Sliding mode control of continuous time systems with reaching law based on exponential function
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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Sliding mode control of continuous time systems with reaching law based on exponential function To cite this article: Piotr Gamorski 15 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - Sliding mode control of direct coupled interleaved boost converter for fuel cell W Y Wang, Y H Ding, X Ke et al. - Lateral mode control in edge-emitting lasers with modified mirrors A Payusov, A Serin, I Mukhin et al. - Two-Link Flexible Manipulator Control Using Sliding Mode Control Based Linear Matrix Inequality Zulfatman, Mohammad Marzuki and Nur Alif Mardiyah This content was downloaded from IP address on /3/1 a5:
2 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 Sliding mode control of continuous time systems with reaching law based on exponential function Piotr Gamorski Institute of Automatic Control Technical University of Lódź, Lódź, Poland pgamorski@wp.pl Abstract. In this paper a pseudo-sliding mode control is proposed by introducing a continuous and smooth input signal in order to guarantee both chattering elimination and boundedness of sliding variable derivative in the presence of non-zero external disturbance. For this purpose, having fixed a suitable sliding manifold, a homogeneous differential equation describing the sliding variable evolution is considered. It is discussed later in this paper that the input signal formed on the basis of this equation provides asymptotic convergence of the sliding variable and its derivative to zero as well as the asymptotic stability of the non-linear system in the absence of external disturbance. The dynamics of the system affected by non-zero external disturbance make the state vector converge to domains in a vicinity of the origin at the exponential rate, as the norm of arbitrary trajectory is limited to decreasing exponential function. In order to expand the variety of controllers based on a reaching law and providing the above-mentioned properties, a certain class of functions is presented. 1. Introduction In many practical applications it is very difficult to obtain an accurate model reflecting a physical phenomenon that needs to be controlled. Discrepancy between the actual dynamics and its mathematical model induced by complexity and non-linearity of the system, timevarying or unknown parameters as well as unmodelled structures, forced engineers to introduce a new control technique combating aforementioned modelling difficulties. One widely adopted approach to both robust and computationally effective control of high-order non-linear plants is the sliding mode control (SMC). The idea of SMC is based on the application of a control signal always moving the state vector toward a region adjacent to a differentiable manifold [1, 11] and allowing the representative point to remain on the sliding surface in its close vicinity thereafter. Once the system state hits a manifold, the dynamics of the system becomes insensitive to some disturbances and uncertainties of the model. The development of these concepts began in the late 195s in the Soviet Union - a significant contribution in the progress of SMC is owed to Utkin [1, ] and Itkis [3]. Since the sliding mode control law is usually not a continuous function of time, thus one needs to take into consideration control chattering problem impeding SMC implementations. Although chattering attenuation received a major boost in the early 199s due to second-order concepts [] and then in s, when higher-order ideas appeared [7], it is still one of the most important aspects in practical applications of SMC. In this paper we consider an alternative control strategy based on a reaching law [, 9]. The designed controller is both continuous and smooth in order to provide complete chattering elimination in the presence of non-zero disturbance. Furthermore, proposed algorithm Content from this work may be used under the terms of the Creative Commons Attribution 3. licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1
