Orbital Stabilization of Nonlinear Systems via the Immersion and Invariance Technique
Romeo Ortega, Bowen Yi, Jose Guadalupe Romero, Alessandro Astolfi
OOrbital Stabilization of Nonlinear Systems via theImmersion and Invariance Technique
Romeo Ortega , Bowen Yi , Jose Guadalupe Romero , and Alessandro Astolfi Laboratoire des Signaux et Syst´emes, CNRS-CentraleSup´elec, France Australian Centre for Field Robotics, The University of Sydney, Australia Departamento Acad´emico de Sistemas Digitales, ITAM, Mexico Department of Electrical and Electronic Engineering, Imperial College London, UK Department of Civil Engineering and Computer Science Engineering, University of Rome “TorVergata”, Italy Department of Control Systems and Informatics, ITMO University, Russia
December 3, 2019
Abstract
Immersion and Invariance is a technique for the design of stabilizing and adaptivecontrollers 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 canalso be used to treat the problem of orbital stabilization , where the final objective is togenerate periodic solutions that are orbitally attractive. The feasibility of our result isillustrated with some classical mechanical engineering examples.
To solve the problems of designing stabilizing and adaptive controllers and state observersfor nonlinear systems a technique, called Immersion and Invariance (I&I), was proposed in[4, 5]. The first step in I&Iis the definition of a target dynamics, which is a lower dimensionalsystem that captures the desired behavior that is to be imposed to the closed-loop system. Inthe second step of the design an invariant manifold in the state space of the system, such thatthe restriction of the system dynamics to this manifold is precisely the target dynamics, isdefined. 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 astabilization problem where it is desired to drive to zero the rest of state, i.e. , the off-the-manifold coordinates, while preserving bounded trajectories. As shown in [21], this latter stepcan also be translated into a contraction problem.In all the examples mentioned above one deals with the problem of stabilization of equilib-rium points—the desired equilibrium for the system in the stabilization and adaptive controlscenarios, or the zero equilibrium for the state estimation error in observer design. In somemodern applications—for example, walking robots, DC-to-AC power converters, electric mo-tors and oscillation mechanisms in biology—the final objective is to induce a periodic orbit .The main objective of this paper is to show that the I&Itechnique can also be applied to1 a r X i v : . [ c s . S Y ] D ec olve this new problem, that is, the generation of attractive periodic solutions. The onlymodification required is in the definition of the target dynamics that, instead of having anasymptotically stable equilibrium, should be chosen with attractive periodic orbits.The problem of designing controllers to ensure orbital stabilization has been studied inthe literature for various applications and with different approaches. For mechanical systemsof co-dimension one, the virtual holonomic constraints (VHC) method has been studied inthe last two decades [13, 16, 22]. As explained in Remark 4, this technique can be viewedas a particular case of the I&Iapproach proposed here. Starting with the pioneering worksof [10, 11, 18], orbital stabilization via energy regulation has been intensively studied, mainlyfor 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-dampingmethod for the stabilization of the up-right equilibrium of pendular systems, yielding analmost globally asymptotically stable equilibrium. See also [3, 8] for more general cases,and [2] for an interesting connection with chaos theory. In [19] the construction of passiveoscillators for Lur’e dynamical systems using “sign-indefinite” feedback static mappings, whichis clearly related with the pumping-and-damping method of [6], has been proposed. A unifiedtreatment of many of these methods has recently been reported in [23, 24].The remainder of the paper is organized as follows. In Section 2 we give the problem for-mulation and present our main result. Section 3 presents some examples, including a simplelinear time-invariant (LTI) system and two models of mechanical system widely studied in theliterature, as well as a power electronics system. The paper is wrapped-up with concludingremarks in Section 4. Notation. I n is the n × n identity matrix. For x ∈ R n , we denote square of the Euclideannorm | x | := x (cid:62) x . All mappings are assumed smooth. Given a function f : R n → R we definethe differential operator ∇ f := (cid:16) ∂f∂x (cid:17) (cid:62) . Given a set A ⊂ R n and a vector x ∈ R n , we denotedist( x, A ) := inf y ∈A | x − y | . We are interested in the generation of attractive periodic solutions for the system˙ x = f ( x ) + g ( x ) u, (1)with state x ( t ) ∈ R n , input u ( t ) ∈ R m , with m < n , and g ( x ) full rank. More precisely, wewant to define a mapping v : R n → R m such that the closed-loop system˙ x = f ( x ) + g ( x ) v ( x ) =: F ( x )has a periodic solution X : R + → R n that is orbitally attractive [Definition 8.2][12]. That is X is such that ˙ X ( t ) = F ( X ( t )) ,X ( t ) = X ( t + T ) , ∀ t ≥ , and the set defined by the closed orbit { x ∈ R n | x = X ( t ) , ≤ t ≤ T } , is attractive and invariant.The main result of the paper is given in the proposition below.2 roposition 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. A1 (Target oscillator) The dynamical system˙ ξ = α ( ξ ) (2)has non-trivial, periodic solutions ξ (cid:63) ( t ) = ξ (cid:63) ( t + T ) , ∀ t ≥
0, which are parameterizedby the initial conditions ξ (0), with ξ ( t ) ∈ R p . A2 (Immersion condition) For all ξ , g ⊥ ( π ( ξ )) (cid:104) f ( π ( ξ )) − ∇ π (cid:62) ( ξ ) α ( ξ ) (cid:105) = 0 , (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 ) = 0 } = { x ∈ R n | x = π ( ξ ) , ξ ∈ R p } . (4) A4 (Attractivity and boundedness) All trajectories of the system˙ z = ∇ φ (cid:62) ( x )[ f ( x ) + g ( x ) v ( x, z )] , ˙ x = f ( x ) + g ( x ) v ( x, z ) , (5)with the initial condition z (0) = φ ( x (0)), z ( t ) ∈ R n − p , and the constraint v ( π ( ξ ) ,
0) = c ( π ( ξ )) , (6)where c ( π ( ξ )) := [ g (cid:62) ( π ( ξ )) g ( π ( ξ ))] − g (cid:62) ( π ( ξ )) (cid:110) ∇ π (cid:62) ( ξ ) α ( ξ ) − f ( π ( ξ )) (cid:111) , (7)are bounded and satisfy lim t →∞ z ( t ) = 0 . (8)Then the system ˙ x = f ( x ) + g ( x ) v ( x, φ ( x )) (9)is such that the periodic solution x (cid:63) ( t ) = π ( ξ (cid:63) ( t )) is orbitally attractive. Proof.
