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of Proceedings the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006 ThB13.3 Robust and Adaptive Partial Stabilization for a Class of Nonlinearly Parameterized Systems Denis V. Efimov, Member, IEEE, Alexander L. Fradkov, Fellow, IEEE Abstract The problem of adaptive stabilization with respect to a set for a class of nonlinear parameterized systems in the presence of external disturbances is considered. A novel adaptive observer-based solution for the case of noisy measurements is proposed. The efficiency of proposed solution is demonstrated via example of swinging a pendulum with unknown parameters. x f ( x, u ) , y h( x ) , (1) is input vector; 0 . Euclidean where x y R p R n is state vector; u R m is output vector; f and h are locally Lipschitz con0 , f ( 0,0 ) tinuous vector functions, h( 0 ) I. INTRODUCTION HE problem of nonlinear adaptive control got a number of solutions during last decades [2], [11], [14], [17], [23], [24]. Most of existing applicability conditions are based on radial unboundedness of the goal functional with respect to the whole state vector of the plant and on the assumption about linear dependence of system equations on adaptation parameters. Such suppositions result in adaptive stabilization with respect to the whole plant state. However, in a number of application problems the stabilization of the plant with respect to a part of variables (i.e. with respect to a set) is needed. For example, such problems arise when stabilizing the desired level of energy for mechanical systems. An additional requirement may consist in the boundedness of control signal [3], [11]. In the case of parametric uncertainty presence the dependence of bounded control law on adjusted parameters leads to the problem of adaptation with nonlinear parameterization. This fact prevents from applying previously mentioned results. There exist a number of solutions for nonlinearly parameterized problems [5], [10], [16], [18], [19], [20], [22], [30] usually based on tuned robust feedbacks. However, they can not be applied to energy control problem due to the requirement of control law boundedness. Besides, they do not deal with the problem of partial stabilization considered here. Therefore there is a demand for a new solution for the problem. Additionally it would be desirable to take into account external disturbances as well as partial and noise measurements. Such a solution is proposed below. II. PRELIMINARIES Let us consider dynamical systems This work is partly supported by grant 05-01-00869 of Russian Foundation for Basic Research, by Russian Science Support Foundation and by Program of Presidium of Russian Academy of Science 22. Authors are with the Control of Complex Systems Laboratory, Institute for problems of mechanical engineering, 61, Bolshoy av. V.O. , StPetersburg, 199178 Russia (efde@mail.rcom.ru, alf@control.ipme.ru). norm will be denoted as x , and u t t denotes the Lm 0, norm of the input ( u( t ) is measurable and locally essentially bounded function u : R u t0 ,T T Rm , R t0 , T R: 0 ): ess sup u( t ) , t , if T then we will write simple u . We will denote as MR m the set of all such Lebesgue measurable inputs u with property u . For initial state x 0 and input u M m let x( t , x 0 , u ) be the unique maximal solution R of (1) (we will use notation x( t ) if all other arguments of solution are clear from the context; y ( t , x 0 , u ) h x( t , x 0 , u ) ), which is defined on some finite interval x0 R n 0, T ; if T for every initial state and u M m , then system is called forward R R n and in- complete. It is said that system (1) has unboundedness observability (UO) property, if for each state x0 put u M m such that T R t T necessarily . lim sup y ( t , x 0 , u ) In other words it is possible to observe any unboundedness of the state. The necessary and sufficient conditions for forward completeness and UO properties were investigated in [1]. Distance in R n from given point x to set A is deand x 0 x noted as x A dist x, A inf x A is standard Euclidean norm. As usual, a continuous function class K if it is strictly increasing and it belongs to class