Teodorescu P.P. Mechanical Systems, Volume III: Analytical Mechanics

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MECHANICAL SYSTEMS, CLASSICAL MODELS

402

21.5 Control Systems

The study of dynamical systems can be effected in closed connection to the study of

control systems; the methodology linked to these latter systems and – especially – the

aspects of optimization of the trajectories can give precious information for the

mechanical systems. All these reasons make necessary and useful the presentation of

some elements concerning the control systems and the optimal trajectories.

21.5.1 Control Systems

After some preliminary considerations concerning the control systems, one presents

Bolza’s problem, as well as some particular cases; we mention that various forms of the

conditions of minimum are put in evidence.

21.5.1.1 Preliminary Considerations

A control system is a mathematical object represented by a system of equations of

state in the form

′

12 12

( , ,..., , , ,..., ; ), 1,2,...,

xfxx xuu uti n,

(21.5.1)

where

=( ), 1,2,...,

xt i n, are variables of state (eventually, generalized co-ordinates

in the space of configurations or in the phase space), while

=(), 1,2,...,

ut j m are

control parameters;

∈

[,]ttt for all these quantities. The fundamental problem

consists in the determination of the control parameters so that a certain functional

(called performance index)

()()

∫

111

(),(); (),(); d

I G xt ut t f xt ut t t,

(21.5.2)

where ≡

() { (), (),..., ()}

ut u t u t u t is the control function, while

≡

( ) { ( ), ( ),..., ( )}

xt x t x t x t is a function of state, does realize a maximum or a

minimum.

For instance, if a control function so that the distance between two points be

travelled through in a minimal time is searched, then we must have

()

111

(),(); 0Gxt ut t ,

()

(), (); 1fxtutt .

We must notice that the state variable cannot be completely arbitrary (e. g., the

velocity cannot pass beyond certain limits); to simplify the problem, we assume that the

domain of definition of the vector

x is

\ . In that concerns the control vector, there

can appear – as well – restrictions (e. g., if

123

,,uuu are direction cosines we have

++=

123

222

1uuu ), superior or inferior limits, discontinuities etc. If U is the domain

of definition, independent on

t , of the vector u , then we have ∈⊆

uU \ . We say

that the functional

u is an admissible control function if:

(i) the function

U is definite and piecewise continuous on the interval

[,]tt;

(ii)

∈uU;

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

403

(iii) for any point of discontinuity we have

→<

=−=

( ) ( 0) lim ( )

uu ut

ττ

ττ ;

(iv)

()ut is a continuous function at the points

t and

t .

≡

U \ , then we say that ()ut is a non-constrained admissible control function.

To the considered control system we associate the initial conditions

( ) , 1,2,...,

xt x i n

(21.5.1')

If the functions

(;)

fxt satisfy the conditions of the theorem of existence and

uniqueness, then the system (21.5.1), (21.5.1') has a unique solution, even if the control

function

()ut is discontinuous. The solution of this system of equations, corresponding

to a given control function

u , will be called trajectory; to an optimal control will

correspond an optimal trajectory.

21.5.1.2 Bolza’s Problem

The most general problem in the calculus of variations is Bolza’s problem, which –

for control systems – asks the determination of the control function

= ()uut on the

interval

[,]tt, realizing a minimum for

[]

()

∫

(), (); (), (); d

Gxt ut t f xt ut t t

(21.5.3)

and which, at any moment, satisfies the equations (equations of motion and constraint

relations)

−== ==<( , ; ) 0, 1,2,..., , ( , ; ) 0, 1,2,...,

ii j

x f xut i nh xut j p m ,

(21.5.3')

and the end conditions

() ()

== ==

00 0

11 1

( ); 0, 1,2,..., , ( ); 0, 1,2,...,

gxtt k rgxtt l r

(21.5.3'')

One assumes that

Gf f and

h are continuous functions with respect to all

arguments, having partial derivatives of a sufficiently great order on an open set; as

well, the variety

S with −

nr dimensions and the variety

S with −

dimensions, defined by the equations (21.5.3''), are supposed to be smooth.

