Time-Optimal Controller of T-S Fuzzy Model

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

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 an

They are absolutely continuous functions [31].

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, 00

H x λ u =λ λ ≥ . (38)

The maximum condition (36) is equivalent to the following:

( ) ( )

^max

( ) ( )

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,

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, [t_a, 0, t_b] [⊂ 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 u corresponds to piecewise constant _j controls 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 of switching.

If T-S fuzzy model is smooth and

(

^{x, ,λ u}

)

is an extremal, then the time It is obvious that the derivatives of the switching functions ψj

( )

t are themselves absolutely 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

}

By Corollary 2, we have

( ( ) ) ( )

{

^{i l j}

}

ⁿ

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 λ^T

( )

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

( )

^T

( )

0 0

j t t bj

ψ =λ

∑

=

and its first derivative is

( ) ( )

0

( )

, ψ_j t =λ^T t ⎡⎣

∑

A x t_i

∑

b_j⎤⎦=0

for all j= …1, ,m, j≠ . If k ψ_k and ψ_k vanish at t= , Since The vector t₀

field

∑

b_k^{, ⎡}⎣

∑

^{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

( )^{k :}=

{ ∑

A xi ^,

∑

bj ∀ =j ^1,…,m

}

LA^.

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.

Remark 11. For

∑

b_j ≠0, the solvable condition (46) can be generalized as

( )^k =ad

( ∑

A xi

)

^k =⁰

L .

For the single input case, Theorem 10 provides the condition that the number of switching is at most p . Similarly, For multiple m vertices of U , the number of switching will not exceed m p⋅ .

3.4 Illustrative Examples

To utilize the time-optimal design techniques, two systems with single input and two inputs respectively will be illustrated.

Example 4.

Consider an articulated vehicle [1] in Fig. 5. The kinematic model of the vehicle is the starting point to model the dynamics of the lateral and orientation motions.

Fig. 5 Articulated vehicle model [1].

The dynamics of articulated vehicle can be formulated as

( ( ) )

x t1 angle difference between truck and trailer;

( )

x t2 angle of trailer;

( )

x t3 vertical position of rear end of trailer;

( )

x t4 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 speed. In this example, let l=1m, L=2.5m, 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 = , i.e., 0

( ) ( ) ( )

1 0, 0, 02 3

x t → x t → x t → .

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 vehicle becomes uncontrollable and the backward motion can not continue any more. To avoid this problem, the analysis of researchable set will be discussed in the following.

For constructing the T-S fuzzy model, assuming that ^{u t}

( )

^, x t2

( )

are small and

( ) ( )

1 / 2, / 2 .

x t ∈ −π π Let X t

( )

= ⎡⎣x t1

( )

x t2

( )

x t3

( )

⎤⎦^T. The dynamics of articulated vehicle can be formulated as:

Rule i : IF x t1

( )

is "Positive" and "Negative", THEN

( )

i

( )

i

( )

X t = A X t +BU t , i=1, 2 (47)

where the membership functions are given in Fig. 6 and the consequent parts are chosen as

Fig. 6 The membership functions of Example 4.

From Corollary 1, we have

0

The matrix

∑ ∑

A_i b_j ^is

The controllability of the fuzzy model can be reformulated by finding the determinant of [W0, , W W1 2]: (i=1, 2), therefore we may conclude that the fuzzy model is controllable and time-optimal solution does exist. To realize time-optimal control, we consider a control as U = + where control input u u^* 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

[^{-0.4 0 0}]

k= . By closed-loop feedback, the consequent parts of the fuzzy model (47) can be reformulated as

1 1 therefore the fuzzy model is concluded to be solvable and the number of switching is at most 2. Let u= , the bang-bang control does exist and the possible control 5 sequence can be concluded as:

{ } { } {

u ^, −u ^{, ,} u −u

} {

^{, , ,}−u u

} {

u, , , , , −u u

} {

−u u −u

}

.

