Trajectory Following Method on Output Regulation of Affine Nonlinear Control Systems with Relative Degree not Well Defined

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1 ITB J. Sci., Vol. 43 A, No., 011, Taectoy Following Method on Output Regulation of Affine Nonlinea Contol Systems with Relative Degee not Well Defined Janson Naibohu Industial Financial Mathematics Goup Faculty of Mathematics and Natual Science, ITB. Abstact. The poblem of output egulation of affine nonlinea systems with the elative degee not well defined by modified steepest descent contol is studied. The modified steepest descent contol is a dynamic feedback contol which is geneated by the taectoy following method. By assuming the system is minimum phase, output of the system can be egulated globally asymptotically. Keywods: minimum phase; modified steepest descent contol; taectoy following method. 1 Intoduction This pape is concened with the poblem of output feedback egulation of nonlinea systems. Many authos have consideed such poblems, using vaious appoaches. Some necessay and sufficient conditions fo egulation via static output feedback ae established in [1], by extending pevious esults of Atstein [] and Sontag [3]. Fo a class of nonlinea systems, Pomet et al. [4] developed sufficient conditions fo the global stabilization of the obseved state via dynamic output feedback by using a Lyapunov-based technique. Vincent and Gantham [5] studied two Lyapunov optimizing appoaches fo designing feedback contolles. One of them is the steepest descent contol. The steepest descent contol coesponds to specifying a pefeed diection of motion, opposite to the gadient of a specified scala-valued descent function. The steepest descent contol is effective at yielding global asymptotic stability [6]. The elated eseach is found in [7-9] which has poposed the diect gadient descent contol. Based on Gantham's idea [5-6] we study the output egulation of affine nonlinea systems (SISO) by using modified steepest descent contol. Fist we define the descent function as a squaed fom. Then we apply the taectoy following method [5] to get modified steepest descent contol which foms a dynamic feedback contolle. This contol is diffeent with contolle in [7-9]. In Received August 30 th, 010, Revised Octobe 19 th, 010, nd Revision Novembe 19 th, 010, Accepted fo publication Decembe 1 st, 010.

2 74 Janson Naibohu [7], Naibohu et al., developed the diect gadient descent contol (dgdc) fo stabilization of nonlinea systems though gateaux diffeential and in [8-9], Shimizu et al., impove the dgdc fo tacking output of nonlinea system with elative degee well defined by added a vitual contol in dgdc. By diect application of necessay condition fo a minimum, we get the static contol law which is the same as the contol law obtained by input-output lineaization technique [10]. This static contol law is effective at yielding local asymptotic stability. In this pape howeve, we show that the modified steepest descent contol egulates the output of the system globally asymptotically. Poblem Statement and Assumptions Conside SISO affine nonlinea contol system whee x x f g u, (1) y h () n R is the state vecto, u R is the contol input and y R is the n n measued output. f : R R is a smooth function with f (0) 0, g : R h R n n R and : n R ae smooth functions. Assume also that h(0) 0. Ou obective is to make the output yt () go to zeo while keeping the state bounded. The main task is to design contol u such that yt ( ) 0 as t. Fo keeping the state bounded we assume that system (1) is minimum phase, i.e. if a nonlinea state feedback can hold the output of the system identically zeo and the intenal dynamics of the system is asymptotically stable [11]. Fo designing contol input u we need an explicit elationship between the output and the input. This elationship is defined as elative degee of the system. In the following we wite the definition of elative degee of the system by Isidoi [10] Definition 1 The nonlinea system (1)-() is said to have a elative degee at a point and ( ) y u a x if 0. ( k ) y u 0 fo all x in a neighbohood of a x and all k 1, Definition 1 means that we need to diffeentiate the output of a system times to geneate an explicit elationship between the output y and input u. Let the

3 Taectoy Following Method on Output Regulation 75 elative degee of the system be. Then is a function of x and u and ( 1) y, y,, y ae functions of x, ( ) y 1 h g 0, g 0,, x x 1 g 0, g 0, x x whee y, 1,, 1. Futhemoe, we have 1 1 ( ) y f g u. (3) x x Conside equation (3). If 1 s ( x ) x s gx ( ) 0, then the elative degee of the s system is not at x. In othe wods, input u can not influence the output y. Thus, we can not do anything to the contol input u so that y becomes zeo. Let us assume that the elative degee of the system (1)-() is. Then we have 1 x Eq. (3) and gx ( ) 0. Fom input-output lineaization technique [10], contol input u is chosen as 1 1 c f c y x 0 u 1 c g x ( ) (4) if 1 gx ( ) 0 x 1 x s A poblem occues if gx ( ) 0 at a point x and the static contol law (4) can not be applied. To avoid this difficulty, many eseaches assume that n system has elative degee well defined, i.e. fo all x R elative degee of the system is. But if this assumption can not be satisfied, Hischon and Davis [1] intoduced the degee of singulaity to solve this poblem. In [10], Isidoi shows that by assuming the zeo dynamics of the system be locally asymptotically stable we can choose a static contol law such that output of the system goes to zeo while keeping the state of the system bounded

