Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor 479 Fig The structure of a multimodel discrete second order sliding mode control (MM-2-DSMC) where md is the number of the partial models The control applied to the system is given by the following relation: u (k) = v1 (k)u1eq (k) + v2 (k)u2eq (k) + + vmd (k)u mdeq (k) + u dis (k); (35) with • vi (k) : the validity of the i th local state model, • u ieq (k) : the partial equivalent term of the 2-DSMC calculated using the i th local state model, • u dis (k) : the discontinuous term of the control u eqi (k) = (C T Bi )−1 α S (k) − C T Ai x (k) + C T ( xd (k + 1) Ai et Bi are the matrixes of the i th partial state model The discontinuous term is given by the following expression: u dis (k) = u dis (k − 1) − M sign (σ(k)) The multimodel discrete second order sliding mode control (MM-2-DSMC) is, then, given by: u (k) = md ∑ vi (k)ueqi (k) + udis (k) i =1 A stability analysis of this last control law is proposed in the following paragraph (36) 480 Sliding Mode Control 3.3 Stability analysis of the MM-2-DSMC Let’s consider the following non stationary system: x ( k + ) = A d x ( k ) + Bd u ( k ) + Γ ( k ) y(k) = Hx (k) (37) Γ (k) represents eventual non linearities and external disturbances md md i =1 i =1 Considering the following notations: Am = ∑ υi Ai and Bm = ∑ υi Bi , we obtain the following model: x ( k + ) = A m x ( k ) + Bm u ( k ) y(k) = Hx (k) (38) which is the multimodel approximation of the system (37) This last system can be, then, written in the following form: x (k + 1) = ( Am + ΔAm ) x (k) + ( Bm + ΔBm ) u (k) + Γ (k) y(k) = Hx (k) (39) Ad = Am + ΔAm Bd = Bm + ΔBm (40) Δ (k) = ΔAm x (k) + ΔBm u (k) + Γ (k) (41) such that: We note: The system (37) can, in this case, be written as follows: x ( k + ) = A m x ( k ) + Bm u ( k ) + Δ ( k ) y(k) = Hx (k) (42) The control law given by (36) is applied to the system In the case of the multimodel, the md equivalent term ∑ vi (k)u eqi (k) of (36) is written as follow: i =1 u eq (k) = (C T Bm )−1 [− β S (k) − C T Am x (k)] (43) In this case, the sliding function dynamics are given by the following expression: C T x (k + 1) = S (k + 1) = − βS (k) + C T Δ (k) + (C T Bm )u dis (k) (44) The sliding function variation [S (k + 1) − S (k)] is given by the following relation: S (k + 1) − S (k) = − β(S (k) − S (k − 1) + C T (Δ (k) − Δ (k − 1)) − (C T Bm ) Msign (S (k) + βS (k − 1)) In what follows, the quantity (C T Bm ) M will be noted M ∗ (45) Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor 481 Which gives: S (k + 1) + βS (k) = (S (k) + βS (k − 1) + C T (Δ (k) − Δ (k − 1)) − M ∗ sign (S (k) + βS (k − 1)) (46) The relation (46) can be written: σ(k + 1) = σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ sign (σ(k)) (47) In discrete time sliding mode control, instead of the sliding mode, a quasi sliding-mode is considered in the vicinity of the sliding surface, such that |σ(k)| < ε, where σ(k) is the sliding function and ε is a positive constant called the quasi-sliding-mode band width Bartoszewicz, in (Bartoszewicz (1998)), gave the following sufficient and necessary condition for a system to satisfy a convergent quasi sliding mode: ⎧ ⎨ σ ( k ) > ε ⇒ − ε ≤ σ ( k + 1) < σ ( k ) (48) σ ( k ) < − ε ⇒ σ ( k ) < σ ( k + 1) ≤ ε ⎩ | σ(k)| < ε ⇒ | σ(k + 1)| ≤ ε ∀k, C T (Δ (k) − Δ (k − 1)) is supposed to be bounded such that: C T (Δ (k) − Δ (k − 1)) < Δ0 (49) with