- Email: [email protected]
Robust Control of a Nonlinear Electrical Oscillator Modeled by Duffing Equation
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Robust Control of a Nonlinear Electrical Oscillator Modeled by Du!ng Equation Manuel Jiménez-Lizárraga, Michael Basin, Pablo Rodriguez-Ramirez Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Nuevo León, San Nicolás de los Garza N. L., México AP 144-F (e-mail: [email protected]). Abstract: This paper studies the control of a nonlinear electrical circuit exemplified by a Du!ng equation, which contains parameter uncertainties in the right hand side of this nonlinear di erential equation. Even though this type of circuit has been object of a variety of control strategies in the past, very few papers have been devoted to the design of an optimal control law with a quadratic performance index and subject to that type of uncertainties. We develop simulation examples that show that not only an optimal control strategy can be applied to such a system, in case of an optimal control strategy without uncertainties, but an optimal control of the mini-max type can also be implemented in case of the uncertainty in the model. Keywords: Nonlinear Electrical Circuit, Mini-max Control, Uncertain Parameters. 1. INTRODUCTION An interesting example of a nonlinear electric oscillator is described by the so called Du!ng equation. Such a circuit has attracted the attention of the control community because it represents a complex chaotic nonlinear system, which has a variety of applications ranging from physics to engineering (see Kapitaniak (1996), Nijmeijer and Harry (1995), Loria et al. (1998)). Given the complex behavior of the circuit, the control task becomes really challenging. Most results found in the literature, regarding the control design, are based on nonlinear control theory (Kapitaniak (1996), Nijmeijer and Harry (1995)). By instance, the Lyapunov design method has been widely used in controlling this circuit. More often the control objective in the mentioned papers is to stabilize the circuit variables around an equilibrium point or follow a given trayectory (tracking problem see Nijmeijer and Harry (1995)). We can also find control laws of the state feedback and observed based design (for more methods of controlling the Du!ng equation, see Chen and Dong (1993)). The well-known electrical circuit with a nonlinear element, which is represented as a nonlinear inductor, an alternating source of voltage, a pure resistive element, and an electrical capacitor is shown in the Figure 1. Applying the node law of the circuit theory, such a circuit can be modelled as g2 1 g 1 3 3 H0 + + + = cos $w gw2 UF gw F F U
(1)
where is the magnetic i oux through the nonlinear inductor, H0 is the alternating source voltage, U and F are the constants of the capacitor, 1 and 3 are some operation constants. The nonlinear term appears because of the nonlinear inductor, which is an inductor with a ferromagnetic core, and is modeled, if an abstraction of the hysteresis phenomenon is made, by an l- nonlinear characteristic. Here l is the electrical current. Such a 978-3-902661-93-7/11/$20.00 © 2011 IFAC
Fig. 1. Electrical Network with a nonlinear element. characteristic is approximated by a constitutive relation of the form (see Hasler and Neirynck (1986)): (2) l = 1 + 3 3 Defining the variables: 1 3 1 H0 s1 = ; s2 = ; s3 = ; (w) = cos $w UF F F U 2 g {1 = ; {˙ 1 = {2 > {˙ 2 = 2 gw (3) the following second order system is obtained: {˙ 1 = {2 (4) {˙ 2 = s1 {2 s2 {1 s3 {31 + (w) = Depending on the choice of these constants, it is known that solutions of (4) exhibit periodic, almost periodic, and chaotic behavior (see Loria et al. (1998)). Note that this system is a nonlinear polynomial system of the third order. Consider this system in a controlled form: {˙ (w) = i (w> {) + Ex (w) + g (w) (5) { (w0 ) = {0 where:
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
¶ µ ¶ {2 0 ; E = 1 s1 {2 s2 {1 s3 {31 (6) | g (w) = ( 0 (w) ) where the a!ne control x (w) is a physical control input, which will be designed later on. Consider also a more general form for the nonlinear term (6) represented as i (w> {> )=d0 +d1 {+d2 {{| + d3 {{{> where d1 is and standard 2D matrix, and d2 and d3 are 3D and 4D tensor (see Basin and Calderon-Alvarez (2009) for details). Such a representacion allows us to design in a compact way the suggested control law. To control such a circuit, taking a di erent approach to that in the above mentioned papers, in this work we propose to set an optimal regulator control problem to drive the nonlinear state dynamic to the origin. Observe that the resulting nonlinear system is actually one of the polynomial type. Although the optimal controller for nonlinear systems has to be determined using the nonlinear filtering theory (see Kushner (1964); Kallianpur (1980)) and the general principles of maximum Kwakernaak and Sivan (1972) or dynamic programming Bellman (1957), which do not provide an explicit form for the optimal control in most cases, there is actually a long tradition of the optimal control design for nonlinear systems (see, for example, Albrekht (1962)—Yoshida and Loparo (1989)) and the optimal closed-form filter design for nonlinear Wonham (1965); Benes (1981); Yau (1994), and in particular, polynomial systems (Basin (2003), Basin et al. (2008)). However, the "optimal" quadratic controller problem for nonlinear, in particular, polynomial, systems with parameters belonging to an uncertain finite set, to the best to the authors knowledge, has not even been consistently treated. All of the above mentioned papers study this problem if the model of the considered dynamics is known with accuracy, but for many applications this assumption is unrealistic, because in practice it is common to find some sort of uncertainties. Hence, it is desirable to develop some kind of robustness for the optimal control problems to deal with such possible uncertainties which may have an impact on the nonlinear dynamics of the system. i (w> {) =