≡

(,;) 0fxut , Bolza’s problem is a problem of Mayer; as well, if

()( )

000

111

(),(); ( ),( );Gxt ut t Gxt ut t , then this one is a problem of Lagrange.

Formulated as an optimization problem, Bolza’s problem can be studied by means of

Bellmann’s principle of optimality (if a trajectory is optimal, then any portion of it is

optimal).

MECHANICAL SYSTEMS, CLASSICAL MODELS

404

We introduce the supplementary variable of state

+1n

x and the supplementary

control variable

+1m

u defined by

+++ ++ +

== =−=

111 11 1

,, (,;)0

nnn mp n

xffuhxGxut ;

(21.5.4)

we denote

≡

{, }

xxx and

≡

{, }

uuu . Bolza’s problem is thus reduced to a

problem of Lagrange type in which one must determine the control function

()

which realizes a minimum for the functional

=++

∑

∫

(,;; )d,

ˆˆ

Fxut tF u f hμμ,

(21.5.5)

with

≡

{, }

μμμ , ≡

{ , ,..., }

μμμ μ, satisfying – at any moment – the equations

−==+ ==+( , ; ) 0, 1,2,..., 1, ( , ; ) 0, 1,2,..., 1

ˆˆ ˆˆ

ii j

x f xut i n h xut j p ,

(21.5.5')

with the restrictions (21.5.3'') at the ends.

Denoting

()

∫

(); min ( , ;; )d

ˆˆˆ

xt t Fxut tΓμ,

(21.5.6)

one obtains

(;) (,;; ) 0

ˆˆˆ

xt Fxut

Γμ;

(21.5.6')

moreover, it results

[]

⎡⎤

+=∈

⎢⎥

⎣⎦

min ( ; ) ( , ; ; ) 0, ,

ˆˆˆ

xt Fxut t t t

Γμ .

(21.5.7)

Because

depends directly on time, as well as by means of the variables

x , which

satisfy the equations (21.5.3'), we can state that we have

[]

⎛⎞

∂∂

++ + + = ∈

⎜⎟

∂∂

⎝⎠

∑∑

min 0, ,

jj i

uf h f ttt

ΓΓ

(21.5.7')

along the optimal trajectories. This condition leads to

∂∂

++ + + =

∂∂

∑∑

jj i

uf h f

ΓΓ

(21.5.8)

∂

∂∂

∂

++ ==

∂∂∂∂

∑∑

0, 1,2,...,

kk k

uuxu

(21.5.8')

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

405

∂

(21.5.8'')

Differentiating the equation (21.5.8) with respect to

+1n

x , we obtain

∂

⎛⎞

⎜⎟

∂

⎝⎠

;

taking into account (21.5.8''), we see that

μ . In this case, from (21.5.8) and

(21.5.8''), we get

∂∂

++ +=

∂∂

∑∑

jj i

fh f

ΓΓ

(21.5.9)

by partial differentiation with respect to

x and to t , respectively, we obtain

()

⎡⎤

∂∂ ∂

⎛⎞

++ − −==

⎜⎟

⎢⎥

∂∂ ∂

⎝⎠

⎣⎦

∑∑

0, 1,2,...,

jj i

fh fl n

tx x x t

ΓΓ

(21.5.10)

()

⎡⎤

∂∂ ∂

++ − −=

⎢⎥

∂∂ ∂

⎣⎦

∑∑

jj i

fh f

tt t x t

ΓΓ

(21.5.10')

Introducing Lagrange’s function

()

=+ − −

∑∑

jiii

fh xfμλL ,

(21.5.11)

where / , 1,2,...,

xi nλΓ=−∂ ∂ = , are Lagrange’s multipliers, the equations

(21.5.8'), (21.5.10) become the Euler–Lagrange equations

∂∂∂

⎛⎞

== −==

⎜⎟

∂∂∂

⎝⎠

0, 12,..., , 0, 1,2,...,

km i n

utxx

LLL

;

