Sliding Mode Control of Constrained Nonlinear Systems

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Sliding Mode Control of Constrained Nonlinear Systems chattering reduction is not the only possible purpose of HOSM, since this latter allows the use of a relative degree greater than one between Gian Paolo Incremona, Matteo Rubagotti, and Antonella Ferrara the discontinuous control input and the sliding variable. In the first-order SMC formulation, input saturations are immediately satisfied if the control variable switches between values Abstract—This technical note introduces the design of sliding mode control algorithms for nonlinear systems in the presence of hard inequal- that are inside the imposed boundaries. When HOSM is used for ity constraints on both control and state variables. Relying on general chattering reduction, specific solutions have been proposed to achieve results on minimum-time higher-order sliding mode for unconstrained the satisfaction of saturation constraints (see, e.g., [19], [20] for the systems, a general order control law is formulated to robustly steer the second-order case). On the other hand, the possible presence of state state to the origin, while satisfying all the imposed constraints. Results on minimum-time convergence to the sliding manifold, as well as on the constraints is usually not taken into account in SMC formulations. maximization of the domain of attraction, are analytically proved for Recently, few solutions have been proposed in order to merge SMC the first-order and second-order sliding mode cases. A general result is and MPC, and combine the constraints satisfaction property of MPC presented regarding the domain of attraction in the general order case, with the robustness of SMC [21]–[25]. As an alternative, in order to while numerical results on the estimation of the domain of attraction and on minimum-time convergence are discussed for the third-order avoid the additional computational burden of MPC, the presence of case, following a procedure applicable to a sliding mode of any order. state constraints has been directly inserted in the SMC law in [26]– [28] for the first-order sliding mode case, in [29]–[32] for the second Index Terms—Sliding mode control, second order sliding mode, higher order sliding mode (HOSM), uncertain systems, constrained control. order sliding mode case, and in [33] for third-order sliding mode with box constraints. In this technical note, an HOSM control law of general order r I.INTRODUCTION is proposed, aimed at guaranteeing the minimum-time convergence Input constraints are present in all practical control implementa- of the state onto the sliding manifold, and, at the same time, tions, mainly in the form of saturations. In order to guarantee an satisfying the imposed hard inequality constraints on input and acceptable performance of the controlled system in their presence, states, in the presence of matched disturbances. More specifically, different solutions have been proposed, mainly in the field of anti- the main contributions of the present work are the following: first windup control (see, e.g., [1], [2]). In addition, in order to avoid of all, the formulation of a control law capable of solving an r- failures or critical conditions of the controlled system, certain regions th order SMC problem for uncertain nonlinear affine systems with of the state space need to be avoided during the execution of many inequality constraints on both input and state variables (note that tasks. In some cases, a conservative tuning of the control laws can this was an open problem, to the best of the authors’ knowledge); lead to the avoidance of these regions. On the other hand, control second, a procedure to select the sliding variable in order to satisfy laws that directly consider the presence of state constraints can reduce the constraints, including two different sufficient conditions which conservativity and improve the overall system performance. The most provide guidance in the choice of the sliding variable; third, the well-known control methodology able to manage both input and state proof of the minimum-time convergence with constraint satisfaction constraints is Model Predictive Control (MPC), for which the reader and maximization of the domain of attraction for the first and second- is referred to [3] and the references therein. order cases (a preliminary result on this aspect, limited to the second- A control technique that naturally handles the presence of some order case, was presented in [30], without proof of the minimum-time classes of uncertainties and disturbances is Sliding Mode Control convergence); fourth, a general result on the domain of attraction (SMC), in which a discontinuous control law steers the state onto a for the r-th order case, along with the specific numerical study of suitably-defined hyper-surface (the so-called sliding manifold), and, the domain of attraction and of the minimum-time convergence for under suitable design conditions, makes the origin of the state space the third-order case. Note that preliminary results on the numerical an asymptotically stable equilibrium point for the closed-loop system. evaluation of the domain of attraction for a third-order sliding mode The convergence to the sliding manifold is guaranteed in a finite- controller (only for the case of box state constraints) were described time interval if the control action is large enough to counteract the in [33], and can be considered now as particular cases of the proposed effect of the uncertain terms. After reaching the sliding manifold, general formulation. the evolution of the state variables is insensitive to the so-called This technical note is organized as follows. After formulating the matched disturbances, i.e., those acting in the same channel as the regulation problem in Section II, Section III describes the design of control variable [4], [5]. The main drawback of SMC is the so-called the proposed control law. Theoretical results are proved in Section chattering [6], [7], which is the high-frequency oscillatory motion IV for the first and second-order cases, while more general results around the sliding manifold due to the discontinuity of the control and possible developments are discussed in Section V. Conclusions law. Higher Order Sliding Mode (HOSM) is a possible solution for are finally drawn in Section VI. For the sake of readability, most of chattering reduction (see, e.g., [8]–[17], and the references therein the theoretical proofs are reported in the Appendix. included). In particular, [16] proposes an algorithm that guarantees a time-optimal reaching of the sliding manifold for arbitrary order II.PROBLEM FORMULATION (i.e., dimension of the sliding manifold, see, e.g., [18]). Note that In this technical note, we consider a class of uncertain nonlinear This is the final version of the accepted paper submitted dynamical systems, defined by to IEEE Transactions on Automatic Control. G.P. Incremona is with the Dipartimento di Ingegneria Civile ed Architettura, x˙(t) = φ(x(t), t) + γ(x(t), t)u(t) (1) University of Pavia, Via Ferrata 3, 27100 Pavia, Italy (e-mail: n [email protected]). where x ∈ R is the state vector, u ∈ R is the control input, n+1 n n+1 n M. Rubagotti is with the Department of Engineering, University φ : R → R and γ : R → R are uncertain smooth of Leicester, University Rd, Leicester LE1 7RH, United Kingdom vector fields, and the whole state vector is assumed to be available for [email protected] ( ). x A. Ferrara is with the Dipartimento di Ingegneria Industriale e feedback. The control objective is the regulation of the state to the dell’Informazione, University of Pavia, Via Ferrata 5, 27100 Pavia, Italy (e- origin. Differently from classical SMC formulations, a set of (possibly mail: [email protected]). mixed) hard input and state inequality constraints is introduced. More 2 specifically, it is required that where X is, in general, a closed set, while α > 0 is the same fixed parameter as in (4), which defines the maximum amplitude of the x¯ ∈ X¯, x¯ x0 u0 ∈ n+1, , R (2) control variable. In order to consider (5) as the general formulation, 0 n+1 where denotes the transposition operator. X¯ ⊆ R is a closed the case m = 0 can be formulated as a particular case of (5), with (possibly unbounded) set that includes the origin in its interior, and xa = x, Φ(·, ·) = φ(·, ·), Γ(·, ·) = γ(·, ·), and Xa = X . is formulated as a system of (possibly nonlinear) algebraic equations. B. Definition of the sliding manifold III.THEPROPOSED HOSM CONTROL LAW After defining the augmented system (5) and the related constraint A. Definition of the augmented system sets in (6), the next step consists of defining a suitable output variable The control law can either be defined as a discontinuous control σ1(t) ∈ R, as a (in general, nonlinear) function of xa. Following the law (classical SMC), or as the result of an m-fold time integration of standard design procedure of HOSM control [10], a vector of time a discontinuous signal w(t) (HOSM aimed at chattering reduction). derivatives of σ1 is defined as 0 The case m > 0, m being an integer number, is considered first. 0 h (r−1)i r σ = σ1 σ2 . . . σr σ σ˙ . . . σ ∈ (8) An integrator-chain dynamics is added to the system, starting from , 1 1 1 R xn+1 , u, as where r ∈ [0 , n+m] is the well defined, uniform and time-invariant x˙ (t) = x (t)  n+1 n+2 relative degree of the system (assuming w as input and σ1 as output).  x˙ n+2(t) = xn+3(t) In the (n + m)-dimensional space defined by the components of (3) . xa, the manifold Σ , {xa : σ(xa) = 0} is referred to as sliding  .  manifold. The control variable will be defined in order to ensure  x˙ n+m(t) = w(t). the finite-time convergence of xa on the sliding manifold, which σ Note that the chain of integrators will be an element of the closed (assuming a correct definition of ) will in turn imply the asymptotic x loop control system for which the stability results proved in Sections convergence of a to zero. With reference to [38, Theorem 13.1], a Ω: n+m → n+m−r × r IV and V hold. diffeomorphism R R R is defined, such that

