A topological view on forced oscillations and control of an inverted pendulum

07/11/2017
Auteurs : Ivan Polekhin
Publication GSI2017
OAI : oai:www.see.asso.fr:17410:22542
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We consider a system of a planar inverted pendulum in a gravitational eld. First, we assume that the pivot point of the pendulum is moving along a horizontal line with a given law of motion. We prove that, if the law of motion is periodic, then there always exists a periodic solution along which the pendulum never becomes horizontal (never falls). We also consider the case when the pendulum with a moving pivot point is a control system, in which the mass point is constrained to be strictly above the pivot point (the rod cannot fall `below the horizon'). We show that global stabilization of the vertical upward position of the pendulum cannot be obtained for any smooth control law, provided some natural assumptions.

A topological view on forced oscillations and control of an inverted pendulum

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A topological view on forced oscillations and control of an inverted pendulum
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A topological view on forced oscillations and control of an inverted pendulum Ivan Polekhin1 Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina str., 119991, Moscow, Russia, ivanpolekhin@mi.ras.ru Abstract. We consider a system of a planar inverted pendulum in a gravitational eld. First, we assume that the pivot point of the pendu- lum is moving along a horizontal line with a given law of motion. We prove that, if the law of motion is periodic, then there always exists a periodic solution along which the pendulum never becomes horizon- tal (never falls). We also consider the case when the pendulum with a moving pivot point is a control system, in which the mass point is con- strained to be strictly above the pivot point (the rod cannot fall `below the horizon'). We show that global stabilization of the vertical upward position of the pendulum cannot be obtained for any smooth control law, provided some natural assumptions. Keywords: inverted pendulum, forced oscillations, global stabilization, control design 1 Introduction Below we consider the following system _ q = p; _ p = u(q; p; t) sinq cosq: (1) Here u is a smooth function. When u =  (t) is a function of time, the system describes the motion of an inverted pendulum in a gravitational eld with a moving pivot point (without loss of generality, we assume that the mass and the length of the pendulum equals 1 and the gravity acceleration is also 1). In this case, the law of motion of the pivot point is de ned by the smooth function (t). We show that for any T-periodic function  (t), there always exists a T-periodic solution such that q(t) 2 (0; ) for all t, i.e., the pendulum never falls. When u is a smooth function from R3 to R, we have a control system. When u is periodic in q and autonomous, it can be shown that the problem of stabilization of the vertical position of an inverted pendulum does not allow continuous control which would asymptotically lead the pendulum to the vertical from any initial position. This follows from the fact that a continuous function on a circle, which takes values of opposite sign, has at least two zeros, i.e., system (1) has at least two equilibria. The following questions naturally arise. First, do the above statements re- main true if we consider the pendulum only in the positions where its mass point is above the pivot point (often there exists a physical constraint in the system which do not allow the rod to be below the plane of support and it is meaningless to consider the pendulum in such positions). Second, is it true that global stabilization cannot be obtained when the control law is a time-dependent function and it is also a non-periodic function of the position of the pendulum? For a relatively broad class of problems, which may appear in practice, we show that the answers are positive for the both questions. The main idea of both proofs goes back to the topological method of Wa_ zewski [1,2] and its developments, which we shortly consider below. 2 Forced oscillations In this section we will discuss the existence of a forced oscillation in the system (1) when u =  (t) is a smooth T-periodic function. For brevity of exposition, we will proof the main result of this section for a slightly modi ed system _ q = p; _ p =  (t) sinq cosq  _ q: (2) This system di ers from (1) by the term  _ q, which describes an arbitrary small viscous friction (we assume that  > 0 can be arbitrarily small). First, we introduce some de nitions and a result from [3,4] which we slightly modify for our use. Let v: R M ! TM be a time-dependent vector eld on a manifold M _ x = v(t; x): (3) For t0 2 R and x0 2 M, the map t 7! x(t; t0; x0) is the solution for the initial value problem for the system (3), such that x(0; t0; x0) = x0. If W  R  M, t 2 R, then we denote Wt = fx 2 M : (t; x) 2 Wg: De nition 1. Let W  R  M. De ne the exit set W as follows. A point (t0; x0) is in W if there exists  > 0 such that (t + t0; x(t; t0; x0)) = 