Predefined-Time Backstepping Control for Tracking a Class of Mechanical Systems

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1 Instituto ecnológico y de Estudios Superiores de Occidente Repositorio Institucional del IESO rei.iteso.mx Departamento de Matemáticas y Física DMAF - Artículos y ponencias con arbitraje 7-7 Predefined-ime Backstepping Control for racking a Class of Mechanical Systems Jiménez-Rodríguez, Esteban; Sánchez-orres, Juan D.; Loukianov, Alexander Jiménez-Rodríguez, E., Sánchez-orres, J.D. and Loukianov, A. 7). "Predefined-ime Backstepping Control for racking a Class of Mechanical Systems". th IFAC World Congress, July 9-4, oulouse, France. Enlace directo al documento: Este documento obtenido del Repositorio Institucional del Instituto ecnológico y de Estudios Superiores de Occidente se pone a disposición general bajo los términos y condiciones de la siguiente licencia: El documento empieza en la siguiente página)

2 Predefined-ime Backstepping Control for racking a Class of Mechanical Systems Esteban Jiménez-Rodríguez Juan Diego Sánchez-orres Alexander G. Loukianov Department of Electrical Engineering, CINVESAV-IPN, Av. del Bosque 45 Col. El Bajío C.P. 459, Guadalajara, Jalisco, México {ejimenezr,louk}@gdl.cinvestav.mx). Department of Mathematics and Physics, IESO, Periférico Sur Manuel Gómez Morín 8585 C.P. 4564, laquepaque, Jalisco, México dsanchez@iteso.mx) Abstract: he predefined-time exact tracking of unperturbed fully actuated mechanical systems is considered in this paper. A continuous second-order predefined-time stabilizing backstepping controller, designed using first-order predefined-time stabilizing functions, is developed to solve this problem. As an example, the proposed solution is applied over a twolink planar manipulator and numerical simulations are conducted to show performance of the proposed control scheme. Keywords: Backstepping, Lyapunov, Mechanical systems, Predefined-time stability, Second-order systems, racking.. INRODUCION he various developments concerning the concept of finite-time stability permit to solve different applications which are characterized for requiring hard time response constraints. Some important works of this topic and its application to control systems have been carried out in Roxin 966); Haimo 986); Utkin 99); Bhat and Bernstein ); Moulay and Perruquetti 5, 6). However, generally this finite time is an unbounded function of the initial conditions of the system. A desired feature is to eliminate this boundlessness, for example, in estimation or optimization problems. his gives rise to a stronger form of stability called fixed-time stability, where the convergence time, as a function of the initial conditions, is bounded. he notion of fixed-time stability have been investigated in Andrieu et al. 8); Cruz- Zavala et al. ); Polyakov ); Fraguela et al. ); Polyakov and Fridman 4). Although the fixed-time stability concept represents a significant advantage over finite-time stability, it is often complicated to find a direct relationship between the tuning gains and the fixed stabilization time. o overcome the above, another class of dynamical systems which exhibit the property of predefined-time stability, have been studied Sánchez-orres et al., 4; Sánchez-orres et al., 5). For this systems, an upper bound sometimes the least upper bound) of the fixed stabilization time appears explicitly in their tuning gains. In this sense, similarly to Jiménez-Rodríguez et al. 6); Sánchez-orres et al. 6), this paper is devoted to the his work was supported by CONACy, México, under grant 545. design of a second-order predefined-time controller, using first-order predefined-time stabilizing functions Sánchez- orres et al., 5) and the backstepping design technique Kokotovic, 99; Krstić et al., 995). Furthermore, this idea is used to solve the problem of predefined-time exact tracking in fully actuated mechanical systems, assuming the availability of the state and the desired trajectory as well as its two first derivatives) measurements. In the following, Section presents the mathematical preliminaries needed to introduce the proposed