galerkin method and weighting functions applied to

Galerkin Method and Weighting Functions Appliedto Nonlinear H� Control with Output Feedback

HENRIQUE CEZAR FERREIRADepartment of Electrical Engineering, University of Brasília, Caixa Postal 04591, CEP70910-900, Asa Norte, Brasília (DF), Brazil ([email protected])

ROBERTO MOURA SALESDepartment of Telecommunication and Control Engineering, University of São Paulo, Av. Prof.Luciano Gualberto, trv. 3, n. 158, CEP 05508-900, São Paulo (SP), Brazil

PAULO HENRIQUE DA ROCHABrazilian Navy Technology Center, Electro-Electronic Project Division, Av. Prof. Lineu Prestes,n. 2468, CEP 05508-000, São Paulo (SP), Brazil

(Received 30 March 2007� accepted 30 March 2008)

Abstract: This paper considers two aspects of the nonlinear H� control problem: the use of weighting func-tions for performance and robustness improvement, as in the linear case, and the development of a successiveGalerkin approximation method for the solution of the Hamilton–Jacobi–Isaacs equation that arises in theoutput-feedback case. Design of nonlinear H� controllers obtained by the well-established Taylor approx-imation and by the proposed Galerkin approximation method applied to a magnetic levitation system arepresented for comparison purposes.

Key words: Galerkin method, magnetic levitation, nonlinear H� control, successive approximation.

1. INTRODUCTION

The development of a systematic analysis of the nonlinear equivalent of the H� controlproblem was initiated by the important contributions of Ball and Helton (1989), Basar andBernhard (1990) and van der Schaft (1991). Although the H� norm is defined as a normon transfer matrices, when translated into the time domain it is nothing else than the L2-induced norm (from the input time functions to the output time functions for initial statezero). Van der Schaft (1992) showed that the solutions of the problem in question, in thecase of state feedback, can be determined from the solution of a Hamilton–Jacobi equation,the Hamilton–Jacobi–Isaacs equation (HJIE) or inequality, which is the nonlinear versionof the Riccati equation considered in the H� control problem for linear systems. The non-linear H� control problem via output feedback was considered, for example, in Isidori and

Journal of Vibration and Control, 16(12): 1817–1843, 2010 DOI: 10.1177/1077546309341140

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1818 H. C. FERREIRA ET AL.

Astolfi (1992), van der Schaft (1993) and Isidori (1994) for systems modeled by equationswhich are affine in both control and disturbance inputs and in Ball et al. (1993), Isidori andKang (1995) and James and Baras (1995) for systems modeled by equations which are notnecessarily affine in the inputs. The nonlinear output-feedback H� controllers have separa-tion structures and necessary and sufficient conditions for the H� control problem to havesolutions involve two HJIEs, associated with the design of a state feedback and an outputinjection gain, respectively.

Although the formulation of the nonlinear theory of H� control has been well developed,solving the HJIE remains a challenge and is the major bottleneck for the practical applicationof the theory (Lu and Doyle, 1995� Beard and McLain, 1998� Voulgaris and Chen, 2000�Aliyu, 2003� Abu-Khalaf et al., 2006). The HJIE is a first-order, nonlinear partial differentialequation not solved analytically in the general case and usually very difficult to be solvedfor specific nonlinear systems. Thus, several numerical methods have been proposed for itssolution.

The first method used to approximate the HJIE was the Taylor series expansion. Thismethod has been the most used in practical applications. Starting with the work of Lukes(1969), who proposed Taylor series approximation of the Hamilton–Jacobi–Belman equa-tion that arises in the nonlinear optimal control problem, many other authors have proposedsimilar approaches to the solution of the HJIE. Huang and Lin (1995) and Voulgaris andChen (2000), for example, found a smooth solution for the state-feedback HJIE by solvingfor the Taylor series expansion coefficients in a very efficient and organized manner. Thedifficulty with series approximations is that it is generally hard to solve for high-order termsin the approximation, and it is difficult to guarantee stability for finite truncations of the se-ries. In addition, the stability region of the approximate control is limited by the region ofconvergence of the underlying Taylor series. This region was investigated in Yazdanpanah etal. (1999) for the state-feedback case, but it is impossible to get an approximation a priori.

The classical methods of finite difference and finite element solutions have also beenused to approximate the HJIE (Xiao and Basar, 1997). For these methods, the computationalload and computer memory required for the approximation grow exponentially with the di-mension of the state of the system. Instead of solving the HJIE directly, Lu and Doyle (1995)studied the convexity of the nonlinear H� control problem in terms of nonlinear matrix in-equalities (NLMIs) that are state-dependent linear matrix inequalities (LMIs). Aliyu (2003)showed that the HJIE can be solved analogously to a scalar quadratic equation with some ad-ditional side conditions and presented a computational procedure for determining symmetricsolutions.

