A Comprehensive Analysis of Chattering in Second Order Sliding Mode Control Systems

A Comprehensive Analysis of Chattering in Second Order Sliding Mode Control Systems Igor Boiko 1 , Leonid Fridman 2 , Alessandro Pisano 3 , and Elio Usai 3 1 University of Calgary, 2500 University Dr. N.W., Calgary, Alberta, Canada i.boiko@ieee.org 2 Department of Control, Engineering Faculty, National Autonomous University of Mexico (UNAM) lfridman@servidor.unam.mx 3 Department of Electrical and Electronic Engineering (DIEE), University of Cagliari, Piazza D’Armi, 09123 Cagliari (Italy) {pisano,eusai}@diee.unica.it 1 Introduction Chattering is the most problematic issue in sliding mode control applications [30], [31], [14], [33], [36]. Among the well known approaches based on smooth approximations of the discontinuities [11],[29] and asymptotic state observers [10, 33], the use of the high order sliding mode control approach can attenuate the chattering phenomenon significantly [15],[21], [2],[4], [5],[3],[22],[27]. There are different approaches to chattering analysis which take into account different causes of it: the presence of fast actuators and sensors [32], [17], [18], [19], [34], [6], time delays and/or hysteresis [35], [32], [33], quantization effects (see for example [23]). The purpose of the present chapter is to present a systematic approach to the chattering analysis in control systems with second-order sliding mode controllers (2-SMC) caused by the presence of fast actuators. We shall follow both the time- domain approach, based on the state-space representation, and the frequency- domain approach. The estimation of the oscillation magnitude in conventional (i.e., first-order) SMC systems with fast actuators and sensors was developed in [32], [18], [19] via the combined use of the singularly-perturbed relay control systems theory and Lyapunov techniques. However, Lyapunov theory is not readily applicable to analyze 2-SMC systems, for which new decomposition techniques are demanded. The Poincar´e maps were successfully used in [24], [13] to study the periodic oscillations in relay control systems. In [17] a decomposition of Poincar´emaps was proposed to analyze chattering in systems with first order sliding modes, which led to Pontryagin-Rodygin [25] like averaging theorems providing sufficient conditions for the existence and stability of fast periodic motions. The describing function (DF) technique [1] offers finding approximate values of the frequency and the amplitude of periodic motions in systems with linear G. Bartolini et al. (Eds.): Modern Sliding Mode Control Theory, LNCIS 375, pp. 23–49, 2008. springerlink.com c  Springer-Verlag Berlin Heidelberg 2008 24 I. Boiko et al. plants driven by the sliding mode controllers [37],[26]. The Tsypkin locus [35] provides an exact solution of the periodic problem, including finding exact values of the amplitude and the frequency of the steady-state oscillation. The above- mentioned frequency-domain methods were developed to analyze relay feedback systems and cannot be used directly for the analysis of 2-SMC systems. In [7],[8] the DF method was adapted to analysis of the Twisting and the Super-twisting 2-SMC algorithms [21]. In [9], a DF based method of parameter adjustment of the generalized sub-optimal 2-SMC algorithm [3],[5] was proposed to ensure the desired frequency and amplitude of the periodic chattering trajectories. In the present Chapter, a systematic approach to analysis of chattering in 2-SMC systems is developed. The presence of parasitic dynamics is considered to be the main cause of chattering, and the corresponding effects are analyzed by means of a few techniques. The treatment is developed by considering the ”Generalized Sub-optimal” (G-SO) algorithm [3],[5]. All the main results can be easily generalized to the Twisting algorithm [21] with minor modifications in the proofs. For a class of nonlinear uncertain systems with nonlinear fast actuators: 1. It is proved that the approximability domain [38] of the 2-SMC G-SO algorithms depends on the actuator time constant μ as O(μ 2 )andO(μ)forthe sliding 2-SMC variable and its derivative respectively. 