ARTICLE. Super-Twisting Algorithm in presence of time and state dependent perturbations

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1 To appear in the International Journal of Control Vol., No., Month XX, 4 ARTICLE Super-Twisting Algorithm in presence of time and state dependent perturbations I. Castillo a and L. Fridman a and J. A. Moreno b a Facultad de Ingeniería, Universidad Nacional Autónoma de México UNAM, 45, CDMX, México; b Instituto de Ingeniería, Universidad Nacional Autónoma de México UNAM, 45, CDMX, México. v. released April 6 In this paper Generalized Super-Twisting Algorithm is applied to a class of systems whose perturbation and uncertain control coefficient are time and state dependent. A non-smooth strict Lyapunov function is used to obtain the conditions for the global finite-time stability and to estimate the time of convergence. Keywords: Sliding Modes, Super-Twisting Algorithm.. Introduction The Sliding Mode Control SMC technique has been known as one of the most efficient tools for theoretically exact compensation of matched uncertainties and perturbations in control systems V. I. Utkin, 99. This properties are achieved due to the discontinuity of control law. Discontinuities in control law in presence of nonidealities e.g. non-modeled dynamics, time delays, hysteresis, etc. produce high-frequency oscillations in the states which is the worst disadvantage of the SMC methodology, the called chattering effect Boiko & Fridman, 5; V. I. Utkin, 99. The Super-Twisting Algorithm STA Levant, 993 was designed as an alternative to conventional Sliding Mode controllers. STA introduces a dynamic extension to the system such that the discontinuous term is hidden behind an integrator, that is why it generates a continuous control signal ensuring theoretically exact compensation of Lipschitz perturbations as well as providing finite-time convergence of the output and its derivative. The properties of the STA are actively discussed see for example Bernuau, Polyakov, Efimov, & Perruquetti 3; Boiko & Fridman 5; Orlov, Aoustin, & Chevallereau ; Polyakov & Poznyak 9; V. Utkin 6; V. I. Utkin 3, as well as the possibilities of its adaptation see for example Shtessel, Taleb, & Plestan ; V. I. Utkin & Poznyak 3. In Levant 993, the proof of convergence was made by means of geometrical arguments for the case when uncertainty in the control coefficient and perturbations only dependent on time. Different Lyapunov functions J. Moreno, ; J. A. Moreno & Osorio, 8; Orlov et al., ; Polyakov & Poznyak, 9; V. I. Utkin, 3 were recently designed in order to get convergence conditions and estimations of the reaching time. Nevertheless, this Lyapunov proofs are made under conservative assumptions: the perturbations are dependent only on time Levant, 993; J. Moreno,. the control coefficient is known Gonzalez, Moreno, & Fridman, ; Guzmán & Moreno, 5; Levant, 993; J. Moreno, ; Picó, Picó-Marco, Vignoni, & Battista, 3. Corresponding author. lfridman@unam.mx In Gonzalez et al. it is considered that the perturbations are state dependent but the total time derivative of the

2 3 perturbations are dependent on state and time, but it is supposed that their total time derivative, i.e. control signal is a priori bounded by some constant Gonzalez et al., ; Shtessel et al.,. In reality for example, any uncertainty in the inertia moment of a mechanical system cause time and state dependent uncertainties/perturbations and also time and state dependent uncertain control coefficient. In this paper the possibility of implementation of General STA GSTA under more general scenarios is considered: the case when both, the perturbation and control coefficient are state and time dependent and, moreover, the control coefficient is uncertain.. Motivation example: Consider the problem of velocity tracking in mechanical system + cos q q + g sin q + b q + arctan q = u, where q R and q R are the state variables and u R the torque control input. The terms of the differential equation represent a varying inertia moment, gravitational term and viscous and dry friction, respectively. It is desired to realize exact velocity tracking to a desired trajectory q d. The dependence of the functions on the variable q can be seen as exogenous functions of time. By defining the error variable e = q q d, the velocity error dynamics are ė = γqu γqg sin q γqbe + q d + arctan e + q d q d, where γq = / + cos q. Here, the time-varying function γqt is an uncertain time-varying control coefficient. Let us remark that γqt is bounded by <.5 γqt, with a bounded time derivative. If the standard STA u st = k e / + z, ż = k e, where a b = a b sign a, is applied to the plant, the closed-loop can be written as ė = k γq e / + γqz g sin q + b e + q d + arctan e + q d q d. γq e Note that the function k e / is not able to compensate any of the non vanishing perturbation terms, so they need to be compensated through the integral action. One of the specific features of the STA lies in the equation of the dynamic extension... ė = k e q d + g cos q γq q d γ q γq + bc q d +bcė, 3 ϕ c = e+ q d + e + q d +, where the discontinuous function k e has to overcome the derivative of the perturbations. Replacing in 3 the value of the last term we get perturbation is assumed a priori to be bounded.

