Time-Optimal Control of T–S Fuzzy Models
via Lie Algebra
Pao-Tsun Lin, Member, IEEE, Chi-Hsu Wang, Fellow, IEEE, and Tsu-Tian Lee, Fellow, IEEE
Abstract—This paper investigates a geometric property of time-optimal problem in the Takagi–Sugeno (T–S) fuzzy model via Lie algebra. We will focus on the existence of a time-optimal solution, singularity of switching function, and number of switching. These inherent problems are considered because of their rich geometric properties. The sufficient condition for the existence of a time-optimal solution reveals the controllability of T–S fuzzy model that can be found by the generalized rank condition. The time-optimal controller can be found as the bang–bang type with a finite number of switching by applying the maximum principle. In the study of the singularity problem, we will focus on the switching function whenever it vanishes over a finite time interval. Finally, we show that the bounded number of switching can be found if the T–S model (also a nonlinear system) is solvable.
Index Terms—Controllability, fuzzy control, Lie algebras, Takagi–Sugeno (T–S) fuzzy model, time-optimal control.
I. INTRODUCTION
I
N RECENT years, fuzzy logic control with human knowl-edge of the plant has witnessed an effective approach to the design of nonlinear control systems. Indeed, there have been many successful applications that are based on fuzzy con-trol [1]–[8]. Takagi and Sugeno [9] proposed an approach to model nonlinear processes. This type of model is known as the T–S model that is further developed in [10]. The T–S fuzzy model blends the dynamics of each fuzzy implication by a lin-ear consequence part [11]–[13]. In this type of fuzzy model, lots of important issues are addressed such as stability [2], [8], [11], performance [13]–[15], and robustness [16]–[18], etc. In [19], a fuzzy approach is used in the design of time-suboptimal feed-back controllers.The T–S fuzzy model has a strong connection with the poly-topic linear differential inclusion (PLDI) [36], [37] that will lead to the relaxed version of T–S fuzzy model defined in this paper. The equivalence between the fuzzy model and the differential inclusion is revealed by the well-known Filippov’s selection lemma [36], [37]. From Filippov’s selection lemma, the set of solutions of T–S fuzzy model coincides with the set of solu-tions of the differential inclusion. By formulating the T–S fuzzy Manuscript received December 30, 2006; revised June 13, 2007 and September 5, 2007; accepted October 15, 2007. First published April 30, 2008; current version published July 29, 2009. This work was supported by the Pro-gram for Promoting Academic Excellence of Universities (Phase II) under Grant NSC96-2752-E-027-001-PAE.
P.-T. Lin and C.-H. Wang are with the Department of Electrical and Control Engineering, National Chiao Tung University, Hsinchu 30050, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]).
T.-T. Lee is with the Department of Electrical Engineering, National Taipei University of Technology, Taipei 106, Taiwan, R.O.C. (e-mail: ttlee@ ntut.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TFUZZ.2008.924321
model as a relaxed version, we can perform some algebraic op-erations on it, such as linear combinations and the Lie bracket product.
The maximum principle has been extensively applied in many time-optimal control problems [20]–[35]. A series of results has been published on the applications of the maximum principle in time-optimal control of finite-dimensional linear systems and certain low-order nonlinear systems [21]–[23]. It is well known that Lie brackets play an essential role in the study of time-optimal control [31]–[35]. In general, the maximum principle can reduce the optimal control problem by the Hamiltonian. However, the Hamiltonian formulation contains no information about the existence of a time-optimal solution. It is better to convert the existence of a time-optimal solution to the study of reachable sets [25], [26], [28]. While the existence of a time-optimal solution is addressed as the compactness of a research-able set, we still have to generalize the analytical process, and this will lead us to the discussion of Lie algebra. An accessible Lie algebra spans a family of analytical vector fields that will imply the controllability of T–S fuzzy model.
Using the maximum principle, a time-optimal trajectory com-bined with the corresponding control is called an extremal. The bounded input is determined by the signs of the associ-ated switching functions. The singularity of the system is a well-known problem in time-optimal control that was explored in [27] and [31]. An optimal trajectory may be singular, i.e., switching functions may vanish along the trajectory. The char-acterization of such trajectories will be investigated in this paper. The existence of extremal will imply that the time-optimal con-troller of the T–S fuzzy model has a finite number for switching, which can be found by Lie algebra in this paper.
This paper is organized as follows. In Section II, we will formulate the time-optimal problem in T–S fuzzy model. In Section III, the T–S fuzzy model is described as a polytopic linear differential inclusion and Lie algebra is adopted to find the controllability of T–S fuzzy model. It can also be shown that if the T–S fuzzy model is controllable, then the time-optimal does exist. Assuming the existence of a time-optimal solution, we will investigate the singular structure in fuzzy model in Section IV. The optimal trajectory is solved by the shooting method, and the numerical illustrations are provided in Section V. Finally, conclusions are included in Section VI.
II. PROBLEMFORMULATION ANDPRELIMINARIES
A. Problem Formulation
Consider a nonlinear control-affine system
˙x = f (x) + g (x) u (1)
where x∈ X is the system state, and u is the control input in an arbitrary set U . The state space X is a smooth differential manifold of dimension n. The vector fields f and g are assumed to be analytic.
In many situations, a fuzzy model with the human knowledge can provide a linguistic description of the nonlinear system in terms of IF–THEN rules. The ith rule of the T–S fuzzy model is described by the following form:
Rule i: IF z1(t) is Mi1· · · and zp(t) is Mip, THEN
˙x = Aix + Biu
where x denotes system states, taking values in an open subset
X of Rn, u∈ Rm is a measurable bounded function on U, i
is the number of IF–THEN rules, zi(t) are some fuzzy input
variables, Mij are fuzzy membership functions in the ith rule,
and ˙x = Aix + Biu is the output from the ith IF–THEN rule.
