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A method for suboptimal design of nonlinear feedback systems
Automatica, Vol. 7, pp. 703-712. Pergamon Press, 1971. Printed in Great Britain.
A Method for Suboptimal Design of Nonlinear Feedback Systems" Une m&hode de calcul sous-optimal de syst mes non-lintaires tt rtaction Eine Methode Zum Suboptimalen Entwurf Nichtlinearer Rtickkoppelungssysteme MeTo)I cy6onTnMaJibrioro paccqera n e y t n v i e f t m , I X CHCTeM C o6paTno~
CBIIabIO
Y. N I S H I K A W A , t N. S A N N O M I Y A t and H. I T A K U R A t
The e-parameter method, with the introduction of a n extended Liapunov equation, pro vides an effective tool for a suboptimal design of nonlinear feedback systems. The method applies also to a coupled systems design. Summary--An approximation method is presented to construct an optimal state regulator for a nonlinear system with quadratic performance index. The nonlinearity is taken to be a perturbation to the system, and a parameter e is introduced to stand for it. By making use of a power-series expansion in e, a sequence of partial differential equations is derived whose solutions form a suboptimal feedback law. Given a polynomial nonlinearity, the partial differential equations are reduced to ordinary differential equations by separation of variables. The zero-order terms yield a well known Riccati equation. Higher-order equations are transformed into conventional type linear equations, owing to a lemma regarding an extended Liapunov equation. It is demonstrated that the l-th order approximation for the feedback law results in the (2•+ 1)th order approximation to the optimal performance. The procedure developed has a wide variety of applications. As one of the straightforward applications, the synthesis of a suboptimal control is discussed for a largescale system as composed of several subsystems of lower dimensions. Three examples attached illustrate several features of the method.
AL'BREKHT [1] has shown a sufficient condition for obtaining the optimal feedback control of a nonlinear analytic system, and developed formally a recursive procedure to construct a suboptimal control in a power series in states. LuKF_s [2] has extended the work of Al'brekht and relaxed the analyticity condition to twice continuous differentiability. However, a closed form solution for the recursive procedure has not been demonstrated in their papers. PEARSON[3] and GARRARDet al. [4, 5] have developed several approximation methods. Their methods are essentially based on the idea utilizing a similar technique to that for the linear systems. This idea much simplifies the calculation; however, the best possible solution can not necessarily be obtained, except for a one-dimensional problem, because of the arbitrariness in the system description. In short, the methods of linearization are not necessarily effective in treating systems of high dimension.
INTRODUCTION THERE have been numerous studies on the optimal feedback control of linear dynamical systems or on the optimal design of linear regulators. On the other hand, however, relatively few works have been reported on nonlinear regulator problems, because of the difficulty in determining the exact optimal feedback law.
In this paper, a systematic procedure is presented to construct a suboptimal state regulator for nonlinear systems in which a parameter e is associated with nonlinearities. The e, parameter, called the perturbation parameter, is either inherent in the system or introduced for convenience. The nonlinearities are assumed analytic in the states. For simplicity, a performance index is considered to be quadratic in the state and control. For a sufficiently small value of e, a desired feedback law may be expanded and sought in a power series in e. The generating solution is the unperturbed solution, which is usually obtained by solving a Riccati type
* Received 27 July 1970; revised 12 February 1971; revised 18 May 1971. The original version of this paper was presented at the IFAC Symposium on Systems Engineering Approach to Computer Control which was held in Kyoto, Japan during August, 1970. It was recommended for publication in revised form by Associate Editor A. Sage. t Department of Electrical Engineering, Kyoto University, Kyoto, Japan. 703
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
704
equation. Correction terms for improving the feedback law are yielded by successive solutions of a sequence of linear partial differential equations. The method of separation of variables is used to reduce the partial differential equation to a linear ordinary differential equation. By using the notion of the Kronecker sum of matrices, the ordinary differential equation is written in a simple conventional form, so that it can readily be solved either analytically or numerically by a computer routine. When the system is time-invariant and a control duration is infinite, and if the unperturbed system is completely controllable, the differential equation is further reduced to an algebraic equation, whose solution is again straightforward. Although the present procedure of power-series expansion resembles, in some respects, that of Al'brekht and Lukes, the introduction of the eparameter is useful to give the recursive formulas in a much nearer fashion. Further, it facilitates the performance analysis to reveal the fact that, if the feedback law is optimal up to the l-th order in 5, the index of performance is minimized up to the (2l+ 1)th order. This is an extension of the theorem given by KOKOTOVI~ and CRUZ [6, 7] for linear systems, and is in accord with the theorem of WERNER and CRUZ [8] for systems with unknown parameters. The method offers also an efficient means for a suboptimal design of a class of large-scale systems. If a system is composed of weakly-coupled subsystems, generally containing nonlinearities, the parameter e could also be associated with the couplings of the subsystems. The basic idea of suboptimal design is discussed for N-coupled systems.
in t. A prime denotes transposition of a vector or a matrix. CONSTRUCTION OF A SUBOPTIMAL CONTROL It is well known that, for the unperturbed linear system, i.e. the system (1) with e = 0 , the optimal feedback control is given by [9]
u = - R - 1B'P(t)x
(3)
where the matrix P, symmetric, is the solution of the Riccati equation
[' = - P A - A'P + P E P - Q,
P(T)=0
(4)
with
E = B R - 1 B ' (: symmetric).
