Generalized Sliding-Mode Control for Multi-Input Nonlinear Systems

1410

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

Generalized Sliding-Mode Control for Multi-Input Nonlinear Systems Ronald M. Hirschorn

Abstract—The purpose of this paper is to present a framework for the design of sliding-mode based controllers for multi-input affine nonlinear systems. In this framework the control effort goes to zero as the state approaches the sliding surface, a key feature for practical implementation. The sliding surface is a submanifold with codimension equal to the dimension of the involutive distribution generated by the controlled vector fields rather than the dimension of their linear span. We also allow nonzero vectors in to be tangent to the sliding surface, the singular case in sliding-mode control. Index Terms—Nonlinear systems, sliding-mode control, stabilization.

I. INTRODUCTION HERE are a number of more or less standard assumptions made in the sliding control literature, either explicitly or implicitly (cf. [24], [26], and the references therein). The sliding surface is assumed to have a codimension equal to the number of independent controlled vector fields. Due to the use of full control authority state trajectories under sliding-mode controllers are solutions to differential inclusions rather than differential equations. Finally, to ensure that state trajectory can be forced to evolve on the sliding surface, at each point in the sliding surface the span of the controlled vector fields are assumed to have a trivial intersection with the tangent space to the sliding surface. The design of such controllers can be problematic for systems whose linearization is not controllable. The purpose of this paper is to present a framework for the design of multi-input nonlinear controllers which act, in some sense, as a practical implementation of sliding-mode control and do not require any of the above assumptions. Our generalized sliding-mode controller is modelled after “sample and hold” controllers used in digital control, hence our state trajectories are continuous piecewise smooth curves. This has some connections with the work of Clarke, Ledyaev, and Sontag on asymptotic controllability [7]. By allowing the state to evolve close to the sliding surface we can accommodate the singular situation in sliding-mode control where the input vector fields become tangent to on some submanifold [13]. In particular we show that for a class of homogeneous systems with this feature our generalized sliding-mode controller achieves global asymptotic stabilization. We note that finite

T

Manuscript received September 13, 2004; revised August 11, 2005, November 30, 2005, and December 14, 2005. Recommended by Associate Editor L. Glielmo. This work was supported by the Natural Sciences and Engineering Research Council of Canada. The author is with the Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada (e-mail: [email protected]. ca). Digital Object Identifier 10.1109/TAC.2006.880959

time stability for homogeneous switched systems has been studied in [17] in the case where the controllers cause the state to evolve on the sliding surface after a finite time interval. One great advantage of sliding-mode control is the reduction of order which comes from keeping the state on or close to the sliding surface . Typically (cf. [13]–[26]) the dimension of is where is the dimension of the state manifold and the number of (independent) input vector fields. For the generalized sliding-mode controllers introduced here the dimension where is the dimension of the involutive disof is tribution generated by the controlled vector fields. This can greatly simplify the problem of designing the sliding surface at the price of a more involved controller to carry the state trajectory to , since we use Lie brackets of the input vector fields to help move the state to the sliding surface. That the state trajectory can be made to closely follow trajectories of an extended system (whose extra input vector fields are Lie brackets of input vector fields) has been noted by Brockett, Pomet, and others [3], [18]. We note that in [10] a quite different way of increasing the codimension of the sliding surface is presented. Here the controller causes the state to approach a submanifold which may depend explicitly on the control and its derivatives. The requirements on the controllers take the form of ordinary differential equations. The assumptions needed are related to those which allow for dynamic feedback linearization. In contrast to the results presented here the controller design and dimension of the sliding surface are independent of the structure of the distribution generated by the controlled vector fields. In the framework developed here the codimension of the sliding surface is not the relative degree of some output function but is determined by the dimension of the involutive distribution generated by the controlled vector fields. Under conventional sliding-mode control the state is moved to the sliding surface in finite time. Practical considerations dictate that the control effort be attenuated as the state trajectory nears the sliding surface and this requires a controller design which accounts for the contribution of the drift vector field to the state trajectory (cf. [24]). Here we do not insist that that the actual trajectory reach the sliding surface in finite time but design our controller to move the state of the corresponding driftless system to the sliding surface in finite time. This can be a relatively straightforward task, especially in the case where the controlled vector fields span an involutive distribution. The paper is organized as follows: in Section II we present our definitions for practical and asymptotic stability for families of controllers. In Section III we define sliding surfaces, our generalized sliding-mode controller, and the -Lyapunov function. Section IV contains Theorem IV.1, our main result on practical

0018-9286/$20.00 © 2006 IEEE

HIRSCHORN: GENERALIZED SLIDING-MODE CONTROL FOR MULTI-INPUT NONLINEAR SYSTEMS

stability for a family of generalized sliding-mode controllers and an illustrative example. In Section V contains results on asymptotic stability for a class of homogeneous systems. II. PRELIMINARIES is a smooth manifold, a smooth Suppose that function and a vector field on . Then denotes the Lie Derivative of along and the integral curve of passing through at , so that . If is a smooth vector field on then denotes the Lie Bracket of and . A dimensional distriis the choice of a dimensional subspace bution on of the tangent space to at for each . is smooth if there exist a neighbourhood of and smooth for each which span at each point of . A vector fields . If is a vector field belongs to if smooth function on then belongs to . is involutive if the Lie Bracket whenever are smooth vector fields which belong to . A continuously differis called positive definite at entiable function if iff and proper if the preimage is compact in . We will consider of every compact set in affine system models of the form (1) where

