Investigaciones sobre colaboración entre maestros 52

Throughout this work, based upon a proposed estimation or control law, we seek to describe the qualitative behavior of the system equilibrium solutions. We say that an equilibrium point is stable if for initial conditions sufficiently close to the equilibrium point, the system states remain in a neighborhood of the equilibrium point for all time. An equilibrium point is said to be asymptotically stable if the system states remain in a neighborhood of the equilibrium point, and converge to the equilibrium point for all initial conditions contained within a set defined as the domain of attraction. In the case where the domain of attraction is the entire space, the equilibrium point is said to be globally asymptotically stable. In the case where the domain of attraction is the entire set except for a subset of Lebesgue measure zero, the equilibrium point is said to be almost globally asymptotically stable. If the domain of attraction can be arbitrarily increased to contain any subset of the entire space, the equilibrium point is said to be semi-globally stable. For more formal stability definitions the reader is referred to Khalil (2002).

Throughout the thesis, we study the performance of closed loop systems (using the proposed estimation and control laws) using Lyapunov based techniques. In general, let x ∈ Rn denote the state of a given system, and D ⊂ Rn denote a subset of the space. We assume thatxis continuously differentiable, where in general

˙

x=f(t, x) where we assume the functionf(t, x) is Lipschitz inD. We denoteV(t, x) :

D → R as a continuously differentiable positive-definite function on the domain D

which contains the origin x = 0, which is referred to as a Lyapunov function. Also, in many cases we may denote V := V(t) := V(t, x), where the arguments t and x

may be intentionally removed, unless they are specifically required. Using Lyapunov functions, in general we analyze the stability of the particular equilibrium pointx= 0

Chapter 2: Background 36

by differentiating V with respect to time. To identify this operation we often denote the time-derivative of a function using the dot notation, for example ˙x := dx/dt

and ¨x := d2x/dt2. Furthermore, when we require derivatives of a higher-order, we use the notation x(n) := dnx/dtn. Fortunately, there are a variety of mathematical tools which are available in the literature which assist us in the stability analysis of equilibrium solutions. One of these tools, is known as Barbalat’s Lemma, which is restated below for convenience.

Lemma 1(Barbalat’s Lemma Khalil (2002), page 323).Letϕ:R→Rbe a uniformly

continuous function on [0,∞). Suppose that limt→∞ ∫_t

0ϕ(τ)dτ exists and is finite.

Then ϕ(t)→0 as t → ∞.

In most cases it can be very difficult to show the time-derivative of Lyapunov functions are negative definite, especially in the case of non-autonomous systems. In these cases, one cannot use invariance set theorems (such as Lasalle’s theorem, Khalil (2002)), and therefore one often relies on the use of Barbalat’s Lemma. In particular, we make frequent use of the following Lemma (which is actually a corollary of Barbalat’s Lemma) which involves the study of non-autonomous systems.

Lemma 2 (Lyapunov-Like Lemma, Slotine and Li (1991), page 125). If a scalar

function V(t, x) satisfies the following conditions

• V(t, x) is lower bounded,

• _V˙_{(t, x) is negative semi-definite,}

• _V˙_{(t, x) is uniformly continuous in time,}

In some situations it may not be straightforward to show ˙V(t, x) is uniformly continuous. In these situations, a sufficient (yet conservative) condition for the uni- form continuity of ˙V(t, x) is to show that ¨V(t, x) is uniformly bounded.

In the case where a system has multiple equilibria, it is sometimes helpful to show that a particular undesired equilibrium point is unstable. One useful Lemma which can be used to demonstrate the instability of a particular equilibrium point is known as Chetaev’s Theorem, which is also restated for convenience.

Theorem 2.1 (Chetaev’s Theorem Khalil (2002)). Letx= 0 be an equilibrium point

for x˙ =f(x). Let Vc :D→ R be a continuously differentiable function on a domain

D ⊂Rn that contains the origin x= 0, such that Vc(0) = 0 and for any ϵ >0 there exists x₀ ∈ B(ϵ,0)∈ D such that Vc(x0)>0. Let Br = {x∈ Rn | ∥x∥ ≤r} denote a ball of radius r >0 and define the set U ={x∈Br | Vc(x)>0}, and suppose that

˙

38

Chapter 3

Attitude Reconstruction and Estimation

In this chapter we explore the challenge of determining the orientation of a rigid-body. We recall the inertial frameI and body frameB (defined in Section 2.1.1), where our primary objective is to determine the rotation matrix R ∈SO(3) or unit-quaternion

Q∈Q which describes the orientation ofB.

To solve our problem we make use of vector-measurements, which refers to the body-frame measurements of vectors which are known in the inertial frame. In some situations, we also require the knowledge of the system angular velocity (measured using a gyroscope which is rigidly attached to B).

In Section 3.1 we describe the so-called attitude reconstruction algorithms, which seek to recover a closed-form solution for the attitude of a rigid-body, without the use of an observer or filter. In the case where the rigid-body is rotating, other estimation schemes have been proposed which combine these attitude reconstructions with the gyroscope measurement, and are known as complementary filters. In Section 3.2, one example of a nonlinear complementary filter is discussed.

Other observers have eliminated the requirement of the reconstruction algo- rithms by applying the vector measurements directly to the estimation laws, which is described in Section 3.3.3. Also, in Section 3.3.4 we describe a similar vector- measurement based observer (that does not require the attitude reconstructions) which also considers low-pass filtering of the vector measurements, and therefore

may be better suited in the case where the measurements are affected by noise or other disturbances.

For the type of observers listed above, the vector measurements are assumed to be constant and known in the inertial frame. However, in many practical situations, two vector measurements most commonly used are obtained using a magnetometer and accelerometer which are rigidly attached to B. The magnetometer is used to measure the ambient magnetic field which is assumed to be known in I, and the accelerometer is used to measure the apparent acceleration of B. However, since the apparent acceleration is not known in I, the previous attitude observer is no longer applicable. This problem is addressed by using an additional filter which uses linear-velocity measurements which are obtained using a GPS. The filtered version of the system velocity can be used to obtain information about the (unknown) system apparent acceleration, and can therefore be used to aid in the estimation of the rigid- body attitude. Two observers of this type are proposed in Section 3.3.5.