ROBUST STABILIZING CONTROLLERS FOR LINEAR FLEXIBLE MECHANICAL SYSTEMS+ Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610 D. Da M. Corless School of Aeronautics & Astronautics Purdue University West Lafayette, Indiana 47907, USA u. q1 and q2 are the displacements of P1 and P2 from fixed points 0 1 and 0 2 , respectively. If the bar B is modelled as a rigid bar of length l o , q1 = q2 and the motion of the system can be described by ABSTRACT In this paper, the problem of stability robustness of linear memoryless feedback controllers applied to linear mechanical systems containing flexible components is treated. In general, if a stabilizing controller is designed based on a ‘rigid’ model, i.e., a model in which some of the flexible components are assumed rigid, it is not true that this controller also stabilizes the ‘real’ flexible system. We present a condition which guarantees that a stabilizing controller whose design is based on a ‘rigid’ model also stabilizes the flexible system, provided that the components, whose flexibilities are neglected in the ‘rigid‘ model, are sufficiently stiff. It is shown that this condition involves the location of the rate sensors and is independent of the location of the displacement sensors. It is also shown that the condition is satisfied when the rate sensors are collocated with the actuators. The results are illustrated by examples. 2&=u Suppose a controller is given by with f l , f i > 0. Then, the feedback-controlled rigid model subject to (1.2) is ( 1.3) which is asymptotically stable. Suppose now the bar is modelled as a flexible bar with linear stiffness coefficient K > O and linear damping coefficient c > 0. With controller (1.2), the feedbackcontrolled flexible model is described by 41 - q2) + 441 - 42) . (1.4a) (1.4b) It can be readily shown by using the root-locus technique that if f2 > 4c, then system (1.4) is unstable for K sufficiently large ; Le., even if the bar is ‘very stiff‘, the behavior of the flexible model (1.4) is qualitatively different from that of the rigid model (1.3). In designing a stabilizing controller for a flexible mechanical system one usually considers a reduced order model in which some of the flexible components m considered rigid. The natural question that arises is whether a controller, whose design is based on the reduced order model, also stabilizes the original system. The following simplc example illustrates that this question is not trivial. A singular perturbation approach is one approach to answer the above robustness question for flexible mechanical systems in which the ‘neglected‘ elements are sufficiently stiff. By ‘neglected’ elements we mean those flexible elements which are considered rigid in the reduced order model. Ficola ef a1 [6] appear to be the first to use a singular perturbation approach to treat the control problem for elastic robots. After that, many papers have been published on utilizing singular perturbation theory to deal with flexible Example. Consider a mechanical system consisting of two particles of unit mass connected by a massless bar B; see Figure 1. The system is constrained to move along an inertially fixed line and particle P1 is subject to a control force Based on research supported by the U.S.National Science Foundation under grants MSM-87-06927 and MSS-90-57079. Copyright c 1991. by American I n s t i t u t e of Aeron a u t i c s & A s t r o n a u t i c s , I n c . A l l r i g h t s reserved = - ~ ( q l - 92) - c(q1 - 42) - f 2 b - flq2 42 = K(ql 1. INTRODUCTION t . 83 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610 mechanical control systems; some of them are Kokotovic [8101 , Khorasani [7] , Marino [ l l ] , Spong 113,141 , McClamroch [121 and Corless [2,3] . A singularly perturbed system is one whose description depends on a scalar parameter p > 0; this dependency is such that setting p = 0 results in a system of lower order than that for p > 0. For small p > 0, it is customary to determine the qualitative behavior of a singularly perturbed system by studying two associated lower order subsystems, namely, the reduced order system (p = 0) and the boundary layer system. The latter subsystem is obtained by letting p = 0 in the 'fast time' scale. One method of controller design for such systems is to base the design on the reduced order model. To guarantee stability robustness with respect to the dynamics which were not taken into account in the controller design, it is customarily required that the boundary layer system be asymptotically stable. In treating a flexible mechanical system