VARIABLE GAIN SUPER-TWISTING CONTROL USING ONLY OUTPUT FEEDBACK

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1 Natal RN 5 a 8 de outubro de 5 VARIABLE GAIN SUPER-TWISTING CONTROL USING ONLY OUTPUT FEEDBACK Paulo Victor N. M. Vidal Timon Asch Keijock Eduardo V. L. Nunes Liu Hsu Dept. of Electrical Eng./COPPE Federal University of Rio de Janeiro Rio de Janeiro Brazil. s: paulovictornmv@poli.ufrj.br timonak9@poli.ufrj.br eduardo@coep.ufrj.br liu@coep.ufrj.br Abstract In this paper an output feedback version of the Variable Gain Super-Twisting Sliding Mode Control is proposed for uncertain plants with relative degree one. This extension is achieved by using first order approximation filters to obtain an upper bound for the norm of unmeasured states. A Lyapunov approach is used to show that the proposed control scheme ensures global stability with exact tracking in finite time. As the control signal is continuous the proposed scheme alleviates the chattering phenomenon. The theoretical results are illustrated by simulations. Keywords higher-order sliding modes global exact tracking output feedback uncertain systems Resumo Neste artigo uma versão via realimentação de saída do controle por modos deslizantes Variable Gain Super-Twisting é proposta para plantas incertas com grau relativo unitário. Essa extensão é alcançada usando aproximação por filtros de primeira ordem para obter um majorante para a norma dos estados não medidos. Uma abordagem baseada em Lyapunov é utilizada para mostrar que o esquema de controle proposto assegura estabilidade global com rastreamento exato em tempo finito. Como o sinal de controle é contínuo o esquema proposto atenua o fenômeno de chattering. Os resultados teóricos são ilustrados por meio de simulações. Palavras-chave sistemas incertos modos deslizantes de ordem superior rastreamento global e exato realimentação de saída Introduction The sliding mode control (SMC) is a robust nonlinear control method that is known to be very effective under uncertainty conditions including unmodeled dynamics parameter variation and external disturbances (Utkin 978; Edwards and Spurgeon 998). Its main advantage is the socalled invariance property i.e. once the sliding mode has been achieved the system becomes insensitive to parameter uncertainties and some disturbance classes. However the discontinuous nature of the control law results in an undesirable chattering effect which consists of a highfrequency control switching that can lead to system instability. The sliding modes concept was generalized in Levant (993) with the introduction of the higherorder sliding modes (HOSM) which preserve the main advantages of classical SMC and provide even greater accuracy. In this case the sliding surface is defined based not only on the sliding variable but also on its derivatives. This new approach results in a control law that is smooth and chattering free (Fridman and Levant ). In particular we highlight the super-twisting algorithm (STA) based on second-order sliding modes which was developed in order to avoid the chattering effect in relative degree one systems without requiring information about the sliding variable derivative (Levant 998). Recently in Moreno and Osorio (8) Dávila et al. () and Gonzalez et al. () a generalization called Variable Gain Super-Twisting Algorithm (VGSTA) was proposed for the STA. Such generalization consists in as its name suggests allow the controller gains to vary with time improving the robustness and ensuring the rejection of a wider class of disturbances. The conventional Super Twisting Algorithm is based on the homogeneity principle and hence its convergence can be slow when the initial errors are large. Another important modification of the VGSTA is the introduction of non-homogeneous higher-order terms which provides for faster convergence with large errors. In some sufficiently small vicinity of the sliding surface the nonhomogeneous higher order terms can be neglected in comparison