Output-Feedback Control of an Underwater Vehicle Prototype: Theory and Experiments

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Copyright © IFAC Robust Control Design Milan, Italy, 2003

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OUTPUT-FEEDBACK CONTROL OF AN UNDERWATER VEHICLE PROTOTYPE: THEORY AND EXPERIMENTS. 1 Alessandro PISANO * Elio USAI *

* Department of Electrical and Electronic Engineering University of Cagliari - Piazza d' Armi - 09123 Cagliari Italy - Tel. +39-070-675.5869 - fax +39-070-675.5900 E-mail: {pisano.eusai}@diee.unica.it

Abstract: This paper deals with the motion control problem for an underwater vehicle (UV) prototype built at the University of Cagliari. The vehicle is equipped with a special propulsion system based on hydro-jets with variable-section nozzles. To cope with the heavy uncertainties affecting the prototype dynamics, the control system has been developed by means of the second-order sliding-mode control (2SMC) approach. Chattering is removed since the 2-SMC approach confines the high-frequency switching to the higher derivatives of the control variable. The UV velocity is estimated in real-time by means of a recently-proposed variablestructure robust differentiator, and the experimental results show that a fast and accurate response can be obtained under several operating conditions. The control system and the main implementation issues are discussed in some detail. Copyright © 2003 IFAC

Keywords: Second-order sliding-modes, underwater vehicles, output-feedback, uncertain systems, nonlinear systems, real-time differentiators.

1. INTRODUCTION

work we refer to the combined use of second-order sliding-mode controllers (2-SMC) (G. Bartolini and Usai, 1999) and robust variable-structure differentiators also based on the 2-SM approach (Levant, 1998).

This paper reports the underlying theory and the main practical implementation issues related to the position control of an underwater vehicle prototype (A utonomous Jet-propelled Object with 1 d.o.j., "AJO-1") developed at the the Dept. of Electrical and Electronic Engineering (DIEE), Cagliari University, as a test bed for high-precision underwater motion control techniques. The considered underwater vehicle (UV) is equipped with a special water-jet propulsion system.

A nonlinear separation principle relevant to the combined use of 2-SM controllers and 2-SM differentiators has been demonstrated in (G. Bartolini and Usai, 2000) for single-input systems combining a 2-SMC and a 2-SM differentiator of order p = 1. By relying on this stability result, we propose an output feedback control system for an UV prototype, which has been successfully implemented in real experiments.

The uncertain and unmodeled dynamics of the vehicle and of the propulsion systems should be both explicitly taken into account in designing the output-feedback control system, and in this

The paper is organized as follows: in Section 2 we recall the basic design principles of outputfeedback 2-SM control. In Sect. 3 a mathematical model of the UV dynamics is given, and in the suc-

I Partially supported by the MURST Project "Navigation, Guidance and Control of Autonomous Underwater Vechicles" .

449

cessive Section 4 the proposed approach is applied to design a robust output-feedback motion controller for the UV prototype. Section 5 describes the experimental setup and discusses the results of the experiments, and in Section 6 some final conclusions are drawn.

and Usai, 2002). It is generally characterized by the three-step procedure described hereafter for the readers' convenience. Step 1 - Sliding manifold design

The sliding variable is expressed as a linear combination of the tracking error e and its first (r - 1) total derivatives

2. NONLINEAR OUTPUT-FEEDBACK CONTROL VIA 2-SM CONTROLLERS AND 2-SM DIFFERENTIATORS

r-2

a

y

= j(x)

= h(x)

+ g(x)u

where e(O) =

(1)

and the coefficients Ci, (i = 1,2, ...,

error space, onto which the output tracking error e converges to zero exponentially. Step 2 - Estimation of the sliding variable

The error derivatives are not available and must be evaluated by some real-time device with possibly finite-time convergence. Recently proposed (Levant, 2003), the variable structure differentiator of order r - 1 based on higher-order sliding modes, reported below, appear to be an effective, yet robust, solution: r-1

Zo = Z1 - Kolzo - e(t)l-r sign(zo - e(t)) r-2 Z1 = Z2 - K11zo - e(t)\-2- sign(zo - e(t)) ... (6) .,..-l-i Zi = Zi - Kdzo - e(t)l-r- sign(zo - e(t))

where L g , L f denote the Lie derivatives. Define

il, ... ,y(r-1)].

Zr-1

The "internal dynamics" (Isidori, 1995), can be expressed in the following form

= -Kr -1

sign(zo - e(t))

where Ki (i = 1,2, ... ,r - 1) are suitable positive constant coefficients. After a finite-time, conditions IZi - e(i) I = 0, (i = 0, 1, ... , r -1) are ideally established (Levant, 2003).

