Unveiling Uncertain Forces in Second-order Driven Oscillators Via Internal Model : A Discrete-time feedback

Unveil uncertain forces acting into (or onto) systems is a very interesting and old problem. Indeed, a lot effort has been devoted to develop procedures which results in the understanding of the uncertain forces and its effects. This contribution deals with 
recovering of the dynamics of the uncertain forces from measurements (time series). The main idea is to construct an internal model of the nonlinear system and design a discretetime feedback in such way that the model/system differences be stabilized at origin. In principle, if the internal model tracks the trajectory of the nonlinear system, then the uncertain force is recovered by the stabilizing command.


INTRODUCTION
Unveiling the forces acting into (or onto) systems is not an easy task.In fact, a lot effort has been devoted to unveil the hidden secrets of nature.The mathematical model development searches knowl- edge in this direction.Several theories have been proposed.There are some alternatives from the probability theory (for instance, Synergetics [1]) whereas another one departs from the determinis- tic and operators theory (see, for example, [2] and [3]).However, mathematical models have some limitations; hence estimation procedures are desired.
Recently, estimation procedures have been taken from control theory.For instance, ob- servability property [4] of the nonlinear systems can be exploited to get estimated values of un- measured states.The observability property strongly depends on the choice of the measured state (which is so-called system output if meas- urements are continuous [4] and time series if measurements are discrete-time [5,6]).The main idea behind observability is the phase-space re- construction; i.e., the history of all states is into the time series [7].The main application of the observability property has been to recon- struct, if exists, the attractor of the dynamical system [5, 6].However, in [4] was discussed that 2. PROBLEM STATEMENT observability property can be exploited to perform chaos synchronization.This is, if the history Consider the following nonlinear dynamical of all states is into the time series and the ob- system servability property conditions are satisfied, then the hidden states can be dynamically re- 5cl,M-XZ,M (1.a) constructed.For example, according to Takens theorem [7], it is possible to reconstruct an em- 22,M FI(XI;pM) + Tl(t; rM) + c(x(t)) bedding of a time series in a phase-space in y Cxt (1.b) absence of noisy measurements [5] by using time- delay coordinates.In addition, the time delay where x4E N2 is a state vector, Fa4 (xM; Pt) is a coordinates can be employed as estimated smooth function (which could be uncertain), values of a dynamical system to yield chaos (t;rt) is an external perturbation term (which suppression [8].Nevertheless, such state re- could be also uncertain) and pro, rm are parameter construction has some constraints, which can set and yME N is the system output (time series).be fundamentally seen as sensitivity tradeoffs Without lost of generality, one can denote c(x(t)) [9].
as the unknown force.On the other hand, from the practical point Note that uncertain forces can be acting into (or of view, there is an important problem.Can onto) the system.For example, let us assume that the uncertain forces be reconstructed from the c(x)=0 and Fa4(x:vi;Pa)or T(t;ra)is uncertain, measured state?.In principle, since the history the force to be unveiled is acting into the system. of the system is in the time series and the dyna-In same way, if Fa4(xac;pa4) and Ta4(t; ra) were mical series are yielded by forces acting into known functions, hence c(x)#0 represents the (or onto) the system, hence the unknown forces uncertain force, which should be unveiled.can be unveiled from the time series.Indeed, un- Let us now take an almost-exact copy of veil uncertain forces is an interesting and old the system (1.a), which is so-called mathematical problem for physicist.Some a priori informa- model tion is often required; for example, the order of the system, smoothness or boundedness.Be-5Cl,s x2  (1.c) sides, some information regarding noise or disturbances can be useful.Nevertheless, the 5C2,s Fs(xs;Ps) + Ts(t; rs) + u more interesting case is to unveil uncertain forces Ys Cxs (1.d) against the least prior knowledge about the system, where u is an external force (control law).A force unveiling procedure is presented in Fs(xs;p,) and T,(t;p,) are known and bounded this contribution.The main idea is to construct a functions; however; Pa P, and rarv.dynamic feedback, which comprises a control The unveiling problem is: given the system law and a discrete-time estimator.The estimator output YM and least prior knowledge of the structure and control cannot be separately designed.This of the nonlinear system (1.a), design an feedback is, the coupling between the control law and the scheme such that the uncertain forces can be discrete-time estimator yields the unveiling of unveiled.In some sense, the unveiling problem the unknown force.The state estimator is based seems as a synchronization one where the on the time-delays coordinates from time series actual system (1.b) can be seen as the so-called whereas the control law structure is a stabilizing "master" whereas the mathematical model (1.c) command, represents the "slave" system.In others words, the synchronization objective is that y tracks YM. which implies that x(t) --0 = Xs(t) --XM(t).This implies that Fs(x, t; Ps) ---+ FM(XM, l; PM) and T(t; r) Ts (t; rs) therefore c (x(t)) u --O.From the control theory viewpoint, the synchronization problem can be interpreted as follows.Let us define x E such that Xi=Xi,M--Xi, for i-- 1,2,3,...,n.In this way, the following dyna- mical system describes the dynamics of the model error 1 =X2 k2 AF(x, t;pl) + AT(t;p2) + c(x) u y= Cx (2) where AF(x; pl) FM(XM; PM) F, (x,;p) and AT(t; P2) TM(I; rM) T,(t, r) are uncertain smooth functions, which represent the mismatches between dynamical system and its model.Now, since the unveiling problem can be seen as a synchronization one and, from the control theory point of view, the synchronization problem can be understood as the stabilization of the system (2) at the origin.Hence, the goal is to find a feedback control law u u(x, t) such that lim x --0 as l--oo. 3. THE PROPOSED UNVEILING SCHEME 3.1.Dynamic Evolution in an Extended Space Following the ideas reported in [10], let us define 7=AF(x;p)+AT(t;pz)+a(x). Then the model error system can be rewritten as follows 21 x2 (3.a) where F(x,7,u,t;Tr) x20(AF(x;p) + AT(t;p2)+ a(x))+[7+u]Oz(AF(x;pl) + 5T(t;p2) + a(x))+ /t, 7r is the parameters set.
It has been proved that, under feedback output control, the a second-order driven oscillator can be stabilized at origin (see Appendix in [10]).In addition, has been proved that the system (3) is externally dynamically equivalent to the system (2).This is, there is a time-invariant manifold, (x, r/, t; r), such that the solution of the system (2) is a projection of the system (3) as long as the initial conditions be (x(0), r/(0), 0; 70 0, which is satis- fied by definition (see Appendix in [10]).In order to illustrate the geometrical interpretation of the augmented state, r/, we have selected the Duffing equation.To perform the numerical simulations, we have defined the augmented state as follows: T] (X 2 + X X --]-Acos(wt).In this way, one has that F(Xl, x2, T], H, t)--Awsin(wt). Figure shows the phase portrait of the systems (2) and (3).The initial conditions were chosen as follows: for system (2) x(0)-(0.0,1.0) while for system (3) (Xl(0),x2(0),/(0))=(0.0,1.0.0.01).The same parameters values were chosen for both systems.
Since systems (2) and (3) are dynamically equivalent, hence a feedback can be designed from the system (3) in such way that trajectories of the system (2) be leaded to origin.In fact, Eq. ( 3) is used as an intermediate system toward the con- struction of the feedback.The main idea is the following.If one is able to stabilize the trajectories of the system (3) neither measurements of the velocity, x2, nor the augmented state, 7, then the trajectories of the system (2) will be leaded to origin (prescribed point) against the uncertain terms.

