optimal robust control design of fuzzy mechanical systems 3/issue 5/ijesit201405_05.pdf ·...

ISSN: 2319-5967

ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT)

Volume 3, Issue 5, September 2014

45

Abstract— We first investigate the fundamental properties of the mechanical system as related to the control design.

Then a new optimal robust control is proposed for mechanical systems with fuzzy uncertainty. Fuzzy set theory is used to

describe the uncertainty in the mechanical system. The desirable system performance is deterministic (assuring the bottom

line) and also fuzzy (enhancing the cost consideration). The proposed control is deterministic and is not the usual if-then

rules-based. The resulting controlled system is uniformly bounded and uniformly ultimately bounded proved via the

Lyapunov minimax approach. A performance index (the combined cost, which includes average fuzzy system.

performance and control effort) is proposed based on the fuzzy information. The optimal design problem associated with

the control can then be solved by minimizing the performance index. The unique closed-form optimal gain and the cost are

explicitly shown. The resulting control design is systematic and is able to guarantee the deterministic performance as well

as minimizing the cost. In the end, a mechanical system is chosen for demonstration.

Index Terms—Fuzzy mechanical systems, fuzzy set theory, robust control, optimal design.

I. INTRODUCTION

For the mechanical system, it is essential to investigate the fundamental properties as related to the control design.

Then, we take advantage of the special structure as well as intrinsic properties of mechanical systems to design the

controller. For example, the skew-symmetric matrix property (Slotine & Li 1991) significantly reduces the work of

control design for mechanical systems. In this paper, we explore more properties of mechanical systems which are

useful in the control design.

There always exist unnoticeable and unknown aspects of the real system in the dynamic model which captures

prominent features of the mechanical system. Researches on mechanical system control have always been very

active, especially on handling uncertainties in the system. Exploring uncertainty and determining what is known

and what is unknown about the uncertainty is very important. Once the bound information of the uncertainty is

clearly identified, we can use this known bound information to develop deterministic control approaches. The

well-known 2H H control (Li et al 1992; Shen & Tamura 1995), the Lyapunov-based control (Chen &

Leitmann 1987; Schoenwald 1994), the sliding mode control (Utkin 1994) and so on contribute to this

deterministic approach. When the known portion cannot be completely isolated from the unknown, one may take

the stochastic control approach. The classic Linear-Quadratic-Gaussian control (Stengel 1994) is in such domain.

The stochastic dynamical systems merge the probability theory with system theory and has been the most

outstanding since 50s. Kalman initiated the effort of looking into the estimation problem and control problem

(Kalman 1960) in the state space framework when a system is under stochastic noise. Although the stochastic

approach is quite self-contained and a impressive arena of practitioners. Concerns on the probability theory’s

validity in describing the real world does exist. That is to say, the link between the stochastic mathematical tool and

the physical world might be loose. Kalman, among others, despite his early devotion to stochastic system theory

now contends that the probability theory might not be all that suitable to describe the majority of randomness

(Kalman 1994).

The uncertainty in engineering is often acquired via observed data and then analyzed by the practitioner. However,

the observed data are, by nature, always limited and the uncertainty is unlikely to be exactly repeated many times.

The earthquake data can be an example for interpretation (Soong 1990). Hence, any interpretation via the

frequency of occurrence, which requires a large number of repetitions, might sometimes be limited due to a lack of

basis. As a result, the fuzzy view via the degree of occurrence may be considered as an alternative in certain

applications. We can find more discussions on the relative advantages of fuzziness versus probability in Bezdek

(1994).

Optimal Robust Control Design of Fuzzy

Mechanical Systems Shengchao Zhen, Kang Huang, Han Zhao, Bin Deng, Ye-Hwa Chen

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Fuzzy theory was initially introduced to describe information (for example, linguistic information) that is in lack of

a sharp boundary with its environment (Zadeh 1965). Most interest in fuzzy logic theory is attracted to fuzzy

reasoning for control, estimation, decision-making, etc . Therefore, the merge between the fuzzy theory and

system theory (fuzzy dynamical system) has been less focused on. Past efforts on fuzzy dynamical systems can be

found in Hanss (2005) and Bede et al (2007). We stress that these are different from the very popular

Takagi-Sugeno model or other fuzzy if-then rules-based models. In this paper, from a different angle, we employ

the fuzzy theory to describe the uncertainty in the mechanical system and then propose optimal robust control

design of fuzzy mechanical systems.

The main contributions are fivefold. First, more properties of mechanical systems are explored to be used in the

control design. Second, we not only guarantee the deterministic performance (including uniform boundedness and

uniform ultimate boundedness), but also explore fuzzy description of system performance should the fuzzy

information of the uncertainty be provided. Third, we propose a robust control which is deterministic and is not the

usual if-then rules-based. The resulting controlled system is uniformly bounded and uniformly ultimately bounded

proved via the Lyapunov minimax approach. Fourth, a performance index (the combined cost, which includes

average fuzzy system performance and control effort) is proposed based on the fuzzy information. The optimal

design problem associated with the control can then be solved by minimizing the performance index. In the end, the

unique closed-form solution of optimal gain and the cost are explicitly presented. The resulting control design is

systematic and is able to guarantee the deterministic performance as well as minimizing the cost.

