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A new approach to control of dynamic systems with unilateral constraints
14th World Congress oflFAC
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH ...
C-2a-06-4
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH
UNILATERAL CONSTRAINTS
Boris M. Miller and Joseph BentslUan
Institute for Information Transmission Problems GSP-4, B. Karetny Per. 19, 101447 Moscow, Russia
Department of 1VIechanical and Industrial Engineering University of Illinois at
Urbana-Champaign 1206 W. Green St. Urbana IL 61801 USA
Abstract: The problem of control for dynamic systems subjected to unilateral constraints is considered. The new approach to the description of generalized (discontinuous) solution based on the idea of discontinuos time transformation was suggesed. This approach gives the opportunity to describe the generalized solution by the differential equtions with a measure and to reduce the original optilllization problem to some problem of nonsmooth optimization. As a result the optimality conditions in the special form of maximum principle can be obtained. CopYrl~ht© 1999 IFAC Keywords: dynamic systems, unilateral constraints, impulses, optimal control, generalized solutions
1. INTRODUCTION
The dynamic systems with constraints can be frequently treated as systems characterized by continuos and discrete behavior. The motion of such systems can be divided into regular and singular parts, i.e., continuous and jumping, respectively. These systems are very typical for various mechanical applications, where the discretecontinuous modes of motions could arise because
of shocks and friction. There has been a significant prog-ress in this area, including the development of the rigorous mathematical framework for the description of these systems and preliminary formulations of the procedures for synthesis of control laws for them. However , the common mathematical feature of these class of systems is the presence of singularities, which manifest themselves in: discontinuities and non smoothness in system motion, jumps in system dimension, the lack of the continuous dependence on initial conditions
1214
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH .. .
and nonuniqiueness of solution of equations of motion (see, for example, (Brogliato, 1996; Jean and M ore au, 1991). Traditionally t.he control of' such systems has been exerted eit.her during the nonsingular phase of the system motion or during the singularity phase, which was induced by the control action itself and did not naturally exist in the system (Miller , 1993; Miller 1996). However, in dynamic systems with constraints the impulsive actions arise due to the interaction of the system with obstacles, therefore, they are of a type of impulsive feedback form. Theory of impulsive controls, which was developed mainly for open loop controls, cannot be directly applied to systems, where impulsive actions arise, when the system achieves the boundary of some set (obstacle). It is also known that unilateral constrains must be weakly violated at the points of impact giving rise to fine spatial structure of the system trajectory in the vicinity of the point of jump. Due to these fine structures on one hand and the slow dynamics of the system to be controJIed on t he other hand the resulting controlled system · possesses the multiscale time-space dynamics. The proper tool for the description of the resulting system is then the multi-resolution time-space analysis such as that provided by a wavelet decomposition or more specifically by a discontinuous time transformation method in impulsive control (Miller, 1993; Miller 1996). The new a.pproach is to consider a jump as a result of some "fictitious motion" along the paths of some auxiliary system, which provides a model of "fast motion" and describes the jump, arising in the motion of dynamic system, in terms of some shift operator. This approach bases on the representation of robust hybrid system, which was obtained in (Miller , 1997), where discret.e-continuos systems are treated as systems with impulsive inputs. However, if we consider these systems as ones with impulsive actions in feedback form, it b ecomes necessary to find a more general mathematical framework , than for standard problems with impulse controls. The main approach is based on the results obtained in (Miller, 1998) and (Bentsmall and Miller , 1998) , where the problem statement and results concerning the generalized solution representation and the existence of the optimal generalized solution were given. The main goal of this paper is to obtain the necessary optimality conditions in the form of generalized maximum principle.
14th World Congress oflFAC
Consider the evolution of discrete-continuous dynamical system with unilateral constraint, whose behavior be described on some interval [0 , T] by variable X(t) E Rn, which satisfies the differential equation X(t)
= F(X(t) , u(t»,
(1)
with given initial condition X(O) = Xo E Rn. The actions of unilateral constraint of the form G(X (t» ~ 0, where GO is continu08 and continuously differentiable, will be described by following intermediate conditions
X(T;)
= X(Ti-) + \If(X(T; -
)),
(2)
which are given for some sequence of instants {Ti ' i = 0, ... , N}, N ~ =, satisfying the recurrence conditions TO
=0 inf{Ti_l 00,
~
T: G(X(t - ) = O} ,
jf the appropria.te set is empty.
