Dynamic second-order sliding mode control of the ... - Semantic Scholar
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 6, NOVEMBER 2002
Dynamic Second-Order Sliding Mode Control of the Hovercraft Vessel Hebertt Sira-Ramírez
Abstract—In this paper, a suitable combination of the differential flatness property and the second-order sliding mode controller design technique is proposed for the specification of a robust dynamic feedback multivariable controller accomplishing prescribed trajectory tracking tasks for the earth coordinate position variables of a hovercraft vessel model. Index Terms—Flat systems, hovercraft, second-order sliding, trajectory planning.
I. INTRODUCTION
T
HE REGULATION of a ship vessel, by means of two independent thrusters located at the aft, has received sustained attention in the last few years. Reyhanoglu [17] uses a discontinuous feedback control law for exponential stabilization toward a desired equilibrium. A feedback linearization approach was proposed by Godhavn [10] for the regulation of the position variables. The scheme, however, did not allow for orientation control. In [13], a time-varying feedback control law is proposed, which exponentially stabilizes the state toward a given equilibrium point. Time-varying quasiperiodic feedback control, developed in [14], has been proposed taking advantage of the homogeneous properties of a suitably transformed model achieving simultaneous exponential stabilization of the position and orientation variables. An interesting experimental setup has been built which is described in the work of Pettersen and Fossen [15]. In their work, the time-varying feedback control, used in [13], is extended to include integral control actions, including excellent experimiental results. High-frequency feedback control signals, in combination with averaging theory and backstepping, have also been proposed by Pettersen and Nijmeijer [16], to obtain practical stabilization of the ship toward a desired equilibrium and also for trajectory tracking tasks. In [18], the ship trajectory tracking control problem was examined from the perspective of Liouvillian systems (a special class of nondifferentially flat systems, i.e., systems which are not equivalent to linear controllable systems by means of endogenous feedback). Our hovercraft model is based on that used in the recent work of Fantoni et al. [5], where the vessel’s dynamics are derived on the basis of the underactuated ship model extensively studied by Manuscript received May 30, 2000. Manuscript received in final form July 23, 2002. Recommended by Associate Editor S. Ge. This work was supported by the Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICITVenezuela), under Research Project 2001001208, and also by the Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV-IPN), México, and by the Consejo Nacional de Ciencia y Tecnologia (CONACYT) under Research Contract 32681-A. The author was with the Departamento de Sistemas de Control of the Universidad de Los Andes, Mérida, Venezuela. He is now with CINVESTAV-IPN, Departamento Ingeniería Eléctrica, Sección de Mecatrónica, 07300 México D.F., México (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2002.804134
Fossen [8]. In [5], a series of interesting Lyapunov-based feedback controllers are derived for the stabilization and trajectory tracking of the hovercraft system. In this paper, we propose a robust dynamic feedback control scheme for the hovercraft system based on offline trajectory planning and dynamic feedback auxiliary trajectory tracking error stabilization to the origin of its phase space coordinates. For both, the trajectory planning and the feedback controller design aspects, use is made of the fact that, contrary to the general surface vessel model [8], the hovercraft system model is indeed differentially flat. The flat outputs are represented by the hovercraft position coordinates with respect to the fixed-earth frame. The sixth-order hovercraft system is shown to be equivalent, under endogenous dynamic feedback, to two fourth-order independent controllable linear systems in Brunovsky’s form. The reader is referred to the work of Fliess et al. [6], [7] for a definition of flatness and a full discussion of this concept with its many theoretical and practical implications. The flatness of the hovercraft model was first established in [19]. Higher order sliding modes appear as a natural extension of first-order sliding modes, thoroughly studied in the work of Utkin [20], and of a vast, and still growing, list of authors (see the survey in [4]). The basic idea, first proposed by Emely’anov et al. [3] in the context of “second-order sliding modes,” is to impose, on a certain auxiliary stabilizing output differential function—or suitably defined tracking error—of arbitrary but well-defined relative degree, the dynamic behavior of a higher order discontinuous (sliding) dynamics. The chosen sliding dynamics, usually of order higher or equal to two, is such that its trajectories globally and robustly converge toward the origin of phase coordinates in finite time. This important robustness characteristic of the chosen “sliding algorithm” is usually guaranteed to be preserved even in the presence of, unmodeled, absolutely bounded disturbances. The reader is referred to the works of Levant [11], [12] and Fridman and Levant [9], for interesting details, extensions, and generalizations of the second-order sliding mode control idea. In the context of uncertain systems, the reader may also benefit from the contents in the recent articles by Bartolini et al. [1], [2]. Section II presents the hovercraft vessel model used in [5]. This model is obtained by taking, as the starting point, the fully actuated, though simplified, ship model found in [8] and also in [13]. In Section II, it is shown that the considered hovercraft system model is differentially flat. In Section III, we pose the trajectory tracking problem and derive a robust dynamic feedback controller based on flatness and second-order sliding modes. These modes are induced on a set of independent auxiliary stable polynomial differential functions of the flat outputs tracking errors. Section IV contains the simulation results for a typical trajectory tracking maneuver and Section V is devoted to some conclusions and suggestions for further research.