II. FUZZY MECHANICAL SYSTEMS

Consider the following uncertain mechanical system:

( ( ), ( ), ) ( ) ( ( ), ( ), ( ), ) ( ( ), ( ), ) ( ( ), ( ), ( ), ) ( ).M q t t t q t V q t q t t t G q t t t T q t q t t t t (1)

Here tR is the time (i.e., the independent variable), nqR is the coordinate, nqR is the velocity, nqR is the

acceleration, pR is the uncertain parameter, and nR is the control input, ( ) n nM q t R is the inertia

matrix, ( ) nV q q t R is the Coriolis/centrifugal force vector, ( ) nG q t R is the gravitational force vector, and

( ) nT q q t R is the friction force and external disturbance (we omit arguments of functions where no confusions

may arise).

Assumption 1. The functions ( )M , ( )V , ( )G , and ( )T are continuous (Lebesgue measurable in t ).

Furthermore, the bounding set is known and compact.

Assumption 2. (i) For each entry of 0q (i.e., 0( )q t ), namely 0iq , 1 2i n , there exits a fuzzy set 0iQ in

a universe of discourse i R characterized by a membership function [0 1]i irrow . That is,

0 0 0 0{( ( )) }ii i i i iQ q q q (2)

Here i is known and compact. (ii) For each entry of the vector ( )t , namely ( )i t , 1 2i p , the

function ( )i is Lebesgue measurable. (iii) For each ( )i t , there exists a fuzzy set iS in a universe of discourse

i R characterized by a membership function [0 1]i i . That is,

{( ( )) }i i i i i iS (3)

Here i is known and compact.

Remark. Assumption 2 imposes fuzzy restriction on the uncertainty 0q and ( )t . We employ the fuzzy

description on the uncertainties in the mechanical system. This fuzzy description earns much more advantage than

the probability avenue which often requires a large number of repetitions to acquire the observed data (always

limited by nature).

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III. FUNDAMENTAL PROPERMITS OF THE MECHANICAL SYSTEM

For the mechanical system, it is essential to explore the fundamental properties as related to the control design. The

special structure and intrinsic properties of mechanical systems are very useful in the control design.

Assumption 3. The inertia matrix ( )M q t in mechanical systems is uniform positive definite, that is, there

exists a scalar constant 0 such that

( )M q t I (4)

for all nq R .

Remark. We emphasize that this is an assumption, not a fact. There are cases that the inertia matrix may be positive

semi-definite (hence, 0 ). One example is documented in McKerrow (1991) where the generalized inertia

matrix

2 2

2 2

2

2

cos 0

0

mlM

ml

(6)

Thus [ ] 0det M if 2 2(2 1) 0 1 2n n That is to say, the generalized inertia matrix M is

singular. When 2 2(2 1) 0 1 2n n the kinetic energy 1

120

TMq forallq which means

the rotation does not bring up kinetic energy.

To choose a legitimate Lyapunov function for control design, it has often been assumed by previous researchers

that the inertia matrix is uniformly bounded above (i.e. M I , where 0 ). However, it is limited to certain

cases where the inertia matrix itself may be state-dependent. We take the two-degree-of-freedom RP manipulator

in Craig (1989) as an example. The inertia matrix is

2 2

1 1 1 2 2 2

2

0

0

m l I I m dM

m

(7)

The generalized coordinate 1 2[ ]q d . Obviously, M is not uniformly bounded above due to the 2

2d term.

So, we should explore more general property of the inertia matrix in the mechanical system.

Assumption 4. There are constants 0ijk , 0 1k s , such that for all ( ) nq t R R , , each

ij -entry ( )ijm q t of the inertial matrix ( )M q t is bounded as

0

( )s

k

ij ijk

k

m q t q

(8)

Unless otherwise stated, always denotes the Euclidean norm (i.e., 2 ). The 1l -norm are sometimes used and

indicated by subscript 1. The number s depends on the term ijm .

Theorem 1. Under Assumption 3 , there always exist constants 0i , 0 1i s such that ( )M q t

is bounded as:

0

( )s

i

i

i

M q t q

(9)

Proof: For a given vector 1 2[ ]T

nx x x x , its 1l norm is

1

1

n

i

i

x x

(10)

and the corresponding induced matrix norm (Desoer & Vidyasagar 1975) is

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1

1 11 1

sup maxn

ijjx i

M Mx m

(11)

Using assumption 4 , we have

1

0

sk

ij k

i j k

M m q

(12)

Where

k ijk

i j

(13)

For any matrixn nM R , its 1l and 2l norms are related by

1

M n M (14)

Let

k k

n (15)

we have

0

sk

k

k

M q

(16)

Q.E.D.

Remark. Theorem 1 is quite generic since it is applicable to most of currently known mechanical systems such as

robotics, aircrafts, automobiles, rail-vehicles and so on. Chen et al (1998) has proved that the upper bound of

the norm of the inertia matrix of serial-link robot manipulators with revolute and prismatic joints also follows this

theorem with 2s . Taking the cylindrical robot in Koivo (1989, pp. 143-146) as an example, since there is the

term ( )tI r in the inertia matrix where r is the generalized coordinate, we can easily get 2s . We will show

that Theory 1 is quite useful in constructing the legitimate Lyapunov function for the control design.

Theorem 2. There always exists the factorization

( ) ( )V q q t C q q t q (17)

such that ( ) 2 ( )M q q t C q q t is skew symmetric (Slotine & Li 1991). Here ( )M q q t is the

time derivative of ( )M q t .

Remark. To satisfy V Cq , the matrix C may not be unique. But if you also want 2M C to be skew

symmetric, then the particular choice of C should be defined as

1 1

1 1

2 2

n nij jkik

ij k k

k kk j i

m mmc q q

q q q

(18)

where ijc is the ij -element of matrix C .

IV. ROBUST CONTROL DESIGN OF FUZZY MECHANICAL SYTSTEM

We wish the mechanical system to follow a desired trajectory 0 1( ) [ ]dq t t t t , with the desired velocity ( )q t .