The entire fuzzy model is formulated as follows: ˙x =
r
i= 1
µi(z(t)) (Aix + Biu) (2)
where r is the total number of rules, µi(z(t)) is the
nor-malized membership function, and µi(z(t)) = αi/
r i= 1αi,
where αi is the firing strength of the ith rule such that
αi=
p
j = 1Mij(zj(t)).
The relaxed version of T–S fuzzy model is described by ˙x∈ Co {[Aix + Biu]| i = 1, . . . , r } (3)
where Co denotes a convex hull [36]. If the T–S fuzzy model is continuous and the control input U is compact, the set of solutions of (2) coincides with the set of solutions of (3) [36], [37], i.e., Co{[Aix + Biu]| i = 1, . . . , r } ⊇ r i= 1 µi(z(t)) (Aix + Biu) .
Therefore, we represent the T–S fuzzy model by (3) as ˙x = r i= 1 µi(t) (Aix + Biu) (4) where µi(t)∈ [0, 1], and r
i= 1µi(t) = 1. To simplify the
no-tion, we adoptAi= r i= 1µi(t)Ai, Bi= r i= 1µi(t)Bi,
and the jth column vector of Bi denoted as
bj =
r
i= 1µi(t)Bij, j = 1, . . . , m and these are assumed to be
lin-early independent. Throughout the rest of this paper, the T–S fuzzy model is denoted as
˙x =Aix +
Biu. (5)
In general, the variable z(t) in (2) sometimes is chosen as the state variable x(t), thus defuzzification µi(z(t)) causes (2) to
become a class of nonlinear systems. This leads to difficultly in performing differential algebra on (2). To avoid this problem, such a T–S fuzzy model (5) is introduced to allow us to perform differential algebra on it.
In this paper, we will make the following assumptions on the control input.
Assumption 1: The control input is given by
U ={u ∈ Rm | aj ≤ uj ≤ bj, j = 1, . . . , m} .
For a given control u(t)⊂ U on a time interval [0, t1] and any
initial point x (t0) = x0 ∈ X, let x (., x0, u) denote the solution
of the fuzzy model (5) with a measurable control u defined on an interval of [0, t1]. For performing optimality on a segment
[0, t1], we introduce a cost functional J (u) =
t1
0
ϕ (x(t), u(t)) dt. (6) Let x0 ∈ X be an initial point and x1 ∈ X be a final point.
We propose the following optimal control problem in terms of the cost functional J .
Problem 1: Find a control u(t)∈ U that minimizes (6) along
the solution of (5) and satisfies the boundary condition
x (t1, x0, u) = x1. (7)
We note that this problem is well posed, i.e., an optimal control does exist. The intuitive interpretation of Problem 1 is clear: find a control that will push the initial state to a given final condition in a given amount of time.
B. Preliminaries
For the nonlinear control-affine system (1), the corresponding
Lie bracket of two smooth vector fields f and g is denoted by
[f, g], and
[f, g] (x) = ∂f
∂xg (x)− ∂g ∂xf (x)
where ∂f /∂x and ∂g/∂x denote the Jacobi matrices of their vector fields. The iterated Lie bracket of f and g is defined as
ad (f )k(g) (x) = [f, ad (f )k−1g] (x) (8) where ad (f )0(g) := g, and k≥ 1. The Lie algebra generated by the vector fields can be expressed as
L = {f, g1, . . . , gm}L A
= span{[gi1· · ·[gik−1, gik]· · ·] | k ≥ 1, 0 ≤ i1, . . . , ik ≤ m} where g0= f. Since the control-affine system can be
repre-sented by a family of vector fields, this will have direct applica-tions to control systems. Considering a T–S fuzzy model with a compact set of control inputs U , the Lie bracket taken at a point of an analytic family of vector fields forms a complete set of its invariants. In particular,L (p0) denotes the space of
tangent vectors at p0 defined by the Lie algebra. Due to the
fact that f = g0 = Aix, g1 = b1, . . . , gm = bm, and
that the Lie bracket of constant vector fields is zero, the iter-ated Lie bracket can be found as
adAix k bj = Aix, ad Aix k−1 bj . (9)
A Lie algebraL is recursively defined by
L(1)= [L, L]
L(2)= [L(1),L(1)], . . . ,L(k ) = [L(k−1),L(k−1)], . . .
is called solvable ifL(k ) = 0 for large k, i.e.,L(k ) ⊃ L(k + 1).
Furthermore, the Lie algebraL is called nilpotent if the sequence ofL is always decreasing with respect to
L1 =L, L2 = [L, L1], . . . ,Lk = [L, Lk−1], . . .
andLk = 0. Any nilpotent Lie algebra is solvable. More details can be found in [38]. In the following, we will introduce notions and results that play a basic role in analyzing the structure of nonlinear control systems. They are directly related to control-lability properties of nonlinear systems. In the following, we denote X as an n-dimensional C∞manifold.
Definition 1: Let TxX be a subspace of the tangent space at
any point x∈ X. A distribution ∆ on X is a map that is
x∈ X → ∆ (x) ⊂ TxX.
The distribution ∆ is a smooth subspace ofRn to each point
x. The dimension of ∆, in general, is not a constant. If the
dimension is constant in a neighborhood of x, then x is said to be a regular point of the distribution. If any point of the distribution is regular with dimension k, the distribution is said to be regular and the dimension of the distribution is k.
