A Discrete Adaptive Variable-structure Controller For MIMO Systems ...

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 5, NO. 3, MAY 1997

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Brief Papers A Discrete Adaptive Variable-Structure Controller for MIMO Systems, and Its Application to an Underwater ROV Maria Letizia Corradini and Giuseppe Orlando

Abstract—This paper addresses discrete-time variable structure control for multiple input–multiple output (MIMO) systems. Two control algorithms are presented, obtained extending to the multivariable case two control techniques recently proposed for single input–single output (SISO) systems [3]. Both techniques consist of variable structure control laws cascaded to a generalized minimum variance controller. The former algorithm refers to a completely known system, while the latter has been designed to deal with parameter uncertainties in the plant. In this case, the connection with a MIMO on-line parameter estimator has been considered. Proofs are provided about the convergence of the proposed control laws. The presented algorithms have been applied to the problem of position and orientation control of an underwater remotely operated vehicle (ROV) used in the exploitation of combustible gas deposits at great water depths. Resulting performances have been tested by simulation, modeling the ROV with a nonlinear differential equations system. Results have been discussed and compared with recent literature. Index Terms—Discrete variable structure control, generalized minimum variance control, uncertain systems, underwater remotely operated vehicles.

I. INTRODUCTION

D

URING the last decade, the increasing need for the exploitation of combustible deposits at oceanic depths has led the oil companies to the development of underwater unmanned vehicles, in order to perform complex tasks such as environmental data gathering, transportation of assembling modules for submarine installations, and inspection of underwater structures. The automatic control of such vehicles presents several difficulties, due to the nonlinearity of the dynamics, to the presence of external unmeasurable disturbances, and to the high uncertainty level in the model [1], [2], [4], [6], [11], [12], [14]. As a consequence, the application of conventional control strategies leads to unacceptable closedloop performances. Recently, two different discrete-time control techniques have been presented for single input–single output (SISO) systems [3], both of which introducing novel variable structure control (VSC) laws cascaded to a generalized minimum variance (GMV) controller, making use of only input–output (I/O) time sequences (the complete measure of the state vector Manuscript received June 5, 1995; revised June 7, 1996. Recommended by Associate Editor, B. Egardt. The work of G. Orlando was supported by Snamprogetti S.p.A.-Fano, Italy. The authors are with the Dipartimento di Elettronica ed Automatica, Universit´a di Ancona, I-60131 Ancona, Italy. Publisher Item Identifier S 1063-6536(97)03268-5.

is not required). The former algorithm refers to a completely known system, while the latter, including the connection with an on-line parameter estimator, has been designed to deal with parameter uncertainties in the plant. It is worth noticing that the adoption of a GMV controller avoids process cancellation and allows to consider also nonminimum phase plants [8], [3]. The aim of this paper is to extend the above results to multivariable systems. Moreover, the multiple input–multiple output (MIMO) algorithms have been applied to the position and orientation control of a remotely operated vehicle (ROV) employed by the Italian company Snamprogetti for the realization of a diverless submarine structure for gas exploitation at great depths. The vehicle is equipped with four thrusts propellers, controlling its position and orientation in planes parallel to the sea surface, and is connected with the surface vessel by a supporting cable which controls the vehicle depth and provides power and communication facilities. The control system is composed of two independent parts: the first part, placed on the surface vessel, monitors the vehicle depth, and the second part controls the position and orientation of the vehicle in the dive plane. In this paper the attention will be focused on this second part of the control system, and the ROV dynamics will be simulated using a three-dimensional (3-D) nonlinear differential equations system. In previous papers [2], [11], the coupling terms among the controlled variables have been neglected during the synthesis, and the control solutions have been designed using three independent SISO controllers based on a decoupled threedimensional (3-D) system model. The neglecting of some coupling terms (particularly as far as the yaw angle dynamics is concerned) in control design appears reasonable from both the theoretical and the numerical viewpoint. On the contrary, a controller synthesis obtained decoupling the dynamics of the position variables yields unsatisfactory performances, in particular if parameter variations of the model are to be taken into account. Hence, complete model decoupling is not fully justified, even if it has been often performed in literature in order to avoid the complexity associated with a multivariable approach to the design problem. The controllers’ performances have been tested by simulation. Reported results show the effectiveness of the first algorithm for the nominal system. In the more realistic situation of the uncertain system, the introduction of the self-tuning variable structure controller provides again satisfactory results,

