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

Consider a nonlinear control-affine system

( ) ( )

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 M_i1 and z_p

( )

t is M_ip, THEN

i 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 x_i +B u_i is the output from the i -th IF-THEN rule. The entire fuzzy model

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.,

Therefore we represent the T-S fuzzy model by (3) as

( )( )

independent. Throughout the rest of this dissertation, the T-S fuzzy model is denoted as

i 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

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

( )( ) ( )( )

ad_Φ f p = ΦT q p , ^{q= Φ−1}

( )

^{p ,}

where TΦ denotes the tangent map of Φ . Note that the coordinate change

( )

p= Φ q transforms the differential equation ^{p= f q}

( )

^{where f =ad}

( )

^{f . If}

the tangent map of Φ is a global diffeomorphism of X , then the operation ad_Φ is a linear operator on the vector fields X . For example, the additive of diffeomorphism is

(

1 1 2 2

)

1

( )

1 2

( )

2

ad_Φ α f +α f =α ad_Φ f +α ad_Φ f . The global diffeomorphism of composition Φ Θ is

( ) ( ) ( )

ad_{Φ Θ} f =ad_Φ f ad_Θ 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:

( )

^{, ,}

( )

[ ]

ad_Φ f ad_Φ g =ad_Φ 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. 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,…,e_k 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.

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.

Definition 4. A subset S ⊂X is called an immersed submanifold of X of

dimension k if 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 preserves this foliation. For any point x∈ , the f locally preserves the foliation S_i

{ }

Si i A_∈ mean that there is a neighborhood U of x such that the image of a piece of 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,

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 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].

Theorem 3. (Chow’s Theorem) [35]

Let F be a set of C^∞ vector fields on X and L=

{

λ λ0, 1,…,λ_k

}

_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

( )

The following well-known theorem of Frobenius is characterized the integrable distribution [38].

Theorem 4. (Generalized Frobenius’ theorem) [38]

If X is a C^ω (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 Sx

Δ =

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 C^ω vector fields on X and L=

{

λ λ0, 1,…,λ_k

}

_LA be the Lie algebra generated by F. For all x∈X , there exists a largest integral manifold of

F passing through x.

The 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 ₀

0 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

(

i

)

^k j i ^,

(

i

)

^{k 1} j

and L^k =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 x∈X have non-empty interior. Similarly, We will call this T-S fuzzy model strongly accessible 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 LA

L (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

( ) )

^{, n}

rank 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 ∀ ∈ R . From (9), it contains an open neighborhood Ω of x ⁿ 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

( )

x0

ζ∈Ω ∩ R then x∈ R

( )

x1 .By symmetry, x1∈ R

( )

ζ , and ζ ∈ R

( )

x0 then

( )

0

x∈ 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 x_{i 0}+

∑

B u_i span the whole space.

This condition means that no vector

∑

B u_i 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

x∈ R , the fuzzy model is controllable. n 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

(

^{W0, ,W1 …, Wn−1}

)

^:=

( ^{∑ ∑ ∑}

^{bj, Ai bj,…,}

( ^∑

^Ai

)

ⁿ⁻¹

^∑

^bj

)

^{, 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 following iterated 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 j}

ad

∑

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= ^… m

L . (11)

If (10) is satisfied, we can conclude that ^dim

( )

La is of full rank n for any t>0

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

In the following, we shall show that the existence of optimal solution of Problem 1

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 ⊆R^m. It is more convenient to consider the T-S fuzzy model in the form

, ,

⎣

∑

⎦

∑

. Since the fuzzy model is controllable, the accessibility Lie algebra L consists of constant vector fields if _a 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

∑

A_i ^{. 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:

( )

0

( )

According to Corollary 1, the corresponding rank of controllability matrix of the fuzzy model is, controllability by the following determinant:

( ) (

1 2

)

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:

Fig. 2 The membership functions of Example 2.

By Corollary 1, the controllability matrix contains the vector fields

( )

If the fuzzy model is controllable then the following condition is satisfied:

( ) (

1 2

)

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 0

x 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

( )

₀^t1

( ( ) ( )

^,

)

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.

Problem 1. Find a control ^{u t}

( )

^∈U that minimizes (14) along the solution of (5)

and satisfies the boundary condition

(

1, 0,

)

1

x 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 ^U Pontryagin’s Minimum Principle. The minimization problem for (1) becomes

( ) ( )

_{( )}

( ) ( )

costates. The superscript (*) denotes the optimal results. The Hamiltonian for system (1) can be written as time-optimal trajectory in minimum time, t . Substituting the equation (18) into the ^* inequality (17), we can obtain

( ) ( )

Since the first two terms are the same on both side of the inequality, therefore the above inequality equation can be simplified as follows

( ) ( ) ( ) ^{( ) ( )} ( )

bang-band control. After applying Pontryagin’s Minimum Principle, we have the following necessary conditions,

, and stationary condition

( ) ( ) ( )

^{, , 0}

H x t⎡⎣ λ t u t ⎤⎦= (25)

for 1, 2,k= … . ,n

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 2n -dimensional vector of initial states and unknown initial costates, p . ₀

(

^{0, , f}

)

^f

e 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 kth iteration have to be corrected in the next iteration using the following formula

1

1/ 2

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 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

[0, t1] there exists one nontrivial (or more) subinterval, [t_a, 0, t_b] [⊂ t1], such that

( )

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

In this section, we will study the properties of time-optimal control using the

在文檔中 T-S 模糊系統之最佳化時間控制 (頁 13-0)