Research Article | Open Access
Carlos Aguilar-Ibañez, Juan C. Martínez-García, Alberto Soria-López, José de Jesús Rubio, "On the Stabilization of the Inverted-Cart Pendulum Using the Saturation Function Approach", Mathematical Problems in Engineering, vol. 2011, Article ID 856015, 14 pages, 2011. https://doi.org/10.1155/2011/856015
On the Stabilization of the Inverted-Cart Pendulum Using the Saturation Function Approach
A simple stabilizing controller for the cart-pendulum system is designed in this paper. Our control strategy describes the underactuated system as a chain of integrators with a high-order smooth nonlinear perturbation and assumes initialization of the system in the upper half plane. The design procedure involves two sequentially associated control actions: one linear and one bounded quasilinear. The first control action brings the nonactuated coordinate near to the upright position and keeps it inside of a well-characterized small vicinity, whereas the second control action asymptotically brings the whole state of the system to the origin. The corresponding closed-loop stability analysis uses standard linear stability arguments as well as the traditional Lyapunov method and the LaSalle's theorem. Our proposed control law ensures global stability of the system in the upper half plane. We illustrate the effectiveness of the proposed control strategy via numerical simulations.
Because of its control-related challenging features, the well-known cart-pendulum system has been extensively studied in recent times by the control community (see, for instance  and the references therein). This control benchmark consists of a free vertical rotating pendulum with a pivot point mounted on a cart horizontally moved by a horizontal force (which corresponds to the system input). The control problem comprises to swinging up the pendulum from its stable hanging position, in order to bring and maintain it to its unstable upright position. What makes this simple mechanical system an interesting control benchmark is the fact that the pendulum angular acceleration cannot be controlled, that is, the cart-pendulum system is a two degrees-of-freedom mechanical underactuated system. Hence, many common stabilizing control techniques developed for fullyactuated systems cannot be directly applied to this system. It must be pointed out that the cart-pendulum system is not input-output (statically or dynamically) feedback linearizable (see for instance ). Moreover, the cart-pendulum system loses controllability and other control-related geometric properties when the pendulum moves through the horizontal plane (see [1, 3]). Since the system is locally controllable around the unstable equilibrium point, closed-loop stabilization by linear pole placement can be used (see, for instance, ).
Stabilizing the cart-pendulum system involves two main aspects: (i) swinging up the pendulum from the stable hanging position to the unstable upright vertical position (see, for instance, [5–12]) and (ii) stabilizing the system around its unstable equilibrium point. For this second aspect it is commonly assumed that the free endpoint of the pendulum is initially located above the horizontal plane or lies inside a well-characterized open vicinity of zero (the vicinity defines the closed-loop stability domain). We focus our attention on the former problem. Let us now review some remarkable works on this aspect. In  a nonlinear controller, based on the backstepping procedure, is used to solve the stabilization problem in the unstable equilibrium point; the proposed controller ensures full state convergence. A controller based on nested saturation functions is proposed in . A similar work is discussed in , where a chain of integrators is considered as a model for the cart-pendulum system. In  a stabilization technique using switching and saturation functions (in addition to the Lyapunov method) is introduced. A control strategy based on controlled Lagrangians is presented in , and a proposal using similar tools is exposed in . A feedback control scheme based on matching conditions is described in , while a simple matching condition is used in  to solve the cart-pendulum regulation problem. A very interesting nonlinear control strategy based on energy shaping techniques combined with input-to-state stability methods is presented in . A solution which exploits power-based passivity properties of the cart-pendulum system is proposed in . A nonlinear controller based on both the fixed point backstepping procedure and saturation functions is proposed in . This list of published works is by no means exhaustive. Let us conclude it by mentioning the work published in , where the challenging nature of the cart-pendulum problem is underscored to the nonlinear control community.
In this work we consider the Inverted Pendulum model introduced in , to propose a simple control strategy, which combines a linear action with a quasilinear action. This strategy is justified by using the traditional Lyapunov method and applying the theorem of LaSalle. We point out that the proposed controller here differs from the obtained in [15, 22]. Because, contrary to these works, we do not need to use sophisticated nonlinear tools to carry out the stability analysis. For instance, we do not use the fixed point control scheme, nor the modified backstepping procedure. Also, our control law has a simple structure than the ones proposed in these two works. Roughly speaking, the linear control action confines, both, the angular position and the angular velocity in a small compact set, which defines the closed-loop stability domain (completely characterized by the cart-pendulum parameters and the controller parameters). The bounded quasilinear control action guarantees, then, full state convergence. We must emphasize that our solution avoids the necessity of solving partial differential equations, nonlinear differential equations, or fixed point control equations.
The paper is organized as follows. Section 2 is concerned with modeling issues as well as the problem statement. We present our proposal in Section 3, which we illustrate with a simulated control scheme. We conclude with some final remarks in Section 4.