K continuous function :R :R 0 R belongs to 0 ; additionally if it is also radially unbounded; and R R is from class KL , if it is from class K for the first argument for any fixed second, and it is strictly decreasing to zero by the second argument for any fixed first one. System (1) is called bounded-input-bounded-state stable 1-4244-0171-2/06/$20.00 2006 IEEE. 3442 45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006 ThB13.3 (BIBS), if for all x 0 erty holds: x( t , x 0 , u ) max Rn , u x0 , M m the following propR u for all x R n inequalities 1 | x |V n 0 V (x) 2 | x |V 0 ,t 0, K. D e f i n i t i o n 1 [13], [29]. An UO system (1) is inputto-output stable (IOS), if there exist KL and K such, that y( t, x0 , u ) x0 , t u ,t 0 holds for all x 0 R n and u are satisfied, where V0 { x R : V ( x ) set. Then system (2) with control u (y) 0 } is a compact has iISS property with respect to set V0 if the system is Vdetectable with respect to output y . B. Positivity in average property The identification ability of adaptation algorithms is one of the most attractive problems in the adaptive control theory. The solution of this problem is closely connected with persistent excitation (PE) property. There exist several equivalent definitions of PE property [10], [21], [24]. Here we will use the following closely connected property. R is called ( r , ) D e f i n i t i o n 4. Function a : R 0, positive in average (PA) if for any t 0 and any r 0, t t M m. R D e f i n i t i o n 2 [9]. Forward complete system (1) is called integral input-to-state stable (iISS) with respect to closed invariant set A if there exist functions K, K and u KL such, that for any x 0 R n and M m , the solutions x( t , x0 , u ) are defined for all R 0 and inequality holds x( t , x 0 , u ) A x0 A , t t 0 t u( ) d , t 0. a( ) d r. A. Robust stabilization with respect to set via passification approach Let us consider the system: x f ( x ) G ( x ) [ u v ] , y h( x ) , (2) where x R n , u Rm , y R p are state, input and output vectors correspondingly; v R m is external disturbances vector on the input of the system; f , h and columns of matrix function G are locally Lipschitz continuous vector functions, h( 0 ) 0 , f ( 0 ) 0 . Let us assume the passivity property of the system (2) from input u to output y [4], [11], [31] with storage function V . Passification approach [11], [27], [28] allows to solve the problem of zero level set of storage function V stabilization for system (2). The key property in substantiation of the approach is detectability assumption [26], [27], [28] proposed in the following definition. D e f i n i t i o n 3. It is said that passive system (2) with storage function V : R n to output y if for all x0 y ( t , x 0 ,0 ) 0, t 0 t Importance of PA property is explained in the following lemma, which slightly modified version was proved in [8]. L e m m a 1. Let us consider time-varying linear dynamical system p a t p b t , t0 0 , R, where p R , p( t0 ) R and functions a : R b:R R are Lebesgue measurable, b is locally essen- tially bounded, function a is ( r , ) PA for some r 0 , 0 and essentially bounded from below, i.e. there exists A R , such, that: ess inf a t , t t 0 A. Then solutions of the system are defined for all t admit the estimate | p( t 0 ) | e r ( t t 0 ) ( A r ) | p( t ) | || b || max{ A 1e A t 0 , r 1e r t 0 }, A | p( t0 ) | e r ( t t 0 ) r || b || max{ , r 1e r t 0 } , 0; t 0 and R is V-detectable with respect R n it holds: A 0. lim V ( x( t , x0 ,0 ) ) 0. The following result [7], [9] presents conditions of iISS with respect to set stabilization by passification. T h e o r e m 1. Let system (2) be passive with continuous storage function V : R n function yT ( y ) :R m C. Adaptive observer design Let us consider the following systems with signal and parametric uncertainties: x A ( y ) x ( y ) B( y ) d1 , y C x , y d y d 2 , (3) where x R n is state vector; y ues belong to compact set R m is output vector; R p is vector of uncertain parameters, which val- R m and smooth non decreasing (0) 0 lim | x|V 0 R, have property . 