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Fuzzy sliding mode controller design for uncertain time-delayed systems with nonlinear input
Fuzzy Sets and Systems 140 (2003) 359 – 374 www.elsevier.com/locate/fss
Fuzzy sliding mode controller design for uncertain time-delayed systems with nonlinear input Fang-Ming Yu, Hung-Yuan Chung∗ , Shi-Yuan Chen Department of Electrical Engineering, National Central University, Chung-Li 32054, Taiwan 320, ROC Received 30 December 1999; received in revised form 1 November 2002; accepted 8 November 2002
Abstract In this paper, the theory of sliding mode control is employed to develop a fuzzy controller for uncertain dynamic time-delayed systems with series nonlinearities. The control method provides a simple way to achieve asymptotic stability of the uncertain time-delayed system. Other attractive features of the method include a minimal realization of the fuzzy controller in spite of the number of state variables, as well as insensitivity to the uncertainties and disturbances, In addition, the method is capable of handling the chattering problem inherent to the sliding mode control easily and e4ectively. Also, this fuzzy sliding mode controller can be extended to handle the MIMO system. Simulation results are presented to demonstrate the power of the method. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy control; Sliding mode control; Series nonlinearity; Uncertain dynamic system; Time-delayed system
1. Introduction Since the work of Mamdani [11] was proposed in 1974, fuzzy control has been an active research topic. In general, a fuzzy control algorithm consists of a set of heuristic decision rules and can be regarded as a nonmathematical control algorithm, in contrast to a conventional feedback control algorithm. Most work in fuzzy control >eld used the error e and the change-of-error e˙ as fuzzy input variables regardless of the complexity of controlled plants [12]. Such fuzzy logic controllers (FLCs) are suitable for simple second order plants. However, in case of complex higher order plants, all process states are taken as fuzzy input variables to implement state feedback FLCs. If all state variables are ∗
Corresponding author. Tel.: +886-3-4227151-4475; fax: +886-3-4255830. E-mail address: [email protected] (H.-Y. Chung).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(02)00529-8
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used to represent the content of the rule antecedent (if part of a rule), it will require a large number of control rules and much e4ort to create. Owing to these facts, a simpler FLCs design is highly desirable. In recent years, some results of the fuzzy sliding mode control (FSMC) [3,6,7,13,15] have been reported. To decrease the number of rules in the rule base, several authors have suggested using the way of a composite state to obtain a fuzzy sliding mode controller described in the previous work. The advantage of such controllers is that the number of rules required is reduced from m n to m 2 in [2,5,6] or nm 2 in Kung [9]. Obviously, the FSMC is one of the reducible fuzzy rules methods. In general, since the FSMC combines both fuzzy control and sliding mode control principles, the performance of closed-loop system is superior to that using only one control theory. In this paper, the main idea of FSMC is to use the reaching condition ss¡0, ˙ and let s and s˙ be fuzzy variables. Simultaneously, according to the Lyapunov stability theorem, the system can thus be ensured asymptotically stable. The rule table is then constructed in a two-dimensional space. If s and s˙ have 7 membership functions respectively, then the fuzzy rules will have 49 rules. As for the single-input system of higher than the second order, the fuzzy rules number of conventional FLC will be much more. However, for FSMC, there are only 49 rules for the input variables of s and s˙ with 7 membership functions each. Moreover, most work of robust control has concentrated on the systems with uncertainties and/or external disturbances. From the viewpoint of design, the uncertain system with linear input is more preferred than that with nonlinear input owing to its simplicity. The assumption of linear input is that the system model is indeed linearizable. However, in practice, due to physical limitation, there do exist nonlinearities in the control input and their e4ect cannot be ignored in realization. In addition, to obtain a more practical system model, the information of delay time should not be discarded. Thereby, it is necessary to develop a new robust control method to deal with the uncertain time-delayed systems with input nonlinearity. In this study, based on an uncertain time-delayed system with nonlinear input, a robust controller is derived through variable structure control (VSC) [4]. Although the method of variable structure control can solve the above-mentioned system, the drawback of the variable structure control is their chattering owing to the sliding control law that has to be discontinuous across the sliding surface. Chattering is undesirable because it involves high control activity and may excite high-frequency dynamics. So, we apply the fuzzy control principle to overcome the drawback. Herein, what we want to demonstrate is that the FSMC can handle this chattering problem e4ectively by adjusting the control input near the sliding hyperplane. Mostly, the FSMC method is applied in the nonlinear system such as the inverted pendulum system [1,5] and tracking control of a nonlinear system [8]. Although the FSMC method can be used to control above-mentioned systems (the inverted pendulum system, tracking control system) very well, those are SISO systems with a linear input, not a nonlinear input which cannot be ignored in reality. Therefore, in this paper, we will adopt the FSMC method to design a robust controller for the uncertain time-delayed system with a nonlinear input. Furthermore, the FSMC can be extended to handle the MIMO system. The rest of the paper is divided into >ve sections. In Section 2, some properties of the uncertain nonlinear time-delayed system with nonlinearities are reviewed. In Section 3, a robust variable structure controller is derived from those properties. In Section 4, the fuzzy sliding mode control is developed and presented. In Section 5, a simulation example is illustrated to demonstrate the features of the FSMC. Finally, we conclude with Section 6.
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2. System description A general description of uncertain time-delayed dynamical systems with nonlinear input [4] is given in the form of ˙ = (A + JA)x(t) + b(u) + Ad x(t − d ) + f(t); x(t) x(t) = X(t);
− d 6 t ¡ 0;
(1)
where x(t) ∈ R n and u ∈ R is the state variable and control input of the system, respectively. A ∈ R n×n is the state matrix, b ∈ R n is the input vector, A d is the delay term matrix including the uncertainty, and (u) : R → R is a continuous function and (0) = 0: JA is the bounded uncertainty matrix of A; d represents a nonzero time delay, X(t) is a continuous vector-valued initial function, and f(t) ∈ R n stands for the disturbance vector. It is also noted that, in the following sections, WT denotes the transpose of W, and W represents the Euclidean norm when W is a vector or the induced norm when W is a matrix. In dealing with this study, the following assumptions are necessary for the sake of convenience: Assumption 1. For the nominal part of the uncertain time-delayed system with series nonlinearities indicated in Eq. (1), matrix pair (A; b) is completely controllable. Assumption 2. The matrix A d and uncertainty matrices JA; f satisfy the following rank conditions [10]: . rank[JA] = rank[JA..b]; . rank[JAd ] = rank[JAd ..b]; . rank[f] = rank[f ..b]:
(2)
Based on Assumption 2, there exist vectors H; G, and scalar d such that the following matching conditions hold [14]: JA = bH;
H 6 1 ;
Ad = bG;
G 6 2 ;
f = bd;
d 6 3 ;
(3)
where H; G, and d are with appropriate dimensions. The above matching conditions (3) are used in the subsequent sections for derivation simplicity. Assumption 3. The nonlinear input (u) applied to the system satis>es the following property: u · (u) ¿ · u2 ; where is a nonzero positive constant, and (u) satis>es (0) = 0.
(4)
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Fig. 1. Series nonlinearity (u) in a single input case.