Definition 2: A distribution ∆ (x) is called involutive if for any
two vector fields f, g∈ ∆ (x), their Lie bracket [f, g] ∈ ∆ (x). For convenience, the following Theorems 1–3 are listed here that are adapted from [35]–[39].
Theorem 1 (Chow’s Theorem) [35]: LetF be a set of C∞
vector fields on X and L = {λ0,λ1, . . . ,λk}L A be the Lie
algebra generated byF. If dim (L (x)) = n for all x ∈ X, then any point of X is reachable by trajectory of the vector fieldsF. Thus
x1 = eλtLL ◦ · · · ◦ eλ 1
t1 (x0)
for some L≥ 1, {λ0,λ1, . . . ,λk} ∈ F and t1, . . . , tL ∈ (0, ∞).
The following well-known theorem of Frobenius is charac-terized by the integrable distribution.
Theorem 2 (Frobenius’ Theorem) [38]: If X is a Cω(regular) manifold of dimension n and ∆ is an involutive distribution, then around any point x∈ X, there exists a largest integral manifold of ∆ passing through x.
Remark 1: A distribution ∆ is said to be integrable if there
exists a submanifold S on X such that for any x∈ X ∆ (x) = TxS
where S is passing through x.
Remark 2: Any analytic involutive distribution ∆ is integrable
[39].
Theorem 3 [39]: LetF be a set of Cωvector fields on X and
L = {λ0,λ1, . . . ,λk}L Abe the Lie algebra generated byF. For
all x∈ X, there exists a largest integral manifold of F passing through x.
The proof of Theorem 3 can be found by using the Campbell– Baker–Hausdorff formula and Theorem 2.
In the following section, the T–S fuzzy model (5) associ-ated with the Lie algebra is derived to show the controllability condition and imply the existence of optimal control.
III. EXISTENCE OFOPTIMALCONTROL IN THET–S FUZZYMODEL
We begin with the formal definition of reachability and controllability.
Definition 3: The reachable setR (x) of the T–S fuzzy model
(5) for time t≥ 0, subject to the initial condition x ∈ X is the set
RT(x) ={x (t, u) : x ∈ X and u : [0, T ] → U} .
Definition 4: The T–S fuzzy model (5) is accessible if its
reachable setRT(x), x∈ X has a nonempty interior. Similarly,
we will call this T–S fuzzy model strongly accessible if the reachable setRT (x) has the nonempty interior for any T > 0.
Definition 5: The T–S fuzzy model (5) is controllable if∀x0
and∀x1in the manifold of X, there exists a finite time T and an
admissible control function u : [0, T ] such that x (T ; x0, u) = x1.
In the following, we shall show that the existence of optimal solution of Problem 1 can be reduced to determine the acces-sibility of the reachable set. The qualitative properties of the reachable sets can be established. One of the basic properties can be shown in the following context.
Definition 6: For the T–S fuzzy model (5), the accessibility Lie algebra is defined as
La := Aix, bj | ∀j = 1, . . . , m LA. (10)
TheLais a finite-dimensional Lie algebra of vector fields that
contains the family {Aix,
bj}. In fact, this accessibility
Lie algebra plays basic role in the controllability of a T–S fuzzy
model.
Theorem 4: If the accessibility Lie algebra of the T–S fuzzy
model in (5) is full rank at x, that is
rank (La(x)) = n ∀x ∈ Rn (11)
then the reachable setup to any time T > 0 has the nonempty interior, and therefore, the fuzzy model is strongly accessible.
Proof: According to Chow’s theorem [35], the reachable set R (x) is the largest integral manifold of Lafor∀x ∈ Rn. From
(11), it contains an open neighborhood Ω of x. This implies that for any x0, its reachable set is an open set. We can conclude that
the reachable setR (x) is arcwise connected and spans into the
Rn space. Q.E.D.
Remark 3: Since the T–S fuzzy model (5) is analytic, using
Chow’s theorem [35] and Frobenius’ theorem [38], the manifold
X represents the maximal-connected reachable manifolds. Each
reachable manifold is the maximal integral manifold ofLa.
Remark 4: By using Chow’s theorem [35], the
control-lable manifolds can be spanned from {Aix,
bj|∀j =
Remark 5: The La implies that the T–S fuzzy model (5) is
accessible from x0if the same collection of vectors together with
Aix0+
Biu span the whole space. This condition means
that no vectorBiu belongs to a proper invariant subspace of
Aix0.
Theorem 5: If a T–S fuzzy model is strongly accessible, then
it is also controllable.
Proof: Using Remark 3, for a T–S fuzzy model, the degree
of largest integral manifold is related to rank of accessibility
Lie algebraLa. As the fuzzy model is strongly accessible, there
exists the nth-degree largest integral manifold. For a given point
x∈ Rn, the fuzzy model is controllable. Q.E.D.
In the following, the generalized rank condition of accessible Lie algebra is derived to show the controllability of the T–S fuzzy model.
Corollary 1: The T–S fuzzy model (5) is controllable if and
only if the following matrix (W0, W1, . . . , Wn−1) := bj, Ai bj, . . . ×Ai n−1 bj , j = 1, . . . , m (12) is of rank n for any t > 0.
Proof: First, we give the proof of sufficient part. Considering
the T–S fuzzy model (5), let f = g0 =
Aix, and g1 =
bj
be a vector field. Then, we have the following iterated Lie brackets: Aix, bj =−Ai bj, Aix, Aix, bj =A2ibj, . . . .
From (9), the iterated Lie brackets are rewritten as adAix l bj = (−1)Ai l bj.
