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Delayed Feedback Asymptotic Stabilization of Rigid Body Attitude Motion for Large Rotations∗
Proceedings of the 12th IFAC Workshop on Time Delay Systems Proceedings of the 12th IFAC Workshop on Time Delay Systems June 28-30, 2015. Ann Arbor, USA Proceedings of the the 12th IFACMI, Workshop on Time Time Delay Delay Systems Systems Proceedings of 12th IFAC Workshop on June 28-30, 2015. Ann Arbor, MI, USA Available online at www.sciencedirect.com June June 28-30, 28-30, 2015. 2015. Ann Ann Arbor, Arbor, MI, MI, USA USA
ScienceDirect IFAC-PapersOnLine 48-12 (2015) 081–086
Delayed Feedback Asymptotic Stabilization Delayed Feedback Asymptotic Stabilization Delayed Feedback Asymptotic Stabilization Delayed Feedback Asymptotic Stabilization of Rigid Body Attitude Motion for Large of Rigid Body Attitude Motion for Large of Rigid Body Attitude Motion for Large of Rigid Body Attitude Motion for Large Rotations Rotations Rotations Rotations
∗ ∗ ∗ Samiei K. Samiei ∗∗ Maziar Maziar Izadi Izadi ∗∗ Amit Amit K. Sanyal Sanyal ∗∗ ∗∗ Samiei Maziar Izadi K. A. ∗∗ Samiei ∗Eric Maziar Izadi ∗ Amit Amit K. Sanyal Sanyal ∗ Eric A. Butcher Butcher ∗∗ Eric A. A. Butcher Butcher ∗∗ Eric ∗ ∗ Department of Mechanical and Aerospace Engineering, New Mexico ∗ Department of Mechanical and Aerospace Engineering, New Mexico of and Aerospace Engineering, New Mexico ∗ Department State University, Cruces, NM, Department State of Mechanical Mechanical andLas Aerospace University, Las Cruces, Engineering, NM, 88003 88003 New Mexico ∗∗ University, Las Cruces, NM, 88003 of Aerospace and Mechanical Engineering, ∗∗ Department State State University, Las Cruces, NM, 88003 University of ∗∗ Department of Aerospace and Mechanical Engineering, University of ∗∗ Department of Aerospace and Mechanical Engineering, Arizona, Tucson, AZ Department of Aerospace and Mechanical Engineering, University University of of Arizona, Tucson, AZ 85721 85721 Arizona, Arizona, Tucson, Tucson, AZ AZ 85721 85721 Abstract: The The delayed delayed feedback feedback control control of of rigid rigid body body attitude attitude motion motion is is addressed addressed where where there there Abstract: Abstract: The delayed feedback control of rigid body attitude motion is addressed where there is an unknown time delay in the feedback measurement. The attitude motion is described on Abstract: The delayed feedback control of rigid body attitude motion is addressed where there is an unknown time delay in the feedback measurement. The attitude motion is described on the the is an delay in the measurement. The attitude motion is described on the tangent bundletime TSO(3) to globally and represent the orientation of body, while is an unknown unknown time delay the feedback feedback measurement. motion is rigid described the tangent bundle TSO(3) to in globally and uniquely uniquely representThe theattitude orientation of the the rigid body,on while tangent bundle TSO(3) to globally and uniquely represent the orientation of the rigid body, while a continuous nonlinear delayed feedback control law is proposed for local asymptotic stabilization tangent bundle TSO(3)delayed to globally and uniquely represent the orientation of the rigidstabilization body, while a continuous nonlinear feedback control law is proposed for local asymptotic a nonlinear delayed feedback control law asymptotic stabilization oncontinuous TSO(3). For For this purpose, purpose, we introduce notion based on on for thelocal Lyapunov technique, which a continuous nonlinear delayed we feedback control law is is proposed proposed for local asymptotic stabilization on TSO(3). this introduce aa notion based the Lyapunov technique, which on TSO(3). For this purpose, we introduce a notion based on the Lyapunov technique, which combines the Morse-Lyapunov method with Lyapunov-Krasovskii technique to yield a suitable on TSO(3). For this purpose, we introduce a notion based on the Lyapunov technique, which combines the Morse-Lyapunov method with Lyapunov-Krasovskii technique to yield a suitable combines the method with technique to aa suitable Morse-Lyapunov-Krasovskii (M-L-K) functional, from conditions proper combines the Morse-Lyapunov Morse-Lyapunov methodfunctional, with Lyapunov-Krasovskii Lyapunov-Krasovskii technique to yield yieldand suitable Morse-Lyapunov-Krasovskii (M-L-K) from which which the the stability stability conditions and proper Morse-Lyapunov-Krasovskii (M-L-K) functional, from which the stability conditions and control gain matrices are obtained in terms of linear matrix inequalities (LMIs). Simulations Morse-Lyapunov-Krasovskii (M-L-K)infunctional, from which stability conditions and proper proper control gain matrices are obtained terms of linear matrixthe inequalities (LMIs). Simulations control are in of matrix illustrategain the matrices performance of the the proposed proposed control scheme. control gain matrices are obtained obtained in terms terms of linear linear matrix inequalities inequalities (LMIs). (LMIs). Simulations Simulations illustrate the performance of control scheme. illustrate the performance of control scheme. illustrate the (International performanceFederation of the the proposed proposed control scheme. © 2015, IFAC of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Attitude control, time-delayed system, LMIs, Lyapunov-Krasovskii functional functional Keywords: Attitude control, time-delayed system, LMIs, Lyapunov-Krasovskii Keywords: Attitude Attitude control, control, time-delayed time-delayed system, system, LMIs, LMIs, Lyapunov-Krasovskii Lyapunov-Krasovskii functional functional Keywords: 1. INTRODUCTION INTRODUCTION Chunodkar 1. Chunodkar and and Akella Akella (2011); (2011); Samiei Samiei and and Butcher Butcher (2013); (2013); 1. INTRODUCTION Chunodkar and Akella (2011); Samiei and (2013); Samiei et al. (2012); Nazari et al. (2013)) who 1. INTRODUCTION Chunodkar and AkellaNazari (2011);etSamiei and Butcher Butcher (2013); Samiei et al. (2012); al. (2013)) who generally generally Samiei aet etlocal al. (2012); (2012); Nazari et al. al. (2013)) (2013)) who generally generally Attitude motion occurs in many practical systems such as utilize coordinate representation of rigid body Samiei al. Nazari et who Attitude motion occurs in many practical systems such as utilize a local coordinate representation of rigid body atatAttitude motion occurs in many many practical systems such as utilize a local coordinate representation of rigid body unmanned vehicles (Sanyal and practical Chyba (2009)) (2009)) orsuch