The switching curves V are shown in Figures 7 and 8. The dotted line is the set V⁻ which is the trajectory by control input

{ }

^−u and the solid line shows the set V⁺ which is the trajectory by control input

{ }

^{u . Let} V denote the set of states which 1

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 from −u to u, the transition must occur on the set V⁺. The set V are ₁ shown in Figures 9 and 10. The dotted line is the set V₁⁻ which is forced by the control sequence

{

^{−u u,}

}

and the solid line shows the set V₁⁺ which is forced by the control sequence

{

^{u, −u}

}

^{. The set} V is the trajectory which can be forced to 2

the origin by the control sequence

{

^{u, , −u u}

}

^or

{

^{−u u, , −u}

}

. To prevent the jackknife phenomenon, the state x should be constrained to be less than ₁ 90 . In

Figures 11 and 12, the ellipses show the reachable set for x₁ ≤90 where the solid ellipses are the set V and the dotted ellipses are the set ₁ V . In fact, ₂ V ⊆ ⊆ .The V₁ V₂ maximal reasonable range of initial positions will be restricted on the reachable set V . 2

Case I

For the initial position, x₀ =240 , x₁=200 , x₂ =40 , x₃ =20m and x₄ =0m, the time-optimal trajectory of x₃ vs. x₄ is depicted in Fig. 13. The corresponding time-optimal control ^{u t*}

( )

is shown in Fig. 14. The shortest time from initial position to the origin is 2.4115 (sec.).

Case II

For the initial position, x₀ =320 , x₁=20 , x₂ =300 , x₃ =20m and x₄ =0m, the time-optimal trajectory of x₃ vs. x₄ is depicted in Fig. 15. The corresponding

time-optimal control ^{u t*}

( )

is shown in Fig. 16. The shortest time from initial position to the origin is 13.6715 (sec.).

Case III

In this case, the control purpose is to realize the forward movement the articulated vehicle along the straight line. For forward speed v=5 /m s, the consequence parts of the system are trajectories are depict in Fig. 21. The corresponding time-optimal control ^{u t*}

( )

^are

shown in Fig. 22.

-80 -60 -40 -20 0 20 40 60 80

-100 -80 -60 -40 -20 0 20 40 60 80 100

-100 -50 0 50 100

-2 -1 0 1 2 3

Fig. 13 Time-optimal trajectory in phase plane (CaseI)

0 0.5 1 1.5 2 2.5

Fig. 14 The corresponded time-optimal control input (Case I).

-15 -10 -5 0 5

Fig. 15 Time-optimal trajectory in phase plane (Case II).

0 2 4 6 8 10 12 14

-80 -60 -40 -20 0 20 40 60 80

-80 -60 -40 -20 0 20 40 60 80

-10 -8 -6 -4 -2 0

Fig. 21 Time-optimal trajectory in phase plane (Case III).

0 1 2 3 4 5

Fig. 22 The correspond time-optimal control input (Case III).

Example 5.

The multiple inputs system is considered here. Consider the following T-S fuzzy model: the consequent parts are chosen as

1 1

The membership functions of the fuzzy model are given in Fig. 23.

0

( )

Fig. 23 The membership functions for Example 5.

The fuzzy model is found to be controllable by Corollary 1. The switching number is at most 2 which is obtained by using Remark 11. Therefore the time-optimal sequences are

{ }

^{1, 1 ,}

{

^{−1, 1−}

}

^,

{

^{1, 1−}

}

^,

{

^{−1, 1}

}

^.

Follow the same analysis in Example 4, the switching curves are explained in the

followings. 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 V be forced by input 1

{

^{1, 1−}

}

^or

{

^{−1, 1}

}

to the origin. The switching curve V is depicted as solid line in

Fig. 24 , the dotted line depicts switching curve V and the time-optimal control ₁ inputs are also shown in Fig. 24. Assume ^R

( )

^{T and} R1

( )

T are reachable sets for V and V respectively that can reach the origin at time ₁ T . Fig. 25 depicts reachable set which is sampled from T =5 to T =20 in every 5 seconds. The dotted line is the reachable set R1

( )

T and the solid line is the reachable set ^R

( )

^{T .}

Case I

For the initial state X0 =[40, 50− ], the time-optimal trajectory is shown in Fig. 26.

