Chapter 2

Theoretical preliminaries

The purpose of this chapter is to present some definitions and back­ground theory that will be used throughout this book. We will firstintroduce some important theorems based on Lyapunov theory. Wewill give some notions and basic concepts of passivity. Then, we willrecall a necessary condition for the existence of a continuously stabiliz­ing control law for non-linear systems, often referred to as Brockett'snecessary condition. The definitions of non-holonomic systems, under­actuated systems and a homoclinic orbit are also given .

2.1 Lyapunov stability

Consider the autonomous system

x = f(x) (2.1)

where f : D -+ JRn is a locally Lipschitz map from a domain D c JRninto JRn . We suppose that the origin x = 0 is an equilibrium point of(2.1), which satisfies

f(O) = 0

Lyapunov theory is the fundamental tool for stability analysis of dy­namic systems. The following definitions and theorems are used tocharacterize and study the stability of the origin (see Khalil [46]).

Definition 2.1 (Khalil, 1996, Definition 3.1) The equilibrium pointx = 0 of system (2.1) is

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I. Fantoni et al., Non-linear Control for Underactuated Mechanical Systems

© Springer-Verlag London 2002

12 CHAPTER 2. THEORETICAL PRELIMINARIES

• stable, if for each 10 > 0, there is ~ = ~(E) > 0 such that

IIx(O)11 < s~ IIx(t)11 < 10, Vt ~ 0

• unstable, if not stable

• asymptotically stable, if it is stable and ~ can be chosen such that

Ilx(O)11 < ~ ~ lim x(t) = 0t-e-co

• exponentially stable, if there exist two strictly positive numbersa and A independent of time and initial conditions such that

Ilx(t)1I ~ allx(O)11 exp( -At)

in some ball around the origin.

Vt> 0 (2.2)

•The above definitions correspond to loeal properties of the systemaround the equilibrium point. The above stability concepts becomeglobal when their corresponding conditions are satisfied for any initialstate.

2.1.1 Lyapunov direct method

Let us consider the following definitions.

Definition 2.2 ((Semi-)definiteness) A scalar continuous functionV(x) is said to be locally positive (semi-) definite if V(O) = 0 andV(x) > 0 (V(x) ~ 0) for x =1= O. Similarly, V(x) is said to be negative(semi- )definite if - V(x) is positive (semi- )definite. •

Definition 2.3 (Lyapunov function) V(x) is called a Lyapunovfunction for the system (2.1) if, in a ball B containing the origin, V(x)is positive definite and has continuous partial derivatives, and if itstime derivative along the solutions of (2.1) is negative semi-definite, i.e.V(x) = (öV/öx)f(x) ~ O. •

The following theorems can be used for local and global analysis ofstability, respectively.

Theorem 2.1 (Local stability) The equilibrium point 0 of system(2.1) is (asymptotieally) stable in a ball B if ihere exists a sealar fune­tion V (x) with eontinuous derivatives sueh that V (x) is positive definiteand V(x) is negative semi-definite (negative definite) in the ball B . •

2.1. LYAPUNOV STABILITY 13

Theorem 2.2 (Global stability) The equilibrium point of system(2.1) is globally asymptotically stable if there exists a scalar funetionV (x) with continuous first order derivatives such that V (x) is positivedefinite, V (x) is negative definite and V (x) is radially unbounded, z. e.

V(x) -t 00 as Ilxll -t 00. •

Krasovskii-LaSalle's invariant set theorem

Krasovskii-LaSalle's results extend the stability analysis of the previoustheorems when V is only negative semi-definite. They are stated asfollows.

Definition 2.4 (Invariant set) A set S is an invariant set for a dy­namic system if every trajectory starting in S remains in S . •

Invariant sets include equilibrium points, limit cycles, as well as anytrajectory of an autonomous system.

