Robust Adaptive Control of Passive Systems with Unknown Disturbances

Copyright 10 IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998

ROBUST ADAPTIVE CONTROL OF PASSIVE SYSTEMS WITH UNKNOWN DISTURBANCES Antonio Loria· Elena Panteley· Henk Nijmeijer* Thor I. Fossent Dept. of Electrical Engineering University of California, Santa Barbara, CA, 93106-9560, USA [email protected] [email protected] *Faculty of Applied Mathematics University of Twente P.O.Box 217, 7500 AE Enschede, The Netherlands [email protected]

*Facu1ty of Mechanical Engineering, Technical University of Eindhoven, POBox 513,5600 MB Eindhoven, The Netherlands t Department

of Engineering Cybernetics Norwegian Univ. of Sc. and Technology O.S. Bragstads plass 8, N 7034 Trondheim, Norway tif@itk .ntnu.no

Abstract: We address the problem of robust stabilisation of nonlinear systems affected by time-varying uniformly bounded affine 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 output of the perturbed system in question. 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. As applications of our results we address the problems of tracking control of the van der Pol oscillator and ship dynamic positioning. Copyright © 1998 IFAC Keywords: Passivity, Mechanical systems, Sliding mode, Tra<;king control.

1

Introduction and problem formulation

noted by e~ . The extended spaces of integrable and square integrable signals x(t) are denoted ere and e~e respectively.

In this note we combine the passivity-based control approach, that is to stabilise a system by rendering it 1.1 Passivity-based control passive, and the use of sliding modes as a technique for robust control of nonlinear perturbed systems. For the Our starting point concerns the stabilisation of passake of clarity, we find it convenient to briefly recall sive systems, for this we recall some results from [1]. Consider the affine nonlinear system some important results from both methods. Notation and basic definitions: For vectors x E 1R,n x = f(x) + g(x)u (1) we denote by IIxll the usual Euclidean norm and by 6

Ixl ~ :E~ lXii, with IXil meaning the absolute value of Xi, the Frobenius norm. A continuous function o : 1R>o -+ R>o is said to be of class K., 0 E K., if o(x f is strictly increasing and 0(0) = O. A dynamical system x = f(x) + g(x)u with input u and output y = h(x) is said to be zero-state detectable if {u(t) = Oand y(t) = 0 'Vt} imply that x(t) -+ 0 as t -+ 00. A continuous function V(x) is said to be proper if V(x) -+ 00 as IIxll -+ 00. The time derivative of Lyapunov function V(x) along the solutions of the differential equation (#) is denoted i{#)(x) . The space of uniformly bounded signals x(t) E lRn is de-

Y = h(x)

where X E En, y, U E R, f, 9 and h are continuous functions of x. Assume further that system (1) defines a passive operator L : u t-+ y with a Cl storage function 6 V = V(x) bounded from below. Under the considerations above, it is well known [8] that the control law u = u· with u· = -~g(x) asymptotically stabilises the origin of (1). A passivity interpretation of the .lurdjevic and Quinn controller u· is contained in the results of Theorem 3.2 of [1]:

841

1.1 Theorem. If system (1) is passive and zercrstate system and certain under-actuated systems. In the last section of this note we present some of these practical detectable then the control input examples. u = -4>(Y) We are interested in a design methodology for a controller u = u( t, x) which guarantees that the solution with 4>(0) = 0 and v4>(y) > 0 for all y i 0, asymp- x(t;to,xo) of (2) with initial conditions to, Xo = x(to) totically stabilises the origin x = O. Further, if the is globally uniformly convergent, that is, storage function V(x) is proper then the origin is globally asymptotically stable. 0 lim x(t) = 0 t-+oo

The result above gives a general solution to the stabilisation of passive systems with measurable output y . Practical applications to which the result above can be applied include Euler-Lagrange systems (in short EL systems) such as robot manipulators, motor drives, electrical circuits, etc. For instance, the main result of [12] concerning the problem of global output feedback set-point control of EL systems follows as a Corollary of Theorem 3.2 of [1] . For other interesting results relating passivity and stability in the sense of Lyapunov, see [2, 16] and the references therein.

1.2

Problem formulation

Our main concern in this paper is to deal with nonlinear systems perturbed by affine disturbances, more precisely, we consider systems of the form

for any initial conditions (to, xo) · Before presenting our main result which is a solution to the problem stated above, we briefly show below how discontinuous control can help in the stabilisation task of a system like (2) .