is a smooth -dimensional manifold, , the drift vector field and the conare smooth, and is an trolled vector fields equilibrium point where vanishes. We will assume that the with constant are complete. vector fields This simplifies the exposition. is an equilibrium state for Definition II.1: Suppose that system (1), a family of (possibly time dependent) feed. Then is practically back controllers indexed by if stabilized by the controllers corresponding to i) the state trajectories exist for all ; of and open ii) given any a compact neighbourhood there exists such subset of containing (cf. [25]). that If instead of ii) we have the stronger condition ii)’ given any of there exists such a compact neighbourhood , we say that is that . If ii)’ holds semiglobally asymptotically stabilized by for all we say that is globally asymptotically sta. bilized by III. A FRAMEWORK FOR SLIDING-MODE CONTROL A. Sliding Surfaces A sliding surface for system (1) is a subset of containing is a smooth submanthe equilibrium and such that ifold of the state manifold and states in can be moved to along piecewise smooth integral curves of the controlled . We let denote the smooth involutive vector fields generated by . To save accounting distribution on

1411

and to present semi-global results we assume that the integral curves for the controlled and drift vector fields are defined for , that is -dimensional , and is the all disjoint union of the leaves of . containing is Definition III.1: A connected subset of called a sliding surface for the system (1) if is an embedded codimension submanifold of a) ; . b) Our notion of the sliding surface differs from the usual one (cf. [13]–[26]) in significant ways. The dimension of . Typically is taken to be our sliding surface is dimensional. The drop in dimension between and is one of the attractions of sliding-mode control and is often (generically in the multi-input the dimension of case) greater than . This can simplify the problem of designing an appropriate sliding surface (see Example IV.2). Another standard assumption in sliding-mode control is that span . This ensures sufficient direct control to keep the state trajectory on the sliding surface (cf. [24][26])). For a significant class of control affine systems this requirement is too restrictive. Definition . III.1 allows for states in such that This lets us use sliding-mode control to generate new robust low-gain controllers to globally asymptotically stabilize a well known class of homogeneous systems (see Section V). We note that the high-gain continuous stabilizer introduced in [19] for these systems causes a globally defined Lyapunov function to decrease along the state trajectory. Since we do impose this requirement we can stabilize these systems using a comparatively low-gain discontinuous feedback controller. evolving on We note that for the linear system , the controlled vector fields are the constant vector fields where are the columns of the matrix . Because we have span Range . of this involutive distribution are the affine The leaves Range . A standard sliding surface is a subspace subsets such that Range . This choice of satisfies Definition III.1 but it is clear that in our framework a great can also act as sliding surfaces for many submanifolds of these linear systems. B. Generalized Sliding-Mode Controllers Two issues arise naturally in sliding-mode control. The first is finding a feedback controller which causes the state trajectory to reach the sliding surface in finite time and thereafter remain on . The second is to guarantee that the resulting trajectory on , the “ sliding motion” (cf. [26]), is stable. Our issues are somewhat different. We wish to allow for the possibility of “singular states” where the controlled vector fields are tangent to . We also want our controller to tends to zero as the state trajectory approaches while being sufficiently forceful so as to keep the state close to in the presence of outside disturbances and unmodelled dynamics. Our philosophy is to design our controller independent of the drift vector field. This implies a certain robustness in the face of unmodelled dynamics. Thus we define our generalized sliding-mode controller so that the state of the

1412

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

driftless system reaches the sliding surface in finite time. The drift vector field only plays a direct role in the choice of the sliding surface. We first consider steering the state of the driftless system to in finite time. Suppose that , where is a leaf of the distribution . Then Chow’s Theorem (cf. [6][12]) implies can be expressed as that every element of

where for and . Thus the leaf is traced out by piecewise smooth curves which are the concatenations of smooth integral curves of the controlled vector fields emanating from . Furthermore, since we are considering a driftless system evolving on , we have small time local controllability (cf. [23]). Supan open set in containing and . We pose that can find an open neighbourhood of contained in such can be reached from in time that every element along a concatenation of integral curves of controlled vector fields which evolve in . Since our system is driftless we can go “backward” along trajectories, so that we can select , vector fields , times defined by and points such that and . Of course are all functions of but, if we are given the point , we can deduce , etc. We can therefore treat as a function . In particular set for . of . If we rescale by the rescaled Then curve has . Here may be thought of as a “hybrid control” which is held constant for a time interval of length 1 i.e., sample and hold with sample interval 1 in the language of digital control theory. Thus, for

for with we have . This reachability property for driftless systems can be refor the stated in terms the existence of “hybrid controllers” driftless system (2) are transfered to in time and such that states of . the controls are constant on each subinterval under this controller have the Furthermore state trajectories local controllability properties discussed above, namely for any there exists a neighbourhood neighbourhood of such that is transfered to in time along trajectories contained in . In light of the above if we define the discussion we have controls appropriately. In particular, since for , we can set some for

