Orbital Stabilization of Nonlinear Systems via the Immersion and Invariance Technique

Transcription

1 Orbital Stabilization of Nonlinear Systems via the Immersion and Invariance Technique Romeo Ortega, Bowen Yi, Jose Guadalupe Romero and Alessandro Astolfi October, 18 arxiv:181.61v1 [cs.sy] 1 Oct 18 Abstract Immersion and Invariance is a technique for the design of stabilizing and adaptive controllers and state observers for nonlinear systems recently proposed in the literature. In all these applications the problem is translated into stabilization of equilibrium points. Motivated by some modern applications we show in this paper that the technique can also be used to treat the problem of orbital stabilization, where the final objective is to generate periodic solutions that are orbitally attractive. The feasibility of our result is illustrated with some classical mechanical engineering examples. 1 Introduction To solve the problems of designing stabilizing and adaptive controllers and state observers for nonlinear systems a new technique, called Immersion and Invariance (I&I), was recently proposed in [4, 5]. The first step in I&I is the definition of a target dynamics, which is a lower dimensional system that captures the desired behavior that is to be imposed to the closed-loop system. In the second step of the design we define an invariant manifold in the state space of the system, such that the restriction of the system dynamics to this manifold is precisely the target dynamics. The design is completed defining a control law that renders this manifold attractive. While the second step of the design involves the solution of a partial differential equation (PDE) corresponding to the Francis-Byrnes-Isidori (FBI) equations [7] the third step is a stabilization problem where it is desired to drive to zero some states, i.e., the offthe-manifold coordinates, while preserving bounded the remaining ones. As shown in [], this latter step can also be translated into a contraction problem. In all the examples mentioned above we deal with the problem of stabilization of equilibrium points the desired equilibrium for the system in the stabilization and adaptive control scenarios, or the zero equilibrium for the state observation error in observer design. In some modern applications for example, walking robots, DC-to-AC power converters, electric motors and oscillation mechanisms in biology the final objective is to induce a periodic orbit to the system. The main objective of this paper is to show that the I&I technique can also be applied to solve this new problem of orbital stabilization, that is, the generation of orbitally attractive periodic solutions for the system. The only modification done to the technique is in the definition of the target dynamics that, instead of having an asymptotically stable equilibrium as before, now should be chosen possessing orbitally attractive periodic orbits. R. Ortega is with LS, CNRS-CentraleSupelec, France. B. Yi is with Department of Automation, Shanghai Jiao Tong University, China. J. G. Romero is with Departamento Académico de Sistemas Digitales, ITAM, Mexico. A. Astolfi is with Department of Electrical and Electronic Engineering, Imperial College London, UK, and also with Department of Civil Engineering and Computer Science Engineering, University of Rome Tor Vergata, Italy. 1

2 The problem of designing controllers to ensure orbital stabilization has been studied in the literature for various applications and with different approaches. For mechanical systems of co-dimension one, the virtual holonomic constraints (VHC) method has been studied in the last two decades [1, 15, 1]. As explained in Remark 4, this technique can be viewed as a particular case of the I&I approach proposed here. Starting with the pioneering works of [9, 1, 17], orbital stabilization via energy regulation has been intensively studied, mainly for pendular systems, where the basic idea is to pump energy into the system to swing-up the pendulum. Such an idea is further elaborated in [6] as the pumping-and-damping method for the stabilization of the up-right equilibrium of pendular systems, with an almost global region of attraction. See also [3, 8] for more general cases, and [] for an interesting connection with chaos. In [18], it is proposed to construct passive oscillators for Lure dynamical systems using sign-indefinite feedback static mappings, which is clearly related with the pumingand-damping method of [6]. A unified treatment of many of these methods has recently been reported in []. The remainder of the paper is organized as follows. In Section we give the problem formulation and present our main result. Section 3 presents some examples, including a simple linear time-invariant (LTI) system and two classical mechanical systems problems widely studied in the literature. The paper is wrapped-up with concluding remarks in Section 4. Notation. I n is the n n identity matrix. For x R n, we denote the Euclidean norm x := x x. All mappings are assumed smooth. Given a function f : R n R we define the ( ) f. differential operator f := x Problem Formulation and Main Result We are interested in this paper in the generation of attractive periodic solutions for the system ẋ = f(x) + g(x)u, (1) with state x R n, input u R m, with m < n, and g(x) full rank. More precisely, we want to define a mapping v : R n R m such that the closed-loop system ẋ = f(x) + g(x)v(x) =: F (x), has a periodic solution X : R + R n that is orbitally attractive [Definition 8.][11]. That is, Ẋ(t) = F (X(t)) X(t) = X(t + T ), t, and the set defined by its associated closed orbit {x R n x = X(t), t T }, is attractive. The main result of the paper is given in the proposition below. Proposition 1. Consider the system (1). Assume we can find mappings α : R p R p, π : R p R n, φ : R n R n p, v : R n R n p R m with p < n, such that the following assumptions hold.

