Adaptive Stabilization of a Class of Nonlinear Systems With Nonparametric Uncertainty

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 11, NOVEMBER Adaptive Stabilization of a Class of Nonlinear Systes With Nonparaetric Uncertainty Aleander V. Roup and Dennis S. Bernstein Abstract We consider adaptive stabilization for a class of nonlinear second-order systes. Interpreting the syste states as position and velocity, the syste is assued to have unknown, nonparaetric position-dependent daping and stiffness coefficients. Lyapunov ethods are used to prove global convergence of the adaptive controller. Furtherore, the controller is shown to be able reject constant disturbances and to asyptotically track constant coands. For illustration, the controller is used to stabilize the van der Pol liit cycle, the Duffing oscillator with ultiple equilibria, and several other eaple systes. Inde Ters Adaptive stabilization, nonlinear systes, nonparaetric uncertainty. I. INTRODUCTION There are any applications of control in which a reliable odel of the dynaical syste is not available. This can occur if the syste is not aenable to analytical odeling due to unknown or unpredictably changing physics, or if identification is not feasible due to instability, disturbances, sensor noise, poor repeatability, or high cost. Under high levels of uncertainty, robust control ay be ineffective and adaptive control is warranted. For ipleentation, adaptive controllers generally require soe knowledge about the plant in the for of paraeter or transfer function estiates, and this knowledge ay be available prior to operation due to analytical odeling or off-line identification, or it ay be deterined during operation through concurrent identification. The forer case is usually tered direct adaptive control, while the latter constitutes indirect adaptive control. In addition, adaptive control ethods often depend on structural assuptions about the plant, for eaple, passivity and relative degree. In this note we consider the proble of adaptive stabilization and constant disturbance rejection for a class of second-order nonlinear systes under full-state feedback. In Section II, we present the adaptive controller and prove convergence of the plant states. The novel aspect of this controller is the fact that global convergence is guaranteed under nonparaetric assuptions about the nonlinearities. Interpreting the syste states as position and velocity, the syste is assued to have unknown, position-dependent daping and stiffness coefficients, which are assued only to be continuous and lower bounded. Furtherore, these lower bounds need not be known. A classical syste satisfying these assuptions is the van der Pol oscillator whose liit cycle is stabilized by our controller without knowledge about the for of the position-dependent, sign-varying daping. The for of our controller is siilar to direct adaptive controllers developed for linear systes. Related theory can be found in [1] [6], where the ephasis is on odel following control. For adaptive stabilization, a self-contained treatent of the relevant ideas and techniques is given in [7], where the stability of the closed-loop syste is proven for linear plants and the controller is applied to nonlinear plants. In [8], Manuscript received Noveber 22, 1999; revised Deceber 18, 2000 and April 20, Recoended by Associate Editor P. Toei. This work was supported in part by the Air Force Office of Scientific Research under Grant