國
立
交
通
大
學
電機與控制工程系
博 士
論
文
T-S 模糊系統之最佳化時間控制
Time-Optimal Control of T-S Fuzzy Models
研 究 生：林保村
指導教授：王啟旭 博士
李祖添 博士
T-S 模糊系統的最佳化時間控制
Time-Optimal Control of T-S Fuzzy Models
研 究 生： 林保村 Student：
Pao-Tsun Lin
指導教授： 王啟旭 Advisor(s)：
Chi-Hsu Wang
李祖添
Tsu-Tian Lee
國 立 交 通 大 學
電機與控制工程系
博 士 論 文
A DissertationSubmitted to Department of Electrical and Control Engineering College of Electrical Engineering and Computer Science
National Chiao Tung University in partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
in
Electrical and Control Engineering June 2008
Hsinchu, Taiwan, Republic of China
T-S 模糊系統之最佳化時間控制
研究生： 林保村
指導教授： 王啟旭 博士
李祖添 博士
國立交通大學電機與控制工程系博士班
摘 要 本論文針對 Takagi-Sugeno (T-S) 模糊模型，設計一最佳化時間控制器， T-S 模 糊 模 型 可 視 為 一 多 面 體 線 性 微 分 包 圍 (polytopic linear differential inclusion) 數學模型，利用李群論（Lie Algebra）的幾何特性推導最佳化時 間之奇異性、存在性及切換次數。 本論文提出最佳化時間之控制器，首先針對控制器之存在性，推導 T-S 模糊模型之可控制性，在 T-S 模糊模型相關論文中，此為首次探討 T-S 模糊 模型之控制性，並提出歸納式秩數(rank)條件式。透過最大值理論(maximum principle)，最佳化控制器為砰-砰(bang-bang)型式，在系統之奇異性之推導中， 證明所提出之歸納式秩數(rank)條件式，可供設計非奇異之控制器使用;亦 即，可控制之 T-S 模糊模型可設計最佳時間控制器，引用可解析李群論 (solvable Lie algebra)於控制器之切換次數證明，此證明可提供於計算最佳演 算解，讓演算法更容易找到最佳解。透過模擬本控制器設計法則均經在聯結 車前進及倒車系統及多輸入系統之可控制性及最佳化時間控制器。Time-Optimal Control of T-S Fuzzy Models
Student： Pao-Tsun Lin
Advisors： Dr. Chi-Hsu Wang
Dr.
Tsu-Tian
Lee
Department of Electrical and Control Engineering
National Chiao Tung University
ABSTRACT
This dissertation investigates geometric property of time-optimal problem in Takagi-Sugeno (T-S) fuzzy model via Lie algebra. We will focus on the existence of time-optimal solution, singularity of switching function and number of switching. These inherent problems are considered because of their rich geometric properties. The necessary condition for the existence of time-optimal solution reveals the controllability of T-S fuzzy model which can be found by the generalized rank condition. The time-optimal controller can be found as the bang-bang type by applying maximum principle. In the study of singularity problem, we will focus on switching function whatever vanished on a finite time interval. The bounded number of switching can be found if the T-S model (also a nonlinear system) is solvable. This feature can be applied to solve the time-optimal problem by numerical approach. Fast response is always a considered property in this dissertation. A notion directly relate to the convergence rate of the state trajectories. A controller design of T-S fuzzy model on maximal convergence rate is introduced by the level set function. The result
of maximizing the convergence rate is characterized from the maximal invariant ellipsoid. The controller is also bang-bang within both the initial states and target states belong to level set.
Acknowledgements
I would like to thank all people who have helped and inspired me during my doctoral study.
I especially want to thank my advisors, Prof. Chi-Hsu Wang and Prof. Tsu-Tian Lee for their guidance during my research and study at National Chiao Tung University. Their perpetual energy and enthusiasm in research had motivated all advisees, including me. In addition, they were always accessible and willing to help me. As a result, research life became smooth and rewarding for me.
I would also like to thanks Prof. Hsin-Hsin Huang for listening to my sometimes very scattered thoughts and helping me channel them in the right direction. Our wonderful friendship is not only a turning point in my life, but also bringing me to the future. Without his encouragements, my ambition to study can hardly be realized.
Special thanks to Prof. Shun-Feng Su, Prof. Ching-Cheng Teng, Prof. Wei-Yen Wang and Prof. Yih-Guang Leu as my thesis committee members.
Fr. Visminlu Vicente L. Chua, S.J., have been my high school teacher. He offers advice and suggestions whenever I need them. Besides, they have set a role model of a typical teacher who cares and loves his students as if they were their own kids. Thank you.
Thanks to all members are at Intelligent Network Laboratory.
My deepest gratitude goes to my family for their unflagging love and support throughout my life; this dissertation is simply impossible without them.
Finally, I would like to thanks all my buddies atthe swimming team of NTUST and D’Vine for making my life unforgettable.
