Brief PaperOn modular backstepping design with second order sliding modes☆
Introduction
The subject of the control of nonlinear systems with uncertainties has been attracting the attention of researchers for years. The two major classes of controllers which are capable of dealing with nonlinear uncertain systems are adaptive and robust controllers. Although, traditionally, these two approaches have been kept separate and even apart from the rest of nonlinear control theory, the recent attitude is to look at each control methodology as a possibly complementary ingredient of a unique design recipe (Bartolini, Ferrara & Utkin, 1995; Freeman & Kokotović, 1993; Liu & Zinober, 1996; Qu, 1994).
Following this attitude, in this paper, a combined adaptive/robust control algorithm is presented, based on the so-called modular backstepping (Krstić & Kokotović, 1995), already taken into account in the preliminary work (Bartolini, Ferrara & Giacomini, 1998b). More precisely, a tracking control objective is considered consisting of forcing the output of a nonlinear uncertain system to track a reference signal with the first n derivatives (n being the system order) known, bounded and piece-wise continuous. The control objective is attained designing a suitable control signal on the basis of a procedure which goes through the construction of a transformed system, characterized by n−2 differential equations analogous to those determinable via a purely modular backstepping design (Krstić et al., 1995), coupled with an uncertain nonlinear second-order auxiliary system. The control is chosen to be a second-order sliding mode control (SOSMC) (Bartolini, Ferrara & Usai, 1998a; Levant, 1997), so as to force the transformed state variables involved in the second-order auxiliary system to zero in finite time. The remainder of the transformed system turns out to be a reduced order system for which results similar to those valid for a purely modular backstepping controller still hold.
Only the choice of the estimation law, among the standard ones, still needs to be specified. In particular, in this paper, we make reference to the classical identifier based on the gradient rule, relying on the extension to nonlinear systems of the so-called swapping technique. It consists of converting a dynamic parametric model into a static form to retrieve a prediction error and to be able to use classical parameter estimation algorithms (Krstić & Kokotović, 1995; Morse, 1980). The proposed estimation-based modular SOSMC controller proves to be input-to-state stable (ISS) (Sontag, 1989), i.e., the system state is bounded, with respect to the parameter error and its derivative, regarded as input signals. On the other hand, the choice of the identifier assures that such signals are bounded, independently of the controller. Transient performance bounds for the transformed state, and, consequently, for the tracking error, are also derived. All the results are confirmed by simulation.
Apart from an increment of robustness due to the generation of a sliding mode, another relevant aspect, discussed in the paper, is the reduction of the computational load provided by the proposed approach. In this sense, our proposal can also be viewed as a contribution to the simplification of controllers designed through a backstepping-like procedure, in alternative to those indicated in Swaroop, Gerdes, Yip and Hedrick (1997), Won and Hedrick (1996) where a basic first-order sliding mode control strategy (Utkin, 1992) is applied and no adaptive components are present. The price of the simplification in our case is the loss of global stability, due to the combination of a globally stable control design (modular backstepping) with an intrinsically semi-globally stable one (SOSMC). Thus, the proposed approach can appear to be significant in applications where the computational burden is a critical parameter, while global stability is not necessary.
Section snippets
Problem statement
A dynamical system, with a single control input, can be described by the system of differential equationswhere , , , , , and are known smooth matrix functions belonging to , while the constant vector represents some parametric uncertainties.
We assume to deal with general systems of type (1), with , transformable via a suitable diffeomorphism in the parametric-strict
The proposed modular second-order sliding mode controller
In recent publications (Krstić & Kokotović, 1995; Krstić, Kokotović & Kanellakopoulos, 1995), the classical backstepping controllers have been reviewed on the basis of the input-to-state stability (ISS) and related concepts (Sontag, 1989). This theory allows the design of backstepping algorithms with a flexible choice of the parameter update law, so to attain better convergence characteristics of the parameter estimates. The modular design is focused on the realization of a controller that is
The swapping-based identifier
Now, to complete the design, the estimation mechanism needs to be specified. Following the line of Krstić and Kokotović (1995) one can, for instance, adopt the same identifier as in the x-swapping scheme based on the extension to nonlinear systems of Morse's linear swapping lemma (Morse, 1980). In our case, all the conditions for the application of Lemma 4.1 in Krstić and Kokotović (1995) are satisfied, so two filters applied to the signals constituting the regressor matrix Φ(x)=[Φ1,…,Φn−2]T
Stability properties of the proposed controller
With reference to the modified transformed system (10), the good behavior of the control module can be theoretically motivated both in convergence and during the transient phase. Theorem 2 Given system (10), ∀xin a connected setcontaining the equilibrium point such that {V≤c}, for somec∈(0,∞), and bounds (14) hold, the control strategy defined by Algorithm 2 with the additional constraint (15) is such that, inthe signals z and y are bounded and. Proof A necessary and
Transients performances
It has been shown that the z-subsystem in (10), with y1=0, is ISS with reference to the ISS-control Lyapunov function V=0.5|z|22. Moreover, by virtue of Lemma 5.8 in Kokotovic̀ et al. (1995).