From (5) with z (0) = φ ( x (0)) we have that z ( t ) = φ ( x ( t )) for all t ≥
0. Replacing in(9), and invoking the boudnedness assumption in A4 ensures x ( t ) ∈ L ∞ . Furthermore, sincelim t →∞ z ( t ) = 0 we conclude that the set M is attractive. Now, (2), (3) and (7) imply˙ x | x = π ( ξ ) ,u = c ( π ( ξ )) = ˙ π, consequently the set M is invariant. From A4 we have thatlim t →∞ z ( t ) = 0 ⇒ lim t →∞ dist { x ( t ) , M (cid:63) } = 0 , where we have defined the attractive set M (cid:63) := { x ∈ R n | x = π ( ξ ) , ξ ∈ Ω } , with Ω := { ξ ∈ R p | ξ ( t ) = ξ (cid:63) ( t ) , ≤ t ≤ T } . The orbital attractivity property is thereforeproved. (cid:3)(cid:3)(cid:3) igure 1: Schematic representation of Remark 2.
Remark 1.
It is important to underscore that the only modification introduced to the mainstabilization result of I&I, that is, [Theorem 2.1][5], is in the definition of the target dynamicsin A1 . Instead of having an asymptotically stable equilibrium, now it possesses orbitallyattractive periodic orbits. Remark 2.
Ideally, we would fix a desired periodic trajectory x (cid:63) ( t ) = x (cid:63) ( t + T ) and thenimpose on the mapping π the additional constraint that ξ (cid:63) ( t ) = π I ( x (cid:63) ( t )) for all t ≥
0, where π I : R n → R p is a left inverse of π , that is, π I ( π ( ξ )) = ξ . But this is a daunting task—evenwhen the desired trajectory is imposed only on some of the state coordinates. Instead, weselect target dynamics that has some periodic orbits, and fix some of the components of themapping π ( · ) to ensure that the coordinates of interest have the same periodic orbit. Noticealso that Proposition 1 does not claim that x converges to a particular periodic orbit π ( ξ (cid:63) ),but only to (a π -mapped) one of the family of periodic orbits of the target dynamics, asillustrated in Fig. 1. Remark 3.
As indicated in [21], the constraint condition (6) is absent in [Theorem 2.1][5].Also, to reduce the number of mappings to be found, we have expressed the FBI equation (3)projecting it into the null space of the input matrix g ( x ). As shown in [Propositions 2 and3][21], the stability condition A4 can be replaced by a contraction condition. Remark 4.
The VHC method of [13, 16] is an alternative technique to induce periodic orbits,which can be viewed as a particular case of the I&Idesign proposed here in the followingsense. First of all, in contrast to our design that is applicable to arbitrary nonlinear systemsof the form (1), the VHC method has been developed mainly for co-dimension one mechanicalsystems with N degrees of freedom. However, see [17] for a recent extension. Second, in VHCthe manifold to be rendered invariant, which is fixed a priori , has the particular form { ( q, ˙ q ) ∈ R N × R N | q = ψ ( ξ ) , q = ψ ( ξ ) , . . . , q N = ξ, ξ ∈ R } , with q the generalized coordinates. Therefore, the choice of target dynamics, which corre-sponds to the zero-dynamics of the system with output q − ψ ( q ), is also restricted. Thirdly,with the notable exception of [14], attention has been centered only on rendering the manifoldinvariant, without addressing the issue of its attractivity, which is the main source of difficultyin I&I.
In this section we present four examples of application of Proposition 1. To illustrate thedesign procedure, we work out first a rather trivial LTI example. Then, we discuss the orbital See point 6 of [Section 2.1][5] for a discussion on the connection between zero-dynamics and I&I.