Beard and McLain (1998) and Abu-Khalaf et al. (2006) used successive approxima-tion, also called policy iteration, to reduce the HJIE to an infinite sequence of linear partialdifferential equations, the solutions of which Beard and McLain (1998) obtained using theGalerkin method and Abu-Khalaf et al. (2006) obtained using a method based on neuralnetworks. The advantage of the successive Galerkin approximation method is that the regionin state space of convergence for the approximate control is known a priori and is usuallydefined explicitly by the designer. Moreover, finite truncations of the algorithm result instable control laws that approximate the true robust solution arbitrarily closely.

This paper has two main purposes. Firstly, it is shown that dynamic weighting func-tions can be used to improve the performance and robustness of the nonlinear H� controlleras in the design of H� controllers for linear plants. In the literature, only static weight-



GALERKIN METHOD AND WEIGHTING FUNCTIONS APPLIED 1819

ing functions have been explored. Secondly, the successive Galerkin approximation methodis used to find approximate solutions for the nonlinear H� control problem in the output-feedback case. In the literature, the above approximate methods have been applied only forstate-feedback control. The results are applied to a magnetic levitation system.

2. NONLINEAR H� OUTPUT-FEEDBACK CONTROLLER

In this section preliminary results on nonlinear H� output-feedback control are presented.Consider the following affine nonlinear state-space system:

�x � f �x�� g1�x�� g2�x�u�

z ��

h1�x�

u

��

y � h2�x�� k21�x�� (1)

where x � �n is the state vector, u � �m is the control input, � � �r is the exogenous inputwhich includes disturbance (to be rejected) and/or references (to be tracked), z � �s is thepenalty variable and y � �p is the measured variable.

The mappings f �x�, g1�x�, g2�x�, h1�x�, h2�x� and k21�x� are smooth mappings (i.e.mappings of class Ck for some sufficiently large k) defined in a neighborhood of the originin �n . It is also assumed that f �0� � 0, h1�0� � 0 and h2�0� � 0.

The control action to system 1 is provided by a controller, which processes the measuredvariable y and generates the appropriate control input u by

�� y��

u � �� (2)

where � is defined on a neighborhood � of the origin in � and � : � � �p � � and

� : � � �m are Ck functions (for some k 1) satisfying ��0� 0� � 0 and ��0� � 0.

The purposes of the controller are: to achieve closed-loop stability and to attenuate theinfluence of the exogenous input � on the penalty variable z. A controller which locallyasymptotically stabilizes the equilibrium �x� � � � �0� 0� of the closed-loop system is saidto be an admissible controller. The requirement of disturbance attenuation is characterizedas follows: given a real number � 0, it is said that the exogenous signals are locallyattenuated by if there is a neighborhood U of the point �x� �� 0� 0� so that for everyT � 0 and for every piecewise-continuous function � : [0� T ] � �

r for which the statetrajectory of the closed-loop system 1 and 2 starting from the initial state �x�0�� 0�� 0� 0� remains in U for all t � [0� T ], then the response z : [0� T ] � �

s satisfies� T

0zT �s�z�s�ds 2

� T

0�T �s��s�ds�




The nonlinear H� control problem using output feedback consists in finding an admissiblecontroller yielding local attenuation of the exogenous input. In order to describe the solu-tion of the nonlinear H� control problem using output feedback, a notion of detectability isnecessary.

Definition 1. Suppose that f �0� � 0 and h�0� � 0. The pair � f� h� is said to be locallydetectable if there exists a neighborhood U of the point x � 0 so that, if x�t� is any integralcurve of �x � f �x� satisfying x�0� � U , then h�x�t�� is defined for all t 0 and h�x�t�� 0for all t 0 implies that limt�� x�t� � 0.

Theorem 1. Consider system 1 and suppose the following:

(H1) The pair � f� h1� is locally detectable.(H2) There is a smooth positive-definite function V �x�, locally defined in a neighborhood of

the origin in �n, which satisfies the HJIE

H �x� Vx� � H�x� V Tx � � � u � � 0� (3)

where

H�x� V Tx � �� u� � Vx� f � g1� � g2u�� hT

1 h1 � 2�T� � uT u� (4)

� � 1

2 2gT

1 V Tx � (5)

u � �1

2gT

2 V Tx � (6)

(H3) There is an n � p matrix G, so that the equilibrium � � 0 of the system

�� f �� g1�� Gh2�� (7)

is locally asymptotically stable.(H4) There is a smooth positive-semidefinite function W �x� ��, locally defined in a neighbor-

hood of the origin �n � �n and so that W �0� � � � 0 for each � �� 0, which solves theHJIE

K �x�W Tx � � � y �� H �x� V T

x � � 0�

W �x�� V �x� � 0� �x �� 0� (8)

where




K �x�W Tx � �� y� � Wx� f � g1�� yT �h2 � k21�� hT

1 h1 � 2�T�� (9)

y �x�Wx�x�� 2 2h2�x�� (10)