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II. HOVERCRAFT MODEL We consider the following hovercraft model, derived in Fantoni et al. in [5]
(1) This system is not exactly linearizable by means of staticstate feedback, as it can be easily verified that the vector relative degree of the position outputs and are not well defined ( and already depend on but not on ). In order to be able to transform the system into the classical Brunovski’s canonical form, with and as the integration heads, flatness indicates that the hovercraft dynamics requires a second-order dynamic extension of the input variable . A. Differential Flatness of the Hovercraft System Definition II.1—[6]: A dynamic system, , with , and , is said to be differentially flat if there exist, , differentially independent variables, called flat outputs (differentially independent meaning that they are not related by any differential equation), which are functions of the state vector and, possibly, of a finite number of time derivatives of the state vector (i.e., derivatives of the inputs may be involved in their definition), such that all system variables (states, inputs, outputs, and functions of these variables) can be, in turn, expressed as functions of the flat outputs and of a finite number of their time derivatives. The flatness property considerably facilitates the offline trajectory planning aspects for the system and trivializes its exact linearization by means of endogenous feedback (i.e., a feedback that only depends on the system variables and, possibly, their time derivatives). It provides a direct way, which involves no need for solutions of differential equations, to establish the open-loop trajectories of all system variables on the basis of nominal trajectories for the flat outputs. The flat outputs, being devoid of any zero dynamics, completely guarantee total internal stability of the system states and outputs, including those outputs which are nonminimum phase.
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We have the following proposition. Proposition II.2: The model (1) is differentially flat, with flat outputs given by and , i.e., all system variables in (1) can be differentially parameterized solely in terms of and , and a finite number of their time derivatives as shown in (2) at the bottom of the page. The proof of this proposition is found in full detail in [19]. B. Invertibility of the Control Parameterization deThe differential parameterization of the input torque pends on the flat outputs, and their time derivatives, up to the fourth order. Note, however, that the corresponding parameterization of the control input only depends on the second-order time derivatives of and . This simple fact, drawn from the flatness property, clearly reveals the existence of an “obstacle” to achieve static feedback linearization and points to the need for a second-order dynamic extension of the control input in order to solve the lack of input–output invertibility and proceed to exactly linearize the system. Use of (1) allows the following (simpler) expressions for the control inputs and , in terms of the system’s state variables, the highest order derivatives of the flat outputs and , and the time derivative of the control input , which acts now as an additional state variable shown in (3) and (4) at the bottom of the next page. Using (2), we have that
Clearly, we are interested in maneuvers for which this quantity is bounded (which is physically reasonable and natural) and it is also bounded away from zero (which somehow limits the class of desired trajectories). Assumption II.3: We assume that the positive quantity, , is uniformly bounded by a constant for all times, and it is nowhere identically zero along the evolution of the system. The previous assumption specifically precludes us from considering trajectories that either lead, or contain, a resting point
(2)
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for the earth position coordinates and of the hovercraft vessel. These may be treated by time reparameterizations, or “control of the clock” techniques (see [7]) which are outside the scope of this paper. On the other hand, straight lines, circles, and many other types of trajectories, which are to be followed at constant speeds, can be handled by the method here proposed. Let and denote two independent auxiliary control inputs. Under the above assumption, the locally defined input coordinate transformation
A. Unperturbed Case Suppose a desired trajectory is given for the position coordinates and in the form of specified time functions: and , respectively. The following proposition proposes a dynamic feedback solution to the trajectory tracking problem, based on flatness and exact tracking error linearization, through the imposition of a second-order sliding dynamics on stable differential polynomials of the position tracking errors. Proposition III.1: Let the set of constant real coefficients and represent independent sets of Hurwitz coefficients, so that the polynomials in the complex variable
(5) yields the transformed system as the following two independent chains of integrations:
The hovercraft system is thus equivalent, under endogenous feedback, to a set of two independent linear systems in Brunovsky’s controllable canonical form. This fact conceptually simplifies the dynamic feedback controller design task to a considerable extent while naturally suggesting a higher order sliding mode controller design approach. III. TRAJECTORY TRACKING FOR THE HOVERCRAFT SYSTEM Second-order sliding modes constitute a useful extension of traditional sliding mode control. The practical advantages of the technique reside in its enhanced robustness, simplicity and the possibilities of simultaneous forced stabilization to zero of a number of higher order time derivatives of the controlled system outputs. While the “sliding surface” is represented by the intersection of these “zero high-order derivative manifolds” the imposed representative transient dynamics converges to this intersection manifold in finite time, with rather controlled features. Thus, chattering effectively occurs when the system relevant velocities are all zero and, not as in the traditional case, while their magnitude is still significant. This gives second-order sliding motions a particular practical relevance for many mechanical systems and some other systems, such as the hovercraft.
have roots located in the left portion of the complex plane. Let and be a given a set of desired trajectories for the position coordinates and which satisfies Assumption II.3. and , define the Associated with the polynomials following two auxiliary differential functions of the position tracking errors:
Then, for any set of real parameters and such and , the dynamic second-order sliding that feedback controller shown in (6) and (7) at the bottom of the page, with
(8) semiglobally stabilizes the auxiliary tracking errors and and to zero, in finite time. As a their first-order time derivatives and consequence, the trajectory tracking errors exponentially asymptotically converge to zero.