In equation (2) X(T;-)
= limX(t), ttT,
(3) and Ti is the
sequence of instants when the system states can change discontinuously on the boundary of constraint. So, the state of system changes continuously in halfintervals [0, TIl, ... h-l' Ti), .. , [TN' T) , and undergoes a sudden change at every instant rj , whose value, due to equation (2), depends on the state preceding the jump. We suppose that control variable in (1) 1..1
(4)
E U C R m,
where U is some compact set, and function F(X, u) be continuous with respect to an variables and continuously differentiable with respect to X. To be sure that solution of (1) is continuable to the right we need some additional assumptions concerning the functions W(X) and G(X). So we suppose that X(rd X(T;-) + \)(X(T;-)) is the result of the action of shift operator along the paths of differential equation
=
y(s)
= B(y(s»
,
sE [0, cc)
(5)
=
with initial condition y(O) X(T;-). Therefore, if (x, s) is the general solution of (5) with initial condition y(O) x, then
=
X{T;) = X(Ti -)
+ "iP"(X(T;-)
=
(6)
2. PROBLEM STATEMENT
1215
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH ...
14th World Congress ofIFAC
and such that the integral constraints
where
s* (X(r;)
f
= inf{ s > 0 : G(4)(X(Ti-)' s) = A},
Tl
(7) and on the interval (O,.~*(X(Td) we have the relation G{4>(X(ri-),s) > O. We assume also that B(y) be continuously differentiable with respect to y. The typical path of the hybrid system is shown in figure 1, where r is the instant of the hitting the boundary of the set G(X) SO. The solid line shows the real motion of the system, "..-hich is processed in ordinary time scale. The fictitious motion, which corresponds to the fast phase of the impulsive control action is shown by dashed line. The optimization problem to be considered is the minimization of performance criterion
J[X(·), tI(.)] ::::
(8)
with some continuously differentiable function
a(s)G+(y(s))ds = 0,
o
(ll)
f
Tl
(1 - a(s))G- (y(s))ds
= 0,
o
where
= max{G(y), O},
G+(y)
G-(y) = min{G(y), a}.
The problem will be considered on nonfixed interval [0, Td, such that Tl < = : 7](Tr) = T with performance criterion
where .po is the same as in (8). Theorem 1. Suppose that the set F(X, U) be convex for any X E Rn, the set of admissible controls of auxiliary problem is non empty, and the set of admissible T11 such that 1](TI) = T is uniformly bounded_ Then the auxiliary problem has the optimal solution {yOO,1]Q(-),a°(-),u~(-)} and the optimal generalized solution of the original problem satisfies the equation
dXO(t)
= F(XO(t), uO(t))dt + B(XO(t»)dp"(dt}+
L:[4>(XO(7"-),/l({r})) - XO(7"-)]J(t -7")dt, T
3. EXISTENCE THEOREM AND
(12)
PROBLEM REDUCTION
satisfying the constraint G(XO(t)) ::::: 0,
Follo...-i.ng the line of paper (Miller, 1998) define on the interval [0, T l ] the auxiliary system
t E [0, TJ.
for any
(1:3)
with r~(t}
y(s)
= a(s)F(y(s), u(t](s))) + (1 -
a{s))B(y(s)),
uO(i)
= u~(rO(i),
.u{[0,
in =
J
(1- (YO(s))ds,
o
1j(s) = 1'1«8) (9)
where
with initial conditions
y(O)
= Xo,
fO(l) = iuf{s : 1]°(s)
7}(O)
> t},
XO(t) = yO(rO(t)).
= o.
Consider the following auxiliary problem for system (9) with controls {a, tld, satisfying
4. NECESSARY OPTIMALITY CONDITIONS
0:(8) E (0,11, (10)
ut(s) E U
a. e.
on
[0, T I ],
and such that y(-) satisfies (11) and T](Td = T,
The auxiliary problelll belongs to a class of non~ smooth optilllization problems due to the nondifferentiability of functions G+ and G-. However, by applying the methods recently obtained for
1216
Copyright 1999 IF AC
ISBN: 008 0432484
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH ...