The Discrete-time Feedback
Via Uncertainties Estimation Now, let us consider the ideal feedback, which can be obtained from system (2) as follows: u=-7+kzx2+klX, where, by definition, AF(x;pl) + AT(t;p2) + c(x).Under such ideal feedback the system (2) becomes stable at origin if polynomial Pz(s)= s2-} kzS-}-kl has its roots at left- hand complex plane.This implies that the system (3) is also stable at origin.Nevertheless, the ideal feedback requires knowledge about the uncertain terms.Here, the following estimation procedure is proposed: The main idea is to estimate the un- measured states of the system (3) from measure- ments of the time series y x.It is clear that, from Eqs. (3.a) and (3.b), the unmeasured states can be estimated by x2 c2=(Xl(ti)--Xl(ti_l))//kt and () (,-) (x(;) + 2x(t_ ) x(_))/zx- u(ti_), where At denotes the sampling rate, (22,)) are estimated values of (Xz, q) and x(ti) is the value of the measured state at time t..In this way the ideal feedback can be modified to get u(ti) -l(ti) + k2x,(ti) + k,(x,(ti) x,(ti_,))/At (4) where il(t) is given by the above estimator.Note that the uncertainties estimator l(ti)=(xl(ti)+ 2x(ti_ ) x(ti_ 2))/At + u(ti_ ) only requires knowledge about the measured and the last control action.This can be seen as a torque balance.In this sense the estimated value )(t.) reconstructs the unknown force acting into (or onto) the system.The feedback (4) and the uncer- tainties estimator cannot be designed separately.
In this way, the unveiling procedure depends on two factors: (a) The feedback parameters should be chosen such that the real part of the roots of the polynomial P2(s) be negative defined and (b) the sample rate At be arbitrarily small.Indeed, as At-+ 0 as the u(h) tends to the ideal feedback, u.Of course, if At=0 the feedback (4) cannot be physically realizable.This is, there is a tradeoff between the estimation and the stabilization capability, which can be resumed as: The rate of the uncertainties estimation is limited by stabilization rate and viceversa.The unveiling tradeoff is illus- trated in next section.

