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A remark on passivity-based and discontinuous control of uncertain nonlinear systems
Automatica 37 (2001) 1481}1487
Technical Communique
A remark on passivity-based and discontinuous control of uncertain nonlinear systems夽 Antonio LormH a *, Elena Panteley, Henk Nijmeijer CNRS, Laboratoire d+Automatique, de Grenoble B.P 46, 38402 St. Martin d'Heres, France Inst. Mech. Engg. Acad. Sc. of Russia, 61, Bolshoi Ave V.O., St. Petersburg, Russia Faculty of Mech. Engg., Technical University of Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands Received 25 January 2000; revised 21 September 2000; received in "nal form 15 March 2000
Abstract We address the problem of robust stabilisation of nonlinear systems a!ected by time-varying uniformly bounded (in time) a$ne perturbations. Our approach relies on the combination of sliding mode techniques and passivity-based control. Roughly speaking we show that under suitable conditions the sliding mode variable can be chosen as the passive output of the perturbed system. Then we show how to construct a controller which guarantees the global uniform convergence of the plant's outputs towards a time-varying desired reference, even in the presence of permanently exciting time-varying disturbances. We illustrate our result on the tracking control of the van der Pol oscillator. 2001 Published by Elsevier Science Ltd. Keywords: Passivity; Mechanical systems; Sliding mode; Tracking control
1. Introduction Motivated by real life applications, robust adaptive control has been a subject of study for many years now. Numerous approaches, sometimes based on the physical structure of the system, have been proposed using techniques as backstepping (Freeman & KokotovicH , 1996), sliding modes (Utkin, 1992; Sira-Ramirez, 1993), passivity-based (Byrnes, Isidori, & Willems, 1991; Ortega, LormH a, Nicklasson, & Sira-RammH rez, 1998), feedback linearisation (Isidori, 1995; Nijmeijer & van der Schaft, 1990) or high gain (Marino & Tomei, 1993). From a theoretical viewpoint, sliding mode control results are very attractive since ideally, they exactly compensate for disturbances (which are bounded in time) and allow convergence in "nite time, to the so-called sliding
夽 A preliminary version of this note was presented in NOLCOS 1998. This paper was recommended for publication in revised form by Associate Editor R. Sepulchre under the direction of Editor Paul Van den Hof. This work was partially carried out while the "rst author was with the Norwegian University of Science and Technology. * Corresponding author. Tel.: #33-4-76826230; fax: #33-476826388. E-mail addresses: [email protected] (A. LormH a), [email protected] (E. Panteley).
manifold. Then, if the zero-dynamics corresponding to this ideal sliding motion is asymptotically stable, perfect tracking is achieved. The fact that the `disturbancesa may represent unmodeled (actuator) dynamics or nonlinearities due to state estimation mismatch in output feedback control (Emelyanov, Korovin, Nersisian, & Nizenson, 1992; Sanchis & Nijmeijer, 1998), makes the "eld of application of this approach potentially very wide. Even though sliding mode control induces an often undesirable chattering in the input which uses the actuators, a popular area of application is in tracking control of mechanical and electromechanical systems (Stepanenko & Yuan, 1993; Slotine & Sastry, 1983; Gorez, 1999; Panteley, Ortega, & Gafvert, 1998). To avoid chattering often precautions as for instance the introduction of a boundary layer are taken, cf. Utkin (1992). In this communique, we highlight some links between two popular control approaches: passivity based (Byrnes et al., 1991; Ortega et al., 1998) and sliding mode. The advantage of the former is that it is an energy-based approach and therefore, it exploits the known physical structure of the system. As pointed out above, discontinuous controls are (theoretically) attractive in robust control applications. Typically, for relative-degree-one systems the control design is based on "rst, tracking the
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system trajectories to a sliding manifold, and then, in designing a control law which keeps the trajectories in it. The usual criteria considered to design the sliding surface is that it should de"ne an asymptotically stable zerodynamics. However, in general, there is no engineering intuition on how to design a suitable sliding surface satisfying the required conditions mentioned. In this note, we will show that robust control of uncertain systems with a$ne disturbances, may be possible when the given nominal system is passi"able via state feedback and then, under appropriate assumptions on the disturbances, a discontinuous term is included in the controller as to compensate for the disturbances. Even though this technique is well understood in the literature of sliding modes, the novelty in our result is to look at it from a passivity viewpoint. Indeed, our proof relies on a passivity approach rather than on a classical Lyapunov analysis. To put our discussion in perspective and to introduce some notation, let us brie#y recall some aspects from passivity-based and discontinuous control.
