Switched Finite Time Control of a Class of Underactuated Systems

Lecture Notes in Control and Information Sciences Editors: M. Thoma · M. Morari

333

Ravi N. Banavar Velupillai Sankaranarayanan

Switched Finite Time Control of a Class of Underactuated Systems With 51 Figures

Series Advisory Board

F. Allg¨ower · P. Fleming · P. Kokotovic · A.B. Kurzhanski · H. Kwakernaak · A. Rantzer · J.N. Tsitsiklis

Authors Dr. Ravi N. Banavar Dr. Velupillai Sankaranarayanan Indian Institute of Technology, Systems and Control Engineering 101 A, ACRE Building IIT Powai Mumbai 400 076 India [email protected] [email protected]

ISSN 0170-8643 ISBN-10 3-540-32799-1 Springer Berlin Heidelberg New York ISBN-13 978-3-540-32799-8 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006923562 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data conversion by authors. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany (www.ptp-berlin.com) Cover-Design: design & production GmbH, Heidelberg Printed on acid-free paper 89/3141/Yu - 5 4 3 2 1 0

Preface

The first author’s venture into the field of nonlinear control and nonholonomic mechanics began nearly 7 years ago with the supervision of the doctoral thesis of Arun Mahindrakar. Since then the subject has continued to hold fascination; we have waded ankle deep into these exciting waters and they are most inviting. The work of many authors has sustained our enthusiasm - notable amongst these are Jerrold Marsden, Richard Murray and Roger Brockett. We are yet some distance away from the elegant geometric approach of these authors, but we believe this to be a modest beginning to our contributions to this challenging field. During these years we have benefitted from a number of discussions with students and colleagues. Sanjay Bhat’s painstaking criticisms have been a great help to better our understanding. Kurien Isaac and Harish Pillai gave a number of suggestions to improve this work. The numerous informal lectures that we organized amongst ourselves - Arun Mahindrakar, Narayan Manjarekar, Rupesh Patayane, Faruk Kazi, Amar Banerji, Vrunda Joshi, Rakesh Singhal and the many other VJTI students inspired by Navdeep Singh - kept the boat moving ahead. The first author gratefully acknowledges the constant support of his parents in all endeavours. The distractions of differential geometry and the absent-minded countenance at the dinner table of the past years were patiently borne by Suparna and Anoushka. The second author wishes to acknowledge the support of his parents - S. Velupillai and V. Amirthajothi - and his wife R. Bharathisundaram. IIT-Bombay, Dec 2005

Ravi N. Banavar V. Sankaranarayanan

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Underactuated Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Linearly Controllable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Not Linearly Controllable . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contributions of This Monograph . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of the Monograph . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 3 4 4 6

2

Mathematical Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nonholonomic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Relative Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Finite-Time Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Finite-Time Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Nonlinear Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Stabilizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 12 14 17 18 18 20

3

A Switched Finite-Time Controller Design Methodology . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Controller Design Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Stabilization of the NI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Stabilization of the ENDI – Wheeled Mobile Robot . . . . . . . . . . 3.4.1 Transformation to the ENDI . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 22 25 25 27 29 31 32 34

VIII

Contents

3.5 Stabilization of an Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . 3.5.1 Feedback Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 41 42 46

4

Alternative Control Strategies for the NI . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Method 1 – Exponential Convergence . . . . . . . . . . . . . . . . . . . . . . 4.3 Method 2 – Bounded Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Method 3 – Time-Varying Control . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 53 54 55 56

5

Switched Stabilization of a Hovercraft . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Circular Shaped Hovercraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Model Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Stabilization of the Velocity . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Stabilization of the Position . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Conventional Type Hovercraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 61 63 63 65 68 68 73 74

6

Output Feedback Stabilization of a Mobile Robot . . . . . . . . . 6.1 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79 79 81 83

A

Examples A.0.1 A.0.2 A.0.3 A.0.4

87 87 90 91 92

and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the NI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of the ENDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of an Underwater Vehicle / Surface Vessel . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Abstract

Underactuated mechanical systems have lesser number of control inputs than their degrees-of-freedom. Examples of such systems include underwater vehicles, surface vessels, mobile robots, satellites and robotic manipulators. This monograph addresses the problem of stabilizing a class of underactuated systems which are not linearly controllable. Systems with drift and without drift are considered. The main contribution is a switched finite-time control algorithm to stabilize this class of systems. The control law ensures global attractivity and relative stability. The notion of relative stability is weaker than conventional Lyapunov stability and is based on the properties of the trajectories of the closed-loop system that originate in a set of the phase-space. The philosophy of the control strategy is to define a set of surfaces whose intersection is a connected set and contains the equilibrium. The control objective then is to move the trajectory from anywhere in the state-space to this set in finite time. The theory of continuous finite-time differential equations is used to ensure that the system reaches this set in finite time and without chattering. The closedloop system is then rendered asymptotically stable in this set. To demonstrate the applicability of our technique, we consider the following three well-studied examples - the first is a drift-less system, while the second and third system possess drift. • Nonholonomic Integrator (NI) - This is a benchmark example of a nonholonomic, underactuated system which mimics the kinematic model of a mobile robot. • Extended Nonholonomic Double Integrator (ENDI) - While the NI mimics the kinematic model, the ENDI mimics the dynamic model of a mobile

X

Abstract

robot. The system is underactuated and has a velocity level nonholonomic constraint. • Underwater vehicle (Surface vessel) - This is an example of an underactuated system with a second order nonholonomic constraint at the acceleration level. Other than the technique mentioned above, we also present a few other techniques to stabilize such systems and an output feedback scheme for one application. Keywords: Underactuated systems, underwater vehicle, nonholonomic integrator, mobile robot, finite-time.

1 Introduction

A large number of mechanical systems have non-integrable constraints. These constraints could be either at the velocity level or the acceleration level. Nonintegrability implies that the dimension of the manifold on which the system evolves is not reduced due to the constraint. This in turn implies that the system can assume any arbitrary configuration in the configuration space inspite of the constraints. Velocity level constraints preclude instantaneous velocities in certain directions of the system. In a similar vein, acceleration level constraints prevent arbitrary accelerations of the system. In this monograph, we shall call all such systems with non-integrable constraints as nonholonomic systems [73]. Though, in a strict sense, the terminology of nonholonomy applies only for non-integrable constraints at the velocity level, we shall classify acceleration level constraints as well in the class of nonholonomic systems. A few instances where nonholonomy arises in mechanical systems are 1. Conservation of angular momentum (free floating multi-body system with no external torque) 2. No slip constraints on rolling (wheeled mobile robot) 3. Underactuation (underwater vehicle / surface vessel) An important result by Brockett holds for many of these systems [15]. This result states that there exists no C 1 state-feedback control law that can asymptotically stabilize the equilibria of these systems. To overcome this, a number of control strategies ( time-varying, discontinuous and hybrid) have been proposed by researchers. For a detailed survey see [12, 37]. It is interesting to note that the main effort by many researchers has been to convert nonholonomic systems into a canonical form or a power form using a coordinate transformation as this facilitates controller design.

R.N. Banavar and V. Sankaranarayanan: Switch. Fin. Time Contrl., LNCIS 333, pp. 1–6, 2006. © Springer-Verlag Berlin Heidelberg 2006

2

1 Introduction

Explicit design of a time-varying controller for a drift-less nonholonomic system is presented by Pomet in [54]. Subsequent to this a number of timevarying controllers have been designed to stabilize nonholonomic systems from the aspect of asymptotic and exponential stability [59, 64, 25]. Even though many such time-varying techniques have been presented in the literature, they suffer from low rate of convergence and oscillating trajectories. Stabilization of nonholonomic systems using discontinuous controllers has been studied by many authors as well [14, 5, 6, 44, 38, 67, 48]. A general procedure for constructing piecewise controllers for nonholonomic systems is presented in [14] with the example of the knife-edge and rolling coin. Sufficient conditions for the stability of a discontinuous nonholonomic systems is presented in [5]. The use of a non-smooth coordinate transformation to overcome the obstruction of stability in Brockett’s theorem is presented by Astolfi in [6]. Later, this rational coordinate transformation using the σ- process, is used to design discontinuous controllers for higher order chained systems systems in [38]. A discontinuous controller is developed for a nonholonomic system in power form based on constructing a series of nested invariant manifolds in [44]. A non-smooth version of a non regular feedback controller is proposed in [67], which is then applied to a nonholonomic chained system to achieve exponential stabilization. Control of higher order chained systems using discontinuous feedback controllers, possibly non-affine in control, is presented in [41]. The σ-process and a dilation are combined to design a discontinuous control in [48]. A switched robust controller is proposed for local robust stabilization of a nonholonomic system in [72]. Stabilization of the nonholonomic integrator using a sliding mode controller is presented by Bloch and co-workers in [13]. In this technique, the closed-loop trajectory enters a manifold in finite-time in which the system is asymptotically stable. A variable structure controller is developed for chained systems based on the extended nonholonomic double integrator in [76] in which a rational transformation and the reaching law method [24] are used to synthesize the control algorithm. Higher-order sliding mode stabilization of nonholonomic systems is presented in [30]. Recently, a time-varying sliding mode technique is proposed for stabilizing uncertain nonholonomic systems in [23]. A logic based switched control scheme is presented for the nonholonomic integrator and is then applied to the kinematic model of the mobile robot with uncertainty in [28]. Tracking of the nonholonomic integrator using a field oriented control method, mainly used for controlling induction motors,

1.1 Underactuated Systems

3

has been presented in [19]. A sliding mode controller has been proposed for tracking of the nonholonomic integrator in [13]. We now focus on underactuated systems.

1.1 Underactuated Systems Underactuation may arise in mechanical systems due to one or more of the following reasons. • Nature of the system • On purpose to reduce actuator cost • Failure of one or more actuators In contrast to controller design for fully actuated systems which is often based on properties such as linear controllability, feedback linearizability and passivity, controller design for underactuated systems is challenging due to the loss of one or more of these properties. Broadly, underactuated systems can be classified into two categories • Those that are linearly controllable • Those that are not linearly controllable 1.1.1 Linearly Controllable Most underactuated systems with potential energy are linearly controllable. The potential field (like gravity and spring stiffness) acts like an additional actuator and imparts linear controllability. A few examples in this class are the pendulum on a cart (POC), the acrobot and the pendubot [65]. Since linear controllability guarantees only local stabilization in a neighborhood of the equilibrium point, it is necessary to design nonlinear controllers to achieve global properties such as swing up of the acrobot and point-to-point control in the POC. Control of these systems using various techniques can be found in [21, 46, 65, 66, 22, 51]. The POC, a benchmark example in this category, has been studied in [21], in which an energy based swing-up algorithm is proposed. Swing-up of the acrobot, a two link manipulator with the second joint actuated, is studied in [46, 65]. Here the control problem is to swing-up the acrobot from any position to its upright position and stabilize it around that point. This is

4

1 Introduction

achieved through an energy based method in [65] and a Lyapunov based technique in [46]. Control of the pendubot, a two link manipulator with the first joint actuated, is dealt in [22, 66] using various control methods. Passivity based methods have also been proposed to stabilize this class of underactuated systems. A general methodology named Interconnection and Damping Assignment - Passivity Based Control (IDA - PBC) has been proposed by Ortega and co-workers in [51]. 1.1.2 Not Linearly Controllable Mechanical systems such as wheeled mobile robots, planar manipulators and underwater vehicles which operate in a horizontal plane (without gravity) fall under this category. Brockett’s result holds for these systems as well. We now briefly summarize the previous work in this area. Reyhanoglu and co-workers [57] have developed a discontinuous controller for stabilizing this class of systems which guarantees global attractivity of the origin. Planar underactuated manipulators and their properties are studied in [42, 3, 32, 43, 47] and a discontinuous controller is designed through a rational coordinate transformation in [47]. Control of underwater vehicles/ surface vessels using various techniques such as discontinuous and time varying controllers are presented in [56, 52, 53, 49, 62, 61]. Time varying controllers for stabilizing a surface vessel have been proposed in [52, 53, 49] from different aspects such as exponential stability, state feedback and output feedback. Stabilization of a surface vessel using a discontinuous control law has been studied by Reyhanoglu [56], in which almost exponential stability is achieved.