µ
2. OPTIMAL CONTROL DESIGN Our objective is to drive the circuit variables to the origin. To that end, consider the following quadratic performance index is set as the control performance measure: 1 j ({ (w) > x (w)) = {| (W ) T{ (W ) + 2 ZW (7) 1 ({| (w) O{ (w) + x| (w) Ux (w)) gw 2 w0
The performance index is given in standard Bolza form where it is assumed that U is a strictly positive definite and symmetric matrix, O and T are two non-negative definite symmetric matrices, W A w0 is a certain time moment, and dW denotes transpose to a vector (matrix) d. The solution of the optimal regulator (control) problem for polynomial systems with linear control input and a quadratic criterion is given by the following feedback control that realizes the optimal control with respect to the quadratic criterion given in (7), for the polynomial system (5): (8) x = U1 E | [P (w) + s(w)]
where the matrix function P is the solution of the Riccati type equation: | P˙ (w) = O + [d1 + 2d2 {(w) + 3d3 {(w){| (w)] P (w)+ | 1 | P (w) [d1 + d2 {(w) + d3 {(w){ (w)] P (w)EU E P (w) (9) with terminal condition P (W ) = T> and the parametrized vector function s is solution of the linear equation: s(w) ˙ = [d1 + 2d2 {(w) + 3d3 {(w){| (w)]| s(w)(10) P (w)EU1 E | s(w)+P (w)g(w) with boundary conditions s (W ) = 0= This result follows from the aplication of the maximum principle, the proof can be found in Basin and Calderon-Alvarez (2009). 3. ROBUST OPTIMAL CASE To solve a regulator problem of the most general type for such a circuit, we introduce a di erent type of control concept for a polynomial system subject to uncertainties. Consider the uncertain circuit as follows: {˙ (w; ) = i (w> {> ) + E (w; ) x (w) + g (w; ) (11) { (w0 ; ) = {0 where is a parameter which belongs to a given parametric set A. In this paper, we consider A as a finite set, that is A = {1 > 2 > ===> Q } > each one representing a possible realization or possible model of the system. The time variable varies in an interval w 5 [w0 > W ]. By instance, the parameters of the polynomial system (11) belongs to: d0 = {d0>1 > d0>2 > ===> d0>Q } E = {E1 > ===> EQ } g = {g1 > ===> gQ } The circuit dynamics is given by a family of Q di erent possible non-linear di erential equations sometimes called Multi-Model problem, with no information about the trajectory that is realized. So, it appears that for this type of problems we have Q possible state dynamic equations, each of them describing a model and there is no a priory information which will be the active one, but of course, it will be at least one.The system (5) is assumed to be uniformly controllable; the definitions of the uniform controllability for nonlinear systems can be found in Isidori (2001). For each fixed parameter , the non-linear function of the circuit i (w> {> ) is a polynomial of 2 variables, which are the components of the state vectors { (w; ) 5 R2 = Following the previous work (see Basin et al. (2006)), a s-degree polynomial of a vector { (w; ) 5 Rq is regarded as a s-linear form of q components of { (w) > that is to say: i (w> {> ) =d0 (w; ) +d1 (w; ) {+d2 (w; ) {{| + · · · (12) +d3 (w; ) {{{= Here, the involved parameters are: d0 is vector of dimension 2, d1 is a matrix of dimension 2 × 2> d2 is a 3D tensor of dimension 2 × 2 × 2> etc= Remark 1. Clearly, the uncertainty on the realized parameters is represented by the value of . Such a parameter belongs, as we said, to the finite set A that contains all the possible scenarios or parametric realizations of the nonlinear plant, which is fixed during the actual process, with no possibility of change once the process has started. So, each trajectory is uniquely determined by such a set of parameters. Nevertheless, there is no information on the trayectory which is realized. In this way, the proposed
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
control should deal with all of the parameters and should provide an acceptable behavior for such a class of systems. The quadratic cost criteria to be minimized is: 1 j ({ (w; ) > x (w) > ) = {| (W ; ) T{ (W ; ) + 2 ZW 1 ({| (w; ) O{ (w; ) + x| (w) Ux (w)) gw= 2
(13)
w1
the criteria with uncertain dynamics, have been studied in the past (see Jimenez-Lizarraga and Poznyak (2007)). The problem is stated and solved applying a min-max concept, that is, taking the worst case scenario of the functional and then minimizing by the control. Therefore the problem to solve is: min max j ({ (w; ) > x> ) (14) x(w) 5A Therefore, the philosophy of design here is really based on the concept of min—max control where the operation of maximization is taken over a set of uncertainties (in our case, a parameter from a finite set) and the operation of minimization is taken over a set of admissible control strategies. Thus, this paper is focused on the design of a control, which exhibits some sort of robustness property for a class of multi-plant polynomial systems given by a system of ordinary di erential equations with parameters from a given finite set. As well-known (Pontryagin et al. (1962)), the Bolza problem can be simplified expressing the original cost function (7) as a Mayer-type functional, that is, a cost function depending only on the terminal values of the state vector; such a transformation involves the extension of the state space. In what follows we define: Zw 1 {q+1 (w; ):= ({| (; ) O{ (; ) +x| () Ux ()) g> 2 w1
