Trajectory Motion Control of Symmetrical Nonlinear Plants

I = I o,/os

o,/oc;

1-

1

(9)

defining appropriate mapping of tangent spaces. Further we fix y* as a point of the given curve and identify the set Y with its neighborhood , where the mentioned diffeomorphism holds and an open arc of the curve is defined as a subset 5

= {y E Y : y* E 5 ,
For smooth curves (arcs) primarily given in Y in some arbitrary manner , the choice of 1j; and

1994a, 1994b). Note that the arcs evidently do not include singular points of transformation (5) , (6) but their limit points may be singular. Further we suppose that such singularities are excluded and , therefore , a neighborhood of the arc 5 defined as

Ye = {y E Y : 11

O}

~i of the form (11 ) and ( such that transformation (12) , (13) is diffeomorphic in X with the inverse one

7f;i,

is not empty and expressed as the Cartesian product S* x R" where

R, = {c; E R m- 1 : 11 c; 11 < cS} Then , in a small neighborhood , it holds

+L

x

= r(a-, e. z )

(14 )

(10)

The result is proved in the main by direct verification of nonsingularity of the appropriate J acobian matrix (see also (Miroshnik , 1990 ; Isidori , 1989) ).

Let us now initially formulate the trajectory control problem for the symmetrical plant as that of providing . (i) a desired mode of the longitudinal motion s(t ) given on the arc S by some conventional requirements for SISO systems , (ii) stability of the arc S as an attractor in the set Y, or that of the point c; = 0 of the set R,.

Under the conditions of the Proposition 1, the submanifold :E is a (n - r (m-I ) )-dimensional manifold with local coordinates a-i and Zi . More~ver , if the probable singularities of the closure :E are excluded , one can define an open neighborhood of the submanifold

M = M*(s)

M'j (s)c;j

j

X e = {x EX : 11~i 11< .6.i, .6. i > O} which is not empty and expressed as the Cartesian product :E* x R e, where

3. LOCAL ANALYSIS OF SPATIAL MOTION

R e = {e E R r(m- 1) : 11 ei 11< .6.,}

Let us introduce m-dimensional vector functions of the form

7f;1

I = 17f;(fo(x)) I 17f; cp(fo (x )) ' ~ 2

t

I

1 ~1

I

2

=

1

The nonlinear transformation of the state vanabIes is used for deriving a transformed plant model in X e which is construct ed in accordan ce with the following result .

Mo(x)l1(x) , . .. ,

Mr- 2 (x)h(x) + .. . + Mo(x)fr-1(X) (11)

Proposition 2.

Let the plant (1) , (2 ) be symmetrical in X e :J :E and the matrix M is nonsingular in v :J S. Then there exist fun ctions 7f;i, rPi of the form (11 ) and ( such t hat for every x E X e the plant can be described by

where

Mo = M(h(x)) ,Mi = Mi(h , 11 , ·· ·, /;-1)'/0 = h(x) , fi = {L ~ hj} , j = [1 , m], and a (n - mr )dimensional vector function «x ). Define ne w state variables a- = {a-;} , e = {ed such that

I I c;s I = e1 I a-1

17f;1(X) ~1( X)

I' I a-,e, I = I Wi~, (x) (X) I (12 ) ' (15)

i=[2 ,r] , and z = {z;} , i = [1, n - rm], z

= «x)

(13)

I ;: I = A(a-, e) I : I +

Let us fix the point x* such that ~i ( X *) = 0, i = [1 , r], h(x*) E S and identify the set X with a neighborhood of x' . Then we can define a simply connected submanifold of X

:E

= {x EX : x* E:E,

+

M fr (a-, e, z)

+ M Gr (a-, e, z) u

(16)

where M = M (-y( a-1, e1)), fr = {L j h,}, G r = {L gj Lj- 1 hi }, i , j = [I , m], and


i

such that y = h( x) E S for each x E :L. Moreover, the following holds.

= f o(a- , e, z) + Go( a-, e, z) u

(17 )

The property is proved in the main by using the conventional procedure of time differentiation of equations (12) , (13) and properties (3 ), (4 ).

