Takagi- Sugeno Robust Control of Uncertain Nonlinear Time- Delay Systems via Integral Sliding Mode Control

Takagi- Sugeno Robust Control of Uncertain Nonlinear Time- Delay Systems via Integral Sliding Mode Control Graciela S. Castro, Alexander G. Loukianov, and B. Castillo- Toledo CINVESTAV del IPN, Unidad Guadalajara, Apto. Postal 31-438 CP. 44550, Guadalajara, Jalisco, México (email: gsandova, louki, toledo @gdl.cinvestav.mx) Abstract: In this paper a novel discontinuous control strategy for robust stabilization of a class of uncertain nonlinear time-delay systems with delays in both the state and control variables is proposed. First, a Takagi- Sugeno technique is used to approximate the nonlinear system by a set of linear subsystems. Then, two predictors are used to compensate the control input delay, and in the following an integral sliding mode control technique is applied to partially compensate the e¤ect of a perturbation term. Finally, a nominal free-delay part of the control input is designed to stabilize the sliding mode dynamics using LMI’s technique. Conditions for the stability of the closed-loop perturbed system are derived. An example illustrates the applicability of the proposed control scheme. Copyright c IFAC 2009 Keywords: Takagi- Sugeno, Sliding Mode Control, Uncertain Nonlinear Systems, Predictor, LMI’s. 1. INTRODUCTION Robust feedback stabilization of time-delay systems remains one of the most challenging problems in control theory because many industrial processes are modeled by delay di¤erential equations (Kolmanovskii and Myshkis (1999)). It is well-known that delays may dramatically limit the performance of the controllers and sometimes destabilize the closed-loop system Malek-Zavarei and Jamshidi (1987). This problem has been extensively studied and several controllers and stability criteria based on optimal control methods (Shen et al. (1991), Phoojaruenchanachai and Furuta (1992)), including LMI’s approaches (Li et al. (1996), Li and Sauza (1997)), used Lyapunov-Krasovskii functional (Fridman and Niculescu (2007), Fridman (2005), Kharitonov and Aguilar (2003), Khusainov (1989), Mondié et al. (2005)), have been proposed. On the other hand, the study of nonlinear systems with these features, the analysis and design of a delay compensator becomes more di¢ cult and complex. One of the techniques that permits to solve this problem is the TakagiSugeno fuzzy approach, which approximates a nonlinear system by a set suitable (usually linear) subsystem in terms of IF-THEN rules, and facilities the design of the control. Furthermore, a sliding mode (SM) control has attractive properties such as decomposition of the original control design procedure and robustness to plant parameter variations and external matched disturbances (Utkin et al. (1999), Drakunov and Utkin (1990) and Young et al. (1999)). In addition, SM control achieves fast transient

response of the closed-loop system. Due to these advantages and simplicity of implementation, sliding mode approach has been used to design robust stabilizing discontinuous controllers for delayed systems (see for example, Shuy and Yan (1997), Koshkouei and Zinober (1990), Choi (1999), Gouaisbaut et al. (1999), Su et al. (2000), Loukianov and Escoto-Hernández (2000), Li and Yurkovich (2001), Gouaisbaut et al. (2002a), Li and Decarlo (2003), Loukianov et al. (2003), Shtessel et al. (1991), and Xia and Jia (2002)). However, most of these controllers were proposed for systems with only delay in the state vector, while the direct implementation of discontinuous control in systems with delays in the control input may cause oscillations (Fridman et al. (1996), Bonnet et al. (1998), Gouaisbaut et al. (2002b)). A possible way to treat this problem is the use of a predictor-based controller (Utkin et al. (1999), Loukianov et al. (2005), Mirkin and Raskin (1999), Mirkin (2003)). A SM predictor controller have been proposed for a class of linear uncertain systems by (Roh and Oh (1999)); however the conditions preserving the matching condition for the uncertainties in the transformed free-delay system, are very restrictive (Nguang (2001)). Moreover, the problem of designing a predictor-based control for a system with delays in both the state and input vectors, to the best knowledge of the authors, is still open. In this paper a Takagi- Sugeno robust control via SM and LMI’s approaches is proposed for robust stabilization of a class of uncertain nonlinear systems with time- delays in both, the state and control variables. First, using the Takagi- Sugeno approach, a nonlinear system is approxi-