14th World Congress ofIFAC
G(X(t» > 0
Figure 1: Path of discrete-continuous system
4. adjoint variable p~ (-), which satisfies the sys-
nonsmooth problems (see, for example (Clarke, 1983), it becomes possible to derive necessary optimality conditions in the form of generalized maximum principle. To formulate the result introduce the notation
tem of equations
p~(t) = -Ao'f'~JX(T)+ T
/ <
p~(T), F~(XO(T), UO(T)) > dT+
T
/ < p~(T), B~(XO(T) > d,ue(T)+ t
and define the subsets DI',D~, and D~, which are supports of measures ,u(dt) , /J-C(dt), and pd(dt), respectively. Theorelll 2. Suppose that {XO(.), uO( .), .uU} be the optimal path, control and corresponding measure, localized on the set {t : O(XO(t) = O} in the problem (1)-(8). Then there exist the set of elements that are non equal to zero simultaneously:
(14)
T
Ji/'~,lG~(XO(T»d>'l(T)+ t T
/
i/>~,:P~(XD(T»d>'2(T)+
L
1. the set of constants AO ~ 0, ~~, ~~,l' ~'~,2;
T>
2. the nonnegative measures '\l(dt) and .\2(dt) which are localized on the set {t : O(XO (t) = O} and being absolutely continuos with respect to Lebesgue measure and measure ,f(dt), respectively;
where
3. the set of nonnegative measures ).Hds), defined for every T E D~ on the corresponding interval [O,,u{ T}], being absolutely continuos with respect to the Lebesgue measure and localized on the sets {w E [O,,u{ T }] : G(-I>(XO(r-),w) = O)};
Ll.P~(T),
t, T E D~
Ap~(T)
be defined by relations
(15)
and PT,,,,(S) satisfies on the interval [0, p{r}J
1217
Copyright 1999 IF AC
ISBN: 008 0432484
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH '"
14th World Congress oflFAC
the follmving equation
REFERENCES
p.,.,x(s) = p~(r)+ p{ T)
f
< PT,x(w),B:(IP(XO(r-),w)) > dw+
I'{T}
f ~'. ,2G~(IP(XO(r-),w)d5.Hw). (16)
such that the following inequalities (generalized maximum principle) are valid on the appropriate subsets, namely:
Bentsman J. and B. M. Miller (1998). -Control of dynamic systems with unilateral constraints and differential equations with measures. In: Proceedings of 4th IFAC Workshop on Nonlinear Control Systems Design, NOLCOS'98. 2. 411 - 416. University of Twente, Enschede, The Netherlands. Brogliato B. (1996). Nonsmooth Impact Mechanics. Models, Dynamics and Control. Lecture Notes in Control and Information Sciences. 220. Springer-Verlag. Clarke F. H. (1983).
Optimization and
Nonsmooth Analysis, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore.
max
1io(p~(t),1P~,O,XO{t),u) ~
uE U
(17)
1il(p~(t), ?/Jj.I,2, XO(t),
which is valid a .e. on the set {t E [0, T] \ Dj.I};
max 1io(p~(t), tP~, 0, XO(t), u) ~ uEU 1il(p~(t),O,XO(t))
(18)
= 0
a.e . with respect to measure /-l(dt) on the set {t E D~};
and
max 1io (PT,X (w), ~~ , 1f1',1, IP(XO(r- ),w), u) ::; uEU
for variables satisfying (16), r E
Dt.
(19)
5. CONCLUSION So the new approach to the description of dynamic systems with unilateral constraints had been developed. The main achievement is the new concept of generalized solution, which can be obtained by time substitution in the solution of standard system, described by ordinary differential equations. This approach give the opportunity to prove the existence theorerrl and to reduce the originally singular control problem to a regular one and to derive the optimality conditions.
Jean M. and J.J. Moreau (1991). Dynamics of elastic or rigid bodies with frictional contact; numerical methods. Publications of Laboratory of :Mechanics and Acoustics Marseille,' April, 124 Miller B. M. (1993). Method of discontinuous time change in problems of control for impulse and discrete-continuous systems. Automation and Remote Control. 54. No 12. 1727-1750. Miller B. M. (1996). The generalized solutions of nonlinear optimization problems with impulse control. SL4.II'f J. Control and Optimization. 34, No 4. 1420-1440. Miller B. M. (1997). Representation of robust and non-robust solutions of nonlinear discrete-continuous systems. In: Proceedings of International Workshop on Hybrid und Real Time Systems. HART'97. Ed.: Oded Maler. Grenoble, March 28-28, France. Miller B. M. (1998). Optimization of generalized solutions of nonlinear hybrid (discrete-continuous) systems. In: Proceedings of First International Workshop on Hybrid Systems: Computation and Control. HSCC '98. Eds.: Tomas Henzinger and Sankar Sastry. Lecture Notes in Computer Science. 1386. 334-345. Berkeley, California, USA.
1218
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH ...
