Some Remarks on the Boundedness and Convergence Properties of Smooth Sliding Mode Controllers

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1 International Journal of Automation an Computing 6(2, May 2009, DOI: /s z Some Remarks on the Bouneness an Convergence Properties of Smooth Sliing Moe Controllers Wallace Moreira Bessa Feeral University of Rio Grane o Norte, Department of Mechanical Engineering, Natal, Brazil Abstract: Conventional sliing moe controllers are base on the assumption of switching control, but a well-known rawback of such controllers is the chattering phenomenon. To overcome the unesirable chattering effects, the iscontinuity in the control law can be smoothe out in a thin bounary layer neighboring the switching surface. In this paper, rigorous proofs of the bouneness an convergence properties of smooth sliing moe controllers are presente. This result corrects flawe conclusions previously reache in the literature. An illustrative example is also presente in orer to confirm the convergence of the tracking error vector to the efine boune region. Keywors: Convergence analysis, Lyapunov methos, nonlinear control, sliing moe. 1 Introuction The sliing moe control theory was conceive an evelope in the former Soviet Union by Emelyanov an Kostyleva [1], Filippov [2], Itkis [3], Utkin [4], an others. However, a known rawback of conventional sliing moe controllers is the chattering phenomenon ue to the iscontinuous term in the control law. In orer to avoi the unesire effects of the control chattering, Slotine [57] propose the aoption of a thin bounary layer neighboring the switching surface, by replacing the sign function by a saturation function. This substitution can minimize or, when esire, even completely eliminate chattering, but turns perfect tracking into a tracking with guarantee precision problem, which actually means that a steay-state error will always remain. This paper presents a convergence analysis of smooth sliing moe controllers. The finite-time convergence of the tracking error vector to the bounary layer is hanle using Lyapunov s irect metho. It is also analytically proven that, once in bounary layer, the tracking error vector exponentially converges to a boune region. This result corrects a minor flaw in Slotine s work, by showing that the bouns of the error vector are ifferent from the bouns provie in [5 7]. Although the bouns propose by Slotine are incorrect, they are until now wiely evoke to establish the bouneness an convergence properties of many control schemes [814]. A simulation example is also presente in orer to illustrate the convergence of the tracking error vector to the efine boune region. 2 Problem statement Consier a class of n-th orer nonlinear systems x (n = f(x + b(xu (1 where u is the control input, the scalar variable x is the output of interest, x (n is the n-th erivative of x with respect Manuscript receive July 24, 2008; revise October 6, 2008 This work was supporte by FAPERJ State of Rio e Janeiro Research Founation (No. E-26/ / aresses: wmbessa@ufrnet.br; wmbessa@ams.org to time t [0,, x = [x, ẋ,, x (n1 ] is the system state vector, an f, b : R n R are both nonlinear functions. In respect of the ynamic system presente in (1, the following assumptions will be mae. Assumption 1. The function f is unknown but boune by a known function of x, i.e., ˆf(x f(x F (x, where ˆf is an estimate of f. Assumption 2. The input gain b(x is unknown but positive an boune, i.e., 0 < b min b(x b max. The propose control problem is to ensure that, even in the presence of parametric uncertainties an unmoele ynamics, the state vector x will follow a esire trajectory x = [x, ẋ,, x (n1 ] in the state space. Regaring the evelopment of the control law, the following assumptions shoul also be mae. Assumption 3. The state vector x is available. Assumption 4. The esire trajectory x is once ifferentiable in time. Furthermore, every element of vector x, as well as x (n, is available an with known bouns. Now, let x = x x be efine as the tracking error in the variable x, an x = x x = [ x, x,, x (n1 ] as the tracking error vector. Consier a sliing surface S efine in the state space by the equation s( x = 0, with the function s : R n R satisfying ( n1 s( x = t + λ x conveniently rewritten as s( x = c T x (2 where c = [c n1λ n1,, c 1λ, c 0], an c i states for binomial coefficients, i.e., ( n 1 (n 1! c i = =, i = 0, 1,, n 1 (3 i (n i 1! i!