Remark 1. In general, if the series nonlinearity is inside sector [1 2 ], i.e., 2 · u · (u) ¿ u · (u) ¿ 1 · u2 ; where (u) denotes a sector bounded function, 2 is often called the gain margin, and 1 is called the gain reduction tolerance. So, we have 2 → ∞ and 1 = in Assumption 3. It is seen that if u increases, then (u) increases and vice versa. An example of a scalar nonlinear function is shown in Fig. 1. In consequence, to design the variable structure control is, >rstly, to choose a proper switching surface for the uncertain nonlinear system with time delay, so that the sliding motion on that switching surface has the desired properties. Next, choose a discontinuous control law, which enforces the motion of the sliding mode to be asymptotically stable. 3. Variable structure control design First, the following lemmas will be employed to derive the variable structure controller [4]. Lemma 1. If the nonlinear input satis5es the property as indicated in Eq. (4), there exists a continuous function (·) : R+ → R+ ; (0) = 0, and (p)¿0 for p¿0. Therefore, if |u(t)| = (q), then · u2 ¿ q · (q);
∀q ¿ 0:
(5)
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Proof. The proof is very straightforward if one chooses (q) = · q;
¿
1
then · u2 = · |u|2 = · 2 (q) ¿ q · (q):
(6)
Usually the VSC design is a two-phase process. Phase one is to choose a switching surface so that the original system, restricted to the intersection of the switching surface (sliding mode), results in the desired behavior. Phase two is to determine a switching control that is able to force the trajectory of the system approaching to and staying on the sliding surface. Hereby, the switching surface is de>ned as s(t) = cT x(t) = 0;
(7)
where c ∈ R n is a constant vector. It is necessary to examine if the property in sliding mode, described in Eq. (7), is valid for the system with nonlinear input, shown in Eq. (1). A necessary condition for the system state trajectory to remain on the sliding surface s is s˙ = 0 namely, ˙ s(t) ˙ = cT x(t) = cT [(A + JA)x(t) + b(u) + Ad xd + f] =0
(8)
where x d denotes x(t − d ). Therefore, the equivalent control eq in the sliding mode s = 0 is given by eq = −(cT b)−1 cT [(A + JA)x(t) + Ad xd + f]:
(9)
Introducing Eqs. (3) and (9) into Eq. (1) produces the equivalent dynamic system with nonlinear input in the sliding mode as s(t) = 0 ˙ = (A + JA)x(t) + beq (u) + Ad xd + f x(t) = (A + bH)x − b(cT b)−1 cT [(A + bH)x(t) + bGxd + bd] + bGxd + bd = [I − b(cT b)−1 cT ]Ax:
(10)
Matching conditions (3) are used in the derivation of Eq. (10). From Eq. (10), it can be seen that the invariance condition also holds even though the time-delayed system is with “nonlinear” input. Remark 2. From the above analysis, it can be concluded that the uncertain time-delayed system with “nonlinear” input in the sliding mode possesses the same property as those with “linear” input in the sliding mode. Accordingly, the design of the switching surface can be achieved through the same design tool for the systems with “linear” input.
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From the analysis mentioned above, it can be seen that how to drive the system trajectories into the sliding mode is the key work for system stabilization. Before stating the scheme of the controller, the condition for reaching the sliding mode is given below: Lemma 2. The motion of the sliding mode (7) is asymptotically stable, if the following condition holds: ss˙ ¡ 0;
∀t ¿ 0:
(11)
To ful>ll the condition stated in Eq. (11), the desired variable structure control is suggested by u(t) = − where (x; t) =
scT b (x; t); scT b
(12)
r {[(cT b)−1 cT A + 1 ]x + 2 xd + 3 };
r ¿ 1:
(13)
The following theorem shows that the proposed variable structure control in Eq. (12) drives the uncertain system with nonlinear input into the sliding mode s(t) = 0. Theorem 1. Consider the uncertain system (1) subjected to Assumptions 1, 2, and inequality of (3). If the variable structure control input u(t) in Eq. (1) is given as that indicated by (12), then the system trajectories asymptotically converge to the sliding mode (7). Proof. Consider the reaching condition of the sliding mode (7). Let V (t) = 12 s 2 be the Lyapunov function of the system described in (1). If Eq. (1) is substituted into the derivative of the states in (11), one can obtain ˙ V˙ (t) = ss˙ = scT x(t) = scT [(A + JA)x(t) + b(u) + Ad x(t − d ) + f(t)] = scT [Ax + b(u) + b(Hx + Gxd + d)]:
(14)
Then, Eq. (3) is applied to the above equation to obtain the following inequality expression: ss˙ 6 scT b(cT b)−1 cT Ax + scT b(u) + scT b(1 x + 2 xd + 3 ):
(15)
From Eqs. (12) and (4), we have u(u) = −
scT b (x; t)(u) ¿ u2 = 2 (x; t): scT b
Therefore, the above expression can be rearranged as scT b · (u) 6 −2 (x; t)
scT b = −(x; t)scT b: (x; t)
(16)
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Inserting Eq. (16) into the right-hand side of the inequality in Eq. (15), one has ss˙ 6 scT b{(cT b)−1 cT Ax − + 1 x + 2 xd + 3 }:
(17)
Thus, once the expression of (x; t) in (13) is valid, one can obtain ss˙ 6 (1 − r)scT b{[(cT b)−1 cT A + 1 ]x + 2 xd + 3 }:
(18)
Since r¿1, it is easy to con>rm that ss˙ ¡ 0
or
V˙ (t) ¡ 0:
(19)
According to the Lyapunov stability theorem, condition (19) ensures that s(t) is toward the switching surface and the sliding mode is asymptotically stable. Then the proof is completed. Remark 3. As expressed in Remark 2, a sector bounded input function is only a special case of series nonlinear function de>ned in (4). Therefore, it is straightforward to ensure that the developed control law of VSC can be applied to control the systems with sector bounded nonlinearities. In summary, variable structure control method has been derived to stabilize the uncertain timedelayed system with nonlinear input. It has been shown that the proposed variable structure control can drive the uncertain system trajectories into the sliding mode even though the time-delayed systems are with “nonlinear” input. Moreover, the uncertain system can also preserve the invariance condition even though the uncertain systems are with delay time and subjected to the “nonlinear” input. 4. Fuzzy sliding mode control As described in the last section, the objective of the SMC is to design a control law (12) such that the reaching condition (11) is satis>ed. In this section, we will develop a fuzzy logic control law in an attempt to accomplish this object. The proposed fuzzy control is called the fuzzy sliding mode control (FSMC) because it is based on the principle of SMC. The proposed FSMC [5] is shown in Fig. 2, where s and s˙ are the inputs and u denote the output of the FSMC. The inputs of the proposed fuzzy controller are S and S˙ which are the fuzzi>ed variables of s and s, ˙ respectively. The output of the fuzzy controller is U which is the fuzzi>ed variable of u. All the universes of discourse of S; S˙ and U are arranged from −1 to 1; thus, the range of nonfuzzy variables s; s˙ and u must be scaled to >t the universe of discourse of fuzzi>ed variable S; S˙ and s(t) .
s( t )
K1
S(t)
K2 .
Fuzzy Control Rules
U
K3
S( t )
Fig. 2. The block diagram of the FSMC.
u(t)
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Fig. 3. Fuzzy variable of triangular type.