1.2 Contributions of This Monograph The main contribution of this monograph is a switched finite-time controller design methodology that is applied to stabilize a class of underactuated systems which are not linearly controllable. The main idea in this approach is to define a set of surfaces whose intersection is a connected set and contains the equilibrium (the origin without loss of generality). Then the controller is designed such that • The closed-loop trajectory reaches the intersection set in finite-time. • The origin of the closed-loop system is asymptotically stable in the intersection set.

1.2 Contributions of This Monograph

5

We use the theory of continuous finite-time differential equations to reach each surface in finite-time. The controllers designed using this technique avoid “chattering” that may result if conventional sliding mode type controllers are employed. The proposed methodology is then applied to the following three wellstudied underactuated systems - the first is a drift-less system, while the second and third system possess drift. 1. Nonholonomic integrator (NI) - This mimics the kinematic model of a mobile robot with three degrees of freedom, one no-slip constraint and two actuators. The control inputs here are at the velocity level. 2. Extended nonholonomic double integrator (ENDI) - This mimics the dynamic model of the mobile robot described earlier. The control inputs here however, are force and torque. This is a benchmark example of an underactuated system with a velocity level nonholonomic constraint. 3. Underwater vehicle / surface vessel - Here the system has three degrees of freedom, two actuators and an acceleration level constraint due to underactuation. In the examples that we present, we adopt the following procedure to select the surfaces. The first surface is selected such that the unactuated dynamics are stable on the corresponding surface. The remaining surfaces are selected such that they aid the trajectory to reach the first sliding surface. We employ the notion of relative stability and extend this to relative asymptotic stability and relative exponential stability. Switching between controllers in certain region of the state space is essential in some examples. We make a few preliminary remarks on our strategies for each of the examples. • Stabilization of the NI The controller designed here ensures exponential convergence of the states to the origin if the initial conditions lie in an open dense set. This kind of property is named almost exponential stability which is a special case of relative stability. • Stabilization of the ENDI The switched control law designed here ensures global attractivity and relative exponential stability of the origin. • Stabilization of an underwater vehicle This is once again a switched control law that ensures global attractivity and relative asymptotically stability of the origin.

6

1 Introduction

Other than the main technique mentioned above, we have presented three alternative control strategies to stabilize the NI. These strategies are partially motivated by the concern for bounded control magnitudes. We also present a control strategy for a symmetrical hovercraft model based on our proposed control philosophy and another switched control law for a more conventional model of the hovercraft. Finally we demonstrate an output feedback scheme for one application.

1.3 Organization of the Monograph This section gives an overview of the chapters in this monograph. Chapter 2 This chapter presents a few mathematical preliminaries on stability, finitetime differential equations and stabilizability. Chapter 3 This chapter presents the main controller design methodology. This methodology is then applied to three examples which includes the NI, the ENDI and the underwater vehicle. This is followed by a detailed comparison between the existing controllers with simulation results. Chapter 4 In this chapter, three alternative control methods are presented to stabilize the NI. Amongst these three methods, two address the problem of large control magnitudes. Comparison between all the four methods including the ones presented in chapter 3 is also made. Chapter 5 The initial part of this chapter presents application of the proposed control philosophy to a symmetrical model of a hovercraft. Later, another switched control strategy is presented for a more conventional and comprehensive model of the hovercraft. Chapter 6 This chapter presents an output feedback scheme to stabilize the wheeled mobile robot. Appendix A Sufficient conditions to check different nonlinear controllability notions are presented. Proofs of several important properties of the NI, the ENDI and underwater vehicle are also presented.

2 Mathematical Preliminaries

This chapter presents a few definitions and notions that are used throughout the monograph.

2.1 Nonholonomic Systems Consider a mechanical system described by (2.1)

q¨ = f (qq , q˙ , u )

where q denotes the generalized coordinates (sometimes called configuration variables ) and takes values in an n-dimensional smooth manifold M called the configuration manifold, f (.) represents the dynamics and u is an mdimensional vector of generalized inputs. Suppose the system is subject to a constraint of the form (2.2)

s(qq , q˙ , q¨) = 0

If the following equivalence holds (2.3)

s(qq , q˙ , q¨) = 0

≡

h(qq ) = 0

then the constraint satisfies the complete integrability property and is called a holonomic constraint and the system is called a holonomic system . If the constraint cannot be expressed as in (2.3), then the system is called a nonholonomic system . We present a few examples.

R.N. Banavar and V. Sankaranarayanan: Switch. Fin. Time Contrl., LNCIS 333, pp. 7–20, 2006. © Springer-Verlag Berlin Heidelberg 2006

8

2 Mathematical Preliminaries

Velocity level constraints Rolling coin The figure(2.1) shows a vertical coin of radius r on a horizontal plane that is allowed to move by pure rolling and steering alone. The configuration variables are q = (x, y, θ, φ) ∈ M = IR1 × IR1 × S 1 × S 1 . The constraints of motion are expre3ssed as x˙ sin θ − y˙ cos θ = 0

No lateral motion

x˙ cos θ + y˙ sin θ = rφ˙

Pure rolling

These constraints can be expressed in a matrix form as   x˙  sin(θ) − cos(θ) 0 0  y˙    =0 cos θ sin θ 0 −r  θ˙  φ˙ This matrix representation of the constraint conveys the fact that permissible motions of the coin are such that allowable velocity directions at any point on the manifold (xp , yp , θp , φp )- described by the vector   x˙  y˙     ˙ θ φ˙ (xp ,yp ,θp ,φp )

are annihilated by the matrix sin(θp ) − cos(θp ) 0 0 cos θp sin θp 0 −r y

φ

θ

Fig. 2.1. Rolling coin

x

2.1 Nonholonomic Systems

9

To explore integrability, we ask the question: Can Ω(qq ) =

sin(θ) − cos(θ) 0 0 cos θ sin θ 0 −r

be expressed as the gradient of two functions λ1 and λ2 with λi (·) : q → IR1 such that   x˙ d(λ1 )  y˙    dt Ω  ˙  = d(λ =0 2) θ dt φ˙ If yes, then λ1 (qq ) = c1

λ2 (qq ) = c2

where c1 and c2 are constants

and this implies that the system evolves on a manifold of reduced dimension (in this case reduced by 2) and the constraints are integrable (or holonomic). However, in this particular case, it is not possible to find such functions λ i s and the constraints are hence non-integrable. We arrive at this conclusive negative answer by employing a result from differential geometry called Frobenius’ theorem [33]. Before we state this, we recast the problem in differential geometric parlance. The constraints can be viewed in terms of annihilator codistributions as   x˙  sin(θ) − cos(θ) 0 0   y˙   ˙=0 cos θ sin θ 0 −r  θ  φ˙ Permissible motions of the coin are such that the vector field is annihilated by the codistribution Ω=

sin(θ) − cos(θ) 0 0 cos θ sin θ 0 −r

To explore integrability, we ask the question: Can Ω be expressed as the gradient of two functions λ1 and λ2 with λi : q → IR1 such that Ω=

dλ1 dλ2

?

If yes, then the constraints are integrable (or holonomic). Frobenius’ result states

10

2 Mathematical Preliminaries

Theorem 2.1. (Frobenius’ theorem) A distribution is integrable if and only if it is involutive. The codistribution in this example is not involutive and hence the constraint is not integrable. Bead in a slot Consider a bead that is restricted to move in a slot on a horizontal plane as shown in figure (2.2). The configuration variables of the bead are (x, y) and the restriction to stay in the slot gives rise to an algebraic constraint of the form g(x, y) = 0 Here g is the equation of the curve describing the slot. The slot thus restricts the possible configurations that the bead can assume. Hence the constraint reduces the dimension of the configuration manifold from 2 to 1. This is a holonomic constraint. While the rolling coin represented a system with nonholonomic constraints at the velocity level, we now explore systems with constraints at the acceleration level. Definition 2.2. (Underactuated mechanical systems) Consider a mechanical system described by (2.1) The system is said to be underactuated if m < n. Acceleration level constraints The Acrobot The acrobot shown in figure (2.3) depicts a two link manipulator moving in a vertical plane with one actuator at its second joint. The equations of the

Fig. 2.2. Bead in a slot

2.1 Nonholonomic Systems

11

motion are of the form d11 (q)q¨1 + d12 (q)q¨2 + h1 (q, q) ˙ + ψ1 (q) = 0 (2.4)

d21 (q)q¨1 + d22 (q)q¨2 + h2 (q, q) ˙ + ψ2 (q) = τ2

where the first equation (with the right hand side being zero) denotes the lack of actuation at the first joint and is a non-integrable acceleration level constraint. τ2 is the torque applied at the second joint. The acrobot can assume any configuration but cannot assume arbitrary accelerations. m1, m

2 l1 , l2 I1 , I 2

= link lengths

= link moments of inertia lc1 , lc2 = centers of masses

link 2

g

= link masses

l c2

q

2 Actuator

l c1

link 1 q 1

Fig. 2.3. Two link underactuated manipulator

Fuel slosh in a launch vehicle Figure (2.4) depicts a launch vehicle with liquid fuel in its tank. The dynamic model of the fuel takes the form of an unactuated pivoted pendulum model. The motion of the pendulum is solely affected by the motion of the outer rigid body. The equations of motion are of the form q¨u + f1 (q, q, ˙ q¨) = 0 (2.5)

q¨a + f2 (q, q, ˙ q¨) = F

12

2 Mathematical Preliminaries

fuel tank

α

δ

Fig. 2.4. Launch vehicle schematic

where q = (qu , qa ), qu corresponds to the configuration variable of the pendulum, qa corresponds to the configuration variable of the outer rigid body and F is the external force. While the acceleration level constraint in the acrobot arises due to purpose of design or loss of actuation, in the case of the launch vehicle it is the inability to directly actuate the fluid dynamics.

2.2 Stability Consider an autonomous system described by (2.6)

x) x˙ = f (x

where x ∈ IRn , f : D −→ IRn is a local map from an open set D ⊂ IRn into xe ) = 0. Without loss of IRn . Let x e ∈ D be an equilibrium point, that is, f (x generality, let us assume x e = 0. Definition 2.3. (Lyapunov stability) The equilibrium point x e = 0 is said to be

2.2 Stability

• stable if, for every

13

> 0, there exists a δ = δ( ) > 0 such that x(0)|| < δ =⇒ ||x x(t)|| < ||x

∀t≥0

• unstable if it is not stable • asymptotically stable if it is stable and δ can be chosen such that x(0)|| < δ =⇒ lim x (t) = 0 ||x t→∞

Definition 2.4. (Exponential stability) The equilibrium point x e = 0 is exponentially stable if there exist positive constants δ, k and λ such that x(t)|| ≤ k||x x(0)||e−λ t, ∀ ||x x(0)|| ≤ δ ∀t ≥ 0 ||x We now state the Lyapunov theorem for stability. Theorem 2.5. (Lyapunov theorem ) Let V : D −→ IR be a continuously differentiable function such that x) > 0 in D \ {0} V (0) = 0 and V (x The equilibrium point x e = 0 is said to be • stable if

x) ≤ 0 ∀ x ∈ D V˙ (x

• asymptotically stable if x) < 0 ∀ x ∈ D \ {0} V˙ (x The above definitions and theorems are valid in a neighbourhood D around an equilibrium point. The following theorem concerns global asymptotic stability. Theorem 2.6. Let V : IRn −→ IR be a continuously differentiable function x) > 0, ∀ x = 0 V (0) = 0 and V (x x) is radially unbounded (||x x|| −→ ∞ =⇒ V (x x) −→ ∞ ) and If V (x x) < 0, ∀ x = 0 V˙ (x then the equilibrium point x e = 0 is globally asymptotically stable. Definition 2.7. (Invariant set) A set S in the phase space is said to be invariant with respect to (2.6) if x(0) ∈ S ⇒ x(t) ∈ S ∀t ∈ IR

14

2 Mathematical Preliminaries

Definition 2.8. (Positively invariant set) A set S in the phase space is said to be positively invariant with respect to (2.6) if x(0) ∈ S ⇒ x(t) ∈ S ∀t ≥ 0 Next we state La Salle’s theorem. Theorem 2.9. (La Salle’s theorem ) Let Ω ⊂ D be a compact set that is positively invariant with respect to (2.6). Let V : D −→ R be a continuously x) ≤ 0 in Ω. Let E be the set of all points differentiable function such that V˙ (x x) = 0. Let M be the largest invariant set in E. Then every in Ω where V˙ (x solution starting in Ω approaches M as t −→ ∞.

