Discontinuous Homogeneous Control

Discontinuous Homogeneous Control Arie Levant and Lela Alelishvili Applied Mathematics Department, Tel-Aviv University {levant,lela}@post.tau.ac.il 1 Introduction Control under heavy uncertainty conditions remains among the main topics of the control theory. The sliding-mode control [50, 53, 10] is one of the main tools in the field. This approach is based on exactly keeping a properly chosen constraint by means of control switching of high (theoretically infinite) frequency. Although very robust and accurate, the approach has two basic restrictions. The direct implementation of standard sliding modes requires the relative degree of the constraint to be 1, i.e. control has to appear explicitly already in the first total time derivative of the constraint function. Also, high-frequency control switching may cause the so-called chattering effect [14, 15, 16, 33]. High-gain control with saturation is used to overcome the chattering effect approximating the sign-function in a boundary layer around the switching manifold [45], also the sliding-sector method [17] was proposed to control disturbed linear time-invariant systems. The sliding-mode order approach [24, 28] considered in this chapter is capable to treat successfully both the chattering and the relative-degree restrictions preserving the sliding-mode features and improving its practical accuracy. High order sliding mode (HOSM) is actually a movement on a discontinuity set of a dynamic system understood in Filippov’s sense [12]. The sliding order characterizes the dynamics smoothness degree in the vicinity of the mode. Let the task be to provide for keeping a constraint given by equality of a smooth function σ to zero. Then the sliding order is a number of continuous total time derivatives of σ (including the zero one) in the vicinity of the sliding mode. Thus, the rth order sliding mode is determined by the equalities σ =˙σ =¨σ = = σ (r−1) =0. (1) forming an r-dimensional condition on the state of the dynamic system. The words ”rth order sliding” are often abridged to ”r-sliding”. The rth derivative σ (r) is mostly supposed to be discontinuous or non-existent, in which case the sliding order is naturally called strict. Standard sliding mode is used with the relative degree 1 of the constraint function σ.Inthatcase˙σ = h(t, x)+g(t, x)u,whereg(t, x) = 0. Then the sliding G. Bartolini et al. (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp. 71–95, 2008. springerlink.com c  Springer-Verlag Berlin Heidelberg 2008 72 A. Levant and L. Alelishvili mode can be locally obtained by means of the simplest control u = −k signσ,ifg is positive and k>0 is sufficiently large. The general case situation is very much alike. The sliding order of the standard sliding mode is 1, for ˙σ is discontinuous. Asymptotically stable HOSMs appear in many systems with traditional sliding-mode control and are deliberately introduced in systems with dynamical sliding modes [43, 36]. While finite-time-convergent arbitrary-order sliding-mode controllers are still theoretically studied [28, 13, 23, 34], 2-sliding controllers are already successfully implemented for the solution of real problems [5, 7, 11, 26, 44, 39, 19, 46, 47, 48, 49]. Construction of r-sliding controllers, r ≥ 3, is more difficult due to the high dimension of the problem. Thus, only a few families of such controllers [27, 28, 23, 30, 29, 35, 41] are known. The finite-time-convergent controllers (r- sliding controllers) [28, 23, 30] require actually only the knowledge of the system relative degree r. The finite-time-stable exact tracking is lost with alternative controllers developed in [49] and [3] for r =3andr = 2 respectively. Many new results on HOSM controllers are based on their homogeneity properties which proved to be very useful. It is natural to construct new finite-time convergent HOSM controllers basing on the homogeneity-based approach [29]. On the other hand, recently published non-homogeneous controllers [23, 41] lack the highest accuracy characterizing other HOSM controllers. The aim of this paper is to sum- marize