1.3

Discontinuorls control to compensate for bounded disturbances

To illustrate the utility of discontinuous control (e.g. the sliding mode approach) in the stabilisation of systems like (2) let us consider for simplicity a scalar affine nonlinear system defined by

x = f(x) + d(t) + u

(5)

where x E m. and assume that there exists an unknown positive constant d such that Id(t)1 < d uniformly in t . Assume further that in the absence of the disturbance where the disturbance d : 1Eho x 1R.n -+ mn satisfies (that is, if d 0) a controller u = u*(x) is known such the following: that the system x = f(x) + u*(x) is globally asympA 1 Each component of the vector function d is uni- totically stable (GAS). From the GAS assumption it follows (see e.g. [7]) that there exists a Lyapunov funcformly bounded in t by tion V = V*(x), radially unbounded such that

x = f(t,x) + g(t, x)[u + d(t,x)]

(2)

=

(6) where 9 1; and 92 ; are unknown non-negative constants and di(lIxll) : ]R.~o -+ R~o is a known continuous function. where 0 E K.. Then a well known result in the literature of sliding mode control (see e.g. [15, 6]) is the following. An obvious implication of (3) is the existence of a con- Define the sliding surface tinuous function d : m.>o -+ 1R.>o and positive con/':" av* stants 91 and 92 such that Ild(t,x)l1 ~ 91 d(llxli) +92 , or (7) u(x) = 8x in compact form T

Ild(t, x)11 ~ 9 ~(llxll)

(4)

then the adaptive control law

u = u·(x) - Jsgn(u) (8) where we have defined 9T ~ (91 , ( 2 ) and ~(lIxll) T ~ (d(llxll), 1). (9) J = sgn(u)u It is important to remark that the disturbance d(t, x) may be the result of unmodeled dynamics, noisy renders the system (5) globally convergent, that is, for measurements, parameter uncertainty or interestingly any initial conditions x(to) we have that enough, thinking of system (2) with d =0 as a paslim x(t) O. sive system, the disturbances d are all the non dist-+oo sipative forces affecting the plant. Physical systems with model (2) are numerous, we can think of Euler- For the sake of completeness we proceed to prove the Lagrange systems or chaotic oscillators such as the pe- claim above, as a matter of fact, the proof of our main riodically forced Duffing equation or the van der Pol result will follow along similar guidelines. First notice

=

where iJ is the estimate of (J, sgn(y) = in our case. It may be also clear that our motivation for [Sgn(Yl, . .. ,sgn(lIm»)T and r = rT > 0, in dosed calling condition (14) "strong" zero-state detectability loop with (2), (3) guarantees that the state x(t) is stems from the fact that this condition is more restricglobally uniformly convergent. 0 tive than the usual definition, It is also worth remarking that in Theorem 2.2 we Proof. We start by noticing that since (15) defines have implicitly assumed that all parameters of our sysan OSP operator with storage function V*(t,x) then tem are known, however, notice that the perturbation there exists a positive constant /3 such that d(t,x) may also contain terms which come from uncertainties in the model. . T 2 (18) V(~5)(t,X)$v 1/-/3llyll· More precisely, consider a nonlinear non-autonomous system defined by Define 6 = d(t,x) -

·T

(J

~(lIxll)sgn(y),

(19)