3 A1 (Target oscillator) The dynamical system ξ = α(ξ) () has non-trivial, periodic solutions ξ (t) = ξ (t + T ), t, which are parameterized by the initial conditions ξ(). A (Immersion condition) For all ξ R p, [ ] g (π(ξ)) f(π(ξ)) π (ξ)α(ξ) =, (3) where g : R n R n m is a full-rank left-annihilator of g(x). A3 (Implicit manifold) The following set identity holds M := {x R n φ(x) = } = {x R n x = π(ξ), ξ R p }. (4) A4 (Attractivity and boundedness) All the trajectories of the system ż = φ (x)[f(x) + g(x)v(x, z)] ẋ = f(x) + g(x)v(x, z) (5) with the initial condition z() = φ(x()) and the constraint v(π(ξ), ) = c(π(ξ)), (6) where { } c(π(ξ)) := [g (π(ξ))g(π(ξ))] 1 g (π(ξ)) π (ξ)α(ξ) f(π(ξ)), (7) are bounded and satisfy Then, the system lim z(t) =, (8) t ẋ = f(x) + g(x)v(x, φ(x)) (9) ensures the periodic solution x (t) = π(ξ (t)) is orbitally attractive. Proof. From (5) with z() = φ(x()) we have that z(t) = φ(x(t)). Replacing in (9), and invoking the boudnedness assumption in A4 ensures x(t) L. Furthermore, since z(t) we conclude that the set M is attractive. Now, (), (3) and (7) imply ẋ x=π(ξ),u=c(π(ξ)) = π, consequently the set M is invariant. implications The proof is concluded with the following chain of Ω asymptotically stable dist{ξ(t), Ω} dist{x(t), M }, where we defined the attractive set M := {x R n x = π(ξ), ξ Ω}. 3

4 Remark 1. It is important to underscore that the only modification introduced to the main stabilization result of I&I, that is, [Theorem.1][5], is in the definition of the target dynamics in A1. Instead of having an asymptotically stable equilibrium as before, now it is chosen possessing orbitally attractive periodic orbits. Remark. Ideally, we would fix a desired periodic trajectory x (t) = x (t + T ) and then impose on the mapping π the additional constraint that ξ (t) = π I (x (t)), where π I : R n R p is a left inverse of π, that is, π I (π(ξ)) = ξ. But this a daunting task even when the desired trajectory is imposed only on some of the state coordinates. Instead, we will select the target dynamics that has some periodic orbits, and fix some of the components of the mapping π( ) to ensure that the coordinates of interest will have the same periodic orbit. Notice also that Proposition 1 does not claim that x will converge to a particular periodic orbit π(ξ ), but only to (a π-mapped) one of the family of periodic orbits of the target dynamics. Remark 3. Notice that, as indicated in [], the necessary constraint condition (6) is absent in [Theorem.1][5]. Also, to reduce the number of mappings to be found, we have expressed the FBI equations (3) projecting them into the null space of the input matrix g(x). As shown in [Propositions and 3][], the stability condition A4 can be replaced by a contraction condition. Remark 4. The VHC method of [1, 15] is an alternative technique to induce periodic orbits, which can be viewed as a particular case of the I&I design proposed here in the following sense. First of all, in contrast to our design that is applicable to arbitrary nonlinear systems of the form (1), the VHC method has been developed only for co-dimension one mechanical systems with N degrees of freedom. See [16] for a recent extension. Second, in VHC the manifold to be rendered invariant, which is fixed a priori, is of the particular form {(q, q) R N R N q 1 = ψ 1 (ξ), q = ψ (ξ),..., q N = ξ, ξ R}. with q the generalized coordinates. Therefore, their choice of target dynamics, which corresponds to the zero-dynamics of the system with output q ψ(q 1 ), is also restricted. 1 Thirdly, with the notable exception of [13], attention has been centered only on rendering the manifold invariant, without addressing the issue of its attractivity, which is the main source of difficulty in I&I. 3 Examples In this section we present three examples of application of Proposition 1. To illustrate the design procedure, we work out first a rather trivial LTI example. Then, we present two classical mechanical systems problems, which have been widely studied in the control literature. 3.1 LTI mechanical system Consider the LTI system ẋ a = x b ẋ b = P x a Rx b + u, x a, x b, u R, R, P R. The control objective is to induce an oscillation of unitary period to the component x a of the state. Towards this end, we will follow step-by-step, the procedure proposed in Proposition 1. 1 See point 6 of [Section.1][5] for a discussion on the connection between zero-dynamics and I&I. 4

