"This paper deals with the chaos suppression for oscillators in canonical form. The underlying idea is to optimize the robust chaos supression by accounting the control cost. The robust chaos supression is attained by the robust asymptotic feedback while the optimization is solved via Riccatti ecuations. A finite horizon is arbitrarily settled and the suppression is achieved at this time by means of optimal control problem. This scheme allows to take into account the energy that is wasted by the controller and the closed-loop performance of states. Some experimental results show the features of the approach when a High-Gain observer is added in order to have available the complete state vector.

Femat Flores, Alejandro Ricardo

English

Repositorio IPICYT

Este artículo puede ser usado únicamente para uso personal o académico. Cualquier otro uso requiere permiso del autor. 1A Note on Optimizationof Chaos Suppression via Robust Asymptoticfeedback: Accounting control cost.C. Jimenez, R. Femat, C. HernandezDivisión de Matemáticas Aplicadas y Sistemas Computacionales, IPICYT, Apdo. Postal3-90, 78231, Tangamanga, San Luis Potosí, México.This paper deals with the chaos suppression for oscillators in canonical form. Theunderlying idea is to optimize the robust chaos supression by accounting the control cost.The robust chaos supression is attained by the robust asymptotic feedback while theoptimization is solved via Riccatti ecuations. A nite horizon is arbitrarily settled andthe suppression is achieved at this time by means of optimal control problem. This schemeallows to take into account the energy that is wasted by the controller and the closed-loopperformance of states. Some experimental results show the features of the approach whena High-Gain observer is added in order to have available the complete state vector.1. IntroductionNowadays the studies of chaotic systems are still having interest from many elds, likephysics, chemistry, electronics, biology, control theory even economy. The chaos suppres-sion problem exhibits an important advance of chaos theory. From the point of view ofcontrol theory, the chaos suppression is treated as stabilization of a nonlinear system anda lot of works dealing with stabilization, robust stabilization, secure communication, etc.had been published in the recent decade. Some of these works exploit the geometric con-trol theory to obtain a (suitable) linear equivalent system and then apply the standardmethods of linear control. As the chaotic system is not fully linearizable an useful way totackle the problem is the denition of the canonical form obtained by Lie-based derivativeoperator [5], [8], [10]. Most of chaotic systems can be transform to this canonical formaided by a nonlinear di¤eomorphic map, see for instance [5] and [10].Currently two questions arises from the chaos suppression problem. The rst one ques-tion is about how to set an arbitrary time in which the stabilization of system be suc-cessfully done. The latter is how to estimate (and possibly to reduce) the control e¤ortwasted in perform the stabilization. Bowong and Kakmeni [3] have reported a scheme forthe compute the "time duration" on robust asymptotic feedback for suppressing chaoticbehavior. Their work opens this research direction, but their method is overcomplex,implementation is di¢ cult, and implies to solve by simulation a given di¤erential equa-tion at each step of control. Moreover, the control cost has not been reported and anexperimental verication shows that high control cost is involved in their approach. Thepresent contribution tackles this two questions and the results shows that is feasible undersome conditions set the time of convergence and take into account the e¤ort wasted.2The rest of the paper is organized as follows. The problem formulation is presented