results. Section 3 states the problem which will be solved in this paper. Section 4 exposes the main result of this paper, which is the second-order predefined-time backstepping controller for tracking of fully actuated mechanical systems. Section 5 describes the model of a planar two-link manipulator, where the proposed controller is applied. he simulation results of the example are shown in Section 6. Finally, Section 7 presents the conclusions of this paper.. MAHEMAICAL PRELIMINARIES Consider the system ẋ = fx; ρ) ) where x R n is the system state, ρ R b represents the parameters of the system and f : R n R n. he initial conditions of this system are x) = x. Definition. Bhat and Bernstein, ; Polyakov, ) he origin of ) is globally finite-time stable if it is globally asymptotically stable and any solution xt, x ) of ) reaches the equilibrium point at some finite time moment, i.e., t x ) : xt, x ) =, where : R n R + {} is called the settling-time function. Remark. he settling time function : R n R + {} for systems with a finite-time stable equilibrium point

3 is usually an unbounded function of the system initial condition. Definition 3. Polyakov, ; Polyakov and Fridman, 4) he origin of ) is fixed-time stable if it is globally finite-time stable and the settling-time function is bounded, i.e. max > : x R n : x ) max. Definition 4. Sánchez-orres et al., 4; Sánchez-orres et al., 5) For the case of fixed time stability when the system ) parameters ρ can be expressed in terms of a bound of the settling-time function max, it is said that the origin of the system ) is predefined-time stable. he following lemma extends the Lyapunov methods given in Sánchez-orres et al. 4); Sánchez-orres et al. 5). his result will be useful in order to apply the approaches presented in the mentioned references to second order systems. Lemma 5. Assume there exist a continuous radially unbounded function V : R n R + {}, and real numbers α >, β >, and < q such that: V ) = V x) >, x, and the derivative of V along the trajectories of the system ) satisfies V α q expβv q )V q. ) hen, the origin is globally predefined-time stable for ) and x ) αβ. Proof. he solution of the differential inequality ) is ) V t) β ln q, αβt + exp βv q ) where V = V x ). Note that V t) = if αt + exp βv q ) =. Hence, the settling-time function is such that x ) exp βv q ). αβ hen, x ) q αβ, since < exp βv ). Remark 6. Lemma 5 characterizes predefined-time stability in a very practical way since the condition ) directly involves a bound on the convergence time. Definition 7. Let h. For x R n, define the function x h x =, 3) h x with x the norm of x. Since lim x x h = for h >, it is defined h =. herefore, the function x h is continuous for h > and discontinuous in x = for h =. heorem 8. he function x h fulfills: i) x h = x h ii) x = x x, a unit vector. iii) x = x = x, iv) d x h dx = I n + h ) xx x h x and d x h dx = h x h, where I n is the n n identity matrix. v) For h, h R, it follows: x h x h = x h+h x h x h = x h+h h vi) For h, h >, then x h h = x h. Definition 9. Sánchez-orres et al., 5) For x R n, the predefined-time stabilizing function is defined as Φ m,q x; c ) = c mq exp x mq ) x mq 4) where m, < q m, and c >. Remark. From Definition 7, the function defined in 4) is continuous for < mq < and discontinuous in x = for mq =. Remark. Using the part iv) heorem 8, the derivative of the predefined-time stabilizing function 4) is given by Φ m,q x; c ) x for x. = exp x mq ) mq xx c mq x + ) I n mq xx x x mq, 5) he following lemma gives meaning to the name predefined-time stabilizing function, introduced in Definition 9. Lemma. Sánchez-orres et al., 5) he origin of the system ẋ = Φ m,q x; c ) 6) with m, < q m, and c > is predefined-time stable with f = c. hat is, xt) = for t > c in spite of the x value. Proof. Consider the radially unbounded Lyapunov function candidate V x) = x m. he time derivative of V along the trajectories of 6) is see heorem 8) V = m x m Φ m,q x; c ) = m c mq exp x mq ) x m q) = c q exp V q ) V q. Hence, applying Lemma 5, the origin of the system 6) is predefined-time stable with f = c. 3. PROBLEM SAEMEN A generic model of second-order, fully actuated mechanical systems of n degrees of freedom has the form Mq) q + Cq, q) q + P q) + γq) = τ, 7) where q, q, q R n are the position, velocity and acceleration vectors in joint space; Mq) R n n is the inertia matrix, Cq, q) R n n is the Coriolis and centrifugal effects matrix, P q) R n is the damping effects vector, usually from viscous and/or Coulomb friction and γq) R n is the gravity effects vector. Defining the variables x = q, x = q and u = τ, the mechanical model 7) can be rewritten in the following state-space form

