K for all t ≥ τ . To prove this result, we proceed by
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contradiction. Assume there is te > τ such that V2 (xt ) < K when t ∈ [τ, te ) and V2 (xte ) = K. Then there exists tc ∈ [τ, te ] such that V˙ 2 (tc ) > 0. Now, (50) implies 2 4JM c2 b2 2 1 ˙ τ V2 (xtc ) |ω(tc )|2 . (52) V2 (tc ) ≤ − + 2 4 Jm − 8b2 τ 2
Combining (59), (60), (62), we obtain V2 (x(τ )) ≤ Kτ with Kτ
Since V2 (xtc ) ≤ K, it follows from (51) that V˙ 2 (tc ) ≤ 0 .
(53)
This yields a contradiction. We deduce that, for all t ≥ τ , V2 (xt ) < K. From this and (48), for all t ≥ τ , we have |ω(t)|2 < K. Then we deduce from (48) that there is cK > 0 such that V˙ 2 (t) ≤
− a8 |q(t)|2 − cK |ω(t)|2 .
(54)
It follows that V2 (xt ) is nonincreasing over [τ, +∞). It follows that w(t), q(t), p(t) are bounded. We deduce that their derivatives are bounded. It follows that these functions are uniformly continuous over [τ, +∞). By integrating (54), we obtain, for all t ≥ τ , Rt Rt V2 (xt ) − V2 (xτ ) ≤ − a8 τ |q(l)|2 dl − cK τ |ω(l)|2 dl, (55) which gives Rt Rt (56) cK τ |ω(l)|2 dl + a8 τ |q(l)|2 dl ≤ V2 (xτ ) R +∞ because V2 is nonnegative. We deduce that τ |ω(l)|2 dl < Rt +∞ and τ |q(l)|2 dl < +∞. Since ω and q are uniformly continuous, Barbalat’s lemma applies: we deduce that ω(t) and q(t) converge to zero.
Q0 (ω(t)) ≤ ≤
2
2
i
(a+bΩ) eτ 2 τ 2 JM Ω + 2Jm (e − 1) 2 2 2b τ a2 τ 3 Ω2 +JM Ω + c22 + J 4c 2 2 2 m −8b τ 2 2 c2 b 3 + 2J 2 −16b JM Ω2 + a + bΩ . 2τ 2 τ m
(64)
IV. N UMERICAL E XAMPLE To further evaluate the sufficient condition for stability in (24), we consider a cube-satellite with inertia matrix: 0.0465 −0.0007 0.0004 J = −0.0007 0.0486 −0.0021 kg · m2 0.0004 −0.0021 0.0482 Here, JM = 0.0504 and Jm = 0.0461. For any specified value of Ω, the control gains (a, b) are related to τ and their feasible values can be computed through the inequality (24).
Fig. 1.
Feasible range of control parameters (a, b) according to (24).
− 1)
We deduce that, for all t ∈ [0, τ ], q |ω(t)| ≤ eτ JJM Ω2 + J12 (a + bΩ)2 (eτ − 1) . m m
h = c1 4a +
Since (24) implies that τ and Ω are such that Kτ < 2 Jm −8b2 τ 2 2 c b2 τ 2 then, bearing in mind (51), we deduce that ω(t) 32JM 2 and q(t) converge to zero. This allows us to conclude.
Step 3. Now, from Lemma 3 in appendix we deduce that, the function Q0 defined in (10) satisfies for all t ∈ [0, τ ], (eτ eτ 12 ω(0)> Jω(0) + (a+bΩ) 2Jm 2 τ (a+bΩ) e 2 (eτ − 1) . 2 JM Ω + 2Jm
(63)
(57) (58)
As shown in Figure 1, smaller values for time-delay τ and initial angular rate upper bound Ω provide a wider range for the control parameters. For the closed-loop control simulation, a = 0.01 and b = 0.024 are selected corresponding to τ = 0.1 sec and Ω = 0.03 rad/sec.
From kp(t)k ≤ 1 and kq(t)k ≤ 1 for all t ≥ τ , we deduce 2 Rt Rt τ2 2 (59) |ω(r)| ˙ drd` ≤ 2J JM Ω2 + a + bΩ 2 t−τ ` m
and
Rt t−τ
Rt `
2 |q(r)| ˙ drd` ≤
τ2 2 2 Ω .
(60)
Moreover, using the fact that |s(t)| = 1 for all t ≥ 0, we deduce that V1 (x(τ )) ≤ c1 V0 (x(τ ))+JM Ω ≤ c1[4a+Q0 (ω(τ ))]+JM Ω . (61) We deduce from (57) that " # JM Ω2 eτ (a + bΩ)2 V1 (x(τ )) ≤ c1 4a + + −1 + JM Ω. 2 2Jm (eτ − 1) (62)
Fig. 2. Time histories of |q(t)| and |ω(t)| under closed-loop control action.
The results in Figure using the initial √ 2 are obtained √ conditions, s(0) = [ 2/2, 0, 2/2, 0]T and ω(0) = [0.03, 0, 0]T . The input torques (not reported here) required for this maneuver appear well in range for the cube-satellite
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class of problems under consideration here. The attitude and angular rate signals converge within approximately 30 seconds indicating satisfactory closed-loop performance. V. C ONCLUSION Through a novel approach of ‘partial strictification’ we determined an estimate of the basin of attraction for rigid-body attitude dynamics stabilized by a linear state feedback with pointwise input delay. Precise value of the time-delay is not necessary for implementing the proposed controller; only an upper bound on the delay-value is adequate. Use of the unitquaternion representation allows us to have no restrictions whatsoever on the initial orientation. The problem of input sampling will be the subject of further research.