The corresponding time-optimal control ^{u t*}

( )

is shown in Fig. 27. The shortest time from initial state to the origin is 9.350 (sec).

Case II

For the initial state X0 =[40, 100], the time-optimal trajectory is depicted in Fig. 28.

The corresponding time-optimal control ^{u t*}

( )

is shown in Fig. 29. The shortest time from initial state to the origin is 25.249 (sec).

-100 -50 0 50 100

Fig. 24 The switching curve and time-optimal control input.

5

-10 0 10 20 30 40

Fig. 26 Time-optimal trajectory in phase plane (Case I).

0 1 2 3 4 5 6 7 8 9 10

Fig. 27 The corresponded time-optimal control input (Case I).

-30 -20 -10 0 10 20 30 40

Fig. 28 Time-optimal trajectory in phase plane (Case II).

0 5 10 15 20 25 30

Chapter 4

The Maximal Convergence Rate of T-S Fuzzy Control

The time-optimal control problem of T-S fuzzy model was discussed in previously section. The time-optimal control is a bang–bang control and implemented successfully by reachable set. If the system is not accessible, the number of switching can not be found and the computation cost is too much under this situation. Fast response is always a considered property in this dissertation. A notion directly relates 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. In this section, 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.

4.1 Problem Formulation

Consider a nonlinear system (1) with zero input. The ellipsoid Ω is invariant for the system if all the trajectories starting from it will stay inside of it. It is contractive invariant if

( )

^{2 T}

( )

⁰

V x = x Pf x < .

The objective is to find a control law with constrained input such that convergence consequence of the maximal convergence control is that it produces the maximal invariant ellipsoid of a given shape. It is easy to see that an ellipsoid can be made invariant if and only if the maximal ^{V x}

( )

on the boundary of the ellipsoid under the maximal convergence control is negative.

In conventional, T-S fuzzy controller design employs the parallel distributed compensation (PDC) via the Lyapunov technique [36]. The PDC is designed by locally feedback gain F as _i

Controller Rule i : IF z t1

( )

is M_i1 and z_p

( )

t is M_ip, THEN u = −F xi

The entire PDC can be formulated as follows:

The entire feedback type of system (4) via PDC are given as following:

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

In general, this type controller is difficult to solve since the coupling relation of

(

^{A xi −B Fi j}

)

. For simplifying the design process, in here, we consider the system (4)

with single controller input and make the following assumptions. For i= … , 1, ,r Bi = and the state feedback controller is given as B

U = −Fx (52)

where F is denoted as the state feedback gain, therefore the feedback T-S fuzzy model can be rewritten as

4.2 On Maximum the Convergence Rate

Consider the feedback fuzzy model (53) under the constraint that U ≤1, we have following definitions.

In here, we give the controller design for maximizing the convergence rate.

The following lemma will illustrate the level set Ω found by Linear Matrix Inequalities (LMIs) [36].

Lemma 1. Consider a T-S fuzzy model (53) with zero input if P>0, α ≥0 and

0 on maximizing the convergence rate.

Proof:

Let ^{V x}

( )

^{=x PxT and P>0}. For a positive number α , the level set associated

with ^{V x}

( )

is ellipsoid,

(

^x,α

) {

^{x n V x}

( )

^{x PxT α}

}

Ω = ∈\ = ≤

Along the trajectory of the system (53),

( )

^T

(

^{iT i}

)

^{2 T 0}

V x =x A P+PA x+ x PB U⋅ < , (57)

(

^,

) { }

^{\ 0}

x x α

∀ ∈Ω . From Definition 15, the controller is minimizing (57), we have ( ^T )

U = −SGN B Px

where ^SGN

( )

^⋅ is sign function. It is clear that the maximal convergence control

produces the maximal invariant ellipsoid of a given ellipsoid ^Ω

(

^x,α

)

. Q.E.D.

Remark 12. The system will have no solution if x=0. This is due to the switching plane B Px^T = . When the system state close to the switching plane, it is easy to 0 have the chattering.