4 76 Janson Naibohu locally. In this pape we want to extend this esult globally by using the modified steepest descent contol. The modified steepest descent contol is a dynamic feedback contol of the fom df u v, (5) du whee F is a descent function and v is an atificial input ( F will be defined in section 3 and v will be detemined such that the descent function F deceases). Thus, in this pape we show that by assuming Assumption 1 System (1) is minimum phase, the modified steepest descent contol (5) will ende the output of the system (1)-() to zeo if time t goes to infinity even if the elative degee of the system not well defined. 3 Taectoy Following Method The taectoy following method [5] is a numeical optimization method based on solving continuous diffeential equations. This method is taectoy following in the sense that we move fom an initial guess fo the solution to a solution point along a taectoy geneated by a set of diffeential equations. Conside fist the poblem of minimizing pefomance function Gu ( ), G : R R, subect to no constaints. Let pefomance function Gu ( ) and use the local minimum point condition fo integating the diffeential equation, * u povide a local minimum fo * u as an initial u f ( u) (6) whee f () is a function at ou disposal, to be detemined shotly. Calculate the time deivative of Gu ( ) along the taectoy geneated by the solution to (6). Then at u u * dg G f u dt u * * u u * ( ) 0. Since we ae inteested in a taectoy that will seach fo a minimum, the above obsevation suggests that we integate (6) by choosing (7) G f( u) u T (8)

5 Taectoy Following Method on Output Regulation 77 and equation (7) becomes dg G f ( u ) 0, u u dt u * Thus, in ode to solve the poblem of minimizing G() in an unconstained contol space, one need only choose an appopiate initial condition fo u and integate T G u u (10) until the equilibium solution is achieved. (9) 4 Modified Steepest Descent Contol We design contol law u though popeties of the solution of highe ode odinay diffeential equation. Conside a diffeential equation c y ( t) c y ( t) c y ( t) c y( t) 0 (11) ( ) ( 1) whee is the elative degee of the system (1)-(). If a polynomial p() s c s c s c s c (1) is Huwitz, then a solution of diffeential equation (11) tends to zeo if t. Remak 1 Constants c can be chosen such that the polynomial (1) is i Huwitz, i.e. the polynomial has all oots stictly in the left-half plane. So we have feedom to choose the ate of convegence that the output goes to zeo. Define a descent function (13) F( y( t), y( t),, y (t)) c y ( t) 0 whee y( t), y ( t),, y ( t) ae bounded. Reasons why we define the descent function as in (13) ae : 1. The contol input u can be designed if thee exists an explicit elationship between input ut () and output yt (). Thus, descent function (13) must be a function of y () t.

6 78 Janson Naibohu. We need that the descent function F( y( t), y ( t),, y ( t)) is a quadatic fom and that F(0,0,,0) must equal zeo. Accodingly, the minimum value of descent function F( y( t), y ( t),, y ( t)) is zeo, then F( y( t), y ( t),, y ( t)) is zeo. If 0 polynomial (1) is Huwitz, then y tends to zeo. c y () t 0. Futhemoe, if Ou obective is to find a contol law ut () fo all t such that the descent function F( y( t), y ( t),, y ( t)) becomes minimum and state xt () is at least bounded. Then ou poblem is fomulated as follows. u() t min F( y( t), y ( t),, y ( t)) (14) sub. to x ( t) f ( x( t)) g( x( t)) u( t) (15) h( x( t)). y t (16) When u is a scala, a necessay condition fo a local minimum is given as follows. whee df y y ( ) (,,, y ) du 0, (17) ( ) 1 df( y, y,, y ) c cy ( ) g x. (18) du 0 x Fom the necessay condition (17) we have o cy 0, (19) 0 1 gx ( ) 0 x (0) By diect application of the necessay condition (19) fo a minimum we get the contol law (4). In the following we use a numeical technique to solve the minimization poblem(14)-(16) at evey point of a taectoy. One numeical techniques is the

