Robust Control of Nonlinear Uncertain Systems via Sliding Mode with Backstepping Design

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1 1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, 1 ThC.5 Robust Control of Nonlinear Uncertain Systems via Sliding Mode with Backstepping Design R. Benayache 1, L. Chrifi-Alaoui 1, A. Benamor, X. Dovifaaz 1 and P. Bussy 1 Laboratoire des Technologies Innovantes (LTI, EA 3899) 1 Université de Picardie Jules Verne, 13 avenue François Mitterrand, F-88 Cuffies, France. Unité de recherche ATSI-ENIM, rue Ibn Jazzar, 519 Monastir, Tunisie. Abstract This paper presents a new combination of sliding mode and backstepping for output tracking of coupled two tank system. The proposed controller can greatly reduce chattering of sliding modes of the system, and it has certain robustness with respect to the external disturbances and good adaptability with respect to the parametric uncertainty. It is proved that the proposed systematic backstepping design method is able to guarantee global uniformly ultimately boundedness of all the signals in the closed-loop system. The tracking error is proven to converge to a small neighborhood of the origin. Finally, some experimental results are demonstrated to validate the proposed controllers. I. INTRODUCTION Sliding mode control () is a powerful and robust control method. methods have been widely studied in the last three decades from theoretical concepts to industrial applications [3, [13, [. A drawback of the methods may be unwanted chattering resulting from discontinuous control. There are many methods which can be employed to reduce chattering, for example, using a continuous approximation of the discontinuous control, and a combination of continuous and discontinuous sliding mode controllers. Chattering may also be reduced using the second order sliding mode control (SO)[1, [6 and dynamic sliding mode control [1, [8. Higher-order sliding mode controllers have recently been addressed to improve the system responses [13. However, when designing a control for a plant it is sometimes more beneficial to use combined techniques, using in conjunction with other methods such as backstepping. Most control design approaches are based upon Lyapunov and linearisation methods. In the Lyapunov approach, it is very difficult to find a Lyapunov function for designing a control and stabilizing the system. The linearisation approach yields local stability. The backstepping approach presents a systematic method for designing a control to track a reference signal by selecting an appropriate Lyapunov function and changing the coordinates [11, [7. The robust output tracking of nonlinear systems has been studied by many authors [-[17. Backstepping technique guarantees global asymptotic stability [4. is a robust control method and backstepping can be considered to be a method of recursive control. The combination of these methods, the so-called backstepping, yields benefits from both approaches. The backstepping sliding mode control (B) approach has been extended to some classes of nonlinear systems which need not be in the the parametric pure feedback (PPF) form or parametric strict feedback (PSF) form [17-[15. A symbolic algebra toolbox allows straightforward design of dynamical backstepping control [18. A backstepping method for designing an for a class of nonlinear system without uncertainties has been presented by Rios- Bolivar and Zinober [18, [16. The adaptive sliding backstepping control of semi-strict feedback systems (SSF) [ has been studied by Koshkouei and Zinober [9. The combination of sliding modes and the backstepping procedure is an attractive approach for developing robust controllers for nonlinear systems with unmatched uncertainties since the stability analysis of sliding mode control fits neatly within the recursive design. In this paper, a systematic design procedure is proposed to combine backstepping control and techniques for a class of nonlinear systems. A controller based on B techniques is designed so that the state trajectories approach a specified hyperplane. These systematic methods do not need any extra condition on the parameters and also any sufficient conditions for the existence of the sliding mode to guarantee the stability of the system. Moreover, in this paper a liquid level system is stabilised via a B obtained by combining the backstepping technique and the sliding mode control theory. This combined method inherits the robustness property from. If