## G kiZi Z2

K (m (sin(z1) (cos(z1) 1)ky) C v ((cos(z1) 1) C ky sin(z1))) (1 C (ky)2)3 2 j This abuse of notation is introduced for simplicity and is possible because and we have shown that the closed loop system is forward complete. We now verify all of the conditions of Lemma 9.1. Verifying Condition C1. We take the function V2 0.5v2 whose time derivative along the solutions of the differential equation v (t, v), see (9.21), satisfies 2 wiimv k2wy kv v vj V2----y- C -Tw (t) m11kumaxMl Ml v2 C llk max C w...

## UotZe sinfe Ud cosfe v

Where k,k2, and 1e are positive constants to be selected later, u0(t,ze) 0, > t0 > 0, and ze e R, is the speed of the virtual ship on the path. Indeed, one can choose this speed to be a constant. However, the time-varying speed and position path-following dependence of the virtual ship on the path is more desirable, especially when the ship starts to follow the path. For example, one might choose uo(t, Ze) < (1 - Xie*2(i-io))e-X3Ze, aU2) where mJ 0, > 0, i 1, 2, 3, 1 < 1. The choice...

## Trajectorytracking Control of Underactuated Ships

This chapter addresses the problem of trajectory-tracking control of underactuated surface ships. In particular, we present a method to design a controller for under-actuated surface ships with only surge force and yaw moment available to globally asymptotically track a reference trajectory generated by a suitable virtual ship. The reference yaw velocity does not have to satisfy a persistently exciting condition as was often required in previous literature. Hence, the reference trajectory is...

## Where a d22m22 ft w11w22 and

It can be seen that the system (4.29) consists of two subsystems, namely (x1 , x2 , x3 , x4) and (x5,x6), connected to each other in a strict feedback form 3 . The control design can be simply carried out in two steps as follows. In this step, the author of 55 considers the first four equations of (4.29), and (x5 ,x6) as controls (v1,v2). With the assumption of x1 0, the coordinate transformation (ct -process) y X1,Z1 X2, Z2 , Z3 (4.31) The feedback control law is designed as V2 -k2iZi -k22Z2...

## Pathtracking Control of Underactuated Ships

This chapter deals with the problem of designing controllers to force an underactuated surface ship with standard dynamics to track a reference path. Both full-state feedback and output feedback cases are considered. In comparison with the preceding three chapters, the requirement that the reference trajectories be generated by virtual ships is relaxed, and the control design is simpler and more amenable for implementation in practice. The control development is based on a series of coordinate...

## Coordinate Transformations

Since designing the control inputs t , t , and Tr to achieve the control directly from (12.1) and (12.2) is difficult, we interpret the tracking errors in a frame attached to the vehicle body as follows cos(f) cos(0) sin(f) cos(0) sin(0) 0 0 sin(0) cos(f) sin(0) sin(f) cos(0) 0 0 cos(f) cos(0) sin(f) cos(0) sin(0) 0 0 sin(0) cos(f) sin(0) sin(f) cos(0) 0 0 Indeed, convergence of (xe, ye, ze, 0e, f e) to the origin implies that of (x xd, y -yd, z zd ,0 0d, f f d) since the matrix cos(f) cos(0)...

## Mathematical Preliminaries

This chapter presents mathematical tools, which will be used in control design and stability analysis in the subsequent chapters. Some standard theorems, lemmas and corollaries, which are available in references, are sometimes given without a proof. Stability theory is important in system theory and engineering. For a controlled system to be usable in practice, the least requirement is that the system is stable under unknown disturbances or noise. There are various types of stability problems...

## R 1 x x yi i y2 y3 A

Where yi, y2, 73 are class functions such that yi( x ) < V(x) < y2( x ), y3( x ) < W(x). Proof. By using (2.68) and (2.70), the first time derivative of V(x) is x Cga k( g)2 > '> 2C TA V 2 2 < W(x) kig g) ' 2C From (2.71), it follows that V is negative whenever W(x) > . Combining this with the second equation of (2.70), we conclude that This means that if x(0) < y3- i ( k1), then If, on the other hand, x(0) > y- i , then V(x) < V(x(0)), which implies x(t) < y - i 1 y2(...

## Modeling of Ocean Vessels

In this chapter, we classify the basic motion tasks for ocean vessels and their mathematical models, which will be used for the design of various control systems in the subsequent chapters. In automatic control, feedback improves system performance by allowing the successful completion of a task even in the presence of external disturbances and initial errors, and inaccuracy of the system parameters. To this end, real-time sensor measurements are used to reconstruct the vehicle state....

## Pz k 2 py

Applying 5.52 to 5.21 and noting Remark 5.3 yields the condition C2 readily. Verifying Condition C3. This condition is verified by showing that the X2e -subsystem is GES. Take the following quadratic function V2 2 z2 C 52 C re2 5.53 Differentiating 5.53 along the solutions of the X2e-subsystem satisfies V2 lt - k3 - 1 5 Ze2 - 1l C - 0 5 2 A12 C - O e2 P2 min 2 3 - 1 5 , 2 V C -0 5 , 2 2 C - AY V V mil V m33 77 Thus it suffices to choose the design constants k3, 1i, and 12 such that Hence...

## Simultaneous Stabilization and Trajectorytracking Control of Underactuated Ships

This chapter examines the problem of designing a single controller that achieves both stabilization and tracking simultaneously for underactuated surface ships without an independent sway actuator and with simplified dynamics. The proposed controller guarantees that stabilization and tracking errors converge to zero asymptotically from any initial values. In comparison with the preceding chapter, a path approaching the origin and a set-point can also be included in the reference trajectory,...

## Control of Ships and Underwater

Design for Underactuated and Nonlinear Marine Systems Other titles published in this series Digital Controller Implementation and Fragility Robert S.H. Istepanian and James F. Whidborne Eds. Optimisation of Industrial Processes at Supervisory Level Doris Saez, Aldo Cipriano and Andrzej W. Ordys Robust Control of Diesel Ship Propulsion Nikolaos Xiros Hydraulic Servo-systems Mohieddine Jelali and Andreas Kroll Model-based Fault Diagnosis in Dynamic Systems Using Identification Techniques Silvio...