a number of recent results in one frame of homogeneous discontinuous control. Most known HOSM controllers possess specific homogeneity properties. The corresponding homogeneity of r-sliding controllers is called the rth-order sliding homogeneity [29]. The homogeneity makes the convergence proofs of the HOSM controllers standard and provides for the highest possible asymptotic accuracy in the presence of measurement noises, delays and discrete measurements [29]. In particular, the controllers provide in finite time for keeping σ ≡ 0, if the measurements of σ are exact, and for σ proportional to the maximal measurement error otherwise. With discrete measurements and τ being the sampling interval, the accuracy σ = O(τ r ) is assured. This asymptotic accuracy is proved to be the best possible with discontinuous control [24]. An output-feedback controller with the same asymptotical accuracy is obtained, when a recently developed robust exact homogeneous differentiator of the order r −1 [25, 28] is included as a standard part of the homogeneous r-sliding controller. HOSM controllers of a new type were recently developed, called quasi- continuous. Such controllers are feedback functions of σ,˙σ, , σ (r−1) , continuous everywhere except the very manifold (1) of the r-sliding mode. The mode σ ≡ 0 is established after a finite-time transient. In the presence of errors in evaluation of the output σ and its derivatives, a motion in some vicinity of (1) takes place. Control practically turns out to be a continuous function of time, for in reality trajectories never hit the manifold (1) with r>1. The proposed controllers [30, 35] are also designed r-sliding homogeneous, which extends all the results of this chapter to their application. They demonstrate significantly less chattering and seem to be generally superior compared with usual r-sliding controllers. Discontinuous Homogeneous Control 73 Effective implementation of the differentiator requires sufficiently high sampling frequency, which is not always available. In the case of the homogeneous controllers successive derivatives can be replaced by means of divided finite differences producing an output-feedback controller possesing the robustness and the asymptotic accuracies [24] of the original controller. With rare measurements such controllers can definitely be a good choice. The r-sliding control is a discontinuous function of the tracking error σ and of its real-time-calculated successive derivatives σ,˙σ, , σ (r−1) .Thecontrol switches with infinite frequency when in the sliding mode. The resulting chattering effect is successfully treated, provided the control derivative is used as a new control input [24, 4, 27, 33], hence artificially increasing the relative degree. The (r + 1)-sliding mode is to be established in that case. Unfortunately, such a simplistic procedure is in general only locally valid, because of the possible interaction between the control and its derivative. The problem can be solved by means of the integral homogeneous sliding mode [34]. The idea is to design in advance a transient trajectory to the (r +1)-sliding mode. In such a way any interaction between the control and its derivative is excluded, since the control becomes some predetermined function of time. The recent results [33] show the robustness of homogeneous sliding mode controllers with respect to the presence of unaccounted for fast stable dynamics of actuators and sensors. Simulation demonstrates the practical applicability of the results. 