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at the cost of a moderate increase in the control activity. This paper is organized as follows. The theorems describing the control algorithms for the nominal and uncertain system are reported in Section II, along with the proofs of their convergence. Section III contains the ROV nonlinear dynamic model and the corresponding discrete-time linear model used for the controller synthesis. Moreover, a thrusters allocation algorithm is proposed in the same section. Implementation details of the control algorithms and numerical simulations results are reported in Section IV. Conclusions are drawn in Section V. II. DISCRETE VARIABLE STRUCTURE CONTROL FOR MIMO SYSTEMS A. Generalized Minimum Variance Sliding Mode Control Let us consider a MIMO nonlinear plant described by the following th-order system of differential equations:

is the solution of the polynomial equation (7) and

is given by (8) is a polynomial matrix of proper degree. where The term in (4) and consequently the control law (6) allows to consider also nonminimum phase plants, since process dynamics is not canceled. Control law (6) ensures the vanishing of the tracking error when we have the following conditions. • Condition a) The polynomial matrix (9)

(1) with where with function. Linearizing

are the output and input vectors, respectively, and is an -dimensional vectorial

around a given point and discretizing the linearized equation with a sampling time the following MIMO I/O map is obtained:

and the matrix

(1)

are such that and

in

for

. • Condition b)

(2) where

is the time backshift operator, , and polynomial matrices are given by

(3) being the identity matrix. In the following, the controller will be synthesized using the linearized model (2), while the plant actually controlled is (1). The control objective can be formalized as the minimization of the squared value of the following quantity: (4) variable and proper degrees

with

and

being the tracking error, with the reference , and polynomial matrices of given by

(5) The above control objective yields the condition which provides the following control law [8], [9]:

have no common zeros outside the unit disk. In fact, manipulating (2) and (6)–(8), one gets (hereafter the dependence on will be omitted) Since because of condition a) and considering (2) we obtain (10) Equation (10) together with conditions a) and b) ensures the vanishing of the tracking error. In general, a straightforward way of imposing condition b) could be to choose a polynomial matrix such that its determinant has assigned stable roots, and then to find two matrices and satisfying Nevertheless, this approach cannot be applied here because the number of the conditions to be imposed is higher than the coefficients number of the unknown matrices, due to the structure of stated in condition a). Under the condition that is a diagonal dominance polynomial matrix, and selecting

(6) where

an assigned Schur polynomial and

with

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condition b) becomes

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B. Self-Tuning Variable Structure Control (11)

and are the th entries of the where polynomial matrices and , respectively. Hence (11) can be satisfied fulfilling the conditions being a factor of As it will be discussed later, the ROV model here considered is such that suitable matrices and can be found satisfying (11). Performances of the control law (6) can be improved by adding an auxiliary input, obtained connecting the GMV controller with a VSC-based block [3], [8]. A novel control algorithm of this kind is proposed in the following theorem:1 Theorem II.1: Given a system of the form (2), the following control law: (12)

When parameter uncertainties are present in the plant, the performances of the control system can be improved updating the control law parameters by on-line estimation [3], [8]. In the following, an implicit identification scheme is considered. The parameter vector to be estimated is constituted by the elements of the matrices The entries of the matrix are assumed known [8]. The estimate of is updated according to

where is the prediction Jacobian with respect to the parameter vector, is the prediction Hessian, and is the smoothing factor at the th time instant. Let’s define as the maximum variation bound on the the elements of (16) and

as

guarantees the achievement of a stable discrete sliding motion on the hyperplane i.e., the vanishing of the tracking error, if is chosen as if if

(13)

and positive scalars, Proof: The insertion of the control law (12) in the expression of provides Defining as the sliding mode existence condition, extending from [7] and [8], can be stated as

with

(17) Theorem II.2: It is given an uncertain system of the form (2), such that model uncertainties represented by parameter variations in the coefficients of and induce parameter variations in and satisfying (16). Let us define n sectors as (18) with

The following control law:

(14) Since

(14) becomes (15)

(19) and are the estimates of the polynomial matrices where and ensures the vanishing of the tracking error if is chosen as if

Equation (15) can be satisfied by fulfilling the inequalities In the case one gets

When

if

scalar

we have

(20)

Proof: Let us consider first the case when Define the following Lyapunov function [8]:

where

is the estimation error, and Since

Remark II.1: The choice of the and parameters affects the duration and the shape of the initial transient of the control and output variables. Hence their setting should be done according to the “best” tradeoff between acceptable initial control efforts and satisfactory transient duration.

with

following the proof given in [8], one gets:

By imposing 1 Since

bold Greek letters are hardly distinguishable from nonbold ones, vectors denoted by Greek symbols will be hereafter underlined for clarity.

