Output feedback stabilization of the angular velocity of a rigid body


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1 Systems & Control Letters 36 (1999) Output feedback stabilization of the angular velocity of a rigid body A. Astol Department of Electrical and Electronic Engineering and Centre for Process Systems Engineering, Imperial College, London SW7 2BT, UK Received 25 November 1997; accepted 5 May 1998 Abstract The problem of stabilization of the angular velocity of a rigid body using only two control signals and partial state information is addressed. It is shown that if any two (out of three) states are measured the system is not asymptotically stabilizable with (continuous) dynamic output feedback. Nevertheless, we prove that practical stability is achievable if the measurable states fulll a certain structural property, and that, under the same structural condition, a hybrid control law yielding exponential convergence can be constructed. Finally, we also study some geometric features of the Euler s equations and the connection between local strong accessibility and local observability. c 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Rigid body; Euler s equations; Output feedback stabilization; Nonlinear systems 1. Introduction The problem of asymptotic stabilization of the angular velocity of a rigid body has been studied by several researchers. In the papers [1, 7] it was shown that the zero solution of Euler s angular velocity equations can be made asymptotically stable by means of two control torques, whereas in the works [2, 3, 13, 14] the same problem has been addressed and solved in the case of only one control torque. Robust stabilization has been studied in [4, 11]. In the rst work, robustness with respect to parameters variation has been discussed, whereas in the second one robustness against exogenous disturbances has been dealt with. All the aforementioned works assume that the whole state vector is available for feedback, whereas, to the best of our knowledge, no result is available for the case of partial state information. In this work we assume that only two out of three states are available for feedback. This problem, of intrinsic mathematical diculty, is also relevant from the applications point of view, as it models the situation of sensor failure. The paper is organized as follows. In Section 2 we present the equations of the system we are dealing with and we formulate precisely the problem under investigation. Section 3 contains a digression devoted to some geometric considerations related to the observability of the Euler s equations with two measured states. The stabilization problem is then addressed. In Section 4 we prove a negative result, namely that asymptotic stabilization by measurement feedback is not achievable with any dynamic smooth control law, whereas in Tel.: ; fax: ; /99/$ see front matter c 1999 Published by Elsevier Science B.V. All rights reserved. PII: S (98)
2 182 A. Astol / Systems & Control Letters 36 (1999) Section 5 we show that, despite this negative result, global practical stability can be achieved by means of smooth dynamic output feedback if the measured states are two appropriately chosen angular velocities. In Section 6 we discuss and interpret the obtained results and we show how they can be used to implement a hybrid control strategy yielding exponential convergence to the zero equilibrium. Finally, Section 7 shows some simulations and Section 8 contains conclusions and outlook on future works. 2. System description and problem statement Consider a rigid body in an inertial reference frame and let! 1,! 2 and! 3 denote the angular velocity components along a body xed reference frame having the origin at the center of gravity and consisting of the three principal axes. The Euler s equations for the rigid body with two independent controls aligned with two principal axes are I 1! 1 =(I 2 I 3 )! 2! 3 + v 1 ; I 2! 2 =(I 3 I 1 )! 3! 1 + v 2 ; I 3! 3 =(I 1 I 2 )! 1! 2 ; (1) where I 1 0, I 2 0 and I 3 0 denote the principal moments of inertia and v = col(v 1 ;v 2 ) the control torques. System (1) can be rewritten as! = f(!)+g 1 u 1 + g 2 u 2 ; (2) where! = col(! 1 ;! 2 ;! 3 ), f(!) = col(a 1! 2! 3 ;A 2! 3! 1 ;A 3! 1! 2 ), A 1 =(I 2 I 3 )=I 1, A 2 =(I 3 I 1 )=I 2, A 3 = (I 1 I 2 )=I 3, g 1 = col(1; 0; 0), g 2 = col(0; 1; 0), u 1 = v 1 =I 1 and u 2 = v 2 =I 2. In what follows, we will be concerned with the following problem. Asymptotic (practical) stabilization problem via output feedback. Given system (2) with output ] ] y = [ y1 y 2 = [!i! j ; (3) where (i; j) {(1; 2); (1; 3); (2; 3)}, nd a nonnegative integer p and a smooth dynamic output feedback control law, described by equations of the form = (; y); u 1 = 1 (; y); u 2 = 2 (; y) (4) with R p, such that the point (! 1 ;! 2 ;! 3 ;)=(0; 0; 0; 0) is a globally asymptotically (practically 1 ) stable equilibrium of closedloop system (2) (4). It must be noted that a necessary condition for global asymptotic or practical stability by state feedback is A 3 0. Therefore, throughout this work we will make often use of the following assumption. Assumption 1. A See [9], where the notion of practical stability was introduced under the denomination of uniform ultimate boundedness. We use the alternative term practical stability as this is more often used.