(5)
Since the exact determination of the true optimal control law is generally impossible for nonlinear systems, the present aim is to obtain a suboptimal feedback control for the nonlinear system (1). For a sufficiently small value of e, it may be reasonable to assume that the optimal feedback control be analytic in 5 for all t~[z, T]. Therefore we try to find a suboptimal control in a power-series form in 5. The Hamiltonian of the problem is given by
H=½x'Qx+½u'Ru+p'(Ax+ef+Bu)
(6)
where p(x, t) is the n-dimensional costate vector satisfying the differential equation
[~: _ t ~ H : _ Q x _ A, p_e(~___f~'p, PROBLEM STATEMENT Consider dynamical systems governed by the equation
Yc= A( t)x + ef(x, t) + B( t)u
(1)
where x is the n-dimensional state vector, u the m-dimensional control vector; A and B are n x nand n x m-matrices, respectively, continuous in t. The n-vector functionf(x, t), continuous in t, is an analytic function in x satisfying f(0, t ) = 0 for any t. 5 is a small scalar parameter. Here and throughout the paper a dot over a quantity denotes differentiation with respect to t. The problem is to find a feedback control law u(x, t) for which the quadratic index of performance
~
T
d(x, z) = ½
[x'Q(t)x + u'R(t)u]dt
(2)
is minimized. In equation (2), Q and R are symmetric positive definite matrices, both continuous
~x
\~x} p(x, T ) = 0 .
(7)
The matrix (dJ'/dx) is defined in such a way that the (i, j)-element of (t3f/dx) is (dfdt3xj). Due to the minimum principle, a necessary condition for the optimality is that the Hamiltonian be minimum with respect to u. Hence the optimal control is given by
u= - R - 1 B ' p .
(8)
Here we develop the costate vector p(x, t) into a power series with respect to e:
p(x, t)=P(t)x+g(x, t; e) =P(t)x+ ~ ~gC~)(x,t).
(9)
k=l
Differentiation of equation (9) with respect to t and use of equations (1) and (8) gives
A method for suboptimal design of nonlinear feedback systems
dO(~) Og(~) Ot +'-~x :ix + S' g(~)
= ekOO(t,) 1~=px+ k~=l := Ot +[p+
oo
705
O,,(k)-I
k~=tek2~x J(Ax+~f-Ep).
_
(10)
Substituting equations (9) and (10) into (7) and equating the coefficients of the like powers of e separately yields a sequence of equations:
X ~1: 63,,0) ~ + C%, ~ (1).Sx+S'90)= - OO_~x'Pf), (11-1) Ot Ox
Ol-
k-lg(O,Eg(k_O]. (13-k) _1
Hence, in each step of the successive calculation, it is required to solve the vector-matrix equation of the form:
~tg(x, t)+[}g(x, t)]S(t)x +S'(t)O(x, t)=fff-h(x, t) (14) Ox
~k :
O~(k) + c),-,(k) ~Ox Sx+S'o(')=-(Of~o('-t)
where #(x, t) is an unknown n-vector and h(x, t) is a given scalar. The function h is assumed to be, for the time being, a homogeneous polynomial of the degree, say, r + 1 in x, and is written as
\Ox]
Og(k-1) f
Ox k - 1 ,qrf(k-i)
+ ~ ~'~ i= 1
Eg(O
OX
(ll-k) (k=2, 3 . . . . ) with the boundary condition g(k)(x, (k = 1, 2 . . . . ). S is the matrix defined by
S=A-EP.
T) = 0 (12)
The zero-order equation is identical with equation (4). Successive solutions of equations (l l-k) with increasing k may determine the functions g(k)(x, t). Though equations (11) are linear in #(~), their solutions are difficult given a general form of the perturbation f. However, if f is a polynomial in x, g(h) are also polynomials in x and their coefficients are exactly determined under appropriate conditions. First the following theorem is established:
Theorem 1 If f (x, t) is a polynomial of the degree q in x, the coefficients being continuous in t, g(k)(X, t) given
h(x, t)=Hjl h.../,+ l(t)xjlxj2.., xi, +1" (15) Here and in what follows we use the summation convention that a twice repeated index is to be interpreted as a summation over that index from 1 to n. The coefficients Hjlj2...j,+l(t ) are assumed to be, without loss of generality, symmetric with respect to any pair of indices.* Hence there are (n + r) !/(n- 1) !(r + 1) ! different coefficients. The i-th component of is given by
Oh/Ox, the gradient of h,
a h = (r + 1)Hut .. " j r x j , . . . axi
xj.
From equation (14) it easily follows that the i-th component of g is a homogeneous polynomial of the degree r in x. Hence the i-th component of g may be written as ~I(X, t)-~G}, ...j,(t)xj,
(17)
. . . xj,
where the coefficients G are symmetric with respect to the lower indices. Substituting equations (16) and (17) into (14) leads to . . . j , d- r S k j l Gkj2 . . . j,"l- S k i G j l , . . . 1,
by equation (1 l-k) is the polynomial of the degree
- (r + 1)Hij , . . j . ] x y , . . , x j, = 0
k(q-1)+ 1 in x whose coefficients are continuous
( i = 1, 2 . . . . .
in t. In particular, g(1)(x, t) is the polynomial of the same degree as f(x, t). Secondly the stepwise solution of equations (11) will be discussed. The right side of equation (11-1) is integrable with respect to x. If (Og(°/Ox) are symmetric for i = I, 2 . . . . . k - 1 , then the right side of equation (l l-k) is also integrable with respect to x, and the equation is rewritten into
(16)
n).
(18)
Equations (18) hold for arbitrary values of x if and
* H are said to be symmetric with respect to any pair of indices, or simply with respect to indices, if the values of H are invariant with any interchange of indices. For example, H j 1 = i , j z = k , j 3 . . . j r + l = H J l = k . h = i , J3 . . . J r + t "
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
706
only ifthe coefficients o f x i , x j , . . , x~, vanish for any combinations of the values of indices. Hence we have n(n+r-1)!/(n-1)!r! simultaneous equations for G~..... j :
Theorem 2 guarantees the symmetry of {.3g(k)/63X) for every k > l . Then the successive solution of equations (11) is completed. By virtue of symmetry of (Og/Ox), equation (14) is equivalent to
i d}, ... j. + Skj,G~j,... j. + S ~ j f i m j , . . . j.