(21)

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Fig. 1. ROV operational configuration. Fig. 2. ROV propellers system.

it results (22) with

Inequality (21) is equivalent to (23)

Expression (23) can be satisfied fulfilling the ities

scalar inequal-

Fig. 3. Control system architecture.

which are equivalent to (24) It can be easily shown that defined in being (20) satisfies (24) and consequently (21) and (22). From this latter it follows that outside the sector This implies that either the th error vanishes or is driven into the th sector. Inside the sector, each component of (19) is simply a self-tuning GMV: its convergence can be easily proved extending to the MIMO case the result given in [8]. III. MATHEMATICAL MODEL

OF THE

ROV

A. ROV Nonlinear Model The equations describing the ROV dynamics have been obtained from classical mechanics [2], [11]. The ROV considered as a rigid body can be fully described with six degrees of freedom, corresponding to the position and orientation with respect to a given coordinate system. Let us consider the inertial frame and the body reference frame [2] shown in Fig. 1. The ROV position with respect to is expressed by the origin of the system while its orientation by the roll, pitch, and yaw angles and respectively. Being the depth controlled by the surface vessel, the ROV is considered to operate on surfaces parallel to the - plane. Accordingly the controllable variables are and the yaw

angle It should be noticed that the roll and pitch angles and will not be considered in the dynamic model: their amplitude, in fact, has been proved to be negligible in a wide range of load conditions, and with different intensities and directions of the underwater current as well [2], [11]. The ROV model is described by the following system of differential equations:

(25) is the vehicle mass, is the addition mass, is the where vehicle inertia moment around the axis, is the addition inertia moment, and is the resistance moment of the cable. are given by the following expressions: (26) They are the forces produced by the cable traction corresponding to a submarine current with a velocity with

(27)

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(a)

353

(b)

(c) Fig. 4. Nominal system: simulations with favorable underwater current error. (c) Propellers’ thrusts.

T c = [0:2 0:2]m/s (condition

V

where is the cable length, the vehicle weight in the the weight for length unit of the cable, the water water, is the drag coefficient of the cable, and is the density, cable diameter. and are the drag forces along the and axes, given by

oc

1). (a) Continuous-time output. (b) Tracking

where is the drag coefficient of rotation, is the packing coefficient of rotation, is the equivalent area of rotation, is the equivalent arm of action, are the vehicle and axes, respectively, and dimensions along the is the angle between the axis and the velocity direction of the current. This model is in agreement with models usually proposed in literature for underwater ROV’s moving in the dive plane [6]. Substituting (26)–(29) in (25), the following equations are obtained:

(28) In (28)

is the drag coefficient of the th side wall the packing coefficient (depending on the is geometrical characteristics of the th side wall and the area of the th side wall and in (25) are the components of the drag torque around the z-axis produced by the vehicle rotation and by the current, respectively, and are given by

(29)

(30) The expressions of coefficients are reported in Table I below. Let us assume that positive known scalars exist, such that

(31)

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(a)

(b)

(c) Fig. 5. Nominal system: simulations with partially favorable underwater current Tracking error. (c) Propellers’ thrusts.

T c = [00:2 0:2]m/s (condition

V

and define EXPRESSIONS

oc

2). (a) Continuous-time output. (b)

TABLE I MODEL PARAMETERS

OF THE

The hypothesis (31) of bounded parameter variations implies that (16) is verified. The quantities and appearing in (30) are the decomposition of the thrust and the torque provided by the four vehicle propellers along the axes of the corresponding decomposition with respect to the axes of will be denoted with and The two triples and are related by the following relationships:

(32) The disposition of the four propellers shown in Fig. 2 gives been

linearized

around

a

working point Let us duce the following state, input, and output variables: (33) with

(see Fig. 2).

B. ROV Linearized Model To the purpose of designing the controllers according to the above theorems, the equations in (30) have

whose units are reported in Table II.

intro-

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(.a)

(b)

(c) Fig. 6. Nominal system: simulations with adverse underwater current VcT = [00:2 error. (c) Propellers’ thrusts.