taking the derivative in time: {˙ q+1 (w; ) = {| (w; ) O{ (w; ) + x| (w) Ux (w) > the new cost function is given by: 1 j ({ (w; ) > x (w) > ) = {| (W ; ) T{ (W ; ) + {q+1 (W ; )= 2 Evidently the term {| (W ; ) T{ (W ; ) does not depends on the new introduced coordinate, that is: C {| (W ; ) T{ (W ; ) = 0 C{q+1 (W ; ) We proceed with finding the necessary condition for minimax optimality, for the new Mayer problem. Introduce the following Hamiltonian function: K (w,{,t,x,) :=t | (w; ) [i (w,{,) +E (w;) x (w) +g (t;)] 1 | +tq+1 (w; ) ({| (w; ) O{ (w; ) + x| (w) Ux (w)) > 2 which depends on the given uncertain vector. Following the general mini-max necessary conditions given in the original work (Boltyanski and Poznyak (1999)), the vectors t (w; ) satisfy the property: g C t (w; ) = [K (w> {> t> x> )] = gw C{(w; µ ) ¶| Ci (w> {> ) | tq+1 (w; ) O{ (w; ) + t (w; ) = C{(w; ) For the last coordenate the time derivative is: t˙q+1 (w; ) = 0
and the transversality conditions for these vectors at the terminal time are: t (W ; ) =- () judg [{| (W ; ) T{ (W ; ) +{q+1 (w; )] = () T{ (W ; ) tq+1 (W ; l ) = () where the constant () is a non-negative real value appearing from the general case of the mini-max problem (for more details on this conditions see also Poznyak et al. (2002)). For each fixed parameter the partial derivative of i (w> {> ) in { is given by: Ci (w> {> ) =d1 (w; ) +2d2 (w; ) {+3d3 (w; ) {{| (15) C{(w; ) Consider the following technique of summation of the individual Hamiltonian functions for each l (l = 1> ===> Q ) > where the summation is taken over all the possible uncertainty parameters. Then, it is still possible to introduce a generalized Hamiltonian function encompassing all of the family plants: Q P K (w,{,t,x) = [(t | (w; l ) [i (w> {> l ) +E (w; l ) x (w) +
l=1 ¤ | (w; l ) 12 ({| (w; l ) O{ (w; l ) +x| (w) Ux (w)) g (w; l )]) +tq+1 which allows us the find the minimax control as: Q X (16) l E | (w; l ) t (w; l ) > x (w) = U1 l=1
where we find the vector of parameters = (1 > 2 > ===> Q )| ) ( Q X Q Q l = 1 (17) V := 5 R : l 0; l=1
Similarly to the linear-quadratic case, the mini-max control appears as a mixture of the controls which are the optimal strategies for each fixed parameter value , and the controls are to be found in a multi-dimensional simplex set (17). Now the mini-max strategies are to be found in a finite dimensional space instead of the original function space. 4. PARAMETRIZED MINI-MAX CONTROL FOR DUFFING EQUATION Let us now introduce the following diagonal block matrices: 5 6 6 5 i (w> {> 1 ) ... 0 O ... 0 9 : 9. .. .. .: f := 7 8 ; L:= 7 .. = .. 8 . = . 0 ... i (w> {> Q ) 0 ... O 5
1 Lq×q 9 .. := 7 . 0
6 5 T ··· 0 : 9 .. .. .. ; Q:= 8 7 . . . 0 · · · 2 Lq×q
··· .. . ···
6 0 .. : . 8 T
L 5 Rq×q l = 1> 2 (18) and the following tensor matrix for the coe!cients of the polynomial: 6 5 dl · · · 0 : 9 al := 7 ... . . . ... 8 0 · · · dl l = 0> ===> v
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B| := [E | (w> 1 ) · · · E | (w> Q )] >
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
{˙ 11 = {12 = 1=1 {11 0=4 {12 ({11 )3 + 2=05 cos(1=8 w) + x {˙ 21 = {22 2 2 {˙ 2 = 1=15 {1 0=45 {22 1=05 ({21 )3 + 2 cos(1=9 w) + x
The extended dynamics for the polynomial systems appears as: x˙ = f (w> x (w)) + B (w) x (w) + d (w) and the extended vector d is defined as d := (g| (w; 1 ) > ===> g| (w; Q ))
Selecting U = 1, T = 1, O = 1 and W = 1=5, we obtain the performance index as a function of the weighting parameter near the extremum point (see Fig. 1). The corresponding state variable dynamics is depicted at Fig. 2 where the blue and green line correspond to the states 1 and 2 of the first plant and the red and light blue represent lines states 1 and 2 of the second plant. The control law is shown in Fig. 3, and the criterion in Fig. 4. Here, the blue line is for plant 1 and the green line is for plant 2. 2.66 2.65 2.64 2.63 Performance index
Note that the dependence on the uncertain parameter has disappeared, the new dynamics includes the complete family of plants, but the control remains the same for all plants. That means that the same control will be applied for all the dynamics simultaneously. Based on the extended system, the mini-max regulator that realizes (14) with respect to the quadratic criterion given in (13), for the polynomial system (11), takes the form: (19) x = U1 B| [M + p ] where the matrix function M is the solution of the following Riccati type equation: ˙ = L + [a1 + 2a2 x+3a3 xx| + === M | +vav x· · · (v 1) wlphv · · ·x] M + | M [a1 + a2 x + a3 xx + === + av x· · · (v 1) wlphv · · ·x] M BU1 B| M (20) with terminal condition M (W ) = Q> and the parametrized vector function p is the solution of the linear equation: p˙ = M a0 + [a1 +2a2 x+3a3 xx| +===+vav x· · · (v 1) wlphv · · ·x]| p M BU1 B| p + M d (21) with terminal condition as p (W ) = 0= The matrix containing the optimal weight parameters = solves the next optimization problem: = min M () 5V Q (22) M () := max j ({ (w; ) > x> )
{˙ 12
2.62 2.61 2.6 2.59 2.58 2.57 0
0.1
0.2
0.3
0.4
0.5 Time
0.6
0.7
0.8
0.9
1
Fig 1. Performance index M().
1.5 1 0.5
5A
0
State
with x (w) given ¡in (19) parametrized by the vector = ¢ | (1 > 2 > ===> Q ) l 5 V Q through (20) and (21). Remark 2. In the case of a fully known plant ( = 1), that is, there is no parametric uncertainty, the above equations collapse into the standard known result of the optimal control for the polynomial system (Basin and CalderonAlvarez (2009)). Remark 3. The dependence on in the cost function can be seen through the dependence of the cost on the solutions of the parametrized equations (20) and (21); an expression for this can be found in Bryson (1998).