(23) ± = f(t,x,(J) + g(t,x)u then the bound (IS) on the storage function V*, co(24) 1/ = h(t,x) incides with the following bound on time derivative of Lyapunov function V· along the closed loop solutions where x E mn, and (J E m,' of constant unknown parameters. Using Theorem 2.2 a controller u = u(t, x) (2), P6) guaranteeing that x(t) .... 0 as t .... 00 can be designed . • 2 T ·T T V(~,16)(t, x, (J) $ -/3llyll - d 1/ - (J ~(I l xll)sgn(1I) y. ' if the dynamics (23) admit the following parameterisa(20) tion x = I'(t,x) + g(t,x)[u + d(t,x,(J») Consider on the other hand the Lyapunov function candidate where the nonlinearity d(t, x, (J) satisfies - 6 1-T 1V(t,x,(J) = V*(t,x) + 2(J r- (J (25) Ild(t,x,(J)1I $ (J~~(lIxll) + (J~ where B ~ iJ - (J, which is clearly positive definite and proper. Using the inequalities (20) and (4), the where (J~ and (J; are unknown constants and ~(llxll) is time derivative of V along the closed loop solutions a continuous known function, Notice furthermore, that the parameters vector (J (x(t), B(t» of system (2), (16), (17) is bounded by does not need to be constant, as long as it defines con2 tinuous uniformly bounded signals, that is, if (J(t) E V(2,16,17) (t, x, 0) $ -/3llyI1 - (JT ~(llxll)(11I1 - 111111) (21) C~ and a bound like (25) can be determined. The claims above will become more clear from the where we recall that Iyl stands for the Frobenius norm (c.f. Notations) . Notice that the subtraction of the last examples we give below. two terms on the right h'a nd side of (21) is non-positive hence . 2 Practical examples (22) 3 V2,16,17(t,X,(J) $ -/3llyll . V

Inequality (22) implies that Vet) is decreasing and hence bounded so the complete state (x, 8) E C~+2 . Furthermore, integrating (22) from to to 00 one concludes that Y E Or . Next, notice from (19), (4) that since (J is constant and ~(lIxll) is continuous, v(t) is also uniformly bounded. From the continuity of h(t,x) it follows that 11 E C: . Then, from the closed loop dynamics (15), (19) we obtain that ± E C~ and since h E Cl we conclude using Barbalat's lemma that yet) .... 0 as t .... 00. The result is finally obtained using the assumption of strong zero-state detectability. •

As pointed out in the previous section our main result applies to the fairly general problem of adaptive trajectory tracking control. We illustrate this fact in the control of chaotic oscillators and ship dynamic positioning [4].

3.1

Chaotic oscillators

It is beyond the scope of this paper to study in detail the control problem of chaotic systems. A fair study can be found in [9] . See also [10], and references therein for the particular cases of the van der Pol and Duffing Interestingly enough the theorem above applies to systems. non-autonomous non-linear systems. In other words, if we think of the state x as a tracking error Theorem 2.2 The van der Pol oscillator can be used to design robust tracking controllers such that the tracking error is guaranteed to converge to Consider an LC electrical network with a nonlinear zero uniformly as t .... 00 . It may clarify the reason for load and all elements connected in parallel. Assume introducing condition (14) that a proof in the spirit of that the load is an active circuit with input voltage p(v) with p(v) -v + the proof of Theorem 3.2 of [1) requires that v 0 (see v and input current i also Lemma 3.2.3 of [16)) which cannot be guaranteed Assume further that the circuit is excited by an AC

=

=

843

=

tv3.

that since 6 is constant the closed loop system (5), (8) is given by :i:

=

J =

f(x)

+ u·(x) -

6sgn(0')

+ d(t)

(10) (11)

sgn(O')O'.

-

~

.

Define the estimation error 6 = 6 - fJ and the Lyapunov - ~ l -2 function V(x, 6) = V·(x)+ 2"fJ , we proceed to calculate the time derivative of the latter along the trajectories of the closed loop system (10), (11) which is defined by the differential inclusion and

F(t,X)~

X = F(t , X) where

(10), (11) { [f(X)+u.(x

X ~ col[x,6)T

-6Q +d(t)],

6

. -.

(jav.j) 10'1- ax

u

=

(12)

for all solutions X( t) such that X E F( t, x) and where is defined in (6) . Using (7) it is clear that the addition of the last two terms on the right hand side of (12) is exactly zero, hence from the assumption that the unperturbed closed loop system :i: = f(x) + u·(x) is G AS we obtain that

V·

2

1-+

11

Main result '

Let us start with the following definition which we will use in order to provide a "passivity-based" proof of our main result.

2.1 Definition. (Strong zero-state detectability)

The nonlinear system (2) with input v ~ u + d and output y = h( t, x) where h is a continuous function of its arguments, is said to be strongly zero-state detectable if

V(x, J) ~ -o(lxl) .