sufficient information about the output is available then the control is accessible and applicable without requiring further knowledge of the system variables. This paper is organised as follows: Section presents the non linear model of two tank system, Section 3 discusses both the designs of the robust backstepping sliding mode controllers, analyses the stability and performance. Section 4 presents the main results and their proofs. Our conclusion is contained in Section 5. II. DYNAMIC MODEL OF THE TWO-TANK SYSTEM A. Dynamic model for the two tank system By using the following physical principal, which is based on the conservation of mass law [1, the rate of the level of fluid in each tank is related directly to the difference between inflow and outflow rates and inversely to tank cross /1/$6. 1 AACC 4695

2 section area. Then the two-tank system can be modeled by where the parameters are defined by the following two differential equations dz A = 1 C 1 = 1 dt A (q A a z 1 S n g (5) 1 q ) dz B = 1 B = 1 dt A (q A b z S L g (6) q 3 ) (1) B. System constraints The exit of the tanks is defined by For the coupled tanks apparatus, the fluid flow, q 1, into tank T A, cannot be negative because the pump can only q = a z1 S n g za z B for z A > z B () pump water into the tank. Therefore, the constraint on the inflow rate is given by q 3 = b z S L gzb for z B > (3) q 1 (7) where At equilibrium, for constant water level set point, the z A, z B liquid levels [m derivatives must be zero, i.e., A section area of cylinder [m q, q 3 outflow rates [m 3 /s ż A = ż B = (8) q 1 inflow rate [m 3 /s Therefore, using Eq.(4) in the steady state, the following S n cross section areas of the connecting pipes between the tanks [m algebraic relationship holds S L maximal cross section area of the leakage [m C 1 sign(z A z B ) z A z B + q 1 a z1 flow coefficient of the interconnecting pipe A = (9) [real values ranging from to 1 C 1 sign(z A z B ) z A z B B zb = b z leak flow coefficient g earth acceleration [m 3 /s where q 1 is the equilibrium inflow rate given by q 1 = AC 1 sign(z A z B ) z A z B (1) From Eq. (1), and to satisfy the constraint in Eq. (7) on the input flow rate, we should have C 1 sign(z A z B ), which implies z A z B In which case the dynamics can be rewritten Fig. 1. Two tank system Valve V 3 was fully closed during the experiments, valve V 1 was fully opened and valve V was partially opened. The valve positions did not change during the experiments. The controlled signal (y) was the height of the liquid level in the second tank T B (y = z B ). This level was controlled by the control voltage of the pump P 1 (u). The system can be considered as a single input single output system (SISO) where the input is inflow q 1 and output is liquid level z B. Then the two-tank system can be modeled by the following two differential equations dz A dt dz B dt = C 1 sign(z A z B ) z A z B + q 1 A = C 1 sign(z A z B ) z A z B B zb (4) Let ż A = C 1 za z B + q1 A ż B = C 1 za z B B zb (11) L 1 = z B, L = z A z B >, u = q 1 and L = [ L1 L The output of the coupled tanks system is taken to be the level of the second tank. Therefore, the dynamic model in (11) can be written as L 1 L = C 1 L B L1 = B L1 C 1 L + u A y = L 1 (1) The dynamic model of the coupled tanks system is highly nonlinear. Therefore, we will define a transformation so that the dynamic model given in (1) can be transformed into a form that[ facilitates the control design. x1 Let x =, and define the transformation x = T(L) x such that x 1 = L 1 x = C 1 L B L1 (13) 4696

3 The derivative of (13) is now expressed as ẋ 1 = C 1 L B L1 = x ẋ = C 1 L L B L 1 L 1 (14) It can be checked that we can write the dynamic model in (14) as ẋ 1 = x (15) ẋ = C1 + B + B C 1 (L 1 L ) + C 1 1 u L1 L A L The objective of the control scheme is to regulate the output y(t) = x 1 (t) = z B (t) to a desired value z Bd. Let x 1d be the desired liquid level for the z B (t) state variable. III. DESIGN OF ROBUST BACKSTEPPING SLIDING MODE CONTROLLER A robust control for a plant with uncertainty may be obtained using a combined method of and backstepping control techniques. A combination of these methods has been studied in [14. The robust backstepping of SSF (semi-strict feedback systems) systems has been studied by Koshkouei and Zinober [9, [1. To provide robustness and reduce chattering, the backstepping algorithm can be modified. The modification is carried out at the final step of the algorithm by incorporating an appropriate