3 A. Astol / Systems & Control Letters 36 (1999) The observability issue This section contains a brief digression devoted to the description of some geometric properties of the Euler s equations. More precisely, we study the local observability issue for system (2) with output (3). See [12] for further detail and denitions. Proposition 1. Suppose system (2) is locally strongly accessible from any! R 3. Then system (2) with output (3) is locally observable at any! R 3. Proof. As remarked in [12], local strong accessibility from any! R 3 is equivalent to the condition A 3 0. To complete the proof we now consider three cases. Case 1: (i; j)=(1; 2), i.e. y = col(! 1 ;! 2 ). To begin with, note that A 3 0 A A 2 2 0; or what is the same, if A 3 is nonzero then at least one of A 1 and A 2 must be nonzero as well. Consider now the functions y 1 =! 1, y 2 =! 2, L g1 L f y 2 = A 2! 3 and L g2 L f y 1 = A 1! 3. Such functions are in the observation space O of system (2) with output (3). As a result, the observability codistribution do has constant rank equal to 3 at any! R 3. Case 2: y = col(! 1 ;! 3 ). Consider the functions y 1 =! 1, y 2 =! 3 and L g1 L f y 2 = A 3! 2. Such functions are in the observation space O of system (2) with output (3). As a result, the observability codistribution do has constant rank equal to 3 at any! R 3. Case 3: y = col(! 2 ;! 3 ). Consider the functions y 1 =! 2, y 2 =! 3 and L g2 L f y 2 = A 3! 1. Such functions are in the observation space O of system (2) with output (3). As a result, the observability codistribution do has constant rank equal to 3 at any! R 3 : It must be noted that local strong accessibility from any! R 3, i.e. Assumption 1, is not necessary for local observability at any! R 3, as expressed in the following result. Proposition 2. Assume A 3 =0. Then the following hold: (i) System (2) with output (3) is locally observable at any! R 3 i (i; j) (1; 2) and A A (ii) System (2) with output (3) is locally observable at any! R 3 ={! 3 =0} if A A (iii) If A 1 = A 2 =0 the system is not locally observable at any! R 3. Proof. The if part of claim (i) can be established with the same arguments used in the proof of Proposition 1. The same holds also for claim (ii). To prove the only if part of claim (i), note that if A 3 = 0 and (i; j)=(1; 3) ((i; j)=(2; 3)) the states (! 1 ; ˆ! 2 ; 0) and (! 1 ;! 2 ; 0) with ˆ! 2! 2 (( ˆ! 1 ;! 2 ; 0) and (! 1 ;! 2 ; 0) with ˆ! 1! 1 ) are indistinguishable for any input functions u 1 (t) and u 2 (t). Finally, claim (iii) results from the observation that, if A 1 = A 2 = A 3 = 0, system (2) with output (3) is a linear unobservable system. We conclude this section remarking that Assumption 1 is sucient either for global asymptotic state feedback stabilizability and for local observability. However, as we will show in the next section, these two properties together are not enough to conclude global (or even local) asymptotic output feedback stabilizability. 4. A negative result This section contains one of the main results of this work, namely the impossibility to solve the global 2 asymptotic stabilization problem by smooth dynamic output feedback. 2 This negative result holds also locally, i.e. the obstruction is not in the size of the region of attraction but on the asymptotic convergence.