Assume 0( ) [ ]d nq t R is of class 2C and ( ) ( )

ddq t tq and ( )d

tq are uniformly bounded. Let

( ) ( ) ( )de t q t q t (19)

and hence ( ) ( ) ( )d

e t q t tq , ( ) ( ) ( )d

e t q t tq . The system (1) can be rewritten as

( )( ) ( )( )d d dd dM e q t e C e q e t eq q q

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( ) ( )dd dG e q t T e q e tq (20)

The functions ( )M , ( )C , ( )G and ( )T can be decomposed as

( ) ( ) ( )d d dM e q t M e q t M e q t

( ) ( ) ( )d d dd d dC e q e t C e q e t C e q e tq q q

( ) ( ) ( )d d dG e q t G e q t G e q t

( ) ( ) ( )d d dd d dT e q e t T e q e t T e q e tq q q (21)

where M , C , G and T are the nominal terms of corresponding matrix/vector and M , C , G , and

T are the uncertain terms which depend on . We now define a vector

( ) ( )( ) ( )( )d d dd de e t M e q t Se C e q e t Seq q q

( ) ( )dd dG e q t T e q e tq (22)

where diag[ ]i n nS s , 0is is a constant, 1 2i n . Obviously 0 if all uncertain terms vanish.

Assumption 5. There are fuzzy numbers ( )d d

k e e tq q ’s and scalars ( )d d

k e e tq q ’s,

1 2k r , such that

1

2

1 2

ˆ

ˆˆ ˆ ˆ ˆ ˆ( ) ( )

ˆ

T

r

r

e e t e e t

(26)

From (23), we have

ˆ ˆ (27)

Remark. One can employ fuzzy arithmetic and decomposition theorem (see the Appendix) to calculate the fuzzy

number based on the fuzzy description of i ’s (Assumption 2 ).

We introduce the following desirable deterministic dynamical system performance.

Definition 1. Consider a dynamical system

0 0( ) ( ( ) ) ( )t f t t t (28)

The solution of the system (suppose it exists and can be continued over 0[ )t ) is uniformly bounded if for any

0r with 0 r , there is ( ) 0d r with ( ) ( )t d r for all 0t t . It is uniformly ultimately bounded

if for 0 r , there are ( ) 0d r and ( ( ) ) 0T d r r such that ( ) ( )t d r for all

0 ( ( ) )t t T d r r .

Let

( ) [ ( ) ( )]Te t e t e t (29)

The control design is to render the tracking error vector ( )e t to be sufficiently small. We propose the control as

2

0

( ) ( ) ( ) ( 1)( ) ( )2

is

id d

i i i i

i

it M Se C Se G T P SD e Pe D eq q

1 1( ) ( 1)

r re Se e Se r (30)

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where diag[ ]ji i n nP p with 0

jip , diag[ ]

ji i n nD d with 0ji

d , 1 2j n , and the scalar

0r . The matrices iP ’s and iD ’s and the scalar and r are all constant design parameters.

Theorem 3. Subject to Assumptions 1-5, the control (30) renders ( )e t of the system (20) to be uniformly bounded

and uniformly ultimately bounded. In addition, the size of the ultimate boundedness ball can be made arbitrarily

small by suitable choices of the design parameters.

Remark. The control ( )t is based on the nominal system, the traking error ( )e t , the bound of uncertainty and

the design parameters. Therefore, this proposed control is deterministic and is not if-then rules-based. Note that s

in the control is the order of the inertia matrix from assumption 4 . If 0 1s r , the control then becomes

2

0 0( ) ( ) ( )d d

M Se C Se G T Pe D e e Seq q (31)

V. PROOF OF THEOREM

The mechanical system with the proposed control is proved to be stable in this section. The chosen Lyapunov

function candidate is shown to be legitimate and then the proof of stability follows via Lyapunov minimax

approach (Corless 1993, Leitmann 1993). The Lyapunov function candidate is chosen as

21

0

1 1( ) ( ) ( ) ( )

2 2

isT T

i i

i

V e e Se M e Se e P SD e

(32)

To prove V is a legitimate Lyapunov function candidate, we shall prove that V is (globally) positive definite and

decrescent.

By (4), we have

22 2 1

0 0

1

1 1 1( ) ( ) ( ( ) )

2 2 2

is

T

m i i

i

V e e Se e P SD e P SD e

2

0 0

1 1( )

2 2

Te Se e P SD e

2 2 22

0 0

1 1

1 1( 2 ) ( )

2 2 i i

n n

i ii i i i i i

i i

s e s e p s d ee e

1

1

2

ni

i iii i

ee e

e

(34)

where is , 0ip ’s and 0i

d ’s are from (22) and (30). ie and ie are the i -th components of e and e , respectively,

and

2

0 0i i ii i

i

i

s p s d s

s

(35)

It can be easily verified that 0i i . Thus by letting 1 112 2

min( ( ) ( ))m m n (hence 0 ),

V is shown to be positive definite:

22 2

1

1( )( )

2

n

im i i

i

V e ee

(36)

By (9), we have

22 21

0 0

( )i

s si i

i i i

i i

V e Se q P SD e

(37)

For the first term on the right-hand side,

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2

( ) ( )Te Se e Se e Se

2 eS S

e eeS I

22

M

S Se

S I

2

s e (41)

Since

( ) max ( )d

dt

q e q t e q t (42)

we have

2

0

si

i

i

e Se q

2

0

( max ( ) )s

i

i dt

i

s e e q t

2

0 0

( )j

s ij

i i

i j

s e e

2

0

( )i

s si

j j

i j i

s e e

2

0

si

i

i

s e e

(43)

For the second term on the right-hand side, by Rayleigh’s principle,

2 22 21 1

0 0

( ) (( ) )i i

s si i

i i M i i

i i

P SD e P SD e e

(44)

With (43) and (44) into (37), we have

22 21

0 0

(( ) )i

s si i

i M i i

i i

V s e e P SD e e

221

0

(( ) )i

si

i M i i

i

e s P SD e

2 2

0 0

s si i

i i

i i

e e e

(45)

Note that i in (45) is a strictly positive constant, which implies that V is decrescent. From (36) and (45), V is

a legitimate Lyapunov function candidate.