� �x�Wx�x�� 1

2 2gT

1 �x�WTx �x�� kT

21�x�h2�x�� (11)

Then, the problem of nonlinear H� control is solved by the output feedback

�� f �� g1�� g2��u �� G�y � h2��

u � u �� (12)

where

�Wx�x�� Vx�x��G � yT �x�W T

x �� (13)

Proof. See Isidori and Astolfi (1992). The HJIE of assumption H4 is presented for the affinesystem 1, as considered in Isidori and Kang (1995). Then, � and y are computed byobserving that in the saddle point (Isidori and Kang, 1995)

K

��x� p� �� y�

�� x�p�y�

� 0� � �x� p� � ��x� p� y ��

K

y�x� p� �� y�

��y�y �x�p�

� 0�

�

Remark 1. The output injection gain G is implicitly defined by means of equation 13. Isidoriand Kang (1995) proposed that the gain G be computed as follows. Let

Wx�x�� Vx�x� � x T R1�x�� (14)

where R1�x� is a matrix which is nonsingular for each x in a neighborhood of x � 0. Theright-hand side of equation 13 can be given a similar expression, say

yT �x�W T

x � � xT L�x�� (15)

and the equation in question is indeed solved by

G�x� � R�11 �x�L�x�� (16)




Proposition 1. (Isidori and Astolfi (1992)). Suppose that the system is linear, given by

�x � Ax � B1� � B2u�

z ��

C1x

u

��

y � C2x � D21�� (17)

Suppose the following:

(L1) The pair �A� B1� is stabilizable.(L2) The pair �A�C1� is detectable.(L3) There is a positive-definite symmetric solution X of the Riccati equation

AT X � X A � CT1 C1 � X B2 BT

2 X � 1

2X B1 BT

1 X � 0� (18)

(L4) There is a positive-definite symmetric solution Y of the Riccati equation

Y AT � AY � B1 BT1 � Y CT

2 C2Y � 1

2Y CT

1 C1Y � 0� (19)

(L5) ��XY � � 2.

Then, hypotheses H1–H4 of Theorem 1 hold, with

G � ZCT2 �

V �x� � xT X x�

W �x� �� 2�x � ��T Z�1�x � ��

Z � Y

�I � 1

2XY

��1

�

Existence of a solution to equation 8 is shown in the conditions of the following propo-sition.

Proposition 2. (Isidori and Astolfi (1992)). Suppose that the system is nonlinear and itslinear approximation at equilibrium �x� �� u� � �0� 0� 0� satisfies the hypotheses L3, L4 andL5 and:

(L3 ) The matrix A � 1 2 B1 BT

1 X � B2 BT2 X has all eigenvalues with negative real part.

(L4 ) The matrix A � 1 2 Y CT

1 C1 � Y CT2 C2 has all eigenvalues with negative real part.




Suppose that the basic controller

�� A � B1 F1 � B2 F2 � GC2�� Gy�

u � F2�� (20)

where F1 � �1� 2�BT1 X , F2 � �BT

2 X and G � ZCT2 , is either controllable or observable.

Then, hypotheses H2–H4 of Theorem 1 hold, with G as in Proposition 1.

3. THE WEIGHTED NONLINEAR H� CONTROL PROBLEM

The penalty variable z comprises any output whose L2 norm is desired to be minimized. Inthe linear H� controller design, the penalty variable can be weighted as required, through sta-tic weighting, frequency (linear) weighting, nonlinear weighting or any combination of these.The penalty variable can include tracking error, actuator effort, sensitivity specifications vianorm bounds, robustness specifications via norm bounds and frequency-response criteria.The selection of weighting functions for the linear design problem is detailed in Zhou andDoyle (1998). On the other hand, in the nonlinear H� controller design, only static weight-ing functions have been explored (Sinha and Pechev, 2004). It is shown here that dynamicweighting functions can also be used in the nonlinear H� controller design. Similar effectsto the linear case are observed for a magnetic levitation system, as shown in Section 5.2.

Consider a nonlinear system modeled by equations of the form

�xp � f p�xp�� gp1�xp�� gp2�xp�u�

y � h p2�x p�� kp21�xp�� (21)

Consider that linear dynamic weighting functions W1�s� and W3�s� with state-space descrip-tion

�x�1 � A�1x�1 � B�1�h p2�x p�� kp21�xp��

z1 � C�1x�1�

�x�3 � A�3x�3 � B�3h p2�xp��

z3 � C�3x�3 � D�3h p2�xp� (22)

are included in the design as in Figure 1. In addition, consider that

z2 � W2u (23)

weights the control effort through the constant gain W2, for simplicity. Combining equations21–23, the system can be expressed as system 1 with




Figure 1. Closed-loop system including weighting functions.

