- Email: [email protected]
On Lyapunov's stability analysis of non-smooth systems with applications to control engineering
International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
On Lyapunov's stability analysis of non-smooth systems with applications to control engineering Q. Wu*, N. Sepehri Department of Mechanical and Industrial Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 Received 1 October 1999; received in revised form 1 June 2000; accepted 1 August 2000
Abstract The extension of Lyapunov's stability theory to non-smooth systems by Shevitz and Paden (Trans. Automat. Control 39 (1994) 1910) is modi"ed with the goal of simplifying the procedure for construction of non-smooth Lyapunov functions. Shevitz and Paden's extension is built upon Filippov's solution theory and Clarke's generalized gradient. One important step in using their extension is to determine the generalized derivative of a non-smooth Lyapunov function on a discontinuity surface, which involves the estimation of an intersection of a number of convex sets. Such a determination is complicated and can become unmanageable for many systems. We propose to estimate the derivative of a non-smooth Lyapunov function using the extreme points of Clarke's generalized gradient as opposed to the whole set. Such a modi"cation not only simpli"es the form, but also reduces the number of the convex sets involved in the estimation of the generalized derivative. This makes the stability analysis for some non-smooth systems practically easier. Three examples, including a mathematical system, a system with stick}slip friction compensator and an actuator having interaction with the environment, are used for demonstration. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Non-smooth dynamic systems, described by differential equations with discontinuous terms, appear naturally and frequently in the control "eld. Typical examples include systems with stick}slip friction, contact task controls, variable structure and optimal control systems. Two issues, involved in the stability analysis of such systems, are "rstly, classical solution theories to ordinary di!erential equations are no longer applicable. Secondly,
* Corresponding author. Tel.: #1-204-474-8843; #1-204-275-7507. E-mail address: [email protected] (Q. Wu). Professor.
fax:
Lyapunov's stability theory was developed for smooth systems. The "rst issue regarding the solution concept can be addressed using Filippov's solution theory [2,3]. This theory is one of the earliest and most conceptually straightforward approaches developed for analysis of non-smooth systems. It has been applied by many researchers [1,4}9]. Consequently, extensions of Lyapunov's second method to non-smooth dynamic systems, based on Filippov's solution theory, have been studied by Hahn [10], Paden and Sastry [5] and most recently by Shevitz and Paden [1] and Wu et al. [11]. In Shevitz and Paden's extension, non-smooth Lyapunov functions were constructed. The important step in constructing a non-smooth Lyapunov function is to examine the sign of its generalized derivative with respect to
0020-7462/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 8 6 - X
1154
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
time on the discontinuity surface. This is done by estimating the sign of each element of a convex set, which is the intersection of a large number of convex sets (see [1] for details). Such estimation requires heavy mathematical machinery and can become unmanageable for systems with more than two state variables or when the discontinuity surface is the intersection of several discontinuity surfaces. The di$culty in determining the generalized derivative may restrict the applications of the above-extended Lyapunov's stability theory. Wu et al. [11] extended Lyapunov's stability theory to non-smooth systems in that smooth Lyapunov functions were used and the conditions for the construction of smooth Lyapunov functions for non-smooth systems have been established. The signi"cance of this work is that it simpli"es the construction of Lyapunov functions and, therefore, makes the stability analysis of non-smooth systems easier. For example, the method has been used for stability control of a base-excited inverted pendulum [9] and its application to the modeling of human trunk movements [12]. However, as pointed out by the authors [11], for many nonsmooth systems, non-smooth Lyapunov functions are the only choices. In this paper, the extended Lyapunov's second method by Shevitz and Paden [1] is revisited with the objective of simplifying the determination of the generalized derivative of a Lyapunov function on a discontinuity surface. First, a brief review of Filippov's solution concept [2], Clarke's generalized gradient [13] and generalized derivative of non-smooth Lyapunov functions [1] is given. Next, a modi"cation to Shevitz and Paden's theorem is proposed which allows the use of only extreme points of Clarke's generalized gradient for the estimation of the above derivative. The modi"ed theorem is then applied to three examples from mathematical and engineering systems. It is shown that not only the form, but also the number of the convex sets involved in the estimation of the generalized derivative is simpli"ed. The signi"cance of the present study is that it makes the stability analysis for at least some nonsmooth systems practically easier in the sense that it does not require much advanced mathematical
machinery. Thus, more non-smooth systems can be studied under Lyapunov's stability theory.