Now, we prove the stability of the mechanical system with the proposed control. For any admissible ( ) , the

time derivative of V along the trajectory of the controlled mechanical system of (20) is given by

1

( ) ( ) ( ) ( )2

T TV e Se M e Se e Se M e Se

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2

0

( 1) ( ) ( )2

isT T

i i i i

i

ie P SD e e P SD e

(46)

by applying d

e q q and equation (1), the first two terms become

1

( ) ( ) ( ) ( )2

T Te Se M e Se e Se M e Se

1

( ) ( ( ))2

dTe Se Mq M MSe M e Seq

1

( ) ( ( ) ( ))2

d dTe Se C e G T M MSe M e Seq q

1

( ) ( ( ) ( ) ( ))2

d dTe Se C Se G T M MSe C e Se M e Seq q

( ) ( ( ) )d dTe Se C Se G T M MSeq q

1

( ) ( )( )2

Te Se M C e Se (47)

With Theorem 2, (21) and (30), we can get

1

( ) ( ) ( ) ( )2


( ) ( ( ) )d dTe Se C Se G T M MSeq q

( ) { ( ) ( ) ( ) ( )d d d dTe Se M Se M Se C Se C Seq q q q

21 1

0

( ) ( 1)( ) ( )}2

is

r ir

i i i i

i

iG G T T e Se e Se P SD e Pe D e

( ) { ( ) ( )d dTe Se M Se C Se G Tq q

21 1

0

( ) ( 1)( ) ( )}2

is

r ir

i i i i

i

ie Se e Se P SD e Pe D e

(48)

By (22)-(27),

1

( ) ( ) ( ) ( )2


21 1

0

( ) [ ( ) ( 1)( ) ( )]2

is

r iT r

i i i i

i

ie Se e Se e Se P SD e Pe D e

21 1

0

( ) ( 1)( ) ( )2

is

r ir T

i i i i

i


21

0

( ) ( ) ( 1)( ) ( )2

is

ir T

i i i i

i


21

0

( ) ( 1)( ) ( )2

is

r iT

i i i i

i

ie Se P SD e Pe D e

(49)

For the first two terms on the right hand, we define the function

1

( )r

f

(50)

Take the first order derivative of f with respect to ,

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( 1)rf

r

(51)

That

0f

(52)

leads to

1

( 1)

r

r

(53)

The second order derivative of f with respect to is

21

2( 1) 0

rfr r

(54)

which suggests that (53) renders a maximum value of f ,

1 1

( )( 1) ( 1)

rr r

maxfr r

1

1

1

1 1

r

r r r

1r

(55)

That means

1r

(56)

Since

2

0

( ) ( 1)( ) ( )2

is

iT

i i i i

i

ie Se P SD e Pe D e

22 2

0

( 1)( ) ( )2

is

i

i i i i

i

iP SD e PS e D e

2

0

( 1)( ) ( )2

is

i T

i i i i

i

iP SD e e P SD e

(57)

we have

1

( ) ( ) ( ) ( )2


22 2

0

( 1)( ) ( )2

is

i

i i i i

i

iP SD e PS e D e

2

0

( 1)( ) ( )2

is

i T

i i i i

i

iP SD e e P SD e

(58)

Substituting (58) into (46), we get

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22 2

0

( 1)( ) ( )2

is

i

i i i i

i

iV P SD e PS e D e

2

0

( 1)( ) ( )2

is

i T

i i i i

i

iP SD e e P SD e

2

0

( 1) ( ) ( )2

isT T

i i i i

i

ie P SD e e P SD e

22 2

0

( 1)( ) ( )2

is

i

i i i i

i

iP SD e PS e D e

22

0

( 1) ( )2

is

i

min i i i

i

iP SD e e

2 2

0 0

s si i

i i

i i

e e e

(59)

where 2

2min ( ) ( ) ( 1) ( )

ii

i min i min i i min i i iPS D P SD

. This in turn means that V is

negative definite for all e such that

2

0

0s

i

i

i

e

(60)

Since all universes of discourse i ’s are compact (hence closed and bounded), is bounded. In addition, both

and i are crisp. Thus, V is negative definite for sufficiently large e . Therefore, by Chen and Leitmann

(1987), the controlled mechanical system is uniformly bounded and uniformly ultimately bounded. The stability of

the mechanical system is guaranteed and tracking error e can be made arbitrarily small by choosing large i

and/or . Q.E.D.

Remark. We have shown that fundamental properties explored in Section 3 are quite useful in constructing the

legitimate Lyapunov function. The first five terms of the control scheme (30) are only for the nominal system (i.e.

the system without uncertainty) while the last term is to compensate the uncertainty. For the last term, if r has been

chosen, the magnitude 1r is still free for we still have freedom on designing to determine the size of the

ultimate boundedness. The larger the value of , the smaller the size. This stands for a trade-off between the

system performance and the cost which suggests an interesting optimal quest for the control design. We will pursue

the optimal design in the following section.