Table of Content
Abstract in Chinese………..i
Abstract in English ………ii
Acknowledgements………iv Contents………...vi List of Figures………vii Chapter 1 Introduction...1 1.1 Time-optimal Control ...1 1.2 Controllability Revisit...2
1.3 On Maximal Convergence Rate...3
1.4 Dissertation Overview ...3
Chapter 2 Controllability of T-S Fuzzy Models ...5
2.1 Takagi-Sugeno (T-S) Fuzzy Models ...5
2.2 Lie Algebras ...8
2.3 Classical Controllability Results Revisited...9
2.4 Lie Algebras of T-S Fuzzy Models ...14
2.5 Controllability of T-S Fuzzy Model...14
2.6 Existence of Optimal Control ...18
2.7 Illustrative Examples ...20
Chapter 3 Time-optimal Control Design ...25
3.1 Problem Formulation ...25
3.2 Introduction of Pontryagin’s Minimum Principle...26
3.2.1 Shooting Method...28
3.3 Time-Optimal Controller of T-S Fuzzy Model ...30
3.4 Illustrative Examples ...35
Chapter 4 The Maximal Convergence Rate of T-S Fuzzy Control...55
4.1 Problem Formulation ...55
4.2 On Maximum the Convergence Rate...57
4.3 Illustrative Examples ...59
Chapter 5 Conclusion ...65
References ...67
Vita ...72
List of Figures
Fig. 1 The membership functions in Example 1. ...21
Fig. 2 The membership functions of Example 2...22
Fig. 3 Reachable tube in Example 2. ...23
Fig. 4 Trajectories of controllable and uncontrollable case. ...24
Fig. 5 Articulated vehicle model [1]. ...35
Fig. 6 The membership functions of Example 4...37
Fig. 7 The projection of the set V on the x1− plane...42 x2 Fig. 8 The projection of the set V on the x2− plane. ...42 x3 Fig. 9 The projection of the set V on the 1 x1− plane...43 x2 Fig. 10 The projection of the set V on the 1 x2− plane. ...43 x3 Fig. 11 The reachable set of V and 1 V on the 2 x1− plane. ...44 x2 Fig. 12 The reachable set of V and 1 V on the 2 x2− plane...44 x3 Fig. 13 Time-optimal trajectory in phase plane (CaseI) ...45
Fig. 14 The corresponded time-optimal control input (Case I). ...45
Fig. 15 Time-optimal trajectory in phase plane (Case II). ...46
Fig. 16 The corresponded time-optimal control input (Case II). ...46
Fig. 17 The projection of the set V on the x1− plane...47 x2 Fig. 18 The projection of the set V on the x2− plane. ...47 x3 Fig. 19 The projection of the set V on the 1 x1− plane...48 x2 Fig. 20 The projection of the set V on the 1 x2− plane. ...48 x3 Fig. 21 Time-optimal trajectory in phase plane (Case III)...49
Fig. 22 The correspond time-optimal control input (Case III)...49
Fig. 23 The membership functions for Example 5. ...50
Fig. 24 The switching curve and time-optimal control input...52
Fig. 25 The reachable sets of Example 5. ...52
Fig. 26 Time-optimal trajectory in phase plane (Case I). ...53
Fig. 27 The corresponded time-optimal control input (Case I). ...53
Fig. 28 Time-optimal trajectory in phase plane (Case II). ...54
Fig. 29 The corresponded time-optimal control input (Case II). ...54
Fig. 30 The level set of Example 6. ...62
Fig. 31 The trajectory in phase plane (Case I). ...62
Fig. 32 Corresponded control input (Case I). ...63
Fig. 33 The trajectory in phase plane (Case II)...63
Fig. 34 Corresponded control input (Case II). ...64
Chapter 1
Introduction
This dissertation deals with the time-optimal control and maximal convergence rate for constrained T-S fuzzy system. In recent years, fuzzy logic control with human knowledge of the plant has witnessed an effective approach to the design of nonlinear control systems. Indeed, there have been many successful applications which are based on fuzzy control [1-8]. In [9], Takagi and Sugeno proposed an approach to model the nonlinear process. This type of models is the so-called T-S model with later further development in [10]. The T-S fuzzy model blends the dynamics of each fuzzy implication by a linear consequence part [11-13]. In this type of fuzzy model, lots of important issues are addressed such as stability [2, 8, 11], H2/H∞ performance [13-15] and robustness [16-18],…, etc. In [19], a fuzzy approach is used in the design of time-suboptimal feedback controllers.
1.1 Time-optimal Control
The maximum principle has been extensively applied in many time-optimal control problems [20-35]. A series of results have been published on the applications of 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 Hamiltonian. However, the Hamiltonian formulation contains no information about the existence of
time-optimal solution. It is better to convert the existence of time-optimal solution to the study of reachable sets [25, 26, 28]. While the existence of time-optimal solution is addressed as the compactness of researchable 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 which will imply the controllability of T-S fuzzy model. Time-optimal control for T-S fuzzy model is a new control problem with its rich geometric properties via Lie algebra.
Using the maximum principle, time-optimal trajectory combined with the corresponding control, is called an extremal. The bounded input is determined by the signs of the associated switching functions. The singularity of the system is a well-known problem in time-optimal control which is explored in [27, 31]. An optimal trajectory may be singular, i.e., switching functions may vanish along the trajectory. The characterization of such trajectories will be investigated in this dissertation. The existence of extremal will imply that the time-optimal controller of the T-S fuzzy model to have finite number of switching, which can be found by Lie algebra in this dissertation.
1.2 Controllability Revisit
Recently, the controllability of systems has also attracted many explorers, such as switched system [41-43], hybrid system [44, 45]. However, the controllability of T-S fuzzy model has not been found in the literature. The controllability of the fuzzy model is a pre-requisite of the proceeding controller design. The effort in this dissertation to design a time-optimal controller via controllable T-S fuzzy model is a new contribution. Since the control-affine system can be represented by a family of
fuzzy model with a compact set of control input U , the Lie bracket taken at a point of an analytic family of vector fields form a complete set of its invariants. By formulating the T-S fuzzy model as a relaxed version, we can perform some algebraic operations on it, such as taking linear combinations and taking a product called Lie bracket.
1.3 On Maximal Convergence Rate
Fast response is always a considered property in this dissertation. A notion directly relate to fast response is the convergence rate of the state trajectories. For a linear system, the convergence rate is determined by the real part of the pole which is closest to the imaginary axis. We will give a controller design of T-S fuzzy model on maximal convergence rate by the introduced level set function. The result of maximizing the convergence rate is characterized from the maximal invariant ellipsoid. The controller is also bang-bang within both the initial states and target states are belong to level set.
1.4 Dissertation Overview
This dissertation is organized as follows. In section 2, we will formulate the time-optimal problem in T-S fuzzy model. In this section, the T-S fuzzy model is described as 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 time-optimal solution, we will investigate the singular structure in fuzzy model in section 3. The optimal trajectory is solved by the numerical illustrations are provided.
By introduced level set, the maximal convergence rate control discuss in section 4. Finally, conclusions are included in section 5.
Chapter 2
Controllability of T-S Fuzzy Models
Controllability properties of a control system are properties related to the following questions. Can the system be steered form a given initial state to a given final states? Can this be done for any pair of initial and final states? How large is the set of points to which the system can be steered from a given initial state? Which trajectories of the system are realizable and how do we find controls realizing them? Such questions can be motivated by practical problems and they are basic for any qualitative study of control systems.