{v(t) E .coo and y(t) == O} ~ Hm x(t) t~oo

The global convergence of x(t) follows using Barbalat's lemma and standard arguments (see e.g. [3)) . The example above illustrates the fact that sliding modes are very useful to compensate for uniformly bounded disturbances. However, a practical "drawback" of the sliding mode approach is that in general, there is no engineering intuition on how to design a suitable sliding surface satisfying the conditions mentioned at the beginning of this section. Roughly speaking, in this note we give a physical meaning to the sliding surface when dealing with passive systems. We finish our Introduction with some preliminary conclusions drawn from the derivations contained in Sections 1.1 and 1.3. 1. Assume that system (5) with zero disturbance defines a passive operator 1: : u 1-+ 11 with respect to the output

= O.

(14)

The above definition differs from the usual one (see for instance [1)) in the fact that we consider all uniformly bounded inputs instead of the trivial one only. In Section 3 we show some practical examples for which this condition is easily verified . As it will become clear later, the condition established by (14) will allow us to prove our main result using pa8aitJity arguments.

2.2 Theorem. Consider the system (2), (3) under the following assumptions

ay·

11 = ax '

+ d(t)

Vu! 0 and a "detect ability condition". The formal statement of this conjecture is strictly contained in our main re0'=0 sult which we present in the next section.

where Q E [-1,1). Thus using Id(t)1 < {) and 10'1 sgn(O')O', the time derivative of V satisfies V(x,fJ)=V·(x)-fJ

anymore to guarantee global uniform asymptotic stability (GUAS). However, we know that the controller (8), (9) guarantees the global convergence of x(t). 3. At this point, notice that if we substitute the output definition (13) in the controller (8), (9) 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 control (8), (9), (13) guarantees the global convergence of x(t) under the assumptions that (5) defines an output strictly passive operator (OSP)

(13)

and assume further that system (5) is zero-state detectable with respect to the same output 11. Under these conditions one can easily construct a globally stabilising control law u = u·(x) using Theorem 1.1. 2. However, in the case when the time varying disturbance d(t) ! 0 then system (5) is nonautonomous and the control u·(x) is not "sufficient"

844

A2 A passifying controller u that the system :i:

= u·(t, x) is known such

= f(t,x) + g(t, x)[u·(t, x) + v(t»)

(15)

defines an OSP operator 1: : v 1-+ y with output y E .cr, y = h(t,x) , where h E Cl and storage function V·(t, x) is positive definite and proper. A3 System (15) is strongly zero-state detectable.

Under these conditions, the controller

u =

·T

u·(t,x) - (J ~(llxll)sgn(1I)

iJ = r~(lIxll)sgn(tI)T y

(16)

(17)

current source. The Kirchorff's current law yields to the loop equation (for a causal circuit)

3.2

Euler-Lagrange systems

In this section we illustrate the utility of our main result in a control problem of Euler-Lagrange systems. = Qsin(wt). We focus on the particular case of dynamic ship podt L 0 Differentiating once with respect to t, multiplying by sitioning however, other results which fit into this L on both sides of the equality above and scaling the framework and which have motivated this research are [13,11]. time-variable t by the factor l/JCL we obtain

dv + -l i t v(r)dr + p(v) c-

..

v+

where we q

V{Lah. C 8v v + v = qcos(wt)

Ship dynamic positioning

> O. Finally, substituting the definition of In this section we present an application to dynamic

p( v) and defining p. = equation ii

~ we obtain

the well known

+ p.(1- v 2 )v + v = qcos(wt),

positioning of a marine vessel in presence of weather disturbances. One of the simplest models (in the Lagrangian formulation) of a ship is [4]

(26)