U with scaling factors K1, K2 and K3, respectively, namely, S = K1 · s(t); S˙ = K2 · s(t); ˙ u(t) = K3 · U: In the implementation, S˙ is approximated with (Sk − Sk −1 )=T where T is the sampling period. For simplicity, a triangular type membership function is chosen for the aforementioned fuzzy variables, as shown in Fig. 3. The reasoning behind the fuzzy control rules for the FSMC, which will be presented later, is explained by the following analysis: By taking the time derivative of both sides of (7), we obtain s˙ = cT x˙ = cT Ax + cT b(u) + cT b(Hx + Gxd + d):
(20)
Then, multiplying both sides of the above equation by s leads to ss˙ = scT Ax + scT b(u) + scT b(Hx + Gxd + d):
(21)
Here, we assume that cT b¿0. In (20), it is seen that s˙ increases as (u) increases and vice versa. In (21), it is seen that if s¿0, then decreasing (u) will make ss˙ decrease and that if s¡0, then increasing (u) will make ss˙ decrease. From the Remark 1, we know the relation between u and (u). So, it is seen that if s¿0, then decreasing u will make ss˙ decrease and that if s¡0, then increasing u will make ss˙ decrease. Now, we choose a Lyapunov-like function V = 12 s2 :
(22)
V˙ = ss:˙
(23)
Then Based on the above qualitative analysis and the Lyapunov stability condition V˙ ¡0 and s = 0 has to be a stable sliding surface, we can ensure that the system is asymptotically stable, simultaneously matching the reaching condition ss¡0, ˙ and the control input u can be designed with condition ss¡0. ˙
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Table 1 Rule table of FSMC S= S˙
NB
NM
NS
ZE
PS
PM
PB
PB PM PS ZE NS NM NB
ZE PS PM PB PB PB PB
NS ZE PS PM PB PB PB
NM NS ZE PS PM PB PB
NB NM NS ZE PS PM PB
NB NB NM NS ZE PS PM
NB NB NB NM NS ZE PS
NB NB NB NB NM NS ZE
Table 2 Look-up table of FSMC S˙
U
−1 S
1 0:84 0:7 0:56 0:42 0:28 0:14 0 −0:14 −0:28 −0:42 −0:56 −0:7 −0:84 −1
−0:84 −0:7
−0:56 −0:42 −0:28 −0:14 0
0 −0:15 −0:27 −0:44 −0:57 0:15 0 −0:11 −0:3 −0:4 0:27 0:11 0 −0:17 −0:3 0:44 0:29 0:19 0 −0:1 0:58 0:41 0:31 0:12 0 0:67 0:53 0:41 0:3 0:19 0:72 0:56 0:52 0:4 0:29 0:9 0:69 0:65 0:53 0:43 0:9 0:72 0:7 0:59 0:5 0:9 0:78 0:78 0:66 0:61 0:9 0:9 0:9 0:76 0:68 0:9 0:9 0:9 0:76 0:76 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9
−0:67 −0:52 −0:4 −0:28 −0:18 0 0:12 0:28 0:42 0:52 0:62 0:68 0:76 0:82 0:9
−0:73 −0:56 −0:52 −0:4 −0:3 −0:12 0 0:15 0:31 0:42 0:5 0:59 0:7 0:73 0:9
−0:9 −0:73 −0:67 −0:56 −0:44 −0:27 −0:15 0 0:15 0:27 0:44 0:56 0:67 0:73 0:9
0.14
0.28
0.42
0.56
0.7
0.84
1
−0:9 −0:73 −0:7 −0:59 −0:5 −0:42 −0:31 −0:15 0 0:15 0:27 0:44 0:56 0:67 0:73
−0:9 −0:82 −0:76 −0:68 −0:62 −0:52 −0:42 −0:28 −0:12 0 0:18 0:28 0:4 0:52 0:67
−0:9 −0:9 −0:9 −0:76 −0:68 −0:61 −0:5 −0:43 −0:29 −0:19 0 0:1 0:3 0:4 0:57
−0:9 −0:9 −0:9 −0:76 −0:76 −0:66 −0:59 −0:53 −0:4 −0:3 −0:12 0 0:17 0:3 0:44
−0:9 −0:9 −0:9 −0:9 −0:9 −0:78 −0:7 −0:65 −0:52 −0:41 −0:31 −0:19 0 0:11 0:27
−0:9 −0:9 −0:9 −0:9 −0:9 −0:78 −0:72 −0:69 −0:56 −0:53 −0:41 −0:29 −0:11 0 0:15
−0:9 −0:9 −0:9 −0:9 −0:9 −0:9 −0:9 −0:9 −0:72 −0:67 −0:58 −0:44 −0:27 −0:15 0
The resulting fuzzy control rules are shown in Table 1. We apply the Mamdani’s min-operation fuzzy implication and adopt the center of gravity as defuzzi>cation method to construct a look-up table based on Table 1. The look-up table is shown in Table 2. For purpose of illustration, a few control rules in Table 1 are described below: • IF S is PB and S˙ is PB, THEN U is NB This control rule states that if S and S˙ are both positive big, which is equivalent to that ss˙ is largely positive, then, by (21), a large negative change of the control input is required to decrease ss˙ quickly. • IF S is PB and S˙ is NB, THEN U is ZE This control rule implies a situation where when ss˙ is largely negative, which is a desired situation, there is no need to change the control input.
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• IF S is NB and S˙ is PB, THEN U is ZE This control rule implies a situation where when ss˙ is largely negative, which is a desired situation, there is no need to change the control input. • IF S is NB and S˙ is NB, THEN U is PB This control rule states that if S and S˙ are both negative big, which is equivalent to that ss˙ is largely positive, then, by (21), a large positive change of the control input is required to decrease ss˙ quickly. Remark 4. As to the fuzzy sliding mode control for MIMO case, a system is given as follows: ˙ = (A + JA)x(t) + B.(u) + Ad x(t − d ) + f(t); x(t) x(t) = X(t); − d 6 t ¡ 0;
(24)
where u ∈ Rm ; B ∈ R n×m ; .(u) : Rm → Rm , and the other matrices or vectors are the same as (1). Similarly, By Assumption 2, the matching conditions of JA = BH; A d = BG, and f = Bd hold. Therefore, we have ˙ = Ax(t) + B.(u) + B[Hx(t) + Gxd + d]: x(t)
(25)
Hereby, the switching surfaces are de>ned as S(t) = Cx(t) = 0;
(26)
where S = [s1 ; s2 ; : : : ; sm ]T ; C = [c1 ; c2 ; : : : ; cn ] with ci = [ci1 ; ci2 ; : : : ; cim ]T , and i = 1; 2; : : : n. Consider the reaching condition of the sliding mode (26), one can obtain ST S˙ = ST CAx + ST CB.(u) + ST CB(Hx + Gxd + d):
(27)
For the sake of convenience, de>ne ˆ .(u) = (CB)−1 .(u);
(28)
ˆ = [ˆ 1 (u1 ); ˆ 2 (u2 ); : : : ; ˆ m (um )]T , then Eq. (27) can be where we let u = [u1 ; u2 ; : : : ; um ]T and .(u) rearranged as ˆ ST S˙ = ST CAx + ST .(u) + ST CB(Hx + Gxd + d) =
m
sj s˙j
j=1
= ST CAx + [s1 ˆ 1 (u1 ) + s2 ˆ 2 (u2 ) + · · · + sm ˆ m (um )] + ST CB(Hx + Gxd + d):
(29)
ˆ Here, assume that CB¿0, and by (28), it is clear that .(u) is proportional to .(u). In (29), for T j = 1; 2; : : : ; m, it is seen that if sj ¿0, then decreasing ˆ j (uj ) will make S S˙ decrease. We also have that if sj ¡0, then increasing ˆ j (uj ) will make ST S˙ decrease. Furthermore, we choose the Lyapunov
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Table 3 jth Rule table of FSMC for MIMO system sj = s˙j
NB
NM
NS
ZE
PS
PM
PB
PB PM PS ZE NS NM NB
ZE PS PM PB PB PB PB
NS ZE PS PM PB PB PB
NM NS ZE PS PM PB PB
NB NM NS ZE PS PM PB
NB NB NM NS ZE PS PM
NB NB NB NM NS ZE PS
NB NB NB NB NM NS ZE
˙ will ensure that S(t) is toward the switching function V = 12 S2 . Then condition of (V˙ = ST S)¡0 surfaces and the fuzzy sliding mode is asymptotically stable. Based on the above qualitative analysis, one can design the control input uj in an attempt to ˙ The resulting fuzzy control rules are shown in Table 3. And one of satisfy the condition ST S¡0. the control rules is given as • IF sj is PB and s˙j is PB, THEN uj is NB According to the above description, we conclude that the FSMC can be extended to handle the MIMO system. 5. Simulation Consider an uncertain time-delayed system with series nonlinearity, which is described by Eq. (1) [4]. It is shown below ˙ = (A + JA)x(t) + b(u) + Ad x(t − d ) + f(t) x(t) = Ax(t) + b.(u) + b[Hx(t) + Gxd + d]; where xd = x(t − d ); H61 ; G62 and d63 . From Eq. (3), we also obtain that JA = bH; A d = bG, and f = bd. The corresponding parameters of the illustrated system are given as follows: 0 1 x1 (t − d ) 0 x1 A= ; xd = ; ; b= ; x= x2 x2 (t − d ) −2 −3 1 (u) = (!esin u + " cos u)u; Hx + Gxd + d = l1 e(1+sin x1 )
! ¿ " ¿ 0; x12 + x22 + l2 cos x2d
x12d + x22d + l3 cos x2 :
For both VSC and FSMC controllers, the switching surface is taken as s(t) = 2x1 + x2 :
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Fig. 4. State variable dynamics for the system under VSC: x1 and x2 .