2. Problem Statement
2.1. The Cart-Pendulum Model
Consider the cart-pendulum system (see Figure 1), described by the following set of normalized differential equations (see for instance ): where is the normalized displacement of the cart; is the actual angle that the pendulum forms with the vertical; is the horizontal normalized force applied to the cart (i.e., the system input), and is a real constant that depends directly on both, the cart and the pendulum masses. In the nonforced case corresponding to and the above system has only one unstable equilibrium point given by , with being constant. Some simple algebra allows us to derive a new control variable : Thus, system (2.1) can be written in a very simple way as Now, we proceed to express system (2.3) as if it were a four-order chain of integrators plus an additional nonlinear perturbation. For this end we define the new coordinates: Then, system (2.3) can be written as where the term is given by and is now the new control variable defined as
Notice that the above set of transformations are well defined for all . That is, the pendulum moves inside the upper half plane. On the other hand, it is easy to verify that function .
Remark 2.1. We emphasize that model (2.5) was proposed and resolved using the stability analysis tool known as the fixed point controller in combination with a version of the backstepping procedure, by Olfati-Saber . On the other hand, it is our opinion that the main contribution of our control strategy is that we solved the stability problem using the Lyapunov method combined with the theorem of LaSalle.
We can now formulate our control problem.
Given the cart-pendulum system described as in (2.5), we want to bring the pendulum to the upright position and, simultaneously, bring the cart to the origin or any other fixed desired position.
We introduce the following useful definitions, used in forthcoming developments.
Definition 2.2. Let . The classical linear saturation function is defined as where constant is strictly positive.
Definition 2.3. By a sigmoidal function , we mean a smooth function that is bounded, strictly increasing with the property that ; and , for all .
We now proceed to propose our control strategy.
3. Regulation of the Cart-Pendulum System
3.1. Linear Transformation
Inspired by what is presented in , we first introduce the following linear transformation: which leads to, when applied to (2.5), In the following section we split the new control input into two control actions. One part of this control, namely, , brings both the state and the state to a small compact set defining the closed-loop stability domain and, consequently, renders the nonlinear terms of system (3.2) to an arbitrarily small vicinity of zero. Simultaneously to the linear control action, a bounded quasilinear control action, namely, , stabilizes the missing states and .
3.2. Stabilization of the States and
In order to guarantee that the states and are bounded, we split as where , with . Thus, after substituting (3.3) into (3.2), we have that We emphasize that, if , with small enough, then the states and converge toward the vicinity (for simplicity .), where the bound can be made small by minimizing and, consequently, all the nonlinear terms in (3.4) can be arbitrarily approximated to zero. This results in the dominance of the linear dynamics over their, respectively, nonlinear dynamics. That is, signal is then selected as a bounding function, where is given latter in our discussion and is a small positive constant fixed as needed.
In order to analyze the boundedness of both the state and the state , we consider the definite positive function given by Differentiating (3.5) and taking into account (3.4) we have that Now, given the assumption we have that is in fact bounded as follows: (Evidently, when , the subsystem (3.4) converges asymptotically to zero, because its time derivative, given by (3.6), is equal to , semidefinite negative and, applying the theorem of LaSalle, we have that, both variables, and , converge to zero.) Note that if , with an arbitrarily small number, then, from (3.7), we have that . Consequently, there exists a finite time after which we have that is is bounded. Even more, because is a nonincreasing function, provided that , then the state is also bounded. However, in order to compute the corresponding bound of , we propose a positive function, , whose time derivative along the of the dynamics of the state , in (3.4), satisfies It implies that there exists a finite time such that
Remark 3.1. Summarizing, after some period of time , we have that variables, and , are bounded. Besides, it holds that
3.3. Stabilization of Both the State and the State
In order to stabilize the missing state variables we propose the bounding control action as follows: with the control parameter being characterized by . Observe that in order to simplify the corresponding bounded analysis, we used the linear saturation function . However, this function can be substituted by any other nonlinear saturation function. On the other hand, we have that Then, after substituting the above controller into the second equation of (3.4) we get We introduce now the following positive definite function: in order to verify the boundedness of the state . Differentiating and using (3.14) we have that Selecting we can assure that , if . Therefore, there is a finite time such that We emphasize that the restriction can be always satisfied. Indeed, from the definition of given in (3.13) we evidently have (just to illustrate how this inequality holds take, for instance, , and , for a given ). Finally, once the state is confined to move inside the region defined by , the linear saturation function no longer acts over this state; that is, . Therefore, turns out to be In the same way, after , we can claim that the model in (3.4) leads to Now, in instead of showing that the state is bounded, we show in what follows that, after a finite period of time , all the states asymptotically converge to zero. Let us first introduce the following useful lemma.
Lemma 3.2. Consider the first two equations in (3.20) and the following positive definite function: After a finite period of time , the following inequality holds: where .
The proof of this Lemma is given in Appendix A.
