Regularization of Second Order Sliding Mode Control Systems Giorgio Bartolini 1 , Elisabetta Punta 2 , and Tullio Zolezzi 3 1 Department of Electrical and Electronic Engineering, University of Cagliari, Piazza d’Armi - 09123 Cagliari, Italy giob@dist.unige.it 2 Institute of Intelligent Systems for Automation, National Research Council of Italy (ISSIA-CNR), Via De Marini, 6 - 16149 Genoa, Italy punta@ge.issia.cnr.it 3 Department of Mathematics, University of Genoa, Via Dodecaneso, 35 - 16146 Genoa, Italy zolezzi@dima.unige.it  1 Introduction The numerical representation of singular systems [1], with index greater than two and inconsistent initial conditions, presents common features with the implementation of higher order sliding motions, [2], [3]. Indeed higher order sliding modes can be viewed as a way to achieve constrained motions, often expressible as an output-zeroing problem after a transient of finite duration [4]. The choice of the sliding output is the first step of a sliding mode design process (e.g. invariance [5]). If the actual control affects the time derivative of the sliding output, starting from a certain order k ≥ 1, the corresponding constrained motion, if attainable, is said to be a k-th order sliding motion [6], [7], [8], [9], [10]. The notion of sliding order is equivalent to the one of relative degree [4]. Discontinuity of the control naturally arises when uncertainties affect the mathematical description of the system be controlled. Indeed the reduction to zero in finite time of the sliding variable implies the stepwise solution of differen- tial inequalities of order greater than one with the aim of ensuring a contractive behaviour, in literature referred to as the Fuller phenomenon [11], [12]. For higher order sliding mode the procedure is much more involved than in the first order case, for which the application of the comparison lemma, [13], suffices, and a variety of algorithms can be identified. In the literature most of the results are related to the development of second order sliding mode algorithms [6], [7], [14], [8], [15], [9].  Work partially supported by MURST, Progetto Cofinanziato “Controllo, ottimiz- zazione e stabilit`a di sistemi non lineari”, by PRAI-FESR Liguria and by MUR-FAR Project n. 630. G. Bartolini et al. (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp. 3–21, 2008. springerlink.com c  Springer-Verlag Berlin Heidelberg 2008 4 G. Bartolini, E. Punta, and T. Zolezzi The finite time transient, the reaching phase, is not at issue in this chap- ter, which is devoted to analyze the motion constrained on the sliding surface, which is ideally represented by differential equations with discontinuous r.h.s., the solution of which is defined according to Filippov’s theory, [16], [17]. The aim of this chapter is that of providing a theoretical framework to analyse the behaviour of the systems controlled by second order sliding mode when, due to non-idealities of unspecified nature, the chosen constraints are only approxi- matively satisfied. A class of regular perturbations needs to be identified, so that all the cor- responding real state trajectories, which fulfill only approximately the sliding conditions due to non-idealities of any nature, converge to the unique solution (if it exists) of the differential algebraic equations representing the exact fulfill- ment of the chosen constraints. This behaviour, called approximability in [18], [19] and regularization in [17], has been analysed for sliding mode control of first order and must be regarded as a basic procedure in the mathematical analysis and validation of sliding mode control techniques. The given system fulfills the approximability property whenever all real states converge uniformly (on the fixed bounded time interval we consider) to the ideal state as the non-idealities disappear. Such a property is of interest since it guar- antees that