On Optimal Predefined-Time Stabilization

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1 Instituto Tecnológico y de Estudios Superiores de Occidente Repositorio Institucional del ITESO rei.iteso.mx Departamento de Matemáticas y Física DMAF - Artículos y ponencias con arbitraje 6- On Optimal Predefined-Time Stabilization Jiménez-Rodríguez, Esteban; Sánchez-Torres, Juan D.; Loukianov, Alexander Jiménez-Rodrguez, E., Sánchez-Torres, J.D. and Loukianov, A. "On Optimal Predefined-Time Stabilization". XVII Latin American Conference of Automatic Control, IFAC, Medellín, Colombia, 6, pp Enlace directo al documento: Este documento obtenido del Repositorio Institucional del Instituto Tecnológico 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 CHAPTER. NONLINEAR SYSTEMS On Optimal Predefined-Time Stabilization Esteban Jiménez-Rodríguez Juan Diego Sánchez-Torres Alexander G. Loukianov Department of Electrical Engineering, CINVESTAV-IPN, Av. del Bosque 45 Col. El Bajío C.P. 459, Guadalajara, Jalisco, México. Department of Mathematics and Physics, ITESO, Periférico Sur Manuel Gómez Morín 8585 C.P. 4564, Tlaquepaque, Jalisco, México. Abstract: This paper addresses the problem of optimal predefined-time stability. Predefinedtime stable systems are a class of fixed-time stable dynamical systems for which the minimum bound of the settling-time function can be defined a priori as a explicit parameter of the system. Sufficient conditions for a controller to solve the optimal predefined-time stabilization problem for a given system are provided. These conditions involve a Lyapunov function that satisfy both a certain differential inequality for guaranteeing predefined-time stability and the steady-state Hamilton-Jacobi-Bellman equation for guaranteeing optimality. Finally, this result is applied to the predefined-time optimization of the sliding manifold reaching phase. Keywords: Hamilton-Jacobi-Bellman Equation, Lyapunov Functions, Optimal Control, Predefined-Time Stability.. INTRODUCTION Finite-time stable dynamical systems provide solutions to applications which require hard time response constraints. Important works involving the definition and application of finite-time stability have been carried out in Roxin 966); Haimo 986); Utkin 99); Bhat and Bernstein ); Moulay and Perruquetti 5, 6). Nevertheless, this finite stabilization time is often an unbounded function of the initial conditions of the system. Making this function bounded to ensure the settling time is less than a certain quantity for any initial condition may be convenient, for instance, for optimization and state estimation tasks. With this purpose, a stronger form of stability, in which the convergence time presents a class of uniformity with respect to the initial conditions, called fixed-time stability was introduced. The notion of fixedtime stability is presented in Andrieu et al. 8) for homogeneous systems and it was proposed in Cruz-Zavala et al. ); Polyakov ); Polyakov and Fridman 4) for systems with sliding modes. When fixed-time stable dynamical systems are applied to control or observation, it may be difficult to find a direct relationship between the gains of the system and the upper bound of the convergence time; thus, tuning the system in order to achieve a desired maximum stabilization time is not a trivial task. A simulation-based approximation to select the values of the tuning parameters is proposed in Fraguela et al. ) under the concept of prescribedtime stability; this method permits to design robust sliding differentiators for noisy signals by expressing the gains as functions of the desired settling time. Therefore, prescribed-time stable systems present a way to surmount the tuning problem. However, this prescribed time usually constitutes a conservative estimation of the upper bound of the convergence time; that is, the prescribed time is commonly larger, maybe quite larger, than the true amount of time the system takes to converge. To overcome the above, another class of dynamical systems which exhibit the property of predefined-time stability, have been studied Sánchez-Torres et al., 4; Sánchez- Torres et al., 5). For this systems the