2.2. Discontinuous control to compensate for bounded disturbances We brie#y recall in this section a well known technique based on discontinuous control. For a more detailed exposition see e.g. Khalil (1996, Chapter 13). For our purposes let us consider the following simple example of a scalar a$ne system x "f (x)#d(t)#u,
(2a)
y"h(x),
(2b)
where f (x) is locally Lipschitz continuous, f (0)"0, x31 and assume that there exists an unknown positive constant such that d(t)( uniformly in t. Assume further that in the absence of the disturbance (that is, if d,0) a controller u"uH(x) is known such that the system x "f (x)#uH(x) is globally asymptotically stable (GAS). From the GAS assumption it follows (see e.g. Hahn (1967)) that there exists a Lyapunov function <"
2. Preliminaries
where 3K (a function : 1 P1 is of class K, V V (3K) if it is continuous, strictly increasing and zero at zero). Then a well known result in the literature of sliding mode control (see e.g. Utkin, 1992; Sira-Ramirez, 1993) is the following. De"ne sgn()"!1 if (0, sgn()"1 if '0, and sgn(0)3[!1,1], and the sliding surface
2.1. Passivity-based control Consider the a$ne nonlinear system x "f (x)#g(x)u, y" : h(x),
(1)
where x31L, y, u31, f, g and h are continuous vector "elds, respectively functions of x. Assume further that system (1) de"nes a passive operator : u C y with a C storage function < " : <(x) which is bounded from below. Under these considerations, it is well known (JurdjevicH & Quinn, 1978) that the control law u"uH with uH"!(
We remind the reader that a system with output y(t)"h(x(t)) is zero-state detectable if y(t),0 implies that x(t)P0 as tPR.
(3)
(4)
then the adaptive control law u"uH(x)!K sgn(),
(5)
QK "sgn(),
(6)
renders the system (2) globally convergent, that is, for any initial conditions x(t ) we have that lim x(t)"0. R For the sake of completeness, we proceed to prove the claim above. As a matter of fact, the proof of our main result will follow along similar lines. First, notice that since is constant the closed loop system (2), (5) is given by x "f (x)#uH(x)!K sgn()#d(t),
(7)
QI "sgn(),
(8)
where we de"ned the estimation error I " : K !. De"ne also the Lyapunov function <(x, I ) " :
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
where " : col[x, I ]? and
y31K,
i (7), (8) g F(t, ) " : j f (x)#uH(x)!K #d(t) g 0 k
∀O0, "0,
where 3[!1,1]. Thus, using d(t)( "sgn(), the time derivative of < satis"es
(9)
and
(10)
for all solutions (t) such that 3F(t, (t)) and where
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(11)
and assume further that system (2a) is zero-state detectable with respect to the same output y. Under these conditions one can construct a globally stabilising control law u"uH(x) using Theorem 1. 2. If we substitute the output de"nition (11) in the controller (5), (6) then the sliding variable happens to be the output of the unperturbed system for which a globally stabilising controller is available. From this, one might conjecture that the controller (5), (6), (11) guarantees the global convergence of x(t) under the assumptions that (2) de"nes an output strictly passive operator (OSP) u#d(t) C y and a `detectability conditiona. The formal statement of this conjecture is strictly contained in our main result which we present below.
3. Main result We consider nonlinear systems perturbed by a$ne disturbances, with control input u31K and output
x "f (t, x)#g(t, x)[u#d(t, x)],
(12a)
y"h(t, x),
(12b)
where the "elds f (t, x), g(t, x), and the function h(t, x) are locally Lipschitz continuous, the input and the disturbance d : 1 ;1LP1L satis"es the following: V A 1 Each component of the vector xeld d is uniformly bounded in t as follows d (t, x)) G (x)# G , ∀ i)n, (13) G G where G and G are unknown non-negative constants and (x) : 1 P1 is a known continuous function. G V V An obvious implication of (13) is the existence of a continuous function : 1 P1 and positive conV V stants and such that d(t, x)) (x)# , or in compact form d(t, x)) ?(x),
(14)
where we have de"ned ? " : ( , ) and ? (x) " : ((x),1). It is important to remark that the disturbance d(t, x) may be the result of unmodeled dynamics, noisy measurements, parameter uncertainty or, thinking of the system (12) with d,0 as a passive system (uCy), the disturbances d are non dissipative forces a!ecting the plant. Physical systems modeled by (12) are numerous, e.g. Euler}Lagrange systems or chaotic oscillators such as the periodically forced Du$ng equation or the van der Pol system and certain under-actuated systems. We are interested in a design methodology for a controller u"u(t, x) which guarantees that the solution x(t; t , x ) of (12) with initial conditions t , x "x(t ) is globally convergent, that is, lim x(t)"0 for any initial condiR tions (t , x ). To that end we will use state discontinuous controllers like (5). As previously discussed, an important implication of this is that the usual regularity conditions for unicity of (closed loop) solutions do not hold. We will not go deeper into this question but refer the reader to Clarke et al. (1998). We will simply notice that for the purposes of this note it is su$cient to observe that the only discontinuity in the closed loop dynamics is due to sgn( ) ) which implies the existence of (limiting) solutions and that they exist for all time, that is, the closed loop system is forward complete (see Clarke et al. (1998)). This said, let us introduce the following de"nition which we will use in order to provide a `passivity-baseda proof of our main result. De5nition 1 (Strong zero-state detectability). Consider the system x "f (t, x)#v where f ( ) , ) ) is such that all existing solutions x( ) , t , x ) are de"ned on [t ,R). Consider also the closed set UL1L. The system, with inputs
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v3U and outputs y"h(t, x) where h( ) , ) ) is continuous, is said to be strongly zero-state detectable if
v3U and y,0 implies that all existing solutions satisfy lim x(t)"0. R Theorem 2. Consider the system (12), (13) under the following assumptions A 2 A passifying controller u"uH(t, x) is known such that the system x "f (t, x)#g(t, x)[uH(t, x)#v]
(15a)
y"h(t, x),
(15b)
where h3C, dexnes an OSP operator : vCy with output y3LK and positive dexnite proper storage function
(16)
QK "(x) sgn(y) y,
(17)
?
