A GLOBAL STABILIZATION STRATEGY FOR AN INVERTED PENDULUM. B. Srinivasan, P. Huguenin, K. Guemghar, and D. Bonvin

Copyright IFAC 15th Triennial World Congress, Barcelona, Spain A GLOBAL STABILIZATION STRATEGY FOR AN INVERTED PENDULUM B. Srinivasan, P. Huguenin, K. Guemghar, and D. Bonvin Labaratoire d Automatique, École Polytechnique Fédérale de Lausanne, CH15Lausanne, Switzerland. e-mail: bala.srinivasan@epfl.ch Abstract: The problems of swinging up an inverted pendulum on a cart and controlling it around the upright position have been traditionally treated as two separate problems. This paper proposes a strategy that is globally asymptotically stabilizing under actuator saturation and, in addition, locally exponentially stabilizing. The proposed methodology performs swing up and control simultaneously and uses elements from input-output linearization, energy control, and singular perturbation theory. Experimental results on a laboratory-scale setup are presented to illustrate the applicability of the method. Keywords: Nonlinear Systems, Stabilization, Feedback Linearization, Energy Control, Singular Perturbation. 1. INTRODUCTION The inverted pendulum has been used as a classical control example for nearly half a century due to its nonlinear, unstable, and nonminimum-phase natures. Many control ideas have been illustrated on this system, e.g., linear feedback stabilization (Schaefer and Cannon, 1967), variable structure control (Yamakita and Furuta, 199), passivitybased control (Fradkov et al., 1995), energy control (Spong and Praly, 1995), predictive control (Ronco et al., 1), and hybrid system control (Guckenheimer, 1995). The interest in the inverted pendulum has been renewed by the recent developments in the field of underactuated mechanical systems (Spong and Praly, 1995; De Luca et al., 1996). All techniques available in the literature stabilize the system locally in a region around the upright position, with the nonlinear techniques being used to enlarge the stabilizing region. Even the papers that address the issue of global stabilization (Mazenc and Praly, 1996) restrict themselves to the region where the pendulum is above the horizontal position. The problem of getting into the stability region, i.e., the problem of swinging up, has always been considered separately (Spong, 1995; Aström and Furuta, ). Thus, an implementation strategy requires an appropriate switching between swing up and control around the upright position. In contrast, the idea in this paper is to design a controller that can do both the swing up and the control together, thereby leading to a globally stabilizing controller. The concepts used here are general enough to be applicable to other underactuated mechanical systems. The proposed methodology is based on inputoutput linearization. However, standard inputoutput linearization when applied to an inverted pendulum has two problems: There is a singularity when the pendulum is horizontal. Actuator saturation, which in general represents an additional complication for control design, is used here to handle the problem of singularity: Around a singularity,

though theoretically the control input goes to infinity, the input is limited by its bound. However, the stability analysis gets complicated due to the saturation. It is resolved by artificially introducing a reference for the angle that takes a value in the set { π,, π}, all of which refer to the same physical point, the upright position. Yet, different choices of the reference provide different ways of approaching the desired equilibrium point. The system is nonminimum phase, which indicates that the input-output linearization leads to unstable internal dynamics. A singular perturbation approach (Khalil, 1996) is used here to stabilize the internal dynamics. The controller gains are chosen so that the system exhibits two-time-scale behavior - fast dynamics for the pendulum and slow dynamics for the cart. Also, two independent controllers are designed such that both subsystems are stable. Under these conditions, it is possible to evoke the basic result of singular perturbation theory regarding the existence of a controller that can stabilize the entire system. The paper is organized as follows: The next section describes the standard model of the inverted pendulum and normalizes it to obtain a simple model that will be used for analysis. In Section 3, the input-output linearization approach is discussed. The stability of the pendulum dynamics is