Δ0 being a positive constant Théorème 0.1 Let’s consider the system (37) to which the MM-2-DSMC given by (36) is applied If the discontinuous term amplitude M is chosen such that: M ∗ > Δ0 (50) where M ∗ = (C T Bm ) M and Δ0 is the external disturbances and system’s parameters’ variation bound given by (49), then, the MM-2-DSMC of (36) results in a convergent quasi sliding mode Proof ε is chosen equal to M ∗ + Δ0 To prove the convergence of the proposed control technique, we must, then, check the following three conditions: σ ( k ) > M ∗ + Δ ⇒ − ( M ∗ + Δ ) ≤ σ ( k + 1) < σ ( k ) (51) σ ( k ) < − ( M ∗ + Δ ) ⇒ σ ( k ) < σ ( k + 1) ≤ M ∗ + Δ (52) ∗ ∗ | σ(k)| < M + Δ0 ⇒ | σ(k + 1)| ≤ M + Δ0 (53) Let’s begin by the condition (51): σ ( k ) > M ∗ + Δ ⇒ − M ∗ − Δ < σ ( k + 1) < σ ( k ) * The inequality σ ( k + 1) < σ ( k ) (54) 482 Sliding Mode Control can be written as follows: σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ sign (σ(k)) < σ(k) (55) Knowing that σ(k) > 0, the inequality (55) becomes: σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ < σ(k) (56) By substructing σ(k) from both sides of this last inequality, we obtain: C T (Δ (k) − Δ (k − 1)) − M ∗ < * This last inequality is true because M ∗ is chosen such that M ∗ > Δ0 The inequality − M ∗ − Δ < σ ( k + 1) (57) (58) can be written as follows: − M ∗ − Δ0 < σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ (59) − Δ0 − C T (Δ (k) − Δ (k − 1)) < σ(k) (60) which gives: This last inequality is true, knowing that σ(k) > and − Δ0 − C T (Δ (k) − Δ (k − 1)) < M∗ + Δ0 > Let’s consider condition (52): σ ( k ) < − M ∗ − Δ ⇒ σ ( k ) < σ ( k + 1) < M ∗ + Δ By replacing σ(k + 1) by its expression, we obtain: σ(k) < σ(k) + C T (Δ (k) − Δ (k − 1)) + M ∗ < M ∗ + Δ0 * The inequality σ(k) + C T (Δ (k) − Δ (k − 1)) + M ∗ < M ∗ + Δ0 (61) (62) can be written as follows: σ(k) + C T (Δ (k) − Δ (k − 1)) < Δ0 (63) σ(k) < Δ0 − C T (Δ (k) − Δ (k − 1)) (64) which gives: − CT * This last inequality is true because Δ0 (Δ (k) − Δ (k − 1)) > and σ(k) < T (Δ (k ) − Δ (k − 1)) + M ∗ , knowing that: Besides, it is evident that σ(k) < σ(k) + C M ∗ > Δ0 > C T (Δ (k) − Δ (k − 1)) Let’s consider condition (53): | σ(k)| < M ∗ + Δ0 ⇒ | σ(k + 1)| < M ∗ + Δ0 (65) Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor * 483 If σ(k) > 0, then, the inequality | σ(k)| < M ∗ + Δ0 becomes: (66) < σ(k) < M ∗ + Δ0 (67) which gives: C T (Δ (k) − Δ (k − 1)) − M ∗ < σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ < M ∗ + Δ0 + C T (Δ (k) − Δ (k − 1)) − M ∗ ⇒ − Δ0 − M ∗ < σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ < M ∗ + Δ0 + Δ0 − M ∗ ⇒ − Δ0 − M ∗ < σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ < Δ0 + Δ0 ⇒ − Δ0 − M ∗ < σ(k) + C T (Δ (k) − Δ (k − 1)) − M ∗ < M ∗ + Δ0 ⇒ − Δ − M ∗ < σ ( k + 1) < M ∗ + Δ Then, * If σ(k) < 0, then, becomes |σ(k + 1)| < M ∗ + Δ0 (68) | σ(k)| < M ∗ + Δ0 (69) − M ∗ − Δ0 < σ(k) < (70) then, C T (Δ (k) − Δ (k − 1)) + M ∗ − M ∗ − Δ0 < σ(k) + C T (Δ (k) − Δ (k − 1)) + M ∗ < C T (Δ (k) − Δ (k − 1)) + M ∗ T (Δ (k ) − Δ (k − 1)) + M ∗ < M ∗ + Δ ⇒ − Δ0 − Δ0 < σ(k) + C ⇒ − Δ0 − M ∗ < σ(k) + C T (Δ (k) − Δ (k − 1)) + M ∗ < M ∗ + Δ0 ⇒ − Δ − M ∗ < σ ( k + 1) < M ∗ + Δ so, |σ(k + 1)| < M ∗ + Δ0 (71) The verification of the three conditions (51), (52) and (53) proves the existence of the convergent quasi sliding mode Therefore, the controller given by (29) is stable Note The bound of Δ (k) can be determined