VI. OPTIMAL GAIN DESIGN

Section 4 and 5 show that a system performance can be guaranteed by a deterministic control scheme. By the

analysis, the size of the uniform ultimate boundedness region decreases as increases. As approaches to

infinity, the size approaches to 0 . This rather strong performance is accompanied by a (possibly) large control

effort, whichis reflected by (assuming r has been chosen). From the practical design point of view, the

designer may be interested in seeking an optimal choice of for a compromise among various conflicting criteria.

This is associated with the minimization of a performance index.

We first explore more on the deterministic performance of the uncertain mechanical system. Define

max 0 1i

i

i s

(61)

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where i is from (45) and i is from (59) and 0 . Note that

2 2

0 0

s si i

ii

i i

e e e e

(62)

Then by (45) and (59), we get

2 2

0 0

1 1s si i

ii

i i

V e e e e V

(63)

with 0 0 0( ) ( ( ))V V t V e t . This is a differential inequality (Hale 1969) whose analysis can be made according

to Chen (1996) as follows.

Definition 2. If ( )w t is a scalar function of the scalars t in some open connected set D, we say a function

0 0( )t t t t t t is a solution of the differential inequality (Hale 1969)

( ) ( ( ) )t w t t (64)

on 0[ )t t if ( )t is continuous on 0[ )t t and its derivative on 0[ )t t satisfies (64).

Theorem 4. Let ( ( ) )w t t be continuous on an open connected set D2R and such that the initial value

problem for the scalar equation (Hale 1969)

0 0( ) ( ( ) ) ( )t w t t t (65)

has a unique solution. If ( )t is a solution of (65) on 0t t t and ( )t is a solution of (64) on 0t t t

with 0 0( ) ( )t t , then ( ) ( )t t for 0t t t .

Instead of exploring the solution of the differential inequality, which is often non-unique and not available, the

above theorems suggest that it may be feasible to study the upper bound of the solution. The reason is, however,

based on that the solution of (65) is unique.

Theorem 5. Consider the differential inequality (64) and the differential equation (65). Suppose that for some

constant 0L , the function ( )w satisfies the Lipschitz condition (Chen 1996)

1 2 1 2( ) ( )w v t w v t L v v (66)

for all points 1 2( ) ( )v t v t D . Then any function ( )t that satisfies the differential inequality (64) for

0t t t satisfies also the inequality

( ) ( )t t (67)

for 0t t t .

We consider the differential equation

0 0

1( ) ( ) ( )r t r t r t V

(68)

The right-hand side satisfies the global Lipschitz condition with 1L . We proceed with solving the

differential equation (68). This results in

0 0

1( ) exp ( )r t V t t

(69)

Therefore

( ) ( )V t r t (70)

or

0 0

1( ) exp ( )V t V t t

(71)

for all 0t t . By the same argument, we also have, for any st and any st ,

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1

( ) exp ( )s sV V t

(72)

where ( ) ( ( ))s s sV V t V e t . The time st is when the control scheme (30) starts to be executed. It does not

need to be 0t .

By (36) 2

( )V e e , the right-hand side of (72) provides an upper bound of 2

e . This in turn leads to an

upper bound of 2

e . For each st , let

1

( ) exp ( )s s st t V t

(73)

( )

(74)

Notice that for each st , ( ) 0st as .

One may relate 0( )t t to the transient performance and ( ) the steady state performance. Since

there is no knowledge of the exact value of uncertainty, it is only realistic to refer to 0( )t t and ( )

while analyzing the system performance. We also notice that both 0( )t t and ( ) are dependent on

. The value of is not known except that it is characterized by a membership function.

Definition 3. Consider a fuzzy set (Chen 2010)

{( ( )) }NN v v N (75)

For any function f N R , the D -operation [ ( )]D f is given by

( ) ( )

[ ( )]( )

NN

NN

f dD f

d

(76)

Remark. In a sense, the D -operation [ ( )]D f takes an average value of ( )f over ( )N . In the special

case that ( )f , this is reduced to the well-known center-of-gravity defuzzification method (Klir & Yuan

1995). Particularly, if N is crisp (i.e., ( ) 1N for all N ), [ ( )] ( )D f f .

Lemma 1. For any crisp constant aR ,

( ) ( )

[ ( )]( )

NN

NN

af dD af

d

( ) ( )

( )

NN

NN

f da

d

[ ( )]aD f (77)

We now propose the following performance index: For any st , let

2 2 2( ) ( ) [ ( )]

ss s

tJ t D t d D

1 2 3( ) ( ) ( )sJ t J J (78)

Where 0 are scalars? The performance index consists of three parts. The first part 1( )sJ t may be

interpreted as the average (via the D -operation) of the overall transient performance (via the integration) from

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time st . The second part 2 ( )J may be interpreted as the average (via the D -operation) of the steady state

performance. The third part 3 ( )J is due to the control cost. Both and are weighting factors. The

weighting of 1J is normalized to be unity. Our optimal design problem is to choose 0 such that the

performance index ( )sJ t is minimized.

Remark. A standard LQG (i.e., linear-quadratic-Gaussian) problem in stochastic control is to minimize a

performance index, which is the average (via the expectation value operation in probability) of the overall state and

control accumulation. The proposed new control design approach may be viewed, loosely speaking, as a parallel

problem, though not equivalent, in fuzzy mechanical systems. However, one cannot be too careful in distinguishing

the differences. For example, the Gaussian probability distribution implies that the uncertainty is unbounded

(although a higher bound is predicted by a lower probability). In the current consideration, the uncertainty bound is

always finite. Also, LQG does not take parameter uncertainty into account.

