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Output-feedback control of an underwater vehicle prototype by higher-order sliding modes
Automatica 40 (2004) 1525 – 1531
www.elsevier.com/locate/automatica
Brief paper
Output-feedback control of an underwater vehicle prototype by higher-order sliding modes Alessandro Pisano∗ , Elio Usai University of Cagliari, Department of Electrical and Electronic Engineering (DIEE), Piazza d’Armi, Cagliari 09123, Italy Received 15 April 2003; received in revised form 17 February 2004; accepted 24 March 2004
Abstract This paper describes some experimental results concerning the practical implementation of a recently proposed nonlinear output-feedback control technique based on the higher-order sliding mode approach. The considered technique is applied to the motion control problem for an underwater vehicle prototype that is equipped with a special propulsion system based on hydro-jets with variable-section nozzles. To cope with the heavy uncertainties a4ecting the prototype dynamics the output-feedback control system has been developed by means of an observer-controller that combines a second-order sliding-mode controller and a second-order sliding-mode di4erentiator. The reported experiments show that the proposed approach is capable of guaranteeing fast and accurate response under several operating conditions. The control system design procedure, and the main implementation issues, are discussed in detail. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Second-order sliding-modes; Underwater vehicles; Output–feedback; Nonlinear uncertain systems; Di4erentiators
1. Introduction This paper details some experiments concerning the motion control of an underwater vehicle (UV) prototype built and developed at the Department of Electrical and Electronic Engineering (DIEE) of the Cagliari University. The considered UV is equipped with a special water-jet propulsion system. Because of both the system realization and the operating conditions its dynamics is a4ected by various uncertainties that should be taken into account in the controller design. A popular approach to nonlinear output-feedback control design under “heavy” uncertainty conditions entails the combined use of Lyapunov-based controllers and high-gain observers (HGOs) to estimate the output derivatives (Teel & Praly, 1995; Atassi & Khalil, 1999). Under the hypothesis that a stabilizing state-feedback control is available, it This paper was presented at the IFAC Workshop on Robust Control Design, ROCOND ’03, Milan, June 2003. This paper was recommended for publication in revised form by Associate Editor Keum-Shik Hong under the direction of Editor Mituhiko Araki. ∗ Corresponding author. Tel.: +39-0706-755760; fax: +39-0706-755782. E-mail addresses: [email protected] (A. Pisano), [email protected] (E. Usai).
0005-1098/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.03.016
was shown in Atassi and Khalil (1999) that using an HGO one can recover the performance achieved under state feedback, thus a rather general nonlinear separation principle was established there. Nevertheless, in real implementation real-time di4erentiators based on high-gain observers can be rather sensitive to the measurement noise a4ecting the output variable. Robust di4erentiators based on the second-order sliding mode control approach (see Levant, 1998; Bartolini, Pisano, & Usai, 2001b), exhibit interesting properties of robustness with respect to the measurement noise. This work validates the theoretical results reported in Bartolini, Levant, Pisano and Usai (2000); Levant (2003) and addresses the relevant practical implementation issues. In the above works stability and separation results about the combined use of second-order sliding-mode controllers (2-SMC) (Bartolini, Ferrara, Levant, & Usai, 1999) and second-order sliding-mode di4erentiators (2-SMD) (Levant, 1998) were dealt with. The paper is organized as follows: in Section 2, we brieKy recall the basic design principles of output-feedback 2-SM control. In Section 3, a mathematical model of the UV dynamics is given, and in the successive Section 4, the combined 2-SMC/2-SMD approach is applied to design a
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A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531
robust output-feedback motion controller for the UV prototype. Section 5 describes the experimental setup and discusses the experimental results, and Section 6 draws some Mnal conclusions. 2. Nonlinear output-feedback control via 2-SM controllers and SM dierentiators
i=0
Consider the nonlinear SISO system x˙ = f(x) + g(x)u; y = h(x)
(1) n
with unavailable state vector x ∈ R , scalar control variable u ∈ R and measurable output y ∈ R. Let f, g and h be unknown, suPciently smooth, vector-Melds of appropriate dimension satisfying proper growth constraints to be speciMed. 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 f, g and h. If conditions Lg Lf h(x)=Lg L2f h(x)=· · ·=Lg Lfr−2 h(x)=0 and Lg Lfr−1 h(x) = 0 hold globally then system (1) possesses a globally-deMned relative degree r (Isidori, 1995) (Lg ; Lf denote the Lie derivatives), and the input–output dynamics can be reduced to y(r) = Lrf h(x) + Lg Lfr−1 h(x)u:
(2)
Let =[y; y; ˙ : : : ; y(r−1) ]T , then it is always possible (Isidori, 1995) to deMne a vector ∈ Rn−r such that the map x = ( ; )
(3) n
is a di4eomorMsm on R and the dynamics, which are referred to as the “internal dynamics” (Isidori, 1995), can be expressed as follows: ˙ = q( ; ):
(4)
If r = n there are no internal dynamics and the system is said to be “fully linearizable” (Isidori, 1995). Assume what follows: Assumption 1. The internal dynamics (4) are input-to-state stable (ISS). Lrf h(x),
Assumption 2. The drift term, and the control gain, Lg Lfr−1 h(x), of the input–output dynamics (2) are globally bounded and Lipschitz. Let yR be a desired smooth output response, deMne the tracking error as e =y −yR and consider the associated error dynamics e(r) = Lrf h(x) − yR(r) + Lg Lfr−1 h(x)u:
Levant, Pisano, & Usai, 2002; Bartolini, Pisano, Punta, & Usai 2003; Levant, 2003) and can be characterized by a three-step procedure: Step 1: Sliding manifold design. The sliding variable is expressed as follows: r−2 ci e(i) ; e(0) = e; (6) = e(r−1) +
(5)
The output-feedback control problem with higher-order sliding modes has been dealt with in recent works (Bartolini,
; r − 2) are chosen such where the coePcients ci (i = 0; 2; : : : r−2 that the polynomial P() = r−1 + i=0 ci i is an Hurwitz one. Step 2: Estimation of the sliding variable. The error derivatives are not available and must be evaluated by means of some accurate real-time device robust against the measurement noise and possibly Mnite-time converging. Recently proposed (Levant, 2003), the arbitraryorder di4erentiator based on higher-order sliding modes appears to be an e4ective, yet robust, solution. If the input/output relative degree is r = 2, only the Mrst derivative of e needs to be estimated, and the arbitrary-order di4erentiator in Levant (2003) reduces to the Mrst-order differentiator (Levant, 1998): (t) = q0 (t) − e(t); q˙0 (t) = q1 (t) − 0 |(t)|1=2 sign((t)); q˙1 (t) = −1 sign((t)):
(7)
The tuning conditions are 1 + C 2 ; (8) 1 − C 2 where C2 is a Lipschitz constant of the error derivative e. ˙ If r ¿ 2 one could implement a cascade of di4erentiators (7) to estimate the higher-order derivatives. However, by performing a noise propagation analysis it results that the higher-order di4erentiator presented in Levant (2003), specially designed for multiple di4erentiation task, is more e4ective. Step 3: Stabilization of the sliding variable. Consider the nonlinear uncertain second-order sliding variable dynamics 1 ¿ C2 ; 02 ¿ 4C2
S = ’( ; R ; ; u) + ( ; )u; ˙
(9)
[yR ; y˙ R ; : : : ; yR(r−1) ]T .