A PPENDIX Lemma 2: Consider the system (29) with continuous initial condition over [−τ, 0]. This system is forward complete i.e. all its solutions are defined over [−τ, +∞). Moreover every solution is continuous. Proof. We proceed by induction. Induction assumption: the solution is continuous and defined over [−τ, iτ ], where i is a nonnegative integer. Step 0: the assumption is satisfied at the step 0 since we consider a solution with a continuous initial condition. Step i: assume that the induction assumption is satisfied to the step i, where i is a nonnegative integer. Let te be the largest number in the interval [iτ, (i + 1)τ ] such that the solution x(t) is defined over [−τ, te ). Then, for all t ∈ [iτ, te ), (29) is satisfied. Let Q0 be the function defined in (10). Then using the fact that ω(t)> ω(t)∗ = 0, we obtain
R EFERENCES
Q˙ 0 (t)
[1] S.P. Bhat, D.S. Bernstein, A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Systems & Control Letters, 39, pp. 63-70, 2000. [2] N. Bekiaris-Liberis and M. Krstic, Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations, Automatica, vol. 49, pp. 1576-1590, 2013. [3] A. A. Chunodkar, M. R. Akella, Attitude Stabilization with Unknown Bounded Delay in Feedback Control Implementation. Journal of Guidance, Control, and Dynamics, vol. 34, No. 2, March-April 2011. [4] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations. (AMS, vol. 99, Springer: New York, 1993). [5] I. Karafyllis, M. Krstic, Nonlinear Stabilization under Sampled and Delayed Measurements, and with Inputs Subject to Delay and ZeroOrder Hold, IEEE Trans. Aut. Contr., vol. 57, pp. 1141-1154, 2012. [6] M. Malisoff, F. Mazenc, Constructions of Strict Lyapunov Functions. Communications and Control Engineering Series, Springer-Verlag London Ltd., London, ISBN: 978-1-84882-534-5 (Print), U.K., 2009. [7] F. Mazenc, P.-A. Bliman, Backstepping Design for Time-Delay Nonlinear Systems, IEEE Trans. Aut. Contr., vol. 51, No.1, 2006, pp.149-154. [8] F. Mazenc, M. Malisoff. Asymptotic Stabilization for Feedforward Systems with Delayed Feedback, Automatica, vol. 49, No. 3, 2013. [9] F. Mazenc, M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement, Automatica, vol. 46, Issue 9 pp. 1428-1436, Sept. 2010. [10] F. Mazenc, M. Malisoff, T. N. Dinh, Robustness of Nonlinear Systems with Respect to Delay and Sampling of the Controls, Automatica, vol. 49, pp. 1925-1931, 2013. [11] F. Mazenc, M. Malisoff, Z. Lin, Further results on input-to-state stability for nonlinear systems with delayed feedbacks, Automatica, vol. 44, No. 9, pp. 2415-2421, 2008. [12] F. Mazenc, D. Nesic, Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem. Mathematics of Control, Signals, and Systems, vol. 19, No 2, pp. 151-182, May 2007. [13] A. Teel, Connections Between Razumikhin-Type Theorems and the ISS Nonlinear Small Gain Theorem, IEEE Trans. Aut. Contr., 43(7), 1998. [14] Wen, J., and Kreutz-Delgado, K., The Attitude Control Problem, IEEE Trans. Aut. Contr., vol. 36, No. 10, 1991, pp. 1148-1162. [15] R. Schlanbusch, A. Loria, R. Kristiansen, P. J. Nicklasson, PD+ Based Output Feedback Attitude Control of Rigid Bodies, IEEE Trans. Aut. Contr., vol. 57, No. 8, 2012, pp. 2146-2152. [16] A. Abdessameud, A. Tayebi, I.-G. Polushin, Attitude Synchronization of Multiple Rigid-Bodies with Communication Delays, IEEE Trans. Aut. Contr., vol. 57, No. 9, 2012, pp. 2405-2411. [17] Y. I. T. Hatanaka, M. Fujita, M. Spong, Passivity-Based Pose Synchronization in Three-Dimensions, IEEE Trans. Aut. Contr., vol. 57, No. 2, 2012, pp. 360-375. [18] S. J. Chung, U. Ahsun, J. J. E. Slotine, Application of Synchronization to Formation Flying Spacecraft: Lagrangian Approach, AIAA J. Guid. Contr., Dyn., vol. 32, No. 2, 2009, pp. 512-526. [19] A. Sarlette, R. Sepulchre, N. E. Leonard, Autonomous Rigid-Body Attitude Synchronization, Automatica, 45(2), 2009, pp. 572-577.
= −ω(t)> [aq(t − τ ) + bω(t − τ )] .
From the triangle inequality and deduce that, for all t ∈ [iτ, te ),
2 Jm Q0 (ω(t))
(65)
≥ |ω(t)|2 , we
1 2 Q0 (ω(t)) + |aq(t − τ ) + bω(t − τ )|2 . (66) Q˙ 0 (t) ≤ Jm 4 By integrating this inequality, we obtain, for all t ∈ [iτ, te ), 2
e Jm (t−iτ ) Q (ω(iτ )) R t 2 (t−`)0 |aq(` − τ ) + bω(` − τ )|d` . e Jm iτ (67) Therefore, for all t ∈ [iτ, te ), Q0 (ω(t)) ≤ +
Q0 (ω(t)) ≤ +
2τ
e Jm Q0 (ω(iτ )) 2τ τ e Jm sup
|aq(`) + bω(`)| .
(68)
`∈[(i−1)τ,iτ ]
Since q and ω are continuous over [(i − 1)τ, iτ ], it follows that sup |aq(m) + bω(m)| is a finite real number. m∈[(i−1)τ,iτ ]
Therefore there exists a constant ω > 0 such that |ω(t)|
≤
ω
,
∀t ∈ [iτ, te ) .