(3)

for for to . We note that the and tend to zero as approaches (as ) functions also have this property. Also has a hence the controllers “memory” of length 1 as depends only on over the . time interval The above discussion serves to motivate our definition of a generalized sliding-mode controller. To simplify notation we (implicitly) assume a somewhat stronger version of controllability for the driftless system (2), namely that one can select a of fixed such that transfers between points in the leaves can be achieved using piecewise smooth trajectories with at most switches. We retain the feature that the the controllers tend to zero as approaches . As we shorten the “sampling time” our controller will act more forcefully to keep the state trajectory close to the sliding surface and hence retain some robustness. We will allow our controllers to have a memory of length as opposed to the memory of length 1 above. In partic, the controls will be ular, over time intervals if functions of . Here , is a sample history of the state trajectory with memory at most . Definition III.2: Let be a sliding surface for system (1) and . A control is called a generalized sliding-mode controller for with memory if on the time interval i) if ; over the time interval ii) for functions if ; iii) trajectories of the driftless system (2) with have and for ; and iv) if is an open neighbourhood of a point there exists an open neighbourhood of such that of the driftless system (2) with trajectories and have and for . We will show that controllable linear systems satisfy Definiand a stronger version of Definition III.2iv) tion III.2 with holds, namely if is an open neighbourhood of a point there exists an open neighbourhood of which is invariant under trajectories of the corresponding driftless system. with For example consider . Then, for the corresponding driftless system , we can steer to using . Then the open where conset of tains and is forward time invariant under trajectories . That is implies the driftless system with . This motivates the following definition. Definition III.3: Let be a sliding surface for system (1). for with memory A generalized sliding-mode controller is called a direct generalized sliding-mode controller for if Definition III.2-iv) holds for an open set forward . invariant under trajectories of the driftless system with In most applications of sampled-data control a high sample rate is desirable. In the discussion so far we have assumed a sample period of , the controller memory. To where define a controller with sample interval

HIRSCHORN: GENERALIZED SLIDING-MODE CONTROL FOR MULTI-INPUT NONLINEAR SYSTEMS

we use the modified state sample history . If is a generalized sliding-mode controller with memory , over the time interval we set

1413

time controllability in [14]. It is a system which does not satisfy the necessary conditions for the construction of a continuous stabilizing feedback controller [3], [22]. Our system model is , with and

(4) We note that the trajectory of the driftless system (2) with is a reparametrization of the trajectory corresponding to . Thus is constant over intervals of length whereas is constant over intervals of length 1. Since our generalized sliding-mode controller is piecewise constant and with constant is complete by assumption, solutions to the resulting state differential equation exist for . Finally, decreasing increases both and the controller’s ability to reject external disturbances. C. S-Lyapunov Functions In sliding-mode control, the sliding surface is designed to ensure the stability of the equivalent dynamics, the dynamics of the state evolution (sliding motion) on the sliding surface. We will show that for any controllable time-invariant linear system there exists a subspace , a quadratic function on with on for some , such that the restriction of to is proper and positive definite at the origin. This motivates the following definition. Definition III.4: Suppose that is a sliding surface for system is (1). A continuously differentiable function if called an -Lyapunov function for system (1) at ; i) ii) on for some ; is a proper positive definite iii) the restriction function at . IV. PRACTICAL STABILIZATION When the distribution generated by the input vector fields has a higher dimension than the number of independent inputs the motion in the state space required to generate the higher dimensional Lie brackets of the input vector fields can make asymptotic stabilization problematic. In this section therefore we consider practical stabilization and establish the following theorem. Theorem IV.1: Suppose that is a sliding surface for system a generalized sliding-mode controller for , and an (1), -Lyapunov function at . Then the equilibrium state is . practically stabilized by Before proving this theorem we present an example and then establish some technical lemmas. Example IV.2: Consider the system model

(5) where and . A lower dimensional version of this system is discussed in the study of local small

for . It follows that the involutive distribution has dimension and is the linear span of . with itself, Thus, with the usual identification of and the leaf of through a point is . Since it is natural , that is to select a codimension 3 siding surface

On

we have

. Since the leaf of through it follows that is a sliding surface for system (5). The function , when restricted to is positive , so that on definite and proper at . Furthermore we have . Since it follows that is an Lyapunov function for system (5) at . Theorem IV.1 asserts that if is a generalized sliding-mode controller . for then system (5) is practically stabilized by For this example finding a generalized sliding-mode controller is fairly straightforward. If we use piecewise constant and are easy to comcontrols the integral curves for pute. There are an infinite number of possible generalized sliding-mode controls for this example. We will construct a rather simple controller based on the following idea. We first nonzero using constant and zero. Next we use make to move to . Then is used to set back to zero. is used to set to zero. This results in a generalFinally . To satisfy ized sliding-mode controller with memory Definition III.2 iv) we need to ensure that when the state is a small amount in step one above. close to we only move To measure the “distance” of a state from we introduce the and define continuous function as follows. If then on ; if then the following hold. we set 1) Over the time interval . Then is and

and the state trajectory over through . Thus

is the integral curve of and

1414

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

2) Over

the

time

interval

we

set .

Then

and the state trajectory over through . Thus

is the integral curve of

and

3) Over

the

time

interval

the state trajectory over through , and

4) Over the time interval Then

we . Then

set

Fig. 1. Practical stabilization using generalized sliding-mode control.

is the integral curve of

we set and

.