2.3 Relative Stability In this section we define a weaker notion of stability called relative stability . This is then extended to relative-asymptotic stability and relative-exponential stability. Motivating example Consider the following second order system (2.7)

x˙ 1 = 2x2

x˙ 2 = −x2 + x1

Let us define x = [x1 , x2 ]T . The origin of this system is unstable. Moreover, the point (0, 0) is a saddle point. The phase plot is shown in figure(2.5). Now x : x2 = −x1 } that contains the origin. If x (0) ∈ O, then define a set O = {x the closed-loop system becomes (2.8)

x˙ 1 = −2x1

x˙ 2 = −2x2

The trajectories that originate in O converge to (0, 0). Now consider a Lyapunov candidate function. x ) = x1 2 + x2 2 V (x Its time derivative on the set O is x) = −4x21 − 4x22 V˙ (x

2.3 Relative Stability

15

5 4 3 2

x2

1 0 −1 −2 −3 −4 −5 −5

−4

−3

−2

−1

0 x1

1

2

3

4

5

Fig. 2.5. Phase plot

x) < ∀ x \ {0} V˙ (x Hence the set O is positively invariant. Even though the origin is not stable in the classical sense but it is stable in a hyperplane. This motivates us to search for some weaker notion of stability which can be used to describe the behaviour of the saddle point. Let us now consider an another example. Consider the following nonsmooth system x˙ 1 = −x1 x2 x˙ 2 = x1 x : x1 = 0, x2 = 0}. Please note that this set does not Let us define a set O = {x contain the point (0, 0). If x (0) ∈ O , then the closed-loop system becomes x˙ 1 = −x1 x˙ 2 = 0 The trajectories originating in the set O converge to the point (0, 0) as time gets large. This is another example that motivates us to define a weaker notion of stability. Furthermore, this example mimics the systems considered in this thesis. Now we define a weaker notion of stability called relative stability.

16

2 Mathematical Preliminaries

D

O Bε ε

δ

U

Bδ O

B

δ

Fig. 2.6. Relative stability

Definition 2.10. (Relative stability) Consider a set O ⊂ D as shown in figure(2.6). The equilibrium point x e = 0 is said to be relatively stable with respect to the set O, if given an > 0 there exists a δ > 0 such that x (0) ∈ O ∩ Bδ =⇒ x (t) ∈ B ∀t ∈ [0, ∞) where, Bδ and B are open balls around the origin of radius δ and tively.

respec-

Definition 2.11. (Relative asymptotic stability) The equilibrium point x e = 0 is said to be relatively asymptotically stable with respect to the set O, if it is relatively stable with respect to the set O and further more x (0) ∈ Bδ ∩ O =⇒ x (t) −→ 0 as t −→ ∞ Definition 2.12. (Relative exponential stability) The equilibrium point x e = 0 is said to be relatively exponentially stable with respect to O if there exist constants δ, k, λ > 0 such that

2.4 Finite-Time Differential Equations

17

x(t)|| ≤ k||x x(0)||e−λt ∀t ≥ 0 x (0) ∈ Bδ ∩ O =⇒ ||x Remark 2.13. Please note that the equilibrium point should be a closure point of the set O.

2.4 Finite-Time Differential Equations In this section we briefly present the theory of continuous finite-time differential equations which we have used to design the finite-time controllers. More elaborate theory of finite-time differential equations and finite-time stability can be found in [26, 11]. The definition of a finite-time differential equation as presented in [26] is now stated. Definition 2.14. (Finite-time differential equations) Differential equations of the form (2.6) with the property that the origin is asymptotically stable and all solutions which converge to zero do so in finite-time are called finite-time differential equations If (2.6) is a finite-time differential equation, then the right hand side of (2.6) will be C 1 everywhere except at zero, where it will be assumed to be continuous. It is straight forward to note that the function f (.) cannot be Lipschitz at zero. Since the function is not Lipschitz it cannot admit a unique solution. But one can construct a finite-time differential equation which admits a unique solution in forward time. To understand this concept, we now explain a first-order finite-time differential equation. Consider the following equation described by x˙ = −xα

(2.9)

where α = ab where a and b are odd and a < b. Note that the function −xα is not Lipschitz. The solution of (2.9) can be written as 1

(2.10)

x(t) =

±[−(1 − α)t + C] 1−α t ≤ T 0 t≥T

where

C 1−α Note that x reaches zero in a finite-time T and the solution is well defined in forward time. The detailed analysis of n-th order finite-time differential equations and finite-time stability has been presented in [11]. Now we state finite-time stability of a dynamical system which admits a unique solution in forward time. C = [x(0)](1−α) and

T =

18

2 Mathematical Preliminaries

2.4.1 Finite-Time Stability Consider a dynamical system described by the equation (2.6). Let x (t) be the solution of (2.6). Definition 2.15. The origin is said to be a finite-time stable equilibrium of (2.6) if there exists a function T and an open neighborhood N ⊆ D such that the following statements hold: 1. Finite-time convergence : For every x (0) ∈ N \ {0}, x (t) is defined on [0, T ), x (t) ∈ N \ {0} for all t ∈ [0, T ), and limt−→T x (t) = 0. 2. Lyapunov stability : For every ball B around the origin there exist a δ ball Bδ around the origin such that, for every x (0) ∈ Bδ \ {0}, x (t) ∈ B for all t ∈ [0, T ). The following theorem [26, 11] is used to prove the finite-time stability of the differential equation. Theorem 2.16. Suppose there exists a continuous function V : D → IR such that the following hold: 1. V is positive definite. 2. There exist real numbers c > 0 and α ∈ (0, 1) and an open neighborhood N ⊆ D of the origin such that x) + c(V (x x))α ≤ 0, x ∈ N/{0} V˙ (x Then the origin is a finite-time stable equilibrium of (2.6). Proof of this theorem is found in [10]

2.5 Nonlinear Controllability Consider a nonlinear affine in control system of the form m

x)ui , gi (x

x˙ = f (x x) +

x(0) = x0

i=1

(2.11)

y = h(x x)

where x = (x1 , . . . , xn ) are local coordinates for a smooth manifold M , and f, g1 , . . . , gm are smooth vector fields on M with f (0) = 0. The control vector u : [0, T ] −→ Ω ⊆ IRm . The output y ∈ IRm is a smooth function of x with h(0) = 0. We now state certain definitions of nonlinear controllability [50].

2.5 Nonlinear Controllability

19

Definition 2.17. (Controllability) The nonlinear system (2.11) is called controllable if and only if for each x 0 , x f ∈ M and for each T > 0, there exists x(t), u (t)) is an admissible input u : [0, T ] −→ Ω with the property that if (x the control trajectory to the initial value problem (2.11), then x (T ) = x f . Linear Controllability Denote by E the set of equilibrium points of (2.11) defined as E=

x, u ) ∈ M × IRm : f (x x) + (x

m

x)ui = 0 . gi (x i=1

xe , u e ) ∈ E. The linearization of (2.11) about (x xe , u e ) is given by Let (x z˙ = Azz + Bvv

(2.12) where

z = x − xe v = u − ue and m

A=

x) x) ∂gi (x ∂f (x + ui x x ∂x ∂x i=1 e

xe ,u ue ) (x

e

x ) . . . gm (x x )]. B = [g1 (x xe , u e ) iff The system (2.11) is said to be linearly controllable at (x rank Γ = n where Γ = B AB . . . An−1 B . The system (2.11) is said to be linearly controllable if the linearized system xe , u e ) is controllable. This (2.12) at any of the system equilibrium points (x property however, does not hold for many underactuated mechanical systems. x0 , T ) deThis motivates us to define other notions of controllability. Let R V (x note the set of reachable points in M from x 0 at time T > 0, using admissible controls u (t) and such that the trajectories remain in the neighborhood V of x0 for all t ≤ T . Furthermore, define (2.13)

x0 , ≤ T ) = RV (x

0≤t≤T

x0 , t) , RV (x x0 ) = RV (x

t>0

x0 , t) RV (x

The other notions of controllability are defined as follows [40]:

20

2 Mathematical Preliminaries

Definition 2.18. Let x 0 ∈ M and int(U ) refer to the interior of the set U . x0 )) = ∅ for any neighbourhood 1. Σ is locally accessible from x 0 if int (RV (x V . If this holds for any x 0 ∈ M then the system is called locally accessible. x0 , T )) = ∅ for each 2. Σ is locally strongly accessible from x 0 if int (RV (x T > 0 and for any neighbourhood V . x0 )) for any neighbour3. Σ is locally controllable from x 0 if x 0 ∈ int (RV (x hood V . 4. Σ is small time locally controllable (STLC) from x0 if there exists T > 0 so x0 , ≤ t)) for each t ∈ (0, T ] and for any neighbourhood that x 0 ∈ int (RV (x V. x0 ) = M . 5. Σ is globally controllable from x 0 if RV (x Sufficient conditions to check different notion of controllability is presented in Appendix A.

2.6 Stabilizability Stabilizability is a property which is related to the existence of a feedback control that makes the closed-loop system stable. Here we present Brockett’s theorem [15] which gives a necessary condition for the existence of a C 1 feedback control law for asymptotic stability. Theorem 2.19. Let x, u ) x˙ = f (x xe , 0) = 0 and f (., .) is continuously differand x ∈ IRn and u ∈ IRm . Let f (x xe , 0). A necessary condition for the entiable function in a neighborhood of (x 1 xe , 0) asymptotically stable is that existence of a C control law which makes (x 1. the linearized system should have no uncontrollable modes associated with eigenvalues whose real part is positive xe , 0) such that for each ξ ∈ N there 2. there exists a neighborhood N of (x exists a control u ξ (.) defined on [0, ∞) such that this control steers the x, uξ ) from x = ξ at t = 0 to x = xe at t = ∞. solution of x˙ = f (x 3. the mapping γ : IRn × IRm −→ IRn x, u ) −→ f (x x, u ) should be onto an open set containing 0. defined by γ : (x

3 A Switched Finite-Time Controller Design Methodology

In this chapter we present our main result. The first section presents the controller design methodology followed by the steps of controller design. Later we apply this technique to three well-studied examples. Simulation results are then presented and a detailed comparison with the existing controllers is listed as well.

3.1 Introduction Our approach here rests on defining a set of surfaces whose intersection is a connected set and contains the equilibrium. The control objective is then to move the closed-loop trajectories from anywhere in the state space to this intersection set in finite-time and make the closed-loop system asymptotically stable in this set. In some applications, it is necessary to switch between the available controllers to reach the intersection set. We reach each surface in finite time since each individual controller is designed using the theory of continuous finite-time differential equations. At this juncture we would like to remark on the similarity to sliding mode control that is a finite-time design technique which falls under the class of variable structure control and has been studied by many authors [71, 31, 24]. But the controller designed here is different from the conventional sliding mode controller where the (sign term appears) and as a result avoids chattering. We consider three well-studied examples to validate the proposed technique. The first one is the stabilization of the NI. The NI is a benchmark example which expresses all the properties of a nonholonomic system and mimics the kinematic model of a mobile robot as well. The transformation

R.N. Banavar and V. Sankaranarayanan: Switch. Fin. Time Contrl., LNCIS 333, pp. 21–50, 2006. © Springer-Verlag Berlin Heidelberg 2006

22

3 A Switched Finite-Time Controller Design Methodology

between a kinematic model to the NI is presented in Appendix A. A finitetime controller is designed to stabilize the NI using the proposed technique and this controller achieves exponential convergence. Next we consider the extended version of the NI, called ENDI, which mimics the dynamic model of a mobile robot. This system is considered as an example of an underactuated system with a velocity level nonholonomic constraint. This system is stabilized using a switched control law. The switching between the two controllers is necessary to achieve the desired objective (stabilization). The third example is an underwater vehicle moving in a plane whose dynamics are similar to a surface vessel. The difference between the mobile robot and the surface vessel is that the surface vessel can freely move in the lateral direction even though this degree-of-freedom is unactuated. The controller design procedure for the underwater vehicle is similar to that of the ENDI. The class of underactuated systems under consideration here are 1. Not linearly controllable 2. Do not satisfy Brockett’s condition 3. Satisfy the STLC property All the above properties for the given examples are proved in Appendix A.