In order to provide bounds on the derivatives of the actual control (ζ, σ) = Ω(xa) (9) variable u, and recalling that the constraints on u = xn+1 are already n+m−r imposed in (2), a set of box constraints is defined, as where ζ ∈ R is the internal state vector. The diffeomorphism allows one to transform system (5) into the normal form w ∈ [−α, α], xn+i ∈ [−βi, βi], i = 2, . . . , m, (4) ζ˙(t) = ψ(ζ(t), σ(t), t) (10a) α, βi > 0 being fixed parameters. The overall augmented system 0 n+m σ˙ i(t) = σi+1(t), i = 1, . . . , r − 1 (10b) dynamics, with state xa , x1 . . . xn+m ∈ R is now represented by σ˙ r(t) = f(ζ(t), σ(t), t) + g(ζ(t), σ(t), t)w(t) (10c) n+m+1 n+m−r n+m+1 n+m+1 x˙ a(t) = Φ(xa(t), t) + Γ(xa(t), t)w(t) (5) with ψ : R →R , f : R →R, g : R →R. The dynamics of system (5) includes uncertain terms, as specified n+m+1 n+m n+m+1 n+m where Φ: R → R and Γ: R → R are when defining the vector fields φ and γ in (1). As a consequence, ψ, smooth vector fields immediately obtainable from (1) and (3). For f and g in (10) also contain uncertain terms, but some information the sake of compactness, the sets of disjoint constraints in (2) and about them is available, as stated in the following assumption. (4) can be merged as Assumption 1: There exist positive constants F , G1, G2, such that