mated by a stet of linear subsystems. Then, the control vector is designed in three parts. The …rst one is designed using integral sliding mode technique (Utkin et al. (1999)). This allows preserving the matching condition for the unknown perturbation, contrary to the results of (Roh and Oh (1999)). In addition, the switching function includes an auxiliary variable that allows compensating the unknown matched perturbation. The second part of the control cancels the known but undesired nonlinear delayed matched dynamics. Finally, the third component of the control, namely the nominal part, is chosen via LMI’s to stabilize the nonlinear nominal delay-free dynamics. Based on this nominal system, two predictors are designed for the state and sliding mode variables to compensate the control input delay ensuring the sliding mode stability and reduce a chattering e¤ect in the sliding mode motion. It can be noted that an integral SM controller for linear perturbed time-delay systems was designed in (M. Basin et al. (2005)), but for the case of control input delay free systems. 2. TIME DELAY SYSTEM DESCRIPTION Consider a nonlinear time-delay system with uncertainties described by the following state equation x(t) _ = f1 (x(t); t) + f2 (x(t 1 ); t) + B(x(t))u(t (1) 2 ) + g(x(t); t) x(t) = '1 (t); u(t) = '2 (t); 8t 2 [t0 ; t0 ]; t0 0; where x 2 Rn , u 2 Rm are the state and control vectors, respectively; the unknown function g 2 Rm represents the system nonlinearities plus any model uncertainties on the system including external disturbances; f1 ; f2 and B are su¢ ciently smooth functions; 1 and 1 ; 2 are known time delays, 1 2. The functions '1 (t) and '2 (t) are the initial continuous functions de…ned on [t0 1 ; t0 ] and [t0 2 ; t0 ], respectively. As it is commonly in a robust SM control system design, we assume that both the undesired known nonlinear term f2 (x(t 1 ); t) and unknown perturbation term g(x(t); t) satisfy the matching condition (Drazenovic (1969)), namely: A1) There are f 2 2 Rm and g 2 Rm such that f2 (x(t

1 ); t)

= B(x(t))f 2 (x(t

1 ); t);

(2)

and g(x(t); t) = B(x(t))g(x(t); t) kg(x(t); t)k (x(t); t) for some known scalar function ( ):

(3) (4)

A2) The perturbation term g(x(t); t) is continuous and Lipschitz.

x(t) _ = f1 (x(t); t) + B(x(t))[f 2 (x(t

1 ); t)

(6)

+u(t 2 ) + g(x(t); t)] the matched known but undesired delay term, f 2 ( ); i.e., u2 (t) =

f 2 (x(t

); t)

=

1

(7)

2

Substituting (5) and (7) into (6) yields x_ (t) = f1 (x(t); t) + B(x(t))[u0 (t +u1 (t

2)

2)

+ g(x(t); t)].

(8)

3.1 Takagi- Sugeno fuzzy model To design a predictor for the nonlinear system (8), this system is approximated by the following rules TakagiSugeno fuzzy model: IF z1 (t) is Mi1 and ... and zp (t) is Mip , THEN x(t) _ = Ai x(t) + Bi [u0 (t 2 ) + u1 (t 2 ) + g(x(t); t)] (9) for i = 1; :::; r, where Mij is the fuzzy set and r is the number of model rules; z1 (t); :::; zp (t) are known premises variables that may be functions of the state variables and/or of time. This representation is obtained using linear approximation while Ai and Bi are the Jacobian matrices with appropriate dimensions evaluated at the arbitrarily operation points x0i . Using nominal part of (9) a predictor is designed having the form

Ai

(t) = e

2

x(t) +

Z0

e

Ai

Bi u0 (t + )d

(10)

2

with a predictive state (t) 2 Rn :Taking the time derivative of (10) results in (t) = Ai (t) + Bi u0 (t) + eAi 2 Bi [u1 (t

2 ) + g(x(t); t)]:

(11) Note that the nominal part u0 in (11) is already delay free, while the matching condition is preserved . De…ne ); t) = 1 now u2 (t) = f 2 (x(t 2 a sliding variable s 2 Rm of the form s(t) = Gi (t) + !(t) (12) where Gi 2 Rm n is a design matrix, and !(t) is de…ned by !(t) _ = Gi [Ai (t) + Bi u0 (t)] (13) with !(0) = Gi (0): Taking the time derivative of (12) and using (11), yields s(t) _ = Gi fAi (t) + Bi u0 (t) + eAi Bi [u1 (t

2)

(14)

+g(x(t); t)]g + !(t): _

3. CONTROLLER DESIGN To preserve the matching conditions (3) the integral SMC technique will be used. For let us …rst rede…ne the control to be u(t) = u0 (t) + u1 (t) + u2 (t); (5) m where u0 2 R is the nominal part of the control, u1 2 Rm is designed to reject the matched perturbation term g( ), and u2 2 Rm is chosen to cancel in

Choosing now Gi = BiT e Ai 2 and using (13), the nominal dynamics are cancelled in (14), i.e. (15) s(t) _ = Mi [u1 (t 2 ) + g(x(t); t)] T with Mi = Bi Bi > 0. To eliminate the delay in (15) the following predictor is used: Z0 (t) = s(t) + Mi u1 (t + )d : (16) 2

Thus, using (15) results in _ (t) = Mi [u1 (t) + g(x(t); t)]: (17) To induce a sliding motion on the manifold (t) = 0; the control component u1 is selected as u1 (t) = (x(t); t)sign( (t)) (18) where (x(t); t) is a positive scalar function for the control gain. Theorem 1. Under the condition: (x(t))

(x(t); t) > 0

(19)

where ( ) represents the bounded of (4). The state of the closed loop system (17) and (18) the surface (t) arises in a …nite time.

