Matching, linear systems, and the ball and beam

Automatica 38 (2002) 2147 – 2152 www.elsevier.com/locate/automatica Brief Paper Matching, linear systems, and the ball and beam
F. Andreeva;1 , D. Aucklyb;1 , S. Gosavic;1 , L. Kapitanskib;∗;1;2 , A. Kelkard;1 , W. Whitec;1 a Department of Mathematics, Western Illinois University, Macomb, IL 61455, USA of Mathematics, Kansas State University, Manhattan, KS 66506, USA c Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS 66506, USA d Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA b Department Received 25 July 2000; received in revised form 22 May 2002; accepted 28 June 2002 Abstract A recent approach to the control of underactuated systems is to look for control laws which will induce some speci ed structure on the closed loop system. In this paper, we describe one matching condition and an approach for nding all control laws that t the condition. After an analysis of the resulting control laws for linear systems, we present the results from an experiment on a nonlinear ball and beam system. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Underactuated systems; Control design; Lambda-method 1. Underactuated systems and the matching condition Over the past 5 years several researchers have proposed nonlinear control laws for which the closed-loop system assumes some special form, see the controlled Lagrangian method of Bloch, Leonard, and Marsden (1997, 2000) and Bloch, Chang, Leonard, and Marsden (2001), the generalized matching conditions of Hamberg (1999, 2000a,b), the interconnection and damping assignment passivity based control of Blankenstein, Ortega, and van der Schaft (2001), the -method of Auckly, Kapitanski, and White (2000), and Auckly and Kapitanski (2002), and the references therein. In this paper we describe the implementation of the -method of Auckly et al. (2000) on a ball and beam system. For the readers convenience we start with the statement of the main theorem on -method matching control laws (Theorem 1). We also present an indicial derivation of the main equations. We then prove a new theorem showing that the family of matching control laws of any linear time invariant system contains all linear state feedback control laws
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Andrew Teel under the direction of Editor Hassan Khalil. ∗ Corresponding author. E-mail address: [email protected] (L. Kapitanski). 1 Supported in part by NSF grant CMS 9813182. 2 Supported in part by NSF grant DMS 9970638. (Theorem 2). We next present the general solution of the matching equations for the Quanser ball and beam system. (Note, that this system is di erent from the system analyzed by Hamberg, 1999.) As always, the general solution contains several free functional parameters that may be used as tuning parameters. We chose these arbitrary functions in order to have a fair comparison with the manufacturer’s linear control law. Our laboratory tests con rm the predicted stabilization. This was our rst experimental test of the -method. We later tested this method on an inverted pendulum cart (Andreev, Auckly, Kapitanski, Kelkar, & White, 2000c). Consider a system of the form grj xj + [jk; r] ẋj ẋk + Cr + @V = ur ; @xr (1) r = 1; : : : ; n, where gij denotes the mass-matrix, Cr the dissipation, V the potential energy, [ij; k] the Christo el symbol of the rst kind 
@gjk 1 @gij @gki [jk; i] = (2) + − 2 @qk @q j @qi and ur is the applied actuation. To encode the fact that some degrees of freedom are unactuated, the applied forces and/or torques are restricted to satisfy Pji g jk uk = 0, where Pji is a g-orthogonal projection. The matching conditions come from this restriction together with the requirement that the 0005-1098/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 1 4 5 - 0 2148 F. Andreev et al. / Automatica 38 (2002) 2147 – 2152 closed-loop system takes the form r]ẋj ẋk + Ĉ r + @V̂ = 0; ĝrj xj + [ jk; @xr notation derivation of Eqs. (6) and (7). Substitute Eqs. (2), (4) for both ijk and ˆ kij into the rst of Eqs. (3) and multiply the result by the scalar 2 