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NOVEMBER 14, 2003 1
Lyapunov-based distributedcontrol of systems on lattices
Mihailo R. Jovanovic Bassam Bamieh
UNDER CONSIDERATION FOR PUBLICATION INIEEE TRANSACTIONS ONAUTOMATIC CONTROL
REVISED VERSION
Abstract
We investigate the properties of distributed parameter systems on discrete spatial domains with spatially distributedsensors and actuators. These systems arise in a variety of applications such as the control of vehicular platoons,formation of unmanned aerial vehicles, arrays of microcantilevers, and satellites in synchronous orbit. We use aLyapunov-based framework as a tool for stabilization/regulation/asymptotic tracking of both linear and nonlinearsystems. We first present results for nominal design and then describe the design of adaptive controllers in thepresence of parametric uncertainties. These uncertainties are assumed to be temporally constant, but are allowed tobe spatially varying. We show that, in most cases, distributed controllers are obtained with the passage of informationdetermined by the interactions between different plant units. We also provide several examples of systems on latticesand validate derived results using computer simulations of systems containing a large number of units.
I. I NTRODUCTION
Systems on lattices are encountered in a wide range of modern technical applications. Typical examples ofsuch systems include: platoons of vehicles ( [1]–[6]), arrays of microcantilevers [7], unmanned aerial vehicles information [8], and satellites in synchronous orbit ( [9]–[11]). These systems are characterized by the interactionsbetween different subsystems which often results in surprisingly complex behavior. A distinctive feature of thisclass of systems is that every single unit is equipped with sensors and actuators. The controller design problem isthus dominated by architectural questions such as localized versus centralized control, and the information passingstructure in both the plant and the controller. This is in contrast with ‘spatially lumped’ control design problems,where the dominant issues are optimal and reduced order controller design.
A framework for considering spatially distributed systems is that of a spatio-temporal system [12]. In the specificcase of systems on discrete spatial domains, signals of interest are functions of time and a spatial variablen ∈ F,whereF is a discrete spatial lattice (any countable set, e.g.Z or N).
A particular class of spatio-temporal systems termed linear spatially invariant systems was considered in [13],where it was shown that optimal controllers (in a variety of norms) for spatially invariant plants inherit this spatiallyinvariant structure. Furthermore, it was shown that optimal controllers with quadratic performance objectives (suchas LQR,H2, andH∞) have an inherent degree of spatial localization. These results can be used to motivate otherdesign procedures (such as adaptive, fixed structure, etc.) for spatio-temporal systems based on certaina priorifixed controller structures which reflect the spatial variation and localization of the plant.
With a priori assumptions on the information passing structure in the distributed controller, sufficient conditionsfor internal stability and strict contractiveness can be expressed using Linear Matrix Inequality (LMI) conditions ([14], [15]). Similarly, control design for linear spatially varying distributed systems has been considered in [16],[17]. Conditions for synthesis are expressed in terms of convex operator inequalities, which in the particular casesof finite and periodic spatial domains simplify to linear matrix inequalities.
Adaptive identification and control are extensively studied for both linear (see, for example [18]–[21]) andnonlinear [22] finite dimensional systems. Several researchers have also considered these problems in the infinitedimensional setting and for spatially interconnected systems. We will only briefly outline some of the recent workhere, and refer the reader to the references below for a fuller discussion.
Decentralized adaptive controllers have been designed for a class of finite dimensional interconnected nonlinearsystems in [23]. The authors establish bounds on the magnitude of deviations of subsystem states for slowly varyingexogenous inputs and sufficiently small state-observer errors.
Decentralized adaptive controllers have been designed for a class of large-scale systems with interconnectionsthat can be bounded by known or unknown polynomials in [24].
M. R. Jovanovic and B. Bamieh are with Department of Mechanical and Environmental Engineering, University of California at Santa Barbara,Santa Barbara, CA 93106, USA (emails: [email protected], [email protected]).
NOVEMBER 14, 2003 2
String stability of a countably infinite interconnection of ‘weakly coupled’ nonlinear systems is studied in [25].Sufficient conditions for asymptotic string stability in the presence of small structural/singular perturbations arederived, and a gradient-based estimator is used to account for parametric uncertainties.
Decentralized adaptive state and output-feedback control of large-scale nonlinear systems with polynomiallybounded interconnections has been considered in [26] and [27], respectively. In [26], the class of systems that canbe transformed to the so-called decentralized strict-feedback form has been identified and geometric conditions underwhich this transformation is possible have been obtained. In [27], the output-feedback controllers that guaranteeglobal uniform boundedness of closed-loop signals in the presence of bounded disturbances have been designed.Additionally, the tracking error can be made arbitrarily small by the appropriate choice of control gain.
The backstepping in combination with an adaptive law with parameter projection has been used for design ofrobust decentralized adaptive controllers in [28]. The boundedness of all signals in the closed-loop and small in themean tracking error have been established for sufficiently weak interactions and subsystem unmodeled dynamics.
A direct adaptive control strategy is employed in [29] to provide stability of a nonlinear platoon system in thepresence of parametric uncertainties. The design procedure results in decentralized controllers that require informa-tion about positions and velocities of two neighboring vehicles and the leading vehicle to provide boundedness ofall signals in the closed-loop and zero steady-state spacing errors.
Adaptive control of a spacecraft formation system is considered in [11], [30]–[32]. In particular, in [11] anonlinear model that describes relative position dynamics of neighboring spacecrafts is developed. The authors havealso designed an adaptive output-feedback controller that guarantees semi-global steady-state tracking of relativepositions between two adjacent units in formation.
Adaptive boundary control of a viscous Burgers’ equation is considered in [33] and [34]. In the former, adecentralized adaptive boundary controller that providesH1 stability and convergence of the system state to zerois designed. A Lyapunov method together with an alternative to the Barbalat lemma is used to prove systemproperties. The authors also show well-posedness of the closed-loop system. On the other hand, [34] designs astabilizing controller for both the nominal Burgers’ equation and the equation with parametric uncertainties. Anonlinear high-gain output-feedback in combination with appropriate parameter update laws are used as a tool forachieving these objectives. This procedure can be generalized to systems in which higher order nonlinearities arepresent.
In this paper, we study distributed control of systems on lattices. We consider both linear and nonlinear modelsand use a Lyapunov-based approach to provide stability/regulation/asymptotic tracking of nominal systems andsystems with parametric uncertainties. In the latter case, we assume that the unknown parameters are temporallyconstant, but are allowed to be either spatially constant or spatially varying. In both of these situations, we designadaptive Lyapunov-based estimators and controllers. As a result of our design, boundedness of all signals in theclosed-loop in the presence of parametric uncertainties is guaranteed. In addition to that, the adaptive controllersprovide convergence of the states of the original system to their desired values. We also show that, in most cases, thedistributed design results in controllers whose information passing structure is similar to that of the original plant.This means, for example, that if the plant has only nearest neighbor interactions, then the distributed controlleralso has only nearest neighbor interactions. The only situation which results in a centralized distributed controlleris for plants with temporally and spatially constant parametric uncertainties when we start our design with oneparameter estimate per unknown parameter. It should be noted however that this problem can be circumvented bythe ‘over-parameterization’ of the unknown temporally and spatially constant parameters. As a result, every singlecontrol unit has its own estimator of unknown parameters and avoids the fully centralized architecture.
Stabilizing controllers are designed using the technique ofbackstepping. Backstepping is a well-studied designtool [22], [35] for finite dimensional systems. In the infinite dimensional setting, a backstepping-like approach canbe used to obtain stabilizing boundary feedback control laws for a class of parabolic systems (see [36], [37] fordetails). Backstepping boundary control can also be used as a tool for vibration suppression in flexible-link gantryrobots [38]. However, backstepping has not been applied to distributed control of infinite dimensional systemson lattices to the best of our knowledge. We remark that backstepping represents a recursive design scheme thatcan be used for systems in strict-feedback form with nonlinearities not constrained by linear bounds [22], [35].At every step of backstepping, a new Control Lyapunov Function (CLF) is constructed by augmentation of theCLF from the previous step by a term which penalizes the error between ‘virtual control’ and its desired value(so-called ‘stabilizing function’). A major advantage of backstepping is the construction of a Lyapunov functionwhose derivative can be made negative definite by a variety of control laws rather than by a specific controllaw [22]. Furthermore, backstepping can be used for adaptive control of ‘parametric pure-feedback systems’ inwhich unknown parameters enter into equations in an affine manner [22].
NOVEMBER 14, 2003 3
Our presentation is organized as follows: in section II, we give an example of systems on lattices, describethe classes of systems for which we design feedback controllers in§ III, discuss well-posedness of the open-loopsystems, and describe different strategies that can be used for control of systems on lattices. In§ IV, we study insome detail the existence and uniqueness of solutions of the closed-loop systems obtained as a result of controldesign. In§ V, we discuss application of controllers developed in§ III, analyze their structure, and validate theirperformance using computer simulations of systems containing a large number of units. In§ VI, we design fullydecentralized nominal controllers for both linear and nonlinear mass-spring systems. In§ VII, we extend the well-known integrator backsteppingdesign tool for finite dimensional nonlinear systems to a more general class ofsystems considered in this paper. We conclude by summarizing major contributions and future research directionsin § VIII.
II. SYSTEMS ON LATTICES
In this section an example of systems on lattices is given. In particular, we consider a mass-spring system ona line. This system is chosen because it represents a simple non-trivial example of an unstable system where theinteractions between different plant units are caused by the physical connections between them. Another exampleof systems with this property is given by an array of microcantilevers. We remark that the interactions betweendifferent plant units may also arise because of a specific control objective that we want to meet. Examples ofsystems on lattices with this property include: a system of cars in an infinite string, aerial vehicles and spacecraftsin formation flights. We also discuss the well-posedness of the unforced systems, describe the classes of systems forwhich we design state and output-feedback controllers in§ III, and briefly outline different approaches to controlof systems on lattices..
A. An example of systems on lattices: mass-spring system
A system consisting of an infinite number of masses and springs on a line is shown in Fig. 1.
Fig. 1. Mass-spring system.
The dynamics of then-th mass are given by
mnxn = Fn−1 + Fn + un, n ∈ Z, (1)
wherexn represents the displacement from a reference position of then-th mass,Fn represents the restoring forceof the n-th spring, andun is the control applied on then-th mass. For relatively small displacements, restoringforces can be considered as linear functions of displacements
Fn = kn(xn+1 − xn), Fn−1 = kn−1(xn−1 − xn), n ∈ Z,
wherekn is then-th spring constant. In this case (1) can be rewritten as
xn =kn−1
mn(xn−1 − xn) +
knmn
(xn+1 − xn) +1mn
un, n ∈ Z,
or equivalently, in terms of its state space representation,
ψ1n = ψ2n
ψ2n =kn−1
mn(ψ1,n−1 − ψ1n) +
knmn
(ψ1,n+1 − ψ1n) +1mn
un
n ∈ Z, (2)
whereψ1n := xn andψ2n := xn.
NOVEMBER 14, 2003 4
We also consider a situation in which the spring restoring forces depend nonlinearly on displacement. One suchmodel is given by the so-calledhardening spring(see, for example [35]) where, beyond a certain displacement,large force increments are obtained for small displacement increments
Fn = kn{(xn+1 − xn) + c2n(xn+1 − xn)3
}=: kn(xn+1 − xn) + qn(xn+1 − xn)3,
Fn−1 = kn−1
{(xn−1 − xn) + c2n−1(xn−1 − xn)3
}=: kn−1(xn−1 − xn) + qn−1(xn−1 − xn)3.