investigated in Section 4, and that of the cart dynamics in Section 5. Section 6 provides experimental results and Section 7 concludes the paper.. INVERTED PENDULUM ON A CART The model of the inverted pendulum on a cart can be found for example in (Mazenc and Praly, 1996), and is given by: m p + µ θ µ θ sin θ = u (1) µ p + J θ µg sin θ = () where p is the position of the cart, θ the angle between the vertical and the pendulum (positive clockwise), m the total mass of the system, µ = m p l p / with m p and l p being the mass and length of the pendulum, respectively, J = J p + m p lp/4 with J p the inertia of the pendulum, g the gravity, and u the force applied to the cart. Note that the friction is neglected in this model. The initial conditions are p() = p o, ṗ() = ṗ o, θ() = θ o, and θ() = θ o. The equations (1)-() can be rearranged to give: θ = mµg sin θ µ (u + µ θ sin θ) (mj µ cos (3) θ) p = J(u + µ θ sin θ) µ g sin θ (mj µ cos (4) θ) The system can be normalized by: (i) considering the input v = p/g, (ii) introducing the normalized time τ = t µg/j, and (iii) introducing the normalized distance y = µp/j. With the first step, the dynamics read: θ = µg J (sin θ v ) and p = gv. The second step consists of normalizing the period of oscillation. In the new time scale, d θ dτ = sin θ v, d p dτ = J µ v and the unforced period of oscillation is π. The third step normalizes the distance, and the system dynamics read: ÿ = v, y() = y o, ẏ() = ẏ o (5) θ = sin θ v, θ() = θ o, θ() = θo (6) Note that the time derivatives in (5)-(6) are with respect to the normalized time τ. This set of equations (5)-(6) is quite simple and descriptive and thus will be used for analysis in this paper. 3. FEEDBACK LINEARIZATION By analyzing the Lie algebra of the system (5)-(6), it can be concluded that the inverted pendulum is not full-state feedback linearizable. So, the inputoutput linearization between θ and v will be considered. From (6), the linearizing feedback is given by: v = k θθ + k ω θ + sin θ (7) This imposes the dynamics θ + k ω θ + kθ θ =, which is stable for any choice of k ω,k θ >. Nevertheless, this methodology has two fundamental problems: The linearizing feedback (7) has a singularity at θ = k+1 π, the horizontal position of the pendulum where =, and The internal dynamics that correspond to ÿ = v are unstable. These two problems will be handled in the next two sections. 4. STABILITY OFTHE PENDULUM DYNAMICS WITH BOUNDED INPUTS The problem of singularity of the linearizing feedback and that of actuator saturation are closely related. Even when but small, v becomes quite large and may exceed its bound v max.in such a case, the control input is: ( ) k θ (θ θ r )+k ω θ + sin θ v = sat,v max, (8) v max if v v max sat(v, v max )= v if v max <v<v max. v max if v v max

The novelty in this paper is the artificial introduction of θ r in a regulation problem and its choice to ensure stability. θ r is chosen in the set {π,, π} so as to guarantee (θ θ r ) θ, which in turn is necessary for stability. Note that all choices of θ r refer to the same physical point, the upright position, but different choices provide different ways of approaching the desired equilibrium point. 4.1 Switching Logic Let θ be the smallest positive value satisfying: v max cos θ = k θ θ + sin θ (9) It follows that if (θ θ r ) < θ, the linearizing feedback will take the system to the desired equilibrium position without saturation. θ r switches from one value to another at θ =, i.e., every time the velocity changes sign. The switching is based on the angle at the instant of switching (denoted as θ i for the i th switch). If θ i is in Zone I of Figure 1, θ r does not change since the linearizing feedback can take it to the upright position. If the pendulum is in Zone II when θ =, then there is not enough energy to go to the upright position by rotating clockwise, so a swing in the anti-clockwise direction is required. This is achieved by giving an extra encirclement to the reference, i.e., the reference is set at (θ r π). A similar argument can be provided for adding π to the reference when θ i is in Zone III. The number of times θ r changes its value depends on the number of swings necessary to bring the pendulum to its upright position. θ Zone II θ r Zone I Fig. 1. Switching strategy for θ r θ Zone III Let θ r,i and θ+ r,i denote the values of the reference before and after the i th switch, respectively. Note that θ r,i+1 = θ+ r,i, indicating that θ r remains constant between