xa ∈ Xa, w ∈ [−α, α], (6) |f(ζ(t), σ(t), t)| ≤ F, (11) with implicit definition of Xa. 0 < G1 ≤ g(ζ(t), σ(t), t) ≤ G2, (12) Remark 1: As m increases, the smoothness of the control signal u also increases [12], together with the number of states of system (5), αr , G1α − F > 0. (13) with a consequent increase in the complexity of the controller. On the other hand, the chattering amplitude does not necessarily decrease as Moreover, the following two hypotheses are valid for the internal m increases [34]. Thus, there is not a “best value” for m in general: dynamics (10a). Firstly, for all initial conditions (ζ(0), σ(0)) ∈ n+m we propose an approach valid for any value of m, and leave the R , for all realizations of the uncertain terms satisfying (11)- decision on its value, for each specific application, to the designer of (12), the internal dynamics (10a) does not present finite time escape phenomena. Secondly, the zero dynamics the control law. Remark 2: The analysis of chattering is outside the scope of this ζ˙(t) = ψ(ζ(t), 0, t). (14) paper. Yet, it is an interesting topic that the reader may deepen making reference to [6], [34]–[37]. Indeed, in systems with chains is globally asymptotically stable. of integrators and certain dynamics of lag type in series, chattering These are typical assumptions in SMC: (11)-(12) require the may exist or not depending on the type of the discontinuous control boundedness of the uncertain terms, (13) ensures that the control law (ideal relay or relay with hysteresis) and the effect of parasitic amplitude is large enough to counteract their effect, the assumption dynamics. on the internal dynamics and on the zero dynamics (14) imply In the particular case when u is directly defined as the discontinuous that, during the reaching phase the internal states remain bounded, control variable (i.e., m = 0 and therefore w = u) the constraints while, once σ has been steered to zero, ζ will also converge to zero in (2) are required to be formulated as disjoint input and state asymptotically. By means of the diffeomorphism Ω(xa) in (9), the constraints, and precisely set Xa can be in general mapped into a new set S, i.e., ¯ n X = {(x, u) ∈ R × R : x ∈ X , u ∈ [−α, α]} , (7) xa ∈ Xa ⇐⇒ (ζ, σ) ∈ S (15) 3 which will be used to enforce the constraints in the coordinate influence the dynamics of σ. Again, by substituting the expression of system defined by (10). The presence of these constraints requires σ into the expression of Xa, the set S will be obtained as a function an additional hypothesis. of σ only. Assumption 2: The diffeomorphism Ω(xa) is defined such that Example 2: Consider as augmented system S = S(σ) ⊂ r, i.e., the expression of the constraint set does not R x˙ (t) = −x (t)e−x1(t) + x5(t) depend on the internal state ζ. 1 1 2 3 Assumption 2 is required by the fact that, in SMC, the control x˙ 2(t) = −x2(t) + w(t) law is defined as a function of σ. As a consequence, it would not with constraint set X = {(x , x ): x ∈ [−1, 3]}. In this case, the be possible to enforce constraint satisfaction on the internal state ζ a 1 2 2 state vector can be divided into two scalar components x = x and during the reaching phase. For this reason, the constraints have to be b 1 x = x . expressed in the r-dimensional space defined by σ. c 2 Choosing ζ = x and σ = x , we obtain a scalar σ ∈ , and the An explicit expression for Ω(x ) is not available in general. 1 1 2 R a dynamics of the system in form (10) is described by However, in the following we provide two different sufficient condi- ˙ −ζ(t) 5 tions, for which Ω(xa) can be immediately defined. We also show ζ(t) = −ζ(t)e + σ1 (t) an example in which neither of these conditions are satisfied, but 3 σ˙ 1(t) = −σ1 (t) + w(t) nonetheless Ω(xa) can be easily found. For the general case, the solution to this problem has to be considered by the designer of the with control system as a specific step of the design process. ζ x1 −1 ζ = Ω(xa) = , xa = Ω (ζ, σ) = . σ1 x2 σ1 C. Design of the sliding manifold The constraint set is defined as S = {σ1 : σ1 ∈ [−1, 3]}, which is The definition of Ω(xa) (or, equivalently, of σ1) such that σ a function of σ only. satisfies Assumption 2 requires, in general, some attention. Guidelines In the general case, Assumption 2 is not straightforwardly verified are provided in the following. Let us start by describing two particular so that a careful choice of σ1 must be made for each case. A simple cases in which Assumption 2 can be easily satisfied. example is shown next. Lemma 1: Consider a given augmented dynamics (5) with asso- Example 3: The augmented system dynamics is given by ciated constraints (6). If σ is defined such that r = n + m, then x˙ 1(t) = −x1(t) + x2(t) + w(t) Assumption 2 is satisfied. Proof: Since r = n + m, Ω(xa) defines a mapping from xa x˙ 2(t) = −x2(t) + w(t) to σ, and no internal state ζ exists. Therefore, each element of xa with associated constraint set Xa = {(x1, x2): |x1 + x2| ≤ 1}. can be expressed as a function of the elements of σ. Substituting the Since Xa depends on both state variables, it is not possible to use corresponding expressions of σ in the formulation of X , the set S a the result in Lemma 2. The constraint set S has to be a function of is obtained, which is a function of σ only. σ only, so we choose σ1 = x1 +x2. The internal state can be chosen Example 1: Consider the augmented system given by as ζ = x1 − x2.

x˙ 1(t) = sin(x2(t)) + u(t) As in the previous example, we obtain a scalar σ ∈ R, and the dynamics of the system in form (10) is x˙ 2(t) = u(t) ˙ 3 1 with constraint set Xa = {(x1, x2): |x1| ≤ 1, |x2| ≤ 1}. Choosing ζ(t) = − ζ(t) + σ1(t) 2 2 2 σ1 = x1 − x2, we obtain a vector σ ∈ R , the dynamics of which 1 1 σ˙ (t) = ζ(t) + σ (t) + 2w(t) is deﬁned by 1 2 2 1

σ˙ 1(t) = σ2(t) with σ˙ 2(t) = cos(x2(t))w(t) ζ x1 − x2 −1 1 ζ + σ1 = Ω(xa) = , xa = Ω (ζ, σ) = . σ1 x1 + x2 2 ζ − σ1 with −1 The constraint set is deﬁned as S = {σ1 : |σ1| ≤ 1}, which is a x1 − x2 −1 σ1 + sin (σ2) σ = Ω(xa) = xa = Ω (σ) = −1 . function of σ only. sin(x2(t)) sin (σ2)

−1 The constraint set is defined as S = {(σ1, σ2): |σ1 + sin (σ2)| ≤ D. Definition of the control law 1, |σ2| ≤ sin(1)}, which is a function of σ only. n+m The control law w(t) has to be defined as a discontinuous function Lemma 2: Assume that the state vector xa ∈ R of the aug- 0 of σ. In [16, Sec. III.C], an HOSM control law for a system mented system is composed of two subvectors, as x = x0 x0 , a b c in form (10b)-(10c) was proposed (i.e., for arbitrary value of r), with x ∈ n+m−p, x ∈ p, p ∈ {1, 2, . . . , n − m}. As a b R c R which guaranteed the reaching of the sliding manifold in minimum consequence, (5) takes the form time, for the worst-case realization of the uncertain terms (i.e.,

x˙ b(t) = Φb(xb(t), xc(t), t) (16) f(ζ(t), σ(t), t) ≡ −F · sgn(w(t)) and g(ζ(t), σ(t), t) ≡ G1). In this worst-case formulation, αr in (13) can be interpreted as x˙ c(t) = Φc(xc(t), t) + Γc(xc(t), t)w(t) (17) a reduced control amplitude (however, notice that αr and α have with implicit deﬁnition of Φb, Φc, and Γc. In addition, assume that different units of measure). We denote the control law proposed in the constraint set Xa is a function of xc only. Then, if σ = σ(xc) [16], which did not take into account the presence of constraints, as r+1 r with r = p, Assumption 2 is satisﬁed. win(σ) , (−1) ·α·c(σ), where function c(σ): R → {−1, +1} Proof: Since r = p, the same development of the proof of is determined as reported in [16]. In general, one obtains an (r − 1)- Lemma 1 can be followed, for subsystem (17). Ω−1(σ) will map dimensional manifold in the r-dimensional space generated by vector σ to xc, and the evolution of the internal state variables ζ will not σ, described by the nonlinear equation s(σ) = 0. This manifold, 4 which includes the origin, divides the σ-space into two half-spaces. constraint set S˜ ⊂ S, so as to prevent the original constraints from By imposing a constant value of c (either −1 or +1) in each half- being violated. space, the control law can be written as