prescribed-time stability coincides with the fixed-time stability when the true settling time is considered. The upper bound for the convergence time of the proposed kind of systems appears explicitly in their dynamical equations; in particular, it equals the reciprocal of the system gain. This bound is not a conservative estimation but truly the minimum value that is greater than all the possible exact settling times. All the mentioned properties of predefined-time stable systems are characterized by a suitable Lyapunov theorem. On the other hand, the infinite-horizon, nonlinear nonquadratic optimal asymptotic stabilization problem was addressed in Bernstein 993). The main idea of the results are based on the condition that a Lyapunov function for the nonlinear system is at the same time the solution of the steady-state Hamilton-Jacobi-Bellman equation, guaranteeing both asymptotic stability and optimality. Nevertheless, returning to the first paragraph idea, the finite-time stability is a desired property in some applications, but optimal finite-time controllers obtained using the maximum principle do not generally yield feedback controllers. In this sense, the optimal finite-time stabilization is studied in Haddad and L Afflitto 6), as an extension of Bernstein 993). Since the results are based on the framework developed in Bernstein993), the controllers obtained are in fact feedback controllers. 37

3 CHAPTER. NONLINEAR SYSTEMS Consequently, as an extension of the ideas presented in Bernstein 993); Sánchez-Torres et al. 5); Haddad and L Afflitto 6), this paper addresses the problem of optimal predefined-time stabilization, namely the problem of finding a state-feedback control that minimizes certain performance measure, guaranteeing at the same time predefined-time stability of the closed-loop system. In particular, sufficient conditions for a controller to solve the optimal predefined-time stabilization problem for a given system are provided. These conditions involve a Lyapunov function that satisfy both a certain differential inequality for guaranteeing predefined-time stability and the steadystate Hamilton-Jacobi-Bellman equation for guaranteeing optimality. Finally, this result is applied to the predefinedtime optimization of the sliding manifold reaching phase. In the following, Section presents the mathematical preliminaries needed to introduce the proposed results. Section 3 exposes the main results of this paper, which are the sufficient conditions for a controller to solve the optimal predefined-time stabilization problem and a particularization to affine systems. Section 4 shows the application of the obtained results to the predefined-time optimization of the sliding manifold reaching phase, and the simulation results are shown in Section 5. Finally, Section 6 presents the conclusions of this paper.. MATHEMATICAL PRELIMINARIES. Predefined-Time Stability Consider the system ẋt) = fxt)), x) = x, ) where x R n is the system state and f : R n R n is a nonlinear function such that f) =, i.e.the origin is an equilibrium point of ). First, the concepts of finite-time, fixed-time and predefined-time stability are reviewed. Definition.. Polyakov, ) The 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 Tx ) : xt,x ) =, where T : R n R + {}. Definition.. Polyakov, ) The origin of ) is fixedtime stable if it is globally finite-time stable and the settling-time function is bounded, i.e. T max > : x R n : Tx ) T max. Remark.. Note that there are several choices for T max. For instance, if the settling-time function is bounded by T m, it is also bounded by λt m for all λ. This motivates the following definition. Definition.3. Sánchez-Torres et al., 4) Let T be the set of all the bounds of the settling time function for the system ), i.e., T = {T max > : x R n : Tx ) T max }. ) The minimum bound of the settling-time function T f, is defined as: T f = inft = sup Tx ). 