by studying the uncertainties of the different partial models, how much they cover the different real system’s dynamics and the method used for the calculation of the validities degrees The linear matrixes inequalities (LMI) approach can be used in these conditions Experimentation on a chemical reactor 4.1 Process description The semi-batch reactor control provides a very challenging problem for the process control engineer, due to the high non linearity that characterizes its dynamic behavior Therefore, we choose to apply the proposed control laws for temperature control of the chemical reactor 484 Sliding Mode Control presented in this section (a photo of the considered reactor is given by figure 2) This process is used to esterify olive oil The produced ester is widely used for the Fig The esterification reactor used for the experiments manufacture of cosmetic products A specific temperature profile sequence must be followed in order to guarantee an optimal exploitation of the involved reagents’ quantities The olive oil contains, essentially, a mono-unsaturated fatty acid that react with alcohol to give water and ester as shown by the following reaction equation: Acid + Alcohol Ester + Water (72) The final solution contains all the reagents and products in certain proportions To drive the reaction equilibrium in the way and, consequently, increase the ester’s proportion, we should take away water from the solution This is done by vaporization The fatty acid (oleic acid) and the ester ebullition temperatures are approximately 300◦ C The chosen alcohol (1-butanol) is characterized by an ebullition temperature of 118◦ C Consequently, heating the reactor to a temperature slightly over 100◦ C will result in the vaporization of water only (which is evacuated through the condenser) The reactor is heated by circulating a coolant fluid through the reactor jacket This fluid is, in turn, heated by three resistors located in the heat exchanger (Figure 3) The reactor temperature control loop monitors temperature inside the reactor and manipulates the power delivered to the resistors It is, also, possible to cool the coolant fluid by circulating cold water through a coil in the heat exchanger Cooling is, normally, done when the reaction is over, in order to accelerate the reach of ambient temperature The process can be considered as a single input - single output system The input is the heating power P (W ) The output is the reactor temperature TR(◦ C ) The interface between the process and the calculator is ensured by a data acquisition card of the type RTI 810 The data acquisition card ensures the conversion of the analog measures of the temperature to digital values and the conversion of the digital control value to an analog electric signal proportional the power applied to the heating resistors Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor 485 TR Col d water Reagents Condensor TF Heat exchanger Heating resistors Agitator Reactor Col d water P omp Fig Synoptic scheme of the real process The control law must carry out the following three stages : • Bring the reactor’s temperature TR to 105◦ C • Keep the reactor’s temperature to this value until the reaction is over (no more water dripping out of the condenser) • Lower the reactor’s temperature We chose, therefore, the set point given by figure We represent, in figure 5, the static characteristic of the