the dynamical behavior of the real states, we obtain in practice, are close to that of the ideal sliding state, which is the main goal of the sliding mode control techniques. This property, discussed in [17], was mathematically formal- ized in [18] and [20] for nonlinear control systems. There, sufficient conditions for approximability were found, generalizing the regularization results of [17], about first order sliding mode control methods. In this chapter the approximability results about first order sliding mode control methods are presented. The analysis is developed for sliding mode control of order two, thus obtaining second order regularization results of sliding mode control systems. A major result we obtain is that first order implies second order approximability (Theorem 4, Section 4) under mild assumptions. Hence the regularization of second order methods is automatically valid as soon as it is guaranteed for first order sliding techniques. Results on first order approximability are summarized in Section 2. In Sec- tion 3 we introduce a new framework to define rigorously the approximability property for systems, the ideal solution of which is achieved by second order sliding mode methods. We compare second order approximability with the one related to first order methods. In Section 4 we prove sufficient conditions for second order approximability. In Section 5 we obtain some sliding error esti- mate. In Section 6 we extend the previous approach to non-idealities occurring on either the sliding output or the control law. Suitable examples are proposed throughout this chapter. A different approach to approximability properties, based on optimization techniques, is considered in [19] and [21] for first and second order sliding methods. Regularization of Second Order Sliding Mode Control Systems 5 2 First Order Approximability: Definitions and Results We consider sliding mode control problems ˙x = f(t, x, u),u∈ U; s(t, x)=0, 0 ≤ t ≤ T, x(0) = x 0 ∈ Z, (1) on a fixed bounded time interval [0,T], where x ∈ IR N , u ∈ U, U is a closed subset of IR K ,ands ∈ IR M , T>0. We assume f ∈ C 2 ([0,T] × IR N × W )with W an open set containing U, Z ⊂ IR N ,ands ∈ C 2 ([0,T] × IR N ). Following the terminology of [17], page 13, and the approach of [18], [20], we deal with ideal states which fulfill exactly the sliding condition s[t, x(t)] = 0 for every t,asoppositetoreal states, fulfilling only approximately such a condition. A parameter ε, belonging to a metric space with a fixed element conventionally denoted by 0, will be used to represent non-idealities of any nature in the real sliding. We write x ε → x to denote convergence of x ε k towards x for each sequence ε k → 0. The notation x ε −→ →x means uniform convergence on [0,T]. We denote by U ∞ the set of all admissible control laws in (1), defined as follows. These are all functions u :[0,T] ×IR N → U which are L ⊗B − measurable, i.e. measurable with respect to the σ − algebra generated by the products of the Lebesgue measurable subset of [0,T]andthe Borel measurable subset of IR N , and which fulfill the following property. For any such u, the differential system in (1) has an almost everywhere or a Filippov solution x on [0,T] such that u[·,x(·)] ∈ L ∞ (0,T). To simplify the notations, we shall write ˙x = f(t, x, u)on[0,T] meaning that u ∈ U ∞ and x is either an almost everywhere or a Filippov solution on [0,T]. The sliding mode control system (1) fulfills the first order approximability property whenever the following is true. For every x 0 ∈ Z such that s(0,x 0 )=0 there exists a unique sliding state y, i.e. for some control u ∗ ∈ U ∞ , not necessarily unique, we have ˙y = f (t, y, u ∗ )on[0,T],y(0) = x 0 , (2) s[t, y(t)] = 0, 0 ≤ t ≤ T. (3) Moreover for every sequence (u ε ,x ε ) such that ˙x ε = f(t, x ε ,u ε )on[0,T]and s(t, x ε ) −→ →0wehavex ε −→ →y provided x ε (0) → y(0). The above