2. Mathematical Review The extension of Lyapunov's stability theory by Shevitz and Paden [1] is based on Filippov's solution theory [2,3] and Clarke's generalized gradient concept [13]. Filippov's solution theory has been reviewed in the previous work by the authors [9]. Clarke's generalized gradient concept was discussed in [1]. Here, we only provide a summary of Filippov's solution concept, Clarke's generalized gradient and the generalized derivative of a nonsmooth function on a discontinuity surface. 2.1. Filippov's solution concept Consider the following vector di!erential equation: x "f (x, t),
(1)
where f: RL;RPRL is measurable and essentially locally bounded. A vector function, x, de"ned on the time interval (t , t ), is a Filippov's solution of Eq. (1) if: (i) it is absolutely continuous and, (ii) for almost all t3(t , t ) and for arbitrary '0, the vector dx(t)/dt belongs to the smallest closed convex set of RL containing all the values of the vector function f (x, t). Point x ranges the entire neighborhood of the point x (with t "xed) except for arbitrary sets of measure zero. More precisely, in the notation adopted above, dx(t) 3K[ f ](x, t), dt
(2)
where K[ f ](x, t), co f (;(x, )!N, t). (3) B I, K f ( ) ) ) is called Filippov's set. co denotes the closure of the convex hull and, N represents the intersection over all sets of Lebesgue measure zero. Note that since f (x, t) is not continuous, the standard existence and uniqueness theorems do not
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
apply. However, since the focus of this work is on stability proof, we assume that there exits a unique Filippov's solution to the non-smooth systems under study. Readers interested in detailed discussions on the existence and uniqueness of Filippov's solutions, are referred to the original literature [2,3].
Consider a locally Lipschitz function, < : RL;RPR. The generalized gradient of < at (x, t) is de"ned as <(x, t)" (4) colim <(x, t) (x , t )P(x, t), (x , t ) , , G G G G 4 where is the set of measure zero in which the 4 gradient of < is not de"ned. The gradient operator,, includes the derivative with respect to time (/t). In this de"nition, Lipschitz means Lipschitz in (x, t), i.e., discontinuities in t are not allowed. For notational brevity, if a function <(x, t) has no explicit t dependence, we shall adopt the convention of dropping the last component of <, which is zero. As an example, Clarke's generalized gradient for function <(x)" x is
<(x)"
x(0,
1,
x'0,
<(x : v)"lim sup (<(y#tv))!<(y)/t (see Ref. [1]). W V Rs Given the above conditions, <(x, t) is absolutely continuous and (d/dt) <(x, t) exists almost everywhere: d <(x, t)"
(6)
Set
2.2. Clarke's generalized gradient
!1,
1155
(5)
co!1, 1, x"0,
where co!1, 1 is equivalent to [!1, 1], i.e., any real number between !1 and 1 belongs to co!1, 1. 2.3. Derivative of non-smooth functions on discontinuity surfaces Let x be a Filippov's solution to x "f (x, t) on an interval containing t. Function <(x, t) : RL;RPR is Lipschitz. Additionally, <(x, t) is a regular function, i.e., (i) for all v, the usual one-sided directional derivative <(x : v) exists, and (ii) for all v, v, <(x : v)"<(x : v) where <(x, v) is the generalized directional derivative and is de"ned as
3. Stability theorem One of the important steps in constructing nonsmooth Lyapunov functions as outlined by Shevitz and Paden [1], is to prove that every element of the set
1156
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
where m is the number of the extreme points of Clarke's generalized gradient. Lyapunov's second method by Shevitz and Paden [1] is now modi"ed and presented below: Theorem. Let x "f (x, t) be essentially locally bounded and 03K[ f ](0, t) in a region QMx3RL x (r;t t )t(R. Also, let <: RL;RPR be a regular function satisfying that (i) V is Lipschitz continuous, positive and dexnite, and (ii) each element of
Discussion. Note that, if the convex set
are involved. Secondly, set
4. Demonstrative examples Three examples on stability analyses of a mathematical system and two control systems, are presented to demonstrate the applicability of the proposed theorem. Example 1. Stability analysis of a non-smooth system [1,5] Consider the following mathematical system:
x 1 !2 !1 sgn(x ) x "! 2 1 1 sgn(x ) . (11) x !1 !1 1 sgn(x ) It is to be proven that the above system is stable about the equilibrium point x , x , x 2" 0, 0, 0. Here we show how to establish the stability by applying the theorem outlined in Section 3. Eq. (9) is to be used for estimating the derivative of a non-smooth Lyapunov function on each discontinuity surface associated with Eq. (11). Here, we only consider the discontinuity surface S:"xo ; x "0, x "0, x O0, which is the intersection of the following two discontinuity surfaces: S :"(x , x , x ); x "0, x O0, x O0, S :"(x , x , x ); x O0, x "0, x O0. (12) Other discontinuity surfaces can be treated in a similar manner. We choose a Lyapunov function as <" x # x # x , which is continuous and positive de"nite. We now need to prove that each element of the convex set
(13)
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
Sets K[ f ] and
1#sgn(x ) !3#sgn(x ) !3!sgn(x ) , !1!sgn(x ) , 2!sgn(x ) !sgn(x )
3#sgn(x ) !1#sgn(x ) 1!sgn(x ) , 3!sgn(x ) !sgn(x ) !2!sgn(x )
and
1
1
sgn(x )
,
1
!1
!1
1
sgn(x )
sgn(x )
(14)
!1
,
!1
sgn(x ) (15)
Vector denotes the extreme points from
1157
Their intersection is K[ f ]"co[!5, !1]. G G For x (0, we have K[ f ]"co[!5, 3], K[ f ]"co[!5, !1], K[ f ]"co[!5, 7], K[ f ]"co[!5, 3]. The intersection of the above sets is
(18)
(19)
(20) K[ f ]"co[!5, !1]. G G From (18) and (20), it is seen that all the elements belonging to set
"
,
(21)
sgn(x ) where variables and take all the values between !1 and 1. Therefore,
1158
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
the same, using
(23)
where x is the location of the mechanism and x* is the location of the environment. The reaction force, r, due to the contact with the environment, is r"k (xA !xH#x ) when xH!x (xA . In the A L L non-contact region, i.e., when xH!x *xA , r"0. L A controller is proposed [14] which controls the position and force of the compliant object during non-contact and contact phases, respectively. The control signal, u(t), is given below:
u(t)"
u(t)"!k (x !xB)!k x , N B free motion (r"0), u(t)"!k k (x !xB)!k k x #rB, ND A BD A contact (r'0). (24)
In (24), xB is the desired position and rB is the desired contact force to be exerted on the environment upon contact. Constants k , k , k and N B ND k are the control gains. Assuming e "x !xB, BD in the non-contact region, the state-space model can be written as e "e , k k e "! N e ! B e . (25) m m In the contact region, however, we have e " x !xB"x !rB/k and the state-space model is A e "e k k (26) e "! A (k #1)e ! BD k e . ND m A m The dynamic system, shown in (25) and (26), is non-smooth. The existence and uniqueness of the solution have been proven, but not shown here. The discontinuity surface is: S :"e; e #xB#xA !xH"0, (27) C L where e"e , e 2. The state-space, , is divided into two parts, \ and >, by the discontinuity surface S : C
>:"e; e 'xH!xB!xA , L (28)
\:"e; e (xH!xB!xA . L When e3 >, the compliant mechanism is in contact mode and when e3 \, it is free to move. A Lyapunov function is now constructed as
1 k < " e # N e 2 2m (k #1)k !k A N (xH!xA !xB) # L 2m for e3 \,
1 k #1 <" < " e # ND k e 2 A 2m
Fig. 1. One-dimensional complaint mechanism.
for e3 >, 1 k #1 < " e # ND k e 2 A 2m for e3S . (29)
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
Note that V is continuous and positive through the entire region under the condition that (k #1)k !k '0. V is also de"nite since V"0 A N (i.e., < "0) if and only if e "0 and e "0. It is easy to show that
k #1 ND k (xH!xA !xB) A L m e
Accordingly, Filippov's set is K[ f ](e3S )"co C
.
k , k ! N (xH!xA !xB)! B e L m m
k #1 . (31) k ! ND k (xH!xA !xB)! BD k e A L m A m For the system under study, we have
k k #1 N ! ND k A m m
k k ;(xH!xA !xB)e ! B e , ! BD k e . L m m A
e k #1 k ! ND k (xH!xA !xB)! BD k e A L m m A
(33)
If we assume k "k k , it is obvious that B BD A
(34)
where x is the displacement of the mass m and F is the acting force. The friction model is de"ned as (35)
where
sgn(x )"
e k , k ! N (xH!xA !xB)! B e L m m
<K[ f ]"co !
F "F sgn(x )#F (1!sgn(x )),
k N (xH!xA !xB) 2 L m co e
;
Similarly, we have
(30)
e
e
1159
k k k #1 "co ! B e , N ! ND k A m m m
k ;(xH!xA !xB)e ! BD k e . L m A
1 0 .