system The different coordinates are taken relatively to the three stages of the process We notice that the system can be considered as a linear one, though, with some approximations According to the step responses of the system, the retained sampling step is equal to 180s A Pseudo-Random, Binary input Signal (PRBS) is applied to the real system An identification of the system structure, based on the instrumental determinants ratio method (Ben Abdennour et al (2001)), led to a discrete second order linear model Due to the nature of the control law to be applied to the reactor, the needed model is a state model The considered state variables are the reactor’s temperature TR(◦ C ) (noted x1 (k)), which is at the same time the system’s output, and the coolant fluid temperature TF (◦ C ) (noted x2 (k)) The state variables sequences x1 (k) and x2 (k) relative to the PRBS excitation input are measured and used for the parametric identification of the system The least square method leads to the following nominal model: 486 Sliding Mode Control T R ( oC ) Reaction stage Cooling stage 105 Heating stage time (mn) 240 120 Fig Desired reactor temperature 110 105 TR(°C) Reaction 100 95 Heating 90 85 80 Cooling 75 70 65 1000 P(W) 1200 1400 1600 1800 2000 2200 Fig Static characteristic of the reactor 0.6158 0.3600 x (k) + u (k) 0.0409 0.9118 0.0033 y(k) = x (k) x ( k + 1) = (73) The application of the multimodel approach and by using the least square method applied on the input/states sequence relative to each reaction stage leads to three partial models of the form: x ( k + ) = A i x ( k ) + Bi u ( k ) y(k) = Hx (k) i ∈ [1, 3] (74) with for: • the heating stage: 0.4712 0.4953 −0.1296 1.0651 H= 10 A1 = ; B1 = 0.0036 (75) Multimodel Discrete Second Order Sliding Mode Control: Stability Analysis and Real Time Application on a Chemical Reactor 487 • the reaction stage: 0.4114 0.5482 −0.0358 0.9787 H= 10 A2 = 0.0034 ; B2 (76) • the cooling stage: 0.6914 0.2877 −0.0386 0.0.9912 H= 10 A3 = 0.0032 ; B3 = (77) The control performance and robustness of the previously mentioned control laws, with respect to the model-system mismatch and external disturbance, are illustrated and compared through the experimental results given in the following paragraph 4.2 Experimental results In this paragraph, the performance of the MM-2-DSMC is shown by an experimentation on the chemical reactor Firstly, the chattering reduction, obtained by exploiting the second order sliding mode control, is illustrated by a comparison between the results obtained by the first order discrete sliding mode control with those realized by the 2-DSMC (Mihoub et al (2009b)) The nominal model (73) is used for both the DSMC and the 2-DSMC 2500 120 u(k) 1−DSMC y(k) 110 1−DSMC 100 2000 90 y (k) c 80 1500 2−DSMC proposée 70 2−DSMC proposée 1000 60 50 40 500 30 20 k 20 40 60 80 100 10 120 (a) Heating power (input) 110 108 k 20 40 60 80 100 120 (b) Reactor temperature (output) y(k) yc(k) 1−DSMC 106 104 102 2−DSMC proposée 100 98 96 94 92 90 30 k 40 50 60 70 80 90 100 (c) Zoom on the reactor temperature evolution Fig Comparison between DSMC and 2-DSMC We observe that the chattering of the control (u (k)) is remarkably reduced (figure 6.a) A better set point tracking is, consequently, obtained as shown by figures 6.b and 6.c, which represent, respectively, the evolution of the reactor temperature and a zooming of this last one in the