(69)
Since s(t) is bounded too, x(t) is bounded over [iτ, te ). We deduce that the finite escape time phenomenon cannot occur. Therefore te = (i + 1)τ and the solution is defined and continuous over [−τ, (i+1)τ ]. Thus the induction assumption is satisfied at the step i + 1. This concludes the proof. Lemma 3: Consider the system J ω(t) ˙ = −ω(t)∗ Jω(t) + u(t) ,
(70)
where J is the matrix in Section II-A, where ω ∈ R3 and u is a continuous function, bounded by the constant u > 0. Then, for all t0 ≥ 0 and t ≥ t0 , the following holds u2 . 1 Q0 (ω(t)) = ω > Jω ≤ et−t0 Q0 (ω(t0 ))+ et−t0 − 1 . 2 2Jm (71) Proof. The derivative Q˙ 0 along trajectories of (70) satisfies Q˙ 0(t) = ω(t)> u(t), since ω(t)> ω(t)∗ = 0. Thus Q˙ 0 (t) ≤ 1 u2 > > −1 u(t) ≤ Q0 (ω(t)) + 2J . By 2 ω(t) Jω(t) + u(t) J m integrating this inequality over [t0 , t], one obtains (71).
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Quaternion-Based Stabilization of Attitude Dynamics Subject to Pointwise Delay in the Input Fr´ed´eric Mazenc,
Maruthi R. Akella
Abstract— The problem of stabilizing rigid-body attitude dynamics in the presence of pointwise time-delay for the input torque is considered. A quaternion-based linear state feedback controller is shown to achieve local stability in addition to the characterization of sufficient condition that depends only on the magnitude of the initial angular rates. More specifically, no restrictions are imposed on the body initial orientation which is a significant contrast with other results from recent literature that adopt three-dimensional representations for the attitude kinematics. Using the quaternion-based linear feedback structure, the closed-loop system is shown to never admit the possibility for finite-time escapes. While the actual magnitude of the time-delay can be unknown, an upper bound on the delay is assumed to be known. The proof relies on the construction of a functional which does not belong to the family of the strict Lyapunov-Krasovskii functionals, but shares important features with the functionals of this family. The stability conditions and results are illustrated through numerical simulations. Key Words: Delay, Attitude Stabilization, Strictification, Lyapunov-Krasovskii functionals.
I. I NTRODUCTION In many engineering applications, pointwise input delays arise from measurements and transport phenomena and may have a destabilizing effect on nonlinear controlled systems, even when they are arbitrarily small, as illustrated in [7]. Thus, the problem of designing for nonlinear systems stabilizing control laws in the presence of delays has been considered in many contributions, see in particular [5], [7], [9], [8], [2], [13] and the references therein. The problem of stabilizing attitude dynamics for a rigidbody with an unknown constant delay in feedback and with a known upper bound is very specific in the sense that the solutions belong to a particular manifold and the objective is not the stabilization of an equilibrium point but of some components of the state variables only. Assuming no delay in the feedback, the attitude control problem has been extensively studied [14] and continues to receive significant attention in recent literature. Specifically it has been established that “simple” proportional-derivative (PD) type controllers can indeed stabilize this system in the absence of delay [15] (and references therein). However, very few of these rigidbody attitude stabilization results carry over to the setting of time-delay systems. Recently, in [3], this problem has F. Mazenc is with EPI INRIA DISCO, L2S-CNRS-Sup´elec, 3 rue Joliot Curie, 91192, Gif-sur-Yvette, France (e-mail: [email protected]). M. Akella is with The University of Texas at Austin, Aerospace Engineering and Engineering Mechanics, W.R. Woolrich Laboratories, E. 24th Street & Speedway, Austin, 78712, Texas, USA (email: [email protected]).
978-1-4799-3271-9/$31.00 ©2014 AACC
been solved locally using the kinematically singular threedimensional Modified Rodrigues Parameter (MRP) representation for the attitude kinematics. In the present work, we revisit this attitude stabilization problem, which is obviously very significant with respect to a wide range of emerging applications, ranging from rigid aircraft and spacecraft systems to coordinated robot manipulators [18], [19], [17], [16]. In the presence of time-delay in feedback, we adopt, for the first time to our best knowledge, the globally non-singular and unit-norm constrained quaternion kinematics for the attitude parameterization. It should be emphasized that the approach taken in [3] does not carry over to this particular context given the fact that the kinematic differential equations governing the time-evolution of the quaternion vector do not lend themselves toward satisfying the Lipschitz-like condition that is enforced in [3]. In this work, we design quaternionbased linear feedback control that is locally stabilizing. The stability analysis is based on construction of a Lyapunov functional which relies on the ‘strictification’ technique (see for instance [6], [12] and the references therein for details on this notion), which consists in transforming a weak Lyapunov function (i.e. a function whose derivative along the trajectories of the studied system is nonpositive) into a strict one (i.e. a function whose derivative along the trajectories of the studied system is smaller than a negative definite function of the state variable). Using this function, we construct a functional which is not Lyapunov-Krasovskii functional (see [4] for the notion of Lyapunov-Krasovskii functional) but shares important features with these functionals. The stability results presented here cannot be derived through the approaches of [10], [11] and [13] because those results apply only to systems that are globally asymptotically stable in the absence of delay, whereas the system we consider cannot be locally asymptotically stabilized (due to now well-understood topological obstructions, see [1]). We prove that finite-time escapes are impossible for the resulting closed-loop system in addition to deriving an estimate of the basin of attraction in terms of the magnitude of the initial angular rates. In accordance with the intuition, the size of these estimates increases with decreasing magnitude of the time-delay and goes to the infinity when the delay goes to zero. In sharp contrast with [3], given the fact that the present work adopts the quaternion representation for the attitude kinematics, our results do not impose any additional restrictions on the initial orientation of the body. The paper is organized as follows. The model is described in Section II and a stabilization result in the absence of delay is presented. The main result is established in Section III