x �

��x p

x�1

x�3

�� f �x� �

��f p�xp�

A�1x�1 � B�1h p21�x p�

A�3x�3 � D�3h p2�xp�

��

g1�x� �

��gp1�xp�

B�1kp21�x p�

0

�� g2�x� �

��gp2�x p�

0

0

��

h1�x� �

��C�1x�1

0

C�3x�3 � D�3h p2�x p�

�� k11�x� �

��0

0

0

��

k12�x� �

��0

W2

0

�� h2�x� � h p2�x p�� k21�x� � kp21�xp��

Suppose that there is a linear H� controller for the linear system obtained from thelinearization of system 21. Then, under the conditions of Proposition 2, there will be anonlinear H� controller for the nonlinear system 21. In other words, if the Riccati equations18 and 19 are solvable for the linearized system, then the HJIEs 3 and 8 are solvable in aneighborhood of the origin in �n .

The same weighting functions used in the linear project are used here for the nonlinearproject. Since for nonlinear systems the frequency-domain properties are very complex, apractical approach is to evaluate the effects of weighting functions specifically for each casestudy.




4. SOLUTION OF HJI EQUATIONS FOR NONLINEAR H� CONTROL

PROBLEMS

In this section a method to solve the HJIE associated with the nonlinear output-feedback H�control problem is developed. The proposed method is similar to the one presented in Beardand McLain (1998) to solve the HJIE that arises in the nonlinear state-feedback H� controlproblem. The basic idea is to reduce the HJIE to an infinite sequence of linear partial dif-ferential equations, the solutions of which are then obtained by the Galerkin approximationmethod.

4.1. Successive Approximation

Successive approximation methods have been widely explored in the control literature sincethe work of Bellman (1957). The application of such methods to transform the HJIE as-sociated with the nonlinear state-feedback control problem in a sequence of linear partialdifferential equations was explored for the first time in Wise and Sedwick (1994).

In Section 4.1.1 a brief review of the state feedback, as proposed in Beard and McLain(1998), is presented and in Section 4.1.2 the output-feedback case is developed.

4.1.1. State Feedback

The HJIE 3 can be rewritten as

V x � f � g1�

� g2u �� hT1 h1 � �u �2 � 2�� 2 � 0� (24)

with

u � �1

2gT

2 V x

T � � � 1

2 2gT

1 V x

T �

The basic idea of successive approximation is that instead of computing V and u si-multaneously, they are iteratively computed, as in the following algorithm proposed in Beardand McLain (1998).

Beard and McLain (1998) and Abu-Khalaf et al. (2006) proved the existence and stabil-ity of u�i� and ��i� j�.

4.1.2. Output Feedback

Substituting equations 9 and 10 into the HJIE 8, it can be rewritten as

Wx� f � g1� �� 2 2hT2 �h2 � k21� �� hT

1 h1 � 2�� 2 � H � 0� (25)

where � is given by equation 11 and H is obtained by substituting equations 5 and 6 intoequation 4, as follows:




Algorithm 1. (Beard and McLain (1998)). Let u�0� be an initial stabilizing control lawfor the system 1 (� � 0) with stability region �.For i � 0 to�

Set ��i�0� � 0For j � 0 to�

Solve for V �i� j� from

V �i� j�x � f � g1�

�i� j� � g2u�i� j�� hT1 h1 � �u�i��2 � 2��i� j��2 � 0

Update the disturbance

��i� j�1� � 1

2 2gT

1 V �i� j�x

T

EndUpdate the control

u�i�1� � �1

2gT

2 V �i� j�x

T

End

H � H�x� V Tx � � � u � � Vx f � 1

4 2Vx g1gT

1 V Tx �

1

4Vx g2gT

2 V Tx � hT

1 h1� (26)

As in the state-feedback case, the HJIE 25 can then be transformed in a sequence oflinear partial differential equations by the following algorithm:

Algorithm 2. Let � be an exogenous input with stability region �.For i � 0 to�Solve for ��i� from

Wx� f � g1��i�� 2 2hT

2 �h2 � k21��i�� hT

1 h1 � 2��i��2 � H � 0

H � Vx f � 14 2 Vx g1gT

1 V Tx � 1

4 Vx g2gT2 V T

x � hT1 h1

Update the exogenous input

��i�1� � 1

2 2gT

1 W Tx � kT

21h2

End.

The proof of convergence of this algorithm is similar to the proof of convergence of Algo-rithm 1 presented in Abu-Khalaf et al. (2006).

In both cases, state and output feedback, the linear partial differential equations are noteasily solved in general. In the state-feedback case, to solve such equations, the characteris-tics method was used in Wise and Sedwick (1994) and a method based on neural networkswas employed in Abu-Khalaf and Lewis (2005). In the next section, the Galerkin approx-imation method is explored to obtain an approximated solution to the corresponding linearpartial differential equations.