3 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 guarantees boundedness of the sliding mode derivative. The price we pay for obtaining such an input signal is a loss of accuracy, as the system trajectories converge to a vicinity of the origin (instead of the origin) in the presence of non-zero external disturbance.. Problem statement We consider the time-varying, nonlinear n-th order dynamic system ẋ 1 = x, ẋ = x 3,..., ẋ n = f(t, x 1,..., x n ) + b u + d (ẋ = dx/dt, t ), (1) where x 1, x,..., x n are the state variables of the system and x = [x 1, x,..., x n ] T is the state vector, t denotes time, u R is the input signal. Function f is a priori known and maps [ ; ) R n into R. Besides, f = f(t, x) is assumed to be bounded and Lipschitz continuous with respect to the second coordinate. Moreover, b is a non-zero constant, while scalar-valued function d defined on [ ; ) is unknown and represents the system external disturbance. Further in this paper, it is assumed that d is continuous almost everywhere (discontinuity set is a measurezero set) [1] and there exists a positive constant D such that Switching function σ : R n R is introduced as follows d D, for all t [ ; ). () σ(x) := c 1 x c n x n, (3) where c 1,..., c n are real constants. Furthermore, it is assumed that the constants c 1,..., c n are chosen so that the eigenvalues of the homogeneous differential equation c 1 x c n x n = () are identical, real and less than zero. Until the end of the next chapter we will be analyzing dynamic system (1), wherein d ( t < ). The differential equation describing the sliding variable evolution is proposed in this brief in the following form ( ) σ = k 1 e σ /σ sign(σ), (5) where k and σ are certain constants greater than zero. Note that the latter equation is satisfied when the following controller is applied u(t, x) = k b ( ) 1 e σ /σ sign(σ) 1 b (c 1x c n 1 x n ) 1 f(t, x). () b The solution of Eq. 5 is determined by the equations ( ) σ = σ ln 1 + e k(t+c)/σ sign(σ), C = σ ( ) k ln e σ()/σ 1 sign[σ( )]. (7) Taking σ as a measure of the distance between a representative point x and a hyperplane H = {x R n : σ(x) = } we obtain, that while representative point is reaching the hyperplane, its velocity of convergence σ is decreasing exponentially. Since the function σ is bounded upper and below by ±k, thus the absolute value of σ never exceeds k. Note that the control function () is both continuous and smooth despite discontinuity of sign function. Although simple calculation shows that lim σ =, lim t σ =, () t
4 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 the smooth feedback controller () cannot provide finite-time convergence of the sliding variable to zero. Convergence of the system trajectories is discussed later in this paper. The use of the controller () for the system carries some mathematical advantage. Indeed, since one can prove that function () has Lipschitz property in the absence of external disturbance d, thus there is no need to understand the solutions of the system in the Filippov sense [5, 1] any longer. Thanks to Picard - Lindel øf theorem [13] the existence and uniqueness of the solution is guaranteed. 3. Asymptotic stability of the system in the absence of external disturbance In this section asymptotic stability of the system including feedback controller () is proven. Let us consider once more dynamic system (1) in the absence of external disturbance d. The idea is to show that every solution of non-homogeneous differential equation c 1 x c n x n = σ, (9) where σ is determined by (7), is asymptotically stable. To do so, one can select a proper Lyapunov - candidate function, however in this very situation it is not an easy task. Therefore, a general law in the theory of ordinary differential equations is proven in this chapter. Definition 1: Consider differential equation ẋ = A x + r, (1) where t (a; b), A : (a; b) L(R n ) is a continuous function, L(R n ) denotes a space of bounded linear operators R n R n and r : (a, b) R n is continuous almost everywhere. Function U : (a; b) L(R n ) defined by the formula where K (t, ) = I and K m+1 (t; ) = s... sm 1 U(t; ) := K m (t; ), (11) m= A(s m )...A(s 1 )A(s)ds m...ds 1 ds = K m (s; )ds (1) is called a resolvent of Eq. 1 [1, 15]. Let φ = φ be a solution of Eq. 1 determined by initial condition φ = φ( ). Note that the function g(t, x) = A x is Lipschitz continuous with respect to the second coordinate, thus every solution can be extended to infinity. Taking into account that U( ; ) = I and U(t; ) = AU(t; ), we obtain that the resolvent is a normalized fundamental matrix of Eq. 1, hence φ = U(t; ) φ. (13) Furhermore, differentiating the following equation φ = U(t; )φ + U(t; s)r(s)ds, (1) one can easily prove that (1) represents a unique solution of (1). Lemma 1: If the trivial solution φ of homogeneous equation associated with Eq. 1 is asymptotically stable, then every solution of Eq. 1 is asymptotically stable. 3