+
-~9( g x, t)+ O=-[xS'o'(Xv,x
k j~-~k+SktGjl...j,
• • • +Skj.G}~...
t)-h(x, t)]=O. (22)
-(r+l)Hij,...j =O (i, j~ . . . . . j , = 1, 2 . . . . .
n).
(19)
By virtue of Lemma in Appendix, a Lemma for an extended Liapunov equation, equations (19) are rewritten into the vector-matrix form:
If S and q in equation (20) are time-invariant, and if all the eigenvalues of S have negative real parts, ~(t) is uniformly asymptotically stable relative to 4o as t ~ - 0% where 40 is the solution of the linear algebraic equation S (' + I)~o= q.
~(t) + ~ {' + l)(t)~(t) = r/(t)
(20)
where S ('+x) is the r + 1 repeated Kronecker sum of S' [10]; ~ and r/are the n'+~-vectors such that the m-th components are G~°... j, and (,+ t)Hjoj,.., i,, respectively, m being
(Jk-- 1)n'-k+ 1. k=0
Now the following theorem can be established.
(23)
Due to Corollary L1 in Appendix, equation (23) is uniquely solvable under the given condition. Thus the following theorem is obtained:
Theorem 3 In equation (14), if S reduces to a stable constant matrix and h reduces to a form free of explicit dependence on t as t ~ - ~ , then the function #(x, t) is uniformly asymptotically stable as t ~ - 0% relative to the polynomial go(x) which satisfies the equation:
x'S'Oo(x)=h(x).
Theorem 2 The matrix (Oglc~x) given by equation (14) is symmetric, and consequently there exists a scalar function v(x, t) such that 0/3
g=-0x
for
te[z, T ] .
(21)
The coefficients of the polynomial 9 are determined by solving equation (20). Proof Since the coefficients H are symmetric with respect to all the indices, the coefficients G are also symmetric with respect to all the indices, including the upper and the lower ones. This is due to Corollary L2 in Appendix. It implies that (Ogi/Oxj)=(Ogj/Oxi) for any pair of i and j. When (O0/Ox) is symmetric, there exists a scalar function v satisfying equation (21) [11].
(24)
The coefficients Gji .... Jr of the polynomial #0(x) are given by equation (23). A direct consequence of Theorem 3 is:
Corollary 1 If the unperturbed linear system is completely controllable and time-invariant, all the eigenvalues of S = A - B R - 1 B ' P ha~e negative real parts as t ~ - ~ . Then, for every k > 1, 9(k)(x, t), the polynomial function in x, is uniformly asymptotically stable as t ~ - 0% relative to the polynomial 9(k)(x) which satisfies the equation: for k = l
x'S'o°)=-x'Pf,
for k ~ 2
x,S,g(k)= _f,g(k-1)+½ ~_~ g(1),Eg(k-i) "
(25-1) k-1
i=1
(25-k)
Q.E.D.
The polynomial function o~k)(x), satisfying equation (25-k), exists and is unique for every k > 1.
In the above discussion, the function h is assumed to be a homogeneous polynomial in x. If h is composed of terms of different degrees, the calculations should be done separately for different degree terms. Due to the linearity of equation (14), the total solution is obtained by simply summing up the resulting 9 polynomials.
Now we have comoleted the procedure for calculating a suboptimal control. The solution of equations (11) or equations (25) up to the order l in e gives the suboptimal control of the l-th order. It is noted that, when T--* oo, the complete controllability of the unperturbed system suffices to yield a suboptimal control of an arbitrary order.
A method for suboptimal design of nonlinear feedback systems
707
[Ovtl)\ x'S'~-~-x )=-x'Pf'
(32-I)
Further the following corollary is useful in the next section.
Corollary 2
~k :
If the equation
x'S' g=O
81-
=
or
g'Sx=O
\
)
0x
k-I
(26)
-½ i=l Z o(°'E0
holds for any value of x with a stable S, and if (OglOx) is known to be symmetric, then g is identically a zero vector.
k-lg(O'E(OV(k-O~ i=1 +Z
(k=2, 3 . . . . PERFORMANCE EVALUATION
Theorem 4 If g(x) is the optimal one, i.e. if g(k)=g *(k) for every k > 1, where g *(k) is the solution of equation (25-k), then J is given by
J=J*=kx'Px+ ~ ekv*(k)[x(~)]
(27)
where gs(x) is a suboptimal feedback function satisfying equations (25) up to a certain order. The value of the performance index J resulting from the feedback control of equation (27) would be a function of x(z) and be written as
Js= ½f ~ (x'Qx + u'sRus)dt =vs[x(z)] .
).
Then the following theorem can be established:
In this section the quality of a suboptimal control law is examined. For simplicity we consider only a completely controllable time-invariant system. As shown in Corollary 1, when T ~ ~ the suboptimal feedback control is free of explicit dependence on t and is given by
us(x ) = - R- 1B' gs(x)
(32-k)
\Ox)
(28)
Evidently vs(0)=0 if gs(0)=0. Now the task is to examine the function vs(x). Differentiation of equation (28) with respect to z and use of equations (1) and (27) gives:
(33)
k=l
where v*(k)(x) is the scalar function satisfying Or*(k)
Ox
= g *(k) for every k ~ 1.
(34)
Theorem 4 can readily be proved inductively by comparing equations (32) with equations (25). Further the following theorem of interest is obtained.
Theorem 5 If g(x) is optimal up to the order I in 8 and is truncated after that, then (i) J is equal to J* up to the order 2/+ 1 in 8.
x Qx+g
Eg+2l~xlk](Ax+ef-Eg)=0.
(29)
In equation (29) all the values are estimated at t = ~. The subscript s is omitted here and throughout the following part of this section, but should be understood. We develop g and v into power series in e:
g(x)= Px + ~ and
k=l oo
ekoCk)(x) 1 (30)
v(x) = ½x'Px + k~=l ekvtk)(x) where the matrix P is the solution of
PA + A ' P - P E P + Q = 0 .