0 for all y Rm { 0} , | h( x ) | / V ( x ) 2 ; d1 M n , d 2 R M m are R Additionally there exist functions 1, K such, that vector signals of external disturbance and measurement T noise, d [ d1 dT ]T ; y d is vector noise measurements of 2 3443 45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006 ThB13.3 system (3) output. Vector function and columns of matrix functions A and B are locally Lipschitz continuous, C is some constant matrix of appropriate dimension. It is necessary to design adaptive observer, which in the absence of disturbances d provides observations of unmeasured components of state space vector of system (3) and identifies values of vector , for any d M n m it should R estimates. T h e o r e m 2 [6]. Let assumptions 1 3 hold and minimum singular value a(t ) of matrix function CT T (t ) be , PA for some 0, 0 . Then solutions of system (3), (5) (8) are bounded for any initial conditions and d M n m and any 0 , for the case of disturbance d R ensures boundedness of the system solutions. In works [6], [8], [12] a solution of this problem is proposed under the following suppositions. A s s u m p t i o n 1 . For any initial condition x0 and any all values of time t x( t , x 0 , , d ) x absence relations hold: lim t t (t ) , lim y ( t ) y (t ) 0 lim L x( t ) L z ( t ) 0 . t , d M n m system (3) is BIBS for almost R 0: 0 ( | x0 III. MAIN RESULT Let us consider uncertain nonlinear system: x A ( y ) x ( y ) B( y ) R ( y ) [ u d3 ] d1 , y C x , yd y d2 , R n is state vector; R q is R p is uncertain vector of |) 0 ( || d || ) , 0 K. The rest suppositions deal with stabilizability by output feedback of linear part of system (3). A s s u m p t i o n 2. There exist matrix L and locally R n m such, Lipschitz continuous matrix function K : R m that there exists continuously differentiable function V : Rn R and 1( | x | ) (9) where (as for system (3) previously) x y R m is output vector; parameters with values from a compact set control; d1 M n , d 2 R M m , d3 R ;u V (x) V (x) x G( y ) x m n 3| Lx | 2 (| x |) ; 2 1, M q is vector sigR nals of external disturbances and measurement noise, | Lx | 2 , | Cx | for any y K R and x R , where are from class and 3 0 , G ( y ) A( y ) K ( y ) C . The assumption ensures uniform asymptotic stability property with respect to output L s [11], [25] for system s Gys r R. R n n (4) T d [ d1 dT dT ]T ; y d is noisy output vector of system (9). 23 and columns of matrix functions A , B , Vector function R are continuous and locally Lipschitz, C is a matrix of appropriate dimension. We will assume the existence of control law u( x ) with the following properties. A s s u m p t i o n 4. There exist locally Lipschitz con- together with system (3) for r property with respect to variable s and r 0 and uniform stability tinuous functions u : R m u kp Rq , : Rn Rr and matrix L with dimension ( k n ) such, that control u( y, L x, ) A s s u m p t i o n 3. For any initial conditions s0 respect to output y : (10) M n , y M m system (4) is BIBS uniformly with R R guarantees for system (9) forward completeness and one of the following properties: to input d . 1. IOS from output | s( t , s 0 , r, y ) | 1( | s0 | ) 1 ( || r || ) , 1 K , t 0 . Let us consider the following equations of adaptive observer: z A( y d ) z ( y d ) B( y d ) y, y ; B( y d ) ; K( yd ) yd G( y ) G( y d ) T Cz; (5) (6) (7) 0, 2. iISS with respect to set Z { x : ( x ) 0 } for input d . It is necessary starting from control (10), depending on unmeasured variables L x and vector of uncertain parameters of the system , to design a new controller using only admissible measured signal y d . The controller should provide boundedness of closed loop system solutions for d M n m q , for the case d 0 it should ensure asympR CT ( y d totic convergence to zero of output or attractiveness of y C ), (8) where z R n is vector of estimate of unmeasured system (3) state vector x ; vector R n and matrix R n p are auxiliary variables, which help to overcome high relative degree obstruction for system (3); R p is vector of the set Z (depending on that part of assumption 4 is satisfied). It is worth to stress, that here two output functions y and appear. The first one defines the measured variables of system (9) while the second one characterizes