as a singularly perturbed system, one lets p = K-% where K > 0 is a stiffness parameter associated with the 'neglected' elements; as K increases the 'neglected' elements become stiffer. The reduced order system corresponds to the 'rigid' model of the system, i.e., the system model obtained by considering the 'neglected' elements rigid. To obtain a boundary layer system (hereafter called 'neglected' elements) of the system are modelled as rigid elements, this can be represented by linear constraints on the generalized coordinates, Le., there exists a matrix s E IR~* with such that sq=o. (2.2) Also we suppose that there are no other possible kinematical constraints on the system. Then the systems under consideration are described by Mq+ Cq+ Kq= Wu + STh Y1 (2.3a) (2.3b) =D1q (2.3~) Y2 = D24 where u E Rm is a vector of control inputs; y1 E IRp' , y2 E RP2 represent measurements which are based on displacement and rate sensors, respectively; M E RN* is the system mass matrix, hence it is symmetric and positive definite, i.e., M~=M>o; (2.4) the matrices C, K E RNfi are such that - Cq - Kq represents all the generalized forces acting on the system except the generalized forces due to the control inputs and the 'neglected' elements mentioned above; W E RNm is the influence matrix; the vector S T h represents the generalized which is asymptotically stable one usually assumes the 'neglected' elements have damping coefficients which are proportional to IC". In this paper, we consider linear flexible mechanical systems in which the damping coefficients of the 'neglected' elements remain constant as K increases. In this case, the boundary layer system is just marginally stable; Le., stable, but not asymptotically stable. We present a condition which guarantees that a stabilizing controller whose design is based on the 'rigid' model, a model in which the 'neglected' elements are assumed rigid, also stabilizes the flexible system provided that the 'neglected' components are sufficiently stiff, It is shown that this condition involves the location of the rate sensors and is independent of the location of the displacement sensors. It is also shown that the condition is satisfied when the rate sensors are collocated with the actnators. Two examples are given to illustrate the results. forces exerted by the 'neglected' elements; and D1 E IRpl*, D2 E RpzXN are determined by the type and location of the displacement and rate sensors, respectively. 2.1. 'Rigid' Model Consider first the situation in which the 'neglected' elements are assumed rigid. Then (2.2) holds. Condition (2.1) guarantees that there exists a matrix U E lRNx(N-L) such that su=o (2.5) and rank (U) = N - L 2. SYSTEM DESCRIPTION Define For the class of flexible mechanical systems under consideration, we let q(t) E lRN denote a vector of generalized coordinates which describe the configuration of the system at time t, We assume that, when some of the flexible elements Then V E lRNxL and 84 . (2.6) F]M[llvI= :] Fz , MkUTMU; (2.8) Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610 Assumption 1. The matrix Fz, given in (3.I), is such that the following inequality is satisfied: (2.lob) (2.1oc) Y2 = (2.1od) (3.2) In the following we present an assumption which guarantees that any stabilizing controller (3.1) for the ‘rigid’ model (2.10) also stabilizes the flexible model (2.13), provided the stiffness parameter K is sufficiently large. (2.10a) =El+ . Mq+(C+STCoS+WF2D2)q+(K+KSTKOS+WF1 D 1)q = 0. (3.3) Under the new coordinates system (2.2),(2.3) can be described by Y1 m1&)= $ 0 If one uses controller (3.1) on the flexible model (2.13), then the feedback controlled flexible model is given by (2.9) O=O RmP2.Then the feedback controlled ‘rigid’ model is a&+ (e+ m 2 6 2 ) &+ (E+ hence the square matrix [u VI is invertible. We introduce a coordinate transformation defined by a&+ e&+ E+= WU E C,+ PTCP + PTWF2D2P> 0 (3.4) where P E RNXL is given by where -A -A C = UT CU ; -KA= UT KU ; W = UT W ; - P 4M-’ST(SM-’ST)-’ . (3.5) A AD1U ; D2 =D2U. Our main result is given by the following theorem. We call this the ‘rigid’ model. Theorem. Suppose the controller (3.1) satisfies Assumption 1 and the feedback controlled ‘rigid‘ model (3.2) is asymptotically stable. Then there exists K* > 0 such that for all K > K* the feedback controlled jlexible model (3.3) is asymptotically stable. 