Consider the LTI system ˙ x a = x b ˙ x b = − P x a − Rx b + u, with x a ( t ) ∈ R , x b ( t ) ∈ R , u ( t ) ∈ R , R ∈ R × and P ∈ R × . The control objective is toinduce an oscillation of unitary period to the component x a of the state. Towards this end,we follow step-by-step the procedure proposed in Proposition 1.For Assumption A1 we pick p = 2 and define the target dynamics as the linear oscillator˙ ξ = J ξ , where J := (cid:20) − (cid:21) . Clearly, ξ ( t ) = e Jt ξ (0) = (cid:20) cos t − sin t sin t cos t (cid:21) ξ (0) . It is easy to verify that the FBI equation (3) in Assumption A2 is satisfied selecting π ( ξ ) = T ξ, T := (cid:20) I J (cid:21) c ( π ( ξ )) = Kπ ( ξ ) , K := (cid:2) P R + J (cid:3) . Also, it is clear that the condition (4) in Assumption A3 holds selecting the mapping φ ( x ) = x b − J x a . Finally, Assumption A4 holds choosing v ( x, z ) = P x a + ( R + J ) x b − z, which satisfies the boundary constraint (6) and yields˙ z = − z ˙ x a = x b ˙ x b = J x b − z. Hence, x ∈ L ∞ and lim t →∞ z ( t ) = 0 ensuring that x converges to (a π -mapped) element ofthe family of periodic orbits of the target dynamics.To verify the validity of the claim of the proposition, consider the control u = v ( x, φ ( x )) = ( P − J ) x a + ( R − J − I ) x b , yielding the closed-loop system ˙ x = A cl x , with A cl := (cid:20) I J J − I (cid:21) , the eigenvalues of which are { i, − i, − , − } . The periodic function X ( t ) := π ( ξ ( t )) = T ξ ( t ) = (cid:20) I J (cid:21) e Jt ξ (0)satisfies ˙ X ( t ) = A cl X ( t ), hence it is a solution of the closed-loop system.5 .2 Inertia Wheel Pendulum Our next example is the model of the inertia wheel pendulum (IWP) shown in Fig. 2. Aftera change of coordinates and a scaling of the input, the dynamic equations of the IWP aregiven by ˙ x = x ˙ x = x ˙ x = m sin( x ) − bu ˙ x = u, (10)where m > , b > x ( t ) ∈ S × S × R × R , with S the unit circle. The control objectiveis to lift the IWP from the hanging position and to induce an oscillation of the link with acenter at the upward position x = 0. x x u Figure 2:
Inertia wheel pendulum
We propose a simple undamped pendulum behavior for the target dynamics, i.e. , p = 2, and˙ ξ = ξ ˙ ξ = − a sin( ξ )with a a constant to be defined. The pendulum has a center at the downright equilibrium if a > a <
0. Consequently, it admits periodic orbits—defined by thelevel sets of the total energy function H ξ ( ξ ) := 12 ξ − a cos( ξ ) , verifying Assumption A1 . Now, motivated by the structure of (10), we propose the mapping π ( ξ ) := π ( ξ ) π ( ξ ) π (cid:48) ( ξ ) ξ π (cid:48) ( ξ ) ξ (11) All the details of the model can be found in [15]. π i ( · ) , i = 1 , , functions to be defined. We note that the first and the second componentsof the FBI equation (3) is satisfied by construction.Consider the choice π ( ξ ) = ξ , π ( ξ ) = kξ , with k a constant to be defined. As a result, we get the linear mapping π ( ξ ) = k
00 10 k ξ =: T ξ. (12)The implicit manifold description in Assumption A3 is satisfied selecting the linear mapping φ ( x ) = (cid:20) − k − k (cid:21) x. (13)After some simple calculations we see that the remaining two components of the FBI equationare solved, for any k (cid:54) = − b , with the choice a := − m bk , (14)and the control c ( π ( ξ )) = − ak sin( ξ ) . (15)To complete our design it only remains to verify Assumption A4 related to the auxiliarysystem (5). First, we compute the dynamics of the off-the-manifold coordinate z = φ ( x ) inclosed-loop with the control u = v ( x, z ) to get˙ z = z ˙ z = − km sin( x ) + (1 + kb ) v ( x, z ) . Let now v ( x, z ) = 11 + kb [ − γ z − γ z + km sin( x )] , γ i > , i = 1 , , which, considering (14) and (15), satisfies the constraint (6). It yields the closed loop dynamics˙ z = z ˙ z = − γ z − γ z ˙ x = x ˙ x = x ˙ x = − a sin( x ) + ε t ˙ x = − ak sin( x ) + ε t , where ε t are exponentially decaying terms stemming from the z -dynamics, which clearlyverifies lim t →∞ z ( t ) = 0 exponentially fast. Now, since x and x live in the unit circle, andthe control v ( x, z ) is a function of sin( x ), these two states are bounded. To complete theproof of boundedness of x , we recall the identity z = (cid:20) − kx + x − kx + x (cid:21) , and consider the change of coordinates x (cid:55)→ ( x , x , z , z ), showing that we only need tocheck boundedness of x . Towards this end, we have the following lemma the proof of which,to enhance readability, is given in Appendix A.7 emma 1. Consider the nonlinear time-varying system˙ x = x ˙ x = − a sin ( x ) + ε t . (16)with ( x , x ) ∈ S × R , where ε t satisfies | ε t ( t ) | ≤ (cid:96) e − (cid:96) t , (17)for some (cid:96) > , (cid:96) >
0. Then, x ( t ) is bounded for all t > (cid:3)(cid:3)(cid:3) Finally, as the unperturbed disk dynamics is given by the pendulum equation ¨ x + a sin( x ) = 0, it has a center at the upright equilibrium if a >
0, or at the downright one if a <
0. Note from Fig. 2 that, unlike the classical pendulum equations, the upright equilib-rium corresponds to x = 0. Since the desired objective is to oscillate the link in the upperhalf plane we impose a >
0, which translates into the constraint k < − b , (18)for k . 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 ofthe simplicity of the statement, and its obvious practical interest, we have not been able tofind a proof of this fact in the literature. Hence, the result is of interest on its own.
In this subsection we present some simulations for the IWP model in (10), with parameters m = 1 . b = 10, in closed-loop with the proposed controller v ( x, φ ( x )) = 11 + kb [ − γ ( − kx + x ) − γ ( − kx + x ) + km sin( x )] , with γ > , γ > k verifying the constraint (18), which ensures that the link oscillationsare in the upper half plane. We concentrate our attention on the link, since the disk has asimilar behavior.In Fig. 3 we show a plot of x vs x for a = 0 . k = − . link hanging , at x (0) = [ π, π, , k . In Fig. 4 we show the transient behavior of x and x for k ∈ {− . , − . , − . , − . } and the initial condition x (0) = [ π, π, x (0) ∈ { π, π, π, π } and retained x (0) = π , x (0) = x (0) = 0,with the same value of a = 0 . a increases (that is, as k decreases). Finally, to evaluatethe effect of the gains γ and γ we carry out a simulation with the same initial conditionsand gain k , but placing the poles of the off-the-manifold coordinate dynamics of the roots ofthe polynomial s + γ s + γ = ( s + p ) , with p ∈ { . , . , . , . , . } . As shown in Fig. 6, the transient degrades for slower rates ofconvergence of the off-the-manifold dynamics—as expected.An animation of the system behavior may be found at .8 emark 3. Lemma 1 proves that the trajectories of an undamped pendulum are bounded, in spite ofthe presence of an exponentially decaying term perturbing its velocity. In spite of the simplicity of thestatement, and its obvious practical interest, we have not been able to find a proof of this fact in theliterature. Hence, the result is of interest on its own.