to obtain r = 1; : : : ; n for some choice of ĝ; Ĉ, and V̂ . The matching conditions read Pkr (ijk − ˆ kij ) = 0; Pkr (gki Ci − ĝ ki Ĉ i ) = 0;   (3) ki @V̂ ki @V r = 0; Pk g − ĝ i i @q @q where ijk is the Christo el symbol of the second kind ijk = gk‘ [ij; ‘]: Pkr ĝ k‘ = Pkr gk‘ Ptr r‘ (4) = Ptr @gij @gjr @gri − Ptr j − Ptr i : r @q @q @q r‘ ĝ‘i = gri (5) Theorem 1. The functions ĝij ; V̂ , and Ĉ satisfy (3) in a neighborhood of x0 if and only if Pkr (gki Ci − ĝ ki Ĉ i ) = 0;   ki @V̂ ki @V r =0 − ĝ Pk g @qi @qi and the following conditions hold. First, there exists a hypersurface containing x0 and transverse to each of the vector elds i‘ Pji @=@x‘ on which ĝij is invertible and symmetric and satis es gki P‘k = kj P‘k ĝji : r‘ Ptr @ĝj‘ @ĝij @ĝ‘i − Ptr r‘ j − Ptr r‘ i ‘ @q @q @q (8) Use Ptr r‘ (@ĝ‘i =@qj ) = @(Ptr r‘ ĝ‘i )=@qj − ĝ‘i (@Ptr r‘ )=@qj and The motivation for this method is that Ĥ = 21 ĝij q̇i q̇j + V̂ is a natural candidate for a Lyapunov function because (d=dt)Ĥ = −Ĉ j q̇j . Following Auckly et al. (2000), introduce new variables ik = gij ĝ jk . We have Second, ji Pkj and ĝij satisfy
@‘ Pks Ptr g‘s rj + [‘j; s]r‘ − [rj; i]si @q  @si i ‘ + gir j + [ij; r]s − [sj; ‘]r = 0; @q @gij @gj‘ @g‘i − Pkr gk‘ j − Pkr gk‘ i : ‘ @q @q @q Multiply by grt and use the self-adjointness of P, i.e., Pik gkj = gik Pjk , to get If the matching conditions (3) hold, the control law will be given by ur = grk (ijk − ˆ kij )q̇i q̇j + (Cr − Ĉ r )   ki @V ki @V̂ + grk g : − ĝ @qi @qi @ĝij @ĝj‘ @ĝ‘i − Pkr ĝ k‘ j − Pkr ĝ k‘ i ‘ @q @q @q (9) in (8) to obtain (7). To derive (6), rst, di erentiate (9) with respect to qj to get r‘ @ĝ‘i @‘ @gri = j − ĝ‘i rj : j @q @q @q Substitute Eq. (10) into Eq. (8) and obtain 
@r‘ @r‘ ‘ @ĝij r Pt ĝ‘i j + ĝ‘j i + r ‘ @q @q @q
 @gjr @gij @grj @gri @gri r = Pt + r : + − j − @qj @qi @q @qi @q Multiply by −Pks si , use (9) and (10) to obtain 
@r‘ ‘ @gjs i @gij s r Pk Pt g‘s j + r ‘ − s r @q @q @q
 i @r‘ ‘ @s s r i = Pk Pt ĝij r ‘ − s ĝ‘j i : @q @q (10) (11) (12) Finally, to obtain (6), add to Eq. (12) an equation obtained from (12) by interchanging k and t; r and s; ‘ and i. 2. Matching and constant coecient linear systems (6) @ĝnm @(r‘ Ptr ) @(r‘ Ptr ) + ĝ + ĝ ‘n ‘m @q‘ @qm @q n In this section, we prove that for linear time invariant systems any linear full state feedback control law is a solution to the matching equations. (7) Theorem 2. When applied to linear, time-invariant systems, the family of matching control laws contains all linear state feedback laws. Although the proof of this proposition may be found in Auckly and Kapitanski (2000, 2002) and Auckly et al. (2000), for convenience, we include an indicial Choose coordinates qi so that the desired equilibrium is at the origin, V = Vij qi qj + vk qk , and Ci = Cij q̇j , where gij ; Vij ; vk ; Cij , and Pkr are constant, and Pkr has rank nu . @gnm = Pt‘ ‘ @q @P ‘ @P ‘ + g‘n mt + g‘m tn : @q @q 2149 F. Andreev et al. / Automatica 38 (2002) 2147 – 2152 Clearly, ĝij ; V̂ = V̂ ij qi qj , and Ĉ i = Ĉ ij q̇j is a solution to the matching equations when ĝij ; V̂ ij , and Ĉ ij are constant provided ĝij and V̂ ij are symmetric, Pkr (gki Vij − ĝ ki V̂ ij ) = 0, and Pkr (gki Cij − ĝ ki Ĉ ij ) = 0. Let uk = vk + aki q̇i + bki q̇i be an arbitrary linear control law, satisfying Pkr gk‘ u‘ = 0. Comparison with Eq. (3) gives grk (gki Vij − ĝ ki V̂ ij ) = arj and grk (gki Cij − ĝ ki Ĉ ij ) = brj : Fig. 1. The ball and beam system. Thus, V̂ ‘j = ĝ‘p gpr (Vrj − arj ) and Ĉ ‘j = ĝ‘p gpr (Crj − brj ): It remains to check that we can nd a symmetric, nondegenerate matrix ĝ ki so that the resulting V̂ ‘j is also symmetric. The symmetry of V̂ ‘j will follow if we have ĝ‘p gpr (Vrj − arj ) − ĝjp gpr (Vr‘ − ar‘ ) = 0; and, therefore, we need to nd a symmetric, nondegenerate matrix ĝ‘p satisfying this equation. The existence of such matrix is guaranteed by the following simple observation. Lemma 1. Given any real n × n matrix R, there is a nondegenerate symmetric matrix X so that RX − X T RT = 0: Indeed, setting X = QYQT , results in the following equation for Y : Q−1 RQY − Y T (Q−1 RQ)T = 0: Hence, without loss of generality we may assume that Q−1 RQ is a real Jordan block (see Horn & Johnson, 1985),    1 0 :::   0  1 :::   ::: or  a    b   0     0  −b 1 0 0 0 a 0 1 0 0 0 a −b 1 0 0 b a 0 1 ::: ::: :::  :::      :  :::     ::: In each case  0 :::  0 :::  Y =   1 0 0 1 ::: ::: 1   0     0 solves the equation. Note that the result of Lemma 1 is true for matrices with coecients in any eld. This is proved in Taussky and Zassenhaus (1959). 