where a0 and b0 are constant function of the design parameters ci,κi,gi, i=1,…,n−2, c0=mini{ci}, and ||·||∞ is the norm. But, y1 is not identically equal to zero for any t, then the above bound must be slightly modified to take into account the y-subsystem
Computational load
Apart from performances, a relevant aspect of the design approach presented in this paper is the reduction of the computational load with respect to a purely backstepping-based design, since two steps of the backstepping procedure are saved. In a previous work (Bartolini, Ferrara, Giacomini & Usai, 1996) it has been made a comparison between the computational load associated with the standard backstepping procedure and a combined backstepping/sliding mode control procedure based on the use of
Numerical example
To complement the theoretical discussion, an academic example is presented in this section.
We consider the augmented systemwhere u=x4 (note that, the anti-chattering effect is introduced). The equilibrium point of the system is {x1,x2,x3,x4}={0,0,0,0}, with the ideal values θ1=θ2=1. Actually, such values are assumed to be unknown. The reference quantities, , come from the linear reference model 60/((s+3)(s+4)(s+5)), fed with a
Conclusions
A modular procedure to design estimation-based second-order sliding mode controllers for uncertain nonlinear systems expressible in parametric-strict feedback form is presented in this paper. It consists of n−2 steps of modular backstepping type to compute all the quantities needed to build a second-order auxiliary system, which is controlled via a VSC algorithm generating a second-order sliding regime. The transformed system, on the whole, is proved to be ISS with respect to the parameter
Acknowledgements
Work partially supported by contract IST 10107-PROTECTOR of the European Community and by MURST Project “Identification and control of industrial systems”.
Antonella Ferrara was born in Genova, Italy in 1963. She received the Laurea degree Magna Cum Laude in Electronic Engineering in 1987, and the Ph.D. in Electronics and Computer Science in 1992 from the University of Genova, Italy. In 1992 she got the position of Assistant Professor at the Department of Communication, Computer and System Sciences of the University of Genova. Since 1998 she is Associate Professor of Automatic Control at the Department of Computer Engineering and Systems Science
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Antonella Ferrara was born in Genova, Italy in 1963. She received the Laurea degree Magna Cum Laude in Electronic Engineering in 1987, and the Ph.D. in Electronics and Computer Science in 1992 from the University of Genova, Italy. In 1992 she got the position of Assistant Professor at the Department of Communication, Computer and System Sciences of the University of Genova. Since 1998 she is Associate Professor of Automatic Control at the Department of Computer Engineering and Systems Science of the University of Pavia, Italy. Her research activity is mainly in the area of variable structure control and adaptive control. She also took part in national and European projects on traffic modelling and control and robotics. At present, she is one of the scientific leaders and member of the management board of the European project IST 1999-10107 PROTECTOR, on automated vehicles. She is author and co-author of more than 130 scientific papers, among which more than 35 appeared in international journals. She serves as a reviewer for numerous scientific journals in the field of automatic control, and has been guest editor of two special issues on sliding mode control, one appeared in the Journal “Control Theory and Advanced Technology” in 1994, the other in the “International Journal of Robust and Nonlinear Control” in 1997.
Luisa Giacomini was born in Varazze, Italy in 1966. She received the Laurea degree in Electronic Engineering in 1991, and the Ph.D. in Electronics and Computer Science in 1999 from the University of Genova, Italy. In 1995 she has been Visiting Research Fellow at the Strathclyde University, Glasgow, and since 1998 she is Contract Research Fellow at the Electronic Engineering Department of the Aston University, UK. Her research interests are in the fields of nonlinear adaptive control, sliding mode control, hybrid systems and flexible manufacturing.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor F. Jabbari under the direction of Editor Roberto Tempo.