spacetitude. (Samiei and (2013)), the stabilization Attitude motion occurs in systems as utilize coordinate representation body atatunmanned vehicles (Sanyal and Chyba or spacetitude. aIn Inlocal (Samiei and Butcher Butcher (2013)), of therigid stabilization unmanned vehicles (Sanyal and Chyba Chyba (2009)) or spacespacetitude. In (Samiei Butcher (2013)), the stabilization craft systems systems (Sanyal(Sanyal and Chaturvedi Chaturvedi (2008); Samiei et al. al. of attitude dynamics in the presence unmanned vehicles and (2009)) or titude. (Samiei and and Butcher (2013)), craft (Sanyal and (2008); Samiei et of rigid rigidInspacecraft spacecraft attitude dynamics in the the stabilization presence of of craft systems (Sanyal andstabilization Chaturvedi (2008); (2008); Samiei et al. external of rigid attitude dynamics in presence (2015a)). Almost global of rigid rigidSamiei body et attistochastic an craft systems (Sanyal and Chaturvedi al. of rigid spacecraft spacecraft attitudeand dynamics in the thetime-varying presence of of (2015a)). Almost global stabilization of body attiexternal stochastic torques torques and an unknown unknown time-varying (2015a)). Almost global stabilization of rigid rigid of body atti- external external stochastic torques and and an an unknownwhere time-varying tude motion motion basedglobal on aa global global representation attitude time in is a (2015a)). Almost stabilization of body attistochastic torques unknown time-varying tude based on representation of attitude time delay delay in the the measurement measurement is studied studied where a linear linear tude motion based based on aainglobal global representation of attitude attitude delay in measurement is studied where linear has attracted attracted attention control systems design, design, see e.g e.g time delayed state feedback controller aa suitable tude motion on representation of time delay in the the measurement is with studied where a a choice linear has attention in control systems see delayed state feedback controller with suitable choice has attracted attention in control systems design, see e.g delayed state feedback controller with a suitable choice (Sanyal and Chaturvedi (2008)). The proposed continuous of Lyapunov-Krasovskii (L-K) functional can guarantee has attracted attention in controlThe systems design, see e.g delayed state feedback controller with a suitable choice (Sanyal and Chaturvedi (2008)). proposed continuous of Lyapunov-Krasovskii (L-K) functional can guarantee (Sanyal and Chaturvedi Chaturvedi (2008)). The aproposed proposed continuous of Lyapunov-Krasovskii (L-K) functional can control strategies strategies in these these(2008)). works have have nonlinearcontinuous structure the local the system (Sanyal and The of (L-K) of functional can guarantee guarantee control in works a nonlinear structure theLyapunov-Krasovskii local asymptotic asymptotic stability stability of the closed-loop closed-loop system control strategies in these these works have aa nonlinear nonlinear structure the local asymptotic stability of the closed-loop system since the state space of the dynamics model is a nonin the mean square sense through the solution of LMIs. control strategies in works have structure the local asymptotic stability of the closed-loop since the state space of the dynamics model is a non- in the mean square sense through the solution ofsystem LMIs. since the state space of the dynamics model is a nonin the mean square sense through the solution of LMIs. linear space, i.e., the tangent bundle TSO(3). In (Sanyal The main drawback with these studies is that the prosince the state space of the dynamics model is a nonin the mean square sense through the solution of linear space, i.e., the tangent bundle TSO(3). In (Sanyal The main drawback with these studies is that theLMIs. prolinear space, i.e., the tangent bundle TSO(3). In (Sanyal The main drawback with these studies is that the and Chaturvedi (2008)) a geometric controller is designed posed controllers can only guarantee the local asymptotic linear space, i.e.,(2008)) the tangent bundlecontroller TSO(3). is Indesigned (Sanyal The drawback these studies is that the proproand Chaturvedi a geometric posedmain controllers can with only guarantee the local asymptotic and Chaturvedi (2008)) geometric controllerand is designed designed posed controllers can guarantee the asymptotic on SO(3) SO(3) to almost almost globally track attitude attitude angular stability of for small maneuvers with and Chaturvedi (2008)) aa geometric controller is posed controllers can only only the local local asymptotic on to globally track and angular stability of the the system system forguarantee small angle angle maneuvers with on SO(3)ofto to almost globally track attitude and angular of system small angle maneuvers with velocity spacecraft in the the presence of and gravity and stability small This is to use on SO(3) globally track attitude angular stability of the thevelocities. system for for angle maneuvers with velocity of aa almost spacecraft in presence of gravity and small angular angular velocities. Thissmall is due due to the the use of of a a local local velocity of a spacecraft in the presence of gravity and small angular velocities. This is due to the use of a local disturbance moments. coordinate representation of attitude which suffers from velocity of amoments. spacecraft in the presence of gravity and small angular velocities. This is due to the use of a local disturbance coordinate representation of attitude which suffers from disturbance moments. moments. coordinate representation of attitude which suffers from mathematical singularities in representing the orientation disturbance coordinate representation which from mathematical singularities of in attitude representing thesuffers orientation On the other hand, the existence of time delay in conmathematical singularities in representing the orientation On the other hand, the existence of time delay in con- of a rigid body. In addition, the proposed delayed feedback mathematical singularities in representing the orientation of a rigid body. In addition, the proposed delayed feedback On the other hand, the existence of time delay trolled systems is inevitable. Time delay occurs in conOn the systems other hand, the existence timeoccurs delay in con- of aa rigid In the proposed delayed feedback trolled is inevitable. Timeofdelay controllers are robust time delay cannot tolerate of rigid body. body. In addition, addition, delayed are not not robust to to the timeproposed delay and and