5 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 Proof: Let ψ = ψ and η = η be arbitrary solutions of (1) determined respectively by initial conditions ψ = ψ( ), η = η( ). From (1) we get ψ η = U(t; )ψ + U(t; s)r(s)ds + (15) = U(t; )η U(t; s)r(s)ds = (1) = U(t; )(ψ η ). (17) The latter equation shows that the difference between any two non-homogeneous system solutions is a solution of a homogeneous system, thus taking into account that the trivial solution of homogeneous equation is asymptotically stable, we obtain the following implication ε> δ> ψ( ) η( ) < δ ψ η < ε, for all t. (1) Obviously lim t ψ η =. This concludes the proof. Now we are in a position to formulate the main result of this chapter. Theorem 1: If d ( t < ), then system (1) including control signal given by () is asymptotically stable. Proof: Simple calculation shows that Eq. 9 might be represented in the following form ẋ = A x + r, (19) where x, r are vectors of dimension (n 1) 1, wherein x = [x 1,..., x n 1 ] T and r = [,..., σ] T, A is in turn matrix of dimension (n 1) (n 1) given by the formula 1 1 A = () 1 c 1 c c 3 c n c n 1 Since homogeneous equation corresponding to (9) is asymptotically stable, therefore in particular trivial solution is asymptotically stable. Relying on lemma 1, we obtain asymptotic stability of (9). The theorem is proven.. Uniform and asymptotic boundedness of the system in the presence of non-zero external disturbance Until now, we have been considering system (1) assuming the absence of disturbance function. Suppose now that d, k > D and control law is designed as in (). Under above assumptions, one can easily notice that sliding variable satisfies the following differential equation ( ) σ = k 1 e σ /σ sign(σ) + d. (1) Define a set Σ D := {σ : σ σ ln (1 D/k)}. () Lemma : If there exists time t 1 [ ; ) such that σ(t 1 ) = σ ln (1 D/k), then σ Σ D for all t t 1. Proof: Suppose that there exists time t > t 1 such that σ(t ) > σ ln (1 D/k). Since disturbance function d is continuous almost everywhere, thus there must exist an interval
6 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 (a; b) (t 1 ; t ) and for all t (a; b) the switching function σ is increasing. Hence, σ > if t (a; b). On the other hand, Eq. 1 yields σ < for all t and σ > σ ln (1 D/k), which ultimately leads to a contradiction. Due to the symmetry of the problem, it is enough to consider the case of σ >. The latter lemma means that if sliding variable reaches boundary set Σ D, then it remains there thereafter. The following Lyapunov - candidate function V (σ) = 1 [σ + sign(σ)σ ln (1 D/k)] (3) guarantees the asymptotic convergence of σ to Σ D. Considering V (σ) = 1 σ as a Lyapunov - candidate function, one can easily prove finite-time convergence. Now we are in a position to formulate the main result of the paper. Theorem : Let φ = φ be an arbitrary solution of system (1) in the presence of non-zero external disturbance, determined by initial condition φ = φ( ). Let λ denotes the eigenvalue of A. Then φ σ ln (1 D/k), if t. () λ Proof: The idea is to show, that every solution of non-homogeneous differential equation c 1 x c n x n = σ, (5) where σ is is related to Eq. 1, satisfies condition (). As we have already seen, Eq. 5 is is equivalent to (19). Taking into account the Ważewski s inequality [15, 17] U(t; ) = e A(t ) e λ(t ) for all t [ ; ), () we obtain that φ U(t; ) φ( ) + U(t; s) r(s) ds (7) e λ(t t) φ( ) + e λ(t s) σ(s) ds = () = e λ(t ) φ( ) + e λt e λs σ(s) ds. (9) Obviously, lim t e λ(t ) φ( ) =. Relying on the generalized de l Hospital theorem in the form of Stolz [15], one can easily prove that lim t eλt e λs σ(s) ds = σ ln (1 D/k). (3) λ Note that the existence of the latter limit is guaranteed only when σ / Σ D. Suppose that there exists time t 1 [ ; ) and σ(t 1 ) = σ ln (1 D/k). Simple calculation shows that if t, then 1 φ lim e λ(t t) φ( ) + lim e λt e λs σ(s) ds + (31) t t + lim e λt e λs σ(s) ds = lim e λt e λs σ(s) ds (3) t t t 1 t 1 lim e λt e λs [ σ ln (1 D/k)] ds = σ ln (1 D/k). (33) t t 1 λ 5