2. Sufficient conditions of the existence of asymptotically orbitally stable pe- riodic solution are obtained in terms of Poincar´emaps. For linear, possibly linearized, dynamics driven by 2-SMC G-SO algorithms, frequency-domain methods of analysis of the periodic solutions are developed, and, in particular: 3. The describing function method is adapted to perform an approximate analysis of the periodic motions. 4. The Tsypkin’s method is modified for the analysis of the systems driven by 2-SMC G-SO algorithms. This modification allows for finding exact values of the parameters of periodic motions, without requiring for the actuator dynamics to be fast. The chapter is organized as follows: in Section 2, a class of nonlinear systems with nonlinear fast actuators is introduced. In Section 3 we show that the 2-SMC G-SO algorithm with suitably chosen parameters steers the system trajectories in finite time towards an invariant vicinity of the second-order sliding set. We also estimate the amplitude of chattering oscillations as a function of the actua- tor time constant. In Section 4 sufficient conditions of the existence and stability of fast periodic motions in a vicinity of the second-order sliding set are derived via the Poincar´e map approach. In Section 5, frequency-domain approaches to chattering analysis are developed. The describing function method is adapted in subsection 5.1 to carry out analysis of periodic motions in systems with linear plants. In Subsection 5.2, the Tsypkin’s method is modified to obtain the pa- rameters of the periodic motion exactly. Examples illustrating the application of the proposed methodologies are spread over the chapter. The proofs of the Theorems are given in the Appendix. A Comprehensive Analysis of Chattering 25 2 2-SMC Systems with Dynamic Actuators We shall consider a nonlinear single-input system: ˙x = a(x, z 1 )(1) with the state vector x =[x 1 ,x 2 , ,x n ] ∈ X ⊂ R n and the scalar “virtual” control input z 1 ∈ Z 1 ⊂ R. The plant input z 1 is modifiable via the dynamic fast actuator μ ˙z = h(z, u), (2) where z =[z 1 ,z 2 , ,z m ] ∈ Z ⊂ R m is the actuator state vector, u ∈ U ⊂ R is the actuator input and μ ∈ R + is a small positive parameter. Let a : X ×Z 1 → R n and h : Z × U → R m be vector-fields satisfying proper restrictions on their growth and smoothness that will be specified later. Let the control task for system (1)-(2) be the finite-time vanishing of the scalar output variable s 1 (x):X → S 1 ⊂ R (3) which defines the sliding manifold s 1 (x) = 0 assigning desired dynamic properties (e.g. stability) to the constrained sliding-mode dynamics. Define s 2 (x, z 1 )= ∂s 1 (x) ∂x a(x, z 1 ):X ×Z 1 → S 2 (4) and assume that the following conditions hold ∀(x, z 1 ) ∈ X ×Z 1 : ∂ ∂z 1 s 2 (x, z 1 )=0 (5) ∂ ∂z 1  ∂ ∂x [s 2 (x, z 1 )a(x, z 1 )]  =0 (6) Laborious but straightforward computations show that conditions (5)-(6) hold if and only if rank J(x, z 1 )=2 ∀(x, z 1 ) ∈ X ×Z 1 (7) with the matrix J defined as follows J(x, z 1 )=  ∂s 1 ∂x 1 ∂s 1 ∂x 2 ∂s 1 ∂x n 0 ∂s 2 ∂x 1 ∂s 2 ∂x 2 ∂s 2 ∂x n ∂s 2 ∂z 1  (8) By virtue of the Inverse Function Theorem, one can explicitly define a vector w ∈ W ⊂ R