Theorem 2.3 (LaSalle's invariance principle) (Khalil, 1996,Theorem 3.4) Let n be a compaet (closed and bounded) set with theproperty that every solution of the system (2.1) that starts in n remainsin n for all future time. Let V : n -t JR be a continuously differentiablefunetion such that V(x) ::; 0 in n. Let E be the set of all points in nwhere V(x) = O. Let M be the largest invariant set in E. Then, everysolution starting in n approaches M as t -t 00. •

When V(x) is positive definite, the following two corollaries extendTheorems 2.1 and 2.2.

Corollary 2.1 (Barbashin-LaSalle) (Khalil, 1996, Corollary 3.1)Let x = 0 be an equilibrium point for (2.1) . Let V : D -t JR be acontinuously differentiable positive definite funetion on a neighborhoodD of x = 0, such that V(x) ::; 0 in D . Let S = {x E DjV(x) = O}, andsuppose that no solution can stay forever in S, other than the trivialsolution. Then, the origin is asymptotically stable . •

Corollary 2.2 (Krasovskii-LaSalle) (Khalil, 1996, Corollary 3.2)Let x = 0 be an equilibrium point for (2.1). Let V : JRn -t JR be acontinuously differentiable, radially unbounded, positive definite func­tion such that V(x) ::; 0 for all x E JRn. Let S = {x E JRnIV(x) = O},and suppose that no solution can stay forever in S, other than the trivialsolution. Then, the origin is globally asymptotically stable . •

14 CHAPTER 2. THEORETICAL PRELIMINARIES

When V{x) is negative definite, S = {O}. Then, Corollaries 2.1 and 2.2coincide with Theorems 2.1 and 2.2, respectively.

2.2 Passivity and dissipativity

Consider the non-linear system

x = f (x) + 9 (x) u

Y = h (x) + j (x) u (2.3)

xE JRn, u,y E JRm, i.s.n.: are smooth. f(O) = h{O) = 0 (see Lozano[56]). Let us call w = w{u, y) the supply rate, such that Vu,Vx{O)

l t

Iw(s)1 ds < 00

i.e. locally integrable.

tE JR+

Definition 2.5 (Dissipative system) The system (2.3) is said tobe dissipative if there exists a storage function V (x) > 0 such thatVu,Vx(O)

V(x{t)) - V(x{O)) ~ l t

w{s)ds

The latter is called a dissipation inequality. •It means that the storage energy function V{x{t)) at a future time tis not bigger than the sum of the available storage function V(x{O)) atan initial time 0 plus the total energy J~ w{s )ds supplied to the systemfrom the external sources in the interval [0,t]. There is no internalcreation of energy.

Passive systems represent an important subset of dissipative systems.

Definition 2.6 A system with input u and output y where u(t), y(t) EJRn is passive if there is a constant ß such that

(2.4)

2.3. STABILIZATION 15

for all functions u and all T ~ O. If, in addition, there are constants8 ~ 0 and E ~ 0 such that

for all functions u, and all T ~ 0, then the system is input strictlypassive if 8 > 0, output strictly passive if E > 0, and very strictlypassive if 8 > 0 and E > 0 •

Obviously ß ~ 0 as the inequality is to be valid for all functionsu and in particular the control u(t) = 0 for all t ~ 0 , which gives0= foT yT(t)u(t)dt ~ ß.

Theorem 2.4 Assume that there is a continuous function V(t) ~ 0such that

V(T) - V(O) ~ l Ty(tfu(t)dt (2.6)

for all functions u, for all T ~ 0 and all V(O). Then, the system withinput u(t) and output y(t) is passive. Assume, in addition, that thereare constants 8 ~ 0 and E ~ 0 such that

V(T) - V(O) ~ lTyT(t)u(t)dt - 81T

uT(t)u(t)dt - ElTyT(t)y(t)dt

(2.7)

for all functions u, for all T ~ 0 and all V(O). Then, the system isinput strictly passive if 8 > 0, output strictly passive if E > 0, and verystrictly passive if 8 > 0 and E > 0 such that the inequality holds . •

2.3 Stabilization

The following theorem gives a necessary condition for the existence ofa continuously differentiable control law for non-linear systems. It waspresented by Brockett [13J (1983) for Cl pure-st äte feedback laws.