(31)
Substitution of equations (30) into (29) results in a sequence of equations:
(ii) v(2t+2), the (2/+2)th term in the expansion of J, is given by xtS,[ -0t)(21+2)
L
g,(21+2)1_ -½a*°+x)'Eg
Proof (i) Again the theorem is proved by induction. First, corresponding to l=0, assume that g(x)=Px. From equations (25-1) and (32-1), it is readily observed that v(1)= v*(~). Second, assume that, if gti)=g.(O for i=l, 2, . . . . l - 1, v(°= v*(° for i= 1, 2 . . . . , 2 1 - 1. Besides if g(t)=g.(t), comparison of equation (32-k) with (25-k) for k = 21leads to x'S'[(Ov(Zt)/Ox) - g,(2t)] _- 0. Due to Corollary 2, it implies that v(2t)= v*(2/). Further, examination of equations (25-k) and (32-k) for k = 21+ 1 leads to v(2t+ l) = v,(2t+ 1). (ii) Comparison of equation (32-k) with (25-k) for k = 2 / + 2 yields equation (35). Q.E.D.
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
708
Theorem 5 is an extension of the theorem obtained by KOKOTOVIt~and CRUZ [6, 7] for linear systems. A similar result has been reported by WERNER and CRUZ [8] regarding an approximate construction of the optimal feedback control for systems with unknown parameters; where the control is expanded in a Taylor series with respect to the unknown parameters. However the present proof is given in a much simpler way by use of induction. It is noted that Theorems 4 and 5 apply also to a time-varying system with a finite T.
Example 1 Consider the third-order system governed by:
~IX2X3
~
~2X3XI
L ~3XIX2
+
u2
iI ul
(38)
u3
with the index of performance APPLICATION
TO A LARGE-SCALE
SYSTEM
The method developed so far has a wide variety of applications. One of the straightforward applications is the synthesis of a suboptimal control for a large-scale system as composed of several subsystems of lower dimensions. Consider the problem of optimizing the system
J = ½ y f (x'Qx+u'Ru)dt, where
(39)
Q=diag(ql, q2, q3), R=diag(rx, r2, r3), and q~, r~>0.
£i=Aixi+efi+Biu~
(i=1, 2 . . . . .
N)
(36)
with respect to the performance index
J = ½ i=t ~
f: (x[Qix~+u~Rtu~)dt
(37)
where x~ is the ncvector of the i-th substate, and u~ is the m~-vector of the i-th subcontrol. An analytic function f~, representing the coupling of the subsystems, may depend on all the substates xi(i= 1, 2 . . . . , N). At, B~, Qi and Ri are matrices of the appropriate dimensions. A particular problem in which all the couplings f~ are linear has been discussed by KOKOTOVI(~et al. [6] and the present authors [12], independently. The method of the preceding sections has much enlarged the range of applications. The first step of the solution involves the solution of lower-order Riccati equation of each subsystem. In the subsequent steps one has only to solve linear equations. This fact greatly relaxes the complexity of computation. Evidently the foregoing theorems apply to the composite system of equations (36). Especially, the complete controllability of all the subsystems is sufficient to guarantee the uniqueness of an improved solution.
ILLUSTRATIVE
It is well known that, if ~l + ~2 + ~3 =0, equation (38) describes the rotational motion of a rigid body. First, the suboptimal control law is calculated by the present procedure up to the third order:
U - - IJ (0) .4- p.lJ (1) 4- ~2~t(2) -1- ~3sn(3) i----i _ _ v _ i __v --i - - v Hi
where
,,I°'= -,/ q,lrlx,, U~ 1) =
C(1)XjXk/ri,
(2)_
Ut
(2)
2
(2) 2
--Xi[C k Xj~I-Cj Xk]/r i,
U(3) __ ~. ~ 1"1.,(3),,2 -1- , . ( 3 ) y 2 +
i
--'~j~'ktJ~i "vi "~'j ~j
c~3)x2]/ri
3
c(')= Z ct,x/q,r,/a, l=l
ctl)[o~i+c(1)/2ri]
(2)
Ci
-
-
tr--4qi/r i
c~3)_ [¢xj+ c (')1r i]c~2) + [gk + C~t)/rk]C~2) a + 2x/qi/rl
EXAMPLES
Three examples are presented to illustrate the application of the present method. The suboptimal controls are compared with those obtained by other methods.
3
a= Z
1=1
x/q,/r, (i,J, k = l , 2, 3; i # j # k # i ) .
(40)
A method for suboptimal design of nonlinear feedback systems It follows from equations (40) that, if e ° ) = 0 , u(k)=0
(41)
709
Example 2
In order to examine the influence of truncation in the power-series solution, consider the secondorder system:
for every k_->l. This implies that u = u (°) is the exact optimal control for this case. KUMAR [13] previously discussed a special case of c ° ) = 0 , i.e.
Xl=X2'
t
(48)
X2=SX~+U
qx =q2 =q3 = 1,
r 1 =r2=ra=K,
and ~x + ~2 '[- 0~3~ O.
(42)
subject to the performance index
Secondly, the method of instantaneous linearization by PEARSON [3] is applied to equation (38). According to the method, the nonlinear system (1) is rewritten into a linearized one such as
d=
q
oo
0
(x 2 + x 2 + u2)dt.
(49)
The suboptimal control of the second order is ~=~(x, t)x+B(t)u
with
4(0,
t)=0 U~ U(0)"1-gU(1)"1-g2U(2)
where .4 is an n x n state-dependent matrix. The state-dependent matrix ,4 for the system (38) is here assumed in the form:
where
U(0) = - x 1-1"7321x2, I
.~=~
0
¢tlVlX3
~2(1 --V2)X3
0
O~3V3X2
gl(1 -- Vl)X2 1 O~2V2X1
~t3(1 -- V3)X1
U(1)= - x ~ - l ' 3 2 3 5 X ~ X z - O ' 6 9 2 3 x l x ~
(50) -0"1323x~,
0 (43)
where v~(i= 1, 2, 3) are arbitrary constants. Then the suboptimal control is given by
//(2)= - 0 " 5 x { - O ' 7 3 9 2 x ~ x 2 - O ' 3 3 2 0 x ~ x ~ +O,O190x~x~+O'O523xlx~
+0"0112x~. (44)
u = - R - 1Px
where P is the positive definite solution of the following quadratic equation: A'P+PA-PR-
1p+ Q = 0 .