innext section. In section 3 the ideal robust controller is constructed. Section 4 is devotedto tackle the problem under estimation of states. Section 5 introduces, by simulations,the features of controller. Section 6 shows the behavior of a real system under this classof controllers. Finally Section 7 present the conclusions of the present document.2. Problem FormulationConsider the chaotic system in the canonical form:8>><>>:_xi = xi+1; i = 1; 2; :::; p  1;_xp = &(x; ) + (x; )u;_ = (x; );y = x1(1)where x 2 Rp,  2 Rn p, u 2 R and y 2 R are the state variables, unobservable vari-ables, system input and output, respectively. &(x; ) and (x; ) are unknown nonlinearfunctions smooth functions.The objective is to design a robust feedback controller for system (1) achieving someoptimal objective and achieving the trajectory y = 0 in a (may be given) nite time.For the uncertain chaotic system (1) we take the following assumptionsAssumption 1. Only the system output y = x1 is available for feedback.Assumption 2. (x; ) is bounded away from zero.Assumption 3. The sign of nonlinear function (x; ) is known and and estimate ̂(x)of (x; ) is available for feedback.Assumption 4. System (1) is minimum phase; i.e., the subsystem _ = (x; ) is stable.By dening (x; ) = (x; )  ̂(x), (x; ; u) = &(x; ) + (x; )u, and  = (x; ; u)[6]. Thus the system (1) takes the form8>>>><>>>>:_xi = xi+1; i = 1; 2; :::; p  1;_xp =  + ̂(x)u;_ = (x; ; ; u; _u)_ = (x; );y = x1(2)First step in our approach is to consider the transitive of states only. To this end, aquadratic performance criterion, qualifying the transient trajectory toward the desiredend point (y = 0), is dened as [4]J(x; u) = x(t)|Qfx(t) +Z Tt0x(t)|Qx(t)dt (3)where t0  0 is the time which control starts and T > t0 is the time when the system (2)achieves the desired trajectory (y = 0); Q > 0, Qf  0 are symmetric matrices.It can be proof that system (2) is dynamically externally equivalent to system (1).33. Ideal robust controlTheorem 1 The control lawu(x) =1̂(x)    12B|P (t)x; t0  t  T (4)where T is given, P (t) is a symmetric positive matrix solving the Riccati equation  _P = A|P + PA  PBB|P +QP (T ) = Qf(5)with A 2 Rpp and B 2 Rpp dened asA =266640 1    0....... . ....0 0    10 0    037775 B =266640...0137775stabilizes the system (2) in the following sense x ! 0 as t ! T < 1; with a suit-able matrices Q and Qf , moreover the closed loop performs a value of the functional (3)J(x; u) = x|0P (0)x0, where x0 is the initial condition of system.Proof. Substituting control law (4) in (2) the closed-loop can be written as_x = A  12BB|P (t)x_ = (x; ; ; u; _u)(6)Dening the Lyapunov functionV (t) = x(t)|P (t)x(t) (7)and evaluate his time derivative_V (t) = _x(t)|P (t)x(t) + x(t)| _P (t)x(t) + x(t)|P (t) _x(t)substituting (5) and (6) and skipping the time dependence_V = x| A  12BB|P|Px+ x|P A  12BB|Px+x(t)| ( A|P   PA+ PBB|P  Q)x(t)nally we have_V (t) =  x(t)|Qx(t) (8)which is negative denite, thus x(t) converges to zero, and Q determines how much fastconverges. Now, Integrating (8) from t0 to T with respect t and using (7) we havex(T )|P (T )x(T )  x(0)|P (0)x(0) =  Z Tt0x(t)|Qx(t)dtthen J(x; u) = x|0P (0)x04Remark 1 The control law (4) is dened only for the interval t0  t  T , and thestabilization is achieved for some matrices Q and Qf if solution of Riccati equation (5)exist in such interval. The proposed feedback requires availability of the complete state.A more complete approach can be settled introducing the following denitions~B =0    0 ̂(x)|uu =   ̂(x)(9)we can call uu the part "unavoidable" of control, that is the control force necessary tocompensate the nonlinear function  in the system. In this way, as a second step ofdesign, the control e¤ort can be added to the cost function. That is, we can redene theperformance criterion to include the "avoidable" control e¤ort in a quadratic criterion (3)to have~J(x; ~u) = x(t)|Qfx(t) +Z Tt0[x(t)|Qx(t) + ~u|R~u] dt (10)with a given symmetric matrix R > 0; ~u := u  uu: Hence the control law becomesu =  ̂(x)+ ~uTherefore the closed loop takes the form_x = Ax+ ~B~u_ = (x; ; ; u; _u)(11)that allows to set the standard LQ problem as [4]minu~J(x; ~u)s:t: _x = Ax+ ~B~uwhose solution for controller is given by~u =  R 1B|P (t)x; t0  t  Twhere P (t) is now the solution of the following Riccati equation  _P = A|P + PA  P ~BR 1 ~B| +QP (T ) = Qf(12)and a value of functional (10) given by~J(x; ~u) = x|0P (0)x0 (13)Notice that, at this point, the approach requires full knowledge of states. Nevertheless,according with Assumption 1, only x1 in available for feedback. In the next section, thefull knowledge situation is relaxed by paying higher control cost due to state estimation.54. Suboptimal robust controlBecause Assumption 1, only x1 is available for feedback. Then, the construction of anobserver becomes suitable to get a suboptimal robust controller. We take the observerfrom [1], [7] and [9].8>>>><>>>>:x̂i = x̂i+1 + Cip+1i(x1   x̂1);i = 1; 2; :::; p  1;x̂p = ̂ + ̂(x̂)u+ Cpp+1p(x1   x̂1);̂ = p+1(x1   x̂1)where (x̂,̂) are estimated values of (x,) respectivelyCip+1 =(p+ 1)!i!(p+ 1  i)!and  is the so-called high-gain parameter. For a su¢ ciently large value of the high-gainparameter , the dynamics of estimation error converge exponentially to zero, see [7].By using the dened estimates, the feedback control law (4) can be written asu(x̂) =1̂(x̂) ̂   12B|P (t)x̂; t0  t  T (14)and we complete the control after transient with the controlu(x̂) =1̂(x̂) ̂   12B| Px̂; t > T (15)where P is a constant positive denite matrix such that the system (2) still in y = 0 fort > T , thus P is also a tuning parameter.Notice that the control cost of the suboptimal robust controller is higher than the idealone. The waste of energy increases due to the estimation, and have the form:~J(x; ~u) = x|0P (0)x0 +Z Tt0(x  x̂)|Q (x  x̂)in consequence, the choice of parameters becomes important and present a trade-o¤ be-tween optimization and estimation.5. Illustrative exampleWe take as example to illustrate the behavior of the proposed controller, the Chuasdynamical system, a system easy to implement physically [2]. Chuas circuit is widelyknow as a system that exhibits a strange attractor and having several congurations ofit. The dynamics of Chuas circuits are given by the di¤erential equations [2]:8<:_x1 = 1 (x2   x1   f(x1)) + u_x2 = x1   x2 + x3_x3 =  2x2(16)6where u is the control input andf(x1)8<:bx1 + a  bax1bx1   a+ bx1 > 1jx1j  1x1 <  1Typical value of the system parameters 1 = 9; 2 = 100=7; a =  8=7 and b =  5=7 cre-ates chaotic behavior in the dynamical system (16). The implementation of the controller(14) on Chuas system allows to show that it is capable to suppress chaos in dynamicalsystems whose Lie derivative could not be dened at some points belonging to its domain.In Chuas system case, the Lie derivative is not dened at (1; 0; 0) [12]. The observercan be written asẑ1 = ̂ + u+ 2(z1   ẑ1)̂ = 2(z1   ẑ1)(17)for this case we can compare the performance with the work of Bowong and Kakmeni [3],They use the same observer but a di¤erent controller based on a Lyapunov methodology.The advantage that they present is that the control time is explicitly computed. In ourcase the control time is arbitrarily settled and the suitable parameters founded.The initial conditions of system (16) in both cases were (x1(0); x2(0); x3(0)) = (0:1; 0; 0)and for the observer (17) (ẑ1(0); ̂(0)) = (0:1; 0:128). The high gain parameter value waschosen as  = 20. The controller parameter wereQ = 1 106; Qf = 1000P = 1000and the nite horizon is settled in T = 0:3. Figure (1) present the behavior of rst stateof Chuas system for the proposed controller (x1 Solid line) and the Bowongs one (xc1Dotted line). Both systems achieves the zero at t = 0:3, that is both converges to theobjective set, but the behavior of the proposed control looks better. A zoom in the timeline shows details and di¤erences of the evolution for both systems. Figure (2) showsthe second and third states for both systems. The system controlled by the proposedcontroller converges faster to the origin.6. Experimental implementationNow, in order to illustrate the real behavior of the approach, the Malasomas oscillatoris physically implemented. The model is based on sprotts work [13], and have three statesand have a nonlinear part due to a multiplicative signal. The analysis of chaos can befound in [11]. The dynamical system is given by:8<:_x1 = x2_x2 = x3_x3 =  x3   x1 + x1x2 + u(18)where  is a parameter that belonging to the set 2:08 <  < 2:51 allows the chaoticbehavior of system. In order to have a physical realization of Malasomas oscillator We70.0 0.5 1.0 1.5 2.0 2.5 3.0-0.020.000.020.040.060.080.100.0 0.1 0.2 0.3 0.4 0.5-0.020.000.020.040.060.080.10x 1 (Solid), xc 1 (Dotted)Time (Seg) x1 xc1Figure 1. x1; xc1 States from two equals Chuas systems for two di¤erent controllers0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.008-0.0040.0000.0040.0080.0 0.5 1.0 1.5 2.0 2.5 3.0-0.04-0.020.000.02x 2 (Solid), xc 2 (Dotted)x 3 (Solid),  xc3 (Dotted)Time(Seg)Figure 2. States for the two systems887651234X1X2Y1Y2AD633JNVs+WZVs-TL084CNTL082CNTL084CNTL084CN+-+-+-+-+12V-12V1149K1K0.1uF0.1uF1nF1nF10K10K20.5M10M1nF-12VIC 1BIC 1AIC 2AIC 1D123 765891014131210KR 110K10K4.7M10KTL084CN+-IC 1C123R 2R 5R 4R 3R 9R 8R 7R 6C 1R 11R 10C 5C 4C 3C 23x&3xα−3xα2x−1x21xx−PFigure 3. Electronic realization of the Malasomas oscillatorused the IC AD633JN, which is a four-quadrant multiplier. It includes high impedance Xand Y inputs and a high impedance input Z (called summing node). The summing nodeallows the addition of two or more multiplier outputs. The di¤erential inputs X and Yare converted into di¤erential currents by means of internal voltage-to-currents converters.The product of these currents is performed by the multiplying core and a buried Zenerreference provides an overall scale factor of 10V . Then the sum of (XY )=10 + Z, whereX = X1   X2 and Y = Y1   Y2 is applied to the amplier output. When the Z is notused, it is grounded. Thus, the output of the multiplier is given byW =(X1  X2)(Y1   Y2)10(R9 +R10)R9+ Zwhere Z is the optional summing input of the multiplier,X1, and Y1 are the inputs enteringto the multiplier and corresponds to the variables to be multiplied, the inputs X2, and Y2are grounded, R9 and R10 are resistors whose value are chosen to compensate the e¤ectof the scaling factor, given by the division of 10. The Figure (3) shows the diagram ofthe circuit. Note that the chaotic circuit also includes ve ampliers for integration andinversion operations The TL082CN and TL084CN are junction eld e¤ect transistors,JFETs, input opamps. Each opamp incorporates well-matched high-voltage JFET andbipolar transistors in the same integrated circuit. The device features are high slewrates, low input bias and o¤set currents, and low o¤set voltage temperature coe¢ cient.This is signicant to reduce sensitivity to circuit parameter values. In this manner, thebifurcation of the circuit can be handled.For a better control the observer is turned onbefore the control acts thus the control law takes the formu(x̂) =8<:01̂(x̂) ̂   12B|P (t)x̂1̂(x̂) ̂   12B| Px̂ t < t0t0  t  Tt > T(19)9Also We used the dSpace Real Time Interface CP1104, reading only state x1 of oscillatorand sending the control signal to the corresponding node. In order to the control signalmust be saturated at 10V . In all experiments the observer is turned on at t = 0 andthe controller at t0 = 10 with a nite horizon of 0:1, that is T = 10:1; using a integrationstep of Ts = 0:001, The weighting matrices are given asQ =0:1 00 0:1; Qf =0:1 00 0:1and the observer gain as  = 135, nally the corresponding tuning parameter for t > T isP =1 10 5 1 10 51 10 5 1 10 5Figure (4) shows the behavior when the observer signals are saturated asx̂! 