3 t ė = k e + ϕ bcγk e / bcγk e τ dτ +bcγ } {{ } ut t ϕτ b k e τ / e τ dτ 4 Here, an algebraic loop occurs in the gains design since the gains themselves become part of the perturbation to overcome, as shown in equation 4. In Shtessel et al. and Gonzalez et al., the control signal ut is assumed to be bounded a priori, and the algebraic loop never occurs. Moreover, a necessary condition of stability of the STA is that k > ϕ + bcė. Is easy to see that the terms in ϕ can be bounded by a constant, nevertheless, non-bounded terms related with the dynamics of the controller itself appear in equation 4. These cannot be globally dominated by the sign function k e. This behavior can be interpreted as perturbations that can grow exponentially in time. Several problems arise when trying to find STA gains k and k that ensure the stability of the closed-loop and dominate the derivative of such perturbation: STA gain k needs to be designed based on functions depending on STA gains k and k! This is an algebraic loop that has not been addressed in any previous work. Although the perturbations ϕ can be bounded by a constant, the terms bcγk e / t bcγk e τ dτ can be very big depending on the initial conditions e and perturbations. 3 The time variation in the control coefficient γqt introduces the term qd γ qt γqt to the total time derivative, and the bounds of γqt and its derivative γqt should be also taken into account in the controller gains design. 4 State dependent perturbations implies state dependent derivatives which makes the perturbation to grow exponentially in time. It is worth noting that the standard STA cannot be designed for the situations and. Moreover, it is not clear if the stability of the system is preserved in spite of the unknown variations of the coefficients in equation due to the uncertain control coefficient γq. The standard gain design for STA Levant, 998 is not applicable here. Evenmore, the nonlinear terms of the Standard STA are not enough to maintain the global stability of the system in presence of uncertain control coefficient and state dependent perturbation. Then, an improved STA will be used to solve this problem. Generalized Super-Twisting Algorithm J. Moreno, ; J. A. Moreno, 9 is defined by u = k φ x + z, ż = k φ x, 5 φ x = x + x, φ x = x + 3 x + x. Note that the Generalized STA GSTA has an extra linear term x in function φ x and growing terms 3 x and x in function φ x, such that φ x = φ xφ x with φ x = +. With the extra linear term, three degrees of freedom in the GSTA gains design are obtained: k, k and. The growing terms help to counteract the effects of state dependent perturbations which can grow exponentially in time. Contribution: This paper proposes a global finite-time stability analysis for the GSTA based on a strict non-smooth Lyapunov function in three different scenarios: x 3

4 state and time dependent uncertain control coefficient and perturbations, known control coefficient with state and time dependent perturbations and, 3 time dependent uncertain control coefficient and perturbation. Section presents the problem statement for the 3 different scenarios and the main results: subsection. considers the general problem of state and time dependent uncertain control coefficient and perturbations, subsection., known control coefficient with state and time dependent perturbations, and subsection.3 time dependent uncertain control coefficient and perturbation. In Section 3, the motivation example is revisited to show the proposed gain design and its effectiveness through simulations. Finally some conclusions are presented in Section 4.. Problem Statements and Main Results The main goal of this paper is to find the conditions for GSTA gains {k, k, } design taking into account the known bounds of the uncertain control coefficient and perturbations as well as its partial derivatives with respect to the time and state, such that the trajectories of the system globally converge to zero in finite-time.. General case Consider the scalar dynamic system represented by the differential equation ẋ = γx, tu + ϕx, t, 6 where x R is the state and u R is the control input. The functions ϕx, t and γx, t are uncertain functions dependent on the state and time. In order to guarantee a continuous control signal, the uncertain control coefficient γx, t and perturbation ϕx, t should be also continuous, since x ẋ, and therefore = γ u + ϕ u eq = ϕ γ. Hence, ϕ and γ are assumed to be Lipschitz continuous functions with respect to t and ϕ, γ C with respect to x. The uncertain control coefficient function is also assumed to be bounded by positive constants < γx, t k M. 7 Following Gonzalez et al., we split the perturbations in two parts ϕx, t = ϕ x, t + ϕ x, t, 8 such that the first term is vanishing at the origin, i.e. ϕ, t = for all t, and bounded by ϕ x, t α φ x. 9 4

