Terminal Control for Nonlinear Systems

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Copyright ~ IFAC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I

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TERMINAL CONTROL FOR NONLINEAR SYSTEMS I.E. Zuber

St .Petersburg State University, Department of Mathematics and Mechanics, Petrodvoretz, Bibliotechnaja, 2, St.Petersburg, Russia [email protected] .edu

Abstract: The problem of terminal control is a well known problem of control system theory. Its solution is known for linear time-stationary systems and is usually received as programmed control. At present paper the solution of terminal control problem is received for nonlinear systems as feedback control. Copyright Cl) 200IIFAC Keywords: control, stability, feedback

1. SOLUTION OF TERMINAL FEEDBACK CONTROL PROBLEM FOR LINEAR TIME-STATIONARY SYSTEMS

ai are coefficients of poly nom det (A - AI) = An + alA n - 1 + ... + an = 0. So the transformed closed loop matrix (1),(2)

b = en = (0, .. . , 1)*

Consider a linear time-stationary system :i;

= Ax + bu,

Xo

=I 0,

x ERn ,

where

(1)

-

A

the pair (A, b) is given.

b = Tob,

Our aim is to construct a feedback control u

= s·x

is the Frobenious matrix with the last line (al," " an) . Suppose eigenvalues ~i(D) are different, Ai =I Aj, i =I j . Then D = CART , A = diag {Ad, C = Ild l . .. dnll, di = (I,Ai, . .. ,A~-I)* are eigenvectors of D corresponding to eigenvalue Ai,

(2)

which provides a fulfillment of condition

X(T) = r, where T is a given moment of time, r given constant vector.

=I

°

(3)

is a

H·

Consider well known similarity transformation providing Frobenious form for closed loop matrix undependly on vector s y

B

-

= Tox,

S=

To = BS.

°

= C- I = IlgI, '" ,gnll·

Consider required condition (3) for solution of transformed closed loop system

Tor =

matrix of controllability of system (1) , 1 al

= ToATo-I ,

Consi~er solution of system if = y = eDtyo for t = T .

... 0

1

r.

Dy,

Yo

=

Toxo,

Condition (3) now is restated as

(4)

an-I ... al 1

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existence n - 1 derivatives for F(x) and b(x) and the total controllability of pairs (A(x), b(x)) , (A(x), en) .

Equality (4) is multiplyed by eigenvector of fr, gi , i = 1,n, then i

= 1,n.

(5)

Such transformation has a form (Zuber, 2000)

Now consider equalities (5) as system of transcidental equations in respect to Ai , i = 1,n. Each solution of (5) with different Ai , i = 1,n is stated a spectre of D providing a fulfillment of condition (4), and hence a fulfil!.: ment of condition (3) . Note, that spectre D with different Ai exists. Each modal control u = s*y corresponding to solution of (5) with different Ai , i = 1,n sets a feedback vector s = TO' -1 S providing fulfillment of (3) . So the following statement is valid:

e~B-1(x)

d

dte~B-1(x)

R(x)

+ e~B-1(x)L1(X)

= n-1

d·

L C;'_1 (d:)j (e~B-1(X))Ln_1_j(X)

j=1 where Lj(x) is a matrix of j-derivative in virtue of system x = A(x)x , B(x) = IIb 1(x) , .. . , bn(x)11, k-1 j d' bk(x) fk(X) - L Ck(dt)Jfk-j(X), bo(x)

Theorem 1. If the pair (A , b) is controllable then exists and can be found in explicit form feedback control which provides a fulfillment of condition (3) for closed loop system (1)-(2) .

)=1

b(x), fk(X)

Cl

Note that choice of system (5) solution can be conditioned by optimization problem.

=

(Lk(x) - dt~.)b(x),

D(y)

+ b(x)u,

Xo

#

0,

x ERn .

(6)

R(r)x(r)

= s*(x)x

g;(y)y(r)

Rewrite the system (6) as

J 1

u

J(ux) du ,

= R(xo)xo ·

= eAiTg;(y)yO,

= S*(y)y

providing such a spectre for D(y) provides a fulfillment of terminal condition (8) so the following statement is valid.

o J(x) - the Jacoby matrix for (6) and perform a similarity transformation

Theorem 2. If vectors F(x) , b(x) are differentiable n - 1 times and pairs (A(x) , b(x)), (A(x) , en ) are totally controllable then exists and can be constructed in explicit form a feedback control providing the fulfillment of terminal condition (3) for closed loop system (6) .

y = R(x)x which provides for a object matrix of transformed system A(y) = R(x)A(x)R- 1(x) - R(x)R(x) the Frobenious form with the last functional line and fulfillment of condition

b(y)

Yo

gi (y) is eigenvector of D* (y) corresponding to eigenvalue Ai = const and consider its solution for Ai # Aj , i # j . Modal control

x = A(x)x + b(x)u, A(x) =

= y(r) = D(y(r))yo ,

(8) Now construct a system of equation in respect to constant spectre of D

which provides a fulfillment of terminal condition (3) for system (6).

where

(7)

Consider terminal condition (3) for system (7)

Our aim is to find conditions of existence and to construct a feedback control u

= A(y) + enS*(y) .