−0.5 −1 −1.5 −2 −2.5 −3 −3.5 0
0.5
1
1.5
Time
Fig. 2. States of the circuit.
3
5. SIMULATION EXAMPLE 2.5
2
Control
In this section we present two numerical examples for controlling of the Du!ng equation. In the first case, we consider the optimal control of the circuit with no uncertainties, the second one presents the numerical results produced by the robust algorithm for a given family of parameters.
1.5
1
0.5
0 0
0.5
1 Time
5.1 Example 1
Fig. 3. Control Signal for W .
Consider the following multi-model circuit 5792
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
6. CONCLUSION
3
2.5
This paper studied the control of a nonlinear electrical circuit modeled by a Du!ng equation which contains parameter uncertainties in the right-hand side of this nonlinear di erential equation. We implemented two type of control laws for this circuit, optimal and robust optimal. This second strategy allows us to solve the problem of controlling the Du!ng equation with uncertainties in the parameters. The simulations examples show the good performance of both controllers for a given set of parameters.
Criterion
2
1.5
1
0.5
0 0
0.5
1
1.5
Time
Fig. 4. Criterion for W .
REFERENCES 5.2 Example 2 Consider the following one-model circuit: {˙ 12
{˙ 11 = {12 = 1=1 {11 0=4 {12 ({11 )3 + 2=1 cos(1=8 w) + x
The parameters of the cost function are: U = 1, T = 1, O = 1 and W = 1=5, we obtain (see Fig. 1) = 0 and m( ) = 0=7167. The corresponding state variable dynamics is depicted at Fig. 5 where the blue and green line correspond to the states 1 and 2. The control law and criterion are shown in Fig. 6 and 7. 1.5 1 0.5 0
State.
−0.5 −1 −1.5 −2 −2.5 −3 −3.5 0
0.5
1
1.5
Time.
Fig. 5. States one model. 4
3.5
3
Control.
2.5
2
1.5
1
0.5
0 0
0.5
1
1.5
Time.
Fig. 6. Control one model. 3
2.5
Criterion.
2
1.5
1
0.5
0 0
0.5
1 Time.
Fig. 7. Criterion.
1.5
Albrekht, E. (1962). On the optimal stabilization of nonlinear systems. J. Appl. Math. Mech, 25, 1254—1266. Basin, M. and Calderon-Alvarez, D. (2009). Optimal controller for uncertain stochastic polynomial systems with deterministic disturbances. International Journal of Control, 82, 1435—1447. Basin, M. (2003). On optimal filtering for polynomial system states. ASME Trans. J. Dynamic Systems, Measurement, and Control, 125, 123—125. Basin, M., Calderon-Alvarez, D., and Skliar, M. (2008). Optimal filtering for incompletely measured polynomial states over linear observations. International J. Adaptive Control and Signal Processing, 22, 482—494. Basin, M., Perez, J., and Skliar, M. (2006). Optimal filtering for polynomial system states with polynomial multiplicative noise. International Journal of Robust and Nonlinear Control, 16, 287—298. Bellman, R. (1957). Dynamic Programming. Princeton University Press, Princeton. Benes, V. (1981). Exact finite-dimensional filters for certain di usions with nonlinear drift. Stochastics, 5, 65—92. Boltyanski, V. and Poznyak, A. (1999). Robust maximum principle in minimax control. International Journal of Control, 72(4), 305—314. Bryson, A. (1998). Dynamic Optimization. Pearson Education. Chen, G. and Dong, X. (1993). On feedback control of chaotic continuos-time systems. IEEE Trans. Circuits Syst. I,, 40, 591—601. Hasler, M. and Neirynck, J. (1986). Nonlinear Circuits. Artech, House. Isidori, A. (2001). Nonlinear Systems. Berlin: Springer. Jimenez-Lizarraga, M. and Poznyak, A. (2007). Robust Nash equilibrium in multi-model LQ di erential games: Analysis and extraproximal numerical procedure. Optimal Control Applications and Methods, 8(2), 117—141. Kallianpur, G. (1980). Stochastic Filtering Theory. Springer. Kapitaniak, T. (1996). Controlling Chaos. New York: Academic,. Kushner, H. (1964). On di erential equations satisfied by conditional probability densities of Markov processes. SIAM J. Control, 12, 106—119. Kwakernaak, H. and Sivan, R. (1972). Linear Optimal Control Systems. Wiley-Interscience, New York. Loria, A., Panteley, E., and Nijmeijer, H. (1998). Control of the chaotic du!ng equation with uncertainty in all parameters. IEEE Transactions on Circuits and
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Systems-I Fundamental Theory and Applications, 45, 1252—1255. Nijmeijer, H. and Harry, B. (1995). On Lyapunov control of the Du!ng equation. IEEE Transactions on Circuits and Systems-I Fundamental Theory and Applications, 42, 473—477. Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., and Mishchenko, E. (1962). The Mathematical Theory of Optimal Processes. Interscience Publishers, New York. Poznyak, A., Duncan, T., Pasik-Duncan, B., and Boltyanski, V. (2002). Robust maximum principle for multimodel LQ-problem. International Journal of Control, 75(15), 1770—1777. Wonham, W. (1965). Some applications of stochastic di erential equations to nonlinear filtering. SIAM J. Control, 2, 347—369. Yau, S. (1994). Finite-dimensional filters with nonlinear drift. I: A class of filters including both Kalman-Bucy and Benes filters. J. Math. Systems, Estimation, and Control, 4, 181—203. Yoshida, T. and Loparo, K. (1989). Quadratic regulatory theory for analytic nonlinear systems with additive controls. Automatica, 25, 531—544.