(3)

where "I E Rn-r is such that the map x = (~, "I) is a diffeomorfism on Rn. Note that if r = n there are no internal dynamics and the system is said to be "fully linearizable" (Isidori, 1995).

See (G. Bartolini and Usai, 2002; Levant, 2003) for the tuning procedure and robustness analysis. If r = 2 then only the first derivative of e needs to be estimated, and system (6) reduces to the first-order sliding differentiator (Levant, 1998)

Let us make the following assumption: Assumption 1: The internal dynamics (3)

e

r-2), are chosen such that the polynomial P()..) = )..(r-1) + L:~,:-g Ci)..(i) is Hurwitz. The vanishing of a defines the so-called "sliding manifold" in the

with unavailable state vector x E Rn, control variable u E R and measurable output y E R. Let j, g and h be unknown, sufficiently smooth, vector-fields of appropriate dimension satisfying proper growth constraints to be specified. The heavy uncertainty of the system prevents immediate reduction of (1) to any normal form by means of standard approaches based on the knowledge of j, g and h. If conditions LgLfh(x) = LgL1h(x) = ... = L gL,-2h(x) = 0 and L gL,-1h(x) =f. 0 hold globally then system (1) possesses a globallydefined relative degree r (Isidori, 1995), and the input-output dynamics can be expressed as

~ = [y,

(5)

i=O

Consider the nonlinear SISO system i;

= e(r-1) + L cie(i)

f(t) = zo(t) - e(t) zo(t) = Z1(t) - Kolf(tW/ 2 sign(f(t))

~s

input-to-state stable (ISS)

Z1(t)

Let YR be the desired output response, and consider the error dynamics

= -K1 sign(f(t))

(7)

The tuning conditions are

(8)

where e

=Y -

YR.

The task is to stabilize the uncertain error dynamics (4) by error feedback. The output-feedback control problem with higher-order sliding modes has been deal with in a recent work (G. Bartolini

The differentiation accuracy is proportional to C~/2 c: 1 / 2, where c: is the maximal input noise magnitude (Levant, 1998).

450

If r > 2, to estimate the higher-order derivatives one could implement a cascade series of differentiators (7). By performing the noise propagation analysis, however, it follows that the cascade of differentiation stages is not satisfactory for multiple differentiation purposes, while it is more effective to implement the higher-order differentiator (6), specially designed for multiple differentiation task.

Remark 2 The actual plant input u(t) is continuous, since it is obtained at the output of an integrator having as input the discontinuous highfrequency-switching signal generated by the Suboptimal controller. Chattering is thus removed. In the particular case of r = 2, it is also proven (G. Bartolini and Usai, 2000) that the outputfeedback control scheme obtained by coupling the sup-optimal controller and the first-order differentiator (7) is stable. The proof of the general case r > 2 is not still available.

Step 3 - Stabilization of the sliding variable Consider the nonlinear uncertain second-order sliding variable dynamics if = cp(~, ~R, 1], u)

+ ,(~, 1])u

3. THE UV PROTOTYPE

(9)

An UV prototype has been recently built at the DIEE-University of Cagliari as a preliminary testbed of a novel propulsion system for underwater vehicles based on water-jets instead of conventional rotating thrusters.

and assume what follows: Assumption 2: If~, ~R, 1], u evolve within an arbitrarily-large compact domain, three positive constants F, rI, r 2 can be found such that the uncertainties cp and , satisfy the boundedness conditions

The vehicle is about 150cm long and 80cm high. It contains a centrifugal pump feeding an hydraulic circuit and two variable-section nozzles located at the opposite edges of the hydraulic circuit and actuated by linear motor drives (Fig. 1).

(10)

The prototype is rigidly connected with a wheeled trolley that "suspends" the UV into a water channel (Fig. 2). This configuration allows the UV to move freely along the channel under the reaction force exerted by the water flow through the nozzles.

If Assumption 2 holds then it is possible to steer to zero both a and if in a finite time by means of a discontinous feedback u = u(a) enforcing a 2SM on the sliding manifold a = 0 (G. Bartolini and Usai, 2001 a; G. Bartolini and Usai, 1999; Levant, 1993). The so-called "Sub-optimal' 2SMC algorithm (G. Bartolini and Usai, 2001a; G. Bartolini and Usai, 1999) is one of the possible solutions:

VARIABLE-SECTION NOZZLES

Sub-optimal algorithm

u(O) = {O

.