win(σ) = −α · sgn(s). (18) IV. ANALYTICAL RESULTS ON FIRSTAND SECOND-ORDER SLIDING MODESWITH BOX CONSTRAINTS As mentioned in [16], the complexity of the expression of s(σ) grows very fast with the order r, and the derivation of efficient methods In this section, the convergence in a finite (minimum) time of (numerical or exact-algebraic) in order to define it, is still an open the sliding variable associated with the proposed closed-loop control problem. A method for determining the expression of s(σ) for generic system is discussed. Note that this result directly implies asymptotic order is proposed in [39]. For relatively low-dimensional cases, the stability of the origin of the closed-loop system since, by assumption, analytical expression of s(σ) is given by the zero dynamics (14) of system (1) transformed via the diffeomorphism Ω(xa) is globally asymptotically stable. 1 σ ∈ R ⇒ s(σ) = σ1 (19) 2 σ2|σ2| σ ∈ ⇒ s(σ) = σ1 + (20) R 2αr A. First-Order Sliding Mode Control 3 2 3 σ3 σ3 sgn(σ3) σ ∈ ⇒ s(σ) = σ1 + 2 + sgn σ2 + × The formulation of the control law (22) for r = 1, using (19), R 3αr 2αr 3 becomes 2 2 2 1 σ3 sgn(σ3) σ3 σ2σ3 × √ sgn σ2 + σ2 + + . (21) αr 2αr 2αr αr w(σ) = −α sgn(σ1) (23)

To be precise, in the problem defined in [16], the expression of win is which coincides with the “unconstrained” control law win in (18). found also in the different sub-cases when σ = 0, which would make The constraint set in this simple case can only take the form the formulation of the control law more complicated. However, as pointed out also in [16], being s(σ) a null-measure set, the definition S = {σ1 : σ1 ∈ [σ1, σ1]}. (24) of the control law for σ = 0 has no practical relevance. For this with σ < 0 and σ1 > 0. Then, the following result can be proved. reason, the value of w (σ) when s(σ) = 0 will not be explicitly 1 in Theorem 1: Given system (10) with r = 1, assuming that Assump- defined, and, when needed in the proofs, will be determined as the tion 1 holds, if the control law (23) is applied, then σ is steered to Filippov solution of the given discontinuous vector field at σ (see 1 the origin with domain of attraction S coinciding with S in (24). [40]). I The set SI is the maximum obtainable domain of attraction for the In order to take the presence of constraints into account, the control given set of constraints, and the convergence takes place in minimum law proposed in this technical note coincides with w (σ) when σ ∈ in time for the worst possible realization of the disturbance terms (i.e., S, while for σ∈ / S is defined as wout(σ) , −α · sgn(σr). Overall, f(ζ(t), σ1(t), t) ≡ −F · sgn(w(t)) and g(ζ(t), σ1(t), t) ≡ G1).. ( −α · sgn(s(σ)), σ ∈ S Proof: See Appendix A. w(σ) = (22) −α · sgn(σr), σ∈ / S. B. Second-Order Sliding Mode Control which is an extremely simple law, yet capable of solving a rather complex nonlinear constrained control problem for uncertain systems. In the case r = 2, the second-order SMC law, according to (20) After the introduction of the proposed control law (22), three and (22), is defined as interesting issues regarding the closed-loop system are addressed ( σ2|σ2| −α sgn σ1 + if (σ1, σ2) ∈ S in the following. The first is the determination of the domain of w(σ) = 2αr (25) attraction SI of the origin for the closed-loop system, i.e., all the −α sgn(σ2) if (σ1, σ2) ∈/ S initial conditions σ(0) for which σ(t) converges to the origin in a finite time without violating the constraints. The second is the while the box constraints can be expressed as problem of determining if the obtained domain of attraction can S = {σ ∈ 2 : σ ∈ [σ , σ ], σ ∈ [σ , σ ]} (26) be enlarged by using a different control law w(t), with the same R 1 1 1 2 2 2 system dynamics, the same bounds on the uncertain terms and on with σ1, σ2 < 0 and σ1, σ2 > 0. A graphical representation of the the control amplitude, and the same state constraints. The third issue switching regions is reported in Figure 1. is related to investigating if w(σ) in (22) still leads to minimum-time Theorem 2: Given system (10) with r = 2, assuming that convergence to σ = 0 for the constrained case, as it was in [16] for Assumption 1 holds, if the control law (25) is applied, then σ is the unconstrained case. In the next section, the cases r = 1 and r = 2 steered to the origin with domain of attraction with box constraints on σ will be considered. For such cases, formal results will be presented. The general case will be tackled in Section SI , S \ {M0 ∪ M1} (27) V. Remark 3: In many practical applications, SMC laws are imple- where mented using a finite sampling time, or one of the so-called pseudo- n σ2|σ2| o M0 , (σ1, σ2): σ1 > − + σ1, σ2 > 0 , (28) sliding techniques (typically, employing a saturation function instead 2αr n σ2|σ2| o of the sign function, or low-pass filtering the discontinuous control M1 (σ1, σ2): σ1 < − + σ , σ2 < 0 . (29) , 2αr 1 signal). All of these techniques typically lead to the convergence to a boundary layer of the sliding manifold, rather than on the The set SI is the maximum obtainable domain of attraction for the sliding manifold itself [41]. If such techniques are directly applied to given set of constraints, and the convergence takes place in minimum approximate the proposed control law, the imposed constraints can time for the worst possible realization of the disturbance terms (i.e., be slightly violated, as the state evolves on a boundary layer around f(ζ(t), σ(t), t) ≡ −F · sgn(w(t)) and g(ζ(t), σ(t), t) ≡ G1). them. The typical solution consists in defining more conservative Proof: See Appendix B. 5

1. 5

0. 5 −α 3 3 α

2 0 σ

2 2

−0.5 1 1

(a) (b) −1

−1.5

−1.5 −1 −0.50 0. 51 1. 5 σ1 3 3 Figure 1. Graphical representation of the switching regions and box constraints in the second order sliding mode case.