definition, compared with that presented in [18] where Z = IR N , does not require either uniqueness of the sliding control law u ∗ or existence of the equivalent control, moreover s is allowed to depend on t as well. Often the sliding manifold is reached at points of a restricted part of it, thus the constraint we introduce on the initial states by the set Z. 6 G. Bartolini, E. Punta, and T. Zolezzi Uniqueness of the sliding state is fulfilled, under standard assumptions, when- ever the equivalent control is available, see [17], [18]. To be more specific, we consider the first order ideal system made up of control-state pairs (v, y) such that  ˙y = f (t, y, v)on[0,T],v∈ U ∞ ; ˙s = s t (t, y)+s x (t, y)f(t, y, v)=0. The first order real control-state pairs are given by pairs (u ε ,x ε ), where x ε is absolutely continuous in [0,T], and such that for almost every t ∈ (0,T)  ˙x ε = f(t, x ε ,u ε ),u ε (t) ∈ U ; s t (t, x ε )+s x (t, x ε )f(t, x ε ,u ε )=m ε (t). (4) About the non-idealities m ε we shall employ the condition m ε → 0inW −1,∞ (0,T)asε → 0, (5) which means that m ε ∈ L 1 (0,T)andsup{|  t 0 m ε (r)dr| :0≤ t ≤ T}→0as ε → 0. Here W −1,∞ (0,T) denotes, as usual, the dual space of the Sobolev space W 1,1 (0,T) (see e.g. [22]) The following conditions will be referred to in the sequel. For every compact set S ⊂ U there exist A 1 , B 1 ∈ L 1 (0,T) such that |f(t, x, u)|≤A 1 (t) |x| + B 1 (t)(6) for a.e. t,everyu ∈ S and x ∈ IR N . For every compact set Z ⊂ IR N × U there exists C 1 ∈ L 1 (0,T) such that |f(t, x  ,u) − f(t, x  ,u)|≤C 1 (t) |x  − x  | (7) for a.e. t,every(x  ,u)and(x  ,u) ∈ Z. Theorem 1. The control system (1) fulfills the first order approximability prop- erty if conditions (6) and (7) are met, U is compact, f(t, x, U ) is convex for a.e. t and every x with s(t, x) = 0, and for every x 0 ∈ Z with s(t, x 0 )=0thereexists a unique sliding state issued from x 0 . For the proof see [19]. The convexity condition on f, which is required by Theorem 1, can be relaxed in some significant cases according to the following theorems. Theorem 2. Let the control system (1) be such that f(t, x, u)=A(x)+B(x)h(u). The first order approximability property holds provided: • U is compact, h(U )isconvex; Regularization of Second Order Sliding Mode Control Systems 7 • for every K>0 there exist constants C 2 , D 2 such that |s(x)|≤K implies |A(x)| + |B(x)|≤C 2 |x| + D 2 ; • s x (x)B(x)=G(x) is nonsingular near the sliding manifold and we have uniqueness in the large for every initial value problem ˙z = A(z)+B(z)ν(z),z(0) given with s [z(0)] , where ν(z)=−G −1 (z)s z (z)A(z). For the proof see [20]. Theorem 3. Let the control system (1) be such that f(t, x, u)=[x 2 ,x 3 , ,x N ,g(x, u)] T , where u ∈ IR. The first order approximability property holds provided: • U is compact; • for every K>0 there exist constants A 2 , B 2 such that |s(x)|≤K implies |g(x, u)|≤A 2 |x| + B 2 ; • we have uniqueness in the large for every initial value problem ˙x i = x i+1 ,i=1, ,N −1, ˙x n ∂s ∂x n +  N−1 j=1 x j+1 ∂s ∂x j =0,s[x(0)] = 0. For the proof see [20]. Recently, in [23], the approximability property has been extended to systems in regular form, which do not necessarily satisfy the convexity condition. Let the control system (1) be such that f(t, x, u)=(f 1 (x) T ,f 2 (x, u) T ) T , where x =(x T 1 ,x T 2 ) T ∈ IR N , x 1 ∈ IR N−K , x 2 ∈ IR K , u ∈ U ⊂ IR K , U compact, f 1 : IR N → IR N−M and f 2 : IR N × U → IR M . Consider s(x)=x 2 − h(x 1 ), where s : IR N → IR M and h : IR N−M → IR M is a C 1 function. The first order approximability property holds provided h x 1 (x 1 )f 1 (x 1 ,h(x 1 )) ∈ cof 2 (x 1 ,h(x 1 ),U). The class of systems for which the approximability holds, includes then any set of k coupled differential equations z (n i ) i = f i (z,u),i=1, ,k, with z =  z 1 , ,z (n 1 −1) 1 , ,z k , ,z (n 1 −1) k  T and u =(u 1 , ,u k ) T . 