where R = From Assumption 2, taking into account that vector R is norm-bounded, it can be concluded that the so-called “equivalent control” (Utkin, 1992) is bounded. Thus, as long as the closed-loop system evolves within a (possibly large) compact domain containing the 2-sliding manifold = ˙ = 0, the following additional assumption can be met. Assumption 3. Three positive constants F, 1 , 2 can be found such that the uncertainties ’ and satisfy the following boundedness conditions: |’| 6 F; 0 ¡
1
6 6
2:
(10)
A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531
Conditions (10), which hold locally, constitute a particular case of more general state-dependent bounds for the uncertain sliding variable dynamics (Bartolini, Ferrara, Pisano, & Usai, 2001a), and allow for a particularly simple constant-parameters realization of the so-called “Sub-optimal” 2-SMC algorithm (Bartolini, Ferrara, Levant, & Usai, 1999): Sub-optimal algorithm u(0) = 0 −UM sign((t) − 12 (0)); u(t) ˙ = −$(t)UM sign((t) 1 − 2 (tMi )); $(t) =
∗
1527
Variable-section nozzles Hydraulic pipe
0 6 t ¡ tM1 ; tMi 6 t ¡ tMi+1 ; i = 1; 2; : : : (11)
1 2 (tMi )) 6 0;
$ ;
(tMi )((t) −
1;
(tMi )((t) − 12 (tMi )) ¿ 0;
Linear motor drives
Pump
Fig. 1. The UV prototype.
(12)
where tMi (i = 1; 2; : : :) is the sequence of the time instants at which =0, ˙ and parameters UM and $∗ are chosen according to the tuning rules $∗ ¿
2
3
1
;
UM ¿ max
(13) 3$∗
4F 1−
2
;
F 1
:
(14)
The above control law steers both and ˙ to zero in a Mnite time (Bartolini et al., 1999, 2001a). The reader is referred to (Bartolini et al., 2000; Levant, 2003) for details regarding the separation principle establishing the closed-loop stability of the above combined 2-SMC/2-SMD scheme. Remark 1. Since ˙ is not known the sequence tMi is unavailable, but can be approximately detected using only sampled measurements of carried out at the time instants tk = k& (k = 0; 1; : : :) (Bartolini et al., 2001a). It has been shown that the resulting approximate implementation of the controller guarantees the reaching of an O(&2 ) boundary layer of the sliding manifold = 0 (Bartolini et al., 2001a, b). 3. An UV prototype with a jet-based propulsion system An UV prototype has been recently built at the DIEE-University of Cagliari as a preliminary test-bed of a novel water-jet-based propulsion system for underwater vehicles. The vehicle is about 150 cm long and 80 cm high. It contains a centrifugal pump feeding an hydraulic pipe and two variable-section nozzles, actuated by means of linear motor drives, located at the opposite ends of the hydraulic circuit (Fig. 1). The prototype is rigidly connected with a wheeled trolley that “suspends” the UV into a water channel (Fig. 2).
Fig. 2. The UV prototype in the water channel.
This conMguration allows the UV to move freely along the channel under the reaction force exerted by the water Kow through the nozzles. The nozzle output sections can be adjusted by moving the corresponding spear valve, directly coupled with a linear electric drive. The spear valve proMle is similar to that in a Pelton turbine, so that the generated thrust is turbulence-free and almost-linearly dependent on the valve position. Fig. 3 shows the hydro-jets in two di4erent opening conditions of the nozzle. The direct mechanical coupling between the valve and the motor, and the large bandwidth of the latter, allow for a very fast control of thrust proMle. 3.1. The UV model The dynamics of the considered jet-propelled UV can be formulated as follows (Fossen, 1994): (Mv + Ma )yS + k1 y| ˙ y| ˙ + k2 y˙ + d(t) = F1 (t) − F2 (t); (15)
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Almost-closed nozzle
Almost-open nozzle
Desired UV position +
Command –
LM position Vehicle controller +
–
Linear motor controllers
Actual UV position Actual LM position Actual UV position
Fig. 5. A schematic representation of the control architecture. Fig. 3. The water jets with open and closed nozzle.
z1(t)
slider
z1 and z2 is r = 2 and the above outlined design procedure yields the following steps. Sliding manifold design. According to (6), the sliding variable is deMned as
Directly-coupled linear motor drive
spear valve Fig. 4. Nozzle position notation.