0 0 2]m/s (condition :

oc

3). (a) Continuous-time output. (b) Tracking

Finally, the discretization of with a sampling time produces a multivariable ARX model of the form (2) with

TABLE II UNITS LIST OF STATE, INPUT, AND OUTPUT VARIABLES

C. Thrusters Allocation Algorithm

The linearization of (30) provides the following transfer matrix in the Laplace domain [2], [11]:

considered in the previous The control variables sections are not the actual thrusts supplied by the propellers, but their decomposition along the axes of However, in order to take into account the physical bounds limiting the power supply of the propellers, it is meaningful to calculate the thrusts corresponding to To this purpose, variables can be calculated by inverting (32). From (33), we get the inverse relation and the thrusts between

(34)

It can be verified that the contributions of and terms are negligible with respect to the remaining matrix entries, and will not be considered hereafter. Therecorresponds to two decoupled fore the transfer matrix systems: an independent system relative to the I/O and a two-dimensional (2-D) MIMO sysmap tem relative to the MIMO I/O map

(35) The independent variable mizing the index

can be determined by mini-

(36)

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(a)

(b)

(c) Fig. 7. Uncertain system (25% parameter variations): simulations with favorable nominal underwater current Continuous-time output. (b) Tracking error. (c) Propellers’ thrusts.

TABLE III MODEL PARAMETERS: NOMINAL VALUES

obtaining (37) can be derived from (35). Finally, the values of Since the maximum thrust each propeller can supply is for the working ROV prototype, a limiting device with saturation equal to has been introduced, in order to calculate the actual values of IV. RESULTS The controllers presented in Theorems II.1 and II.2 have been tested by simulation, using the ROV model (30) as the controlled plant (Fig. 3). The initial condition of the system has been chosen as and the set point as In order to smooth the control activity during the transient, each component of has been replaced by a third order polynomial connecting the initial

T c = [0:2 0:2]m/s (condition

V

1). (a)

oc

point to the desired point in the first seconds of simulation, with s for and and s for Simulations have been performed in three different operative conditions (oc’s) of the nominal submarine current: • oc1) Favorable current • oc2) Partially favorable current • oc3) Adverse current Parameters appearing in the model (30) are reported in Table III. Equation (30) has been linearized in the neighborhood of The chosen working point, replaced in (30), gives Then, the transfer matrix of the model (34) has been discretized with a sampling period s. As a consequence, the matrices of the multivariable ARX model are

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(a)

357

(b)

(c) Fig. 8. Uncertain system (25% parameter variations): simulations with partially favorable nominal underwater current 2). (a) Continuous-time output. (b) Tracking error. (c) Propellers’ thrusts.

oc

and the smoothing factor was

Restricting the attention to the nominal system, Figs. 4–6 show the performances of the controller defined in Theorem II.1, in the three operative conditions , respectively. The control algorithm was set as follows:

In each figure, subplot depicts the output of the nonlinear continuous-time model, subplot the corresponding tracking error, while subplot shows the propellers’ thrusts deduced through the algorithm of Section III-C. Figs. 7–9 refer to the uncertain system. In these plots, a parameter variation of 25% with respect to the nominal value has been applied to all the parameters appearing in the model (30) and to the submarine current. The figures show the performances of the self-tuning controller defined in Theorem II.2, in the three operative conditions The control algorithm was set with As far as the MIMO on-line parameter estimator is concerned, the Hessian matrix had the following initial condition:

T c = [00:2 0:2]m/s (condition

V

with

Subplot depicts the output of the nonlinear continuoustime model and subplot the corresponding tracking error, while subplot shows the propellers’ thrusts. In the above simulations, the polynomial matrices and satisfying (11) have been chosen as

Due to the structure of the choice satisfies condition a) of Section II-A. Moreover, condition b) is satisfied, too, and the hypothesis of diagonal dominance of is fulfilled. As far as simulation results are concerned, Figs. 4–6, relative to the nominal case, show a very satisfactory transient, both for its duration and for control efforts. This behavior is particularly relevant if the very large inertia of the system is considered, in comparison with previous literature. In particular, the self-tuning controller presented in [11] and [12] and the algorithm [1], [12] applied to

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(a)

(b)

(c) Fig. 9. Uncertain system (25% parameter variations): simulations with adverse nominal underwater current (a) Continuous-time output. (b) Tracking error. (c) Propellers’ thrusts.

