A new controller for the inverted pendulum on a cart

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL Int. J. Robust Nonlinear Control 2008; 18:1607–1621 A new Controller for the Inverted Pendulum on a Cart F. Gordillo∗, J. Aracil Departamento de Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla,, Spain. Email:{gordillo, aracil}@esi.us.es SUMMARY This paper presents a complete solution to the problem of swinging-up and stabilization of the inverted pendulum on a cart, with a single control law. The resulting law has two parts: first, an energy-shaping law is able to swing and maintain the pendulum up. Then, the second part introduces additional control to stop the cart and it is based on forwarding control with bounded input. The resulting control law is the sum of both parts and does not commute between different laws although there exist switches inside the controller. Copyright c 2008 John Wiley & Sons, Ltd. key words: Control of pendulums, Swing up, Stabilization methods, Energy control, Stability analysis. ∗Correspondence to: Francisco Gordillo, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n. 41092 - Sevilla, Spain. Contract/grant sponsor: MEC-FEDER; contract/grant number: DPI2006-07338 Copyright c 2008 John Wiley & Sons, Ltd. 1608 F. GORDILLO, J. ARACIL 1. Introduction The inverted pendulum is an interesting device that has attracted the attention of the nonlinear control community for many years (see [2, 8, 9, 16, 22, 23, 26, 27, 28], to mention only a few references). It is a simple underactuated system that serves as a benchmark for nonlinear control techniques. The control problems associated with it can be found in many applications, such as attitude control of a space booster on takeoff and the stability of walking robots. Two are the main problems that are usually considered: 1) swing the pendulum up from the downward position (or, even, any arbitrary initial position and velocity); and 2) stabilization around the upper vertical position once the pendulum is in a neighborhood of it. The second problem can be easily solved by means of linear methods since only small perturbations are considered. For the swing-up problem several control strategies have been presented [7, 10, 16]. Nevertheless, these solutions are not able to simultaneously solve the second problem: they maintain the saddle nature of the desired equilibrium and what they achieve is asymptotic stabilization of its stable manifold. Hence, in order to solve the full problem, an hybrid control strategy is implemented: the swing-up law is committed to carrying the pendulum to the neighborhood of the desired position (global law) and, then, the control law is switched into a stabilizing one (local law). This hybrid control works quite well in experimental frameworks. However, it has been a main theoretical challenge in nonlinear control to merge both control laws into a single one. In [19] an interesting approach is presented, which merges smoothly both local and global solutions to the problem, guarantying the stability of the resultant system. In [22] a single controller is also proposed but it requires a strategy for commutation of the reference value. The problem addressed in [18] is similar but as the pendulum swing-up controller does not guarantee stability, the techniques presented in that paper are not directly Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls A NEW CONTROLLER FOR THE INVERTED PENDULUM ON A CART 1609 applicable. In this paper, we present a new control law that it is not the combination of different laws designed for solving the two subproblems separately. Instead, the new law solves simultaneously both problems. It is based on energy-shaping methods [17] but instead of just introducing damping, a combination of damping and energy injection (pumping) is needed giving rise to a pumping-damping [6] strategy. The resultant control law drives the pendulum to the desired upright position from any initial position and/or at any velocity (except a set of zero measure), and it is able to stabilize it. Previous results have been reported in [11, 3, 4]. A similar idea is used in [6]. Furthermore, in a second part of the paper, the carrying element of the pendulum is also taken into account and the full pendulum-on-a-cart system is considered. A new objective is added: the cart is desired to stop once the pendulum reaches the upright position. This problem is solved by means of the addition to the previous