where K is the estimate of , sgn(y)"[sgn(y ),2, sgn(y )]? and "?'0, in closed loop with (12), (13) K guarantees that the state trajectory x(t) is globally convergent and the system is uniformly globally stable. Remark 1. Noticing that since (15a) de"nes an OSP operator with storage function
(19)
then the bound (18) on the storage function
where y stands for the Frobenius norm, i.e. y " : Ly . G G Notice that the subtraction of the last two terms on the right hand side of (21) is non-positive hence
(22)
hence the system is uniformly globally stable. The inequality (22) also implies that all <(t, x(t), I (t)) are decreasing and hence bounded so the complete state (x, I )3LL>. Furthermore, integrating (22) from t to R one concludes that y3LK. Next, notice from (19), (14) that since is constant and ( ) ) is continuous, v(t) is also uniformly bounded. From the continuity of h(t, x) it follows that all y3LK . Then, from the closed loop dynam ics (15a), (19) we obtain that x 3LL and since h3C we conclude using Barbalat's lemma, that each y(t)P0 as tPR. The result is "nally obtained using the assumption of strong zero-state detectability with U being the set of all bounded signals v(t, x(t)) de"ned in (19) with y,0. 䊐
4. Discussion (a) From a Lyapunov stabilisation viewpoint, our controller uH(t, x) which is designed to render OSP the nominal system, may be compared to the `equivalent controla of the sliding mode approach. However, in contrast to the latter technique, our control design is passivity-based and therefore, exploits the natural structure of the system instead of forcing the trajectories to remain on a sliding manifold for which, in general, there is no physical motivation behind. In our interpretation, the sliding manifold (x)"0 corresponds to zeroing the passive output. However, notice that uH(t, x) is not designed with the aim at achieving "nite time convergence to the sliding manifold but asymptotic convergence of the passive output. This has important consequences since a known practical disadvantage of discontinuous controllers achieving "nite time convergence is chattering. (b) Some other interesting remarks concerning conditions under which the sliding variable (x) can be considered as a passive output are found in Sira-Ramirez (1999). In this reference, su$cient conditions are given to transform an a$ne nonlinear system into a Hamiltoninan one. Then conditions on the resulting dynamics are given so that the system de"nes a passive map with output y"(x) and storage
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
condition
(23)
y"h(t, x),
(24)
where x31L, and 31J of constant unknown parameters. Using Theorem 2 a controller u"u(t, x) guaranteeing that x(t)P0 as tPR can be designed if the dynamics (23) admit the following parameterisation x " f (t, x)#g(t, x)[u#d(t, x, )] where the nonlinearity d(t, x, ) satis"es d(t, x, )) (x)# (25) where and are unknown constants and ( ) ) is a continuous known function. Notice furthermore, that the parameters vector does not need to be constant, as long as it de"nes continuous uniformly bounded signals, that is, if (t)3LJ and a bound like (25) be determined. 4.1. Example: the van der Pol oscillator The van der Pol oscillator is known to be the model of an LC electrical network with a nonlinear load and all elements connected in parallel (see e.g. Kapitaniak (1996)). If we assume that the load is an active circuit with input voltage v and input current i"(v) with (v)"!v#v. Assuming further that the circuit is excited by an AC current source of frequency and amplitude Q " : q, with q'0 we can express the dynamic equations of the circuit by the well known equation vK #(1!v)v #v"q cos(t),
(26)
which van der Pol used to model an electrical circuit with a triode valve. We remark that for the particular case when "5, q"5 and "2.463 the van der Pol oscillator exhibits a chaotic behaviour (Parlitz & Lauterborn, 1987). Consider now the controlled unperturbed van der Pol equation with the voltage v delivered to the active load as the output to be controlled, that is vK #(1!v)v #v"uH#w,
(27)
where w is a bounded external input which includes the term q cos(t) and uH(v)"!k v !k v #(2vv v !v v )#v ,
(28)
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where v is the desired constant output voltage and v "v!v . Then, the closed loop system (27), (28) de"nes an output strictly passive operator : w C y with respect to the output y"v #v where '0 is small enough to ensure that the storage function 1 1 <(v,v )" v # (k #1)v 2 2 1 # v v # (k !)v # v , 4 2 is positive de"nite. Furthermore, by choosing k *2## and k *2 the time derivative of <(v,v ) N yields
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Fig. 1.