3) x R n Definition.4. Sánchez-Torres et al., 4) For the case of fixed time stability when the time T f defined in 3) can be tuned by a particular selection of the parameters of the system ), it is said that the origin of the system ) is predefined-time stable. The following Lyapunov-like lemma provides a characterization of predefined-time stability. Lemma.. Sánchez-Torres et al., 4) Assume there exist a continuous radially unbounded function V : R n R + {}, and real numbers α > and < p, such that: V) = 4) Vx) >, x, 5) andthederivativeofv alongthetrajectoriesofthesystem ) satisfies V α p expv p )V p. 6) Then, the origin is globally predefined-time stable for ) and T f = α.. Optimal Control Consider the controlled system ẋt) = fxt),ut)), x) = x, 7) where x R n is the system state, u R m is the system input,whichisrestrictedtobelongtoacertainsetu R m of the admissible controls, and f : R n R m R n is a nonlinear function with f,) =. The system 7) is to be controlled to minimize the performance measure Jx,u )) = tf Lxt),ut))dt, 8) where L : R n R m R is a continuous function, assumed to be convex in u. To this end, define the minimum cost function as J xt),t) = min u U { tf t } Lxτ),uτ))dτ. 9) Defining the Hamiltonian, for p R n usually called the costate) Hx,u,p) = Lx,u)+p T fx,u), ) the Hamilton-Jacobi-Bellman HJB) equation can be written as = min u U { H x,u, J x,t) )} T + J x,t), ) t and it provides a sufficient condition for optimality. For infinite-horizon problems limit as t f ), the cost does not depend on t anymore and the partial differential equation ) reduces to the steady-state HJB equation = min u U H x,u, J x) T ). ) 3. OPTIMAL PREDEFINED-TIME STABILIZATION The main result of this paper is presented in this section. First, the notion of optimal predefined-time stabilization is defined. 38

4 CHAPTER. NONLINEAR SYSTEMS Definition 3.. Consider the optimal control problem for the system 7) where min Jx,u )) = u UT c) Lxt), ut))dt 3) UT c ) = {u ) : u ) stabilizes 7) in a predefined time T c }. This problem is called the optimal predefined-time stabilization problem for the system 7). The following theorem gives sufficient conditions for a controller to solve this problem. Theorem 3.. Assume there exist a continuous radially unbounded function V : R n R + {}, real numbers T c > and < p, and a control law φ : R n R m such that: V) = 4) Vx) >, x, 5) φ ) = 6) fx,φ x)) T c p expv p )V p 7) ) H x,φ x), T = 8) H x,u, Then, with the feedback control u ) = φ x )) = arg min u UT c) H T ), u UT c ). 9) x,u, T ), ) the origin of the closed-loop system ẋt) = fxt),φ xt))) ) is predefined-time stable with T f = T c. Moreover, the feedbackcontrollaw)minimizesjx,u ))inthesense that Jx,φ x ))) = min Jx,u )) ) u UT c) = Vx ), 3) i.e., the feedback control law ) solves the optimal predefined-time stabilization problem for the system 7). Proof. Predefined-time stability with predefined time T c follows directly from the conditions 4)-7) and applying the Lemma. to the closed-loop system ). Toprove3),letxt)bethesolutionofsystem).Then, Vxt)) = fxt),φ xt))). From the above and 8) it follows Lxt),φ xt))) = Lxt),φ xt)))+ fxt),φ xt))) Vxt)) Hence, = H = Vxt)). xt),φ xt)), T ) Vxt)) Jx,φ x ))) = Vxt))dt = lim t Vxt))+Vx ) = Vx ). Now, to prove ), let u ) UT c ) and let xt) be the solution of 7), so that Vxt)) = fxt),ut)). Then Lxt),ut)) = Lxt),ut))+ fxt),ut)) Vxt)) ) = H xt),ut), T Vxt)). Since u ) stabilizes 7) in predefined time T c, using 8) and 9) we have ) ] Jx,u )) = H xt),ut), T Vxt)) dt = lim Vxt))+Vx )+ t ) H xt),ut), T dt ) = Vx )+ H xt),ut), T dt Vx ) = Jx,φ x ))). Remark 3.. The conditions 8) and 9) together are exactly the steady-state Hamilton-Jacobi-Bellman equation ). Remark 3.. It is important that the optimal predefinedtime stabilizing controller u = φ x) characterized by Theorem 3. is a feedback controller. Although Theorem 3. provides sufficient conditions for a controller to solve the optimal predefined-time stabilization problem for a given system, it does not provide a closed form expression for the feedback controller. Instead, the feedback controller is obtained by solving ). To obtain a closed form expression for the controller, the result of Theorem 3. is specialized to affine systems of the form ẋt) = fxt))+bxt))ut), x) = x, 4) where x R n is the system state, u R m is the system control input, f : R n R n is a nonlinear function with f) = and B : R n R n m. The performance integrand is also specialized to Lx,u) = L x)+l x)u+u T R x)u, 5) where L : R n R, L : R n R m and R : R n R m m is a positive definite matrix function. The following corollary of Theorem 3. provides an inverse optimal controller which solves the optimal predefinedtime stabilization problem for the affine system 4) with performance integrands of the form 5). 39