2 Homogeneous Inclusions A differential inclusion ˙x ∈ F(x) is further called a Filippov differential inclusion, if the vector set F (x) is non-empty, closed, convex, locally bounded and upper-semicontinuous [12]. The latter condition means that the maximal dis- tance of the points of F(x)fromthesetF(y) vanishes when x → y. Solutions are defined as absolutely-continuous functions of time satisfying the inclusion almost everywhere. Such solutions always exist and have most of the well-known standard properties except uniqueness [12]. It is said that a differential equation ˙x = f (x) with a locally-bounded Lebesgue-measurable right-hand side is understood in the Filippov sense, if its solutions are defined as solutions of a specially constructed Filippov differential inclusion ˙x ∈ F(x). In the most usual case, when f is continuous almost everywhere, the procedure is to take F(x) being the convex closure of the set of all possible limit values of f at a given point x, obtained when its continuity point y tends to x. In the general case approximate-continuity [42] points y are taken (one of the equivalent definitions by Filippov [12]). Values of f on any set of the measure 0 do not influence the Filippov solutions. Note that with continuous f the standard definition is obtained. The presence of small delays and measere- ment errors results in the solutions, which converge to the Filippov solutions, when these imperfections vanish [12]. 74 A. Levant and L. Alelishvili A function f : R n → R (respectively a vector-set field F (x) ∈ R n , x ∈ R n , or a vector field f : R n → R n ) is called homogeneous of the degree q ∈ R with the dilation d κ :(x 1 ,x 2 , , x n ) → (κ m 1 x 1 ,κ m 2 x 2 , , κ m n x n )[2],wherem 1 , , m n are some positive numbers (weights, homogeneity degrees), if for any κ>0 the identity f (x)=κ −q f(d κ x) holds (respectively F (x)=κ −q d −1 κ F (d κ x), or f(x)=κ −q d −1 κ f(d κ x). The non-zero homogeneity degree q of a vector field can always be scaled to ±1 by an appropriate proportional change of the weights m 1 , , m n . Note that the homogeneity of a vector field f(x) (a vector-set field F(x)) can equivalently be defined as the invariance of the differential equation ˙x = f(x) (differential inclusion ˙x ∈ F (x)) with respect to the combined time-coordinate transformation G κ :(t, x) → (κ p t, d κ x), p = −q,wherep might naturally be considered as the weight of t. Examples. Let the weights of x 1 , x 2 be 3 and 2 respectively. Then the function x 2 1 + x 3 2 is homogeneous of the weight (degree) 6: (κx 1 ) 2 +(κx 2 ) 3 = κ 6 (x 2 1 + x 3 2 ). The differential inequality |˙x 1 | +˙x 2 2 ≤|x 1 | 2/3 + |x 2 | corresponds to the homogeneous differential inclusion (˙x 1 , ˙x 2 ) ∈  (z 1 ,z 2 ):|z 1 |+ z 2 2 ≤|x 1 | 2/3 + |x 2 |  of the degree -1. Also the system of differential equations and the inclusion  ˙x 1 = x 2 ˙x 2 = −x 1 1/3 −|x 2 | 1/2 sign x 2 ,  ˙x 1 = x 2 ˙x 2 ∈ x 1 1/3 [−1, 1] are of the degree -1, the system being finite-time stable. 1 ◦ .Adifferentialinclusion ˙x ∈ F (x)(equation ˙x = f(x)) is further called globally uniformly finite-time stable at 0, if it is Lyapunov stable and for any R>0thereexistsT>0 such that any trajectory starting within the disk ||x|| <Rstabilizes at zero in the time T . 2 ◦ .Adifferentialinclusion ˙x ∈ F (x)(equation ˙x = f(x)) is further called globally uniformly asymptotically stable at 0, if it is Lyapunov stable and for any R>0, ε>0existsT>0 such that any trajectory starting within the disk ||x|| <Renters the disk ||x|| <εin the time T to stay there forever. AsetD is called dilation retractable if d κ D ⊂ D for any 0 ≤ κ<1. 3 ◦ . A homogeneous differential inclusion ˙x ∈ F (x)(equation ˙x = f(x)) is further called contractive if there are 2 compact sets D 1 , D 2 and T>0 such that D 2 lies in the interior of D 1 and contains the origin; D 1 is dilation-retractable; and all trajectories starting at the time 0 within D 1 are localized in D 2 at the time moment T . Theorem 1. [29] Let ˙x ∈ F (x) be a homogeneous Filippov inclusion with a negative homogeneous degree −p,thenproperties1 ◦ ,2 ◦ and 3 ◦ are equivalent and the maximal settling time is a continuous homogeneous function of the initial conditions of the degree p. As an important consequence obtain that, due to the obvious robustness of the property 3 ◦ , the finite-time stability of a homogeneous differential inclusion with Discontinuous Homogeneous Control 75 negative homogeneous degree is insensitive with respect to small homogeneous perturbations of the right-hand side. For the case of continuous differential equations the equivalence of 1 ◦ and 2 ◦ was proved in [8]. The equivalence of 1 ◦ and 2 ◦ was also