(45)
In order for equation (45) to have the exact solution with the parameters of equations (42), it is required that
iv1 iv 0iill 1 -vx
0
v3
~2
0
v2
1 - v3
a3
----0. (46)
Equation (46) holds for nontrivial values of the parameters st, if and only if arbitrary constants v~ satisfy the following relation: V1p2 -~t-p2p3 -~--k'aVl--[-1 = P1 -~t-V2-]--V3.
(47)
It is noted that the form of ~ should skillfully be chosen in order to obtain a good solution.
Now the stabilization property of the controller is examined. With u = u t°), the origin of equations (48) is globally asymptotically stable with e<0, while it is only locally asymptotically stable with 8 > 0. In fact, for e > 0, there exist unstable singular points at (+x/l/e, 0), and trajectories passing through either of these points give the stability boundaries. With u = u ( ° ) + e u (1), the origin becomes locally stable for e < 0, so that the first-order approximation gives a better result than the zeroorder one only in a limited region around the origin. On the other hand, with e > 0, the first-order controller stabilizes the origin in the large. The second-order controller is of a locally stabilizing nature for both signs of e. Figure 1, showing the loci of the initial states for which J = 5, reflects the stabilization property mentioned above. The curves are shown only on the upper-half plane, because they are symmetrical about the origin. Within the regions illustrated, the higher-order approximations improve the performance with e= 1, but it is not necessarily with e=-l.
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
710
Xl =X2,
I ..... -
-
-
-
-
~
:
by the zero-order control
:
by the first-order control
:
by the second-order control
/t,,t<
X2 = g'CX4 + gX4X 6 + U 1 -{" 13X3 U 3,
!
"~3= X4, ~~'4 =
3
--
(51)
~X2X6 + 112 -- ~X1123,
X6 = EX2X4 "F 123 + ~X 1 ~12 •
1 ,
0 -3
-2
\',,
\\
-1
0 X1
The quantities xl, x 3 and x 5 represent the roll, yaw and pitch motion, respectively, of the body about its principal axes. c is a constant associated with the orbiting angular velocity. The performance index is taken to be
,
1
2
=
(a) e= 1
'
-2
-I
-- 0
i
When e = 0 , the system of equations (51) is decoupled into three subsystems, each of them being a double-integral system. The method for composite systems can be applied then. Although equations (51) contain the product terms of x and u, the method still works effectively. In fact, if an additional term eB(')(x, t)u exists on the right side of equation (1), there appear terms containing B (~t and 9 (t) on the right side of equation (ll-k) ( / < k ; k = 1, 2 , . . . ); while its left side is kept unchanged. Then the recursive calculation proceeds in much the same way.
2
X I
(b) e - - : FIG. 1.
~CX2
X5 = X 6 ,
2
0-3
--
The loci of the initial states resulting J =
5.
Figure 2 illustrates examples of variation of J with e for various approximations.
Example 3
Table 1 compares values of J obtained by the zero- and the first-order control for various values of e when c = l . The first-order control affords better results even for moderately large e.
The following equations approximately describe a three-axis attitude-control system of an orbiting space body [14]:
,.8
/
I
1"4
....
:
by the zero-order control
J--J
i
by the f i r s t - o r d e r
:
by the second-order
I I t /
control
~o.trol
I
I
///
,/,
. ,/./
js p sI
1,0
0.E -i'0
1 -0-5
I 0"5
2. Variation of Jwith e for various approximations of the control (Initial condition: x 1= 1, x2=0).
FIG.
1.0
A m e t h o d for suboptimal design o f nonlinear feedback systems TABLE 1. COMPARISON OF VALUES OF PERFORMANCE INDEX jr. J0 : J BY THE ZERO-ORDER CONTROL ; J l : J BY THE FIRST-ORDER CONTROL
(a) Initial condition: xt =x3 =x5=2, X 2 : X 4 : X 6 : 0 0"1 0"2 0'5 1"0 2"0 5'0 J0
10.26
10"14
J1
10"23 10"06
9 " 9 6 11"28 17"84 9"78 10-26 11'61
14"58
(b) Initial condition: xl=xa=xs=O, X2=X4=X6:2 e 0-1 0-2 0"5 1"0 2'0 5'0 J0
10"53 10"61 10'79
J1
10"51 10"54 10"43
10"82 10'44 9"92
8'95
11-28 7"74
CONCLUSION A systematic procedure has been developed for the suboptimal design of a nonlinear state regulator. When the nonlinearity is characterized by a polynomial function in the state, the definite way to determine a suboptimal feedback control o f a power-series form is established by Theorems l and 2. The stability property o f the suboptimal feedback function is discussed in T h e o r e m 3. The contents o f T h e o r e m 3 as well as o f L e m m a in Appendix could also be used to construct a desired polynomial function, e.g. a desired Liapunov function. Theorems 4 and 5 clarify the approximation property o f the suboptimal policy qualitatively. However the quantitative estimation o f the performance degradation is generally difficult. When the nonlinearity is large, the convergence o f the series-form solution is not assured, and higher-order approximations do not necessarily give better results. The m e t h o d developed provides a promising tool for a variety o f problems, including linear or nonlinear, deterministic or stochastic systems optimization. A n application to a composite system, whether it is linear or non linear, is straightforward. The effectiveness o f the method is illustrated by examining typical examples. Since the procedure is completely systematic, it can efficiently be used in a c o m p u t e r - p r o g r a m m e d computation.