5 For Assumption A1, [ we pick ] p = and define the target dynamics as the linear oscillator 1 ξ = Jξ, where J :=. Clearly, 1 [ ] ξ(t) = e Jt cos t sin t ξ() = ξ(). sin t cos t It is easy to verify that the FBI equations (3) of Assumption A are satisfied selecting [ ] I π(ξ) = T ξ, T := J c(π(ξ)) = Kπ(ξ), K := [ P R J ]. Also, it is clear that the condition (4) of Assumption A3 holds with the mapping φ(x) = x b + Jx a. Finally, Assumption A4 is completed choosing v(x, z) = P x a + (R J)x b z, which satisfies the boundary constraint (6) and yields ż = z ẋ a = x b ẋ b = Jx b z. Hence, x L and z(t) ensuring that x will converge to (a π-mapped) element of the family of periodic orbits of the target dynamics. To verify the validity of the claim of the proposition, let us apply the control u = v(x, φ(x)) = (P J)x a + (R J I )x b, yielding the closed-loop system ẋ = A cl x, where [ ] I A cl :=, J (J + I ) whose eigenvalues are {i, i, 1, 1}. The periodic function [ ] I X(t) := π(ξ(t)) = T ξ(t) = e Jt ξ() J satisfies Ẋ(t) = A clx(t), hence is a solution of the closed-loop system. 3. Inertia Wheel Pendulum Our next example is the inertia wheel pendulum (IWP) shown in Fig. 1. After a change of coordinates and a scaling of the input, the dynamic equations of the IWP are given by ẋ 1 = x 3 ẋ = x 4 ẋ 3 = m sin(x 1 ) bu ẋ 4 = u, (1) where m, b > and x {S S R R}, with S the unit circle. The control objective is to lift the IWP from the hanging position and to induce an oscillation of the link with a center at the upward position x 1 =. All the details of the model can be found in [14]. 5

6 x u x 1 Figure 1: Inertia wheel pendulum 3..1 I&I controller design We propose a simple pendulum behavior for the target dynamics, i.e., p =, and ξ 1 = ξ ξ = a sin(ξ 1 ) with a a constant to be defined. The pendulum has a center at the downright equilibrium if a > or at the upright one if a <. Consequently, it admits periodic orbits defined by the level sets of the total energy function H ξ (ξ) := 1 ξ a cos(ξ 1 ), verifying Assumption A1. Now, motivated by the structure of (1), we propose the mapping π(ξ) := π 1 (ξ 1 ) π (ξ 1 ) π 1 (ξ 1)ξ π (ξ 1)ξ (11) with π i ( ), i = 1,, functions to be defined. We note that the first and second elements of the FBI equations (3) are satisfied by construction. Consider the simple choice π 1 (ξ) = ξ 1, π (ξ 1 ) = kξ 1, with k a constant to be defined. In this case, we get a simple linear mapping 1 π(ξ) = k 1 ξ =: T ξ. (1) k The implicit manifold description of Assumption A3 is satisfied with the linear mapping [ ] k 1 φ(x) = x. (13) k 1 6

7 After some simple calculations, we see that the remaining two FBI equations are solved, for any k 1 b, with the choice a := m 1 + bk, (14) and the control c(π(ξ)) = ak sin(ξ 1 ). (15) To complete our design it only remains to verify Assumption A4 related to the auxiliary system (5). First, we compute the dynamics of the off-the-manifold coordinate z = φ(x) in closed-loop with the control u = v(x, z) to get Choose the control law ż 1 = z ż = km sin(x 1 ) + (1 + kb)v(x, z). v(x, z) = kb [ γ 1z γ z 1 + km sin(x 1 )], γ i >, i = 1,, which, considering (14) and (15), satisfies the constraint (6). It yields the closed loop dynamics ż 1 = z ż = γ 1 z γ z 1 ẋ 1 = x 3 ẋ = x 4 ẋ 3 = a sin(x 1 ) + ε t ẋ 4 = ak sin(x 1 ) + ε t, where ε t are exponentially decaying terms stemming from the z-dynamics, which clearly verifies z(t) exponentially fast. Now, since x 1 and x leave on the unit circle, and the control v(x, z) is a function of sin(x 1 ) these two states are bounded. To complete the proof of boundedness of x, we recall the identity [ ] kx1 + x z =, kx 3 + x 4 and consider the change of coordinates x (x 1, x 3, z 1, z ), showing that we only need to check the boundedness of x 3. Towards this end, we have the following lemma whose proof, to enhance readability, is given in Appendix A. Lemma 1. Consider the nonlinear time-varying system ẋ 1 = x 3 ẋ 3 = a sin (x 1 ) + ε t. (16) with (x 1, x 3 ) S R, where ε t satisfies ε t (t) l 1 e l t for some l 1, l >. Then, x 3 (t) is bounded for t [, ). (17) 7