4 ẋ = x ẋ = fx, x ) + Bx, x )u, where fx, x ) = M x )Cx, x )x +P x )+γx ), Bx, x ) = M x ) are continuous maps and the initial conditions are x ) = x,, x ) = x,. Remark 3. he matrix function Mx ) is, in fact, invertible since Mx ) = M x ) is positive definite. A common problem in mechanical systems control is to track a desired time-dependent trajectory described by the triplet q d t), q d t), q d t)) of desired position q d t) = q d t) q dn t) R n, velocity q d t) = q d t) q dn t) R n and acceleration q d t) = q d t) q dn t) R n, which are all assumed to be known. 8) he task is to design a state-feedback, second-order, predefined-time controller to track the desired trajectory. 4. PREDEFINED-IME BACKSEPPING RACKING CONROLLER o be consequent with the state-space notation used in 8), redefine the desired position as x,d = q d. Step : let the position error be e = x x,d. hen, using 8), the dynamics of e are ė = x ẋ,d. 9) Let m and consider the Lyapunov function candidate V e ) = e m. ) Differentiating ) with respect to time, along the trajectories of 9), it yields V = m e m x ẋ,d. ) Let e = x x,d, ) with x,d = Φ m,q e ; c ) + ẋ,d, where < q < m and c >. Replacing ), ) becomes V = m e m Φm,q e ; c ) + m e m e = exp V q c q ) V q + m e m e. 3) Step : using 8), the dynamics of ) are ė = fx, x )+Bx, x )u+ dφ m,q e ; c ) ẍ,d. 4) dt Let m and consider the Lyapunov function candidate V e, e ) = e m + e m = V + e m. 5) Differentiating 5) with respect to time, and substituting 3) and 9) it yields V = m e m Φm,q e ; c )+m e m e + m e m fx, x ) + Bx, x )u+ dφ m,q e ; c ) dt ẍ,d. 6) hus, the control variable u is designed as u = B x, x ) fx, x ) dφ m,q e ; c ) + ẍ,d dt m e m e m e Φ m,q m e ; c ), 7) where < q < m and c >. Finally, substituting 7) in 6) it yields V = m e m Φm,q e ; c ) m e m Φm,q e ; c ) <. 8) From 8), the error system 9) and 4) closed-loop with the control signal 7) is asymptotically stable. Moreover, the predefined-time stability is stated in heorem 4 and proved in Appendix B. heorem 4. he origin of the error system 9) and 4), closed-loop with 7) is globally predefined-time stable with f r c, where c = max{ c, c }, r = q M qm, q m = min{q, q } and q M = max{q, q }. Remark 5. Note that selecting q = q, the number r is minimized and becomes r =. In the same manner, the number c is minimized selecting c = c = c. 5. EXAMPLE: RAJECORY RACKING FOR A WO-LINK MANIPULAOR Consider a planar, two-link manipulator with revolute joints as the one exposed in Example. of Utkin et al. 9). he manipulator link lengths are L and L, the link masses concentrated in the end of each link) are M and M. he manipulator is operated in the plane, such that the gravity acts along the z axis. Examining the geometry, it can be seen that the endeffector the end of the second link, where the mass M is concentrated) position x w, y w ) is given by x w = L cosq ) + L cosq + q ) 9) y w = L sinq ) + L sinq + q ), where q and q are the joint positions angular positions). Applying the Euler-Lagrange equations, a model according to 7) is obtained, with m = L M + M ) + L M + L L M cos q ) L M m = m = L M + L L M cos q m = L M h = L L M sin q c = h q c = h q + q ) c = h q c =, m m Mq) = c c, Cq, q) = m m c c P q) =, γq) =. For this example, the end-effector of the manipulator is required to follow a circular trajectory of radius r d and