In this case, the state space representation of (1) takes the following form:
ψ1n = ψ2n
ψ2n =
ψ1,n−1 − ψ1n
ψ1,n+1 − ψ1n
(ψ1,n−1 − ψ1n)3
(ψ1,n+1 − ψ1n)3
∗
1mn
kn−1
kn
qn−1
qn
+1mn
un
n ∈ Z. (3)
In the particular situation in which all masses and springs are homogeneous, that is,
mn = m = const., kn = k = const., cn = c = const., ∀n ∈ Z,
(2) and (3) can be rewritten as
ψ1n = ψ2n
ψ2n =k
m{ψ1,n−1 − 2ψ1n + ψ1,n+1} +
1mun
n ∈ Z, (4)
andψ1n = ψ2n
ψ2n =
[ψ1,n−1 − 2ψ1n + ψ1,n+1
(ψ1,n−1 − ψ1n)3 + (ψ1,n+1 − ψ1n)3
]∗1m
[k
q
]+
1mun
n ∈ Z, (5)
respectively. If the values ofm and k in (4) are exactly known, then (4) represents a linearspatially invariantsystem. This implies that it can be analyzed using the tools of [13]. The other mathematical representations of amass-spring system are either nonlinear or spatially-varying when masses and spring constants are unknown. Themain purpose of the present study is to design feedback controllers for this broader class of systems.
B. Classes of systems
In this subsection, we briefly summarize the classes of systems for which we design feedback controllers in§ III. Inparticular, we considerm-th order subsystems over discrete spatial latticeF with finite number of interconnectionswith other plant units. We assume that all subsystems satisfy thematching condition[22]. Clearly, the modelspresented in§ II-A belong to this class of systems, as well as the model of an array of microcantilevers [7].Furthermore, we remark that our results can be also used for control of fully actuated systems in two and threespatial dimensions.
We consider systems without parametric uncertainties of the form
ψ1n = ψ2n, n ∈ F, (6a)
ψ2n = ψ3n, n ∈ F, (6b)...
ψmn = fn(ψ1, · · · , ψm) + κnun, n ∈ F, (6c)
whereψk := {ψkn}n∈F, k ∈ {1, . . . ,m}, andκn’s are the so-calledcontrol coefficients[22].If, on the other hand, we have a situation where all parameters have constant but unknown values∀ t ∈ R+ and
∀n ∈ F, it is convenient to rewrite (6) as
ψ1n = ψ2n, n ∈ F, (7a)
ψ2n = ψ3n, n ∈ F, (7b)...
ψmn = νn(ψ1, · · · , ψm) + g∗n(ψ1, · · · , ψm)θ + κun, n ∈ F, (7c)
NOVEMBER 14, 2003 5
whereθ represents a vector of unknown parameters and ‘∗’ denotes the transpose of vectorgn.The most general case that we discuss in this paper is the one in which all parameters have unknown spatially
varying values that do not depend on time. In other words, the parameters are allowed to be a function ofn ∈ Fbut not of t ∈ R+. In this particular situation, it is convenient to rewrite (6) in the following form:
ψ1n = ψ2n, n ∈ F, (8a)
ψ2n = ψ3n, n ∈ F, (8b)...
ψmn = τn(ψ1, · · · , ψm) + h∗n(ψ1, · · · , ψm)θn + κnun, n ∈ F. (8c)
The major difference between models (7) and (8) is in the number of unknown parameters. Namely, in the formercase the number of unknown parameters is finite, and in the latter case the number of unknown parameters can beinfinite.
We also consider output-feedback design for nominal systems of the form
ψ1n = ψ2n, n ∈ F, (9a)
ψ2n = ψ3n, n ∈ F, (9b)...
ψmn = fn(ψ1) + κnun, n ∈ F, (9c)
yn = ψ1n, n ∈ F, (9d)
and systems with parametric uncertainties of the form
ψ1n = ψ2n, n ∈ F, (10a)
ψ2n = ψ3n, n ∈ F, (10b)...
ψmn = τn(ψ1) + h∗n(ψ1)θn + κnun, n ∈ F, (10c)
yn = ψ1n, n ∈ F, (10d)
wherey := {yn}n∈F = {ψ1n}n∈F represents the distributed output of (9) and (10), respectively.We introduce the following assumptions about the systems under study:Assumption 1:The number of interconnections between different plant units is uniformly bounded. In other
words, there existM ∈ N, M 6= M(n), such thatfn, gn, hn, νn, and τn depend on at mostM elements ofψ1, . . . , ψm.
Assumption 2:fn, gn, hn, νn, andτn are known, continuously differentiable functions of their arguments.Assumption 3:gn andνn are bounded by polynomial functions of their arguments. Furthermore, these polyno-
mials are equal to zero at the origin.Assumption 4:The sign ofκ in (7c) is known.Assumption 5:The signs ofκn, ∀n ∈ F, in (8c) and (10c) are known.These assumptions are used in the sections related to the distributed control design and the well-posedness of
both open and closed-loop systems.Remark 1:For notational convenience, both the well-posedness and the control design problems are solved for
second order systems over discrete spatial latticeF, that is form = 2.
C. Well-posedness of open-loop systems
In this subsection we briefly analyze the well-posedness of the open-loop systems form = 2. In particular, weconsider the systems of§ II-B as the abstract evolution equations either on a Hilbert spaceH := l2 × l2 or on aBanach spaceB := l∞ × l∞. Either representation is convenient for addressing the question of the existence anduniqueness of solutions. With this in mind, we prove the well-posedness of the open-loop systems onH and remarkthat similar argument can be used if the underlying state-space isB rather thanH.
The unforced systems of§ II-B can be rewritten as the abstract evolution equations of the form
ψ = Aψ + p(ψ). (11)
NOVEMBER 14, 2003 6
The state of system (11), a linear operatorA : D(A) ⊂ H → H, and a (possibly) nonlinear mappingp : H → Hare defined as
ψ :=[ψ1
ψ2
]∈ H, A :=
[0 I0 0
], p(ψ) :=
[0
p2(ψ)
],
whereD(A) represents thedomain of A, and p2(ψ) denotes{fn(ψ)}n∈F, {νn(ψ) + g∗n(ψ)θ}n∈F, {τn(ψ) +h∗n(ψ)θn}n∈F, {fn(ψ1)}n∈F, or {τn(ψ1) + h∗n(ψ1)θn}n∈F depending on whether systems (6), (7), (8), (9), or(10) are considered.
One can show thatA is a bounded operatorand, therefore, it generates auniformly continuous semigroup. Thelatter property automatically implies that the semigroup generated byA is strongly continuous(C0)1. If in additionto that p is locally Lipschitz then system (11) has a uniquemild solutionon [0, tmax) (see [41], Theorem 2.73).Moreover, if p is continuously (Frechet) differentiable then (11) has a uniqueclassical solutionon [0, tmax) (see[41], Theorem 2.74). Iftmax < ∞ then limt→tmax ‖ψ(t)‖ = ∞. We remark that a functionp : H → H is Frechetdifferentiable at a pointψ ∈ H if there exists abounded linear operatorD : H → H such that
lim‖ϕ‖→0
‖p(ψ + ϕ) − p(ψ) − Dϕ‖‖ϕ‖
= 0.
If such a linear operator exists, we denote it by∂p∂ψ (ψ), and refer to it as the Frechet derivative ofp.It can be readily shown that for the systems described in§ II-B that satisfy Assumption 1 and Assumption 2,p is
a continuously Frechet differentiable function of its arguments which in turn guarantees existence and uniquenessof classical solutions for these systems. To illustrate this, we consider a nonlinear mass-spring system (5) anddetermine the Frechet derivative ofp as
D :=∂p
∂ψ(ψ) =
[0 0D21 0
],
with D21 being a tridiagonal operator determined by
D21 :=
...
d∗n−1
d∗n
d∗n+1
...
,
andd∗n :=
[· · · 0 dn,n−1 dn,n dn,n+1 0 · · ·
],
dn,n−1 :=k
m+
3qm
(ψ1,n−1 − ψ1,n)2,
dn,n := − 2km
− 3qm
{(ψ1,n−1 − ψ1,n)2 + (ψ1,n+1 − ψ1,n)2
},
dn,n+1 :=k
m+
3qm
(ψ1,n+1 − ψ1,n)2.
The tridiagonal structure ofD21 and the fact that
ψ ∈ H := l2 × l2 ⇒ ψ ∈ l∞ × l∞,
imply boundedness ofD21. Therefore, it follows thatD is a bounded operator, that is,
Dϕ ∈ H, ∀ϕ ∈ H.
It is noteworthy that boundedness ofD implies thatp is Frechet differentiable onH. Moreover, the continuity ofD (with respect toψ) implies thatp is continuously Frechet differentiable onH (see, for example, [42] for a fullerdiscussion).
1The definitions of a bounded operator, a uniformly continuous semigroup, and aC0–semigroup can be found, for example, in [39] and [40].
NOVEMBER 14, 2003 7
CENTRALIZED: LOCALIZED : FULLY DECENTRALIZED:
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-G1
-G2
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K
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6
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6
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-G1
-G2
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-K0
-K1
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6
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6
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6
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K0 K1 K2
6
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Fig. 2. Distributed controller architectures for centralized, localized (with nearest neighbor interactions), and fully decentralized controlstrategies.
D. Distributed controller architectures
Fig. 2 illustrates different control strategies that can be used for distributed control of systems on lattices:centralized, localized, and fully decentralized. Centralized controllers require information from all plant units forachieving the desired control objective. On the other hand, in fully decentralized strategies control unitKn usesonly information from then-th plant unitGn on which it acts. An example of a localized control architecture withnearest neighbor interactions is shown in Fig. 2.
In the next section we design stabilizing controllers for nominal systems (6) and (9). We also discuss the adaptivecontrol schemes for systems (7), (8), and (10) with unknown parameters and compare them.
III. L YAPUNOV-BASED DISTRIBUTED CONTROL DESIGN
In this section, we address the problem of designing controllers that provide stability/regulation/asymptotictracking of systems described in§ II-B. Assuming that the full state information is available and that every unit isequipped with sensors and an actuator, we use the Lyapunov-based approach to solve this problem. The Lyapunovdesign is very suitable because it leads, in most cases, to controllers that are not centralized. This feature is ofparamount importance for practical implementation. It is also remarkable that the basic ideas of finite dimensionalLyapunov-based adaptive design [22] are easily extendable to this infinite dimensional setting. Furthermore, theassumption on full state information can be relaxed for systems in which nonlinearities depend only on the distributedoutput variable, as illustrated in§ III-C and § III-D. One such example is system (3), with output being the infinitevector of the mass displacements from their reference positions. Clearly, in this case, the nonlinearities are only afunction of the outputψ1 := {ψ1n}n∈Z.
A. Nominal state-feedback design
We first consider Lyapunov design for systems without any parametric uncertainties. We observe that system (6)is amenable to be analyzed by the backstepping design methodology. Even though a stabilizing controller can bedesigned using various tools (because of the matching condition), we choose backstepping because it gives both astabilizing feedback law and a CLF for a system under consideration. Once CLF is constructed its derivative canbe made negative definite using a variety of control laws.
In the first step of backstepping, equation (6a) is stabilized∀n ∈ F by consideringψ2 = {ψ2n}n∈F as its control.Sinceψ2 is not actually a control, but rather, a state variable, the error betweenψ2 and the value which stabilizes(6a) must be penalized in the augmented Lyapunov function at the next step. In this way, a stabilizing control lawis designed for the overall system.