5 The total time derivative of the second term divided by the control coefficient γx, t can be represented as d dt ϕ x, t γx, t = ϕ γ t ϕ γ γ t δ x,t System 6 in closed-loop with GSTA 5 has the form + γ ϕ ϕ γ x γ x δ x,t ẋ a = δ x, t + δ x, tẋ. b ẋ = k γx, tφ x + ϕ x, t + γx, t z + ϕ x, t, γx, t and defining x = x and x = z + ϕx,t γx,t, we obtain ẋ = γx, t k φ x + ϕ x, t ẋ = k φ x + d dt ϕ x, t γx, t γx, t + x, a. b Moreover, according to the previous discussion we assume that the perturbation terms are such that the functions δ x, t = γx, t ϕ x, t t ϕ x, t γ x, t γx, t, δ x, t = t γx, t ϕ x, t x ϕ x, t γx, t γ x, t x are bounded by positive constants δ x, t δ, δ x, t δ. 3 Theorem : Suppose that γx, t and ϕx, t of system 6 satisfy 7 and 3. Then, the states x and x converge to zero and z converges to ϕx, t, globally and in finite-time, if GSTA gains k, k and > are designed as follows: for any ɛ > : k > + ɛ 4ɛ α k M + δ ɛ k + ɛ M Λk + α, 4 4ɛ Λ k = α k M + δ ɛ δ k α M ɛkM km + δ. k k M hɛ c c k + δ, k M hɛ c + c k + δ, 5 5

6 k min k min k k a b Figure.: Decrease of of k min and k min vs. + ɛkm + ɛ k = 6 hɛ 8 m m kk kk h = δ δ α + δ + δ ɛ + α. + ɛ k α, c = k + ɛk M, k = k α km. Moreover, a system trajectory starting at x = x, x reaches the origin in a time smaller than T x = µ ln V x, x +. 6 µ µ Remark : When the perturbation in the system is state dependent, the function δ x, t appears in the total time derivative, and it is worth noting that the extra terms in functions φ x and φ x related with the gain, are necessary to guarantee the global stability of the equilibrium point. This behavior can be observed in 4 and 5, where the gain appears in the ratio δ. This allows to arbitrarily decrease the effects of the perturbation δ by increasing the gain. GSTA gain does not only allow to guarantee the global stability of the system. It can be used to reduce the chattering effect by reducing the minimum values of gains k and k see Figs. a and b. Once the gain is fixed, the possible set of gains k and k are depicted in Fig. a. The parameter ɛ >, also allows to adjust the minimum values k min and k min. Fig. a shows how the parameter ɛ can be selected such that k min and k min are in their lowest values.. Case of state dependent perturbations and known coefficient of control The second case of analysis is taking into account time and state dependent perturbations with known coefficient of control, ẋ = u + ϕx, t, 7 where x R is the state, u R is the control input, and ϕ a Lipschitz continuous function with respect to t and ϕ C with respect to x. 6