This defines over the time interval . The above steps are repeated to define over , etc. The controller defined previously satisfies Definition III.2. Conditions i) . Simand ii) of Definition III.2 hold by our construction of of the ilarly we have demonstrated above that trajectories driftless system (2) with have and for so that Definition III.2 iii) holds. To show that Definition III.2 iv) holds we suppose that is any and . From our construction open set containing of we see that both and tend to approaches . Furthermore we can easily write zero as for (we have done so down explicit formulae for above) and verify that the distance from to for can be made arbitrarily small by choosing sufficiently close to . In particular we can choose an open neighbourhood of with such that trajectories of the driftless and have and system (2) with for so that Definition III.2 is a generalized sliding-mode controller and iv) holds. Thus Theorem IV.1 asserts that system (5) is practically stabilizable for sufficiently small. by

Fig. 2 shows the results of a simulation performed using Simnon/PCW for Windows Version 2.01 (SSPA Maritime Consulting AB, Sweden). The simulation results were and using a for initial condition Runge-Kutta-Fehlberg 4/5 scheme. For the system (5) is not clear how one would go about designing a standard sliding-mode controller which requires the state to slide on a sliding submanifold of codimension 2. a natural sliding For the linear system and acts as a surface is -Lyapunov function. For let denote the region in containing and bounded by the lines and . Then, on , we have . Thus is proper and we have . positive definite at More generally, suppose that is a sliding surface for system . (1), is an -Lyapunov function, and containing is Definition IV.3: A closed subset -extension of if called a a) ; b) the restriction of to is proper and positive definite at ; . c) Lemma IV.4 shows that -extensions of exist. Lemma IV.4: Suppose that is a sliding surface for system (1) and is an -Lyapunov function for system (1) at . Then -extension of . there exists a Proof: Here is an -Lyapunov function hence there exsuch that for all . Choose ists and . Then there exists an open neighbourof on which and hood . This ensures that does not vanish on . is metrizable we can further shrink if necessary to Since to is at most ensure that the distance from each point in . Let be the closure of the union of and all such

HIRSCHORN: GENERALIZED SLIDING-MODE CONTROL FOR MULTI-INPUT NONLINEAR SYSTEMS

1415

Fig. 2. Global stabilization of a mechanical system using generalized sliding-mode control.

sets. Then , the restriction of to is pos(because restricted to is proper and itive definite at does not vanish on the closure of the sets ), and . Thus i) and iii) of Definition III.4 are satisfied and is positive definite. To show that this map is is proper, so for any closed also proper we note that , we know that bounded interval is closed and bounded. Since is continuous is closed in hence in . By construction the distance from any point in the bounded set to a point is at most . Since is the closure of the and the sets it follows union of directly that is bounded as well. In particular is compact hence a proper map. Finally, by hence is positive deficonstruction, is nonzero on and ii) of Definition III.4 is satisfied as well. nite at Lemma IV.5 shows that, as the sample rate increases (i.e., decreases), the state trajectory for the system (1) with approaches the sliding surface . a Lemma IV.5: Suppose that is a sliding surface, compact neighbourhood of any open set containing , and a generalized sliding-mode controller with memory for such that for all system (1). Then there exists and the sufficiently small, the state trajectory for system (1) with and reaches in time . be a compact neighbourhood of an Proof: Let . Since is a generalopen set containing , and ized sliding-mode controller with memory the state trajectory for the driftless system (2) with initial condition has . By definition the -th com-

is over the time interval , hence ponent of for is the integral curve of the trajectory . Thus, the smooth vector field . Similarly, for will where be the integral curve of the smooth vector field . It follows that . We now consider the state trajectory for the system (1) with and over . As above, for , the state trais the integral curve of so that jectory hence approaches as tends to , 0. A similar reasoning applies on each interval approaches hence we conclude that as . Thus there exists and an open neighbourhood such that for and . We repeat this construction at each get an open covering by sets of the form and point in corresponding constants such that for and . Since is compact and is an open covering of , we can find a finite subcovering and a finite . We set . set of constants Then for the state trajectory for system (1) with and reaches in time . Suppose that is the -extension of for the linear considered above. It is straightforward system to verify that, for sufficiently small, is invariant under . A more general state trajectories corresponding to version of this invariance in the linear case is established in Corollary V.3. The following Lemma shows that a local version

1416

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

of this invariance holds for driftless control affine systems under generalized sliding-mode controllers. Lemma IV.7: Suppose that is a sliding surface for system (1), an -Lyapunov function at a -extension a generalized sliding-mode controller of , and denote the trajectory of the with memory . Let and driftless system (2) with and let denote the trajectory for system (1) with . Then there exists a -extension of such that ; a) b) if then for and for ; such that, for c) if and . Proof: Let be a generalized slidinga -extension of . mode controller with memory , and Then Definition III.2 iv) implies that there exists an open set such that and such that for all we have and as a consequence be the closure of the set which of Definition III.2 iii). Let and the subsets for all . is the union of satisfies i) and ii) above. To show that is Then the set a -extension we note that, by construction, is a closed containing in its interior. Since is a subset of -extension of Definition IV.3 hold for , hence holds for of . In particular is a -extension of . the subset To establish iii) we note that from ii) above if then for and . But in the proof of Lemma IV.5 we showed that, for each approaches as tends to zero. Thus for sufficiently small and , it follows that . , we can ensure that Furthermore, since by further shrinking . We know from Lemma IV.5 that a generalized sliding-mode controller will move states of system (1) into any open neigh-extension of bourhood of a sliding surface , but a need not be an open neighbourhood of (only of ). using an The following Lemma provides a way to enlarge open set so that contains in its interior. This will be needed in the proof of Theorem IV.1. Lemma IV.7: Suppose that is a sliding surface for system a set containing in its interior, a gener(1), alized sliding-mode controller with memory , and an open with compact closure. Let deneighbourhood of note the trajectory for system (1) with for . Then there exists an open set and such that a) ; and then for b) if and . Proof: From our definition of a generalized sliding-mode controller with memory [Definition III.2 iii) and iv)] there exists an open neighbourhood of in such that for all we have and , where denotes the trajectory of the driftless system (2) with and . By definition is an open neighbourhood of . In the proof of Lemma IV.5 we showed approaches as . Choose . that

