Marino, R.

Respondek, W.

Schaft, A.J. van der

Tomei, P.

English

Sufficient geometric conditions are given which lead to the explicit construction of a state feedback tracking control for single-input single-output nonlinear systems with bounded unmodelled disturbances entering nonlinearly. For any initial condition the output asymptotically tracks a bounded reference signal with bounded time derivatives with an arbitrary attenuation of the influence of the disturbance. The sufficient conditions are weaker than those presented previously and the technique of proof is also different

Dissertations of the University of Groningen

   University of GroningenTracking Control of Nonlinear Systems with Disturbance AttenuationMarino, R.; Respondek, W.; Schaft, A.J. van der; Tomei, P.Published in:Proceedings of the 31st IEEE Conference on Decision and Control, 1992IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1992Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Marino, R., Respondek, W., Schaft, A. J. V. D., & Tomei, P. (1992). Tracking Control of Nonlinear Systemswith Disturbance Attenuation. In Proceedings of the 31st IEEE Conference on Decision and Control, 1992(pp. 2469-2474). University of Groningen, Johann Bernoulli Institute for Mathematics and ComputerScience.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 04-06-2022FA1 -8:50 Tracking Control of Nonlinear Systems with Disturbance Attenuation R. Marino*, W. Respondek**, A.J. van der Schaft# and P. Tomei* * Dept. of Electrical Eng., University of Roma, 'Tor Vergata', Via della Ricerca Scientifica, 00133 Roma, Italy. ** Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-950 Warsaw, Poland. # Dept. Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. Abstract Sufficient geometric conditions are given which lead to the explicit construction of a state feedback track- ing control for single-input single-output nonlinear systems with bounded unmodelled disturbances enter- ing nonlinearly. For any initial condition the output asymptotically tracks a bounded reference signal with bounded time derivatives with an arbitrary attenua- tion of the influence of the disturbance. The sufficient conditions are weaker than those presented in [6] and the technique of proof is also different. 1 Introduction This paper provides sufficient conditions for the ex- plicit construction of a state feedback control capa- ble of forcing the output y to  track a bounded ref- erence signal Yd(t  with an arbitrary attenuation of a bounded unmode 1' led time varying disturbance O(t)  for nonlinear systems 3 : =  y = h ( z ) ,  y E  R.  f ( 2 )  + g(z)u + !?(XI qq), 2 E R", 21 E R, (1) In (1) f : R" -+ R", g : R" -+ R", q : R" x cj -+ R", h : R" + R are smooth functions, g(z) # 0, Vx E R", x is the state, U is the input, 8 : R+ -+ Cl E RP is the disturbance, y is the output which is required to track a reference signal Yd(t) .  This problem is also called al- most disturbance decoupling, following the terminol- ogy introduced in [ll] for linear systems: it is posed when the well understood disturbance decoupling (or rejection) problem, that is the design of a state feed- back control which makes the output insensitive to unmodelled disturbances, is not solvable. Necessary and sufficient geometric conditions for the solvability of the disturbance decoupling problem are obtained in 41 and 5 ,  generalizing the results established in [12\ and [lf lor linear systems. Equiv- alent conditions based on the notion of characteris- tic indexes are given in [2] for linear systems and in [S] for nonlinear ones. When the disturbance decou- pling problem is not solvable, it is natural to look for conditions which guarantee the attenuation of the influence of the disturbance on the output with any desired degree of accuracy. This problem, called al- most disturbance decoupling, was posed and solved in 111 for linear systems in terms of necessary and suf- k cient geometric conditions. In particular, the prob- lem turns out to be always solvable for single-input, single-utput linear systems of type x = F x + g u + Q O  y = h3: (2) where F and Q are n x n and n x p constant matrices, g