7 k k and k k min k min a Set of GSTA gains {k, k }, when and parameter ɛ are fixed. k b Minimum values of k and k vs ɛ. Figure.: Theorem. Set of possible selection of gains {k, k, }. ǫ In this case the total time derivative of the second term of perturbation 8 is given by d dt ϕ x, t = ϕ t δ x,t ϕ + x δ x,t ẋ 8a = δ x, t + δ x, tẋ. 8b Note that functions δ x, t and δ x, t in 8 are a particular case of when control coefficient is γ =. System 7 in closed-loop with GSTA takes the form ẋ = k φ x + z + ϕ x, t + ϕ x, t. Defining x = x and x = z + ϕ x, t, we have ẋ = k φ x + x + ϕ x, t, 9a ẋ = k φ x + d dt ϕ x, t. 9b Moreover, the perturbation terms are such that the functions δ x, t = ϕ x, t, δ x, t = ϕ x, t t x are bounded by positive constants δ x, t δ, δ x, t δ. Corollary : Suppose that ϕx, t of the system 7 satisfy. Then, the states x and x converge to zero and z converges to ϕx, t, globally and in finite-time, if GSTA gains k, k and > are designed as follows: for any ɛ > : 7

8 k > + ɛ δ + α, k > + ɛ δ α 4 hɛ k α + δ + δ ɛ + α + δ 3 Remark : When the control coefficient is known or it is unknown but constant, the partial derivatives of γ with respect to the state and time are zero γ t = γ x, = GSTA is able to ensure the global finite-time stability even in the case of unbounded perturbations ϕ..3 Case of time dependent uncertain control coefficient and perturbation The last case of analysis is taking into account an uncertain control coefficient and a perturbation, both Lipschitz continuous functions of time, ẋ = γtu + ϕt, 4 where x R is the state and u R is the control input. The uncertain control coefficient is assumed to be bounded by positive constants < γt k M. 5 The total derivative of the second term in the perturbation 8 has the form δ t := d dt System 4 in closed-loop with GSTA has the form and defining x = x and x = z + ϕt γt, ẋ ϕ t = dϕ γt γ dt ϕ dγ γ dt. 6 = k γtφ x + ϕ t + γt ẋ = γt k φ x + ϕ t ẋ = k φ x + d dt ϕ t γt z + ϕt γt γt + x,. The total time derivative of the perturbation is assumed to be bounded by a positive constant, δ t δ. 7 Corollary : Suppose that γt and ϕt of system 4 satisfy 5 and 7. Then, the states x and x converge to zero and z converges to ϕt γt, globally and in finite-time, if GSTA gains k, k and are designed as follows: for any ɛ > : 8

9 k > + ɛ 4ɛ α + ɛ k M + Λk + α. 4ɛ 8a Λ k = α k M + 6ɛk M δ. k k M ɛ c c k + δ, k M ɛ c + c k + δ, 8b + ɛkm + ɛ α k = 6ɛ 8 δ + + m m kk kk c = m kk + ɛk M, = k α. k. Example : Let be ϕt = t and γt = t+ +. The time derivatives are ϕt = and γt = t+ which are bounded by, and k m =, k M =. Using 6 and 7, ϕt + ϕt γt.3. Following 8, we design the GSTA gains ɛ =, k = 3, k = 5 and =. Fig. 3a shows the functions ϕt and γt. Fig. 3b shows the behavior of d dt, and the value of the bound δ. It ϕt γt can be seen how the perturbation term never exceeds the calculated bound. The Sliding Mode is achieved almost at the beginning of the simulation and remains there for all future time Fig. 3c. Note that the perturbation grows without limit, and the GSTA is able to compensate it globally and for all time. GSTA compensates uncertainties coming from both, uncertain control coefficient and perturbation, as the ratio ϕ. Note that boundedness in 7 and 3 implies that the the product dγ dt γ ϕ should be bounded. This is an example where the term ϕ can be growing in time while decreases such that bounds 7 and 3 exist. Example : ϕt = sint + and γt = +.5 cost. The time derivatives are φt = cost, γt =.5 sint.5, =.5, k M =.5. Using 6 and 7 the derivative dγ dt δ t = cost.5 sintsint cost +.5 cost, δ t 6. In Figure 4, it is possible to observe the set of possible selection of gains following 8, for example k =, k = 3 and =. Also the set of possible gains following Levant 993 is depicted. It is possible to see that the obtained gain conditions for GSTA in presence of time dependent uncertain control coefficient and perturbations is much less restrictive. 9

10 5.4 ϕt. γt.8.6 d dt δ ϕt γt Time s a Perturbation and uncertain control coefficient b d dt ϕt γt vs δ. State x Control u Time s c Finite-time convergence to Sliding Mode and control signal. Figure 3.: Example. 3 5 Levant,93 Corollary k k Figure 4.