1

0::; t < tMI

-UMSlgn(u(t) - 2u(O»

u(t)

.(t)

=

=

(11)

1

-a(t)UMsign(u(t) - 2u(tM,) tM; ::; t

t

< tMi+1

= 1,2, ...

i

1

U(tMi )(u(t) - 2u(tMi» ::; 0 (12)

PUMP

where tMi (i = 1,2, ...) is the sequence of the time instants at which if = 0 (approximately detected using only measurements of a (G. Bartolini and Usai, 2001a)), and the controller parameters UM and 0:* are chosen according to the tuning rules

r2

0:*

UM

> 3r 1

Fig. 1. The DV prototype The nozzle sections can be varied by moving two spear valves by means of directly-coupled linear electric drives (Fig. 5). The spear valve profile is designed so that the generated thrust is turbulence-free and almost-linearly dependent on the valve position. Fig. 3 shows the hydro-jets in two different nozzle opening conditions. The direct mechanical coupling between the valve and

(13)

4F

> max{ 30:* r 1

-

F r ' -r } 2

1

(14)

451

With a good approximation it can be assumed that F 1 (t) and F 2 (t) depends instantaneously on the positions, Yl and Y2, of the spear valves by some nonlinear function h (16)

The positions of the two spear valves are defined according to the notation represented in Fig. 5: Yl (Y2) is zero when the nozzle is closed and increases while the spear valve is opening. Thus, Yl and Y2 remains always non-negative.

Fig. 2. The DV prototype in the water channel the motor, and the large bandwidth of the latter, allow for a very fast control of thrust profile. ALMOST-CLOSED NOZZLE

~"iIiE.I. Directly-eoupled linear motor drive

ALMOST-OPEN NOZZLE

to"

spear valve

Fig. 5. Fig. 3. The water jets with open and closed nozzle

ozzle position notation

Function h is very difficult to determine, and therefore is considered uncertain in the present treatment. Obviously it is strictly positive, monotonically increasing and zero when its argument is zero.

A cascade compensation scheme for the control system is implemented (Fig. 4): the "high-level" vehicle controller feeds the LM controller with reference profiles for the linear motor positions.

The system parameters Mu, Ma, k 1 , k 2 as well as the disturbance ~(t) are unknown. Collecting the uncertainties affecting the DV dynamics and setting g = l/(Ma + Mu) the system dynamics can be rearranged as follows

i = f(x, t)

+ g[h(yd

- h(Y2))

(17)

with implicit definition of function f(x, t) Fig. 4. A schematic representation of the control architecture

5. CONTROLLER DESIGN Let x· be a smooth reference profile for the vehicle position, and let X;, X; be a-priori defined constants such that

4. DYNAMICAL DV MODELING

Ix·'

The dynamics of the considered jet-propelled DV can be formulated as follows (Fossen, 1994)

~

X;

Ix·'

~

X;

(18)

Define the tracking error and its derivative (15)

e = x - x·

e = x-x·

(19)

By (17), the relative degree between the position error e and the control variables Yl and Y2 is r = 2.

where x is the vehicle position, Mu is the vehicle mass, Ma represents the added mass effect, k 1 and k 2 are the viscous friction and drag coefficients, ~(t) accounts for the external disturbances (e.g. ocean currents, border effects) and F 1 (t), F 2 (t) are the control thrusts exerted on the DV by the two opposite jets.

Sliding manifold design: according to (5), the sliding variable is defined as

a(t) = e(t)

452

+ ce(t)

c>o

(20)

where e = x-x· and ~ = Zl is computed by using differentiator (7). The reference position profiles for the linear motors are set according to (28), where is defined according to the Sub-optimal 2-SMC algorithm (11)-(14).

Estimation of the sliding variable: the actual vehicle velocity error e is estimated by means of the differentiator (7), with the parameters "'0 and "'I set on the basis of (8) where Cz is sufficiently large so that the followig condition is met

a;

(21)

By relying on the separation principle demonstrated in (G. Bartolini and Usai, 2000) it can be asserted that the following conditions are ideally simultaneously fulfilled after a finite time

Stabilization of the sliding variable: a desired profile for the spear-valves coordinates Yl and yz is defined to guarantee that the sliding variable (J is reduced to zero in finite time.

I~

The first-order dynamics of the sliding variable can be expressed as.

a=

x· + ce(t) + gA(Yl, yz)

f(±, t) -

(22)

Define the dummy control variable

IJ - el ::; Cl lal ::; cz

(24)

subjected to the aforementioned constraints Yl 2: 0

(26) -h(oy)

c5y

6. THE EXPERIMENTAL SETUP: IMPLEMENTATION ISSUES AND TEST RESULTS

<0

Differentiating further (22) and considering (23)(26) one obtains ..

(J

(. r)

= cp x, x,

U

y, t

dA r

+ gJ8uy

In this section the experimental setup is described in some detail.

(27)

y

The linear motor drives (factored by Linmot™) are rated with bandwidth between 15Hz and 20Hz. A dedicated driver module allows for setting the force w applied to the slider by means of a reference input w· (the force-loop is internal to the driver). Furthermore, the driver module outputs the slider positions Yl and yz an encoderlike signals. A PID position loop has been closed externally to the LM controller (Fig. 4).