cannotfeedback tolerate trolled systems due is inevitable. inevitable. Time occurs in to the the existence existence of delay delay in sensors, sensors, trolled is Time delay delay occurs in concon- controllers controllers are not robust to time delay and cannot tolerate systems due to of in large values of delay, which makes the implementation of controllers are not robust to time delay and cannot tolerate large values of delay, which makes the implementation of trolled systems due to the existence of delay in sensors, communication channels, or actuators. Controlled systems trolled systems channels, due to the delay in systems sensors, the large values of delay, which makes the implementation of communication or existence actuators.ofControlled control design impractical. large values of delay, which makes the implementation of the control design impractical. communication channels, or actuators. Controlled systems designed based based on on nonlinear feedback schemes schemes are systems usually the control design impractical. communication channels, or actuators. Controlled designed nonlinear feedback are usually the control designcontrol impractical. designed based on nonlinear feedback schemes are usually robust to a small amount of time delay. However, if the Delayed feedback design for arbitrary attitude madesigned on amount nonlinearoffeedback schemes are usually robust tobased a small time delay. However, if the Delayed feedback control design for arbitrary attitude marobust to aa increases small amount amount of time delay. delay. However, if the the neuvers Delayed feedback control design arbitrary attitude matime delay delay in aa of controlled system due to toif the of is demanding robust to small time However, Delayed on feedback controlbundle design for for arbitrary matime increases in controlled system due neuvers on the tangent tangent bundle of SO(3) SO(3) is aaattitude demanding time delay increases in aa controlled controlled system due then to the the neuversproblem on the the tangent tangent bundle ofsystem SO(3)must is aa be demanding failure of system components or external sources, a control since the control designed time delay increases in system due to neuvers on bundle of SO(3) is demanding failure of system components or external sources, then a control problem since the control system must be designed failure offeedback system components components or external external sources, then aa control problem control system must be delayedof control scheme scheme is essential. essential. In addition, addition, on space TSO(3), there failure system or sources, then control problem since since the thestate control system mustwhile be designed designed delayed feedback control is In on the the non-contractible non-contractible state space TSO(3), while there is is delayed feedback control scheme is essential. In addition, on the non-contractible state space TSO(3), while there is the existence of time delay in a controlled system with time delay in the feedback loop. However, if probdelayed feedback control scheme essential. system In addition, thedelay non-contractible state loop. space However, TSO(3), while is the existence of time delay in a iscontrolled with on time in the feedback if this thisthere probthe existence of time delay in a controlled system with time delay in the feedback loop. However, if this probrotational dynamics can cause chattering or oscillatory lem can be properly tackled on TSO(3), those drawbacks the existence of timecan delay in achattering controlledorsystem with time delay in the feedback However, if drawbacks this probrotational dynamics cause oscillatory lem can be properly tackled loop. on TSO(3), those rotational dynamics can cause Thus chattering or oscillatory oscillatory lem can can be be for properly tackled on TSO(3), TSO(3), those drawbacks drawbacks behavior (Ailon (Ailon et al. al.can (2004)). it appears appears that the the lem mentioned delayed feedback control based rotational dynamics cause chattering or properly on those behavior et (2004)). Thus it that mentioned for delayedtackled feedback control schemes schemes based on on behavior (Ailon et etcontrol al. (2004)). (2004)). Thus itattitude appearsmotion that the the mentioned for delayed feedback control schemes based on delayed feedback design for is a local attitude representation will be eliminated. behavior (Ailon al. Thus it appears that mentioned for delayed feedback control schemes basedThe on delayed feedback control design for attitude motion is a local attitude representation will be eliminated. The delayed feedback control design for attitude motion is a local attitude representation will be eliminated. The practical and crucial. Unfortunately, this problem has been conventional technique to address the nonlinear control delayed control design forthis attitude is aconventional local attitude representation willthe be nonlinear eliminated. The practical feedback and crucial. Unfortunately, problemmotion has been technique to address control practical and crucial. crucial. Unfortunately, problem been conventional technique to the nonlinear control only studied studied by aa few few researchers this (Ailon et al. al.has (2004); dynamics evolving TSO(3) is practical and Unfortunately, this problem has been design conventional technique to address address the on nonlinear only by researchers (Ailon et (2004); design for for attitude attitude dynamics evolving on TSO(3)control is the the only studied studied by by aa few few researchers researchers (Ailon (Ailon et et al. al. (2004); (2004); design design for attitude attitude dynamics dynamics evolving on(Milnor TSO(3)(1963)), is the the use of Morse-Lyapunov (M-L) functions only for evolving on TSO(3) is use of Morse-Lyapunov (M-L) functions (Milnor (1963)), support from the National Science Foundation under Financial Financial support from the National Science Foundation under use of Morse-Lyapunov (M-L) functions (Milnor (1963)), while L-K or Lyapunov-Razumikhin (L-R) techniques use of Morse-Lyapunov (M-L) functions (Milnor (1963)), Financial support from the National Science Foundation under Grant No. CMMI-1131646 is gratefully acknowledged. while L-K or Lyapunov-Razumikhin (L-R) techniques are are Financial support from is thegratefully Nationalacknowledged. Science Foundation under Grant No. CMMI-1131646 while Grant No. CMMI-1131646 is gratefully acknowledged. while L-K L-K or or Lyapunov-Razumikhin Lyapunov-Razumikhin (L-R) (L-R) techniques techniques are are Grant No. CMMI-1131646 is gratefully acknowledged.