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Robust Control of a Nonlinear Electrical Oscillator Modeled by Du!ng Equation Manuel Jiménez-Lizárraga, Michael Basin, Pablo Rodriguez-Ramirez Facultad de Ciencias Físico-Matemáticas, Universidad Autónoma de Nuevo León, San Nicolás de los Garza N. L., México AP 144-F (e-mail: [email protected]). Abstract: This paper studies the control of a nonlinear electrical circuit exemplified by a Du!ng equation, which contains parameter uncertainties in the right hand side of this nonlinear di erential equation. Even though this type of circuit has been object of a variety of control strategies in the past, very few papers have been devoted to the design of an optimal control law with a quadratic performance index and subject to that type of uncertainties. We develop simulation examples that show that not only an optimal control strategy can be applied to such a system, in case of an optimal control strategy without uncertainties, but an optimal control of the mini-max type can also be implemented in case of the uncertainty in the model. Keywords: Nonlinear Electrical Circuit, Mini-max Control, Uncertain Parameters. 1. INTRODUCTION An interesting example of a nonlinear electric oscillator is described by the so called Du!ng equation. Such a circuit has attracted the attention of the control community because it represents a complex chaotic nonlinear system, which has a variety of applications ranging from physics to engineering (see Kapitaniak (1996), Nijmeijer and Harry (1995), Loria et al. (1998)). Given the complex behavior of the circuit, the control task becomes really challenging. Most results found in the literature, regarding the control design, are based on nonlinear control theory (Kapitaniak (1996), Nijmeijer and Harry (1995)). By instance, the Lyapunov design method has been widely used in controlling this circuit. More often the control objective in the mentioned papers is to stabilize the circuit variables around an equilibrium point or follow a given trayectory (tracking problem see Nijmeijer and Harry (1995)). We can also find control laws of the state feedback and observed based design (for more methods of controlling the Du!ng equation, see Chen and Dong (1993)). The well-known electrical circuit with a nonlinear element, which is represented as a nonlinear inductor, an alternating source of voltage, a pure resistive element, and an electrical capacitor is shown in the Figure 1. Applying the node law of the circuit theory, such a circuit can be modelled as g2 1 g 1 3 3 H0 + + + = cos $w gw2 UF gw F F U
(1)
where is the magnetic i oux through the nonlinear inductor, H0 is the alternating source voltage, U and F are the constants of the capacitor, 1 and 3 are some operation constants. The nonlinear term appears because of the nonlinear inductor, which is an inductor with a ferromagnetic core, and is modeled, if an abstraction of the hysteresis phenomenon is made, by an l- nonlinear characteristic. Here l is the electrical current. Such a 978-3-902661-93-7/11/$20.00 © 2011 IFAC
Fig. 1. Electrical Network with a nonlinear element. characteristic is approximated by a constitutive relation of the form (see Hasler and Neirynck (1986)): (2) l = 1 + 3 3 Defining the variables: 1 3 1 H0 s1 = ; s2 = ; s3 = ; (w) = cos $w UF F F U 2 g {1 = ; {˙ 1 = {2 > {˙ 2 = 2 gw (3) the following second order system is obtained: {˙ 1 = {2 (4) {˙ 2 = s1 {2 s2 {1 s3 {31 + (w) = Depending on the choice of these constants, it is known that solutions of (4) exhibit periodic, almost periodic, and chaotic behavior (see Loria et al. (1998)). Note that this system is a nonlinear polynomial system of the third order. Consider this system in a controlled form: {˙ (w) = i (w> {) + Ex (w) + g (w) (5) { (w0 ) = {0 where:
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10.3182/20110828-6-IT-1002.02142
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
¶ µ ¶ {2 0 ; E = 1 s1 {2 s2 {1 s3 {31 (6) | g (w) = ( 0 (w) ) where the a!ne control x (w) is a physical control input, which will be designed later on. Consider also a more general form for the nonlinear term (6) represented as i (w> {> )=d0 +d1 {+d2 {{| + d3 {{{> where d1 is and standard 2D matrix, and d2 and d3 are 3D and 4D tensor (see Basin and Calderon-Alvarez (2009) for details). Such a representacion allows us to design in a compact way the suggested control law. To control such a circuit, taking a di erent approach to that in the above mentioned papers, in this work we propose to set an optimal regulator control problem to drive the nonlinear state dynamic to the origin. Observe that the resulting nonlinear system is actually one of the polynomial type. Although the optimal controller for nonlinear systems has to be determined using the nonlinear filtering theory (see Kushner (1964); Kallianpur (1980)) and the general principles of maximum Kwakernaak and Sivan (1972) or dynamic programming Bellman (1957), which do not provide an explicit form for the optimal control in most cases, there is actually a long tradition of the optimal control design for nonlinear systems (see, for example, Albrekht (1962)—Yoshida and Loparo (1989)) and the optimal closed-form filter design for nonlinear Wonham (1965); Benes (1981); Yau (1994), and in particular, polynomial systems (Basin (2003), Basin et al. (2008)). However, the "optimal" quadratic controller problem for nonlinear, in particular, polynomial, systems with parameters belonging to an uncertain finite set, to the best to the authors knowledge, has not even been consistently treated. All of the above mentioned papers study this problem if the model of the considered dynamics is known with accuracy, but for many applications this assumption is unrealistic, because in practice it is common to find some sort of uncertainties. Hence, it is desirable to develop some kind of robustness for the optimal control problems to deal with such possible uncertainties which may have an impact on the nonlinear dynamics of the system. i (w> {) =