with the standard homogeneous ones and thus the asymptotic accuracy and finitetime convergence are preserved. Furthermore such terms also allow the rejection of disturbances growing together with the sliding variable which is a necessary feature in control of systems whose linear part is not exactly known. Unfortunately the VGSTA assumes that the state vector is available for control purposes restricting its applicability. In this paper we propose an output feedback version of the VGSTA for minimum phase uncertain plants with relative degree one. To this end we use first order approximation filters (FOAF) (Hsu et al. 3) to generate a norm bound for the unmeasured state so that we can obtain upper bounds for the disturbances that are necessary for 999

2 the VGSTA implementation. Simulation results show the robustness and finite-time convergence of the proposed control scheme. Preliminaries In what follows all κ s denote positive constants. π(t) denotes an exponentially decaying function i.e. π(t) Ke λt t K possibly depends on the system initial conditions and λ is a (generic) positive constant. stands for the Euclidean norm for vectors or the induced norm for matrices. Here Filippov s definition for the solution of discontinuous differential equations is assumed (Filippov 964). For precise meaning of mixed time domain (state-space) and Laplace transform domain (operator) representations the following notations are adopted. The output y of a linear timeinvariant system with transfer function H(s) and input u is denoted H(s)u. Then the following notation is adopted y(t) = H(s)u(t). Pure convolution operations h(t) u(t) h(t) is the impulse response of H(s) is denoted by H(s) u. The stability margin of a p m rational transfer function matrix G(s) = C(sI A) B is given by γ := min { Re(p j )} j p j is a pole of G(s). The system with transfer matrix G(s) is BIBO stable if and only if γ >. The stability margin of a real matrix A is given by λ := min j { Re(λ j )} λ j is an eingenvalue of A. If λ > then A is Hurwitz. 3 Problem Statement The following uncertain linear time-invariant SISO plant is considered ẋ = Ax + B[u + d(x t) y = Cx () x R n u R y R and d(x t) R is a state dependent uncertain nonlinear disturbance. The following assumptions are made: (A) The transfer function G(s) = C(sI A) B is minimum phase that is all its zeros are in the open left-half plane (Khalil p. 55). (A) The system is controllable and observable. (A3) G(s) has relative degree that is lim s sg(s) = k p = CB. (A4) The sign of the high-frequency gain (sign (k p )) is known and supposed positive for simplicity and without loss of generality since if k p < one can invert the controller sign and replace k p by k p. From assumptions (A) and (A3) and the fact that CB = k p it can be showed that applying the following linear transformation: [ η B = x B y C B = the system () can be transformed into the following normal form η = A η + A y () ẏ = A η + a y + k p (u + d) (3) η R n is an unmeasured state and the zero dynamics given by η = A η is stable since G(s) is minimum phase from assumption (A). The main objective is to determine a control law u such that y tracks in some finite time a reference signal y m. Then the tracking error is given by e = y y m and its dynamics can be described by ė = k p u + f(η e t) f(η e t) = A η +a e+k p d(η e t)+a y m ẏ m which can be considered as a disturbance rewritten as g (ηet) g (ηt) {}}{{}}{ f(η e t) = [f(η e t) f(η t) + f(η t) = g (η e t) + g (η t) g (η e t) = if e =. that Thus it follows g (η e t) = a e + k p {d(η e t) d(η t)} (4) and g (η t) = A η + k p d(η t) + a y m ẏ m. (5) For the purpose of developing a control scheme based on the super-twisting algorithm some upper bounds for the elements of the disturbance and its derivative are needed. In particular as η is an unmeasured state its necessary to be available a norm bound for it. Such norm bound can be obtained through a first order approximation filter (FOAF) (Hsu et al. 3).