4877
followed by an illustrative numerical example in Section IV. Concluding remarks in Section V end the paper. Preliminaries: The notation will be simplified whenever no confusion can arise from the context. Let n ∈ N be arbitrary. We let In denote the identity matrix in Rn×n , and | · | is the usual Euclidean norm of matrices and vectors. For any continuous function ϕ : [−τ, +∞) → Rn and all t ≥ 0, we define the function ϕt by ϕt (θ) = ϕ(t+θ) for all θ ∈ [−τ, 0]. Let C 1 denote the set of all continuously differentiable functions, and C 0 denote the set of all continuous functions, where the domains and ranges of the functions will be clear from the context. Given any constant τ > 0, we let C([−τ, 0], Rn ) denote the set of all continuous Rn -valued functions defined on a given interval [−τ, 0]. We often abbreviate this set as Cin , and we call it the set of initial functions. Let M ∈ Rn×n be a symmetric positive definite matrix. Then for all vectors µ ∈ Rn , ν ∈ Rn and all constant c > 0, the inequality 1 > ν Mν 4c is satisfied. Let v = (v1 , v2 , v3 ) ∈ R3 . Then v ∗ denotes the skew-symmetric matrix defined as 0 −v3 v2 0 −v1 . v ∗ = v3 −v2 v1 0 µ> M ν ≤ cµ> M µ +
The following will be used throughout our work [14]: when |s(0)| = 1, together with the fact that the timederivative of $(s) = s> s along the trajectories of (1) equals zero for all t ≥ 0, it follows that |s(t)| = 1 ,
|q(t)| ≤ 1 ,
(1)
with ω ∈ R3 , where u ∈ R3 is the input torque, τ ≥ 0 the time-delay, and T T s = p, q T ∈ R4 , E(s) = [−q, (pI3 − q ∗ )] , (2) where the moment-of-inertia matrix J is constant and symmetric, positive definite, i.e., J = J T > 0. Let us introduce the notation x = (s, ω) ∈ R7 . Throughout the paper, we consider initial conditions such that |s(θ)| = 1 for all θ ∈ [−τ, 0]. Let Jm be the smallest eigenvalue of J. Since J is symmetric positive-definite, Jm > 0. Note for later use that |J −1 | = 1/Jm . Let JM = |J|, the largest eigenvalue of J.
(4)
u(q, ω) = −aq − bω ,
(5)
which is s(t) ˙ = 21 E(s(t))ω(t) J ω(t) ˙ = −ω(t)∗ Jω(t) − aq(t) − bω(t) ,
(6)
are such that lim |ω(t)| = 0 ,
lim |q(t)| = 0
t→+∞
(7)
t→+∞
when |s(0)| = 1. Moreover, the derivative of the function V1 (x) = c1 a q > q + (p − 1)2 + c1 Q0 (ω) + q > Jω (8)
II. P ROBLEM STATEMENT AND PRELIMINARY RESULT
We consider the system s(t) ˙ = 21 E(s(t))ω(t) J ω(t) ˙ = −ω(t)∗ Jω(t) + u(t − τ ) ,
∀t ≥ 0 .
,
In this section, we state and prove the following result: Lemma 1: Let a and b be two positive real numbers. Then the solutions of (1) in closed-loop with the linear feedback
A. The model
|p(t)| ≤ 1
B. Stabilization of the model in the absence of delay
c1 = max In this section, we introduce the attitude kinematics based on the quaternion representation and Euler’s rotational dynamics. We first consider a classical controller that uses linear feedback for this problem in the delay-free case and present our ‘partial strictification’ approach. In Section III, we will show that this same control law still gives a local stability property to the system when it is affected by a sufficiently small pointwise delay.
(3)
which implies that
with
Note that for all v ∈ R3 , the inequality |v ∗ | ≤ |v| holds.
∀t ≥ 0
√ JM b JM + 1 2 + + √ , Jm b 2a 2a
(9)
and
1 > ω Jω 2 along the trajectories of (6) satisfies Q0 (ω) =
V˙ 1 (t) ≤
(10)
− a2 |q(t)|2 − |ω(t)|2 .
(11)
Moreover
√ √ a 2 a JM − Jm + √ |q|2 + |ω|2 , ∀ x ∈ R7 . Jm Jm 2JM (12) Remark 1: The system (6) admits two equilibria: (p = 1, q = 0, ω = 0) and (p = −1, q = 0, ω = 0). Therefore it does not admit a globally asymptotically stable equilibrium point. Moreover, neither (p = 1, q = 0, ω = 0) nor (p = −1, q = 0, ω = 0) is a locally exponentially stable equilibrium of (6). This can be readily checked by analyzing the linear approximation of (6) around these equilibria. Proof. The function V1 defined in (8) satisfies V1 (x) = c1 a q > q + (p − 1)2 + c21 ω > Jω + q > Jω ≥ Jc1Ma q > Jq + c21 ω > Jω + q > Jω . (13) Since (c1 /2 − 1/Jm ) > 0, triangle inequality implies that
4878
V1 (x) ≥
V1 (x) ≥
=
c1 a > c1 > JM q Jq + 2 ω Jω − 2c −1 4 q > Jq − c21 1 Jm
h
c1 a JM
−
1 2c1 − J4m
i
>
−
q Jq +
1 Jm
ω > Jω
1 > Jm ω Jω
.
(14)
From (9), we deduce that (12) is satisfied. Next, let us introduce a function V0 (x) = a q > q + (p − 1)2 + Q0 (ω) .
(15)
Then (3) implies that, along the trajectories of (1) we consider, V0 satisfies V0 (x(t)) = 2a [1 − p(t)] + Q0 (ω(t)) .
V˙ 0 (t) = −2ap(t) ˙ + ω(t)> J ω(t) ˙ = −b|ω(t)|2 .
(17)
where we used ω(t)> ω(t)∗ = 0. Therefore V0 is nonincreasing along the trajectories of the considered system. Moreover, V0 is nonnegative. However, stability properties cannot be easily derived from these facts. Indeed, one cannot apply the Lyapunov theorem because the right-hand-side of (17) is not a negative definite function of x(t). However, we manage to prove (7) by partially ‘strictifying’ V0 , i.e. by considering V1 whose derivative contains extra nonpositive terms which enable us to prove (7). More importantly, V1 will be used in the next section as an important building block to establish a stability result in the presence of a time-delay in the input u. To ease the analysis of V˙ 1 , consider the function: N (x) = q > Jω .
