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An approximation theorem for linear optimal regulators
JOURNAL
OF MATHEMATICAL
ANALYSIS
An Approximation
AND
Theorem
APPLICATIONS
21, 249-252 (1969)
for Linear Optimal
Regutators”
P. V. KOKOTOVIC AND J. B. CRUZ, JR. Coordinated
Science Laboratory and Department of Electrical Engineering, University of Illinois, Urbana, Illinois Submitted by J. P. LaSalle
1. INTRODUCTION Taylor series expansions represent a convenient tool for sensitivity analysis and near optimum design of dynamical systems depending on parameters [ 1,2]. The theorem presented in this note establishes a property of expansions of optimal feedback matrices for linear state regulator systems with quadratic performance indices. It extends a recent theorem for expansions of optimal controls [2]. Since the theory of linear systems with quadratic performance indices is widely used in optimal control, estimation, and Lyapunov stability, the theorem in this note has a variety of applications. 2. STATEMENT OF THE PROBLEM Consider the well-known linear state-regulator problem [3] for the plant k=Ax$Bu
(1)
with performance index J = ; /;(x’Qx
+ u’Ru) dt,
(2)
where x is the n-dimensional state vector and u is the r-dimensional control vector, dot denotes differentiation with respect to time, and a prime denotes a transpose. Let the matrices A, B, Q, and R be analytic functions of a scalar parameter OLin some neighborhood of 01= 0. These matrices may be time-varying, but they must satisfy the usual assumptions of the linear state regulator problem [3]. * This work was supported Contract DAAB-07-67-C-0199 under Grant AFOSR 931-68.
by the Joint Services Electronics Program under and by the Air Force Office of Scientific Research 249
KOKOTOVI6 AND CRUZ
250
The optimum matrix K used in the state feedback regulator u = -R-IB’Kx
(3)
is an analytic function of 01,K = K(t, a). This follows from the theorem on differentiability of solutions of ordinary differential equations with respect to a parameter when applied to the Riccati equation K=
-KA-A~K+KSK-Q,
K(T+)=o,
(4)
where S = BR-‘B’. Let M = M(t, LX)b e a matrix defined by
The problem considered here is to compare the performance JM of the system $=(A-SM)x (6) with the state feedback regulator u = -R-IB’Mx,
(7)
to the performance Jg of the system R = (A - SK)x,
(8)
with the optimum state feedback regulator (3).
3. APPROXIMATION THEOREM The first 2m terms of the Taylor’s series for Jnr are equal to the corresponding 2m terms of the Taylor series for JK,
a”JK @Jhf 32 &)=aOLd &)’
i = 0, l,..., 2m - 1.
(9)
PROOF. The expression for Jg is well known to be [3]
JK=;x’Ks/t
t. -0
Similarly it can be shown (see for example, Ref. 4, pp. 81-84) that the performance JM is Jhr=;xfPx~
, t--to
(11)
AN
APPROXIMATION
THEOREM
FOR LINEAR
OPTIMAL
REGULATORS
251
where P = P(t, a) is the solution of the linear symmetric matrix equation I.’ = -P(A
- SM) - (A’ - MS) P - MSM - Q, P(T, a) = 0. (12)
Define r as r==p-K.
(13)
Then to prove (9) it is equivalent to prove that at (Y = 0 and t := to , r and its first 2m - 1 derivatives are zero matrices,
air(t, g ___ =o, ad t=tO
i = 0, 1, 2,..., 2m - 1.
(14)
a=0
From (5) K - M = cum/l,
(15)
where
Subtracting (4) from (12) and using (13) and (15),
I'=
-r~
- ix-dyrsA
+Asr)-2mAsA,
r(T,ol) = 0, (17)
where G = A - SK. This equation shows that (14) holds not only at t = to but also for all t E [to , T]. To see this, first check (14) for i = 0. That is, find the solution r(t, a) of (17) for (Y = 0. At a: = 0 (17) reduces to a homogeneous linear equation
I'=-m-m,
r(T,o) = 0,
(18) whose solution I’(t, 0) must be zero for all t, since it is zero at t = T. Next, denote derivatives by subscripts, ri = air/M, Gj = 8G/a& etc., i,j=O,1,2 ,... . To verify (14) for i = 1 differentiate (17) with respect to 01 and let 01= 0,
I', = -I-',G, -Girl
- FoGI - Gil', - mOm-l(roSoAo + A,S,r,),
where mom-l is 1 when m = 1 and is 0 when m > 1. Since the result of the verification for i = 0 is r, = 0, (19) becomes homogeneous and hence r, = 0. Using induction, verify whether the assumption of I’, = 0, for j = 0, I,..., i, makes the equation for I’,+l homogeneous, and determine the maximum i for which this is true. Differentiating (17) i + 1 times and letting 01= 0,
pi+, = --ri+,Go - G;ri,, -F,
ri+JT,O) = 0.