Ehsan Ehsan Ehsan Ehsan
Copyright IFAC 2015 81 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, IFAC 2015 81 Copyright IFAC 2015 81 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 81 Control. 10.1016/j.ifacol.2015.09.357
IFAC TDS 2015 82 June 28-30, 2015. Ann Arbor, MI, USA
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usually used to address the stabilization of delay differential equations in Euclidean space. This work combines these two approaches by using the concept of a M-LK functional to address the delayed feedback control of attitude dynamics evolving on TSO(3). This manuscript is a continuation of a preliminarily work, presented by the same authors in (Samiei et al. (2015b)). In this prior work, a continuous delayed feedback controller was designed for the attitude motion based on LMI stability conditions. The nonlinear terms in the dynamics model were considered after solving LMIs and then a condition in which the derivative of the proposed M-L-K functional is negative definite was obtained. Thus, the validity of the controller is only guaranteed inside the region in which the derivative of the M-L-K functional is negative definite. However, in this paper, a continuous full-state delayed feedback control law is chosen and a M-L-K functional candidate is formulated. Application of Lyapunov stability techniques using the ML-K functional candidate results in the solution of LMIs for the controller gain matrices using convex optimization. Furthermore, the nonlinear terms in the dynamics model are analyzed within the LMI stability conditions. The norm of the obtained control gain matrices are also minimized in order to have an admissible control effort for large maneuvers. Finally, the effectiveness of the proposed controller is demonstrated through numerical simulations.
where yi , i = 1, 2, 3 are the elements of y. In addition, (·)× can also denote the cross product, i.e., y × z = y × z for any y, z ∈ R3 . The canonical basis {e1 , e2 , e3 } for the body frame B can form a basis for the Lie algebra so(3) 3 such that y × = yi e × i . The principal rotation elements i=1
in terms of the rotation matrix R are given as 1 1 (R − RT ), (3a) φ = cos−1 ( (tr(R) − 1)), × = 2 2 sin Φ where φ represents the principal rotation angle, ∈ S2 denotes the principal rotation axis, S2 is the 2−dimensional unit sphere embedded in R3 , and Ri,j , i, j = 1, 2, 3 are the elements of the rotation matrix R. More information about the global representation of attitude dynamics can be found in (Sanyal and Chaturvedi (2008)). The attitude dynamics in the body frame B is obtained ˙ using the Euler’s equations of motion as J Ω(t) = −Ω(t) × JΩ(t)+u(t), where u(t) ∈ R3 is the external torque applied to the rigid body in B and J = J T > 0 is the known inertia matrix. We assume that three actuators provide torques about the body frame axes. Thus, the rigid body rotational equations of motion is defined on the state space TSO(3) ≈ SO(3) × R3 as ˙ (4a) R(t) = R(t)Ω(t)× , × ˙ (4b) J Ω(t) = −Ω(t) JΩ(t) + u(t).
2. NOTATION AND PROBLEM FORMULATION
The state variables are expressed as the pair (R, Ω) ∈ TSO(3). The equilibrium subspace of the attitude dynamics model of Eq.(4) is obtained as E = {(R, Ω) ∈ TSO(3)|Ω = 0, R ∈ SO(3)}.
Notation In this paper, we denote R+ as the set of nonnegative real numbers. Rn represents the n−dimensional Euclidean space and Rm×n denotes a vector space of real matrices which maps Rn to Rm . det(·) and tr(·) stands for the determinant and the trace of a square matrix, respectively. n The Euclidean norm of y ∈ R is denoted as y = T y y ∈ R, while the Frobenius norm of A ∈ Rm×n is represented as A F = tr(AT A) ∈ R.
3. ROBUST ASYMPTOTIC DELAYED FEEDBACK CONTROL DESIGN In this section, we aim to design a continuous full-state feedback controller to stabilize the attitude motion of Eq.(4) such that (R, Ω) → (I3 , 0) as t → ∞ in the presence of an unknown constant time delay in the feedback path. The main contribution of this paper in comparison with current studies based on a local representation of attitude, as discussed in introduction, is that the proposed delayed feedback controller is on the nonlinear (non-Euclidean) space TSO(3), and thus the topological difficulty in obtaining the almost global stabilization of attitude (see e.g., Bhat and Bernstein (2000) for more details on such obstruction) is eliminated and the system can be stabilized from almost every initial orientation. In addition, the other obstructions resulting from the feedback control design based on traditional practice such as unwinding will be naturally addressed.