7 Taectoy Following Method on Output Regulation 79 taectoy following method which have been eviewed in section 3. The idea of the taectoy following method is to solve the numeical optimization poblem only once, at the initial point of the taectoy. Fo all othe points x along a taectoy, the contol u is detemined fom the diffeential equation df u du c ( ) 1 cy g x. (1) 0 x The contol law in equation (1) is called the steepest descent contol. Calculate the time deivative of descent function (13) along the taectoy of the extended system Then we have x f g u, () F y y 1 u c ( ). cy g x (3) 0 x ( ) (,,, y ) d. dt ( ) c y c 1y c y (4) 0 1 By substituting (3) and (1) into (4), we have F y y ( ) (,,, y ) 1 d c y c 1y c f x 0 1 dt x ( ) 1 1 d g u 4 c cy ( ). g x dt x (5) 0 x Fom equation (5), the value of time deivative of descent function (13) along the taectoy of ()-(3) can not be guaanteed to be less than zeo fo t 0. Conside the extended system ()-(3) and time deivative of descent function (5). We do not have a vaiable which will be used to push the time deivative of descent function (5) less than zeo.

8 80 Janson Naibohu Now we modify the steepest descent contol (3) by adding an atificial input v. Then the extended system ()-(3) becomes x f g u, (6) 1 u c ( ). cy g x v (7) 0 x The contol law in equation (7) is called as modified steepest descent contol. The simila method using atificial input can be seen in [8]. By the same way, let us calculate time deivative of descent function (13) along the taectoy of (6)-(7) yielding F y y ( ) (,,, y ) 1 d c y c 1y c 0 1 dt x 1 1 ( ) ( ) g u 4 c cy ( ) g x 0 d x x dt x x 1 cy 0 x c g v. (8) Conside equation (8). We can choose the atificial input v such that ( ) F ( y, y,, y ) be less than zeo. If we take 1 k( x, y,, y, u) if g 0 x v 1 0 if g 0 x whee k( x, y,, y, u) d g 1 x 1 dt x c c y c f (9)

9 Taectoy Following Method on Output Regulation 81 d 1 ( x ) g ( x ) u dt x then ( ) F ( y, y,, y ) (30) 1 4 c ( ) cy g x (31) 0 x if 1 x gx ( ) 0 and less than zeo fo nonzeo cy 0. Conside the descent function (13) and its time deivative (31). The descent ( ) function (13) is bounded because y, y,, y ae bounded. If we take v as in (9) then the descent function (13) will decease until cy 0 becomes zeo. Adding atificial input v into dynamic contolle (3) is used to guaantee the descent function (13) decease to zeo ( 0 is ealistic to put the value of v becomes zeo if equation (9) becomes 1 k x y y u if g x o c 0 y v 1 o c 0 y cy (,,,, ) x ( ) 0 0 if x g 0 0 ). By this eason, it cy 0. Thus, the (3) Remak Taking v as in (9) is useful to push down the time deivative of descent function (8) so that it becomes less than zeo fo all xu, and ( ) except fo y, y,, y x s x whee 1 s ( x ) x s gx ( ) 0. Theoem 1 Assume that system (1)-() has elative degee and satisfies Assumptions 1. Choose constants c i such that the polynomial p() s c c s c s c s (33) 1 o 1 1 is Huwitz. Then, by using the dynamic contol law

10 8 Janson Naibohu 1 u c c y g v, (34) 0 x with v as in (3), y and x tend to zeo if time t goes to infinity. Poof Let Conside equation (31) and what happens if c y 0, t t 0. Fom Eq. (31) o (8), 0 1 cy is zeo. 0 F y y ( ) (,,, y ) becomes zeo. Fom equation (13) (the definition of descent function F), ( ) F ( y, y,, y ) becomes zeo. It means that ou goal to minimize ( ) F ( y, y,, y ) is achieved. Thus, if we choose c, 0,, such that polynomial (1) is Huwitz then y goes to zeo as t. By Assumption 1), if the modified steepest descent contol is applied, the output of the system goes to zeo and state x becomes bounded. 5 Example Conside the nonlinea system equation (SISO): x 1 x x x. 1 1 x x u (35) y Fo this system we have y x 3x x u and elative degee of the system is 1 0 at any point x 0 ( elative degee of system is not well defined). System (35) is minimum phase [10]. Define descent function (,, ) ( c y c y c y). (36) F y y y Then accoding to (4), the static contol law is 1 1 u x 3x 1 c y c y 0 1 x c (37) and accoding to (34) and (9), the modified steepest descent contol is u 4 x c ( c y c y c y) v, (38) 0 1