Therefore, the accessibility Lie algebraLa consists of
con-stant vector fields only
La= Span Ai l bj l ≥ 0, j = 1 ,..., m. (13) If (12) is satisfied, we can conclude that dim (La) is of full
rank n for any t > 0, and then, the fuzzy model is controllable. Next, we show the necessary condition. From the Frobenius’ theorem [38] and Remark 3, it follows that if the T–S fuzzy model (5) is controllable, then there exits the nth-degree largest integral manifold for x∈ X. If (12) is satisfied from Theorem 3 and Remark 1, there exists a largest integral nth-order sub-manifold S that is unique and contained in the largest integral
manifold. Q.E.D.
Remark 6: In analyzing controllability properties of the
fuzzy model (5), we can replace the set of G (x) =
{Aix + Biu : u∈ U, i = 1 , . . . , r} by its convex hull, and the
trajectories of convexified system can be approximated by the trajectories of the original fuzzy model (2). In particular, if 0∈ intCo {G (x)} for all x ∈ X, then the fuzzy model is con-trollable.
Fig. 1. Membership functions in Example 1.
Remark 7: Obviously, for the single-rule T–S fuzzy model,
Corollary 1 degenerates to the Kalman controllability matrix of the linear system.
Remark 8: If all the subsystems are controllable, whereas the
overall system cannot be concluded as controllable, then the overall system can be called local controllable.
The membership functions obviously play the critical roles in the controllability of the system. In the following examples, the local controllability and controllability of the T–S fuzzy model will be illustrated. The nonlinear system will be modeled with the distinct membership functions.
Example 1: Consider a nonlinear system
˙x = tan (u) ˙
y = 10 sin(x) cos (x) .
Assume that x(t)∈ [−π/2, π/2]. Then, the T–S fuzzy model of the nonlinear system can be formulated as follows:
Rule i: IF x(t) is about “Positive” and “Negative,” THEN ˙ X(t) = AiX(t) + Biu, i = 1, 2 (14) where X(t) = [x(t) y(t)]T A1 = 0 0 10β 0 , B1 = 1 0 A2 = 0 0 −10β 0 , B2 = 1 0
and β = cos (88◦). The membership functions are shown in Fig. 1. According to Corollary 1, the corresponding rank of controllability matrix of the fuzzy model is
Rankbj, Ai bj where W0 = bj = 1 0 W1 = Ai bj = µ1 0 0 0.349 0 1 0 + µ2 0 0 −0.349 0 1 0 = 0 0.349 (µ1− µ2) .
Fig. 2. Membership functions in Example 2.
The fuzzy model is controllable if Rank ([W0, W1]) = 2. We
can check the controllability by the following determinant:
10 0.349 (µ0
1− µ2)
= 0.349(µ1− µ2) .
Unfortunately, the rank of [W0, W1] for µ1 = µ2 = 0.5 is 1.
From the membership functions, we can observe that the fuzzy model is uncontrollable if x(t) = 0. Although x(t) = 0 is one of the equilibrium points; however, the fuzzy model is concluded to be uncontrollable when x(t) = 0 and y(t) = 0.
In following example, we redesign the nonlinear system with different membership functions.
Example 2: Consider the nonlinear system in Example 1. If
the membership functions are chosen as Fig. 2, then the conse-quence parts of the fuzzy model can be formulated as
A1 = 0 0 10 0 , B1 = 1 0 A2 = 0 0 10β 0 , B2 = 1 0 .
By Corollary 1, the controllability matrix contains the vector fields W0 = bj = 1 0 W1 = Ai bj = µ1 0 0 10 0 + µ2 0 0 0.349 0 1 0 = 0 10 (µ1+ 0.0349µ2) .
If the fuzzy model is controllable, then the following condi-tion is satisfied: 10 10 (µ 0 1+ 0.0349µ2) = 10(µ1+ 0.0349µ2) = 0.
Since the firing strengths µi∈ [0, 1] and µ1+ µ2 = 1, then
10 (µ1+ 0.0349µ2) = 0 for ∀t. Then, we can conclude that the
overall T–S fuzzy model is controllable.
Remark 9: An important and natural question arises in the
design of a feedback controller using local controllability. The controllability of a physical system is a prerequisite for proceed-ing with the controller design.
The following theorem discusses the existence of the optimal solution for Problem 1.
Corollary 2: If the T–S fuzzy model in (5) is controllable,
then there exists an optimal control for any bounded input.
Proof: Consider the T–S fuzzy model with bounded input u(t)∈ U ⊆ Rm. It is more convenient to consider the T–S fuzzy
model in the form
˙x =Aix + v, v∈ V
where V is the image of U under the map b :Rm → Rn.
Thus, the Lie brackets are
Aix, v
=Ai· v, v∈ V.
Let the set W ={v− v| v, v∈ V }. The Lie algebra of
the T–S fuzzy model contains the vector fields Aix + v− Aix + v = v− v ∈ W. Consider all constant vector fieldsf = w, w∈ W. Thus, it contains the Lie brackets [w,Aix + v] =
Aiw. Since the
fuzzy model is controllable, the accessibility Lie algebra La
consists of constant vector fields if
La= dim span Ai l w0 ≤ i ≤ n − 1,w ∈ W = n (15) for l = 0, . . . , n− 1 ∀t > 0. This condition means that if the bounded input U is nonempty, then the controllability rank condition implies that the system can be spanned the whole
space. Q.E.D.
The condition of Corollary 2 means that there exists no vector
v = v− v∈ U, j = k, such that no image of U belongs to an
invariant subspace of matrixAi. In the next section, we shall
design the time-optimal controller for the T–S fuzzy model with the maximum principle.