2.1 Attitude Motion Formulation
The rigid body attitude dynamics are represented in the body fixed coordinate frame B with its origin assigned at the center of mass of the rigid body. The attitude of the rigid body is expressed by the rotation matrix R from the body frame B to the inertial frame N as an element of the real special orthogonal matrix Lie group SO(3) such that SO(3) = {R ∈ R3×3 | RT R = I3 and det(R) = 1}, (1) with corresponding matrix Lie algebra so(3), where I3 ∈ R3×3 is the identity matrix. The rotation matrix R uniquely and globally describes the orientation of a rigid body from the body frame B to the inertial frame N with no mathematical singularities. The attitude kinematic differential equation in terms of R ∈ SO(3) can ˙ be represented as R(t) = R(t)Ω(t)× , where Ω(t) ∈ R3 is the angular velocity represented in the body frame B and y × : R3 → so(3) is the map from the Euclidean space R3 to the Lie algebra so(3) which is defined as 0 −y3 y2 (2) y × = y3 0 −y1 , −y2 y1 0
We assume that the unknown constant time delay satisfies 0 ≤ τ ≤ τmax , where τmax ∈ R+ is known. Furthermore, due to existence of time delay in the measurement, current information of the state variables is unavailable for the purpose of control design. 3.1 Preliminaries Results A Morse function on SO(3) is defined as a smooth realvalued function from SO(3) to R+ with non-degenerate isolated critical points. An appropriate choice of the Morse 82
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Delayed Feedback Asymptotic Stabilization Delayed Feedback Asymptotic Stabilization Delayed Feedback Asymptotic Stabilization Delayed Feedback Asymptotic Stabilization of Rigid Body Attitude Motion for Large of Rigid Body Attitude Motion for Large of Rigid Body Attitude Motion for Large of Rigid Body Attitude Motion for Large Rotations Rotations Rotations Rotations
∗ ∗ ∗ Samiei K. Samiei ∗∗ Maziar Maziar Izadi Izadi ∗∗ Amit Amit K. Sanyal Sanyal ∗∗ ∗∗ Samiei Maziar Izadi K. A. ∗∗ Samiei ∗Eric Maziar Izadi ∗ Amit Amit K. Sanyal Sanyal ∗ Eric A. Butcher Butcher ∗∗ Eric A. A. Butcher Butcher ∗∗ Eric ∗ ∗ Department of Mechanical and Aerospace Engineering, New Mexico ∗ Department of Mechanical and Aerospace Engineering, New Mexico of and Aerospace Engineering, New Mexico ∗ Department State University, Cruces, NM, Department State of Mechanical Mechanical andLas Aerospace University, Las Cruces, Engineering, NM, 88003 88003 New Mexico ∗∗ University, Las Cruces, NM, 88003 of Aerospace and Mechanical Engineering, ∗∗ Department State State University, Las Cruces, NM, 88003 University of ∗∗ Department of Aerospace and Mechanical Engineering, University of ∗∗ Department of Aerospace and Mechanical Engineering, Arizona, Tucson, AZ Department of Aerospace and Mechanical Engineering, University University of of Arizona, Tucson, AZ 85721 85721 Arizona, Arizona, Tucson, Tucson, AZ AZ 85721 85721 Abstract: The The delayed delayed feedback feedback control control of of rigid rigid body body attitude attitude motion motion is is addressed addressed where where there there Abstract: Abstract: The delayed feedback control of rigid body attitude motion is addressed where there is an unknown time delay in the feedback measurement. The attitude motion is described on Abstract: The delayed feedback control of rigid body attitude motion is addressed where there is an unknown time delay in the feedback measurement. The attitude motion is described on the the is an delay in the measurement. The attitude motion is described on the tangent bundletime TSO(3) to globally and represent the orientation of body, while is an unknown unknown time delay the feedback feedback measurement. motion is rigid described the tangent bundle TSO(3) to in globally and uniquely uniquely representThe theattitude orientation of the the rigid body,on while tangent bundle TSO(3) to globally and uniquely represent the orientation of the rigid body, while a continuous nonlinear delayed feedback control law is proposed for local asymptotic stabilization tangent bundle TSO(3)delayed to globally and uniquely represent the orientation of the rigidstabilization body, while a continuous nonlinear feedback control law is proposed for local asymptotic a nonlinear delayed feedback control law asymptotic stabilization oncontinuous TSO(3). For For this purpose, purpose, we introduce notion based on on for thelocal Lyapunov technique, which a continuous nonlinear delayed we feedback control law is is proposed proposed for local asymptotic stabilization on TSO(3). this introduce aa notion based the Lyapunov technique, which on TSO(3). For this purpose, we introduce a notion based on the Lyapunov technique, which combines the Morse-Lyapunov method with Lyapunov-Krasovskii technique to yield a suitable on TSO(3). For this purpose, we introduce a notion based on the Lyapunov technique, which combines the Morse-Lyapunov method with Lyapunov-Krasovskii technique to yield a suitable combines the method with technique to aa suitable Morse-Lyapunov-Krasovskii (M-L-K) functional, from conditions proper combines the Morse-Lyapunov Morse-Lyapunov methodfunctional, with Lyapunov-Krasovskii Lyapunov-Krasovskii technique to yield yieldand suitable Morse-Lyapunov-Krasovskii (M-L-K) from which which the the stability stability conditions and proper Morse-Lyapunov-Krasovskii (M-L-K) functional, from which the stability conditions and control gain matrices are obtained in terms of linear matrix inequalities (LMIs). Simulations Morse-Lyapunov-Krasovskii (M-L-K)infunctional, from which stability conditions and proper proper control gain matrices are obtained terms of linear matrixthe inequalities (LMIs). Simulations control are in of matrix illustrategain the matrices performance of the the proposed proposed control scheme. control gain matrices are obtained obtained in terms terms of linear linear matrix inequalities inequalities (LMIs). (LMIs). Simulations Simulations illustrate the performance of control scheme. illustrate the performance of control scheme. illustrate the (International performanceFederation of the the proposed proposed control scheme. © 2015, IFAC of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Attitude control, time-delayed system, LMIs, Lyapunov-Krasovskii functional functional Keywords: Attitude control, time-delayed system, LMIs, Lyapunov-Krasovskii Keywords: Attitude Attitude control, control, time-delayed time-delayed system, system, LMIs, LMIs, Lyapunov-Krasovskii Lyapunov-Krasovskii functional functional Keywords: 1. INTRODUCTION INTRODUCTION Chunodkar 1. Chunodkar and and Akella Akella (2011); (2011); Samiei Samiei and and Butcher Butcher (2013); (2013); 1. INTRODUCTION Chunodkar and Akella (2011); Samiei and (2013); Samiei et al. (2012); Nazari et al. (2013)) who 1. INTRODUCTION Chunodkar and AkellaNazari (2011);etSamiei and Butcher Butcher (2013); Samiei et al. (2012); al. (2013)) who generally generally Samiei aet etlocal al. (2012); (2012); Nazari et al. al. (2013)) (2013)) who generally generally Attitude motion occurs in many practical systems such as utilize coordinate representation of rigid body Samiei al. Nazari et who Attitude motion occurs in many practical systems such as utilize a local coordinate representation of rigid body atatAttitude motion occurs in many many practical systems such as utilize a local coordinate representation of rigid body unmanned vehicles (Sanyal and practical Chyba (2009)) (2009)) orsuch spacetitude. (Samiei and (2013)), the stabilization Attitude motion occurs in systems as utilize coordinate representation body atatunmanned vehicles (Sanyal and Chyba or spacetitude. aIn Inlocal (Samiei and Butcher Butcher (2013)), of therigid stabilization unmanned vehicles (Sanyal and Chyba Chyba (2009)) or spacespacetitude. In (Samiei Butcher (2013)), the stabilization craft systems systems (Sanyal(Sanyal and Chaturvedi Chaturvedi (2008); Samiei et al. al. of attitude dynamics in the presence unmanned vehicles and (2009)) or titude. (Samiei and and Butcher (2013)), craft (Sanyal and (2008); Samiei et of rigid rigidInspacecraft spacecraft attitude dynamics in the the stabilization presence of of craft systems (Sanyal andstabilization Chaturvedi (2008); (2008); Samiei et al. external of rigid attitude dynamics in presence (2015a)). Almost global of rigid rigidSamiei body et attistochastic an craft systems (Sanyal and Chaturvedi al. of rigid spacecraft spacecraft attitudeand dynamics in the thetime-varying presence of of (2015a)). Almost global stabilization of body attiexternal stochastic torques torques and an unknown unknown time-varying (2015a)). Almost global stabilization of rigid rigid of body atti- external external stochastic torques and and an an unknownwhere time-varying tude motion motion basedglobal on aa global global representation attitude time in is a (2015a)). Almost stabilization of body attistochastic torques unknown time-varying tude based on representation of attitude time delay delay in the the measurement measurement is studied studied where a linear linear tude motion based based on aainglobal global representation of attitude attitude delay in measurement is studied where linear has attracted attracted attention control systems design, design, see e.g e.g time delayed state feedback controller aa suitable tude motion on representation of time delay in the the measurement is with studied where a a choice linear has attention in control systems see delayed state feedback controller with suitable choice has attracted attention in control systems design, see e.g delayed state feedback controller with a suitable choice (Sanyal and Chaturvedi (2008)). The proposed continuous of Lyapunov-Krasovskii (L-K) functional can guarantee has attracted attention in controlThe systems design, see e.g delayed state feedback controller with a suitable choice (Sanyal and Chaturvedi (2008)). proposed continuous of Lyapunov-Krasovskii (L-K) functional can guarantee (Sanyal and Chaturvedi Chaturvedi (2008)). The aproposed proposed continuous of Lyapunov-Krasovskii (L-K) functional can control strategies strategies in these these(2008)). works have have nonlinearcontinuous structure the local the system (Sanyal and The of (L-K) of functional can guarantee guarantee control in works a nonlinear structure theLyapunov-Krasovskii local asymptotic asymptotic stability stability of the closed-loop closed-loop system control strategies in these these works have aa nonlinear nonlinear structure the local asymptotic stability of the closed-loop system since the state space of the dynamics model is a nonin the mean square sense through the solution of LMIs. control strategies in works have structure the local asymptotic stability of the closed-loop since the state space of the dynamics model is a non- in the mean square sense through the solution ofsystem LMIs. since the state space of the dynamics model is a nonin the mean square sense through the solution of LMIs. linear space, i.e., the tangent bundle TSO(3). In (Sanyal The main drawback with these studies is that the prosince the state space of the dynamics model is a nonin the mean square sense through the solution of linear space, i.e., the tangent bundle TSO(3). In (Sanyal The main drawback with these studies is that theLMIs. prolinear space, i.e., the tangent bundle TSO(3). In (Sanyal The main drawback with these studies is that the and Chaturvedi (2008)) a geometric controller is designed posed controllers can only guarantee the local asymptotic linear space, i.e.,(2008)) the tangent bundlecontroller TSO(3). is Indesigned (Sanyal The drawback these studies is that the proproand Chaturvedi a geometric posedmain controllers can with only guarantee the local asymptotic and Chaturvedi (2008)) geometric controllerand is designed designed posed controllers can guarantee the asymptotic on SO(3) SO(3) to almost almost globally track attitude attitude angular stability of for small maneuvers with and Chaturvedi (2008)) aa geometric controller is posed controllers can only only the local local asymptotic on to globally track and angular stability of the the system system forguarantee small angle angle maneuvers with on SO(3)ofto to almost globally track attitude and angular of system small angle maneuvers with velocity spacecraft in the the presence of and gravity and stability small This is to use on SO(3) globally track attitude angular stability of the thevelocities. system for for angle maneuvers with velocity of aa almost spacecraft in presence of gravity and small angular angular velocities. Thissmall is due due to the the use of of a a local local velocity of a spacecraft in the presence of gravity and small angular velocities. This is due to the use of a local disturbance moments. coordinate representation of attitude which suffers from velocity of amoments. spacecraft in the presence of gravity and small angular velocities. This is due to the use of a local disturbance coordinate representation of attitude which suffers from disturbance moments. moments. coordinate representation of attitude which suffers from mathematical singularities in representing the orientation disturbance coordinate representation which from mathematical singularities of in attitude representing thesuffers orientation On the other hand, the existence of time delay in conmathematical singularities in representing the orientation On the other hand, the existence of time delay in con- of a rigid body. In addition, the proposed delayed feedback mathematical singularities in representing the orientation of a rigid body. In addition, the proposed delayed feedback On the other hand, the existence of time delay trolled systems is inevitable. Time delay occurs in conOn the systems