F and the François-Xavier Bagnoud Foundation Fellowship. The authors are with the Departent of Aerospace Engineering, the University of Michigan, Ann Arbor, MI USA (e-ail: aroup@uich.edu; dsbaero@uich.edu). Publisher Ite Identifier S (01) the controller presented in [7] is applied to otion control eperients. The ain difference between the controller of this note and [7] is a condition on the sign of the (1; 2) entry of the Lyapunov atri P. In [7] this sign condition is iplicit in the solution of the Lyapunov equation for second-order systes in copanion for. Nuerical eperients show that violation of this condition can destabilize the closed-loop syste. Since we assue full-state feedback control in copanion coordinates, that is, position and velocity easureents, our controller is a direct adaptive controller, and thus paraeter estiates are not needed. In addition, full-state feedback availability avoids the need for positivity assuptions. Etensions to output feedback, nonconstant disturbance rejection, and odel reference adaptive control will be considered in future work. II. ADAPTIVE STABILIZATION We wish to deterine a feedback control law for the nonlinear syste q(t) +g(q(t)) _q(t) +f (q(t))q(t) =bu(t) +d (1) where f : IR! IR; g: IR! IR, and ; b; d 2 IR, such that q(t)! 0 and _q(t)! 0 as t!1. We assue that (1) is uncertain in the following sense. The functions f and g are known to be locally Lipschitz on IR and lower bounded but are otherwise uncertain, the constant is known to be positive but is otherwise uncertain, the constant b is known to be nonzero with known sign but is otherwise uncertain, and the constant d is uncertain. Under the above assuptions, the control law u(t) =k 1 (t)q(t)+k 2 (t)_q(t) +(t) (2) where the gains k 1(t); k 2(t) and the paraeter (t) are adapted, will be used to obtain q(t)! 0 and _q(t)! 0 as t!1. Note that if u 0 and d = 0, then (q; _q) = (0; 0) is an equilibriu of (1) but not necessarily the only equilibriu. Furtherore, if u 0 but d 6= 0, then (q; _q) =(0; 0) is not an equilibriu of (1). Define the state and the gain atri (t) = 1(t) 2 (t) 1 = q(t) _q(t) (3) K(t) 1 =[k 1(t) k 2(t)]: (4) Dynaic variables will henceforth be written without a tie dependence arguent. The state equation for (1), (2) is _ = 2 (1=)[bK + b + d 0 2 g( 1 ) 0 1 f ( 1 ) Let P =[ 1 p 1 p 12 ] be positive definite, with p 12 > 0. Let p 12 p 2 and define the set 1 = inf q2ir f (q) K s = [k 1s k 2s] :bk 1s <; : (5) = 1 inf g(q) (6) q2ir bk 2s <0 p1 2 + p12 : (7) p 2 4p 12 ( 0 bk 1s ) /01$ IEEE

2 1822 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 11, NOVEMBER 2001 Lea 1: K s is not epty. Now, let K s = [k1s k 2s] 2Ks and define R 2 IR 222 ; ~ f : IR! IR, and ~g : IR! IR by R 1 = p ( 0 bk 1s) p p 1 p ( 0 bk ; (8) 2s) 0 p 12 ~f (q) 1 = f (q) 0 bk 1s ; ~g(q) 1 = g(q) 0 bk 2s : (9) Then R is positive definite, and ~ f (q) > 0 and ~g(q) > 0 for all q 2 IR. Furtherore, with K = K s and = 0d=b, the origin of (5) is a globally asyptotically stable equilibriu. Proof: The first inequality of (7) is an upper bound on bk 1s. The second inequality of (7) is an upper bound on bk 2s in ters of bk 1s. Since bk 1s and bk 2s are only bounded above, K s is not epty. The atri R is positive definite if and only if p 12 ( 0 bk1s) > 0; (10) p 2 ( 0 bk 2s) 0 p 12 > 0; (11) p 12 ( 0 bk 1s) p 2 ( 0 bk 2s) 0 p 12 > 1 4 p2 1: (12) The first inequality of (7) iplies (10), while the second inequality of (7) iplies (12). Furtherore, (10) and (12) iply (11). Therefore, since K s 2K s ;Ris positive definite. To show that f ~ (q) > 0 and ~g(q) > 0 for all q 2 IR, note that for all q 2 IR and ~f (q) =f (q) 0 bk 1s 0 bk 1s > 0; (13) ~g(q) =g(q) 0 bk 2s 0 bk 1s > p1 2 + p 12 > 0: (14) p 2 4p 12( 0 bk 1s) Net we show that if K = K s 2K s and = 0d=b, then the origin of (5) is a globally asyptotically stable equilibriu. The closed-loop syste (5) can be written in the for _ = (1=)[0 2 ~g( 1 ) 0 ~ : (15) 1 f ( 1 )] Note that =0is the unique equilibriu of (15) since f ~ (q) > 0 and ~g(q) > 0 for all q 2 IR. Consider the Lyapunov candidate for (5) given by V () = 1 2 T P+ p ~ f () d + p 12 0 ~g() d (16) which is positive definite and radially unbounded. The derivative of V along the syste trajectory is _V () = T P _ + p 2 1 f ~ ( 1)_ 1 + p 12 1~g(1)_1 = 0 p 12 ~f ( 1) p ~g(1) 0 p p 12 ( 0 bk 1s) p ( 0 bk 2s) 0 p = 0 T R: (17) Since R is positive definite, the origin of (5) is globally asyptotically stable. Now, consider the syste (5) with the adaptation law 2 _ = (18) (1=)[bK + b + d 0 2 g( 1 ) 0 1 f ( 1 )] _K = 00B T 0 P T 3 (19) _ = 0B T 0 P (20) where 0 2 IR; P 2 IR 222 ; 3 2 IR 222, and 2 IR. Let 0 > 0; 3 be positive definite, >0, and define B 0 1 =[0 sign(b)] T. The equilibria set of the closed-loop syste (18) (20) is E = f(; K; ) 2 IR 2 2 IR IR : =[0 0] T ; K 2 IR 122 ; = 0d=bg: (21) Define the subset of equilibria E s = f(; Ks;) 2E: K s 2K sg: (22) Theore 1: Every eleent of E s is a Lyapunov stable equilibriu of the closed-loop syste (18) (20). Furtherore, the functions ; K, and satisfying (18) (20) are bounded, and (t)! [0 0] T and (t)! 0d=b as t!1. Proof: Let ([0 0] T ;K s ; 0d=b) 2E s, where K s = [k 1s k 2s ]. Define 1 ~k 1 = k1 0 k 1s; k ~ 1 2 = k2 0 k 2s; K ~ = 1 K 0 K s; ~ = 1 + d=b; (23) the atri R as in (8), and the functions ~ f and ~g as in (9). Note that since [k 1s k 2s ] 2K s it follows fro Lea 1 that R is positive definite and ~ f (q) > 0 and ~g(q) > 0 for all q 2 IR. The closed-loop syste (18) (20) can be written in the for _ = 2 (1=)[b ~ K + b ~ 0 2 ~g( 1 ) 0 1 ~ f ( 1 )] ; (24) _~K = 00B T 0 P T 3; (25) _~ = 0B T 0 P : (26) The Lyapunov analysis that follows concerns the stability of the equilibriu point (; ~ K; ~ ) = ([0 0] T ; [0 0]; 0) of (24) (26). Note that the equilibriu point (; ~ K; ~ ) = ([0 0] T ; [0 0]; 0) of (24) (26) corresponds to the equilibriu point (; K; ) = ([0 0] T ;K s ; 0d=b) of (18) (20) through the coordinate transforation (23). Consider the Lyapunov candidate for the syste (24) (26) V (; K; ~ )= ~ 1 2 T P+ p 2 f ~ () d + p 12 ~g() d 0 0 jbj + 2 tr 001 K3 ~ 01 K ~ T jbj + 2 tr ~ 01 ~ T (27) which is positive definite and radially unbounded. The derivative of V along the syste trajectory is _V (; K; ~ )= ~ T P _ + p 2 1 f ~ ( 1 )_ 1 + p 12 1~g( 1 )_ 1 + jbj tr 001 ~ K3 01 _~ K T + jbj tr ~ 01 _~ T =( 1 p p 12 ) ( 1p p 2 ) 2 (b K ~ + b ~ 0 2~g( 1) 0 ~ 1 f ( 1)) + p 2 12 f ~ ( p 12 1)+ 12 ~g(1) jbj + tr 001 K3 ~ 01 K _~ T jbj + tr ~ 01 _~ T =( 1 p p 12 ) ( 1p p 2 ) 2 (~g( 1 ) 2 + ~ 1 f ( 1 )) + p 2 1 ~ 2 f ( 1 ) + p ~g( jbj 1 )+ tr K( ~ T PB K _~ T 0 01 jbj )+ tr ( ~ T PB _~ T ) = 0 p 12 ~f ( 1 )1 2 + p ~g( 1) 0 p (28)