11 Remark 3: It is worth noting that system perturbations in Example cannot be covered by conditions found in Levant 993. Moreover, in Example, the STA gain conditions 8 is much less restrictive than those found in Levant Motivation Example revisited Consider again the system and the desired speed as q d = a sin ωt. System parameters are b =, g =, a =, ω =, =.5, k M =. The closed-loop with the GSTA, and perturbations split in two parts ė = k γqφ e γqbe +γqz g sin q b q d + arctan e + q d q d. 9 γq ϕ e,t ϕ e,t γq Then, the system with dynamic extension e = z + ϕe,t γq, can be rewritten in the form ϕe, t γt The partial derivatives are ė = γqk φ e + γqe + ϕ e, t 3a ė = k φ e + d ϕ e, t, 3b dt γq = g sin q b q d + arctan e + q d q d γq. γq cosq sinq = t + cos q, γq =, e ϕe, t t ϕe, t e = = γq g sinq + b q d + b arctane + q d γq g cosq + b q d + bγq e + q d +. Following the proposed analysis, from 9, we get that and the bounds 3 are e + q d + bγqe < α e = 6 > bk M and α =., δ x, t aω + b + ω + g = 4 δ x, t bk M =.... q d, 3c Then, using 4 and 5 we can select ɛ = and k = 4 > k min = 9.8, and k = 7 > k min = 7.59.

12 Tracking q d q gsinq b q 3 Control u Time s a Finite-time convergence of the states and control signal. γq Time s b State dependent perturbation and the uncertain control coefficient. Figure 5.: System trajectories with initial condition q =, q =. GSTA gains are designed with the proposed method. The state and control converge to zero in finite-time. Fig. 5a shows the speed variable converging to the desired reference in finite-time and the continuous control signal. Fig. 5b shows the state dependent perturbation and the uncertain control coefficient. With the proposed design the problem of the algebraic loop, uncertain control coefficient and state dependent perturbations can be solved. 4. Conclusions In this paper it is shown that during the STA design for systems with the state dependent perturbation or/and uncertain control coefficient the problem of algebraic loop appears and classic STA is unable to solve this problem. To overcome the problem of the algebraic loop the GSTA is proposed. Based on strict non-smooth Lyapunov function J. Moreno, the sufficient conditions for global finite-time stability for three scenarios are found. Moreover, it turns that in case when perturbations and control coefficient are time-dependent the analysis performed in this paper provides much less restrictive conditions for global finite-time convergence. Acknowledgement The authors are grateful for the financial support of Consejo Nacional de Ciencia y Tecnología CONACyT CVU 4555,Project 47; PAPIIT-UNAM Programa de Apoyo a Proyectos de Investigacíon e Innovación Tecnológica IN 36; Programa de Cooperación FI-II, Project IISGBAS--5. References Bernuau, E., Polyakov, A., Efimov, D., & Perruquetti, W. 3. Verification of ISS, IISS and IOSS properties applying weighted homogeneity. Systems and Control Letters, 6, Retrieved from