8 G. Bartolini, E. Punta, and T. Zolezzi 3 Second Order Approximability: Definitions The second order ideal system of control-state pairs (u, z), both absolutely con- tinuous in [0,T], such that for almost every t ∈ (0,T)  ˙z = f(t, z, u),u(t) ∈ U; ¨s = P (t, z, u)+Q(t, z, u)˙u =0, (8) where P = s tt +2s tx f + f T s xx f + s x f t + s x f x f, Q = s x f u and f T s xx f denotes the vector of components f T s jxx f, j =1, ,M. We model the non-idealities acting on the second order ideal system (8) by using two different terms. The first, denoted by b ε = b ε (t) ∈ IR M , takes into ac- count second order sliding non-idealities, so that in the real second order system we have ¨s = b ε . The second, denoted by c ε = c ε (t) ∈ IR K , takes into account non-idealities in obtaining ˙u ε , so that in the real second order system we work with w ε =˙u ε + c ε instead of w ε =˙u ε . Specific properties of the non-idealities acting on the second order system may depend on the particular control problem at hand. Therefore we fix a nonempty subset N 0 ⊂ L 1 (0,T) of sequences (b ε ,c ε ), and we consider non-idealities be- longing to N 0 . The second order real control-state pairs are thereby given by pairs (u ε ,x ε ), both absolutely continuous in [0,T], such that for almost every t ∈ (0,T) ⎧ ⎨ ⎩ ˙x ε = f(t, x ε ,u ε ),u ε (t) ∈ U ; P (t, x ε ,u ε )+Q(t, x ε ,u ε )w ε = b ε (t); w ε (t)= ˙u ε (t)+c ε (t). (9) About the non-idealities c ε we shall employ the condition c ε → 0inW −1,∞ (0,T)asε → 0, (10) which means that c ε ∈ L 1 (0,T)andsup{|  t 0 c ε (r)dr| :0≤ t ≤ T }→0as ε → 0. About b ε we consider two different ways b ε can vanish as ε → 0, namely b ε → 0inW −2,∞ (0,T), (11) either b ε → 0inW −1,∞ (0,T). (12) By (11) we mean that b ε ∈ L 1 (0,T)andsup{|θ ε (t)| :0≤ t ≤ T }→0where ¨ θ ε = b ε almost everywhere in (0,T), θ ε (0) = ˙ θ ε (0) = 0. Here W −1,∞ (0,T)andW −2,∞ (0,T) denote, as usual, the dual spaces of the Sobolev spaces W 1,1 (0,T)andW 2,1 (0,T) respectively (see e.g. [22]). Accordingly, we formulate two definitions of second order approximability of (1) within N 0 . Both second order approximability properties identify second Regularization of Second Order Sliding Mode Control Systems 9 order sliding mode control systems such that real states are uniformly close to the unique sliding state, as the non-idealities acting on the system are suitably small. Roughly speaking, first kind approximability means that such a robust behaviour of the control system is guaranteed whenever ˙s is uniformly small. Second kind approximability means that the same behaviour is present whenever s is small. This property is similar to that required in the definition of first order approximability (by the quite different first order methods). In each definition we require the following condition. Condition A: For every x 0 ∈ Z such that s(0,x 0 ) = 0 there exists a unique sliding state y corresponding to some continuous control u ∗ , i.e. (2), (3) are fulfilled. Definition 1. Second order approximability of the first kind within N 0 means the following. First, condition A is fulfilled. Second, given any sequences b ε , c ε in N 0 we require that x ε −→ →y provided x ε (0) → y (0), u ε (0) → u ∗ (0) for every sequences x ε satisfying (9) such that (10) and (12) are fulfilled. Definition 2. Second order approximability of the second kind within N 0 means the same as in Definition 1, except that (12) is replaced by (11). Since strong convergence in W −1,∞ (0,T) implies the same in W −2,∞ (0,T), we have that second kind implies first kind approximability. To define N 0 , we shall consider the following conditions about sequences b ε and c ε : sup ε  T 0 (|b ε (t)| + |c ε (t)|)dt < +∞, (13) sup ε  T 0 | ˙ Q[t, x ε (t),u ε (t)]|dt < +∞. (14) Condition (13) is needed here for technical reasons (however see Example 3). Condition (14) (again needed for technical reasons) is fulfilled provided U is compact and |f(t, x, u)|≤a(t)+b(t)|x| for almost every t,allx and u,witha, b ∈ L 1 (0,T). This follows in a standard way by Gronwall’s lemma. We emphasize that the approximability properties we have defined are in- dependent of the particular algorithm used to enforce any second order sliding motion. 