4 184 A. Astol / Systems & Control Letters 36 (1999) Proposition 3. System (2) with output (3) cannot be asymptotically stabilized by smooth dynamic output feedback. Proof. The closedloop system (2) (4) is described by equations of the form! = f(!)+g 1 1 (; y)+g 2 2 (; y); = (; y); where y = col(! i ;! j ) for (i; j) {(1; 2); (1; 3); (2; 3)}, and R p for some p 0. By smoothness 3 of the controller we infer that, for any choice of (i; j), the origin is not an isolated equilibrium point of the closedloop system, i.e. in any neighborhood of the origin there are innite equilibrium points, which proves the claim. Remark 1. Asymptotic stabilization via discontinuous and=or timevarying output feedback is not ruled out by Proposition 3 and it is an active area of research. 5. Main results Despite the negative result proved in Section 4, it is possible to achieve global practical stability via dynamic output feedback if the measured states are two appropriately chosen angular velocities, as expressed in the following statement. Proposition 4. Consider system (2) with output (3). Suppose Assumption 1 holds. Assume, moreover, that the indexes (i; j) are such that I i I k I j (5) or I i 6I k 6I j (6) with k {1; 2; 3} such that k i and k j. Then, the output feedback practical stabilization problem is solvable. Proof. To begin with, observe that by conditions (5) and (6) it is possible to conclude A i A j 0: For simplicity of exposition we now carry on the proof in the case (i; j)=(1; 3), i.e. y = col(! 1 ;! 3 ). Moreover, we also assume, without loss of generality, that A 3 0. Let be a real number. We show that, for any 0 it is possible to nd a dynamic output feedback control law, described by equations of the form (4) with p = 1, such that the point (; 0; 0; 0) is a globally asymptotically (exponentially) stable equilibrium of the closedloop system. Consider the positive denite and proper function: W (;! 2 ;! 3 ;)= 1 ( ) 2 + +! 2! (2 +(! 2 +! 3 ) 2 +! 3) 2 3 Continuity at the origin is what we really need.
5 A. Astol / Systems & Control Letters 36 (1999) with =! 1, R, (0; 1) and such that 0. Let and = (; y) = A A 3 3! 3 + A k 2! 3 + A 1! A 1! A 1! 2 3 ( 1+) 2 A 3 3! 3 + A 3! A 1! k 2! 3 + A 1! 3 + A 3! 3 ( 1+) 2 k 2 + A 3! 3 A 1! A 3 2A 3 + A 3! 3 2 ( 1+) 2 2A 3! 3 + A 3 2A 3 k 2 + k 2! 3 A 3! 3 3 ( 1+) 2 2 A 3! 3 +! 3 k A 3! A A 3 + A 1! 3 3 ( 1+) 2 A 3! 3 3! 3k 1 A A 3 2! 3 2A 3 +2A 3 ( 1+) 2 3A 3 2! 3 A 3 2 2A 1! 3 2A 3! 3 2A 3! 3 ( 1+) 2 ; u 1 = 1 (; y)= k 1 + (A 1 + A 3 )! 2 3 u 2 = 2 (; y) = A 2! 3 + A 3 2! 3 A 1! 3 + A 3 2! 3 A 2! 3 A 3! 3 A 3! 3 ( + + (A 3 + A 3 + k 2 ) +! 3 +! 3 ); 1 with k 1 0 and k 2 0. Simple but tedious calculations show that ( ) 2 W = k k 2 +! 3 A 3! ( ) 2 (A 1! A 3 ( + ) )! 2! 3 ; 1 which proves the claim. The proof for the case (i; j)=(2; 3) is analogous to the one above, whereas the statement for (i; j)=(1; 2) can be proved using the candidate (control) Lyapunov function W (;! 2 ;! 3 ;)= 2( ( 2 +1 )) (! 3! 2( + )+ 1 2! ) (2 +1) with =! 1, R, ( ) 2 0; 2 +1 and such that 0: (2 +(! 2 +! 3 ) 2 +! 2 3)
6 186 A. Astol / Systems & Control Letters 36 (1999) Remark 2. It is worth noting that the result proved in Proposition 4 is somewhat stronger than practical stability. We have proved that there exist points arbitrarily close to the equilibrium (! 1 ;! 2 ;! 3 ;)=(0; 0; 0; 0) which can be made globally asymptotically stable (and locally exponentially stable). This property implies, but it is not implied by, global practical stability. Remark 3. By Assumption 1, in conditions (5) and (6) at least one inequality is strict. Remark 4. Conditions (5) or (6) imply that the measured velocities are the ones around the principal axes having the largest and the smallest moments of inertia. This is a structural property: it describes the geometry of the measurement devices needed to achieve output feedback (practical) stabilization. Remark 5. The result summarized in Proposition 4 can be easily adapted to the