(21.5.12)

as well, the equations (21.5.3') take the form

∂∂

== ==

∂∂

0, 12,..., , 0, 1,2,...,

in j p

λμ

(21.5.12')

Eliminating ∂∂/ tΓ between (21.5.9) and (21.5.10'), it results

∂

⎛⎞

−+ =

⎜⎟

∂

⎝⎠

∑

λ 

(21.5.13)

MECHANICAL SYSTEMS, CLASSICAL MODELS

406

Hence, if

L does not depend explicitly on time (

= 0



), we get the first integral

−+ =

∑

const

xλ L ;

(21.5.14)

we can also write

−+ =

∑

const

fxλ  ,

(21.5.14')

because we have



along the optimal trajectory.

21.5.1.3 Conditions of Minimum

It is sufficient that the derivative of first order of the function (21.5.9) with respect to

u be zero, so that this function does admit a minimum with respect to u ; besides

these conditions, it is necessary that Legendre’s condition

∂

∂∂

∑∑

αα

(21.5.15)

be fulfilled for any unit vector

α of components

, ,...,

αα α.

This condition does not exclude the possibility of existence of a relative minimum.

To have an absolute minimum (with respect to the whole set

U ) it is necessary that

Weierstrass’s condition

()

∂

≥= − − −

∂

∑

0, ,,,,, ,,,,,

E E xut x xut x x x

μλ μλ





(21.5.16)

where we have introduced Weierstrass’s function for any set

()

≠,,, ,,,xutx xutx



which satisfies the equations (21.5.12'),

∀∈

[,]ttt, be verified. From (21.5.16) one

obtains Clebsch’s (or the Legendre–Clebsch) condition, according to which the

inequality (21.5.15) must take place for

, ,...,

αα α, which verify the equations

∂

∑

0, 1,2,...,

(21.5.17)

The performance index of the optimal control function is realized by passing the

system from a point of the manifold

S to a point of the manifold

S ; one determines

the moments

t and

t , so that the equations (21.5.3'') which define a

(

+−

1nr)-dimensional or a ( +−

1nr)-dimensional manifold, respectively, be

verified. Taking into account the conditions at the ends, one obtains the transversality

conditions

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

407

⎡⎛ ⎞ ⎤

+− + = =

⎜⎟

⎢⎥

⎣⎝ ⎠ ⎦

∑∑

ddd0,0,1

ii i i

Gf ft x sλλ ,

(21.5.18)

which give

+− +

22( )nrr equations; associating the conditions (21.5.3''), one

obtains

+22n

equations at the ends. The first equations, so that the theorem of

implicit functions can determine the unknowns

u and

μ as function of

λ and

x .

Substituting in the rest of the equations (21.5.12), (21.5.12'), it results a system of

differential equations of first order in the unknowns

λ and

x ; the general solution of

this system depends on

+22n constants (including

t and

t ), which are specified by

the

+22n conditions at the ends. The problem is thus entirely solved.

τ is a point of discontinuity of the control function ()ut , then it is shown that the

Weierstrass-Erdmann conditions

=− =+

∂∂

⎡⎤⎡⎤

−+ =−+

⎢⎥⎢⎥

∂∂

⎣⎦⎣⎦

∑∑

ττ



(21.5.19)

must be verified.

21.5.2 Optimal Trajectories

In what follows we consider the case of problems of Lagrange type without

constraints, being thus led to Pontryagin’s principle of maximum; we present then some

applications.