3.2 Controller Design Methodology We now enumerate the steps in the controller design. Consider the affine-inthe-control system m

(3.1)

x) + x˙ = f (x

x)ui gi (x i=1

x), . . . , Sj (x x) (j ≤ m, Sj (x x) : IRn −→ 1. Define scalar valued functions S1 (x IR ) called switching functions such that the set x : {S1 (x x) = 0} ∩ . . . ∩ {Sj (x x) = 0}} K = {x x : Sj (x x) = 0} is is a connected set and contains the equilibrium. The set {x called the switching surface associated with the switching function S j (·) 2. Design the control law such that the trajectory of the closed-loop system reaches the set K in finite time 3. Ensure that the closed-loop dynamics is asymptotically stable in the set K

3.2 Controller Design Methodology

23

The switching functions are selected such that the closed-loop system is stable on the corresponding switching surface. The intersection set K can be reached in finite-time by reaching each surface in finite-time. The controller design to x) be a switching function, reach a switching surface is now presented. Let S(x assume that the control appears in the first derivative of the switching function x) such that the dynamics of S(x x) is and we synthesize a control law u = h(x given by ˙ x) = −ks S(x x )α S(x where α ∈ (0, 1) and ks > 0. Now consider a Lyapunov candidate function V (·) : IR → IR

V =

x )2 S(x 2

Its rate of change is given by x)α+1 V˙ = −ks S(x Choose α + 1 such that V˙ < 0. Say for α = x)4/3 < 0 V˙ = −ks S(x

1 3

this implies

x) = 0 ∀S(x

x : S(x x) = 0} is attractive globally. To prove that the Hence the surface {x x) = trajectory starting from any point in the state space reach the surface S(x 0 in finite time, we use theorem (2.16) which states that if there exists a β ∈ (0, 1) and k > 0 such that V˙ + kV β ≤ 0 then the trajectory reaches the surface in finite time. For β = we have V˙ + kV

2 3

x)4/3 (ks − = −S(x

k 2

23

2 3

2

and k < 2 3 ks

) 0.

u=

−k1 x1 1/3

x −k3 x31

− k 1 x2

26

3 A Switched Finite-Time Controller Design Methodology

Proof Exponential convergence : If x1 (0) = 0, then the system (3.2) with controller (3.3) becomes 1/3

(3.4)

x˙ 1 = −k1 x1

x˙ 2 =

−k3 x3 x1

− k 1 x2

1/3

x˙ 3 = −k3 x3

System (3.4) can be written as x˙ 1 = −k1 x1 x) = where d(x

1/3

x) x˙ 2 = −k1 x2 + d(x

x˙ 3 = −k3 x3

1/3

−k3 x3 x1

. Since the solution of x3 (t) can be written as  3/2  ± − 23 k3 t + C 0≤t≤T x3 (t) =  0 t≥T

where C = x3 (0)2/3 and T = 32 C, and the solution of x1 (t) can be written as x1 (t) = x1 (0)e−k1 t we have d(t) =

  

3 ± x1k(0) − 32 k3 t + C

1/2

ek 1 t 0 ≤ t ≤ T

0

t≥T

It is straightforward to see that for every initial condition there exist a η, γ > 0 such that |d(t)| ≤ ηe−γt The trajectories x1 (t), x2 (t) and x3 (t) can hence be bounded by decaying exponentials and the trajectories of the closed-loop system exponentially converges to the origin. Relative stability : Since k1 , k3 > 0, the dynamics of x3 implies finite time stability [26] while that of x1 implies exponential stability. So there exists a finite time T ≥ 0 such that x3 (t) ≡ 0 ∀t ≥ T . The system is then governed by the set of equations (3.5)

x˙ 1 = −k1 x1

x˙ 2 = −k1 x2

x˙ 3 ≡ 0

which implies that x1 and x2 go to zero. Furthermore, if x (0) ∈ O then the closed-loop system is (3.5), and is relatively exponentially stable with respect to the set O. Remark 3.2. If x1 (0) = 0, we can apply any open loop control to steer the system to a non-zero value of x1 .

3.3 Stabilization of the NI

27

3.3.2 Simulation The controllers u1 and u2 are simulated with the controller parameters k1 = k3 = 1, α = 1/3 and the initial conditions x1 (0) = 3,x2 (0) = −2,x3 (0) = 1. The simulation results are shown in figures(3.1-3.2). 3

x1 x2 x3

2.5 2 1.5

x

1 0.5 0 −0.5 −1 −1.5 −2 0

2

4

time in sec

6

Fig. 3.1. Stabilization to the origin

8

10

3 A Switched Finite-Time Controller Design Methodology 2

u 1 u

1.5

2

1 0.5

2

0 1

u ,u

28

−0.5 −1 −1.5 −2 −2.5 −3 0

2

4

time in sec

6

Fig. 3.2. Control inputs

8

10

3.4 Stabilization of the ENDI – Wheeled Mobile Robot

29

3.4 Stabilization of the ENDI – Wheeled Mobile Robot A car like wheeled mobile robot of the unicycle type moving in a plane as shown in figure (3.3), is one of the benchmark problems in underactuated systems. The vehicle has two identical parallel rear wheels which are controlled by two independent motors. We assume that the wheels do not slip with respect to the ground. The masses and inertias of the wheels are negligible as compared to the mass of the vehicle and the center of mass of the robot is located in the middle of the axis connecting the real wheels. M is the mass of the vehicle, I is the moment of inertia, L is the distance between the center of mass and the wheel, τ1 is the left wheel motor torque, τ2 is the right wheel L (τ1 − τ2 ) motor torque, R is the radius of the rear wheel, the net torque τ = R 1 and the force F = R (τ1 + τ2 ) The kinematic equations are x˙ = v cos θ y˙ = v sin θ θ˙ = ω

(3.6)

y

τ1

θ

τ2 x Fig. 3.3. Mobile robot

30

3 A Switched Finite-Time Controller Design Methodology

where the triple (x, y, θ) denotes the position and orientation of the vehicle with respect to the inertial frame and v, ω are the linear and angular velocities of the body respectively. The dynamic model of the mobile robot can be written as (3.7)

M v˙ = F I ω˙ = τ

By assuming that the wheels do not slide, a first order nonholonomic constraint on the motion of the mobile robot of the form (x˙ sin θ − y˙ cos θ = 0 ) is imposed. The resulting dynamic model is a fifth order system with drift given as         0 θ˙ ω 0 0 x˙ v cos θ 0 y˙ = v sin θ + 0 F+ 0 τ 1 0  v˙   0   M  1 0 ω˙ 0 I Control of such mobile robots using discontinuous, time-varying and hybrid controllers has been discussed in [8, 1, 17, 60, 35, 18, 34, 13]. Most of the control laws are based on the kinematic model of the mobile robot and it is not applicable for systems where forces and torques are considered as the inputs since these controllers are not differentiable. Stabilization of the dynamic wheeled mobile robot has received lesser attention in the community. A discontinuous controller has been developed in [8] for a kinematic and dynamic model of the mobile robot using a polar co-ordinate system and it has been shown that kinematic system is almost exponentially stable. Time-varying controllers have been designed to stabilize mobile robot in [60, 18, 35, 34]. Among these a global exponential set point control is proposed in [18], tracking of a mobile robot in [35] and bounded tracking control in [34]. Even though these controllers give global properties , they have low rates of convergence and oscillating trajectories. In [17] a piecewise continuous controller was developed to exponentially stabilize the mobile robot, but this cannot stabilize the mobile robot from any arbitrary initial orientation. Almost exponential stability of the mobile robot is addressed through an discontinuous controller in [39]. In this approach initial condition of the system is restricted by a line in the phase plane between the angular position and the angular velocity. A variable structure controller has been developed in [77] for the stabilization of a chained system and this is applied to a kinematic model of a car-like robot.

3.4 Stabilization of the ENDI – Wheeled Mobile Robot

31

3.4.1 Transformation to the ENDI We now define a sequence of transformations to change the representation into a form that resembles the extended nonholonomic double integrator (ENDI) [1]. The standard ENDI form facilitates the application of our algorithm. Step 1 - State transformation     z1 θ z2 x cos θ + y sin θ z3 = x sin θ − y cos θ ω  z4    z5 v − (x sin θ − y cos θ)ω The dynamic model in the z variables is     z˙1 z4 z˙2 z5 z˙3 = z2 z4 τ z ˙   4  I F τ 2 − z − z z z˙5 3 2 4 M I We define alternate inputs as u1 =

τ I

u2 = −z42 z2 −

τ F z3 + I M

Step 2 - State transformation and affect another state transformation from (z1 , z2 , z3 , z4 , z5 ) to (x1 , x2 , x3 , y1 , y2 ) as x1 = z1 x2 = z2 x3 = −2z3 + z1 z2 x˙ 1 = y1 = z4 x˙ 2 = y2 = z5 The system equations are now in the form x ¨ 1 = u1 x ¨ 2 = u2 x˙ 3 = x1 x˙ 2 − x2 x˙ 1 Define x = [x1 , x2 , x3 , y1 , y2 ]T ∈ S 1 ×IR4 as the state vector and u = [u1 , u2 ] ∈ IR2 as the control and we have 2

(3.8)

x)ui gi (x

x) + x˙ = f (x i=1

32

3 A Switched Finite-Time Controller Design Methodology

where      0 0 y1 0 0 y2 x) = 0 x) = 0 ; g2 (x x) = x1 y2 − x2 y1 ; g1 (x f (x 0 0 1   1 0 0 

This is the ENDI and the salient features of this system are x ∈ S 1 × IR4 : y1 = y2 = 0} and • The equilibria are of the form x e = {x satisfy the following properties 1. Not linearly controllable. 2. xe is not stabilized by any C 1 feedback control laws. 3. The ENDI is locally strongly accessible from any x ∈ S 1 × IR4 . 4. The ENDI is small time locally controllable (STLC) from any x e . Proof of the above properties are presented in the Appendix A. 3.4.2 Controller Design The control objective is to move the mobile robot from any position and orientation (x, y, θ) to a target position and orientation (xd , yd , θd ). Without loss of generality the problem of point-to-point control can be converted in to the problem of stabilizing the system (6.1) at the origin. Now we define the following functions x) = kx1 + y1 S1 (x x) = x1 y2 − x2 y1 + k3 x3 S2 (x and the set K as (3.9)

x ∈ S 1 × IR4 : y1 = −kx1 and x1 y2 − x2 y1 = −k3 x3 } K = {x

x) is selected such that the unactuated dynamics is stable The function S2 (x x) is selected such that the controller used on this surface. The function S1 (x x) = 0 is well defined on the surface S1 (x x) = 0. This to reach the surface S2 (x x : S2 (x x) = motivates us to define the sequence of the sliding surfaces as K = {x x) = 0} ⊂ M1 = {x x : S1 (x x) = 0}. Now we state the following proposition 0, S1 (x

3.4 Stabilization of the ENDI – Wheeled Mobile Robot

33

Proposition 3.3. The origin of the ENDI is globally attractive and relatively exponentially stable with respect to the set O = K \ {x1 = y1 = 0} with the following control law u1 = −ky1 − S1 (x x)1/3 if (x1 , y1 ) = (0, 0)

(3.10)

1/3

(3.11)

u2 =

−S2 x1

− k3 y2 − (k3 − k)kx2 x ∈ M1 \ {x1 = y1 = 0} −x2 − y2 otherwise

where k3 > k > 0. Proof Global attractivity The closed-loop dynamics for (x1 (0), y1 (0)) = 0 becomes x˙ 1 = y1 x˙ 2 = y2 x˙ 3 = x1 y2 − x2 y1 x)1/3 y˙ 1 = −ky1 − S1 (x y˙ 2 = −x2 − y2 x) becomes The dynamics of S1 (x x) = −S1 (x x)1/3 S˙ 1 (x x) = 0 is finite time stable. So there exists which implies that the surface S1 (x a finite-time T1 ≥ 0 such that the closed-loop trajectories for any permissible initial condition reaches the set M1 and stays there for all future time. The control law u1 allows both x1 and y1 to converge to zero as time gets large. At the same time as the system is being driven towards M1 , the PD control law u2 with unity gain makes states x2 and y2 converge to zero. The closed-loop system on M1 is x˙ 1 = −kx1 x˙ 2 = y2 x˙ 3 = x1 y2 − x2 y1 y˙ 1 = −ky1 1/3

y˙ 2 =

−S2 x1

− k3 y2 − (k3 − k)kx2

34

3 A Switched Finite-Time Controller Design Methodology

x) becomes The dynamics of S2 (x x) = −S2 (x x)1/3 S˙ 2 (x So there exists a time T2 ≥ T1 ≥ 0 such that the closed loop trajectories reach the set K and stay there for all future time. The closed-loop system on K is x˙ 1 x˙ 2 x˙ 3 y˙ 1 y˙ 2

= −kx1 = y2 = −x3 = −ky1 = −k3 y2 − (k3 − k)kx2

which is exponentially stable. If the system starts with the initial condition x1 = y1 = 0, then an open-loop control law can perturb the system from this initial condition. This ends the proof of global attractivity. Relative stability Furthermore, if x (0) ∈ O, then the closed-loop system equations are (3.12) and the system is relatively exponentially stable. To draw comparisons with similar efforts we consider the discontinuous time-invariant controllers synthesized in [8, 39]. In [8] the intial condition is restriced to zero intial velocities (vehicle starts from rest) of the vehicle. In [39] the intial condition is restricted to a linear relationship between angular velocity and orientation of the vehicle shown as in figure 3.4(a). But in our proposed control law, the only intial condition that is excluded is (x1 (0), y1 (0)) = (0, 0) which is shown in figure 3.4(b). The proposed controller gives exponential stability in a set which contains the origin and global convergence to the set from any initial condition in finite-time. 3.4.3 Simulation We simulate the controller for k = 0.5, k3 = 1, α = 1/3. The initial position and the orientation of the vehicle is x(0) = −1.5m, y(0) = 4m and θ(0) = −2.3rad. The initial velocities of the vehicle are ω(0) = 1rad/sec and v(0) = −1m/sec. Simulation results are shown in figures(3.5-3.7).