5 CHAPTER. NONLINEAR SYSTEMS Corollary 3.. Assume there exist a continuous radially unbounded function V : R n R + {}, and real numbers T c > and < p such that V) = 6) Vx) >, x, 7) fx)+bx) R x) L x)+ Bx) ] T ]] T c p expv p )V p 8) L ) = 9) L x)+ fx) L x)+ ] 4 Bx) R x) L x)+ ] T Bx) =. 3) Then, with the feedback control u = φ x) = R x) L x)+ ] T Bx), 3) the origin of the closed loop system ẋt) = fxt))+bxt))φ xt)) 3) is predefined-time stable with T f = T c. Moreover, the performance measure Jx,u )) is minimized in the sense of ) and Jx,φ x ))) = Vx ), 33) i.e., the feedback control law 3) solves the optimal predefined-time stabilization problem for the system 4). Proof. We can see that the hypotheses of Theorem Theorem 3. are satisfied. The control law 3) follows from u H x,u, )] = with Lx,u) specialized to 5). Then, setting u = φ x) as in 3), the conditions 6), 7) and 8) become the hypotheses 4), 5) and 7), respectively. The hypothesis 6) follows from 9). )] Since φ x) satisfies u H x,u, =, and u=φ x) noticing that 3) can be rewritten in terms of φ x) as L x)+ fx) φ T x)r x)φ x) =. 34) the hypothesis 8) is directly verified. Finally, from 8), 3) and the positive definiteness of R x) it follows H x,u, ) = Lx,u)+ fx)+bx)u] = Lx,u)+ fx)+bx)u] Lx,φ x)) fx)+bx)φ x)] = L x)+ ] Bx) u φ x))+ u T R x)u φ T x)r x)φ x) = φ T x)r x)u φ x))+ u T R x)u φ T x)r x)φ x) = u φ x)] T R x)u φ x)], which is the hypothesis 9). Applying Theorem 3., the result is obtained. Remark 3.3. The feedback controller 3) provided by Corollary 3. is an inverse optimal controller in the following sense: instead of solving the steady-state HJB equation directly to minimize some given performance measure, it is defined a family of predefined-time stabilizing controllers that minimize a certain cost function. In this case, one can flexibly specify L x) and R x), while from 34) L x) is parametrized as L x) = φ T x)r x)φ x) fx). 35) Remark 3.4. It is not always easy to satisfy the hypotheses 6)-3) of Corollary 3.. However, for affine systems of relative degree one the functions L x) and R x) can be easily chosen to fulfill these conditions. This motivates the following section. 4. INVERSE OPTIMAL PREDEFINED-TIME STABLE REACHING LAW In this section, first, some basic concepts corresponding to integral manifolds and sliding mode manifolds are reviewed. For this purpose, consider again the autonomous unforced system ) ẋt) = fxt)), x) = x, Definition 4.. Drakunov and Utkin, 99) Let σ : R n R m be a smooth function, and define the manifold S = {x R n : σx) = }. If for an initial condition x S, the solution xt,x ) S for all t, the manifold S is called an integral manifold. Definition 4.. Drakunov and Utkin, 99) If there is a nonempty set N R n S such that for every initial condition x N, there is a finite time t s > in which the system state reaches the manifold S then the manifold S is called an sliding mode manifold. Remark 4.. A sliding mode on a certain sliding manifold can only appear if f is a non-smooth possibly discontinuous) function. For this case, the solutions of ) are understood in the Filippov sense Filippov, 988). With the above definitions, the main objective of the controller is to optimally drive the trajectories of affine system 4) to the set S in a predefined time. The function σ : R n R m is selected so that the motion of the system 4) restricted to the sliding manifold σx) = has a desired behavior. The dynamics of σ are described by σt) = axt))+gxt))ut), σx)) = σ, 36) where ax) = σ σ fx) and Gx) = Bx). It is assumed that σx) is selected such that the matrix Gx) R m m has inverse for all x R n. It means that the system 36) has relative degree one. Now, consider the optimal predefined time stabilization problem 3) for the system 36). The aim is to choose the functions V, L and R such that the hypotheses of Corollary 3. are satisfied. To this end, assume that Vσ) 3