µ
2. OPTIMAL CONTROL DESIGN Our objective is to drive the circuit variables to the origin. To that end, consider the following quadratic performance index is set as the control performance measure: 1 j ({ (w) > x (w)) = {| (W ) T{ (W ) + 2 ZW (7) 1 ({| (w) O{ (w) + x| (w) Ux (w)) gw 2 w0
The performance index is given in standard Bolza form where it is assumed that U is a strictly positive definite and symmetric matrix, O and T are two non-negative definite symmetric matrices, W A w0 is a certain time moment, and dW denotes transpose to a vector (matrix) d. The solution of the optimal regulator (control) problem for polynomial systems with linear control input and a quadratic criterion is given by the following feedback control that realizes the optimal control with respect to the quadratic criterion given in (7), for the polynomial system (5): (8) x = U1 E | [P (w) + s(w)]
where the matrix function P is the solution of the Riccati type equation: | P˙ (w) = O + [d1 + 2d2 {(w) + 3d3 {(w){| (w)] P (w)+ | 1 | P (w) [d1 + d2 {(w) + d3 {(w){ (w)] P (w)EU E P (w) (9) with terminal condition P (W ) = T> and the parametrized vector function s is solution of the linear equation: s(w) ˙ = [d1 + 2d2 {(w) + 3d3 {(w){| (w)]| s(w)(10) P (w)EU1 E | s(w)+P (w)g(w) with boundary conditions s (W ) = 0= This result follows from the aplication of the maximum principle, the proof can be found in Basin and Calderon-Alvarez (2009). 3. ROBUST OPTIMAL CASE To solve a regulator problem of the most general type for such a circuit, we introduce a di erent type of control concept for a polynomial system subject to uncertainties. Consider the uncertain circuit as follows: {˙ (w; ) = i (w> {> ) + E (w; ) x (w) + g (w; ) (11) { (w0 ; ) = {0 where is a parameter which belongs to a given parametric set A. In this paper, we consider A as a finite set, that is A = {1 > 2 > ===> Q } > each one representing a possible realization or possible model of the system. The time variable varies in an interval w 5 [w0 > W ]. By instance, the parameters of the polynomial system (11) belongs to: d0 = {d0>1 > d0>2 > ===> d0>Q } E = {E1 > ===> EQ } g = {g1 > ===> gQ } The circuit dynamics is given by a family of Q di erent possible non-linear di erential equations sometimes called Multi-Model problem, with no information about the trajectory that is realized. So, it appears that for this type of problems we have Q possible state dynamic equations, each of them describing a model and there is no a priory information which will be the active one, but of course, it will be at least one.The system (5) is assumed to be uniformly controllable; the definitions of the uniform controllability for nonlinear systems can be found in Isidori (2001). For each fixed parameter , the non-linear function of the circuit i (w> {> ) is a polynomial of 2 variables, which are the components of the state vectors { (w; ) 5 R2 = Following the previous work (see Basin et al. (2006)), a s-degree polynomial of a vector { (w; ) 5 Rq is regarded as a s-linear form of q components of { (w) > that is to say: i (w> {> ) =d0 (w; ) +d1 (w; ) {+d2 (w; ) {{| + · · · (12) +d3 (w; ) {{{= Here, the involved parameters are: d0 is vector of dimension 2, d1 is a matrix of dimension 2 × 2> d2 is a 3D tensor of dimension 2 × 2 × 2> etc= Remark 1. Clearly, the uncertainty on the realized parameters is represented by the value of . Such a parameter belongs, as we said, to the finite set A that contains all the possible scenarios or parametric realizations of the nonlinear plant, which is fixed during the actual process, with no possibility of change once the process has started. So, each trajectory is uniquely determined by such a set of parameters. Nevertheless, there is no information on the trayectory which is realized. In this way, the proposed
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
control should deal with all of the parameters and should provide an acceptable behavior for such a class of systems. The quadratic cost criteria to be minimized is: 1 j ({ (w; ) > x (w) > ) = {| (W ; ) T{ (W ; ) + 2 ZW 1 ({| (w; ) O{ (w; ) + x| (w) Ux (w)) gw= 2
(13)
w1
the criteria with uncertain dynamics, have been studied in the past (see Jimenez-Lizarraga and Poznyak (2007)). The problem is stated and solved applying a min-max concept, that is, taking the worst case scenario of the functional and then minimizing by the control. Therefore the problem to solve is: min max j ({ (w; ) > x> ) (14) x(w) 5A Therefore, the philosophy of design here is really based on the concept of min—max control where the operation of maximization is taken over a set of uncertainties (in our case, a parameter from a finite set) and the operation of minimization is taken over a set of admissible control strategies. Thus, this paper is focused on the design of a control, which exhibits some sort of robustness property for a class of multi-plant polynomial systems given by a system of ordinary di erential equations with parameters from a given finite set. As well-known (Pontryagin et al. (1962)), the Bolza problem can be simplified expressing the original cost function (7) as a Mayer-type functional, that is, a cost function depending only on the terminal values of the state vector; such a transformation involves the extension of the state space. In what follows we define: Zw 1 {q+1 (w; ):= ({| (; ) O{ (; ) +x| () Ux ()) g> 2 w1