3.4 Stabilization of the ENDI – Wheeled Mobile Robot y

1

Restricted intial condition

x1

(a) Controller[39] y

1

Excluded intial condition

x1

(b) Proposed controller

Fig. 3.4. Restriction on the initial conditions

35

3 A Switched Finite-Time Controller Design Methodology 5 4

initial configuration

3 2

y in meters

1 0

final configuration

−1 −2 −3 −4 −5 −5

−4

−3

−2

−1

0 x in meters

1

2

3

4

5

Fig. 3.5. Stabilization to the origin 5

x y θ

4 3 x,y in meters, θ in rad

36

2 1 0 −1 −2 −3 0

10

20

time in sec

30

40

Fig. 3.6. Position and orientation of the vehicle

50

3.4 Stabilization of the ENDI – Wheeled Mobile Robot 1.5

ω v

v in m/sec, ω in rad/sec

1

0.5

0

−0.5

−1 0

10

20

time in sec

30

40

50

Fig. 3.7. Velocities profile 0.15

τ 1 τ2

0.1

τ1,τ2 in Nm

0.05 0 −0.05 −0.1 −0.15 −0.2 0

10

20

time in sec

30

Fig. 3.8. Input to the motors

40

50

37

3 A Switched Finite-Time Controller Design Methodology 1.4

1.2

1

1 2

0.8

s ,s

38

s

2

0.6

0.4

0.2

0

−0.2

s 0

1 1

2

3

4

5

6

time in sec

7

8

9

x), S2 (x x) Fig. 3.9. Finite-time convergence of the functions S1 (x

10

3.5 Stabilization of an Underwater Vehicle

39

3.5 Stabilization of an Underwater Vehicle The control of the position and orientation of underwater vehicles / surface vessels has received much attention in recent years [52, 56] due to various applications. The system we consider is an underwater rigid body moving in a horizontal plane (neutrally buoyant). A schematic is illustrated in Figure (3.10). The configuration space is Q = IR2 × 1 and is parameterized by the co-ordinates (x, y, θ) represented in inertial frame. The triple (x, y, θ) represents the position of the center of mass and orientation of the body in the inertial frame. The corresponding linear and angular velocities in the body frame are denoted by (vx , vy , ωz ). The inertial velocities and the body velocities are related by the equations !

x˙ = vx cos θ − vy sin θ y˙ = vx sin θ + vy cos θ θ˙ = ωz The equations of motion are (for detail see [52, 56]) given by (3.12)

m11 v˙ x − m22 vy ωz + d11 vx = Fx m22 v˙ y + m11 vx ωz + d22 vy = Fy m33 ω˙ z + (m22 − m11 )vx vy + d33 ωz = τz

where mii , dii , i = 1, 2, 3, are positive constants that represent the elements of the inertia matrix including added masses and the elements of the damping

Y Yb

Xb θ

(x, y)

X Fig. 3.10. Schematic of underwater vehicle

40

3 A Switched Finite-Time Controller Design Methodology

matrix respectively. Typically the vehicles are actuated by only two control forces and the inputs are the yaw (τz ) and forward thrust Fx . Since the surface vessel is actuated in the yaw-axis and forward thrust only, it is underactuated. The unactuated dynamics constitute a second-order nonholonomic constraint which renders it unsuitable to apply the full-state feedback linearizing technique. The absence of potential terms causes the linearized model of the nonlinear system uncontrollable. The controllability properties of this class of systems is studied in [75]. This system does not admit time-invariant smooth state-feedback control that can locally asymptotically stabilize the equilibrium (Brockett, [15]). A much weaker notion of controllability called STLC holds for this system. But the existence of timevarying/periodic or discontinuous control that can asymptotically stabilize the equilibrium is guaranteed in view of the STLC property [57]. All the above properties are proved in Appendix A. Stabilization of this system has been studied by many authors from the point of time-varying, discontinuous and hybrid controllers. We briefly summarize the previous work in this area. Asymptotic stabilization of the surface vessel’s position but not orientation using a continuous feedback controller is presented in [74]. Time-varying controllers are synthesized to stabilize the surface vessel in [52, 53, 49] . Exponential stabilization is presented in [52], a semi-global exponential stabilization in [53] and global asymptotic stabilization in [49]. As mentioned in the previous section, the time-varying controllers suffer from low convergence rate and oscillating trajectories since these controllers employ a sinusoidal term that generates an oscillatory motion. Recently a time-varying controller is designed for stabilizing a non-holonomic system without a sinusoidal function in the controller and has been applied to a surface vessel [69]. In [56], a discontinuous control law has been presented to achieve asymptotic stabilization to an equilibrium configuration with exponential convergence rates. The discontinuous nature of the control law is due to the rational transformation introduced for the states of the system. The control law has atmost one switching and is valid for a restricted set of initial conditions. A discontinuous controller is developed for a higher order chained nonholonomic system and then applied to a surface vessel/underwater vehicle in [39] with a restriction on the initial condition that is less restrictive than [56]. The detailed comparison of our proposed method with these controllers is given in the last section.

3.5 Stabilization of an Underwater Vehicle

41

3.5.1 Feedback Linearization We define the state vector x = (x1 , x2 , x3 , x4 , x5 , x6 ) as [52]   θ x cos θ + y sin θ −x sin θ + y cos θ x= vy   ωz vx The ordering of the states is motivated by a desire to split the state-vector into actuated and unactuated subsystems. The control inputs after a partial feedback linearization are defined as u1 = (τz − d33 ωz + (m11 − m22 )vx vy )/m33 u2 = (Fx + m22 vy ωz − d11 vx )/m11 . which yields the state-space representation of (5.2) with Fy = 0, defined on the manifold M = S 1 × IR5 as (3.13) where

x) + g1 (x x)u1 + g2 (x x)u2 x˙ = f (x  x5 x6 + x 3 x5 x4 − x 2 x5 x) = f (x ; −αx4 − βx5 x6   0 0     0 0 0 0 0 0 x) = x) = g1 (x ; g2 (x ; 0 0 1 0 0 1 

and α = d22 /m22 , β = m11 /m22 . The equilibrium configuration denoted by x e is all of Q since the motion of the vehicle is in the horizontal plane.

42

3 A Switched Finite-Time Controller Design Methodology

3.5.2 Controller Design The control objective of the surface vessel is to move from any position and orientation (x, y, θ) to a target position and orientation (xd , yd , θd ). Without loss of generality the problem of point-to-point control can be converted into the problem of stabilizing the system (3.13) to the origin. For the system under consideration we define the following functions x ) = x1 + x5 S1 (x x) = αx3 + x4 S2 (x

x) is selected such that the unactuated dynamics are stable The function S2 (x x) is selected such that the on the corresponding surface. The function S1 (x x) = 0 is valid in a domain around the controller to reach the surface S2 (x x x) = 0, its derivative surface S1 (x ) = 0. To reach and remain on the surface S2 (x also has to be made zero. This motivates us to define an another function and the set K as x) = (kx2 + x6 ) S3 (x x ∈ M : x1 = −x5 , x4 = −αx3 , kx2 = −x6 } K = {x where k=

α β

Before stating our proposition we define the following sets x ∈ M|(x5 < 0 K1 = {x

x) ≥ 0) S1 (x

x ∈ M : S1 (x x) = 0} M1 = {x

(x5 > 0

x) ≤ 0)} S1 (x

x ∈ M : S2 (x x) = 0} M2 = {x

ME = M1 ∩ M2 x ∈ K \ x1 = x5 = 0} O = {x We now state the following proposition Proposition 3.4. The origin of (3.13) is globally attractive and relatively asymptotically stable with respect to the set O under the following control law u1 = −(x1 + x5 )1/3 − x5 if (x1 , x5 ) = 0 u2 =

uf +u1 (αx2 +βx6 ) −x5

−α β (x6 + x3 x5 ) x ∈ K1 \ {x1 = x5 = 0} 0 elsewhere

3.5 Stabilization of an Underwater Vehicle

43

where x)1/3 − S4 (x x)1/3 uf = −S2 (x and x) = −βx5 S3 (x x) S4 (x

Proof: Global attractivity x) becomes For (x1 (0), x5 (0)) = 0 under the control law u1 , the dynamics of S1 (x x) = −S1 (x x)1/3 S˙1 (x x) = 0 is finite-time stable. So there exists a time T1 ≥ 0 The surface S1 (x such that the closed-loop trajectories enter K1 and stay there. The phase plot between x1 and x5 is shown in figure (3.11). After the trajectory enters K1 there exists a finite-time T2 ≥ T1 such that the trajectory enters the manifold M1 . It is easy to note that the manifold M1 is positively invariant and the only largest positively invariant set in M1 is the origin. So x1 (t), x5 (t) → 0 as t −→ ∞ in M1 . The controller u2 is well defined in K1 and it is switched on once the trajectory enters the set K1 . Under this control action the dynamics x) and S4 (x x) becomes of the functions S2 (x x) = 0 S1 (x

K1

x5

x) ≥ 0 S1 (x x1

x) ≤ 0 S1 (x

K1

Fig. 3.11. Phase plot between x1 and x5

44

3 A Switched Finite-Time Controller Design Methodology

x) = S4 (x x) S˙ 2 (x x) = −S2 (x x)1/3 − S4 (x x)1/3 S˙ 4 (x x) ≡ S4 (x x) ≡ 0. It is Hence, there exists a finite-time T3 ≥ T1 such that S2 (x easy to note that the manifold M2 is positively invariant along the closed-loop x) ≡ 0 =⇒ S3 (x x) ≡ 0 since x5 system. Furthermore, on the manifold M2 , S4 (x goes asymptotically to zero in the set K1 . So the dynamics of the closed-loop x) ≡ S3 (x x) ≡ 0. Therefore, system on the manifold has the property that S2 (x from both the control actions u1 , u2 there exists a finite-time T4 ≥ T1 such that the closed-loop trajectory enters the set K. Since the manifolds M1 , M2 are positively invariant, the set K is also positively invariant for the trajectories of the closed-loop system. The dynamics of the closed-loop system in K is x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5 x˙ 6

(3.14)

= −x1 = −kx2 − x3 x1 = −αx3 + x2 x1 = −αx4 − αx1 x2 = −x5 = −kx6 + kx3 x1

Now consider a Lyapunov candidate function x) = V (x

x2 x2 x2 x2 x2 x21 + 2 + 3 + 4 + 5 + β2 6 2 2 2 2 2 2

Take V˙ along the closed-loop solution, we have x) = −x21 − kx22 − x1 x2 x3 − αx23 + x1 x2 x3 V˙ (x −αx24 − αx1 x2 x4 − x25 − kβ 2 x26 + kβ 2 x1 x3 x6 = −x21 − kx22 − αx23 − αx24 −αx1 x2 (−αx3 ) − x25 − kβ 2 x26 + kβ 2 x1 x3 (−kx2 ) = −x21 − kx22 − αx23 − αx24 − x25 − kβ 2 x26 0 is a predefined constant where S(x and d is any non zero constant. Proof: Case 1 Let us assume x (0) ∈ M1 , then the NI exponentially converges to the origin as mentioned in proposition 4.2.1 and the manifold M1 is positively invariant. Case 2 If x ∈ M2 , then the closed-loop system becomes x˙ 1 = ks x˙ 2 = 0

x)1/3 S(x x2 + c

x˙ 3 = −x2 ks

x)1/3 S(x x2 + c

The dynamics of S becomes x)1/3 S˙ = −ks S(x

54

4 Alternative Control Strategies for the NI

and it is finite-time stable. So there exists a finite-time T1 such that S(t) = 0∀t ∈ [T1 , ∞). So all the trajectories will reach the boundary of the set M1 in some finite-time after which case 1 follows. Remark 4.4. Note that the value of x2 remains constant here. Case 3 If x ∈ M3 , then the dynamics of x2 becomes x˙ 2 = d So there exists a finite-time T2 such that |x2 (T2 )+k| = 1. Then Case 2 follows. When x1 (0) = 0, we use an open-loop control law to perturb the system away from this condition. Remark 4.5. In this method our notion of boundedness is not in terms of the control magnitude being bounded by a pre-defined constant but rather based on the limitation of the value | xx13 |.