4 Second Order Approximability: Results Let N 0 be defined by the property that sup ε |b ε |∈L 1 (0,T), a stronger bound- edness condition on b ε than (13); then the two corresponding approximability properties are indeed equivalent, as a corollary of the following proposition. Proposition 1. Let sup ε |b ε |∈L 1 (0,T). Then convergence of b ε in W −2,∞ (0,T) implies convergence in W −1,∞ (0,T). 10 G. Bartolini, E. Punta, and T. Zolezzi For the proof see [24]. The main result of this section is the following. Theorem 4. Let N 0 be defined by (13) and (14). If system (1) fulfills first order approximability and, for all x 0 ∈ Z with s(0,x 0 ) = 0, the corresponding sliding state can be generated by a continuous control, then (1) fulfills both second order approximability properties within N 0 . For the proof see [24]. Continuity of a sliding control is not a very restrictive assumption, being fulfilled whenever the equivalent control is available, as it often happens in sliding mode control applications, see [17]. According to Theorem 4, if first order sliding mode control methods cannot give rise to ambiguous motions, no further ambiguous behaviour can be induced by any second order control algorithm. Hence the validation of second order methods relies on checking known criteria yielding first order approximability, as those known from [18], [20] and [19]. Remark: A slightly weaker definition of first and second order approximability properties can be given by requiring that x ε −→ →y for all the sequences (u ε ,x ε )of real control-state pairs such that sup u ε  ∞ < +∞,where· ∞ denotes the L ∞ (0,T) norm. Thus we require convergence to the ideal state y only when the control laws are uniformly bounded. Then the proof of Theorem 4 shows that, with these weaker definitions, first order approximability still implies both second order corresponding properties. The same proof (see [24]) shows that second order real states of (1) are also first order real states. The next example shows that the converse to Theorem 4 fails. A sliding mode control system may fulfill second kind approximability of second order, and may fail to possess first order approximability. Example 1. Consider ˙x 1 = u 1 , ˙x 2 = u 2 , ˙x 3 = u 1 u 2 , 0 ≤ t ≤ 1, |u 1 |≤1, |u 2 |≤1,s 1 (x)=x 1 ,s 2 (x)=x 2 , with Z = IR 3 . The only sliding control is the equivalent control u ∗ = 0 and the only sliding state, given y(0), is the constant y(t)=y(0). This system does not fulfill first order approximability, see [18]. Let us show that second kind approximability of second order is however fulfilled within N 0 defined by (13) and (14). Given b ε , c ε as in (11), (10) respectively, and satisfying (13), let x ε (0) → y(0), u ε (0) → 0. Since P =0andQ =  10 01  we have x jε (t)=x jε (0) +  t 0 u jε dr, j =1, 2, u ε (t)=u ε (0) +  t 0 b ε dr −  t 0 c ε dr. Regularization of Second Order Sliding Mode Control Systems 11 Hence  t 0 u ε dr −→ →0, whence x jε (t) −→ →y j (0). We compute x 3ε (t) − x 3ε (0) =  t 0 u 1ε u 2ε dr = u 2ε (t)  t 0 u 1ε dr −  t 0   r 0 u 1ε dα  (b 2ε − c 2ε ) dr. We have u 2ε (t)  t 0 u 1ε dr −→ →0, and remembering (13) we get     t 0   r 0 u 1ε dα  (b 2ε − c 2ε ) dr    ≤  T 0 (|b 2ε |+ |c 2ε |) dr max r    r 0 u 1ε dα   ≤ (constant) max r    r 0 u 1ε dα   → 0. It follows that x 3ε −→ →y 3 (0) hence x ε −→ →y, yielding second order approximability. The following example presents a case of applicability of the second order ap- proximability. Example 2. Let y d (t) ∈ C 2 ([0,T]) be an available signal such that |¨y d (t)|≤L for every t. Consider the sliding mode