Consider a nonlinear control system x= f x u
(
,)
, where x∈ ⊂ R and X n u is control in set U . This system can be viewed as collection of dynamical systems parameterized by control input. In study of controllability properties of systems, the set of available velocities F x( )
={
f x u(
, :)
u∈U}
by its convex hull, the trajectories of the convexified system can be approximated by the trajectories of the original system. In particular, if 0∈int co F x( )
for all x∈X , then the system is completely controllable.2.1 Takagi-Sugeno (T-S) Fuzzy Models
( )
( )
x= f x +g x u (1)
where x∈ ⊆ RX n is system state and u is control input is an arbitrary set U . The state space X is a smooth differential manifold of dimension n and U the control set. The vector fields f and g are assumed to be analytic.
In many situations, fuzzy model with the human knowledge can provide a linguistic description of the nonlinear system in terms of IF-THEN rules. The i -th
rule of the T-S fuzzy model is described by the following form: Rule i : IF z t1
( )
is Mi1 and zp( )
t is Mip, THENi i
x =Ax +B u
where x is system states, taking values in an open subset X of \ , n u∈ R is a m
measurable bounded function on U , i is the number of IF-THEN rules, z ti
( )
are some fuzzy input variables, M are fuzzy membership functions in the ij i-thrule, and x=A xi +B ui is the output from the i -th IF-THEN rule. The entire fuzzy modelis formulated as follows:
( )
(
)
(
)
1 r i i i i x μ z t A x B u = =∑
+ (2)where r is the total number of rules, μi
(
z t( )
)
is the normalized membership function and(
( )
)
1 / r i i i i z t μ α α ==
∑
and αi is the firing strength of i-thrule and( )
(
)
1 p i ij j j M z t α = =∏
.The T-S fuzzy model has strong connection with the polytopic linear differential inclusion (PLDI) [36, 37] which will lead to the relaxed version of T-S fuzzy model defined in this dissertation. 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 solutions of the differential inclusion.
The relaxed version of T-S fuzzy model is described by
[
]
{
i i 1, ,}
x∈Co A x+B u i= … r (3)
where Co denotes as convex hull [36]. If the T-S fuzzy model is continuous and control input U is compact, the set of solutions of (2) coincides with the set of solutions of (3) [36, 37], i.e.,
[
]
{
}
(
( )
)
(
)
1 1, , r i i i i i i Co A x B u i r μ z t A x B u = + = … ⊇∑
+ .Therefore we represent the T-S fuzzy model by (3) as
( )(
)
1 r i i i i x μ t A x B u = =∑
+ (4) where μi( )
t ∈[ ]
0, 1 and( )
1 1. r i i t μ = =∑
To simplify the notion, we adopt( )
1 r i i i i A μ t A = =∑
∑
,( )
1 r i i i i B μ t B = =∑
∑
and the j-th column vector of∑
Bi aredenoted as
( )
1 r j i ij i b μ t B = =∑
∑
, j= …1, ,m and are assumed to be linearly independent. Throughout the rest of this dissertation, the T-S fuzzy model is denoted asi i
x=
∑
A x+∑
B u. (5)In general, the variable z t
( )
in (2) sometimes is chosen as the state variables x t( )
, thus de-fuzzification μi(
z t( )
)
causes (2) to become a class of nonlinear systems.This lead to difficultly perform differential algebra on (2). To avoid this problem, such T-S fuzzy model (5) is introduced to allow us to perform differential algebraic on it.
2.2 Lie Algebras
The nonlinear control-affine system (1) can be viewed as a collection of dynamical system with control input. It is typical to expect that basic properties of such a system depend on interconnections between the different dynamical systems corresponding to different controls. The Lie bracket of two vector fields is another vector field which measures noncommutativeness of the flows of the vector fields.
Let f and g be vector fields on X , the corresponding Lie bracket of two smooth vector fields is denoted by
[
f g,]
, and[
f g,]
( )
x f g x( )
g f x( )
x 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
( ) ( )( )
( )
1( )
,
k k
ad f g x = ⎣⎡f ad f − g⎤⎦ x (6) where ad f
( ) ( )
0 g := and g k≥1. The Lie algebra generated by the vector fieldscan be expressed as
{
}
{
1 1}
1 1 , , , span , , , 1, 0 , , k k m LA i i i k f g g g g − g k i i m = ⎡ ⎡ ⎤ ⎤ = ⎣ ⎣ ⎦ ⎦ ≥ ≤ ≤ … … L where g0 = f.To study the coordinate change, consider a global diffeomorphism Φ: X →X as tangent vectors are transformed through the Jacobian map. Consider a diffeomorphism is defined as
( )( )
( )( )
adΦ f p = ΦT q p , q= Φ−1
( )
p ,where TΦ denotes the tangent map of Φ . Note that the coordinate change
( )
the tangent map of Φ is a global diffeomorphism of X , then the operation adΦ is a linear operator on the vector fields X . For example, the additive of diffeomorphism is
(
1 1 2 2)
1( )
1 2( )
2adΦ α f +α f =α adΦ f +α adΦ f . The global diffeomorphism of composition Φ Θ is
( )
( )
( )
adΦ Θ f =adΦ f adΘ f .
From the definition of Lie bracket that
[
f g,]
transforms with coordinate changes like a vector field which is via the Jacobian map. If the tangent map of Φ is a diffeomorphism of X , the basic property of equivariance of Lie bracket with coordinate changes are as following:( )
, ,( )
[
]
adΦ f adΦ g =adΦ f g
⎡ ⎤
⎣ ⎦ .
2.3 Classical Controllability Results Revisited
In analyzing controllability properties of systems, the follow theorems are introduced. In the following, we will introduce notions and results which play a basic role in analyzing the structure of nonlinear control systems. They are directly related to controllability properties of nonlinear system. In the following, we denote X as a n dimensional C∞ manifold.
Definition 1. Let T X be a subspace of the tangent space at any point x x∈X . A distribution Δ on X is a map which is
( )
xx∈ → ΔX x ⊂T X .
The distribution Δ is a smooth subspace of \ to each point n x. The dimension of Δ , 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.