other hand, the existence timeoccurs delay in con- of aa rigid In the proposed delayed feedback trolled is inevitable. Timeofdelay controllers are robust time delay cannot tolerate of rigid body. body. In addition, addition, delayed are not not robust to to the timeproposed delay and and cannotfeedback tolerate trolled systems due is inevitable. inevitable. Time occurs in to the the existence existence of delay delay in sensors, sensors, trolled is Time delay delay occurs in concon- controllers controllers are not robust to time delay and cannot tolerate systems due to of in large values of delay, which makes the implementation of controllers are not robust to time delay and cannot tolerate large values of delay, which makes the implementation of trolled systems due to the existence of delay in sensors, communication channels, or actuators. Controlled systems trolled systems channels, due to the delay in systems sensors, the large values of delay, which makes the implementation of communication or existence actuators.ofControlled control design impractical. large values of delay, which makes the implementation of the control design impractical. communication channels, or actuators. Controlled systems designed based based on on nonlinear feedback schemes schemes are systems usually the control design impractical. communication channels, or actuators. Controlled designed nonlinear feedback are usually the control designcontrol impractical. designed based on nonlinear feedback schemes are usually robust to a small amount of time delay. However, if the Delayed feedback design for arbitrary attitude madesigned on amount nonlinearoffeedback schemes are usually robust tobased a small time delay. However, if the Delayed feedback control design for arbitrary attitude marobust to aa increases small amount amount of time delay. delay. However, if the the neuvers Delayed feedback control design arbitrary attitude matime delay delay in aa of controlled system due to toif the of is demanding robust to small time However, Delayed on feedback controlbundle design for for arbitrary matime increases in controlled system due neuvers on the tangent tangent bundle of SO(3) SO(3) is aaattitude demanding time delay increases in aa controlled controlled system due then to the the neuversproblem on the the tangent tangent bundle ofsystem SO(3)must is aa be demanding failure of system components or external sources, a control since the control designed time delay increases in system due to neuvers on bundle of SO(3) is demanding failure of system components or external sources, then a control problem since the control system must be designed failure offeedback system components components or external external sources, then aa control problem control system must be delayedof control scheme scheme is essential. essential. In addition, addition, on space TSO(3), there failure system or sources, then control problem since since the thestate control system mustwhile be designed designed delayed feedback control is In on the the non-contractible non-contractible state space TSO(3), while there is is delayed feedback control scheme is essential. In addition, on the non-contractible state space TSO(3), while there is the existence of time delay in a controlled system with time delay in the feedback loop. However, if probdelayed feedback control scheme essential. system In addition, thedelay non-contractible state loop. space However, TSO(3), while is the existence of time delay in a iscontrolled with on time in the feedback if this thisthere probthe existence of time delay in a controlled system with time delay in the feedback loop. However, if this probrotational dynamics can cause chattering or oscillatory lem can be properly tackled on TSO(3), those drawbacks the existence of timecan delay in achattering controlledorsystem with time delay in the feedback However, if drawbacks this probrotational dynamics cause oscillatory lem can be properly tackled loop. on TSO(3), those rotational dynamics can cause Thus chattering or oscillatory oscillatory lem can can be be for properly tackled on TSO(3), TSO(3), those drawbacks drawbacks behavior (Ailon (Ailon et al. al.can (2004)). it appears appears that the the lem mentioned delayed feedback control based rotational dynamics cause chattering or properly on those behavior et (2004)). Thus it that mentioned for delayedtackled feedback control schemes schemes based on on behavior (Ailon et etcontrol al. (2004)). (2004)). Thus itattitude appearsmotion that the the mentioned for delayed feedback control schemes based on delayed feedback design for is a local attitude representation will be eliminated. behavior (Ailon al. Thus it appears that mentioned for delayed feedback control schemes basedThe on delayed feedback control design for attitude motion is a local attitude representation will be eliminated. The delayed feedback control design for attitude motion is a local attitude representation will be eliminated. The practical and crucial. Unfortunately, this problem has been conventional technique to address the nonlinear control delayed control design forthis attitude is aconventional local attitude representation willthe be nonlinear eliminated. The practical feedback and crucial. Unfortunately, problemmotion has been technique to address control practical and crucial. crucial. Unfortunately, problem been conventional technique to the nonlinear control only studied studied by aa few few researchers this (Ailon et al. al.has (2004); dynamics evolving TSO(3) is practical and Unfortunately, this problem has been design conventional technique to address address the on nonlinear only by researchers (Ailon et (2004); design for for attitude attitude dynamics evolving on TSO(3)control is the the only studied studied by by aa few few researchers researchers (Ailon (Ailon et et al. al. (2004); (2004); design design for attitude attitude dynamics dynamics evolving on(Milnor TSO(3)(1963)), is the the use of Morse-Lyapunov (M-L) functions only for evolving on TSO(3) is use of Morse-Lyapunov (M-L) functions (Milnor (1963)), support from the National Science Foundation under Financial Financial support from the National Science Foundation under use of Morse-Lyapunov (M-L) functions (Milnor (1963)), while L-K or Lyapunov-Razumikhin (L-R) techniques use of Morse-Lyapunov (M-L) functions (Milnor (1963)), Financial support from the National Science Foundation under Grant No. CMMI-1131646 is gratefully acknowledged. while L-K or Lyapunov-Razumikhin (L-R) techniques are are Financial support from is thegratefully Nationalacknowledged. Science Foundation under Grant No. CMMI-1131646 while Grant No. CMMI-1131646 is gratefully acknowledged. while L-K L-K or or Lyapunov-Razumikhin Lyapunov-Razumikhin (L-R) (L-R) techniques techniques are are Grant No. CMMI-1131646 is gratefully acknowledged.
