Dynamic Integral Sliding Mode Control of Nonlinear SISO Systems with States Dependent Matched and Mismatched Uncertainties

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1 Milano (Italy) August 28 - September 2, 2 Dynamic Integral Sliding Mode Control of Nonlinear SISO Systems with States Dependent Matched and Mismatched Uncertainties Qudrat Khan*, Aamer Iqbal Bhatti,* Qadeer Ahmed* * Controls & Signal Processing Research (CASPR) group Mohammad Ali Jinnah University (M.A.J.U), Islamabad. Abstract: In this paper, the authors propose a novel dynamic integral sliding mode controller for state dependent matched and unmatched uncertainties. An output feedback methodology based designed dynamic controller attenuate the effects of the both matched and unmatched uncertainties along with considerable reduction in chattering phenomena. An integral manifold is used which provides sliding mode without reaching phase and further enhances the robustness of the controller against uncertainties. A Lyapunov candidate is used for the stability analysis against uncertainties. The feasibility of the control law is illustrated via a kinematic car model. Keywords: Sliding Mode Control, Chattering, Mismatched Uncertainty, Robustness.. INTRODUCTION Sliding mode control (SMC) performs a significant role in robust stabilization of uncertain systems. Indeed, a salient is that the system behavior on the sliding surface is unaffected by uncertainties satisfying the matching conditions (Utkin 992). However, there are many systems affected by uncertainties which do not satisfy the matching condition. To solve this problem, various methods are being proposed by Kwan (995), Shyu et al. (998), Zinober (996) and Swaroop et al. (2). Kwan used an adaptive Sliding mode technique for the reduction of the effects of mismatched uncertainties. This method is based on the fact that the matching conditions are required for the reduced order system in sliding mode. Shyu focused on a method which ensures the asymptotic stability of the reduced order system in the presence of mismatched uncertainties with- out Kwan s assumptions. Zinober and Swaroop put forwarded backstepping based SMC design to relax the matching conditions. However, since this technique suffers from explosion of terms, an additional procedure is needed to solve this problem (Swaroop et al. 2). For a system experiencing mismatched uncertainty, Cao and Xu (24), proposed a nonlinear integral-type sliding surface, in the presence of both matched and unmatched uncertainties. The stability of the controlled system with unmatched uncertainties depends on the controlled nominal system, nature and size of the equivalent unmatched uncertainties. Zak et.al. (993), developed a generic condition for the existence and stability of the reduced order sliding motion. Based on the work in (Zak et.al. 993), some dynamic output feedback control schemes have been proposed for robust control of uncertain systems (Kwan 2; Shyu et.al. 2). Bartolini et.al (996), identified some conditions under which, even in the presence of uncertainty, the convergence to the sliding manifold is ensured via the application of a multi-input control. Han Ho Choi (22), proposed a linear matrix inequality (LMI)-based sliding surface design method for integral sliding-mode control of systems which may have mismatched norm bounded uncertainties in the state matrix as well as the input matrix. Further, Park (27), extended Choi s method (22) and proposed a dynamical output feedback variable structure control law to deal with the same problem. Since, the dynamic of sliding surface is always unmeasured states, the high gain control (Kwan 2; Choi 22 and Park 27), is introduced to achieve global convergence. Xiang et al (26), applied an iterative LMI method to avoid high gain problem. Although, these papers have demonstrated the use of use switching control to guarantee the convergence, the system states are not assured to be driven to the sliding surface in finite time. In the aforementioned approaches, robustness is ensured but with a compromise on chattering phenomenon. In this context, Da Silva et.al (29), developed an algorithm in which the existence and the reachability problems have been formulated using a polytopic description in order to tackle mismatched uncertainties with reduced chattering. Wang et al. (29), presented a method to estimate the dynamics of sliding mode control for a class of uncertain system in presence of unmatched parameter variations and external disturbances. Recently, Jeang-Lin Chang (29), proposed a dynamic output feedback sliding-mode control methodology for linear MIMO systems with mismatched norm-bounded uncertainties along with disturbances and matched nonlinear perturbations. The proposed methodology guarantees robust stabilization and sustain the nature of performing disturbance attenuation when the solutions to two algebraic Riccati inequalities can be found. The work of Da selve et al (29), and Chang (29), were developed for linear systems. Khan et.al (2), proposed a control law for uncertain nonlinear systems which robustly compensate against uncertainties and the desired stability point is achieved in finite time. Keeping in view the attributes of output feedback techniques applications; the proposed work is carried out for uncertain Copyright by the International Federation of Automatic Control (IFAC) 3932