Matrix D(y) as Frobenious matrix is totally defined by its spectre. If its spectre is constant then D(y) is a constant matrix. According to (Zuber, 2000) the choice of modal control providing constant spectre for D(y) provides similarity for linear time-stationary system and system (7) .

Consider system

F(x)

1, n - 1,

Consider transformed closed loop system (6)

2. SOLUTION OF TERMINAL FEEDBACK CONTROL PROBLEM FOR NONLINEAR SYSTEMS

x=

k

is a binomial coefficient.

3. AN EXAMPLE

= R(x)b(x) = en = (0, . . . ,0, 1)* .

According to (Zuber , 1998; Zuber, 2000) sufficient conditions for existence such a transformation is

Consider an equation of pendulum with outher action

498

x + ax + bsinx = m-I F(x) f(x), where (3(x) =

bs~nx ,

or X + ax + (3(x)x f(x) = m-I F(x) .

=

desired traectory of vehicle set an assembly of supporting points Pj j = 1, ... , N . Now we find feedback control which providing the solution of terminal problem for each pairs of points Pi ,PH 1, i = 1, .. . , N - 1.

Our aim is to define F(x) which for given Xo =1= 0, r1 =1= 0, r2 provides the fulfillment of condition x(r) = rI, x(r) = r2 .

xo,

Perform similarity transformation Y1 = X, Y2 = X, then considered equation will be rewrited as nonlinear closed loop system with Frobenious matrix. Consider the last line of this matrix as polynomial coefficients -a, -(3(x) + f(x) and rewrite the terminal conditions as Y1 (r) = rI, Y2 (r) = r2 ·

ACKNOWLEDGEMENTS The work is supported by the Grant Board of President of RF and of the State Support of Leading Science Schools (Project 00-15-96028) and by Fond of French-Russian Lyapunov Institute, 2000-2001 , and by RFFI, project 00-01-00083 .

Now we suppose to choice the constant roots of polynom A2 + ((3(x) - f(x))A + 0: = O. Then Y = eDtyo,

REFERENCES

Yo = (xo, 0)"

and we came to considering system (5) for n = 2.

Zuber, I.E. (1998). Stabilization of non linear systems by similarity transformation, Journ. of Applied Mathematics and Stochastic Analysis, 11, No. 4, 519-526. Zuber, I.E. (2000) . Spectral stabilization of nonlinear systems based on special similarity transformation [in Russian] . Vestnik St. Petersburg Univ., ser. 1, 2 , No. 8, 8-13 . Zuber, I.E. and K.Y. Petrova (2001). Design of regulator for nonstationary model of an autonomous vehicle Differential 'nie uravneniya i protsessi upravleniya (Electronic Journal), (accepted)(in Russian) .

Now gl = (A2' -1)*, g2 = (AI, -1)* and solution of transcidental equations

can be received by replacing e>.,T its expansion for n = 2. Received values of Ai, i = 1, 2 define f(x) .

4. CONCLUSIONS So terminal feedback control for considered class of non linear system is defined. Note that received conditions of existence terminal feedback control for linear systems differ for such a conditions for terminal program control by following points:

(1) (2) (3) (4)

Xo =1= 0,

x(r) =1= 0, the feedback vector s(x) depends on xo, total controllability of both pairs (A (x), b( x)) , (A(x) , en) is required.

Note, that condition Xo =1= 0 is not very essential. Consider transformed system (6) for Xo = 0 i.e. Yo = 0

if

= A(y)y + enu(y) ,

and consider control u(y)

={~

Then solution of this system is found in explicit form and is considered in moment t = tl as y6 . Now we can find the feedback control providing the fulfillment of terminal condition for system and r1 = r - t1' (7) with y(O) =

Y6

Obtained solution of terminal control problem for non linear time-stationary system is almost literally repeated for nonlinear time-varying systems. Then this solution can be used for motor vehicle control. Consider linear time-varying description of motor vehicle motion with observer and compensator (Zuber and Petrova, 2001) . Considering

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