case of a rigid body with three control inputs and two measured outputs, provided that the moments of inertia associated with the measured velocities fullls either condition (5) or condition (6). Remark 6. System (2) with output (3) belongs to the class of nonlinear systems which are linear in the unmeasured states. Such a class of systems has been widely studied in the last years, see e.g. [5, 6]. Therein, it has been shown that asymptotic stability via measurement feedback is attainable if the state feedback and output injection problems are both solvable. For system (2) with output (3) the state feedback problem has been solved in [1], whereas the output injection problem has not been discussed yet. As a byproduct of the discussion in this paper, we conclude that the (smooth) output injection problem for system (2) with output (3) is not solvable. Finally, we mention that the Lyapunov construction contained in Proposition 4 takes inspiration from the results in [6]. Remark 7. It must be noted that the point (! 1 ;! 2 ;! 3 )=(; 0; 0) which is a globally asymptotically (exponentially) stable equilibrium of the closedloop system is also an equilibrium for the openloop system with zero control. Such an equilibrium is (openloop) unstable if (i; j)=(2; 3), i.e. if the measured state are! 2 and! 3, whereas it is (simply) stable if (i; j)=(1; 3) or (i; j)=(1; 2). Remark 8. The proof of Proposition 4 could be simply modied to design a dynamic output feedback control law rendering the point 4 (! 1 ;! 2 ;! 3 )=(0;;0), with 0, a globally asymptotically (exponentially) stable equilibrium. However, the technique used in the proof of Proposition 4 cannot be used to design a dynamic output feedback control law rendering the point (! 1 ;! 2 ;! 3 )=(0; 0;), a globally asymptotically (exponentially) stable equilibrium if! 3 is not one of the measured states. This could be proved observing that a state feedback control law rendering the point (! 1 ;! 2 ;! 3 )=(0; 0;) a globally asymptotically stable equilibrium could be designed using the control Lyapunov function V (! 1 ;! 2 ;)= 1 2 ((! 1 1 ) 2 +(! ) ) with =! 3 and 1 2 A 3 0. However, if! 3 is not measured this function is not quadratic in the unmeasured states, and this prevents the applicability of the techniques used in the proof of Proposition 4, as discussed in detail in [5]. 6. Discussion and further results The results presented so far lend themselves to several interesting theoretical interpretations and practical uses, as detailed hereafter. 4 Observe that such a point is an equilibrium for the uncontrolled system.
7 A. Astol / Systems & Control Letters 36 (1999) Theoretical interpretation The results summarized in Propositions 3 and 4 set the boundary of performance that can be achieved by smooth output feedback control. Hence, they must be regarded as purely theoretical contributions. Their practical use can be questionable (and we thank one of the anonymous reviewers for this remark) and will be elaborated in what follows Asymptotic stabilization at a nonzero equilibrium As observed in Remarks 7 and 8 the methodology of this paper, namely the use of the nonlinear separation principle reported in [5] together with a shift of the equilibrium point, could be applied to stabilize some of the nonzero equilibria of the Euler s equations, provided the measured output fullls one of the structural conditions (5) or (6) and that! 3 is measured if one wishes to stabilize the equilibrium (! 1 ;! 2 ;! 3 )=(0; 0;) with 0. A nonzero equilibrium corresponds to the rigid body spinning around one of its principal axes with a predened angular velocity. Such a conguration is of interest for specic applications Achieving asymptotic convergence We nally show how, despite the negative result proved in Proposition 3, the result expressed in Proposition 4 can be used to design a hybrid control law achieving exponential convergence of the states (! 1 ;! 2 ;! 