sliding surface in terms of the error coordinates. In this section, a systematic design procedure is proposed to combine backstepping control and techniques for a class of nonlinear systems. Step 1: Define z 1 = x 1 x 1d (16) as the first tracking error variable. Its derivative, taking into account the model equation (16), is found to be ż 1 = ẋ 1 ẋ 1d = x ẋ 1d (17) We added and subtracted an auxiliary control ξ to and from the right hand side of (17) and rewrite (17) as ż 1 = z + ξ ẋ 1d (18) where z is an auxiliary tracking error variable defined as z = x ξ (19) Based on the tracking error dynamics (18), the corresponding stabilizing function ξ is designed as ξ = k 1 z 1 + ẋ 1d () where k 1 is a positive constant control gain. Now, substituting () into (17) yields the closed-loop, liquid level error dynamics ż 1 = k 1 z 1 + z (1) The first Lyapunov function is chosen as V 1 = 1 z 1 () The derivative of V 1 is V 1 = z 1 z 1 = k 1 z1 + z 1 z (3) Step : The derivative of z is now expressed as ż = C1 + B + C 1B (L 1 L ) + k 1 ż 1 ẍ 1d L1 L + C 1 u (4) A L Now, using (17) in (4) and isolating the control input u yields ż = ˆF + F + Ĉ1 u (5) A L where ˆF = Ĉ 1 + ˆB + Ĉ1 ˆB (L 1 L ) L1 L + k 1 (Ĉ1 L ˆB L1 ẋ 1d ) ẍ 1d ˆF is a known function. The expression of unknown function F is given by F = C 1 C 1 Ĉ 1 + B + B ˆB + λ C 1 L + B Ĉ 1 + C 1 B + ˆB C1 + C 1 u + d A L (L 1 L ) L1 L λ B L1 where d is the external disturbance. To design the BSM control system, the lumped uncertainty is assumed to be bounded, i.e. F < F, and define the following Lyapunov function V = 1 z s (6) with the sliding surface s = c 1 z 1 + z (7) Using (18) and (5), the derivative of V can be derived as follows V = z 1 z 1 + sṡ [ = z 1 ż 1 + s c 1 (z k 1 z 1 ) + ˆF + F + Ĉ1 u A L [ = k 1 z1 + z 1 z + s c 1 (z k 1 z 1 ) + ˆF + F + Ĉ1 u A L (8) For greater accuracy of the design, one can design a backstepping sliding mode. The BSM occurs if s = ṡ = and the sufficient condition ṡ = hs K s p sgn(s) (9) where h, K > and < p.5, is satisfied. The following control law satisfies the condition (9) u = A L {z 1 + ˆF } + hs + K s p sgn(s) (3) Ĉ

4 In our case we choose p =.5. In the following subsection, the stability analysis of the closed-loop dynamics (18), (5) is considered. A. Stability Analysis Theorem 1: With the developed nonlinear backstepping sliding-mode controller (3) and a stable sliding surface (7), the condition V < is satisfied. Thus, all components of z(t), z 1 and z, are exponentially convergent to the origin. Proof 1: To prove the Theorem, compute the time derivative of (6) V = z 1 ż 1 + sṡ (31) Substituting (9) into (31), the following equation can be obtained V = k 1 z 1 + z 1 z + s F s F Ks s sign(s) hs k 1 z 1 + z 1 z hs K s s + s ( F F) k 1 z 1 + z 1 z hs K s s (3) We can rewrite formula (3) as V z T Qz K s s (33) where Q is symmetric matrix with following form [ k1 + hc Q = 1 hc 1 1 hc 1 1 h In order to make certain V <, Q must be a positive symmetric matrix. Then ( Q = h(k 1 + hc 1 ) hc 1 1 ) = h(k 1 + c 1 ) 1 4 > (34) By choosing right value of h, k 1 and c 1 we can make certain Q >. Q will be a positive symmetric matrix. Now define the following term g = z T Qz + K s s V (35) Then t g(τ)dτ V (z 1 (), z ()) V (z 1 (t), z (t)) because V () is bounded and V (t) is non increasing and bounded, the following result can be obtained lim t t g(τ)dτ < (36) So V (t) + t g(τ)dτ = V (). In particular, t g(τ)dτ t V (). Therefore, lim t g(τ)dτ exists. According to Barbalat s lemma lim t g(t) = which guarantees the sliding mode stability. Since s and ṡ tend to zero, (7) implies that z 1 and z also tend to zero. Then, from (3), one can conclude the trajectories approach an equilibrium point along the sliding surface s = ṡ =. IV. EXPERIMENTAL RESULTS The application used to illustrate the above design techniques and demonstrate the performances of the system, is a two tank system which is in our laboratory (Fig. 1). The experimental schemes have been done under Matlab/Simulink, using Real-Time Interface, and run on the DS11 DSPACE system, which is equipped by a power PC processor. The control algorithm is implemented on DSP (TMS 3C31). The bidirectional information flux between the physical part and the computer is supported by a data acquisition interface. Control Desk is used to visualize the functional parameters of the simulated two-tank-system and to acquire data in real time. The numerical values of the parameters for the two tank, liquid level system are provided in TABLE 