!1
In Eq. (35), slip friction, F , is a positive constant and stick friction, F , is de"ned as
F>, F*F>'0, 1 1 F (F)" F, F\(F(F>, 1 1 F\, F)F\(0, 1 1
(32)
(36)
where F> and F\ are the positive and negative 1 1 limits on the static friction forces. Southward et al. [7] proposed the following non-linear control law for F to keep the system having a unique
1160
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
equilibrium point at x "0 and x "0: F"!k x !k x !k N B N 0, x 'x , & (x !x ), 0(x )x , & & x "0, (37) ; 0, (x !x ), x )x (0, * * 0, x (x , * where x "!F>/k and x "!F\/k . * 1 N & 1 N The state-space model shown in Eq. (34) together with Eqs. (35) and (37) describe a non-smooth dynamic system. The discontinuity is caused by (i) the model of the dry friction which is discontinuous with respect to the velocity, x , and (ii) the control law which is discontinuous at x "0. The discon tinuity surfaces are
S :"x; x "0, x O0, S :"x; x "0, x O0, (38) S :"x; x "0 and x "0, where x"x , x 2. The discontinuity surface S is the intersection of surfaces S and S . Southward et al. [7] suggested a non-smooth Lyapunov function to prove the stability of the above control system. The derivative of their Lyapunov function on the discontinuity surface, however, was replaced with a Dini derivative and was discussed in a somewhat illustrative manner. In this paper, the derivative of the Lyapunov function on the discontinuity surface is replaced with Clarke's generalized gradient and is estimated quantitatively. The special case for the discontinuity surface of S is only studied. Other cases of S and S can be investigated in a similar manner. The Lyapunov function candidate is shown below: 1 1 <(x , x )" k x # k x 2 N 2 B 1 k x , x 'x , & 2 N & 1 k x x ! x , 0)x )x , N & 2 & # 1 k x x ! x , x )x )0, N * 2 * 1 k x , x (x . * 2 N * (39)
Following the framework described in Section 3, we need to prove that each element of the convex set
k x k x N * , N & mx mx
,
(40)
K[ f ](x3S )" co
x , 1 (!k x !k x !F sgn(x )) B N * m
x 1 (!k x !k x !F sgn(x )) B N & QJGN m
.
(41)
We only need to prove that all the elements of the convex set
"co k x mx N *
x ; 1 , (!k x !k x !F sgn(x )) B N * m k x mx N *
x ; 1 (!k x !k x !F sgn(x )) B N & m "co[!k x !F x , B
(42)
!k x !F x !k x (x !x )]. B N & * Similarly, K[ f ]"co[!k x !F x !k x B N "(x !x ), !k x !F x ]. (43) * & B We now prove that
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
intersection of sets K[f] K[ f ] is
1161
References [1] D. Shevitz, B. Paden, Lyapunov stability theory of nonsmooth systems, IEEE Trans. Automat. Control 39 (1994) 1910. [2] A.F. Filippov, Di!erential equations with discontinuous right-hand side, Math. Sb. 51 (1960) 99 (English Translation: Am. Math. Soc. Translations 42 (1964) 191). [3] A.F. Filippov, Di!erential equations with second members discontinuous on intersecting surfaces, Di!erentsial'nye Uravneniya 15 (1979) 1814 (English Translation: Di!erential Equations 15 (1980) 1292). [4] J.J. Slotine, S.S. Sastry, Tracking control of nonlinear systems using sliding surfaces with application to robot manipulators, Int. J. Control 38 (1983) 465. [5] B.E. Paden, S.S. Sastry, Acalculus for computing Filippov's di!erential inclusion with application to the variable structure control of robot manipulators, IEEE Trans. Circuits Systems 34 (1987) 73. [6] R.A. DeCarlo, S.H. Zak, P. Matthews, Variable structure control of nonlinear multivariable systems: a tutorial, Proc. IEEE 76 (1988) 212. [7] S.C. Southward, C.J. Radcli!e, C.R. MacCluer, Robust nonlinear stick-slip friction compensation, ASME J. Dyn. Systems Meas. Control 113 (1991) 639. [8] M. Corless, Control of uncertain nonlinear systems, ASME J. Dyn. Systems Meas. Control 115 (1993) 362. [9] Q. Wu, A.B. Thornton-Trump, N. Sepehri, Lyapunov stability control of inverted pendulums with general base point motion, Int. J. Non-Linear Mech. 33 (1998) 801. [10] W. Hahn, Theory and Application of Lyapunov's Direct Method, Prentice-Hall, Englewood Cli!s, NJ, 1963. [11] Q. Wu, S. Onyshko, N. Sepehri, A.B. Thornton-Trump, On construction smooth Lyapunov functions for nonsmooth systems, Int. J. Control 69 (1998) 443. [12] Q. Wu, N. Sepehri, A.B. Thornton-Trump, M. Alexander, Stability and control of human trunk movement during walking, Comput. Methods Biomech Biomed. Eng. 1 (1998) 247. [13] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, NY, 1983. [14] S. Payandeh, A method for controlling robotic contact tasks, Robotica 14 (1996) 281.