IV. DESIGNINGTIME-OPTIMALCONTROLLER FOR ACONTROLLABLET–S FUZZYMODEL
In this section, we will study the properties of time-optimal control using the maximum principle [20], [27]. In general, Problem 1 can be formulated as a Hamiltonian by the maximum principle. The Hamiltonian for Problem 1 can be described as
H (x,λ, u) := λT Aix +λT
Biu (16)
whereλ : [0, t1] is a costate satisfying the adjoint equation
as-sociated with (5) ˙ λ = −∂H ∂x =−λ T A i. (17)
By using the maximum principle [20], Problem 1 becomes
H (x,λ, u) = max
v∈U H (x,λ, v) . (18)
Definition 7: Trajectories of (5), (16), and (17) that satisfy
the maximum principle are called extremal (x,λ, u) : [0, t1]→ Rn× Rn\ {0} × U. When the constant λ
0is zero, the extremal
Definition 8: For j = 1 , . . . , m, the switching functions ψj(·) along an extremal (x, λ, u) are defined by
ψj : [0, t1]→ R, ψj(t) :=λT
bj. (19)
They are absolutely continuous functions [31].
The necessary condition for optimality provided by the max-imum principle states that u : [0, t1] must pointwise maximize H (x(t),λ(t), ·) for the costate λ associated with the optimal
trajectory. Moreover, the Hamiltonian is constant along the so-lutions of (16) and must satisfy
H (x,λ, u) = λ0, λ0 ≥ 0. (20)
The maximum condition (18) is equivalent to the following:
uj(t)ψj(t) = max vj∈U
vj(t)ψj(t), j = 1, . . . , m. (21)
Obviously, the functions ψj(t) play a crucial role in the study
of optimal trajectories. Under Assumption 1, the time-optimal control must satisfy the following conditions almost everywhere:
uj = bj, if ψj(t) > 0
uj = aj, if ψj(t) < 0 (22)
for j = 1 , . . . , m. In such a case, switching functions having zeros have to be carefully analyzed.
Remark 10: Determination of optimal control sequence of
(22) is related to the trajectory of costates. This introduces other problems as the initial costates and final time are unknown. This kind of problem is called two-point boundary value problems (TPBVP). The shooting method [40], however, has been used to solve this problem. The optimal solution can be obtained by solving (5), (17), (20), and (22) simultaneously. For TPBVP, no practical method has been developed yet to compute the time-optimal feedback control.
Supposing that in the time interval [0, t1] there exists one
nontrivial (or more) subinterval [ta, tb]⊂ [0, t1] such that ψj(t)
is identically zero, then the corresponding extremal is called
singular. If ψj(t) = 0 for almost all t ∈ [0, t1], the maximum
principle implies that the control uj corresponds to piecewise
constant controls taking values in the set of m vertices of U called bang–bang. An extremal is said to be normal if control
uj is bang–bang with at most a finite number of switching.
If T–S fuzzy model is smooth and (x, λ, u) is an extremal, then the time derivative of the absolutely continuous function
ψj(t) is given by ˙ ψj(t) =λT −Aix(t), bj +λT bk, bj uj(t) =λT −Aix(t), bj . (23)
Since bj, j = 1 , . . . , m and j = k are constant
terms, therefore, [bk,
bj] = 0. It is obvious that the
deriva-tives of the switching functions ψj(t) are themselves absolutely
continuous functions, and therefore, we can perform further derivatives of it. In the next theorem, Lie brackets will be
cru-cial in establishing a bound on the number of switches when
bang–bang controls are derived.
Theorem 6: If the T–S fuzzy model is controllable, then the extremal is normal.
Proof: Let (x,λ, u) be extremal in t ∈ [0, t1]. We shall prove
the theorem by contradiction. Suppose there exists a sequence of infinite distinct singular sets
S ={s0 , . . . , si, . . .}
where siis the ith time interval [ta, tb]isuch that ψj(t) = 0∀t ∈
[ta, tb]i, j = 1, . . . , m. Assume t0∈ si. Then, we have the
fol-lowing relation:
ψj(t) =λT (t0)
bj = 0, j = 1, . . . , m. (24)
From (24), we have the first derivation of ψj(t)
˙ ψj(t) =λT (t0) Aix(t), bj = 0. (25)
Indeed, the lth derivative of ψj(t) can be expressed as
ψjl(t) =λT (t0) ad Aix(t) l bj = 0 (26)
for l = 1 , . . . , n− 1. By Corollary 2, we have span adAix(t) l bj ∈ Rn, l = 1, . . . , n− 1.
Hence, we haveλ (t0) = 0, which contradicts the necessary
condition of the maximum principle. Therefore, we can con-clude that the set S is finite. Outside the set S, the switching functionλT(t)bj attains the maximum on U at one vertex;
thus, the optimal control u(t) is bang–bang on [0, t1]\t0.
Q.E.D. If the T–S fuzzy model is extremal, then the system will also simultaneously establish a bounded number of switching for bang–bang optimal controls. Further, consider the trajectories for which m control vectors are simultaneously singular. From the proof of Corollary 2, we also know the the set of all vector fields{[Aix,
bj]} is linear independent, and therefore, we
have the following result.
Theorem 7: If an extremal of the T–S fuzzy model in (5) is
normal, then the switching function ψj(t), j = 1 , . . . , m will
not vanish for any t.
Proof: Assume that k is a fixed element of{1, . . . , m} and
(x,λ, u) is extremal with a common accumulation point of zeros at t = t0. From (24) and (25), we have
ψj(t) =λT (t0)
bj = 0
and its first derivative is ˙ ψj(t) =λT (t0) Aix(t), bj = 0
for all j = 1, . . . , m, j = k. If ψk and ˙ψk vanish at t = t0, then
the vector fieldbk, [
Aix,
bj] for j = 1, . . . , m is linear
independent. This yields a contradiction with the nonvanishing condition for costate in the maximum principle. Q.E.D. The solvable Lie algebra is defined for the T–S fuzzy model (5) as follows.
