Switching through Singularitiesl

Transcription

1 Proceedings of the 36th Conference on Decision & Control San Diego, California USA * December 1997 Switching through Singularitiesl C. J. Tomlin and S. S. Sastry Department of Electrical Engineering and Computer Sciences University of California at Berkeley, CA { clairet, sastry}@eecs. berkeley. edu WAOI 1O:OO Abstract Asymptotic tracking is studied for systems which are not regular, that is th.e relative degree is not well defined. Hence the control law derived from exact inputoutput linearization has singularities in the state space. We propose a tracking control law which switches between an approximate tracking law [l] close to the singularities, and an exact tracking law away from the singularities, and we study the applicability of this law based on the behavior of the system s zero dynamics at the switching boundary. As in [l], the ball and beam example is used to motivate the study. 1 Introduction Nonlinear control techniques have built up a fair level of sophistication with the use of input-output and full state linearization, approximate linearization, and bounded tracking for nonminimum phase systems. One area in which results have been hard to come by is the tracking of szngular or non regular nonlinear control systems, i.e. those that fail to have relative degree. While the problem of trying to track trajectories that go through singularities was begun by Hirschorn and Davis [2] who limited the class of outputs that could be tracked, the first set of practical schemes for approximate asymptotic tracking through singularities was given by Hauser, Sastry and KokotoviC in [l], using an approximation technique. This in turn spurred the development of a result by Grizzle, Di Benedetto and Lamnabhi-Lagarr igue [3] which proved the necessity of regularity for asymptotically exact tracking. In parallel with this activity has been the interest in using switching control laws for adaptive control ([4]), for steering and stabilization ([5],[6], [7]), and for tracking ([SI). In this paper, we combine ideas from switching along with our results on exact tracking of slightly nonminimum phase systems (developed in [!)] building on the techniques and ideas of [lo] and [ll]) to describe our results in approxzrnate tracking for sarigular or nonregular nonlinear systems. The outline of our paper is as follows. in Section 2 we describe the ball and beam example of [l] with some added new insights about the zero dynamics. In Sec- ~~ ~ lresearch supported by NASA under grant NAG , and by NSERC and ZONTA jgraduate fellowships. tion 3, we state our result for approximate tracking of non regular systems with unstable zero dynamics, and we propose a tracking control law for nonlinear SISO systems which switches between an approximate and exact tracking law, so as to avoid the singularities of the exact law while increasing the domain of attraction of the approximate law. In Section 4 we discuss the nature of the zero dynamics of these systems which we classify into three cases, and in Section 5 we give our current results about tracking using switched controllers. The basic result is that slow switching is acceptable for the third case, where the equilibrium structure is preserved between the exact and approximate internal dynamics. In the first and second cases (in fact a generalization of the ball and beam example), the change in equilibrium in the internal dynamics across the switching boundary is likely to destabilize the switched control law. 2 Motivating Example: the ball and beam Consider the following system, which describes the motion of a ball rolling along a rotating beam [l]. The state variables are where r is the distance of the ball from the pivot of beam, and 0 is the angle of the beam. The assumptions made in the development of this system are that the ball always remains in contact with the beam (even if the beam flips upside down), and that the rolling occurs without slipping. In the following, G is the acceleration due to gravity, and B := M/( Jb/R2 + M ) where M and Jb are the mass and moment of inertia of the ball, and R is the radius of the ball. The input T is the torque applied at the center of rotation: a preliminary change of coordinates in the input space, as in [l], is performed to define a new input variable U which appears linearly in the system equations: The output equation is y = XI - ro, where ro is a given position of the ball along the beam., Thus f(2) = [22,B(slz:- Gsins3),~4,0]~, g(x) = 0-7ao al97 $10.00 Q 1997 IEEE 1