In this subsection we present some simulations of the inertia wheel pendulum (10) in closed-loop with theproposed controller v ( x, φ ( x )) = 11 + kb [ − γ ( − kx + x ) − γ ( − kx + x ) + km sin( x )] , γ i > , i = 1 , , for the system (10) with the parameters m = 1 .
962 and b = 10 and k verifying the constraint (18), whichensure the link oscillations are in the upper half plane. We concentrate our attention on the link, since thedisk has a similar behavior.In Fig. 2 we show a plot of x vs x for a = 0 . k = − . i.e. , x (0) = [180 , , , k . In Fig. 3 we show the transient behavior of x and x forthe values of k = [ − . , − . , − . , − . , − .
0] and the initial condition x (0) = [135 , , ,
0] . Third,the effect of the initial conditions is shown in Fig. 4, where we used the initial conditions x (0) =[30 , , , , ] , x (0) = [60 , ,
0] and kept the same value of a = 0 . a increases ( k decreases). Finally, to evaluate theeffect of the gains γ , γ we carry out a simulation with the same initial conditions and gain k but placingthe poles of the polynomial s + γ s + γ = ( s − p ) at p = [0 . , . , . , . , . −100 −50 0 50 100 150 200−250−200−150−100−50050 x (degrees) x ( deg r ee s / s e c ) Figure 2:
Plot of x vs x starting with the link hanging and lifting it to oscillate in the upper half plane.We need to show in the animation: C1.
Start hanging and oscillate in the upward position.
C2.
Show the effect of IC’s, k and γ i Figure 3: (IWP.) Plot of x vs x starting with the link hanging and lifting it to oscillate inthe upper half plane. −100 −50 0 50 100 150−200−150−100−50050 x (degrees) x ( deg r ee s / s e c ) k=−1.4k=−1.6k=−1.8k=−2.0k=−2.2 Figure 3:
Transient behavior of x and x with different gains k and the same initial conditions. −100 −50 0 50 100 150−200−150−100−50050 x (degrees) x ( deg r ee s / s e c ) x (0)=30 x (0)=60 x (0)=90 x (0)=120 x (0)=150 Figure 4:
Transient behavior of x and x (use degrees in the axes) with different initial conditions of x (0) and the same k . −150 −100 −50 0 50 100 150−300−200−1000100 x (degrees) x ( deg r ee s / s e c ) p=0.5p=1.0p=2.0p=3.0p=4.0 Figure 5:
Transient behavior of the state x and x with different gains γ and γ . Figure 6:
Pendulum on a cart system7
Figure 4: (IWP.) Transient behavior of x and x with different gains k and the same initialconditions. −100 −50 0 50 100 150−200−150−100−50050 x (degrees) x ( deg r ee s / s e c ) k=−1.4k=−1.6k=−1.8k=−2.0k=−2.2 Figure 3:
Transient behavior of x and x with different gains k and the same initial conditions. −100 −50 0 50 100 150−200−150−100−50050 x (degrees) x ( deg r ee s / s e c ) x (0)=30 x (0)=60 x (0)=90 x (0)=120 x (0)=150 Figure 4:
Transient behavior of x and x (use degrees in the axes) with different initial conditions of x (0) and the same k . −150 −100 −50 0 50 100 150−300−200−1000100 x (degrees) x ( deg r ee s / s e c ) p=0.5p=1.0p=2.0p=3.0p=4.0 Figure 5:
Transient behavior of the state x and x with different gains γ and γ . Figure 6:
Pendulum on a cart system7
Figure 5: (IWP.) Transient behavior of x and x with different initial conditions x (0).9
100 −50 0 50 100 150−200−150−100−50050 x (degrees) x ( deg r ee s / s e c ) k=−1.4k=−1.6k=−1.8k=−2.0k=−2.2 Figure 3:
Transient behavior of x and x with different gains k and the same initial conditions. −100 −50 0 50 100 150−200−150−100−50050 x (degrees) x ( deg r ee s / s e c ) x (0)=30 x (0)=60 x (0)=90 x (0)=120 x (0)=150 Figure 4:
Transient behavior of x and x (use degrees in the axes) with different initial conditions of x (0) and the same k . −150 −100 −50 0 50 100 150−300−200−1000100 x (degrees) x ( deg r ee s / s e c ) p=0.5p=1.0p=2.0p=3.0p=4.0 Figure 5:
Transient behavior of the state x and x with different gains γ and γ . Figure 6:
Pendulum on a cart system7
Figure 6: (IWP.) Transient behavior of the state x and x with different gains γ and γ . x x Figure 7:
Pendulum on a cart systemIn this subsection we consider the model of a cart-pendulum system as depicted in Fig. 7.After a partial feedback linearization and normalization of the dynamical model, we obtainthe dynamics ˙ x = x ˙ x = x ˙ x = a sin( x ) − a cos( x ) u ˙ x = u, (19)where ( x ( t ) , x ( t )) ∈ S × R are the pendulum angle with the upright vertical and the cartposition, respectively, x ( t ) ∈ R , x ( t ) ∈ R are their corresponding velocities, u ( t ) ∈ R isthe input, and a > a > upper-half plane , to induce an oscillation of the link with a center at theupward position x = 0. Note that, for reasons to be explained below—unlike the IWP—wedo not attempt to lift the pendulum from the hanging position. See [1, 20] for further detail. .3.1 Controller design Similarly to the example in Subsection 3.2, we select a two-dimensional target dynamics, i.e. , p = 2. In this case we consider a more general mechanical system of the form˙ ξ = ξ ˙ ξ = α ( ξ ) , (20)with α ( · ) a function to be defined. The system has a total energy function H ξ ( ξ ) := 12 ξ + U ( ξ ) , where U ( ξ ) := − (cid:90) ξ α ( s ) ds is its potential energy. Since the system is undamped, the derivative of its energy function iszero. Consequently, if the potential energy has a minimum at zero, which is implied by theconditions α (0) = 0 α (cid:48) (0) < , (21)then the target dynamics (20) admits periodic orbits—defined by the level sets of H ξ ( ξ ), andverifies Assumption A1 .We propose the mapping (11), with π i ( · ) , i = 1 , , functions to be defined. From thethird and the fourth components of the FBI equation (3) of Assumption A2 we see that thesefunctions must satisfy a sin( π ( ξ )) − a cos( π ( ξ )) (cid:20) π (cid:48)(cid:48) ( ξ ) ξ + π (cid:48) ( ξ ) α ( ξ ) (cid:21) = π (cid:48)(cid:48) ( ξ ) ξ + π (cid:48) ( ξ ) α ( ξ ) . (22)Factoring the elements depending on ξ we conclude that π (cid:48)(cid:48) ( ξ ) = π (cid:48)(cid:48) ( ξ ) = 0, which impliesthat these functions should be linear . Therefore, we select the mapping (12), with k a constantto be defined. The implicit manifold description of Assumption A3 is satisfied with the linearmapping (13).Replacing the expressions of (12) in (22) we obtain α ( ξ ) = a sin( ξ )1 + ka cos( ξ ) , while the control must be chosen such that c ( π ( ξ )) = ka sin( ξ )1 + ka cos( ξ ) . (23)To ensure that the potential energy has a minimum at zero we must verify the conditions(21). Hence, we compute α (cid:48) (0) = a ka , and we must impose on k the constraint − a > k. (24)11ith this choice, singularities are avoided in the interval cos( ξ ) > − ka , which contains theorigin. Note that the interval above is, unfortunately, strictly contained in the upper-halfplane and controller singularities may appear during the transient—stymying the possibilityto 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 auxiliarysystem (5). First, we compute the dynamics of the off-the-manifold coordinate z = φ ( x ) inclosed-loop with the control u = v ( x, z ) to get˙ z = z ˙ z = − ka sin( x ) + [1 + ka cos( x )] v ( x, z ) . Let the control law be v ( x, z ) = 11 + ka cos( x ) [ − γ z − γ z + ka sin( x )] , γ i > , i = 1 , , (25)which, considering (23), satisfies the constraint (6). It yields the closed loop dynamics˙ z = z ˙ z = − γ z − γ z ˙ x = x ˙ x = x ˙ x = a sin( x ) − a cos( x ) ε t ka cos( x )˙ x = a sin( x ) + ε t ka cos( x ) , which is such that lim t →∞ z ( t ) = 0. Now, since x lives in the unit circle, and the control v ( x, z ) is a function of sin( x ) and cos( x ), this state is bounded. Similarly to the inertia wheelpendulum example, we only need to verify boundedness of x . For, we have the followinglemma, the proof of which is given in Appendix B. Lemma 2.
Consider the nonlinear time-varying system˙ w = w ˙ w = a sin( w ) + ε t ka cos( w ) (26)with ( w ( t ) , w ( t )) ∈ S × R , a , a > k verifying (24) and ε t satisfying (17). If the initialstate satisfies w (0) ∈ ( − β (cid:63) , β (cid:63) )with β (cid:63) := arccos (cid:18) − ka (cid:19) , then, there exists (cid:96) min2 > (cid:96) ≥ (cid:96) min2 = ⇒ w ( t ) ∈ ( − β (cid:63) , β (cid:63) ) and | w ( t ) | ≤ M. (cid:3)(cid:3)(cid:3)
12o complete the proof we note that, with a suitable definition of ε t , the right-hand side of˙ x may be written in the form (26) and observing that the exponential decay ratio of the z dynamics—and consequently the parameter (cid:96) —can be made arbitrarily large with a suitableselection of the gains γ and γ . Remark 6.