two switches. With θr, =, π θ o π, the switching law is given by: θ r,i π π (θ i θ r,i ) < θ θ + r,i = θ r,i θ r,i +π θ (θ i θ r,i ) + θ (1) + θ <(θ i θ r,i ) +π 4. Analysis of the Switching Logic The stability of the θ-dynamics with such a controller can be analyzed using the Lyapunov approach. However, the main difficulty arises from the switching of θ r. To analyze the switching of θ r, concepts from energy control are used next. Lemma 1. Consider the dynamics of θ given by (6), along with the controller (8)-(1). If k θ (1+ k ω ), k ω >, then the switching of θ r will stop in finite time after a finite number of switches. Proof: Consider the energy in the pendulum and its time derivative: E = 1 θ + + 1 (11) Ė = θ θ θ sin θ = v θ (1) The total energy consists of two parts: the normalized kinetic energy 1 θ and the normalized potential energy (+1). When the pendulum rests at the downward position (θ = π, θ = ), the energy E = and the problem of swinging the pendulum up is to pump in sufficient energy so as to achieve E = that corresponds to (θ =, θ = ). Also, note that E does not have a discontinuity even when θ r is discontinuous. The switching will stop once E cos θ + 1. This results from the fact that, when E cos θ +1 and θ =, then cos(θ i θ r ) cos θ θ i θ r θ and in such a case, it can be seen from (1) that θ r does not change. From (8), independent of whether or not the input is saturated, v and (k θ (θ θ r )+k ω θ + sin θ) will have the same sign. The idea is to make k θ (θ θ r ) larger than k ω θ + sin θ by a proper choice of k θ. An upper bound on θ can be obtained from the energy E, θ (E 1 ). Since, during switchings, the energy is E (cos θ +1), θ (1 ) = (1 cos(θ θ r ) (θ θ r ). Also, sin θ = sin(θ θ r ) (θ θ r ). So, k ω θ + sin θ (1 + kω ) (θ θ r )). When k θ (1+k ω ), k θ (θ θ r ) k ω θ+sin θ, and thus v will have the same sign as k θ (θ θ r ), i.e., v (θ θ r ). Since θ r is chosen to satisfy (θ θ r ) θ, (θ θ r ) and θ have opposite signs. So, v θ and Ė. Thus, the above controller constantly pumps energy into the system and leads to the following two possibilities: (i) the energy is sufficient (E cos θ + 1) and there is no more switching, or (ii) the energy is insufficient and θ = before reaching the upright position. In the latter case, θ r switches but the energy at the current switching instant is larger than that in the previous one, E i+1 >E i.

This monotonic increase of energy leads to the lemma. 4.3 Lyapunov Analysis v = k θ(θ θ r )+k ω θ + sin θ + w The dynamics then read: (16) Theorem 1. Consider the dynamics of θ given by (6), along with the controller (8)-(1). If k θ (1+ k ω ), k ω >, then θ = θ r, θ = is a globally asymptotically stable equilibrium point. Proof: Consider the Lyapunov function candidate and its derivative: V = 1 ( k θ (θ θ r ) + (13) V = k θ (θ θ r ) θ + θ sin θ v θ (14) Note that V (t) is discontinuous as θ r (t) switches. In fact, V increases with every switch. However, from Lemma 1, after a finite time, there are no more switchings. The Lyapunov analysis presented here is for the interval after all switchings have occurred. If v is not saturated, v = k θ (θ θ r )+k ω θ + sin θ and V = k θ ω. If v is saturated, (14) can be rearranged as follows: V =(1 α) ( k θ (θ θ r ) + sin θ ) θ αkω θ (15) v v us where α = with v us the unsaturated input. The switching law for θ r guarantees (θ θ r ) θ. As in the proof of Lemma 1, k θ (θ θ r ) sin θ when k θ 1+k ω. So, ( k θ (θ θ r ) + sin θ ) θ. Since V in (15) is a convex combination of two negative quantities, V even when v is saturated. Using LaSalle s invariance theorem (Khalil, 1996), the invariant set corresponds to θ =, θ =, which requires v = tan θ, and from (6) and (8), θ = θ r. So, with this controller, (θ = θ r, θ =)is a globally asymptotically stable equilibrium point of the dynamics (6). 5. STABILITY OFTHE CART DYNAMICS USING A SINGULAR PERTURBATION APPROACH The dynamics of the cart resulting from inputoutput linearization (internal dynamics) are unstable. Once the inverted pendulum is in its upright position, there is no reason why the cart velocity needs to be zero; and a non-zero cart velocity makes the cart drift away. In this paper, stabilizing these internal dynamics is approached using a singular perturbation framework. 