the same ROV model provided a transient duration of nearly 50 s. This latter controller produced a relevant overshoot (about 40%) which is not present in the reported simulations [see Figs. 4(a)–6(a)], due also to the already mentioned choice of the reference trajectory. Comparing the simulations relative to the different operative conditions (see Figs. 4–6), a short initial undershoot can be noticed in the variables subject to adverse submarine current [Fig. 6(a) and 6(b)]. Correspondingly, the control activity arises during both the transient and the steady-state condition, yet remaining by far below the saturation threshold of the propellers [Fig. 6(c)]. The simulations relative to the uncertain system (Figs. 7–9) show an increased absolute value of the steady-state tracking error [e.g., for the yaw angle this value arises from 0.005 rad to 0.01 rad in the worst condition as shown in Figs. 6(b) and 9(b)], a higher transient control activity (e.g., thrust grows from in Fig. 6(c) to the saturation value in Fig. 9c for the worst condition and an increased steadystate control variable (e.g., thrust changes from in in Fig. 9(c) for the worst condition ). It Fig. 6(c) to is worth noticing that the saturation occurs only in the adverse condition and only for thruster V. CONCLUSIONS In this paper, the issue of discrete-time VSC theory for MIMO systems was addressed. Two different control tech-

T c = [00:2

V

0 0 2]m/s (condition :

3).

oc

niques were proposed, both introducing novel VSC laws cascaded to a GMV controller. The presented control laws were derived extending to multivariable systems two techniques recently proposed for SISO plants [3]. The algorithm presented in Theorem II.1 refers to a completely known MIMO system, while the algorithm of Theorem II.2 is designed to deal with parameter uncertainties in the plant. In this latter case the connection with a MIMO on-line parameter estimator is considered. Presented theoretical results were applied to the problem of the position and orientation control of an underwater ROV, used in the exploitation of combustible gas deposits at great water depths. The vehicle dynamics was modeled by a system of nonlinear differential equations. Simulation results provided satisfactory performances with respect to the transient behavior and the tracking error, both for the nominal and for the uncertain plant. As expected, in this latter case the control activity arises during both the transient and the steady-state condition, yet remaining noticeably below the propellers’ saturation threshold for most of the simulation time. From a practical viewpoint, the presented control algorithms can be relevant in applications, mainly due to their discretetime character and their easy implementability. Moreover, no state feedback is needed, but only output measurement are fed to the controller. The robustness of the algorithm is ensured both by the VSC mechanism and by the adaptive

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characteristics of the control law. Finally, the control activity, besides remaining noticeably below the actuators’ threshold, does not show any oscillating behavior.

REFERENCES

H1

control of a remotely operated under[1] G. Conte and A. Serrani, “ water vehicle,” in Proc. ISOPE ’94, Osaka, Japan. [2] A. Conter, S. Longhi, and C. Tirabassi, “Dynamic model and selftuning control of an underwater vehicle,” in Proc. 8th Int. Conf. Offshore Mechanics Arctic Eng., The Hague, The Netherlands, 1989, pp. 139–146. [3] M. L. Corradini and G. Orlando, “Discrete variable structure control for nonlinear systems,” in Proc. European Contr. Conf. (ECC95), Rome, vol. 2, Sept. 1995, pp. 1465–1470. [4] R. Cristi, A. P. Fotis, and A. J. Healey, “Adaptive sliding mode control of autonomous underwater vehicles in the dive plane,” IEEE J. Oceanic Eng., vol. 15, pp. 152–160, 1990. [5] R. A. De Carlo, S. H. Zak, and G. P. Matthews, “Variable structure control of nonlinear multivariable systems: A tutorial,” Proc. IEEE, vol. 76, no. 3, pp. 212–232, 1988.

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[6] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York: Wiley, 1994. [7] K. Furuta, “Sliding mode control of a discrete system,” Syst. Contr. Lett., vol. 14, pp. 145–152, 1990. , “VSS type self-tuning control,” IEEE Trans. Ind. Electron., vol [8] 40, pp. 37–44, 1993. [9] R. Isermann, Digital Control Systems. New York: Springer-Verlag, 1981. [10] L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987. [11] S. Longhi and A. Rossolini, “Adaptive control for an underwater vehicle: Simulation studies and implementation details,” in Proc. IFAC Wkshp. Expert Syst. Signal Processing Marine Automa., Copenhagen, Denmark, 1989, pp. 271–280. [12] S. Longhi, G. Orlando, A. Serrani, and A. Rossolini, “Advanced control strategies for a remotely operated underwater vehicle,” in Proc. 1st World Automat. Congr. (WAC’94), Maui, HI, 1994, pp. 105–110. [13] J. J. Slotine and S. S. Sastry, “Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulators,” Int. J. Contr., vol. 38, no. 2, pp. 465–492, 1983. [14] D. R. Yoerger and J. J. E. Slotine, “Robust trajectory control of underwater vehicles,” IEEE J. Oceanic Eng., vol. OE-10, pp. 462–470, 1985.