(3) (4)
(6) (7)
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 6, NOVEMBER 2002
Proof: Subtracting the controller expression, for in (7), from the open-loop expression in (4), we obtain, after some algebraic manipulations
(9)
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absolutely uniformly bounded.1 Thus, for some strictly positive and , the perturbed transformed constant parameters , system can be assumed to be of the form: with and , for all . Following the same steps in the proof of the previous proposition, we find that the perturbed version of the dynamics of the auxiliary tracking error functions and are now governed by
Proceeding in a similar fashion with respect to the corresponding closed- and open-loop expressions for , one finds (14)
(10) Then, clearly, the tracking errors functions ideal second-order sliding dynamics
and
satisfy the
(11) As a consequence, the variables and , as well as their corresponding time derivatives, and , converge to zero in finite time, thus imposing the following asymptotically exponentially stable dynamics on the flat outputs tracking errors:
According to the results in [19], the perturbed evolutions of and converge to zero in finite time, provided, , and , with with . This implies that for a set of suit, and for an absolutely uniable controller parameters, , the trajectory formly bounded perturbation input signal , and , still asymptotically tracking errors exponentially converge to zero in spite of the influence of the perturbations. A similar conclusion can be reached for the case of unmatched perturbations affecting the nonactuated sway velocity dynamics. For absolutely uniformly bounded perturbations with similarly bounded first-order time derivatives, acting on the , nonactuated sway velocity equation, as the input coordinate transformation (3), (4) yields the following perturbed Brunovsky canonical forms:
B. Perturbed Case A simple tracing of the influence of unmodeled perturbations, in the open-loop system (1), reveals that a hypothesized perturbation in either the surge velocity equation, or the sway dynamics, affects all the state variables of the system, except the orientation angle . On the contrary, a similar perturbation affecting the yaw-rate dynamics, propagates to all of the states in the system. We consider first the latter case. , Consider an unmodeled matched perturbation input, which is absolutely uniformly bounded by a strictly positive . This perturbation affects the constant , i.e., ship’s yaw-rate dynamics in the form (12) The input coordinate transformation (3), (4) on the perturbed system (12) results now in the following set of perturbed Brunovsky canonical forms:
(13) Evidently, from Assumption II.3, and for absolutely uniand , the formly bounded surge and sway velocities perturbation terms affecting the transformed system are also
The perturbations affecting the right-hand sides of the Brunovsky forms are evidently absolutely bounded for bounded yaw rates, , and absolutely bounded perturbation , with an absolutely bounded first and second-order inputs, time derivatives. Thus, expressions similar to (14) are also valid. In the simulations presented below, we test the proposed nominal dynamic second-order sliding mode controller of Proposition III.1 with an unmatched perturbation input signal of the form just discussed. IV. SIMULATION RESULTS Simulations were carried out to evaluate the performance of the proposed dynamic feedback controller for a rather common trajectory tracking task: the tracking of a circular trajectory, defined in the earth fixed coordinate frame, of radius , centered around the origin. 1Note that the sway velocity dynamics are a linear time-invariant dynamics, with a strictly negative eigenvalue, excited by the product of the surge velocity u and the yaw rate r . Since it is physically plausible to assume that these two velocities are absolutely uniformly bounded, then it follows that the absolute value of the sway velocity is also uniformly bounded.
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A. Tracking a Circular Trajectory A circular trajectory, or radius , is to be followed in a clock, with a given constant angular vewise sense in the plane locity of value . In other words, the flat outputs are nominally specified as (15) For this particular choice of is given by angle
and , the nominal orientation
Fig. 1. Simplified hovercraft system.
(16) . with According to the system dynamics (1) one easily obtains that
(17) Fig. 2. Feedback controlled position coordinates for circular path tracking.
, the nominal surge and From this fact, and the fact that sway velocities and the nominal yaw angular velocity are given by the following constant values: (18) Similarly, one easily verifies that (19) , we obtain that the nominal From this and the fact that applied inputs are given by the following constant values: (20)
Fig. 3. Feedback controlled velocity variables for circular path tracking.
Note that for the chosen trajectory, the nominal value of the , appearing in the denominator of the controller quantity expression for , is given by The only system parameter was set to be . We have chosen the following parameters for the circular reference trajectory:
which result in rad, controller parameters were set to be
8.304
10
. The
A simplified hovercraft system is depicted in Fig. 1. Fig. 2 depicts the controlled evolution of the hovercraft position coordinates when the vessel motions are started significantly far away from the desired trajectory. Fig. 3 shows the corresponding surge, sway, and yaw angular velocities. Fig. 4 depicts the apand . Fig. 5 shows the circular plied external inputs path tracking performance under unmodeled sustained perturbations.
Fig. 4. Applied control inputs for circular path tracking.