n−2 and a diffeomorfic state coordinate change x = Φ(s, w):S × W → X ,withs =[s 1 ,s 2 ] ∈ S, which is one to one at any point where condition (7) holds [12]. Assume that vector w can be selected in such a way that its dynamics does not depend on the plant input variable z 1 , i.e., let the transformed system (1),(2) dynamics in the (w, s) coordinates be defined as follows 26 I. Boiko et al. ˙w = g(w, s), (9) ˙s 1 = s 2 , ˙s 2 = f (w, s, z 1 ), (10) μ ˙z = h(z,u), (11) where g : W × S → R n−2 , f : W × S ×Z 1 → R, h : Z × U → R m are smooth functions of their arguments such that f ∈ C 2 [ ¯ W × ¯ S × ¯ Z 1 ],g∈ C 2 [ ¯ W × ¯ S]and h ∈ C 2 [ ¯ Z × ¯ U], where upper bar means the closure of domain. This means that the “sliding variable” s 1 has a well–defined relative degree r = 2 with respect to the plant input variable z 1 over the whole domain of analysis. We consider here the case when the actuator output z 1 has the full relative degree m, equal to the order of the actuator dynamics, with respect to the discontinuous control u. Remark 3. The special form (9) for the internal dynamics can be always achieved if the original dynamics (1) has affine dependence on z 1 [20]. We are considering in this paper the subclass of non-affine systems (1) for which such a special choice of vector w can be found. 3 Generalized Suboptimal Algorithm: Convergence Conditions Consider system (9)-(11) driven by the “Generalized Suboptimal” 2-SMC algo- rithm [5] u = −Usign(s 1 − βs 1Mi ) (12) where U and β are the constant controller parameters and s 1Mi is the latest “singular point” of s 1 , i.e., the value of s 1 at the most recent time instant t Mi (i =1, 2, ) such that ˙s 1 (t Mi ) = 0. Our analysis is semi-global in the sense that the initial conditions w(0), s(0), z(0)areassumedtobelongtotheknown, arbitrarily large, compact domains W 0 , S 0 ,andZ 0 , respectively. The solutions of the system (9)-(12) are understood in the Filippov sense [16]. Remark 4. Since the relative degree between the sliding output s 1 and the discontinuous control u is m + 2, only sliding modes of order m + 2, occurring onto the following sliding set 1 [21], can take place. s 1 = 0 (13) ˙s 1 = s 2 = 0 (14) ¨s 1 (w, 0,z 1 )=f(w, 0,z 1 ) = 0 (15) (16) s (m) 1 (w, 0,z) = 0 (17) s (m+1) 1 (w, 0,z) = 0 (18) 1 The successive total time derivatives of s 1 must be evaluated along the trajectories of system (9)-(11) in the usual way. A Comprehensive Analysis of Chattering 27 The internal dynamics in the (m + 2)-th order sliding-mode is described by equation 2 ˙w = g(w, 0). (19) Suppose that for all w ∈ W and z ∈ Z there exists a unique isolated value of u = u 0 (z,w) as a solution of equation s (m+2) 1 (w, 0,z,u) = 0 (20) which maintains the system trajectories onto the (m+2)-th order sliding domain (13)-(18). Note that the actuator input u appears explicitly as an argument of (20), but not of (13)-(18), according to the fact that the relative degree of s 1 with respect to u is m +2. Then, the system equilibrium point can be computed as the unique solution w 0 , z 0 , u 0 (w 0 ,z 0 ) of the system of equations (15)-(20). The knowledge of the equilibrium point will be used later in the Chapter to define a local linearization for the system (9)-(11). Assumption 1. The internal dynamics (9) and the actuator dynamics (11) meet the following input-to-state stability properties for some positive constants ξ 1 ,ξ 2 [28] w(t)≤w(0) + ξ 1 sup 0≤τ ≤t s(τ) (21) z(t)≤z(0)+ ξ 2 sup 0≤τ≤t |u(τ)| (22) Assumption 2. There exist positive constants H 0 , H 1 , H 2 G m , G M such that function f is bounded as follows: z 1 ≤ 0: − ˜ H(s, w)+G M z 1 ≤ f(w, s 1 ,s 2 ,z 1 ) ≤ ˜ H(s, w)+G m z 1 z 