5.1 Separation into a Two-time-scale System Discarding saturation, consider the control law (8) with an additional input w: ÿ = k θ(θ θ r )+k ω θ + sin θ + w (17) θ = k θ (θ θ r ) k ω θ w (18) When k θ and k ω are relatively large (compared to the gains that will be used to control the y- dynamics), then θ corresponds to the fast dynamics that will quickly reach their stable equilibrium: θ = θ r w k θ and θ =. Plugging this in (17) gives: ÿ = tan w k θ (19) Thus, a two-time-scale system is artificially created. 5. Stability of the Slow Dynamics The dynamics (19) can be stabilized using the following feedback: Lemma. Consider the dynamics of y given by (19), along with the controller w = k θ tan 1 (k y y + k v ẏ) () In the absence of saturation, if k y,k v >, then y =ẏ = is exponentially stable. Proof: Consider the Lyapunov function candidate W and its derivative with [ (ky + ky + k ] Q = v) k v k v (1 + k y ) W = 1 [ ] [ ] ẏ y ẏ Q y (1) Ẇ =(k y + ky + kv) yẏ + k v (yÿ +ẏ )+(1+k y )ẏÿ = k y k v (y +ẏ ) λ 3 z () with z = [y ẏ] T. W is positive definite and it can be checked that λ 1 z W λ z, where λ 1, are the eigenvalues of the matrix Q. Since W and Ẇ are bounded by z, the origin is locally exponentially stable. 5.3 Stability of the Overall system The standard results in singular perturbation theory (Khalil, 1996) are used to prove the stability of the overall system. The only difficulty is that the standard results require exponential stability of the two subsystems. However, exponential convergence can only be guaranteed in the neighborhood of the equilibrium

point, where the input is not saturated. So, the following result is based on the fact that, after a certain finite time, the system enters the exponential region where the standard results can be applied. Theorem. Consider the overall dynamics of the inverted pendulum (5)-(6) along with the controller v us = k θ(θ θ r )+k ω θ + sin θ + kθ tan 1 (k y y + k v ẏ) v = sat (v us,v max ) (3) θ θ + r,i π π (θ i θ r,i ) < θ r,i = θ r,i θ (θ i θ r,i ) + θ (4) θ r,i +π + θ <(θ i θ r,i ) +π where θ is the smallest positive value satisfying v max cos θ = k θ θ + sin θ +k θ tan 1 (k y y max + k v ẏ max ) (5) where y max = π (π + v max ) and ẏ max = πv max. If k θ > (1 + k ω ), and k θ,k ω k y,k v >, then (θ = θ r, θ = y = ẏ = ) is globally asymptotically stable. In the region where the input is not saturated, the convergence is locally exponential. Proof: If k y,k v, the asymptotic stability of the pendulum dynamics is proved in Theorem 1, i.e., (θ θ r ) and θ will converge to zero. With such a controller, an upper bound on the position and the speed of the cart can be derived. Note that the normalized period of oscillation is π and thus either θ or cos(θ) changes sign within π normalized seconds. Since the maximum acceleration is v max, the maximum cart velocity is ẏ max = πv max. The maximum number of switches required to swing up the pendulum is given by π+v max v max (Aström and Furuta, ). So, an upper bound on the position can be obtained as the product of maximum speed and maximum time taken to swing up: y max =ẏ max π + v max v max π = π (π + v max )(6) If the term tan 1 (k y y + k v ẏ) in (3) is dealt with the introduction of a margin in the computation of θ as in (5), then after a finite time, the system will stay in a region where the input is not saturated. In the absence of saturation, the exponential stability of the slow (reduced) cart dynamics is proven in Lemma. The exponential stability of the fast (boundary-layer) pendulum dynamics in the unsaturated case works with a Lyapunov function similar to (1), where (y, ẏ, k y,k v ) are replaced by (θ, θ, k θ,k ω ) and therefore the proof is not repeated. Considering these facts, the proof of the theorem follows directly from the stability result of the singularly perturbed system (refer to (Khalil, 1996), Theorem 9.3, p 38). The present result is existential though, in principle, a bound on k y, k v as a function of k θ, k ω can be obtained. Such an exercise is not undertaken here since the expressions are quite involved. Instead, a necessary condition for stability, i.e., stability of the linearized system around the equilibrium point, is used to design the various gains. The gains are so designed that the state matrix of the dynamics linearized around the upright position has all its eigenvalues in the closed left half plane. 