and hT are n x 1 constant vectors. As pointed out in [ll] the almost disturbance decoupling problem is related to high-gain feedback design since, when it is not (exactly) solvable, the higher the gains are the higher the disturbance attenuation results. In fact in [7] a parameterized state feedback control is explicitly obtained when the almost disturbance decoupling is solvable; for square and minimum-phase systems a parameterized output-feedback control is given in [lo]. At the moment it is not known whether, as in the linear case, the almost disturbance decoupling prob- lem is always solvable for single-input, single-output nonlinear systems (1). Sufficient conditions are ob- tained in [6] using differential geometric tools and sin- ular perturbation techniques. The example (given in i1 = z2 + O l ( t )  i2 = .;e,(t) + U ( t )  (3)  k1) Y = 21 fails to satisfy the sufficient conditions in [6] and shows that the almost disturbance decoupling problem can- not be solved on the basis of linear approximations. The non-local nature of the problem is also pointed out by the example (also given in [SI) $1 = arctan22 + O ( t )  2 2  = u ( t )  (4) Y = 2 1  where disturbances lO(t)l > r/2 cannot be attenuated. In this paper we provide sufficient conditions for disturbance attenuation which generalize those given CH3229-2/92/0000-2469$1.00  1992 IEEE 2469 in [6]: for instance, the example (3) satisfies the con- ditions given here. The control algorithm we develop generalizes the one given in [6] and coincides with it in special simpler cases even though the techniques of proof are entirely different. We do not use singular perturbation techniques; we determine special coordi- nates in which the control algorithm and a Lyapunov function are recursively built. 2 Main Result Definition 2.1 The control characteristic index of system (1) is defined as the integer p such that L,L)h(x)  = 0 ,  0 5 i I p -  2,Vx E R", L,L;-'h(x) # 0 ,  V x E  R". If L,L)h(x) = 0,  Vi,Vx E Rn, then p = CO. 0 Definition 2.2 The disturbance characteristic index v of system (1) is defined as the integer such that L ,L )h ( t )  = 0 ,  0 5 i L V -  2,Vx E R", L,L;-'h(x) # 0,  for some 8 E R ,  some x E R". 0 As shown in [5] and [8] the exact disturbance decou- pling problem is locally solvable if and only if v > p .  Assumption 1 We assume in the following that p is well defined and that v I p < 00, that is that the exact disturbance decoupling problem is not solvable. Definition 2.3 The tracking problem with distur- bance attenuation is said to be solvable for system (1 , if for any smooth bounded reference trajectory y,j(tj, with bounded time derivatives y y ) ,  . . . , &), for any bounded disturbance 9( t )  E R c R P ,  and for any ini- tial condition z(0) a parameterized stale feedback con- and for every c > 0 and T > 0 there exists k ( c , T )  such that x ,  E ,  t )  ezists such that Il+(t)! and the t )  = y ( t )  - y , j ( t )  are bounde Vt 2 0 ,  Ile(t)ll I 6, V t  2. T(c) ,Vk > k ( e , T ) .  0 Under Assumption 1 we can locally define a change of coordinates zn = 4n(x)  with $i(z), p +  1 5 i 5 n, such that < d d j , g  >= 0. In new coordinates we have i l  = 22 + L,h(x)  i,-l = zp + L,L?-'h(x) i p + j  = LjSp+j(z) + Lq4p+j(z,e(t)) i, = L$h(x)  + L,L;-'h(x) + L,L;-'h(x)u A = Pj(z ,e ( t ) ) ,  1 L j I - P (6) Y = 21 in which L,L;-'h(x) # 0, Vx  E R", according to Assumption 1 and Definition 2.1. Denoting .zr = ( zp+l , .  . , zn)  and p = (PI , .  . . , Pn-,) the dynamics i r  =P(zr , z l ( t ) , .  . . , t p ( t ) , B  l ( t ) , . . . , e p ( t ) )  (7) is called the tracking dynamics where z l ( t ) ,  . . . , z,(t),Bl(t), . . . , O p ( t )  are the inputs. When 21 = . - . =  zp = 0 and 0 = 0 the tracking dynamics is the zero dynamics. Assumption 2 The tracking dynamics (7) is bounded input bounded state. Theorem 2.1 Assume in addition to Assumptions 1 and 2 that the following conditions are satisfied for system (1): (i) there exist p - v + 1 smooth functions a i (x ) ,  v - 1,s z 5 p - 1, satisfying dai E span{dh,dLjh,. . . ,dL)h) ,  such that Vx  E R", ve E R, J L , L ) h l I  ai, v -  15 i I p -  1; (ii) the vector fields - 1 1 L;h,  j = -  9 f=f-- L,  L;-'h L,  L5-l h are complete. Then, the problem of tracking with disturbance atten- uation is solvable. Proof We consider the general case in which v = 1. By virtue of Assumption 1 and the additional con- dition (ii) the chan e of coordinates (5) is globally defined (see [SI and73 and system (1) can be glob- ally transformed into (16,. We introduce a new control variable w,  defined as v = L,L?