Fig. 2.
output y"v #v with " : /1#v , on the other hand, this output was found in experiments to perform best in robot control applications (see Gorez (1999) and references therein).
5. Concluding remarks We have highlighted some links between passivitybased, and discontinuous control design for uncertain nonlinear systems. Most importantly, we have provided with a di!erent insight to the sliding mode technique. The main di!erence with the common sliding mode controllers is that the control design starts at a passivation step and not at the de"nition of a sliding regime. The second part of the control design is made with the aim at compensating for the disturbances and not with the aim at tightening the dynamics of the system to a manifold.
References Byrnes, C., Isidori, A., & Willems, J. C. (1991). Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transactions on Automatic Control, AC-36(11), 1228}1240. Clarke, F. H., Ledyaev, Yu. S., Stern, R. J., & Wolenski, P. R. (1998). Nonsmooth analysis and control theory. Graduate texts in mathematics. Berlin: Springer. Desoer, C. A., & Vidyasagar, M. (1975). Feedback systems: Input}output properties. New York: Academic Press. Emelyanov, S. V., Korovin, S. K., Nersisian, A. L., & Nizenson, Yu. E. (1992). Output feedback stabilization of uncertain plants: A variable structure systems approach. International Journal of Control, 55(1), 61}81. Freeman, R. A., & KokotovicH , P. V. (1996). Robust nonlinear control design: State-space and Lyapunov control techniques. Boston: BirkhaK user. Gorez, R. (1999). Conclusions of "ve-year investigations in sliding mode control of manipulators. Proceedings of the xfth European control conference, Karlsruhe, Germany. Hahn, W. (1967). Stability of motion. New York: Springer. Isidori, A. (1995). Nonlinear control systems. (3rd ed). Berlin: Springer.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
JurdjevicH , V., & Quinn, J. P. (1978). Controllability and stability. Journal of Diwerential Equations, 28, 381}389. Kapitaniak, T. (1996). Controlling chaos. New York: Academic Press. Khalil, H. (1996). Nonlinear systems. (2nd ed.). New York: Macmillan Publishing Co. Marino, R. L., & Tomei, P. (1993). Robust stabilization of feedback linearizable time-varying uncertain nonlinear systems. Automatica, 29, 181}189. Nijmeijer, H., & van der Schaft, A. J. (1990). Nonlinear dynamical control systems. New York: Springer. Ortega, R., LormH a, A., Nicklasson, P. J., & Sira-RammH rez, H. (1998). Passivity-based control of Euler}Lagrange systems: Mechanical, electrical and electromechanical applications. Communications and control engineering. London: Springer, ISBN 1-85233-016-3. Panteley, E., Ortega, R., & Gafvert, M. (1998). An adaptive friction compensator for global tracking in robot manipulators. Systems & Control Letters, 33(5) 307}313. Parlitz, U., & Lauterborn, W. (1987). Period-doubling cascades and devil's staircases of the driven van der pol equation. Physical Review A, 36(3), 1428}1434.
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Sanchis, R., & Nijmeijer, H. (1998). Sliding controller-sliding observer design for nonlinear systems. European Journal of Control, 4, 208}234. Sira-Ramirez, H. (1993). On the dynamical sliding mode control of nonlinear systems. International Journal of Control, 57(5), 1039}1061. Sira-Ramirez, H. (1999). A general canonical form for sliding mode control of nonlinear systems. Proceedings of the xfth European control conference, Karlsruhe, Germany. Slotine, J. J., & Li, W. (1988). Adaptive manipulator control: A case study. IEEE Transactions on Automatic Control, AC-33, 995}1003. Slotine, J. J., & Sastry, S. (1983). Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulators. International Journal of Control, 38(2), 465}492. Stepanenko, Y., & Yuan, J. (1993). Variable structure control of robot manipulators with nonlinear sliding manifolds.. International Journal of Control, 58(2), 285}300. Utkin, V. I. (1992). Sliding modes in control optimization. Berlin, Heidelberg: Springer.