Definition 2. A distribution Δ
( )
x is called involutive if for any two vector fields( )
,
f g∈ Δ x , their Lie bracket
[
f g,]
∈ Δ( )
x .The involutive plays the basic role in following is well-know Forbenius theorem.
Theorem 1. (Frobenius’ theorem) [35].
If distribution Δ is involutive distribution of class C∞ and of dimension k on X then, locally around any points in X , there exists a smooth change of
coordinates with transforms the distribution Δ to the following constant distribution
(
1, , k)
,span e … e
where e1,…,ek are the constant vector with 1 at the i -th place.
In order to introduce a global version of Frobenius’ theorem, we have following definitions.
Definition 3. A subset S⊂ X is called regular submanifold of X with dimension k if for any x⊂S there exists a neighborhood U of x and a diffeomorphism
:U V n
Φ → ⊂ R onto an open subset V such that
(
U V)
{
x(
x1, ,xn)
V xk+1 0, ,xn 0}
Φ ∩ = = … ∈ = … = .
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. In other words, a regular submanifolds of dimension k is a subset which locally looks like a piece of subspace of dimension k with changing of coordinates. A weaker version of a submanifold is introduced in the following definition.
dimension k if 1 i i S S ∞ = = ∪ , where S1⊂S2 ⊂ S
and S are regular submanifolds of i X of dimension k.
In fact, if subset S itself is regular submanifold, then Si = and S S is also an immersed submanifold. From geometric view, if two vectors field f and g are tangent to an immersed submanifold S then also their Lie bracket
[
f ,g]
is tangent to this submanifold.Remark 1. This is geometric definition of Lie bracket. If vectors field f is tangent to submanifold S, the fact that it is flow transforms points of S into points for any time t sufficiently small. With respect to t , the
[
f ,g]
is gives a tangent vector toS.
Definition 5. A foliation
{ }
Si i A∈ of X of dimension k is a partitioni i A
X S
∈
= ∪
of X into arc-wise connected (immersed) submanifolds Sα. In here, Sα is called
leaves.
Let g is a vector field of tangent to a foliation
{ }
Si i A∈ , that is, it is tangent to its leaves. The Lie bracket[
f ,g]
is tangent to this foliation, if the flow of f locallypreserves this foliation. For any point x∈ , the f locally preserves the foliation Si
{ }
Si i A∈ mean that there is a neighborhood U of x such that the image of a pieceof a leaf is contained in a neighborhood of leaf of the foliation, for any time t sufficiently small.
Definition 6. Consider a set of vector fields F=
{ }
fu u U∈ , the orbit of a point x∈X is the set of points of X which and be reached by piecewisely by trajectories of vector fields,( )
{
1 1}
1 1 1, ,1 , , ,1 , k k k k u u u t t t k k Orb x γ γ − γ k u u U t t − + = ≥ … ∈ … ∈R ,where γtu is denoted the flow of the vector field f . u
Theorem 2. For all x∈X , the orbit S =Orb x( ) of a set of vector fields
{ }
fu u U∈=
F is an immersed submanifold. Further, the tangent space of this
submanifold is given by the distribution T Sx = Δ
( )
x .Corollary 1. If the vector fields f are analytic, then the tangent space of the orbit u
can be obtained as
( )
{ }
{
}
x u u U
T S = g x g∈L f ∈ ,
where L f
{ }
u u U∈ denotes smallest set of vector fields which contains the set F and is closed under taking linear combinations and Lie bracket.Denote X be an open subset on R or a differentiable manifold of dimension n n. We have the following definition.
For convenience, the following Theorems 2 ~ 4 are listed here which are adapted from [35-37].
Let F be a set of C∞ vector fields on X and L=
{
λ λ0, 1,…,λk}
LA be the Lie algebra generated by F. If dim(
L( )
x)
=n for all x∈X , then any point of X is reachable by trajectory of the vector fields F. Thus( )
1 1 1 0 L L t t x =eλ eλ xfor some L≥ , 1
{
λ λ0, 1,…,λk}
∈F and t1,…,tL∈(
0, ∞)
.The following well-known theorem of Frobenius is characterized the integrable distribution [38].
Theorem 4. (Generalized Frobenius’ theorem) [38]
If X is a Cω (regular) manifold of dimension n and Δ is an involutive distribution then around any point x∈X , there exists a largest integral manifold of
Δ passing through x.
Remark 2. A distribution Δ is said to be integrable if there exists a submanifold S on X such that for any x∈X
( )
x T SxΔ =
where S is passing through x.
Remark 3. Any analytic involutive distribution Δ is integrable [39].
Theorem 5.[39]
Let F be a set of Cω vector fields on X and L=
{
λ λ0, 1,…,λk}
LA be the Lie algebra generated by F. For all x∈X , there exists a largest integral manifold ofThe proof of Theorem 5 can be found by using the Campbell-Baker-Hausdorff formula and Theorem 4.
2.4 Lie Algebras of T-S Fuzzy Models
Since the control-affine system can be represented by a family of vector fields, this will have direct applications to control systems. Consider a T-S fuzzy model with a compact set of control input U, the Lie bracket taken at a point of an analytic family of vector fields form a complete set of its invariants. In particular, L
( )
p0 denotes the space of tangent vectors at p defined by the Lie algebra. Due to the fact that 00 i
f =g =
∑
A x, g1=∑
b1,…,gm =∑
bm , and that Lie bracket of constant vector fields is zero, the iterated Lie bracket can be found as(
)
(
)
1,
k k
i j i i j
ad
∑
A x∑
b = ⎢⎡⎣∑
A x ad∑
A x −∑
b ⎤⎥⎦ (7) A Lie algebra L is recursively defined by( )1
[
]
( )2 ( )1 ( )1 ( ) ( 1) ( 1), , ⎡ , ⎤, , k ⎡ k− , k− ⎤,
= =⎣ ⎦ … =⎣ ⎦ …
L L L L L L L L L ,
is called solvable if L( )k =0 for large k, i.e., L( )k ⊃L(k+1). Furthermore, Lie algebra
L is called nilpotent if the sequence of L is always decreasing with respect to
1 2 1 1
, , ⎡ ⎤, , k ⎡ , k− ⎤,
= =⎣ ⎦ … =⎣ ⎦ …
L L L L L L L L ,
and Lk =0
. Any nilpotent Lie algebra is solvable. More details can be found in [38].