2 Milano (Italy) August 28 - September 2, 2 nonlinear systems operating under matched and mismatched uncertainties. In this work, the authors have presented an output feedback based designed dynamic controller for nonlinear systems to attenuate the effects of matched and mismatched uncertainties in the sliding mode. The proposed methodology provides sliding mode without reaching phase with increased robustness from the very beginning of the process, chattering free control input and the other benefit of this methodology is that the control design, (Castanos et.al. 26), is not required for the mismatched uncertainties rejection. The rest of the paper is organized in the following sequence. In Section 2, the problem is formulated and in Section 3 the design methodology of the new control law is discussed. In Section 4, the stability analysis is carried out to ensure the sliding mode convergence. A standard car model is simulated, in Section 5, to certify the attenuation of the effects of both matched and unmatched uncertainties. Section 6 contains the comprehensive concluding remarks followed by references. 2. PROBLEM FORMULATION Consider a nonlinear SISO dynamic system described by the state equation:,,,, (),, where is the state vector, is scalar control input,, and, are smooth vector fields, and, are matched uncertainties,, is mismatch uncertainty and, is sufficiently smooth output function. Assumption 2..,, and, are continuous and bounded with continuous bounded time derivatives,. i.e.,,, where is some positive constant, and,. The aforementioned system in locally differential Input Output (I-O) form) or in companion form (Khan et. al. 2) can be written as follows,,,, φ,u,,,, (2) The representation in (2) is called Local Generalized Controllable Canonical Form (LGCCF). The terms,,, and,, are the uncertainties and their time derivatives. Assumption 2.2. Assume that,,,,,,,, and,, ;,2,,, where, and are positive contants. Furthermore, consider that,,,,,,,, and,,. The nominal system corresponding to system (2) is given by φ,u,, (3) where χ,u, χ,u, φ, u. Definition 2.. The differential input output (I-O) form (or LGCCF) is termed as proper if ). It is single input single output, 2).,, 3). The following regularity condition is satisfied,, (4) A wide class of nonlinear systems can be put into LGCCF form with the addition of compensator term which appears as a chain of integrators (Fliess 99). Definition 2.2. The zero dynamics of the system in (3) are defined as,, (5) The system in (3) is called minimum phase if the zero dynamics defined in (5) are uniformly asymptotically stable. Fliess (988). 3. CONTROL LAW DESIGN The proposed control law for the system (3) is of dynamic nature which can be expressed as (6) The first part, is continuous which stabilizes the system at the equilibrium point and the second part is discontinuous in nature which is named as the dynamic integral control. This effectively rejects the uncertainties. In the next two subsections, the design of, and is demonstrated.. 3. Design of the continuous controller Consider the nominal system in (3) and assume that: Assumption 3.. The system in (2) is considered to be independent of nonlinearities i.e., χ,, and uncertainties in the very beginning and we also suppose that the system operates under only. Then the system (2) results in the following form (Khan et. al 2): 3933

3 Milano (Italy) August 28 - September 2, 2 where For the sake of simplicity, the input is designed as an optimal linear state feedback control (LQR) (7) where state space sector. 3.2 Design of discontinuous control law Before the design of the discontinuous control law it is assumed: Assumption 3.2. The system in (3) is either minimum phase in sense defined in Definition 2.2 or the zero dynamics are stable. Now in order to achieve the desired performance, robust compensation of the uncertainties for all times the discontinuous part is designed using the integral sliding manifold in (8). (8) where is conventional sliding surface which is mathematically defined by with and is the integral term. The time derivative of (8) along (3) yields χ,, (9) Now, choose with the following expression, () Then, (9) becomes, () This initial condition is adjusted in such a way to meet the requirement. The benefit of this manifold enhances the robustness of the controller by eliminating the reaching phase. For the proposed controller the reachability condition is defined as follows, (2) that satisfies the following conditions Now, by comparing () and (2), the expression of dynamic discontinuous controller becomes:, (3) This control law enforces sliding mode along the sliding manifold defined in (8). The constant can be selected according to the subsequent stability analysis and,, can be selected according to the desired application. Thus the final control law can be obtained by inserting (7) and (3) in (6). The resulting control law can be implemented by integrating the derivative of the control k times. 4. STABILITY ANALYSIS The aforementioned dynamic control law effectively rejects the uncertainties and establishes sliding mode without reaching phase. The stability analysis is demonstrated in the subsequent subsections in the presence of both matched and mismatched uncertainties. 