On Lyapunov's stability analysis of non-smooth systems with applications to control engineering Q. Wu*, N. Sepehri Department of Mechanical and Industrial Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 Received 1 October 1999; received in revised form 1 June 2000; accepted 1 August 2000
Abstract The extension of Lyapunov's stability theory to non-smooth systems by Shevitz and Paden (Trans. Automat. Control 39 (1994) 1910) is modi"ed with the goal of simplifying the procedure for construction of non-smooth Lyapunov functions. Shevitz and Paden's extension is built upon Filippov's solution theory and Clarke's generalized gradient. One important step in using their extension is to determine the generalized derivative of a non-smooth Lyapunov function on a discontinuity surface, which involves the estimation of an intersection of a number of convex sets. Such a determination is complicated and can become unmanageable for many systems. We propose to estimate the derivative of a non-smooth Lyapunov function using the extreme points of Clarke's generalized gradient as opposed to the whole set. Such a modi"cation not only simpli"es the form, but also reduces the number of the convex sets involved in the estimation of the generalized derivative. This makes the stability analysis for some non-smooth systems practically easier. Three examples, including a mathematical system, a system with stick}slip friction compensator and an actuator having interaction with the environment, are used for demonstration. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Non-smooth dynamic systems, described by differential equations with discontinuous terms, appear naturally and frequently in the control "eld. Typical examples include systems with stick}slip friction, contact task controls, variable structure and optimal control systems. Two issues, involved in the stability analysis of such systems, are "rstly, classical solution theories to ordinary di!erential equations are no longer applicable. Secondly,
* Corresponding author. Tel.: #1-204-474-8843; #1-204-275-7507. E-mail address: [email protected] (Q. Wu). Professor.
fax:
Lyapunov's stability theory was developed for smooth systems. The "rst issue regarding the solution concept can be addressed using Filippov's solution theory [2,3]. This theory is one of the earliest and most conceptually straightforward approaches developed for analysis of non-smooth systems. It has been applied by many researchers [1,4}9]. Consequently, extensions of Lyapunov's second method to non-smooth dynamic systems, based on Filippov's solution theory, have been studied by Hahn [10], Paden and Sastry [5] and most recently by Shevitz and Paden [1] and Wu et al. [11]. In Shevitz and Paden's extension, non-smooth Lyapunov functions were constructed. The important step in constructing a non-smooth Lyapunov function is to examine the sign of its generalized derivative with respect to
0020-7462/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 8 6 - X
1154
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
time on the discontinuity surface. This is done by estimating the sign of each element of a convex set, which is the intersection of a large number of convex sets (see [1] for details). Such estimation requires heavy mathematical machinery and can become unmanageable for systems with more than two state variables or when the discontinuity surface is the intersection of several discontinuity surfaces. The di$culty in determining the generalized derivative may restrict the applications of the above-extended Lyapunov's stability theory. Wu et al. [11] extended Lyapunov's stability theory to non-smooth systems in that smooth Lyapunov functions were used and the conditions for the construction of smooth Lyapunov functions for non-smooth systems have been established. The signi"cance of this work is that it simpli"es the construction of Lyapunov functions and, therefore, makes the stability analysis of non-smooth systems easier. For example, the method has been used for stability control of a base-excited inverted pendulum [9] and its application to the modeling of human trunk movements [12]. However, as pointed out by the authors [11], for many nonsmooth systems, non-smooth Lyapunov functions are the only choices. In this paper, the extended Lyapunov's second method by Shevitz and Paden [1] is revisited with the objective of simplifying the determination of the generalized derivative of a Lyapunov function on a discontinuity surface. First, a brief review of Filippov's solution concept [2], Clarke's generalized gradient [13] and generalized derivative of non-smooth Lyapunov functions [1] is given. Next, a modi"cation to Shevitz and Paden's theorem is proposed which allows the use of only extreme points of Clarke's generalized gradient for the estimation of the above derivative. The modi"ed theorem is then applied to three examples from mathematical and engineering systems. It is shown that not only the form, but also the number of the convex sets involved in the estimation of the generalized derivative is simpli"ed. The signi"cance of the present study is that it makes the stability analysis for at least some nonsmooth systems practically easier in the sense that it does not require much advanced mathematical
machinery. Thus, more non-smooth systems can be studied under Lyapunov's stability theory.