6 CHAPTER. NONLINEAR SYSTEMS is a Lyapunov function candidate. Its derivative along the trajectories of the system 36) closed loop with 3) σ = ax)+gx)φ x), is calculated as V = σ ax)+gx)φ x)] ax)+gx) = σ Then, choosing )] R x)gt x) T. σ R x)lt x) L x) = a T x) G x) ] T R x), 37) R x) = T cp exp V p σx))) G T x)gx) ], 38) the derivative of V becomes V = T c p expv p ) σ. Finally, V must be chosen such that σ = V p. 39) It can easily be checked that Vσ) = c i σ T σ) i >, σ 4) with i = p+ and 4i c = 4c p+) =, satisfies 39). Example 4.. Consider a pendulum system with Coulomb friction ẋ = x, ẋ = g L sinx ) V s J x P s J signx )+ u, 4) J where x is the angular position, x is the angular velocity, u is the input torque, J is the moment of inertia, g is the gravity acceleration, L is the length of the pendulum, and P s and V s are friction constants. Due to the structure of the model 4), a good candidate for σ is σx) = x +kx with k >. The dynamics of σ are described by the equation 36) with ax) = g L sinx ) V s J x Ps J signx )+kx and Gx) = J. The functions V, R and L are selected according to 37)-4) as Vσ) = c p+ σ p+, R x) = T cp J exp c p p+ σ p p+ ), and L x) = T cp J exp c p p p+ σ p+ ) g L sinx ) V s J x ] P s J signx )+kx, and u = φ x) is implemented as in 3). Finally, according to 35) the resulting function L is L x) = J 4T c p expc p p p+ σ p+ ) L x)+ ] J σ σ ax). 5. SIMULATION RESULTS The simulation results of the Example 5. are presented in this section. The pendulum parameters are shown in Table. Table. Parameters of the pendulum model 4). Parameter Values Unit M kg L m J = ML kg m V s. kg m s P s.5 kg m s g 9.8 m s The simulations were conducted using the Euler integration method, with a fundamental step size of 3 s. The initial conditions for the system 4) were selected as: x ) = π/rad and x ) = rad/s. In addition, the controller gains were adjusted to: T c =, k = and p = /. sigmat) time s] Figure. Function σxt)). Jt) Jt) Vs)) time s] Figure. Function Jt) = t L +L u+u T R u]dτ. Note that σt) = for t.87 s < T c = s Fig. ). Once the system states slide over the sliding manifold σx) =, this motion is governed by the reduced order system ẋ t) = kx t) = x. This imply that the system state tends exponentially to zero at a rate of k Fig. 3). Fig. 4 shows the control signal torque) versus time. Finally, from Fig., it can be seen 3

7 CHAPTER. NONLINEAR SYSTEMS xt) x t) x t) time s] Figure 3. Evolution of the states. ut) time s] Figure 4. Control input. thatthecostasafunctionoftimegrowsquicklytoasteady state value, corresponding to Vσ)). 6. CONCLUSION In this paper, the problem of optimal predefinedtime stability was addressed. Sufficient conditions for a controller to guarantee both predefined-time stability and optimality were provided. The results were applied to the predefined-time optimization of the sliding manifold reaching phase. This application was illustrated by an example, which was simulated. ACKNOWLEDGEMENTS Esteban Jiménez acknowledges to CONACyT, 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), Bernstein, D.S. 993). Nonquadratic Cost and Nonlinear Feedback Control. International Journal of Robust and Nonlinear Control, 3), 9. 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. Drakunov, S.V. and Utkin, V.I. 99). Sliding mode control in dynamic systems. International Journal of Control, 55, Filippov, A.F. 988). Dierential equations with discontinuous righthand sides. Kluwer Academic Publishers Group, Dordrecht. 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, Haddad, W.M. and L Afflitto, A. 6). Finite-Time Stabilization and Optimal Feedback Control. IEEE Transactions on Automatic Control, 64), doi:.9/tac Haimo, V. 986). Finite time controllers. SIAM Journal on Control and Optimization, 44), 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 Transactions 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-Torres, 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-Torres, 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. Utkin, V.I. 99). Sliding Modes in Control and Optimization. Springer Verlag. 3