where y(t) is the vehicle position, Mv is the vehicle mass, Ma represents the added mass e4ect, k1 and k2 are the viscous friction and drag coePcients, d(t) accounts for the external disturbances (e.g. currents, border e4ects) and F1 (t), F2 (t) are the control thrusts exerted on the UV by the two opposite jets. With a good approximation it can be assumed that F1 (t) and F2 (t) depend instantaneously on the positions, z1 and z2 , of the spear valves (Fig. 4) by a nonlinear function h F1 = h(z1 );
F2 = h(z2 ):
(16)
The positions of the two spear valves are deMned according to the notation represented in Fig. 4: z1 (z2 ) is zero when the nozzle is closed and increases while the spear valve is opening. Thus, z1 and z2 remain always nonnegative. Function h is quite diPcult to determine, and therefore it is considered uncertain in the present treatment. Obviously h is strictly positive, monotonically increasing and zero when its argument is zero. The system parameters Mv , Ma , k1 , k2 as well as the disturbance d(t) are unknown. Collecting the uncertainties affecting the UV dynamics, system (15) can be rearranged as follows: yS = f(y; ˙ t) + g[h(z1 ) − h(z2 )];
(17)
where g = 1=(Ma + Mv ) and with implicit deMnition of function f(y; ˙ t). 4. Controller design Let yR be a smooth reference proMle for the vehicle position and let e be the tracking error. By (17), the relative degree between the position error e and the control variables
(t) = e(t) ˙ + ce(t); c ¿ 0:
(18)
Estimation of the sliding variable. The actual vehicle velocity error e˙ is estimated in real-time by means of the di4erentiator (7), with the parameters 0 and 1 set on the basis of (8) with C2 suPciently large so that condition |e| S 6 C2 is met. Stabilization of the sliding variable. A desired proMle for the spear-valves coordinates z1 and z2 is deMned so that the sliding variable is reduced to zero in Mnite time. The Mrst-order dynamics of the sliding variable can be represented by the following di4erential equation: ˙ = f(y; ˙ t) − yS R + ce(t) ˙ + gA(z1 ; z2 );
(19)
where A(z1 ; z2 ) = h(z1 ) − h(z2 ):
(20)
To minimize the energy consumption, the two nozzles should not be both opened at the same time. This corresponds to keep either z1 or z2 to zero at each time instant. DeMne the dummy control variable +z = z1 − z2
(21)
subjected to the aforementioned constraints z1 ¿ 0;
z2 ¿ 0; z1 · z2 = 0:
By (21) and (22), Eq. (20) reduces to +z ¿ 0; h(+z ); A ≡ A(+z ) = −h(−+z ); +z ¡ 0:
(22)
(23)
Di4erentiating (19) and considering (20)–(23) one obtains dA ˙ S = ’(y; y; ˙ +z ; t) + g (24) +z : d+z Thanks to the imposed constraints (22), dynamics (24) is formally equivalent to (9) with u = +z . By analyzing the physical structure of the system, it can be asserted that function ’ is bounded and A(+z ) is strictly increasing, then conditions (10) hold for some constants F, 1 and 2 . A cascade compensation scheme is employed for the control system (Fig. 5): the “high-level” vehicle controller
A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531 25
20
20
15
15
[cm]
[cm]
25
10
10
5
5
0
0
10
20
30
40
0
0
10
Time [sec]
20 Time [sec]
1529
30
40
Fig. 6. The actual and desired position proMle with 2-SMC (left) and PID-control (right).
drives each LM controller with a reference proMle for the LM position. Given a command proMle +zR for +z , the desired proMles z1R and z2R for the spear valve positions are obtained by inverting (21) under conditions (22): z1R = max{+zR ; 0}; z2R = −min{+zR ; 0}:
(25)
The inner loop in Fig. 5 might cause the actual proMles of z1 and z2 to be largely di4erent from the prescribed ones z1R and z2R . Nevertheless, considering the large bandwidth of the LM drive control system (over 15 Hz) and the considerable inertia of the UV it is possible to regard the LM dynamics as a singular perturbation suPciently fast to preserve the sliding mode stability (Fridman, 2003). The robustness against fast unmodeled dynamics is indeed one of the most important features of the SMC approach as far as the practical implementation issues are dealt with. Let us summarize the overall controller. The sliding manifold is (t) ˆ = eˆ˙ + ce;
c ¿ 0;
(26)
where e = y − yR and eˆ˙ is computed by using di4erentiator (7). The reference position proMles for the linear motors are set according to (25), where +˙zR is a discontinuous signal deMned according to the Sub-optimal 2-SMC algorithm (11)–(14). By relying on the separation principle demonstrated in Bartolini et al. (2000), and taking into account the actual implementation e4ects like noise (Levant, 2000) and discretization (Bartolini et al., 2001b), it can be asserted that the following conditions are simultaneously fulMlled after a Mnite time: |eˆ˙ − e| ˙ = 1 ;
(27)
|| ˆ = 2
(28)
with 1 ; 2 ≈ 0. The exponential convergence of e toward a small neighbor of zero follows from trivial arguments.