3. Example: the ball and beam In order to demonstrate the approach described above, we have implemented one of the control laws from the family of control laws described in the rst section on a ball and beam system, Fig. 1 (this system is commercially available from Quanser Consulting, Ontario, Canada). The s-coordinate is unactuated, the -coordinate is actuated by the servo, and the objective is to bring the ball to the center of the beam. The physical parameters of the system are given in Table 1. One can express as a concrete function of  from the kinematic relation (‘b (1 − cos( )) − rg (1 − cos()))2 +(‘b sin( ) + ‘l − rg sin())2 = ‘l2 : The kinetic energy of the system is
2 1 1 1 1 1 T = mb s2 ˙2 + IB ˙ + ṡ + Ib ˙2 + Is ˙ 2 : 2 2 rB 2 2 The potential energy is V = 12 ml grg sin() + 21 (mb + ml )g‘b sin( ) +mB gs sin( ) ˙ After rescaling, we and the dissipation is C1 = 0; C2 = c0 . get (1 − cos( ) − a2 (1 − cos()))2 + (sin( ) + a1 − a2 sin())2 = a21 ; 2150 F. Andreev et al. / Automatica 38 (2002) 2147 – 2152 Table 1 The physical parameters of the system IB = 25 mB rB2 = ball inertia = 4:25 × 10−6 kg m2 Ib = inertia of the beam = 0:001 kg m2 Is = e ective servo inertia = 0:002 kg m2 g = gravitational acceleration = 9:8 m=s2 s0 = desired equilibrium position = 0:22 m c0 = inherent servo dissipation = 9:33 × 10−10 kg m2 =s ‘b = length of the beam = 0:43 m ‘l = length of the link = 0:11 m rg = radius of the gear = 0:03 m rB = radius of the ball = 0:01 m mB = mass of the ball = 0:07 kg mb = mass of the beam = 0:15 kg ml = mass of the link = 0:01 kg T = 21 ṡ2 + 12 (a4 + (a3 + 52 s2 )( ′ ())2 )˙ 2 + ′ ˙ ()ṡ; () d; y = ( )s − s0 + 0 V = a5 sin() + (s + a6 ) sin( ()) ( ) = exp −5 and 0 C1 = 0; ˙ C2 = a7 ;  1 () d : 1′ () Here h(y); w(y); 1 ( ) are arbitrary functions of one where the ak are dimensionless parameters rg ‘l (Ib + IB ) Is a1 = ; a2 = ; a3 = ; a4 = ; ‘b ‘b IB IB variable, and Ĉ 2 is an arbitrary function which is odd in velocities. For details see Andreev et al. (2000a,b). ml rg ‘b (mb + ml ) ; a6 = ; 2mB rB 2mB rB
1=2 5 c0 a7 = : 3 mB 2rB g 4. Experimental results a5 = The notation ′ is used to denote a derivative of a function of one variable. For general underactuated systems, the use of the powerful -method to solve the matching equations is discussed in Auckly and Kapitanski (2002) and Auckly et al. (2000). For systems with two degrees of freedom, the -method produces the general solution to the matching equations in an explicit form (Auckly & Kapitanski, 2000). When applied to the ball and beam system, the explicit family of control laws is given by Eq. (5) with the following expressions for ĝ; V̂ , and Ĉ, where
 d’ 2 ĝ11 = ( ) h(y(s; )) + 10 ; ′ 2 (’) 0 1 (’) ĝ12 = 1 (g11 − ĝ11 );  ĝ22 = V̂ (s; ) = w(y) + 5(y + s0 ) 0 −5 0 i1 −1 Ĉ 1 = (g1i ĝ ) sin(’) (’) 1′ (’) j2 1 (g12 − ĝ12 );  sin(’) d’ 1′ (’) (’) ’ () d d’; 0 (C1 − g1j ĝ Ĉ 2 ); where ′ ( ()) (s; ) = 1 ; 5sg12 1 (s; ) = 1 ( ) − 1′ ( ); 5s Our experiments were conducted on the Quanser ball and beam system. The control signal is a voltage supplied to the servo and the sensed output of the system is s and  sampled at 300 Hz. A Quanser MULTIQJ data acquisition card is used for the analog signal input and output. The velocities are computed via numerical di erentiation using the forward di erence algorithm. The control law produces a voltage signal and is supplied through the D/A converter to the DC servomotor via an ampli er. The relation between the control voltage, vin , and the torque, u (=u2 in −1 2 2 ˙ Eq. (5)), is u = Ng Km R−1 m vin − Rm Km Ng , where Rm = armature resistance = 2:6 ; Ng = gear ratio = 70:5; Km = motor torque constant = 0:00767 V s. Any stabilizing linear control law for this system is speci ed by four constants. The nonlinear control laws in our family are speci ed by the four arbitrary functions: ˙ We chose 1 ( ); h(y); w(y), and Ĉ 2 (s; ; ṡ; ). 