13 Boiko, I., & Fridman, L. 5, Sept. Analysis of chattering in continuous sliding-mode controllers. IEEE Transactions on Automatic Control, 5 9, Gonzalez, T., Moreno, J. A., & Fridman, L., Aug. Variable gain super-twisting sliding mode control. IEEE Transactions on Automatic Control, 57 8, -5. Guzmán, E., & Moreno, J. A. 5. Super-twisting observer for second-order systems with time-varying coefficient. IET Control Theory Applications, 9 4, Levant, A Sliding order and sliding accuracy in sliding mode control. International Journal of Control, 58 6, Retrieved from Levant, A Robust exact differentiation via sliding mode technique*. Automatica, 34 3, Retrieved from Moreno, J.. Lyapunov approach for analysis and design of second order sliding mode algorithms. In L. Fridman, J. A. Moreno, & R. Iriarte Eds., Sliding modes after the first decade of the st century Vol. 4, p Springer-Verlag Berlin Heidelberg. Moreno, J. A. 9, Jan. A linear framework for the robust stability analysis of a generalized super-twisting algorithm. In Electrical engineering, computing science and automatic control,cce,9 6th international conference on p. -6. Moreno, J. A., & Osorio, M. 8, Dec. A lyapunov approach to second-order sliding mode controllers and observers. In Decision and control, 8. cdc 8. 47th ieee conference on. cancun, méxico p Orlov, Y., Aoustin, Y., & Chevallereau, C., March. Finite time stabilization of a perturbed double integrator-part i: Continuous sliding mode-based output feedback synthesis. IEEE Transactions on Automatic Control, 56 3, Picó, J., Picó-Marco, E., Vignoni, A., & Battista, H. D. 3. Stability preserving maps for finitetime convergence: Super-twisting sliding-mode algorithm. Automatica, 49, Retrieved from Polyakov, A., & Poznyak, A. 9, Aug. Reaching time estimation for super-twisting second order sliding mode controller via lyapunov function designing. IEEE Transactions on Automatic Control, 54 8, Shtessel, Y., Taleb, M., & Plestan, F.. A novel adaptive-gain supertwisting sliding mode controller: Methodology and application. Automatica, 48 5, Retrieved from Utkin, V. 6, March. Discussion aspects of high-order sliding mode control. IEEE Transactions on Automatic Control, 6 3, Utkin, V. I. 99. Sliding modes in control and optimization. Springer-Verlag Berlin Heidelberg. Utkin, V. I. 3, Aug. On convergence time and disturbance rejection of super-twisting control. IEEE Transactions on Automatic Control, 58 8, 3-7. Utkin, V. I., & Poznyak, A. S. 3. Adaptive sliding mode control with application to super-twist algorithm: Equivalent control method. Automatica, 49, Retrieved from Appendix A. Proofs Proof of the Theorem 3

14 A. Separation of Perturbations Consider the first order system ẋ = γx, tu + ϕx, t. A The terms ϕx, t and γx, t are uncertain functions dependent on the state and time. The control coefficient function is assumed to be bounded by positive constants Terms of perturbation are < γx, t k M. A ϕx, t = ϕ x, t + ϕ x, t such that the first term is vanishing at the origin bounded by and the second term has its total derivative such that d ϕ x, t = ϕ dt γx, t γ t ϕ γ γ t δ x,t ϕ x, t α φ x A3 + γ ϕ ϕ γ x γ x δ x,t ẋ A4 = δ x, t + δ x, tẋ A5 Moreover, the partial derivatives of the perturbation are bounded by positive constants δ x, t δ, δ x, t δ. A6 A. Closed-Loop System A closed-loop with the Generalized STA can be written as ẋ = k γx, tφ x + ϕ x, t + γx, t z + ϕx,t γx,t and expressing the system with the change of variables x = x and x = z + ϕx,t γx,t, with x = x x T, The first equation can be rewritten as ẋ = γx, t k φ x + ϕ x, t γx, t + x, A7a ẋ = k φ x + d ϕ x, t A7b dt γx, t ẋ = γx, t k φ x + x, 4

15 where k = k ϕ x, t γx, tφ x, A8 and note that from A and 9, ϕ x, t γx, tφ x From, we rewrite the second equation as α φ x γx, t φ x α. ẋ = k φ x + δ x, t + δ x, tẋ = k φ x + δ x, t + δ x, tγx, t k φ x + x = k φ x + δ x, t δ x, tγx, t k φ x + δ x, tγx, tx. Due to the fact that φ x = φ x φ x, ẋ = φ x = γx, tφ x k φ x + δ x, t φ x δ x, t k γx, t φ x k γx, t + δ x, t φ x k φ x + φ + δ x, tγx, t φ x x δ x, t γx, tφ x + δ x, t φ x x. A.3 Representation of Lyapunov candidate derivative in quadratic form Now consider the candidate Lyapunov function V = ξ T P ξ, P = Whose derivative along the trajectories of the system V = φ x p φ x x γx, t k φ x + x + + p x φ x γx, tφ k x γx, t + δ x, t φ x k { k φ x + x = γx, tφ x + p x φ x p φ x x k γx, t + δ x, t φ x k p, ξ p T = φ x x A9 + φ x + φ x + After several algebraic manipulations, it is possible to group the terms δ x, t γx, tφ x + δ x, t φ x x δ x, t γx, tφ x + δ } x, t φ x x. 5