21.5.2.1 Problems of Lagrange Type Without Constraints

In case of a problem of Lagrange type without constraints (for which = 0

h ,

= 1,2,...,jp), Lagrange’s function (21.5.11) has a much more simple form

()

=+ −

∑

ii i

fxfλ L

;

(21.5.20)

the equations (21.5.12), (21.5.12') are reduced to

∂

0, 1,2,...,

(21.5.21)

∂∂ ∂

⎛⎞

−= ==

⎜⎟

∂∂ ∂

⎝⎠

0, 0, 1,2,...,

ii i

tx x

λ

LL L

(21.5.21')

The equations (21.5.21') become

∂∂

==−=

∂∂

, , 1,2,...,

xin



(21.5.22)

MECHANICAL SYSTEMS, CLASSICAL MODELS

408

where we have introduced Hamilton’s function

=− +

∑

ffλH ;

(21.5.23)

as well, the equations (21.2.21) read

∂

0, 1,2,...,

(21.5.22')

Weierstrass’s condition (21.5.16) takes the form

()()

≤,;; ,;;xut xutλλ

(21.5.24)

for any

u which satisfies the first equations (21.5.3') and

() (), [ , ]ut ut t t t≠∀∈.

21.5.2.2 Principle of Maximum

The conditions (21.5.22') and (21.5.24) can be written together in the form

()

[

]

∈

(),();;() sup (),;;()

xt ut t t xt ut tλλ=

;

(21.5.25)

the necessary condition of optimality expresses Pontryagin’s principle of maximum.

A control autonomous system is described by equations of the form

==( , ), 1,2,...,

xfxui n ,

(21.5.26)

where the time does not appear explicitly; the functions

and ∂∂/

fx are considered

as defined on the set

U×

\ . Thus, for a given control function = ()uut and for

given initial conditions

()xt x , the system (21.5.26) admits a unique solution. If

the trajectory

= ()xxt is definite for ∈

[,]ttt and =

()xt x , then we say that the

control function

()ut transfers the system from the position

x to the position

x .

Considering the performance index (

f has the same properties as

f )

∫

(,)d

Ifxut,

(21.5.27)

we can enounce the problem of optimality is the following form: Being given two points

∈

xS and ∈

xS and the admissible control functions = ()uut (if there exist

such functions), which transfer the material system from the position

x to the position

x , one asks to determine that admissible control function for which the index of

performance (21.5.27) takes the minimal value.

The moment

t is determined by the condition =

()xt x and by the second

condition (21.5.3'').

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

409

We introduce the co-ordinate

x definite by the equation =

(,)xfxu , so that the

differential system (21.5.26) does take place for

= 0i too; we denote ≡x

{,}xx. Let

be also the system

∂

=− =

∂

∑

, 1,2,...,

λλ



(21.5.28)

which defines the vector function

≡

( ) { ( ), ( ), ( ),..., ( )}

tttttλλλλ λ for the

trajectory

()xt and an admissible control function ()ut . By means of Hamilton’s

function

()

∑

,, (,)

ufuλλH ,

(21.5.29)

the equations (21.5.26), (21.5.28) are written in the canonical form (21.5.22) (for

= 0i

too). We may thus state

Theorem 21.5.1 If ()ut is an optimal control functions, then there exists a continuous

and non-zero vector function

()tλ so that:

(i)

x()t and ()tλ are solutions of the canonical system (21.5.22);

(ii)

∀∈

[,]ttt the function

()

x(), , ()tu tλ

, with respect to the variable

∈uU

, reaches its maximum for ()ut (principle of maximum)

()()

∈

xx( ), ( ), ( ) sup ( ), ( ), ( )

tut t tut tλλ=

H ;

(21.5.30)

∀∈

[,]ttt we have

()

=≤ =x

() const 0, (), (), () 0ttuttλ λ

(21.5.30')

The equations thus established allow the complete determination of the unknown

functions

==( ), ( ), 1,2,..., , ( ), 1,2,...,

ii j

xt t i nut j mλ , and of the moment

In case of non-autonomous systems one can make a convenient substitution, which

formally reduces the problem to that corresponding to autonomous systems.