1 > 0: − ˜ H(s, w)+G m z 1 ≤ f(w, s 1 ,s 2 ,z 1 ) ≤ ˜ H(s, w)+G M z 1 (23) ˜ H(s, w)=H 0 + H 1 s + H 2 w (24) Assumption 3. Consider the actuator dynamics (11) with the constant input u(t)= U, t ≥ t 0 .Then,∀ε ∈ (0, 1) there is γ ∈ [γ m ,γ M ] and N(ε) > 0 such that |z 1 − γU|≤εγU ∀t ≥ t 0 + N(ε)μ (25) Assumption 1 prescribes a linear growth of w(t) and z(t) w.r.t.s(t) and |u|, respectively. Assumption 2 guarantees that the virtual plant control input z 1 , with large enough magnitude, can set the sign of f (see Fig. 1). The knowl- edge of constants ξ 1 ,ξ 2 , H 0 , , G M is mainly a technical requirement. With sufficiently large U,andβ ∈ [0.5, 1) sufficiently close to 1, stability can be in- sured regardless of ξ 1 , , G M . Assumption 3 requires a “non-integrating” stable 2 In this case the (m+2) th order sliding dynamics do not depend on the control, i.e. they do not depend on the definition of solutions in (m+2)-th order sliding mode. 28 I. Boiko et al. actuator dynamics whose settling time in the step response is O(μ). Note that γ and N are considered uncertain. Assumption 3 is always satisfied in the special case of a linear asymptotically stable actuator dynamics. Assume, only temporarily, that there exists a certain constant H such that | ˜ H(s, w)|≤H, then conditions (23)-(24) reduce to the following: z 1 ≤ 0 ⇒−H + G M z 1 ≤ f(w, s 1 ,s 2 ,z 1 ) ≤ H + G m z 1 z 1 > 0 ⇒−H + G m z 1 ≤ f (w, s 1 ,s 2 ,z 1 ) ≤ H + G M z 1 (26) which can be represented graphically as in Fig. 1. The dashed lines limit the “admissible region” for the uncertain function f . H -H H + G M z 1 −H+G M z 1 f z 1 H + G m z 1 −H+G m z 1 Fig. 1. Bounding curves for the function f Consider the following tuning rules: U = ηH (1 −ε)γ m G m ,β∈  1 − q(η − 1) η +1+Δ , 1  , (27) Δ = γ M G M (1 + ε) γ m G m (1 −ε) ,η>1,ε∈ (0, 1),q∈ (0, 1) (28) In the next Theorem it is proven that with sufficiently large H and suf- ficiently small μ, control (12), (27)-(28) actually guarantees that condition | ˜ H(·)|≤H holds, and furthermore, it is also shown that a positively invari- ant μ-neighborhood of the second order sliding set s 1 = s 2 = 0, defined as O μ ≡  (s 1 ,s 2 ):|s 1 |≤ρ 1 μ 2 , |s 2 |≤ρ 2 μ  (29) attracts in finite time the system trajectories. Theorem 1. Consider system (9)-(11), satisfying Assumptions 1-3 and driven by the Generalized Sub-Optimal controller (12), (27)-(28). Then, if H is suffi- ciently large and μ is sufficiently small, the closed loop system trajectories enter in finite time the invariant domain O μ defined by (29), where ρ 1 and ρ 2 are proper constants independent of μ. A Comprehensive Analysis of Chattering 29 Proof. See the appendix. Simulation example. Consider system ˙w = −sin(w)+s 1 + s 2 , ˙s 1 = s 2 , ˙s 2 = s 2 1+s 2 2 + s 1 + w +[2+cos(z 1 + s 2 )]z 1 , μ ˙z 1 = z 2 ,μ˙z 2 = −z 1 − z 2 + u. (30) The initial conditions and the controller parameters are s 1 (0) = 20, s 2 (0) = 5, w(0) = 5, z 1 (0) = z 2 (0) = 0, U = 80, β =0.8. Figure 2 shows the time evolution of s 1 and s 2 when μ =0.001. The amplitude of chattering was evaluated as the maximum of |s 1 | and |s 2 | in the steady state, yielding |s 1 |≤7E-4 and |s 2 |≤0.4. We performed a second test using μ =0.01. The accuracy of s 1 changed to 7E − 2, and the accuracy of s 2 changed to 4, in perfect accordance with (29). Simulations show high-frequency periodic motions for s 1 , s 2 and w. Below those “chattering” trajectories are investigated in further detail. 