7 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 The last equality follows from the generalized de l Hospital theorem in the form of Stolz. The proof of the theorem is complete. It is worth noting that the record lim t φ is formally incorrect as the considered limit does not need to exist, though the statement () remains valid. Corollary 1 : Let φ = φ be an arbitrary solution of system (1) in the presence of non-zero external disturbance, determined by initial condition φ = φ( ). Let λ denotes the eigenvalue of A. Then φ φ( ) e λ(t t) + σ λ ln (1 D/k) for all t [; ). (3) Proof : Since d t e λ(t s) σ(s) ds = σ (35) dt and the function of the upper limit is increasing, hence φ φ( ) e λ(t t) + lim e λ(t s) σ(s) ds. (3) t Computing the latter limit, we obtain the thesis. Corollary : Let φ = φ be an arbitrary solution of system (1) in the presence of non-zero external disturbance, determined by initial condition φ = φ( ). Let λ denotes the eigenvalue of A. If det(a), then for all t [ ; ). Proof : Note that φ φ( ) e λ(t t) + σ( ) A 1 sup ( U(t; ) I ) (37) t [ ; ) φ( ) e λ(t ) + σ( ) A 1 n, (3) φ φ( ) e λ(t t) + e A(t s) σ( ) ds = (39) = φ( ) e λ(t t) A k (t s) k + σ( ) ds. () k! k= As the assumptions of the Fubini s theorem [1] are satisfied, thus A k+1 (t ) k+1 A k (t ) k = I sup ( U(t; ) I ) (1) (k + 1)! k! k= k= t [ ; ) ( ) e λ(t t) + I n, () which concludes the proof. sup t [ ; ) 5. Generalization on the control law In this section a generalization on the differential equation (5) is proposed. Consider function z : R R of variable σ such that z, z is continuous and even, z is differentiable or differentiable excep,
8 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 z σ is continuous or continuous excep, z reaches global minimum in and z() =, z [; ) is increasing, lim σ z(σ) =, lim σ a z(σ)/σ z σ = g <, where a > 1. Let C,1 a,l denotes the class of all functions satisfying the upper conditions. The following controller can be used instead of the conventional control law () u(t, x) = k b ( ) 1 a z(σ)/σ sign(σ) 1 b (c 1x c n 1 x n ) 1 f(t, x). (3) b It is easy to observe that appliance of controller (3) to the system makes the sliding variable satisfy the following differential equation ( ) σ = k 1 a z(σ)/σ sign(σ) + d. () Depending on the choice of z, homogeneous equation associated with the Eq. may be impossible to solve analytically, yet the existence and uniqueness is guaranteed by Picard - Lindel øf theorem. Indeed, if σ >, then P σ = [ ( ) k 1 a z(σ)/σ ] ln(a) sign(σ) = k a z(σ)/σ z σ (5) σ σ and z C,1 a,l, thus P σ is continuous on [; ), which implies that lim σ P σ =, lim σ P σ = k ln(a) g/σ. Hence P σ is bounded and as a result Lipschitz continuous. The same is true for σ <. Selecting a proper Lyapunov - candidate function, one can easily prove that in the absence of external disturbance lim σ =, lim t σ =. () t From the foregoing considerations, we obtain that controller (3) makes the sliding variable and its derivative possess the characteristics and properties of the variable that satisfies (5). Therefore, all the lemmas and theorems discussed so far may be prescribed to an accuracy of some constants. Indeed, consider a set σ min := inf {σ (; ) : z(σ) > σ log a (1 D/k)}, (7) where k > D. Note that z C,1 a,l implies the existence of σ 1 [, ) such that function z min = z [σ1 ; σ min ] is strictly increasing and as a result injective. Hence, Σ a D := { σ R : σ z 1 min ( σ log a (1 D/k)) } () is used instead of Σ D. Obviously, if the function z is injection on [; ), then it is enough to consider z 1 instead of z 1 min Example For the mass-spring-damper system with friction [1] modeled by mẍ + b ẋ + b 1 sign(ẋ) + k 1 x = u + d, (9) 7