4.4 Method 3 – Time-Varying Control In this method we propose a sliding mode controller which avoids large values of u2 due to small initial values of x1 in (3.3). Proposition 4.6. The origin of the NI is globally attractive and the trajectories locally exponentially converge to the origin with the following control law  −k1 x1 x(t)) ≥ 0 ∀ s ∈ [t, ∞) if S(x  −x x1/3 − k x 1 3 1 2 u= x1 x(t)) < 0 if S(x  x2 x) = x21 − 3k1 x2/3 where S(x where k1 > 0 is a predefined constant, 3 Proof x(0)) ≥ 0, then we have Case 1 If S(x x˙ 1 = −k1 x1 1/3 x˙ 2 = −x1 x3 − k1 x2 1/3 x˙ 3 = −x21 x3

4.5 Discussion

55

The evolution of x1 (t) is given by x1 (t) = x1 (0)e−k1 t and the evolution of x3 by x1 (0)2 e−2k1 t − S(x(0)) 3k1 x3 (t) = 0 ∀t ≥ T1 x3 (t) = ±

3/2

∀t ∈ [0, T1 ]

The positive (negative) sign corresponds to the solution for an initial condition x3 (0) > 0 (x3 (0) < 0). The closed-loop system after time T1 is x˙ 1 = −k1 x1 x˙ 2 = −k1 x2 x˙ 3 ≡ 0 and trajectories converge to the equilibrium exponentially. The arguments for exponential stability are similar to that presented for Method 3. x(0)) < 0 , then we have Case 2 If x1 (0) = 0 and S(x x˙ 1 = x1

x˙ 2 = x2

x˙ 3 = 0

x(t)) as time The above dynamics leads to an increase in the value of S(x increases. Note that x3 remains a constant and x1 has an exponential growth. x(t)) ≥ 0. Then Case 1 follows. So there exists a finite-time such that S(x case 3 x(0)) < 0, then any open loop control can steer x1 = 0 to If x1 (0) = 0 and S(x a nonzero value, then either case 1 or case 2 follows.

4.5 Discussion We now briefly summarize the four control laws that we have proposed to stabilize the NI. Method 1 guarantees relative exponential stability in an open dense set. It resembles the σ process (or rational transformation) which has been widely used in the literature [9, 7, 8, 39]. But we would like to point out that we do not initially transform the system into a set of new coordinates. Similar to controllers designed with the σ process philosophy, the control law suffers from the problem of large control magnitudes when the system is close to the surface x1 = 0. Method 2 alleviates this problem and in the process

56

4 Alternative Control Strategies for the NI

guarantees global attractivity. In this method we take care only of the boundedness of the variable xx13 and not boundedness of the control magnitude in a strict sense. In the control law presented in chapter 3, we use time-varying control that guarantees exponential convergence. The controller design is different from [13] since we have used a power rate reaching law proposed in [24]. This control law is similar to Method 1 but the crucial difference is the variable structure nature of the control law which reduces the order of the system after a certain stage. Method 3 reduces large control magnitudes in the controller(3.3) proposed in the previous chapter when the system is close to the surface x1 = 0. This is achieved by using the state x1 as a gain for the 1/3 convergence of x3 to zero in finite-time. So we have the term x1 x3 instead of 1/3

x3 x1

. But since this control law is valid in a local domain, we have introduced a switching strategy to move from anywhere to this domain.

4.6 Simulation In this section we simulate the controllers for the following initial condition and we compare the control magnitudes. The initial conditions are x1 (0) = 0.01, x2 (0) = −2, x3 (0) = 4 and the following controller parameters k1 = 0.5, k3 = 1, k4 = 0.5, ks = 1, d = 1. For method three we simulate different x)(0) < 0. The simulation results are shown in initial conditions such that S(x the figures (4.1 - 4.8).

4.6 Simulation 50

x 1 x 2 x3

0 −50

x

−100 −150 −200 −250 −300 0

10

20

time in sec

30

40

50

Fig. 4.1. Convergence of x - Method 1 100

u 1 u2

50 0 −50

u

−100 −150 −200 −250 −300 −350 −400 0

10

20

time in sec

30

Fig. 4.2. Control input u - Method 1

40

50

57

4 Alternative Control Strategies for the NI 4

x1 x 2 x

3

3

x

2

1

0

−1

−2 0

10

20

time in sec

30

40

50

Fig. 4.3. Convergence of x -Method 2 3

u2 u 1

2 1 0 u

58

−1 −2 −3 −4 −5 0

10

20

time in sec

30

Fig. 4.4. Control input u - Method 2

40

50

4.6 Simulation 10

x1 x2 x3

0 −10

x

−20

S(x)>0

−30 −40 −50 −60 −70 0

20

40

time in sec

60

80

100

Fig. 4.5. Convergence of x - Method 3-case1 10

u1 u2

0 −10 S(x)>0

u

−20 −30 −40 −50 −60 0

20

40

time in sec

60

80

Fig. 4.6. Control input u - Method 3-case1

100

59

4 Alternative Control Strategies for the NI 6

x 1 x2 x 3

5

4

S(x) 1. Proof Under the control law (5.8) and for v(0) = 0, the dynamics of S(X) becomes ˙ S(X) = −S(X)1/3

(5.9)

So there exists a finite-time T such that the trajectory enters the set K = {X ∈ IR3 : S(X) = 0}. The closed-loop system after a finite-time T becomes x˙ 2 = −kx2 + (5.10)

where c =

x23 (k−1)x2

x˙ 3 = −cx3 v˙ = −cv k k−1 .

Consider the Lyapunov function

5.2 Circular Shaped Hovercraft

V =

67

x3 2 v2 x2 2 + + 2 2 2

Time derivative of V along the closed-loop solution after time T becomes V˙ = −kx22 +

1 x2 − cx23 − cv 2 k−1 3

V˙ < 0 ∀X \ {0} Moreover if X(0) ∈ O, then closed-loop system equations are (5.10). Hence the system is relatively asymptotically stable in O. Simulation We simulate the closed-loop system with the controller parameters k = 2 and the initial conditions are x2 (0) = −2, x3 (0) = −3, v(0) = 6. Simulations results are shown in figures(5.4,5.5) 6

x 2 x

5

v

3

4 3

x2,x3,v

2 1 0 −1 −2 −3 −4

0

1

2

3

4

5

time in sec

6

Fig. 5.4. position

7

8

9

10

68

5 Switched Stabilization of a Hovercraft 18

u r

16 14 12

u,r

10 8 6 4 2 0 −2

0

1

2

3

4

5

time in sec

6

7

8

9

10

Fig. 5.5. Velocity inputs

5.3 Conventional Type Hovercraft In this section we present an alternate switched controller to move the hovercraft, which resembles the surface vessel as shown in figure(5.6), from any given position and orientation to a target position and orientation. The logic for switching here differs from the previous one. The entire objective of moving from the given position and orientation to the desired position and orientation is converted to smaller objectives. The strategy is feedback in each stage but is crucially dependant on certain initial conditions at the beginning of the stage. 5.3.1 Controller Design The dynamic model remains the same as in Chapter 4. The control philosophy here is “ In the first step, regulate the unactuated velocity to zero in finite time. At all subsequent steps (stages), appropriately set one of the actuated velocities to zero since this will ensure that unactuated velocity remains unaffected (this

5.3 Conventional Type Hovercraft

69

Y

y

x θ

(x,y)

F 1 F 2

X

Fig. 5.6. Hovercraft

is possible due to the dynamics of the system). This mode of switching is continued till all the states are regulated to zero in finite time”. This necessitates the splitting-up of the system into subsystems through state and input transformations, each of which is regulated using finite-time controllers. The controller architecture is shown in figure(5.7) Without loss of generality we can assume the initial values of x5 and x6 as non-zero. If they are zero, they can be steered to nonzero states using the controls u1 and u2 . Let x5 (0) = c1 and x6 (0) = c2 where c1 , c2 are arbitrary constants. The remaining states can take on any arbitrary initial values.

Stage 1 : In this stage our objective is to bring the unactuated velocity vy = x4 to zero. For this stage we set u2 = 0. Consequently x˙ 6 = 0

⇒ x6 (t) = c2 ∀ t ≥ 0.

Then the system of equations for this stage becomes

(5.11)

x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5

= x5 = c 2 + x 3 x5 = x 4 − x 2 x5 = −αx4 − βc2 x5 = u1

70

5 Switched Stabilization of a Hovercraft S1

Contoller 1

✂! ✂!

x=f(x)

S2

I/O

Controller 2 State dependent decision maker

O/P

S3 Controller 3

System

S4

Hovercraft

Controller 4 S5

Controller 5

G(x)U

Fig. 5.7. Switched finite-time controller architecture

We focus on the last two equations and define a new state vnew1 = −x5 βc2 − αx4 and a new input unew1 that is related to the actual input u1 by the relationship unew1 = v˙ new1 = u1 βc2 − αx˙ 4 . In terms of vnew1 and unew1 , the system (3.13) becomes

(5.12)

x˙ 1 x˙ 2 x˙ 3 x˙ 4 v˙ new1

−αx4 = −vnew1 βc2 −αx4 = x6 + x3 ( −vnew1 ) βc2 −vnew1 −αx4 = x4 − x2 ( ) βc2 = vnew1 = unew1

Now consider the subsystem x˙ 4 = vnew1 v˙ new1 = unew1 The following finite-time control law [10] (5.13)

1/3

5/3

unew1 = −vnew1 + (x4 + (3/5)vnew1 )1/3

5.3 Conventional Type Hovercraft

71

moves the states x4 and vnew1 to zero in finite-time t1 . At the end of the first stage we have x4 (t1 ) = x5 (t1 ) = 0 and x6 (t1 ) = c2 . Stage 2 Here, we regulate the state x6 to zero. For this stage we set u1 = 0. Recall x5 (t1 ) = 0 and consequently x˙ 5 = 0

⇒ x5 (t) = 0 ∀ t ≥ t1 .

From x4 (t1 ) = 0 and the dynamics of x4 we can also conclude that x4 (t) = 0 ∀ t ≥ t1 . We next focus on the subsystem x˙ 2 = x6 x˙ 6 = u2 1/3

and apply the control u2 (t) = −x6 (power rate reaching law [24]) to regulate x6 to zero in finite-time t2 . Unlike the previous stage where both the states of the double integrator are brought to zero, here we do not choose to bring x2 to zero as well; the reason for this is that x2 = 0 makes the dynamics of x3 uncontrollable. Note that this control law will however affect x2 . At time t2 we have x4 (t2 ) = x5 (t2 ) = x6 (t2 ) = 0. Stage 3 In this stage we zero the state x3 . For this we set u2 = 0. Recall x6 (t2 ) = 0 and consequently x˙ 6 = 0

⇒ x6 (t) = 0 ∀ t ≥ t2 .