4.2. Galerkin Approximation Method

Consider a partial differential equation ��V � � 0 with boundary conditions V �0� � 0. TheGalerkin method assumes a complete set of basis functions ��[ j]��j�1, so that �[ j]�0� � 0,for all j , and V �x� � �

j�1 c[ j]�[ j]�x�, where the sum is assumed to converge point-wise in some set �. An approximation to V is formed by truncating the series to VN �x� � N

j�1 c[ j]�[ j]�x�. The coefficients c[ j ] are obtained by solving the algebraic equations��

��VN �x��[ j]�x�dx � 0� j � 1� � � � � N � (27)

For a more rigorous and complete treatment see, for example, Fletcher (1984).

4.2.1. State Feedback

Assume that u : � � �m is a feedback control law that asymptotically stabilizes system

1 on a compact set �. Also assume that the set ��[ j]��j�1 is a complete basis set for thedomain of the HJIE 3. Then, an approximate solution is given by

VN �x� �N�

j�1

c�[ j]�� [ j]�x� � ��c� � (28)

with

c� � [ c� [1] � � � c� [N ] ]T � ��x� � [ �� [1] � � � �� [N ] ]T �

where the coefficients satisfy the equation��

��VN �x� f � g1� � g2u�� hT

1 h1 � �u�2 � 2��2�� [ j]dx � 0� (29)

with

�VN �x � VN

x�

N�j�1

c� [ j ] �� [ j]

x� ��T

� c� � (30)

After algebraic manipulations (see Beard and McLain (1998)), equation 29 can be rewrit-ten as

�A1 � A2�� A3�u��c� � b1 � 2b2�� b3�u�� (31)

where

A1 ��

�

�� f T��T� dx� A2��

��

��T gT

1 ��T� dx�




A3�u� ��

�

��uT gT2 ��T

� dx� b1 � ��

�

��hT1 h1dx�

b2��

�

��2dx� b3�u� ��

�

��u�2dx �

We do not need to recompute the integrals A2��, A3�u�, b2�� and b3�u� at each stepof the algorithm. Using VN to compute and to update control u and exogenous input � , oneobtains

� � 1

2 2gT

1

VN

x� 1

2 2gT

1

N�j�1

c�[ j] �� [ j]

x� 1

2 2gT

1 ��T� c� � (32)

u � �1

2gT

2

VN

x� �1

2gT

2

N�j�1

c� [ j] �� [ j]

x� �1

2gT

2 ��T� c� � (33)

Then,

A2�c�� 1

2 2

��

��cT� ��g1gT

1 ��T� dx

� 1

2 2

N�j�1

c� [ j]��

��

�T� [ j]

xg1gT

1 ��T� dx � 1

2 2

N�j�1

c�[ j]G1[ j]�

A3�c�� 1

2

��


2 ��T� dx

� �1

2

N�j�1

c�[ j]��

��

�T� [ j]

xg2gT

2 ��T� dx � �1

2

N�j�1

c�[ j]G2[ j]�

b2�c�� 1

4 4

��


1 ��T� c�dx

� � 1

4 4

N�j�1

c�[ j]��

��

�T� [ j ]

xg1gT

1 ��T� dxc� � � 1

4 4

N�j�1

c� [ j]G1[ j]c� �

b3�c�� 1

4

��


1 ��T� c�dx

� 1

4

N�j�1

c� [ j]��

��

�T� [ j]

xg1gT

1 ��T� dxc� � 1

4

N�j�1

c�[ j]G1[ j]c� �

Therefore, the coefficients c� pull outside the integrals and A2��, A3�u�, b2�� and b3�u�can be computed iteratively once the matrices �G1[ j]�N

j�1 and �G2[ j]�Nj�1 have been calcu-

lated.




The Galerkin approximation to the HJIE can be combined with Algorithm 1 to producethe following algorithm for computing an approximation to the HJIE:

Algorithm 3. (Beard and McLain (1998)). Let u�0� be an initial stabilizing control lawand� = 0. Pre-compute the integrals A1, A3�u�0��, b1, b3�u�0��, �G1[ j]�N

j�1 and �G2[ j]�Nj�1.

For i � 0 to�If i � 0

A�i� � A1 � A3�u�0��

b�i� � �b1 � b3�u�0��

Else

A�i� � A1 � 1

2

Nk�1 c�i�1�

� [k]G2[k]

b�i� � �b1 � 1

4

Nk�1 c�i�1�

� [k]G2[k]c�i�1�� [k]

EndFor j � 0 to�

If j � 0A � A�i�

b � b�i�

Else

A � A�i� � 1

2 2

Nk�1 c�i� j�1�

� [k]G1[k]

b � b�i� � 1

4 2

Nk�1 c�i� j�1�

� [k]G1[k]c�i� j�1�� [k]

Endc�i� j�� A�1b

EndEnd

u � �1

2gT

2 ��T� c�

� � 1

2 2gT

1 ��T� c�

The stopping criterion used is �c�i� j�1�� c�i� j�� and �c�i�1��

� � c�i�� for �small (typically 0.001).