Obviously, cT b¿0. This meets the assumption cT b¿0 in (21). For this system, the following parameters are given: T = 1:0;
l1 = 0:04;
l2 = 0:3;
l3 = −0:6;
r = 0:3:
In addition, the following initial values are arbitrarily chosen, that is x1 (0) = −1;
x2 (0) = 1:
5.1. VSC method The computer simulation for this uncertain time-delayed system with series nonlinearity is carried out with sampling step of 0:01 s. For computer simulation, the corresponding parameters are r = 1:1; = 0:7; 1 = 0:3; 2 = 0:3; 3 = 0:3, respectively. The delay time constants are taken as d = 0:2. Fig. 4 shows the transient response of x1 and x2 under the VSC controller. Fig. 5 shows the phase plane between x1 and x2 under the VSC. From the simulation result, it shows that the VSC works well for the uncertain time-delayed system with nonlinear input. But there exists chattering phenomenon that is not desired. 5.2. FSMC method For this uncertain time-delayed system with series nonlinearity, the sampling step and delay time is the same as VSC method. In this case study, a (15 × 15)-matrix of the look-up table and two-degree
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Fig. 5. The phase plane between x1 and x2 : x2 (x1 ) under the VSC.
nesting are chosen (Table 2). Accordingly, the scaling factors are chosen as K1 = 0:5; K2 = 0:5 and K3 = 50. The simulation results are shown in Figs. 6 and 7. Fig. 6 shows the transient response of x1 and x2 under the FSMC controller. Fig. 7 shows the phase plane between x1 and x2 under the FSMC. From the simulation result, it is shown that the FSMC not only works well for the uncertain time-delayed system with nonlinear input, but also eliminates the chattering phenomenon. Remark 5. In sliding mode control, state trajectories chattering along a sliding hyperplane is inevitable in practice, because the control input is discontinuous in the vicinity of the sliding hyperplane. Chattering is a serious drawback of sliding mode control and has been an important problem in the >eld of sliding mode control. Here, we want to demonstrate that the FSMC can handle this chattering problem e4ectively by adjusting the control input near the sliding hyperplane. The simulation results in Fig. 7 show the state trajectory of system response for uncertain time-delayed system with series nonlinearity. No signi>cant chattering along the sliding hyperplane is observed in the trajectory. Remark 6. Because the switching function is de>ned only by the state variable of x(t), that is, s(t) = cT x(t) = 0, free of the time delay d ; d a4ect the system stability very little. From the simulation result, it is found that the robustness of system is little susceptible to time delay d . 6. Conclusions We have shown that the FSMC developed in this work can be easily applied to uncertain timedelayed dynamical systems with nonlinear input as described in Eq. (1). The scheme of this work
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Fig. 6. State variable dynamics for the system under FSMC: x1 and x2 .
Fig. 7. The phase plane between x1 and x2 : x2 (x1 ) under the FSMC.
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is to design a fuzzy sliding mode controller to stabilize uncertain time-delayed systems with series nonlinearity. Moreover, the proposed FSMC can be extended to handle the MIMO system. Several important features of fuzzy sliding mode are summarized in the following: (1) No exact mathematical model for the nonlinear systems is required for the controller design. The only information required to design the controller is the qualitative inRuences of u on ss. ˙ (2) An inherent chattering problem of sliding mode control can be handled e4ectively without involving sophisticated mathematics. (3) By the Lyapunov stability theorem and the reaching condition of (V˙ = ss)¡0, ˙ we only need to adjust the control input u to satisfy ss¡0, ˙ the construction of rule base is simple and clear. Then, the asymptotic stability for uncertain time-delayed systems with series nonlinearities can be guaranteed. (4) The minimum numbers of fuzzy control rules can be obtained. Because in each fuzzy implication only s and s˙ are taken as inputs of the fuzzy sliding mode controller in spite of the number of state variables in the system. (5) From the simulation result, it is shown that the FSMC indeed can be applied to the uncertain time-delayed system with a nonlinear input. The robustness for the uncertainty and perturbation is still very satisfactory. The e4ect is the same as that of systems with a linear input. We will further investigate that the fuzzy control combines the MIMO VSC to generate a new FSMC method for uncertain time-delayed system with a nonlinear input in the future. Acknowledgements The authors would like to thank the >nancial support of the National Science Council of the Republic of China under contract NSC88-2213-E008-043. References [1] H. Allamehzadeh, J.Y. Cheung, Design of a stable and robust fuzzy controller for a class of nonlinear system, Proc. Fifth IEEE Internat. Conf., Vol. 3, 1996, pp. 2150 –2154. [2] G. Bartolini, A. Ferrara, Multivariable fuzzy sliding mode control by using a simplex of control vectors, in: S.G. Tzafestas, A.N. Venetsanopoulos (Eds.), Fuzzy Reasoning in Information, Decision, and Control Systems, Kluwer, Amsterdam, The Netherlands, 1994, pp. 307–328. [3] J.S. Glower, J. Munighan, Design fuzzy controllers from a variable structures standpoint, IEEE Trans. Fuzzy Systems 5 (1) (1997) 138–144. [4] K.-C. Hsu, Adaptive variable structure control design for uncertain time-delayed systems with nonlinear input, Dynamics and Control 8 (1998) 341–354. [5] G.-C. Hwang, S.-C. Lin, A stability approach to fuzzy control design for nonlinear systems, Fuzzy Sets and Systems 48 (1992) 279–287. [6] Y.R. Hwang, M. Tomizuka, Fuzzy smoothing algorithms for variable structure systems, IEEE Trans. Fuzzy Systems 2 (1994) 277–284. [7] S.-W. Kim, J.-J. Lee, Design of a fuzzy controller with fuzzy sliding surface, Fuzzy Sets and Systems 71 (1995) 359–367.
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[8] C.-C. Kung, C.-C. Liao, Fuzzy-sliding mode controller design for tracking control of non-linear system, Proc. American Control Conf., June 1994, pp. 180 –184. [9] C.C. Kung, S.C. Lin, Fuzzy controller design: a sliding mode approach, in: S.G. Tzafestas, A.N. Venetsanopoulos (Eds.), Fuzzy Reasoning in Information, Decision, and Control Systems, Kluwer, Amsterdam, The Netherlands, 1994, pp. 277–306. [10] N. Luo, De La, M. Sen, State feedback sliding mode control of a class of uncertain time delay systems, IEE Proc., Part D, Vol. 140, 1993, pp. 261–274. [11] E.H. Mamdani, Applications of fuzzy algorithms for simple dynamic plants, Proc. IEE 121 (1974) 1585–1588. [12] R.K. Mudi, N.R. Pal, A robust self-tuning scheme for PI- and PD-type fuzzy controllers, IEEE Trans. Fuzzy Systems 7 (1) (1999) 2–16. [13] R. Palm, Designs of fuzzy controllers, in: H.T. Nguyen, M. Sugeno (Eds.), Fuzzy Systems Modeling and Control, The Handbooks of Fuzzy Sets, Kluwer Academic, Boston, 1998, pp. 227–272. [14] K. Shyu, J. Yan, Variable-structure model following adaptive control for systems with time-varying delay, Control theory Adv. Technol. 10 (1994) 513–521. [15] H.B. Verbruggen, H.J. Zimmermann, R. Babuska, Fuzzy Algorithms for Control, Kluwer Academic, Boston, 1999.