1. TABLE I NUMERICAL VALUES FOR PHYSICAL PARAMETERS OF THE TWO TANK SYSTEM Symbol Value Meaning A.154m tank section of valve S n m cross section a z < a z 1.7 flow coefficient of the interconnecting pipe b z < b z 1. leak flow coefficient g 9.81ms gravity constant z max.6m maximum water level in each tank q 1max m 3 /s maximum inflow through pump The objective is to control the liquid level of tank T B by introducing a leakage (external disturbance) in the outflow pipe of tank T B. For this application, two types of control are applied to a physical laboratory plant consisting of the two tank system namely, sliding mode control () with saturation [19 and robust backstepping sliding mode control (B). The control law is obtained from equation (3), where u max = m 3 /s, A =.154m, S n = S L = m and g = 9.81m/s. A comparison is carried out between them in order to show advantages and drawbacks of each one. In all the cases, the same process configurations are chosen with same operation conditions in the two tank namely the initial water levels and the set points. During the experimentation, the following control parameters were used: c 1 = 3, k 1 =.7, h = 1, K = 5 and p =.5. Figs., 3, 4 and 5 show the experimental results of the two controllers without external disturbances. It can be seen that both controller can track the liquid level very well. The results obtained using the proposed control strategy is shown in Fig.(dashed line). The system response showed perfect tracking with no overshoots and oscillations. It can be seen from these responses that the output converges to its desired value. The figure 3 shows how the control signals adjusted themselves so that the output converges to its desired value. 4698

5 Liquid level zb(m) Fig.. The control input (m 3 /s) B refrence Stabilisation of water level in tank T B using and B. 1. x B Fig. 3. Inflow rate into Tank T A using and B. from these responses that the output converges to its desired value. In this case, the proposed BSM controller performs Liquid level zb(m) B Reference Time (s) Fig. 6. Stabilisation of water level z B when the external disturbance is introduced in tank T B. The zoom of liquid level zb(m) B Reference Fig.4 and 5 portray the sliding variable and the tracking error when the and the B are applied. The sliding variable s The tracking errors (m) x 1 3 B Fig. 4. The sliding variable s B Fig. 5. The tracking errors The next performance test is to test the system in the rejection of external disturbances. To introduce a disturbance, the valve V 3 of Tank T B was opened at t =. It can be seen Fig. 7. Stabilisation of water level z B when the external disturbance is introduced in tank T B. very well and there is no discernable drop in the water level due to the leak, which was introduced at t = sec (Fig. 6). It is evident that the proposed controller has improved the dynamic and steady-state performance significantly. The results obtained using the proposed control strategy is shown in Fig. 7 (dashed line). The system output was able to return to the steady state value with no oscillatory behavior. The control action again was smooth, non-chattering, and it was able to drive the system states to the sliding surface s =. The sliding mode control with saturation is optimal in settling time Fig. 3, but when there are uncertainties and external disturbances Fig. 7, the performance are quite different, classical sliding mode controller has large error and the asymptotic stability of s(t) is undefined in the boundary layer [1. The figure 8 shows how the control signals adjusted themselves so that the output converges to its desired value. The performances of the controlled system are studied under variations in system parameters and in the presence of external disturbance. The robust control characteristics of this controller versus the external disturbance can be observed in the figure below (figure 8). Fig.9 and 1 portray the sliding variable and the tracking error when the external disturbance is introduced in tank T B. The presented results are obtained without changing the 4699