2.2. Flexible Model We suppose that the effect of the flexibilities of the ‘neglected’ elements can be represented by letting h = - KKO(Sq) - CO(S4) (2.11) Proof. We outline the proof in the next section. A complete proof will be given in a forthcoming paper. where KO E RLxLis symmetric and positive definite, Le., K;f=&>O, (2.12) Remark 1. The theorem shows thaL the satisfaction of Assumption 1 is the only condition for a stabilizing controller to be robust with respect to unmodelled flexibilities. The term C, + PTCP in (3.4) represents the damping of the ‘neglected’ flexible components which is not related to the controller. The term PTWF2D2P represents damping exerted by the controller. Inequality (3.4) guarantees that the total damping related to the ‘neglected’ components is positive. and C, E IRLxL; K E (0, -) is a parameter characterizing the stiffness of these elements; see [I] . Clearly, the larger K is, the stiffer these elements are. Then the flexible model is given bY M i + (C + sTC,S)q + (K + KST&s)q = WU (2.13a) Y1 =D1q (2.13b) Y2 =D24. (2.13c) 3.2. Collocation of Rate Sensors and Actuators 3. MAIN RESULTS If the rate sensors of the system are collocated with the actuators, we have 3.1. The Main Theorem D2 = WT Suppose one designs a stabilizing controller U=--lYl -F2Y2 and hence F2 E R” (3.1) is a square matrix. Suppose C, + P T C P > 0 . for the ‘rigid’ model (2.10), where F1 E Rmpl and 85 (3.6) Then we have the following corollary for this special case. (4.3e) Corollary. Suppose (3.6) and (3.7) hold, and F2 2 0. Then any stabilizing controller (3.1) for the ‘rigid‘ model (2.10) QISO stabilizes the jfexible model (2.13) provided K is sujjiciently large. Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610 Remark 2. By considering (3.4) and (3.6) we can see that the collocation of the rate sensors and actuators guarantees that the feedback control does not decrease the (4.30 The reduced order system associated with (4.3) represents the feedback controlled ‘rigid’ model (3.2) and the associated boundary layer system which is described by (4.4) damping related to the ‘neglected’ components. 4. OUTLINE OF THE PROOF OF THE MAIN THEOREM The flexible mechanical system (3.3) can be represented by a singularly perturbed system by letting & K-v2 5. TWO EXAMPLE§ 5.1. A Simple Mechanical System and introducing the coordinate transformation (2.9), which yields Z $ = G ~ ~ ~ , + G ~ ~ ~(4.la) +H 8 = (3214, + G226 + Hzl I$+ H22 8 - p-’kO is just marginally stable. The remainder of the proof proceeds by applying the results in [4,5] to (4.3). (4.1b) where Consider the simple mechanical system introduced in Section 1; also see Figure 1. Suppose the measured output y form+(2.3b),(2.3~), has ~ the ~ ~ H ~ ~ where O D1 = [dii di21 ; Dz = [dzi, dzzl ; q = ~1 ; (5.1) and dij E IR ;i,j = 1,2. If the bar B is modelled as a rigid bar of length l o , we obtain a linear constraint on q1 and q2, Le., S q A= [ l - l ] q = O , (5.2) and the system can be described by 24, = u Let y1 = (dl1 + d l d 4 y2 = (d21 + d2214, A (4.2) Then, system (3.3) can be represented by X + A12(p)z (4.3a) = A21 x + A22 (P)z (4.3b) = Allx A where 4, = q1 = 92. We call this the rigid model. Suppose now the bar is modelled as a flexible bar with linear stiffness coefficient K > 0 and linear damping coefficient c > 0. Then the system can be described by (2.13) with M = I ; C=K=O ; W= where (4.3c) (5.3a) (5.3b) (5.3c) ; &=l We call this the flexible model. Suppose there is a stabilizing controller (4.3d) for the rigid model (5.3). Since ; CO=C. [] I2 -11 p A M-~ST(SM-~ST)-~ = ’ (5.8) Assumption 1 is satisfied iff 1 Co + PTWf2D2P= c + -f2(d21 - d22) > 0 . 4 and the system can be described by (5.5) JT$ Hence by the theorem in Section 3, the feedback controlled flexible system Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610 M i = - WflDlq - Wf2D2q - KSTSq - cSTSq or JT = J1 + 352 + J3 + J4 M=diag (J1, 3J2, 53, J4) C=K=O W=[l 0 0 op &)=I3 5.2. Flexible Dual Rudder Steering Mechanism Co = c13 . Consider the system illustrated in Figure 2. Each part of the system is constrained to rotate about a line parallel to the Consider inertially fixed direction g. PI is a rotor with moment of inertia J1 about L1 and its angular displacement is 91. P2 is a gear assembly composed of three identical gears, each of which has moment of inertia J2 about its aiis of rotation. We let qz be the angular displacements of the gears as shown in Figure 2. PI and P2 are connected by a massless shaft B1. Two rudders P3 and P4 with moments of inertia J3 and J4 about and L3 are connected to P2 by two massless shafts J1 = J3 = J4 = 1, now J2 system parameters = -, 1 and suppose there is a controller 3 u = - fIY1 -f2y2 (5.10) which stabilizes rigid model (5.9). Since PAM-~sT(sM- B2 