3 4 First Order Approximation Filter FOAF In order to obtain a norm bound for η we make the following assumption: (A5) A lower bound λ > for the stability margin of A is known. Lemma Consider system () under assumption (A5). Let γ be the stability margin of the transfer function matrix H(s) := (si A ) A and γ := γ δ δ > is an arbitrary positive constant. Then c c > such that the impulse response h(t) of the system () satisfies h(t) c exp ( γt) t and the following inequalities hold: h(t) y(t) c exp ( γt) y(t) η(t) for all t. c exp ( γt) y(t) + +c exp [ (λ δ)t η() Proof: see Hsu et al. (3 Lemma ). Considering the result obtained in Lemma it is possible to show that η(t) ˆη(t) + π η (t) ˆη(t) := c y(t) (6) s + γ with c γ > being appropriate constants that can be computed by the optimization methods described in Cunha et al. (8). The exponentially decaying term π η accounts for the system initial conditions. 5 Output Feedback Variable Gains Super-Twisting Algorithm Consider the Variable Gain Super-Twisting Algorithm (VGSTA) proposed in Dávila et al. () and Gonzalez et al. () given by u = k (t ˆη e)φ (e) t k (t ˆη e)φ (e)dt (7) φ (e) = e sign (e) + k3 e k 3 > φ (e) = sign (e) + 3 k 3 e sign (e) + k 3e being this definition a variable gains generalization of the conventional super-twisting algorithm which is obtained by making k and k constants and k 3 =. In this paper in order to propose an output feedback version of the VGSTA considering the presence of the FOAF (6) in the closed-loop system we make the following assumption regarding the disturbance which is slightly different from the one made in Dávila et al. () and Gonzalez et al. (): (A6) The disturbance f(η e t) satisfies the following inequalities: g (η e t) { (t ˆη e) + π (t) } φ (e) d dt g (η t) { (t ˆη e) + π (t) } φ (e) (8) (t ˆη e) and (t ˆη e) are known continuous functions. Considering (7) the closed-loop system can be rewritten as η = A η + A (e + y m ) ė = k p k (t ˆη e)φ (e) + z + g (η e t) ż = k p k (t ˆη e)φ (e) + d dt g (η t) (9) and the main result of the paper is formulated as follows. Theorem Consider the system (6)(9) under assumptions (A) (A6). Suppose that the disturbance satisfies (8) for some known continuous functions (t ˆη e) (t ˆη e). Then for any initial conditions [ η T e T z the sliding surface ė = e = will be reached in finite time if the variable gains are selected as k (t ˆη e) δ + k p β { 4ǫ [4ǫ + ǫ + +ǫ( + k p β) + ǫ + (k p β + 4ǫ )} k (t ˆη e) = β + ǫk (t ˆη e) () β > ǫ > and δ > are arbitrary positive constants. Proof: see Appendix A. 6 Simulation Results Consider the following uncertain LTI plant: [ ẋ = x + [u + d(x t) y = [ x with transfer function G(s) = C(sI A) B = s + (s ) and high-frequency gain k p = CB =. The statedependent disturbance is given by d(x t) = Cx +.5 sin (8πt) = y +.5 sin (8πt).

4 Applying the linear transformation [ η B = x = x y C the system can be transformed into the following normal form: η = η + y ẏ = 4η + 3y + u + d(x t) and a norm bound for the unmeasured state η can be obtained by means of the FOAF ˆη(t) = s +.5 y(t). The objective is to ensure that the plant output y tracks in some finite time a reference signal given by y m = sin (πt). In this case considering equations (4) and (5) we have g (η e t) = 3e + e + y m y m g (η t) = 4η + y m +.5 sin (8πt) + 3y m ẏ m and δ = 3 ǫ = β = 35 and k i =.5 is a lower bound for k p. Further we set k 3 =.5. In this simulation we chose the plant initial conditions as x() = [.5.5 T while the initial conditions of the VGSTA and FOAF integrators were set to zero. The