8 Finally, as the unperturbed disk dynamics is given by the pendulum equation ẍ 1 + a sin(x 1 ) =, it has a center at the upright equilibrium if a >, or at the downright one if a <. Notice from Fig. 1 that, unlike the classical pendulum equations, the upright equilibrium corresponds to x 1 =. Since the desired objective is to oscillate the link in the upper half plane we impose a >, which translates into the constraint for the free gain k. k < 1 b, (18) Remark 5. Lemma 1 proves that the trajectories of an undamped pendulum are bounded, in spite of the presence of an exponentially decaying term perturbing its velocity. In spite of the simplicity of the statement, and its obvious practical interest, we have not been able to find a proof of this fact in the literature. Hence, the result is of interest on its own. 3.. Simulation results In this subsection we present some simulations of the IWP (1), with parameters m = 1.96, b = 1, in closed-loop with the proposed controller v(x, φ(x)) = kb [ γ 1( kx 3 + x 4 ) γ ( kx 1 + x ) + km sin(x 1 )], with γ 1, γ > and k verifying the constraint (18), which ensures the link oscillations are in the upper half plane. We concentrate our attention on the link, since the disk has a similar behavior. In Fig. we show a plot of x 1 vs x 3 for a =.138, (that is, k = 1.6), starting with the link hanging, at x() = [18, 6,, ], and lifting it to oscillate in the upper half plane. Second, we illustrate the effect of the parameter k. In Fig. 3 we show the transient behavior of x 1 and x 3 for the values of k { 1.4, 1.6, 1.8,.} and the initial condition x() = [135, 6,, ]. Third, the effect of the initial conditions is shown in Fig. 4, where we used the following values for the link position x 1 () {3, 6, 9, 1, 15 } and retained x () = 6, x 3 () = x 4 () =, with the same value of a =.138. As expected from the analysis of the pendulum dynamics the link oscillates with an amplitude determined by the initial conditions and a frequency increasing when the magnitude of a increases (that is, as k decreases). Finally, to evaluate the effect of the gains γ 1, γ we carry out a simulation with the same initial conditions and gain k but placing the poles of the off-the-manifold coordinate dynamics polynomial s + γ 1 s + γ = (s p), at p {.5, 1.,., 3., 4.}. As shown in Fig. 5, the transient degrades for slower rates of convergence of the off-the-manifold dynamics as expected. An animation of the system behavior may be found at youtube.com/watch?v=q5w9kxqbfo&t=9s. 8

9 effect of the gains γ 1, γ we carry out a simulation with the same initial conditions and gain k but placing the poles of the polynomial s + γ 1 s + γ = (s p) at p = [.5, 1.,., 3., 4.]. As shown in Fig. 6, the transient degrades for slower rates of convergence of the off-the-manifold dynamics as expected. An animation of the system behavior may be found at... 5 x 3 (degrees/sec) x 1 (degrees) Figure: : Plot (IWP.) of x 1 vs Plot x 3 of starting x 1 vswith x 3 starting the link hanging with theand linklifting hanging it to oscillate and lifting in the it to upper oscillate half plane. in the upper half plane. We need to show in the animation: C1. Start hanging and oscillate in the upward position. C. Show the effect of IC s, k and γ i 5 x 3 (degrees/sec) x 3 (degrees/sec) k= 1.4 k= 1.6 k= 1.8 k=. 6 1 k=. k= k= 1.6 x 1 (degrees) 15 k= 1.8 k=. k=. Figure Figure 3: 3: (IWP.) Transient Transient behavior behavior of x 1 andof x 31 with anddifferent x 3 withgains different k andgains the same k andinitial the same conditions. initial conditions. x 1 (degrees) Figure 3: Transient behavior 5 of x 1 and x 3 with different gains k and the same initial conditions. x 3 (degrees/sec) x 3 (degrees/sec) x 1 ()=3 x ()=6 1 x 1 ()=9 x ()=1 1 1 ()=15 x 1 ()=3 x ()= x x 1 (degrees) 15 1 ()=9 x 1 ()=1 x Figure 4: Transient behavior of 1 ()=15 and x 3 (use degrees in the axes) with different initial conditions of x 1 () and the same k x 1 (degrees) Figure 4: 4: Transient (IWP.) behavior Transient of xbehavior 1 and x 3 of (use x 1 degrees and x 3 inwith the axes) different withinitial different conditions initial conditions x 1 (). of x 1 () and the same k. 1 rees/sec) x 3 (degrees/sec) p=.5 p=1. p=. 9