2 W. M. Bessa / Some Remarks on the Bouneness an Convergence Properties of Smooth Sliing Moe Controllers 155 which makes c n1λ n1 + +c 1λ+c 0 a Hurwitz polynomial. From (3, it can be easily verifie that c 0 = 1, for n 1. Thus, for notational convenience, the time erivative of s will be written in the following form: ṡ = c T x = x (n + c T x (4 where c = [0, c n1λ n1,, c 1λ]. Now, let the problem of controlling the uncertain nonlinear system (1 be treate via the classical sliing moe approach, efining a control law compose of an equivalent control û = ˆb 1 ( ˆf + x (n c T x an a iscontinuous term Ksgn(s: u = ˆb 1 ( ˆf + x (n c T x Ksgn(s (5 where ˆb = b maxb min is an estimate of b, K is a positive gain, an sgn( is efine as 1, if s < 0 sgn(s = 0, if s = 0 1, if s > 0. Base on Assumptions 1 an 2 an consiering that β 1 ˆb/b β, where β = b max/b min, the gain K shoul be chosen accoring to K βˆb 1 (η + F + (β 1 ˆb1 ( ˆf + x (n c T x (6 where η is a strictly positive constant relate to the reaching time. Therefore, it can be easily verifie that (5 is sufficient to impose the sliing conition 1 2 t s2 η s which, in fact, ensures the finite-time convergence of the tracking error vector to the sliing surface S an, consequently, its exponential stability. However, the presence of a iscontinuous term in the control law leas to the well-known chattering effect. To avoi these unesirable high-frequency oscillations of the controlle variable, Slotine [57] propose the aoption of a thin bounary layer, S, in the neighborhoo of the switching surface: S = { x R n s( x } (7 where is a strictly positive constant that represents the bounary layer thickness. The bounary layer is achieve by replacing the sign function by a continuous interpolation insie S. It shoul be emphasize that this smooth approximation, which will be calle here ϕ(s,, must behave exactly like the sign function outsie the bounary layer. There are several options to smooth out the ieal relay but the most common choices are the saturation function: sat ( s = sgn(s, s, an the hyperbolic tangent function tanh(s/. if if s 1 s (8 < 1 In this way, the smooth sliing moe control law can be state as follows: u = ˆb ( 1 ˆf + x (n c T x Kϕ(s,. (9 3 Convergence analysis The attractiveness an invariant properties of the bounary layer are establishe in the following theorem. Theorem 1. Consier the uncertain nonlinear system (1 an Assumptions 1 4. Then, the smooth sliing moe controller efine by (6 an (9 ensures the finite-time convergence of the tracking error vector to the bounary layer S, efine accoring to (7. Proof. Let a positive-efinite Lyapunov function caniate V be efine as V (t = 1 2 s2 where s is a measure of the istance of the current error to the bounary layer, an can be compute as follows: ( s s = s sat. (10 Noting that s = 0 in the bounary layer, one has V (t = 0 insie S. From (8 an (10, it can be easily verifie that ṡ = ṡ outsie the bounary layer an, in this case, V becomes V (t =s ṡ = s ṡ = (x (n x (n + c T xs = ( f + bu x (n + c T x s. Consiering that outsie the bounary layer, the control law (9 takes the following form: u = ˆb ( 1 ˆf + x (n c T x Ksgn(s an noting that f = ˆf ( ˆf f, we have V (t = [ f + bˆb 1 ( ˆf + x (n c T x bksgn(s x (n + c T x ] s = [ ( ˆf f bˆb 1 ( ˆf + x (n c T x+ ( ˆf + x (n c T x + bksgn(s ] s. Therefore, consiering Assumptions 1 an 2 an efining K accoring to (6, V becomes V (t η s which implies V (t V (0 an that s is boune. From the efinition of s an s, respectively (2 an (10, it can be verifie that x is boune. Thus, Assumption 4 an (4 imply that ṡ is also boune. The finite-time convergence of the tracking error vector to the bounary layer can be shown by recalling that V (t = 1 2 t s2 = s ṡ η s. Then, iviing by s an integrating both sies between 0 an t gives t 0 s s ṡ τ t 0 η τ