control law, a new term based on forwarding-with-bounded-input techniques [12, 14, 13, 24, 25]. For this, it is very useful to start with the (local) Lyapunov function provided in the first stage of the design. The resulting control law is the sum of both parts and avoids any commutation between different sub-controllers. The rest of the paper is organized as follows. In Section 2 the problem of the pendulum on a cart is solved in two steps. This approach is based on dealing first with the simple pendulum, disregarding the cart for the moment. A controller is proposed that swings up the pendulum and stabilizes it at the upright position. In Section 3 this controller is enlarged to use in the full system in such a way that the cart is also stopped. The paper closes with a Section of conclusions. Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls 1610 F. GORDILLO, J. ARACIL 2. Control of the pendulum subsystem The model of the pendulum on a cart after partial linearization and normalization [21] is ˙x1=x2 ˙x2= sin x1−cos x1u ˙x3=u, (1) where x1is the angular position of the pendulum with the origin at the upright position, and x2and x3are the velocities of the pendulum and the cart respectively and uis the force applied to the cart. Therefore, the system is defined on a cylindrical state space: (x1, x2, x3)∈S×R2. If only the pendulum is considered, the equations (1) reduce to ˙x1=x2 ˙x2= sin x1−cos x1u. (2) −1 0 1 −4 −2 0 2 4 −2 0 2 4 6 8 x1/π x2 Figure 1. Shape of the energy for the simple pendulum. In the absence of forcing uthe only stable equilibrium point is the hanging position. The upright position is a saddle point. This fact can be illustrated using energy considerations. The Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls A NEW CONTROLLER FOR THE INVERTED PENDULUM ON A CART 1611 energy of the pendulum subsystem with u= 0, represented in Fig. 1, is the sum of potential and kinetic energies: H(x1, x2) = cos x1−1 + x2 2 2. Our goal is to design a controller that is able to swing up the pendulum from (almost) all initial conditions and to maintain the pendulum at the upright position. We will base the derivation on the potential energy shaping method, choosing as desired Hamiltonian functions of the form Hd(x1, x2) = Vd(x1) + x2 2 2,(3) where the desired potential energy Vdshould have a single minimum at the desired upright position. A generalized Hamiltonian target system with Hdas a Hamiltonian function is     ˙x1 ˙x2    =    0 1 −1−ka        Dx1Hd Dx2Hd    ,(4) where kais a damping coefficient (or even it can be a function of x1and x2). With Hdas given by (3), (4) yields ˙x1=x2 ˙x2=−V′ d(x1)−kax2. (5) It is well known that the problem with choosing an appropriate Vd(x1) function is related to the term cos x1, affecting to the control signal, u, in the second equation of (2). For instance, the most elementary choice is Vd=−cos x1, which has an appropriate shape (a single minimum at the desired upright position), but it leads to the control law u= 2 tan x1(for the case ka= 0) which cannot be implemented in the full range |x1| ≤ πbecause for x1=±π/2 the feedback law is unbounded. Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls 1612 F. GORDILLO, J. ARACIL To solve the matching problem of the open (2) and closed (5) loop behaviors, and in order to avoid the division by cos x1, a good choice of V′ dis V′ d=−sin x1+ cos x1β(x1),(6) and then, for ka= 0, u=β(x1) (the case ka6= 0 will be discussed latter). Some additional conditions should be imposed on β(·). First, β(0) = 0 to guarantee that the origin (0,0) is an equilibrium of the closed-loop system. Just as the pendulum behaves in a cylindrical state space, the closed-loop system should display some periodicity. Then, it is reasonable to make β(x1) = sin x1¯ β(cos x1). This choice facilitates the integration of (6) to get Vd. We must also impose that Vd(x1) = Vd(−x1) and, therefore, V′ d(x1) = −V′ d(−x1). A family of functions, Vd, that fulfill these conditions is given by Vd=a0+ cos x1−a2cos2x1−a3cos3x1− · · · ,(7) which yields V′ d=Dx1Vd=−sin x1+ 2a2sin x1cos x1+ 3a3sin x1cos2x1+... =−sin x1+ sin x1cos x1(2a2+ 3a3cos x1+...), (8) which clearly allows us to determine β(x1) to match this last expression with (6). Therefore, we have a family of functions, Vd, which solves the matching problem for the pendulum. The simplest case of this family is obtained by taking a0=a−1, a2=aand ak= 0,∀k > 2, which leads to Vd(x1) = cos x1−acos2x1+a−1.