10 x 1 () and the same k x 3 (degrees/sec) 1 p=.5 p=1. p=. p=3. p= x 1 (degrees) Figure 5: Figure (IWP.) 5: Transient behavior of of the the state state x 1 and x 1 and x 3 with x 3 with different different gains γgains 1 andγ 1. and γ. Figure 6: Pendulum on a cart system x 1 7 x Figure 6: Pendulum on a cart system 3.3 Cart-pendulum system In this subsection we consider the classical cart-pendulum system depicted in Fig. 6. After a partial feedback linearization stage and normalization this yields the dynamics 3 ẋ 1 = x 3 ẋ = x 4 ẋ 3 = a 1 sin(x 1 ) a cos(x 1 )u ẋ 4 = u, (19) where (x 1, x ) S R are the pendulum angle with the upright vertical and the cart position, respectively, x 3, x 4 R are their corresponding velocities, u R is the input, and a 1 > and a > are some physical parameters. The control objective is to, starting with the link in the upper-half plane, to induce an oscillation of the link with a center at the upward position x 1 =. Notice that, for reasons to be explained below unlike the IWP we do not attempt to lift the pendulum from the hanging position. 3 See [1, 19] for further details. 1

11 3.3.1 Controller design Similarly to the example of Subsection 3., we select a two-dimensional target dynamics, i.e., p =. But in this case, we consider a more general mechanical system of the form ξ 1 = ξ ξ = α (ξ 1 ), () with α ( ) a function to be defined. The system has a total energy function H ξ (ξ) := 1 ξ + U(ξ 1 ), where ξ1 U(ξ 1 ) := α (s)ds is its potential energy. Since the system is undamped, the derivative of its energy function is zero. Consequently, if the potential energy has a minimum at zero, which is implied by the conditions α () = α () <, (1) then the target dynamics () admits periodic orbits defined by the level sets of H ξ (ξ), and verifies Assumption A1. We propose the mapping (11), with π i ( ), i = 1,, functions to be defined. From the third and the fourth element of the FBI equations (3) of Assumption A we see that these functions must satisfy [ ] a 1 sin(π 1 (ξ 1 )) a cos(π 1 (ξ 1 )) π (ξ 1 )ξ + π (ξ 1 )α (ξ 1 ) = π 1(ξ 1 )ξ + π 1(ξ 1 )α (ξ 1 ). () Factoring the elements depending on ξ we conclude that π 1 (ξ 1) = π (ξ 1) =, which implies that these functions should be linear. Therefore, we select the mapping (1), with k a constant to be defined. The implicit manifold description of Assumption A3 is satisfied with the linear mapping (13). Replacing the expressions of (1) in () we obtain that while the control must be chosen as α (ξ 1 ) = a 1 sin(ξ 1 ) 1 + ka cos(ξ 1 ), c(π(ξ)) = ka 1 sin(ξ 1 ) 1 + ka cos(ξ 1 ). (3) To ensure that the potential energy has a minimum at zero we must verify the conditions (1). Hence, we compute α () = a ka, and we must impose on k the constraint 1 a > k. (4) 11

12 With this choice, singularities are avoided in the interval cos(ξ 1 ) > 1 ka, which contains the origin. Notice that the interval above is, unfortunately, strictly contained in the upper-half plane and controller singularities may appear during the transient stymying the possibility to lift the pendulum for the lower-half plane and making local our stability result. To complete our design it only remains to verify Assumption A4 related to the auxiliary system (5). First, we compute the dynamics of the off-the-manifold coordinate z = φ(x) in closed-loop with the control u = v(x, z) to get Choose the control law v(x, z) = ż 1 = z ż = ka 1 sin(x 1 ) + [1 + ka cos(x 1 )]v(x, z) ka cos(x 1 ) [ γ 1z γ z 1 + ka 1 sin(x 1 )], γ i >, i = 1,, (5) which, considering (3), satisfies the constraint (6). It yields the closed loop dynamics ż 1 = z ż = γ 1 z γ z 1 ẋ 1 = x 3 ẋ = x 4 ẋ 3 = a 1 sin(x 1 ) a cos(x 1 )ε t 1 + ka cos(x 1 ) ẋ 4 = a 1 sin(x 1 ) + ε t 1 + ka cos(x 1 ), which verifies z(t). Now, since x 1 leaves on the unit circle, and the control v(x, z) is a function of sin(x 1 ) and cos(x 1 ), this state is bounded. Similarly to the inertia wheel pendulum example, we only need to verify the boundedness of x 3. For, we have the following lemma, whose proof is given in Appendix B. Lemma. Consider the nonlinear time-varying system ẇ 1 = w ẇ = a 1 sin(w 1 ) + ε t 1 + ka cos(w 1 ) (6) with (w 1, w 3 ) S R, a 1, a >, k verifying (4) and ε t satisfying (17). If the initial state satisfies w 1 () ( β, β ) with ( β := arccos 1 ), ka then, there exists l min > such that l l min = w 1 (t) ( β, β ) and w 3 (t) M. 1