One can show that

2( )

ss

tt d

2

2exp ( )

ss s

tV t d

2

2exp ( )

2s

s s

t

V t d

2

2sV

(79)

Taking the D -operation yields

2 ( )

ss

tD t d

2

2sD V

2

2 22

s sD V V

22 2

2[ ] 2 [ ] [ ]

2s sD V D V D

(80)

The last equality is due to Lemma 1. Next, we analyze the cost 2 ( )J . Again by Lemma 1, we have

2[ ( )]D

2

D

22

2[ ]D

(81)

With (80) and (81) into (78), we also have

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2 22 2 2 2

2 2( ) [ ] 2 [ ] [ ] [ ]

2s s sJ t D V D V D D

232 4

1 2 2

(82)

where 2

1 ( 2) [ ]sD V , 2

2 [ ]sD V , 3 2

3 ( 2) [ ]D , 2 2

4 [ ]D .

The optimal design problem is then equivalent to the following constrained optimization problem: For any st ,

min ( ) subject to 0sJ t

(83)

By using the performance index in (82), we will then pursue the optimal solution in the next section.

VII. THE CLOSED-FORM SOLUTON OF OPTIMAL GAIN

For any st , taking the first order derivative of J with respect to

32 4

2 3 32 2 2

J

4

2 3 43

1( 2 2 2 )

(84)

That

0J

(85)

leads to

4

2 3 42 2 2 0 (86)

or

4

2 3 42 2( ) (87)

Equation (87) is a quartic equation.

Theorem 6. Suppose [ ] 0D . For given 1 2 3 4 , the solution 0 to (87) always exists and is

unique, which globally minimizes the performance index (82).

Proof. Let 4

2( ) 2 . Then (0) 0 and ( ) is continuous in . In addition, since 2 0 and

0 , ( ) is strictly increasing in . Since [ ] 0D , we have [ ] 0D , 2[ ] 0D , 3 4 0 , and

therefore 3 42( ) 0 (notice that 0 ). As a result, the solution 0 to (87) always exists and is

unique. For the unique solution 0 that solves (87),

24 3

2 3 4 22 4 3

3 1( 2 2 2 ) ( 8 )

J

3

23

1( 8 )

0 (88)

Therefore the positive solution 0 of the quartic equation (87) solves the constrained minimization problem

(83). Q.E.D.

Remark. In the special case that the fuzzy sets are crisp, [ ]D , 2 2[ ]D . The current setting still

applies. The optimal design can be found by solving (87).

The solutions of the quartic equation (87) depend on the cubic resolvent (Bronshtein & Semendyayev 1985)

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59

3 2

1 2( 4 ) 0z r z r (89)

where

1 3 4

1( )r

(90)

22

2r

(91)

Let 2

1 1 2 24p r p r . The discriminant H of the cubic resolvent is given by

3 2

1 2

3 2

p pH

(92)

Since 0 0r H . The solutions of the cubic resolvent are given by

1z u w (93)

2

( ) 3( )

2 2

u wz u w i

(94)

3

( ) 3( )

2 2

u wz u w i

(95)

where

13

2

2

pu H

(96)

13

2

2

pw H

(97)

The cubic resolvent possesses one real solution and two complex conjugate solutions. This in turn implies that the

quartic equation has two real solutions and one pair of complex conjugate solutions. The maximum real solution,

which is positive and is therefore the optimal solution to the constrained optimization problem, of the quartic

equation is given by

1 2 3

1( )

2opt z z z (98)

By using (87), the cost J in (82) can be rewritten as

232 4

1 2 2J

4 23 4

1 22 2 2

1( 2 ) 3

23 4

1 3 42 2 2

1[2( )] 3

23 4

1 2 23

4

1 3 42

1( 3 )

(99)

With (98), the minimum cost is given by

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4

min 1 3 4 1 2 32

1 2 3

4 3

16J z z z

z z z

(100)

Remark. Combining the results of Sections 4 - 7 , the robust control scheme (30) using the optimal design of

0 renders the tracking error e of the closed-loop mechanical system uniformly ultimately bounded (with the

initial state ( )se t ). In addition, the performance index J in (82) is globally minimized.

The optimal design procedure is summarized as follows:

Step 1: For a given inertia matrix M , obtain i ’s and the order s of (9).

Step 2: According to in (22), obtain and in (27).

Step 3: Based on the ( )V e in (32), solve for the i ’s in (45). For given S , iP ’s, and iD ’s, solve for the i ’s in

(59). Thus, is given in (61).

Step 4: For a chosen r , using the obtained in Step 2 and the sV in (72), calculate 1 2 3 4 in (82) based

on the D -operation.

Step 5: For given and , solve for the opt in (98) and the minimum cost given in (100).

Step 6: The optimal robust control scheme is given in (30).

VIII. SOME LMITING PEFORMANCE

As was shown earlier, the tracking error e of the controlled system enters the uniform ultimate boundedness

region after a finite time and stays within the region thereafter. Thus it is interesting to consider, in the limiting case,

only the cost associated with this portion of performance.

In the limiting case, the transient performance cost 1( )sJ t is not considered. The cost is then dictated by that of

the steady state performance and the control gain:

2 3( ) ( ) ( )J J J

24

2

(101)

The quartic equation (87) is reduced to

4

4 (102)

Its positive solution is given by

14

4opt

(103)

With (102) into (101),

4

42

1( )J

4

2

2

(104)

Using (103), the minimum cost is then

min 42J (105)

If the weighting , then in the quartic equation, the positive solution . This simply means that the

relative cost of the ultimate boundedness region (as is given by 2 ( )J ) is high and the control gain, which is

relatively cheap, is turned high.