neighborhood of 105◦ C As mentioned above, the reaction takes place essentially during this phase If the temperature reactor overshoots 105◦ C, a large amount of alcohol is evaporated 488 Sliding Mode Control and wasted and if it does not reach 105◦ C, the reaction kinetics are slowed down So, the 2-DSMC results in a better efficiency relatively to the first order DSMC 10 110 S(k) y(k) 100 90 MM−2−DSMC 80 2−DSMC 2−DSMC y (k) 70 c 60 MM−2−DSMC 50 40 30 20 k −2 20 40 60 80 100 k 10 120 10 (a) Sliding function 20 30 40 50 60 70 80 90 100 110 120 (b) Reactor temperature (output) 108 106 y(k) 104 yc(k) 102 MM−2−DSMC 100 98 96 94 2−DSMC 92 k 40 50 60 70 80 90 100 (c) Zoom of the reactor temperature evolution Fig Comparison between 2-DSMC and MM-DSMC Secondly, the multimodel approach is combined with the 2-DSMC in order to enhance the reaching phase The MM-2-DSMC and the 2-DSMC are represented together in figure (Mihoub et al (2009a)) It can be observed that the sliding function overshoots due to a bad reaching phase in the case of the 2-DSMC are reduced thanks to the multimodel approach (see figure 7.a) A better set point tracking is then obtained, as shown by figures 7.b and 7.c An amelioration of the efficiency of the chemical reactor is, consequently, obtained Conclusion In this work, the problems of the discrete sliding mode control are discussed A solution to the chattering problem can be given by the second order sliding mode To enhance the reaching phase, the multimodel approach is exploited A combination of the 2-DSMC and the multimodel approach is, then, used A stability analysis of the multimodel second order discrete sliding mode control is proposed in this work An experimentation on a chemical reactor is considered On the one hand, a comparison between the results obtained by the first order DSMC and those obtained by the 2-DSMC showed the chattering reduction offered by the second order approach On the other hand, a comparison between the results of the 2-DSMC and those of the MM-2-DSMC, illustrated both an enhancement of the reaching phase and a notable reduction of the chattering phenomenon A better efficiency of the reactor is, therefore, obtained 499 Two Dimensional Sliding Mode Control h ⎡C h B1 C h B2 ⎤ ⎡us ( i , j )⎤ ⎞ 1⎛ ⎜ S00 + ⎢ ⎥⎢ v ⎥⎟ = ⎟ 2⎜ ⎢C v B3 C v B4 ⎥ ⎢us ( i , j )⎥ ⎠ ⎣ ⎦⎣ ⎦ ⎝ h h ⎡ h ⎤ ⎡us (i , j )⎤ T C B1 C B2 = S00 + S00 ⎢ ⎥⎢ v ⎥ ⎢C v B3 C v B4 ⎥ ⎢us (i , j )⎥ ⎣ ⎦⎣ ⎦ (31) h h h ⎛ ⎡C B1 C B2 ⎤ ⎡us (i , j )⎤ ⎞ + ⎜⎢ ⎥ ⎟ < S00 ⎥⎢ v ⎜ ⎢C v B C v B ⎥ ⎢u (i , j )⎥ ⎟ ⎣ 4⎦⎣ s ⎦⎠ ⎝ Therefore, T S00 h h h h ⎡C h B1 C h B2 ⎤ ⎡us (i , j )⎤ ⎛ ⎡C B1 C B2 ⎤ ⎡us (i , j )⎤ ⎞ ⎥ < − ⎜⎢ v ⎥⎟ ⎢ v ⎥⎢ v ⎥⎢ v ⎜ ⎢C B3 C v B4 ⎥ ⎢us (i , j )⎥ ⎟ ⎢C B3 C v B4 ⎥ ⎢us (i , j )⎥ ⎦ ⎦⎠ ⎣ ⎦⎣ ⎦⎣ ⎝⎣ (32) Now with respect to the switching control laws (28) we can write T S00 ⎡ k hC h B1 ⎢ v ⎢ C B3 ⎣ h h C h B2 ⎤ T ⎡ k C B1 ⎥ S00 + S00 ⎢ v k vC v B4 ⎥ ⎢ C B3 ⎦ ⎣ C h B2 ⎤ ⎥ k vC v B4 ⎥ ⎦ T ⎡ k hC h B1 ⎢ v ⎢ C B3 ⎣ C h B2 ⎤ ⎥ S00 < k vC v B4 ⎥ ⎦ (33) This completes the proof 3.5 Robust control design In this section, assume that the 2-D system in RM (3) is not given exactly and we have ⎡ x h (i + 1, j )⎤ ⎡ x h ( i , j )⎤ ⎡ u h ( i , j )⎤ ⎢ v ⎥ = ( A + ΔA) ⎢ v ⎥ + ( B + ΔB) ⎢ v ⎥ ⎢ u ( i , j )⎥ ⎢ x (i , j + 1)⎥ ⎢ x ( i , j )⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (34) where ΔA and ΔB are denoted as the uncertainties in the system Assume that ΔA = pA ΔB = pB (35) where p is an unknown constant number and there exists a known positive real number, α , such that p Then, by replacing h(t ) in (9) with hsim (t ) from (12), a new corrective control law can be defined as uc (t ) = khsim (t ) = ky p Sy p (t ) (14) where k y p = k ⋅ ksim is a positive constant 3.2 Neural Networks for equivalent control To avoid the requirement of the thorough knowledge of the parameters and dynamics of the nominal plant (1), we use a feedforward NN, which consists of an input layer, a hidden layer, and an output layer as in (Yasser et al., 2006 b), to construct the equivalent control input ueq (t ) of the SMC in (7) The equivalent control input ueq (t ) is described as ueq (t ) = α uNN (t ) = α f ZOH ( uNN ( k )) (15) where α is a positive constant, uNN (t ) is a continuous-time output of the NN, uNN ( k ) is a discrete-time output of the NN, and f ZOH ( ⋅) is a zero-order hold function As in (Yasser et al., 2006 b), we implement a sampler in front of the NN with an appropriate sampling period to obtain the discrete-time input of the NN, and a zero-order hold is 513 Sliding Mode Control Using Neural Networks implemented to transform the discrete-time output uNN ( k ) of the NN back to the continuous-time output uNN (t ) of the NN The input i( k ) of the NN is given as i( k ) = ⎡ ey p ( k − 1), ⎣ , ey p ( k − n )⎤ ⎦ (16) where ey p ( k ) is the discrete-time form of ey p (t ) in (11) And the dynamics of the NN are given as (Yasser et al., 2006 b) hq ( k ) = ∑ ii ( k )miq ( k ) (17) i uNN ( k ) = o( k ) = ∑ S1 ( hq ( k ))mqj ( k ) (18) i where ii ( k ) is the input to the i -th neuron in the input layer ( i = 1, , ni ), hq ( k ) is the input to the q -th neuron in the hidden layer ( q = 1, , nq ), o( k ) is the input to the single neuron in the output layer, ni and nq are the number of neurons in the input layer and the hidden layer, respectively, miq ( k ) are the weights between the input layer and the hidden layer, mqj ( k ) are the weights between the hidden layer and the output layer, and S1 (⋅) is a sigmoid function The sigmoid function is chosen as S1 ( X ) = −1 + exp( − μ X ) (19) where μ > The objective of the NN training is to minimize the error function Ey p ( k ) described as n Ey p ( k ) = j e y p ( k ) = ∑ ⎡ y m ( k ) − y p ( k )⎤ ⎦ 2 j ⎣ (20) where ey p ( k ) is the discrete-time form of ey p (t ) in (11) The NN training is done by adapting miq ( k ) and mqj ( k ) using the method in (Yasser et al., 2006 b) as follows Δmqj ( k ) = −c ⋅ ∂E( k ) ∂mqj ( k ) (21) ⎡ ⎤ = c ⋅ ⎣ ym ( k ) − y p ( k )⎦ ⋅ J plant ⋅ S1 ( hq ( k )) Δmiq ( k ) = −c ⋅ ∂E( k ) ∂miq ( k ) μ (22) = c ⋅ ⎡ ym ( k ) − y p ( k )⎤ ⋅ J plant ⋅ mqj ( k ) ⋅ (1 − S1 ( X )) ⋅ ii ( k ) ⎣ ⎦ where c is a learning parameter, and J plant represents the plant Jacobian estimated using ... design a 2-D control system Sliding mode control of 2-D systems In this section, we review some prominence of the 1-D sliding mode control and then present the 2-D sliding mode control for RM... discrete time sliding mode control, instead of the sliding mode, a quasi sliding- mode is considered in the vicinity of the sliding surface, such that |σ(k)| < ε, where σ(k) is the sliding function... Applications, Vol 35, 1496? ?150 3 Young, K D.; Utkin, V I & Ozguner, U.(1999) A Control Engineer’s Guide to Sliding Mode Control IEEE Trans Control Systems Technology, Vol 27 Sliding Mode Control Using Neural