Theorem 2.5 Let x = f(x, u) be given with f(xo,O) = 0 and f( .,.)continuously differentiable in a neighborhood of (xo ,O). A necessarycondition for the existence of a continuously differentiable control lawwhich makes (xo ,O) asymptotically stable is that :

16 CHAPTER 2. THEORETICAL PRELIMINARIES

(i) The linearized system should have no uncontrollable modes asso­ciated with eigenvalues whose real part is positive.

(ii) There exists a ne ighborhood N of (xQ,0) such that for each ( E Nthere exists a control u( (.) defined on [0, 00 [ such that this controlsteers the solution of x = f(x, U() from x = ( at t = 0 to x = XQat t = 00 .

(iii) The mapping 'Y : A X JRrn -t JRn defined by'Y : (x, u) H f(x, u)should be onto an open set containing O. •

The first condition refers to the rank condition of a linear control sys­tem. Note that in the linear case, the rank condition is necessary andsufficient for a linear system x = Ax + Bu to be controllable and toprovide the existence of a continuously differentiable control law for thelinear system.

The second condition refers to the controllability property in thenonlinear case. On the other hand, this condition is not sufficient sincewe want a control law with some smoothness. In general, we needsomething more than just a controllability condition. Therefore, weneed to introduce condition (iii), which corresponds to the necessarycondition of this theorem.

The third condition means that the mapping should be locally sur­jective or that the image of the mapping (x, u) H f(x, u), for x and uarbitrarily elose to 0, should contain a neighborhood of the origin.

To make this more elear, let us consider the following example:

x u = Cl

y v = C2

i yu - xv = C3

The question to ask is if there exists a continuous control law (u, v) =

(u(x , y , z) , v(x , y, z)) that makes the origin asymptotically stable for theabove system. The third condition of Brockett's theorem means thatthe system equations should contain a solution (x,y,z,u ,v) for everyci( i = 1,2,3) in a neighborhood of the origin. This is not the case here ,since the system does not have a solution for C3 -# 0 and Cl = 0, cl = O.

Contrary to the above, the following example satisfies the third con­dition

x u = cl

Y V = C2

z xy = C3

2.4. NON-HOLONOMIC SYSTEMS 17

Therefore, for this particular system there exists a continuous controllaw, which makes the origin asymptotically stable.

2.4 Non-holonomic systems

Definition 2.7 (Holonomic systems) (Goldstein [33]) Consider asystem of generalized coordinates q

ij = f(q,q,u) (2.8)

where f (.) is the vector field representing the dynamics and u is a vectorof external generalized inputs. Suppose that some constraints limit themotion of the system. If the conditions of constraint can be expressedas equations connecting the coordinates (and possibly the time) havingthe form

h(q,t) = 0 (2.9)

then the constraints are said to be holonomic. This type of constraintis a so-called holonomic constraint, since it can be integrated. •

A simple and suitable example illustrates weIl the concept of holo­nomic systems and was proposed by Lefeber [50J . Let us consider thesystem

(2.10)

where (Xl, X2) is the state and u is the input. This system contains aconstraint on the velocities as follows

XIXI + X2X2 = 0

Since this constraint can be integrated to obtain

1 2 1 22"XI + 2"X2 = C

(2.11)

(2.12)

where c is a constant, the constraint (2.11) is called a holonomic con­straint.

Definition 2.8 (Non-holonomic systems) (Goldstein [33]). On theother hand, when it is not possible to reduce them furt her by meansof equations of constraint of the form (2.9), they are then called non­holonomic. With non-holonomic systems, the generalized coordinatesare not independent of each other. •

18 CHAPTER 2. THEORETICAL PRELIMINARIES

Let us consider the kinematic equations of a vehicle

x = vcosO

y vsinO (2.13)

The constraint on the velocities of the model is given by

Xsin 0 - iJ cos 0 = 0 (2.14)

However, contrary to (2.11), the constraint (2.14) cannot be integrated,i.e. the constraint (2.14) cannot be written as a time derivative of somefunction of the state. It is called a non- holonomic constraint.