Euler s method was used with fixed integration step of 6. The results are presented in Figure. For comparison purposes it is also presented the results obtained by the state-feedback VGSTA the FOAF state ˆη is replaced by the plant state in the implementation of the VGSTA majorant functions. Though the use of the FOAF may lead to a more conservative control signal in this case the results obtained by the output-feedback VGSTA are quite similar to the ones obtained by the state-feedback version. Note the finite time convergence of the output tracking error to zero despite the plant uncertainties and the state-dependent disturbance. This result shows the effectiveness and robustness of proposed control strategy. Furthermore Figure also shows the smoothness of the control signal which is free of chattering. dg dt = 4η 4y + ẏ msign (y m ) + 4π cos (8πt)+ + 3ẏ m ÿ m. which implies that g 3 e + e dg dt 4 k 3 φ (e) 4 η + 4 e + y m + 4π + 4 ẏ m + ÿ m 4 [ˆη + π η (t) + 4 e + 4 y m + π + 4π { } 4 max k 3 8ˆη+8 y m +4π+8π + π (t) φ (e) π (t) = 8π η (t) and the upper bounds ẏ m π and ÿ m 4π were considered. Since the system is uncertain only upper and lower bounds for the elements of A B C and k p are available for control purposes. Therefore the VGSTA majorant functions are chosen more conservatively in order to account for the uncertainties in system matrices and high-frequency gain and given by (t ˆη e) = 6 k 3 { } 9 (t ˆη e) = max k3 8ˆη + 8 y m + 5π + 8π Then the VGSTA gains k and k are chosen such that the inequalities () hold as follows: k (t ˆη e) = δ + k i β { 4ǫ [4ǫ + ǫ + +ǫ( + k i β) + ǫ + (k i β + 4ǫ )} k (t ˆη e) = β + ǫk (t ˆη e) e u yym Figure : VGSTA performance: output tracking error e control signal u and plant output y with ( ) output-feedback scheme; ( ) state-feedback scheme. (- -) reference signal y m. 7 Conclusion We conclude in this paper that first order approximation filters can be effectively used to implement a modified Variable Gain Super-Twisting controller based only on output feedback. As in t t t

5 the state feedback case the modified controller can also provide chattering alleviation and global stability properties with finite time convergence of the tracking error to zero. The result is obtained considering a similar Lyapunov approach used to establish the stability and robustness properties of the state feedback VGSTA. The proposed control scheme can be applied to uncertain minimum phase plants with relative degree one and can exactly compensate a class of uncertainties/disturbances. Simulation results were presented to validate the theoretical findings and to illustrate the effectiveness of the proposed control strategy. Acknowledgements This work was supported in part by CAPES CNPq and FAPERJ. References Cunha J. P. V. S. Costa R. R. and Hsu L. (8). Design of first-order approximation filters for sliding-mode control of uncertain systems IEEE Trans. on Ind. Electronics 55(): Dávila A. Moreno J. and Fridman L. (). Variable gains super-twisting algorithm: A Lyapunov based design Proc. American Contr. Conf. Baltimore MD. Edwards C. and Spurgeon S. K. (998). Sliding Mode Control - Theory and Applications Taylor and Francis. Filippov A. F. (964). Differential equations with discontinuous right-hand side American Math. Soc. Translations 4(): Fridman L. and Levant A. (). Higher order sliding modes in W. Perruquetti and J. Barbot (eds) Sliding Mode Control in Engeneering Marcel Dekker Inc. pp. 53. Gonzalez T. Moreno J. A. and Fridman L. (). Variable gain super-twisting sliding mode control IEEE Trans. Aut. Contr. 