711
[7] P. V. KOKOTOVIt~and J. B. CRUZ, JR. : An approximation theorem for linear optimal regulators. J. Math. Anal. Applic. 27, 249-252 (1969). [8] R. A. WERNERand J. B. CRUZ, JR.: Feedback control which preserves optimality for systems with unknown parameters. 1EEE Trans. Aut. Control AC-13, 621-629 (1968). [9] R. E. KALMAN:Contributions to the theory of optimal control. Bol. Soc. Math. Mex. 5, 102-119 (1960). [10] R. BELLMAN: Introduction to Matrix Analysis, pp. 227-231. McGraw-Hill, New York (1960). [11] P. M. MORSEand H. FESHBACH: Methods of Theoretical Physics, pp. 14-15. McGraw-Hill, New York (1953). [121 Y. NISHIKAWA, N. SANNOMIYA, H. ITAKURA and T. FUKUDA" Suboptimal feedback control of weakly coupled systems. Preprints of Symp. on Systems Engng., Soc. of Instrument and Control Eng. of Japan, Tokyo, pp. 9-12 (in Japanese, 1969). [13] K. S. P. KUMAR: On the optimum stabilization of a satellite. IEEE Trans. Aerospace and Electronic Systems AES-1, 82-83 (1965). [14] Y. NISmKAWA,C. HAYASHIand N. SANNOMIYA: Fuel and energy minimization in three dimensional attitude control of an orbiting satellite. Proc. IFAC Symp. on Peaceful Uses of Automation in Outer Space, Stavanger, Norway, pp. 287-298 (1965). [15] Y. NlSmKAWA, N. SANNOMIYAand H. ITAKURA: Suboptimal design of a nonlinear feedback system. Memoirs of the Faculty of Engineering, Kyoto University, Voi. XXXII, pp. 334-347 (1970). APPENDIX Lemma.for deriving equation (20) The following lemma regarding an extended L i a p u n o v equation provides a basis for rewriting equation (19) into equation (20): Lemma Consider the n ' algebraic equations for n ' quantities Xjt.,.jr: Ckj,Xkj2... jr + Ckj2Xj~k,,3... Jr + • • • + CkjrXj~...Jr-,k = Dy,...Jr
(A. 1)
where Cij and Dj~... jr are given quantities. All the indices run from 1 to n. Equations (A.1) can be rewritten into the vectormatrix f o r m : C~'J~ ~')= q(') (A.2) where t~(')=C ' ~ C ' ~ . . . ~ C ' . (A,3) r
REFERENCES [1] E. G. AL'BREKHT: On the optimal stabilization of nonlinear systems. J. Appl. Math. Mech. (PMM) 25, 1254-1266 (1961). [2] D.L.LuKES: Optimal regulation ofnonlinear dynamical systems. SIAMJ. Control 7, 75-100 (1969). [3] J. D. PEARSON: Approximation methods in optimal control, I. suboptimal control. J. Electron. Control 13, 453--469 (1962). [4] W. L. GARRARD,N. H. McCLAMROCHand L. G. CLARK: An approach to sub-optimal feedback control of nonlinear systems. Int. J. Control 5, 425-435 (1967). [5] W. L. GARRARD: Additional results on sub-optimal feedback control of non-linear systems. Int. J. Control 10, 657-663 (1969). [6] P. V. KOKOTOVIt~,W. R. PERKINS,J. B. CRUZ, JR. and G. D'ANS: e-coupling method for near-optimal design of large-scale linear systems. Proc. IEE 116, 889-892 (1969).
C is the n x n-matrix such that the (i, j)-element is Cij; ~ ' ) and r/(r) are the n'-vectors o f which the m-th components are X j , . . . j , and D j , . . . j r , respectively, rn being (jk--1)nr-k + l . k=l
The symbol @ denotes the Kronecker sum of matrices.
eroo/ Let us proceed inductively. equations (A.1) reduce to
First, for r = 2 ,
Ckjl Xkj2 "~ CkjlXjl k = Dj,j2.
(A,4)
712
Y. NISHIKAWA, N. SANNOMIYA a n d H. ITAKURA
By i n t r o d u c i n g the matrices X a n d D whose ( i , j ) elements are X~j a n d D~j, respectively, equations (A.4) are rewritten into
C ' X + X C = D.
(A.5)
In terms o f ¢(2) a n d ~/~2) as defined in L e m m a , e q u a t i o n (A.5) is equivalent to [10] (C,(~Ct)~(2)=
(A.6)
q(2}.
Secondly we show t h a t L e m m a is true for r + 1 if it is true for r. F o r r + 1, e q u a t i o n s (A.1) are written as [Ckj~Xkj~
• • . Jr+t + C k j z X j l R
+ CkjrXjt...
• • • Jr+~ + " " "
J r - ,kjr+ 1] + C k j . + I X j l . . .
jrk
=Dj . . . . ~,+~.
(A.7)
T h e terms in the b r a c k e t are identical, for a fixed value of.j,+ t, with those on the left side o f e q u a t i o n s (A.1). Hence, with use o f ~('+~) and r/("+t) equations (A.7) are rewritten into [1~") ® t ~ t ' ) + C ' ®I("')]~'+l)=q (~+~)
(A.8)
where the s y m b o l ® denotes the K r o n e c k e r p r o d u c t o f matrices [10], a n d I ~k) is the k x k identity matrix. D u e to the definition o f the K r o n e c k e r sum [10], e q u a t i o n (A.8) is equivalent to (~(r+ 1)~(r + 1)=/,](, + 1).
(A.9) Q.E.D.
The following c o r o l l a r i e s were used to establish T h e o r e m s 2 a n d 3:
Corollary L1--Uniqueness of the solution Let 2~ ( i = l, 2, . . . , n) be the eigenvalues o f C' o r equivalently o f C. T h e n if a n d only if a n y sum o f r eigenvalues o f C, 2 ~ + 2 ~ + ... +2~, (Jx, J2 . . . . rJ, = 1, 2 . . . . . n), is n o t zero, X s . . . . ~, o f e q u a t i o n s (A.1) are uniquely d e t e r m i n e d . In p a r t i c u l a r , if C is a stable matrix, X~, ...~, are unique.