4.. The System Operating Under Matched Uncertainties In this case it is considered that the system (2) operates only under matched uncertainties only. To show that this system is asymptotically stabilized in the presence of matched uncertainties; the following theorem is established. Theorem 4.: Consider the nonlinear system in (2) subject to Assumptions 2., 2.2, 3. and 3.2. If the sliding surface is chosen according to (8) and the control law is selected according to (6) with the switching gain satisfying 2 Then, the existence of sliding mode is guaranteed in the presence of matched uncertainties. Proof: To prove that the above controller can maintain sliding mode, differentiating (8) with respect to time using (2), and then substituting (6) and (), one obtains [,,, (4) Let us consider a Lyapunov candidate function. The time derivative of this Lyapunov function along (4) becomes,,, In view of Assumption 2.2, the above expression can be written as (5) The expression (5) holds only if (6) Thus, sliding mode can be guaranteed when the gains are selected according to (6) for the controller given in (6). Proposition 4.: The dynamics of the system (3) with matched uncertainties, with control law (6) and integral manifold (8), in sliding mode is governed by the linear control law. Proof: The time derivative of (8) along (2) with matched uncertainties yields 3934

4 Milano (Italy) August 28 - September 2, 2 φ, u,, (7) Substituting (), the equivalent control law becomes [,,,] (8) Now, using (8) in (2) with matched uncertainties, ones has (9) Thus, it is proved that the system in sliding mode operates under the linear control law. The subscript in (9) shows that the system is in sliding mode The System Operating Under both Matched and Mismatched Uncertainties In this study it is considered that the system (2) operates under both matched and unmatched uncertainties. The objective is to regulate the output of the system in the presence of these uncertainties. To prove that the proposed control law nullifies the effects of these uncertain terms, the following theorem is established. Theorem 4.2: Consider the nonlinear system in (2) subject to Assumptions2., 2.2, 3., and 3.2. If the sliding surface is chosen according to (8) and the control law is selected according to (6), with the switching gain satisfying 2 then the existence of sliding mode is guaranteed in the presence of both matched and unmatched uncertainties. Proof: Consider the time derivative of (6) along (2), and then substituting (6) and (), one obtains,,, Φ,, (2) Let us consider the Lyapunov candidate function and computing the time derivative of this Lyapunov function along (2), yields, Therefore, when the gains of the control law (6) are selected according to (22), sliding mode is guaranteed in the presence of matched and unmatched uncertainties. Proposition 4.2: The dynamics of the system (2), with control law (6) and integral manifold (8), in sliding mode is governed by the linear control law. Proof: Following the same procedure as mentioned in Proposition 4., one can easily come to the result that the nonlinear system operating under matched and unmatched uncertainties is governed by the linear dynamic control law in sliding mode. 5. KINEMATIC CAR MODEL Consider a simple kinematic car model Levant (2) cos, sin, tan,,, (23) where and are the Cartesian coordinates of the rear-axle middle point, x the orientation angle and x the steering angle, u the control input. is the longitudinal velocity (w ms ), and is the distance between the two axles ( 5). The terms,, are matched uncertainties and, are components of the mismatched uncertainty which satisfy Assumptions 2. and 2.2 and these terms contributes to the system uncertainty with the following mathematical expressions.,.3,,.2,.3 3,,. 3, It is known that,, ;,2,3,4 and,.3. Furthermore, consider that,.3, where is some positive constant. The objective is to regulate the output of the car from some initial position to the equilibrium point (origin) in the presence of these uncertainties. The output of interest is with relative degree 3. Consequently, the 4 th, derivative of the output function becomes:,, Φ,, (2) (2) will hold only when 2 (22) Fig.. Kinematic Car Model. This system in differential I-O (LGCCF) form becomes 3935

5 Milano (Italy) August 28 - September 2, 2,,,, φ,u,,,, (24) where,u 2 and 2,,,,,, and and the control law Output x 2 3 2, W. Simulations are carried out to confirm the aforementioned claim of state dependent mismatched uncertainty rejection. The simulation results are displayed in Fig.2 and 3. Form Fig.2 (a), It is clear that the controller effectively corrects the deviation of the system s output in the presence of matched uncertain terms. In addition, the controller effort is chatter free as shown in Fig.2 (b). The sliding surface convergence is displayed in Fig.2 (c). The trajectories of the system s output in the presence of mismatched uncertainties are depicted in Fig.3 (a). These simulation results confirm the robust and chatter free nature of the proposed controller as illustrated in Fig.3 (b). The convergence of the sliding surface is shown in Fig.3 (c). In the very beginning of the process the controller bears a little bit oscillations but in real applications such small oscillation are not harmful for actuators health. The asymptotic