2 [0, O,O, 1]*, h(x) = 21. The ball and beam system as modeled by (1) is non-regular, meaning that the relative degree of the system is not well defined globally over the state space. Indeed, L,L;h(x) = 2Bx1x4, so that the feedback linearizing control is undefined when 51x4 = 0. In [l], an approximate input-output linearization scheme is defined using the coordinates El = z1 - ro, E2 = x2, 53 = Bxlxi - BGsinx3, E4 = Bx2xi - BGx4 cos x3. The system dynamics in mixed x and 5 coordinates is 51 = E2 E2 = E3 E3 = [4 + ~BXIX~U (2) (4 = B(Bx~x: - BGsinx3)x: + BGx; sin53 + (~Bxzx~ - BGCOSX~)U Ignoring the coefficient of U in (3 gives rise to the control law: [-B(Bxlxi - BGsinx3)xi - BGx:sinxS +U] U= ~BxZX~ - BG COS 23 (3) for auxiliary input U. The zero dynamics are zerodimensional, and consist of the equilibrium points (To, 0,0, 0), (To, 0, -T, 0), (To, 0, T, 0) with uo = 0, which correspond to the beam being held in a stationary horizontal position. This approximate scheme is exact when 2Bx1x4 = 0, and approximates the exact scheme well when 12Bx1x41 is small. However, when 12Bx1x41 becomes large (which is when the amplitude of the desired trajectory becomes large) the approximate tracking scheme is unstable. We introduce a switching scheme which uses this approximate scheme in a neighborhood of the singularity, and switches to a tracking control based on exact input-output linearization outside of this neighborhood. To do this, partition the state space into the regions: MO = {x E R4 : lxlx4l < S}, M- = {x E R~ : ~1x4 < -S}, M+ = {x E R4 : ~1x4 > S} for small 6 E R+. In MO, we use the control law (3). In the regions M+ and M-, we define a new coordinate system: [I = xl-ro, E2 = x2, [3 = Bxlx%-BGsinx3, 7 = x4, where the nonlinear coordinate transformation defined by : x + (t,~) is a local diffeomorphism away from x3 = g. The system dynamics are $1 = 52?j= U The input-output linearizing U is therefore given by U=- 1 (-Bz~x~ + BGx~ COS 53 + v) ~BXIX~ (5) for an auxiliary input w. The zero dynamics are obtained by setting &, &, &, and their derivatives to zero, for which the solutions are: 24 = uo : x4 = uo : uo = 0 and (x3,xq) = (0,O) or (T, 0) uo = Gcosxa and rex: = G sin x3 2 ~ 0 (6) The first solution above is trivial, since it corresponds to holding the ball and beam at the equilibrium position and applying zero input. The second solution is more interesting, since the zero dynamics evolve on the one-dimensional surface SI, which causes the beam to do cartwheels! Now consider the problem of tracking a desired output trajectory yo(t) corresponding to a periodic motion of the ball along the beam. The state trajectory which implements yo(t) traverses the region in which ~ 1x4 = 0, since x4 must necessarily change sign. We define a 6-region around Xlx4 = 0 by {x : 1x1~415 S} and in this region, the approximate tracking law of equation (3) may be used to stabilize the system to yo(t), with v = Y y - 4 2= 1 (4) a,(& - yo where the az are selected so that s4 + a4s3 + a3s2 + a2s + a1 is Hurwitz. We expect the system to stabilize nicely in this &region (around x3 = 0) since the zero dynamics correspond to a single equilibrium. In the region {x : 1x1~41 2 S}, the exact tracking law of equation (5) may be used to stabilize the output of the system to yo(t): 3 v = y p - E aa(& - y p ) a=1 where s3+a3s2+a2s+a1 is Hurwitz. Because the zero dynamics in this region are constrained to lie on SI, we cannot expect such a nice behavior of the inverse dynamics of the system in this region. Figure 1 shows the result of tracking yo(t) = l.gsin(1.3t) +3 using control based on approximate linearization only, and Figure 2 shows the result of tracking the same trajectory using a control law which switches between approximate and exact tracking. The approximate tracking scheme is unstable, due to the neglect of the 2Bx1xg term away from the singularity where the magnitude of x1 is large. The switched scheme is stable, yet the driven dynamics of x3 cause the beam to continually flip upside down and right side up. 3 General Formulation Consider single-input single-output controllable nonlinear systems of the form i = f(x) +g(x)u Y = h(x) (7) where x E R, U E R, y E R, f(x) and the columns of g(x) are analytic vector fields, and h(x) is an analytic