Ehsan Ehsan Ehsan Ehsan
Copyright IFAC 2015 81 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, IFAC 2015 81 Copyright IFAC 2015 81 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 81 Control. 10.1016/j.ifacol.2015.09.357
IFAC TDS 2015 82 June 28-30, 2015. Ann Arbor, MI, USA
Ehsan Samiei et al. / IFAC-PapersOnLine 48-12 (2015) 081–086
usually used to address the stabilization of delay differential equations in Euclidean space. This work combines these two approaches by using the concept of a M-LK functional to address the delayed feedback control of attitude dynamics evolving on TSO(3). This manuscript is a continuation of a preliminarily work, presented by the same authors in (Samiei et al. (2015b)). In this prior work, a continuous delayed feedback controller was designed for the attitude motion based on LMI stability conditions. The nonlinear terms in the dynamics model were considered after solving LMIs and then a condition in which the derivative of the proposed M-L-K functional is negative definite was obtained. Thus, the validity of the controller is only guaranteed inside the region in which the derivative of the M-L-K functional is negative definite. However, in this paper, a continuous full-state delayed feedback control law is chosen and a M-L-K functional candidate is formulated. Application of Lyapunov stability techniques using the ML-K functional candidate results in the solution of LMIs for the controller gain matrices using convex optimization. Furthermore, the nonlinear terms in the dynamics model are analyzed within the LMI stability conditions. The norm of the obtained control gain matrices are also minimized in order to have an admissible control effort for large maneuvers. Finally, the effectiveness of the proposed controller is demonstrated through numerical simulations.
where yi , i = 1, 2, 3 are the elements of y. In addition, (·)× can also denote the cross product, i.e., y × z = y × z for any y, z ∈ R3 . The canonical basis {e1 , e2 , e3 } for the body frame B can form a basis for the Lie algebra so(3) 3 such that y × = yi e × i . The principal rotation elements i=1
in terms of the rotation matrix R are given as 1 1 (R − RT ), (3a) φ = cos−1 ( (tr(R) − 1)), × = 2 2 sin Φ where φ represents the principal rotation angle, ∈ S2 denotes the principal rotation axis, S2 is the 2−dimensional unit sphere embedded in R3 , and Ri,j , i, j = 1, 2, 3 are the elements of the rotation matrix R. More information about the global representation of attitude dynamics can be found in (Sanyal and Chaturvedi (2008)). The attitude dynamics in the body frame B is obtained ˙ using the Euler’s equations of motion as J Ω(t) = −Ω(t) × JΩ(t)+u(t), where u(t) ∈ R3 is the external torque applied to the rigid body in B and J = J T > 0 is the known inertia matrix. We assume that three actuators provide torques about the body frame axes. Thus, the rigid body rotational equations of motion is defined on the state space TSO(3) ≈ SO(3) × R3 as ˙ (4a) R(t) = R(t)Ω(t)× , × ˙ (4b) J Ω(t) = −Ω(t) JΩ(t) + u(t).
2. NOTATION AND PROBLEM FORMULATION
The state variables are expressed as the pair (R, Ω) ∈ TSO(3). The equilibrium subspace of the attitude dynamics model of Eq.(4) is obtained as E = {(R, Ω) ∈ TSO(3)|Ω = 0, R ∈ SO(3)}.
Notation In this paper, we denote R+ as the set of nonnegative real numbers. Rn represents the n−dimensional Euclidean space and Rm×n denotes a vector space of real matrices which maps Rn to Rm . det(·) and tr(·) stands for the determinant and the trace of a square matrix, respectively. n The Euclidean norm of y ∈ R is denoted as y = T y y ∈ R, while the Frobenius norm of A ∈ Rm×n is represented as A F = tr(AT A) ∈ R.
3. ROBUST ASYMPTOTIC DELAYED FEEDBACK CONTROL DESIGN In this section, we aim to design a continuous full-state feedback controller to stabilize the attitude motion of Eq.(4) such that (R, Ω) → (I3 , 0) as t → ∞ in the presence of an unknown constant time delay in the feedback path. The main contribution of this paper in comparison with current studies based on a local representation of attitude, as discussed in introduction, is that the proposed delayed feedback controller is on the nonlinear (non-Euclidean) space TSO(3), and thus the topological difficulty in obtaining the almost global stabilization of attitude (see e.g., Bhat and Bernstein (2000) for more details on such obstruction) is eliminated and the system can be stabilized from almost every initial orientation. In addition, the other obstructions resulting from the feedback control design based on traditional practice such as unwinding will be naturally addressed.
2.1 Attitude Motion Formulation
The rigid body attitude dynamics are represented in the body fixed coordinate frame B with its origin assigned at the center of mass of the rigid body. The attitude of the rigid body is expressed by the rotation matrix R from the body frame B to the inertial frame N as an element of the real special orthogonal matrix Lie group SO(3) such that SO(3) = {R ∈ R3×3 | RT R = I3 and det(R) = 1}, (1) with corresponding matrix Lie algebra so(3), where I3 ∈ R3×3 is the identity matrix. The rotation matrix R uniquely and globally describes the orientation of a rigid body from the body frame B to the inertial frame N with no mathematical singularities. The attitude kinematic differential equation in terms of R ∈ SO(3) can ˙ be represented as R(t) = R(t)Ω(t)× , where Ω(t) ∈ R3 is the angular velocity represented in the body frame B and y × : R3 → so(3) is the map from the Euclidean space R3 to the Lie algebra so(3) which is defined as 0 −y3 y2 (2) y × = y3 0 −y1 , −y2 y1 0
We assume that the unknown constant time delay satisfies 0 ≤ τ ≤ τmax , where τmax ∈ R+ is known. Furthermore, due to existence of time delay in the measurement, current information of the state variables is unavailable for the purpose of control design. 3.1 Preliminaries Results A Morse function on SO(3) is defined as a smooth realvalued function from SO(3) to R+ with non-degenerate isolated critical points. An appropriate choice of the Morse 82
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