The dynamics (27) is formally equivalent to (9) with u = ay. Since by physical arguments it can be asserted that function cp is bounded and A(c5y ) is strictly increasing it follows that conditions (10) hold for some constants F, r land r z.

a;

Given a command profile for c5y , the desired profiles Yi and Y2 for the spear valve positions are obtained by inverting (24) under conditions (25):

~; {

=

ma~{o;~o}

The control system has been implemented on a PC-based platform (Pentium2 processor at 350Mhz). The computational burden of the control system is limited, and much less computing power would be sufficient. The controller and the differentiator have been discretized by the classical backward-difference method with a sampling step of 2ms (the sample-and-hold effect was analyzed in (G. Bartolini and Usai, 2001b».

(28)

Yz = -mm{oy,O}

Let us summarize the overall controller. The sliding manifold is a(t) =

J+ ce

(33)

ay 2: 0

h(c5y) {

(32)

with cl, cz ~ o. Under mild smoothness requirements the system behaviour is "regular", and the tracking error e exponentially converge toward a small neighbor of zero.

Yl . yz = 0 (25)

By combining (23) and (24)-(25) it can be written

A == A(c5y ) =

(31)

Remark 3 Due to actual implementation effects (e.g. noise (Levant, 2000), discretization (G. Bartolini and Usai, 2001b», conditions (30)-(31) are guaranteed only approximately

To minimize the energy consumption, the two nozzles should not be both opened at the same time. This corresponds to keep either Yl or yz to zero at each time instant.

yz 2: 0

(30)

which mean that (J = e+ce vanishes in finite time. The exponential convergence to zero of e follows from trivial arguments.

where

ay = Yl - yz

- el =0 /al=O

c>O

(29)

453

7. CONCLUSIONS

The parameters of the controller and of the differentiator are set as follows:

Sub-optimal algorithm: 2-SM difJerentiator:

KO

UM

This paper dealt with the output-feedback control problem of a jet-propelled UV prototype.

= 10, a* = 1.

Because of the system uncertainties, an observer/ controller scheme based on second-order sliding modes has been designed and experimentally verified. The mild computational demand allows for the digital implementation on a lowcost platform. Experiments pointed out the good performance and robustness of the proposed controller, and, in particular, the performance improvement with respect to a classical PID controller.

= 12 Kl = 20

The proposed 2-SM controller/observer scheme has been implemented using the above parameter set, and its performance has been compared with that of a classical PID controller with gains Kp = 2, K] = KD = 1. A first regulation test with a piecewise-constant reference profile has been carried out. Fig. 6 reports the actual and desired trajectory obtained using the two different approaches, and puts into evidence that the VSC is more accurate and has finite convergence time. Fig. 7 reports the time evolution of the sliding variable.

8. REFERENCES Fossen, T. (1994). Guidance and Control of Ocean vechicles. Wiley. UK. G. Bartolini, A. Ferrara, A. Levant and E. Usai (1999). On second order sliding mode controllers. Lecture Notes in Control and Information Sciences 247,329-350. G. Bartolini, A. Ferrara, A. Pisano and E. Usai (2001a). On the convergence properties of a 2-sliding control algorithm for nonlinear uncertain systems. Int. J. Control 74, 718731. G. Bartolini, A. Levant, A. Pisano and E. Usai (2000). On the robust stabilization of nonlinear uncertain systems with incomplete state availability. ASME J. of Dyn. Syst. Meas. and Contr. 122, 738-745. G. Bartolini, A. Levant, A. Pisano and E. Usai (2002). Higher-order sliding modes for output-feedback control of nonlinear uncertain systems. Lecture Notes in Control and Information Sciences 274, 83-108. G. Bartolini, A. Pisano and E. Usai (2001b). Digital second order sliding mode control for uncertain nonlinear systems. Automatica 37, 1371-1377. Isidori, A. (1995). Nonlinear Control Systems. Third Edition. Springer Verlag. Berlin. Levant, A. (1993). Sliding order and sliding accuracy in sliding mode control. Int. J. Control 58, 1247-1263. Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica 34,379384. Levant, A. (2000). Variable measurement step in 2-sliding control. Kybernetica 36, 77-93. Levant, A. (2003). Higher order sliding modes, differentiation and output-feedback control. Int. J. Control.

Fig. 6. Regulation test with 2-SMC (left) and PID-control (right). The actual and desired position profile.

.

.. .

--. .

. . .

Fig. 7. Regulation test with 2-SMC. The sliding variable time history. A tracking test using a sinusoidal reference profile has been carried out. Fig. 8 shows that the actual trajectory converges to the desired one after a very short transient.

Fig. 8. Tracking test with 2-SMC. The actual and desired position profiles.

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