16 { V = γx, tφ x k + k δ x, t φ x p δ x, t φ x The last two terms can be rewritten as p δ x, t γx, tφ x x then, substituting, p δ x, t φ x k γx, t k + p p γx, t δ x, t φ x x + p δ x, t γx, tφ x x δ x, t γx, tφ x φ x = p δ x, t γx, tφ x φ x x φ x + φ x x δ x, t γx, tφ x φ x } δ x, t γx, tφ x φ x. V = γx, tφ x p k δ x, t φ x k δ x, t γx, t φ x φ x + p k + k p δ x, t φ x p δ x, t φ x h h x. + p δ x, t φ x } {{ } p From functions on GSTA controller φ x = we have x p k δ x, t γx, t φ x φ x x p k φ x ; φ x = sign x + 3 x + x. +, and φ x = x + 3 x + x Several terms can be grouped as perturbed bounded terms, p = p δx,t φ x k = γx,t k = k ϕx,t k δx,t φ x p p, p k k, k γx,tφ x k k, k = h = pδx,t φ x h h, h = = = p δ, p + δ k M k δ, k + δ, k α, k + α p δ, + p δ,,. A 6

17 and rewritten in a matrix form, V = γx, tφ x k p k q t φ x p k k h + p φ x x } {{ } q 3t h q t x Aa = γx, tφ x ξ T Qtξ Ab where q t q Qt = 3 t = q 3 t q t k p k p k k h + p p k k h + p h. A If matrix Qt is positive definite then the time derivative V is negative definite. A.4 Positiveness of matrix Qt The determinant of matrix Qt can be expressed as det Qt = a p p + b p p + c p, A3 where a p = 4, b p = h k + k p, c p = 4 h k k p h k. Note that the coefficient a p is negative, then, in order to get the positiveness of det Qt, the discriminant of the second degree equation det Qt = must be positive, and the constant value p should be selected within the interval defined by the two real roots of det Qt =. p x, t = b p 4a p c p = h k k p >, A4 For the h > factor > δ x, tp φ x p > δ. A5 A6 For k > factor k > δ. 7

18 For k p > factor, we get p > k, and taking into account A p > p > k α k α Then, a possible selection of p is p = + ɛ, for ɛ >. A7 k α Finally, from A6 and A7 we get kkm α +ɛ > δ, and consequently k > δ + ɛ + α. The two real roots of the quadratic equation det Qt = are p + = h k + k p + δ x, t φ x + p x, t. p = h k + k p + δ x, t φ x } {{ } p c p x, t. A8 A9a A9b The two real roots define the endpoints of an interval wich is a set of all possible values for p such that Qt >. However, the endpoints are dependent on perturbed bounded terms h, k, k and δ x, t/φ x which make each endpoint to move from a minumum value p + min, p min to a maximum value p + Max, p Max. A valid selection of a constant p must belong to an intersection set between the minor endpoint at its maximum value p Max and the maximum end point at its minumum value p + min as shown in Figure A. p p Max, p+ min. A Some possible bounds for the terms in A9 are < p c p c p c, and < p x, t p x, t p x, t, where p c = h k + k p δ, pc = h k + k p + δ, p x, t = h k k p, p x, t = k p. h k 8

19 det Qt Intersection p min p Max p + min p + Max Figure A.: Intersection set. and note that possible upper and lower bounds for p Max, p+ min are p Max h k + k p δ h k k p p + min h k + k p + δ + h k k p. A A Then p should be selected within p h k + k p δ h k k p, + p + δ h k k + h k k p. A3 A.5 Intersection set In order to guarantee a non-empty intersection set the next condition must hold p + min > p Max. A4 Then, A4 can be rewritten as + p x, t > p c p x, t. pc After several algebraic manipulations, the last inequality can be expressed as where a k K + b k K + c k >, A5 +ɛkm km a k =, kkm b k = 4 hɛ, c k = +ɛ and with a change of variable K = k. kkm δα + δ δɛ + α, 9