sufficiently small, we have for Then, for and . Thus, we can find of such that for an open neighbourhood and for . if necessary the above holds for all , Shrinking and is compact the set the closure of . Since is also compact and we can find a finite covering by sets . Set . Thus i) and ii) hold using and . be a compact a compact Proof (Theorem IV.1): Let neighbourhood of and an open subset of containing . We must show that there exist such that for any we have for . a genSuppose that is a sliding surface for system (1), eralized sliding-mode controller for , and an -Lyapunov function at . Let be a -extension of (whose existence denote the trajecis guaranteed by Lemma IV.4). Let and . tory of system (1) with be a -extension of satisfying the hypotheses of Let Lemma IV.6. by adjoining an open neighOur first task is to enlarge bourhood of to create a set which contains in its interior. and is compact we know that is a Since compact neighbourhood of . Lemma IV.7 with and implies that there exists an open set and such that and for all and if then for and . Set . We have established that, for all for and . We now show that all states in will be steered to in finite time, so that, without loss of generality, we can assume that our initial state is in a compact subset of . Since is contained in the interior of Lemma IV.5 implies that, for suffor all . Furthermore the ficiently small, image of under our state trajectory map is compact. Thus, after one “sample interval” of length , the to . state is transfered from the state will be in . This is a bounded Thus at time . Since the restriction of to the -exsubset of tension is proper and positive definite, we can assume that for the initial state for system (1) lies in . Thus any bounded subset of is contained in a some for some . This means that bounded set . Thus we can assume that the state of system (1) starts off from . Furthermore, since the restriction of to the -extenfor if and only if , sion is proper and there exists such that and . Since we have two cases or where is the compact set to consider,

We note that the intersection of and need not be empty. we have shown that for In the case and . To complete our proof it suffices the trajectory of system to show that for an initial state (1) reaches in finite time and then stays in thereafter. From

HIRSCHORN: GENERALIZED SLIDING-MODE CONTROL FOR MULTI-INPUT NONLINEAR SYSTEMS

Lemma IV.6iii) we know that if there exists such that, for all , and . In fact we can choose an open neighbourhood of such that for we have and . The sets form an open cover of the compact set . Choosing a finite such that, for all , subcover we can find and for any . To complete our proof we use that fact that the -Lyapunov is decreasing along the system trajectories which function causes the state trajectory to reach . From the definition of an Lyapunov function, we have and . Thus, while the state trajectory evolves in , the -Lyapunov function decreases. In particular

(6) . Thus, after the “initial sample period” of we have . If then and we can repeat the above argument over the time interval to show that and . Continuing in this manner we see that eventually we have for some positive integer . Thus after some time we have hence . As we noted we have for and above if . Of course can increase on . Set . If then and over the this cycle repeats and the state next time interval of length ( ) then stays in . If, on the other hand, we must ensure that over the following time interval of length the state does not leave . Recall that was picked so that and and we have established that for initial states in the function decreases along the state trajectory over a time interval of length . Suppose that . Then over the next sample period (of duration ) decreases along the state and the state stays in . It foltrajectory so stays less than if we can ensure lows that the state stays in for all that whenever the state trajectory has the property that ). This is easy to do by decreasing . The rate of change of along the state trajectory was shown to be . But and the closure of is compact. Thus we for the rate at which can increase have an upper bound is an upper along state trajectories in . This means that bound on the increase of over a sample interval. If we choose then we will have . We have established that, for sufficiently small, for all there exists such that . This implies that for in some open neighbourhood of . These neighbourhoods for duration

1417

is compact we can are an open covering of . Since . Let be the maxfind a finite subcover . Then for any we have imum of . This completes the proof. V. ASYMPTOTIC STABILIZATION is a direct generalized sliding-mode controller one can If -extension of such that, given show that there exists a any open neighbourhood of and initial state , such that for there exists if for . Assumption A1i) below is a stronger version of this forward invariance which has implications for asymptotic stability. Assumption A1ii) below is a somewhat stronger version of the controllability result of Lemma IV.5 and is needed to establish global results. A1. is a sliding surface, a direct generalized sliding-mode controller for an -Lyapunov function -extension of such that, for system (1) at and a for sufficiently small a) is forward invariant under trajectories of the ; system (1) for carries b) the state trajectory of system (1) with in finite time. any initial state to The following local version of A1 suffices for semiglobal stabilization. A2. is a sliding surface, a compact neighbourhood of a direct generalized sliding-mode controller for an -Lyapunov function for system (1) at and a -extension of such that, for sufficiently small i) trajectories of the system (1) with stay in for all ; carries ii) the state trajectory of system (1) with in time . any initial state in to Theorem V.1: Suppose that is a sliding surface for system (1), a direct generalized sliding-mode controller for an -Lyapunov function at , and a -extension of such that A1 (A2) holds. Then is globally (semi-globally) . asymptotically stabilized by Before proving this theorem we introduce two corollaries which show that Theorem V.1 can be used to stabilize a class of single-input homogeneous systems and controllable multi-input linear systems. Consider the single-input affine system model on in triangular form described by

.. . (7) where are odd positive integers, is , and is homogeneous of degree 0 with respect , where .A to the dilation vector field on is said to be homogeneous

1418

of degree if, for all

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

with respect to the dilation ,

(cf [4][8])