the distance to 3444 45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006 ThB13.3 the goal set. Although the vector of unknown parameters appears in linear fashion in the right hand side of system (9), the right hand side of the closed loop system (9), (10) may depend on nonlinearly since Assumption 4 does not specify the form of dependence of function u on its arguments. The form of system (9) is close system to (3) for which it is possible to design adaptive observer (5) (8). Substituting estimates of vectors L x and obtained from the observer, into the control law (10), it is possible to solve posed problem (we assume that matrixes L in assumptions 2 and 4 are identical). The principal difference of the solved problem from the problem of adaptive observer construction from Theorem 2 consists in appearance of control input in the right hand side of system (9). Thus generically in the absence of control (10) system can possess unbounded solutions. Fortunately this obstacle does not prevent from design of adaptive observer similarly to (5) (8): z A( y d ) z ( y d ) B( y d ) y,y Cz; assumption 3 is satisfied for any Lebesgue measurable inputs y ; minimum singular value a(t ) of matrix function CT T (t ) is , u PA for some 0, 0; B( y d ( t ) ) B for all t 0 . Then control u( y d , L z, ) ensures for system (9) forward completeness property, solutions of the system (11) (14) and variable (t ) are bounded for all initial conditions, d M n m q and any R 0 provided that: 1) the first part of assumption 4 holds and d 2 ( t ) 0 for all t 0; 2) the first part of assumption 4 holds and control u is globally Lipschitz function with respect to the first argument, function is globally Lipschitz, A( y ) A , B( y ) B and R ( y ) R; R( yd ) u K( yd ) yd (11) (12) (13) G( y ) G( y d ) T ; B( y d ) ; 3) the second part of assumption 4 holds and d( t ) 0 for all t 0 . Additionally for d( t ) 0 for all t 0 relations lim t t (t ) 0 (for the first part of assumption 4) and CT ( y d y C ), 0, (14) lim | x( t ) |Z 0 (for the second part of assumption 4) where all symbols save their meaning. Since matrix function R depends on output vector only and control u is produced by the controller, then their appearance does not change dynamics of state estimation error e x z and auxiliary error e G( yd ) e d1( t ) e (y) ( ): B( y d ) ( y d ) B( y ) K ( y d ) d 2 ( t ) R ( y ) d3 R( y ) R( y d ) u (15) A( y ) A( y d ) x , G( yd ) d1( t ) (y) ( yd ) B( y ) B ( y d ) R( y ) R( y d ) u K ( y d ) d 2 ( t ) R ( y ) d3 (16) A( y ) A( y d ) x . 0 systems (15), For the case of absence of disturbances d (16) can be rewritten as follows e G( y ) e B( y ) [ G( y ) . ], (17) (18) Form of equations (17), (18) are the same as calculated for the observer (5) (8) and, therefore, the proof of workability of observer (11) (14) can be borrowed from the proof of theorem 2 with minimal modifications dealing with a-prior absence of assumption 1 for system (9). In the case of noise d 2 presence the dependence of right hand sides of equations (15), (16) on vectors u and x makes difficulties for application of the theorem 2 proof. It is the reason why this case will be considered under special conditions below. T h e o r e m 3. Let for system (9) assumption 2 holds and hold. The proofs are omitted due to space limitations. In the theorem it is required that assumption 3 should hold for all (not necessary bounded as in theorem 2) inputs y . Such more stronger requirement is originated by possible unboundedness of system (9) solutions. For the first part of assumption 4 the theorem provides workability conditions for cases of external disturbances and noise presence. The noise case needs in additional structural restrictions. For the second part of assumption 4 the theorem 3 does not propose constructive conditions providing operating of the system in the presence of disturbance d . Robust properties of control (10) in this case are oriented on parametric uncertainty presence and partial state measurements compensation. It is possible to weaken requirements of theorem 3 for the second part of assumption 4, supposing boundedness and asymptotic convergence to zero of disturbance