Fuzzy sliding mode controller design for uncertain time-delayed systems with nonlinear input Fang-Ming Yu, Hung-Yuan Chung∗ , Shi-Yuan Chen Department of Electrical Engineering, National Central University, Chung-Li 32054, Taiwan 320, ROC Received 30 December 1999; received in revised form 1 November 2002; accepted 8 November 2002
Abstract In this paper, the theory of sliding mode control is employed to develop a fuzzy controller for uncertain dynamic time-delayed systems with series nonlinearities. The control method provides a simple way to achieve asymptotic stability of the uncertain time-delayed system. Other attractive features of the method include a minimal realization of the fuzzy controller in spite of the number of state variables, as well as insensitivity to the uncertainties and disturbances, In addition, the method is capable of handling the chattering problem inherent to the sliding mode control easily and e4ectively. Also, this fuzzy sliding mode controller can be extended to handle the MIMO system. Simulation results are presented to demonstrate the power of the method. c 2002 Elsevier B.V. All rights reserved. Keywords: Fuzzy control; Sliding mode control; Series nonlinearity; Uncertain dynamic system; Time-delayed system
1. Introduction Since the work of Mamdani [11] was proposed in 1974, fuzzy control has been an active research topic. In general, a fuzzy control algorithm consists of a set of heuristic decision rules and can be regarded as a nonmathematical control algorithm, in contrast to a conventional feedback control algorithm. Most work in fuzzy control >eld used the error e and the change-of-error e˙ as fuzzy input variables regardless of the complexity of controlled plants [12]. Such fuzzy logic controllers (FLCs) are suitable for simple second order plants. However, in case of complex higher order plants, all process states are taken as fuzzy input variables to implement state feedback FLCs. If all state variables are ∗
Corresponding author. Tel.: +886-3-4227151-4475; fax: +886-3-4255830. E-mail address: [email protected] (H.-Y. Chung).
c 2002 Elsevier B.V. All rights reserved. 0165-0114/03/$ - see front matter doi:10.1016/S0165-0114(02)00529-8
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used to represent the content of the rule antecedent (if part of a rule), it will require a large number of control rules and much e4ort to create. Owing to these facts, a simpler FLCs design is highly desirable. In recent years, some results of the fuzzy sliding mode control (FSMC) [3,6,7,13,15] have been reported. To decrease the number of rules in the rule base, several authors have suggested using the way of a composite state to obtain a fuzzy sliding mode controller described in the previous work. The advantage of such controllers is that the number of rules required is reduced from m n to m 2 in [2,5,6] or nm 2 in Kung [9]. Obviously, the FSMC is one of the reducible fuzzy rules methods. In general, since the FSMC combines both fuzzy control and sliding mode control principles, the performance of closed-loop system is superior to that using only one control theory. In this paper, the main idea of FSMC is to use the reaching condition ss¡0, ˙ and let s and s˙ be fuzzy variables. Simultaneously, according to the Lyapunov stability theorem, the system can thus be ensured asymptotically stable. The rule table is then constructed in a two-dimensional space. If s and s˙ have 7 membership functions respectively, then the fuzzy rules will have 49 rules. As for the single-input system of higher than the second order, the fuzzy rules number of conventional FLC will be much more. However, for FSMC, there are only 49 rules for the input variables of s and s˙ with 7 membership functions each. Moreover, most work of robust control has concentrated on the systems with uncertainties and/or external disturbances. From the viewpoint of design, the uncertain system with linear input is more preferred than that with nonlinear input owing to its simplicity. The assumption of linear input is that the system model is indeed linearizable. However, in practice, due to physical limitation, there do exist nonlinearities in the control input and their e4ect cannot be ignored in realization. In addition, to obtain a more practical system model, the information of delay time should not be discarded. Thereby, it is necessary to develop a new robust control method to deal with the uncertain time-delayed systems with input nonlinearity. In this study, based on an uncertain time-delayed system with nonlinear input, a robust controller is derived through variable structure control (VSC) [4]. Although the method of variable structure control can solve the above-mentioned system, the drawback of the variable structure control is their chattering owing to the sliding control law that has to be discontinuous across the sliding surface. Chattering is undesirable because it involves high control activity and may excite high-frequency dynamics. So, we apply the fuzzy control principle to overcome the drawback. Herein, what we want to demonstrate is that the FSMC can handle this chattering problem e4ectively by adjusting the control input near the sliding hyperplane. Mostly, the FSMC method is applied in the nonlinear system such as the inverted pendulum system [1,5] and tracking control of a nonlinear system [8]. Although the FSMC method can be used to control above-mentioned systems (the inverted pendulum system, tracking control system) very well, those are SISO systems with a linear input, not a nonlinear input which cannot be ignored in reality. Therefore, in this paper, we will adopt the FSMC method to design a robust controller for the uncertain time-delayed system with a nonlinear input. Furthermore, the FSMC can be extended to handle the MIMO system. The rest of the paper is divided into >ve sections. In Section 2, some properties of the uncertain nonlinear time-delayed system with nonlinearities are reviewed. In Section 3, a robust variable structure controller is derived from those properties. In Section 4, the fuzzy sliding mode control is developed and presented. In Section 5, a simulation example is illustrated to demonstrate the features of the FSMC. Finally, we conclude with Section 6.
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2. System description A general description of uncertain time-delayed dynamical systems with nonlinear input [4] is given in the form of ˙ = (A + JA)x(t) + b(u) + Ad x(t − d ) + f(t); x(t) x(t) = X(t);
− d 6 t ¡ 0;
(1)
where x(t) ∈ R n and u ∈ R is the state variable and control input of the system, respectively. A ∈ R n×n is the state matrix, b ∈ R n is the input vector, A d is the delay term matrix including the uncertainty, and (u) : R → R is a continuous function and (0) = 0: JA is the bounded uncertainty matrix of A; d represents a nonzero time delay, X(t) is a continuous vector-valued initial function, and f(t) ∈ R n stands for the disturbance vector. It is also noted that, in the following sections, WT denotes the transpose of W, and W represents the Euclidean norm when W is a vector or the induced norm when W is a matrix. In dealing with this study, the following assumptions are necessary for the sake of convenience: Assumption 1. For the nominal part of the uncertain time-delayed system with series nonlinearities indicated in Eq. (1), matrix pair (A; b) is completely controllable. Assumption 2. The matrix A d and uncertainty matrices JA; f satisfy the following rank conditions [10]: . rank[JA] = rank[JA..b]; . rank[JAd ] = rank[JAd ..b]; . rank[f] = rank[f ..b]:
(2)
Based on Assumption 2, there exist vectors H; G, and scalar d such that the following matching conditions hold [14]: JA = bH;
H 6 1 ;
Ad = bG;
G 6 2 ;
f = bd;
d 6 3 ;
(3)
where H; G, and d are with appropriate dimensions. The above matching conditions (3) are used in the subsequent sections for derivation simplicity. Assumption 3. The nonlinear input (u) applied to the system satis>es the following property: u · (u) ¿ · u2 ; where is a nonzero positive constant, and (u) satis>es (0) = 0.