A function on is said to be homogeneous of degree with respect to the dilation if, for all , . The state equivalence of a nonlinear system to a triangular form is considered in [5]. It is well known that for many systems of the form (7) (for example systems whose linearization is not stabilizable via linear state feedback) which cannot be stabilized by means of smooth state feedback (cf [1], [2], [15], [22], and the references therein). Corollary V.2: There exists a sliding surface for system , a direct generalized (7), an -Lyapunov function at , and -extension of such sliding-mode controller that A1 holds. In particular system (7) is globally asymptotically . stabilized by is a controllable Corollary V.3: Suppose that and rank . Then there exists linear system with an dimensional subspace which is a sliding surface , for this system, a quadratic -Lyapunov function at , and a -exa direct generalized sliding-mode controller tension of such that A1 holds. In particular the controllable linear system can be asymptotically stabilized . by We note that in the proof of Corollary V.2 we obtain a simple formula (13) for a stabilizing direct generalized sliding-mode controller. Before proving the above we present the following example studied in [21]. Example V.4: Consider the system model

(8) introduced in [21]. These equations model an under-actuated two degree of freedom mechanical system composed of a mass on a horizontal surface connected on one side to a wall by a linear spring and on the opposite side to an inverted pendulum by a nonlinear spring (spring force proportional to the cube of the displacement). The control is a force acting on the mass. Here is the angular displacement of the inverted pendulum, its angular velocity, the displacement of the mass and its velocity, is the length of the pendulum, the acceleration due to gravity. After a nonlinear the above equations of motion feedback to cancel terms in result (cf. [21]). As in [21] we take . Here . This homogeneous system has , and thus is homogeneous with respect to the dilation . Here the functions in system (7) are . It is straightforward to verify from the proof of Corollary V.2 that

is a sliding surface for this system where for constants arising from the backstepping construction and estimates . These are very in [19]. For example conservative estimates and smaller values of can usually ( holds for any be used. In any event here single input system) and, from the proof of Corollary V.2

is an

-Lyapunov function for this example, where . The direct generalized sliding-mode controller from the proof of Corollary V.2 ((13)) is

for . Corollary V.2 asserts that, for sufficiently small, the generalized controller achieves global asymptotic stability for system (8). Fig. 2 shows the results of a simulation performed using Simnon/PCW for Windows Version 2.01 (SSPA Maritime Consulting AB, Sweden), where the simulation has . Proof: (Theorem V.1) Suppose that is a sliding surface for system (1), a direct generalized sliding-mode controller for an -Lyapunov function at , and is a -extension of such that A1 holds. Then, for sufficiently small, A1ii) asserts that the state trajectory of system (1) with carries any initial state to in finite time. A1i) asserts that is invariant under trajectories for system (1) where . This implies that such that . Because is a -extension of we have proper and positive definite at and . It follows that for . In particular hence . Because restricted to is positive definite at it follows that . In the case where A2 holds we proceed in the same manner as above. Suppose that is a compact neighbourhood of and . For sufficiently small A2ii) asserts that the state of system (1) with carries any trajectory initial state to a point in time less than or equal to . Since is compact the set of all such points is contained in a compact neighbourhood of . Shrinking if necessary A2i) asserts that is invariant under trajectories for system (1) where . We can repeat the above argument to conclude that . Before proving Corollary V.2 we present the following lemma. denote the affine system Lemma V.5: Let function and smooth (7). Then there exist a functions on such that the following hold. is proper and positive i) The restriction of to definite at 0.

HIRSCHORN: GENERALIZED SLIDING-MODE CONTROL FOR MULTI-INPUT NONLINEAR SYSTEMS

ii) for , where . iii) for iff . Proof: (Lemma V.5) It is straightforward to verify (cf. [4]) in (7) implies that, that the homogeneity of the functions after a simple homogeneous feedback, the system (7) can be expressed as some

(9) where

1419

. Thus, by inand setting and we satisfy i) and ii). Conclusion iii) follows from the definition of . One can incorporate the change in coordinates and homogeneous feedback to show that this result also holds for system (7). Proof: (Corollary V.2) From Lemma V.5 there exist funcdefined on such that Lemma V.5 i)-iii) tions hold. In light Lemma V.5ii) we define our sliding surface for where system (7) to be and . From (12) we and have

where duction,

for some constant

if if and

on . System (7) has

hence the integral curve . By definition is the graph

of the function We only need a slight modification of the proof of Theorem 3.1 in [19]. Here we have the inequality (10) where the functions can be chosen to be . Now constants, namely we repeat the backstepping constructions of [19] which result in proper positive definite functions . Begin a sequence of proper positive definite function of , with the . As in the proof of [19, Th. 3.1] where for . Now suppose that we have found constants functions and defined by

(11) where tions

for

and proper positive definite funcsuch that

. Then

is a proper positive definite

function and

(12)

and thus, given , there exists a unique such that . In particular, where is the is leaf of containing . By construction is a nonzero constant. Thus a smooth function and 0 is a regular value for and also for . This implies that is a codicontaining mension one embedded submanifold of , and thus is a sliding surface for system (7). To show that is an Lyapunov function for system (7) at it suffices to verify that for some and that the restriction of to is positive definite at 0. But is the graph of , hence the restriction of to is positive definite as a consequence of for some we Lemma V.5 i). To see that use homogeneity. First, we note that on we have established that and is positive definite at 0. Thus, given any , there exists such that . Shrinking of necessary we can choose an open neighbourhood of in such that . Consider the compact set where is the “homogeneous -sphere” corresponding to the dilation . Here is a “diand the homogeneous lated norm” with respect to . Then, corre“ -sphere” is sponding to to each , there exists an open set such that . This gives rise to an open covering of . Thus there exists a finite subcover of by open . Setting we have sets on . In particular on . In light of (9) it . straightforward to verify that , a vector We note that, for a given dilation is homogeneous of degree iff is homogeneous field