Before we illustrate the distributed Lyapunov-based design we introduce the following assumption:Assumption 6:The initial distributed state is such that bothψ1(0) ∈ l2 andψ2(0) ∈ l2.
Step 1The recursive design starts with subsystem (6a) by proposing a CLF of the form
V1(ψ1) =12〈ψ1, ψ1〉 :=
12
∑n∈ F
ψ21n. (12)
The derivative ofV1(ψ1) along the solutions of (6a) is given by
V1 =⟨ψ1, ψ1
⟩= 〈ψ1, ψ2〉 =
∑n∈ F
ψ1nψ2n. (13)
NOVEMBER 14, 2003 8
In particular, the choice of a ‘stabilizing function’ψ2nd of the form
ψ2nd = − anψ1n, an > 0, ∀n ∈ F, (14)
clearly rendersV1(ψ1) negative definite. Sinceψ2 is not actually a control, but rather, a state variable, we introducethe change of variables
ζ2n := ψ2n − ψ2nd = ψ2n + anψ1n, ∀n ∈ F, (15)
which adds an additional term on the right-hand side of (13)
V1 = −∑n∈ F
anψ21n +
∑n∈ F
ψ1nζ2n. (16)
The sign indefinite term in (16) will be taken care of at the second step of backstepping.Step 2Coordinate transformation (15) renders (6c) into a form suitable for the remainder of our design
ζ2n = anψ1n + ψ2n = anψ2n + fn(ψ1, ψ2) + κnun, n ∈ F.
Augmentation of the CLF from Step 1 by a term which penalizes the error betweenψ2 andψ2d yields a function
V2(ψ1, ζ2) := V1(ψ1) +12〈ζ2, ζ2〉 = V1(ψ1) +
12
∑n∈ F
ζ22n, (17)
whose derivative along the solutions of
ψ1n = − anψ1n + ζ2n, n ∈ F, (18a)
ζ2n = anψ2n + fn(ψ1, ψ2) + κnun, n ∈ F, (18b)
is determined by
V2 = V1 +∑n∈ F
ζ2nζ2n
= −∑n∈ F
anψ21n +
∑n∈ F
ζ2n {ψ1n + anψ2n + fn(ψ1, ψ2) + κnun} .(19)
With a control law of the form
un = − 1κn
{ψ1n + anψ2n + fn(ψ1, ψ2) + bnζ2n}
= − 1κn
{(1 + anbn)ψ1n + (an + bn)ψ2n + fn(ψ1, ψ2)} , n ∈ F, (20)
wherebn’s are positive design parameters∀n ∈ F, V2 becomes a negative definite function, that is
V2 = −∑n∈ F
anψ21n −
∑n∈ F
bnζ22n < 0.
Remark 2:Control law (20) achieves global asymptotic stability of the origin of system (6) form = 2 on H :=l2 × l2 by completely altering its dynamics through the process of cancellation at the second step of backstepping.As a result of feedback design, a closed-loop system that contains infinite number ofdecoupledsecond orderlinear subsystems is obtained. The designed controller is essentially a ‘feedback linearization type’ controller [43].However, as a design tool, backstepping is less restrictive than feedback linearization [22]. Furthermore, the resultspresented here are easily generalizable to the adaptive setting, as we show in§ III-B.1 and § III-B.2. We alsoremark that, by a careful analysis of the dynamical properties of a particular system, the problem of unnecessarycancellations can be circumvented. This is illustrated in§ VI where we design fully decentralized nominal controllersfor both linear and nonlinear mass-spring systems.
Remark 3: If the initial state of system (6) does not satisfy Assumption 6, controller (20) can still be used toattain global asymptotic stability of its origin, but on a Banach spaceB := l∞× l∞, rather than on a Hilbert spaceH := l2 × l2. This follows from the fact that for everyfixedn ∈ F, the derivative of
V2n(ψ1n, ψ2n) :=12ψ2
1n +12(ψ2n + anψ1n)2 > 0, ∀ψn :=
[ψ1n ψ2n
]∗ ∈ R2 \ {0},
NOVEMBER 14, 2003 9
along the solutions of the ‘n-th subsystem’ of (6,20) is determined by
V2n = − anψ21n − bn(ψ2n + anψ1n)2 < 0, ∀ψn ∈ R2 \ {0}.
Therefore, we conclude global asymptotic stability of the origin of the ‘n-th subsystem’ of (6,20) for everyn ∈ F,which in turn guarantees global asymptotic stability of the origin of (6,20) on a Banach spaceB := l∞ × l∞.Furthermore, Assumptions 1–2 imply boundedness ofun for every n ∈ F. However, we note thatV2(ψ1, ψ2)defined by (17) no longer represents a Lyapunov function for system (6,20) if Assumption 6 is violated.
Remark 4: If a control objective is to asymptotically track a reference outputrn(t), with output of system (6)being defined asyn := ψ1n, ∀n ∈ F, then it can be readily shown that the following control law
un = − 1κn
{(1 + anbn)(ψ1n − rn(t)) + (an + bn)(ψ2n − rn(t)) + fn(ψ1, ψ2) − rn(t)} , n ∈ F,
fulfills this objective. We assume that, for everyn ∈ F, rn, rn, andrn are known and uniformly bounded, and thatrn is piecewise continuous.
B. Adaptive state-feedback design
In this subsection we consider adaptive state-feedback design. We study systems with both spatially constant andspatially varying unknown parameters. We assume that these parametric uncertainties do not depend on time. Thedynamic controllers that guarantee boundedness of all signals in the closed-loop and achieve ‘regulation’ of theplant state are obtained using Lyapunov-based approach.
1) Temporally and spatially constant parametric uncertainties:The first step of the backstepping design for thisclass of systems is the same as in§ III-A. However, at the second step not only do we have to account for theerror between a virtual control and its desired value, but also for our lack of knowledge of parameters in (7c). Inother words, we need to estimate the values of unknown parameterθ and reciprocal of unknown control coefficientκ, % := 1
κ , in order to avoid the division with an estimate ofκ which can occasionally assume zero value.Step 2 We start the second step of our design by augmenting CLF (17) by two terms that account for the errorsbetween unknown parametersθ and% and their estimatesθ and %
Va1(ψ1, ζ2, θ, %) := V2(ψ1, ζ2) +12θ∗Γ−1θ +
|κ|2β%2, (21)
whereθ(t) := θ− θ(t), %(t) := %− %(t), Γ is a positive definite matrix (Γ = Γ∗ > 0), andβ is a positive constant.The derivative ofVa1 along the solutions of
ψ1n = − anψ1n + ζ2n, n ∈ F, (22a)
ζ2n = anψ2n + τn(ψ1, ψ2) + g∗n(ψ1, ψ2)(θ + θ) + κun, n ∈ F, (22b)
is determined by
Va1 = −∑n∈ F
anψ21n +
∑n∈ F
ζ2n
{ψ1n + anψ2n + τn(ψ1, ψ2) + g∗n(ψ1, ψ2)θ + κun
}
+ θ∗
{Γ−1 ˙
θ +∑n∈ F
ζ2ngn(ψ1, ψ2)
}+
|κ|β% ˙%. (23)
We can eliminateθ from (23) with the following parameterθ update law
˙θ =
∑n∈ F
ζ2nΓgn(ψ1, ψ2). (24)
A choice of control law of the form
un = − %{ψ1n + anψ2n + τn(ψ1, ψ2) + g∗n(ψ1, ψ2)θ + bnζ2n
}, n ∈ F, (25)
together with (24) and the relationship
κ% = κ(%− %) = 1− κ%,
NOVEMBER 14, 2003 10
renders (23) into
Va1 = −∑n∈ F
anψ21n −
∑n∈ F
bnζ22n
+|κ|β%
{˙% + β sign(κ)
∑n∈ F
ζ2n
{ψ1n + anψ2n + νn(ψ1, ψ2) + g∗n(ψ1, ψ2)θ + bnζ2n
}}.(26)
With the following choice of update law for the estimate%
˙% = β sign(κ)∑n∈ F
ζ2n
{ψ1n + anψ2n + νn(ψ1, ψ2) + g∗n(ψ1, ψ2)θ + bnζ2n
}, (27)
(26) simplifies to
Va1 = −∑n∈ F
anψ21n −
∑n∈ F
bnζ22n =: − W (ψ1, ζ2) ≤ 0.
Using the definition ofζ2n given by (15), one can rewrite (25), (24), and (27) as
un = − %{
(1 + anbn)ψ1n + (an + bn)ψ2n + νn(ψ1, ψ2) + g∗n(ψ1, ψ2)θ}, n ∈ F,
˙θ =
∑n∈ F
(ψ2n + anψ1n)Γgn(ψ1, ψ2),
˙% = β sign(κ)∑n∈ F
(ψ2n + anψ1n){
(1 + anbn)ψ1n + (an + bn)ψ2n + νn(ψ1, ψ2) + g∗n(ψ1, ψ2)θ}.
(28)Negative semi-definiteness ofVa1 implies thatVa1 is a non-increasing function of time. Hence, based on the
definition ofVa1 and its boundedness att = 0 (see Assumption 6), we conclude thatψ1, ψ2, θ, and % are globallyuniformly bounded, that is,
‖ψ1(t)‖2 :=∑n∈ F
ψ21n(t) < ∞
‖ψ2(t)‖2 :=∑n∈ F
ψ22n(t) < ∞
‖θ(t)‖2 := θ∗(t)θ(t) < ∞
‖%(t)‖2 := %2(t) < ∞
∀ t ≥ 0.
In view of the last equation, properties ofgn andνn (see Assumptions 1–2), and definition ofun, it follows that
θ ∈ L∞, % ∈ L∞, {ψ1n ∈ L∞, ψ2n ∈ L∞, un ∈ L∞, ∀n ∈ F} ,
which in turn implies
ψ1n ∈ L∞, ψ2n ∈ L∞, ∀n ∈ F.
Using Assumption 3 and the fact that{ψ(t) ∈ l2 × l2, ∀ t ≥ 0}, we also conclude that{ ˙θ(t) < ∞, ˙%(t) <
∞, ∀ t ≥ 0}. This follows from the properties of functionsνn and gn (see Assumptions 1–3) and a simpleobservation that{ψ(t) ∈ lp × lq, ∀ t ≥ 0, ∀ p, q ∈ (2, ∞)} whenever{ψ(t) ∈ l2 × l2, ∀ t ≥ 0}. Furthermore, sinceVa1(ψ1(t), ζ2(t), θ(t), %(t)) is a non-increasing non-negative function, it has a limitVa1∞ as t → ∞. Therefore,integration of (28) yields∫ ∞
0
W (ψ1(t), ζ2(t)) dt =∑n∈ F
an
∫ ∞
0
ψ21n(t) dt +
∑n∈ F
bn
∫ ∞
0
ζ22n(t) dt
≤ Va1(ψ1(0), ζ2(0), θ(0), %(0)) − Va1∞
< ∞,
which together with (15) implies
ψ1n ∈ L2, ψ2n ∈ L2, ∀n ∈ F.
NOVEMBER 14, 2003 11
Therefore, we have shown that
ψ1n, ψ2n ∈ L2 ∩ L∞, ψ1n, ψ2n ∈ L∞, ∀n ∈ F.