1 ( ) = 1:0849 exp(4:7845 sin( )); h(y) = 1:1031; w(y) = 0:0023y2 ; ˙ = −ĝ12 (1 + ṡ2 + 10˙ 2 )(−ṡ + ): ˙ Ĉ 2 (s; ; ṡ; ) These functions produce the control law, u, in rescaled units. The values of the constants a1 through a7 are as follows a1 = 0:2547; a5 = 0:1889; a2 = 0:0588; a6 = 42; a3 = 236:294; a7 = 5 × 10−6 ; a4 = 471:126: F. Andreev et al. / Automatica 38 (2002) 2147 – 2152 Fig. 2. Ball position response. The nal control signal is obtained by converting back into MKS units and using the formula in the preceding paragraph to get the input voltage. These choices were made from the following considerations. The form of the function 1 was chosen to simplify the integrals in the expressions for y; , and ĝ11 . The form of Ĉ 2 was chosen to ensure that Ĉ 1 ṡ + Ĉ 2 ˙ would be positive(for Ĥ to be a Lyapunov function). Finally, the coecients in these functions were chosen so that the linearization of the nonlinear control law would agree with the linear control law provided by the manufacturer. Extensive numerical simulations done using MatlabJ con rm that the nonlinear control law stabilizes the system. The linear control law appears to stabilize the system for a wider range of initial conditions than the nonlinear control law. This is an empirical observation, not a mathematical fact. Finding an adequate mathematical framework to compare di erent control laws is a very interesting unresolved problem, see Auckly and Kapitanski (2000). Usually, given two locally stabilizing control laws, there exist initial conditions stabilized by one but not by the other. For example, one set of physically unrealistic initial conditions with a large angular velocity ˙ = 3:6 (or 158 rad=s in physical units) is stabilized by our nonlinear control law but not by the linear one. We have implemented the nonlinear control law in the laboratory. The laboratory tests con rm the predicted behavior of the nonlinear controller. Figs. 2 and 3 show a comparison of the time histories of the ball position (s) and angular displacement () for the linear and nonlinear control laws. In both cases the control signal reached the saturation limit for a short duration during the initial rise of the response. The di erence in the steady-state values of the responses is attributed to a lack of sensitivity of the resistive strip used to measure the ball position. 2151 Fig. 3. Angular displacement response. 5. Conclusions The -method produces explicit in nite-dimensional families of control laws and simultaneously provides a natural candidate for a Lyapunov function. When this method is applied to linear time-invariant systems, the resulting family contains all linear state feedback control laws (Proposition 2). In this paper we also present the results of the rst implementation of a -method matching control law on a concrete physical device, the ball and beam system. The experimental results agree with theoretical predictions and numerical simulations. In our experiments we observe that the linear control law performs better than our nonlinear control law for the ball and beam system. However, in a later experiment with an inverted pendulum cart (Andreev et al., 2000c) we found that a properly tuned -method matching control law performed better than the corresponding linear one. At the moment, it is not known for which systems matching control laws will perform better. This is an important problem that must be resolved. References Andreev, F., Auckly, D., Kapitanski, L., Kelkar, A., & White, W. (2000a) Matching, linear systems, and the ball and beam, http://arXiv.org/abs/math.OC/0006121. Preprint. Andreev, F., Auckly, D., Kapitanski, L., Kelkar, A., & White, W. (2000b). Matching control laws for a ball and beam system. Proceedings of the IFAC workshop on Lagrangian and Hamiltonian methods for nonlinear control, Princeton (pp. 161–162). Andreev, F., Auckly, D., Kapitanski, L., Kelkar, A., & White, W. (2000c). Matching and digital control implementation for underactuated systems. Proceedings of the American control conference, Chicago, IL (pp. 3934 –3938). Auckly, D., & Kapitanski, L. (2000). Mathematical problems in the control of underactuated systems. CRM Proceedings and Lecture Notes, 27, 41–52. 