Proposition 1. Let the plant (1) , (2) be symmetrical in X J ~ and the ma-

The transformed model proposed includes controlled multi connected model (15 ), (16 ) of transversal and longitudinal dynamics which is disturbed by a so called internal dynamics model

trix M be nonsingular in the neighborhood Y of the smooth submanifold S. Then there exist functions 609

(17) through the functions Ir and G r. For solving the trajectory control problem, the closed loop control law u = u( 0", e, z) must be chosen to provide desired properties of the model (15) , (16). Among them there is longitudinal dynamics prescribed by the identity s = s* (t) E S and, therefore , 0" O"*(t) , where 0" {O"n is the demand vector which , for the sake of definiteness , is supposed to be generated by a reference model of the form

=

is the longitudinal control and U e is the (m-1)dimensional vector of transversal control. The description of the basis control law (20) is known (Miroshnik et al. , 1986,1990; Isidori , 1989 , 1991). It provides, partially, plant nonlinearity compensation as well as removing the disturbance influence of zero dynamics (supposed to be bounded). The control law (20) is determined by the structure and dynamic properties of the plant only and does not depend on the curve. Equation (21) introduces a smooth control variable transformation that , in contrast , depends only on the curve properties. Us

=

(18) u;(t) is the smooth model input signal.

Substituting (20)-(21) into equation (16) we obtain

However! the main desired property is stability of the system about the attractor ~ , all trajectories on which satisfy the identity e = 0 and, therefore , E: = O. The motion in the manifold ~., determined by variables 0" and z is described by equations (18) and z=le(O"* ,O, z) (19)

I

I~: =

A(O", e)·I:

I+ I ~: I

(22)

Equations (15) and (22) specify the models of longitudinal and transversal dynamics interacting through the matrix

where le = lo(O"* , O,z) + Go(O"*,O,z)u(O"*,O,z) . The latter characterizes the behavior of the system uncontrollable part caused by the plant zero dynamics (Isidori , 1989) . Minimal requirements for its properties follow from the necessity to bound the mentioned above influence to the transversal and longitudinal dynamics as well as to satisfy conventional system design principles. They involve boundedness of the forced motion of uncontrollable model (19) that is reached if the reference trajectories in ~* given by the variable O"*(t) are bounded and internal model (19) is input to state stable (ISS , see (Byrnes et al., 1991 ; Sontag, 1990)).

aT

A(O", e)

= I As

The model structure predetermines final steps of the synthesis procedure. However,here we would like to consider an indirect way that leads to a somewhat simplified control law . In accordance with item (ii) of the problem statement (see Part 3) the transversal control law has to be derived in order to stabilize the transversal motion model which takes the form

Dnder these assumptions , we can correctly enough pose the trajectory control problem as that of spatial motion control in Xe reduced to (i) the desired smooth variations of 0" established by model (18) , (ii) asymptotic stability of the point e = 0 of the set R e .

This is provided by (24)

where the choice of the feedback gain matrix K e = K e (0") insures asymptotic stability of transversal motion and any desired performance of the transient processes in Re . In particular , the choice Ke = Ae + Ko ensures complete independence from longitudinal dynamics.

4. CONTROL LAW DESIGN Let us consider a plant symmetrical in X e and an arc S C YE Then the structure of the transformed plant model (15) - (16) as well as its component properties (4) and (10 ) make evident the following design procedure .

Let us now consider the motion along ~ , where e = 0 and the model of longitudinal motion takes the form (25)

The baSIS control law is chosen in the form

In accordance with item (i) of the control problem statement , the longitudinal control law has to provide tracking for the reference model (18) that is reached by

(20) where

u is the trajectory control vector defined as (21)

Us = u; 610

+ (a -

as f

0"*

+

k:

~O"

(26)

where k'[ = k'[ (0') is the row matrix of feed back gains on the tracking error ~O' = 0'* - 0'. On the submanifold L such a choice provides stability of the solution ~O' = O.

and the basis control law can be found as

+

u=G-1'{-f

M-ll~: I}

Then model (27) takes the form

Thus the control law is described by basis algorithm (20), control variable transformation (21) and local control algorithms (24), (26). The properties of the closed loop control system in Xe can be verified by using an error model derived from (18), (23) , (25). After linearization and appropriate substitutions it takes the form

s

=

E

Us ,

=U e

In order to completely solve the control problem . the signals Us and U e are chosen as Us

= U .,

u e = -kF:

where k is the feedback gain. The motions of the plant considered along the variety of plane and space curves have been simulated. Here we present the most typical results relating to an ellipse initially described in the plane in the implicit form

where

A} = uO' A , A c (") {UO'i 0' = I (as - 0 ksf

2

!!..l2 +

The structure of the matrix Ac( 0'*) and the properties of its elements confirm that the appropriate choice of the feedback matrices Ke and ks can ensure asymptotic stability of the solutions ~O' = 0, e = 0, uniform for bounded variations of 0'" E L".