2.5 Controllability of T-S Fuzzy Model
We begin with the formal definition of reachability and controllability. In this section, T-S fuzzy model (5) associated with Lie algebra is derived to show the controllability condition and imply the existence of optimal control.
Definition 7. The reachable set R
( )
x of T-S fuzzy model (5) for time t≥0, subject to the initial condition x∈X is the set( )
{
( )
, : and : 0,[
]
}
T x = x t u x∈X u T U
R .
Definition 8. The T-S fuzzy model (5) is accessible if its reachable set RT
( )
x , x∈X have non-empty interior. Similarly, We will call this T-S fuzzy model stronglyaccessible if the reachable set RT
( )
x has nonempty interior for any T >0.Definition 9. The T-S fuzzy model (5) is controllable if ∀ and x0 ∀ in the x1 manifold of X , there exists a finite time T and admissible control function
[
]
: 0,
u T such that x T x u
(
; 0,)
=x1.Definition 10. For T-S fuzzy model (5), the accessibility Lie algebra is defined as
{
}
: , 1, , .
a =
∑
A xi∑
bj ∀ =j … m LAL (8)
The L is a finite-dimensional Lie algebra of vector fields which contains the family a
{
∑
A xi ,∑
bj}
. In fact, this accessibility Lie algebra plays basic role in the controllability of a T-S fuzzy model.Theorem 6. If the accessibility Lie algebra of the T-S fuzzy model in (5) is full rank
at x, that is
( )
(
a)
, nrank L x =n ∀ ∈ Rx (9)
then the reachable set up to any time T >0 has the nonempty interior and so the fuzzy model is strongly accessible.
Proof:
According to Chow’s theorem [35], the reachable set R
( )
x is the largest integral manifold of L for a nx
∀ ∈ R . From (9), it contains an open neighborhood Ω of x. This implies that for any x , its reachable set is an open set. We shall prove the 0
theorem by contradiction. We claim that R
( )
x0 is closed and is denoted as( )
(
0)
.cl R x Therefore, there exists a x1∈cl
(
R( )
x0)
\R( )
x0 . Hence R( )
x1 contains an open neighborhood Ω of x1 , then Ω∩ R( )
x0 ≠φ. Let( )
x0ζ∈Ω ∩ R then x∈ R
( )
x1 .By symmetry, x1∈ R( )
ζ , and ζ ∈ R( )
x0 then( )
0x∈ R x . Therefore Ω ⊂ R
( )
x0 , which is contradiction. We can conclude that the reachable set R( )
x is arc-wise connected and span into R space. n Q.E.D.Remark 4. Since T-S fuzzy model (5) is analytic, using Chow’s theorem [35] and Frobenius’ theorem [38], the manifold X is maximal connected reachable manifolds. Each reachable manifold is the maximal integral manifold of L . a
Remark 5. By using Chow’s theorem [35], the controllable manifolds can be spanned from
{
∑
A xi , 1,∑
bj ∀ =j …,m}
.Remark 6. The L implies that The T-S fuzzy model (5) is accessible form a x if 0
the same collection of vectors together with
∑
A xi 0+∑
B ui span the whole space. This condition means that no vector∑
B ui belongs to a proper invariant subspace of∑
A x .Theorem 7. If T-S fuzzy model is strongly accessible, then it’s also controllable.
Proof:
Using Remark 4, for a T-S fuzzy model the degree of largest integral manifold is related to rank of accessibility Lie algebra L . Due to the fuzzy model is strongly a
accessible, there exists the n-th degree largest integral manifold. For a given point
n
x∈ R , 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 T-S fuzzy model.
Corollary 1. The T-S fuzzy model (5) is controllable if and only if the following
matrix
(
)
(
(
)
1)
0, ,1 , 1 : , , , n n j i j i j W W … W − =∑ ∑ ∑
b A b …∑
A −∑
b , j= …1, ,m (10)is of rank n for any t>0
Proof:
Firstly, we give the proof of sufficient part. Consider the T-S fuzzy model (5), let
0 i
f =g =
∑
A x and g1 =∑
bj to be a vector filed. Then we have the followingiterated Lie brackets,
2 , = , , , , , i j i j i i j i j A x b A b ⎡ A x A x b ⎤ A b ⎡ ⎤ − ⎡ ⎤ = ⎣
∑
∑
⎦∑ ∑
⎣∑
⎣∑
∑
⎦⎦∑ ∑
… .From (7), the iterated Lie brackets are rewrote as
(
i)
l j(
( )
1 i)
l jad
∑
A x∑
b = −∑ ∑
A b .Therefore, the accessibility Lie algebra L consists of constant vector fields only, a
(
)
(
)
{
l 0, 1, ,}
a =Span
∑
Ai∑
bj l≥ j= … mL . (11)
then the fuzzy model is controllable.
From the Frobenius’ theorem [38] and Remark 4, it follows that the T-S fuzzy model (5) is controllable, there exits the n-th degree largest integral manifold for x∈X . If (10) is satisfied, from Theorem 5 and Remark 2, there exists a largest integral submanifold S which is unique and contained in the largest integral manifold. Q.E.D.
Remark 7. In analyzing controllability properties of the fuzzy model (5) we can replace the set of G x
( ) {
= A xi +B u ui : , ∈U i= …1, ,r}
by its convex hull, the trajectories of convexified system can be approximated by the trajectories of the original fuzzy model (2). In particular, if 0∈int Co G x{
( )
}
for all x∈X , then the fuzzy model is controllable.Remark 8. Obviously, for single rule T-S fuzzy model, Corollary 1 degenerates to the Kalman controllability matrix of linear system.
Remark 9. If all the subsystems are controllable, whereas the overall system can not concluded controllable, then the overall system can be called local controllable. The membership functions obviously play the critical roles in the controllability of system. In the following examples, the local controllability and controllability of T-S fuzzy model will be illustrated. The nonlinear system will be modeled with the distinct membership functions.