As indicated in Lemma 2 stability of the closed-loop system is only establishedfor large gains γ i > i = 1 , To enlarge the domain of attraction of the periodic orbit and remove the restriction of usinghigh gains explained in Remark 6, we propose in this subsection an alternative controllerdesign. For, we take the nonlinear mapping π ( ξ ) = ξ k ( ξ ) ξ k (cid:48) ( ξ ) ξ . The implicit manifold description of Assumption A3 is satisfied selecting the mapping φ ( x ) = (cid:20) x − k ( x ) x − k (cid:48) ( x ) x (cid:21) . (27)Some simple calculations prove that the FBI equations of Assumption A2 are solved selectingthe controller c ( π ( ξ )) = k (cid:48)(cid:48) ( ξ ) ξ + a k (cid:48) ( ξ ) sin( ξ )1 + a k (cid:48) ( ξ ) cos( ξ ) , (28)together with the target dynamics ˙ ξ = ξ , ˙ ξ = ρ ( ξ ) + β ( ξ ) ξ , (29)where ρ ( ξ ) := a sin( ξ )1 + a k (cid:48) ( ξ ) cos( ξ ) ,β ( ξ ) := − a k (cid:48)(cid:48) ( ξ ) cos( ξ )1 + a k (cid:48) ( ξ ) cos( ξ ) . To enlarge the range of x for which singularities are avoided we propose to select k ( · ), suchthat the denominator of the control (28) is constant , that is as the solution of the ordinarydifferential equation 1 + a k (cid:48) ( s ) cos( s ) = − a, (30)with a a constant to be defined. The solution of (30) is given by k ( s ) = − aa ln (cid:18) s )cos( s ) (cid:19) + a , (31)13here we have added a constant a that allows setting the center of the cart at any desiredposition. Notice that the function k ( · ) is well-defined in the interval ( − π , π ). With this choiceof k ( ξ ), the functions ρ ( ξ ) and β ( ξ ) become ρ ( ξ ) = − a a sin( ξ ) , β ( ξ ) = − aa tan( ξ ) . (32)Now, the target dynamics (29), is an undamped mechanical system with total energy function H ξ ( ξ ) = m ( ξ )2 ξ + U ( ξ ) , (33)inertia m ( ξ ) := exp (cid:26) (cid:90) ξ (1 + a ) a tan( s ) ds (cid:27) = (cid:12)(cid:12)(cid:12) cos( ξ ) (cid:12)(cid:12)(cid:12) − a ) (34)and potential energy U ( ξ ) := a a (cid:90) ξ sin( s ) m ( s ) ds = a a + 2 cos( ξ ) − (1+ a ) for cos( ξ ) >
0. From (34) we conclude that there exist constants m min and m max such that0 < m min ≤ m ( s ) ≤ m max , ∀ s ∈ (cid:18) − π , π (cid:19) . It is obvious that, with a >
0, the potential energy U ( ξ ) has a minimum at zero, ensuring As-sumption A1 . To verify Assumption A4 we define from (27) the off-the-manifold coordinates z = x − k ( x ) ,z = x − k (cid:48) ( x ) x , the dynamics of which are˙ z = z ˙ z = (cid:2) a k (cid:48) ( x ) cos( x ) (cid:3) u − (cid:2) k (cid:48)(cid:48) ( x ) x + a k (cid:48) ( x ) sin( x ) (cid:3) = − au − (cid:2) k (cid:48)(cid:48) ( x ) x + a k (cid:48) ( x ) sin( x ) (cid:3) , where we have used (30) to get the second identity. We design the feedback law as v ( x, z ) = − a (cid:18) k (cid:48)(cid:48) ( x ) x + a k (cid:48) ( x ) sin( x ) − γ z − γ z (cid:19) , which satisfies (6) and ensures z ( t ) → x , x in closed-loop with the control given above, which is given by˙ x = x ˙ x = ρ ( x ) + β ( x ) x − a a cos( x )( γ z + γ z ) . (35)Computing the derivative of the energy function H ξ ( x , x ), defined in (33), along the trajec-tories of (35) we get that˙ H ξ = − m ( x ) x a a cos( x )( γ z + γ z ) ≤ m max a ( γ + γ ) a | x || z (0) | exp( − (cid:96) t ) , x ∈ (cid:18) , π (cid:19) . U ( x ) ≥ U (0) = 0 , x ∈ (cid:18) − π , π (cid:19) , we obtain the following inequality | x | ≤ (cid:114) m min H ξ ( x ) , from which we obtain the bound ˙ H ξ ≤ (cid:96) (cid:113) H ξ ( x ) exp( − (cid:96) t ) , (36)where we have used the definition (cid:96) := m max a | z (0) | ( γ + γ ) a (cid:114) m min . Finally, consider the auxiliary dynamics˙ p = (cid:96) √ p exp( − (cid:96) t ) , with p (0) ≥
0, the solution of which is (cid:112) p ( t ) = (cid:96) (cid:96) (1 − exp( − (cid:96) t )) + (cid:112) p (0) . Clearly, p ( t ) is bounded thus, applying the Comparison Lemma [12] to (36), we conclude that H ξ ( x ( t )), and consequently x and x , are bounded. Remark 7.
The main advantage of the controller proposed in this subsection is that thependulum can now move in the whole upper-half plane. Another advantage is that stabilityis ensured for all gains γ , γ > In this subsection we first present some simulations of the cart-pendulum system (19) with a = 9 . a = 1, in closed-loop with the controller proposed in Subsection 3.3.1, namely v ( x, φ ( x )) = 11 + ka cos( x ) [ − γ ( − kx + x ) − γ ( − kx + x ) + km sin( x )] , with γ > , γ > k verifying the constraint (24).In Fig. 8 we show a plot of x vs x for k = − γ = γ = 2, with initial conditions x (0) = [ π, , π, k is illustrated in Fig. 9, with the values of k ∈ {− , − , − } and the sameinitial condition as before. As shown in the figure, the parameter k affects the period of theoscillation in a direct manner. Fig. 10 illustrates the effect of the gains γ and γ .We now give simulation results for the second design for the cart-pendulum system, thatis, the controller v ( x, φ ( x )) = − a (cid:18) k (cid:48)(cid:48) ( x ) x + a k (cid:48) ( x ) sin( x ) − γ ( x − k ( x )) − γ ( x − k (cid:48) ( x ) x ) (cid:19) , with k ( x ) given by (31). Fig. 11 displays the plot of x vs x for a = 2 , a = 0 and γ = γ = 1, starting with the link closer to the horizontal position and without any initialvelocity, i.e. , x (0) = [ π, − π, , a is illustrated in Fig. 12.An animation of the system behavior may be found at .15 -60 -40 -20 0 20 40 60-150-100-50050100150 Figure 8: (Cart pendulum.) Plot of x vs x starting with the link in the upper-half planeand a non-zero velocity. -60 -40 -20 0 20 40 60-150-100-50050100150 Figure 9: (Cart pendulum.) Transient behavior of the state x and x with different gains k and the same initial conditions. -60 -40 -20 0 20 40 60-100-50050100 Figure 10: (Cart pendulum.) Transient behavior of the state x and x with different gains γ and γ and the same initial conditions. 