0 2 4 6 −30 −20 −10 0 10 20 30 Time [sec] s 1 (t) s 2 (t) Fig. 2. The steady-state evolution of s 1 and s 2 4Poincar´e Map Analysis We are going to derive conditions for the existence of stable periodic motions, in the vicinity O μ of the second order sliding set, in terms of the properties of some associated Poincar´e maps. We will also give a constructive procedure to compute the parameters of such periodic chattering motions. Introducing the new variables y 1 = μ −2 s 1 , y 2 = μ −1 s 2 , rewrite system (9)-(11) in the form ˙w = g(w, μ 2 y 1 ,μy 2 ), (31) μ ˙y 1 = y 2 ,μ˙y 2 = f(w, μ 2 y 1 ,μy 2 ,z 1 ), (32) μ ˙z = h(z,u(μ 2 y 1 )) (33) Note that the Generalized Sub-Optimal algorithm (12) is endowed by the homogeneity property u(μ 2 y 1 )=u(y 1 ). Consider the Original System in the Fast Time (OSFT) 30 I. Boiko et al. dw/dτ = μg(w, μ 2 y 1 ,μy 2 ), (34) dy 1 /dτ = y 2 ,dy 2 /dτ = f(w, μ 2 y 1 ,μy 2 ,z 1 ), (35) dz/dτ = h(z,u(y 1 )) (36) and the Fast Subsystem (FS) with the “frozen” slow dynamics (w ∈ W is con- sidered here as a fixed parameter): d¯y 1 /dτ =¯y 2 ,d¯y 2 /dτ = f (w, 0, 0, ¯z 1 ), (37) d¯z/dτ = h(¯z,u(¯y 1 )) (38) Consider the solution of system (37)-(38) with initial condition ¯y + 1 (0,w)=¯y 0 1 , ¯y + 2 (0,w)=0 ¯z + (0,w)=¯z 0 ,f(w, ¯y 0 1 , 0, ¯z 0 1 ) < 0 (39) such that f (w, ¯y 0 1 , 0, ¯z 0 1 ) < 0 for all w ∈ W. Suppose that for all w ∈ W there exists the smallest positive root of equation τ = T (w)forwhich¯y + 2 (T (w),w)=0, f(w, ¯y + 1 (T (w),w), 0, ¯z + 1 (T (w),w)) < 0. Now we can define for all w ∈ W the Poincar´emap: (¯y 0 1 , ¯z 0 ) → Ξ(w, ¯y 0 1 , ¯z 0 )=  ¯y 1 (T (w),w) ¯z(T (w),w)  (40) of the of the domain f(w, y 1 , 0,z 1 ) < 0onthesurfacey 2 = 0 into itself, generated by system FS (37)-(38) (the details of this mapping are described in Appendix B). 4.1 Sufficient Conditions for the Periodic Solution Existence Let us suppose that the FS (37)-(38) has a nondegenerated isolated periodic solution and the following conditions hold: Condition 1. ∀w ∈ W the FS (37)-(38) has an isolated T 0 (w)-periodic solution (¯y 10 (τ,w), ¯y 20 (τ,w), ¯z 0 (τ,w)). (41) Condition 2. ∀w ∈ W the Poincar´emapΞ(w, y 0 1 , z 0 ) has an isolated fixed point (¯y ∗ 1 (w),z ∗ (w)) corresponding to the periodic solution (41). Condition 3. ∀w ∈ W the eigenvalues λ i (w)(i =1, ,m+1)ofthematrix ∂Ξ ∂(y 1 ,z) (w, ¯y ∗ 1 (w), ¯z ∗ (w)) (42) are such that |λ i (w)| =1. Condition 4. The averaged systems dw dt = p(w)= 1 T 0 (w)  T 0 (w) 0 g(w, ¯y 10 (τ,w), ¯y 20 (τ,w), ¯z 0 (τ,w))dτ (43) has an isolated, nondegenerated, equilibrium point w 0 such that A Comprehensive Analysis of Chattering 31 p(w 0 )=0,det     dp dw (w 0 )     =0. (44) The following Theorem is demonstrated: Theorem 2. Under conditions 1 −4, system (31)-(33) has an isolated periodic solution near the cycle (w 0 , ¯y 10 (t/μ, w 0 ), ¯y 20 (t/μ, w 0 ), ¯z 0 (t/μ, w 0 )) (45) with period μ(T 0 (w 0 )+O(μ)). Proof. See the Appendix. 4.2 Sufficient Conditions of Stability of Periodic Solution Let ν j (w 0 )(j =1, , n − 2) be the eigenvalues of the matrix dp dw (w 0 ). Suppose that the periodic solution of FS (37)-(38) is exponentially orbitally stable and the equilibrium point of the averaged equations is exponentially stable, i.e. : Condition 5. |λ i (w)| < 1, (i =1, ,m+1). Condition 6. ν j (w 0 ) are real negative, i.e., ν j (w 0 ) < 0, ∀j =1, , n − 2. Theorem 3. Under conditions 1 − 6, the periodic solution (45) of the system (31)-(33) is orbitally asymptotically stable. Proof. See the Appendix. 