2.6 Existence of Optimal Control
can be reduced to determine the accessibility of reachable set. The qualitative properties of the reachable sets can be established. One of the basic properties can be shown in the following context. The following theorem discusses the existence of the optimal solution for Problem 1.
Corollary 2. If 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
, ,
i
x=
∑
A x+v v V∈where V is the image of U under the map
∑
b: Rm→Rn.Thus, the Lie brackets is , , i i A x v A 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(
)
' '' ' ''
i i
A x v+ − A x v+ = − ∈v v W
∑
∑
.Consider all constant vector fields f =w w W, ∈ .Thus, it contains the Lie brackets
, i i
w A x v A w
⎡ + ⎤=
⎣
∑
⎦∑
. Since the fuzzy model is controllable, the accessibility Liealgebra L consists of constant vector fields if a
(
)
{
}
dim span l 0 1,
a =
∑
Ai w ≤ ≤ −i n w W∈ =nL (12)
for 0,l= …,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= − ∈ , j kv v U ≠
such that, no image of U belongs to a invariant subspace of matrix
∑
Ai . In the next section, we shall design the time-optimal controller for T-S fuzzy model with maximum principle.2.7 Illustrative Examples
Example 1. Consider a nonlinear system:
( )
( )
tan 10 sin( ) cos . x u y x x = =Assume that x t
( )
∈ −[
π/ 2, / 2π]
. Then the T-S fuzzy model of the nonlinear system can be formulated as:Rule i : IF x t
( )
is about "Positive" and "Negative", THEN( )
i( )
i X t =A X t +B u, i=1, 2 (13) where X t( )
= ⎡⎣x t( )
y t( )
⎤⎦ , T 1 1 2 2 0 0 1 , 10 0 0 0 0 1 , 10 0 0 A B A B β β ⎡ ⎤ ⎡ ⎤ =⎢ ⎥ =⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ =⎢ ⎥ =⎢ ⎥ − ⎣ ⎦ ⎣ ⎦0
( )
x t 0 1 Positive Negativeπ/ 2 − π/ 2 Rule 1 Rule 2Fig. 1 The membership functions in Example 1.
According to Corollary 1, the corresponding rank of controllability matrix of the fuzzy model is,
(
j, i j)
Rank∑ ∑ ∑
b A b where(
)
0 1 1 2 1 2 1 0 0 0 0 1 0 0 1 . 0.349 0.349 0 0 0.349 0 0 j i j W b W A b μ μ μ μ ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ ⎛ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ = =⎜⎜ ⎢ ⎥ ⎢ ⎥+ ⎢− ⎥ ⎢ ⎥=⎢ − ⎥⎟⎟ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠∑
∑ ∑
The fuzzy model is controllable if Rank W
(
[
0, 2W1]
)
= . We can check thecontrollability by the following determinant:
(
)
(
1 2)
1 2 1 0 0.349 0 0.349 μ μ μ μ ⎡ ⎤ = − ⎢ − ⎥ ⎣ ⎦ .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 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 consequence parts of fuzzy model can be formulated as: 1 1 2 2 0 0 1 , 10 0 0 0 0 1 , . 10 0 0 A B A B β ⎡ ⎤ ⎡ ⎤ =⎢ ⎥ =⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ =⎢ ⎥ =⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 0
( )
x t 0 1 Positive Negativeπ/ 2 − π/ 2 Rule 1 Rule 2 Rule 2Fig. 2 The membership functions of Example 2.
By Corollary 1, the controllability matrix contains the vector fields
(
)
0 1 1 2 1 2 1 0 0 0 0 0 0 1 . 10 0.0349 10 0 0.349 0 0 j i j W b W A b μ μ μ μ ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ ⎛ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎞ ⎡ ⎤ = =⎜ ⎢ ⎥+ ⎢ ⎥ ⎢ ⎥⎟ =⎢ ⎥ + ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠∑
∑ ∑
If the fuzzy model is controllable then the following condition is satisfied:
(
)
(
1 2)
1 2 1 0 10 0.0349 0 0 10 μ 0.0349μ μ μ ⎡ ⎤ = + ≠ ⎢ + ⎥ ⎣ ⎦ .for ∀t. Then we can conclude that the overall T-S fuzzy model is controllable. Since the Example 2 is controllable, the reachable set for t=
[
0 2]
is plotted in Fig. 3.Fig. 3 Reachable tube in Example 2.
Example 3. Choose the closed-loop eigenvalues
[
−1 −1]
for Example 1 and 2. The stabilizable controller is designed by Parallel Distributed Compensation (PDC) [2]. Fig. 4 shows the response of the controllable and uncontrollable system. The dotted lines show the responses of locally controllable case (Example 1). The solid lines indicate the responses of controllable case (Example 2). The controllable case is no surprising to stable the system. From Example 1, we know that the system is not controllable in x t( )
=0. The dotted lines show that the system can not converge to zero. This is due to the controllability of system is disappeared.0 1 2 3 4 5 6 7 8 9 10 -0.2 -0.1 0 0.1 0.2 t (sec.) x 0 1 2 3 4 5 6 7 8 9 10 -0.8 -0.6 -0.4 -0.2 0 0.2 t (sec.) y
Fig. 4 Trajectories of controllable and uncontrollable case.
Remark 10. An important and natural question arises in the design of feedback controller using local controllability. The controllability of a physical system is a pre-requisite of the proceeding controller design.
Chapter 3
Time-optimal Control Design
Now, we give the Time-optimal control via Pontryagin’s Minimum Principle. The controller is derived as bang-bang and the number of switching will be shown as below section.
3.1 Problem Formulation
We will make the following assumption on the control input.
Assumption 1. The control input is given by
{
m j j j, 1, ,}
U = u∈R a ≤u ≤b j= … m .
For a given control u t
( )
⊂U on a time interval[
0, t1]
and any initial point( )
0 0x t = ∈x X , let x
(
.,x u0,)
denote the solution of the nonlinear control-affine (5) with an measurable control u defined on a interval of[
0, t1]
. For performing optimality on a segment[
0, t1]
, we introduce a cost functional( )
1(
( ) ( )
)
0 ,
t
J u =
∫
ϕ x t u t dt (14)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.
and satisfies the boundary condition
(
1, 0,)
1x t x u =x . (15)
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.