MiI + Dv == u +.TT (1])b(t) (30) which van der Pol used to model an electrical circuit ~ = J(1])v (31) with a triode valve. Note that for the particular case when p. == 5, q = 5 and w = 2.463 the van der Pol where M E ]Ra is the constant inertia matrix which inoscillator exhibits a chaotic behaviour [14]. cludes hydrodynamic added inertia. For a straight-line Consider now the controlled unperturbed van der Pol stable ship [5] the eigen-values of the constant dampequation with the voltage v delivered to the active load ing matrix D are strictly positive, however D :f:. D T. as the output to be controlled, that is The generalized positions are collected in the vector 2 ii + p.(1 - v )v + V = u· + w, (27) 1] ~ col [x, y, 1/1], where the x, y coordinates correwhere w is a bounded external input which includes spond to the position of the centre of mass of the vessel referred to the · Earth's frame (these variables are the term qcos(wt) and measured with a GPS system), while v corresponds to u·(v) = -kpv - kdt + P.(2VVdV - v~iJ) + Vd (28) the ship-fixed velocities. The angle 1/1 is the so called where Vd == Vd(t) is the desired output voltage and "heading" angle of the ship which describes its orienv = V - Vd . Then, the closed loop system (27), (28) tation. Notice that only rotations about yaw are condefines an output strictly passive operator r: : wHy sidered, that is, the ship is metacentric stable. Under with respect to the output y == t + cV where c > 0 is these considerations, the ship .lacobian J(1]) which relates the Earth-fixed frame to the ship's frame is a pure small enough to ensure that the storage function rotation about yaw: 1 v:.2 1 k -2 -:. 1 ck -2 cp. -4 V( V, V.) == 2 v + 2 pV + cVV + 2 dV + 4 cos1/1 - sin 1/1 0 is positive definite. The proof of this claim is straightJ(1])=J(1/1)= sin1/1 cos 1/1 0 [ o forward. Taking the time derivative of V along the 0 1 trajectories of the closed loop (27), (28) we obtain hence, clearly J is orthogonal, that is, .TT J = I. V(v,v) :5 -(kd - c)~2 - ckpv2 + wy. The control inputs are collected in the vector u E :n3 Notice further that by choosing kd ~ 2 + c and k p ~ 2c and finally, the perturbationb(t) E :n3 corresponds one can write to all the slowly-varying environmental disturbances that we refer to as bias. Typically, b(t) corresponds V(V, ti) :5 _y2 + wy, (29) to bounded Gaussian white noise of unknown maximal the OSP property follows by integrating (29) on both amplitude. sides of the inequality from 0 to t. Hence assumption To show that system (30), (31) lies within the frameA2 is met. work of our main result we start by defining the timeSecondly, the disturbance q C06(wt) ~ q so assump- varying disturbance tion Al is met with 81 q and 6 == O. Finally the strong zero-state detectability condition is met as well observing that the output equation t + cV = y constitutes a strictly proper and stable filter with input y Notice that since II.T(1])11 is uniformly bounded in 1] and hence v -+ 0 as t -+ 00 if y == 0, regardless of w(t). by assumption, the bias is bounded white noise, there Thus, using Theorem 2.2 we can easily construct an exists a constant d such that d ~ 1I.T(t'](t»Tb(t)ll. Also, adaptive robust control law for the controlled and per- the unperturbed dynamics turbed van der Pol oscillator for the case when none of the parameters w, p. nor q is known. MiI+Dv =u (32)

1

=

845

constitutes an Output Strictly Passive map 'E : u I-t v. In order to enhance this passivity property and to stabilise the ship at the desired position 1]d we can use a simple PD control law u· = -Kp .T(1]) T fi - KtJiJ, where Kp and Kd are positive definite matrices. Define u = u· + v, with v an additional external input then the closed loop dynamics is MiI

+ Dv + K p.J(1]) T fi + KdfJ = v.

(33)

It is easy to see that system (33) also defines an Output Strictly Passive map 'E : V I-t v if we use the storage function V(fi,v)

= ~v T Mv + ~fiT Kpfi

which is obviously positive definite in v and fJ. Derivating V with respect to time along the trajectories of (33) and integrating again from 0 to T we have

loT v(t)v(t)dt ~ ~v

T

(D

+ DT + Kd)v.

Hence in order to stabilise the perturbed system (30), (31) we only need to add the discontinuous term v = -dsgn(v) to the control input u· then using Theorem 2.2 it follows that the solutions [v(t),fi(t)] of the closed loop system (30), (31) with

= d =

u

-Kp.J(1]) T fJ

- KdfJ - dsgn(v)

sgn(v) T v.

(34) (35)

Using (33) and the fact that .J(1]) is full rank, A3 holds. Acknowledgements

This work was partially carried out while the first author was with the Norwegian Univ. of Science and Technology.

4

Concluding remarks

We have illustrated in this note possible applications of adaptive control of nonlinear systems with uniformly (in time) bounded disturbances. For a given nonlinear perturbed system the control task is divided in two steps: 1) a passifying control law is designed for the system without disturbance. 2) A discontinuous control term is added which depends on the output of the resulting passive map of the first step. We have illustrated our contribution with the tracking control problem of a van der Pol oscillator and dynamic positioning of a marine vessel.

References

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