3 function. We assume that x = ZO is an equilibrium point, that is f (ZO) = 0, and without loss of generality we assume that h(z0) = 0. In the following we need the definitions of robust relative degree and uniformly hzgher order [I]: Definition 1 (Robust relative degree) The nonlanear system (7) is said to have robust relative degree y at zo zf there exist smooth functions 4%(z), i = 1... y such that lo Time (sec) *O where the functions &(x, U), i = 0... y are sums of terms of O(z), O(Z, U), or O(u) (denoted O(x, u)~), where a(.) and b(z) are smooth, and a(xo) # 0. Figure 1: Tracking yr,(t) = I.gsin(1.3t) -t 3 (shown as a dashed curve) using a tracking law based on approximate linearization only. The parameters used in this simulation are B = , b = 3, (ai,az, ~~3,021) = (0.6,6,4,4). Definition 2 (Uniformly higher order) A function $ : Rn x E% -+ E% is said to be uniformly higher order on B, x B, if for some E > 0, U > 0 there exists a monotone increasing function of E, K(E), such that B, and B, are balls of radius E, U centered at the origin. Tracking using switched control 6, I Assume that the system (7) has robust relative degree y. Adopting the notation of [I], we define new coordinates ([,q) by = $i(z), i = l,...,y and qi = qi(z), i = 1,...,n- y where the functions r]i(x) are chosen so that L,qi(~) = 0. The system (7) is written in mixed coordinates as: lo- - lo l5 Time (sec) Figure 2: Tracking y ~(t) = 1.9*sin(1.3t) +3 (shown as a dashed curae) using the switched tracking law for the same parameters as in the previous figure. The system is stable, although the driven dynamics zg = 6 flip between 6 = 0 and 6 = T, as expected. 3 with output y = (1 + &(z,u). The approximate system corresponding to (9) is obtained by dropping $~(z,u), i = 1...y. Since we are interested in tracking a desired trajectory yd(t), which is assumed to be at least y times continuously differentiable, define <D(t) E [yd(t)t$d(t),... 7yk- (t)lt. We consider four possible scenarios for the system (9), which lead to four different nonlinear tracking control laws: the exact system has stable zero dynamics; the exact system has unstable zero dynamics; the approximate system has stable zero dynamics; the approximate system has unstable zero dynamics. In the first

4 scenario, feedback linearization results in exact tracking of yo(t) with stable zero dynamics. In the second, Devasia-Chen-Paden s nonlinear inversion technique [lo] may be used to calculate the stable solution to the zero dynamics, and exact tracking results. For the third, hounded tracking and O(E) tracking er- ror are proven in [l] when the functions $jz(z,u),i = 1,..., y - 1 are uniformly higher order in x and U, the zero dynamics of the approximate system (G = q(0, q)) are exponentially stable, and the desired output and its derivatives ( y, ~,...,) y; are sufficiently small. In the long version of this paper, [la], we have extended the result of [l] to the fourth scenario, using the nonlinear inversion technique of [lo]: Theorem 3 (Approximate Tracking for Unstable Zero Dynamics) Consider the system (7) with normal form (9). Assume that in a neighborhood B, of 20 i. the zero dynamics of the approximate system, 7j =a q(o,q), are hyperbolic (the Jacobian Q = g(0,qo) has no eigenvalues on the jw-axis); 2. the error in Jacobian linearization of the corresponding driven dynamics, defined by r([,v) := q((,q) - Qq, is Lipschitz continuous in both of its arguments: with Lipschitz constants IC1, IC2 small enough; 3. the functzons $i(x,u),z = I,,.., y - 1 are uniformly higher order; and 4. the desired trajectory a,nd its derivatives are sufficiently small. Then, for x E B,, U E B, there exists a# unique bounded solution to the driven dynamics, called 770; the states of the closed loop system defined by (9) with control law (9) hecomes (,-I E1 = E2 = L;-lh(z) + LgL;-2h(x)u b(x) 4.1 i = q(l77) We also assume that L Ly-2h(zo) = 0, meaning that 9 f the singular surface contains xo. We have shown in this section that for the system (11) with approximate control law (lo), if the dynamics 7j = q(0, q) are unstable hut hyperbolic, the states of the system remain bounded and the tracking error is small if the desired trajectory and its derivatives (YO, YO,..., yo (7)) remain small. However, we would like to track trajectories in which the desired output and its derivatives may force the state through the singular surface of L,L;-2h(x) = 0, and also far away from this singular surface. Away from this singular surface, JL,L;- h(z)j is not small, and the approximate control law fails. We propose a switching scheme which uses the approximate control law close to the singular surface, and switches to a control law derived from exact input-output linearization far away from the singular surface. 