Recall that x4 (t2 ) = 0 from the previous stage and from the dynamics of x4 we have x˙ 4 = 0 ⇒ x4 (t) = 0 ∀ t ≥ t2 To regulate x3 we use the following kinematic equation x1 = θ x2 = x cos θ + y sin θ x3 = −x sin θ + y cos θ From the inverse kinematics, let the value of x1 that assigns x3 = 0 be denoted by x1ref , which is

72

5 Switched Stabilization of a Hovercraft

x1ref = tan−1 (y/x) and further define x1e = x1ref − x1 . Now to reach x1ref , apply the following finite-time control law 1/3

u1 = −x5

5/3

− (x1e + (3/5)x5 )1/3

to the subsystem x˙ 1 = x5 x˙ 5 = u1 which results in x1e = 0 which further imply that x3 = 0 and vnew2 go to zero in finite-time t3 and at the end of which we have x3 (t3 ) = x4 (t3 ) = x5 (t3 ) = x6 (t3 ) = 0. Stage 4 In this stage we zero the state x2 . For this stage we set u1 = 0. Recall x5 (t3 ) = 0 and consequently x˙ 5 = 0

⇒ x5 (t) = 0 ∀ t ≥ t3 .

A similar argument holds for x4 . We focus on the subsystem x˙ 2 = x6 x˙ 6 = u2 for which we apply the finite-time controller [10] given by 1/3

u2 (t) = −x2

5/3

− (x2 + (3/5)x6 )1/3

The closed-loop system is stabilized in finite-time t4 . At the end of fourth stage we have x2 (t4 ) = x3 (t4 ) = x4 (t4 ) = x5 (t4 ) = x6 (t4 ) = 0. Stage 5 Towards the end, we steer x1 to zero. For this stage we set u2 (t) = 0 and now stabilize the rotational double integrator system x˙ 1 = x5 x˙ 5 = u1

5.3 Conventional Type Hovercraft

73

and use the following finite-time controller proposed in [10] γ

u1 = −sign(x5 )|x5 |γ − sign(sin(φ(x1 , x5 ))) | sin(φ(x1 , x5 ))| 2−γ , γ ∈ (0, 1) 1 where φ(x1 , x5 ) = x1 + 2−γ sign(x5 )|x5 |2−γ . This controller avoids the unwinding phenomenon which cannot be avoided using (5.13). We can now summarize the switching strategy as follows, where ui , i = 1, .., 5, corresponds to the control law in the ith stage. −unew1 −αx˙4 βc2

u1 =

0

0 1/3 −x6

u2 =

1/3

−x5

u3 =

u4 =

1/3

−x2

5/3

− (x1e + (3/5)x5 )1/3 0

0 5/3 − (x2 + (3/5)x6 )1/3

γ

5

u =

−sign(x5 )|x5 |γ − sign(sin(φ(x1 , x5 ))) | sin(φ(x1 , x5 ))| 2−γ 0

The stage-by-stage switching between two actuated velocities is shown in figure (5.8). 5.3.2 Discussion We first note that each of the subsystems is stabilized by a partial-feedback controller. This is a result of splitting up of the system into subsystems composed of double integrators. An immediate outcome of this closed-loop structure is that any disturbance that acts on the already regulated states results in additional switchings, with no upper bound on the number of switchings. However, in the absence of any disturbances on the states, the control law has

74

5 Switched Stabilization of a Hovercraft

x6 Stage 1

Stage 2 Stage 4

x Stage 3 & Stage 5

5

Fig. 5.8. Schematic of switching between actuated velocities

at most 5 switchings. In comparison with the controllers presented in [39], the proposed control law is valid for all initial conditions. We also note that we can interpret the stages 1 to 5 as Stage Stage Stage Stage Stage

1 2 3 4 5

: : : : :

Regulating the unactuated velocity to zero. Bringing the vehicle to rest. Homing the vehicle towards the target location. Moving the vehicle to the target point. Rotating the vehicle to the desired orientation.

5.3.3 Simulation The system is simulated using the following parameters [56] : m11 = 200 kg, m22 = 250 kg, m33 = 80 kg.m2 , d11 = 70 kgs−1 , d22 = 100 kgs−1 , d33 = 50 kgm2 s−1 . The control parameter γ = 1/3 and the initial conditions are x(0) = −3.7142 m, y(0) = −3.8993 m, θ(0) = 2 rad, vx (0) = 1 m/s, vy (0) = 3 m/s, ωz (0) = −4 rad/s. The time-response of the states and the control inputs is as shown in the figures (5.9-5.11). The path traced by the UUV and its motion is as shown in figures (5.12) and (5.13).

5.3 Conventional Type Hovercraft 20

θ x y

15

10

θ,x,y

5

0

−5

−10

−15

0

20

40

60

time in sec

80

100

120

Fig. 5.9. position and orientation 3

v y ω z v x

2

0

x y

v ,v ,ω

z

1

−1

−2

−3

−4

0

20

40

60

time in sec

Fig. 5.10. Velocities

80

100

120

75

5 Switched Stabilization of a Hovercraft 3500

Fx τz

3000

2500

F ,τ

x z

2000

1500

1000

500

0

−500

−1000

0

20

40

60

80

time in sec

100

120

Fig. 5.11. Input force and torque 20

15

10

5 y in meters

76

final configuration 0

initial configuration

−5

−10

−15

−20 −20

−15

−10

−5

0 x in meters

5

Fig. 5.12. Vehicle motion

10

15

20

5.3 Conventional Type Hovercraft 2 finla configuration

0

y in meters

−2

−4

initial configuration

−6

−8

−10

−12 −5

0

5

x in meters

10

Fig. 5.13. Path traced by the vehicle

15

20

77

6 Output Feedback Stabilization of a Mobile Robot

In all the previous chapters, we assumed full-state information for the purpose of controller synthesis. Here we propose an observer scheme for one of the applications presented before - the wheeled mobile robot model. By using position measurements alone, we stabilize the model at its equilibrium.

6.1 Observer Design The controller design phase is split into two parts. The first one is an observer design in which the non-measurable states are predicted through a nonlinear observer whose error dynamics converges to zero in finite time. Nonlinear observers are designed in the literature using various techniques [70, 29, 16, 4] and the convergence of the closed-loop system (with observer) is proved using the ISS (Input-to-State Stability) property of the system. But here we design an observer whose error dynamics converges to zero in finite-time which helps to prove the convergence of the closed-loop system without resorting to the ISS property of the system. Later a discontinuous controller, on the same lines as described before, is designed using the observed states. The block diagram of the closed loop system is as shown in figure(3). Consider a nonlinear observer of the form u, y , xˆ ) xˆ˙ = f1 (ˆ x ) + l(u where xˆ is the observer state and is defined as xˆ = [ξ1 ξ2 x ˆ4 x ˆ5 ] where ξ1 and ξ2 denote dummy variables and x ˆ4 and x ˆ5 denote the estimates of x4 and x5 respectively. The proposed observer dynamics are

R.N. Banavar and V. Sankaranarayanan: Switch. Fin. Time Contrl., LNCIS 333, pp. 79–86, 2006. © Springer-Verlag Berlin Heidelberg 2006

80

6 Output Feedback Stabilization of a Mobile Robot

Controller

Control Input

Mobile robot

Position

Velocity

Observer

Fig. 6.1. Controller and Observer

ξ˙1 = x ˆ4 + (x1 − ξ1 )1/3 ˆ˙x4 = u1 + (x1 − ξ1 ) ξ˙2 = x ˆ5 + (x2 − x ˆ5 )1/3 ˆ˙x5 = u2 + (x2 − x ˆ5 ) Notice that the observer has a non-conventional structure in the sense that we have two states ξ1 and ξ2 that come in mainly from the dynamic structure associated with the observer. Defining the error variables as e1 = x 1 − ξ 1 e2 = x 2 − ξ 2 e4 = x 4 − x ˆ4 e5 = x 5 − x ˆ5 The error dynamics becomes 1/3

e˙ 1 = e4 − e1 e˙ 4 = −e1

1/3

e˙ 2 = e5 − e2 (6.1)

e˙ 5 = −e2

As can be seen, the error dynamics has a finite-time stable property.

6.2 Controller Design

81

6.2 Controller Design Incorporating the observer dynamics, the augmented system can be written as x¯˙ = f¯(¯ x) +

(6.2)

2

u g¯(¯ x )u i=1

where x¯ = (x1 , x2 , x3 , x4 , x5 , ξ1 , ξ2 , x ˆ4 , x ˆ5 )      0 0 x4 0 0 x5 0 0 x1 x5 − x 2 x4 0 1 0 x¯) = 1 ; g¯1 (¯ x ) = 0 ; g¯2 (¯ f¯(¯ x¯) = 0 0 0 x ˆ4 + (x1 − ξ1 )1/3 0 1 (x1 − ξ1 ) 1/3 0 0 x ˆ5 + (x2 − x ˆ5 )  1 0 (x2 − x ˆ5 ) 

and x¯ ∈ L = S 2 × R7 The control technique, as mentioned before, is to reach a connected set in finite-time in which the closed-loop system is asymptotically stable. This connected set is defined as a intersection of three sets. To proceed we define the following functions S1 (¯ x¯) = kx1 + x4 S2 (¯ x¯) = x1 x5 − x2 x4 + k3 x3 and the sets E, M1 , K E = {¯ x¯ ∈ L : x1 = ξ1 , x2 = ξ2 , x4 = x ˆ 4 , x5 = x ˆ5 } M1 = {¯ x ∈ E : x4 = −kx1 } K = {¯ x ∈ E : x4 = −kx1 and x1 x5 − x2 x4 = −k3 x3 } Notice that on the set E, the state has been estimated perfectly and we have the following inclusion K ⊂ M1 ⊂ E

82

6 Output Feedback Stabilization of a Mobile Robot

Proposition 6.1. The origin of (6.2) is globally attractive with the following control law (6.3) (6.4) u2 =

u1 = −kˆ x4 − (kx1 + x ˆ4 )1/3 if (x1 , x ˆ4 ) = (0, 0) −(x1 x ˆ5 +x2 x ˆ4 +x3 )1/3 x1

− k3 x ˆ5 − (k3 − k)kx2 x¯ ∈ M1 \ {(x1 , x ˆ4 ) = (0, 0)} −x2 − x ˆ5 otherwise

Proof: (Global attractivity ) From the estimator dynamics (6.1), there exists a finite-time T1 ≥ 0 such that the trajectories enters the set E and stay there for all future time. The closed-loop dynamics for (x1 , x ˆ4 ) = 0 and x¯ ∈ / M1 becomes x˙ 1 = x4 x˙ 2 = x5 x˙ 3 = x1 x5 − x2 x4 x˙ 4 = −kˆ x4 − (kx1 + x ˆ4 )1/3 x˙ 5 = −x2 − x ˆ5 ξ˙1 = x ˆ4 + (x1 − ξ1 )1/3 ˆ˙x4 = u1 + (x1 − ξ1 ) ξ˙2 = x ˆ5 + (x2 − x ˆ5 )1/3 ˆ˙x5 = u2 + (x2 − x ˆ5 ) The dynamics of S1 (¯ x¯) is S˙ 1 (¯ x¯) = −S1 (¯ x¯)1/3 + g(e4 ) For t ≥ T1 the dynamics of S1 (¯ x¯) becomes S˙ 1 (¯ x¯) = −S1 (¯ x¯)1/3 which implies finite-time stability. So there exists a time T2 ≥ T1 such that the closed-loop trajectory for any permissible initial condition reaches the set M1 and stays there for all future time. The control law u1 allows both x1 and x4 to converge to zero as time gets large. At the same time as the system is being driven towards M1 , the PD control law u2 with unity gain makes states x2 and x5 converge to zero.

6.3 Simulation

83

The closed-loop system on M1 is x˙ 1 = −kx1 x˙ 2 = x5 x˙ 3 = x1 x5 − x2 x4 x˙ 4 = −kx4 x ¯)1/3 −S2 (¯ − k3 x5 − (k3 − k)kx2 x˙ 5 = x1 The dynamics of S2 (¯ x¯) is S˙ 2 (¯ x¯) = −S2 (¯ x¯)1/3 So there exists a time T3 ≥ T2 such that the closed-loop trajectories reach the set K and stay there for all future time. The closed-loop system on K is x˙ 1 = −kx1 x˙ 2 = x5 x˙ 3 = −x3 x˙ 4 = −kx4 x˙ 5 = −k3 x5 − (k3 − k )kx2 All trajectories exponentially converge to the origin. (If the system starts with the initial condition x1 = x ˆ4 = 0, then an open-loop control law can perturb the system from this initial condition.).