5 center in the origin. o solve this problem the controller exposed in Section 4 is applied. 6. SIMULAION RESULS he simulation results of the example in Section 5 are presented in this section. he two-link manipulator parameters used are shown in able. able. Parameters of the two-link manipulator model. Parameter Values Unit M. kg M. kg L. m L. m he simulations were conducted using the Euler integration method, with a fundamental step size of 4 s. he initial conditions for the two-link manipulator were selected as: x ) = 3π 4 π 4 and x ) =. In addition, the controller gains were adjusted to: c = c =.5, m = m =, and q = q = 6. he desired circular trajectory in the joint coordinates is described by the equations π qd t) t π q d t) = x,d t) = = q d t) π, ) and it corresponds to a circumference of radius.88 m. he following figures show the behavior of the proposed controller. Note that V t) = for t.3 s < c = 3 s Fig. ), which implies that the error variables are exactly zero at the same time Fig. ). Fig. 3 shows the control signal torque) versus time. Finally, from Fig. 4, it can be seen the reference tracking in rectangular coordinates. Error variables st comp. e t) rad nd comp. e t) rad st comp. e t) rad/s nd comp. e t) rad/s Fig.. Error variables. First component of e dark gray and thick), second component of e black and dashed), first component of e light gray and solid) and second component of e black and solid). Control signals ime s ime s Fig. 3. Control signal. First component gray and solid) and second component black and solid) Start of desired trajectory Lyapunov function V x 4 y... Start of actual trajectory ime s Fig.. Lyapunov function V. Note that V t) = for t > c = 3 s Fig. 4. Actual trajectory x w, y w ) black and solid) and desired trajectory x w,d, y w,d ) black and dashed). x

6 7. CONCLUSION In this paper the problem of predefined-time exact tracking in fully actuated mechanical systems was solved by means of a continuous second-order predefined-time backstepping controller. his controller was constructed as an application of continuous first-order predefinedtime stabilizing functions. o show the feasibility of the proposed controller, it was implemented over a two-link planar manipulator. he numerical simulations showed a good performance. ACKNOWLEDGEMENS Esteban Jiménez acknowledges to CONACy, México for the MSc scholarship number REFERENCES Andrieu, V., Praly, L., and Astolfi, A. 8). Homogeneous approximation, recursive observer design, and output feedback. SIAM Journal on Control and Optimization, 474), Bhat, S. and Bernstein, D. ). Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 383), Cruz-Zavala, E., Moreno, J., and Fridman, L. ). Uniform second-order sliding mode observer for mechanical systems. In Variable Structure Systems VSS), th International Workshop on, 4 9. Fraguela, L., Angulo, M., Moreno, J., and Fridman, L. ). Design of a prescribed convergence time uniform robust exact observer in the presence of measurement noise. In Decision and Control CDC), IEEE 5st Annual Conference on, Haimo, V. 986). Finite time controllers. SIAM Journal on Control and Optimization, 44), Hardy, G.H., Littlewood, J.E., and Pólya, G. 934). Inequalities. Cambridge University Press, edition. Jiménez-Rodríguez, E., Sánchez-orres, J.D., Gómez- Gutiérrez, D., and Loukianov, A.G. 6). Predefinedtime tracking of a class of mechanical systems. In 6 3th International Conference on Electrical Engineering, Computing Science and Automatic Control CCE), 5. doi:.9/iceee Kokotovic, P.V. 99). he joy of feedback: nonlinear and adaptive. IEEE Control Systems, 3), 7 7. doi:.9/ Krstić, M., Kanellakopoulos, I., and Kokotović, P. 995). Nonlinear and adaptive control design. Adaptive and learning systems for signal processing, communications, and control. Wiley. Mitrinovic, D.S. 97). Analytic Inequalities. Springer- Verlag Berlin Heidelberg, edition. doi:.7/ Moulay, E. and Perruquetti, W. 5). Lyapunovbased approach for finite time stability and stabilization. In Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC 5, doi:.9/cdc Moulay, E. and Perruquetti, W. 6). Finite-time stability and stabilization: State of the art. In C. Edwards, E. Fossas Colet, and L. Fridman eds.), Advances in Variable Structure and Sliding Mode Control, volume 334 of Lecture Notes in Control and Information Science, 3 4. Springer Berlin Heidelberg. Polyakov, A. ). Nonlinear feedback design