16 -50 0 50-300-200-1000100200300 Figure 11: (Cart pendulum, the second controller.) Plot of x vs x starting with the linkin the upper-half plane and zero velocity. -80 -60 -40 -20 0 20 40 60 80-300-200-1000100200300 Figure 12: (Cart pendulum, the second controller.) Transient behavior of the state x and x with different gains a and the same initial conditions.17 .4 DC-AC Converter The last example is a three-phase DC-AC converter with a pure resistive load—see Fig. 13.The system dynamics can be described as [9]˙ x = − RC x + 1 C x ˙ x = − RC x + 1 C x ˙ x = − L x + EL u ˙ x = − L x + EL u with the inductance L >
0, the capacitance
C >
0, the capacitor voltages ( x , x ), and theinductor currents ( x , x ) in αβ coordinates, where E is the DC source voltage. The controlobjective is to generate a sinusoidal signal in the voltages ( x , x ). Figure 13:
DC-AC converterWe propose a target oscillator with p = 2 described by the equation˙ ξ = (cid:20) − ( | ξ | − A ) ω − ω − ( | ξ | − A ) (cid:21) ξ, (37)with A > , ω >
0. It should be underscored that the target system admits a unique attractive orbit { ξ ∈ R || ξ | = A } according to the analysis in [24]. Such a case is slightly different from A1 where a family of orbits are parameterized by initial conditions.Fixing the first two components of π ( x ) as π ( ξ ) = ξ , π ( ξ ) = ξ and solving the FBI equation, we get π ( ξ ) = 1 R ξ − C ( | ξ | − A ) ξ + Cωξ π ( ξ ) = 1 R ξ − Cωξ − C ( | ξ | − A ) ξ . The implicit manifold assumption A3 can be verified using M := (cid:40) x ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:20) x x (cid:21) − β ( x , x ) = 0 (cid:41) β : R → R defined as β ( x , x ) = col( π ( x , x ) , π ( x , x )). We then choose theoff-the-manifold coordinate z := (cid:20) x x (cid:21) − β ( x , x ) , the dynamics of which is ˙ z = EL v ( x, z ) − L (cid:20) x x (cid:21) − ∇ β (cid:62) ( x , x ) F ( x ) , with the feedback law v ( x, z ) to design and the mapping F ( x ) := (cid:20) − RC x + C x − RC x + C x (cid:21) . Hence, we can construct the feedback law as v ( x, z ) = 1 E col( x , x ) + LE ∇ β (cid:62) ( x , x ) F ( x ) − γz (38)with a tunable parameter γ >
0. The off-the-manifold coordinate z has a globally expo-nentially stable equilibrium of zero, and the target oscillator is almost globally exponentiallyorbitally stable. It is relatively trivial to show the boundedness of the closed-loop dynamicswith the aid of some basic perturbation analysis, unlike the case with the undamped targetoscillators, for instance, the examples of cart-pendulum and inertial wheel pendulum systems.Finally, due to the boundedness of the state there always exist E ∗ and γ ∗ > E > E ∗ and the gain γ ∈ (0 , γ ∗ ), we can guarantee u ( t ) ∈ [ − ,
1] for all t >
0, thus satisfying the physical constraints. The obtained periodic signals, at the steadystate, are x ( t ) = A sin( ωt + φ ) , x ( t ) = A cos( ωt + φ ) , where the phase φ ∈ R depends on initial conditions. We have shown that, by selecting the target dynamics in the well-known I&Imethod [5] topossess periodic orbits—instead of an asymptotically stable equilibrium—it is possible to solvethe task of inducing orbitally attractive oscillations to general nonlinear systems. As usualwith the I&Imethod, a large flexibility exists in the selection of the target dynamics and thedefinition of the manifold that is rendered attractive and invariant, which can be exploited tosimplify the controller design. The result has been illustrated with some classical examples ofmechanical and power electronics systems.tCurrent research is under way to develop a systematic procedure to apply the techniquethat, at this stage, is used on a case-by-case basis. Towards this end, we plan to considera “more structured” class of systems, for instance port-Hamiltonian systems, or a class ofphysically motivated systems like power converters and electric motors.
A . Proof of Lemma 1
Define the energy of the unperturbed pendulum (16) as r ( x ) := 12 x − a cos( x ) , r = x ε t . (39)Note that r ( x ) ≥ − a, ∀ x ∈ R , (40)and | r ( x ( t )) | ∈ L ∞ ⇒ | x ( t ) | ∈ L ∞ . (41)Thus, we only need to prove that r ( x ( t )) is bounded.From the bound | ˙ x | ≤ | a sin x | + | ε t | ≤ a + (cid:96) , we have, for any t ≥ | x ( t ) | − | x (0) | ≤ (cid:12)(cid:12) x ( t ) − x (0) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t ˙ x ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) t (cid:12)(cid:12) ˙ x ( s ) (cid:12)(cid:12) ds ≤ ( a + (cid:96) ) t, thus | x ( t ) | ≤ | x (0) | + ( a + (cid:96) ) t. Recalling (39), we have ˙ r ≤ | x || ε t | = (cid:96) e − (cid:96) t + (cid:96) te − (cid:96) t , with (cid:96) := (cid:96) | x (0) | and (cid:96) := (cid:96) ( a + (cid:96) ). Then, r ( x ( t )) − r ( x (0)) ≤ (cid:90) t ( (cid:96) e − (cid:96) s + (cid:96) se − (cid:96) s ) ds = (cid:96) (cid:96) (cid:16) − e − (cid:96) t (cid:17) + (cid:96) (cid:96) (cid:16) − (cid:96) te − (cid:96) t − e − (cid:96) t (cid:17) . As a result lim t →∞ r ( x ( t )) ≤ r ( x (0)) + (cid:96) (cid:96) + (cid:96) (cid:96) , implying r ( x ( t )) ∈ L ∞ for all t ≥
0, which completes the proof.