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 11, NOVEMBER p 12 ( 0 bk1s)2 1 + p ( 0 bk2s) 0 p = 0 T R 0 (29) for all (; ~ K; ~ ) 2 IR 2 2 IR IR. Therefore, ([0 0] T ;K s ; 0d=b) is a Lyapunov stable equilibriu of (18) (20). Since V (; ~ K; ~ ) > 0 and _ V (; ~ K; ~ ) 0 it follows that V (; ~ K; ~ ) is bounded. Since, in addition, V (; ~ K; ~ ) is radially unbounded, it follows that ; K, and are bounded. Net, to prove (t)! [0 0] T and (t)! 0d=b as t! 1, assue ((0); ~ K(0); ~ (0)) 6= ([0 0] T ; [0 0]; 0) and let 1 = V ((0); ~ K(0); ~ (0)) > 0 and N 1 = f(; ~ K; ~ ): _ V (; ~ K; ~ )=0;V(; ~ K; ~ ) g: (30) Note that because R is positive definite, V _ (; K; ~ )=0iplies ~ that = [0 0] T. Conversely, substituting = [0 0] T into (28) gives _V (; K; ~ )=0. ~ Therefore V _ (; K; ~ )=0if ~ and only if =[00] T. Hence N = f(; ~ K; ~ ): =[0 0] T ;V(; ~ K; ~ ) g: (31) Fig. 1. Syste trajectory in the q; _q plane for Eaple 1, the van der Pol oscillator. Initial conditions are q = 01; _q = 1; k = 0; k = 0, and = 0. Uncontrolled response is shown with thin line. Controlled response is shown with thick line. Substituting [0 0] T into (24) (26) it can be seen that _ =[0 0] T if and only if ~ =0. It follows that the largest invariant subset of N is M = f(; ~ K; ~ ) 2N : ~ =0g: (32) Now, LaSalle s theore ([9, Th. 3.4]) iplies that (; ~ K; ~ )!Mas t!1. It follows that (t)! [0 0] T and (t)!0d=b as t!1. Note that the lower bounds and for f and g defined by (6) are used only in the proof and need not be known to ipleent the adaptive controller (2), (19), (20). For the case in which (1) is linear, Theore 1 specializes to [7, Cor. 3.1]. In [7], the atri P was obtained as the solution to the Lyapunov equation 0 =A T s P + PA s + R, where A s = A + BK s and R is an arbitrary positive definite atri. It can be seen that when A s is in canonical for, the (1, 2) entry of P is always positive. Hence the requireent p 12 > 0 represents no loss of generality when Theore 1 is applied to linear plants. The adaptive controller (2), (19), (20) can be used to asyptotically track constant position references. Define the position error e(t) 1 = q(t) 0 r (33) where r is a constant reference. In ters of e and r, the syste equation (1) has the for e(t) +g(e(t)+r)_e(t) +f (e(t)+r)(e(t)+r) Now defining (34) can be written as = bu(t) +d: (34) f 1 1(e) = 1 [f (e + r)(e + r) 0 f (r)r]; (35) e g 1 (e) 1 = g(e + r) (36) d 1 1 = d 0 f (r)r (37) e(t) +g 1 (e(t)) _e(t) +f 1 (e(t))e(t) =bu(t)+d 1 (38) which is identical in for to (1). The adaptive controller (2), (19), (20) can be applied to (38) using the state definition = [e _q] T to give q! r and _q! 0 as t!1. Fig. 2. Tie history of V _ for Eaple 1, the van der Pol oscillator. Control syste is activated at t = 100, which is arked with a vertical dashed line. III. NUMERICAL EXAMPLES Eaple 1: Consider the van der Pol oscillator with constant disturbance q +10(q 2 0 1) _q + q = u +0:9: (39) For this syste, g(q) = 10(q 2 0 1) and f (q) = 1, which are both bounded fro below. Note that with the constant disturbance ter d = 0:9, the open-loop syste does not have an equilibriu point at (q; _q) = (0; 0). However, the closed-loop syste (39), (2), (19), (20) has the equilibria set E = f([q _q] T ;K;):q =0; _q =0;K 2 IR 122 ; = 00:9)g. Choose P = 1 0:5 ; 0=1; 3= 1 0 0: ; =1: (40) Fig. 1 shows the syste trajectory in the q; _q plane. The uncontrolled syste is allowed to approach a liit cycle, and then the adaptive control syste is activated at t = 100. V (; ~ k; ~ ) was calculated using (27) with the paraeters k 1s =0and k 2s = 015 selected to satisfy (7). Fig. 2 shows the tie history of _ V, and Fig. 3 shows the tie history of k 1 ;k 2, and before and after control syste activation. The states q and _q of the closed-loop syste converge to q =0and _q =0.