6.3 Simulation We simulate the controller for k = 0.5, k3 = 1, α = 1/3. The initial position and the orientation of the vehicle is x(0) = −1.5m, y(0) = 4m and θ(0) = −2.3rad. The initial velocities of the vehicle are ω(0) = −3rad/sec and v(0) = −1m/sec. The initial conditions of the observer are ξ1 (0) = 3 ,ξ2 (0) = −2, x ˆ4 (0) = 4, x ˆ5 (0) = 1. The vehicle parameters are M = 10Kg, I = 2Kgm2 , L = 6cm, R = 3cm. The simulations results are shown in figures (6.2- 6.6)

6 Output Feedback Stabilization of a Mobile Robot 8

θ x y

6 4 2

x,y,θ

0 −2 −4 −6 −8 −10 0

10

20

time in sec

30

40

50

Fig. 6.2. Stabilization to the origin 1.5

τ1 τ2

1

0.5 τ1,τ2

84

0

−0.5

−1

−1.5 0

10

20

time in sec

30

Fig. 6.3. Input to the motors

40

50

6.3 Simulation 15

10

y in meters

5 initial configuration 0

final configuration

−5

−10

−15 −15

−10

−5

0 x in meters

5

10

15

Fig. 6.4. Path traced by the vehicle 4

ξ1 ξ2 hat x4 hat x5

3 2

ξ1,ξ2,hat x4,hat x5

1 0 −1 −2 −3 −4 −5 −6 0

10

20

time in sec

30

Fig. 6.5. Observer states

40

50

85

6 Output Feedback Stabilization of a Mobile Robot 1

e1 e2

0

−1

−2 e1,e2

86

−3

−4

−5

−6 0

5

10

15

20

25 30 time in sec

Fig. 6.6. Observer errors

35

40

45

50

A Examples and Proofs

Here we present sufficient conditions to check certain nonlinear controllability notions. Later we prove important properties of the examples considered. A.0.1 Nonlinear Controllability In this section we present sufficient conditions to check different nonlinear controllability notions Lie algebraic conditions Lie algebraic sufficient conditions exist for the types of controllability presented in chapter 2. We first define a few definitions which involve some basic concepts from Lie algebra. Definition A.1. V ∞ (M ) : It is the linear space of C ∞ vector fields on M with the bilinear operation being the Lie bracket of vector fields. In fact V ∞ (M ) with the Lie bracket is an infinite dimensional Lie algebra. Consider the vector fields f, g on IR2 given by f=

x21 0

;g =

0 x21

Then, [f, g] =

0 −2x31

; [f, [f, g2 ]] =

0 6x41

.

It can be shown by induction that the vector field (0, xn1 ) is contained in V ∞ (M ). Therefore, V ∞ (M ) is an infinite dimensional algebra of vector fields.

R.N. Banavar and V. Sankaranarayanan: Switch. Fin. Time Contrl., LNCIS 333, pp. 87–92, 2006. © Springer-Verlag Berlin Heidelberg 2006

88

A Examples and Proofs

Consider the following system m

x)ui , gi (x

x) + x˙ = f (x

x (0) = x 0

i=1

(A.1)

x) y = h(x

where x = (x1 , . . . , xn ) are local coordinates for a smooth manifold M , and f, g1 , . . . , gm are smooth vector fields on M with f (0) = 0. The control vector u : [0, T ] −→ Ω ⊆ IRm . The output y ∈ IRm is a smooth function of x with h(0) = 0. ¯ Definition A.2. Accessibility algebra C ¯ is the smallest subalConsider the system (A.1). The accessibility algebra C gebra of V ∞ (M ) that contains f, g1 , . . . , gm . Definition A.3. Accessibility distribution C ¯ It is the distribution generated by the accessibility algebra C. ¯ x ∈ M }. C(x) = span{X(x) : X ∈ C, The sufficient condition for local accessibility is presented in the following proposition. Proposition A.4. If dim(C(x)) = 2n

∀x ∈ M,

then the system is locally accessible at x. The above criterion is termed accessibility rank condition ¯ which contains Definition A.5. Define Cˆ0 as the smallest subalgebra of C ˆ ˆ g1 , . . . , gm and satisfies [f, X] ∈ C0 , ∀X ∈ C0 . Define the distribution C0 (x) = span{X(x) : X ∈ Cˆ0 , x ∈ M }. ˆ 0 and C0 are called the strong accessibility algebra and strong accessibility C distribution respectively. We now state the theorem concerning the strong accessibility. Proposition A.6. Consider the system (A.1). Suppose that (A.2)

dim(C0 (x0 )) = 2n,

then the system is locally strongly accessible from x0 . Condition (A.2) is called the strong accessibility rank condition at x 0 .

A Examples and Proofs

89

Small-time local controllability Sussmann [68] showed a general sufficient condition for STLC of systems with drift. A particular case of this result known as the Bianchini and Stefani condition has been used by Reyhanoglu et al. [58] to prove the STLC property for underactuated systems. We now present their results which center around identifying the so called good and bad brackets and their relationship. ¯ . Let δ 0 (B), δ 1 (B), . . . , δ m (B) denote the Let B denote any bracket in C number of times f, g1 , . . . , gm occur in the bracket B respectively. Define m δ(B) = i=0 δ i (B ) and let an admissible weight vector be l = (l0 , l1 , . . . , lm ) such that li ≥ l0 ≥ 0, i = 1, . . . , m. The l-degree of B is defined as lm degree(B) = i=0 li δ i (B). A bracket which has δ 0 (B) odd and δ i (B) even for each i = 1, . . . , m is called bad. For example the bracket [g1 , [f, g1 ]] is a bad bracket since δ 0 (B) is 1 and δ 1 (B) = 2. The above concepts are made clear by considering a two-input system. For admissible weights l0 = 2, l1 = 2, l2 = 2, the good brackets along with their l-degree are tabulated as shown in Table (A.1). The corresponding table for some of the bad brackets is as shown in Table (A.2). From the tables it clearly follows that δ(B) for a bad bracket is odd and the l-degree for the

Table A.1. Good brackets δ 0 (B) δ 1 (B) δ 2 (B) δ(B) l-degree(B)

Vector B g1

0

1

0

1

2

g2

0

0

1

1

2

[f, g1 ]

1

1

0

2

4

[f, g2 ]

1

0

1

2

4

[g1 , [f, g2 ]]

1

1

1

3

6

[f, [g1 , [f, g2 ]]]

2

1

1

4

8

Table A.2. Bad brackets Vector B f

δ 0 (B) δ 1 (B) δ 2 (B) δ(B) l-degree(B) 1

0

0

1

2

[g1 , [f, g1 ]]

1

2

0

3

6

[g2 , [g2 , [g1 , [f, g1 ]]]]

1

2

2

5

10

90

A Examples and Proofs

spanning good brackets is < 10. For bad brackets with δ(B) ≥ 5, the l-degree is ≥ 10. The Bianchini and Stefani condition for STLC for a strongly accessible system is that the bad brackets must be a linear combination of good brackets of lower l-degree at the equilibrium. Then the bad brackets are said to be lneutralized. We now state a general theorem by Sussmann for STLC of an affine control system. Theorem A.7. A sufficient condition for the STLC at the equilibrium of an affine system (A.1) is 1. The system satisfies the Lie algebra rank condition at the equilibrium. 2. Every bad bracket is a linear combination of good brackets of lower l-degree at the equilibrium. A.0.2 Properties of the NI Linear controllability Let x e , u e denote the equilibrium point of the NI. Linearization around this point with ue = 0 can be written as x + Bu u x˙ = Ax where A= The matrix A, B are

∂f ∂f xe , u e ) B = xe , u e ) (x (x x u ∂x ∂u



 000   A = 0 0 0, 000



 1 0   B= 0 1  −x2e x1e

The linear system is controllable iff the controllability matrix C = [B AB A2 B] has rank 3. But the matrix

 1 0 0000   C= 0 1 0 0 0 0 −x2e x1e 0 0 0 0 

has rank two. So the NI is not linearly controllable.

A Examples and Proofs

91

Brockett’s condition Consider the third condition of Brockett’s theorem. Consider a point of the form (0, 0, ) ( = 0) contained in any arbitrary neighborhood (open set) of the x, u ) −→ (u1 , u2 , x1 u2 − origin. This point is not in the image of the mapping (x x2 u1 ). STLC The NI can be rewritten as x)u1 + g2 (x x)u2 x˙ = g1 (x where x) = (1, 0, −x2 )T g1 (x

g2 (x ) = (0, 1, x1 )T

The Lie bracket ∂g2 ∂g1 g1 (x) − g2 (x) x x ∂x ∂x = (0, 0, 2)T

[g1 , g2 ] =

Since the strong accessibility distribution C0 = span{g1 , g2 , [g1 , g2 ]} has rank 3, the NI is STLC. Please note that for drift-less systems, the strong accessability condition and STLC condition are same. A.0.3 Properties of the ENDI The proof of the properties listed in chapter 3 is presented here. Linear controllability It is trivial as seen from the NI case Brockett’s condition Consider a point of the form (0, 0, , 0, 0) ( = 0) belonging to any neighborhood of the origin. It is easy to note that this point is not in the image of the x, u ) −→ (x1 , y1 , x1 y2 − x2 y1 , u1 , u2 ). Thus Brockett’s necessary condimap (x tion not satisfied. Strong accessibility Consider the Lie brackets       1 0 0        0   1   0             [g1 , f ] =   −x2  , [g2 , f ] =  x1  , [g2 , [f, [g1 , f ]]] =  −2         0   0   0  0 0 0

92

A Examples and Proofs

The strong accessability distribution [g1 , g2 , [g1 , f ], [g2 , f ], [g2 , [f, [g1 , f ]]] has rank 5, hence the ENDI is strongly accessible from any x . STLC The necessary condition for STLC is that the bad brackets of the strong accessibility distribution are a linear combination of the good brackets of lower degree. All the brackets in the accessibility distribution are good and height order is 4. The bad brackets of degree lesser then 4 are f and [g1 , [f, g1 ]], [g2 , [f, g2 ]]. All these brackets vanish at the equilibrium. So it can be expressed by the linear combination of the good brackets. Hence the ENDI is STLC. A.0.4 Properties of an Underwater Vehicle / Surface Vessel The proof of the properties mentioned in chapter 3 is presented here. Brockett’s condition Consider a point of the form (0, 0, , 0, 0, 0) ( = 0) in any neighbourhood of x, u ) −→ (x1 , x6 + x3 x5 , x4 − the origin. It is not in the image of the mapping (x x2 x5 , −αx4 −βx5 x6 , u1 , u2 ). Hence the underwater vehicle does not satisfy the Brockett’s necessary condition. Strong accessibility Consider the Lie brackets [f, g1 ], [f, g2 ], [g2 , [f, g1 ]], [[f, g2 ], [f, g1 ]] and the following distribution [g1 , g2 , [f, g1 ], [f, g2 ], [g2 , [f, g1 ]], [[f, g2 ], [f, g1 ]]]. This has dimension 6. Hence the underwater vehicle is locally strongly accessible. STLC To prove STLC, consider the bad brackets of order less than 4. All the bad brackets are vanishes at the equilibrium point. Thus the underwater vehicle is STLC from any equilibrium point.

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Index

acceleration level constraints, 10 acrobot, 11 asymptotically stable, 13

locally accessible, 20 Lyapunov stability, 12 Lyapunov theorem, 13

bead in a slot, 10

nonholonomic system, 7

exponentially stable, 13

relative asymptotic stability, 16 relative exponential stability, 16 relative stability, 14 rolling coin, 8

finite-time convergence, 18 finite-time differential equation, 17 finite-time stability, 17 global asymptotic stability, 13 holonomic constraint, 7 holonomic system, 7 La Salle’s theorem, 14 Lipschitz, 12

slosh, 11 small time locally controllable, 20 stabilizability, 20 underactuated, 10 velocity level constraints, 7

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