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Technical Communique
A remark on passivity-based and discontinuous control of uncertain nonlinear systems夽 Antonio LormH a *, Elena Panteley, Henk Nijmeijer CNRS, Laboratoire d+Automatique, de Grenoble B.P 46, 38402 St. Martin d'Heres, France Inst. Mech. Engg. Acad. Sc. of Russia, 61, Bolshoi Ave V.O., St. Petersburg, Russia Faculty of Mech. Engg., Technical University of Eindhoven, P.O. Box 513, 5600 MB Eindhoven, Netherlands Received 25 January 2000; revised 21 September 2000; received in "nal form 15 March 2000
Abstract We address the problem of robust stabilisation of nonlinear systems a!ected by time-varying uniformly bounded (in time) a$ne perturbations. Our approach relies on the combination of sliding mode techniques and passivity-based control. Roughly speaking we show that under suitable conditions the sliding mode variable can be chosen as the passive output of the perturbed system. Then we show how to construct a controller which guarantees the global uniform convergence of the plant's outputs towards a time-varying desired reference, even in the presence of permanently exciting time-varying disturbances. We illustrate our result on the tracking control of the van der Pol oscillator. 2001 Published by Elsevier Science Ltd. Keywords: Passivity; Mechanical systems; Sliding mode; Tracking control
1. Introduction Motivated by real life applications, robust adaptive control has been a subject of study for many years now. Numerous approaches, sometimes based on the physical structure of the system, have been proposed using techniques as backstepping (Freeman & KokotovicH , 1996), sliding modes (Utkin, 1992; Sira-Ramirez, 1993), passivity-based (Byrnes, Isidori, & Willems, 1991; Ortega, LormH a, Nicklasson, & Sira-RammH rez, 1998), feedback linearisation (Isidori, 1995; Nijmeijer & van der Schaft, 1990) or high gain (Marino & Tomei, 1993). From a theoretical viewpoint, sliding mode control results are very attractive since ideally, they exactly compensate for disturbances (which are bounded in time) and allow convergence in "nite time, to the so-called sliding
夽 A preliminary version of this note was presented in NOLCOS 1998. This paper was recommended for publication in revised form by Associate Editor R. Sepulchre under the direction of Editor Paul Van den Hof. This work was partially carried out while the "rst author was with the Norwegian University of Science and Technology. * Corresponding author. Tel.: #33-4-76826230; fax: #33-476826388. E-mail addresses: [email protected] (A. LormH a), [email protected] (E. Panteley).
manifold. Then, if the zero-dynamics corresponding to this ideal sliding motion is asymptotically stable, perfect tracking is achieved. The fact that the `disturbancesa may represent unmodeled (actuator) dynamics or nonlinearities due to state estimation mismatch in output feedback control (Emelyanov, Korovin, Nersisian, & Nizenson, 1992; Sanchis & Nijmeijer, 1998), makes the "eld of application of this approach potentially very wide. Even though sliding mode control induces an often undesirable chattering in the input which uses the actuators, a popular area of application is in tracking control of mechanical and electromechanical systems (Stepanenko & Yuan, 1993; Slotine & Sastry, 1983; Gorez, 1999; Panteley, Ortega, & Gafvert, 1998). To avoid chattering often precautions as for instance the introduction of a boundary layer are taken, cf. Utkin (1992). In this communique, we highlight some links between two popular control approaches: passivity based (Byrnes et al., 1991; Ortega et al., 1998) and sliding mode. The advantage of the former is that it is an energy-based approach and therefore, it exploits the known physical structure of the system. As pointed out above, discontinuous controls are (theoretically) attractive in robust control applications. Typically, for relative-degree-one systems the control design is based on "rst, tracking the
0005-1098/01/$ - see front matter 2001 Published by Elsevier Science Ltd. PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 0 9 7 - 8
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system trajectories to a sliding manifold, and then, in designing a control law which keeps the trajectories in it. The usual criteria considered to design the sliding surface is that it should de"ne an asymptotically stable zerodynamics. However, in general, there is no engineering intuition on how to design a suitable sliding surface satisfying the required conditions mentioned. In this note, we will show that robust control of uncertain systems with a$ne disturbances, may be possible when the given nominal system is passi"able via state feedback and then, under appropriate assumptions on the disturbances, a discontinuous term is included in the controller as to compensate for the disturbances. Even though this technique is well understood in the literature of sliding modes, the novelty in our result is to look at it from a passivity viewpoint. Indeed, our proof relies on a passivity approach rather than on a classical Lyapunov analysis. To put our discussion in perspective and to introduce some notation, let us brie#y recall some aspects from passivity-based and discontinuous control.