5. The experimental setup: implementation issues and test results The linear motor drives (by Linmot TM ) have a rated bandwidth between 15 Hz and 20 Hz. A dedicated driver module allows for setting the force w applied to the slider by means of a reference input wR (the force-loop is embedded in the driver module). Furthermore, the driver module gives the slider positions y1 and y2 as incremental encoder-like signals, and a PID position-force loop has been closed externally to the LM controller. The control system has been implemented on a PC-based platform (Pentium2 processor at 350 Mhz). The computational burden of the control system is limited, and much less computing power would be suPcient. Discretization has been performed by the classical backward-di4erence method with a sampling step of 2 ms (the sample-and-hold e4ect was analyzed in Bartolini et al. (2001b)). The parameters of the controller and of the di4erentiator are: UM = 10, $∗ = 1 (Sub-optimal 2-SMC), 0 = 12, 1 = 20 (2-SMD). The performance of the proposed 2-SM controller/observer scheme with the above parameter set has been compared with that of a classical PID controller with gains KP = 2, KI = KD = 1. In a Mrst test a piece-wise constant reference position was used. Fig. 6 reports the actual and desired trajectory obtained using the two di4erent approaches, and evidences that the VSC is more accurate. Fig. 7 (left) shows the time evolution of the sliding variable, while Fig. 7 (right) reports the discontinous signal +˙∗z . Fig. 8 reports the actual and desired position of the two linear motors. It can be seen that the two nozzles are never both opened at the same time instant. A tracking test using a sinusoidal reference proMle has been also carried out. Fig. 9 shows that the actual trajectory converges to the desired one after a very short transient. 6. Conclusions This paper dealt with the 2-SMC approach to the output-feedback control problem for nonlinear uncertain
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A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531 10
20 15 10 [cm s-1]
[cm s-1]
5 0 -5
5 0 -5 -10 -15
-10
0
10
20 30 Time [sec]
-20
40
0
10
20 30 Time [sec]
40
5
5
4
4
3
3 [mm]
[mm]
Fig. 7. The sliding variable time history (left) and the discontinuous signal +˙zR (t) (right) in the regulation test.
2
2
1
1
0
0 -1
-1 0
10
20 Time [sec]
30
40
0
10
20 Time [sec]
30
40
Fig. 8. The actual and desired position proMles for the two linear motors in the regulation test with 2-SMC. Left plot: motor n. 1. Right plot: motor n. 2.
5
platform. Experiments pointed out the good performance and robustness of the proposed controller, and, in particular, an improvement of accuracy with respect to a classical PID controller.
0
Acknowledgements
[cm]
10
Partially supported by the MIUR (former MURST) project “Navigation, Guidance and Control for Underwater Vehicles”.
-5
-10
0
10
20 Time [sec]
30
40
Fig. 9. The actual and desired position proMles in the tracking test with 2-SMC.
systems. A challenging application is considered: the motion control for a jet-propelled underwater-vehicle prototype by output feedback. Because of the heavy system uncertainties, it is sensible to use an observer/controller scheme based on second-order sliding modes. Its mild computational load allows for the digital implementation on a low-cost hardware
References Atassi, N. A., & Khalil, H. K. (1999). A separation principle for the stabilization of a class of nonlinear systems. IEEE Transactions on Automatic Control, 44, 1672–1687. Bartolini, G., Ferrara, A., Levant, A., & Usai, E. (1999). On second order sliding mode controllers. In K. D. Young, & U. Ozguner (Eds.), Variable structure systems, sliding mode and nonlinear control, Lecture Notes in Control and Information Sciences, Vol. 247 (pp. 329 –350). Berlin: Springer. Bartolini, G., Levant, A., Pisano, A., & Usai, E. (2000). On the robust stabilization of nonlinear uncertain systems with incomplete state availability. ASME Journal of Dynamic Systems, Measurement and Control, 122, 738–745.
A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531 Bartolini, G., Ferrara, A., Pisano, A., & Usai, E. (2001a). On the convergence properties of a 2-sliding control algorithm for nonlinear uncertain systems. International Journal of Control, 74, 718–731. Bartolini, G., Pisano, A., & Usai, E. (2001b). Digital second order sliding mode control for uncertain nonlinear systems. Automatica, 37(9), 1371–1377. Bartolini, G., Levant, A., Pisano, A., & Usai, E. (2002). Higher-order sliding modes for output-feedback control of nonlinear uncertain systems. In X. Yu, & J. X. Xu (Eds.), Variable structure systems: Towards the 21st century, Lecture Notes in Control and Information Sciences, Vol. 274 (pp. 83–108). Berlin: Springer. Bartolini, G., Pisano, A., Punta, A., & Usai, E. (2003). A survey of applications of second-order sliding mode control to mechanical systems. International Journal of Control, 76(9/10), 875–892. Fossen, T. (1994). Guidance and control of ocean vehicles. UK: Wiley. Fridman, L. (2003). Chattering analysis in sliding mode systems with inertial sensors. International Journal of Control, 76(9/10), 906–912. Isidori, A. (1995). Nonlinear control systems (3rd ed.). Berlin: Springer. Levant, A. (1998). Robust exact di4erentiation via sliding mode technique. Automatica, 34, 379–384. Levant, A. (2000). Variable measurement step in 2-sliding control. Kybernetica, 36, 77–93. Levant, A. (2003). Higher order sliding modes, di4erentiation and output-feedback control. International Journal of Control, 76, 924–941. Teel, A., & Praly, L. (1995). Tools for semi-global stabilization via partial state and output feedback. SIAM Journal of Control and Optimization, 33, 1443–1488.
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Utkin, V. I. (1992). Sliding modes in control and optimization. Berlin: Springer. Alessandro Pisano was born in Sassari, Italy, in 1972. He graduated in Electronic Engineering in 1997 at the Department of Electrical and Electronic Engineering (DIEE) of the Cagliari University (Italy), where he received the Ph.D. degree in Electronics and Computer Science in 2000. He is currently a research associate at DIEE. His current research interest include nonlinear and robust control, variable-structure systems and sliding-mode control implementation in mechanical and electromechanical systems. Elio Usai was born in Sassari, Italy, in 1960. He graduated in Electrical Engineering at the University of Cagliari, Italy, in 1985. Up to 1994 he has been working for international industrial companies. Since 1994 he is at the Department of Electrical and Electronic Engineering (DIEE), University of Cagliari, where currently he is associate professor of automatic control. Current research interests are in the Meld of control engineering, variable structure systems, optimal control and modeling. He is a member of IEEE and of the Associazione Elettrotecnica ed Elettronica Italiana.