(20)
252
KOKOTOVIt
AND
CRUZ
For i < 2m - 1, each term in F is linear in some rj , j = 0, I,..., i. Since Pi = 0, for j = 0, l,..., i, F = 0 and the linear equation (2) is homogeneous. Thus ri+Jt, 0) = 0 because it is zero at t = T. This proves (14) and therefore (9).
REMARKS
The main content of the theorem is in establishing the order to which the system performance approximates the optimal performance if instead of the optimum gain matrix its truncated series M is used. It is sufficient that the order of the truncated series 1M is one-half of the desired order of the approximation of the optimal performance. Moreover, the parameter cr can be selected to make the system dimensional&y lower when a = 0, than when a > 0 (singular perturbation of a [5,6]). Although in this case some elements of the system matrices A and B have a pole at a = 0, it can be shown [7] that under mild conditions, K and P are analytic in a at a = 0. Therefore, the theorem of this paper extends to that class of “singular perturbation” problems.
1. H. J. KELLEY. An optimal guidance approximation theory. IEEE Trans. Automatic Control AC-9 (1964), 375-380. 2. R. A. WERNER AND J. B. CRUZ, JR. Feedback control which preserves optimality for systems with unknown parameters. IEEE Trans. Automatic Control AC-13 (1968), 621-629. 3. R. E. KALMAN. Contribution to the theory of optimal control. Bol. Sot. Matemutica Mexicanu, 1960, 102-I 19. 4. J. LASALLE AND S. LEFSCHETZ. “Stability of Liapunov’s Direct Method,” Academic Press, New York, 1961. 5. A. N. TIKHONOV. On the dependence of solutions of differential equations on a small parameter. Mat. Sbo7nik 22 (64), 193-204, Moscow, 1948. [See also, A. N. TIKHONOV. Systems of differential equations containing small parameters muftiplying some of the derivatives. Mat. Sbornik 31 (73), 575-586, Moscow, 1952.1 6. A. B. VASILEVA. Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivative. Russian Matk. Swweys 18 (1963), 13-81. 7. P. SANNUTI AND P. KOKOTOVIC. Near-optimum design of linear systems by a singular perturbation method. IEEE Trans. Automatic Control AC-14 (1969) (to appear).
OF MATHEMATICAL
ANALYSIS
An Approximation
AND
Theorem
APPLICATIONS
21, 249-252 (1969)
for Linear Optimal
Regutators”
P. V. KOKOTOVIC AND J. B. CRUZ, JR. Coordinated
Science Laboratory and Department of Electrical Engineering, University of Illinois, Urbana, Illinois Submitted by J. P. LaSalle
1. INTRODUCTION Taylor series expansions represent a convenient tool for sensitivity analysis and near optimum design of dynamical systems depending on parameters [ 1,2]. The theorem presented in this note establishes a property of expansions of optimal feedback matrices for linear state regulator systems with quadratic performance indices. It extends a recent theorem for expansions of optimal controls [2]. Since the theory of linear systems with quadratic performance indices is widely used in optimal control, estimation, and Lyapunov stability, the theorem in this note has a variety of applications. 2. STATEMENT OF THE PROBLEM Consider the well-known linear state-regulator problem [3] for the plant k=Ax$Bu
(1)
with performance index J = ; /;(x’Qx
+ u’Ru) dt,
(2)
where x is the n-dimensional state vector and u is the r-dimensional control vector, dot denotes differentiation with respect to time, and a prime denotes a transpose. Let the matrices A, B, Q, and R be analytic functions of a scalar parameter OLin some neighborhood of 01= 0. These matrices may be time-varying, but they must satisfy the usual assumptions of the linear state regulator problem [3]. * This work was supported Contract DAAB-07-67-C-0199 under Grant AFOSR 931-68.
by the Joint Services Electronics Program under and by the Air Force Office of Scientific Research 249
KOKOTOVI6 AND CRUZ
250
The optimum matrix K used in the state feedback regulator u = -R-IB’Kx
(3)
is an analytic function of 01,K = K(t, a). This follows from the theorem on differentiability of solutions of ordinary differential equations with respect to a parameter when applied to the Riccati equation K=
-KA-A~K+KSK-Q,
K(T+)=o,
(4)
where S = BR-‘B’. Let M = M(t, LX)b e a matrix defined by
The problem considered here is to compare the performance JM of the system $=(A-SM)x (6) with the state feedback regulator u = -R-IB’Mx,
(7)
to the performance Jg of the system R = (A - SK)x,
(8)
with the optimum state feedback regulator (3).