4.2.2. Output Feedback

In this section, an algorithm for output-feedback control design is proposed. In fact, theproposed algorithm can be seen as a dual version of the state-feedback algorithm originallyproposed in Beard and McLain (1998). Assume that � : ��

s is the exogenous input for




the system 1 in a compact set �. Also assume that the set �� [ j]��j�1 is a complete basis setfor the domain of the HJIE 8. Then, according to equation 27, an approximate solution toequation 25 is given by WN �x� � �W �x�cW , where the coefficients satisfy the equation�

�

��WN �x� f � g1� �� 2 2hT

2 �h2 � k21� �

� hT1 h1 � 2�� 2 � H �x� �VN �x�

�� [ j]dx � 0� (34)

with

H �x� �VN �x� � �VN �x f � 1

4 2�VN �x g1gT

1 �VN �Tx �

1

4�VN �x g2gT

2 �VN �Tx � hT

1 h1 (35)

and

�WN �x � WN

x�

N�j�1

c� [ j] �� [ j]

x� ��T

�c�� (36)

After some algebraic manipulations, it can be rewritten as

�A1 � A2�� c� � 2 2b1 � 2 2b2�� 2b3�� b4 � 1

4 2b5 � 1

4b6� (37)

where

A1 ��

�

�� f T��T�dx� A2��

��

��T g

T1 ��T

�dx�

b1 ��

�

��hT2 h2dx� b2��

��

��T k

T21h2dx�

b3��

�

��T � dx� b4 �

��

�� f T��T� c�dx�

b5 ��

�


1 ��T� c�dx� b6 �

��


2 ��T� c�dx �

Note that c� is computed using algorithms. Moreover, it is not necessary to recompute theintegrals A2��, b2�� and b3�� at each step of the algorithm. Substituting WN �x� intoequation 11, one obtains

� �x�Wx�x�� 1

2 2gT

1 ��c� � kT21h2� (38)

Substituting equation 37, we have




�A1 � 1

2 2A2�c�n��

�c�n�1�� 2b1 � b2 � 1

4 2A2�c�n�� c

�n�� 1

4 2b3 � 1

4b4� (39)

where

A1 ��

�

�� f T��T�dx� A2�c��

��

��cT��g1gT

1 ��T�dx�

b1 ��

�

��hT2 h2dx� b2 �

��

�� f T��T� c�dx�

b3 ��

�


1 ��T� c�dx� b4 �

��


2 ��T� c�dx �

Note that only the integral A2 depends on the coefficients c� and it can be rewritten as

A2�c�� N�

j�1

c� [ j]��

��

�T� [ j]

xg1gT

1 ��T�dx �

N�j�1

c� [ j]X [ j]�

The coefficients c� pull outside the integrals, and A2�c�� can be computed iteratively oncethe matrices �X [ j]�N

j�1 have been calculated.In order to calculate the output injection G, VN � ��c� and WN � ��c� are substituted

into equation 14, resulting in

xT R1�x� � cT�� cT

� �� (40)

In addition, from equations 15 and 10,

x T L�x� � 2 2h2� (41)

Therefore, according to Remark 1, G�x� can be calculated by equation 16.The convergence of Algorithms 3 and 4 has not been proved yet. The convergence

of the Galerkin approximation to the Hamilton–Jacobi–Bellman equation that arises in thenonlinear optimal control problem via state feedback is shown in Beard et al. (1997). In orderto prove the convergence of these two algorithms, the HJIEs that arise in state-feedback andoutput-feedback problems can be rewritten as generalized Hamilton–Jacobi equations, theconvergence of which was proved in Beard et al. (1997).

5. NONLINEAR H� CONTROL DESIGN FOR A MAGNETIC

LEVITATION SYSTEM

In this section, nonlinear H� controllers are designed and applied to a magnetic levitationsystem. In Section 5.2 a nonlinear controller based on the Taylor approximation method, asproposed in van der Schaft (1992), is designed to illustrate the benefits produced by dynamic




Algorithm 4. Let W �0� be an initial stabilizing solution to system 1. Let c� be thecoefficients obtained by Algorithm 2 to approximate equation 3. Pre-compute theintegrals A1, A2��, b1, b2, b3, b4 and �X [ j ]�N

j�1.

For i � 0 to�If i � 0

A�i� � A1 � 1

2 2A2�W �0�

x �

b�i� � 2b1 � 1

4 2A2�W �0�

x �c�0�� b2 � 1

4 2 b3 � 14 b4

Else

A�i� � A1 � 1

2 2

Nj�1 c�i�1�

� [ j]X [ j]

b�i� � 2b1 � 1

4 2

Nj�1 c�i�1�

� [ j]X[ j]c�i�1�� [ j]� b2 � 1

4 2 b3 � 14 b4

End

c�i�� A�i��1

b�i�

End

Extract R1�x� from xT R1�x� � cT�� cT

� ��

Extract L�x� from x T L�x� � 2 2h2

Compute the output feedback with G�x� � R�11 �x�L�x�.

weighting functions. In Section 5.3 a controller based on the proposed Galerkin approxima-tion method is designed and compared to the Taylor approximation controller. In order tofocus on the controller performance, weighting functions are not employed here.