www.elsevier.com/locate/automatica
Brief paper
Output-feedback control of an underwater vehicle prototype by higher-order sliding modes Alessandro Pisano∗ , Elio Usai University of Cagliari, Department of Electrical and Electronic Engineering (DIEE), Piazza d’Armi, Cagliari 09123, Italy Received 15 April 2003; received in revised form 17 February 2004; accepted 24 March 2004
Abstract This paper describes some experimental results concerning the practical implementation of a recently proposed nonlinear output-feedback control technique based on the higher-order sliding mode approach. The considered technique is applied to the motion control problem for an underwater vehicle prototype that is equipped with a special propulsion system based on hydro-jets with variable-section nozzles. To cope with the heavy uncertainties a4ecting the prototype dynamics the output-feedback control system has been developed by means of an observer-controller that combines a second-order sliding-mode controller and a second-order sliding-mode di4erentiator. The reported experiments show that the proposed approach is capable of guaranteeing fast and accurate response under several operating conditions. The control system design procedure, and the main implementation issues, are discussed in detail. ? 2004 Elsevier Ltd. All rights reserved. Keywords: Second-order sliding-modes; Underwater vehicles; Output–feedback; Nonlinear uncertain systems; Di4erentiators
1. Introduction This paper details some experiments concerning the motion control of an underwater vehicle (UV) prototype built and developed at the Department of Electrical and Electronic Engineering (DIEE) of the Cagliari University. The considered UV is equipped with a special water-jet propulsion system. Because of both the system realization and the operating conditions its dynamics is a4ected by various uncertainties that should be taken into account in the controller design. A popular approach to nonlinear output-feedback control design under “heavy” uncertainty conditions entails the combined use of Lyapunov-based controllers and high-gain observers (HGOs) to estimate the output derivatives (Teel & Praly, 1995; Atassi & Khalil, 1999). Under the hypothesis that a stabilizing state-feedback control is available, it This paper was presented at the IFAC Workshop on Robust Control Design, ROCOND ’03, Milan, June 2003. This paper was recommended for publication in revised form by Associate Editor Keum-Shik Hong under the direction of Editor Mituhiko Araki. ∗ Corresponding author. Tel.: +39-0706-755760; fax: +39-0706-755782. E-mail addresses: [email protected] (A. Pisano), [email protected] (E. Usai).
0005-1098/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.03.016
was shown in Atassi and Khalil (1999) that using an HGO one can recover the performance achieved under state feedback, thus a rather general nonlinear separation principle was established there. Nevertheless, in real implementation real-time di4erentiators based on high-gain observers can be rather sensitive to the measurement noise a4ecting the output variable. Robust di4erentiators based on the second-order sliding mode control approach (see Levant, 1998; Bartolini, Pisano, & Usai, 2001b), exhibit interesting properties of robustness with respect to the measurement noise. This work validates the theoretical results reported in Bartolini, Levant, Pisano and Usai (2000); Levant (2003) and addresses the relevant practical implementation issues. In the above works stability and separation results about the combined use of second-order sliding-mode controllers (2-SMC) (Bartolini, Ferrara, Levant, & Usai, 1999) and second-order sliding-mode di4erentiators (2-SMD) (Levant, 1998) were dealt with. The paper is organized as follows: in Section 2, we brieKy recall the basic design principles of output-feedback 2-SM control. In Section 3, a mathematical model of the UV dynamics is given, and in the successive Section 4, the combined 2-SMC/2-SMD approach is applied to design a
1526
A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531
robust output-feedback motion controller for the UV prototype. Section 5 describes the experimental setup and discusses the experimental results, and Section 6 draws some Mnal conclusions. 2. Nonlinear output-feedback control via 2-SM controllers and SM dierentiators
i=0
Consider the nonlinear SISO system x˙ = f(x) + g(x)u; y = h(x)
(1) n
with unavailable state vector x ∈ R , scalar control variable u ∈ R and measurable output y ∈ R. Let f, g and h be unknown, suPciently smooth, vector-Melds of appropriate dimension satisfying proper growth constraints to be speciMed. 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 f, g and h. If conditions Lg Lf h(x)=Lg L2f h(x)=· · ·=Lg Lfr−2 h(x)=0 and Lg Lfr−1 h(x) = 0 hold globally then system (1) possesses a globally-deMned relative degree r (Isidori, 1995) (Lg ; Lf denote the Lie derivatives), and the input–output dynamics can be reduced to y(r) = Lrf h(x) + Lg Lfr−1 h(x)u:
(2)
Let =[y; y; ˙ : : : ; y(r−1) ]T , then it is always possible (Isidori, 1995) to deMne a vector ∈ Rn−r such that the map x = ( ; )
(3) n
is a di4eomorMsm on R and the dynamics, which are referred to as the “internal dynamics” (Isidori, 1995), can be expressed as follows: ˙ = q( ; ):
(4)
If r = n there are no internal dynamics and the system is said to be “fully linearizable” (Isidori, 1995). Assume what follows: Assumption 1. The internal dynamics (4) are input-to-state stable (ISS). Lrf h(x),
Assumption 2. The drift term, and the control gain, Lg Lfr−1 h(x), of the input–output dynamics (2) are globally bounded and Lipschitz. Let yR be a desired smooth output response, deMne the tracking error as e =y −yR and consider the associated error dynamics e(r) = Lrf h(x) − yR(r) + Lg Lfr−1 h(x)u:
Levant, Pisano, & Usai, 2002; Bartolini, Pisano, Punta, & Usai 2003; Levant, 2003) and can be characterized by a three-step procedure: Step 1: Sliding manifold design. The sliding variable is expressed as follows: r−2 ci e(i) ; e(0) = e; (6) = e(r−1) +
(5)
The output-feedback control problem with higher-order sliding modes has been dealt with in recent works (Bartolini,
; r − 2) are chosen such where the coePcients ci (i = 0; 2; : : : r−2 that the polynomial P() = r−1 + i=0 ci i is an Hurwitz one. Step 2: Estimation of the sliding variable. The error derivatives are not available and must be evaluated by means of some accurate real-time device robust against the measurement noise and possibly Mnite-time converging. Recently proposed (Levant, 2003), the arbitraryorder di4erentiator based on higher-order sliding modes appears to be an e4ective, yet robust, solution. If the input/output relative degree is r = 2, only the Mrst derivative of e needs to be estimated, and the arbitrary-order di4erentiator in Levant (2003) reduces to the Mrst-order differentiator (Levant, 1998): (t) = q0 (t) − e(t); q˙0 (t) = q1 (t) − 0 |(t)|1=2 sign((t)); q˙1 (t) = −1 sign((t)):
(7)
The tuning conditions are 1 + C 2 ; (8) 1 − C 2 where C2 is a Lipschitz constant of the error derivative e. ˙ If r ¿ 2 one could implement a cascade of di4erentiators (7) to estimate the higher-order derivatives. However, by performing a noise propagation analysis it results that the higher-order di4erentiator presented in Levant (2003), specially designed for multiple di4erentiation task, is more e4ective. Step 3: Stabilization of the sliding variable. Consider the nonlinear uncertain second-order sliding variable dynamics 1 ¿ C2 ; 02 ¿ 4C2
S = ’( ; R ; ; u) + ( ; )u; ˙
(9)
[yR ; y˙ R ; : : : ; yR(r−1) ]T .