3 156 International Journal of Automation an Computing 6(2, May 2009 s (t s (0 η t. In this way, consiering t reach as the time require to hit S an noting that s (t reach = 0, one has t reach s (0 η which guarantees the convergence of the tracking error vector to the bounary layer in a time interval smaller than s (0 /η. Therefore, the value of the positive constant η can be properly chosen in orer to keep the reaching time, t reach, as short as possible. Fig. 1 shows that the time evolution of s is boune by the straight line s (t = s (0 η t. Fig. 1 Time evolution of s Finally, the proof of the bouneness of the tracking error vector relies on Theorem 2. Theorem 2. Let the bounary layer S be efine accoring to (7. Then, once insie S, the tracking error vector will exponentially converge to an n-imensional box efine accoring to x (i ζ iλ in+1, i = 0, 1,, n1}, with ζ i satisfying 1, for i = 0 ζ i = 1 + i1 (11 ζj, for i = 1, 2,, n 1. j=0 ( i j Proof. From the efinition of s, (2, an consiering that s(x may be rewritten as s(x, we have c 0 x (n1 + c 1λ x ( + + c n1λ n1 x. (12 Multiplying (12 by e λt yiels e λt n1 t n1 ( xeλt e λt (13 Integrating (13 between 0 an t gives λ eλt + λ t ( xeλt t ( xeλt λ eλt λ. (14 (14 can be conveniently rewritten as ( λ eλt t ( xeλt + λ t ( xeλt ( λ eλt + t ( xeλt + λ. (15 The same reasoning can be repeately applie until the (n 1-th integral of (13 is reache: ( λ n1 eλt t ( xeλt + t λ (n 2! ( x(0 + xe λt λ n1 λ n1 eλt + ( t ( xeλt + ( t λ (n 2! + + x(0 + λ n1 (16 Furthermore, iviing (16 by e λt, it can be easily verifie that, for t, x(t (17 λn1 λ n1 Consiering the (n 2-th integral of (13 ( λ eλt t ( xeλt + t n3 λ (n 3! ( x(0 + λ t ( xeλt λ eλt + ( t ( xeλt + t n3 λ (n 3! + + ( x(0 + (18 λ an noting that ( xe λt /t = xe λt + xλe λt, by imposing the bouns (17 to (18 an iviing it again by e λt for t, we have 2 λ x(t 2. (19 λ Now, applying the bouns (17 an (19 to the (n3-th integral of (13 an iviing it once again by e λt for t, we have 6 λ x(t 6. (20 n3 λn3 The same proceure can be successively repeate until the bouns for x (n1 are achieve: ( ( n ζ i x (n1 i i=0 ( ( n ζ i (21 i where the coefficients ζ i (i = 0, 1,, n 2 are relate to the previously obtaine bouns of each x (i an can be summarize as in (11. In this way, by inspection of the integrals of (13, as well as (17, (19 (21 an the other omitte bouns, the tracking error will be confine within the limits x (i ζ iλ in+1, i = 0, 1,, n 1, where ζ i is efine by (11. Remark 1. Theorem 2 corrects a minor error in [5 7]. Slotine propose that the bouns for x (i coul be summarize as x (i 2 i λ in+1, i = 0, 1,, n 1. Although both results lea to the same bouns for x an x, they i=0

4 W. M. Bessa / Some Remarks on the Bouneness an Convergence Properties of Smooth Sliing Moe Controllers 157 start to iffer from each other when the orer of the erivative is higher than one, i > 1. For example, accoring to the metho of Slotine, the bouns for the secon erivative woul be x 4λ 3n not x 6λ 3n, as emonstrate in Theorem 2. Remark 2. It must be note that the n-imensional box efine accoring to the aforementione bouns is not completely insie the bounary layer. Consiering the attractiveness an invariant properties of S emonstrate in Theorem 1, the region of convergence can be state as the intersection of the bounary layer an the n-imensional box efine in Theorem 2. Therefore, the tracking error vector will exponentially converge to a close region Φ = {x R n s( x an x (i ζ iλ in+1, i = 0, 1,, n 1}, with ζ i efine by (11. Fig. 2 illustrates the region of convergence Φ, efine accoring to Remark 2, for a secon-orer system (n = 2. Fig. 3 Phase portrait of the unforce Van er Pol oscillator Fig. 2 Bouns of x (i for a secon-orer system 4 Illustrative example In orer to confirm the convergence of the tracking error vector to the boune region efine in Remark 2, consier a controlle Van er Pol oscillator ẍ µ(1 x 2 ẋ + x = bυ (22 The simulation stuy was performe with an implementation in C, with sampling rates of 500 Hz for control system an 1 khz for the Van er Pol oscillator, an the ifferential equations were numerically solve using the fourth-orer Runge-Kutta metho. The chosen parameters for the Van er Pol oscillator were b = 1 an µ = 1. Regaring the controller esign, to ratify its robustness against both structure an unstructure uncertainties, an uncertainty of ±20% over the value of b was taken into account, i.e., b min = 0.8 an b max = 1.2, an the ea-ban was treate as moeling imprecision, i.e., not consiere in controller esign. In this way, for a secon-orer system with state vector x = [x, ẋ] an s = x+λ x, a smooth sliing moe controller can be chosen as follows: u = ˆb 1 [µ(1 x 2 ẋ + x + ẍ λ x] Ksat ( s. The following parameters were aopte for the controller: β = 1.22, ˆb = 0.98, η = 0.1, λ = 0.6, = 0.1, an F = Consiering that the initial state an initial esire state are not equal, x(0 = [2.0, 0.4], Figs. 4 6 show the corresponing results for the tracking of x = [sin t, cos t]. As