3. APPROXIMATION THEOREM The first 2m terms of the Taylor’s series for Jnr are equal to the corresponding 2m terms of the Taylor series for JK,
a”JK @Jhf 32 &)=aOLd &)’
i = 0, l,..., 2m - 1.
(9)
PROOF. The expression for Jg is well known to be [3]
JK=;x’Ks/t
t. -0
Similarly it can be shown (see for example, Ref. 4, pp. 81-84) that the performance JM is Jhr=;xfPx~
, t--to
(11)
AN
APPROXIMATION
THEOREM
FOR LINEAR
OPTIMAL
REGULATORS
251
where P = P(t, a) is the solution of the linear symmetric matrix equation I.’ = -P(A
- SM) - (A’ - MS) P - MSM - Q, P(T, a) = 0. (12)
Define r as r==p-K.
(13)
Then to prove (9) it is equivalent to prove that at (Y = 0 and t := to , r and its first 2m - 1 derivatives are zero matrices,
air(t, g ___ =o, ad t=tO
i = 0, 1, 2,..., 2m - 1.
(14)
a=0
From (5) K - M = cum/l,
(15)
where
Subtracting (4) from (12) and using (13) and (15),
I'=
-r~
- ix-dyrsA
+Asr)-2mAsA,
r(T,ol) = 0, (17)
where G = A - SK. This equation shows that (14) holds not only at t = to but also for all t E [to , T]. To see this, first check (14) for i = 0. That is, find the solution r(t, a) of (17) for (Y = 0. At a: = 0 (17) reduces to a homogeneous linear equation
I'=-m-m,
r(T,o) = 0,
(18) whose solution I’(t, 0) must be zero for all t, since it is zero at t = T. Next, denote derivatives by subscripts, ri = air/M, Gj = 8G/a& etc., i,j=O,1,2 ,... . To verify (14) for i = 1 differentiate (17) with respect to 01 and let 01= 0,
I', = -I-',G, -Girl
- FoGI - Gil', - mOm-l(roSoAo + A,S,r,),
where mom-l is 1 when m = 1 and is 0 when m > 1. Since the result of the verification for i = 0 is r, = 0, (19) becomes homogeneous and hence r, = 0. Using induction, verify whether the assumption of I’, = 0, for j = 0, I,..., i, makes the equation for I’,+l homogeneous, and determine the maximum i for which this is true. Differentiating (17) i + 1 times and letting 01= 0,
pi+, = --ri+,Go - G;ri,, -F,
ri+JT,O) = 0.
(20)
252
KOKOTOVIt
AND
CRUZ
For i < 2m - 1, each term in F is linear in some rj , j = 0, I,..., i. Since Pi = 0, for j = 0, l,..., i, F = 0 and the linear equation (2) is homogeneous. Thus ri+Jt, 0) = 0 because it is zero at t = T. This proves (14) and therefore (9).
REMARKS
The main content of the theorem is in establishing the order to which the system performance approximates the optimal performance if instead of the optimum gain matrix its truncated series M is used. It is sufficient that the order of the truncated series 1M is one-half of the desired order of the approximation of the optimal performance. Moreover, the parameter cr can be selected to make the system dimensional&y lower when a = 0, than when a > 0 (singular perturbation of a [5,6]). Although in this case some elements of the system matrices A and B have a pole at a = 0, it can be shown [7] that under mild conditions, K and P are analytic in a at a = 0. Therefore, the theorem of this paper extends to that class of “singular perturbation” problems.
1. H. J. KELLEY. An optimal guidance approximation theory. IEEE Trans. Automatic Control AC-9 (1964), 375-380. 2. R. A. WERNER AND J. B. CRUZ, JR. Feedback control which preserves optimality for systems with unknown parameters. IEEE Trans. Automatic Control AC-13 (1968), 621-629. 3. R. E. KALMAN. Contribution to the theory of optimal control. Bol. Sot. Matemutica Mexicanu, 1960, 102-I 19. 4. J. LASALLE AND S. LEFSCHETZ. “Stability of Liapunov’s Direct Method,” Academic Press, New York, 1961. 5. A. N. TIKHONOV. On the dependence of solutions of differential equations on a small parameter. Mat. Sbo7nik 22 (64), 193-204, Moscow, 1948. [See also, A. N. TIKHONOV. Systems of differential equations containing small parameters muftiplying some of the derivatives. Mat. Sbornik 31 (73), 575-586, Moscow, 1952.1 6. A. B. VASILEVA. Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivative. Russian Matk. Swweys 18 (1963), 13-81. 7. P. SANNUTI AND P. KOKOTOVIC. Near-optimum design of linear systems by a singular perturbation method. IEEE Trans. Automatic Control AC-14 (1969) (to appear).