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If , then in the quartic equation, the positive solution 0 . This shows the other extreme case when

the control is very expensive.

IX. ILLUSTRATIVE EXAMPLE

Consider a spiral mechanism with a sliding rod (Figure 1). The nut is fixed while the screw is on a spiral motion

(i.e., upward movement and also rotation) with variable pitch (Gillie 1978; Su & Lin 1993; Yu et al 1997). We

use y to denote the upward moving distance and the rotating angle. Because the screw has variable pitch, we

assume 3y . 1 2m m are the masses of the screw and sliding rod, respectively. 1 2I I are the inertia tensors.

We use x to denote the moving distance of the sliding rod.

Fig 1. Spiral mechanism with a sliding rod

We choose two generalized coordinates 1 2[ ] [ ]T Tq q q x to describe the mechanical system. The two

coordinates are independent of each other. The kinetic energy of the mechanical system is

2 2 24 22

1 2 1 2 2 2

1 9 1 1( ) ( )

2 2 2 2T I I m m m m xx (106)

The potential energy 0V . Then the Lagrange’s equation

( )d L L

dt q q

(107)

can be written out where the Lagrangian L T V , is the external control force. The equation of motion can

be written in matrix form (1) by using Lagrange’s equation as

4 2

1 2 1 2 2

2

9( ) 0( )

0

I I m m m xM q

m

(109)

3

1 2 2 2

2

18( )( )

0

m m m xx m xC q q

m x

(111)

( ) ( ) 0G q T q q (112)

Since there is 4 in the entry of inertia matrix M , from Assumption 4 , we know 4s . The masses 1 2m are

uncertain with 1 21 2 1 2 ( )m m tm , where 1 2 0m are the constant nominal values and 1 2m are the

uncertainty. So, we have

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4 21 2 21 2

2

9( ) 0

0

xm m mI IM

m

(114)

31 2 2 2

2

18( )

0

xx xm m m mC

xm

(116)

4 2

1 2 1 2 2

2

9( ) 0

0

I I m m m xM

m

(118)

3

1 2 2 2

2

18( )

0

m m m xx m xC

m x

(120)

The desired trajectory ( )dq t , the desired velocity and acceleration ( ) ( )d d

t tq q are given by

sin cos sin

cos sin cos

d ddd dd

d d d

t t tq q q

t t tx x x

(127)

By using d ddq e q q e q eq q , (113) can be rewritten as

4 21 2 21 2 1 2

2

9( )( sin ) ( cos ) 0

0

e t e tm m mI IM

m

31 2 1 2 2 2 11 2 2

2 12

18( )( sin ) ( cos ) ( cos )( sin ) ( cos )( cos )

( cos )( cos ) 0

e t t e t t e t tm m e m e m eC

e t tm e

4 2

1 2 1 2 1 2 2

2

9( )( sin ) ( cos ) 0

0

I I m m e t m e tM

m

11 2 2 1

12 2

( cos )( cos )

( cos )( cos ) 0

C m e t teC

m e t te

(132)

where 3

1 211 1 2 1 2 218( )( sin ) ( cos ) ( cos )( sin )C m m e t t m e t te e . We choose S to be

a 2 2 identity matrix. Therefore, we can get

( ) ( )( ) ( )( )d d dd de e t M e q t Se C e q e t Seq q q (133)

ˆˆ ˆ ˆ( ) ( )T

e e t e e t (134)

where

1 2 1

1 2 2

2 3

ˆ ˆ

I I

m m

m

(139)

and

11 sin te

4 3

1 12 1 1 19 ( sin ) ( sin ) 18 ( sin ) ( cos )( cos )e t t e t t e te e

2

1 2 23 2 2 1( cos ) ( sin ) ( cos )( sin )( cos ) cose t t e t t e t te e e

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1 12 2 2 1( cos )( cos )( sin ) ( cos )( cos )( cos )e t t e t e t t e te e (140)

For the mechanical system, we choose 1 2 1 21 1m I and assume the uncertainties 1 2 1 2m I are all “close

to 0 " and governed by the membership function

1 2 1 2

1 10 0 1 0

1 10 0 0 1m I

(141)

Set the design parameters 0 1 2 3 4P , and 0 1 2 3 4D to be the identity matrix 2 2I . Follow the design procedure, we

have 0 24 33 , 1 1 , 2 16 56 , 3 1 , 4 29 00 , 0 1 , 1 2 12 , 2 4 , 3 7 07 ,

4 12 , and 24 33 . For simplicity, we choose 1r . By using the fuzzy arithmetic and decomposition

theorem, we obtain 1 5761 30 , 2 48 56 , 3 0 24 , 4 0 02 . By selecting five sets of weighting

and , the optimal gain opt and the corresponding minimum cost minJ are summarized in Table 1.

Table 1. Weighting/optimal gain/minimum cost

opt minJ

(1,1) 1 3.3097 5793

(1,10) 0.1 1.5386 5832

(1,100) 0.01 0.71655 5915

(10,1) 10 3.3128 5794

(100,1) 100 3.3428 5795

We choose 0st and the initial condition (0) [1 0 5 0 51]Te . For numerical simulation, we choose the

uncertainties as 1 0 1m , 2 0 1sin(10 )m t , 1I is a rectangular pulse jumping between 1 and 1 every

second and 2 0 1cos(2 )I t . Four different classes of uncertainties (i.e., constant, low frequency, high

frequency and rectangular pulse) are used to test the proposed control scheme. Simulation results are as follows.