Using the Barbalat lemma (see, for example [21], [35], [44]), we conclude that bothψ1n(t) and ψ2n(t) go tozero ast → ∞, for all n ∈ F. Therefore, the dynamical controller obtained as a result of our design guaranteesboundedness of all signals in the closed-loop system (7,28) and asymptotic convergence of the state of (7) form = 2 to zero.
Remark 5: If Assumption 6 is not satisfied we are not able to guarantee that{ψ(t) ∈ l2×l2, ∀ t > 0}, which may
lead to unboundedness of both˙θ(t) and ˙%(t) even when Assumptions 1–3 hold. Therefore, we need Assumption 6to be satisfied in order to be able to guarantee boundedness of all signals in the closed-loop for systems withtemporally and spatially constant parametric uncertainties with one estimate per unknown parameter. In§ III-B.2,we show that this assumption can be relaxed if we allow for ‘over-parameterization’ of unknown parameters.
2) Temporally constant spatially varying parametric uncertainties:Here we consider state-feedback design forsystems with time independent spatially varying unknown parameters. An example of such a system is given by(8).
The first step of the backstepping design is the same as in§ III-A. However, in the second step, we have toconstruct an Adaptive Control Lyapunov Function (ACLF) to account for the error between a virtual control andits desired value, and for our lack of knowledge of parameters in (8c). In other words, we need to estimate thevalues of unknown parametersθn and reciprocals of unknown control coefficientsκn, %n := 1
κn, in order to avoid
the division with an estimate ofκn which can occasionally assume zero value.Step 2 We start the second step of our design by augmenting CLF (17) by two terms that account for the errorsbetween unknown parametersθn and%n and their estimatesθn and %n
Va2(ψ1, ζ2, θ, %) := V2(ψ1, ζ2) +12
∑n∈ F
θ∗nΓ−1n θn +
12
∑n∈ F
|κn|βn
%2n, (29)
whereθn(t) := θn− θn(t), %n(t) := %n− %n(t), Γn is a positive definite matrix, andβn is a positive constant. Weassume thatVa2(ψ1(0), ζ2(0), θ(0), %(0)) is finite. The derivative ofVa2 along the solutions of
ψ1n = − anψ1n + ζ2n, n ∈ F, (30a)
ζ2n = anψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)(θn + θn) + κnun, n ∈ F, (30b)
is determined by
Va2 = −∑n∈ F
anψ21n +
∑n∈ F
ζ2n
{ψ1n + anψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn + κnun
}+
∑n∈ F
θ∗n
{Γ−1n
˙θn + ζ2nhn(ψ1, ψ2)
}+
∑n∈ F
|κn|βn
%n ˙%n. (31)
We can eliminateθn from (31) with the parameter update laws of the form
˙θn = ζ2nΓnhn(ψ1, ψ2), n ∈ F. (32)
A choice of control law of the form
un = − %n
{ψ1n + anψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn + bnζ2n
}, n ∈ F, (33)
together with (32) and the relationship
κn%n = κn(%n − %n) = 1− κn%n,
renders (31) into
Va2 = −∑n∈ F
anψ21n −
∑n∈ F
bnζ22n
+∑n∈ F
|κn|βn
%n
{˙%n + βn sign(κn)ζ2n
{ψ1n + anψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn + bnζ2n
}}.
NOVEMBER 14, 2003 12
With a choice of update law for the estimate%n
˙%n = βn sign(κn)ζ2n{ψ1n + anψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn + bnζ2n
}, n ∈ F, (34)
Va2 simplifies to
Va2 = −∑n∈ F
anψ21n −
∑n∈ F
bnζ22n =: W (ψ1, ζ2) ≤ 0.
Using the definition ofζ2n given by (15), one can rewrite (33), (32), and (34) for everyn ∈ F as
un = − %n{
(1 + anbn)ψ1n + (an + bn)ψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn},
˙θn = (ψ2n + anψ1n)Γnhn(ψ1, ψ2),
˙%n = βn sign(κn)(ψ2n + anψ1n){
(1 + anbn)ψ1n + (an + bn)ψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn}.
(35)Since Va2 ≤ 0, we conclude thatVa2 is a non-increasing function of time. Thus, using (29) and the original
assumption thatVa2(ψ1(0), ζ2(0), θ(0), %(0)) < ∞, we conclude global uniform boundedness ofψ1, ψ2, θn, and%n, that is,
‖ψ1(t)‖2 :=∑n∈ F
ψ21n(t) < ∞
‖ψ2(t)‖2 :=∑n∈ F
ψ22n(t) < ∞
‖θn(t)‖2 := θ∗n(t)θn(t) < ∞
‖%n(t)‖2 := %2n(t) < ∞
∀ t ≥ 0.
Using the last equation, properties ofhn andτn (see Assumptions 1–2), and (33), it follows that
{θn ∈ L∞, %n ∈ L∞, ψ1n ∈ L∞, ψ2n ∈ L∞, un ∈ L∞, ∀n ∈ F},
which in turn implies
{ ˙θn ∈ L∞, ˙%n ∈ L∞, ψ1n ∈ L∞, ψ2n ∈ L∞, ∀n ∈ F}.
Moreover, sinceVa2(ψ1(t), ζ2(t), θ(t), %(t)) is a non-increasing function of time bounded from below by zero, ithas a limitVa2∞ := limt→∞ Va2(ψ1(t), ζ2(t), θn(t), %n(t)). Hence, integration ofVa2 gives∫ ∞
0
W (ψ1(t), ζ2(t)) dt =∑n∈ F
an
∫ ∞
0
ψ21n(t) dt +
∑n∈ F
bn
∫ ∞
0
ζ22n(t) dt
≤ Va2(ψ1(0), ζ2(0), θ(0), %(0)) − Va2∞
< ∞,
which in combination with the definition ofζ2 implies
ψ1n ∈ L2, ψ2n ∈ L2, ∀n ∈ F.
Therefore, we have shown that
ψ1n, ψ2n ∈ L2 ∩ L∞, ψ1n, ψ2n ∈ L∞, ∀n ∈ F.
Application of the Barbalat lemma implies that the state of (8) converges to zero ast→∞. In other words, bothψ1n(t) andψ2n(t) asymptotically go to zero for alln ∈ F. Therefore, the dynamical controller obtained as a resultof the backstepping design guarantees boundedness of all signals in the closed-loop system (8,35) and asymptoticconvergence of the state of (8) form = 2 to zero.
Remark 6:The assumption on finiteness ofVa2(ψ1(0), ζ2(0), θ(0), %(0)) is not natural. Namely, it is reasonableto assume that the initial distributed state belongs to the underlying state-space (in this casel2 × l2), but, since wewant to consider systems with infinite number of unknown parameters it is somewhat artificial to assume that att = 0 we know most of them. If we have somea priori information about values that the unknown parameters can
NOVEMBER 14, 2003 13
assume we can choose sequences{Γn}n∈F and{βn}n∈F such thatVa2(ψ1(0), ζ2(0), θ(0), %(0)) is finite. However,this would lead to parameter update laws with very large gains since elements of these two sequences have toincrease their values asn → ∞. Clearly, this is not desirable from a practical point of view. Here we show thatcontroller (35) guarantees boundedness of all signals in the closed-loop, and convergence of the state of (8) form = 2 to zero ast→∞ even whenVa2(ψ1(0), ζ2(0), θ(0), %(0)) = ∞. This follows from the fact that for everyfixed n ∈ F the derivative of
Va2n(ψ1n, ζ2n, θn, %n) :=12ψ2
1n +12ζ22n +
12θ∗nΓ
−1n θn +
12|κn|βn
%2n,
along the solutions of the ‘n-th subsystem’ of (30,35) is determined by
Va2n = −anψ21n − bnζ
22n =: Wn(ψ1n, ζ2n) ≤ 0.
Using similar argument as before we conclude that
θn ∈ L∞, %n ∈ L∞, ˙θn ∈ L∞, ˙%n ∈ L∞, ψ1n ∈ L2 ∩ L∞, ψ2n ∈ L2 ∩ L∞, un ∈ L∞, ∀n ∈ F,
which in turn implies boundedness of all signals in the closed-loop system (8,35) and convergence of bothψ1n(t)and ψ2n(t) to zero ast → ∞, for all n ∈ F. It is noteworthy that when
∑n∈ F Va2n(t = 0) < ∞ it might
be advantageous to use∑n∈ F Va2n as a CLF for the entire infinite dimensional system rather than to consider
Va2n as a CLF for then-th plant unit. By performing analysis of this type both beneficial interactions betweendifferent subsystems and beneficial nonlinearities can be identified and unnecessary cancellations can be avoidedin the process of control design.
Remark 7:The results of this subsection can also be applied for the control of systems with temporally andspatially constant parametric uncertainties. Clearly, the design of§ III-B.1 yields a centralized controller (28) withthe number of parameter estimates equal to the number of unknown parameters. On the other hand, controller(35) has a different parameter update law in every single control unitKn, even when all parameters are bothspatially and temporally constant. It is noteworthy that this ‘over-parameterization’ is advantageous in applicationsbecause it allows for implementation of localized adaptive distributed controllers, as illustrated in§ V-B.2. Thisuseful property cannot be achieved with controller (28) since it requires information about entire distributed stateto estimate unknown parameters (see§ V-B.1 for details). Furthermore, the design procedure described here doesnot require Assumption 3 to hold and, consequently, can be applied to a broader class of systems.
Remark 8:One can show that the dynamical controller of the form
un = − %nsn, n ∈ F,
˙θn = {(ψ2n − rn(t)) + an(ψ1n − rn(t))}Γnhn(ψ1, ψ2), n ∈ F,
˙%n = βn sign(κn){(ψ2n − rn(t)) + an(ψ1n − rn(t))}sn, n ∈ F,
(36)
guarantees boundedness of all signals in the closed-loop system (8,36) form = 2 and convergence ofψ1n(t) torn(t) as t→∞, for all n ∈ F, where
sn :={
(1 + anbn)(ψ1n − rn(t)) + (an + bn)(ψ2n − rn(t)) + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn − rn(t)}.
It is assumed that, for everyn ∈ F, the reference signalrn, and its first two derivativesrn, and rn are known anduniformly bounded, and thatrn is piecewise continuous.
C. Nominal output-feedback design
The controllers of§ III-A, § III-B.1, and § III-B.2 provide stability/regulation/asymptotic tracking of the closed-loop systems under the assumption that the full state information is available. Here, we study a more realisticsituation in which only a distributed output variable is measured. We show that, as in the case of finite dimensionalsystems [22], the observer backstepping can be used as a tool for fulfilling the desired control objective for systemson lattices in which nonlinearities depend only on the measured signals. The starting point of the output-feedbackapproach is a design of an observer which guarantees the exponential convergence of the state estimates to theirreal values. Once this is accomplished, the combination of backstepping and nonlinear damping is used to accountfor the observation errors and provide closed-loop stability.
NOVEMBER 14, 2003 14
We rewrite (9) in a form suitable for observer design
ψn = Aψn + ϕn(y) + κne2un, n ∈ F, (37a)
yn = Cψn, n ∈ F, (37b)
where, form = 2,
ψn :=[ψ1n
ψ2n
], ϕn(y) :=
[0
fn(y)
], A :=
[0 10 0
], e2 :=
[01
], C :=
[1 0
].