2. Mathematical Review The extension of Lyapunov's stability theory by Shevitz and Paden [1] is based on Filippov's solution theory [2,3] and Clarke's generalized gradient concept [13]. Filippov's solution theory has been reviewed in the previous work by the authors [9]. Clarke's generalized gradient concept was discussed in [1]. Here, we only provide a summary of Filippov's solution concept, Clarke's generalized gradient and the generalized derivative of a nonsmooth function on a discontinuity surface. 2.1. Filippov's solution concept Consider the following vector di!erential equation: x "f (x, t),
(1)
where f: RL;RPRL is measurable and essentially locally bounded. A vector function, x, de"ned on the time interval (t , t ), is a Filippov's solution of Eq. (1) if: (i) it is absolutely continuous and, (ii) for almost all t3(t , t ) and for arbitrary '0, the vector dx(t)/dt belongs to the smallest closed convex set of RL containing all the values of the vector function f (x, t). Point x ranges the entire neighborhood of the point x (with t "xed) except for arbitrary sets of measure zero. More precisely, in the notation adopted above, dx(t) 3K[ f ](x, t), dt
(2)
where K[ f ](x, t), co f (;(x, )!N, t). (3) B I, K f ( ) ) ) is called Filippov's set. co denotes the closure of the convex hull and, N represents the intersection over all sets of Lebesgue measure zero. Note that since f (x, t) is not continuous, the standard existence and uniqueness theorems do not
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
apply. However, since the focus of this work is on stability proof, we assume that there exits a unique Filippov's solution to the non-smooth systems under study. Readers interested in detailed discussions on the existence and uniqueness of Filippov's solutions, are referred to the original literature [2,3].
Consider a locally Lipschitz function, < : RL;RPR. The generalized gradient of < at (x, t) is de"ned as <(x, t)" (4) colim <(x, t) (x , t )P(x, t), (x , t ) , , G G G G 4 where is the set of measure zero in which the 4 gradient of < is not de"ned. The gradient operator,, includes the derivative with respect to time (/t). In this de"nition, Lipschitz means Lipschitz in (x, t), i.e., discontinuities in t are not allowed. For notational brevity, if a function <(x, t) has no explicit t dependence, we shall adopt the convention of dropping the last component of <, which is zero. As an example, Clarke's generalized gradient for function <(x)" x is
<(x)"
x(0,
1,
x'0,
<(x : v)"lim sup (<(y#tv))!<(y)/t (see Ref. [1]). W V Rs Given the above conditions, <(x, t) is absolutely continuous and (d/dt) <(x, t) exists almost everywhere: d <(x, t)"
(6)
Set
2.2. Clarke's generalized gradient
!1,
1155
(5)
co!1, 1, x"0,
where co!1, 1 is equivalent to [!1, 1], i.e., any real number between !1 and 1 belongs to co!1, 1. 2.3. Derivative of non-smooth functions on discontinuity surfaces Let x be a Filippov's solution to x "f (x, t) on an interval containing t. Function <(x, t) : RL;RPR is Lipschitz. Additionally, <(x, t) is a regular function, i.e., (i) for all v, the usual one-sided directional derivative <(x : v) exists, and (ii) for all v, v, <(x : v)"<(x : v) where <(x, v) is the generalized directional derivative and is de"ned as
3. Stability theorem One of the important steps in constructing nonsmooth Lyapunov functions as outlined by Shevitz and Paden [1], is to prove that every element of the set
1156
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
where m is the number of the extreme points of Clarke's generalized gradient. Lyapunov's second method by Shevitz and Paden [1] is now modi"ed and presented below: Theorem. Let x "f (x, t) be essentially locally bounded and 03K[ f ](0, t) in a region QMx3RL x (r;t t )t(R. Also, let <: RL;RPR be a regular function satisfying that (i) V is Lipschitz continuous, positive and dexnite, and (ii) each element of
Discussion. Note that, if the convex set
are involved. Secondly, set
4. Demonstrative examples Three examples on stability analyses of a mathematical system and two control systems, are presented to demonstrate the applicability of the proposed theorem. Example 1. Stability analysis of a non-smooth system [1,5] Consider the following mathematical system:
x 1 !2 !1 sgn(x ) x "! 2 1 1 sgn(x ) . (11) x !1 !1 1 sgn(x ) It is to be proven that the above system is stable about the equilibrium point x , x , x 2" 0, 0, 0. Here we show how to establish the stability by applying the theorem outlined in Section 3. Eq. (9) is to be used for estimating the derivative of a non-smooth Lyapunov function on each discontinuity surface associated with Eq. (11). Here, we only consider the discontinuity surface S:"xo ; x "0, x "0, x O0, which is the intersection of the following two discontinuity surfaces: S :"(x , x , x ); x "0, x O0, x O0, S :"(x , x , x ); x O0, x "0, x O0. (12) Other discontinuity surfaces can be treated in a similar manner. We choose a Lyapunov function as <" x # x # x , which is continuous and positive de"nite. We now need to prove that each element of the convex set
(13)
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
Sets K[ f ] and
1#sgn(x ) !3#sgn(x ) !3!sgn(x ) , !1!sgn(x ) , 2!sgn(x ) !sgn(x )
3#sgn(x ) !1#sgn(x ) 1!sgn(x ) , 3!sgn(x ) !sgn(x ) !2!sgn(x )
and
1
1
sgn(x )
,
1
!1
!1
1
sgn(x )
sgn(x )
(14)
!1
,
!1
sgn(x ) (15)
Vector denotes the extreme points from
1157
Their intersection is K[ f ]"co[!5, !1]. G G For x (0, we have K[ f ]"co[!5, 3], K[ f ]"co[!5, !1], K[ f ]"co[!5, 7], K[ f ]"co[!5, 3]. The intersection of the above sets is
(18)
(19)
(20) K[ f ]"co[!5, !1]. G G From (18) and (20), it is seen that all the elements belonging to set
"
,
(21)
sgn(x ) where variables and take all the values between !1 and 1. Therefore,
1158
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
the same, using
(23)
where x is the location of the mechanism and x* is the location of the environment. The reaction force, r, due to the contact with the environment, is r"k (xA !xH#x ) when xH!x (xA . In the A L L non-contact region, i.e., when xH!x *xA , r"0. L A controller is proposed [14] which controls the position and force of the compliant object during non-contact and contact phases, respectively. The control signal, u(t), is given below:
u(t)"
u(t)"!k (x !xB)!k x , N B free motion (r"0), u(t)"!k k (x !xB)!k k x #rB, ND A BD A contact (r'0). (24)
In (24), xB is the desired position and rB is the desired contact force to be exerted on the environment upon contact. Constants k , k , k and N B ND k are the control gains. Assuming e "x !xB, BD in the non-contact region, the state-space model can be written as e "e , k k e "! N e ! B e . (25) m m In the contact region, however, we have e " x !xB"x !rB/k and the state-space model is A e "e k k (26) e "! A (k #1)e ! BD k e . ND m A m The dynamic system, shown in (25) and (26), is non-smooth. The existence and uniqueness of the solution have been proven, but not shown here. The discontinuity surface is: S :"e; e #xB#xA !xH"0, (27) C L where e"e , e 2. The state-space, , is divided into two parts, \ and >, by the discontinuity surface S : C
>:"e; e 'xH!xB!xA , L (28)
\:"e; e (xH!xB!xA . L When e3 >, the compliant mechanism is in contact mode and when e3 \, it is free to move. A Lyapunov function is now constructed as
1 k < " e # N e 2 2m (k #1)k !k A N (xH!xA !xB) # L 2m for e3 \,
1 k #1 <" < " e # ND k e 2 A 2m
Fig. 1. One-dimensional complaint mechanism.
for e3 >, 1 k #1 < " e # ND k e 2 A 2m for e3S . (29)
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
Note that V is continuous and positive through the entire region under the condition that (k #1)k !k '0. V is also de"nite since V"0 A N (i.e., < "0) if and only if e "0 and e "0. It is easy to show that
k #1 ND k (xH!xA !xB) A L m e
Accordingly, Filippov's set is K[ f ](e3S )"co C
.
k , k ! N (xH!xA !xB)! B e L m m
k #1 . (31) k ! ND k (xH!xA !xB)! BD k e A L m A m For the system under study, we have
k k #1 N ! ND k A m m
k k ;(xH!xA !xB)e ! B e , ! BD k e . L m m A
e k #1 k ! ND k (xH!xA !xB)! BD k e A L m m A
(33)
If we assume k "k k , it is obvious that B BD A
(34)
where x is the displacement of the mass m and F is the acting force. The friction model is de"ned as (35)
where
sgn(x )"
e k , k ! N (xH!xA !xB)! B e L m m
<K[ f ]"co !
F "F sgn(x )#F (1!sgn(x )),
k N (xH!xA !xB) 2 L m co e
;
Similarly, we have
(30)
e
e
1159
k k k #1 "co ! B e , N ! ND k A m m m
k ;(xH!xA !xB)e ! BD k e . L m A
1 0 .