Fig. 3. Articulated vehicle model [1].
Definition 9: For T–S fuzzy model (5), the solvable Lie alge-bra is defined as L(k ) := A ix, bj|∀j = 1, . . . , m L A. (27)
If the derived seriesL(k ) vanishes for large k, then the T–S
fuzzy model is called solvable.
In the next theorem, the solvable Lie algebra, which will be crucial in establishing a bound on the number of switching for bang–bang control will be derived.
Theorem 8: If the controllable T–S fuzzy model (5) is
solv-able, then the total number of switching is bounded.
Proof: The controllable T–S fuzzy model (5) will imply L = span adAix k bj , for k = 1, . . . , n− 1.
If L is solvable Lie algebra, i.e., L(k ) = ad (Aix)k
bj = 0, for k≥ p ≥ n − 1. From (26), we have
ψkj(t) =λT(t0) ad Aix k bj , for k≥ p (28) is identically zero as the T–S fuzzy model is solvable. In (28), if
ψk
j(t) vanishes for k≥ p, then the polynomial degree of
switch-ing function ψj(t) does not exceed p. Q.E.D.
Remark 11: Forbj = 0, the solvable condition (28) can be
generalized asL(k ) = ad (A
ix)k = 0.
V. ILLUSTRATIVEEXAMPLES
To utilize the time-optimal design techniques, two systems with single input and two inputs, respectively, will be illustrated.
Example 3: Consider an articulated vehicle [1] in Fig. 3. The
kinematic model of the vehicle is the starting point to model the dynamics of the lateral and orientation motions. The dynamics of the articulated vehicle can be formulated as
˙x0 = v l tan (u(t)) x1 = x0− x2 ˙x1 = ˙x0− ˙x2 = v l tan (u(t))− v Lsin (x1(t))
Fig. 4. Membership functions in Example 3.
˙x2= v Lsin (x1(t)) ˙x3= v cos (x1(t))· sin (x2(t)) ˙x4=−v cos (x1(t))· cos (x2(t)) where x0(t) angle of truck;
x1(t) angle difference between truck and trailer; x2(t) angle of trailer;
x3(t) vertical position of rear end of trailer; x4(t) horizontal position of rear end of trailer; u(t) steering angle.
l is the length of truck, L is the length of trailer, and v is
the constant backward speed. In this example, let l = 1 m, L = 2.5 m, and v =−5 m/s.
The control purpose is to find the steering angle with constant backward speed so that the articulated vehicle will reach the straight line x3= 0, i.e.,
x1(t)→ 0, x2(t)→ 0, x3(t)→ 0.
If the angle difference between the truck and trailer expands to 90◦, i.e.,|x1| = 90◦, this phenomenon is called “jackknife.”
When a jackknife phenomenon happens, an articulated ve-hicle becomes uncontrollable and the backward motion cannot continue anymore. To avoid this problem, the analysis of the researchable set will be discussed in the following.
For constructing the T–S fuzzy model, assume that
u(t), x2(t) are small and x1(t)∈ (−π/2, π/2) . Let X(t) =
[x1(t) x2(t) x3(t)]T. The dynamics of the articulated vehicle
can be formulated as:
Rule i: IF x1(t) is “Positive” and “Negative,” THEN
˙
X(t) = AiX(t) + BiU (t), i = 1, 2 (29)
where the membership functions are given in Fig. 4 and the consequent parts are chosen as
A1 = −v/L 0 0 v/L 0 0 0 v 0 , B1 = v/l 0 0 A2 = −v/L 0 0 v/L 0 0 0 β· v 0 , B2 = v/l 0 0 and β = cos (88◦).
From Corollary 1, we have W0 = bj = v/l 0 0 . The matrixAi bj is W1= µ1 −v/L 0 0 v/L 0 0 0 v 0 v/l 0 0 +µ2 −v/L 0 0 v/L 0 0 0 βv 0 v/l 0 0 = −v2/ (lL) (µ 1+ µ2) v2/ (lL) (µ 1 + µ2) 0 . The matrixA2 i bj is W2= −v3/ lL2 (µ 1+ µ2) v3/ lL2 (µ1+ µ2) −v3/ (lL) (µ 1+ βµ2) .
The controllability of the fuzzy model can be reformulated by finding the determinant of [W0, W1, W2]
v/l −v2/ (lL) −v3/ lL2 (µ 1+ µ2) 0 −v2/ (lL) v3/ lL2 (µ 1+ µ2) 0 0 −v3/ (lL) (µ 1+ βµ2) . (30)
The determinant of (30) can be found as (v/l)· −v2/ (lL)!·
"
−v3/ [(lL) (µ
1+ βµ2)]
#
. Since the determinant of (30) can-not be zero for∀µi∈ [0, 1] with
µi= 1 (i = 1, 2), therefore,
we may conclude that the fuzzy model is controllable and a time-optimal solution does exist. To realize time-time-optimal control, we consider a control as U = ˜u + u∗, where the control input ˜
u =−kx can be designed by the pole assignment and
time-optimal control u∗(steering angle) is constrained in [5◦,−5◦]. Choose the closed-loop eigenvalues as [ 0 0 0 ], and we have
k = [−0.4 0 0 ]. By closed-loop feedback, the consequent
parts of the fuzzy model (29) can be reformulated as
A1 = 0 0 0 −2 0 0 0 −5 0 , B1 = −5 0 0 A2 = 0 0 0 −2 0 0 0 −0.1745 0 , B2 = −5 0 0 . Due toBi = 0 for ∀t ≥ 0, by using Remark 11, we have
L(0) = µ 1A1+ µ2A2 = µ1 0 0 0 −2 0 0 0 −5 0 + µ2 0 0 0 −2 0 0 0 −0.1745 0
Fig. 5. Projection of the set V on the x1− x2plane.