4 Classification of the Internal Dynamics As we did with the ball and beam, we classify the behavior of the internal dynamics of (11) to determine when such a switching law may he used. These internal dynamics are the (n-?+i)-dimensional exact internal dynamics in the region in which L,LT-2h(x) # 0: i, = L;h(z) + L,L;-lh(z)ueX(x) il = Q(J,77) (12) and the (n - 7)-dimensional approximate internal dynamics when L,L;-2h(z) = 0: (where f1 E E%T and f2 E Rn-Y are chosen to stabilize the closed loop system) remain bounded; and the tracking error is O(c2). In order to keep the notation as simple as possible, we concentrate on an example system in which the difference between the robust relative degree and the relative degree is 1. That is, $1 = $2 =... = +,-2(z,u) = 0 and $jt-l(x,u) # 0. Under this assumption, system 4 i = dj,rl) (13) Exact Internal Dynamics Assuming that L,L;-2h(z) # 0, consider the exact control law U: (.) required to hold the output and its derivatives identically at zero: L;- h(z) + LgL:-2h(x)~F(x) = 0 (14) As the state trajectory approaches xo, this control law becomes

5 Note that for the class of smgular systems that we are considering both the numerator L;- h(xo) and denominator L,L;-2h(Izo) of this limit vanish, since f(z0) = 0, and zo E MO. Because the numerator and denominator are both analytic functions, we can rewrite (14) as where at(z) is O(lz -- xolr), a,(z> is O(1x - ZOI ), ql(zo) # 0, qz(z0) # 0, and T > 0 (again since f ( ~ = ) 0). For the driven dynamics, if the desired output is yo(t), the required input uex(t) may be calculated from Figure 3: MO, M-, and M+ The exact zero dynamics of the system may be classified into three cases, depending on the limits U;~(ZO) and uex(zg): 1. Exact zero dynamics do not exist: s > T, u;x(zo) = Exact zero dynaimics exist, but zo is not an equilibrium: s = T, uy(z0) = --% $: Exact zero dynamics exist, and zo is an equilibrium: s < T, uy(z0) = 0. Refer to [12] for more details of this classification. In Case 3 the zero dynamics exist and have an equilibrium point at zo, and the driven dynamics include the term which, depending on yo(t), may cause the driven dynamics to become unstable on one side of the singular surface. Approximate Internal Dynamics Now consider the approximate control law U, () required to hold the approximate system s output and its derivatives at zero: so that L;h(x) + L,L;-lh(z)u:PP(x) := 0 (18) For the approximate driven dynamics, if the desired output is yo(t), the required input uapp(t) may be calculated from w(4q:(e) + ql(z)u PP(.) = &(t) (21) Thus, u PP(x0) is always finite. A control scheme which switches between exact and approximate control laws calculated from (17) and (21) is not likely to work for Cases 1 and 2 above, when there is a change in equilibrium between the exact and approximate internal dynamics. As we saw in Section 2, the switching control in the ball and beam example triggered higher order zero dynamics (a limit cycle with no equilibria) resulting in cyclic oscillations of the beam from 0 to x and back. Our conclusion is that for these classes of singular systems, it is best to not use a switching control law, but to use the approximate control law (22) for small enough values of y~(.) and its derivatives. Thus, for this class of nonlinear systems there are intrinsic limitations to the magnitude of the inputs and their derivatives that can be approximately tracked. In the next section, we prove approximate tracking for slow switching, for Case 3 above. 5 Tracking yo(t) for Case 3 Consider the system (11) for the case in which both the approximate and the exact zero dynamics have an equilibrium at $0. Partition the state space into MO, M-, M+ as in the ball and beam example, illustrated in Figure 3. As in Section 3, we define in analogy to the control (10): an approximate control law Rewrite (18) as ari(z)q;(ic) + q;(ic)u;pp(ic) = 0 (20) where w(x) is O(lz - x0lr ), $(SO) # 0, ql(z0) # 0, and T > 0 (note tlhat L,L;- h(z0) :f 0 since the robust relative degree is 7). The only solution to this is u ~pp(z~) = 0, so thatt zo is always an equilibrium of the approximate zero dynamics. in the region MO, and an exact control law in the region M- and M+. In each case, XD is obtained from either [:p and r)gpp, or [Ex and QF,