13 To complete the proof we notice that, with a suitable definition of ε t, the right-hand side of ẋ 3 may be written in the form (6) and observing that the exponential decay ratio of the z dynamics and consequently the parameter l can be made arbitrarily large with a suitable selection of the gains γ 1 and γ. Remark 6. As indicated in Lemma stability of the closed-loop system is only established for large gains γ i > (i = 1, ), that ensure a sufficiently fast convergence to the invariant manifold. Interestingly, although this requirement is imposed by the stability proof, we have not been able to observe instability even for extremely small gains in our simulations An alternative controller design To enlarge the domain of attraction of the periodic orbit and remove the restriction of using high gains explained in Remark 6, we propose in this subsection an alternative controller design. For, we take the nonlinear mapping π(ξ) = ξ 1 k(ξ 1 ) ξ k (ξ 1 )ξ The implicit manifold description of Assumption A3 is satisfied with the mapping [ ] x k(x φ(x) = 1 ) x 4 k. (7) (x 1 )x 3 Some simple calculations prove that the FBI equations of Assumption A are solved with the following control c(π(ξ)) = k (ξ 1 )ξ + a 1k (ξ 1 ) sin(ξ 1 ) 1 + a k, (8) (ξ 1 ) cos(ξ 1 ) together with the target dynamics. ξ 1 = ξ ξ = ρ(ξ 1 ) + β(ξ 1 )ξ, (9) where a 1 sin(ξ 1 ) ρ(ξ 1 ) := 1 + a k (ξ 1 ) cos(ξ 1 ) β(ξ 1 ) := a k (ξ 1 ) cos(ξ 1 ) 1 + a k (ξ 1 ) cos(ξ 1 ). To enlarge the range of x 1 where singularities are avoided we propose to select k( ), such that the denominator of the control (8) is constant, that is as the solution of the following ordinary differential equation 1 + a k (s) cos(s) = a, (3) with a a constant to be defined. The solution of (3) is given by k(s) = 1 + a ( ) 1 + sin(s) ln + a, (31) a cos(s) 13

14 where we have added a constant a that allows to set the center of the cart at any desired position. Notice that the function k( ) is well-defined in the interval ( π, π ). With this choice of k(ξ 1 ), the functions ρ(ξ 1 ) and β(ξ 1 ) become ρ(ξ 1 ) = a 1 a sin(ξ 1), β(ξ 1 ) = 1 + a a tan(ξ 1 ). (3) Now, the target dynamics (9), is an undamped mechanical system with total energy function H ξ (ξ) = m(ξ 1) ξ + U(ξ 1 ), (33) with the inertia and the potential energy { ξ1 } (1 + a) m(ξ 1 ) := exp tan(s)ds a U(ξ 1 ) := a 1 a ξ1 sin(s)m(s)ds. From (34) we conclude that there exist constants m min and m max such that ( < m min m(s) m max, s π, π ). We proceed now to prove that, with a >, the potential energy has a minimum at zero, ensuring Assumption A1. For, we observe that U (ξ 1 ) < for ξ 1 ( π, ), U () =, and U (ξ 1 ) > for ξ 1 (, π ), thus arg min U(ξ 1) =. ( π, π ) To verify Assumption A4 we define from (7) the off-the-manifold coordinates whose dynamics is given as z 1 = x k(x 1 ) z = x 4 k (x 1 )x 3, ż 1 = z ż = [ 1 + a k (x 1 ) cos(x 1 ) ] u [ k (x 1 )x 3 + a 1 k (x 1 ) sin(x 1 ) ] = au [ k (x 1 )x 3 + a 1 k (x 1 ) sin(x 1 ) ], where we have used (3) to get the second identity. We design the feedback law as v(x, z) = 1 ) (k (x 1 )x 3 + a 1 k (x 1 ) sin(x 1 ) γ 1 z 1 γ z, a which satisfies (6) and ensures z(t) exponentially fast. Similarly to the analysis of the previous subsection, we only need to prove the boundedness of the subsystem x 1, x 3 in closed-loop with the control given above, which is given by (34) ẋ 1 = x 3 ẋ 3 = ρ(x 1 ) β(x 1 )x 3 a a cos(x 1)(γ 1 z 1 + γ z ). (35) 14

15 Computing the derivative of the energy function H ξ (x 1, x 3 ), defined in (33), along the trajectories of (35) we get that Ḣ ξ = m(x 1 )x 3 a a cos(x 1)(γ 1 z 1 + γ z ) m maxa (γ 1 + γ ) x 3 z() exp( l t), x 1 a (, π ). Now,from the fact that U(x 1 ) U() =, x 1 ( π, π ), we obtain the following inequality x 3 m min H ξ (x), from which we obtain the bound where we used the following definition Ḣ ξ l 3 H ξ (x) exp( l t), (36) l 3 := m maxa z() (γ 1 + γ ) a Finally, consider the auxiliary dynamics with p(), whose solution is ṗ = l 3 p exp( l t), m min. p(t) = l 3 l (1 exp( l t)) + p(). Clearly, p(t) is bounded thus, applying the Comparison Lemma [11] to (36), we conclude that H ξ (x(t)), and consequently x 1 and x 3, are bounded. Remark 7. The main advantage of the controller proposed in this subsection is that the pendulum can now move in the whole upper-half plane. Another advantage is that stability is ensured for all gains γ 1, γ > this is in contrast with the controller of the previous subsection as indicated in Remark 6. Of course, the prize that is paid for these goodies is a significant increase in the controller complexity Simulation results In this subsection we first present some simulations of the cart-pendulum system (19) with a 1 = 9.8 and a = 1, in closed-loop with the controller proposed in Subsection 3.3., namely v(x, φ(x)) = ka cos(x 1 ) [ γ 1( kx 3 + x 4 ) γ ( kx 1 + x ) + km sin(x 1 )], with γ 1, γ > and k verifying the constraint (4). 15