L(1)= µ 1A1A1+ µ2A2A2= µ1 0 0 0 0 0 0 10 0 0 + µ2 0 0 0 0 0 0 0.3490 0 0 L(2)= µ 1A1A1A1+ µ2A2A2A2 = 0 0 0 0 0 0 0 0 0 L(3)=L(4)=· · · = 0 0 0 0 0 0 0 0 0 . For∀µi∈ [0, 1] with µi = 1 (i = 1, 2), and k≥ 2, L(k )is
identically zero; therefore, the fuzzy model is concluded to be solvable and the number of switching is at most 2. Assuming u = 5◦, the bang–bang control does exist and the possible control sequence can be concluded as
{u}, {−u}, {u, −u}, {−u, u}, {u, −u, u}, {−u, u, −u}.
The switching curves V are shown in Figs. 5 and 6. The dotted line is the set V−, that is, the trajectory by control input
{−u}, and the solid line shows the set V+, that is, the trajectory
by control input{u}. Let V1 denote the set of states that can be
forced to the origin by the control sequence{u, −u} or {−u, u}. The transition from the control input u to−u must occur on the set V . If the control sequence is from−u to u, the transition must occur on the set V+. The set V
1−is shown in Figs. 7 and 8. The
dotted line is the set V1−, which is forced by the control sequence
{−u, u}, and the solid line shows the set V+
1 , which is forced by
the control sequence{u, −u}. The set V2 is the trajectory that
can be forced to the origin by the control sequence{u, −u, u} or{−u, u, −u}. To prevent the jackknife phenomenon, the state
Fig. 6. Projection of the set V on the x2–x3plane.
Fig. 7. Projection of the set V1on the x1–x2plane.
Fig. 8. Projection of the set V1on the x2–x3plane.
the ellipses show the reachable set for|x1| ≤ 90◦, where the
solid ellipses are the set V1, and the dotted ellipses are the set V2. In fact, V ⊆ V1 ⊆ V2. The maximal reasonable range of
initial positions will be restricted on the reachable set V2.
Fig. 9. Reachable set of V1and V2on the x1–x2plane.
Fig. 10. Reachable set of V1and V2on the x2–x3plane.
Fig. 11. Time-optimal trajectory in phase plane (Case I).
Case I: For the initial position, x0 = 240◦, x1 = 200◦, x2 =
40◦, x3 = 20 m, and x4 = 0 m, the time-optimal trajectory of x3 versus x4 is depicted in Fig. 11. The corresponding
time-optimal control u∗(t) is shown in Fig. 12. The shortest time from the initial position to the origin is 2.4115 s.
Fig. 12. Corresponded time-optimal control input (Case I).
Fig. 13. Time-optimal trajectory in phase plane (Case II).
Fig. 14. Corresponded time-optimal control input (Case II).
Case II: For the initial position, x0 = 320◦, x1= 20◦, x2 =
300◦, x3 = 20 m, and x4 = 0 m, the time-optimal trajectory of x3 versus x4 is depicted in Fig. 13. The corresponding
time-optimal control u∗(t) is shown in Fig. 14. The shortest time from the initial position to the origin is 13.6715 s.
Fig. 15. Membership functions in Example 4.
Fig. 16. Switching curve and time-optimal control input.
Example 4: The multiple inputs system is considered here.
Consider the following T–S fuzzy model
Rule i: IF x1(t) is “Positive” and “Negative,” THEN
˙
X(t) = AiX(t) + BiU (t), i = 1, 2 (31)
where X(t) = [x1(t) x2(t)]T, U (t) = [u1(t) u2(t)]T, |u1(t)| ≤ 1, |u2(t)| ≤ 1, and the consequent parts are chosen
as A1 = 0 0 0.18 0 , B1 = 4 0.5 0.5 −4 A2 = 0 0 0.2 0 , B2= 4 0.5 0.5 −4 .
The membership functions of the fuzzy model are given in Fig. 15. The fuzzy model is found to be controllable by Corollary 1. The switching number is at most 2 that is obtained by using Remark 11. Therefore, the time-optimal sequences are
{1, 1}, {−1, −1}, {1, −1}, {−1, 1}.
Following the same analysis in Example 3, the switching curves are explained in the following. There are two possible switching curves in this example. Let the set of states V be forced by input{1, 1} or {−1, −1} and V1be forced by input{1, −1}
or{−1, 1} to the origin. The switching curve V is depicted as a solid line in Fig. 16, the dotted line depicts switching curve V1,
Fig. 17. Reachable sets in Example 4.
Fig. 18. Time-optimal trajectory in phase plane (Case I).
Fig. 19. Corresponded time-optimal control input (Case I).
Assume R(T ) and R1(T ) are reachable sets for V and V1,
respectively, that can reach the origin at time T . Fig. 17 depicts a reachable set that is sampled from T = 5 to T = 20 every 5 s. The dotted line is the reachable setR1(T ), and the solid line is
the reachable setR(T ).
Fig. 20. Time-optimal trajectory in phase plane (Case II).
Fig. 21. Corresponded time-optimal control input (Case II).
Case I: For the initial state X0 = [40,−50], the
time-optimal trajectory is shown in Fig. 18. The corresponding time-optimal control u∗(t) is shown in Fig. 19. The shortest time from initial state to the origin is 9.350 s.