-'h(x)u + L;h(x)  - y y )  (8) which substituted in (6) gives el = e2 +L,h(x)  2470 where ei = zi - yy-'), 1 5 i 5 p. Note that if v > p the control (8) solves the exact disturbance decoupling problem. Define e; = -Le1 - e l p l ( t )  (10) where p1 is a smooth function yet to be defined and k > 0. Consider the function 1 2 (11) VI = -e: The time derivative of VI, with e2 = e; in (9), is given ii = -kef - e:p1 + e lL,h(z)  (12) (13) by and, according to the inequality in (i), we have VI I -ke? - e:pi + lellao(z1) Since a0 is a smooth function and Yd is bounded, we can write ao(z1) = a d e l  + Yd) = Q ' o ( Y ~ )  + &(e l ,  yd)el (14) in which (15) Q'O(z1) - Q'O(Yd) e l  GO(e1, Yd) = Hence, we choose p1 as a smooth function satisfying PI 2 Go(e1, yd). (16) (17) v = e;. (18) From (13), we obtain Vl 5 -ke; + IelIIw(yd)I Therefore, if p = 1 the thesis is proved with In fact, we can write Recalling that Yd is bounded, Icro(yd)l < y ,  y > 0; for every le11 2 €12, we obtain vi - < -2k + 2y/t, Vl - which implies VI ( t ) 5 v1 (0 )e(- + '7' ' I t .  For any E > 0, T > 0 there exists IC which solves the problem. If p > 1, we prove the following Claim. Claim. Assume that for a given index i, 1 5 i 5 p ,  for the system there exist i functions the function has time derivative, with ei+l = ing the inequality in (19), satisfy- with r]i a suitable smooth function such that ( l - l ) ,  k) = 0. (24) r]i(Yd,. .  . l  Yd k lim k - o o  Then, for the system 4 = e z + L , h ( z )  (25) ei+l = ei+2 + LqL>h(z) there exists a function such that in new coordinates (M,+l > 0) the function . i+ l  has time derivative, with ei+2 = ing the inequality in (25), satisfy- 2471 where qi+l is a suitable smooth function such that Proof of the Claim. following notations: For convenience we adopt the %(i) = [ % I , .  . . , Z i I T ,  26) = [yd,. . . , y$-l)]T e ( i )  = [ e l , .  . . , eiIT, ~ ( i )  = [ ~ l , .  . . ,e'iIT From (21), (22) and (25) we obtain k+1 = -kllE(i)l12 + [lg(i)llqi(z$), k )  + Mi+lEiEi+l (31) +gi+ 1 &+ 1 Since by (21) and (25), we define (33) so that (32) with ei+2 = becomes = -kEi+l + pi+l&+l - &Mi+l +- 1 (LqL;h  - Cj,, 2 L q L $ - ' h )  Mi+l Substituting (34) into (31), we have (with ei+2 = By assumption (ii), we have Therefore, we can write (37) i+l i I i  I 2472 which exists since 3/d and its derivatives are bounded, so that (45) From (43), we obtain Therefore, choosing the thesis is proved with U Since we have shown that the hypotheses of the claim are true for i = 1, applying ( p  - 1)-times the claim we can construct a function We also construct a change of coordinates such that the function . P  has time derivative satisfying the inequality (with k suitably redefined) with = 0. (53) qp(Yd, . . . > &-')I  k) k Therefore, &,, 1 5 i I p,  are bounded and conse- quently, since the reference signal yd(t) is bounded with its p - 1 time derivatives, r i ( t ) ,  1 _< i 5 p,  are bounded. Since Assumption 2 holds llzr(t)ll is also bounded. By virtue of (53), we have lim k-m IrlPl 5 Y > 0, so that for every le11 2 c/2 which implies e ? ( t )  5 ~ , ( t )  5 V p ( 0 ) e ( - 2 k + 2 ~ l c ) * .  For any c > 0, T > 0 there exists k which solves the problem. Therefore, the problem is solved by the feed- back controller given by (8) and (49) with a suitable choice of k .  U Remark. Theorem 2.1 generalizes the main result in [6] in several ways. The disturbances are only al- lowed to enter linearly in [6] and are required to have bounded time derivatives while in this paper they may enter nonlinearly and no requirement is made on their time derivatives. While condition (ii) is common to both theorems the most important difference lies in condition (i) which considerably weakens the corre- sponding condition in two ways. The result in [6] re- quires for v - 1 < i < p - 1,  d ( ~ , ~ ) h )  E span{dh,. . . , d ( ~ ; - ' h ) } ,  while condition (i) in Theorem 2.1 only requires d ( ~ , ~ ) h )  E span { d h , .  . . , d ( ~ ) h ) ) ,  v - 1 5  i 5 p-1, (55) or even the weaker condition on some bounding func- tions (54) IL,L)hl 5 ai, 1/ - 1 , s  i 5 p - 1, with dai E span {dh ,  . . . , d ( ~ ) h ) }  For instance condition (55) applies to system (3) while the stronger condition (54) does not. 3 Example Consider the system i1 = xz + e l ( t )  iz = +,(t) (56) Y =  2 1  with l B l ( t ) l  < 1 and 10z(t)l  5 1, where B ( t )  = [61(t), 6 2 ( t ) l T  is a disturbance signal. It is easily seen 2473 that system (31) satisfies the conditions of Theorem 2.1 with Suppose that yd( t )  is the desired output reference to be tracked. Define el = 2 1  - yd, e2 = x2 - yd. (58) From (31) and (32), we have el  = e2 + i2 = .;e,(t) + - jid = z;e2(t)  + w .  A Define as in (10) e; = -kel - e l p l ( t )  Since by (16) p1 L &del ,Yd)  = 0 we choose p1 = 0 so that e; = - k e l .  Define as in (21) and (33) w = -Mz(ke'z + &pZ(t) + M z e l )  - be2 According to (44), we have Mz = k and according to (42) I'o = # + k  rl = -2k = 2le;+3ezid+ 3621 k lx3e2'I which imply 2 k 112 = - ( e ;  + 3e2Yd + 3Y;)'. The final control U is given by U = yd - 2k2(e'2 + e l )  - 2~2(e;  + 3e2yd + 3 ~ : ) ~ .  (59) If we considered as in [6 the problem of stabilizing the using the same design technique, the following control law linear approximation o 2 system (56) we would obtain, U = - k ( k i z  + k ~ 1 )  - kr2 = - 2 k ( ~ 2  + k ~ l )  which is very similar to the one obtained in [6] (60) U =I -- ,i( € 2 2  + 21) 1 k once we define =: -. In [6] it was shown that the control algorithm (60) does not guarantee almost disturbance decoupling for any bounded disturbance. Therefore, the nonlinear part of the control law (59) is crucial in order to obtain disturbance attenuation. Acknowledgement This work was  supported in part by Minister0 dell'Universitd e della Ricerca Scientifica e Tecnolog- ica. References [l] G. Basile and G. Marro. Controlled and condi- tioned invariant subsDaces in linear svstems the- ory. J. of Opt.  Theory and Applicatlons, 3:306- 315, 1969. S.P. Bhattacharyya. Disturbance rejection in lin- ear systems. Inl. J. of Syst. Science, 5:633-637, 1974. C.I. Byrnes and A. Isidori. Global feedback stabi- lization of nonlinear systems. In IEEE 24th Conj. on Decision and Control, pages 1031-1037, Ft. Lauder dale, 1985. R.M. Hirschorn. (A,B)-invariant distributions and disturbance decoupling of nonlinear systems. SIAM J. Control Optimiz., 19:l-19, 1981. A. Isidori, A.J. Krener, C. Gori-Giorgi, and S. Monaco. Nonlinear decoupling via feedback: a differential geometric approach. IEEE Trans. Au- tomatic Control, 26:331-345, 1981. R. Marino, W. Respondek, and A.J. van der Schaft. Almost disturbance decoupling for single-input single-output nonlinear systems. IEEE Trans. Automatic Control, 34:1013-1017, 1989. R. Marino , W. Respondek, and A.J. van der Schaft. A direct approach to almost disturbance and almost input-output de- coupling. Int. J. of Control, 48~353-383, 1988. H. Nijmeijer and K. Tchon. A n  Input-Output Characterization of Nonlinear Disturbance De- coupling. Technical Report 502, Dept. Math. Twente Univ. Technol., Feb. 1985. H. Nijmeijer and A. van der Schaft. Nonlin- ear Dynamical Control Systems. Springer-Verlag, Berlin, 1990. A. Saberi. Output feedback control with almost disturbance decoupling property: a singular per- turbation approach. Int. J. of Control, 45:1705- 1722, 1987. J.C. Willems. Almost invariant subspaces: an approach to high-gain feedback design-Part I: Al- most controlled invariant subspaces. IEEE Trans. Automatic Control, 26:235-252, 1982. W.M. Wonham and AS.  Morse. Decoupling and pole assignment in linear multivariable systems. SIAM J. Control Optimit . ,  8:l-18, 1970. 2474 

Tracking Control of Nonlinear Systems with Disturbance Attenuation

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Tracking Control of Nonlinear Systems with Disturbance Attenuation

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