20 Solutions of the last inequality belong to the interior of the interval defined by K = b k ± k a k a k = hɛ k + ɛk M ± k + ɛk M k where + k = b ɛkm + ɛ k 4a k c k = 6 hɛ 8 m m kk kk Finally, gain k can be selected within the interval k δ α + δ + δ ɛ + α. k M hɛ c c k + δ, k M hɛ c + c k + δ, A6 h = δ + ɛ k α, c = k + ɛk M, k = k α km. A.6 Guarantee the existence of the interval for gain k In order to have an interval between two real roots, the discriminant k to zero k. It is possible to express the last inequality as must be greater or equals a k k + b k k + c k > A7 where a k = 6ɛ, b k = 8+ɛ α k M + δ ɛ c k = 8+ɛ k k m M δα + δ. k M +, It is worth it to note that, the coefficient a k is positive and the discriminant k = b k 4a k c k of A7 is always positive k = 64 + ɛ k m α k M + δ ɛ δ k α M + + 8ɛkM km Λ k + δ >. Then, there is always possible to find a gain k to ensure the existence of a valid selection interval for the gain k. Selecting the k gain, is possible to ensure that the interior set of parables p is not empty, and therefore, to ensure the positiveness of det Qt.

21 Gain k can be selected as k > b k + a k > + ɛ 4ɛ b k 4a k c k + α, a k α k M + δ ɛ k M ɛ 4ɛ Λk + α. A8 A9 We have shown that if A9 and A6 are satisfied, det Qt >. However the condition A9 does not imply A8 for every case. Condition A8 and A9 when δ x, t =, α = are k > δ + ɛ + ɛ δ, k > k M + by modifying the factor k M + to k M + + we achieve δ + ɛ δ + δ ɛ δ + ɛ δ km km δ ɛ + δ + ɛ δ A3 A3 km +. A3 then A9 always imply A8. Finally, condition A9 ensures that the term h = q t >. It can be seen that if det Qt = q tq t q 3 t > and q t >, q t is also positive definite. Therefore, matrix Qt is positive definite. A.7 Finite-time convergence and Convergence Time Based on J. A. Moreno 9, recall the standard inequality for quadratic forms and λ min {P } ξ ξ T P ξ λ max {P } ξ, ξ = ξ + ξ = φ x + x = x + x 3 + x + x A33 is the Euclidean norm of ξ, and note that the inequality x ξ V x, x λ min {P } is satisfied. Since Qt is definite positive and all its elements are bounded, then Qt εi >, for ε > and I the identity matrix, we can rewrite

22 where V γx, tφ x ξ T Qtξ φ x ξ T εiξ εφ x ξ ε ξ x ε ξ ελ min {P } λ max {P } V x, x ε λ max {P } V x, x µ V x, x µ V x, x, µ = ελ min {P } λ max {P }, µ = ε λ max {P }. that shows that the system trajectories converge in finite time. Since the solution of the differential equation v = µ v µ v, v = v si given by vt = exp µ t v + µ µ exp µ t, if µ and µ >. If follows from A34 and the comparison principle that V t vt, when V x, x v. Therefore, x t and x t converge to zero in finite-time and reaches that value at most after a time given by T = µ ln V x, x + µ µ A34 Proof of the Corollary When there is no uncertain control coefficient, the bounds of the function γx, t are, with loss of generality k M = = and quadratic term in inequality A5 become zero. Then, the condition for gain k is k > 6 hɛ c k + δ + ɛ k α > 4 hɛ Condition from quadratic inequality A7 reduces to A35a δ α + δ + δ ɛ + α + δ A35b k > + ɛ δ + α, A36

23 the same as in A8. Proof of the Corollary When there is no state dependent perturbations, the function δ x, t =, then the conditions are simply reduced as k k M ɛ c c k + δ, k M ɛ c + c k + δ, A37 + ɛkm + ɛ k = 6ɛ 8 δ + m m kk kk α. h = δ + ɛ k α, c = k + ɛk M, k = k α km. Gain k can be selected as k > + ɛ 4ɛ α + ɛ k M + Λk + α. 4ɛ A38 Λ k = α k M + 6ɛk M δ. 3