taking the derivative in time: {˙ q+1 (w; ) = {| (w; ) O{ (w; ) + x| (w) Ux (w) > the new cost function is given by: 1 j ({ (w; ) > x (w) > ) = {| (W ; ) T{ (W ; ) + {q+1 (W ; )= 2 Evidently the term {| (W ; ) T{ (W ; ) does not depends on the new introduced coordinate, that is: C {| (W ; ) T{ (W ; ) = 0 C{q+1 (W ; ) We proceed with finding the necessary condition for minimax optimality, for the new Mayer problem. Introduce the following Hamiltonian function: K (w,{,t,x,) :=t | (w; ) [i (w,{,) +E (w;) x (w) +g (t;)] 1 | +tq+1 (w; ) ({| (w; ) O{ (w; ) + x| (w) Ux (w)) > 2 which depends on the given uncertain vector. Following the general mini-max necessary conditions given in the original work (Boltyanski and Poznyak (1999)), the vectors t (w; ) satisfy the property: g C t (w; ) = [K (w> {> t> x> )] = gw C{(w; µ ) ¶| Ci (w> {> ) | tq+1 (w; ) O{ (w; ) + t (w; ) = C{(w; ) For the last coordenate the time derivative is: t˙q+1 (w; ) = 0
and the transversality conditions for these vectors at the terminal time are: t (W ; ) =- () judg [{| (W ; ) T{ (W ; ) +{q+1 (w; )] = () T{ (W ; ) tq+1 (W ; l ) = () where the constant () is a non-negative real value appearing from the general case of the mini-max problem (for more details on this conditions see also Poznyak et al. (2002)). For each fixed parameter the partial derivative of i (w> {> ) in { is given by: Ci (w> {> ) =d1 (w; ) +2d2 (w; ) {+3d3 (w; ) {{| (15) C{(w; ) Consider the following technique of summation of the individual Hamiltonian functions for each l (l = 1> ===> Q ) > where the summation is taken over all the possible uncertainty parameters. Then, it is still possible to introduce a generalized Hamiltonian function encompassing all of the family plants: Q P K (w,{,t,x) = [(t | (w; l ) [i (w> {> l ) +E (w; l ) x (w) +
l=1 ¤ | (w; l ) 12 ({| (w; l ) O{ (w; l ) +x| (w) Ux (w)) g (w; l )]) +tq+1 which allows us the find the minimax control as: Q X (16) l E | (w; l ) t (w; l ) > x (w) = U1 l=1
where we find the vector of parameters = (1 > 2 > ===> Q )| ) ( Q X Q Q l = 1 (17) V := 5 R : l 0; l=1
Similarly to the linear-quadratic case, the mini-max control appears as a mixture of the controls which are the optimal strategies for each fixed parameter value , and the controls are to be found in a multi-dimensional simplex set (17). Now the mini-max strategies are to be found in a finite dimensional space instead of the original function space. 4. PARAMETRIZED MINI-MAX CONTROL FOR DUFFING EQUATION Let us now introduce the following diagonal block matrices: 5 6 6 5 i (w> {> 1 ) ... 0 O ... 0 9 : 9. .. .. .: f := 7 8 ; L:= 7 .. = .. 8 . = . 0 ... i (w> {> Q ) 0 ... O 5
1 Lq×q 9 .. := 7 . 0
6 5 T ··· 0 : 9 .. .. .. ; Q:= 8 7 . . . 0 · · · 2 Lq×q
··· .. . ···
6 0 .. : . 8 T
L 5 Rq×q l = 1> 2 (18) and the following tensor matrix for the coe!cients of the polynomial: 6 5 dl · · · 0 : 9 al := 7 ... . . . ... 8 0 · · · dl l = 0> ===> v
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B| := [E | (w> 1 ) · · · E | (w> Q )] >
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
{˙ 11 = {12 = 1=1 {11 0=4 {12 ({11 )3 + 2=05 cos(1=8 w) + x {˙ 21 = {22 2 2 {˙ 2 = 1=15 {1 0=45 {22 1=05 ({21 )3 + 2 cos(1=9 w) + x
The extended dynamics for the polynomial systems appears as: x˙ = f (w> x (w)) + B (w) x (w) + d (w) and the extended vector d is defined as d := (g| (w; 1 ) > ===> g| (w; Q ))
Selecting U = 1, T = 1, O = 1 and W = 1=5, we obtain the performance index as a function of the weighting parameter near the extremum point (see Fig. 1). The corresponding state variable dynamics is depicted at Fig. 2 where the blue and green line correspond to the states 1 and 2 of the first plant and the red and light blue represent lines states 1 and 2 of the second plant. The control law is shown in Fig. 3, and the criterion in Fig. 4. Here, the blue line is for plant 1 and the green line is for plant 2. 2.66 2.65 2.64 2.63 Performance index
Note that the dependence on the uncertain parameter has disappeared, the new dynamics includes the complete family of plants, but the control remains the same for all plants. That means that the same control will be applied for all the dynamics simultaneously. Based on the extended system, the mini-max regulator that realizes (14) with respect to the quadratic criterion given in (13), for the polynomial system (11), takes the form: (19) x = U1 B| [M + p ] where the matrix function M is the solution of the following Riccati type equation: ˙ = L + [a1 + 2a2 x+3a3 xx| + === M | +vav x· · · (v 1) wlphv · · ·x] M + | M [a1 + a2 x + a3 xx + === + av x· · · (v 1) wlphv · · ·x] M BU1 B| M (20) with terminal condition M (W ) = Q> and the parametrized vector function p is the solution of the linear equation: p˙ = M a0 + [a1 +2a2 x+3a3 xx| +===+vav x· · · (v 1) wlphv · · ·x]| p M BU1 B| p + M d (21) with terminal condition as p (W ) = 0= The matrix containing the optimal weight parameters = solves the next optimization problem: = min M () 5V Q (22) M () := max j ({ (w; ) > x> )
{˙ 12
2.62 2.61 2.6 2.59 2.58 2.57 0
0.1
0.2
0.3
0.4
0.5 Time
0.6
0.7
0.8
0.9
1
Fig 1. Performance index M().
1.5 1 0.5
5A
0
State
with x (w) given ¡in (19) parametrized by the vector = ¢ | (1 > 2 > ===> Q ) l 5 V Q through (20) and (21). Remark 2. In the case of a fully known plant ( = 1), that is, there is no parametric uncertainty, the above equations collapse into the standard known result of the optimal control for the polynomial system (Basin and CalderonAlvarez (2009)). Remark 3. The dependence on in the cost function can be seen through the dependence of the cost on the solutions of the parametrized equations (20) and (21); an expression for this can be found in Bryson (1998).