(18)
Its derivative along the trajectories of (6) satisfies: N˙ (t) = −q(t)> ω(t)∗ Jω(t) − a|q(t)|2 bq(t)> ω(t) + 12 [q ∗ (t)ω(t) + p(t)ω(t)]> Jω(t) = −a|q(t)|2 − bq(t)> ω(t) + 21 p(t)ω(t)> Jω(t) − 12 q(t)> ω ∗ (t)Jω(t) ,
N˙ (t) ≤ −a|q(t)| − bq(t) ω(t) + JM |ω(t)|
2
Moreover, (21) and (17) imply that h 2 i V˙ 1 (t) ≤ − a2 |q(t)|2 + b2a + JM − c1 b |ω(t)|2 .
(20)
(22)
From (9), the inequality (11) is satisfied. The inequality (22) implies that ω is bounded. Since s is bounded, we deduce that ω and q are uniformly continuous. Then Barbalat’s lemma allows us to conclude that the equalities (7) hold. III. M AIN RESULT In this section we establish the main result of the paper. Let us introduce the constant:
and let us state and prove the following result:
2 Jm −8b2 τ 2 2 c b2 τ 2 32JM 2
3
(23)
,
2
2 2 with ξ1 (τ ) = Jm c2 τ / 2 Jm − 8b2 τ , 2 32ac21 + 72 b2 τ 2 ≤ Jm
(25)
and 2 2 2 Jm a + 4b4 c2 τ 2 ≤ Jm − 8b2 τ 2 /4 ,
(26)
with c1 defined in (9) and c2 defined in (23). Then the solutions of (1) in closed-loop with the feedback u(t) = −aq(t − τ ) − bω(t − τ )
(27)
with initial conditions in Cin satisfying |s(0)| = 1 and |ω(t)| ≤ Ω for all t ∈ [−τ, 0] are such that lim |ω(t)| = 0 ,
lim |q(t)| = 0 .
t→+∞
t→+∞
(28)
Remark 2: The conditions imposed in Theorem 1 seem algebraically complicated, however they can be easily checked. J2 Moreover, by observing that (25) implies that 8b2 τ 2 ≤ 9m we deduce the simpler sufficient conditions: i h 2 τ τ (e − 1) + JM Ω c1 4a + e2 JM Ω2 + (a+bΩ) 2J m h 2 2 i 9c2 τ 3 1 2 2 2 + a Ω + J 2 JM Ω + a + bΩ b 16 m
is satisfied. From the triangle inequality, it follows that 2 a b N˙ (t) ≤ − |q(t)|2 + + JM |ω(t)|2 . (21) 2 2a
c2 = c21 + 2/a
ω ∗ = ω > (q ∗ )> . Since |s(t)| = 1 for all t ≥ 0, it follows that the inequality >
m
(16)
The derivative of V0 along the trajectories of (1) in closedloop with (5) satisfies
2
Theorem 1: Let a and b be two positive real numbers. Let τ > 0 and Ω > 0 be two constants such that the following inequalities hold: i h τ 2 τ + JM Ω c1 4a + e2 JM Ω2 + (e −1)(a+bΩ) 2Jm h i 2 (24) + a2 Ω2 + J12 JM Ω2 + a + bΩ b2 ξ1 (τ )
> = − 2 |q(t)| − |ω(t)| + [c1 ω(t) + q(t) ]r(xt ) . (32) Next, we define a functional: Rt Rt 2 V2 (xt ) = V1 (x(t)) + γ1 t−τ ` |q(r)| ˙ drd` Rt Rt (33) 2 +γ2 t−τ ` |ω(r)| ˙ drd` , where γ1 > 0 and γ2 > 0 are constants to be selected later. Then elementary calculations give V˙ 2 (t) ≤ − a2 |q(t)|2 − |ω(t)|2 + |c1 ω(t) + q(t)||r(xt )| Rt 2 +τ γ1 |q(t)| ˙ 2 + τ γ2 |ω(t)| ˙ − γ1 t−τ |q(l)| ˙ 2 dl Rt 2 −γ2 t−τ |ω(l)| ˙ dl. (34) From (30), we deduce that |q(t)| ˙ 2 ≤ |ω(t)|2 ≤ |ω(t)∗ Jω(t) + aq(t) + bω(t) − r(xt )|2 2 |ω(t)|4 + |r(xt )|2 + a2 |q(t)|2 + b2 |ω(t)|2 . ≤ 4 JM (35) Combining (34) and (35), we obtain 2 2 |ω(t)| ˙ Jm
V˙ 2 (t) ≤ − a2 |q(t)|2 − |ω(t)|2 + |c1 ω(t) + q(t)||r(xt )| + τ γ1 |ω(t)|2 2 +τ ς2 JM |ω(t)|4 + |r(xt )|2 + a2 |q(t)|2 + b2 |ω(t)|2 Rt Rt 2 −γ1 t−τ |q(l)| ˙ 2 dl − γ2 t−τ |ω(l)| ˙ dl 2 2 + ς1 τ + τ ς2 JM |ω(t)| − 1 |ω(t)|2 +c1 |ω(t)||r(xt )| + |q(t)||r(xt )| + τ ς2 |r(xt )|2 Rt Rt 2 −γ1 t−τ |q(l)| ˙ 2 dl − γ2 t−τ |ω(l)| ˙ dl , 2 2 2 with ς1 = γ1 Jm + 4γ2 b2 /Jm , ς2 = 4γ2 /Jm . From the triangle inequality, we deduce that V˙ 2 (t) ≤ τ ς2 a − 12 a|q(t)|2 4τ γ J 2 + ς1 τ + J22 M |ω(t)|2 − 1 |ω(t)|2 m
+ + −
c21 1 a 2 2 2 2 |ω(t)| + 2 |r(xt )| + 4 |q(t)| 4τ γ2 1 2 2 2 |r(xt )| a |r(xt )| + Jm Rt Rt 2 2 γ1 t−τ |q(l)| ˙ dl − γ2 t−τ |ω(l)| ˙ dl
(36) .