16 Figure 7: (Cart pendulum.) Plot of x 1 vs x 3 starting with the link in the upper-half plane and a non-zero velocity Figure 8: (Cart pendulum.) Transient behavior of the state x 1 and x 3 with different gains k and the same initial conditions Figure 9: (Cart pendulum.) Transient behavior of the state x 1 and x 3 with different gains γ 1 and γ and the same initial conditions. 16

17 In Fig. 7 we show a plot of x 1 vs x 3 for k = 4 and γ 1 = γ =, with initial conditions x() = [36,, 18, ]. Notice that a non-zero initial velocity is assumed for the link. The effect of the parameter k is illustrated in Fig. 8, with the values of k { 3, 4, 6} and the same initial condition as before. As shown in the figure, the parameter k affects the period of the oscillation in a direct manner. Fig. 9 shows the effect of the gains γ 1 and γ. We now give the simulation results of the second design for the cart-pendulum system, that is, the controller v(x, φ(x)) = 1 ( ) k (x 1 )x 3 + a 1 k (x 1 ) sin(x 1 ) γ (x k(x 1 )) γ 1 (x 4 k (x 1 )x 3 ), a with k(x 1 ) given by (31). In Fig. 1 we give the plot of x 1 vs x 3 for a =, a = and γ 1 = γ = 1, starting with link closer to the horizontal position and without any initial velocity, i.e., x() = [54, 5,, ]. The effect of the parameter a is illustrated in Fig Figure 1: (Cart pendulum, the second controller.) Plot of x 1 vs x 3 starting with the link in the upper-half plane and zero velocity Figure 11: (Cart pendulum, the second controller.) Transient behavior of the state x 1 and x 3 with different gains a and the same initial conditions. An animation of the system behavior may be found at youtube.com/watch?v=q5w9kxqbfo&t=9s. 17

18 4 Concluding remarks We have shown in this paper that, by selecting the target dynamics in the well-known I&I method [5] to possess periodic orbits instead of an asymptotically stable equilibrium it is possible to solve the task of inducing orbitally attractive oscillations to general nonlinear systems. As usual with the I&I method, a large flexibility exists in the selection of the target dynamics and the definition of the manifold that is rendered attractive and invariant, which can be exploited to simplify the controller design. The result was illustrated with some classical examples of mechanical systems. Current research is under way to develop a more systematic procedure to apply the technique that, at this stage, was done on a case-by-case basis. Towards this end, we plan to consider a more structured class of systems, for instance port-hamiltonian, or a class of physical systems like power converters and electric motors. A Proof of Lemma 1 Define the energy of the unperturbed pendulum (16) whose derivative satisfies and We make the following observations r(x) := 1 x 3 a cos(x 1 ), ṙ = x 3 ε t. (37) r(x) a, x R, (38) r(x(t)) L x 3 (t) L. (39) Thus, we only need to prove that r(x(t)) is bounded.. From the bound ẋ 3 a sin x 1 + ε t a + l 1, we get for any t > thus Recalling (37), it yields x 3 (t) x 3 () x 3 (t) x 3 () t = ẋ 3 (s)ds t ẋ3 (s) ds (a + l 1 )t, x 3 (t) x 3 () + (a + l 1 )t. ṙ x 3 ε t = l 3 e l t + l 4 te l t, with l 3 := l 1 x 3 () and l 4 := l 1 (a + l 1 ). Then, r(x(t)) r(x()) t (l 3 e l s + l 4 se l s )ds = l 3 l ( 1 e l t ) + l 4 l 18 (1 l te l t e l t )