independently proved for the Filippov discontinuous differential equations in [40]. Let ˙x ∈ F (x) be a homogeneous Filippov differential inclusion. Consider the case of “noisy measurements” of x i with the noise magnitude ε i τ m i ˙x ∈ F(x 1 + ε 1 τ m 1 [−1, 1], , x n + ε n τ m n [−1, 1]),τ>0. Taking successively the convex hull at each point x and the closure of the right- hand-side graph, obtain some new Filippov differential inclusion ˙x ∈ F τ (x). Theorem 2. [29] Let ˙x ∈ F (x) be a globally uniformly finite-time-stable homogeneous Filippov inclusion with the homogeneity weights m 1 , , m n and the degree −p<0,andletτ>0. Suppose that a continuous function x(t) be defined for any t ≥−τ p and satisfy some initial conditions x(t)=ξ(t), t ∈ [−τ p , 0]. Then if x(t) is a solution of the disturbed inclusion ˙x(t) ∈ F τ (x(t +[−τ p , 0])), 0 <t<∞ ,theinequalities|x i |≤γ i τ m i are established in finite time with some positive constants γ i independent of τ and ξ. Note that Theorem 2 covers the cases of retarded or discrete noisy measurements of all or some of the coordinates and any mixed cases. In particular, infinitely extendible solutions certainly exist in the case of noisy discrete measurements of some variables or in the constant time-delay case. The Theorem conditions do not impose any restrictions on the real noises and delays to be observed in reality. Indeed, in any practical case one has some concrete noises or delay magnitudes. Then the choice of parameters ε i and τ is not unique. Indeed, one may always increase τ or each one of ε i keeping the same fixed real system parameters. Mark also that this Theorem provides for the asymptotic accuracy of all known finite-time stable continuous homogeneous differential equations with negative degrees [2]. 3 “Black-Box” Control Problem and Its Sliding-Mode Solution Let a Single-Input-Single-Output (SISO) system to be controlled have the form ˙x = a(t, x)+b(t, x)u, x ∈ R n ,u∈ R, (2) σ :(t, x) −→ σ(t, x) ∈ R, where σ is the measured output of the system, u is the control. Smooth functions a, b, σ are assumed unknown, the dimension n can also be uncertain. The task is to make σ vanish in finite time by means of a possibly discontinuous feedback and to keep σ ≡ 0. The solutions are understood in the Filippov sense, and system trajectories are supposed to be infinitely extendible in time for any bounded 76 A. Levant and L. Alelishvili Lebesgue-measurable input. In real applications σ can be a deviation of a system output from some command signal available in real time, or from any auxiliary constraint chosen by the system designer. Although it is formally not needed, the weakly minimum-phase property is usually required in practice. It is assumed that the relative degree r [18] of the system is constant and known.That means [18] that the equation σ (r) = h(t, x)+g(t, x)u, g(t, x) =0, (3) holds, with some uncertain h(t, x)=σ (r) | u=0 , g(t, x)= ∂ ∂u σ (r) . The uncertainty prevents immediate reduction of (2) to the standard form (3). Suppose that the inequalities |σ (r) | u=0 |≤C, 0 <K m ≤ ∂ ∂u σ (r) ≤ K M . (4) hold for some K m ,K M ,C > 0. These conditions are satisfied at least locally for any smooth system (2) having a well-defined relative degree at a given point with σ =˙σ = = σ (r−1) = 0. Assume that (4) holds globally. Then (3), (4) imply the differential inclusion σ (r) ∈ [−C, C]+[K m ,K M ]u. (5) The problem is solved in two steps. First a bounded feedback control u = Ψ(σ, ˙σ, , σ (r−1) ), (6) is constructed, such that all trajectories of (5), (6) converge in finite time to the origin of the r-sliding phase space σ,˙σ, , σ (r−1) . At the next step the lacking derivatives are real-time evaluated, producing an output-feedback controller. Here and further the right-hand sides of all differential inclusions are enlarged at discontinuity points of control producing Filippov inclusions. If it is not mentioned explicitly, the minimal enlargement is taken. Here the Filippov set of limit values of (6) is substituted for u in (5) (see Section 2). The function Ψ is assumed to be a Borel-measurable