(4)
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Fig. 1. Series nonlinearity (u) in a single input case.
Remark 1. In general, if the series nonlinearity is inside sector [1 2 ], i.e., 2 · u · (u) ¿ u · (u) ¿ 1 · u2 ; where (u) denotes a sector bounded function, 2 is often called the gain margin, and 1 is called the gain reduction tolerance. So, we have 2 → ∞ and 1 = in Assumption 3. It is seen that if u increases, then (u) increases and vice versa. An example of a scalar nonlinear function is shown in Fig. 1. In consequence, to design the variable structure control is, >rstly, to choose a proper switching surface for the uncertain nonlinear system with time delay, so that the sliding motion on that switching surface has the desired properties. Next, choose a discontinuous control law, which enforces the motion of the sliding mode to be asymptotically stable. 3. Variable structure control design First, the following lemmas will be employed to derive the variable structure controller [4]. Lemma 1. If the nonlinear input satis5es the property as indicated in Eq. (4), there exists a continuous function (·) : R+ → R+ ; (0) = 0, and (p)¿0 for p¿0. Therefore, if |u(t)| = (q), then · u2 ¿ q · (q);
∀q ¿ 0:
(5)
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Proof. The proof is very straightforward if one chooses (q) = · q;
¿
1
then · u2 = · |u|2 = · 2 (q) ¿ q · (q):
(6)
Usually the VSC design is a two-phase process. Phase one is to choose a switching surface so that the original system, restricted to the intersection of the switching surface (sliding mode), results in the desired behavior. Phase two is to determine a switching control that is able to force the trajectory of the system approaching to and staying on the sliding surface. Hereby, the switching surface is de>ned as s(t) = cT x(t) = 0;
(7)
where c ∈ R n is a constant vector. It is necessary to examine if the property in sliding mode, described in Eq. (7), is valid for the system with nonlinear input, shown in Eq. (1). A necessary condition for the system state trajectory to remain on the sliding surface s is s˙ = 0 namely, ˙ s(t) ˙ = cT x(t) = cT [(A + JA)x(t) + b(u) + Ad xd + f] =0
(8)
where x d denotes x(t − d ). Therefore, the equivalent control eq in the sliding mode s = 0 is given by eq = −(cT b)−1 cT [(A + JA)x(t) + Ad xd + f]:
(9)
Introducing Eqs. (3) and (9) into Eq. (1) produces the equivalent dynamic system with nonlinear input in the sliding mode as s(t) = 0 ˙ = (A + JA)x(t) + beq (u) + Ad xd + f x(t) = (A + bH)x − b(cT b)−1 cT [(A + bH)x(t) + bGxd + bd] + bGxd + bd = [I − b(cT b)−1 cT ]Ax:
(10)
Matching conditions (3) are used in the derivation of Eq. (10). From Eq. (10), it can be seen that the invariance condition also holds even though the time-delayed system is with “nonlinear” input. Remark 2. From the above analysis, it can be concluded that the uncertain time-delayed system with “nonlinear” input in the sliding mode possesses the same property as those with “linear” input in the sliding mode. Accordingly, the design of the switching surface can be achieved through the same design tool for the systems with “linear” input.
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From the analysis mentioned above, it can be seen that how to drive the system trajectories into the sliding mode is the key work for system stabilization. Before stating the scheme of the controller, the condition for reaching the sliding mode is given below: Lemma 2. The motion of the sliding mode (7) is asymptotically stable, if the following condition holds: ss˙ ¡ 0;
∀t ¿ 0:
(11)
To ful>ll the condition stated in Eq. (11), the desired variable structure control is suggested by u(t) = − where (x; t) =
scT b (x; t); scT b
(12)
r {[(cT b)−1 cT A + 1 ]x + 2 xd + 3 };
r ¿ 1:
(13)
The following theorem shows that the proposed variable structure control in Eq. (12) drives the uncertain system with nonlinear input into the sliding mode s(t) = 0. Theorem 1. Consider the uncertain system (1) subjected to Assumptions 1, 2, and inequality of (3). If the variable structure control input u(t) in Eq. (1) is given as that indicated by (12), then the system trajectories asymptotically converge to the sliding mode (7). Proof. Consider the reaching condition of the sliding mode (7). Let V (t) = 12 s 2 be the Lyapunov function of the system described in (1). If Eq. (1) is substituted into the derivative of the states in (11), one can obtain ˙ V˙ (t) = ss˙ = scT x(t) = scT [(A + JA)x(t) + b(u) + Ad x(t − d ) + f(t)] = scT [Ax + b(u) + b(Hx + Gxd + d)]:
(14)
Then, Eq. (3) is applied to the above equation to obtain the following inequality expression: ss˙ 6 scT b(cT b)−1 cT Ax + scT b(u) + scT b(1 x + 2 xd + 3 ):
(15)
From Eqs. (12) and (4), we have u(u) = −
scT b (x; t)(u) ¿ u2 = 2 (x; t): scT b
Therefore, the above expression can be rearranged as scT b · (u) 6 −2 (x; t)
scT b = −(x; t)scT b: (x; t)
(16)
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365
Inserting Eq. (16) into the right-hand side of the inequality in Eq. (15), one has ss˙ 6 scT b{(cT b)−1 cT Ax − + 1 x + 2 xd + 3 }:
(17)
Thus, once the expression of (x; t) in (13) is valid, one can obtain ss˙ 6 (1 − r)scT b{[(cT b)−1 cT A + 1 ]x + 2 xd + 3 }:
(18)
Since r¿1, it is easy to con>rm that ss˙ ¡ 0
or
V˙ (t) ¡ 0:
(19)
According to the Lyapunov stability theorem, condition (19) ensures that s(t) is toward the switching surface and the sliding mode is asymptotically stable. Then the proof is completed. Remark 3. As expressed in Remark 2, a sector bounded input function is only a special case of series nonlinear function de>ned in (4). Therefore, it is straightforward to ensure that the developed control law of VSC can be applied to control the systems with sector bounded nonlinearities. In summary, variable structure control method has been derived to stabilize the uncertain timedelayed system with nonlinear input. It has been shown that the proposed variable structure control can drive the uncertain system trajectories into the sliding mode even though the time-delayed systems are with “nonlinear” input. Moreover, the uncertain system can also preserve the invariance condition even though the uncertain systems are with delay time and subjected to the “nonlinear” input. 4. Fuzzy sliding mode control As described in the last section, the objective of the SMC is to design a control law (12) such that the reaching condition (11) is satis>ed. In this section, we will develop a fuzzy logic control law in an attempt to accomplish this object. The proposed fuzzy control is called the fuzzy sliding mode control (FSMC) because it is based on the principle of SMC. The proposed FSMC [5] is shown in Fig. 2, where s and s˙ are the inputs and u denote the output of the FSMC. The inputs of the proposed fuzzy controller are S and S˙ which are the fuzzi>ed variables of s and s, ˙ respectively. The output of the fuzzy controller is U which is the fuzzi>ed variable of u. All the universes of discourse of S; S˙ and U are arranged from −1 to 1; thus, the range of nonfuzzy variables s; s˙ and u must be scaled to >t the universe of discourse of fuzzi>ed variable S; S˙ and s(t) .
s( t )
K1
S(t)
K2 .
Fuzzy Control Rules
U
K3
S( t )
Fig. 2. The block diagram of the FSMC.
u(t)
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Fig. 3. Fuzzy variable of triangular type.