2 W for all t 2 (0; ). De nition 2. We call W  R  M a Wa_ zewski block for the system (3) if W and W are compact. De nition 3. A set W  [a; b]  M is called a simple periodic segment over [a; b] if it is a Wa_ zewski block with respect to the system (3), W = [a; b]  Z, where Z  M, and Wt1 = Wt2 for any t1; t2 2 [a; b). De nition 4. Let W be a simple periodic segment over [a; b]. The set W = [a; b] Wa is called the essential exit set for W. In our case, the result from [3,4] can be presented as follows. Theorem 1. Let W be a simple periodic segment over [a; b]. Then the set U = fx0 2 Wa : x(t a; a; x0) 2 Wt nWt for all t 2 [a; b]g is open in Wa and the set of xed points of the restriction x(b a; a; )jU : U ! Wa is compact. Moreover, the xed point index of x(b a; a; )jU can be calculated by means of the Euler-Poincar e characteristics of W and Wa as follows ind(x(b a; a; )jU ) = (Wa) (Wa ): In particular, if (Wa) (Wa ) 6= 0 then x(b a; a; )jU has a xed point in Wa. Theorem 2. Suppose that the function   : R ! R in (2) is T-periodic, then for any  > 0 there exists q0 and p0 such that for all t 2 R 1. q(t; 0; q0; p0) = q(t + T; 0; q0; p0) and p(t; 0; q0; p0) = p(t + T; 0; q0; p0) , 2. q(t; 0; q0; p0) 2 (0; ). Proof. First, in order to apply Theorem 1, we show that a periodic Wa_ zewski segment for our system can be de ned as follows W = f(t; q; p) 2 [0; T] R=2Z R: 0 6 q 6 ; p0 6 p 6 p0g; where p0 satis es p0 > 1  sup t2[0;T ] (j j+ 1): (4) It is clear that W is compact. Let us show that W is compact as well and W =f(t; q; p) 2 [0; T] R=2Z R: q = 0; p0 6 p 6 0g[ f(t; q; p) 2 [0; T] R=2Z R: q = ; 0 6 p 6 p0g: If p = p0, then from (2) and (4) we have _ p < 0. Therefore, (t; q; p) = 2 W for 0 < q < , t 2 [0; T], p = p0. When q = 0 and t 2 [0; T], we have (t; q; p) 2 W for p0 6 p < 0 and (t; q; p) = 2 W for any p > 0. Moreover, it can be proved that (t; q; p) 2 W when p = 0. Indeed, for q = p = 0, we have  q = _ p = 1: Therefore, any solution starting at q = p = 0 leaves W. The cases p = p0 and q =  can be considered in a similar way. Finally, we obtain (W0) (W0 ) = 1 and Theorem 1 can be applied. As we said above, this result can be proved without the assumption of the presence of friction. To be more precise, the following holds Theorem 3. Suppose that the function u =   : R ! R in (1) is T-periodic, then there exists q0 and p0 such that for all t 2 R 1. q(t; 0; q0; p0) = q(t + T; 0; q0; p0) and p(t; 0; q0; p0) = p(t + T; 0; q0; p0) , 2. q(t; 0; q0; p0) 2 (0; ). In this case, the proof is basically the same, but the Wa_ zewski block W has a di erent form. However, it can be continuously deformed to a simple periodic segment and an extended version of Theorem 1 can be applied (Fig. 1). h '0(t) W [0;T] W0 W Fig. 1. Wa_ zewski block W. W is in gray. 3 Global stabilization The key observation in the proof of Theorem 2 was that the solutions, that reach the points where q = 0 or q =  and p = 0, are externally tangent to W. From this we obtain that if a solution leaves W, then all solutions with close initial data also leave W at close points. It can be seen that this property also holds for system (1) for any u. From this we immediately obtain Theorem 4. For any smooth control function u: R3 ! R in (1), there exists q0 and p0 such that q(t; 0; q0; p0) 2 (0; ) on the interval of existence of the solution. Proof. Consider an arbitrary line segment
in the hyperplane t = 0 which con- nects the set fq; p; t: q = 0; p 6 0g with the set fq; p; t: q = ; p > 0g. Suppose that all solutions starting at
reach the hyperplane q = 0 or the hyperplane q = . As it was mentioned before, if some solution reaches the set q = 0, then all solutions with close initial data also reach this set at close points. In other words, we have a continuous map from
to the above hyperplanes. Therefore, we can construct a continuous map from the line segment to its boundary points. This contradiction proves the theorem. Similar arguments can be applied to the case when we try to stabilize our system in a vicinity of the vertical upward position. Suppose that we are looking for a control that would stabilize system (1) in a vicinity of a certain equilibrium position in the following sense. Let M be a subset of the phase space of the system such that the points of M correspond to the positions of the pendulum in which the rod is above the horizontal line (in our case, M = f0 < q < g) and  = (=2; 0) 2 M is the equilibrium for a given control u. We assume that the control function u is chosen in such a way that there exists a compact subset U  M,  2 U n@U and a C1-function V : U ! R) with the following properties L1. V () = 0 and V > 0 in U n. L2. Derivative _ V with respect to system (1) is negative in U n for all t. Since the function V can be considered as a Lyapunov function for our system, the equilibrium  is stable. For instance, such a function exists in the following case. Suppose that for a given u, system (1) can be written as follows in a vicinity of  _ x = Ax + f(x; t); where x = (q; p), A is a constant matrix and its eigenvalues have negative real parts, f is a continuous function and f(t; x) = o(kxk) uniformly in t. Then there exists [5] a function V satisfying properties L1, L2 (in this case,  is asymptotically stable). Theorem 5. For a given control u(q; p; t), suppose there exists a function V satisfying L1 and L2 for system (1). Then there exists an initial condition (q0; p0) and a neighborhood B  M of  such that on the interval of existence the solution (q(t; 0; q0; p0); p(t; 0; q0; p0)) stays in M nB. Proof. The proof is similar to the one in Theorem 4. It can be shown that for " > 0 small enough, the level set V = " is a circle (topologically). Let B = fq; p: V (q; p) 6 "g. Let