2.2. Discontinuous control to compensate for bounded disturbances We brie#y recall in this section a well known technique based on discontinuous control. For a more detailed exposition see e.g. Khalil (1996, Chapter 13). For our purposes let us consider the following simple example of a scalar a$ne system x "f (x)#d(t)#u,
(2a)
y"h(x),
(2b)
where f (x) is locally Lipschitz continuous, f (0)"0, x31 and assume that there exists an unknown positive constant such that d(t)( uniformly in t. Assume further that in the absence of the disturbance (that is, if d,0) a controller u"uH(x) is known such that the system x "f (x)#uH(x) is globally asymptotically stable (GAS). From the GAS assumption it follows (see e.g. Hahn (1967)) that there exists a Lyapunov function <"
2. Preliminaries
where 3K (a function : 1 P1 is of class K, V V (3K) if it is continuous, strictly increasing and zero at zero). Then a well known result in the literature of sliding mode control (see e.g. Utkin, 1992; Sira-Ramirez, 1993) is the following. De"ne sgn()"!1 if (0, sgn()"1 if '0, and sgn(0)3[!1,1], and the sliding surface
2.1. Passivity-based control Consider the a$ne nonlinear system x "f (x)#g(x)u, y" : h(x),
(1)
where x31L, y, u31, f, g and h are continuous vector "elds, respectively functions of x. Assume further that system (1) de"nes a passive operator : u C y with a C storage function < " : <(x) which is bounded from below. Under these considerations, it is well known (JurdjevicH & Quinn, 1978) that the control law u"uH with uH"!(
We remind the reader that a system with output y(t)"h(x(t)) is zero-state detectable if y(t),0 implies that x(t)P0 as tPR.
(3)
(4)
then the adaptive control law u"uH(x)!K sgn(),
(5)
QK "sgn(),
(6)
renders the system (2) globally convergent, that is, for any initial conditions x(t ) we have that lim x(t)"0. R For the sake of completeness, we proceed to prove the claim above. As a matter of fact, the proof of our main result will follow along similar lines. First, notice that since is constant the closed loop system (2), (5) is given by x "f (x)#uH(x)!K sgn()#d(t),
(7)
QI "sgn(),
(8)
where we de"ned the estimation error I " : K !. De"ne also the Lyapunov function <(x, I ) " :
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where " : col[x, I ]? and
y31K,
i (7), (8) g F(t, ) " : j f (x)#uH(x)!K #d(t) g 0 k
∀O0, "0,
where 3[!1,1]. Thus, using d(t)( "sgn(), the time derivative of < satis"es
(9)
and
(10)
for all solutions (t) such that 3F(t, (t)) and where
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(11)
and assume further that system (2a) is zero-state detectable with respect to the same output y. Under these conditions one can construct a globally stabilising control law u"uH(x) using Theorem 1. 2. If we substitute the output de"nition (11) in the controller (5), (6) then the sliding variable happens to be the output of the unperturbed system for which a globally stabilising controller is available. From this, one might conjecture that the controller (5), (6), (11) guarantees the global convergence of x(t) under the assumptions that (2) de"nes an output strictly passive operator (OSP) u#d(t) C y and a `detectability conditiona. The formal statement of this conjecture is strictly contained in our main result which we present below.
3. Main result We consider nonlinear systems perturbed by a$ne disturbances, with control input u31K and output
x "f (t, x)#g(t, x)[u#d(t, x)],
(12a)
y"h(t, x),
(12b)
where the "elds f (t, x), g(t, x), and the function h(t, x) are locally Lipschitz continuous, the input and the disturbance d : 1 ;1LP1L satis"es the following: V A 1 Each component of the vector xeld d is uniformly bounded in t as follows d (t, x)) G (x)# G , ∀ i)n, (13) G G where G and G are unknown non-negative constants and (x) : 1 P1 is a known continuous function. G V V An obvious implication of (13) is the existence of a continuous function : 1 P1 and positive conV V stants and such that d(t, x)) (x)# , or in compact form d(t, x)) ?(x),
(14)
where we have de"ned ? " : ( , ) and ? (x) " : ((x),1). It is important to remark that the disturbance d(t, x) may be the result of unmodeled dynamics, noisy measurements, parameter uncertainty or, thinking of the system (12) with d,0 as a passive system (uCy), the disturbances d are non dissipative forces a!ecting the plant. Physical systems modeled by (12) are numerous, e.g. Euler}Lagrange systems or chaotic oscillators such as the periodically forced Du$ng equation or the van der Pol system and certain under-actuated systems. We are interested in a design methodology for a controller u"u(t, x) which guarantees that the solution x(t; t , x ) of (12) with initial conditions t , x "x(t ) is globally convergent, that is, lim x(t)"0 for any initial condiR tions (t , x ). To that end we will use state discontinuous controllers like (5). As previously discussed, an important implication of this is that the usual regularity conditions for unicity of (closed loop) solutions do not hold. We will not go deeper into this question but refer the reader to Clarke et al. (1998). We will simply notice that for the purposes of this note it is su$cient to observe that the only discontinuity in the closed loop dynamics is due to sgn( ) ) which implies the existence of (limiting) solutions and that they exist for all time, that is, the closed loop system is forward complete (see Clarke et al. (1998)). This said, let us introduce the following de"nition which we will use in order to provide a `passivity-baseda proof of our main result. De5nition 1 (Strong zero-state detectability). Consider the system x "f (t, x)#v where f ( ) , ) ) is such that all existing solutions x( ) , t , x ) are de"ned on [t ,R). Consider also the closed set UL1L. The system, with inputs
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v3U and outputs y"h(t, x) where h( ) , ) ) is continuous, is said to be strongly zero-state detectable if
v3U and y,0 implies that all existing solutions satisfy lim x(t)"0. R Theorem 2. Consider the system (12), (13) under the following assumptions A 2 A passifying controller u"uH(t, x) is known such that the system x "f (t, x)#g(t, x)[uH(t, x)#v]
(15a)
y"h(t, x),
(15b)
where h3C, dexnes an OSP operator : vCy with output y3LK and positive dexnite proper storage function
(16)
QK "(x) sgn(y) y,
(17)
?