B. Robustness With Respect to Unmodeled Unmatched Perturbations In order to test the robustness of the proposed controller, we introduced in the nonactuated dynamics (i.e., in the sway acceleration equation) an unmodeled external perturbation force, simulating a rather strong “wave field” effect, of the form
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the object of ongoing research by many authors. The problem certainly deserves attention from the flatness viewpoint using time-reparameterizations and other suitable techniques. Similarly, consideration of more complex models of ships dynamics, including parasitic and actuator dynamics, should be the object of future studies. ACKNOWLEDGMENT The author wishes to thank Dr. G. Silva-Navarro of CINVESTAV-IPN for discussions and also Dr. A. Levant and Prof. L. Fridman for e-mail discussions. Their criticism and encouragement is gratefully acknowledged. REFERENCES
Fig. 5. Circular path tracking performance under unmodeled sustained perturbations.
with and . The parameters of the second-order sliding dynamics and the auxiliary function were set to be
In spite of the unmatched nature of the perturbation signal, the proposed dynamic feedback controller, with the same controller parameters used before, efficiently corrects the undesirable deviations due to the persistent perturbation. It manages to accomplish the trajectory tracking task with satisfactory precision. V. CONCLUSION In this paper, we have illustrated how the property of differential flatness can be advantageously combined with the robustness and simplicity of higher order sliding modes. We have carried out this combined controller design option in the context of the trajectory tracking regulation of an underactuated hovercraft system model, derived through some simplifying assumptions from the general surface vessel model. This model is shown to be differentially flat. The flatness property immediately allows to establish the equivalence of the model, by means of dynamic state feedback, to a set of two decoupled controllable linear systems. A trajectory planning, combined with a second-order sliding mode trajectory tracking scheme, allows us to obtain a direct feedback controller synthesis for arbitrary position trajectory following. The design was shown to be robust with respect to significant perturbation input forces even when they affect the nonactuated portion of the hovercraft velocity dynamics. The characteristics, and simplicity, of higher order sliding mode controllers, beyond those of the second-order type treated here, seem to be a natural, and remarkably robust, alternative for the efficient regulation and trajectory tracking tasks of perturbed nonlinear systems which are nominally differentially flat. The more difficult problem of hovercraft position regulation toward trajectories that include a resting equilibrium point is
[1] G. Bartolini and P. Pidynowsky, “An improved chattering free VSC scheme for uncertain dynamic systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 1220–1226, Aug. 1996. [2] G. Bartolini, A. Ferrara, A. Levant, and E. Usai, “On second order sliding mode controllers,” in Variable Structure Systems, Sliding Mode and Nonlinear Control, K. D. Young and Ü. Özgüner, Eds. London, U.K.: Springer-Verlag, 1999, vol. 247, Lecture Notes in Control and Information Sciences, pp. 329–350. [3] S. V. Emel’yanov, S. K. Korovin, and L. V. Levantovsky, “Higher order sliding modes in the binary control systems,” Sov. Phys., vol. 31, no. 4, pp. 291–293, Dokady 1986. [4] S. V. Emelyanov, Titles in the Theory of Variable Structure Control Systems. Irmis, Moscow, Russia: Int. Res. Inst. Management Sci., 1990. [5] I. Fantoni, R. Lozano, F. Mazenc, and K. Y. Pettersen, “Stabilization of a nonlinear underactuated hovercraft,” Int. J. Robust Nonlinear Contr., vol. 10, no. 8, 2000. [6] M. Fliess, J. Lévine, Ph. Martín, and P. Rouchon, “Flatness and defect of nonlinear systems: Introductory theory and examples,” Int. J. Contr., vol. 61, no. 6, pp. 1327–1361, 1995. , “A Lie–Bäcklund approach to equivalence and flatness,” IEEE [7] Trans. Automat. Contr., vol. 44, pp. 922–937, May 1999. [8] T. I. Fossen, Guidance and Control of Ocean Vehicles. Chichester, U.K.: Wiley, 1994. [9] L. Fridman and A. Levant, “Higher order sliding modes as a natural phenomenon in control theory,” in Robust Control via Variable Structure and Lyapunov Techniques, F. Garofalo and L. Glielmo, Eds. London, U.K.: Springer-Verlag, 1996, vol. 217, Lecture Notes in Control and Information Science, pp. 107–133. [10] J. M. Godhavn, “Nonlinear tracking of underactuated surface vessels,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1996, pp. 987–991. [11] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” Int. J. Contr., vol. 58, no. 6, pp. 1247–1263, 1993. [12] A. Levant. Controlling output variables via higher order sliding modes. presented at Proc. European Contr. Conf. ECC’97 [13] K. Y. Pettersen and O. Egeland, “Exponential stabilization of an underactuated surface vessel,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1996, pp. 967–971. [14] , Robust control of an underactuated surface vessel with thruster dynamics. presented at Proc. 1997 Amer. Contr. Conf. [15] K. Y. Pettersen and T. L. Fossen. Underactuated ship stabilization using integral control: Experimental results with cybership I. presented at Proc. 1998 IFAC Symp. Nonlinear Contr. Syst. NOLCOS’98 [16] K. Y. Pettersen and H. Nijmeijer, “Global practical stabilization and tracking for an underactuated ship—A combined averaging and backstepping approach,” in Proc. 1998 IFAC Conf. Syst. Structure Contr., Nantes, France, July 1998, pp. 59–64. [17] M. Reyhanoglu, “Control and stabilization of an underactuated surface vessel,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, pp. 2371–2376. [18] H. Sira-Ramírez. On the control of the underactuated ship: A trajectory planning approach. presented at Proc. 37th IEEE Conf. Decision Contr. [19] H. Sira-Ramírez and C. Aguilar-Ibáñez, “On the control of the hovercraft system,” Dynamics Contr., vol. 10, pp. 151–163, 2000. [20] V. I. Utkin, Sliding Modes and Their Applications in the Theory of Variable Structure Systems. Moscow, Russia: MIR, 1977.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 6, NOVEMBER 2002
Dynamic Second-Order Sliding Mode Control of the Hovercraft Vessel Hebertt Sira-Ramírez
Abstract—In this paper, a suitable combination of the differential flatness property and the second-order sliding mode controller design technique is proposed for the specification of a robust dynamic feedback multivariable controller accomplishing prescribed trajectory tracking tasks for the earth coordinate position variables of a hovercraft vessel model. Index Terms—Flat systems, hovercraft, second-order sliding, trajectory planning.