−0.5 −1 −1.5 −2 −2.5 −3 −3.5 0
0.5
1
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Time
Fig. 2. States of the circuit.
3
5. SIMULATION EXAMPLE 2.5
2
Control
In this section we present two numerical examples for controlling of the Du!ng equation. In the first case, we consider the optimal control of the circuit with no uncertainties, the second one presents the numerical results produced by the robust algorithm for a given family of parameters.
1.5
1
0.5
0 0
0.5
1 Time
5.1 Example 1
Fig. 3. Control Signal for W .
Consider the following multi-model circuit 5792
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
6. CONCLUSION
3
2.5
This paper studied the control of a nonlinear electrical circuit modeled by a Du!ng equation which contains parameter uncertainties in the right-hand side of this nonlinear di erential equation. We implemented two type of control laws for this circuit, optimal and robust optimal. This second strategy allows us to solve the problem of controlling the Du!ng equation with uncertainties in the parameters. The simulations examples show the good performance of both controllers for a given set of parameters.
Criterion
2
1.5
1
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0 0
0.5
1
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Time
Fig. 4. Criterion for W .
REFERENCES 5.2 Example 2 Consider the following one-model circuit: {˙ 12
{˙ 11 = {12 = 1=1 {11 0=4 {12 ({11 )3 + 2=1 cos(1=8 w) + x
The parameters of the cost function are: U = 1, T = 1, O = 1 and W = 1=5, we obtain (see Fig. 1) = 0 and m( ) = 0=7167. The corresponding state variable dynamics is depicted at Fig. 5 where the blue and green line correspond to the states 1 and 2. The control law and criterion are shown in Fig. 6 and 7. 1.5 1 0.5 0
State.
−0.5 −1 −1.5 −2 −2.5 −3 −3.5 0
0.5
1
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Time.
Fig. 5. States one model. 4
3.5
3
Control.
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2
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1
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0 0
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1
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Time.
Fig. 6. Control one model. 3
2.5
Criterion.
2
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1
0.5
0 0
0.5
1 Time.
Fig. 7. Criterion.
1.5
Albrekht, E. (1962). On the optimal stabilization of nonlinear systems. J. Appl. Math. Mech, 25, 1254—1266. Basin, M. and Calderon-Alvarez, D. (2009). Optimal controller for uncertain stochastic polynomial systems with deterministic disturbances. International Journal of Control, 82, 1435—1447. Basin, M. (2003). On optimal filtering for polynomial system states. ASME Trans. J. Dynamic Systems, Measurement, and Control, 125, 123—125. Basin, M., Calderon-Alvarez, D., and Skliar, M. (2008). Optimal filtering for incompletely measured polynomial states over linear observations. International J. Adaptive Control and Signal Processing, 22, 482—494. Basin, M., Perez, J., and Skliar, M. (2006). Optimal filtering for polynomial system states with polynomial multiplicative noise. International Journal of Robust and Nonlinear Control, 16, 287—298. Bellman, R. (1957). Dynamic Programming. Princeton University Press, Princeton. Benes, V. (1981). Exact finite-dimensional filters for certain di usions with nonlinear drift. Stochastics, 5, 65—92. Boltyanski, V. and Poznyak, A. (1999). Robust maximum principle in minimax control. International Journal of Control, 72(4), 305—314. Bryson, A. (1998). Dynamic Optimization. Pearson Education. Chen, G. and Dong, X. (1993). On feedback control of chaotic continuos-time systems. IEEE Trans. Circuits Syst. I,, 40, 591—601. Hasler, M. and Neirynck, J. (1986). Nonlinear Circuits. Artech, House. Isidori, A. (2001). Nonlinear Systems. Berlin: Springer. Jimenez-Lizarraga, M. and Poznyak, A. (2007). Robust Nash equilibrium in multi-model LQ di erential games: Analysis and extraproximal numerical procedure. Optimal Control Applications and Methods, 8(2), 117—141. Kallianpur, G. (1980). Stochastic Filtering Theory. Springer. Kapitaniak, T. (1996). Controlling Chaos. New York: Academic,. Kushner, H. (1964). On di erential equations satisfied by conditional probability densities of Markov processes. SIAM J. Control, 12, 106—119. Kwakernaak, H. and Sivan, R. (1972). Linear Optimal Control Systems. Wiley-Interscience, New York. Loria, A., Panteley, E., and Nijmeijer, H. (1998). Control of the chaotic du!ng equation with uncertainty in all parameters. IEEE Transactions on Circuits and
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Systems-I Fundamental Theory and Applications, 45, 1252—1255. Nijmeijer, H. and Harry, B. (1995). On Lyapunov control of the Du!ng equation. IEEE Transactions on Circuits and Systems-I Fundamental Theory and Applications, 42, 473—477. Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., and Mishchenko, E. (1962). The Mathematical Theory of Optimal Processes. Interscience Publishers, New York. Poznyak, A., Duncan, T., Pasik-Duncan, B., and Boltyanski, V. (2002). Robust maximum principle for multimodel LQ-problem. International Journal of Control, 75(15), 1770—1777. Wonham, W. (1965). Some applications of stochastic di erential equations to nonlinear filtering. SIAM J. Control, 2, 347—369. Yau, S. (1994). Finite-dimensional filters with nonlinear drift. I: A class of filters including both Kalman-Bucy and Benes filters. J. Math. Systems, Estimation, and Control, 4, 181—203. Yoshida, T. and Loparo, K. (1989). Quadratic regulatory theory for analytic nonlinear systems with additive controls. Automatica, 25, 531—544.
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