control system ˙x 1 = x 2 , ˙x 2 = u, |u|≤L; s(t, x)=x 1 − y d (t) , 0 ≤ t ≤ T. (15) Here N =2andK = M =1.SinceQ = s x f u = 0 everywhere, the approximabil- ity criteria developed in [18] do not apply. We check first order approximability via Corollary 4.1 of [19]. The required properties of linear growth and local Lipschitz continuity of the dynamics are obviously fulfilled, as well as convex- ity of f(t, x, U). Given x 0 ∈ IR 2 such that s(t, x 0 ) = 0, i.e. x 01 (0) = y d (0), if x 02 (0) = ˙y d (0) there exists a unique sliding state y issued from x 0 ,namely y =(y d , ˙y d ) T , which corresponds to the continuous control u =¨y d (in the almost everywhere sense). Hence, by Theorem 4 with Z =  (y d (0) , ˙y d (0)) T  , second order approximability holds. Then x 1ε copies y d and x 2ε copies ˙y d . This happens for every non-idealities b ε , c ε acting on the system and fulfilling (10), either (11) or (12), (13), (14), independently of the particular second order sliding mode algorithm employed to control the system. Let us consider system (15) in the second order mode (9) affected by the non- ideality b ε =sin  t ε  , c ε =0,ε>0. Since the initial values converge and both s ε and ˙s ε converge uniformly to 0, as ε → 0, if the actual ε is small enough, x 1ε copies y d and x 2ε copies ˙y d (Figures 1 and 2). The following example shows the possible implications of the failure of condition (13) for the second order approximability. Example 3. We consider the same sliding mode control system of Example 2, namely ˙x 1 = x 2 , ˙x 2 = u; s(t, x)=x 1 − y d (t) , 0 ≤ t ≤ T, except the constraint |u|≤L, with non-idealities given by 12 G. Bartolini, E. Punta, and T. Zolezzi 0 5 10 15 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 t [sec] x 1 ε and y d 0 5 10 15 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 t [sec] s ε Fig. 1. The signal x 1ε copies y d (ε ≈ 10 −3 √ 2π) 0 5 10 15 −3 −2 −1 0 1 2 3 t [sec] x 2 ε and d y d /d t 0 5 10 15 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 t [sec] d s ε /d t Fig. 2. The signal x 2ε =˙x 1ε copies ˙y d (ε ≈ 10 −3 √ 2π) b ε (t)= 1 ε cos  t ε  ,c ε (t)=0,ε>0, 0 ≤ t ≤ T. Here the control region is the whole real line, b ε → 0inW −2,∞ (0,T), b ε → 0in W −1,∞ (0,T). Let (u ε ,x ε ) be any second order real control-state pair such that x 1ε (0) → y d (0), x 2ε (0) → ˙y d (0). Then we have ˙s ε (t)= ˙s ε (0) + sin  t ε  , ˙s ε (0) = x 2ε (0) − ˙y d (0), s ε (t)=t ˙s ε (0) + ε  1 − cos  t ε  . Since the initial values converge and s ε converges uniformly to 0, as ε → 0, if the actual ε is small enough, x 1ε copies y d (Figure 3). However, since ˙s ε does not converge uniformly to 0, as ε → 0, not even point-wise, then, even if the actual ε is small, x 2ε =˙x 1ε does not copy ˙y d (Figure 4). Here the point is that unbounded non-idealities are allowed and condition (13) fails. In such a case first order approximability may fail, even if, as checked in Example 2, the same property is present, provided the control region is bounded. Indeed consider the first order real control-state pairs ( u ε , x ε ) such that [...]... further (direct) proof of second order approximability of second kind for sliding mode control systems the dynamics of which are affine in the control variables (for the corresponding wellknown result in the first order setting see [17], [18], and [19]) 6 Extensions of the Approximability Results In this section we consider some variants of the previous model (9), describing the second order real system... Utkin, V.: Sliding Modes in Control and Optimization Springer, Berlin (1992) 18 Bartolini, G., Zolezzi, T.: Control of nonlinear variable structure systems J Math An Appl 118, 42–62 (1986) 19 Zolezzi, T.: Well-posedness and sliding mode control ESAIM Contr Optim Calc Var 11, 219–228 (2005) Regularization of Second Order Sliding Mode Control Systems 21 20 Bartolini, G., Zolezzi, T.: Behaviour