At this point however, we have to analyze the behavior of the original predicted variable x(t + 2 ), instead of merely that of the system (23). De…ning (t) = x(t+ 2 ) and using (10), this variable can be de…ned on (t) = 0 as (t) = (t) +

;eq (t

+ )d :

(26)

;eq (t

+

2 );

(27)

where g ;eq (t + 2 ) = u1eq (t) + u1eq (t 2 ) is the predicted value for g ;eq (t 2 ) de…ned in (24), and it can be expressed as ;eq (t

+

2 ) = g(x(t

+

2 ); t)

= g( (t); t)

g(x(t); t)

g( (t

2 ); t):

(28)

3.3 SM motion stability analysis using LMI approach Based on Takagi- Sugeno fuzzy technique, and using (27) and (25), the …nal outputs of the fuzzy systems are inferred as follows r X

h2i Nii (t) + 2

Imposing _ (t) = 0 in (17) we have

r X r X

Mi [u1eq (t) + g(x(t); t)] = 0; (20) from where, the equivalent control u1eq (t) is obtained as u1eq (t) = g(x(t); t) (21) or g(x(t (22) u1eq (t 2 ); t): 2) =

r r X X

hi hj Bi K0j

i=1 i
+

r X

Nij + Nji 2

hi hj

i=1 i
i=1

Substituting this equivalent control into (11), the sliding mode motion for (t) on (t) = 0 is described by the following perturbed system: _ (t) = Ai (t) + Bi u0 (t) + eAi 2 Bi g ;eq (t); (23)

Bi g

_ (t) = Ai (t) + Bi u0 (t) + Bi g

_ (t) =

3.2 Sliding mode dynamics

Ai

The dynamics for (t) on (t) = 0 are thus given by

g

if we choose the control gain (x(t); t) under the condition (19), V_ (t) is negative, (t) vanishes and a sliding mode motion occurs on the manifold (t) = 0 in a …nite time.

e

2

Proof. Proposing a Lyapunov Functional candidate V (t) = 1 T (t)Mi 1 (t) and taking the time derivative along the 2 the trajectories of (17) with control (18) and using (4), yields V_ (t) [ (x(t); t) (x(t); t)] k (t)k :

Z0

Z0

e

A1j

Bj g

;eq (t

(t)

+ )d

2

hi B i g

;eq (t

+

2)

(29)

i=1

where hi = hi (x (t)) and Nij = Ai + Bi Kj ; i < j r hi \ hj 6= 0; Tanaka and Wang (2001). Theorem 2. The solution of the fuzzy system such as (29) is ultimately bounded if there is a common positive de…nite matriz P such that the following conditions are satis…ed

where g

;eq (t)

=

g(x(t

2 ); t)

+ g(x(t); t)

NiiT P + P Nii

(24)

Q

(30)

The previous equation implies that: 1. The predictor (10) enables to preserve the matching condition (3) with respect to the control part u1 : 2. In the sliding mode, the equivalent control u1eq (t) (21) cancels the perturbation term g( ) in the system (11) or (23) at time (t 2 ) but not at time t (this is a natural situation). However, due to this cancellation at time (t 2 ) the system (23) contains the perturbation term up to the order O( 2 ) only. The nominal component of the control, u0 ; in (23) can be now selected as u0 (t) = K0i (t); (25) where K0i 2 Rn n .

Nij + Nji 2

T

P +P

Nij + Nji 2

0

(31)

for i = 1; :::; r, j = 1; :::; r and i < j r such that hi \ hj 6= 0 where Q is a positive de…nite matrix. Proof. To study the behavior of (t) we use a Lyapunov function V (t) = T (t)P (t) with P positive de…nite solution of the Lyapunov equation. Di¤erentiating V along the trajectories of (29) yields

V_ (t) =

T

+2

(t)

T

r X

h2i NiiT P + P Nii

i=1 r X r X

(t)

hi hi

i=1 i
Nij + Nji 2

+P

2

T

(t)

r X r X

"

+2

(t) T

Nij + Nji 2 T

(t)

r X

hi hi P Bi K0j

i=1 i
i=1 Z0

P (t)

h2i P Bi g

e

Aj

;eq (t

Bj g

+

;eq (t

2)

+ )d :

2

(32) By assumption A2 there is a constant q1 > 0 such that q2 e max (Ai )t ; k g ;eq (t)k q1 2 . Moreover, e Ai t 0 Z e Ai g ;eq (t + )]d q2 > 0 and therefore K0i

Fig. 1. Comparison between the state x(t) and the predictor (t); with g(t) = 3. is ultimately bounded by (34) and (35).