C-2a-06-4
Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH
UNILATERAL CONSTRAINTS
Boris M. Miller and Joseph BentslUan
Institute for Information Transmission Problems GSP-4, B. Karetny Per. 19, 101447 Moscow, Russia
Department of 1VIechanical and Industrial Engineering University of Illinois at
Urbana-Champaign 1206 W. Green St. Urbana IL 61801 USA
Abstract: The problem of control for dynamic systems subjected to unilateral constraints is considered. The new approach to the description of generalized (discontinuous) solution based on the idea of discontinuos time transformation was suggesed. This approach gives the opportunity to describe the generalized solution by the differential equtions with a measure and to reduce the original optilllization problem to some problem of nonsmooth optimization. As a result the optimality conditions in the special form of maximum principle can be obtained. CopYrl~ht© 1999 IFAC Keywords: dynamic systems, unilateral constraints, impulses, optimal control, generalized solutions
1. INTRODUCTION
The dynamic systems with constraints can be frequently treated as systems characterized by continuos and discrete behavior. The motion of such systems can be divided into regular and singular parts, i.e., continuous and jumping, respectively. These systems are very typical for various mechanical applications, where the discretecontinuous modes of motions could arise because
of shocks and friction. There has been a significant prog-ress in this area, including the development of the rigorous mathematical framework for the description of these systems and preliminary formulations of the procedures for synthesis of control laws for them. However , the common mathematical feature of these class of systems is the presence of singularities, which manifest themselves in: discontinuities and non smoothness in system motion, jumps in system dimension, the lack of the continuous dependence on initial conditions
1214
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH .. .
and nonuniqiueness of solution of equations of motion (see, for example, (Brogliato, 1996; Jean and M ore au, 1991). Traditionally t.he control of' such systems has been exerted eit.her during the nonsingular phase of the system motion or during the singularity phase, which was induced by the control action itself and did not naturally exist in the system (Miller , 1993; Miller 1996). However, in dynamic systems with constraints the impulsive actions arise due to the interaction of the system with obstacles, therefore, they are of a type of impulsive feedback form. Theory of impulsive controls, which was developed mainly for open loop controls, cannot be directly applied to systems, where impulsive actions arise, when the system achieves the boundary of some set (obstacle). It is also known that unilateral constrains must be weakly violated at the points of impact giving rise to fine spatial structure of the system trajectory in the vicinity of the point of jump. Due to these fine structures on one hand and the slow dynamics of the system to be controJIed on t he other hand the resulting controlled system · possesses the multiscale time-space dynamics. The proper tool for the description of the resulting system is then the multi-resolution time-space analysis such as that provided by a wavelet decomposition or more specifically by a discontinuous time transformation method in impulsive control (Miller, 1993; Miller 1996). The new a.pproach is to consider a jump as a result of some "fictitious motion" along the paths of some auxiliary system, which provides a model of "fast motion" and describes the jump, arising in the motion of dynamic system, in terms of some shift operator. This approach bases on the representation of robust hybrid system, which was obtained in (Miller , 1997), where discret.e-continuos systems are treated as systems with impulsive inputs. However, if we consider these systems as ones with impulsive actions in feedback form, it b ecomes necessary to find a more general mathematical framework , than for standard problems with impulse controls. The main approach is based on the results obtained in (Miller, 1998) and (Bentsmall and Miller , 1998) , where the problem statement and results concerning the generalized solution representation and the existence of the optimal generalized solution were given. The main goal of this paper is to obtain the necessary optimality conditions in the form of generalized maximum principle.
14th World Congress oflFAC
Consider the evolution of discrete-continuous dynamical system with unilateral constraint, whose behavior be described on some interval [0 , T] by variable X(t) E Rn, which satisfies the differential equation X(t)
= F(X(t) , u(t»,
(1)
with given initial condition X(O) = Xo E Rn. The actions of unilateral constraint of the form G(X (t» ~ 0, where GO is continu08 and continuously differentiable, will be described by following intermediate conditions
X(T;)
= X(Ti-) + \If(X(T; -
)),
(2)
which are given for some sequence of instants {Ti ' i = 0, ... , N}, N ~ =, satisfying the recurrence conditions TO
=0 inf{Ti_l 00,
~
T: G(X(t - ) = O} ,
jf the appropria.te set is empty.