3.4. Asymptotic Convergence to the Origin of the Whole State
From the above discussion we conclude that, after the finite time , the states , , and are bounded in some compact set, which defines the closed-loop stability domain. To guarantee that all the states asymptotically converge to zero we propose the following candidate Lyapunov function: where and were previously defined in (3.5) and (3.21), respectively. Since functions and are strictly positive definite function, with their respective arguments, we can claim that qualifies as a candidate Lyapunov function. So, in case that we have that the time derivative of satisfies the following inequality (see Lemma 3.2 and Remark 3.1): Selecting we have that (e.g., , for a given .) From Lyapunov's direct method we ensure the stability of the whole state in the Lyapunov sense. In order to prove now asymptotic stability, we use the well-known LaSalle's theorem . In the region defined as we have that , and . Thus, in the set , we also have . Now, from the four chained integrators model (3.4) we have , in the set . Therefore, the largest invariant set is given by . Thus, according to LaSalle's theorem all the trajectories of system (3.20) asymptotically converge towards to the largest invariant set .
We summarize our previous discussion with the next proposition, which corresponds to our main result.
Proposition 3.3. Consider the closed-loop cart-pendulum system as described by model (2.5) with: Then the closed-loop system is globally asymptotically stable and locally exponentially stable, provided that the parameters and satisfy the inequalities
Remark 3.4. In order to simplify as much as possible the previous stability analysis we used the proposed , which is formed using a linear saturation function. However, nonlinear saturation functions can also be used, as we showed in Appendix B.
3.5. An Illustrative Example
In order to show the effectiveness of the proposed nonlinear control strategy we developed an experiment that allows us to compare the behavior of the strategy, in the presence and the absence of a damping force. The damping force was added to the left-hand side of the first equation in (2.1), as . We chose the controller parameter values to be . As far as the initial conditions are concerned we take . Figure 2 shows the results coming out from the numerical simulations. As can be seen we have, as expected, a quite effective performance for the controller. Even when the sustained damping force is present, the closed-loop response is still quite well. Observe that in order to compensate the damping force effect the cart has to make larger displacements. This numerical result is of importance, because, contrarily to our strategy, many others developments for the same porpoises are very sensitive to the presence of an unmatched damping force in the nonactuated coordinate.
4. Concluding Remarks
In this paper a new control strategy is proposed in order to solve the well-known cart-pendulum regulation problem, assuming that the pendulum is initialized in the upper half plane. The control strategy used a control-oriented model of the considered system (a model consisted of a nonlinearly perturbed chain of four integrators), previously introduced in . The model choice lets us design a simple stabilizer consisted of two parts. The first part characterizes a linear controller, devoted to bring the nonactuated coordinate (i.e., both the angular position and the angular velocity) near to the unstable vertical position and keep it inside of a small vicinity which defines the closed-loop stability domain. The other part is a bounded controller which, in conjunction with the linear part, ensures that the closed-loop whole state asymptotically converges to the origin. The combined control law ensures then the regulation of the system. Our stability analysis was carried out using standard arguments from linear systems theory in conjunction with the traditional Lyapunov method and the famous LaSalle's theorem. We strongly believe that many other nonlinear underactuated dynamical systems can be stabilized using our simple control approach. We must point out that a main advantage of this work is that we did not need to solve PDE, nonlinear differential equations, and nested saturation functions. Finally, the numerical experiments carried out with an academic example illustrated how effective is our control strategy when an unknown damping force is present.
A. Proof of Lemma 3.2
Proof. We must remark that the time derivative of (3.21) around the trajectories defined by the first two equations of (3.20) is given by Then after we must have . Therefore, can be expressed as and evidently can be bounded by Substituting (A.2) and (A.3) into (A.1), we get inequality (3.22), which concludes this proof.
B. Proof of Remark 3.4
Proof. For simplicity we use as nonlinear saturation function the following: . That is, is formed as:
Selecting and the bound for as discussed in Section 3 and taking into account expressions (3.3) and (3.13), we guarantee that there exists a time , such that
Therefore, the first and the second equations of (3.20) become
To analyze the boundedness of , we use the positive definite function , whose time derivative can be bounded as
Hence, selecting and , such that
therefore, there is a time such that
Indeed, it follows because, if , then . Notice that the state can be confined to move inside of a compact set relying on the bound . Note that this bound can be manipulated, almost, as desired. Then we can select, for instance, to make
hold. Simple geometric arguments can be applied to prove this inequality. Until now we have only provided sufficient conditions to guarantee that , , and are bounded (with the corresponding bounds being freely fixed). Now we are in conditions to proof that the whole state asymptotically converges to the origin.
We first choose a positive function (similar to the one used in Lemma 3.2) defined as Differentiating the above equation with respect to (B.3), we have, after using simple algebra as in Lemma 3.2, the following inequality: Notice that can be expressed as Now, under the assumption , selecting , and taking into account (B.7) in the above expression, we have Thus, can be bounded as We built now the candidate Lyapunov function , with defined as in (3.5). Then, using some simple algebra and Remark 3.1 it is easy to show that can be bounded as Selecting (as in Proposition 3.3), we have that is semidefinite negative. Hence all the states are bounded. Finally, invoking the LaSalle's theorem and following standard arguments, we can show that the whole state of the closed-loop system asymptotically converges to the origin. This concludes the proof.
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