I. INTRODUCTION
T
HE REGULATION of a ship vessel, by means of two independent thrusters located at the aft, has received sustained attention in the last few years. Reyhanoglu [17] uses a discontinuous feedback control law for exponential stabilization toward a desired equilibrium. A feedback linearization approach was proposed by Godhavn [10] for the regulation of the position variables. The scheme, however, did not allow for orientation control. In [13], a time-varying feedback control law is proposed, which exponentially stabilizes the state toward a given equilibrium point. Time-varying quasiperiodic feedback control, developed in [14], has been proposed taking advantage of the homogeneous properties of a suitably transformed model achieving simultaneous exponential stabilization of the position and orientation variables. An interesting experimental setup has been built which is described in the work of Pettersen and Fossen [15]. In their work, the time-varying feedback control, used in [13], is extended to include integral control actions, including excellent experimiental results. High-frequency feedback control signals, in combination with averaging theory and backstepping, have also been proposed by Pettersen and Nijmeijer [16], to obtain practical stabilization of the ship toward a desired equilibrium and also for trajectory tracking tasks. In [18], the ship trajectory tracking control problem was examined from the perspective of Liouvillian systems (a special class of nondifferentially flat systems, i.e., systems which are not equivalent to linear controllable systems by means of endogenous feedback). Our hovercraft model is based on that used in the recent work of Fantoni et al. [5], where the vessel’s dynamics are derived on the basis of the underactuated ship model extensively studied by Manuscript received May 30, 2000. Manuscript received in final form July 23, 2002. Recommended by Associate Editor S. Ge. This work was supported by the Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICITVenezuela), under Research Project 2001001208, and also by the Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional (CINVESTAV-IPN), México, and by the Consejo Nacional de Ciencia y Tecnologia (CONACYT) under Research Contract 32681-A. The author was with the Departamento de Sistemas de Control of the Universidad de Los Andes, Mérida, Venezuela. He is now with CINVESTAV-IPN, Departamento Ingeniería Eléctrica, Sección de Mecatrónica, 07300 México D.F., México (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2002.804134
Fossen [8]. In [5], a series of interesting Lyapunov-based feedback controllers are derived for the stabilization and trajectory tracking of the hovercraft system. In this paper, we propose a robust dynamic feedback control scheme for the hovercraft system based on offline trajectory planning and dynamic feedback auxiliary trajectory tracking error stabilization to the origin of its phase space coordinates. For both, the trajectory planning and the feedback controller design aspects, use is made of the fact that, contrary to the general surface vessel model [8], the hovercraft system model is indeed differentially flat. The flat outputs are represented by the hovercraft position coordinates with respect to the fixed-earth frame. The sixth-order hovercraft system is shown to be equivalent, under endogenous dynamic feedback, to two fourth-order independent controllable linear systems in Brunovsky’s form. The reader is referred to the work of Fliess et al. [6], [7] for a definition of flatness and a full discussion of this concept with its many theoretical and practical implications. The flatness of the hovercraft model was first established in [19]. Higher order sliding modes appear as a natural extension of first-order sliding modes, thoroughly studied in the work of Utkin [20], and of a vast, and still growing, list of authors (see the survey in [4]). The basic idea, first proposed by Emely’anov et al. [3] in the context of “second-order sliding modes,” is to impose, on a certain auxiliary stabilizing output differential function—or suitably defined tracking error—of arbitrary but well-defined relative degree, the dynamic behavior of a higher order discontinuous (sliding) dynamics. The chosen sliding dynamics, usually of order higher or equal to two, is such that its trajectories globally and robustly converge toward the origin of phase coordinates in finite time. This important robustness characteristic of the chosen “sliding algorithm” is usually guaranteed to be preserved even in the presence of, unmodeled, absolutely bounded disturbances. The reader is referred to the works of Levant [11], [12] and Fridman and Levant [9], for interesting details, extensions, and generalizations of the second-order sliding mode control idea. In the context of uncertain systems, the reader may also benefit from the contents in the recent articles by Bartolini et al. [1], [2]. Section II presents the hovercraft vessel model used in [5]. This model is obtained by taking, as the starting point, the fully actuated, though simplified, ship model found in [8] and also in [13]. In Section II, it is shown that the considered hovercraft system model is differentially flat. In Section III, we pose the trajectory tracking problem and derive a robust dynamic feedback controller based on flatness and second-order sliding modes. These modes are induced on a set of independent auxiliary stable polynomial differential functions of the flat outputs tracking errors. Section IV contains the simulation results for a typical trajectory tracking maneuver and Section V is devoted to some conclusions and suggestions for further research.