57(8): 5. Hsu L. Costa R. R. and Cunha J. P. V. S. (3). Model-reference output-feedback sliding mode controller for a class of multivariable nonlinear systems Asian Journal of Control 5(4): Khalil H. (). Nonlinear Systems 3 edn Prentice Hall. Levant A. (993). Sliding order and sliding accuracy in sliding mode control Int. J. Robust and Nonlinear Contr. 58(6): Levant A. (998). Robust exact differentiation via sliding mode technique Automatica 34(3): Moreno J. A. and Osorio M. (8). A lyapunov approach to second-order sliding mode controllers and observers IEEE Conference on Decision and Control CDC 8 pp Utkin V. I. (978). Sliding Modes and Their Application in Variable Structure Systems MIR. A Proof of Theorem Proof: Consider the following Lyapunov function candidate: ζ = φ (e) P = z V (e z) = ζ T P ζ () (kp β + 4ǫ ) ǫ ǫ β > and ǫ >. Note that the function V (e z) is every continuous and differentiable every except on the subspace S = { (e z) R e = }. Now consider the derivative φ (e) = dφ de = e / + k 3 φ R have the following property: ( ) φ = φ φ = e + k / 3 φ. In this case it follows that φ (e){ k p k (t ˆη e)φ (e) + z + g (η e t)} ζ = k p k (t ˆη e)φ (e) + d dt g (η t) = φ (e)a(t ˆη e)ζ + γ(η e t) kp k (t ˆη e) A(t ˆη e) = k p k (t ˆη e) [ φ (e)g (η e t) γ(η e t) = d dt g (η t) every in R \S the derivative exists. One can calculate the derivative of V (e z) on the same subset as V (e z) = ζ T P ζ + ζ T P ζ = φ (e)ζt ( A(t ˆη e) T P + P A(t ˆη e) ) ζ +ζ T P γ(η e t) = φ (e)ζt Q(t ˆη e)ζ + ζ T P γ(η e t) 3

6 Q(t ˆη e) = = = (A(t ˆη e) T P + P A(t ˆη e)) [ βk p k + 4ǫk p (ǫk k ) k p (k ǫk β) 4ǫ [ βk p k 4ǫk p β 4ǫ 4ǫ being the last equality given considering the gain k in (). Furthermore from ζ T P γ(η e t) = { (k p β + 4ǫ )φ ǫz } φ g + { ǫφ + z} d dt g (k p β + 4ǫ )φ φ g + ǫφ z g + {ǫ φ + z } d dt g and ζ = [ φ z T. Considering the inequality ζ T Qζ ζ T Q ζ it follows that and V (e z) φ { ζt Q(t ˆη e) ζ ζ T Π(t) ζ } Q ǫi = Q(t ˆη e) = Q(t ˆη e) Γ(t ˆη e) [ 4ǫ βk p k (kpβ+4ǫ ) 4ǫ( +k pβ) ǫ (4ǫ + +ǫ ) Selecting k as in () then the matrix Q ǫi is positive definite. Thus ( ) {ǫ ζ V (e z) e + k / 3 ζ T Π(t) ζ } () once ζ = ζ. Since π (t) and π (t) are bounded ( ) V (e z) + k ζt 3 Π(t) ζ ( e / ) + k e / 3 ǫ φ {c φ φ + c z z } ( + 3 k 3 e / + k 3 e ) {c ζ ζ } c ζ + c ζ. ζ = φ + z = e + k 3 e 3/ + k 3 e + z is the Euclidean norm of ζ. It follows from that λ min {P } ζ ζ T P ζ λ max {P } ζ (3) V (e z) c V (e z) + c V (e z) and V (e z) cannot escape in finite time. Further ζ T Π(t) ζ λ max {Π(t)} ζ πζ (t) ζ and equation () implies that ( ) V (e z) (ǫ π ζ (t) ) e + k / 3 ζ. {[ φ ( + π )(k p β + 4ǫ ) + ǫ ( + π ) φ Since π ζ (t) is an exponentially decaying function after some finite time t + [( + π ) + ǫ( + π ) z φ } follows that (ǫ π ζ (t) ) µ t t ζ T P γ φ { ζt Γ(t ˆη e) ζ + ζ T Π(t) ζ } µ < ǫ is a positive constant. Then ( ) [ (k p β + 4ǫ V (e z) µ ) + 4ǫ e + k / 3 ζ t t. Γ(t ˆη e) = + ǫ [ π (k p β + 4ǫ ) + 4ǫ π Π(t) = π + ǫ π From equation (3) and e / ζ V (e z) λ min {P } follows that V κ V (e z) κ V (e z) t t κ = µλ min {P } λ max {P } κ = µk 3 λ max {P }. (4) Observing that the trajectories of the VGSTA cannot stay on the subset S = { (e z) R e = } one concludes by Zubov s stability theorem that the equilibrium point (e z) = is reached in finite time for any initial conditions. Since the solution of v = κ v κ v t t v(t ) = v t is given by v(t) = [ v t + κ κ ( exp [ κ (t t ) ) exp [κ (t t ) it follows that (e(t) z(t)) converges to zero at most after some finite time T = t + ( ) κ ln V (σt z t ) +. κ κ 4