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Terminal Control for Nonlinear Systems
IFAC
Copyright ~ IFAC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I
c:
0
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Publications www.elsevier.comllocatelifac
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
497
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
499
Copyright ~ IFAC Nonlinear Control Systems, SI. Petersburg, Russia, 200 I
c:
0
C>
Publications www.elsevier.comllocatelifac
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
497
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
499