By grouping the terms, we obtain V˙ 2 (t) ≤ ς2 τ a − 14 a|q(t)|2 2 2 + ς1 τ + ς2 JM τ |ω(t)| − 12 |ω(t)|2 +
−
c21 2
1 a
+ ς2 τ |r(xt )|2 Rt Rt 2 ˙ dl . γ1 t−τ |q(l)| ˙ 2 dl − γ2 t−τ |ω(l)| +
Now, observe, that, for all t ≥ τ , the equality holds: Rt Rt r(xt ) = a t−τ q(l)dl ˙ + b t−τ ω(l)dl ˙ .
We deduce that V˙ 2 (t) ≤ ς2 aτ − 14 a|q(t)|2 2 + ς1 τ + ς2 τ JM |ω(t)|2 − 12 |ω(t)|2 Rt Rt 2 +ξ2 (τ ) a2 t−τ |q(l)| ˙ 2 dl + b2 t−τ |ω(l)| ˙ dl Rt R t 2 2 −γ1 t−τ |q(l)| ˙ dl − γ2 t−τ |ω(l)| ˙ dl ,
2 with ξ2 (τ ) = c2 τ + 8τ 2 γ2 /Jm , where c2 is the constant defined in (23) and the choice
γ1 = (c2 + 2ς2 τ ) a2 τ
(38)
From Cauchy-Schwarz’s inequality, we have for all t ≥ τ , Rt Rt 2 |r(xt )|2 ≤ 2a2 τ t−τ |q(l)| ˙ 2 dl + 2b2 τ t−τ |ω(l)| ˙ dl . (39)
(41)
gives 1 2 V˙ 2 (t) ≤ ς aτ − a|q(t)| 2 4 2 2 2 + ξ2 (τ )a + ς2 b τ + ς2 τ JM |ω(t)|2 − 21 |ω(t)|2 R R t t 2 2 + c22 + ς2 τ 2b2 τ t−τ |ω(l)| ˙ dl − γ2 t−τ |ω(l)| ˙ dl . (42) By grouping the terms, we obtain V˙ 2 (t) ≤ ς2 τ a − 41 a|q(t)|2 1 2 2 2 (43) h+ ξ3 (τ )τ+ ς22 τ2JM |ω(t)| i R− 2 |ω(t)| t 2 + c2 b2 τ + 8bJ 2τ − 1 γ2 t−τ |ω(l)| ˙ dl , m
where ξ3 (τ ) = (c2 + 2τ ς2 ) a2 τ + ς2 b2 . Since (25) implies 2 2 > 0, we can choose − 8b2 τ 2 /Jm that Jm 2 2 γ2 = c2 b2 Jm τ / Jm − 8b2 τ 2 . (44) This choice yields 2 2 1 2b τ V˙ 2 (t) ≤ J4ac a|q(t)|2 2 −8b2 τ 2 − 4 m (45) + ξ4 (τ )τ 2 + ξ5 (τ )τ 2 |ω(t)|2 − 21 |ω(t)|2 , 4 2 2 τ 2b with ξ4 (τ ) = 1 + J 28b c2 a2 + J 2 4c 2 2 2 τ 2 , ξ5 (τ ) = −8b τ −8b m m 2 2 2 2 2 4JM c2 b / Jm − 8b τ . The inequality (25) implies that 2 ξ4 (τ )τ 2 = 4ac2 b2 τ 2 / Jm − 8b2 τ 2 ≤ 1/8 . (46) The inequality (26) implies that 2 2 2 c2 τ 2 Jm a + 4b4 / Jm − 8b2 τ 2 ≤ 1/4 .
(47)
It follows that a 1 2 2 2 ˙ V2 (t) ≤ − |q(t)| + ξ6 (τ )τ |ω(t)| − |ω(t)|2 (48) 8 4 2 2 − 8b2 τ 2 . From (12) and with ξ6 (τ ) = 4JM c2 b2 / Jm the definition of V2 , we deduce that |ω(t)|2
(37)
(40)
≤ V1 (x(t)) ≤ V2 (xt ) .
(49)
It follows that, for all t ≥ τ , the inequality a 1 4J 2 c2 b2 V˙ 2 (t) ≤ − |q(t)|2 + − + 2 M 2 2 τ 2 V2 (xt ) |ω(t)|2 8 4 Jm − 8b τ (50) is satisfied. Step 2. Let K be a real number such that 2 2 0 < K < Jm − 8b2 τ 2 / 16JM c2 b2 τ 2 (51) and assume that V2 (xτ ) < K. Then let us show that V2 (xt ) < K for all t ≥ τ . To prove this result, we proceed by
4880
contradiction. Assume there is te > τ such that V2 (xt ) < K when t ∈ [τ, te ) and V2 (xte ) = K. Then there exists tc ∈ [τ, te ] such that V˙ 2 (tc ) > 0. Now, (50) implies 2 4JM c2 b2 2 1 ˙ τ V2 (xtc ) |ω(tc )|2 . (52) V2 (tc ) ≤ − + 2 4 Jm − 8b2 τ 2
Combining (59), (60), (62), we obtain V2 (x(τ )) ≤ Kτ with Kτ
Since V2 (xtc ) ≤ K, it follows from (51) that V˙ 2 (tc ) ≤ 0 .
(53)
This yields a contradiction. We deduce that, for all t ≥ τ , V2 (xt ) < K. From this and (48), for all t ≥ τ , we have |ω(t)|2 < K. Then we deduce from (48) that there is cK > 0 such that V˙ 2 (t) ≤
− a8 |q(t)|2 − cK |ω(t)|2 .