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II. HOVERCRAFT MODEL We consider the following hovercraft model, derived in Fantoni et al. in [5]
(1) This system is not exactly linearizable by means of staticstate feedback, as it can be easily verified that the vector relative degree of the position outputs and are not well defined ( and already depend on but not on ). In order to be able to transform the system into the classical Brunovski’s canonical form, with and as the integration heads, flatness indicates that the hovercraft dynamics requires a second-order dynamic extension of the input variable . A. Differential Flatness of the Hovercraft System Definition II.1—[6]: A dynamic system, , with , and , is said to be differentially flat if there exist, , differentially independent variables, called flat outputs (differentially independent meaning that they are not related by any differential equation), which are functions of the state vector and, possibly, of a finite number of time derivatives of the state vector (i.e., derivatives of the inputs may be involved in their definition), such that all system variables (states, inputs, outputs, and functions of these variables) can be, in turn, expressed as functions of the flat outputs and of a finite number of their time derivatives. The flatness property considerably facilitates the offline trajectory planning aspects for the system and trivializes its exact linearization by means of endogenous feedback (i.e., a feedback that only depends on the system variables and, possibly, their time derivatives). It provides a direct way, which involves no need for solutions of differential equations, to establish the open-loop trajectories of all system variables on the basis of nominal trajectories for the flat outputs. The flat outputs, being devoid of any zero dynamics, completely guarantee total internal stability of the system states and outputs, including those outputs which are nonminimum phase.
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We have the following proposition. Proposition II.2: The model (1) is differentially flat, with flat outputs given by and , i.e., all system variables in (1) can be differentially parameterized solely in terms of and , and a finite number of their time derivatives as shown in (2) at the bottom of the page. The proof of this proposition is found in full detail in [19]. B. Invertibility of the Control Parameterization deThe differential parameterization of the input torque pends on the flat outputs, and their time derivatives, up to the fourth order. Note, however, that the corresponding parameterization of the control input only depends on the second-order time derivatives of and . This simple fact, drawn from the flatness property, clearly reveals the existence of an “obstacle” to achieve static feedback linearization and points to the need for a second-order dynamic extension of the control input in order to solve the lack of input–output invertibility and proceed to exactly linearize the system. Use of (1) allows the following (simpler) expressions for the control inputs and , in terms of the system’s state variables, the highest order derivatives of the flat outputs and , and the time derivative of the control input , which acts now as an additional state variable shown in (3) and (4) at the bottom of the next page. Using (2), we have that
Clearly, we are interested in maneuvers for which this quantity is bounded (which is physically reasonable and natural) and it is also bounded away from zero (which somehow limits the class of desired trajectories). Assumption II.3: We assume that the positive quantity, , is uniformly bounded by a constant for all times, and it is nowhere identically zero along the evolution of the system. The previous assumption specifically precludes us from considering trajectories that either lead, or contain, a resting point
(2)
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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 6, NOVEMBER 2002
for the earth position coordinates and of the hovercraft vessel. These may be treated by time reparameterizations, or “control of the clock” techniques (see [7]) which are outside the scope of this paper. On the other hand, straight lines, circles, and many other types of trajectories, which are to be followed at constant speeds, can be handled by the method here proposed. Let and denote two independent auxiliary control inputs. Under the above assumption, the locally defined input coordinate transformation
A. Unperturbed Case Suppose a desired trajectory is given for the position coordinates and in the form of specified time functions: and , respectively. The following proposition proposes a dynamic feedback solution to the trajectory tracking problem, based on flatness and exact tracking error linearization, through the imposition of a second-order sliding dynamics on stable differential polynomials of the position tracking errors. Proposition III.1: Let the set of constant real coefficients and represent independent sets of Hurwitz coefficients, so that the polynomials in the complex variable
(5) yields the transformed system as the following two independent chains of integrations:
The hovercraft system is thus equivalent, under endogenous feedback, to a set of two independent linear systems in Brunovsky’s controllable canonical form. This fact conceptually simplifies the dynamic feedback controller design task to a considerable extent while naturally suggesting a higher order sliding mode controller design approach. III. TRAJECTORY TRACKING FOR THE HOVERCRAFT SYSTEM Second-order sliding modes constitute a useful extension of traditional sliding mode control. The practical advantages of the technique reside in its enhanced robustness, simplicity and the possibilities of simultaneous forced stabilization to zero of a number of higher order time derivatives of the controlled system outputs. While the “sliding surface” is represented by the intersection of these “zero high-order derivative manifolds” the imposed representative transient dynamics converges to this intersection manifold in finite time, with rather controlled features. Thus, chattering effectively occurs when the system relevant velocities are all zero and, not as in the traditional case, while their magnitude is still significant. This gives second-order sliding motions a particular practical relevance for many mechanical systems and some other systems, such as the hovercraft.
have roots located in the left portion of the complex plane. Let and be a given a set of desired trajectories for the position coordinates and which satisfies Assumption II.3. and , define the Associated with the polynomials following two auxiliary differential functions of the position tracking errors:
Then, for any set of real parameters and such and , the dynamic second-order sliding that feedback controller shown in (6) and (7) at the bottom of the page, with
(8) semiglobally stabilizes the auxiliary tracking errors and and to zero, in finite time. As a their first-order time derivatives and consequence, the trajectory tracking errors exponentially asymptotically converge to zero.