3 ;) to the zero equilibrium. Let B(c; r) denote a (open) ball in R 4 with center at (c; 0; 0; 0) and radius r, i.e. B(c; r)={(! 1 ;! 2 ;! 3 ;) R 4 (! 1 c) 2 +! 2 2 +! r 2 } and let x = col(! 1 ;! 2 ;! 3 ;). The control law described in Proposition 4, i.e. the functions (; y), 1 (; y) and 2 (; y), depends (among others) upon the design parameters and. In what follows we exploit this dependence and we show how a judicious schedule of and yields a hybrid control law achieving exponential convergence to the zero equilibrium. This dependence will be explicitly indicated in the sequel, i.e. control law (4) will be written as = (; y; ; ); u 1 = 1 (; y; ; ); u 2 = 2 (; y; ; ); (7) with ( ), 1 ( ) and 2 ( ) as in the proof of Proposition 4. The controlled closedloop system (2) (7) is locally exponentially stable around the equilibrium (! 1 ;! 2 ;! 3 ;)=(; 0; 0; 0) for any choice 5 of and such that 0. Let 0? 1,? 0 and n 1 be given. Let denote the largest eigenvalue of the linear approximation of the closedloop system around the equilibrium (? ; 0; 0; 0). Observe that is a positive number. Consider the closedloop system with control law (7) for =? and =?. There exists a nite time instant t 1 0, depending possibly on the initial condition of the system, such that x(t 1 ) B 1 = B(? ;? =2): At time t = t 1 consider a new control law described by Eq. (7) with =? 2 and =? : 5 It must be noted that, k 1 and k 2 must be also selected as in Proposition 4. They are not scheduled by the hybrid controller.
8 188 A. Astol / Systems & Control Letters 36 (1999) The resulting closedloop system has a new equilibrium at (! 1 ;! 2 ;! 3 ;)=(? =2; 0; 0; 0) which is globally asymptotically stable and locally exponentially stable. Therefore, at time t 2 = t 1 + n ; one has x(t 2 ) B 2 = B (?2 ) ; 2? e n=2 and, moreover, x(t) B (?2 ) ; 2? for all t [t 1 ;t 2 ]. At time t = t 2 the control parameters and are changed again and set as =? 4 and =? : The closedloop system has a new equilibrium at (? =4; 0; 0; 0). Therefore, at time t 3 = t 2 + n = t 1 +2 n ; one has ( )? x(t 3 ) B 3 = B 4 ;? e n=4 and, moreover, ( )? x(t) B 4 ;? for all t [t 2 ;t 3 ]. This switching procedure continues for all t. As a result the state of the closedloop system remains bounded for all time and converges to the point (! 1 ;! 2 ;! 3 ;)=(0; 0; 0; 0): It must be noted that the origin is exponentially attractive, as the center and the volume of the balls B i approach zero exponentially. Remark 9. It must be observed that the proposed hybrid control strategy is not the only possible and several others could be thought. It has been selected to reduce, as much as possible, the integral of! 1 (t). Remark 10. It is worth noting that the scheduling procedure described above is somewhat reminiscent of the general construction for discontinuous stabilization of nullcontrollable nonlinear systems developed in [8]. 7. Simulations In this section we present some simulation results showing the performance of the controlled system. From [10] we adopt the moments of inertia I 1 =9: kg m 2, I 2 =9: kg m 2, and I 3 =7: kg m 2, which were assumed to be the data of a space power tower and we suppose that! 1 and! 3 are the measured variables. Simple algebra shows that condition (5) holds with i =1, k = 2 and j = 3. Finally, we set =0:05, = 10 and k 1 = k 2 =1. To begin with we show the performance of the closedloop system with a dynamic output feedback controller designed on the basis of Proposition 4. Figs. 1 and 2 display the state trajectories of the closedloop system and the control actions from the initial condition (! 1 (0);! 2 (0);! 3 (0);(0)) = (0; 0; 0:5; 0), which corresponds to a stable equilibrium for the openloop system.
9 A. Astol / Systems & Control Letters 36 (1999) Fig. 1. State histories from the initial state (! 1 (0);! 2 (0);! 3 (0);(0)) = (0; 0; 0:5; 0). The variables! i are in rad=s. The dashed line in the top left gure indicates the value of. Fig. 2. Control actions. The variables u i are in s 1.