U with scaling factors K1, K2 and K3, respectively, namely, S = K1 · s(t); S˙ = K2 · s(t); ˙ u(t) = K3 · U: In the implementation, S˙ is approximated with (Sk − Sk −1 )=T where T is the sampling period. For simplicity, a triangular type membership function is chosen for the aforementioned fuzzy variables, as shown in Fig. 3. The reasoning behind the fuzzy control rules for the FSMC, which will be presented later, is explained by the following analysis: By taking the time derivative of both sides of (7), we obtain s˙ = cT x˙ = cT Ax + cT b(u) + cT b(Hx + Gxd + d):
(20)
Then, multiplying both sides of the above equation by s leads to ss˙ = scT Ax + scT b(u) + scT b(Hx + Gxd + d):
(21)
Here, we assume that cT b¿0. In (20), it is seen that s˙ increases as (u) increases and vice versa. In (21), it is seen that if s¿0, then decreasing (u) will make ss˙ decrease and that if s¡0, then increasing (u) will make ss˙ decrease. From the Remark 1, we know the relation between u and (u). So, it is seen that if s¿0, then decreasing u will make ss˙ decrease and that if s¡0, then increasing u will make ss˙ decrease. Now, we choose a Lyapunov-like function V = 12 s2 :
(22)
V˙ = ss:˙
(23)
Then Based on the above qualitative analysis and the Lyapunov stability condition V˙ ¡0 and s = 0 has to be a stable sliding surface, we can ensure that the system is asymptotically stable, simultaneously matching the reaching condition ss¡0, ˙ and the control input u can be designed with condition ss¡0. ˙
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Table 1 Rule table of FSMC S= S˙
NB
NM
NS
ZE
PS
PM
PB
PB PM PS ZE NS NM NB
ZE PS PM PB PB PB PB
NS ZE PS PM PB PB PB
NM NS ZE PS PM PB PB
NB NM NS ZE PS PM PB
NB NB NM NS ZE PS PM
NB NB NB NM NS ZE PS
NB NB NB NB NM NS ZE
Table 2 Look-up table of FSMC S˙
U
−1 S
1 0:84 0:7 0:56 0:42 0:28 0:14 0 −0:14 −0:28 −0:42 −0:56 −0:7 −0:84 −1
−0:84 −0:7
−0:56 −0:42 −0:28 −0:14 0
0 −0:15 −0:27 −0:44 −0:57 0:15 0 −0:11 −0:3 −0:4 0:27 0:11 0 −0:17 −0:3 0:44 0:29 0:19 0 −0:1 0:58 0:41 0:31 0:12 0 0:67 0:53 0:41 0:3 0:19 0:72 0:56 0:52 0:4 0:29 0:9 0:69 0:65 0:53 0:43 0:9 0:72 0:7 0:59 0:5 0:9 0:78 0:78 0:66 0:61 0:9 0:9 0:9 0:76 0:68 0:9 0:9 0:9 0:76 0:76 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9 0:9
−0:67 −0:52 −0:4 −0:28 −0:18 0 0:12 0:28 0:42 0:52 0:62 0:68 0:76 0:82 0:9
−0:73 −0:56 −0:52 −0:4 −0:3 −0:12 0 0:15 0:31 0:42 0:5 0:59 0:7 0:73 0:9
−0:9 −0:73 −0:67 −0:56 −0:44 −0:27 −0:15 0 0:15 0:27 0:44 0:56 0:67 0:73 0:9
0.14
0.28
0.42
0.56
0.7
0.84
1
−0:9 −0:73 −0:7 −0:59 −0:5 −0:42 −0:31 −0:15 0 0:15 0:27 0:44 0:56 0:67 0:73
−0:9 −0:82 −0:76 −0:68 −0:62 −0:52 −0:42 −0:28 −0:12 0 0:18 0:28 0:4 0:52 0:67
−0:9 −0:9 −0:9 −0:76 −0:68 −0:61 −0:5 −0:43 −0:29 −0:19 0 0:1 0:3 0:4 0:57
−0:9 −0:9 −0:9 −0:76 −0:76 −0:66 −0:59 −0:53 −0:4 −0:3 −0:12 0 0:17 0:3 0:44
−0:9 −0:9 −0:9 −0:9 −0:9 −0:78 −0:7 −0:65 −0:52 −0:41 −0:31 −0:19 0 0:11 0:27
−0:9 −0:9 −0:9 −0:9 −0:9 −0:78 −0:72 −0:69 −0:56 −0:53 −0:41 −0:29 −0:11 0 0:15
−0:9 −0:9 −0:9 −0:9 −0:9 −0:9 −0:9 −0:9 −0:72 −0:67 −0:58 −0:44 −0:27 −0:15 0
The resulting fuzzy control rules are shown in Table 1. We apply the Mamdani’s min-operation fuzzy implication and adopt the center of gravity as defuzzi>cation method to construct a look-up table based on Table 1. The look-up table is shown in Table 2. For purpose of illustration, a few control rules in Table 1 are described below: • IF S is PB and S˙ is PB, THEN U is NB This control rule states that if S and S˙ are both positive big, which is equivalent to that ss˙ is largely positive, then, by (21), a large negative change of the control input is required to decrease ss˙ quickly. • IF S is PB and S˙ is NB, THEN U is ZE This control rule implies a situation where when ss˙ is largely negative, which is a desired situation, there is no need to change the control input.
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F.-M. Yu et al. / Fuzzy Sets and Systems 140 (2003) 359 – 374
• IF S is NB and S˙ is PB, THEN U is ZE This control rule implies a situation where when ss˙ is largely negative, which is a desired situation, there is no need to change the control input. • IF S is NB and S˙ is NB, THEN U is PB This control rule states that if S and S˙ are both negative big, which is equivalent to that ss˙ is largely positive, then, by (21), a large positive change of the control input is required to decrease ss˙ quickly. Remark 4. As to the fuzzy sliding mode control for MIMO case, a system is given as follows: ˙ = (A + JA)x(t) + B.(u) + Ad x(t − d ) + f(t); x(t) x(t) = X(t); − d 6 t ¡ 0;
(24)
where u ∈ Rm ; B ∈ R n×m ; .(u) : Rm → Rm , and the other matrices or vectors are the same as (1). Similarly, By Assumption 2, the matching conditions of JA = BH; A d = BG, and f = Bd hold. Therefore, we have ˙ = Ax(t) + B.(u) + B[Hx(t) + Gxd + d]: x(t)
(25)
Hereby, the switching surfaces are de>ned as S(t) = Cx(t) = 0;
(26)
where S = [s1 ; s2 ; : : : ; sm ]T ; C = [c1 ; c2 ; : : : ; cn ] with ci = [ci1 ; ci2 ; : : : ; cim ]T , and i = 1; 2; : : : n. Consider the reaching condition of the sliding mode (26), one can obtain ST S˙ = ST CAx + ST CB.(u) + ST CB(Hx + Gxd + d):
(27)
For the sake of convenience, de>ne ˆ .(u) = (CB)−1 .(u);
(28)
ˆ = [ˆ 1 (u1 ); ˆ 2 (u2 ); : : : ; ˆ m (um )]T , then Eq. (27) can be where we let u = [u1 ; u2 ; : : : ; um ]T and .(u) rearranged as ˆ ST S˙ = ST CAx + ST .(u) + ST CB(Hx + Gxd + d) =
m
sj s˙j
j=1
= ST CAx + [s1 ˆ 1 (u1 ) + s2 ˆ 2 (u2 ) + · · · + sm ˆ m (um )] + ST CB(Hx + Gxd + d):
(29)
ˆ Here, assume that CB¿0, and by (28), it is clear that .(u) is proportional to .(u). In (29), for T j = 1; 2; : : : ; m, it is seen that if sj ¿0, then decreasing ˆ j (uj ) will make S S˙ decrease. We also have that if sj ¡0, then increasing ˆ j (uj ) will make ST S˙ decrease. Furthermore, we choose the Lyapunov
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369
Table 3 jth Rule table of FSMC for MIMO system sj = s˙j
NB
NM
NS
ZE
PS
PM
PB
PB PM PS ZE NS NM NB
ZE PS PM PB PB PB PB
NS ZE PS PM PB PB PB
NM NS ZE PS PM PB PB
NB NM NS ZE PS PM PB
NB NB NM NS ZE PS PM
NB NB NB NM NS ZE PS
NB NB NB NB NM NS ZE
˙ will ensure that S(t) is toward the switching function V = 12 S2 . Then condition of (V˙ = ST S)¡0 surfaces and the fuzzy sliding mode is asymptotically stable. Based on the above qualitative analysis, one can design the control input uj in an attempt to ˙ The resulting fuzzy control rules are shown in Table 3. And one of satisfy the condition ST S¡0. the control rules is given as • IF sj is PB and s˙j is PB, THEN uj is NB According to the above description, we conclude that the FSMC can be extended to handle the MIMO system. 5. Simulation Consider an uncertain time-delayed system with series nonlinearity, which is described by Eq. (1) [4]. It is shown below ˙ = (A + JA)x(t) + b(u) + Ad x(t − d ) + f(t) x(t) = Ax(t) + b.(u) + b[Hx(t) + Gxd + d]; where xd = x(t − d ); H61 ; G62 and d63 . From Eq. (3), we also obtain that JA = bH; A d = bG, and f = bd. The corresponding parameters of the illustrated system are given as follows: 0 1 x1 (t − d ) 0 x1 A= ; xd = ; ; b= ; x= x2 x2 (t − d ) −2 −3 1 (u) = (!esin u + " cos u)u; Hx + Gxd + d = l1 e(1+sin x1 )
! ¿ " ¿ 0; x12 + x22 + l2 cos x2d
x12d + x22d + l3 cos x2 :
For both VSC and FSMC controllers, the switching surface is taken as s(t) = 2x1 + x2 :
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Fig. 4. State variable dynamics for the system under VSC: x1 and x2 .