Fig 2. Comparison of fuzzy mechanical system performances

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Figure 2 shows comparison of the tracking error norm e trajectory with the proposed control (under

3 3097opt when 1 ), without any control and with the nominal PD control (without the part of

control that governs the uncertainty i.e., 0 ). The trajectory e with the proposed control enters a much

smaller region around 0 after some time (hence ultimately bounded) than the trajectory only with the nominal PD

control. The uncontrolled trajectory e moves faraway from 0 . Figure 3 shows the corresponding trajectories of

the proposed control 1 2[ ]Tu u .

Fig 3. Control histories

Figure 4 shows the tracking error norm trajectories for all five opt ’s (by using different ( ) combinations).

The use of 0 (not optimal) is also shown for a comparison. Figure 5 shows the corresponding histories of

control efforts ( )t . In order to show the results explicitly, we list the area S between the tracking error norm

trajectory (under different opt ) and the coordinate axes, and also the area R between the control effort trajectory

(under different opt ) and the coordinate axes in Table 2.

Fig 4. Comparison of system performances under different opt

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Fig 5. Comparison of control efforts under different opt

Table 2. S and R under different opt

opt 3.3097 1.5386 0.7165 3.3128 3.3428 0

S 1.6502 1.7140 1.7852 1.6488 1.6471 3.2715

R 80.8343 81.5461 84.2688 80.8033 80.7629 90.3669

X. CONLUSION

Properties of the mechanical system (especially, the upper bound property of the inertia matrix) are explored. We

have shown that the properties are very useful in the control design. Fuzzy description of uncertainties in

mechanical systems is employed and we incorporate fuzzy uncertainty and fuzzy performance into the control

design. A new robust control scheme is proposed to guarantee the deterministic performance (including uniform

boundedness and uniform ultimately boundedness). The control is deterministic and is not if-then rules-based. The

resulting controlled system is stable proved via the Lyapunov minimax approach. A performance index is proposed

and by minimizing the performance index, the optimal design problem associated with the control can be solved.

The solution of the optimal gain is unique and closed-form. The resulting control design is systematic and is able to

guarantee the deterministic performance as well as minimizing the cost.

APPENDIX

We briefly review some preliminaries regarding fuzzy numbers and their operations (Klir & Yuan 1995):

. Fuzzy number. Let G be a fuzzy set in R , the real number. G is called a fuzzy number if: (i) G is normal,

(ii) G is convex, (iii) the support of G is bounded, and (iv) all -cuts are closed intervals in R .

Throughout, we shall always assume the universe of discourse of a fuzzy set number to be its 0 -cut.

. Fuzzy arithmetic. Let G and H be two fuzzy numbers and [ ]G g g

, [ ]H h h

be their

-cuts, [0 1] . The addition, substraction, multiplication, and division of G and H are given by,

respectively,

( ) [ ]G H g h g h

(142)

( ) [min( ) max( )]G H g h g h g h g h

(143)

( ) [min( )G H g h g h g h g h

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max( )]g h g h g h g h

(144)

( ) [min( )G H g h g h g h g h

max( )]phag h g h g h g h

(145)

. Decomposition theorem. Define a fuzzy set V in U with the membership function ( )

V VI x

where ( ) 1V

I x

if x V and ( ) 0alphaV

I x if x U V . Then the fuzzy set V is obtained as

[0 1]

V V

(146)

where is the union of the fuzzy sets (that is, sup over [0 1] ).

Based on these, after the operation of two fuzzy numbers via their -cuts, one may apply the decomposition

theorem to build the membership function of the resulting fuzzy number.

ACKNOWLEDGMENT

Here we show thanks and appreciations sincerely to Professor Jie Tian of Hefei University of Technology (China)

for his instructions and help during the process of research. The research is also supported in part by the Hefei

University of Technology Chunhua(Primavera) Project ( 2013HGCH0010). The project name is Electric vehicle

integrated powertrain design and control system.

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AUTHOR BIOGRAPHY

Shengchao Zhen born in 1988, is currently a Ph.D. candidate in Hefei University of Technology, China and a visiting scholar in Georgia

Institute of Technology, U.S.A. His research interests include analytic mechanics, dynamics of multi-body systems, optimal control, robust

control, adaptive control, fuzzy engineering, and uncertainty management.

Tel: +1-404-9531295; E-mail: [email protected]

Kang Huang is currently a vice-dean and doctoral supervisor in Hefei University of Technology, China. His research interests include

mechanical transmission, magnetic machine, vehicles, digital design and manufacturing, information system, dynamics and control.

Tel: +86-13665606407; E-mail: [email protected]

Han Zhao born in 1957, is currently a vice-president and doctoral supervisor in Hefei University of Technology, China. He is a committee

member of international IFToMM education commission and also an editorial member of Chinese Journal of Mechanical Engineering. His

research interests include mechanical transmission, magnetic machine, vehicles, digital design and manufacturing, information system,

dynamics and control.


Bin Deng born in 1990, is currently a Ph.D. candidate in Hefei University of Technology. His research interests include electric vehicles,

hybrid electric vehicles, vehicle stability control for HEVs, robust control.


Ye-Hwa Chen got his Ph.D. in the University of California, Berkely in 1985, is currently a professor in Georgia Institute of Technology,

U.S.A. He is a member of IEEE, ASME and Sigma Xi. He is also a regional editor (North America) of Nonlinear Dynamic sand System Theory

and an associate editor of International Journal of Intelligent Automation and Soft Computing. His research interests include advanced control

methods for mechanical manipulators, neural networks and fuzzy engineering, adaptive robust control of uncertain systems, uncertainty

management.

Tel: +1-404-894-3210; E-mail: [email protected]

mailto:[email protected]

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