and B3, respectively; q3 and 94 are the angular displacements of P3 and P4, respectively. u is a control torque exerted on PI. We suppose the measured output has the form s 1 T -1 -- ’1 - 1 (5.11) 4 -1 1-1 -1 Assumption is satisfied iff i 1 - 1611 D2 = [d21 d22 d23 d241. and di, E R ;i=1,2 ;j=1, ...,4. The rigid model of the system is obtained by considering the shafts Bi, i = 1, 2, 3, rigid. This results in 9 (5.9b) Suppose now the shafts are considered flexible. We model each Bi, i = 1, 2, 3, as a flexible shaft of linear torsional stiffness coefficient K > 0 and linear damping coefficient c > 0. Thus the flexible model of the system can be described by (2.13) with then, for any f2 2 0, (5.5) holds; hence, provided the feedback controlled rigid model is asymptotically stable, the feedback controlled flexible model (5.6) is asymptotically stable for sufficiently large K. A . (5.6) d21 = 1 , d22=O, @ = q 1=q2 =93 =q4 (5.9a) where is asymptotically stable for sufficiently large K if (5.5) holds. If the rate sensor is collocated with the actuator, i.e., D2= WT =u 1 -1612 where 11 = W 2 1 - d22 - d23 - d d f 2 12 = (3d23 - d2i - d22 - d d f 2 (5.7) 13 = (3d3 - d21 - d22 - du)f2 which puts a linear constraint on q, Le., . Two special cases arc considered in the following. 07 31 marginally stable boundary layer systems, Proceedings of the 28st Annual Allerton Conference on Communication, Control, and Computing, 1990. ( 1 ) A single rate sensor which is placed on Pd. Thus D2=[O 0 0 11. It can readily be obtained that Assumption 1 is satisfied if ;:: [;O 0 1. 5. D. Da and M. Corless, Lyapunov functions for linear singularly perturbed systems with marginally stable boundary layer systems, Proceedings of the American Control Conference, 1991. 6. A. Ficola, R. Marino, and S. Nicosia, A singular perturbation approach to the dynamic control of elastic robots, Proceedings of the 21st Annual Allerton Conference on Communication, Control, and Computing, 1983. 7. K. Khorasani, A slow manifold approach to linear equivalents of nonlinear singulariy perturbed systems, Automatica Vol. 25 pp. 301-306. 1989. 8. P. V. Kokotovic, Applications of Singular Perturbation Techniques to Control Problems, SIAM Review Vol. 26 pp. 396-416, 1984. 9. P.V. Kokotovic and H.K. Khalil, Singular Perturbations in Systems and Control, IEEE Press, New York , 1986. 10. P.V. Kokotovic, H.K. Khalil, and J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design, Academic Press , 1986. 11. R. Marino and S. Nicosia, On the feedback control of industrial robots with elastic joints: a singular perturbation approach, University of Rome: R-84.01 , 1984. 12. N. H. McClamroch, A singular perturbation approach to modeling and control of manipulators constrained by a stiff environment, ZEEE Conference on Decision and Control, 1989. 13. M. W. Spong, Modeling and control of elastic joint robots, Journal of Dynamic Systems, Measurement and Control Vol. 109 pp. 310-319, 1987. 14. M. W. Spong, K. Khorasani, and P. V. Kokotovic, An integral manifold approach to the feedback control of flexible joint robots, IEEE Journal of Robotics and Automation Vol. RA-3 pp. 291-300, 1987. G 9-; 1- 30 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610 where 0 A= f2/16c; and as a result, a condition which assures the satisfaction of Assumption 1 is given by O<G<- 1 8 or O<f2<2c By the theorem in Section 3, this guarantees that controller (5.10) also stabilizes the flexible model of the system for sufficiently large K. (2) A single rate sensor which is placed on P I . In this case, the rate sensor and actuator are collocated, hence D 2 = W T = [ 1 0 0 01 and 9 -3 - 3 Co + PTWf2D2P = CI-thence Assumption 1 is satisfied for any f2 2 0. REFERENCES 1. 2. 3. 4. M. Corless, Modelling ’flexible constraints’ in mechanical systems, Twentieth Midwestern Mechanics Conference, 1987. M. Corless, Stability robustness of linear feedback controlled mechanical systems in the presence of unmodelled flexibilities, Proceedings of the Conference on Decision and Control , 1988. M. Corless, Controllers which guarantee robustness with respect to unmodelled flexibilities for a class of uncertain mechanical systems, International Journal of Adaptive Control and Signal Processing Vol. 4 pp. 565-579, 1990. M. Corless and D. Da, On the stability of singularly perturbed linear systems which have 88 Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2610 Figure 1: Two Masses Connected by a Massless Flexible Bar Figure 2: Dual Rudder Steering Mechanism 89

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