a

2

Y2 _ 1

b2

=0

where a = 4, and b = 2, and a space spiral given in the parametric form Yl = a cos ws , Y2 = asifl ws , Y3 = cs

where a = 4, c = 3 , W = 1. s E Rl Note that for sufficiently small e and jectory control law takes the form

~O'

the traTrajectories of the plane motion of the 2nd order plant from the neighborhood of the ellipse with a given longitudinal velocity u· = 2 are shown in Fig. l. Those of the 3d order plant space motion along the spiral are demonstrated in Fig.2. Simulation confirms the stability of the designed system about the prescribed at tractors for a sufficiently large domain of the initial conditions.

that , being substituted into algorithm (20) , can be used for verifying the ISS property of the internal model (19) 5. EXAMPLES AND SIMULATION RESULTS

4 To illustrate the main principles of control law design and present the vivid simulation examples, we analyze the trajectory motion of the simplest symmetrical plants of the form

if

~~----+-----~~-+----~~

2

o r-~--~-----r----r----T~

= f(y) + G(y)u

where r = 4, m = n = 2,3 and the matrix G is invertible.

-4 1---+-----+------r----+------+--1

The trajectory control problem is to stabilize the plant with respect to typical smooth plane or space curves and to provide the proportional longitudinal motion prescribed by the model 5* = u' = const.

-4

+

MGu.

0

2

4

6. CONCLUSION

For such a case, the transformed model of the plant is

Mf

-2

Fig . ! Stabilization about the ellipse

In this paper we discussed one of the widespread problems of nonlinear plant spatial motion connected with trajectory control in the output space.

(27) 611

The approach was based in the main on a proper local transformation of the output , state and control variables that gave an opportunity to propose the clarified structure of the control system and to reach desirable properties of its transversal and longitudinal dynamics. The results can be directly extended to the more general class of nonlinear plants and that of controlled motion along given hypersurfaces.

20 18 16 14 12 10

8

1

/'

~

V 1\-"'>

~

6

7. REFERENCES

2

Krut'ko , P.D. (1987) . Inverse Problems of the Control System Dynamics. Nauka Phys. and Math . Pub!. , Moscow (in Russian). Miroshnik, I. V . (1990) . Coordinating Control of Multivariable Systems. Energoatomizdat, Leningrad (in Russian) . Morozovsky, V .T. (1970). Multiconnected Automatic Control Systems. Energia, Moscow (in Russian) Miroshnik , LV., A.V. Ushakov (1977) . Synthesis of an algorithm for simultaneous control of quasisimilar plants. A utomation and Remote Control, no .11 , pp. 22-29 Miroshnik, LV. (1986) . On stabilization of the motion about a manifold . Automatica (Kiev) , no.4, pp. 65-68 Miroshnik, I.V. , V.O Nikiforov (1991). Adaptive Control of N onlinear Plant Spatial Motion. Automation and Remote Control, no .9, pp. 78-87. Miroshnik, LV., V.O Nikiforov (1994a). Adaptation and self-learning of multivariable nonlinear plants. Preprints Vol. of the 2nd IFAC Symp. on Intelligent Comp. and Instr. for Control Application, pp. 322-327 Miroshnik , I. V ., V. O. Nikiforov (1994 b ). Coordinating control and self-learning of robot trajectory motion. Preprints Volume of the 4th IFAC Symp. on Robot Control, pp. 811-812 Isidori, A ., A .J . Krener , C . Gori-Giorgi and S. Monaco (1981). Nonlinear de coupling via feedback: A differential geometric approach. IEEE Trans . Autom. Control, v.26, no .2 , pp. 331-345 Isidori , A. (1989). Nonlinear Control Systems. Springer-Verlag , l\".Y. Byrnes , C.I., A. Isidori (1991). Asimptotic Stabilization of Minimum Phase Nonlinear Systems. IEEE Trans. Autom. Control, v.36, no.10, pp . 1122-1137 Sontag, E .D . (1990). Further facts about input to state stabilization. IEEE Trans. Autom. Control, v .35, pp . 473-476

........

Ib

..-

u

o

J>

-;::

r/'

4

---- "'-

-~

I" ---

---k'

o--"P"

............ b

~

-4

-2

o

2

4

-4

-2

0

2

4

2

o -2 -4

Fig .2 Stabilization about space spiral (projections)

612