11 Taectoy Following Method on Output Regulation 83 whee 1 c y c y c x x v ( x u)(6x ) 0 1 cx ( ), x 0 o c y ( ) 0, x 0 o 0 c y 0 Conside the static contol law (37). This static contol law is fail if x becomes zeo. Simulation esults ae shown in Figue 1 fo constants: c 6; 0 c 5; c 1 and in Figue 3 fo constants: c 30; c 11; c 1. In Figue 1, afte time about t 0,335 and in Figue 3, afte time about t 0,051 computation stopped because the value of static contol law (37) becomes infinity. Simulation esults fo the modified steepest descent contol ae shown in Figue and in Figue 4. Thus, fom simulation esults, the modified steepest descent contol success to handle the poblems that aise if the elative degee of the nonlinea system not well defined. 4 3 x 1(t) 1 0 x (t) ,05 0,1 0,15 0, 0,5 0,3 0,35 time (t) Figue 1 Static Contol Law. Initial value: x (0) 3; x (0) ; Constants: 1 c 6; c 5; c

12 84 Janson Naibohu u(t) 1 x 1(t) 0-1 x (t) time (t) Figue Modified Steepest Descent Contol. Initial value: x (0) 3; x (0) ; u(0) 5; Constants: c 6; c 5; c x 1(t) 1 0 x (t) ,01 0,0 0,03 0,04 0,05 0,06 time (t) Figue 3 Static Contol Law. Initial value: x (0) 3; x (0) ; Constants: 1 c 30; c 11; c

13 Taectoy Following Method on Output Regulation u(t) 0 x 1(t) x (t) time (t) Figue 4 Modified Steepest Descent Contol. Initial value: x (0) 3; x (0) ; u(0) 5; Constants : c 30; c 11; c Conclusions We have applied the taectoy following method fo egulation output of affine nonlinea contol system. By using this method we geneate the modified steepest descent contol which egulate the output of the system globally asymptotically. The modified steepest descent contol is a dynamic feedback contol and does not need elative degee of the system well defined. This is a supeioity of the modified steepest descent contol if we compae with the static contol law which is obtained by input-output lineaization technique. Refeences [1] Tsinias, J. & Kalouptsidis, N., Output Feedback Stabilization, IEEE Tans. Automat. Contol, 35, , [] Atstein, Z., Stabilization with Relaxed Contols, Nonlinea Anal., 7, , [3] Sontag, E.D., A Univesal Constuction of Atstein's Theoem on Nonlinea Stabilization, Syst. Cont. Lett.,13(), , [4] Pomet, J.B., Hischon, R.M. & Cebuha, W.A., Dynamic Output Feedback Regulation fo a Class of Nonlinea Systems, Math. Contol Signals Systems, 6, , [5] Vincent, T.L. & Gantham, W.J., Nonlinea and Optimal Contol Systems, John Wiley & Sons, Inc., New Yok, 1997.

14 86 Janson Naibohu [6] Gantham, W.J. & Chingcuanco, A.O., Lyapunov Steepest Descent Contol of Constained Linea Systems, IEEE Tans. Automat. Contol, AC-9(8), , [7] Naibohu, J. & Shimizu, K., Diect Gadient Descent Contol fo Global Stabilization of Geneal Nonlinea Contol Systems, IEICE Tans. Fundamental, E83-A(3), , 000. [8] Shimizu, K., Otsuka, K. & Naibohu, J., Impoved Diect Gadient Descent Contol of Geneal Nonlinea Systems, Poc. of Euopean Contol Confeence, [9] Shimizu, K., Nukumi, H. & Ito, S., Tacking Contol of Geneal Nonlinea Systems by Diect Gadient Descent Method, Pepints of the 4th Nonlinea Contol Systems Design Symposium 1998, Enschede, The Nethelands, pp , 1-3 July [10] Isidoi, A., Nonlinea Contol Systems: An Intoduction, Thid Edition, Spinge-Velag Belin, Heidelbeg [11] Khalil, H.K., Nonlinea Systems, Thid Edition, Pentice Hall, New Jesey, 00. [1] Hischon, R.M. & Davis, J.H., Output Tacking fo Nonlinea Systems with Singula Points, SIAM J. Contol and Optimization, 5(3), , 1988.