3.2 Introduction of Pontryagin’s Minimum Principle
The system (1) under bounded controls u t
( )
≤ can be formulated by using the UPontryagin’s Minimum Principle. The minimization problem for (1) becomes
( )
( )
( )( ) ( )
* * * ( ), , T min ( ), , T u t U H x t λ t u t H x t λ t u t ∈ ⎡ ⎤= ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ (16)for t∈
[
0 t1]
, or, equivalently,( )
( )
( ) ( )
* * *
( ), , T ( ), , T
H x t⎣⎡ λ t u t ⎤⎦≤H x t⎡⎣ λ t u t ⎤⎦ (17)
where H is called Hamiltonian, * ( )
x t is optimal trajectories and λT
is a vector of costates. The superscript (*) denotes the optimal results. The Hamiltonian for system (1) can be written as
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( ) ( )
( ), , 1 1 , 1 , , T T T T T H x t t u t x t f x g x u t f x t g x u t λ λ λ λ λ ⎡ ⎤ = + ⋅ ⎣ ⎦ = + + = + + (18)Suppose that u t*
( )
is a time-optimal control and x t*( )
is the resultant of time-optimal trajectory in minimum time, t . Substituting the equation (18) into the *( )
( )
( )
( )
( )
( )
* * * * * 1 , , g 1 , , . T T T T f x x u t f x t g x u λ λ λ λ + + ≤ + + (19)Since the first two terms are the same on both side of the inequality, therefore the above inequality equation can be simplified as follows
( ) ( )
( )
( ) ( )
( )
* * *
g g
T T
u t λ t x ≤u t λ t x . (20)
By defining ψj: 0,
[
t1]
→R, ψj( )
t :=λT g( )
x* , we can conclude that( )
( )
*
j j
u t ψ ≤u t ψ . (21)
From Assumption 1, u t
( )
≤U, therefore, time-optimal controller can be generalized as( )
{ }
*
j
u t = −SGN ψ U . (22)
In (14), it is obvious that if time-optimal control, u t*
( )
, exists then there is a unique bang-band control. After applying Pontryagin’s Minimum Principle, we have the following necessary conditions,Optimal state trajectory:
( ) ( )
( )
( )
* * , , H x t t u t x t λ λ ⎡ ⎤ ∂ ⎣ ⎦ = ∂ (23) Costate equation:( )
t H x t( ) ( ) ( )
, ,( )
t u t x t λ λ = −∂ ⎡⎣ ⎤⎦ ∂ (24), and stationary condition
( ) ( ) ( )
, , 0H x t⎡⎣ λ t u t ⎤⎦= (25)
3.2.1 Shooting Method
The shooting method [40] is used to solve this problem. The shooting method can be used to determine the time-optimal control problem as described in what follows. In T-S fuzzy model, equations (5), (35), (38) and (40) can be rewritten as
( ) ( )
, X = ⎡F X t⎣ u t ⎤⎦ (26)( )
0[
0, 0]
T X t = x p (27)(
, , f)
f 0 e X p t⎡⎣ t ⎤ =⎦ (28)( )
{ }
j u t = −SGN ψ U (29)where X =
[
x p,]
T is a vector of 2n variables, which are the states, x, and costates, p . F X u[
,]
is combined with a vector of fuzzy system states and costates.( )
0 0
p = p t is an n-dimensional vector of unknown initial costates, X t
( )
0 is2n -dimensional vector of initial states and unknown initial costates, p . 0
(
0, , f)
fe x p⎡⎣ t t ⎤⎦ is an l-dimensional vector, where l≥n, representing the error at the target point. This vector includes the final conditions of states, and the extra condition for Hamiltonian (25) to be met at the target point. q X
( )
is a switch function. In order to reduce e x p⎣⎡(
0, , tf)
tf⎤⎦ to zero, the values pk, tkf in the kthiteration have to be corrected in the next iteration using the following formula 1 1 k k k k k k f f f p p p t t t + + ⎡ ⎤ ⎡ ⎤ ⎡Δ ⎤ = + ⎢ ⎥ ⎢ ⎥ ⎢Δ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦. (30)
The correction terms Δ and pk Δtkf can be computed by minimizing a norm of e given by
1/ 2 2 1 l i i e e = ⎛ ⎞ = ⎜ ⎟ ⎝
∑
⎠ . (31)Obtaining an analytical expression may not always be possible. However, the stationary condition (25) offers the gradient along which the decision variables can be corrected. In [40], Newton’s method is adopted. The vector corrections is defined as
k k k k k f f p p t t δ α δ ⎡Δ ⎤ ⎡ ⎤ = − ⎢Δ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ (32)
where δpk and δ can be calculated using following expression: tkf
, , , k k k k k f f k k f k f f e p t e p t p e p t t p t δ δ ⎡∂ ⎡⎣ ⎤⎦ ∂ ⎡⎣ ⎤⎦⎤⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥= ⎣ ⎦ ∂ ∂ ⎢ ⎥ ⎢⎣ ⎥⎦ ⎣ ⎦ . (33)
The scalar αk is chosen in the range 0≤αk ≤ . The initial gauss of the set of the 1 values ⎡⎣p t0, 0f⎤⎦ are required. Due to the fact that the costates don’t have a physical meaning, initial gauss are difficult to obtain. For more complex problems, forward-backward method (FBM) was proposed in [25], which offered a good guess of the initial costates.
Determination of optimal control sequence of (29) is related to the trajectory of costates. This introduces other problems in that the initial costates and finial 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 obtain by solving equations (5), (35), (38) and (40) simultaneously. For TPBVP, no practical method has been developed yet for computing the time-optimal feedback control. The main reason is that it is generally impossible to characterize the switching surface. Suppose that in the time interval
( )
j t
ψ is identically zero, then the shooting method is fail. We will give more details
for this case in following section.