Corollary L2--Symmetry with respect to the indices I f D ~ . . . ~ . are symmetric with respect to the indices, Xj . . . . j , are also symmetric. F o r m o r e details o f L e m m a a n d Corollaries, refer to Ref. [15]. E q u a t i o n (A.1) is a linear e q u a t i o n to determine the n-index quantities X given the two-index quantities C a n d the n-index quantities D. F o r the special case o f n = 2 , the so called L i a p u n o v e q u a t i o n , i. e., e q u a t i o n (A.5), is obtained. Hence equation ( A . I ) is considered an extended version o f the L i a p u n o v equation. In fact it can p r o v i d e a L i a p u n o v function o f the n-form for linear systems. Rrsumr--L'article prrsente une mrthode d'approximation pour construire un rrgulateur d'rtat optimal pour un systrme non-linraire avec indice de performance quadratique. La non-linrarit6 est considrrre comme une perturbation du systrme et un paramrtre s est introduit pour la reprrsenter.
En utilisant un drveloppement en srrie selon les puissances de ~, on obtient un systrme d'rquations diffelentielles aux d6rivbes partielles dont les solutions fournissent une Ioi de rraction sous-optimale. Etant donnre une non-linrarit6 polynomiale, les 6quations differentielles aux drrivres partielles sont reduites ~ des 6quations differentielles ordinaires par srparation des variables. Les termes d'ordre zrro fournissent une 6quation bien connue de Riccati. Les 6quations d'ordres sup6rieurs sont transformdes en 6quations linraires du type conventionnel grace /tun lemme se rapportant /t une 6quation grnrralisre de Liapunov. I1 est montr6 que l'approximation du l-irme ordre pour le loi de rraction correspond/t l'approxirnation du (2•+ 1)-i~me ordre pour la performance optimale. Le procrd6 developp6 possrde une large variet6 d'applications. A titre d'une des approximations directes, I'article discute la synthese d'une commande sous-optimale pour un systrme/t grande 6chelle considrr6 comme 6tant compos6 de plusieurs sous-syst~mes de dimensions infrrieures. Trois exemples inclus illustrent plusieurs caractrristiques de la mrthode. Zummmenfassmalg--Es wird eine Anniiherungsmethode zum Aufbau eines optimalen Zustandsreglers f'tir ein nicht lineares System mit quadratischem Leistungsindex dargestellt. Die Nichtlinearit~tt wird als StSrung des Systems angesehen, und es wird ein Parameter e dafilr eingef'tihrt. Unter Verwendung einer Potenzreihenentwicklung in ~ wird eine Reihe partieller Differentialgleichungen abgeleitet, deren Lrsungen ein suboptimales Rtickkoppelungsgesetz formen. Bei gegebener polynomischer Nonlinearit/~t werden die partiellen Differentialgleichungen auf gewrhnliche Differentialgleichungen durch Trennung von Veriinderlichen zurtickgefiihrt. Die Null-Grad Ausdrticke fiihren auf eine gut bekannte Riccati Gleichung. Gleichungen hSheren Grades werden infolge eines Hilfsatzes in Bezug auf eine erweiterte Liapunov Gleichung in lineare Gleichungen normaler Art umgewandelt. Es wird gezeigt, dass die Ann~iherung l-ten Grades ftir das Riickkoppelungsgesetz die Ann~iherung (21+ 1) ten Grades fiir die optimale Leistung ergibt. Das entwickelte Verfahren hat eine grosse Auswahl von Anwendungszecken. Die Synthese einer suboptimalen Regelung wird als einer der einfachen Anwendungszwecke fiJr ein System yon weiten Ausmassen besprochen, das sich aus mehreren Subsystemen kleinerer Ausmasse zusammensetzt. Drei beigeftigte Beispiele illustrie;en mehrere Merkmale der Methode. Ve31oMe---CTaTbR npemIaraer MeToJI npx6rmxceHaa mm KOHCTpyHpOBaHlifl peryJ~Topa OnTHMa.rmHOrO COCTOmma
JLrDIHeJI~He]TDIO~CE[CTeMblC KBa~paTIItleCKHMiloxa3aTeJleM pa6oTbL HeJIHHefiHOCTb paCCMaTpHBaeTca KaK BO3MyLKeHHe CltCTeMbl H BBOJIIITCHnapaMeTp ~ JIJI~ e r o IIQHHIITHII B ytler.
Idcr~oJib3y~ pa3noxeHHe no p~/laM no eTeHeH~M ~, nony,laeTca C14CTeMa/m~claepemma.rm141axypaBrmnalt c qacrHbIM14 nporI3t~O/mbXM14, pemea-mt roTOpSlx ~IatOT 3aKOH cy6OIITHMaJIbHO~ orpaT14Oit CBa3H.
B IIp14cyTCTBHIt MItOFO-
qnenHot~ nen14rleflHOCT14,~lrtqb~epemma.rmrlueypaane1414~ c qaCTH/~IMHnpoH3BOBMbIM14CBO~ITCil K O 6 ~ ~14qbqbepeH1JjlaJIb14blM ypaBHe14FmM nyTeM pa3~eBeFn~R HepeMeHHblX. LIJIeHbl HyJleBOrO nopa~xa ~ta~T o6tueH3BecT8oe ypaB14eHne PrmgaTm YpaBHerma BbICIIIItXnopKLiKOB npeBpama~Tea B
HltHe~trible ypaBHeI~I~l o6binuoro THIIa 6maroztapM neMMe OT~OCaRteilca x O606OmeHHOMy ypaBHermto JI~ny~oaa. Hora3smacTca qTO np146nvoKen14e l-oro nopa~Ka )Inn 3aroHa o6paT8Ol~ CBa3H COOTBeTCTByeT np1463m~re1414vo (21+l)-oro nopmlra /L~IS orrraManbHoli pa6oTbI. Pa313HTbn~ MeTO~ HMee~r maporoe qacao 14prrMeae1414~. B Ka~eCTBe o~noro 143 HenOCpeJlcTaenltblX 14pmvte14eH14i~ CTaTbll orcyx~aer C14HTe3 cy6-O14TnMa.rlbHOrO ynpaB~eHH~ /IH~ m14pOKOMaeLUTarHoi:l CHCTeMbI paccMaTp14BaeMoi~i r a t COCTOIIUleIi I43 p~.Jm cyr-c14c'reM Me14bm14Xpa3MepOB.