function, which provides for the Lebesgue measurability of composite functions to be obtained in the presence of Lebesgue- measurable noises. Actually all functions used in the sliding-mode control theory are Borel measurable. Indeed, any superposition of Borel-measurable functions is Borel-measurable; the sign function and all continuous functions are Borel measurable. Note that the function Ψ has to be discontinuous at the origin. Otherwise u is close to the constant Ψ(0, 0, , 0) in a small vicinity of the origin, and, taking c ∈ [−C, C]andk ∈ [K m ,K M ]sothatc + kΨ(0, 0, , 0) =0,achieve that (6) cannot stabilize the dynamic system σ (r) = c + ku.Thus,σ (r) is to be discontinuous along the trajectories of the original system (2), (6), which means that the r-sliding mode σ ≡ 0 is to be established. All known r-sliding controllers [4, 7, 23, 28, 29, 30, 35, 41] may be considered as controllers for (5) steering σ,˙σ, , σ (r−1) to 0 in finite time. Inclusion (5) does not “remember” the original system (2). Thus, such controllers are obviously robust with respect to any perturbations preserving the system relative degree and (4). Discontinuous Homogeneous Control 77 4 Homogeneous Sliding-Mode Control Suppose that feedback (6) imparts homogeneity properties to the closed-loop inclusion (5), (6). Due to the term [−C, C], the right-hand side of (5) can only have the homogeneity degree 0 with C>0. Indeed, with a positive degree the right hand side of (5), (6) approaches zero near the origin, which is not possible with C>0. With a negative degree it is not bounded near the origin, which contradicts the local boundedness of Ψ . Thus, the homogeneity degree of the right-hand side of (5) is to be 0, and the homogeneity degree of σ (r−1) is to be opposite to the degree of the whole system. Scaling the system homogeneity degree to -1, achieve that the homogeneity weights of t, σ,˙σ, , σ (r−1) are 1, r, r −1, , 1 respectively. This homogeneity is further called the r-sliding homogeneity. Denote σ =(σ, ˙σ, , σ (r−1) ). Trajec- tories of (5), (6) are preserved by the combined time-coordinate transformation G κ :(t, σ) → (κt, d κ σ), where d κ σ =(κ r σ, κ r−1 ˙σ, , κσ (r−1) )(7) with any κ>0. Respectively the corresponding controller (6) is called r-sliding homogeneous if Ψ(κ r σ, κ r−1 ˙σ, , κσ (r−1) )=Ψ(σ, ˙σ, , σ (r−1) ). (8) Such a homogeneous controller is inevitably discontinuous at the origin (0, , 0), unless Ψ is a constant function. It is also uniformly bounded, since it is locally bounded and takes on all its values in any vicinity of the origin. Almost all known r-sliding controllers, r ≥ 2, are r-sliding homogeneous. Also the sub-optimal 2-sliding controller [4, 7] is homogeneous in the sense that (7) preserves trajectories. The following recursively built r-sliding homogeneous controllers u = −αΨ r−1,r (σ, ˙σ, , σ (r−1) )(9) solve the general problem stated in Section 3. The parameters of the controllers can be chosen in advance for each relative degree. Only the magnitude parameter α is to be adjusted for any fixed system, most conveniently by computer simulation, avoiding redundantly large estimations of the bounds C, K m , K M . Obviously, α is to be negative with ∂ ∂u σ (r) < 0. Nested-sliding-mode (nested-SM) controller [27, 28] This controller is based on a complicated switching motion, which can be qualita- tively described by a sequence of nested sliding modes. Let p>r, i =1, , r −1, β 1 , , β r−1 be some positive numbers. It is defined by the procedure N 1,r = |σ| (r−1)/r ,N i,r =(|σ| p/r + |˙σ| p/(r−1) + + |σ (i−1) | p/(r−i+1) ) (r−i)/p , Ψ 0,r =signσ, ϕ i,r = σ (i) + β i N i,r Ψ i−1,r ,Ψ i,r =signϕ i,r . A list of such controllers is presented in [27, 28]. Obviously, N i,r and Ψ i,r are r-sliding homogeneous functions of the weights r − i and 0 respectively, N i,r is 78 A. Levant and L. Alelishvili also a positive-definite function of σ, ˙σ, , σ (i−1) . The idea of the convergence proof is that (9) provides in finite time for the approximate keeping of ϕ r−1,r =0 (the exact 1-sliding mode is impossible, since Ψ r−2,r is discontinuous). The latter