(3) (4)
(6) (7)
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 10, NO. 6, NOVEMBER 2002
Proof: Subtracting the controller expression, for in (7), from the open-loop expression in (4), we obtain, after some algebraic manipulations
(9)
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absolutely uniformly bounded.1 Thus, for some strictly positive and , the perturbed transformed constant parameters , system can be assumed to be of the form: with and , for all . Following the same steps in the proof of the previous proposition, we find that the perturbed version of the dynamics of the auxiliary tracking error functions and are now governed by
Proceeding in a similar fashion with respect to the corresponding closed- and open-loop expressions for , one finds (14)
(10) Then, clearly, the tracking errors functions ideal second-order sliding dynamics
and
satisfy the
(11) As a consequence, the variables and , as well as their corresponding time derivatives, and , converge to zero in finite time, thus imposing the following asymptotically exponentially stable dynamics on the flat outputs tracking errors:
According to the results in [19], the perturbed evolutions of and converge to zero in finite time, provided, , and , with with . This implies that for a set of suit, and for an absolutely uniable controller parameters, , the trajectory formly bounded perturbation input signal , and , still asymptotically tracking errors exponentially converge to zero in spite of the influence of the perturbations. A similar conclusion can be reached for the case of unmatched perturbations affecting the nonactuated sway velocity dynamics. For absolutely uniformly bounded perturbations with similarly bounded first-order time derivatives, acting on the , nonactuated sway velocity equation, as the input coordinate transformation (3), (4) yields the following perturbed Brunovsky canonical forms:
B. Perturbed Case A simple tracing of the influence of unmodeled perturbations, in the open-loop system (1), reveals that a hypothesized perturbation in either the surge velocity equation, or the sway dynamics, affects all the state variables of the system, except the orientation angle . On the contrary, a similar perturbation affecting the yaw-rate dynamics, propagates to all of the states in the system. We consider first the latter case. , Consider an unmodeled matched perturbation input, which is absolutely uniformly bounded by a strictly positive . This perturbation affects the constant , i.e., ship’s yaw-rate dynamics in the form (12) The input coordinate transformation (3), (4) on the perturbed system (12) results now in the following set of perturbed Brunovsky canonical forms:
(13) Evidently, from Assumption II.3, and for absolutely uniand , the formly bounded surge and sway velocities perturbation terms affecting the transformed system are also
The perturbations affecting the right-hand sides of the Brunovsky forms are evidently absolutely bounded for bounded yaw rates, , and absolutely bounded perturbation , with an absolutely bounded first and second-order inputs, time derivatives. Thus, expressions similar to (14) are also valid. In the simulations presented below, we test the proposed nominal dynamic second-order sliding mode controller of Proposition III.1 with an unmatched perturbation input signal of the form just discussed. IV. SIMULATION RESULTS Simulations were carried out to evaluate the performance of the proposed dynamic feedback controller for a rather common trajectory tracking task: the tracking of a circular trajectory, defined in the earth fixed coordinate frame, of radius , centered around the origin. 1Note that the sway velocity dynamics are a linear time-invariant dynamics, with a strictly negative eigenvalue, excited by the product of the surge velocity u and the yaw rate r . Since it is physically plausible to assume that these two velocities are absolutely uniformly bounded, then it follows that the absolute value of the sway velocity is also uniformly bounded.
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A. Tracking a Circular Trajectory A circular trajectory, or radius , is to be followed in a clock, with a given constant angular vewise sense in the plane locity of value . In other words, the flat outputs are nominally specified as (15) For this particular choice of is given by angle
and , the nominal orientation
Fig. 1. Simplified hovercraft system.
(16) . with According to the system dynamics (1) one easily obtains that
(17) Fig. 2. Feedback controlled position coordinates for circular path tracking.
, the nominal surge and From this fact, and the fact that sway velocities and the nominal yaw angular velocity are given by the following constant values: (18) Similarly, one easily verifies that (19) , we obtain that the nominal From this and the fact that applied inputs are given by the following constant values: (20)
Fig. 3. Feedback controlled velocity variables for circular path tracking.
Note that for the chosen trajectory, the nominal value of the , appearing in the denominator of the controller quantity expression for , is given by The only system parameter was set to be . We have chosen the following parameters for the circular reference trajectory:
which result in rad, controller parameters were set to be
8.304
10
. The
A simplified hovercraft system is depicted in Fig. 1. Fig. 2 depicts the controlled evolution of the hovercraft position coordinates when the vessel motions are started significantly far away from the desired trajectory. Fig. 3 shows the corresponding surge, sway, and yaw angular velocities. Fig. 4 depicts the apand . Fig. 5 shows the circular plied external inputs path tracking performance under unmodeled sustained perturbations.
Fig. 4. Applied control inputs for circular path tracking.
B. Robustness With Respect to Unmodeled Unmatched Perturbations In order to test the robustness of the proposed controller, we introduced in the nonactuated dynamics (i.e., in the sway acceleration equation) an unmodeled external perturbation force, simulating a rather strong “wave field” effect, of the form
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the object of ongoing research by many authors. The problem certainly deserves attention from the flatness viewpoint using time-reparameterizations and other suitable techniques. Similarly, consideration of more complex models of ships dynamics, including parasitic and actuator dynamics, should be the object of future studies. ACKNOWLEDGMENT The author wishes to thank Dr. G. Silva-Navarro of CINVESTAV-IPN for discussions and also Dr. A. Levant and Prof. L. Fridman for e-mail discussions. Their criticism and encouragement is gratefully acknowledged. REFERENCES
Fig. 5. Circular path tracking performance under unmodeled sustained perturbations.