2.5 Underactuated systems

In this book, we will simply define an underactuated system as one hav­ing less control inputs than degrees of freedom. The precise definitionis given below. In some underactuated systems, the lack of actuationon certain directions can be interpreted as constraints on the accelera­tion. This is in fact the case for the underactuated hovercraft, treatedin Chapter 11, for which the lateral acceleration is nil.

Definition 2.9 (Underaetuated systems) Consider systems thatcan be written as

ij = f(q , q) + G(q)u (2.15)

where q is the state vector of independent generalized coordinates, f( .)is the vector field representing the dynamics of the systems, q is thegeneralized velocity vector, G is the input matrix, and u is a vector ofgeneralized force inputs. The dimension of q is defined as the degrees offreedom of (2.15). System (2.15) is said to be underactuated ifthe exter­nal generalized forces are not able to command instantaneous accelera­tions in all directions in the configuration space, i.e, rank(G) < dim(q).

•This definition is connected to the one used by Oriolo and Nakamura[80], which says that underactuated systems are systems with fewerindependent cont rol actuators than degrees of freedom to be controlled.

Several examples of underactuated systems will be considered in thisbook. In order to illustrate the above definition, let us consider themodel ofthe pendubot system (see Figure 5.1), which will be developed

2.6. HOMOCLINIC ORBIT 19

in Chapter 5. The dynamic equations of the system in standard formare given by

where

D(q)ij + C(q, q)q + g(q) = 7 (2.16)

(2.18)

and 7 = [ ~ ] (2.19)

In this system, there is only one actuator acting on the first link (i.e.the angle qt}, while the second link (Le. the angle q2) is free. Indeed,in the vector 7, there is only one term 71 on the first line. Therefore,this system is underactuated since it has two degrees of freedom withonly one actuator.

Remark 2.1 Let us note that for the case of the mobile robot in kine­matic equations (2.13) the generalized coordinates are (x,y,8) and areof three-dimensional while the system has two control inputs (the for­ward acceleration and the angular momentum) . Tlius , the mobile robotis an underactuated system in view of Definition 2.9. Actually as shownin equation (2.14), the generalized coordinate components are not inde ­pendent. This comes from the fact that the system cannot move later­ally. Furthermore. an actuator to move the system laterally is besidethe point. Note, houieuer, that in this case there are only two degrees offreedom to be controlled: the forward (or backward) displacement andthe angular position. In some sense, the system is fully aetuated. Thisshows the lim itations of Definition 2.9. •

2.6 Homoclinic orbit

The not ions and examples developed in this seetion are related to thebook of Jackson [44] . Let us consider autonomous systems of the form

x = F(x;c) xE JRn (autonomous) (2.20)

20 CHAPTER 2. THEORETICAL PRELIMINARIES

The solutions of (2.20) involving different initial conditions generate thelamily 01 oriented phase curves in the phase space, which is called thephase portrait of the system. The phase portrait consists of orientedcurves through all points ofthe phase space, where the functions F(x; c)in (2.20) are defined. These curves are called trajectories or orbits.

One important example is the limit cycle, which is a closed (periodic)orbit, whose neighboring orbits tend asymptotically towards (or away)from it .

Definition 2.10 (Homoclinic orbit) A homoclinic orbit is a singleorbit where a stable manifold and an unstable manifold intersect. Thisorbit leaves the saddle point in one direction, and returns in anotherdirection. It converges to the same saddle point. •

The example of a ball rolling without friction on a curved surfacewith maxima of different heights is a simple example of a homoclinicorbit. In this example, there is a single orbit that converges to the samesaddle point both when t -t +00 and when t -t -00. This situationis illustrated in Figure 2.1, where the height of the surface is shown in(a).

z

..... \

x

(a)

xHomoclinicorbit

x

(b)

Figure 2.1: Example of a homoclinic orbit