1420

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

of degree for all homogeneous of degree . Thus is homogeneous of degree 0 with respect to the dilation . Similarly the functions are ho. mogeneous of degree 1 with respect to the dilation Now, by construction, is homogeneous of degree with respect to the dilation and, since is homogeneous of degree 0 with respect to will . Thus, for be homogeneous of degree and , we have which implies that

This implies that holds on all of , hence is an -Lyapunov function. To define a direct generalized sliding-mode controller we to be first note that we defined For we set and define our direct generalized sliding-mode controller to be

. We set . Then Lemma V.5ii) implies that the rate of change along the system trajectories is where of . we have Since But are defined to be even powers of hence . Sup. Then . pose that we choose To see that this is the case we suppose that . From our definition of and Lemma V.5iii) it follows that . Since we and . This contradicts have . It follow that by further shrinking we can ensure that . Thus given any we can choose and an open neighbourhood of in such that . Since with the relative topology is coman open covering of pact and there exists such that . For we saw above that for and . It is easy to verify that the functions are homogeneous of dewith respect to the dilation . Thus gree

(13) We note that is a function of only and if and only if . It is straightforward to verify that is homogeneous of degree with respect to the dilation . Over the time interval the trajectory of the driftless system (2) with is and . In particular

hence . Furthermore, if , then and for . It is straightforward to check satisfies all of the remaining conditions necessary to that qualify as a direct generalized sliding-mode controller. for which A1 holds. Let We now define a -extension and denote the homogeneous sphere in . Set so that . Since and is homogeneous of degree 1 with respect to the dilation we know that then for all if . We set

so that some on

. Then

iff

). We now show that, for . Let . Then

for (that is and sufficiently small, for

if . In particular on . With this fact it is is a -extension of . straightforward to verify that To complete the proof we must show that with the above , and that assumption A1 holds. That is choices for sufficiently small we have i) is invariant under trafor , and ii)the state jectories for system (7) with carries any initial state to trajectory of system (1) with in finite time. . We first consider the case where . i) Let over the first sample period and Then for . We we must show that first suppose that (so that the dilated and ) and examine , norm the integral curve for the homogeneous drift vector field . Since for the system (7) through there exists such that for all . From the compactness of it follows such that for all that there exists and . More generally suppose that . Then for some . Because is homogeneous we of degree 0 with respect to the dilation where have . But we showed above that for all and . From our definition of as for a “homogeneous extension” it follows that all . In particular for . Now

HIRSCHORN: GENERALIZED SLIDING-MODE CONTROL FOR MULTI-INPUT NONLINEAR SYSTEMS

consider the case where is arbitrary. Let denote the trajectory of system (7) with and . The controlled vector field for the system (7) can be represented as . In light of the above discussion we can choose sufficiently small . Set to ensure that , a compact set. If then and for . In for . Since particular tends to zero as as approaches and is a constant for and vector field it follows that in some open neighbourhood of in . If then trajectories of the driftless system with and stay in for as a straightforward consequence of our constructions. Thus, as in the proof of Lemma IV.7, we can choose sufficiently small and an of in such that open neighbourhood for and . Since is compact we such that for and can find . More precisely we have shown that

1421

have hence the trajectory of where we set the system (7) will have . Thus is suffices to show that trajectory of the system (7) with initial states in reach in finite . For the drifttime. We first suppose that we have . less system with Thus for sufficiently small it follows that the trajectory of the system (7) will have for all in . From the some open neighbourhood of in we can conclude that, for compactness of sufficiently small, for all . As above we can use the homogeneity of with respect to the initial condition and the homogeneity to conclude that for used to generate . Since we indicated above that all states are all steered to in finite time it follows that A1 holds. That folsystem (7) is asymptotically stabilized by lows from Theorem V.1. Proof: (Corollary V.3) We begin by considering a single input system in Brunovsky form, that is (15)

for

(14)

Now is a homogeneous vector field of degree 0 with respect to the dilation . In particular we have for . Here , and , a homogeneous function of degree with respect to the . It follows that is homogedilation hence, for neous of degree 0 with respect to

Thus if

and

From (14) we have

it follows that

where . This implies

hence . Since for is the set of points we have established i). all of ii) To show that the state trajectory of system (7) with carries any initial state to in finite time we first of the driftless note that the state trajectory with carries any initial state system to in time . Of course initial states of the form

This is a homogeneous system of the form (7) with . The relevant dilation is thus . Corollary V.2 yields a sliding surface , a direct generalized sliding-mode controller, an -Lya-extension such that A1 is satisfied. punov function , and Furthermore the sliding surface is a subspace of codimension 1. We then apply the linear static state feedback and linear change of coordinates in the state and input spaces to convert the Brunovsky form back to the original system. The resulting satisfy A1 and our linear system is asymptoti. For cally stabilized via the direct sliding-mode controller independent inputs in Brunovsky the multi-input case with form we repeat the above on each single-input single-output sub-block and let the sliding surface be the intersection of the hyperspaces which act as sliding surface for each of the single-input subsystems. Similarly is the sum of the -Lyapunov functions created for each of the single-input sub-systems, is the intersection of the -extensions and the vector controller whose components are the direct generalized sliding-mode controllers for the single-input subsystems. We then apply the linear static state feedback and linear change of coordinates in the state and input spaces to convert back to the original system from the Brunovsky form. It is is a direct generstraightforward to show that the resulting alized sliding-mode controller, is a codimension subspace of and satisfy A1. Theorem V.1 implies that . our linear system is asymptotically stabilized by ACKNOWLEDGMENT The author would like to thank the anonymous referees for their very careful reading of this paper and their constructive comments which guided the significant revisions to the manuscript. Thanks also to N.H. McClamroch for suggesting the physical system considered in Example 2.