Obviously, cT b¿0. This meets the assumption cT b¿0 in (21). For this system, the following parameters are given: T = 1:0;
l1 = 0:04;
l2 = 0:3;
l3 = −0:6;
r = 0:3:
In addition, the following initial values are arbitrarily chosen, that is x1 (0) = −1;
x2 (0) = 1:
5.1. VSC method The computer simulation for this uncertain time-delayed system with series nonlinearity is carried out with sampling step of 0:01 s. For computer simulation, the corresponding parameters are r = 1:1; = 0:7; 1 = 0:3; 2 = 0:3; 3 = 0:3, respectively. The delay time constants are taken as d = 0:2. Fig. 4 shows the transient response of x1 and x2 under the VSC controller. Fig. 5 shows the phase plane between x1 and x2 under the VSC. From the simulation result, it shows that the VSC works well for the uncertain time-delayed system with nonlinear input. But there exists chattering phenomenon that is not desired. 5.2. FSMC method For this uncertain time-delayed system with series nonlinearity, the sampling step and delay time is the same as VSC method. In this case study, a (15 × 15)-matrix of the look-up table and two-degree
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Fig. 5. The phase plane between x1 and x2 : x2 (x1 ) under the VSC.
nesting are chosen (Table 2). Accordingly, the scaling factors are chosen as K1 = 0:5; K2 = 0:5 and K3 = 50. The simulation results are shown in Figs. 6 and 7. Fig. 6 shows the transient response of x1 and x2 under the FSMC controller. Fig. 7 shows the phase plane between x1 and x2 under the FSMC. From the simulation result, it is shown that the FSMC not only works well for the uncertain time-delayed system with nonlinear input, but also eliminates the chattering phenomenon. Remark 5. In sliding mode control, state trajectories chattering along a sliding hyperplane is inevitable in practice, because the control input is discontinuous in the vicinity of the sliding hyperplane. Chattering is a serious drawback of sliding mode control and has been an important problem in the >eld of sliding mode control. Here, we want to demonstrate that the FSMC can handle this chattering problem e4ectively by adjusting the control input near the sliding hyperplane. The simulation results in Fig. 7 show the state trajectory of system response for uncertain time-delayed system with series nonlinearity. No signi>cant chattering along the sliding hyperplane is observed in the trajectory. Remark 6. Because the switching function is de>ned only by the state variable of x(t), that is, s(t) = cT x(t) = 0, free of the time delay d ; d a4ect the system stability very little. From the simulation result, it is found that the robustness of system is little susceptible to time delay d . 6. Conclusions We have shown that the FSMC developed in this work can be easily applied to uncertain timedelayed dynamical systems with nonlinear input as described in Eq. (1). The scheme of this work
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Fig. 6. State variable dynamics for the system under FSMC: x1 and x2 .
Fig. 7. The phase plane between x1 and x2 : x2 (x1 ) under the FSMC.
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is to design a fuzzy sliding mode controller to stabilize uncertain time-delayed systems with series nonlinearity. Moreover, the proposed FSMC can be extended to handle the MIMO system. Several important features of fuzzy sliding mode are summarized in the following: (1) No exact mathematical model for the nonlinear systems is required for the controller design. The only information required to design the controller is the qualitative inRuences of u on ss. ˙ (2) An inherent chattering problem of sliding mode control can be handled e4ectively without involving sophisticated mathematics. (3) By the Lyapunov stability theorem and the reaching condition of (V˙ = ss)¡0, ˙ we only need to adjust the control input u to satisfy ss¡0, ˙ the construction of rule base is simple and clear. Then, the asymptotic stability for uncertain time-delayed systems with series nonlinearities can be guaranteed. (4) The minimum numbers of fuzzy control rules can be obtained. Because in each fuzzy implication only s and s˙ are taken as inputs of the fuzzy sliding mode controller in spite of the number of state variables in the system. (5) From the simulation result, it is shown that the FSMC indeed can be applied to the uncertain time-delayed system with a nonlinear input. The robustness for the uncertainty and perturbation is still very satisfactory. The e4ect is the same as that of systems with a linear input. We will further investigate that the fuzzy control combines the MIMO VSC to generate a new FSMC method for uncertain time-delayed system with a nonlinear input in the future. Acknowledgements The authors would like to thank the >nancial support of the National Science Council of the Republic of China under contract NSC88-2213-E008-043. References [1] H. Allamehzadeh, J.Y. Cheung, Design of a stable and robust fuzzy controller for a class of nonlinear system, Proc. Fifth IEEE Internat. Conf., Vol. 3, 1996, pp. 2150 –2154. [2] G. Bartolini, A. Ferrara, Multivariable fuzzy sliding mode control by using a simplex of control vectors, in: S.G. Tzafestas, A.N. Venetsanopoulos (Eds.), Fuzzy Reasoning in Information, Decision, and Control Systems, Kluwer, Amsterdam, The Netherlands, 1994, pp. 307–328. [3] J.S. Glower, J. Munighan, Design fuzzy controllers from a variable structures standpoint, IEEE Trans. Fuzzy Systems 5 (1) (1997) 138–144. [4] K.-C. Hsu, Adaptive variable structure control design for uncertain time-delayed systems with nonlinear input, Dynamics and Control 8 (1998) 341–354. [5] G.-C. Hwang, S.-C. Lin, A stability approach to fuzzy control design for nonlinear systems, Fuzzy Sets and Systems 48 (1992) 279–287. [6] Y.R. Hwang, M. Tomizuka, Fuzzy smoothing algorithms for variable structure systems, IEEE Trans. Fuzzy Systems 2 (1994) 277–284. [7] S.-W. Kim, J.-J. Lee, Design of a fuzzy controller with fuzzy sliding surface, Fuzzy Sets and Systems 71 (1995) 359–367.
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