19 We have lim r(x(t)) r(x()) + l 3 + l 4, t l implying r(x(t)) L, which completes the proof. B Proof of Lemma Define an energy-like function H w (w) := 1 w + a 1 ka ln ( 1 + ka cos(w 1 ) ), which is a first integral of the system (6) in the absence of the decaying term. This function is lower bounded as H w (w) H min w := a 1 ka ln( 1 ka ), for w 1 ( β, β ), We note that, in view of the constraint (4), 1 ka >, hence Hw min Moreover, lim H w (w) = +. w 1 β We also have the following bounds regarding the function H w (w) w l is well-defined. (H w (w) + H min w ) (4) and Clearly, we have 1 + ka cos(w 1 ) { } ka exp H w (w). (41) a 1 w ε t Ḣ w = 1 + ka cos(w 1 ) w 1 + ka cos(w 1 ) l 1 exp( l t) l 1 (H w (w) + Hw min ) exp { ka a 1 H w (w) } exp( l t), where the last inequality has used (4) and (41). For the application of the Comparison Lemma we study the boundedness of the following one-dimensional auxiliary system { ṙ = l 1 (r + Hw min ) exp ka } r exp( l t), (4) a 1 Notice that, [ Hw min, + ) is an invariant set for the differential equation (4), with r = Hw min an equilibrium point. Therefore, we are interested only in the trajectories satisfying r(t) + Hw min >. In which case, ṙ >, and consequently r(t) is a strictly increasing function of time. Define a function F (r) as { F (r) := exp ka } r (r + Hw a min ). 1 19

20 In view of the monotonicity, it is clear that there exists r > such that and F (r ) =, F (r) r > r. Therefore, there are two possible scenarios for system (4): 1) r(t) < r for all t > ; ) there exists a moment t 1 such that r(t) r for all t t 1. For Case 1), the boundedness of r(t) follows immediately. For the second case, the dynamics (4) yields [ { ṙ = l 1 exp ka } ] { r F (r) exp ka } r exp( l t) a 1 a 1 { l 1 exp ka } r exp( l t), t t 1, a 1 where the first identity has used the definition of F (r), and the second inequality has used the fact F (r(t)) > for t t 1 in the second case. For the second case, by applying the Comparison Lemma we construct the second auxiliary system { v = l 1 exp ka } v exp( l t) (43) a 1 with the initial condition v() r. For the system (43), we have v exp { ka } v = l 1 exp( l t) a 1 thus integrating via variable separation we get t { v(s) exp ka } t v(s) ds = l 1 exp( l s)ds a 1 v(t) { } t exp k v(s) dv = l 1 exp( l s)ds, v() with k := k a a 1 >. After some straightforward calculations, we then get exp( k v(t)) exp( k v()) = k l 1 l ( exp( l t) 1 ). (44) According to (44), if ( ) l 1 exp( k v()) + k exp( l t) 1 >, (45) l we have v(t) = 1 k ln [ ( l 1 exp( k v()) + k exp( l t) 1) ]. l

21 Using in (45) the following inequality ( ) l 1 l 1 k < k exp( l t) 1, l l we conclude that, if l > k l 1 exp { k v() } := l, (46) the condition (45) holds for all t >. Implying that the solutions of (43) are bounded. Specifically, 4 lim v(t) = 1 { ln exp ( k v() ) } l 1 k < +. t k l Now, we return to the first auxiliary system (4) combining Case 1), if the l is large enough, we can obtain the boundedenss of r(t) in terms of the Comparison Lemma and the boundedness of v(t) for the auxiliary system (43). Using the Comparison Lemma again and selecting { l min := k l 1 exp k max { r, Hw min + H w (w()) }}, for l > l min, the energy-like function H w (w(t)) for the system (6) is bounded for all t >. Invoking the inequalities (4) and (41), we complete the proof. References [1] J. Acosta, R. Ortega, A. Astolfi and A. Mahindrakar, Interconnection and damping assignment passivity based control of mechanical systems with underactuation degree one, IEEE Trans. Automatic Control, vol. 5, no. 1, pp. 1-18, 5. [] E.A. Androulidakis, A.T. Alexandridis and G.C. Konstantopoulos, Studying complexity for a modified dissipative Hamitonian systems: From Lyapunov stability at the origin to a limit cycle and chaos, IFAC Symposium on System Structure and Control, Grenoble, France, Feb. 4-6, pp , 13. [3] J. Aracil, F. Gordillo and E. Ponce, Stabilization of oscillations through backstepping in high-dimensional systems, IEEE Trans. Automatic Control, vol. 5, pp , 5. [4] A. Astolfi and R. Ortega, Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems, IEEE Trans. Automatic Control, vol. 48, pp , 3. [5] A. Astolfi, D. Karagiannis and R. Ortega, Nonlinear and Adaptive Control with Applications, Springer, London, 7. [6] K.J. Astrom, J. Aracil and F. Gordillo, A family of smooth controllers for swinging up a pendulum, Automatica, vol. 44, pp , 8. [7] C. I. Byrnes, F. D. Priscoli and A. Isidori, Output Regulation of Uncertain Nonlinear Systems, Springer, New York, We would like to point out here that for the auxiliary system (43), if l = l then lim v(t) = ; t and for the case l (, l ) the system (43) has finite escaping time. 1

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