Case II: For the initial state X0 = [40, 100], the
time-optimal trajectory is depicted in Fig. 20. The corresponding time-optimal control u∗(t) is shown in Fig. 21. The shortest time from initial state to the origin is 25.249 s.
VI. CONCLUSION
This paper presents a new design of time-optimal controller for a controllable T–S fuzzy model in which the maximum prin-ciple is applied. In particular, the subsystems of T–S fuzzy model are blended by a set of firing strengths, which leads it to a class of nonlinear systems. First, we proposed the proof of the existence of optimal control in the T–S fuzzy model that can be addressed as the compactness of the reachable set. The generalized rank condition of the accessible Lie algebra is also applied in this paper for the proof of the existence of optimal controller for the T–S fuzzy model. This also results in the controllability of the T–S fuzzy model. According to the maximum principle, the time-optimal control of the T–S fuzzy model is bang–bang that
is determined by the switching function. By investigating the singular structure of the switching functions of the controllable T–S fuzzy model, we can yield the conditions for the existence, i.e., if the extremal is normal, then there exists the time-optimal
controller for the T–S fuzzy model. In other words, the
time-optimal control of controllable T–S fuzzy model is bang–bang with a finite number of switchings over all trajectories for all
t. The bounded number of switching is related to the
polyno-mial degree of switching function that is obtained by introduc-ing the solvable Lie algebra. Several examples are fully illus-trated to show the conditions for the existence of a time-optimal controller with their optimal trajectories found by numerical simulation.
ACKNOWLEDGMENT
The authors would like to thank the reviewers and the Asso-ciate Editor for their helpful and detailed comments that have helped to improve the presentation of this paper.
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Pao-Tsun Lin (S’02–M’03) was born in Hsinchu,
Taiwan, in 1976. He received the B.S. and M.S. de-grees in electrical engineering from the National Tai-wan University of Science and Technology, Taipei, Taiwan, in 1999 and 2001, respectively. He is cur-rently working toward the Ph.D. degree with the De-partment of Electrical and Control Engineering, Na-tional Chiao Tung University, Hsinchu.
His current research interests include the fields of fuzzy control, optimal control, support vector ma-chines, and intelligent transport systems.
Chi-Hsu Wang (M’92–SM’93–F’08) was born in
Tainan, Taiwan, in 1954. He received the B.S. degree in control engineering from the National Chiao Tung University, Hsinchu, Taiwan, the M. S. degree in computer science from the National Tsing Hua University, Hsinchu, and the Ph.D. de-gree in electrical and computer engineering from the University of Wisconsin, Madison, in 1976, 1978, and 1986, respectively.
He was an Associate Professor and a Professor with the Department of Electrical Engineering, Na-tional Taiwan University of Science and Technology, Taipei, Taiwan, in 1986 and 1990, respectively. He is currently a Professor with the Department of Elec-trical and Control Engineering, National Chiao Tung University. His current research interests include the areas of digital control, fuzzy neural networks, intelligent control, adaptive control, and robotics.
Prof. Wang is a member of the Board of Governors and Webmaster of the IEEE Systems, Man, and Cybernetics Society. He is currently serving as an Associate Editor of the IEEE TRANSACTIONS ONSYSTEMS, MAN,AND
CYBERNETICS, PARTB.
Tsu-Tian Lee (M’87–SM’89–F’97) was born in
Taipei, Taiwan, in 1949. He received the B.S. degree in control engineering from the National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 1970 and the M.S. and Ph.D. degrees in electri-cal engineering from the University of Oklahoma, Norman, in 1972 and 1975, respectively.
During 1975, he was an Associate Professor with NCTU, where during 1978, he was a Professor and the Chairman of the Department of Control Engineer-ing. During 1981, he was a Professor and the Direc-tor of the Institute of Control Engineering, NCTU. In 1986, he was a Visiting Professor and, during 1987, a Full Professor of electrical engineering with the University of Kentucky, Lexington. During 1990, he was a Professor and the Chairman of the Department of Electrical Engineering, National Taiwan Uni-versity of Science and Technology (NTUST). During 1998, he was a Professor and Dean of the Office of Research and Development, NTUST. Since 2000, he has been with the Department of Electrical and Control Engineering, NCTU, where he is currently a Chair Professor. His professional activities include serv-ing on the Advisory Board of the Division of Engineerserv-ing and Applied Science, National Science Council, serving as the Program Director of the Automatic Control Research Program and the National Science Council and serving as an Advisor of the Ministry of Education, Taiwan. He also held numerous consult-ing positions. He is the author or coauthor of more than 200 refereed journal and conference papers in the areas of automatic control, robotics, fuzzy systems, and neural networks. His current research interests include motion planning, fuzzy and neural control, optimal control theory and application, and walking machines.
Prof. Lee was the Fellow of the New York Academy of Sciences (NYAS) in 2002. He has been actively involved in many IEEE activities. He has served as the Member of Technical Program Committee and Member of Advisory Committee for many IEEE-sponsored international conferences. He is now the Vice President—Membership of the IEEE Systems, Man, and Cybernetics Society. He was the recipient of the Distinguished Research Award from the National Science Council, R.O.C., during 1991–1992, 1993–1994, 1995–1996, and 1997–1998, respectively, and the Academic Achievement Award in engi-neering and applied science from the Ministry of Education, R.O.C., in 1997, the National Endow Chair from the Ministry of Education, R.O.C., and the TECO Science and Technology Award from the TECO Technology Foundation in 2003.
![Fig. 3. Articulated vehicle model [1].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7964213.158259/7.891.458.818.98.257/fig-articulated-vehicle-model.webp)