6 where Sgpp = (yo,..., yd (7-1)), qgpp(t) = JTT Gapp(t - ~)rapp([~pp, qgpp)d~, <$x = (yo,..., ~g-~ ), and qg(t) = JTT Gex(t - ~)r (tg, qf)d~. For both cases, the choice of T is discussed below. As was shown in Section 4 the internal dynamics (12) may be nonminimum phase either in M- or in M+ or both. If so, the tracking control law that is used in (23) would have to be calculated using the technique of Devasia, Paden and Chen [lo]. The main drawback of the computations involved in the Devasia et al. scheme is that it involves computation of the steady state response as discussed in [9]. To address this, we will assume that the rate of switching is slow compared to the zero dynamics of the exact and approximate system. We formalize this assumption in the following definition, and prove the following theorem in [12]: Definition 4 (Slow Switching) Let {ti}z,=i be the sequence of switching times indexed by the set I. The switching between control laws (22) and (23) is called slow if infaci{ta+l - ta} > 2T where T is such that 7)r(ED, qd)dt (24) qd(t) = lt is an t-approximation to the infinite time computation of 00 vo(t) = T)?-(ED, Vl))dT -K! i.e. that SUPtE[-T,T] IqD(t) - qd(t)( < 6. Theorem 5 (Approximate Tracking Using Switched Control) Consider the system (7) with normal form (ll), and with the control law switching between (22) and (23) at the surfaces {z E R : L,L?- h(z) = &a} Assume that if the exact and/or approximate internal dynamics (12),(13) are unstable, then conditions 1,2,3 of Theorem 3 hold for these dynamics, and the desired trajectory yo(.) is such that the conditions of slow switching are satisfied for small enough E. Under these assumptions, the switching control law results an asymptotic approximate tracking with output error less than or equal to KS with all state variables bounded. Remarks: Theorem 5 clearly holds if the number of switches is finite provided that the switches are placed far enough apart. In light of the discussion before Theorem 5 about the need for steady state calculations in the Devasia-Paden-Chen scheme, we feel that the foregoing switching control is not a good alternative to approximate linearization if the switching is fast. Even though the last step of the proof of Theorem 5 (see [12]) uses the method of [l], it is important to note that the theorem is different and may yield better asymptotic tracking because of the switching. 6 References [I] J. Hauser, S. Sastry, and P. KokotoviC, Nonlinear control via approximate input-output linearization: The ball and beam example, IEEE Transactions on Automatic Control, vol. 37, no. 3, pp , [2] R. Hirschorn and J. Davis, Output tracking for nonlinear systems with singular points, SIAM Journal of Control and Optimization, vol. 25, no. 3, pp , [3] J. W. Grizzle, M. D. D. Benedetto, and F. Lamnabhi-Lagarrigue, Necessary conditions for asymptotic tracking in nonlinear systems, IEEE Transactions on Automatic Control, vol. 39, no, 9, pp , [4] A. S. Morse and F. M. Pait, MIMO design models and internal regulators for cyclicly switched parameter-adaptive control systems, IEEE Transactions on Automatic Control, vol. 39, no. 9, pp , [5] J. Guckenheimer, A robust stabilization strategy for equilibria, IEEE Transactions on Automatic Control, vol. 40, no. 2, pp , [6] J. Malmborg, B. Bernhardsson, and K. J. Astrom, A stabilizing switching scheme for multi controller systems, in Proceedings of the International Federation of Automatic Control, (San F rancisco, CA), [7] M. Branicky, Stability of switched and hybrid systems, in Proceedings of the IEEE Conference on Decision and Control, pp , [8] S. Devasia, B. Paden, and C. Rossi, Exactoutput tracking theory for systems with parameter jumps, International Journal of Control, vol. 67, no. I, pp ,1997. [9] C. Tomlin and S. Sastry, Bounded tracking for nonminimum phase nonlinear systems with fast zero dynamics, tech. rep., UCB/ERL Memo M96/46, Electronics Research Laboratory, UC Berkeley, CA 94720, Longer version to appear in the International Journal of Control. [lo] S. Devasia, D. Chen, and B. Paden, Nonlinear inversion-based output tracking, IEEE Transactions on Automatic Control, vol. 41, no. 7, pp , [Ill G. Meyer, I;. R. Hunt, and R. Su, Nonlinear system guidance, in Proceedings of the IEEE Conference on Decision and Control, (New Orleans, LA), pp , [la] C. Tomlin and S. Sastry, Switching through singularities, tech. rep., UCB/ERL Memo M96/54, Electronics Research Laboratory, UC Berkeley, CA 94720, 1996.