4 1824 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 11, NOVEMBER 2001 Fig. 3. Tie history of k ( ); k (- -), and (0 1) for Eaple 1, the van der Pol oscillator. Control syste is activated at t =100, which is arked with a vertical dashed line. (a) (b) Fig. 4. Syste trajectory in the q; _q plane for Eaple 2, the Duffing oscillator. Initial conditions are q = 01; _q = 1;k = 0;k = 0, and = 0. Uncontrolled response is shown with thin line. Controlled response is shown with thick line. Fig. 5. Randoly generated piecewise linear functions f and g used in Eaple 3. Eaple 2: Net, consider the Duffing oscillator with constant disturbance q +(1=4) _q +(q 2 0 4)q = u +1: (41) The uncontrolled syste has stable foci at (01:86; 0) and (2.11, 0) and a saddle at (00:25; 0). For this syste, g(q) =1=4 and f (q) =q 2 04, which are bounded fro below. Choose controller paraeters as in (40). Fig. 4 shows the syste trajectory in the q; _q plane. The uncontrolled syste is allowed to approach a stable focus, and the adaptive control syste is activated at t =30. The states q and _q of the closed-loop syste converge to q =0 and _q = 0. Eaple 3: Net consider the nonlinear syste with randoly generated piecewise linear stiffness and daping (1) with = 1;b = 1, and d = 0. The stiffness and daping functions f and g are the randoly generated piecewise linear functions shown in Fig. 5. These nonlinear functions can be viewed as interpolations of lookup table data. Choose controller paraeters as in (40). Fig. 6 shows the syste trajectory in the q; _q plane. The uncontrolled syste is allowed to diverge, Fig. 6. Syste trajectory in the q; _q plane for Eaple 3. Initial conditions are q = 01; _q = 01;k = 0;k = 0, and = 0. Uncontrolled response is shown with thin line. Controlled response is shown with thick line. and the adaptive control syste is activated at t =0:4. The states q and _q of the closed-loop syste converge to q =0and _q =0.

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 11, NOVEMBER Fig. 7. Function g for Eaple 4. Fig. 9. Syste trajectory in the q; _q plane for Eaple 5. Initial conditions are q =0:1; _q =0:1;k =0;k =0, and =0. Uncontrolled response is shown with thin line. Controlled response is shown with thick line. 01 and _q =0, the uncontrolled syste approaches a liit cycle. The adaptive control syste is activated at t =19. The states q and _q of the closed-loop syste converge to q =1and _q =0. Eaple 5: Consider the nonlinear syste q + (q 2 +! 02 _q 2 0 a 2 )_q +! 2 q = u: (47) Fig. 8. Syste trajectory in the q; _q plane for Eaple 4. Initial conditions are q = 01; _q =0;k =0;k =0, and =0. Uncontrolled response is shown with thin line. Controlled response is shown with thick line. Eaple 4: Consider the nonlinear syste Uncontrolled trajectories of (47) with nonzero initial conditions approach a sinusoidal liit cycle with aplitude a and frequency!. The paraeter adjusts the rate of convergence to the liit cycle. Theore 1 does not apply to this eaple because g is a function of _q as well as q. For this eaple choose paraeters a =1;! =1, and =0:5. Choose controller paraeters as in (40). Figure 9 shows the syste trajectory in the q; _q plane. Beginning fro the initial condition q =0:1 and _q =0:1, the uncontrolled syste approaches a liit cycle. Then the adaptive control syste is activated at t =35. The state q and _q of the closed-loop syste converge to q =0and _q =0. q + g(q)_q + q = u (42) where g(q) = 2:2jq3 j03:3q 2 +0:1; jqj < 1 3:3(jqj 01) 2 0 1; jqj 1: (43) The daping function g is shown in Fig. 7. We wish to drive the state q to q =1. With the reference input r =1, choose controller paraeters as in (40) and use the odified control u = k 1 (q 0 r) +k 2 _q + (44) _K = 00B T 0 P _ = 0B T 0 P q 0 r _q q 0 r _q [q 0 r _q]3 (45) : (46) Note that with r 0, (44) (46) are identical to (2), (19), (20). Choose controller paraeters as in (40). Fig. 8 shows the syste trajectory in the q; _q plane. Beginning fro the initial condition q = REFERENCES [1] K. S. Narendra and A. M. Annasway, Stable Adaptive Systes. Upper Saddle River, NJ: Prentice-Hall, [2] S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness. Upper Saddle River, NJ: Prentice-Hall, [3] K. J. Åströ and B. Wittenark, Adaptive Control, 2nd ed. Reading, MA: Addison-Wesley, [4] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. New York, NY: Wiley, [5] P. A. Ioannou and J. Sun, Robust Adaptive Control. Upper Saddle River, NJ: Prentice-Hall, [6] H. Kaufan, I. Barkana, and K. Sobel, Direct Adaptive Control Algoriths: Theory and Applications, 2nd ed. New York: Springer-Verlag, [7] J. Hong and D. S. Bernstein, Adaptive stabilization of nonlinear oscillators using direct adaptive control, Int. J. Control, vol. 74, no. 5, pp , [8], Eperiental application of direct adaptive control laws for adaptive stabilization and coand following, in Proc. Conf. Decision Control, Phoeni, AZ, Dec. 1999, pp [9] H. K. Khalil, Nonlinear Systes, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996.