We proceed by designing an equivalent ofKrener-Isidori observer(see, for example, [22], [45]) for (37)
˙ψn = Aψn + Ln(yn − yn) + ϕn(y) + κne2un, n ∈ F, (38a)
yn = Cψn, n ∈ F, (38b)
whereLn :=[l1n l2n
]∗is chosen such thatA0n := A − LnC is a Hurwitz matrix for everyn ∈ F. Clearly,
this is going to be satisfied if and only iflin > 0, ∀ i = {1, 2}, ∀n ∈ F. In this case, an exponentially stable systemof the form
˙ψn = A0nψn, n ∈ F, (39)
is obtained by subtracting (38) from (37). The properties ofA0n imply the exponential convergence ofψn :=ψn − ψn to zero and the existence of the positive definite matrixP0n that satisfies
A∗0nP0n + P0nA0n = − I, ∀n ∈ F. (40)
We are now ready to design an output-feedback controller that guarantees stability of (9) form = 2.Step 1The observer-backstepping design starts with subsystem (9a) by rewriting it as
ψ1n = ψ2n + ψ2n, n ∈ F, (41)
and consideringψ2n as a virtual control andψ2n as a disturbance generated by (39). We propose a CLF for the‘n-th subsystem’ of (41)
V1n(ψ1n, ψn) :=12ψ2
1n +1d1n
ψ∗nP0nψn,
whereP0n is a positive definite matrix that satisfies (40), andd1n > 0 is a design parameter. The derivative ofV1n
along the solutions of (41,39) for everyn ∈ F is determined by
V1n = ψ1n(ψ2n + ψ2n) − 1d1n
‖ψn‖22 ≤ ψ1n(ψ2n + d1nψ1n) +1
4d1nψ2
2n − 1d1n
‖ψn‖22
≤ ψ1n(ψ2n + d1nψ1n) − 34d1n
‖ψn‖22, (42)
where Young’s Inequality (see [22], expression (2.254)) is used to upper boundψ1nψ2n. In particular, the choiceof a ‘stabilizing function’ψ2nd of the form
ψ2nd = − (an + d1n)ψ1n, an > 0, ∀n ∈ F,
clearly rendersV1n(ψ1n, ψn) negative definite. Sinceψ2n is not actually a control, but rather, an estimate of a statevariable, we introduce the change of variables
ζ2n := ψ2n − ψ2nd = ψ2n + (an + d1n)ψ1n, ∀n ∈ F, (43)
which adds an additional term on the right-hand side of (42)
V1n ≤ − anψ21n − 3
4d1n‖ψn‖22 + ψ1nζ2n. (44)
The sign indefinite term in (44) will be taken care of at the second step of backstepping.
NOVEMBER 14, 2003 15
Step 2We express our system into new coordinates as
ψ1n = − (an + d1n)ψ1n + ζ2n + ψ2n, n ∈ F, (45a)
ζ2n = (an + d1n)(ψ2n + ψ2n) + l2n(ψ1n − ψ1n) + fn(y) + κnun, n ∈ F, (45b)
˙ψn = A0nψn, n ∈ F, (45c)
and propose the following CLF for its ‘n-th subsystem’
V2n(ψ1n, ζ2n, ψn) := V1n(ψ1n, ψn) +12ζ22n +
1d2n
ψ∗nP0nψn,
with d2n > 0. The derivative ofV2n along the solutions of (45) for everyn ∈ F is determined by
V2n = V1n + ζ2nζ2n − 1d2n
‖ψn‖22 ≤ − anψ21n − 3
4(
1d1n
+1d2n
)‖ψn‖22 +
ζ2n{ψ1n + (an + d1n)ψ2n + l2n(ψ1n − ψ1n) + fn(y) + d2n(an + d1n)2ζ2n + κnun}.
A control law of the form
un = − 1κn{ψ1n + (an + d1n)ψ2n + l2n(ψ1n − ψ1n) + fn(y) + d2n(an + d1n)2ζ2n + bnζ2n}, (46)
with bn > 0 for everyn ∈ F, guarantees negative definiteness ofV2n, that is
V2n ≤ − anψ21n − bnζ
22n − 3
4(
1d1n
+1d2n
)‖ψn‖22.
Therefore, we conclude that our design guarantees global asymptotic stability of the origin of the closed-loop system(9,39,46) form = 2 on B := l∞ × l∞ × l∞ × l∞.
D. Adaptive output-feedback design
We rewrite (10) in a form suitable for adaptive output-feedback design
ψn = Aψn + ηn(y) +r∑j=1
θjnϕjn(y) + κne2un, n ∈ F, (47a)
yn = Cψn, n ∈ F, (47b)
where, form = 2,
ψn :=[ψ1n
ψ2n
], ηn(y) :=
[0
τn(y)
], ϕjn(y) :=
[0
hjn(y)
],
A :=[
0 10 0
], e2 :=
[01
], C :=
[1 0
].
We proceed by designing filters which provide ‘virtual estimates’ of unmeasured state variables (see [22],§ 7.3)
ξ(0)n = A0nξ(0)n + Lnyn + ηn(y), n ∈ F, (48a)
ξ(j)n = A0nξ(j)n + ϕjn(y), 1 ≤ j ≤ r, n ∈ F, (48b)
vn = A0nvn + e2un, n ∈ F, (48c)
whereLn :=[l1n l2n
]∗is chosen such thatA0n := A − LnC is Hurwitz for everyn ∈ F. Clearly,A0n is
going to be Hurwitz if and only iflin > 0, ∀ i = {1, 2}, ∀n ∈ F. In this case, an exponentially stable system ofthe form
εn = A0nεn, εn := ψn − {ξ(0)n +r∑j=1
θjnξ(j)n + κnvn}, n ∈ F, (49)
is obtained by combining (47) and (48). The properties ofA0n imply the exponential convergence ofεn to zeroand the existence of the positive definite matrixP0n that satisfies (40).
We are now ready to design an adaptive output-feedback controller for (10) using backstepping.
NOVEMBER 14, 2003 16
Step 1The adaptive observer-backstepping design starts with subsystem (10a) by rewriting it as
ψ1n = ξ(0)2n +
r∑j=1
θjnξ(j)2n + κnv2n + ε2n, n ∈ F, (50)
and consideringv2n as a virtual control andε2n as a disturbance generated by (49). Ifv2n were control, and allparameters were known, then (50) could be stabilized by
v2n = − 1κn
{ξ(0)2n + (an + d1n)ψ1n
}−
r∑j=1
θjnκn
ξ(j)2n , n ∈ F, (51)
wherean andd1n are positive design parameters. To account for parametric uncertainties we add and subtract theright-hand side of (51) tov2n in (50) to obtain
ψ1n = − (an + d1n)ψ1n + κn
{v2n + ω(1)∗
n ϑ(1)n
}+ κnω
(1)∗
n ϑ(1)n + ε2n, n ∈ F, (52)
where
ω(1)n :=
ξ(0)2n + (an + d1n)ψ1n
ξ(1)2n
...
ξ(r)2n
, ϑ(1)n :=
1κnθ1nκn...
θrnκn
, ϑ(1)
n := ϑ(1)n − ϑ(1)
n ,
with ϑ(1)n denoting the vector of unknown parameters.
We propose a CLF of the form
Va1n(ψ1n, ϑ(1)n , εn) :=
12ψ2
1n +|κn|2ϑ(1)∗
n Γ−1n ϑ(1)
n +1d1n
ε∗nP0nεn,
whereP0n is a positive definite matrix that satisfies (40),d1n > 0 is a design parameter, andΓn = Γ∗n > 0. Thederivative ofVa1n along the solutions of (52,49) for everyn ∈ F is determined by
Va1n ≤ − anψ21n + κnψ1n(v2n + ω(1)∗
n ϑ(1)n ) + |κn|ϑ(1)∗
n
{Γ−1n
˙ϑ(1)n + sign(κn)ψ1nω
(1)n
}− 3
4d1n‖εn‖22,
where we used Young’s Inequality (see [22], expression (2.254)) to upper boundψ1nε2n. In particular, the followingchoices of a ‘stabilizing function’v2nd and update law for the estimateϑ(1)
n
v2nd = −ω(1)∗
n ϑ(1)n , ∀n ∈ F,
˙ϑ
(1)n = sign(κn)ψ1nΓnω
(1)n , ∀n ∈ F,
clearly renderVa1n(ψ1n, ϑ(1)n , εn) negative semi-definite. Sincev2n is not actually a control, we introduce the
second error variable as
ζ2n := v2n − v2nd = v2n + ω(1)∗
n ϑ(1)n , ∀n ∈ F, (53)
which adds an additional term on the right-hand side ofVa1n
Va1n ≤ − anψ21n − 3
4d1n‖εn‖22 + κnψ1nζ2n. (54)
The sign indefinite term in the last equation will be taken care of at the second step of backstepping.Step 2The differentiation ofζ2n with respect to time yields
ζ2n = v2n + ω(1)∗
n ϑ(1)n + ω(1)∗
n˙ϑ(1)n
= − l2nv1n + un + ω(1)∗
n ϑ(1)n + ω(1)∗
n˙ϑ(1)n .
NOVEMBER 14, 2003 17
We now use the definition ofω(1)n to rewrite ω(1)∗
n ϑ(1)n as
ω(1)∗
n ϑ(1)n = µ∗nϑ
(1)n + (an + d1n)ϑ
(1)1n ψ1n
= µ∗nϑ(1)n + (an + d1n)ϑ
(1)1n (ξ(0)2n + ε2n) + ϑ
(1)1nω
(2)∗
n ϑ(2)n ,
whereϑ(1)1n represents the first element of vectorϑ(1)
n and
µn :=
ξ(0)2n
ξ(1)2n
...
ξ(r)2n
, ω(2)n := (an + d1n)
ξ(1)2n
...
ξ(r)2n
v2n
, ϑ(2)n :=
θ1n
...
θrn
κn
.
Hence,ζ2n can be expressed as
ζ2n = σn + ϑ(1)1nω
(2)∗
n ϑ(2)n + (an + d1n)ϑ
(1)1n ε2n + un,
whereσn := − l2nv1n + ω
(1)∗
n˙ϑ
(1)n + µ∗nϑ
(1)n + ϑ
(1)1n
{(an + d1n)ξ
(0)2n + ω
(2)∗
n ϑ(2)n
},
ϑ(2)n := ϑ
(2)n − ϑ
(2)n .
The CLF from Step 1 is augmented by the three terms that penalizeζ2n, ϑ(2)n , andεn, respectively, to obtain
Va2n(ψ1n, ζ2n, ϑ(1)n , ϑ(2)
n , εn) := Va1n(ψ1n, ϑ(1)n , εn) +
12ζ22n +
12ϑ(2)∗
n ∆−1n ϑ(2)
n +1d2n
ε∗nP0nεn,
whered2n > 0 and∆n = ∆∗n > 0. The derivative ofVa2n is determined by
Va2n = Va1n + ζ2nζ2n + ϑ(2)∗
n ∆−1n
˙ϑ(2)n − 1
d2n‖εn‖22
≤ − anψ21n + ϑ(2)∗
n
{∆−1n
˙ϑ(2)n + ζ2n
(ψ1ner+1 + ϑ
(1)1nω
(2)n
)}− 3
4(
1d1n
+1d2n
)‖εn‖22
+ ζ2n
{σn + ψ1ne
∗r+1ϑ
(2)n + d2n
((an + d1n)ϑ
(1)1n
)2
ζ2n + un
},
with er+1 being the(r + 1)-st coordinate vector inRr+1. In particular, the following choices of a control lawunand update law for the estimateϑ(2)
n
un = −{σn + ψ1ne
∗r+1ϑ
(2)n + d2n
((an + d1n)ϑ
(1)1n
)2
ζ2n + bnζ2n
}, ∀n ∈ F,
˙ϑ
(2)n = ζ2n∆n
{ψ1ner+1 + ϑ
(1)1nω
(2)n
}, ∀n ∈ F,
with bn > 0, transformVa2n(ψ1n, ζ2n, ϑ(1)n , ϑ
(2)n , εn) into a negative semi-definite function of the form
Va2n ≤ − anψ21n − bnζ
22n − 3
4(
1d1n
+1d2n
)‖εn‖22 ≤ 0, ∀n ∈ F.