2 2

1 1 V. GENERAL CASE:HIGHER ORDER SLIDING MODESWITH ARBITRARY CONSTRAINTS (c) (d)

In Section IV, the properties of the proposed scheme have been Figure 2. Numerical estimate of the domain of attraction SI for Example 4: analyzed for constrained first-order and second-order sliding modes (18) with α = 1 (a), (25) with α = 1 (b), (18) with α = 5 (c), and (25) with α = 5 (d). with box constraints. These represent very common and relevant cases (for instance, a mechanical system with constraints on position and velocity). Yet, sometimes one has to design higher-order sliding mode controllers and/or to deal with more general constraints. We provide domains of attraction amount at 3% and 86% of the sphere volume, now general considerations and results for the higher-order case with respectively. Two observations are in place: arbitrary constraints on σ. • The domain of attraction using the constrained control law (22) In the cases analyzed in Section IV, it has been proved that the is not reduced (actually, it is enlarged) with respect to that obtained domain of attraction is maximized for the given set of obtained by using (18), as expected from Lemma 3. constraints. The following result can be proved for the general case. • The difference between the two domains of attraction seems to Lemma 3: Given system (10) with generic r, assuming that widen as α increases: this is due to the ability of the proposed Assumption 1 holds, if the control law (22) is applied, then the control law (22) to impose a state trajectory that, when possible, domain of attraction SI , for the given set of constraints, includes moves on the boundary of S before moving to its interior. 0 the domain of attraction SI obtained by applying the unconstrained The result regarding minimum-time convergence also in the presence control law (18). of constraints, formally proved in the previous section for lower- Proof: The result immediately follows by observing that, if order cases, becomes analytically intractable for higher sliding mode 0 0 σ(0) ∈ SI , then w(0) = win(0). Being SI by definition a robust orders. However, in order to show that the proposed method might positively invariant (RPI) set [42, Def. 4.3] when applying (18), the be achieving the minimum-time convergence for the general case, we time evolution of σ will coincide with that obtained by applying (18). show the following example. 0 This implies that SI ⊆ SI . Example 5: Consider again the system of Example 4, this time with Example 4: In order to show the domain of attraction in a constraints defined as kσk2 ≤ 0.1. In order to test the minimum- rather general case, we consider the third-order SMC, for which the time converge properties of the controller, we set α = αr = 0.5. dynamics of the sliding variable is defined as Considering an initial condition σ = [0.05 0.05 0]0, the control law (22) is simulated with MATLAB using the Ode4 solver with a σ˙ 1(t) = σ2(t) fixed discretization interval of 10−3 s. With the same initial condition σ˙ 2(t) = σ3(t) and the same constraints, a minimum-time constrained optimization

σ˙ 3(t) = w(t). problem is solved numerically by using CVX [43], by implementing a bisection routine which runs a feasibility problem at every step, Note that, in order to be able to make comparisons, the uncertain for a fixed time interval. In Figures 3-4 it is possible to compare the terms have been fixed as f(ζ(t), σ(t), t) ≡ 0 and g(ζ(t), σ(t), t) ≡ time evolution of the two control laws, and the trajectories of the two 1, which implies α = αr. The set of constraints is defined as kσk2 ≤ state vectors. From 0 to about 0.6 s, both control laws are equal to 1, which is a sphere centered at the origin. By fixing α = 1, we −α. The two state trajectories are indistinguishable, and they both numerically obtain the domains of attraction shown in Figures 2a approach the boundary of the sphere. At about 0.6 s, the boundary (63% of the sphere volume) and 2b (76% of the sphere volume) by is reached: the proposed control law switches between −α and +α, applying the unconstrained control law (18) and the new proposed while the numerical solver determines a solution that is continuous control law (22), respectively. Increasing the control amplitude to between the boundaries. The two state trajectories slide along the α = 5, the corresponding domains of attraction are shown in Figures surface of the sphere, and are again indistinguishable. After 2 s, in 2c and 2d, again for (18) and (22), respectively. The corresponding both cases the control law keeps the value +α, while at about 3.3 6 s it switches to −α (with a quick but continuous transition in the APPENDIX A numerical case). The two state trajectories are still indistinguishable, PROOFOF THEOREM 1 and converge to the origin in about 4 s. Assume that σ1(0) ∈ S, i.e., σ1(0) ∈ [σ1 , σ1]. Specifically, from . Assumption 1 it follows that A similar procedure has been repeated for different systems and different sets of constraints, always obtaining indistinguishable state 0 < σ1(t) ≤ σ1 ⇒ σ˙ 1(t) ≤ F − G1α = −αr < 0, (30) trajectories, and the same time interval for convergence to the origin. σ1 ≤ σ1(t) < 0 ⇒ σ˙ 1(t) ≥ −F + G1α = αr > 0. (31) This can lead us to the conjecture that the proposed control law As a consequence, σ (t) is steered to zero in finite time without achieves minimum-time convergence in the general r-order case. The 1 leaving S for all σ (0) ∈ S ≡ S. Being S the set of state formal proof of such a result can be a topic for further research. 1 I constraints, a larger set SI cannot be obtained, which proves the maximization of the domain of attraction. Also, being signals w(σ(t))

0.15 and win(σ(t)) coinciding for all t, the controller solves a minimum-

(22) time problem for the worst-case disturbance, as proved in [16]. 0.1 numerical

0.05 APPENDIX B

) PROOFOF THEOREM 2 t

( 0 w Deﬁne the external perimeter ∂SI of SI as the union of the −0.05 segments AB, CD, EF , GH, BC, DE, FG, and HA (see Figure 5). The extreme points (e.g., A and B in AB) are not considered as −0.1 part of the segment for the ﬁrst four ones, while they are taken into

0 0.5 1 1.5 2 2.5 3 3.5 4 account for the other segments. Figure 5 shows a realization of the t (s) invariant set SI in the simple case σ1 = σ2 = 1, σ1 = σ2 = −1, α = 1. The white region is S , while the black regions represent Figure 3. Comparison between the time evolutions of the proposed control r I law (22) and of the numerical minimum-time control law, for Example 5. the set {M0 ∪ M1}. The so-called ‘switching line’ of equation

σ2|σ2| σ1 = − (32) 2αr is also shown as a solid blue line.