equation provides in finite time for the approximate keeping of ϕ r−2,r =0,etc. The last approximate equality is ϕ 1,r =˙σ + β 1 |σ| (r−1)/r signσ = 0. Obviously, there is an attracting vicinity of the origin (σ, ˙σ, , σ (r−1) ) = 0. Thus, the closed loop inclusion (5), (9) is contractive, and, therefore, finite-time stable according to Theorem 1. Quasi-continuous controller [30] In order to reduce the chattering, a controller is designed, which is continuous everywhere except the r-sliding set σ =˙σ = = σ (r−1) = 0. Such a controller is naturally called quasi-continuous, for in practice, in the presence of measurement noises, singular perturbations and switching delays, the motion takes place in some vicinity of the r-sliding set and the control actually is a continuous function of time.Leti =1, , r − 1. Denote ϕ 0,r = σ, N 0,r = |σ|,Ψ 0,r = ϕ 0,r /N 0,r =signσ, ϕ i,r = σ (i) + β i N (r−i)/(r−i+1) i−1,r Ψ i−1,r ,N i,r = |σ (i) |+ β i N (r−i)/(r−i+1) i−1,r , Ψ i,r = ϕ i,r /N i,r , where β 1 , , β r−1 are positive numbers. The following proposition is easily proved by induction. Proposition 1. Let i =0, , r − 1. N i,r be positive definite, i.e. N i,r =0iff σ =˙σ = = σ (i) =0.Theinequality|Ψ i,r |≤1 holds whenever N i,r > 0.The function Ψ i,r (σ, ˙σ, , σ (i) ) is continuous everywhere (i.e. it can be redefined by continuity) except the point σ =˙σ = = σ (i) =0. Also here the idea of the convergence proof is that the control successively causes the aproximate keeping of the equations ϕ r−1,r = 0, , ϕ 0,r =0. Theorem 3. Provided β 1 , , β r−1 , α>0 are chosen sufficiently large in the list order, both above designs result in the r-sliding homogeneous controller (9) providing for the finite-time stability of (5), (9) with any sufficiently large α. The finite-time stable r-sliding mode σ ≡ 0 is established in the system (2), (9). Thus, one does not need to know the exact values of K m , K M , C to apply the controllers in practice. Each proper choice of β 1 , , β r−1 determines a controller family applicable to all systems (2) of the relative degree r,providedα is large enough. Here and further the maximal possible transient time is a locally bounded function of initial conditions (Section 2). Following are quasi-continuous controllers with r ≤ 4 and simulation-tested β i . Note that the same parameters β i canbeusedforthenestedSMcontrollers. Discontinuous Homogeneous Control 79 1. u=−α sign σ, 2. u=−α(˙σ + |σ| 1/2 sign σ)/(|˙σ|+ |σ| 1/2 ), 3. u=−α[¨σ +2(|˙σ| + |σ| 2/3 ) −1/2 (˙σ + |σ| 2/3 signσ)]/[|¨σ| +2(|˙σ| + |σ| 2/3 ) 1/2 ], 4. ϕ 3,4 = σ +3[¨σ +(|˙σ| +0.5|σ| 3/4 ) −1/3 (˙σ +0.5|σ| 3/4 sign σ)] [|¨σ| +(|˙σ| +0.5|σ| 3/4 ] −1/2 , N 3,4 = | σ | +3[|¨σ| +(|˙σ| +0.5|σ| 3/4 ) 2/3 ] 1/2 , u = −αϕ 3,4 /N 3,4 . While the control is a continuous function of time everywhere except the r- sliding set, it may have infinite derivatives when certain surfaces are crossed. Another quasi-continuous controller family is constructed in [29], generalized controllers are introduced in [35] containing arbitrary functional parameters. The following Theorems are standard consequences [29] of the r-sliding homogeneity of controller (9) and Theorems 1, 2. Theorem 4. Let the control value be updated at the moments t i ,witht i+1 −t i = τ = const > 0, t ∈ [t i ,t i+1 ) (the discrete sampling case). Then controller (9) provides in finite time for keeping the inequalities |σ| <μ 0 τ r , |˙σ| <μ 1 τ r−1 , , |σ (r−1) | <μ r−1 τ with some positive constants μ 0 , μ 1 , , μ r−1 . That is the best possible accuracy attainable with discontinuous σ (r) [24]. The following result shows robustness of controller (9) with respect to measurement errors. Theorem 5. Let σ(i) be measured with accuracy η i ε (r−i)/r for some fixed η i > 0, i =1, , r − 1. Then with some positive constants μ i the inequalities |σ (i) |≤ μ i ε (r−i)/r , i =0, , r −1, are established in finite time for any ε>0. The convergence time may be reduced changing coefficients β j . In particular, one can substitute λ −j σ (j) for σ (j) and λ r α for α , λ>0, causing convergence time to be diminished approximately by λ times. A set of parameters β j satisfies the above Theorems, if the differential equations ϕ 1,r = 0, , ϕ r−1,r = 0 are finite- time stable [35]. That indicates the recursive way of choosing the parameters. Note that these equations do not contain uncertainties. 