One can establish boundedness of all signals in the closed-loop adaptive system and asymptotic convergence ofψ1n, ζ2n, andεn to zero for everyn ∈ F, using similar proof technique to the one presented in§ III-B.
IV. W ELL-POSEDNESS OF CLOSED-LOOP SYSTEMS
The results established in§ III are valid only if the solution to the resulting system of equations exists. In thissection, we show that the designed controllers yield the well-posed closed-loop systems.
NOVEMBER 14, 2003 18
Well-posedness of closed-loop systems with temporally constant spatially varying parametric uncertainties
The closed-loop systems obtained using adaptive feedback design of§ III-B.2 can be considered as the abstractevolution equations on a Banach spaceB := l∞× l∞× lr+1
∞ , wherer denotes the dimension of the vectorθn. Thisrepresentation is most convenient for addressing the question of the existence and uniqueness of solutions.
The closed-loop system (8,35) can be rewritten as the abstract evolution equation of the form
φ = Aφ + p(φ). (55)
The state of system (55), a linear operatorA, and a nonlinear mappingp are defined as
φ :=
ψ1
ψ2
θ
%
∈ B, A :=
0 I 0 00 0 0 00 0 0 00 0 0 0
, p(φ) :=
0
p2(φ)
p3(ψ1, ψ2)
p4(ψ1, ψ2, θ)
,whereψ1 := {ψ1n}n∈F, ψ2 := {ψ2n}n∈F, θ := {θn}n∈F, % := {%n}n∈F, andpi are infinite vectors∀ i = 2, 3, 4whose elements are determined by
p2n := τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn − κn%nsn(ψ1, ψ2, θn), n ∈ F,p3n := (ψ2n + anψ1n)Γnhn(ψ1, ψ2), n ∈ F,p4n := βn sign(κn)(ψ2n + anψ1n)sn(ψ1, ψ2, θn), n ∈ F,
where
sn(ψ1, ψ2, θn) = {(1 + anbn)ψ1n + (an + bn)ψ2n + τn(ψ1, ψ2) + h∗n(ψ1, ψ2)θn}, n ∈ F.
Clearly, A : B → B is a bounded operator and, thus, it generates aC0–semigroup. Moreover, for the systemsdescribed in§ II-B, p is a continuously differentiable function of its arguments and Lyapunov analysis of§ IIIguarantees boundedness of‖φ(t)‖ for all times. Therefore, we conclude that the closed-loop system (8,35) form = 2 has a unique classical solution on[0, ∞) (see [41], Theorem 2.74).
Remark 9:The nominal feedback designs of§ III-A and § III-C yield closed-loop systems with infinite numberof completely decoupled second and fourth order linear subsystems, respectively. Therefore, the existence anduniqueness of solutions is guaranteed in these two cases.
Remark 10:The existence and uniqueness of solutions of the closed-loop systems obtained using adaptive state-feedback design of§ III-B.1 can be established on a Hilbert spaceH := l2 × l2 × Rr+1, wherer denotes thedimension of the vectorθ, using similar results.
Remark 11:The well-posedness of the closed-loop systems obtained as a result of the adaptive output-feedbackdesign of§ III-D can be established using similar results. The underlying state space in this case isB := l4r+8
∞ ,whereθn ∈ Rr, for everyn ∈ F.
V. A N EXAMPLE OF LYAPUNOV-BASED DISTRIBUTED CONTROL DESIGN: MASS-SPRING SYSTEM
In this section, we discuss application of controllers developed in§ III to the systems described in§ II-A.Furthermore, we analyze the structure of these controllers and validate their performance using computer simulationsof systems containing a large number of units.
A. Nominal state-feedback design
In this subsection, we illustrate that the nominal Lyapunov-based state-feedback design leads to static distributedcontrollers with the same localization as the original plant which is advantageous for practical implementation.
In the particular case of a linear mass-spring system (2) without parametric uncertainties, control law (20) takesthe following form
u1n = − mn
{(1 + anbn − kn−1 + kn
mn)xn + (an + bn)xn +
kn−1
mnxn−1 +
knmn
xn+1
}, n ∈ Z. (56)
For a spatially invariant system (4) controller (56) simplifies to
u2n = − m
{(1 + anbn − 2k
m)xn + (an + bn)xn +
k
m(xn−1 + xn+1)
}, n ∈ Z. (57)
NOVEMBER 14, 2003 19
-G0
-G1
-G2
-����
-K0
-K1
-K2
-����
6
?
6
?
6
?
(a) Mass-spring system with localizedcontrollers of§ V-A, § V-B.2, § V-C, and§ V-D.
-G0
-G1
-G2
-����
K
6
?
6
?
6
?
(b) Mass-spring system with centralizedadaptive controllers (60) and (61).
-G0
-G1
-G2
-����
K0 K1 K2
6
?
6
?
6
?
(c) Mass-spring system with fully decen-tralized controllers (64) and (71).
Fig. 3. Controller architectures of mass-spring system.
When a nonlinear mass-spring system (3) without parametric uncertainties is considered control law (20) can berewritten as
u3n = u1n − qn−1(xn−1 − xn)3 − qn(xn+1 − xn)3, n ∈ Z, (58)
whereu1n is given by (56). For system (5) controller (58) simplifies to
u4n = u2n − q{(xn−1 − xn)3 + (xn+1 − xn)3
}, n ∈ Z, (59)
with u2n being defined by (57).Fig. 3(a) illustrates controller architecture of mass-spring system with the aforementioned controllers. We observe
that the nominal output-feedback design of§ V-C and adaptive distributed designs of§ V-B.2 and§ V-D result inthe closed-loop systems with similar passage of information. Remarkably, in all these cases, Lyapunov-based designyields localized distributed controllersKn, ∀n ∈ Z, that require only measurements from then-th plant unitGnand its immediate neighborsGn−1 andGn+1, to achieve desired objective.
In applications, we clearly have to work with systems on lattices that contain large but finite number of units.All considerations related to infinite dimensional systems are applicable here, but with minor modifications. Forexample, if we consider the mass-spring system shown in Fig. 4 withN masses (n = 1, 2, . . . , N ) both the equationspresented in§ II-A and the control laws of this section are still valid with appropriate ‘boundary conditions’ of theform xj = xj = uj ≡ 0, ∀ j ∈ Z \ {1, 2, . . . , N} .
Fig. 4. Finite dimensional mass-spring system.
Simulation results of uncontrolled (left) and controlled (right) linear mass-spring system withN = 100 massesandm = k = 1 are shown in Fig. 5. The initial state of the system is randomly selected. The right plot is obtainedusing control law (57) withan = bn = 1, ∀n = 1, 2, . . . , N . Simulation results illustrate that the nominal linearlocalized distributed controller (57) achieves desired control objective in an effective manner.
B. Adaptive state-feedback design
1) Temporally and spatially constant parametric uncertainties:In this section we discuss application of con-trollers developed in§ III-B.1 to some of systems described in§ II-A with temporally and spatially constantunknown parameters. We illustrate that the adaptive Lyapunov design in this particular situation leads to centralizedcontrollers. Namely, information about entire state of the original system is used in the equations for parameterupdate laws. This is clearly an obstacle for practical control implementations. Nevertheless, we illustrate that thedesigned controller achieves the desired objective successfully by performing computer simulations of systemscontaining a large number of units.
NOVEMBER 14, 2003 20
0 5 10 15 20−1.5
−1
−0.5
0
0.5
1
1.5
t
ψ1(t
)
0 2 4 6 8 10−1
−0.5
0
0.5
1
t
ψ1(t
)
0 2 4 6 8 10−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
t
ψ2(t
)
0 2 4 6 8 10−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
t
u(t)
Fig. 5. Simulation results of uncontrolled (far left) and controlled linear mass-spring system withN = 100 masses andm = k = 1. Theinitial state of the system is randomly selected. Control law (57) is used withan = bn = 1, ∀n = 1, 2, . . . , N .
In the particular case of a linear mass-spring system (4) with parametersθ = km andκ = 1
m > 0 being constantbut unknown, control law (28) takes the following form
u1n = − %{
(1 + anbn)xn + (an + bn)xn + θ(xn−1 − 2xn + xn+1)}, n ∈ Z,
˙θ = γ
∑n∈Z
(xn + anxn)(xn−1 − 2xn + xn+1),
˙% = β∑n∈Z
(xn + anxn){
(1 + anbn)xn + (an + bn)xn + θ(xn−1 − 2xn + xn+1)},
(60)
whereβ, γ > 0. When a nonlinear mass-spring system (5) with constant but unknown parametersθ1 = km , θ2 = q
m ,andκ = 1
m > 0 is considered control law (28) can be rewritten as
u2n = − %sn, n ∈ Z,[ ˙θ1˙θ2
]=
∑n∈Z
(xn + anxn)Γ[
xn−1 − 2xn + xn+1
(xn−1 − xn)3 + (xn+1 − xn)3
],
˙% = β∑n∈Z
(xn + anxn)sn,
(61)
whereΓ = Γ∗ > 0, β > 0, and
sn := (1 + anbn)xn + (an + bn)xn +[
xn−1 − 2xn + xn+1
(xn−1 − xn)3 + (xn+1 − xn)3
]∗ [θ1θ2
].
Clearly, both (60) and (61) are centralized controllers, as illustrated in Fig. 3(b).In applications, we still can use the above expressions, but with minor modifications. For example, if we consider
mass-spring system shown in Fig. 4 withN masses (n = 1, 2, . . . , N ) the appropriate formulae for controller (60)are given by
u1n = − %{
(1 + anbn)xn + (an + bn)xn + θ(xn−1 − 2xn + xn+1)}, 1 ≤ n ≤ N,
˙θ = γ
N∑n=1
(xn + anxn)(xn−1 − 2xn + xn+1),
˙% = βN∑n=1
(xn + anxn){
(1 + anbn)xn + (an + bn)xn + θ(xn−1 − 2xn + xn+1)},
with the ‘boundary conditions’ of the formx0 = x0 = xN+1 = xN+1 = 0.Fig. 6 shows simulation results of linear mass-spring system withN = 100 masses and unknown parameters
m = k = 1, using control law (60) withγ = β = an = bn = 1, ∀n = 1, 2, . . . , N , and θ(0) = ρ(0) = 0.5. Theinitial state of the mass-spring system is randomly selected. Clearly, centralized adaptive scheme drives state ofthe original linear system to zero and provides boundedness of parameter estimates with a reasonable amount ofcontrol effort. We remark that the states of the estimators converge to constant values which do not correspond tothe values of unknown parameters.
NOVEMBER 14, 2003 21
0 2 4 6 8 10−1
−0.5
0
0.5
1
t
ψ1(t
)
0 2 4 6 8 10
−1
−0.5
0
0.5
1
t
ψ2(t
)
0 1 2 3 4 5−1
0
1
2
3
4
5
t
estim
ates
of θ
, ρ
θ
ρ
0 2 4 6 8 10−6
−4
−2
0
2
4
6
t
u(t)
Fig. 6. Simulation results of linear mass-spring system withN = 100 masses and unknown parametersm = k = 1, using control law (60)with γ = β = an = bn = 1, ∀n = 1, 2, . . . , N , and θ(0) = ρ(0) = 0.5. The initial state of the mass-spring system is randomly selected.