Remark 13. It becomes obvious that the maximal convergence control is also a bang-bang control.

4.3 Illustrative Examples

In this section, we demonstrate the application of the proposed maximal convergence rate for T-S fuzzy model.

Example 6.

Consider a nonlinear mass-spring-damper mechanical system that can be formulated as

We use the following mass-spring-damper and fuzzy model formulated in [15]:

3 3 the consequent parts are chosen as

1 1

In this example, the system is not solvable therefore there are no information about the number of switching. In this situation, the numerical reachable set is difficult obtained and computation cost is high. We design the controller by purposed controller on maximal convergency rate. With all the ellipsoids satisfying the set invariance condition in Lemma 1, we have

0.0252 -0.0131

In the case, the saturation control in [46] is introduced to compare our results. The initial point is x=[1, -0.5]^T. In Fig. 31, the states are converged by saturation control over 35 (sec.). The saturation control input is depicted in Fig. 32.

Case II

Let the initial point as Case I, the maximal convergency rate control is considered in this Case. The states converge at 1.2 (sec.) and depicted in Fig. 33. The corresponded control input is depicted in Fig. 34. The convergence rate of states is expected faster then Case I. We can conclude that the system has faster response by the maximal convergency rate control. Obviously, the sign function is sensitivity when the states approach the original. This phenomenon is called chartering. Since

that the sign function is sensitivity when the states approach the original (switching plane). To overcome this phenomenon, we combine two approach in Case I and II and demonstrated in the following.

Case III

In this case, the mixed control is applied for overcoming the chartering phenomenon.

At first, the he maximal convergency rate control is adopted for fast response and then the saturation control is applied when the states approach the switching plane. In this case, we consider the following control strategy:

( ) ( )

( )

, x 0.01 , .

T

T

SAT B PX U t

SGN B PX other

⎧− <

= ⎨⎪

⎪⎩−

The trajectory is depicted in Fig. 35 and control input is depicted in Fig. 36. We can conclude that the system has fast property by the maximal convergence rate control and smooth when approach the switching plane.

-1.5 -1 -0.5 0 0.5 1 1.5 -1.5

-1 -0.5 0 0.5 1 1.5

x1

x 2

Fig. 30 The level set of Example 6.

0 5 10 15 20 25 30 35

-0.5 0 0.5 1

t (sec.)

X

0 5 10 15 20 25 30 35 -0.2

0 0.2 0.4 0.6 0.8 1 1.2

t (Sec.)

U

Fig. 32 Corresponded control input (Case I).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.5 0 0.5 1

t (Sec.)

X

Fig. 33 The trajectory in phase plane (Case II).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

t (Sec.)

U

Fig. 34 Corresponded control input (Case II).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.5 0 0.5 1

t (Sec)

X

0 1 2 3 4 5 0

0.2 0.4 0.6 0.8 1

t (Sec.)

U

Fig. 36 Corresponded control input (Case III).

Chapter 5 Conclusion

This dissertation presents a new design of time-optimal controller for controllable Takagi-Sugeno (T-S) fuzzy model in which the maximum principle 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 system. First, we proposed the proof of the existence of optimal control in T-S fuzzy model, which can be addressed as the compactness of reachable set. The generalized rank condition of accessible Lie algebra is also applied for the proof of the existence of optimal controller for 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 T-S fuzzy model is bang-bang which is determined by 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 finite number of switching over all trajectories for all t . The bounded number of switching is related to the polynomial degree of switching function which is obtained by introducing solvable Lie algebra. Several examples are fully illustrated to show the conditions for the existence of time-optimal controller with their optimal trajectories found by numerical simulation. Further, the feedback controller design of T-S fuzzy model on maximal convergence rate is introduced by level set function. The result of maximizing the convergence rate is characterized from the maximal invariant ellipsoid. The controller is also bang-bang with a simple switching strategy. To handle the chartering phenomenon, a two stages control of saturation and maximizing the convergence rate is also demonstrated. Numerical simulations show the system response is fast and control input is smooth.

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在文檔中 T-S 模糊系統之最佳化時間控制 (頁 40-0)