21.5.2.3 Applications

As applications to the results presented above we can study the motion of a rocket,

modeled as a particle of variable mass (see Sect. 10.3.1.1). We consider thus

Meshcherskiĭ’s equation

(), 0mm m=+ − <





vF uv ,

(21.5.31)

where

is the velocity of the particle of mass m with respect to an inertial frame of

reference,

u is the absolute velocity of the detached mass and

is the resultant of the

given forces; obviously,

=−wuv

is the relative velocity of the detached mass with

respect to a non-inertial frame of reference, attached to the particle in motion, while

=Rwm is the reactive force. We assume that the motion of the rocket can be guided

(iii)

MECHANICAL SYSTEMS, CLASSICAL MODELS

410

by modifying the force of propulsion (which includes the reactive force), after an

imposed program; because

< 0m

, the force of propulsion is guided in the sense of the

motion, accelerating it. We denote by

123

,,uuu the direction cosines corresponding to

this sense (with respect to an inertial frame), by

=− >

0um

the velocity of variation

of mass with changed sign and by

==−wuv

||| |u the magnitude of the relative

velocity; these variables are control variables. The variables of state are the co-ordinates

123

,,xxx of the particle (of the rocket), the components ===

41 263

,,xvxvxv of

its velocity and the mass

xm. With respect to an inertial frame of reference, the

equation (21.5.31) becomes

[]

=−

,1,2,3,

(,) ,

xxi

xFtuuu



(21.5.32)

We suppose that

F depends only on the variables of state and time. The problem

which is put leads to the definition of the set

U by the conditions

++= ≤≤ ≤≤

123

222

1, 0 , 0uuu uu uu.

(21.5.33)

Among the problems which can be put, we mention: the optimization of the force of

propulsion (in particular, in case of the motion in a homogeneous gravitational field),

the finding of an optimal control function to realize a maximal distance of flight, the

determination of an optimal control function to realize a minimal consumption of

combustible in case of a rocket of constant power etc.

Another interesting problem which leads to optimization problems is that of airplane

flight. One can mention other problems concerning the modern industry and

technology, in connection also with the idea of automatization. All these aspects have

led to a chapter in the theory of differential equations, which is now – by its importance

and its development – independent.

Chapter 22

Dynamics of Non-holonomic Mechanical Systems

The theory of non-holonomic mechanical systems appeared when it was seen that the

classical Lagrangian formalism (corresponding to the holonomic mechanical systems)

cannot be applied in case of some very simple problems (e.g., the rolling without sliding of

a rigid solid on a fixed plane). After that Lindelöf’s error in this direction has been detected

by Chaplygin, one has obtained many formulations and there have been made many studies

by Appell, Bobylev, Chaplygin, Tsenov, Hamel, Hertz, Maggi, Voronets, Zhukovskiĭ and

others; the actual research in this direction is very rich.

The development of dynamics of holonomic mechanical systems puts in evidence the

closed connection between the study of these systems and the study of the holonomic ones;

moreover, the geometric treatment of these problems led to the creation of the

non-holonomic geometry by Vrănceanu, Scouten, Wagner and others. On the other hand,

there appeared also many specific aspects (e.g., non-linear non-holonomic constraints

considered by Gibbs, Appell, Chetaev, Hamel, Johnson, Novoselov etc.); as well, other

types of non-classical constraints have been put in evidence. We mention also that, in the

last time, an analog between these systems and the electromechanical ones has been

developed.

After passing in review some elements of kinematical nature, one presents Lagrange’s

equations, as well as other equations of motion (especially the Gibbs-Appell equations and

Chaplygin’s equations) and various applications; one considers then also other problems

concerning the dynamics of non-holonomic mechanical systems (Neĭmark, Ju.I. and

Fufaev, N.A., 1972; Pars, L., 1965).

22.1 Kinematics of Non-holonomic Mechanical Systems

After some general considerations, one presents some conditions of holonomy and

one introduces the notion of quasi-co-ordinate. A special attention is given to the

geometrization of the problem of non-holonomic mechanical systems.

22.1.1 General Considerations

One presents, in the following, some preliminary aspects concerning the differential

forms, the number of degrees of freedom of the mechanical systems, the representative

spaces which are used etc. One considers then various cases, especially the rolling of a rigid

solid over another rigid solid.

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P.P. Teodorescu, Mechanical Systems, Classical Models,