where R = From Assumption 2, taking into account that vector R is norm-bounded, it can be concluded that the so-called “equivalent control” (Utkin, 1992) is bounded. Thus, as long as the closed-loop system evolves within a (possibly large) compact domain containing the 2-sliding manifold = ˙ = 0, the following additional assumption can be met. Assumption 3. Three positive constants F, 1 , 2 can be found such that the uncertainties ’ and satisfy the following boundedness conditions: |’| 6 F; 0 ¡
1
6 6
2:
(10)
A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531
Conditions (10), which hold locally, constitute a particular case of more general state-dependent bounds for the uncertain sliding variable dynamics (Bartolini, Ferrara, Pisano, & Usai, 2001a), and allow for a particularly simple constant-parameters realization of the so-called “Sub-optimal” 2-SMC algorithm (Bartolini, Ferrara, Levant, & Usai, 1999): Sub-optimal algorithm u(0) = 0 −UM sign((t) − 12 (0)); u(t) ˙ = −$(t)UM sign((t) 1 − 2 (tMi )); $(t) =
∗
1527
Variable-section nozzles Hydraulic pipe
0 6 t ¡ tM1 ; tMi 6 t ¡ tMi+1 ; i = 1; 2; : : : (11)
1 2 (tMi )) 6 0;
$ ;
(tMi )((t) −
1;
(tMi )((t) − 12 (tMi )) ¿ 0;
Linear motor drives
Pump
Fig. 1. The UV prototype.
(12)
where tMi (i = 1; 2; : : :) is the sequence of the time instants at which =0, ˙ and parameters UM and $∗ are chosen according to the tuning rules $∗ ¿
2
3
1
;
UM ¿ max
(13) 3$∗
4F 1−
2
;
F 1
:
(14)
The above control law steers both and ˙ to zero in a Mnite time (Bartolini et al., 1999, 2001a). The reader is referred to (Bartolini et al., 2000; Levant, 2003) for details regarding the separation principle establishing the closed-loop stability of the above combined 2-SMC/2-SMD scheme. Remark 1. Since ˙ is not known the sequence tMi is unavailable, but can be approximately detected using only sampled measurements of carried out at the time instants tk = k& (k = 0; 1; : : :) (Bartolini et al., 2001a). It has been shown that the resulting approximate implementation of the controller guarantees the reaching of an O(&2 ) boundary layer of the sliding manifold = 0 (Bartolini et al., 2001a, b). 3. An UV prototype with a jet-based propulsion system An UV prototype has been recently built at the DIEE-University of Cagliari as a preliminary test-bed of a novel water-jet-based propulsion system for underwater vehicles. The vehicle is about 150 cm long and 80 cm high. It contains a centrifugal pump feeding an hydraulic pipe and two variable-section nozzles, actuated by means of linear motor drives, located at the opposite ends of the hydraulic circuit (Fig. 1). The prototype is rigidly connected with a wheeled trolley that “suspends” the UV into a water channel (Fig. 2).
Fig. 2. The UV prototype in the water channel.
This conMguration allows the UV to move freely along the channel under the reaction force exerted by the water Kow through the nozzles. The nozzle output sections can be adjusted by moving the corresponding spear valve, directly coupled with a linear electric drive. The spear valve proMle is similar to that in a Pelton turbine, so that the generated thrust is turbulence-free and almost-linearly dependent on the valve position. Fig. 3 shows the hydro-jets in two di4erent opening conditions of the nozzle. The direct mechanical coupling between the valve and the motor, and the large bandwidth of the latter, allow for a very fast control of thrust proMle. 3.1. The UV model The dynamics of the considered jet-propelled UV can be formulated as follows (Fossen, 1994): (Mv + Ma )yS + k1 y| ˙ y| ˙ + k2 y˙ + d(t) = F1 (t) − F2 (t); (15)
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A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531
Almost-closed nozzle
Almost-open nozzle
Desired UV position +
Command –
LM position Vehicle controller +
–
Linear motor controllers
Actual UV position Actual LM position Actual UV position
Fig. 5. A schematic representation of the control architecture. Fig. 3. The water jets with open and closed nozzle.
z1(t)
slider
z1 and z2 is r = 2 and the above outlined design procedure yields the following steps. Sliding manifold design. According to (6), the sliding variable is deMned as
Directly-coupled linear motor drive
spear valve Fig. 4. Nozzle position notation.