A. Positive invariance of SI

As a preliminary result, it will be shown that SI is a robust positively invariant (RPI) set [42, Def. 4.3] for the closed-loop system, which is proved by checking that for each σ ∈ ∂SI , the 0 vector ﬁeld σ˙ = [σ ˙ 1, σ˙ 2] never points outside SI [42, Theorem 4.10]. Case 1 (σ ∈ HA ∨ σ ∈ DE): Assume σ ∈ DE so that σ˙ = 0 [σ2, f − gα] . Notice that, from this point and for all the remainder of this proof, the dependency of f and g from their arguments will be omitted for the sake of readability. Consider that σ ∈ DE implies σ2 < 0, while −F − G2α ≤ f − gα ≤ F − G1α < 0. Then, the

1 ABC Figure 4. Comparison between the state trajectories using the proposed control 0.8 law (22) and the numerical minimum-time control law, for Example 5. 0.6 SI 0.4

0.2

VI.CONCLUSIONS 2 0 σ H D This technical note has presented a new approach to design −0.2

HOSM control laws for nonlinear uncertain systems with arbitrary −0.4 relative degree subject to input and state constraints. The presence of −0.6 constraints, while being a topic of paramount importance in practical applications, has not been dealt with extensively in the SMC literature −0.8 G F E up to now. The proposed control laws, apart from maintaining the −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 system state and the control variable always within the admissible σ1 domain, are aimed at achieving the minimum-time convergence on the sliding manifold, and the maximization of the domain of attraction. Analytical proofs together with numerical results have been reported Figure 5. Graphical representation of a possible invariant region SI in the second order case. in order to show the potential of the proposed method. 7

vector field is pointing down-left, that is towards the interior of SI . σ2 < 0, obtaining Analogous considerations can be done if σ ∈ HA, where the vector 2 σ2 − ∇ σ1 − · σ˙ field is always pointing up-right. 2αr + σ˙ = 2 σ˙ σ2 − + Case 2 (σ ∈ BC ∨ σ ∈ FG): Assume σ ∈ FG \{G} so ∇ σ1 − · (σ ˙ − σ˙ ) 0 2αr that σ˙ = [σ2, f + gα] , with σ2 = σ2 < 0 and 0 < −F + G1α ≤ 2 σ2 + f+gα ≤ F +G2α. Then, the vector field σ˙ is always pointing up-left, ∇ σ1 − 2α · σ˙ − r σ˙ − which means towards the interior of S . Analogous considerations 2 I σ2 − + ∇ σ1 − · (σ ˙ − σ˙ ) can be done if σ ∈ BC, where the vector field is always pointing 2αr down-right. f + gα − αr + f − gα − αr − 0 = σ˙ − σ˙ = [σ2, αr] Case 3 (σ ∈ CD ∨ σ ∈ GH): Assume σ ∈ GH so that σ˙ = 2gα 2gα 0 [σ2, f + gα] . One has always that σ2 < 0 is on this segment, while which is always tangent to the switching curve so that the state moves 0 < −F + G1α = αr ≤ f + gα ≤ F + G2α. It is easy to notice towards the origin, and converges to it in a finite time. The same σ2|σ2| that, since all the points on this segment verify σ1 = − 2α + σ1, holds for σ(0) ∈ {σ : σ1 = −σ2|σ2|/2αr, σ2 > 0, for which r 0 σ˙ can be at most tangent to the line but never points outside. The σ˙ = [σ2, −αr] . Combining the two obtained vector fields we can same considerations can be stated for σ ∈ CD. also obtain that σ˙ = [0, 0]0 for σ = 0. Case 4 (σ ∈ AB ∨ σ ∈ EF ): Assume σ ∈ AB. The control law We have therefore proved that SI is a domain of attraction for the is discontinuous on AB (see Figure 1). The vector field is therefore origin, with finite-time convergence. generated as the Filippov solution (see, e.g., [4], [5]) of the state space equations (10b)-(10c) for the second-order case. More precisely, σ˙ C. Maximal region of attraction + 0 − belongs to the convex hull of σ˙ = [σ2, f −gα] and σ˙ = [σ2, f + Assume that σ(0) ∈ M0. Considering that σ1(0), σ2(0) > 0, then gα]0 . The solution is obtained as σ˙ = µσ˙ + + (1 − µ)σ ˙ −. Finding σ1 will continue to increase in time until σ2 > 0. The quickest way 0 µ from condition ∇σ2 · σ˙ = 0, with ∇σ2 = [0, 1] , one has that to make σ2 decrease is to use the control variable w(σ) = −α. In this case, the system will move on a parabolic arc, the equation of − + which, in the worst case, taking into account the uncertain terms, is ∇σ2 · σ˙ + ∇σ2 · σ˙ − σ˙ = − + σ˙ − − + σ˙ ∇σ2 · (σ ˙ − σ˙ ) ∇σ2 · (σ ˙ − σ˙ ) σ2|σ2| σ1 = − + σ1 + ε (33) f + gα + f − gα − 0 2αr = σ˙ − σ˙ = [σ2, 0] 2gα 2gα with ε > 0. It is immediate to see that this arc intersects the σ1-axis outside S. One can easily see that any other realization of the control which is always tangent to the segment AB and pointing left. An variable will also lead to the same outcome. As a consequence, analogous procedure has been used to analyze the case σ ∈ EF . M0 cannot be part of the region of attraction for any realization of the control variable, given the constraints imposed on w, and on We can therefore conclude that SI is an RPI set for the considered σ. Analogous considerations hold if σ(0) ∈ M1. In conclusion SI closed-loop system. is the largest achievable region of attraction.

D. Minimum-time convergence The proof of the minimum-time convergence to the origin of the B. Finite-time convergence to the origin space {σ1, σ2} follows from [44, Chapter 8] and [45]. Considering the worst-case realization of the disturbance terms, it is possible to The convergence property will be proved in three parts, showing express the system dynamics as that obtained by applying the control input (25) to the double integrator plant that, for any initial condition σ(0) ∈ SI , σ(t) reaches the origin in a ﬁnite time. σ˙ (t) = σ (t) 1 2 (34) Case 1: Assume that one has σ(0) ∈ SI ∩ {σ : σ1 > σ˙ 2(t) = wr(t).