(54)
It follows that V2 (xt ) is nonincreasing over [τ, +∞). It follows that w(t), q(t), p(t) are bounded. We deduce that their derivatives are bounded. It follows that these functions are uniformly continuous over [τ, +∞). By integrating (54), we obtain, for all t ≥ τ , Rt Rt V2 (xt ) − V2 (xτ ) ≤ − a8 τ |q(l)|2 dl − cK τ |ω(l)|2 dl, (55) which gives Rt Rt (56) cK τ |ω(l)|2 dl + a8 τ |q(l)|2 dl ≤ V2 (xτ ) R +∞ because V2 is nonnegative. We deduce that τ |ω(l)|2 dl < Rt +∞ and τ |q(l)|2 dl < +∞. Since ω and q are uniformly continuous, Barbalat’s lemma applies: we deduce that ω(t) and q(t) converge to zero.
Q0 (ω(t)) ≤ ≤
2
2
i
(a+bΩ) eτ 2 τ 2 JM Ω + 2Jm (e − 1) 2 2 2b τ a2 τ 3 Ω2 +JM Ω + c22 + J 4c 2 2 2 m −8b τ 2 2 c2 b 3 + 2J 2 −16b JM Ω2 + a + bΩ . 2τ 2 τ m
(64)
IV. N UMERICAL E XAMPLE To further evaluate the sufficient condition for stability in (24), we consider a cube-satellite with inertia matrix: 0.0465 −0.0007 0.0004 J = −0.0007 0.0486 −0.0021 kg · m2 0.0004 −0.0021 0.0482 Here, JM = 0.0504 and Jm = 0.0461. For any specified value of Ω, the control gains (a, b) are related to τ and their feasible values can be computed through the inequality (24).
Fig. 1.
Feasible range of control parameters (a, b) according to (24).
− 1)
We deduce that, for all t ∈ [0, τ ], q |ω(t)| ≤ eτ JJM Ω2 + J12 (a + bΩ)2 (eτ − 1) . m m
h = c1 4a +
Since (24) implies that τ and Ω are such that Kτ < 2 Jm −8b2 τ 2 2 c b2 τ 2 then, bearing in mind (51), we deduce that ω(t) 32JM 2 and q(t) converge to zero. This allows us to conclude.
Step 3. Now, from Lemma 3 in appendix we deduce that, the function Q0 defined in (10) satisfies for all t ∈ [0, τ ], (eτ eτ 12 ω(0)> Jω(0) + (a+bΩ) 2Jm 2 τ (a+bΩ) e 2 (eτ − 1) . 2 JM Ω + 2Jm
(63)
(57) (58)
As shown in Figure 1, smaller values for time-delay τ and initial angular rate upper bound Ω provide a wider range for the control parameters. For the closed-loop control simulation, a = 0.01 and b = 0.024 are selected corresponding to τ = 0.1 sec and Ω = 0.03 rad/sec.
From kp(t)k ≤ 1 and kq(t)k ≤ 1 for all t ≥ τ , we deduce 2 Rt Rt τ2 2 (59) |ω(r)| ˙ drd` ≤ 2J JM Ω2 + a + bΩ 2 t−τ ` m
and
Rt t−τ
Rt `
2 |q(r)| ˙ drd` ≤
τ2 2 2 Ω .
(60)
Moreover, using the fact that |s(t)| = 1 for all t ≥ 0, we deduce that V1 (x(τ )) ≤ c1 V0 (x(τ ))+JM Ω ≤ c1[4a+Q0 (ω(τ ))]+JM Ω . (61) We deduce from (57) that " # JM Ω2 eτ (a + bΩ)2 V1 (x(τ )) ≤ c1 4a + + −1 + JM Ω. 2 2Jm (eτ − 1) (62)
Fig. 2. Time histories of |q(t)| and |ω(t)| under closed-loop control action.
The results in Figure using the initial √ 2 are obtained √ conditions, s(0) = [ 2/2, 0, 2/2, 0]T and ω(0) = [0.03, 0, 0]T . The input torques (not reported here) required for this maneuver appear well in range for the cube-satellite
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class of problems under consideration here. The attitude and angular rate signals converge within approximately 30 seconds indicating satisfactory closed-loop performance. V. C ONCLUSION Through a novel approach of ‘partial strictification’ we determined an estimate of the basin of attraction for rigid-body attitude dynamics stabilized by a linear state feedback with pointwise input delay. Precise value of the time-delay is not necessary for implementing the proposed controller; only an upper bound on the delay-value is adequate. Use of the unitquaternion representation allows us to have no restrictions whatsoever on the initial orientation. The problem of input sampling will be the subject of further research.