where K is the estimate of , sgn(y)"[sgn(y ),2, sgn(y )]? and "?'0, in closed loop with (12), (13) K guarantees that the state trajectory x(t) is globally convergent and the system is uniformly globally stable. Remark 1. Noticing that since (15a) de"nes an OSP operator with storage function
(19)
then the bound (18) on the storage function
where y stands for the Frobenius norm, i.e. y " : Ly . G G Notice that the subtraction of the last two terms on the right hand side of (21) is non-positive hence
(22)
hence the system is uniformly globally stable. The inequality (22) also implies that all <(t, x(t), I (t)) are decreasing and hence bounded so the complete state (x, I )3LL>. Furthermore, integrating (22) from t to R one concludes that y3LK. Next, notice from (19), (14) that since is constant and ( ) ) is continuous, v(t) is also uniformly bounded. From the continuity of h(t, x) it follows that all y3LK . Then, from the closed loop dynam ics (15a), (19) we obtain that x 3LL and since h3C we conclude using Barbalat's lemma, that each y(t)P0 as tPR. The result is "nally obtained using the assumption of strong zero-state detectability with U being the set of all bounded signals v(t, x(t)) de"ned in (19) with y,0. 䊐
4. Discussion (a) From a Lyapunov stabilisation viewpoint, our controller uH(t, x) which is designed to render OSP the nominal system, may be compared to the `equivalent controla of the sliding mode approach. However, in contrast to the latter technique, our control design is passivity-based and therefore, exploits the natural structure of the system instead of forcing the trajectories to remain on a sliding manifold for which, in general, there is no physical motivation behind. In our interpretation, the sliding manifold (x)"0 corresponds to zeroing the passive output. However, notice that uH(t, x) is not designed with the aim at achieving "nite time convergence to the sliding manifold but asymptotic convergence of the passive output. This has important consequences since a known practical disadvantage of discontinuous controllers achieving "nite time convergence is chattering. (b) Some other interesting remarks concerning conditions under which the sliding variable (x) can be considered as a passive output are found in Sira-Ramirez (1999). In this reference, su$cient conditions are given to transform an a$ne nonlinear system into a Hamiltoninan one. Then conditions on the resulting dynamics are given so that the system de"nes a passive map with output y"(x) and storage
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condition
(23)
y"h(t, x),
(24)
where x31L, and 31J of constant unknown parameters. Using Theorem 2 a controller u"u(t, x) guaranteeing that x(t)P0 as tPR can be designed if the dynamics (23) admit the following parameterisation x " f (t, x)#g(t, x)[u#d(t, x, )] where the nonlinearity d(t, x, ) satis"es d(t, x, )) (x)# (25) where and are unknown constants and ( ) ) is a continuous known function. Notice furthermore, that the parameters vector does not need to be constant, as long as it de"nes continuous uniformly bounded signals, that is, if (t)3LJ and a bound like (25) be determined. 4.1. Example: the van der Pol oscillator The van der Pol oscillator is known to be the model of an LC electrical network with a nonlinear load and all elements connected in parallel (see e.g. Kapitaniak (1996)). If we assume that the load is an active circuit with input voltage v and input current i"(v) with (v)"!v#v. Assuming further that the circuit is excited by an AC current source of frequency and amplitude Q " : q, with q'0 we can express the dynamic equations of the circuit by the well known equation vK #(1!v)v #v"q cos(t),
(26)
which van der Pol used to model an electrical circuit with a triode valve. We remark that for the particular case when "5, q"5 and "2.463 the van der Pol oscillator exhibits a chaotic behaviour (Parlitz & Lauterborn, 1987). Consider now the controlled unperturbed van der Pol equation with the voltage v delivered to the active load as the output to be controlled, that is vK #(1!v)v #v"uH#w,
(27)
where w is a bounded external input which includes the term q cos(t) and uH(v)"!k v !k v #(2vv v !v v )#v ,
(28)
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where v is the desired constant output voltage and v "v!v . Then, the closed loop system (27), (28) de"nes an output strictly passive operator : w C y with respect to the output y"v #v where '0 is small enough to ensure that the storage function 1 1 <(v,v )" v # (k #1)v 2 2 1 # v v # (k !)v # v , 4 2 is positive de"nite. Furthermore, by choosing k *2## and k *2 the time derivative of <(v,v ) N yields
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Fig. 1.