of variable-structure... integrably bounded A similar weakening of the assumption f ∈ C 2 is also possible Regularization of Second Order Sliding Mode Control Systems 15 5 Sliding Error Estimates In this section we consider sliding mode control systems (1) the dynamics of which are affine in the control variable, namely f (t, x, u) = A(t, x) + B(t, x)u (17) where A(t, x) and B(t, x) are of the appropriate dimension, and A, B... have established second order regularization results of sliding mode control systems An attained major result is that first order implies second order approximability under mild assumptions Hence the regularization of second order methods is automatically valid as soon as it is guaranteed for first order sliding techniques Moreover conditions have been found, under which second order approximability holds... on first order approximability results The real state trajectories of second order sliding mode control systems have been compared to the ideal state trajectories up to point-wise sliding error estimates between real and ideal states based on the norms of the non idealities and on the gap of initial data General situations, involving possible combination of first and second order sliding mode control, ... N R We obtain an explicit estimate of the point-wise sliding error between second order real states and the ideal sliding state, based on the norms of the nonidealities bε , cε and the gap of the initial values Such a result extends known estimates of [17], chapter 2, Section 3, to second order sliding mode control systems Let (uε , xε ) be any second order real control- state pair, thereby fulfilling... Notes in Control and Information Science, vol 217 Springer, Berlin (1996) 8 Fridman, L., Levant, A.: Higher order sliding modes In: Perruquetti, W., Barbot, J.P (eds.) Sliding Mode Control in Engineering Control Engineering Series, vol 3197 Marcel Dekker, New York (2002) 9 Bartolini, G., Pisano, A., Punta, A., Usai, E.: A survey of applications of secondorder sliding mode control to mechanical systems. .. → Gronwall’s lemma we get yε − xε − Hence second order approximability of any →0 kind, within any N0 , of the control system (1) implies that → →y yε − where y is the unique sliding state of (1), whenever the non-idealities bε , cε disappear in the sense corresponding to the approximability property of (1) Regularization of Second Order Sliding Mode Control Systems 4 19 3 3 2 2 d σε/d t d /d t and... (α, xε , uε ) cε ] dα ˙ sε (t) = s [0, xε (0)] + ˙ 0 (21) Regularization of Second Order Sliding Mode Control Systems 17 Thus (21) holds provided (10), (12) and (14) are fulfilled, and xε (0) → y (0) (see the proof of Theorem 4), where y denotes the unique sliding state Suppose now that the control system (1) fulfills second order approximability of the first kind within N0 defined by (13) and (14) Then... Control Systems, 3rd edn Springer, New York (1995) 5 Drazenovic, B.: The invariance conditions for variable structure systems Automatica 5, 287–295 (1969) 6 Levant, A.: Sliding order and sliding accuracy in sliding mode control Int J Contr 58, 1247–1263 (1993) 7 Fridman, L., Levant, A.: Higher order sliding modes as a natural phenomenon in control theory In: Garofalo, F., Glielmo, L (eds.) Robust control . developed for sliding mode control of order two, thus obtaining second order regularization results of sliding mode control systems. A major result we obtain is that first order implies second order approximability. for first and second order sliding methods. Regularization of Second Order Sliding Mode Control Systems 5 2 First Order Approximability: Definitions and Results We consider sliding mode control problems ˙x. properties identify second Regularization of Second Order Sliding Mode Control Systems 9 order sliding mode control systems such that real states are uniformly close to the unique sliding state, as