4.3 Example of Poincar´e Map Analysis Consider the following linear dynamics ˙w = −w + y 1 ˙y 1 = y 2 , ˙y 2 = z μ ˙z = −z + u, u = −sign  y 1 − 1 2 y 1M  (46) We have shown that analysis of periodic solutions can be performed by referring to the decomposition into fast and slow subsystem dynamics. The FS dynamics dy 1 /dτ = y 2 ,dy 2 /dτ = z, dz/dτ = −z + u (47) generates the following Poincar´emapΞ + (y 1 ,z)=(Ξ + 1 (y 1 ,z),Ξ + 2 (y 1 ,z)) of the domain z<0onthesurfacey 2 = 0 into the domain z>0ofthesamesurface (see the Appendix for the detailed derivation): 32 I. Boiko et al. Ξ + 1 (y 1 ,z)= 1 2 y 1 +  (z + 1)(1 − e −T + sw ) − T + sw  T + p + +  (z +1)e −T + sw − 2  (T + p + e −T + p − 1) + 1 2 T + 2 p Ξ + 2 (y 1 ,z)=1+((z +1)e −T + sw − 2)e −T + p (48) where T + sw = T + sw (y 1 ,z)andT + p = T + p (y 1 ,z) are the smallest positive roots of the following equations: 1 2 y 1 +(z +1)(e −T + sw + T + sw − 1) − 1 2 T + sw 2 = 0 (49) (z + 1)(1 −e −T + sw )+  (z +1)e −T + sw − 2  (1 −e −T + p )+T + p − T + sw = 0 (50) Taking into account the symmetry of dynamics (47) with respect to origin (0, 0, 0) we can skip the computation of the map Ξ − (y 1 ,z) and rewrite condition for the periodicity of system (47) trajectory in the form: Ξ + 1 (y 0 1p ,z 0 p )=−y 0 1p ,Ξ + 2 (y 0 1p ,z 0 p )=−z 0 p (51) The fixed points are y 0 1p ≈ 3.95,z 0 p ≈−0.96, (52) and the switching times are T + sw ≈ 2.01, T + p ≈ 3.93, T + 0 = T + sw + T + p ≈ 5.94. (53) The Frechet derivatives entering the Jacobian matrix are given by J = ∂Ξ ∂(y 1 ,z) =  ∂Ξ 1 ∂y 1 | (y 0 1p ,z 0 p ) ∂Ξ 1 ∂z | (y 0 1p ,z 0 p ) ∂Ξ 2 ∂y 1 | (y 0 1p ,z 0 p ) ∂Ξ 2 ∂z | (y 0 1p ,z 0 p )  =  −0.4894 1.5273 0.0095 −0.0147  (54) The eigenvalues of matrix J are eig(J)=[−0.5182, 0.0141], i.e. they are both lying within the unit circle of the complex plane, which implies that the periodic solution of the fast subsystem (47) is orbitally asymptotically stable. The averaged equations for the internal dynamics has the form ˙w = −w.Now from Theorems 1-3 it follows that: i) system (46) has an orbitally asymptotically stable periodic solution lying in the O μ boundary layer (29) of the second-order sliding set y 1 = y 2 = 0. ii) in the steady state the internal dynamics w variable features a O(μ) deviation from the equilibrium point w = 0 of the averaged solution. It is also expected from Theorem 2 that the period of oscillation is O(μ). The period and amplitude of the periodic solutions of (46) can be easily inferred from (52) and (53) via proper μ-dependent scaling. The above results have been checked by means of computer simulations. The initial conditions are w(0) = y 1 (0) = y 2 (0) = z(0) = 1. The value μ =0.1was used in the first test. It is expected, on the basis of the previous considerations, that y 1 exhibits a steady oscillation with amplitude μ 2 y 0 1p ≈ 0.0395 and period 2μT + p ≈ 1.18s. The plots in Fig. 3 highlight the convergence to the periodic solution starting from initial conditions outside from the attracting O μ domain. [...]... vicinity of the sliding surface is estimated • Sufficient conditions of the existence and orbital asymptotic stability of the fast periodic motions are obtained in terms of the properties of corresponding Poincar´ maps e A Comprehensive Analysis of Chattering 41 For linear plants and linear actuators: • A methodology of approximate analysis of the amplitude and the frequency of chattering via application... Fridman, L.: Analysis of Chattering in Continuous Sliding- Mode Controllers IEEE Trans Aut Contr 50, 1442–1446 (2005) 9 Boiko, I., Fridman, L., Iriarte, R., Pisano, A. , Usai, E.: Parameter tuning of second- order sliding mode controllers for linear plants with dynamic actuators Automatica 42, 833–839 (2006) 10 Bondarev, A. G., Bondarev, S .A. , Kostylyeva, N.Y., Utkin, V.I.: Sliding Modes in Systems with Asymptotic... out in the state-space and frequency domains for linear and nonlinear plants It is shown that the system motions always converge to the Oμ domain (29) of approximation with respect to the fast