(9) Other choices can be found in [5]. Figure 2 shows that Vdhas a minimum at the origin. It can be easily shown that this happens for a > 0.5. With this Vdwe obtain V′ d=−sin x1+ 2acos x1sin x1, Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls A NEW CONTROLLER FOR THE INVERTED PENDULUM ON A CART 1613 −1 −0.5 0 0.5 1 −2 −1 0 1 2 3 4 5 x1/π Vd Figure 2. Shape of Vdfor a= 5. and then with the feedback law u= 2asin x1,(10) which is defined everywhere, the matching of the open and closed loop systems is solved. Another interesting case [11] is obtained with Vd(x1) = −cos 3x1 3+1 3,(11) which yields V′ d=−sin x1+ 4 cos2x1sin x1, and the feedback law u= 4 cosx1sin x1= 2 sin 2x1.(12) Many other Vdbelonging to the same family can be conceived. However, and for the sake of simplicity, in the sequel we will be only concerned with the desired potential Vdgiven by (9). Nevertheless, most of the results developed here can be extended to other Vdbelonging to family (7), such as (11). Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls 1614 F. GORDILLO, J. ARACIL As it is shown in Fig. 2, function Vdgiven by (9) has other minima, which are undesirable. The same happens with every function of class (7). To overcome this problem, we begin adopting a damping term for the desired closed loop of the form −kx2cos x1(that is, in Eq. (5) ka=kcos x1). With this term, for values of x1such that π/2<|x1|< π, and due to the sign of cos x1, energy injection is produced instead of damping. With the injection of energy the pendulum tends towards the region above the horizontal. In that region, there is only a single minimum for Vd, the desired upright position. Therefore, the term −kx2cos x1causes the equilibrium at the bottom of the additional minima to change from stable to unstable. It should be noted that kadoes not have a definite sign and, consequently, the closed-loop system loses the generalized Hamiltonian structure and the stability of the system has to be analyzed by other methods. In summary, for the moment we propose a target system with the structure:     ˙x1 ˙x2    =    0 1 −1−kcos x1        Dx1Hd Dx2Hd    ,(13) that is, ˙x1=x2 ˙x2=−V′ d(x1)−kx2cos x1, (14) where, if Hdis given by Hd(x1, x2) = cos x1−acos2x1+x2 2 2,(15) the target system is ˙x1=x2 ˙x2= sin x1−2acos x1sin x1−kx2cos x1, (16) which can clearly be matched to system (2) with the control law u= 2asin x1+kx2.(17) Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls A NEW CONTROLLER FOR THE INVERTED PENDULUM ON A CART 1615 The local stability achieved with this control law can be proved using (15) as Lyapunov function. Law (17) has been also proposed in [9], but without justification; as reported there, its actual application requires careful analysis as parameters aand kvary. We can distinguish between two versions of the swing up problem. The first is the strict one, where only the transition from the hanging position (that is, for initial conditions x1≈πand x2= 0) is considered. This problem is solved with controller (17) as can be shown by means of simulations. There is a more complex version of the swing up problem, the global one, where almost any initial condition is taken into account. For this second version, controller (17) has a big drawback. It injects energy in any case when the pendulum is below the horizontal. This means that even in the case when the pendulum has enough energy to rotate, and then to go above the horizontal, we are still injecting more energy. In such a case, the energy injection can lead the system to a rotating limit cycle [1]. This is a very interesting phenomenon that deserves thorough analysis, but that is outside of the scope of this paper. When a trajectory is in the basin of an undesirable minimum, we need to inject energy to force the trajectory to leave it. This means that we have to inject energy only when the energy of the pendulum is not great enough to leave this region. Recalling that Vdhas the shape of Fig. 2, this undesirable well can be visualized as enclosed by the curve represented in Fig. 3. This curve is the Hd−level curve corresponding to Hd=H∗△ =1 4a+a−1. But when the system is outside this region we must damp it. To that end, we define the following function: ϕ(x1, x2)△ =       −kif Hd(x1, x2)≤H∗and cos x1<1 2a kelsewhere, (18) where k > 0 is a tuning parameter. That is, ϕ(x1, x2) is negative only inside the regions where Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls 1622 F. GORDILLO, J. ARACIL −1.5 −1 −0.5 0 0.5 1 1.5 −2 0 2 x1/π x2 0 10 20 30 40 50 −5 0 5 Time [s] x1 0 10 20 30 40 50 −5 0 5 Time [s] u Figure 5. Results of a simulation with a= 1, ε=.25 and k= 0.2. Initial conditions: hanging position at rest. −1.5 −1 −0.5 0 0.5 1 1.5 −2 0 2 x1/π x2 0 2 4 6 8 10 12 14 16 18 20 −2 0 2 4 6 Time [s] x1 0 2 4 6 8 10 12 14 16 18 20 −2 0 2 4 Time [s] u Figure 6. Results of a simulation with a= 1, ε=.25 and k= 0.6. Initial conditions: hanging position at rest. Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls A NEW CONTROLLER FOR THE INVERTED PENDULUM ON A CART 1623 −0.5 00.5 11.5 2 −4 −2 0 2 4 0 10 20 30 40 50 60 70 80 x1/π x2 Time [s] Figure 7. Trajectory leaving the point x1=πand x2= 0 and reaching the up-right position, for k= 0.2. When this control law is applied to system (1), we obtain ˙x1=x2(26) ˙x2= sin x1−2asin x1cos x1−eϕεx2cos2x1−cos x1v(27) ˙x3= 2asin x1+eϕεx2cos x1+v. (28) The need for term vis clear because, without it, variable x3is not fed back and it would evolve without control, driven by the motions of x1and x2produced while the pendulum is reaching the upright position. Once the pendulum has reached the inverted position, then x1=x2= 0, but x36= 0. In effect, x3=Z(2asin x1+eϕεx2cos x1)dt 6= 0 and, therefore, as x3is not equal to zero, the cart suffers a drift which leaves its motion unbounded. System (26)–(28) is a cascade with feedforward structure. In effect, making [x1x2]T=ξ Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls 1624 F. GORDILLO, J. ARACIL and x3=z, system (26)–(28) can be written in the standard feedforward form ˙z=h(ξ) + g1(ξ)v ˙ ξ=f(ξ) + g2(ξ)v, (29) where the equations have been reordered as it is usual. The upper equation in (29) is the one of the cart, and the lower one corresponds to the pendulum. Moreover, it should be noted that ˙ ξ=f(ξ) is almost GAS, with a Lyapunov function obtained in the previous section. The problem now is to determine a feedback v(x1, x2, x3) such that the speed of the cart is controlled at the same time that the upright position of the pendulum maintains its stability. The feedforward form of system (26)–(28) suggests applying conventional forwarding to get the control law v(x1, x2, x3). Unfortunately, the application of this method in this case leads to a partial differential equation that it is hard to solve. One way to overcome this difficulty is to use saturation functions in the forwarding design method [24, 25]. Here we use the approach proposed by Astolfi and Kaliora [12, 14, 13]. Furthermore, it has the advantage of having a meaningful physical interpretation. Following Astolfi and Kaliora’s suggestion, we consider the control law v=−ε2σλx3 ε2,(30) where σis the saturating function σ(y) = sgn(y) min{|y|,1},λis a design parameter, and ε2is a small positive constant. Here we use ε2to distinguish it from the εintroduced in the previous section when the switching function eϕεwas defined. Therefore, u= 2asin x1+eϕεx2cos x1−ε2σλx3 ε2.(31) The purpose of the new term vis explained in the following. Notice that all the nonlinearities of the closed-loop system, except the saturation that appears in v, do not depend on x3. First of Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls A NEW CONTROLLER FOR THE INVERTED PENDULUM ON A CART 1625 all, consider the case when the pendulum is near the upright position with small velocity –i.e. x1and x2are close to zero– with any value of the cart speed x3. In this situation, and due to the former fact, the system can be linearized around (x1, x2) = (0,0) resulting a linear system combined with a saturation. This kind of system is well-known (it belongs to the class known as Lure’s systems) and, thus, parameter λcan be tuned in order to stabilize this system. Then, if the pendulum reaches a neighborhood of (x1, x2) = (0,0) and it remains there, the cart will eventually stop. The problem now is to see if there is any guarantee that the pendulum will eventually arrive at this neighborhood and that it will stay there. Notice that with v= 0 this is true and notice also that |v| ≤ ε2so it can be very small by design. Law (31) will work if the behavior of the pendulum for small values of vis similar to the one corresponding to v= 0. Fortunately, this question has been rigourously formulated fifteen years ago (see [20]). This desired characteristic of the pendulum subsystem is fulfilled if it is input to state stable (ISS). There exist nice and powerful Lyapunov-like results that help us to guarantee