with and . The parameters of the second-order sliding dynamics and the auxiliary function were set to be
In spite of the unmatched nature of the perturbation signal, the proposed dynamic feedback controller, with the same controller parameters used before, efficiently corrects the undesirable deviations due to the persistent perturbation. It manages to accomplish the trajectory tracking task with satisfactory precision. V. CONCLUSION In this paper, we have illustrated how the property of differential flatness can be advantageously combined with the robustness and simplicity of higher order sliding modes. We have carried out this combined controller design option in the context of the trajectory tracking regulation of an underactuated hovercraft system model, derived through some simplifying assumptions from the general surface vessel model. This model is shown to be differentially flat. The flatness property immediately allows to establish the equivalence of the model, by means of dynamic state feedback, to a set of two decoupled controllable linear systems. A trajectory planning, combined with a second-order sliding mode trajectory tracking scheme, allows us to obtain a direct feedback controller synthesis for arbitrary position trajectory following. The design was shown to be robust with respect to significant perturbation input forces even when they affect the nonactuated portion of the hovercraft velocity dynamics. The characteristics, and simplicity, of higher order sliding mode controllers, beyond those of the second-order type treated here, seem to be a natural, and remarkably robust, alternative for the efficient regulation and trajectory tracking tasks of perturbed nonlinear systems which are nominally differentially flat. The more difficult problem of hovercraft position regulation toward trajectories that include a resting equilibrium point is
[1] G. Bartolini and P. Pidynowsky, “An improved chattering free VSC scheme for uncertain dynamic systems,” IEEE Trans. Automat. Contr., vol. 41, pp. 1220–1226, Aug. 1996. [2] G. Bartolini, A. Ferrara, A. Levant, and E. Usai, “On second order sliding mode controllers,” in Variable Structure Systems, Sliding Mode and Nonlinear Control, K. D. Young and Ü. Özgüner, Eds. London, U.K.: Springer-Verlag, 1999, vol. 247, Lecture Notes in Control and Information Sciences, pp. 329–350. [3] S. V. Emel’yanov, S. K. Korovin, and L. V. Levantovsky, “Higher order sliding modes in the binary control systems,” Sov. Phys., vol. 31, no. 4, pp. 291–293, Dokady 1986. [4] S. V. Emelyanov, Titles in the Theory of Variable Structure Control Systems. Irmis, Moscow, Russia: Int. Res. Inst. Management Sci., 1990. [5] I. Fantoni, R. Lozano, F. Mazenc, and K. Y. Pettersen, “Stabilization of a nonlinear underactuated hovercraft,” Int. J. Robust Nonlinear Contr., vol. 10, no. 8, 2000. [6] M. Fliess, J. Lévine, Ph. Martín, and P. Rouchon, “Flatness and defect of nonlinear systems: Introductory theory and examples,” Int. J. Contr., vol. 61, no. 6, pp. 1327–1361, 1995. , “A Lie–Bäcklund approach to equivalence and flatness,” IEEE [7] Trans. Automat. Contr., vol. 44, pp. 922–937, May 1999. [8] T. I. Fossen, Guidance and Control of Ocean Vehicles. Chichester, U.K.: Wiley, 1994. [9] L. Fridman and A. Levant, “Higher order sliding modes as a natural phenomenon in control theory,” in Robust Control via Variable Structure and Lyapunov Techniques, F. Garofalo and L. Glielmo, Eds. London, U.K.: Springer-Verlag, 1996, vol. 217, Lecture Notes in Control and Information Science, pp. 107–133. [10] J. M. Godhavn, “Nonlinear tracking of underactuated surface vessels,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1996, pp. 987–991. [11] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” Int. J. Contr., vol. 58, no. 6, pp. 1247–1263, 1993. [12] A. Levant. Controlling output variables via higher order sliding modes. presented at Proc. European Contr. Conf. ECC’97 [13] K. Y. Pettersen and O. Egeland, “Exponential stabilization of an underactuated surface vessel,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1996, pp. 967–971. [14] , Robust control of an underactuated surface vessel with thruster dynamics. presented at Proc. 1997 Amer. Contr. Conf. [15] K. Y. Pettersen and T. L. Fossen. Underactuated ship stabilization using integral control: Experimental results with cybership I. presented at Proc. 1998 IFAC Symp. Nonlinear Contr. Syst. NOLCOS’98 [16] K. Y. Pettersen and H. Nijmeijer, “Global practical stabilization and tracking for an underactuated ship—A combined averaging and backstepping approach,” in Proc. 1998 IFAC Conf. Syst. Structure Contr., Nantes, France, July 1998, pp. 59–64. [17] M. Reyhanoglu, “Control and stabilization of an underactuated surface vessel,” in Proc. 35th IEEE Conf. Decision Contr., Kobe, Japan, pp. 2371–2376. [18] H. Sira-Ramírez. On the control of the underactuated ship: A trajectory planning approach. presented at Proc. 37th IEEE Conf. Decision Contr. [19] H. Sira-Ramírez and C. Aguilar-Ibáñez, “On the control of the hovercraft system,” Dynamics Contr., vol. 10, pp. 151–163, 2000. [20] V. I. Utkin, Sliding Modes and Their Applications in the Theory of Variable Structure Systems. Moscow, Russia: MIR, 1977.