1422

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 9, SEPTEMBER 2006

REFERENCES [1] D. Aeyels, “Stabilization of a class of nonlinear systems by smooth feedback control,” Syst. Control Lett., vol. 5, pp. 289–294, 1985. [2] R. W. Brockett, , R. W. Brockett, R. S. Millman, and H. J. Sussmann, Eds., “Asymptotic stability and feedback stabilization,” in Differential Geometric Control Theory. Boston, MA: Birkhäuser, 1983, pp. 181–191. [3] R. W. Brockett, “Pattern generation and the control of nonlinear systems,” IEEE Trans. Autom. Control, vol. 48, no. 10, pp. 1699–1711, Oct. 2003. [4] S. Celikovsky and E. A. Bricaire, “Constructive nonsmooth stabilization of triangular systems,” Syst. Control Lett., vol. 36, pp. 21–37, 1999. [5] S. Celikovsky and H. Nijmeijer, “Equivalence of nonlinear systems to triangular form: The singular case,” Syst. Control Lett., vol. 27, pp. 135–144, 1996. [6] W. L. Chow, “Uber systeme von linearen partiellen differentialgleichungen erster ordnung,” Math. Ann., vol. 117, pp. 98–105, 1939. [7] F. Clarke, Y. Ledyaev, E. Sontag, and A. I. Subbotin, “Asymptotic controllability implies feedback stabilization,” IEEE Trans. Autom. Control, vol. 42, no. 10, pp. 1394–1407, Oct. 1997. [8] J. M. Coron and L. Praly, “Adding an integrator for the stabilization problem,” Syst. Control Lett., vol. 17, pp. 89–104, 1991. [9] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides. Dordrecht, The Netherlands: Kluwer, 1988. [10] L. Fridman and A. Levant, , W. Perruquetti and J.-P. Barbout, Eds., “Higher order sliding-modes,” in Sliding Mode Control in Engineering. New York: Marcel Dekker, 2002, pp. 53–102. [11] L. Grune, “Homogeneous state feedback stabilization of homogeneous systems,” SIAM J. Control Optim., vol. 38, no. 4, pp. 1288–1308, 2000. [12] R. Hermann, “On the accessibility problem in control theory,” in International Symposium Nonlinear Differential Equations and Nonlinear Mechanics. New York: Academic, 1963, pp. 325–332. [13] R. Hirschorn, “Singular sliding-mode control,” IEEE Trans. Autom. Control, vol. 46, no. 2, pp. 276–286, Feb. 2001. [14] R. Hirschorn and A. Lewis, Geometric local controllability: secondorder conditions 2006. [15] M. Kawski, “Homogeneous stabilizing feedback laws,” in Control Theory Adv. Technol., Tokyo, Japan, 1990, vol. 6, pp. 497–516. [16] H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems. New York: Springer-Verlag, 1990. [17] Y. Orlov, “Finite-time stability and robust control synthesis of uncertain switched systems,” SIAM J. Control Optim., vol. 43, no. 4, pp. 1253–1271, 2005.

[18] J.-B. Pomet, “On the curves that may be approached by trajectories of a smooth affine system,” Syst. Control Lett., vol. 36, pp. 143–149, 1999. [19] C. Qian and W. Lin, “Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization,” Syst. Control Lett., vol. 42, pp. 185–200, 2001. [20] C. Qian and W. Lin, “A continuous feedback approach to global strong stabilization of nonlinear systems,” IEEE Trans. Autom. Control, vol. 46, no. 7, pp. 1061–1079, Jul. 2001. [21] C. Rui, M. Reyhanoglu, I. Kolmanovsky, S. Cho, and N. H. McClamroch, “Nonsmooth stabilization of an under-actuated two degree of freedom mechanical system,” in Proc. 36th IEEE Conf. Decision and Control. New York: IEEE Publications, 1997, pp. 3998–4003. [22] E. P. Ryan, “On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback,” SIAM J. Control Optim., vol. 32, pp. 1597–1604, 1994. [23] S. Sastry, “Nonlinear systems, analysis stability and control,” in Interdisciplinary Applied Mathematics. New York: Springer-Verlag, 1999. [24] J.-J. E. Slotine, Applied Nonlinear Control. Upper Saddle River, NJ: Prentice-Hall, 1991. [25] J. Tsinias, “Planar nonlinear systems: Practical stabilization and Hermes controllability condition,” Syst. Control Lett., vol. 17, pp. 291–296, 1991. [26] V. I. Utkin, Sliding Modes in Control Optimization. Berlin, Germany: Springer-Verlag, 1992, CCES.

Ronald M. Hirschorn received the B.A.Sc. degree in engineering from the University of Toronto, Toronto, ON, Canada, the S.M. degree in aeronautics and astronautics from the Massachusetts Institute of Technology, Cambridge, and the Ph.D. degree in applied mathematics from Harvard University, Cambridge, MA, in 1968, 1970, and 1973, respectively. Since 1973, he has been a Faculty Member with the Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada. His research interests have included controllability and disturbance rejection for nonlinear systems, output tracking, model-based friction compensation, and sliding-mode control.