6. EXPERIMENTAL RESULTS Experimental results on a laboratory-scale set up are presented in this section. The unnormalized version of the control law (3) is used: v = 1 ( J k θ (θ θ r )+ µg k θ ω + sin θ+ )) µ gj k vẏ k θ tan 1 ( µ J k yy + (7) u us = vg(mj µ cos θ)+µ g sin θ Jµ θ sin θ J V m = sat(u us /K m,v max ) (8) where V m is the voltage applied to the motor, V max the maximum voltage, and K m the ratio between the voltage applied to the motor and the linear force acting on the cart. The parameters used for implementation are given in Table 1. m.335 kg k θ 13 µ 1.365 1 3 kg m k ω 1.6 J 1.565 1 4 kg m k y.5 g 9.81 m/s k v.4 K m.3 kg m/vs Table 1. Parameters of the system and controller gains Two cases for the bounds are considered: a) V max =3.5 V and b) V max =V. The former is the true physical limit, while the latter is artificially imposed to test the performance of the controller. The simulations with V max =3.5 V are shown in Figures and 3, and with V max =V in Figures 4 and 5. Calculations show that even 3.5 V for V max is not sufficient to swing up the pendulum in one go. So, the switching of θ r is inevitable in both cases. θ r switches 4 times before reaching its final value with V max = 3.5 V, and 9 times with = V. Note that the input changes sign V max

.5 -.5 θ/π.5 Pendulum Angle.1 p [m] -.1 Cart Position tuned to the particular system at hand, some of the ideas can be generalized for the control of nonlinear nonminimum-phase systems. The possibility of using a reference to handle constraints and a two-time-scale approach to handle the internal dynamics shows promise. - 4 6 8 1 1 -. 4 6 8 1 1 Fig.. Pendulum & Cart Positions (V max =3.5) 3 1 V m [V] - -3 Input 4 6 8 1 1 -.5 θ r /π.5 Reference - 4 6 8 1 1 Fig. 3. Input & Reference (V max =3.5 V ) every time the horizontal position is crossed due to the division by in (7). In the model that has been used for control, there are many effects that have not been included. In particular, the static friction that is quite significant has been neglected. The presence of static friction in this case can be seen from the fact that the cart is stuck after 1s, though there is a non-zero voltage on the motor (Figures and 3). Despite such modeling errors, the controller works satisfactorily, which illustrates the robustness of the proposed approach. θ/π -.5.5 - Pendulum Angle -.5 4 6 8 1..15.1.5 p [m] -.5 Cart Position -.1 4 6 8 1 Fig. 4. Pendulum & Cart Positions (V max =V ) 3 1 V m [V] - -3 Input 4 6 8 1 θ r /π -.5.5 Reference - 4 6 8 1 Fig. 5. Input & Reference (V max =V ) 7. CONCLUSION A global stabilizing strategy for the inverted pendulum was proposed in this paper. It uses actuator saturation to handle singularity, switching of references to solve the stability problem, and a singular perturbation approach to stabilize the internal dynamics. Though the proposed methodology is ACKNOWLEDGMENTS The authors would like to thank Prof. K. J. Aström for useful discussions and initiation to this subject. Also, Mr. Philippe Cuanillon of EPFL is thanked for his timely help with the prototype. 8. REFERENCES Aström, K. J. and K. Furuta (). Swinging up a pendulum by energy control. Automatica 36(), 87 98. De Luca, A., R. Mattone and G. Oriolo (1996). Control of underactuated mechanical systems: Application to the planar r robot. In: Proceedings of the 35th IEEE CDC. Kobe, Japan. pp. 1455 146. Fradkov, A. L., P. Y. Guzenko, D. J. Hill and A. Y. Pogromsky (1995). Speed gradient control and passivity of nonlinear oscillators. In: Proceedings of IFAC Symposium on Control of Nonlinear Systems. Lake Tahoe. pp. 655 659. Guckenheimer, J. (1995). A robust hybrid stabilisation strategy for equilibria. IEEE Transactions on Automatic Control 4(), 31 36. Khalil, H. K. (1996). Nonlinear Systems. Prentice Hall, New Jersey. Mazenc, F. and L. Praly (1996). Adding integrations, saturated controls, and stabilisation of feedforward systems. IEEE Transactions on Automatic Control 41(11), 1559 1578. Ronco, E., B. Srinivasan, J. Y. Favez and D. Bonvin (1). Predictive control with added feedback for fast nonlinear systems. In: ECC. Porto. pp. 3167 317. Schaefer, J. F. and R. H. Cannon (1967). On the control of unstable mechanical systems. In: Automatic Remote Control III, Proceedings of the 3 rd IFAC World Congress. pp. 6C.1 6C.13. Spong, M. W. (1995). The swing up control problem for the acrobot. IEEE Control Systems Magazine 15(1), 49 55. Spong, M. W. and L. Praly (1995). Energy based control of underactuated mechanical systems using switching and saturation. In: Preprints of the Block Island Workshop on Control using Logic-based Switching. Rhode Island. pp. 86 95. Yamakita, M. and K. Furuta (199). VSS adaptive control based on nonlinear model for titech pendulum. IEEE Control Systems Magazine pp. 1488 1493.