6 The control input (m 3 /s) 1. x B Fig. 8. Inflow rate into Tank T A when the external disturbance is introduced in tank T B. The sliding variable s x 1 3 B Fig. 9. The sliding variable s when the external disturbance is introduced in tank T B. The tracking errors (m) B Time(sec) Fig. 1. The tracking errors when the external disturbance is introduced in tank T B. control gains value. It is evident that the proposed controller has improved the dynamic and steady-state performances significantly. V. CONCLUSION This paper presented the design of robust output-tracking controller based on backstepping and second order sliding modes. The BSO results are compared with the nonlinear classical using smooth function. We validated this method on a nonlinear model (two-tank system). The obtained results using the latter prove the viability of this control method and presented bad performances in term of robustness to the system disturbances and uncertainties. A new algorithm based on the BSO has been proposed. The stability of the controller was proved with proper selection of the Lyapunov function. From presented equation, the controller should have a good adaptability with respect to the system unknown parameters, and it also retain certain robustness as in conventional sliding mode control. The results obtained demonstrate that this new approach can be easily implemented and its performance and robustness are better than the classical. REFERENCES [1 N. B. Almutairi and M. Zribi. Sliding mode control of coupled tanks. Mechatronics, 16(47-441), 6. [ S Behtash. Robust output tracking for nonlinear systems. Int. J. Contr, 5( ), 199. [3 R Benayache, L. Chrifi-Alaoui, P. Bussy, and J-M. Castelain. Design and implementation of sliding mode controller with varying boundary layer for a coupled tanks system. Mediterranean Conference on Control and Automation, (115-1), 9. [4 R. Benayache, L. Chrifi-Alaoui, X. Dovifaaz, and P. Bussy. Realtime nonlinear adaptive backstepping liquid level control for a state coupled three tank system. Europeen control conference, ( ), 9. [5 R. Benayache, S. Mahieddine Mahmoud, and L. Chrifi-Alaoui. Application of a robust second order sliding mode control to eliminate chattering for a non lineair uncertain system. STA Analyste et commande des systmes, (1-14), 7. [6 R. Benayache, S. Mahieddine Mahmoud, L. Chrifi-Alaoui, P. Bussy, and J-M. Castelain. Controller design using second order sliding mode algorithm with an application to a coupled-tank liquid-level system. International Conference on Control and Automation, ( ), 9. [7 I Kanellakopoulos, P.V Kokotovic, and A.S Morse. Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Aut. Contr, 36( ), [8 A.J Koshkouei, K Burnham, and A.S.I Zinober. Dynamic sliding mode control for nonlinear systems. IEE Proc. Control Theory and Applications, 15(39-396), 5. [9 A.J Koshkouei and A.S.I Zinober. Adaptive sliding backstepping control of nonlinear semi-strict feedback form systems. In: Proc. 7th IEEE Mediterranean Control Conf, [1 A.J. Koshkouei and A.S.I. Zinober. Adaptive output tracking backstepping sliding mode control of nonlinear systems. n: Proc. 3rd IFAC Symposium on Robust Control Design,. [11 M Krstic, I Kanellakopoulos, and P.V. Kokotovic. Adaptive nonlinear control without overparametrization. Syst. Contr. Lett, 19( ), 199. [1 A. Levant. Full real-time control of output variables via higher order sliding modes. In: Proc. European Control Conf. ECC, (Karlsruhe, Germany), [13 A. Levant. Construction principles of -sliding mode design. Automatica, 43( ), 7. [14 M. Rios-Bolvar and A.S.I. Zinober. Dynamical adaptive sliding mode output tracking control of a class of nonlinear systems. Int. J. Rob. Nonlin. Contr., 7(387-45), [15 M Rios-Bolyvar and A.S.I Zinober. Sliding mode control for uncertain linearizable nonlinear systems: A backstepping approach. In: Proc. IEEE Workshop on Robust Control via Variable Structure and Lyapunov Techniques, [16 M Rios-Bolyvar and A.S.I Zinober. Dynamical adaptive backstepping control design via symbolic computation. In: Proc 3rd European Control Conference ECC, [17 M Rios-Bolyvar and A.S.I Zinober. Dynamical adaptive sliding mode output tracking control of a class of nonlinear systems. Int. J. Rob. Nonlin. Contr., 7(387-45), [18 M Rios-Bolyvar and A.S.I Zinober. A symbolic computation toolbox for the design of dynamical adaptive nonlinear control. Appl. Math. and Comp. Sci, 8(73-88), [19 J. J. E Slotine and K Hedrick. Robust input output feedback linearization. Int. J. Control, 57( ), [ V. I Utkin. Sliding modes in control and optimization. Springer, 199. [1 V. I Utkin, J Guldner, and J Shi. Sliding modes control in electromecha-nical systems. Taylor-Francis, [ B Yao and M Tomizuka. Adaptive robust control of siso nonlinear systems in a semi-strict feedback form. Automatica, 33(893-9), [3 A.S.I Zinober. Variable Structure and Lyapunov Control. Springer,