In equation (2) X(T;-)
= limX(t), ttT,
(3) and Ti is the
sequence of instants when the system states can change discontinuously on the boundary of constraint. So, the state of system changes continuously in halfintervals [0, TIl, ... h-l' Ti), .. , [TN' T) , and undergoes a sudden change at every instant rj , whose value, due to equation (2), depends on the state preceding the jump. We suppose that control variable in (1) 1..1
(4)
E U C R m,
where U is some compact set, and function F(X, u) be continuous with respect to an variables and continuously differentiable with respect to X. To be sure that solution of (1) is continuable to the right we need some additional assumptions concerning the functions W(X) and G(X). So we suppose that X(rd X(T;-) + \)(X(T;-)) is the result of the action of shift operator along the paths of differential equation
=
y(s)
= B(y(s»
,
sE [0, cc)
(5)
=
with initial condition y(O) X(T;-). Therefore, if (x, s) is the general solution of (5) with initial condition y(O) x, then
=
X{T;) = X(Ti -)
+ "iP"(X(T;-)
=
(6)
2. PROBLEM STATEMENT
1215
Copyright 1999 IFAC
ISBN: 0 08 043248 4
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH ...
14th World Congress ofIFAC
and such that the integral constraints
where
s* (X(r;)
f
= inf{ s > 0 : G(4)(X(Ti-)' s) = A},
Tl
(7) and on the interval (O,.~*(X(Td) we have the relation G{4>(X(ri-),s) > O. We assume also that B(y) be continuously differentiable with respect to y. The typical path of the hybrid system is shown in figure 1, where r is the instant of the hitting the boundary of the set G(X) SO. The solid line shows the real motion of the system, "..-hich is processed in ordinary time scale. The fictitious motion, which corresponds to the fast phase of the impulsive control action is shown by dashed line. The optimization problem to be considered is the minimization of performance criterion
J[X(·), tI(.)] ::::
(8)
with some continuously differentiable function
a(s)G+(y(s))ds = 0,
o
(ll)
f
Tl
(1 - a(s))G- (y(s))ds
= 0,
o
where
= max{G(y), O},
G+(y)
G-(y) = min{G(y), a}.
The problem will be considered on nonfixed interval [0, Td, such that Tl < = : 7](Tr) = T with performance criterion
where .po is the same as in (8). Theorem 1. Suppose that the set F(X, U) be convex for any X E Rn, the set of admissible controls of auxiliary problem is non empty, and the set of admissible T11 such that 1](TI) = T is uniformly bounded_ Then the auxiliary problem has the optimal solution {yOO,1]Q(-),a°(-),u~(-)} and the optimal generalized solution of the original problem satisfies the equation
dXO(t)
= F(XO(t), uO(t))dt + B(XO(t»)dp"(dt}+
L:[4>(XO(7"-),/l({r})) - XO(7"-)]J(t -7")dt, T
3. EXISTENCE THEOREM AND
(12)
PROBLEM REDUCTION
satisfying the constraint G(XO(t)) ::::: 0,
Follo...-i.ng the line of paper (Miller, 1998) define on the interval [0, T l ] the auxiliary system
t E [0, TJ.
for any
(1:3)
with r~(t}
y(s)
= a(s)F(y(s), u(t](s))) + (1 -
a{s))B(y(s)),
uO(i)
= u~(rO(i),
.u{[0,
in =
J
(1- (YO(s))ds,
o
1j(s) = 1'1«8) (9)
where
with initial conditions
y(O)
= Xo,
fO(l) = iuf{s : 1]°(s)
7}(O)
> t},
XO(t) = yO(rO(t)).
= o.
Consider the following auxiliary problem for system (9) with controls {a, tld, satisfying
4. NECESSARY OPTIMALITY CONDITIONS
0:(8) E (0,11, (10)
ut(s) E U
a. e.
on
[0, T I ],
and such that y(-) satisfies (11) and T](Td = T,
The auxiliary problelll belongs to a class of non~ smooth optilllization problems due to the nondifferentiability of functions G+ and G-. However, by applying the methods recently obtained for
1216
Copyright 1999 IF AC
ISBN: 008 0432484
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH ...