Fig. 2.
output y"v #v with " : /1#v , on the other hand, this output was found in experiments to perform best in robot control applications (see Gorez (1999) and references therein).
5. Concluding remarks We have highlighted some links between passivitybased, and discontinuous control design for uncertain nonlinear systems. Most importantly, we have provided with a di!erent insight to the sliding mode technique. The main di!erence with the common sliding mode controllers is that the control design starts at a passivation step and not at the de"nition of a sliding regime. The second part of the control design is made with the aim at compensating for the disturbances and not with the aim at tightening the dynamics of the system to a manifold.
References Byrnes, C., Isidori, A., & Willems, J. C. (1991). Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems. IEEE Transactions on Automatic Control, AC-36(11), 1228}1240. Clarke, F. H., Ledyaev, Yu. S., Stern, R. J., & Wolenski, P. R. (1998). Nonsmooth analysis and control theory. Graduate texts in mathematics. Berlin: Springer. Desoer, C. A., & Vidyasagar, M. (1975). Feedback systems: Input}output properties. New York: Academic Press. Emelyanov, S. V., Korovin, S. K., Nersisian, A. L., & Nizenson, Yu. E. (1992). Output feedback stabilization of uncertain plants: A variable structure systems approach. International Journal of Control, 55(1), 61}81. Freeman, R. A., & KokotovicH , P. V. (1996). Robust nonlinear control design: State-space and Lyapunov control techniques. Boston: BirkhaK user. Gorez, R. (1999). Conclusions of "ve-year investigations in sliding mode control of manipulators. Proceedings of the xfth European control conference, Karlsruhe, Germany. Hahn, W. (1967). Stability of motion. New York: Springer. Isidori, A. (1995). Nonlinear control systems. (3rd ed). Berlin: Springer.
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JurdjevicH , V., & Quinn, J. P. (1978). Controllability and stability. Journal of Diwerential Equations, 28, 381}389. Kapitaniak, T. (1996). Controlling chaos. New York: Academic Press. Khalil, H. (1996). Nonlinear systems. (2nd ed.). New York: Macmillan Publishing Co. Marino, R. L., & Tomei, P. (1993). Robust stabilization of feedback linearizable time-varying uncertain nonlinear systems. Automatica, 29, 181}189. Nijmeijer, H., & van der Schaft, A. J. (1990). Nonlinear dynamical control systems. New York: Springer. Ortega, R., LormH a, A., Nicklasson, P. J., & Sira-RammH rez, H. (1998). Passivity-based control of Euler}Lagrange systems: Mechanical, electrical and electromechanical applications. Communications and control engineering. London: Springer, ISBN 1-85233-016-3. Panteley, E., Ortega, R., & Gafvert, M. (1998). An adaptive friction compensator for global tracking in robot manipulators. Systems & Control Letters, 33(5) 307}313. Parlitz, U., & Lauterborn, W. (1987). Period-doubling cascades and devil's staircases of the driven van der pol equation. Physical Review A, 36(3), 1428}1434.
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Sanchis, R., & Nijmeijer, H. (1998). Sliding controller-sliding observer design for nonlinear systems. European Journal of Control, 4, 208}234. Sira-Ramirez, H. (1993). On the dynamical sliding mode control of nonlinear systems. International Journal of Control, 57(5), 1039}1061. Sira-Ramirez, H. (1999). A general canonical form for sliding mode control of nonlinear systems. Proceedings of the xfth European control conference, Karlsruhe, Germany. Slotine, J. J., & Li, W. (1988). Adaptive manipulator control: A case study. IEEE Transactions on Automatic Control, AC-33, 995}1003. Slotine, J. J., & Sastry, S. (1983). Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulators. International Journal of Control, 38(2), 465}492. Stepanenko, Y., & Yuan, J. (1993). Variable structure control of robot manipulators with nonlinear sliding manifolds.. International Journal of Control, 58(2), 285}300. Utkin, V. I. (1992). Sliding modes in control optimization. Berlin, Heidelberg: Springer.
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