where y(t) is the vehicle position, Mv is the vehicle mass, Ma represents the added mass e4ect, k1 and k2 are the viscous friction and drag coePcients, d(t) accounts for the external disturbances (e.g. currents, border e4ects) and F1 (t), F2 (t) are the control thrusts exerted on the UV by the two opposite jets. With a good approximation it can be assumed that F1 (t) and F2 (t) depend instantaneously on the positions, z1 and z2 , of the spear valves (Fig. 4) by a nonlinear function h F1 = h(z1 );
F2 = h(z2 ):
(16)
The positions of the two spear valves are deMned according to the notation represented in Fig. 4: z1 (z2 ) is zero when the nozzle is closed and increases while the spear valve is opening. Thus, z1 and z2 remain always nonnegative. Function h is quite diPcult to determine, and therefore it is considered uncertain in the present treatment. Obviously h is strictly positive, monotonically increasing and zero when its argument is zero. The system parameters Mv , Ma , k1 , k2 as well as the disturbance d(t) are unknown. Collecting the uncertainties affecting the UV dynamics, system (15) can be rearranged as follows: yS = f(y; ˙ t) + g[h(z1 ) − h(z2 )];
(17)
where g = 1=(Ma + Mv ) and with implicit deMnition of function f(y; ˙ t). 4. Controller design Let yR be a smooth reference proMle for the vehicle position and let e be the tracking error. By (17), the relative degree between the position error e and the control variables
(t) = e(t) ˙ + ce(t); c ¿ 0:
(18)
Estimation of the sliding variable. The actual vehicle velocity error e˙ is estimated in real-time by means of the di4erentiator (7), with the parameters 0 and 1 set on the basis of (8) with C2 suPciently large so that condition |e| S 6 C2 is met. Stabilization of the sliding variable. A desired proMle for the spear-valves coordinates z1 and z2 is deMned so that the sliding variable is reduced to zero in Mnite time. The Mrst-order dynamics of the sliding variable can be represented by the following di4erential equation: ˙ = f(y; ˙ t) − yS R + ce(t) ˙ + gA(z1 ; z2 );
(19)
where A(z1 ; z2 ) = h(z1 ) − h(z2 ):
(20)
To minimize the energy consumption, the two nozzles should not be both opened at the same time. This corresponds to keep either z1 or z2 to zero at each time instant. DeMne the dummy control variable +z = z1 − z2
(21)
subjected to the aforementioned constraints z1 ¿ 0;
z2 ¿ 0; z1 · z2 = 0:
By (21) and (22), Eq. (20) reduces to +z ¿ 0; h(+z ); A ≡ A(+z ) = −h(−+z ); +z ¡ 0:
(22)
(23)
Di4erentiating (19) and considering (20)–(23) one obtains dA ˙ S = ’(y; y; ˙ +z ; t) + g (24) +z : d+z Thanks to the imposed constraints (22), dynamics (24) is formally equivalent to (9) with u = +z . By analyzing the physical structure of the system, it can be asserted that function ’ is bounded and A(+z ) is strictly increasing, then conditions (10) hold for some constants F, 1 and 2 . A cascade compensation scheme is employed for the control system (Fig. 5): the “high-level” vehicle controller
A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531 25
20
20
15
15
[cm]
[cm]
25
10
10
5
5
0
0
10
20
30
40
0
0
10
Time [sec]
20 Time [sec]
1529
30
40
Fig. 6. The actual and desired position proMle with 2-SMC (left) and PID-control (right).
drives each LM controller with a reference proMle for the LM position. Given a command proMle +zR for +z , the desired proMles z1R and z2R for the spear valve positions are obtained by inverting (21) under conditions (22): z1R = max{+zR ; 0}; z2R = −min{+zR ; 0}:
(25)
The inner loop in Fig. 5 might cause the actual proMles of z1 and z2 to be largely di4erent from the prescribed ones z1R and z2R . Nevertheless, considering the large bandwidth of the LM drive control system (over 15 Hz) and the considerable inertia of the UV it is possible to regard the LM dynamics as a singular perturbation suPciently fast to preserve the sliding mode stability (Fridman, 2003). The robustness against fast unmodeled dynamics is indeed one of the most important features of the SMC approach as far as the practical implementation issues are dealt with. Let us summarize the overall controller. The sliding manifold is (t) ˆ = eˆ˙ + ce;
c ¿ 0;
(26)
where e = y − yR and eˆ˙ is computed by using di4erentiator (7). The reference position proMles for the linear motors are set according to (25), where +˙zR is a discontinuous signal deMned according to the Sub-optimal 2-SMC algorithm (11)–(14). By relying on the separation principle demonstrated in Bartolini et al. (2000), and taking into account the actual implementation e4ects like noise (Levant, 2000) and discretization (Bartolini et al., 2001b), it can be asserted that the following conditions are simultaneously fulMlled after a Mnite time: |eˆ˙ − e| ˙ = 1 ;
(27)
|| ˆ = 2
(28)
with 1 ; 2 ≈ 0. The exponential convergence of e toward a small neighbor of zero follows from trivial arguments.