observe in Fig. 6, the tracking error vector is riven to the propose region of convergence an remains insie Φ as t, even in the presence of moeling imprecisions. with a ea-ban in the control input efine accoring to u + 0.4, if u 0.4 υ = 0, if 0.4 < u < 0.4 u 0.4, if u 0.4. (23 The unforce Van er Pol oscillator, i.e., by consiering u = 0, exhibits a limit cycle. The control objective is to let the state vector x = [x, ẋ] track a esire trajectory x = [sin t, cos t] situate insie the limit cycle. Fig. 3 shows the phase portrait of the unforce Van er Pol oscillator with the limit cycle, two convergent orbits an the esire trajectory. Fig. 4 Tracking performance

5 158 International Journal of Automation an Computing 6(2, May 2009 Fig. 6 Fig. 5 5 Conclusions Control action Phase portrait of the tracking In this paper, a convergence analysis of smooth sliing moe controllers was presente. The attractiveness an invariant properties of the bounary layer as well as the exponential convergence of the tracking error vector to a boune region were analytically proven. This last result correcte flawe conclusions previously reache in the literature. Numerical simulations with a Van er Pol oscillator confirm the convergence of the tracking error vector to the propose region of convergence. Acknowlegements The author woul like to thank Prof. Roberto Barrêto an Prof. Gilberto Corrêa for their insightful comments an suggestions. References [1] S. V. Emelyanov, N. E. Kostyleva. Design of Variable Structure Systems with Discontinuous Switching Function. Engineering Cybernetics, vol. 21, no. 1, pp , [2] A. F. Filippov. Differential Equations with Discontinuous Right-han Sies. American Mathematical Society Translations, vol. 42, no. 2, pp , [3] U. Itkis. Control Systems of Variable Structure, Wiley, New York, USA, [4] V. I. Utkin. Variable Structure Systems with Sliing Moes. IEEE Transactions on Automatic Control, vol. 22, no. 2, pp , [5] J. J. E. Slotine. Sliing Controller Design for Non-linear Systems. International Journal of Control, vol. 40, no. 2, pp , [6] J. J. E. Slotine, J. A. Coetsee. Aaptive Sliing Controller Synthesis for Non-linear Systems. International Journal of Control, vol. 43, no. 6, pp , [7] J. J. E. Slotine, W. Li. Applie Nonlinear Control, Prentice Hall, New Jersey, USA, pp , [8] T. Sharaf-Elin, M. W. Dunnigan, J. E. Fletcher, B. W. Williams. Nonlinear Robust Control of a Vector-controlle Synchronous Reluctance Machine. IEEE Transactions on Power Electronics, vol. 14, no. 6, pp , [9] D. Q. Zhang, S. K. Pana. Chattering-free an Fastresponse Sliing Moe Controller. IEE Proceeings of Control Theory an Applications, vol. 146, no. 2, pp , [10] C. Y. Liang, J. P. Su. A New Approach to the Design of a Fuzzy Sliing Moe Controller. Fuzzy Sets an Systems, vol. 139, no. 1, pp , [11] X. S. Wang, C. Y. Su, H. Hong. Robust Aaptive Control of a Class of Nonlinear Systems with Unknow Dea-zone. Automatica, vol. 40, no. 3, pp , [12] H. M. Chen, J. C. Renn, J. P. Su. Sliing Moe Control with Varying Bounary Layers for an Electro-hyraulic Position Servo System. The International Journal of Avance Manufacturing Technology, vol. 26, no. 1 2, pp , [13] Q. Wang, C. Y. Su. Robust Aaptive Control of a Class of Nonlinear Systems Incluing Actuator Hysteresis with Prant Ishlinskii Presentations. Automatica, vol. 42, no. 5, pp , [14] T. P. Zhang, Y. Yi. Aaptive Fuzzy Control for a Class of MIMO Nonlinear Systems with Unknown Dea-zones. Acta Automatica Sinica, vol. 33, no. 1, pp , Wallace Moreira Bessa receive the B. Sc. egree at the State University of Rio e Janeiro, Brazil, in 1997, the M. Sc. egree at the Military Institute of Engineering, Rio e Janeiro, in 2000, an the Ph. D. egree at the Feeral University of Rio e Janeiro, Brazil, in 2005, all in mechanical engineering. Part of his octoral research was evelope at the Institute for Mechanical an Ocean Engineering of the Hamburg University of Technology, from 2002 to He is currently an associate professor at the Feeral University of Rio Grane o Norte. He is a member of the International Physics an Control Society (IPACS, American Mathematical Society (AMS, Brazilian Mathematical Society (SBM, Brazilian Society for Applie an Computational Mathematics (SBMAC, an Brazilian Association for Mechanical Engineering an Sciences (ABCM. His research interests inclue control theory, fuzzy logic, nonlinear ynamics, an robotics.