2152 F. Andreev et al. / Automatica 38 (2002) 2147 – 2152 Auckly, D., & Kapitanski, L. (2002). On the -equations for matching control laws, SIAM Journal on Control and Optimization (accepted). Auckly, D., Kapitanski, L., & White, W. (2000). Control of nonlinear underactuated systems. Communications on Pure Applied Mathematics, 53, 354–369. Blankenstein, G., Ortega, R., & van der Schaft, A. J. (2001). The matching conditions of controlled Lagrangians and interconnection and damping assignment passivity based control. Preprint. Bloch, A., Chang, D., Leonard, N., & Marsden, J. E. (2001). Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping. IEEE Transactions on Automatic Control, 46, 1556–1571. Bloch, A., Leonard, N., & Marsden, J. (1997). Stabilization of mechanical systems using controlled Lagrangians. Proceedings of the IEEE conference on decision and control, San Diego, CA (pp. 2356 –2361). Bloch, A., Leonard, N., & Marsden, J. (2000). Controlled Lagrangians and a stabilization of mechanical systems I: The rst matching theorem. IEEE Transactions on Automatic Control, 45, 2253–2270. Hamberg, J. (1999). General matching conditions in the theory of controlled Lagrangians. In Proceedings of the IEEE conference on decision and control, Phoenix, AZ. Hamberg, J. (2000a). Controlled Lagrangians symmetries and conditions for strong matching. Proceedings of the IFAC workshop on Lagrangian and Hamiltonian methods for nonlinear control, Princeton, NJ (pp. 62– 67). Hamberg, J. (2000b). Simpli ed conditions for matching and for generalized matching in the theory of controlled Lagrangians. Proceedings of the ACC, Chicago, Illinois (pp. 3918–3923). Horn, R., & Johnson, C. (1985). Matrix analysis. Cambridge: Cambridge University Press. Taussky, O., & Zassenhaus, H. (1959). On the similarity transformation between a matrix and its transpose. Paci c Journal of Mathematics, 9, 893–896. Fedor Andreev received his Ph.D. in Mathematics from the Steklov Institute of Mathematics, St. Petersburg, Russia, in 1997. He has worked at the Steklov Institute of Mathematics and at Kansas State University. He is currently an Assistant Professor at Western Illinois University. Dave Auckly received his Ph.D. in Mathematics from the University of Michigan in 1991. He has worked at the University of Texas in Austin, UC Berkeley, and MSRI. He is now an Associate Professor at Kansas State University. Shekhar Gosavi received his B.S. from University of Pune, India, in 1995, and M.S. from Kansas State University in 2000. He is currently working on his Ph.D. in mechanical engineering at Kansas State University. Lev Kapitanski received his Ph.D. and D.Sc. degrees in Mathematics from the Steklov Institute of Mathematics, St. Petersburg, Russia, in 1981 and 1991, respectively. He has worked at the Steklov Institute of Mathematics, Princeton University, and Brown University. He is currently a Professor of Mathematics at Kansas State University. Atul G. Kelkar received his Ph.D. degree in Mechanical Engineering from Old Dominion University, Norfolk, Virginia, in 1993. He has worked at Old Dominion University, NASA Langley Research Center, Hampton, Virginia, and at Kansas State University. He is currently an Associate Professor at the Department of Mechanical Engineering at Iowa State University, Ames, Iowa. Warren White received the Bachelors in Electrical Engineering from Tulane University, the Masters in Electrical Power Engineering from Rensselaer Polytechnic Institute, and the Ph.D. in Mechanical Engineering from Tulane University in 1974, 1977, and 1985, respectively. Dr. White is currently an Associate Professor of Mechanical and Nuclear Engineering at Kansas State University.