Magnetic levitation systems have been widely used and studied, especially because mag-netic actuators present some advantages over other types of actuators, such as noncontactactuation, avoiding mechanical wear. Such systems are inherently nonlinear and open-loopunstable and the development of high-performance position controllers for levitated objectshas led to the publication of many papers in the last decade. Sinha and Pechev (2004) de-signed a nonlinear H� controller for a similar system studied in the present paper. However,they used the well-established Taylor approximation method and weighting functions werenot considered in the design.

5.1. Nonlinear Model for the Magnetic Levitation System

The magnetic levitation system considered is schematically represented in Figure 2, wherei�t� indicates the current through the magnetic bearing that produces the attraction forceF�t�, m is the mass to be levitated and x1�t� represents the deviation from the desired levi-tation gap �x1. Defining u�t� � i2�t� as the control action and considering �1 a disturbancein the force F�t�, �2 a disturbance in the gap x�t� and g the gravity force, the system isdescribed by the following equations:




Figure 2. Magnetic levitation system.

�x1 � x2�

�x2 � g � g�x2

1

� �x1 � x1�2� k

m

u

� �x1 � x1�2� 1

m�1�

y � x1 � �2� (42)

which may be put in the affine form given by equation 21, where

f p�x� �� x2

g � g�x2

1

� �x1 � x1�2

�� gp1�x� �� 0 0

1

m0

��

gp2�x� �� 0

� k

m

1

� �x1 � x1�2

�� h p1�x� �

��x1

x2

0

�� kp11�x� � 0�

kp12�x� �

��0

0

1

�� h p2�x� � x1� kp21 ��

0 1��

The adopted parameter values for the system are

k � 1�6� 10�4 N m2�A2� m � 0�240 kg� �x1 � 5 mm� g � 9�8 m�s2�




Figure 3. Magnetic levitation system.

5.2. Taylor Approximation Design Including Dynamic Weighting Functions

The weighting function W1 is specified to produce an integral effect at low frequencies andW�1

3 is a low-pass filter to attenuate noise signals, as can be seen in Figure 3.We take

W1�s� � 100

s � 0�001� W2�s� � 1�0� 10�2� W3�s� � s � 0�001

s � 1000�

These weighting functions are included as shown in Figure 1, and the global system iswritten in the affine form as in Section 3.

Since this section is focused on the effect of weighting functions, the HJIEs are solvedby Taylor series approximation only. Thus, the first-order Taylor approximation to the cor-responding HJIE 3 produces the following state-feedback law:

u � 5201�47x1 � 127�44x2 � 10099�51x3 � 6�85x4� (43)

In order to get an observer with simpler expressions, W1 and W3 were implementedand the corresponding states, x3 and x4, directly measured. Thus, the first-order Taylorapproximation to the corresponding HJIE 8 produces the following output injection:

G ��

206�29

12916�01

�� (44)




The lowest value of for which positive-definite solutions were found to equations 18and 19, which correspond to the first-order Taylor approximations to equations 3 and 8, is � 200. Figure 4(a) shows the integral effect produced by W1� a unit-step disturbance,applied at t � 0�1 s, produces a steady-state error when W1 � 1 and zero steady-state errorfor W1 as specified. Figure 4(b) shows the noise attenuation effect produced by W3� for awhite noise with standard deviation 0.1 N, added to �1, the output variance for W3 � 1 is4�67� 10�5 mm and 4�10� 10�5 mm for W3 as specified.

5.3. Successive Galerkin Approximation Design

In this section nonlinear output-feedback controller designs based on Taylor approximationand on the proposed Galerkin approximation are presented. Weighting functions are notincluded here. In this case, the Taylor approximation design is given by

u � 4706�12x1 � 125�09x2�

G ��

206�29

12916�01

�� (45)

In the Galerkin approximation design, the basis functions

�� x1 � x1�

2x1x2

12� �x1 � x1�

2x22

��

�x3

1 � 103x22

x21 � 103x3

2

�

are proposed to the state-feedback HJIE 3 and to the output-feedback HJIE 8, respectively.The state-feedback basis �� is intentionally chosen so that when substituted in equation 33a linear state-feedback control law is obtained. The corresponding coefficients are computedby Algorithm 3. Thus, Algorithm 3, initialized with the control law u�0� obtained by Taylorapproximation given by equation 45, produces

u � 6197�37x1 � 162�70x2� (46)