5 Output-Feedback Sliding-Mode Control Any r-sliding homogeneous controller can be complemented by an (r −1)th order differentiator [1, 5, 22, 25, 28, 21, 52] producing an output-feedback controller.Due to the demonstrated robustness of the described controllers with respect to the measurement errors, the resulting output feedback controller will localy provide for approximate real [24] r-sliding mode. In order to preserve the demonstrated exactness, finite-time stability and the corresponding asymptotic properties, the natural way is to calculate ˙σ, , σ (r−1) in real time by means of a robust finite-time convergent exact homogeneous differentiator [28]. Its application is possible due to the boundedness of σ (r) provided by the boundedness of the feedback function Ψ in (6). Following is the short description of the differentiator. 80 A. Levant and L. Alelishvili Arbitrary-order real-time exact robust differentiation Suppose that it is known that the input signal is compounded of a smooth signal f 0 (t) to be differentiated and a noise being a bounded Lebesgue-measurable function of time. Both signals are unknown and only their sum is available. It is proved that if the base signal f 0 (t)has(r-1)th derivative with Lipschitz’s constant L>0, the best possible kth order differentiation accuracy is d k L k/r ε (r−k)/r , where d k > 1 may be estimated [20, 25]. Moreover, it is proved that such a robust exact differentiator really exists [25, 28]. The aim is to find real-time robust estimations of f 0 (t),f 0 (t), , f (p) 0 (t), being exact in the absence of measurement noise and continuously depending on the noise magnitude. The differentiator is recursively constructed and has the form ˙z 0 = ν 0 ,ν 0 = −λ 0 L 1 p+1 |z 0 − f (t)| p p+1 sign(z 0 − f (t)) + z 1 , ˙z i = ν i ,ν i = −λ i L 1 p−i+1 |z i − ν i−1 | p−i p−i+1 sign(z i − ν i−1 )+z i+1 , ˙z p = −λ p L sign(z p − ν p−1 ) (10) The coefficients are easily found by simulation, since the pth order differentiator requires only one parameter to be chosen, if the lower-order differentiators are already built. A set of such parameters is listed further. The proof is based on the introduction of new variables σ i = z i −f (i) 0 (t). Taking f (p+1) 0 (t) ∈ [−L, L] obtain a differential inclusion. Assigning the weight p −i to σ i = z i −f (i) 0 (t)obtain a homogeneous differential inclusion of the degree -1. With properly chosen parameters the inclusion is finite-time stable. Following Theorem 2 the following accuracy is obtained |z i − f (i) 0 (t)|≤μ i ε (p−i+1) (p+1) ,i=0, , p; |v i − f (i+1) 0 (t)|≤ν i ε (p−i) (p+1) ,i=0, , p −1. Exact differentiation is provided with ε = 0. Using recursive high-order differentiators the noise propagation is obviously counteracted as compared with the cascade implementation of first-order differentiators. Consider the discrete- sampling case, when z 0 (t j )−f(t j ) is substituted for z 0 −f(t), with t j ≤ t<t j+1 , t j+1 − t j = τ>0. Then the following accuracy is obtained |z i − f (i) 0 (t)|≤μ i τ p−i+1 ,i=0, , p; |v i − f (i+1) 0 (t)|≤ν i τ p−i ,i=0, , p − 1. Implementation of the differentiator with an r-sliding homogeneous controller The proposed output-feedback dynamical feedback has the form u = Ψ(z 0 ,z 1 , , z r−1 ), (11) [...]