1 and
2 be two line segments in the plane t = 0 connecting the sets fq; p; t: q = 0; p 6 0g and fq; p; t: q = ; p > 0g with boundary @B, correspondingly. Suppose that any solution starting at
1[
2[@B leaves M n B. From the same arguments as in Theorem 4, we conclude that if the considered solution leaves this set, then all solutions with close initial data also leave it. Therefore, we obtain a continuous map between
1 [
2 [@B and a disconnected set (@B and two boundary points of
1 and
2). The contradiction proves the theorem. Remark 1. Actually, as it can be seen from the proofs of Theorems 4 and 5, we obtain not a single solution, that does not leave the considered sets, but a one- parameter family of such solutions. This family can be constructed by varying the line segments considered in the proofs. 4 Conclusion In this note, we have presented topological ideas which can be used for study- ing forced oscillations and global stabilization of an inverted pendulum with a moving pivot point. In both cases, the results can be generalized and extended to a broader class of systems. For instance, the result on the existence of a forced oscillation can be proved for the case when we consider a mass point on a manifold with a boundary in a periodic external eld [6,7]. In particular, similar result holds for the spherical inverted pendulum with a moving pivot point (here manifold with a boundary is the upper semi-sphere). Moreover, similar result can be proved for groups of interacting nonlinear systems [8]. As an illustration, we can consider the following system. Let us have a nite number of planar pendulums moving with viscous friction (can be arbitrarily small) in a gravitational eld (Fig. 2). Let ri be a radius-vector of the massive point of the i-th pendulum. Suppose that their pivot points are moving along a horizontal line in accordance with a T-periodic law of motion h: R=TZ ! R, which is the same for all pendulums. Let us also assume the following: for any two pendulums there is a repelling force Fij acting on the mass point of the i-th pendulum from the j-th pendulum (Fij is parallel to ri rj ). It is possible to prove that in this system with non-local g Fi 1;i Fi;i+1 'i 1 Fig. 2. When the i-th pendulum is horizontal, the repelling forces acting on it are directed downward. interaction (each pendulum is in
uenced by all other pendulums), there always exists a forced oscillation and along this solution the pendulums never become horizontal. Our simple results on global stabilization can also be proved for various simi- lar systems. One of the main possible generalizations is the system of an inverted pendulum on a cart, which is more correct from the physical point of view. It is also possible to consider multidimensional systems or systems with friction. We can also omit the requirement of the existence of a Lyapunov-type function V satisfying L1 and L2 (we just need the existence of a `capturing' set containing the point ). Moreover, systems without the assumption on the uniqueness of the solutions can also be considered, since only the right-uniqueness is impor- tant for our considerations above. For instance, we can consider systems with set-valued right-hand sides (Filippov-type systems), including systems with dry friction. References 1. Wa_ zewski, T.: Sur un principe topologique de l'examen de l'allure asymptotique des int egrales des  equations di  erentielles ordinaires. Ann. Soc. Polon. Math 20 279313 (1947) 2. Reissig, R., Sansone, G., Conti, R.: Qualitative Theorie nichtlinearer Di erential- gleichungen, Edizioni Cremonese (1963) 3. Srzednicki, R.: Periodic and bounded solutions in blocks for time-periodic nonau- tonomous ordinary di erential equations. Nonlinear Analysis: Theory, Methods & Applications 22, 707{737 (1994) 4. Srzednicki, R., W ojcik, K., Zgliczy nski, P.: Fixed point results based on the Wa_ zewski method. Handbook of topological xed point theory, 905{943, Springer (2005) 5. Demidovich, B.P.: Lectures on the mathematical theory of stability, Nauka, Moscow (1967) 6. Polekhin, I.: Forced oscillations of a massive point on a compact surface with a boundary. Nonlinear Analysis: Theory, Methods & Applications 128, 100{105 (2015) 7. Bolotin, S.V., Kozlov V.V.: Calculus of variations in the large, existence of tra- jectories in a domain with boundary, and Whitney's inverted pendulum problem. Izvestiya: Mathematics 79.5, 894{901 (2015) 8. Polekhin, I.: On forced oscillations in groups of interacting nonlinear systems. Non- linear Analysis: Theory, Methods & Applications 135, 120-128 (2016)