for fixedtime stabilization of linear control systems. IEEE ransactions on Automatic Control, 578), 6. Polyakov, A. and Fridman, L. 4). Stability notions and Lyapunov functions for sliding mode control systems. Journal of the Franklin Institute, 354), Special Issue on - Advances in Variable Structure Systems and Sliding Mode Algorithms. Roxin, E. 966). On finite stability in control systems. Rendiconti del Circolo Matematico di Palermo, 53), Sánchez-orres, J.D., Sánchez, E.N., and Loukianov, A.G. 4). A discontinuous recurrent neural network with predefined time convergence for solution of linear programming. In IEEE Symposium on Swarm Intelligence SIS), 9. Sánchez-orres, J.D., Sanchez, E.N., and Loukianov, A.G. 5). Predefined-time stability of dynamical systems with sliding modes. In American Control Conference ACC), 5, Sánchez-orres, J.D., Jiménez-Rodríguez, E., Gómez- Gutiérrez, D., and Loukianov, A.G. 6). Non- Singular Predefined-ime Stable Manifolds. In 6 XVII Latin American Conference in Automatic Control, Medellín, Colombia. Utkin, V.I. 99). Sliding Modes in Control and Optimization. Springer Verlag. Utkin, V.I., Guldner, J., and Shi, J. 9). Sliding Mode Control in Electro-Mechanical Systems, Second Edition Automation and Control Engineering). CRC Press, edition. Appendix A. SOME IMPORAN INEQUALIIES In this appendix, some important inequalities are reviewed. hese results were taken from Hardy et al., 934) and Mitrinovic, 97). A. Chebyshev s inequality heorem 6. Let n N. If a = a,..., a n ) and b = b,..., b n ) are two similarly ordered real sequences, i.e., a a n and b b n or a a n and b b n. hen, the following inequality holds n a i b i n n a i b i. n i= i= i= Corollary 7. Let x, x R be two real numbers and f, g : R R be two similarly ordered functions, i.e., if x x, then fx ) fx ) and gx ) gx ), or fx ) fx ) and gx ) gx ). hen, fx )gx ) + fx )gx ) fx ) + fx )) gx ) + gx )).

7 A. Inequality of arithmetic and geometric means heorem 8. Let n N. If a = a,..., a n ) is a sequence of positive numbers, then n ) n a i n a i. n i= i= Corollary 9. Let x, x R + be two positive real numbers, then x + x x x ). Appendix B. PROOF OF HEOREM 4 Some lemmas are to be stated and proved before proving heorem 4. Lemma. Let < q and consider the function f q : R + {} R + {} defined by f q x) = exp x q ) x q. hen, the function f q is increasing with respect to q, i.e., if < q q, then for x R + {} f q x) f q x). Proof. Note that for any < q, f q ) =. hen, for < q q f q ) = = f q ). Furthermore, f q x) q In any case fqx) q with respect to q. = exp x q ) x x q lnx) { >, < x < =, x = >, x >., i.e., the function f q is increasing Lemma. Let < q and x, x R + {}. hen the following inequality holds exp x q ) x q + exp x q ) x q exp x q ) + exp xq ) x q + x q. Proof. Define the functions fx) = exp x q ) and gx) = x q. Note that dfx) = q exp xq ) x q > and dgx) dx = q)x q >. dx hen, since both functions f, g are increasing, the result is a direct consequence of Corollary 7. Lemma. Let q and x, x R + {}. hen, x q + xq x + x ) q. Proof. Consider the function ϵ : {R + {}} {R + {}} R, defined by ϵx, y) = x q + y q x + y) q. It is to be proved that ϵx, x ). First of all, note that ϵx, ) = ϵ, y) = for all x, y R + {}. Furthermore, since q, the partial derivatives ϵx, y) = q x q x + y) q, x and ϵx, y) = q y q x + y) q, y for x, y R +. Hence, ϵx, x ) and the proof is concluded. With the above results, Proof. Of heorem 4) From 8) V = c q exp e mq ) e m q) c q exp e mq ) e m q). Defining c = max{ c, c } and q M = max{q, q }, the following holds V exp e mq ) e m q) + c q M exp e mq ) e m q). Now, defining q m = min{q, q } and applying Lemma to the right side of the above inequality, it yields V exp e mqm ) e m qm) + c q M exp e mqm ) e m qm). Using Lemma, the following inequality is obtained V exp e mqm ) + exp e mqm ) c q M e m qm) + e. m qm) Since expx) > for all x R, a direct application of Corollary 9 yields V exp c q M e mqm + e mqm ) e m qm) + e ). m qm) Finally, defining r = q M qm and, since < q m q < <, using Lemma V exp r c q m e m + e m ) qm e m + e m ) qm) = ) exp r c q m V qm V qm). Hence, heorem 4 follows from Lemma 5.