B . Proof of Lemma 2
Define the energy-like function H w ( w ) := 12 w + a ka ln (cid:0) | ka cos( w ) | (cid:1) . which is a first integral of the system (26) in the absence of the decaying term.This function is lower bounded as H w ( w ) ≥ H min w := a ka ln( − − ka ) , for w ∈ ( − β (cid:63) , β (cid:63) ) ,
20e note that, in view of the constraint (24), − − ka >
0, hence H min w is well-defined.Moreover, lim | w |→ β (cid:63) H w ( w ) = + ∞ . We also have the following bounds (cid:12)(cid:12) w (cid:12)(cid:12) ≤ (cid:113) H w ( w ) + H min w ) (42)and (cid:12)(cid:12) ka cos( w ) (cid:12)(cid:12) ≥ exp (cid:26) ka a H w ( w ) (cid:27) . (43)Clearly, ˙ H w = w ε t ka cos( w ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) w ka cos( w ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:96) exp( − (cid:96) t ) ≤ (cid:96) (cid:113) H w ( w ) + H min w ) exp (cid:26) − ka a H w ( w ) (cid:27) exp( − (cid:96) t ) , where the last inequality has used (42) and (43). To apply the Comparison Lemma we studythe boundedness of the following one-dimensional auxiliary system˙ r = (cid:96) (cid:113) r + H min w ) exp (cid:26) − ka a r (cid:27) exp( − (cid:96) t ) . (44)Note that [ −H min w , + ∞ ) is an invariant set for the differential equation (44), with r = −H min w an equilibrium point. Therefore, we are interested only in the trajectories satisfying r ( t ) + H min w >
0. In which case, ˙ r >
0, and consequently r ( t ) is a strictly increasing function of time.Define a function F ( r ) as F ( r ) := exp (cid:26) − ka a r (cid:27) − (cid:113) r + H min w ) . In view of the monotonicity, it is clear that there exists r > F ( r ) = 0 , and F ( r ) ≥ ∀ r > r . Therefore, there are two possible scenarios for system (44):1) r ( t ) < r for all t > t ≥ r ( t ) ≥ r for all t ≥ t .For Case 1), the boundedness of r ( t ) follows immediately. For the second case, the dy-namics (44) yields˙ r = (cid:96) (cid:20) exp (cid:26) − ka a r (cid:27) − F ( r ) (cid:21) exp (cid:26) − ka a r (cid:27) exp( − (cid:96) t ) ≤ (cid:96) exp (cid:26) − ka a r (cid:27) exp( − (cid:96) t ) , ∀ t ≥ t , F ( r ), and the second inequality has usedthe fact that F ( r ( t )) > t ≥ t .For the second case, by applying the Comparison Lemma we construct the auxiliary system˙ v = (cid:96) exp (cid:26) − ka a v (cid:27) exp( − (cid:96) t ) (45)with the initial condition v (0) ≥ r . For the system (45), we have˙ v exp (cid:26) ka a v (cid:27) = (cid:96) exp( − (cid:96) t ) , thus integrating via variable separation we get (cid:90) t ˙ v ( s ) exp (cid:26) ka a v ( s ) (cid:27) ds = (cid:90) t (cid:96) exp( − (cid:96) s ) ds ⇒ (cid:90) v ( t ) v (0) exp (cid:26) k v ( s ) (cid:27) dv = (cid:90) t (cid:96) exp( − (cid:96) s ) ds, with k := − k a a > − k v ( t )) − exp( − k v (0)) = k (cid:96) (cid:96) (cid:18) exp( − (cid:96) t ) − (cid:19) . (46)According to (46), if exp( − k v (0)) + k (cid:96) (cid:96) (cid:18) exp( − (cid:96) t ) − (cid:19) > , (47)we have v ( t ) = − k ln (cid:34) exp( − k v (0)) + k (cid:96) (cid:96) (cid:18) exp( − (cid:96) t ) − (cid:19)(cid:35) . Using in (47) the following inequality − k (cid:96) (cid:96) < k (cid:96) (cid:96) (cid:18) exp( − (cid:96) t ) − (cid:19) , we conclude that, if (cid:96) > k (cid:96) exp (cid:8) k v (0) (cid:9) := (cid:96) , (48)the condition (47) holds for all t >
0, implying that the solutions of (45) are bounded.Specifically, lim t →∞ | v ( t ) | = − k ln (cid:26) exp (cid:0) − k v (0) (cid:1) − k (cid:96) (cid:96) (cid:27) < + ∞ . We would like to point out that for the auxiliary system (45), if (cid:96) = (cid:96) thenlim t →∞ v ( t ) = ∞ ;and for the case (cid:96) ∈ (0 , (cid:96) ) the system (45) has finite escape time. (cid:96) is large enough, wecan obtain the boundedenss of r ( t ) in terms of the Comparison Lemma and the boundednessof v ( t ) for the auxiliary system (45). Using the Comparison Lemma again and selecting (cid:96) min2 := k (cid:96) exp (cid:26) k max (cid:8) r , H min w + H w ( w (0)) (cid:9)(cid:27) , for (cid:96) > (cid:96) min2 , the energy-like function H w ( w ( t )) for the system (26) is bounded for all t > Acknowledgement
The authors are grateful to two anonymous reviewers for their insightful remarks that helpedto improve the quality of the paper. This paper is supported by Ministry of Science andHigher Education of the Russian Federation, project unique identifier RFMEFI57818X0271,and by the European Union’s Horizon 2020 Research and Innovation Programme under grantagreement No 739551 (KIOS CoE).
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