10̂ ! 0:5notice that stabilization around the origin is achieved in the given horizon. In the Figure(4) is also plot a zoom of the controller interval in order to view the acceptable behaviortaking into account that the system was observed from state x1. Finally the Figure (4)shows the control action, the dotted lines pointed the control horizon. After t = 10:1 thecontrol signal is used only for conne near to zero the state. Figure (5) shows the phaseportrait of oscillator when the control action is applied, notice that only x1 versus x2 isplotted, but from the equation (18) we know that x3 is also near to zero.Figure (6) showsthe behavior when the observer signals are not saturated but the controller is saturated at4V . Notice that stabilization around the origin is also achieved in the given horizon. Inthe Figure (6) is also plot a zoom of the controller interval in order to view the acceptablebehavior. Finally the Figure (6) shows the control action, the dotted lines pointed thecontrol horizon. Figure (7) shows the phase portrait of oscillator when the control actionis applied, notice that only x1 versus x2 is plotted and there are near to zero.7. ConclusionsIn this contribution, an optimal robust feedback control is designed to suppress chaoticbehavior on dynamical system in strict-feedback form. The problem is solved for the fullknowledge of states and for the estimation of states as well, both consider the partiallinearization of the dynamical system. The controller takes into account the behavior oftransient response and the controller e¤ort. The proposed strategy leads to construct acontroller that optimizes the control cost and at the same time allows to set the timefor suppress the chaotic behavior. Two examples, shows the features of the proposedcontroller by simulation and by physical realization of system. The construction of thecontroller requires minimal information of system and have several tuning parametersthat shows the compromises between time of suppress an cost, and between estimation ofstates and cost.109.98 10.00 10.02 10.04 10.06 10.08 10.10 10.12-2-10128 9 10 11 12 13 14-10-8-6-4-202469.95 10.00 10.05 10.10 10.15 10.20-10-8-6-4-20246Control ActionTime (Seg)Control OnState x 1Finite HorizonFigure 4. Chaos Suppression in the rst state of Malasomas Oscillator, zoom at nitehorizon and controller behavior.Figure 5. Phase portriat of Malasomas Oscillator when controller is applied.119.98 10.00 10.02 10.04 10.06 10.08 10.10 10.12-4-3-2-1012348 9 10 11 12 13 14-10-8-6-4-202469.95 10.00 10.05 10.10 10.15 10.20-10-8-6-4-20246Control ActionTime (Seg)Control OnState x 1Finite HorizonFigure 6. Behavior of Malasomas under the Control action plottedFigure 7. Malasomas phase portrait under control12REFERENCES1. ALVAREZ RAMIREZ, J., VARGAS VILLAMIL, F., State estimation for a class ofnonlinear oscillators with chaotic attractor, Phys. Letts. A, 197 (1995) 116.2. BARTISSOL, P., CHUA, L.O., The Doule Hook, IEEE Trans. Circuits and Systems,34 (1988) 1512.3. BOWONG, S., MOUKAM-KAKMENI F.M.: Chaos control and duration time of aclass of uncertain chaptic systems, Phys. Letts. A, 316 (2003) 206.4. 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A note on optimization of chaos suppression via robust asymptotic feedback: accounting control cost

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A note on optimization of chaos suppression via robust asymptotic feedback: accounting control cost

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