!1
In Eq. (35), slip friction, F , is a positive constant and stick friction, F , is de"ned as
F>, F*F>'0, 1 1 F (F)" F, F\(F(F>, 1 1 F\, F)F\(0, 1 1
(32)
(36)
where F> and F\ are the positive and negative 1 1 limits on the static friction forces. Southward et al. [7] proposed the following non-linear control law for F to keep the system having a unique
1160
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
equilibrium point at x "0 and x "0: F"!k x !k x !k N B N 0, x 'x , & (x !x ), 0(x )x , & & x "0, (37) ; 0, (x !x ), x )x (0, * * 0, x (x , * where x "!F>/k and x "!F\/k . * 1 N & 1 N The state-space model shown in Eq. (34) together with Eqs. (35) and (37) describe a non-smooth dynamic system. The discontinuity is caused by (i) the model of the dry friction which is discontinuous with respect to the velocity, x , and (ii) the control law which is discontinuous at x "0. The discon tinuity surfaces are
S :"x; x "0, x O0, S :"x; x "0, x O0, (38) S :"x; x "0 and x "0, where x"x , x 2. The discontinuity surface S is the intersection of surfaces S and S . Southward et al. [7] suggested a non-smooth Lyapunov function to prove the stability of the above control system. The derivative of their Lyapunov function on the discontinuity surface, however, was replaced with a Dini derivative and was discussed in a somewhat illustrative manner. In this paper, the derivative of the Lyapunov function on the discontinuity surface is replaced with Clarke's generalized gradient and is estimated quantitatively. The special case for the discontinuity surface of S is only studied. Other cases of S and S can be investigated in a similar manner. The Lyapunov function candidate is shown below: 1 1 <(x , x )" k x # k x 2 N 2 B 1 k x , x 'x , & 2 N & 1 k x x ! x , 0)x )x , N & 2 & # 1 k x x ! x , x )x )0, N * 2 * 1 k x , x (x . * 2 N * (39)
Following the framework described in Section 3, we need to prove that each element of the convex set
k x k x N * , N & mx mx
,
(40)
K[ f ](x3S )" co
x , 1 (!k x !k x !F sgn(x )) B N * m
x 1 (!k x !k x !F sgn(x )) B N & QJGN m
.
(41)
We only need to prove that all the elements of the convex set
"co k x mx N *
x ; 1 , (!k x !k x !F sgn(x )) B N * m k x mx N *
x ; 1 (!k x !k x !F sgn(x )) B N & m "co[!k x !F x , B
(42)
!k x !F x !k x (x !x )]. B N & * Similarly, K[ f ]"co[!k x !F x !k x B N "(x !x ), !k x !F x ]. (43) * & B We now prove that
Q. Wu, N. Sepehri / International Journal of Non-Linear Mechanics 36 (2001) 1153}1161
intersection of sets K[f] K[ f ] is
1161
References [1] D. Shevitz, B. Paden, Lyapunov stability theory of nonsmooth systems, IEEE Trans. Automat. Control 39 (1994) 1910. [2] A.F. Filippov, Di!erential equations with discontinuous right-hand side, Math. Sb. 51 (1960) 99 (English Translation: Am. Math. Soc. Translations 42 (1964) 191). [3] A.F. Filippov, Di!erential equations with second members discontinuous on intersecting surfaces, Di!erentsial'nye Uravneniya 15 (1979) 1814 (English Translation: Di!erential Equations 15 (1980) 1292). [4] J.J. Slotine, S.S. Sastry, Tracking control of nonlinear systems using sliding surfaces with application to robot manipulators, Int. J. Control 38 (1983) 465. [5] B.E. Paden, S.S. Sastry, Acalculus for computing Filippov's di!erential inclusion with application to the variable structure control of robot manipulators, IEEE Trans. Circuits Systems 34 (1987) 73. [6] R.A. DeCarlo, S.H. Zak, P. Matthews, Variable structure control of nonlinear multivariable systems: a tutorial, Proc. IEEE 76 (1988) 212. [7] S.C. Southward, C.J. Radcli!e, C.R. MacCluer, Robust nonlinear stick-slip friction compensation, ASME J. Dyn. Systems Meas. Control 113 (1991) 639. [8] M. Corless, Control of uncertain nonlinear systems, ASME J. Dyn. Systems Meas. Control 115 (1993) 362. [9] Q. Wu, A.B. Thornton-Trump, N. Sepehri, Lyapunov stability control of inverted pendulums with general base point motion, Int. J. Non-Linear Mech. 33 (1998) 801. [10] W. Hahn, Theory and Application of Lyapunov's Direct Method, Prentice-Hall, Englewood Cli!s, NJ, 1963. [11] Q. Wu, S. Onyshko, N. Sepehri, A.B. Thornton-Trump, On construction smooth Lyapunov functions for nonsmooth systems, Int. J. Control 69 (1998) 443. [12] Q. Wu, N. Sepehri, A.B. Thornton-Trump, M. Alexander, Stability and control of human trunk movement during walking, Comput. Methods Biomech Biomed. Eng. 1 (1998) 247. [13] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, NY, 1983. [14] S. Payandeh, A method for controlling robotic contact tasks, Robotica 14 (1996) 281.