2) Temporally constant spatially varying parametric uncertainties:Here we discuss application of controllersdeveloped in§ III-B.2 to some of systems described in§ II-A with time independent spatially varying unknownparameters. We illustrate that the adaptive Lyapunov design in this particular situation leads to localized distributedcontrollers as in§ V-A. This is of paramount importance for practical implementations since there is no need forcentralized computer due to the fact that design procedure yields a controller with both localized actuation andlocalized passage of information. We illustrate the effectiveness of the adaptive scheme by performing computersimulations of systems containing a large number of units.
In the particular case of a linear mass-spring system (2) with temporally constant spatially varying unknownparametersθn1 = kn−1
mn, θn2 = kn
mn, andκn = 1
mn> 0, control law (35) takes the following form
u1n = − %n{
(1 + anbn)xn + (an + bn)xn +[xn−1 − xnxn+1 − xn
]∗ [θn1
θn2
]},
˙θn = (xn + anxn)Γn
[xn−1 − xnxn+1 − xn
],
˙%n = βn(xn + anxn){
(1 + anbn)xn + (an + bn)xn +[xn−1 − xnxn+1 − xn
]∗ [θn1
θn2
]},
(62)
whereΓn = Γ∗n > 0 and βn > 0 for all n ∈ Z. Formulae for adaptive state-feedback controller for a nonlinearmass-spring system (3) with temporally constant spatially varying parametric uncertainties can be readily obtainedcombining (3) and (35). In both cases the design procedure yields localized distributed controllers with thearchitecture shown in Fig. 3(a).
Simulation results of linear mass-spring system withN = 100 masses and unknown parametersmn = kn = 1,∀n = 1, 2, . . . , N , using control law (62) withγn = βn = an = bn = 1, and θn(0) = ρn(0) = 0.5, ∀n =1, 2, . . . , N are shown in Fig. 7. The initial state of the mass-spring system is randomly selected. We observe thatlocalized adaptive controller (62) provides boundedness of all parameter estimates and convergence of the state ofthe error system to zero.
C. Nominal output-feedback design
The nominal Lyapunov-based output-feedback design for mass-spring system leads to localized dynamic dis-tributed controllers of the form
˙ψ1n = ψ2n + l1n(ψ1n − ψ1n)˙ψ2n = l2n(ψ1n − ψ1n) + fn + κnun
un = − 1κn{ψ1n + (an + d1n)ψ2n + l2n(ψ1n − ψ1n) + fn
+ (d2n(an + d1n)2 + bn)(ψ2n + (an + d1n)ψ1n)}
n ∈ Z, (63)
where, for example, for system (5)fn is determined by
fn =k
m{ψ1,n−1 − 2ψ1n + ψ1,n+1} +
q
m{(ψ1,n−1 − ψ1n)3 + (ψ1,n+1 − ψ1n)3}, n ∈ Z.
Fig. 8 illustrates simulation results of linear mass-spring system using output-feedback control law (63). Clearly,numerical results show that the nominal output-feedback distributed controller (63) achieves desired control objectivein an effective manner with a reasonable amount of control effort.
NOVEMBER 14, 2003 22
0 5 10 15−1
−0.5
0
0.5
1
t
ψ1(t
)
0 5 10 15−1
−0.5
0
0.5
1
t
ψ2(t
)
0 5 10 15
−1.5
−1
−0.5
0
0.5
1
1.5
t
u(t)
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t
esti
mat
es o
f θ
0 5 10 150.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
t
esti
mat
es o
f ρ
Fig. 7. Simulation results of linear mass-spring system withN = 100 masses and unknown parametersmn = kn = 1, ∀n = 1, 2, . . . , N ,using control law (62) withγn = βn = an = bn = 1, and θn(0) = ρn(0) = 0.5, ∀n = 1, 2, . . . , N . The initial state of the mass-springsystem is randomly selected. We remark thatx andv denote states of the error system, that is,x := ψ1, v := ψ2.
0 2 4 6 8 10−1
−0.5
0
0.5
1
t
ψ1(t)
0 2 4 6 8 10−1.5
−1
−0.5
0
0.5
1
1.5
t
ψ2(t)
0 2 4 6 8 10−6
−4
−2
0
2
4
6
t
u(t)
Fig. 8. Simulation results of linear mass-spring system withN = 100 masses andmn = kn = 1, ∀n = 1, 2, . . . , N , using output-feedbackcontrol law (63) withan = bn = 1, d1n = d2n = 0.2, l1n = 5, l2n = 6, and ψ1n(0) = ψ2n(0) = 0, ∀n = 1, 2, . . . , N . The initial stateof the mass-spring system is randomly selected.
D. Adaptive output-feedback design
Formulae for adaptive output-feedback controllers for mass-spring systems can be readily obtained combiningresults of§ II-A and § III-D. We remark that as a result of our design we obtain localized distributed dynamiccontrollers whose architecture is shown in Fig. 3(a).
VI. D ESIGN OF FULLY DECENTRALIZED NOMINAL CONTROLLERS FOR MASS-SPRING SYSTEM
In this section we consider the problem of designing fully decentralized controllers for mass-spring system withoutparametric uncertainties. We illustrate that backstepping can be employed to obtain a sequence of controllersKn
that guarantees global asymptotic stability of the origin of the overall system using only measurements from thecorrespondingn-th plant unitGn, ∀n ∈ Z. This is accomplished by a careful analysis of the interactions in theunderlying system, and feedback domination rather than feedback cancellation of ‘unfavorable’ interactions. Fornotational convenience we consider a situation in which all masses and springs are homogeneous. However, wenote that results presented here can be also applied in the nonhomogeneous case, as long as all parameters haveknown values.
NOVEMBER 14, 2003 23
A. Linear mass-spring system
We first consider a linear spatially invariant mass-spring system (4). The backstepping design closely follows theprocedure described in§ III-A. In particular, expression (19) can be rewritten as
V2 = −∑n∈Z
anψ21n +
∑n∈Z
ζ2n
{(1 − 2k
m)ψ1n + anψ2n +
1mun
}+
∑n∈Z
k
m{ζ2nψ1,n−1 + ζ2nψ1,n+1} .
We now invoke Young’s Inequality (see [22], expression (2.254))
ζ2nψ1i ≤ pζ22n +
14pψ2
1i, p > 0, ∀n ∈ Z, ∀ i := {n− 1, n+ 1} ,
and choose a control law of the form
un = −m{
(1 − 2km
)ψ1n + anψ2n + bnζ2n
}= −m
{(1 − 2k
m+ anbn)ψ1n + (an + bn)ψ2n
}, n ∈ Z, (64)
to obtain
V2 ≤ −∑n∈Z
(an −
k
2mp
)ψ2
1n −∑n∈Z
(bn −
2kpm
)ζ22n.
Therefore, a fully decentralized controller (64), whose architecture is shown in Fig. 3(c), guarantees global asymp-totic stability of the origin of the closed-loop system if the design parametersan andbn satisfy
an >k
2mp, bn >
2kpm
, ∀n ∈ Z.
Since (4) is a spatially invariant system it might be of interest to preserve this property under feedback. This canbe achieved by assigning constant valuesa andb to all design parametersan andbn, respectively, that is
an := a >k
2mp, bn := b >
2kpm
, ∀n ∈ Z.
We can also analyze properties of system (4) using the tools of [13]. Application of the ‘bilateralZ–transform’evaluated on the unit circlez = ejθ transforms (4) into theθ–parameterized system of the form ˙
ψ1θ
˙ψ2θ
=
0 12km
(cos θ − 1) 0
[ψ1θ
ψ2θ
]+
01m
uθ =: Aθψθ + Bθuθ, θ ∈ [0, 2π], (65)
where, for example,
uθ :=∑n∈Z
une−jθn.
Properties of system (4) are completely determined by properties of its transformed equivalent (65). Namely, (4)is exponentially stable (stabilizable) if and only if (65) is exponentially stable (stabilizable) for everyθ ∈ [0, 2π](see [13], Corollary 3). Since the transformed system is clearly exponentially stabilizable, the original systemcan be stabilized by a fully decentralized state-feedback controller if and only if there exist a constant matrixK :=
[k1 k2
]such thatAθ − BθK is Hurwitz for everyθ ∈ [0, 2π]. It is readily shown that this is the
case when both design parameters assume positive values, that isk1 > 0 and k2 > 0. Therefore, based on thisanalysis we conclude that Lyapunov-based design yields a somewhat conservative estimates on parameters thatguarantee stability of the closed-loop system. This is not surprising and we certainly do not recommend the use ofbackstepping for control of linear spatially invariant systems since, for example, optimal controllers with respectto quadratic performance criteria can be easily obtained for this class of systems [13]. However, we remark thatbackstepping procedure does not give one fixed controller, but rather a CLF that can be used to generate a varietyof different control laws. Furthermore, backstepping can be applied as a constructive design tool for a broad classof systems, including spatially varying and nonlinear systems. In particular, we illustrate in§ VI-B that it can be
NOVEMBER 14, 2003 24
0 2 4 6 8 10−1
−0.5
0
0.5
1
t
ψ1(t)
0 2 4 6 8 10
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
t
ψ2(t)
0 2 4 6 8 10−4
−3
−2
−1
0
1
2
3
4
t
u(t)
Fig. 9. Simulation results of linear mass-spring system withN = 100 masses andmn = kn = 1, ∀n = 1, 2, . . . , N , using control law (64)with an = bn = 2, ∀n = 1, 2, . . . , N . The initial state of the mass-spring system is randomly selected.
used to obtain a fully decentralized stabilizing state-feedback controller for a nonlinear mass-spring system. Thisfurther demonstrates the power of backstepping and its flexibility as a design tool.
Theoretical results of this subsection are confirmed numerically by performing simulation on a system withN = 100 masses. Fig. 9 illustrates that fully decentralized control law (64) drives the state of a linear mass-springto zero in an effective manner. By comparing these results to the ones shown in Fig. 5 we observe that the massvelocities have slightly better transient response in the latter case. Furthermore, as expected, the fully decentralizedcontroller requires larger amount of control effort to fulfill the desired control objective. We note that this can beadjusted by the different choice of design parametersan andbn.
B. Nonlinear mass-spring system
We consider a nonlinear mass-spring system (5) and employ backstepping to obtain a fully decentralized stabilizingstate-feedback controller.Step 1We start the recursive design with subsystem (6a) by proposing a CLF (12). The derivative of this functionalong the solutions of (6a) is given by (13). However, we now choose a ‘stabilizing function’ψ2nd of the form
ψ2nd = − anψ1n − bnψ31n, an > 0, bn > 0, ∀n ∈ Z,
which clearly rendersV1(ψ1) negative definite. Sinceψ2 is not actually a control, but rather, a state variable, weintroduce the change of variables
ζ2n := ψ2n − ψ2nd = ψ2n + anψ1n + bnψ31n, ∀n ∈ Z, (66)
to obtain
V1 = −∑n∈Z
anψ21n −
∑n∈Z
bnψ41n +
∑n∈Z
ψ1nζ2n. (67)
The sign indefinite term in (67) will be taken care of in the second step of backstepping.Step 2Coordinate transformation (66) renders the second equation of (5) into a form suitable for the remainder ofour design
ζ2n = (an + 3bnψ21n)ψ1n + ψ2n (68)
= (an + 3bnψ21n)ψ2n +
[ψ1,n−1 − 2ψ1n + ψ1,n+1
(ψ1,n−1 − ψ1n)3 + (ψ1,n+1 − ψ1n)3
]∗1m
[k
q
]+
1mun, n ∈ Z.