A PPENDIX Lemma 2: Consider the system (29) with continuous initial condition over [−τ, 0]. This system is forward complete i.e. all its solutions are defined over [−τ, +∞). Moreover every solution is continuous. Proof. We proceed by induction. Induction assumption: the solution is continuous and defined over [−τ, iτ ], where i is a nonnegative integer. Step 0: the assumption is satisfied at the step 0 since we consider a solution with a continuous initial condition. Step i: assume that the induction assumption is satisfied to the step i, where i is a nonnegative integer. Let te be the largest number in the interval [iτ, (i + 1)τ ] such that the solution x(t) is defined over [−τ, te ). Then, for all t ∈ [iτ, te ), (29) is satisfied. Let Q0 be the function defined in (10). Then using the fact that ω(t)> ω(t)∗ = 0, we obtain
R EFERENCES
Q˙ 0 (t)
[1] S.P. Bhat, D.S. Bernstein, A topological obstruction to continuous global stabilization of rotational motion and the unwinding phenomenon. Systems & Control Letters, 39, pp. 63-70, 2000. [2] N. Bekiaris-Liberis and M. Krstic, Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations, Automatica, vol. 49, pp. 1576-1590, 2013. [3] A. A. Chunodkar, M. R. Akella, Attitude Stabilization with Unknown Bounded Delay in Feedback Control Implementation. Journal of Guidance, Control, and Dynamics, vol. 34, No. 2, March-April 2011. [4] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations. (AMS, vol. 99, Springer: New York, 1993). [5] I. Karafyllis, M. Krstic, Nonlinear Stabilization under Sampled and Delayed Measurements, and with Inputs Subject to Delay and ZeroOrder Hold, IEEE Trans. Aut. Contr., vol. 57, pp. 1141-1154, 2012. [6] M. Malisoff, F. Mazenc, Constructions of Strict Lyapunov Functions. Communications and Control Engineering Series, Springer-Verlag London Ltd., London, ISBN: 978-1-84882-534-5 (Print), U.K., 2009. [7] F. Mazenc, P.-A. Bliman, Backstepping Design for Time-Delay Nonlinear Systems, IEEE Trans. Aut. Contr., vol. 51, No.1, 2006, pp.149-154. [8] F. Mazenc, M. Malisoff. Asymptotic Stabilization for Feedforward Systems with Delayed Feedback, Automatica, vol. 49, No. 3, 2013. [9] F. Mazenc, M. Malisoff, Stabilization of a chemostat model with Haldane growth functions and a delay in the measurement, Automatica, vol. 46, Issue 9 pp. 1428-1436, Sept. 2010. [10] F. Mazenc, M. Malisoff, T. N. Dinh, Robustness of Nonlinear Systems with Respect to Delay and Sampling of the Controls, Automatica, vol. 49, pp. 1925-1931, 2013. [11] F. Mazenc, M. Malisoff, Z. Lin, Further results on input-to-state stability for nonlinear systems with delayed feedbacks, Automatica, vol. 44, No. 9, pp. 2415-2421, 2008. [12] F. Mazenc, D. Nesic, Lyapunov functions for time varying systems satisfying generalized conditions of Matrosov theorem. Mathematics of Control, Signals, and Systems, vol. 19, No 2, pp. 151-182, May 2007. [13] A. Teel, Connections Between Razumikhin-Type Theorems and the ISS Nonlinear Small Gain Theorem, IEEE Trans. Aut. Contr., 43(7), 1998. [14] Wen, J., and Kreutz-Delgado, K., The Attitude Control Problem, IEEE Trans. Aut. Contr., vol. 36, No. 10, 1991, pp. 1148-1162. [15] R. Schlanbusch, A. Loria, R. Kristiansen, P. J. Nicklasson, PD+ Based Output Feedback Attitude Control of Rigid Bodies, IEEE Trans. Aut. Contr., vol. 57, No. 8, 2012, pp. 2146-2152. [16] A. Abdessameud, A. Tayebi, I.-G. Polushin, Attitude Synchronization of Multiple Rigid-Bodies with Communication Delays, IEEE Trans. Aut. Contr., vol. 57, No. 9, 2012, pp. 2405-2411. [17] Y. I. T. Hatanaka, M. Fujita, M. Spong, Passivity-Based Pose Synchronization in Three-Dimensions, IEEE Trans. Aut. Contr., vol. 57, No. 2, 2012, pp. 360-375. [18] S. J. Chung, U. Ahsun, J. J. E. Slotine, Application of Synchronization to Formation Flying Spacecraft: Lagrangian Approach, AIAA J. Guid. Contr., Dyn., vol. 32, No. 2, 2009, pp. 512-526. [19] A. Sarlette, R. Sepulchre, N. E. Leonard, Autonomous Rigid-Body Attitude Synchronization, Automatica, 45(2), 2009, pp. 572-577.
= −ω(t)> [aq(t − τ ) + bω(t − τ )] .
From the triangle inequality and deduce that, for all t ∈ [iτ, te ),
2 Jm Q0 (ω(t))
(65)
≥ |ω(t)|2 , we
1 2 Q0 (ω(t)) + |aq(t − τ ) + bω(t − τ )|2 . (66) Q˙ 0 (t) ≤ Jm 4 By integrating this inequality, we obtain, for all t ∈ [iτ, te ), 2
e Jm (t−iτ ) Q (ω(iτ )) R t 2 (t−`)0 |aq(` − τ ) + bω(` − τ )|d` . e Jm iτ (67) Therefore, for all t ∈ [iτ, te ), Q0 (ω(t)) ≤ +
Q0 (ω(t)) ≤ +
2τ
e Jm Q0 (ω(iτ )) 2τ τ e Jm sup
|aq(`) + bω(`)| .
(68)
`∈[(i−1)τ,iτ ]
Since q and ω are continuous over [(i − 1)τ, iτ ], it follows that sup |aq(m) + bω(m)| is a finite real number. m∈[(i−1)τ,iτ ]
Therefore there exists a constant ω > 0 such that |ω(t)|
≤
ω
,
∀t ∈ [iτ, te ) .
(69)
Since s(t) is bounded too, x(t) is bounded over [iτ, te ). We deduce that the finite escape time phenomenon cannot occur. Therefore te = (i + 1)τ and the solution is defined and continuous over [−τ, (i+1)τ ]. Thus the induction assumption is satisfied at the step i + 1. This concludes the proof. Lemma 3: Consider the system J ω(t) ˙ = −ω(t)∗ Jω(t) + u(t) ,
(70)
where J is the matrix in Section II-A, where ω ∈ R3 and u is a continuous function, bounded by the constant u > 0. Then, for all t0 ≥ 0 and t ≥ t0 , the following holds u2 . 1 Q0 (ω(t)) = ω > Jω ≤ et−t0 Q0 (ω(t0 ))+ et−t0 − 1 . 2 2Jm (71) Proof. The derivative Q˙ 0 along trajectories of (70) satisfies Q˙ 0(t) = ω(t)> u(t), since ω(t)> ω(t)∗ = 0. Thus Q˙ 0 (t) ≤ 1 u2 > > −1 u(t) ≤ Q0 (ω(t)) + 2J . By 2 ω(t) Jω(t) + u(t) J m integrating this inequality over [t0 , t], one obtains (71).
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