−σ2|σ2|/2αr, σ2 > σ2}. The vector field in this case is σ˙ (0) = 0 in which wr = (αr/α)w implicitly takes into account the effect of [σ2(0), f − gα] , being w(σ(0)) = −α, and it is possible to state 0 0 the disturbance terms. Let [0, 0] = [σ1(tc), σ2(tc)] , where tc is the that f −gα < 0. As a consequence, for all the considered σ(0), σ˙ (0) time needed to reach the origin from the given initial condition for a has a vertical component which is always strictly negative. When given control law. Following the approach discussed in [45], given a f − gα < −αr, σ(t) will reach in finite time either EF , or the linear system subject to convex state constraints and strongly convex σ2|σ2| switching line σ1 = − . An analogous proof is obtained for 2αr control constraints, if a covariant function ψ(t) and functions η(t) σ(0) ∈ SI ∩ {σ : σ1 < −σ2|σ2|/2αr, σ2 < σ2}: in that case, the and φ(t) can be found such that sliding variable will reach in a finite time the segment AB, or the switching line. φ˙(t) = ψ(t) + ξ(t)η(t) (35) Case 2: Assume that σ(0) ∈ EF . As proved in Case 4 in Appendix for some time-optimal control law in the absence of state constraints B-A, the trajectory of the system is kept on the line σ2 = σ2. Since σ˙ wnc(σ), and for any 0 ≤ t ≤ tc, then the obtained evolution of the has a strictly negative horizontal component during this time interval, sliding variable σ(t) is the one and only minimal time path and σ2|σ2| point F (which is on the switching line σ1 = − ) is reached 2αr ( wnc(σ) if σ < σ2 < σ2 in finite time. Analogous considerations hold for σ(0) ∈ AB. w(σ) = 2 (36) 0 if σ2 = σ or σ2 = σ2 Case 3: Assume that σ(0) ∈ {σ : σ1 = −σ2|σ2|/2αr, σ2 < 0}. 2 Since the control law is discontinuous on the switching line, we need is the one and the only minimal time control connecting points 2 h i0 σ2 σ2 to use the Filippov solution, with ∇ σ1 − = 1, − and (σ1(0), σ2(0)) and (0, 0). In our case, the defined set of state 2αr αr 8