14th World Congress ofIFAC
G(X(t» > 0
Figure 1: Path of discrete-continuous system
4. adjoint variable p~ (-), which satisfies the sys-
nonsmooth problems (see, for example (Clarke, 1983), it becomes possible to derive necessary optimality conditions in the form of generalized maximum principle. To formulate the result introduce the notation
tem of equations
p~(t) = -Ao'f'~JX(T)+ T
/ <
p~(T), F~(XO(T), UO(T)) > dT+
T
/ < p~(T), B~(XO(T) > d,ue(T)+ t
and define the subsets DI',D~, and D~, which are supports of measures ,u(dt) , /J-C(dt), and pd(dt), respectively. Theorelll 2. Suppose that {XO(.), uO( .), .uU} be the optimal path, control and corresponding measure, localized on the set {t : O(XO(t) = O} in the problem (1)-(8). Then there exist the set of elements that are non equal to zero simultaneously:
(14)
T
Ji/'~,lG~(XO(T»d>'l(T)+ t T
/
i/>~,:P~(XD(T»d>'2(T)+
L
1. the set of constants AO ~ 0, ~~, ~~,l' ~'~,2;
T>
2. the nonnegative measures '\l(dt) and .\2(dt) which are localized on the set {t : O(XO (t) = O} and being absolutely continuos with respect to Lebesgue measure and measure ,f(dt), respectively;
where
3. the set of nonnegative measures ).Hds), defined for every T E D~ on the corresponding interval [O,,u{ T}], being absolutely continuos with respect to the Lebesgue measure and localized on the sets {w E [O,,u{ T }] : G(-I>(XO(r-),w) = O)};
Ll.P~(T),
t, T E D~
Ap~(T)
be defined by relations
(15)
and PT,,,,(S) satisfies on the interval [0, p{r}J
1217
Copyright 1999 IF AC
ISBN: 008 0432484
A NEW APPROACH TO CONTROL OF DYNAMIC SYSTEMS WITH '"
14th World Congress oflFAC
the follmving equation
REFERENCES
p.,.,x(s) = p~(r)+ p{ T)
f
< PT,x(w),B:(IP(XO(r-),w)) > dw+
I'{T}
f ~'. ,2G~(IP(XO(r-),w)d5.Hw). (16)
such that the following inequalities (generalized maximum principle) are valid on the appropriate subsets, namely:
Bentsman J. and B. M. Miller (1998). -Control of dynamic systems with unilateral constraints and differential equations with measures. In: Proceedings of 4th IFAC Workshop on Nonlinear Control Systems Design, NOLCOS'98. 2. 411 - 416. University of Twente, Enschede, The Netherlands. Brogliato B. (1996). Nonsmooth Impact Mechanics. Models, Dynamics and Control. Lecture Notes in Control and Information Sciences. 220. Springer-Verlag. Clarke F. H. (1983).
Optimization and
Nonsmooth Analysis, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, Singapore.
max
1io(p~(t),1P~,O,XO{t),u) ~
uE U
(17)
1il(p~(t), ?/Jj.I,2, XO(t),
which is valid a .e. on the set {t E [0, T] \ Dj.I};
max 1io(p~(t), tP~, 0, XO(t), u) ~ uEU 1il(p~(t),O,XO(t))
(18)
= 0
a.e . with respect to measure /-l(dt) on the set {t E D~};
and
max 1io (PT,X (w), ~~ , 1f1',1, IP(XO(r- ),w), u) ::; uEU
for variables satisfying (16), r E
Dt.
(19)
5. CONCLUSION So the new approach to the description of dynamic systems with unilateral constraints had been developed. The main achievement is the new concept of generalized solution, which can be obtained by time substitution in the solution of standard system, described by ordinary differential equations. This approach give the opportunity to prove the existence theorerrl and to reduce the originally singular control problem to a regular one and to derive the optimality conditions.
Jean M. and J.J. Moreau (1991). Dynamics of elastic or rigid bodies with frictional contact; numerical methods. Publications of Laboratory of :Mechanics and Acoustics Marseille,' April, 124 Miller B. M. (1993). Method of discontinuous time change in problems of control for impulse and discrete-continuous systems. Automation and Remote Control. 54. No 12. 1727-1750. Miller B. M. (1996). The generalized solutions of nonlinear optimization problems with impulse control. SL4.II'f J. Control and Optimization. 34, No 4. 1420-1440. Miller B. M. (1997). Representation of robust and non-robust solutions of nonlinear discrete-continuous systems. In: Proceedings of International Workshop on Hybrid und Real Time Systems. HART'97. Ed.: Oded Maler. Grenoble, March 28-28, France. Miller B. M. (1998). Optimization of generalized solutions of nonlinear hybrid (discrete-continuous) systems. In: Proceedings of First International Workshop on Hybrid Systems: Computation and Control. HSCC '98. Eds.: Tomas Henzinger and Sankar Sastry. Lecture Notes in Computer Science. 1386. 334-345. Berkeley, California, USA.
1218
Copyright 1999 IFAC
ISBN: 0 08 043248 4