d . The proof of theorem 3 remains valid with minimal modifications. However, even under such restrictive conditions exactly the second part of assumption 4 is the most important for practical applications since this part allows to design adaptive control systems for mechanical Hamiltonian systems (passive systems). Theorem 1 presents results for iISS stabilization of passive systems with respect to set. An example of situation where there exists such problem is the problem of energy levels (Hamiltonian levels) stabilization for mechanical systems. If parameters of the plant are unknown, then Hamiltoin nian can depend on vector of uncertain parameters 3445 45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006 ThB13.3 complex nonlinear fashion, that prevents to applying of conventional adaptation techniques oriented on convex parameterization of system equation. Combining results of theorems 1 and 3 it is possible to propose a solution of this problem. C o r o l l a r y 1 . Let the following series of properties be satisfied: 1. System (9) for d 0 is passive with respect to output LR ( y )W ( x )T and input u with smooth storage function W : Rn 1 [26]. If H * 2 2 , then the set of storage function zeros is compact. The value H * 2 2 corresponds to stabilization of the upper energy level of the pendulum. In [11] for energy control of this system the following control law was proposed u () 2 ( x2 [ 0.5 x2 2 (1 cos( x1 ) ) H * ] ) , where during simulation we choose ( ) tanh( ) . Such control and storage function possess all conditions of the corollary. The equations (11) (14) of adaptive observer for the example take form: z1 z2 K ( x1 z1 ) ; K 0 ; z2 K ( x1 z1 ) 2 ( z2 [ 0.5 z2 1 2 R and | x |W 0 W (x) 2 | x |W , 0 1, 2 K, where W0 {x : W ( x ) 0 } is a compact set; system (9) is , lim | x|V 0 W-detectable for output | ( x ) | /W ( x ) . sin( x1 ) ( 1 cos( x1 ) ) 2 ] ) ; 1 2 2. For system (9) assumption 2 holds and assumption 3 is satisfied for any Lebesgue measurable inputs y ; minimum singular value a (t ) of matrix function CT , t 0. T K K 1 1 2; K K 1) 1 1 2 2 1 ; (t ) is sin( x1 ) ; 1 ( x1 ; PA for some 0, 0 ; B( y d ( t ) ) R q for all B for all z1 . 3. Smooth function sesses inequality holds for function u T : Rq () ( (x)) R q { 0 } pos- According to the result of corollary 1 the observer with control (1 cos( x1 ) ) 2 ] ) provides stabilization of upper energy level of the penduK 1 and lum. Simulation result of the system with u 2 ( z2 [ 0.5 z2 0 and the following relation u ( y , L x, ) . : Then control law u u( y , L z, ) provides for system (9), (11) (14) global boundedness of solutions for case d 0 and any 0 , additionally t lim | x( t ) |W 0 0. Further let us consider example of the corollary results application. IV. ADAPTIVE SWINGING THE PENDULUM all zeros initial condition except x1 ( 0 ) 0.1 is presented in Fig. 1. Trajectories in state space of the pendulum (solid line) and in coordinates of the adaptive observer ( z1, z2 ) (dot line) are shown in Fig. 1, . The observation error is presented in Fig. 1,c separately. In figures 1,b and 1,d plots of variables (t ) and H (t ) are shown. Simulation results confirm that the proposed controller solves the problem of adaptive swinging the pendulum to the desired energy level. V. CONCLUSION Adaptive control algorithms are proposed for stabilization with respect to a set for a class of nonlinearly parameterized systems. Applicability of the algorithms is established in the presence of external disturbances and partial observations with measurement noise. Efficiency of the proposed solution is demonstrated via example of computer simulation for swinging the pendulum. REFERENCES [1] Angeli, D. and E.D. Sontag, Forward completeness, unboundedness observability, and their Lyapunov characterizations . Systems and Control Letters, 38, 1999, pp. 209 217. Astolfi A., Ortega R. Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems. IEEE Trans. Aut. Contr., 48, 2003, pp. 590 606. Bullo F. Stabilization of relative equilibria for underactuated systems on Riemannian manifolds. Automatica, 36, 2000, pp. 1819 1834. 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