9 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 where x is the displacement from equilibrium, u is the external force applied to system, m is the mass of the block, b is damping constant, b 1 is friction, k 1 is force constant of spring, d denotes unknown external disturbance, the control law u ( u(x) = km 1 e σ) sign(σ) mẋ + b ẋ + b 1 sign(ẋ) k 1 x (5) applied to (9) makes the sliding variable satisfy () (for the purposes of this example, we will call (5) non-conventional control law), if the switching function is given by σ(x) = x + ẋ and a = exp, z(σ) = σ. The control system is simulated with m = 1 [kg], b =.1 [kg/s], b 1 =.5 [kg/s ], k 1 =.5 [kg/s], k =, σ = 1, d = sin, D = 1, x() = 7 [m] and ẋ() = 1 [m/s]. Furthermore, we demonstrate the results of the simulation with the same control system simulated with conventional sliding mode control law given by u(x) = x 3 sign(x + ẋ). (51) Lines located above and below the hyperplane σ(x) = (see Fig. 1) are given respectively by 1 1 x x x x 1 Figure 1: Phase portrait obtained using non-conventional control law Figure : Phase portrait obtained using conventional control law sigma sigma Figure 3: Sliding variable obtained by using non-conventional control law Figure : Sliding variable obtained by using conventional control law x = x 1 σ ln (1 D/k) = x 1 ln (.5), (5) wherein x 1 = x and x = ẋ. Lines located in Fig. 3 are determined respectively by σ = ln (1 D/k) = ln (.5). (53)
10 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 1 5 u 1 u Figure 5: Control signal generated according to non-conventional sliding mode control Figure : Control signal generated according to conventional sliding mode control x 1 x Figure 7: System output x 1 obtained using non-conventional control law Figure : System output x 1 obtained using conventional control law 1 1 x x Figure 9: System output x obtained using non-conventional control law Figure 1: System output x obtained using conventional control law. Conclusions In this paper an alternative sliding mode control for continuous time systems has been proposed. The method employs a reaching law - the strategy has been designed so that the rate of change of the sliding variable is always bounded and decreases exponentially with the sliding variable 9
11 1th European Workshop on Advanced Control and Diagnosis (ACD 15) Journal of Physics: Conference Series 59 (15) 1 doi:1.1/17-59/59/1/1 decrease. On the one hand, presented algorithm enforces the convergence of the representative point to the origin while considering unperturbed systems. On the other hand, for systems affected by external disturbance it has been proven that sliding variable enters in finite time a specified set and never leaves it. As a result, every trajectory starting from any initial position converges to a vicinity of the origin and the system becomes uniformly and asymptotically bounded. Furthermore, application of the proposed reaching law has resulted in chattering elimination. Techniques demonstrated in this article can be adapted for discrete time systems. References [1] Utkin V Variable structure systems with sliding modes 1977 IEEE Trans. on Automatic Control, vol., pp. 1- [] Utkin V Sliding mode and their applications in variable structure control 197 MIR [3] Itkis U Control systems of variable structure 197 John Wiley & Sons [] DeCarlo R S, Żak S and Mathews G Variable structure control of nonlinear multivariable systems 19 Proc. of IEEE, vol. 7, pp. 1-3 [5] Shtessel Y, Edwards C, Fridman L and Levant Arie Sliding mode control and observation 1 Springer [] Levant A and Levantowsky L V Sliding order and sliding accuracy in sliding mode control 1993 Int. J. Control, pp [7] Levant A Universal SISO sliding mode controllers with finite-time convergence 1 IEEE Trans. on Automatic Control, vol., pp [] Gao W and Hung J C Variable structure control of nonlinear systems: a new approach 1993 IEEE Trans. on Industrial Electronics, vol., pp [9] Bartoszewicz A A new reaching law for sliding mode control of continuous time systems with constraints 15 Trans. of the Institute of Measurement and Control, vol. 37, pp [1] Lee J M Introduction to topological manifolds Springer [11] Hirsch M W Differential topology 197 Springer [1] Joó I and Tallos P The Filipov-Ważewski relaxation theorem revisited 1999 Acta Math pp [13] Gerald T Ordinary Differential equations and dynamical systems 1 American Mathematical Society [1] Grimmer R C Resolvent operators for integral equations in a Banach space 19 American Mathematical Society vol. 73 pp [15] Demidovich B P Lectures on mathematical stability theory 197 Moscow [1] Bruckner A M, Bruckner J B and Thomson B S Real analysis 1997 Prentice-Hall [17] Gil M I On Generalized Wazewski and Lozinski inequalities for semilinear abstract differential-delay equations 199 J. oflnequal. & Appl., vol., pp.55-5 [1] Halliday D, Resnick Robert, Walker J Fundamentals of physics extended 13 Wiley 1
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