constraints in (26) is convex, while the input constraint set [−αr, αr] [19] A. Ferrara and M. Rubagotti, “A sub-optimal second order sliding mode is strongly convex. Also, relying on the linear system (34), it is controller for systems with saturating actuators,” IEEE Trans. Automat. possible to define ψ(t) ≡ 0, ξ(t) = |σ (t)| + σ (t)w(t)/α , Control, vol. 54, no. 5, pp. 1082–1087, Jun. 2009. 2 2 r [20] I. Castillo, M. Steinberger, L. Fridman, J. A. Moreno, and M. Horn, η(t) = sgn(σ2(t)) if σ(t) ∈/ S, and η(t) ≡ 0 if σ(t) ∈ S. Moreover, “Saturated super-twisting algorithm: Lyapunov based approach,” in Proc. it is known that the time-optimal control law in the absence of state Int. Workshop on Variable Structure Systems, Nanjing, China, Jun. 2016, constraints would be the bang-bang control law pp. 269–273. [21] M. Rubagotti, D. Raimondo, A. Ferrara, and L. Magni, “Robust σ2|σ2| model predictive control with integral sliding mode in continuous-time wnc(σ) = −αr sgn σ1 + . (37) 2αr sampled-data nonlinear systems,” IEEE Trans. Automat. Control, vol. 56, no. 3, pp. 556–570, Mar. 2011. Condition (35) is therefore satisfied, as φ˙(t) = ψ(t) + ξ(t)η(t) = [22] A. Ferrara, G. P. Incremona, and L. Magni, “A robust MPC/ISM wnc(σ(t)) hierarchical multi-loop control scheme for robot manipulators,” in Proc. sgn(σ2(t)) |σ2(t)| + σ2(t) , from which it follows that αr 52th IEEE Conf. Decision Control, Florence, Italy, Dec. 2013, pp. 3560– σ2(t)|σ2(t)| φ(t) = σ1(t) + . Given the definition of S in (26), the 3565. 2αr effect of the application of (25) to the uncertain system (10b)-(10c) [23] A. Chakrabarty, V. Dinh, G. T. Buzzard, S. H. Zak, and A. E. Rundell, would be the same of applying (36) to system (34), with w defined “Robust explicit nonlinear model predictive control with integral sliding nc mode,” in Proc. American Control Conf., Portland, OR, USA, Jul. 2014, in (37). This proves that the proposed control law (25) drives σ to pp. 2851–2856. the origin in minimum time, for the worst-case realization of the [24] D. M. Raimondo, M. Rubagotti, C. N. Jones, L. Magni, A. Ferrara, and disturbance terms. M. Morari, “Multirate sliding mode disturbance compensation for model predictive control,” Int. J. Robust Nonlin. Control, vol. 25, no. 16, pp. 2984–3003, 2015. REFERENCES [25] A. Ferrara, G. P. Incremona, and L. Magni, “Model-based event- [1] M. V. Kothare, P. J. Campo, M. Morari, and C. N. Nett, “A unified triggered robust MPC/ISM,” in Proc. European Control Conf., Stras- framework for the study of anti-windup designs,” Automatica, vol. 30, bourg, France, Jun. 2014, pp. 2931–2936. no. 12, pp. 1869–1883, Dec. 1994. [26] M. Innocenti and M. Falorni, “State constrained sliding mode con- [2] S. Galeani, S. Tarbouriech, M. Turner, and L. Zaccarian, “A tutorial on trollers,” in Proc. American Control Conference, Philadelphia, PA, Jun. modern anti-windup design,” in Proc. European Control Conf., Budapest, 1998, pp. 104–108. Hungary, Jul. 2009, pp. 306–323. [27] H. Tanizawa and Y. Ohta, “Sliding mode control under state and control [3] J. Rawlings and D. Mayne, Model Predictive Control: Theory and constraints,” in IEEE Conf. on Control Applications, Singapore, Oct. Design. Nob Hill Pub, Llc, 2009. 2007, pp. 1173–1178. [4] V. I. Utkin, Sliding Modes in Optimization and Control Problems. New [28] J. Fu, Q.-X. Wu, and Z.-H. Mao, “Chattering-free smc with unidirec- York: Springer Verlag, 1992. tional auxiliary surfaces for nonlinear system with state constraints,” Int. [5] C. Edwards and S. K. Spurgeon, Sliding Mode Control: Theory and J. Innov. Comput., Inf. Control, vol. 9, no. 12, pp. 4793–4809, 2013. Applications. London, UK: Taylor and Francis, 1998. [29] F. Dinuzzo, “A second order sliding mode controller with polygonal [6] I. Boiko, L. Fridman, A. Pisano, and E. Usai, “Analysis of chattering constraints.” in Proc. 48th IEEE Conf. Decision Control, Shanghai, in systems with second-order sliding modes,” IEEE Trans. Automat. China, Dec. 2009, pp. 6715–6719. Control, vol. 52, no. 11, pp. 2085–2102, Nov. 2007. [30] M. Rubagotti and A. Ferrara, “Second order sliding mode control of a [7] I. Boiko, “Analysis of chattering in sliding mode control systems with perturbed double integrator with state constraints,” in Proc. American continuous boundary layer approximation of discontinuous control,” in Control Conf., Baltimore, MD, Jun. 2010, pp. 985–990. Proc. American Control Conf., San Francisco, CA, USA, Jul. 2011, pp. [31] M. Tanelli and A. Ferrara, “Switched second-order sliding mode control 757–762. with partial information: Theory and application,” Asian J. Control, [8] G. Bartolini, A. Ferrara, and E. Usai, “Output tracking control of vol. 15, no. 1, pp. 20–30, Jan. 2013. uncertain nonlinear second-order systems,” Automatica, vol. 33, no. 12, [32] ——, “Enhancing robustness and performance via switched second order pp. 2203 – 2212, Dec. 1997. sliding mode control,” IEEE Trans. Automat. Control, vol. 58, no. 4, pp. [9] ——, “Chattering avoidance by second-order sliding mode control,” 962–974, Apr. 2013. IEEE Trans. Automat. Control, vol. 43, no. 2, pp. 241–246, Feb. 1998. [33] A. Ferrara, G. P. Incremona, and M. Rubagotti, “Third order sliding [10] G. Bartolini, A. Ferrara, A. Levant, and E. Usai, “On second order mode control with box state constraints,” in Proc. 52th IEEE Conf. sliding mode controllers,” in Variable structure systems, sliding mode Decision Control, Los Angeles, CA, USA, Dec. 2014, pp. 4727–4732. and nonlinear control, ser. Lecture Notes in Control and Information, [34] A. Swikir and V. I. Utkin, “Chattering analysis of conventional and K. Young and Ü. Özgüner, Eds. London, UK: Springer-Verlag, 1999, super twisting sliding mode control algorithm,” in Proc. Int. Workshop pp. 329–350. on Variable Structure Systems, Nanjing, China, Jun. 2016, pp. 98–102. [11] G. Bartolini, A. Ferrara, E. Usai, and V. Utkin, “On multi-input [35] A. Levant, “Chattering analysis,” IEEE Trans. Automat. Control, vol. 55, chattering-free second-order sliding mode control,” IEEE Trans. Au- no. 6, pp. 1380–1389, 2010. tomat. Control, vol. 45, no. 9, pp. 1711–1717, Sep. 2000. [36] I. Boiko and L. Fridman, “Analysis of chattering in continuous sliding- [12] A. Levant, “Higher-order sliding modes, differentiation and output- mode controllers,” IEEE Trans. Automat. Control, vol. 50, no. 9, pp. feedback control,” Int. J. Control, vol. 76, no. 9-10, pp. 924–941, Jan. 1442–1446, Sept 2005. 2003. [37] I. Boiko, “Oscillations and transfer properties of relay servo systems— [13] T. Floquet, J. P. Barbot, and W. Perruquetti, “Higher-order sliding the locus of a perturbed relay system approach,” Automatica, vol. 41, mode stabilization for a class of nonholonomic perturbed systems,” no. 4, pp. 677–683, 2005. Automatica, vol. 39, no. 6, pp. 1077 – 1083, Jun. 2003. [38] H. K. Khalil, Nonlinear Systems, 3rd ed. Prentice Hall, 1996. [14] A. Levant, “Quasi-continuous high-order sliding-mode controllers,” [39] U. Walther, T. T. Georgiou, and A. Tannenbaum, “On the computation of IEEE Trans. Automat. Control, vol. 50, no. 11, pp. 1812 – 1816, 2005. switching surfaces in optimal control: a Gröbner basis approach,” IEEE [15] F. Plestan, A. Glumineau, and S. Laghrouche, “A new algorithm for high- Trans. Autom. Control, vol. 46, no. 4, pp. 534–540, 2001. order sliding mode control,” Int. J. Robust Nonlin. Control, vol. 18, no. [40] A. F. Filippov, Differential Equations with Discontinuous Right-Hand 4-5, pp. 441–453, 2008. Sides, ser. Mathematics and its Applications, F. Arscott, Ed. Springer [16] F. Dinuzzo and A. Ferrara, “Higher order sliding mode controllers with Netherlands, 1988, vol. 18. optimal reaching,” IEEE Trans. Automat. Control, vol. 54, no. 9, pp. [41] K. D. Young, V. I. Utkin, and U. Ozguner, “A control engineer’s guide to 2126 –2136, Sep. 2009. sliding mode control,” IEEE Trans. Control Syst. Technol., vol. 7, no. 3, [17] Y. Shtessel, C. Edwards, L. Fridman, and A. Levant, “Higher-order pp. 328–342, May 1999. sliding mode controllers and differentiators,” in Sliding Mode Control [42] F. Blanchini and S. Miani, Set-theoretic methods in control. Springer and Observation, ser. Control Engineering. Springer New York, 2014, Science & Business Media, 2007. pp. 213–249. [43] M. Grant, S. Boyd, and Y. Ye, “CVX: Matlab software for disciplined [18] A. Levant, “Sliding order and sliding accuracy in sliding mode control,” convex programming,” 2008. Int. J. Control, vol. 58, no. 6, pp. 1247–1263, Dec. 1993. [44] M. Athans and P. L. Falb, Optimal Control. New York: McGraw Hill, 1966. 9

[45] S. S. L. Chang, “Optimal control in bounded phase space,” Automatica, vol. 1, no. 1, pp. 55–67, Jan. 1963.