10 190 A. Astol / Systems & Control Letters 36 (1999) Fig. 3. Time history of the state! 1 using the hybrid control. Fig. 4. Logarithm of (! 1 ;! 2 ;! 3 ;).
11 A. Astol / Systems & Control Letters 36 (1999) Fig. 5. Position of the projection of the balls B 1 ;B 2 ;:::;B 4 into the space (! 1 ;! 2 ;! 3 ). At a second stage we implement the hybrid, dynamic, output feedback control law. The switching of the parameters and is performed every 40 s after t = 120 s. Fig. 3 shows the convergence to zero of the state! 1, through successive switching, whereas Fig. 4 shows the logarithm of the norm of the state. Observe the exponential decay rate. Finally, Fig. 5 displays the projection of the balls B 1 ;B 2 ;:::;B 4 (see Section 6.3) into the space (! 1 ;! 2 ;! 3 ). 8. Conclusions and outlook The problem of output feedback stabilization of the angular velocity of a rigid body has been addressed. It has been shown that asymptotic stability cannot be achieved by smooth dynamic output feedback. Nevertheless, global practical stability is attainable if the measured velocities are appropriately selected, as expressed in Proposition 4. If neither condition (5) nor condition (6) are met, we are unable to give a positive or negative answer to the stabilization problem, and we regard this as an open problem worth investigation. Moreover, under the hypotheses of Proposition 4 a hybrid, dynamic, output feedback control law achieving (global) exponential convergence has been derived. Finally, we point out that this work is intended as a starting point to study the much more challenging problem of output feedback stabilization with one control action and one measured variable. Acknowledgements The author is indebted to F. Mazenc (Imperial College) for pointing out the result in Proposition 3 and to A. Tornambe (Terza Universita of Rome) for several useful discussions. Part of this work has been performed while the author was visiting the Terza Universita of Rome (Italy). Financial support of the European Science
12 192 A. Astol / Systems & Control Letters 36 (1999) Foundation Programme on Control of Complex Systems (COSY) is gratefully acknowledged. Finally, insights and suggestions from the anonymous reviewers are thanked. References [1] D. Aeyels, Stabilization of a class of nonlinear systems by smooth feedback control, Systems Control Lett. 5 (1985) [2] D. Aeyels, Stabilization by smooth feedback of the angular velocity of a rigid body, Systems Control Lett. 5 (1985) [3] D. Aeyels, M. Szafranski, Comments on the stabilizability of the angular velocity of a rigid body, Systems Control Lett. 10 (1988) [4] A. Astol, A. Rapaport, Robust stabilization of the angular velocity of a rigid body, Systems Control Lett., 34 (1998) [5] S. Battilotti, Global output regulation and disturbance attenuation with global stability via measurement feedback for a class of nonlinear systems, IEEE Trans. Automat. Control 41 (1996) [6] S. Battilotti, A note on reduced order dynamic output feedback stabilizing controllers, Systems Control Lett. 30 (1997) [7] R.W. Brockett, Asymptotic stability and feedback stabilization, in: Dierential Geometry Control Theory, Birkhauser, Boston, 1993, pp [8] F.H. Clarke, Y.S. Ledyaev, E.D. Sontag, A.I. Subbotin, Asymptotic controllability implies feedback stabilization, IEEE Trans. Automat. Control 42 (1997) [9] M.J. Corless, G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Automat. Control 26 (1981) [10] P. Hattis, Predictive momentum management for the space station, J. Guidance Control. Dyn. 9 (1986) [11] P. Morin, Robust stabilization of the angular velocity of a rigid body with two controls, European J. Control 2 (1996) [12] H. Nijmeijer, A.J. Van der Schaft, Nonlinear Dynamical Control Systems, Springer, Berlin, [13] R. Outbib, G. Sallet, Stabilizability of the angular velocity of a rigid body revisited, Systems Control Lett. 18 (1992) [14] E.D. Sontag, H.J. Sussmann, Further comments on the stabilizability of the angular velocity of a rigid body, Systems Control Lett. 12 (1989)
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