convergence of the system trajectories make sure that any type of state dependent mismatched uncertainties can be handled-effectively. The controller gains and the sliding manifolds coefficients are listed in Table. (a) Trajectory with matched uncertainy Nomianl trajectory Control Input u (b) (c).5 s for nominal system s for system with matched uncertainy u for nominal system u for system with matched uncertainy Sliding Surface s Fig. 2. (a).output regulation of the nominal system and of the uncertain system in the presence of matched uncertainty. (b). Control effort of both nominal and uncertain system. (c). Integral sliding manifold convergence of the nominal and uncertain system (a) Output with mismatch uncertainty Output with no uncertainty Output with match uncertainty Output x Control Input u (b) (c).3 s with mismatch uncertainty 6.2 s of nominal system s with matched uncertainty u with match uncertainty -.2 u of nominal system u with mismatch uncertainty Sliding Surface s Fig. 3. (a).output regulation of the nominal system and of the uncertain system in the presence of both matched and mismatched uncertainty. (b). Control effort of both nominal and uncertain system. (c). Integral sliding manifold convergence of the nominal and uncertain system. 3936

6 Milano (Italy) August 28 - September 2, 2 Table. The parameters values of the control law used in the simulations. Constants Values CONCLUSIONS In this work, a novel output feedback dynamic integral sliding mode (DISM) approach is used for a class of SISO nonlinear systems with both matched and mismatched state dependent uncertainties. The uncertain system is asymptotically stabilized in the presence of the above discussed uncertainties. The control law incorporates an integral sliding manifold which guarantee the elimination of the reaching phase. Consequently, it enhances the robustness, considerably eliminates chattering at the system input. The simulation results confirm that the DISM control law effectively corrects the deviations of the outputs of the uncertain system. REFERENCES Andrade-Da Silva, J.M., Edwards, C., and Spurgeon, S.K. (29). Sliding-Mode Output-Feedback Control Based on LMIs for Plants With Mismatched Uncertainties, IEEE Transaction on Ind. Electronics, Vol. 56, No. 9. Bartolini, G., and Ferrara, A. (996). Multi-Input Sliding Mode Control of a Class of Uncertain Nonlinear Systems, IEEE Transaction on Automatic Control, 4, No.. Cao, W. J. and Xu, J. Xin. (24). Nonlinear Integral-Type Sliding Surface for Both Matched and Unmatched Uncertain Systems, IEEE Transaction on Automatic Control, 49, No. 8. Castanos, F. and Fridman, L. (26). Analysis and design of integral sliding manifolds for systems with unmatched perturbations, IEEE Transaction on Automatic Control, vol.5, No.5. Chang, J. Lin. (29). Dynamic Output Integral sliding mode control with disturbance attenuation, IEEE Transaction on Automatic Control, vol.54, No.. Choi, H.H. (22). Variable Structure output feedback control design for a class of uncertain dynamics systems, Automatica, vol.38, pp Fliess, M. (988). Nonlinear control theory and differential algebra. In: Byrnes, C., Kurzhanski, A. (Eds), Modeling and Adaptive Control. Vol. 5 of Lecture notes in Control and Information Sciences. Springer-Verlag, New York, pp Fliess, M. (99). Generalized controller canonical form for linear systems and nonlinear dynamics. IEEE tran. on Autmaictic control, 35(9), Isidori, A. (995). Nonlinear Control Systems, 3 rd Edition, Springer-Verlag. Khan, Q., Bhatti, A.I., Iqbal, S., and Iqbal, M. (2). Dynamic Integral Sliding Mode Control of Uncertain MIMO Nonlinear Systems, International Journal of Control Automation and Systems. Kwan, C.-M. (995). Sliding Mode Control fo Linear Systems with Mismatch Uncertainites, Automatica, vol.3, No.2, pp Kwan, C.M. (2). Further results on variable output feedback controllers, IEEE Transaction on Automatic Control, 46, Levant, A. (2). Universal siso sliding mode controller with finite convergence, IEEE Transaction on Automatic Control, 49, Park, P., Choi, D.J. and Kong, S.G. (27). Output feedback variable structure control for linear systems with uncertainties and disturbances, Automatica, vol.43, PP Shyu, K.-K. Tasi, Y.-W. and Lai, C.-K. (998). Sliding Mode Control for Mismatch Uncertain Systems, Electron. Lett., vol. 34, No.24, PP Shyu, K.K., Tasi, Y.W. and Lai, C.K.(2). A dynamic output feedback controller for mismatch uncertain variable structure systems, Automatica, 37, Swaroop, P. Hindrik, J.K. Yip, P.P and Gerdes, J.C. (2). Dynamic Surface Control for a Class of Nonlinear Systems, IEEE Transaction on Automatic Control, vol.45, No., pp Utkin, V. I. (992). Sliding Modes in Control Optimization, Berlin, Germany, Springer-Verlag. Wang Xiaoyan, Q.S. Sha, Li. and Wen-hui, T. (29). Sliding mode control for a class of uncertain systems, Control and Decision Conference, pp Xiang, J., Wei, W. and Su, H. (26). An ILMI approach to robust output feedback sliding mode control, Int. J. Control, vol. 79, pp Zak, S.H. and Hui, S. (993). On variable structure output feedback controllers for uncertain dynamic systems, IEEE Transaction on Automatic Control, 38, Zinober, A.S.I. and Liu, P. (996). Robust Control of Nonlinear Uncertain Systems via Sliding Mode with Backstepping Design, UKKAC International Conference on Control, pp