that system (26)–(27) is ISS (with restrictions) as stated the following proposition. Proposition 3. System (26)–(27) is locally ISS for ||x|| <0.095π. Sketch of the proof: Choose as Lyapunov function candidate the function V(x1, x2) = cos x1−acos2x1+x2 2 2+a−1 + εvx1x2. It is straightforward to see that it fulfills the conditions of Theorem 5.2 in [15] for ||x|| <0.095π. As we are interested in a global law that is even able to swing up the pendulum, we have to include another discontinuity: we choose v= 0 in order to reach the region for which system (26)–(27) is locally ISS. Once this region is reached, we use control law (31) in order to also Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls 1626 F. GORDILLO, J. ARACIL stabilize (28) according to the procedure proposed in [13]. In summary, the resultant control law is given by: u=ue+v where ueis given in (25) and vis given by v=       0 for ||x|| ≥ 0.095π ε2σλx3 ε2for ||x|| <0.095π (32) Since ueintroduces a discontinuity, the final control signal will have two points of discontinuity: the first one when cos x1= 1/(2a) (due to the discontinuity of eϕε) and the second one when ||x|| = 0.095π(due to the discontinuity introduced in v). Notice that the gap of both discontinuities can be made arbitrarily small reducing εand ε2. However, making ε2 small deteriorates the performance of the system, in the sense that it may take a long time to stop the cart. In any case, this discontinuity can be avoided modulating (30) by a function that goes to zero when kxkis larger than a certain value. It should be realized that the ideas of [14] can be recursively applied and, thus, the position of the cart could also be controlled in a further step. Nevertheless, for simplicity, this step is omitted here. The performance of the proposed strategy can be observed in the simulation that appears in Fig. 8 which corresponds to the values x(0) = (0.99π, 0,0), a= 1, ε= 0.5, k= 0.5, ε2= 0.1 and λ= 0.1. The top left graph shows the projection of the trajectory into the (x1, x2) plane. It can be seen that the trajectory tends to the origin of this plane. The time evolutions of x1and x3are plotted in the top-right and the bottom-left graphs respectively. It can be seen these two variables (and, consequently, also x2) eventually tend to zero. The bottom right graph Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls A NEW CONTROLLER FOR THE INVERTED PENDULUM ON A CART 1627 shows the evolution of u. The vertical dotted lines represent the instant when the switch in ˜ϕεoccurs (left line) and when the ISS region is entered (right line). −1 0 1 −3 −2 −1 0 1 2 3 0 10 20 30 40 50 −1 0 1 2 3 4 5 Time [s] 0 10 20 30 40 50 −2 −1 0 1 2 Time [s] 0 10 20 30 40 50 −2 −1 0 1 2 Time [s] x1 x1/π x2 x3 u Figure 8. Simulation results for the full controller. As it was expected, the pendulum evolves towards a neighborhood of the desired position and, then, the cart slowly decelerates (so the pendulum does not fall) until it eventually stops. This two-time-scale behavior can be regarded as a natural solution of the problem considered. 4. Conclusions In this paper, we have presented a single controller able to swing up the pendulum on a cart from the hanging position to the upright position in an inverted pendulum. A design procedure Copyright c 2008 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2008; 18:1607–1621 Prepared using rncauth.cls 1628 F. GORDILLO, J. ARACIL with two main steps has been introduced. In the first one, a control law has been obtained that drives the pendulum to the desired upright position, disregarding the cart for the moment. This control law belongs to the family of the energy shaping methods. The proposed control law includes a pumping-damping term in such a way that instead of adding positive damping in all the state space, negative damping (that is, energy injection) will be added inside some undesirable regions. In this way, the system tends to leave these regions and carried on to the region where the desired equilibrium point stands. In the second step, this control law is extended to take into account the cart. To accomplish this, forwarding design via saturation functions has been used. Thus, a full controller has been obtained that is able to control both the pendulum and the cart. ACKNOWLEDGEMENTS The authors appreciate the valuable comments by the anonymous reviewers of this paper. REFERENCES 1. A. A. 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