3.3 Time-Optimal Controller of T-S Fuzzy Model
In this section, we will study the properties of time-optimal control using the maximum principle [20], [27]. The Time-optimal controller is designed via a controllable T-S fuzzy model. In general, Problem 1 can be formulated as a Hamiltonian by maximum principle. The Hamiltonian for Problem 1 can be described as
(
, ,)
: T i T iH x λ u =λ
∑
A x+λ∑
B u (34)where λ: 0, t
[
1]
is a costate satisfying the adjoint equation associated with (5):T i H A x λ= −∂ = −λ ∂
∑
. (35)By using the maximum principle [20], the Problem 1 becomes
(
, ,)
max(
, ,)
v U
H x λ u H x λ v
∈
= . (36)
Definition 11. Trajectories of (5), (34) and (35) that satisfy the maximum principle
is called extremal
(
x, ,λ u)
: 0, \ 0[
t1]
Rn×Rn{ }
×U . When the constant λ0 is zero, the extremal is said to be abnormal [31].Definition 12. For j= …1, ,m , the switching functions ψj
( )
⋅ , along anextremal
(
x, ,λ u)
are defined by[
1]
( )
: 0, , : T
j t j t bj
ψ →R ψ =λ
∑
. (37)The necessary condition for optimality provided by the maximum 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 solutions of (34) and must satisfy(
, ,)
0, 00H x λ u =λ λ ≥ . (38)
The maximum condition (36) is equivalent to the following:
( ) ( )
max( ) ( )
j j j j j v U u t ψ t v t ψ t ∈ = , j= …1, ,m. (39)Obviously, the functions ψj
( )
t play a crucial role in the study of time-optimal trajectories. Under Assumption 1, the time-optimal control must satisfy the following conditions almost everywhere,( )
( )
if 0 if 0 j j j j j j u b t u a t ψ ψ = > = < (40)for 1,j= …,m. In case, switching functions having zeros have to be carefully
analyzed.
Suppose that in the time interval
[
0, t1]
there exists one nontrivial (or more) subinterval,[
ta, 0, tb] [
⊂ t1]
, such that ψj( )
t is identically zero, then thecorresponding extremal is called singular. If ψj
( )
t ≠0 for almost all t∈[
0, t1]
, the maximum principle implies that the control u corresponds to piecewise constant jcontrols taking values in the set of m vertices of U , is called bang-bang. An extremal is said to be normal if control u is bang-bang with at most a finite number j
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( )
( )
( )
( )
, , , . T T j i j k j j T i j t A x t b b b u t A x t b ψ λ λ λ ⎡ ⎤ ⎡ ⎤ = ⎣− ⎦+ ⎣ ⎦ ⎡ ⎤ = ⎣− ⎦∑
∑
∑ ∑
∑
∑
(41)Since
∑
bj, 1,j= …,m and j≠ are constant terms, therefore k ⎡⎣∑ ∑
bk, 0bj⎤ =⎦ . It is obvious that the derivatives of the switching functions ψj( )
t are themselvesabsolutely continuous function, and therefore we can perform further derivatives of it. In the next theorem, Lie brackets will be crucial in establishing a bound on the number of switches for bang-bang controls will be derived.
Theorem 8. 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 set
{
0, , ,i}
S = s … s … ,
where s is the i-th time interval i
[
ta, tb i]
such that ψj( )
t =0, ∀ ∈t[
ta, tb i]
,1, ,
j= … m. Assume t0∈ . Then we have the following relation: si
( )
( )
0 0T
j t t bj
ψ =λ
∑
= , j= …1, ,m (42)From (42), we have the first derivation of ψj
( )
t :( )
( )
0( )
, ψj t =λT t ⎡⎣
∑
A x ti∑
bj⎤⎦=0. (43) Indeed, l-th derivative of ψj( )
t can be expressed as:( )
( )
(
( )
)
l(
)
0l T
t t ad A x t b
By Corollary 2, we have
( )
(
)
(
)
{
l}
n i j span ad∑
A x t∑
b ∈R , l=1,…,n−1.Hence, we have λ
( )
t0 =0, which contradicts to the necessary condition of maximum principle. So we can conclude that the set S is finite. Outside the set S, the switching function λT( )
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 set of all vector fields{
⎡⎣∑
A xi ,∑
bj⎤⎦}
are linear independent, so we have the following result.Theorem 9. If an extremal of the T-S fuzzy model in (5) is normal, then the switching
function ψj
( )
t , j= …1, ,m will not be vanished 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= . From (42) and (43) we have t0( )
( )
0 0T
j t t bj
ψ =λ
∑
=and its first derivative is
( )
( )
0( )
, ψj t =λT t ⎡⎣
∑
A x ti∑
bj⎤⎦=0field
∑
bk, ⎡⎣∑
A xi ,∑
bj⎤⎦ for j= …1, ,m, are linear independent. This yields a contradiction with the non-vanishing condition for costate in the maximum principle.Q.E.D. The solvable Lie algebra is defined for the T-S fuzzy model (5) as following.
Definition 13. For T-S fuzzy model (5), the solvable Lie algebra is defined as
( )
{
}
: , 1, , . k i j LA A x b j m =∑
∑
∀ = … L (45)if derived series L( )k is vanished for larger k. Then the T-S fuzzy model is called
solvable.
In the next theorem, solvable Lie algebra will be crucial in establishing a bound on the number of switching for bang-bang control will be derived.
Theorem 10. If the controllable T-S fuzzy model (5) is solvable, then the total number
of switching is bounded.
Proof:
The controllable T-S fuzzy model (5) will imply
(
)
{
}
span ad A xi k bj
=
∑
∑
L , for k=1,…,n−1.
If L is solvable lie algebra, .i.e., L( )k =ad
(
∑
A xi)
k∑
bj =0 for k ≥ ≥ − . p n 1 Form (44), we have( )
( )
0(
)
(
)
k k T j t t ad A xi bj ψ =λ∑
∑
, for k≥ , (46) pis identically zero due to the T-S fuzzy model is solvable. In (46), ψkj
( )
t is vanishedfor k≥ , then the polynomial degree of switching function p ψj
![Fig. 5 Articulated vehicle model [1]. The dynamics of articulated vehicle can be formulated as](https://thumb-ap.123doks.com/thumbv2/9libinfo/7944625.157650/45.892.294.610.734.1004/fig-articulated-vehicle-model-dynamics-articulated-vehicle-formulated.webp)