Augmentation of the CLF from Step 1 by a term which penalizes the error betweenψ2 andψ2d yields a function
V2(ψ1, ζ2) := V1(ψ1) +12〈ζ2, ζ2〉 = V1(ψ1) +
12
∑n∈Z
ζ22n, (69)
NOVEMBER 14, 2003 25
whose derivative along the solutions of (6a,68) is determined by
V2 = −∑n∈Z
anψ21n −
∑n∈Z
bnψ41n
+∑n∈Z
ζ2n
{(1 − 2k
m− 2q
mψ2
1n)ψ1n + (an + 3bnψ21n)ψ2n +
1mun
}+
∑n∈Z
k
m{ζ2nψ1,n−1 + ζ2nψ1,n+1}
+∑n∈Z
q
m
{ζ2nψ
31,n−1 − 3ζ2nψ1nψ
21,n−1 + 3ζ2nψ2
1nψ1,n−1
}+
∑n∈Z
q
m
{ζ2nψ
31,n+1 − 3ζ2nψ1nψ
21,n+1 + 3ζ2nψ2
1nψ1,n+1
}.
(70)
We now employ Young’s Inequality (see [22], expressions (2.253) and (2.254)) to bound the interactions betweenGn and its immediate neighborsGn−1 andGn+1
ζ2nψ1i ≤ pζ22n +
14pψ2
1i, p > 0, ∀n ∈ Z, ∀ i := {n− 1, n+ 1} ,
ζ2nψ31i ≤ r4
4ζ42n +
34r4/3
ψ41i, r > 0, ∀n ∈ Z, ∀ i := {n− 1, n+ 1} ,
− ζ2nψ1nψ21i ≤ lζ2
2nψ21n +
14lψ4
1i, l > 0, ∀n ∈ Z, ∀ i := {n− 1, n+ 1} ,
ζ2nψ21nψ1i ≤ sζ2
2nψ41n +
14sψ2
1i, s > 0, ∀n ∈ Z, ∀ i := {n− 1, n+ 1} ,
and choose a control law of the form
un = − m
{(1 − 2k
m− 2q
mψ2
1n)ψ1n + (an + 3bnψ21n)ψ2n
}− m
{cnζ2n + dnζ
32n + αnζ2nψ
21n + βnζ2nψ
41n
}, n ∈ Z, (71)
to obtain
V2 = −∑n∈Z
(an − k
2mp− 3q
2ms)ψ2
1n −∑n∈Z
(bn − 3q2mr4/3
− 3q2ml
)ψ41n −
∑n∈Z
(cn − 2kpm
)ζ22n
−∑n∈Z
(dn − qr4
2m)ζ4
2n −∑n∈Z
(αn − 6qlm
)ζ22nψ
21n −
∑n∈Z
(βn − 6qsm
)ζ22nψ
41n.
Therefore, a fully decentralized state-feedback controller (71), whose architecture is shown in Fig. 3(c), providesglobal asymptotic stability of the origin of a nonlinear mass-spring system (5) if the design parametersan, bn, cn,dn, αn, andβn satisfy
an >k
2mp+
3q2ms
, dn >qr4
2m
bn >3q
2mr4/3+
3q2ml
, αn >6qlm
cn >2kpm
, βn >6qsm
∀n ∈ Z. (72)
Simulation results obtained using control law (71) for a nonlinear mass-spring system withN = 100 masses andmn = kn = qn = 1, ∀n = 1, 2, . . . , N , are shown in Fig. 10. Clearly, the fully decentralized controller requires bigamount of initial effort to account for the lack of information about interactions between different subsystems. Weremark that there is some room for improvement of these large initial excursions of control signals by the differentchoice of design parametersan, bn, cn, dn, αn, andβn. In particular, these parameters are chosen to satisfy (72)for p = r = l = s = 1, for everyn = 1, 2, . . . , N . Clearly, one can assign different values top, r, l, ands, andinvestigate corresponding changes in the performance of fully decentralized controller (71).
NOVEMBER 14, 2003 26
0 2 4 6 8 10−1
−0.5
0
0.5
1
t
ψ1(t)
0 2 4 6 8 10−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
t
ψ2(t)
0 2 4 6 8 10−40
−30
−20
−10
0
10
20
30
40
t
u(t)
Fig. 10. Simulation results of nonlinear mass-spring system withN = 100 masses andmn = kn = qn = 1, ∀n = 1, 2, . . . , N , usingcontrol law (71) withan = cn = 2.1, bn = 3.1, dn = 0.6, αn = βn = 6.1, ∀n = 1, 2, . . . , N . The initial state of the mass-spring systemis randomly selected.
VII. I NTEGRATOR BACKSTEPPING
In this section we present results that allow for constructive design of stabilizing controllers for systems on latticeswithout parametric uncertainties. In particular, we show how to extend previously described nominal state-feedbackcontrollers to a more general situation in which a system of the form
ψn = fn(ψ) + gn(ψ)un, fn(0) = 0, n ∈ F, (73)
is augmented by the infinite number of integrators. We assume thatfn andgn satisfy Assumptions 1–2, and denotethe state and control vectors byψ := {ψn}n∈F andu := {un}n∈F, respectively, withψn ∈ Rm andun ∈ R for alln ∈ F. The results summarized in Lemma 1 and Lemma 2 represent natural extensions of the well known finitedimensionalintegrator backsteppingdesign tool ( [22], [35]), and they are respectively based on the followingassumptions:
Assumption 7:There exists a continuously differentiable state-feedback control law
un = αn(ψ), αn(0) = 0, (74)
and a smooth positive definite radially unbounded functionVn : Rm → R for the ‘n-th subsystem’ of (73) suchthat
∂Vn∂ψn
(ψn) {fn(ψ) + gn(ψ)αn(ψ)} ≤ − Wn(ψn) < 0, ∀ψn ∈ Rm \ {0}, (75)
whereWn : Rm → R is a positive definite function for everyn ∈ F.Assumption 8:There exists a continuously differentiable state-feedback control law
u = {αn(ψ)}n∈F , αn(0) = 0, ∀n ∈ F, (76)
and a smooth positive definite radially unbounded functionalV (ψ) :=∑n∈ F Vn(ψn), V : l2 → R, for system (73)
such that ∑n∈ F
∂Vn∂ψn
(ψn) {fn(ψ) + gn(ψ)αn(ψ)} ≤ − W (ψ) < 0, ∀ψ ∈ l2 \ {0}, (77)
whereW : l2 → R is a positive definite functional.We remark that condition (75) guarantees global asymptotic stability of the origin of the ‘n-th subsystem’ of
(73) for everyn ∈ F, which in turn guarantees global asymptotic stability of the origin of (73) onlm∞. If, on theother hand, the conditions of Assumption 8 are satisfied then global asymptotic stability of the origin of (73) onlm2 can be concluded. In addition to that, Assumption 8 guarantees existence of a CLF for (73) which can be usedto obtain distributed controllers with favorable information passing properties, as illustrated in§ VI.
In the remainder of this section, we consider system (73) augmented by the infinite number of integrators, thatis,
ψn = fn(ψ) + gn(ψ)ξn, n ∈ F, (78a)
ξn = un, n ∈ F. (78b)
NOVEMBER 14, 2003 27
Lemma 1:Suppose that, for everyn ∈ F, the ‘n-th subsystem’ of (78a) satisfies Assumption 7 withξn ∈ R asits control. Then, the augmented function of the form
Van(ψ, ξn) := Vn(ψn) +12(ξn − αn(ψ))2,
represents a CLF for the ‘n-th subsystem’ of (78). Therefore, there exists a state-feedback control lawu ={αan(ψ, ξ)}n∈F which guarantees global asymptotic stability of the origin of system (78) onlm+1
∞ . One suchcontrol law is given by
αan(ψ, ξ) = − cn(ξn − αn(ψ)) +∞∑
j=−∞
∂αn∂ψj
{fj(ψ) + gj(ψ)ξj} − ∂Vn∂ψn
(ψn)gn(ψ).
Lemma 2:Suppose that (78a) satisfies Assumption 8 withξ := {ξn}n∈F, ξn ∈ R, ∀n ∈ F, as its control. Then,the augmented function of the form
Va(ψ, ξ) :=∑n∈ F
Vn(ψn) +12
∑n∈ F
(ξn − αn(ψ))2,
represents a CLF for system (78). Therefore, there exists a state-feedback control lawu = {αan(ψ, ξ)}n∈F whichguarantees global asymptotic stability of the origin of system (78) onlm+1
2 . One such control law is given by
αan(ψ, ξ) = − cn(ξn − αn(ψ)) +∞∑
j=−∞
∂αn∂ψj
{fj(ψ) + gj(ψ)ξj} − ∂Vn∂ψn
(ψn)gn(ψ).
Results of Lemma 1 (Lemma 2) can be also applied to a more general class of systems of the form
ψn = fn(ψ) + gn(ψ)ξn, n ∈ F, (79a)
ξn = fan(ψ, ξ) + gan(ψ, ξ)un, n ∈ F, (79b)
where we assume thatgan(ψ, ξ) 6= 0 for all n ∈ F. In this case, the input transformation
un :=1
gan(ψ, ξ){uan − fan(ψ, ξ)} ,
renders (79) into (78) which allows for the application of Lemma 1 (Lemma 2).For example, either of the above Lemmata can be applied to obtain a stabilizing control law for a nonlinear mass
spring system with a dynamical actuator of the form
ψ1n = ψ2n
ψ2n =
[ψ1,n−1 − 2ψ1n + ψ1,n+1
(ψ1,n−1 − ψ1n)3 + (ψ1,n+1 − ψ1n)3
]∗1m
[k
q
]+
1mξn
ξn = − 1Tξn + un
n ∈ Z.
VIII. C ONCLUDING REMARKS
This paper has dealt with the distributed control of spatially discrete infinite dimensional systems. It has beenillustrated that Lyapunov-based approach can be successfully used to obtain state and output-feedback controllersfor both nominal systems and systems with parametric uncertainties. It has been also shown that the control problemcan always be posed in such a way to yield controllers of the same structure as the original plant. Therefore, asa result of Lyapunov-based design control systems with an intrinsic degree of decentralization are obtained. In aparticular case of a nominal mass-spring system we have illustrated that fully decentralized stabilizing feedbackcontrollers can be obtained by a careful analysis of nonlinearities and interactions between different subsystems.Furthermore, within the Lyapunov framework desired control objectives can be achieved irrespective of whetherthe plant dynamics is linear or nonlinear.
Our current efforts are directed towards development of modular adaptive schemes in which parameter updatelaws and controllers are designed separately. The major advantage of using this approach rather than the Lyapunov-based design is the versatility that it offers. Namely, adaptive controllers of this paper are limited to Lyapunov-basedestimators. From a practical point of view it might be advantageous to use the appropriately modified standardgradient or least-squares type identifiers.
NOVEMBER 14, 2003 28
ACKNOWLEDGMENT
The first author would like to thank Professor Petar Kokotovic for his encouragement, inspiring discussions, anduseful suggestions.
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