2

q3 2 with q3 = Pr knowing that i V_ (t)

q1 q2 max (Ai )

max (Ai ) 2

kK0i k (1 + e 1 hi (x(t)) = 1, hi (x(t))

min (Q) k

): Indeed, 0 , thus

2

(t)k + 2 k (t)k 2 Z0 24 P Bi g ;eq (t) K0i e

Ai

g

;eq (t

2

(1

2

) k (t)k

+2 k (t)k P Bi2 (q1 + q3 ) (1 8 k (t)k

2

k (t)k

3

+ )d 5

2

2

) k (t)k ;

(33)

with min (Q) > 0 and 2 2q0 2 P Bi = ; 0< <1

(34)

where q0 = q1 +q3 . Therefore, the solution of the perturbed system (29) is ultimately bounded by Khalil (1996) s max (P ) k (t)k ; = : (35) min (P ) The results obtained above are stated as follows. Based on a LMI approach, it is necessary to …nd a common P for all linear subsystems. Then, we can reduce the inequalities (30) and (31) to the following linear matrix inequality: V X >0 (36) X W with W > 0; X > 0; where X = P

1

3.4 Example Consider the following …rst order nonlinear perturbed system: x(t) _ = x(t) cos(x(t)) + x2 (t 1 ) + u(t 2 ) + g(t); (39) with a time- delays 1 = 0:3 s and 2 = 0:11 s, and two distinct disturbances: g(t) = 3 and g(t) = 3 + 0:5 sin(t). Applying the proposed control design procedure and choosing u2 (t) = x2 (t ), = 1 2 , yields x(t) _ = x(t) cos(x(t)) + u0 (t 2 ) + u1 (t 2 ) + g(t): (40) Now, the operations points for (40) are chosen arbitrary such as x(t)01 = 20 ; x(t)02 = 30 and x(t)03 = 80 . The nominal parameters and membership functions for these are shown in Table 1. Table 1. Nominal Parameters A1 = 0:8202705058

B1 = 1

A2 = 0:6042242364

B2 = 1

M2 (x(t)) =

A3 =

B3 = 1

M3 (x(t)) =

1:201483808

M1 (x(t)) =

M4 (x(t)) =

30

x(t) 50 x(t)+20 50 80 x(t) 50 x(t) 30 50

Finally, the resulting control for each linear subsystem is summarized in u(t) = K0i (t) (x(t); t)sign( (t)) x2 (t ): (41)

; W = Q, and

4. SIMULATIONS RESULTS

T

Using the control tool box LMIEDIT of MATLAB we obtain the positive parameter P = 0:0155 and the gains K0i showed in Table 2.

V = (Ai X Bi Mi ) + (Ai X Bi Mi ) 1 + f(Ai X Bi Mj ) + (Aj X Bj Mi )g 2 1 T + f(Ai X Bi Mj ) + (Aj X Bj Mi )g 2 where Mi =K0i X; for i = 1; :::; r and j = 1; :::; r such that hi \ hj 6= 0. Therefore, if conditions (4), (3) and (19) are satis…ed, then the solution of the closed-loop system (1) with the following control: u(t) = K0i (t) (x(t))sign( (t)) f 2 (x(t ); t) (38)

Table 2. Gains K01 =

1:8253

K02 =

1:6092

K03 =

0:2333:

Figures (1) and (4) show the comparison between the state variable x(t) and the evolution of the predictor (t). The variable x(t) converges to a neighborhood of zero, that is, it remains bounded despite the presence of the persistent

Fig. 2. Response of surface (t)

Fig. 3. Comparision between the disturbance g(t) = 3 and control u1 (t) = (x(t); t)sign( (t)):

Fig. 5. Response of surface (t) with g(t) = 3 + 0:5sin(t)

Fig. 6. Comparison between the disturbance g(t) = 3 + 0:5sin(t) and the control u1 (t) = (x(t); t)sign( (t)): 5. CONCLUSION A robust delay compensation method based on the predictor design and integral sliding mode control techniques, has been formulated for nonlinear time-delay systems with uncertainties using the Takagi - Sugeno approach. The proposed control allows to reduce the e¤ect of perturbations and guarantees boundedness of the closed-loop system solution. REFERENCES

Fig. 4. Comparison between the state x(t) and the predictor (t); with g(t) = 3 + 0:5sin(t).

perturbation signal. Figures (2) and (5) show the behavior of the sliding variable (t) used in the discontinuous control u1 (t) = (x(t); t)sign( (t)). Figures (3) and (6) show the comparison of u1 (t) with the disturbance g(t) under the condition (x(t); t) (x(t)) where (x(t)) is the perturbation boundary.

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