actuator’s time constant μ, and, under some conditions, a stable attracting limit cycle fully contained in this domain exists For nonlinear plants and nonlinear actuators: • The amplitude of chattering in a small... (2001) 5 Bartolini, G., Pisano, A. , Punt, E., Usai, E.: A survey of applications of second order sliding mode control to mechanical systems Int J Contr 76, 875–892 (2003) 6 Boiko, I.: Analysis of sliding modes in the frequency domain Int J Contr 78, 969–981 (2005) 7 Boiko, I., Fridman, L., Castellanos, M.I.: Analysis of Second Order Sliding Mode Algorithms in the Frequency Domain IEEE Trans Aut Contr... UNAM, grant no 107006-2, and by Italian MURST Project “Novel control systems in high–speed railways” References 1 Atherton, D.P.: Nonlinear Control Engineering - Describing Function Analysis and Design Workingam Beks, UK (1975) 2 Bartolini, G., Ferrara, A. , Usai, E.: Chattering avoidance by second order sliding mode control IEEE Trans Aut Contr 43, 241–246 (1998) 3 Bartolini, G., Ferrara, A. , Levant,... right-hand sides Kluwer Academic, Dordrecht (1988) 17 Fridman, L.: An averaging approach to chattering IEEE Trans Aut Contr 46, 1260–1264 (2001) 18 Fridman, L.: Singularly Perturbed Analysis of Chattering in Relay Control Systems IEEE Trans Aut Contr 47, 2079–2084 (2002) 19 Fridman, L.: Chattering analysis in sliding mode systems with inertial sensors Int J Contr 76, 906–912 (2002) 20 Isidori, A. : Nonlinear... Structure Systems Analysis Nauka, Moscow (1974) (in Russian) 38 Zolezzi, T.: A variational approach to second- order approximability of sliding mode control system Optimization 53, 641–654 (2004) A Comprehensive Analysis of Chattering 43 Appendices Proof of Theorem 1 Step A There exists a sequence of time instants tMj , j ≥ 1 such that s2 (tMj ) = 0 Let t0 = 0 be the initial time instant and assume, without... Levant, A. , Usai, E.: On Second Order Sliding Mode Controllers In: Young, K.D., Ozguner, U (eds.) Variable Structure Systems, Sliding Mode and Nonlinear Control Lecture Notes in Control and Information Sciences, vol 247 Springer, New York (1999) 4 Bartolini, G., Ferrara, A. , Pisano, A. , Usai, E.: On the convergence properties of a 2 -sliding control algorithm for nonlinear uncertain systems Int J Contr... Tsypkin Analysis of the elastic arm example - 1/N(Ay) ω=400 2.5 Imaginary Axis -0.1 -4 Real Axis Φ(ω) Real: - 0.0017 Imag: 0.0022 Chattering frequency: 442 85 rad/sec Chattering Amplitude: 0.0028 2 ω=500 1.5 1 0.5 0 -3 -2.5 -2 -1.5 Real Axis -1 -0.5 0 -3 x 10 Fig 7 The elastic arm example Upper plot: DF analysis Lower plot: Modified Tsypkin analysis 40 I Boiko et al The sliding variable s1 in the steady... Therefore, the use and adaptation of frequency methods for the chattering analysis in control systems with the fast actuators driven by 2-SMC G-SO algorithms seems expedient However, this approach applies to linear dynamics For this reason in this section we assume that the plant plus actuator dynamics are either linear or linearized in the conventional sense in a small vicinity of the domain (29) Subsection . Comprehensive Analysis of Chattering 41 For linear plants and linear actuators: • A methodology of approximate analysis of the amplitude and the frequency of chattering via application of the DF. conditions, a stable attracting limit cycle fully contained in this domain exists. For nonlinear plants and nonlinear actuators: • The amplitude of chattering in a small vicinity of the sliding surface. analysis of chattering in 2-SMC systems is developed. The presence of parasitic dynamics is considered to be the main cause of chattering, and the corresponding effects are analyzed by means of