5. The experimental setup: implementation issues and test results The linear motor drives (by Linmot TM ) have a rated bandwidth between 15 Hz and 20 Hz. A dedicated driver module allows for setting the force w applied to the slider by means of a reference input wR (the force-loop is embedded in the driver module). Furthermore, the driver module gives the slider positions y1 and y2 as incremental encoder-like signals, and a PID position-force loop has been closed externally to the LM controller. The control system has been implemented on a PC-based platform (Pentium2 processor at 350 Mhz). The computational burden of the control system is limited, and much less computing power would be suPcient. Discretization has been performed by the classical backward-di4erence method with a sampling step of 2 ms (the sample-and-hold e4ect was analyzed in Bartolini et al. (2001b)). The parameters of the controller and of the di4erentiator are: UM = 10, $∗ = 1 (Sub-optimal 2-SMC), 0 = 12, 1 = 20 (2-SMD). The performance of the proposed 2-SM controller/observer scheme with the above parameter set has been compared with that of a classical PID controller with gains KP = 2, KI = KD = 1. In a Mrst test a piece-wise constant reference position was used. Fig. 6 reports the actual and desired trajectory obtained using the two di4erent approaches, and evidences that the VSC is more accurate. Fig. 7 (left) shows the time evolution of the sliding variable, while Fig. 7 (right) reports the discontinous signal +˙∗z . Fig. 8 reports the actual and desired position of the two linear motors. It can be seen that the two nozzles are never both opened at the same time instant. A tracking test using a sinusoidal reference proMle has been also carried out. Fig. 9 shows that the actual trajectory converges to the desired one after a very short transient. 6. Conclusions This paper dealt with the 2-SMC approach to the output-feedback control problem for nonlinear uncertain
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A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531 10
20 15 10 [cm s-1]
[cm s-1]
5 0 -5
5 0 -5 -10 -15
-10
0
10
20 30 Time [sec]
-20
40
0
10
20 30 Time [sec]
40
5
5
4
4
3
3 [mm]
[mm]
Fig. 7. The sliding variable time history (left) and the discontinuous signal +˙zR (t) (right) in the regulation test.
2
2
1
1
0
0 -1
-1 0
10
20 Time [sec]
30
40
0
10
20 Time [sec]
30
40
Fig. 8. The actual and desired position proMles for the two linear motors in the regulation test with 2-SMC. Left plot: motor n. 1. Right plot: motor n. 2.
5
platform. Experiments pointed out the good performance and robustness of the proposed controller, and, in particular, an improvement of accuracy with respect to a classical PID controller.
0
Acknowledgements
[cm]
10
Partially supported by the MIUR (former MURST) project “Navigation, Guidance and Control for Underwater Vehicles”.
-5
-10
0
10
20 Time [sec]
30
40
Fig. 9. The actual and desired position proMles in the tracking test with 2-SMC.
systems. A challenging application is considered: the motion control for a jet-propelled underwater-vehicle prototype by output feedback. Because of the heavy system uncertainties, it is sensible to use an observer/controller scheme based on second-order sliding modes. Its mild computational load allows for the digital implementation on a low-cost hardware
References Atassi, N. A., & Khalil, H. K. (1999). A separation principle for the stabilization of a class of nonlinear systems. IEEE Transactions on Automatic Control, 44, 1672–1687. Bartolini, G., Ferrara, A., Levant, A., & Usai, E. (1999). On second order sliding mode controllers. In K. D. Young, & U. Ozguner (Eds.), Variable structure systems, sliding mode and nonlinear control, Lecture Notes in Control and Information Sciences, Vol. 247 (pp. 329 –350). Berlin: Springer. Bartolini, G., Levant, A., Pisano, A., & Usai, E. (2000). On the robust stabilization of nonlinear uncertain systems with incomplete state availability. ASME Journal of Dynamic Systems, Measurement and Control, 122, 738–745.
A. Pisano, E. Usai / Automatica 40 (2004) 1525 – 1531 Bartolini, G., Ferrara, A., Pisano, A., & Usai, E. (2001a). On the convergence properties of a 2-sliding control algorithm for nonlinear uncertain systems. International Journal of Control, 74, 718–731. Bartolini, G., Pisano, A., & Usai, E. (2001b). Digital second order sliding mode control for uncertain nonlinear systems. Automatica, 37(9), 1371–1377. Bartolini, G., Levant, A., Pisano, A., & Usai, E. (2002). Higher-order sliding modes for output-feedback control of nonlinear uncertain systems. In X. Yu, & J. X. Xu (Eds.), Variable structure systems: Towards the 21st century, Lecture Notes in Control and Information Sciences, Vol. 274 (pp. 83–108). Berlin: Springer. Bartolini, G., Pisano, A., Punta, A., & Usai, E. (2003). A survey of applications of second-order sliding mode control to mechanical systems. International Journal of Control, 76(9/10), 875–892. Fossen, T. (1994). Guidance and control of ocean vehicles. UK: Wiley. Fridman, L. (2003). Chattering analysis in sliding mode systems with inertial sensors. International Journal of Control, 76(9/10), 906–912. Isidori, A. (1995). Nonlinear control systems (3rd ed.). Berlin: Springer. Levant, A. (1998). Robust exact di4erentiation via sliding mode technique. Automatica, 34, 379–384. Levant, A. (2000). Variable measurement step in 2-sliding control. Kybernetica, 36, 77–93. Levant, A. (2003). Higher order sliding modes, di4erentiation and output-feedback control. International Journal of Control, 76, 924–941. Teel, A., & Praly, L. (1995). Tools for semi-global stabilization via partial state and output feedback. SIAM Journal of Control and Optimization, 33, 1443–1488.
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Utkin, V. I. (1992). Sliding modes in control and optimization. Berlin: Springer. Alessandro Pisano was born in Sassari, Italy, in 1972. He graduated in Electronic Engineering in 1997 at the Department of Electrical and Electronic Engineering (DIEE) of the Cagliari University (Italy), where he received the Ph.D. degree in Electronics and Computer Science in 2000. He is currently a research associate at DIEE. His current research interest include nonlinear and robust control, variable-structure systems and sliding-mode control implementation in mechanical and electromechanical systems. Elio Usai was born in Sassari, Italy, in 1960. He graduated in Electrical Engineering at the University of Cagliari, Italy, in 1985. Up to 1994 he has been working for international industrial companies. Since 1994 he is at the Department of Electrical and Electronic Engineering (DIEE), University of Cagliari, where currently he is associate professor of automatic control. Current research interests are in the Meld of control engineering, variable structure systems, optimal control and modeling. He is a member of IEEE and of the Associazione Elettrotecnica ed Elettronica Italiana.