... respect to small homogeneous perturbations, if the homogeneity degree is negative The corresponding r-sliding homogeneity notion is introduced, simplifying and standardizing design and convergence proofs of r-sliding mode controllers The asymptotic accuracy is calculated Discontinuous Homogeneous Control 93 A new type of high-order sliding mode controllers is introduced, which features control continuous... state variable u and the new control v = u ˙ Implementation of r-sliding controller when the relative degree is less than r Introducing successive time derivatives u, u, , u(r−k−1) as new auxiliary vari˙ ables and u(r−k) as a new control, achieve different modifications of each r-sliding controller intended to control systems with relative degrees k = 1, 2, , r The resulting control is (r − k − 1)-smooth... differences in homogeneous discontinuous control IEEE Trans Aut Contr 52, 1208–1217 (2007) Discontinuous Homogeneous Control 95 33 Levant, A.: Chattering Analysis In: Proc ECC 2007, Kos, GR (2007) 34 Levant, A., Alelishvili, L.: Integral high-order sliding modes IEEE Trans Aut Contr 52, 1278–1282 (2007) 35 Levant, A., Pavlov, Y.: Generalized Homogeneous Quasi-Continuous Controllers Int J Rob Nonlin Contr... the actual control θ is about 16◦ and the vibration frequency is about 0.5s−1 , which is quite feasible Mark that τ = 0.2s is close to the typical human reaction time Discontinuous Homogeneous Control 89 Fig 2 Constant sampling step τ = 10−4 , noise magnitude ε = 0 Fig 3 Constant sampling step τ = 0.2s, noise magnitude ε = 0.1m Raising the relative degree Consider a variable-length pendulum control problem... supplies estimates of σ , σ Here L > sup|ϕ(4) | is required), y is an additional ¨ auxiliary variable approximating , y(0) = s2 (0) = s3 (0) = 0 Discontinuous Homogeneous Control 91 Denote zi = si − ξ (i) , i = 0, 1, 2, 3 The 4-sliding homogeneous quasicontinuous controller [30] Ψ4 = −{z3 + 3[|z2 | + (|z1 | + 0.5|z0 |3/4 )−1/3 |z1 + 0.5|z0 |3/4 sign z0 |] [|z2 | + (|z1 | + 0.5|z0 |3/4 )2/3 ]−1/2 }/{|z3|.. .Discontinuous Homogeneous Control 81 z0 = v0 , v0 = −λ0 L1/r |z0 − σ|(r−1)/r sign(z0 − σ) + z1 , ˙ z1 = v1 , v1 = −λ1 L1/(r−1) |z1 − v0 |(r−2)/(r−1) sign(z1 − v0 ) + z2 , ˙ zr−2 = vr−2 , vr−2 = −λr−2 L1/2 |zr−2 − vr−3 |1/2 sign(zr−2 − vr−3 ) + zr−1 , ˙ zr−1 = −λr−1 L sign(zr−1 − vr−2 ), ˙ (12) where Ψ is an r-sliding homogeneous controller, L ≥ C + sup |Ψ |KM , and... interaction will be automatically removed, since the control and its first k − r − 1 derivatives are predefined by the trajectory Choose the function ϕ now satisfying the conditions ϕ(t, σ) = (t − T (σ))r (c0 (σ) + c1 (σ)t + + ck−1 (σ)tk−1 ), (26) i ∂ ϕ(0, σ) = σ (i) , i = 0, , k − 1 ∂ti (27) Discontinuous Homogeneous Control 87 The function T (σ) is k-sliding homogeneous, positive-definite and continuous function... weight 1 Theorem 10 [34] Let the initial conditions t0 , x(t0 ), u(t0 ), , u(k−r−1) (t0 ) belong to some compact set in Rn+k−r+1 , and controller (6) be one of the finitetime convergent k-sliding homogeneous controllers from [24, 27, 29, 30, 31, 35] Then with sufficiently large α controller (20) establishes the k-sliding mode σ ≡ 0 with the transient time T (σ(t0 )) The equality σ(t, x(t)) = ϕ(t− t0 , σ(t0... of a predefined controller, and the system performance is robust with respect to the presence of bounded sampling noises of any nature and unaccounted-for fast stable actuators and sensors Global/local validity of (4) corresponds to the global/local applicability of the controller Stability features and asymptotic accuracy are studied of homogeneous differential inclusions with negative homogeneous degree... 3-sliding homogeneous controllers from the above lists may be applied here The quasi-continuous 3sliding controller is applied here with α = 1 The 3-sliding homogeneity implies the identity Ψ (σ i , (σ i − σ i−1 )/τ , (σ i − 2σi−1 + σ i−2 )/τ 2 )) = = Ψ (σ i τ 6 , (σ i − σ i−1 )τ 3 , (σ i − 2σ i−1 + σ i−2 )) (31) which allows to avoid division by small numbers The resulting outputfeedback controller . finite-time stability of a homogeneous differential inclusion with Discontinuous Homogeneous Control 75 negative homogeneous degree is insensitive with respect to small homogeneous perturbations. any perturbations preserving the system relative degree and (4). Discontinuous Homogeneous Control 77 4 Homogeneous Sliding-Mode Control Suppose that feedback (6) imparts homogeneity properties. Respectively the corresponding controller (6) is called r-sliding homogeneous if Ψ(κ r σ, κ r−1 ˙σ, , κσ (r−1) )=Ψ(σ, ˙σ, , σ (r−1) ). (8) Such a homogeneous controller is inevitably discontinuous at the

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