ILLUSTRATIVE EXAMPLES
We have selected two interesting systems: The former is the magneto-elastic beam.In this system, we illustrate that forces acting onto the system can be reconstructed via the proposed scheme.The latter is a magnetic bearing.In this case, the uncertainties source is into the systems.The unknown function is the force between the magnet and a class of high-temperature superconductor.
4.1.Unveiling External Perturbing Force Let us consider the magneto-elastic beam system.This system was experimentally studied by Moon and Holmes (see Chapter 2 in [2]) and can be modelled by the a driven second-order oscillator (Duffing equation): 5 + 5c-x + x re(t) + u, where is a damping coefficient, u denotes the stabilizing command and re(t) is a periodic func- tion, which represents the perturbing external force, e.g., re(t)= Acos(wt).Note that in this case the uncertain force a(x(t)) =_ 0 for all t, see Eq. (1 .b).
Let us assume that only the perturbing force, re(t), is unknown and unmeasured.In this way, the augmented state is defined by ./:re(t)and the extended uncertain system becomes where 7 and its time-derivative are unknown.Thus, the internal model feedback is given by Figure 2 shows the performance of the time discrete feedback for several sampling rate, At.
The sampling rate were chosen as follows: At=0.0005 (dashed line), At--0.01 (solid line), At=0.1 (dotted line) and At=0.5 dash-dotted line.The unveiling feedback scheme was activated at t-50.
If the uncertain force is unveiled (which means that is close to the unknown force), the control action compensates the perturbation onto the system.On contrary, as the At increases the esti- mated value of the uncertain term the control action increases and stabilization is lost.For instance, if At =0.5 (see dash-dot line in Fig. 2) then the estimation error is around 1.0, the position, x, cannot be stabilized at origin and the stabilizing command is larger, u.This is, the uncertain force can be unveiled if the position error is stabilized at origin and viceversa.

Unveiling the Magnetic Force in a Levitation System
Consider the dynamical system given by + + x + t) + where x means the position, is a damping factor, -(t)=Acos(t) represents a periodic perturbing force, a(x, k, t) is a nonlinear function acting into the system, which represents the force between a magnet supported by the high temperature Type-II superconductor [11].
In principle, the feedback (6) can be experimen- tally implemented in a magnetic bearing device.
Figure 3 shows the performance of the uncertain force for several sampling rates, At.In this case, if sampling rate At =0.5 the closed-loop system is Time (Seg) FIGURE 3 Performance of the discrete-time unveiling scheme.The force acting into the magnetic-bearing equation is unveiled.
unstable; hence the sampling rate were chosen as follows: At 0.0005 (dashed line), At 0.01 (solid line) and At=0.1 (dotted line).The unveiling feedback was activated for > 100.Note that as the sampling rate decreases the stabilizing com- mand increases, in consequence the unveiling error increases.This is, if At-0 the stabilizing com- mand approach is not physically realizable.Hence, the hidden forces cannot be unveiled.

CONCLUDING REMARKS
A procedure to construct an unveiling scheme was proposed in this letter.The unveiling is a discrete- time feedback scheme and comprises an estimation procedure and a stabilizing command.The proce- dure departs defining an augmented state to get the construction of an extended system, which is an uncertain nonlinear system.After that, the augmented state is estimated via backward finite differences.In this way, the resulting scheme only requires measurements of one available state at time ti, ti-1 and ti-2 and the knowledge about last stabilizing command, ui_ 1.Two illustrative exam- ples were presented.In first one an uncertain force acting onto the system is unveiled whereas in the second one an unknown force acting into the system is unveiled.In principle, the unveiling scheme allows to get an estimated value of any uncertain force from on-line measurements.Hence, experimental im- plementation can be expected.Previously, the dis- crete-time unveiling scheme should be designed.Then, the following configuration for experimental unveiling can be proposed: The measurements from the experimental apparatus are entered to any computing machine, where the model and the discrete-time unveiling are programmed.The goal is to synchronize the experimental system and the programmed model via the discrete-time scheme.If synchronous behavior is attained, the estimated

2 FIGURE
FIGUREBlue line is the space-phase of the Duffing sequation whereas black line represents the trajectories of the Duffing extended space.

FIGURE 2
FIGURE 2 Performance of the discrete-time unveiling scheme.The force acting onto the Duffing equation is unveiled.