Note that this function basis is chosen to result in a linear control law for two reasons: (1) inorder to simplify the controller� (2) in order to compare the performance with the linear state-feedback control law obtained by the Taylor series approximation. Obviously, for anotherfunction basis, a nonlinear state-feedback control law could be obtained. The basis for theoutput-feedback design is chosen so that matrix R1 from equation 40 is nonsingular whenthe error is zero. Moreover, the gain must be sufficiently high for a small steady-state error.Note that this choice is not a simple task. Algorithm 4, initialized with W and � , obtainedby the Taylor series approximation, produces the following output injection:

G � 1

�

�1�00� 105 � 3�95� 102x2 � 7�81x1 � 7�80� 102x2

1

1�49� 102 � 5�95� 102x1 � 8�92� 104x21 � 3�90x2 � 7�81� 102x1x2

�� (47)




Figure 4. Performance and robustness simulation results.




Figure 5. Convergence of algorithms based on the Galerkin method.




Figure 6. Position and velocity for output feedback for the Taylor and Galerkin approximation controllers.




Figure 7. Noise attenuation for the Taylor and Galerkin approximation controllers for a white noise withstandard deviation 0.1 N added to �1.

� � 187�5x1 � 0�72x1x2 � 60�875x21 � 555x3

1 � 330� 1�3x2 � 4�35x21 x2

� 33125x41 � 290x3

1 x2� (48)

Figures 5(a) and 5(b) illustrate the fast convergence rate of J � �c�i�1�]� � c�i�� J �

0�001 is used as stopping criterion for Algorithms 3 and 4. The computation time for Algo-rithm 3 was 0.6 s and for Algorithm 4 it was 0.2 s. As seen in Section 4.2, the way in whichthe integrals of the Galerkin method are computed decreases the number of calculations forthe Algorithms.

The lowest value for for which Algorithm 4 converges is � 20, indicating thepossibility of a higher noise attenuation. Notwithstanding, for comparison purposes with theTaylor approximation design, � 200 was used.

The adopted stability domain � is x1 � �1�0� 10�3� and x2 � �0�3�. In this domain thecontrol law used for initializing Algorithm 4 stabilizes the closed-loop system. It is worthobserving that, while the Galerkin method does not decrease the stability domain, in theTaylor approximation method the stability domain is not known a priori, either.

Figure 6 shows position x1 and velocity x2 in the levitation transient for the Taylor andGalerkin controllers. As observed, the Galerkin controller produces a smoother transient.Figure 7 shows the higher noise attenuation produced by the Galerkin controller.




Figure 8. Effect of stability domain on system response.




Figure 9. Stability domain �x1� � a� a � 1� 2� 3� 4 mm and �x2� � b� b � 0�1� 0�2� 0�3� 0�4 m/s.




5.4. Stability Region Analysis

The purpose of this section is to investigate the effect of the stability domain assumed in thenonlinear H� controller project on the magnetic levitation system response. In this analysisthe HJIE is solved using the successive Galerkin approximation and the weighting functionsare assumed to be unitary.

Firstly, for �x1� � 3 mm and x2 � 0�1 m/s, x2 � 0�2 m/s, x2 � 0�3 m/s and x2 � 0�4 m/s,four output-feedback controllers were designed, corresponding to each one of these stabilitydomains, respectively. For each controller the initial position was set to x0 � 1 mm. InFigure 8(a) it can be seen that the overshoot response increases and the rise time decreaseswhen the x2 range increases.

Controllers were also designed for �x2� � 0�2 m/s and x1 � 1 mm, x1 � 2 mm, x1 � 3mm and x1 � 4 mm. Figure 8(b) shows the response of each controller to initial conditionx0 � 1 mm. While overshoot and settling time increase, rise time decreases by increasingthe x1 range.

Finally, controllers were designed for stability domains with x1 between 1 and 4 mm andx2 between 0.1 and 0.4 m/s. Time responses for initial condition x0 � 1 mm were obtainedfor each controller. Figures 9(a) and 9(b) show overshoot and settling time values, respec-tively, for each stability domain. It can be seen that overshoot increases both by increasingthe �x1� range and by decreasing the �x2� range� settling time decreases by increasing the �x2�range but changes in the �x1� range do not significantly change the settling time.

6. CONCLUSION

Benefits of dynamic weighting functions are here explored in nonlinear H� control design,similarly to the linear case. Although frequency-domain properties for nonlinear systemsare very complex, similar effects to the linear case can be observed using linear weightingfunctions in a nonlinear H� controller for a magnetic levitation system.

An algorithm based on the successive Galerkin approximation for design of nonlinearoutput-feedback H� controllers is also proposed and shown to be more efficient than thealready established Taylor approximation method in the case of a magnetic levitation system.The existence and stability of control laws as well as convergence of this algorithm are thesubject of future works.

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galerkin method and weighting functions applied to

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