Tp~ BKn~oqem~b~X npnMepa EmnV0CTp14py~oTnecgo~'~bgo xapaxTep14eTnK MeTO~a.
A Method for Suboptimal Design of Nonlinear Feedback Systems" Une m&hode de calcul sous-optimal de syst mes non-lintaires tt rtaction Eine Methode Zum Suboptimalen Entwurf Nichtlinearer Rtickkoppelungssysteme MeTo)I cy6onTnMaJibrioro paccqera n e y t n v i e f t m , I X CHCTeM C o6paTno~
CBIIabIO
Y. N I S H I K A W A , t N. S A N N O M I Y A t and H. I T A K U R A t
The e-parameter method, with the introduction of a n extended Liapunov equation, pro vides an effective tool for a suboptimal design of nonlinear feedback systems. The method applies also to a coupled systems design. Summary--An approximation method is presented to construct an optimal state regulator for a nonlinear system with quadratic performance index. The nonlinearity is taken to be a perturbation to the system, and a parameter e is introduced to stand for it. By making use of a power-series expansion in e, a sequence of partial differential equations is derived whose solutions form a suboptimal feedback law. Given a polynomial nonlinearity, the partial differential equations are reduced to ordinary differential equations by separation of variables. The zero-order terms yield a well known Riccati equation. Higher-order equations are transformed into conventional type linear equations, owing to a lemma regarding an extended Liapunov equation. It is demonstrated that the l-th order approximation for the feedback law results in the (2•+ 1)th order approximation to the optimal performance. The procedure developed has a wide variety of applications. As one of the straightforward applications, the synthesis of a suboptimal control is discussed for a largescale system as composed of several subsystems of lower dimensions. Three examples attached illustrate several features of the method.
AL'BREKHT [1] has shown a sufficient condition for obtaining the optimal feedback control of a nonlinear analytic system, and developed formally a recursive procedure to construct a suboptimal control in a power series in states. LuKF_s [2] has extended the work of Al'brekht and relaxed the analyticity condition to twice continuous differentiability. However, a closed form solution for the recursive procedure has not been demonstrated in their papers. PEARSON[3] and GARRARDet al. [4, 5] have developed several approximation methods. Their methods are essentially based on the idea utilizing a similar technique to that for the linear systems. This idea much simplifies the calculation; however, the best possible solution can not necessarily be obtained, except for a one-dimensional problem, because of the arbitrariness in the system description. In short, the methods of linearization are not necessarily effective in treating systems of high dimension.
INTRODUCTION THERE have been numerous studies on the optimal feedback control of linear dynamical systems or on the optimal design of linear regulators. On the other hand, however, relatively few works have been reported on nonlinear regulator problems, because of the difficulty in determining the exact optimal feedback law.
In this paper, a systematic procedure is presented to construct a suboptimal state regulator for nonlinear systems in which a parameter e is associated with nonlinearities. The e, parameter, called the perturbation parameter, is either inherent in the system or introduced for convenience. The nonlinearities are assumed analytic in the states. For simplicity, a performance index is considered to be quadratic in the state and control. For a sufficiently small value of e, a desired feedback law may be expanded and sought in a power series in e. The generating solution is the unperturbed solution, which is usually obtained by solving a Riccati type
* Received 27 July 1970; revised 12 February 1971; revised 18 May 1971. The original version of this paper was presented at the IFAC Symposium on Systems Engineering Approach to Computer Control which was held in Kyoto, Japan during August, 1970. It was recommended for publication in revised form by Associate Editor A. Sage. t Department of Electrical Engineering, Kyoto University, Kyoto, Japan. 703
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
704
equation. Correction terms for improving the feedback law are yielded by successive solutions of a sequence of linear partial differential equations. The method of separation of variables is used to reduce the partial differential equation to a linear ordinary differential equation. By using the notion of the Kronecker sum of matrices, the ordinary differential equation is written in a simple conventional form, so that it can readily be solved either analytically or numerically by a computer routine. When the system is time-invariant and a control duration is infinite, and if the unperturbed system is completely controllable, the differential equation is further reduced to an algebraic equation, whose solution is again straightforward. Although the present procedure of power-series expansion resembles, in some respects, that of Al'brekht and Lukes, the introduction of the eparameter is useful to give the recursive formulas in a much nearer fashion. Further, it facilitates the performance analysis to reveal the fact that, if the feedback law is optimal up to the l-th order in 5, the index of performance is minimized up to the (2l+ 1)th order. This is an extension of the theorem given by KOKOTOVI~ and CRUZ [6, 7] for linear systems, and is in accord with the theorem of WERNER and CRUZ [8] for systems with unknown parameters. The method offers also an efficient means for a suboptimal design of a class of large-scale systems. If a system is composed of weakly-coupled subsystems, generally containing nonlinearities, the parameter e could also be associated with the couplings of the subsystems. The basic idea of suboptimal design is discussed for N-coupled systems.
in t. A prime denotes transposition of a vector or a matrix. CONSTRUCTION OF A SUBOPTIMAL CONTROL It is well known that, for the unperturbed linear system, i.e. the system (1) with e = 0 , the optimal feedback control is given by [9]
u = - R - 1B'P(t)x
(3)
where the matrix P, symmetric, is the solution of the Riccati equation
[' = - P A - A'P + P E P - Q,
P(T)=0
(4)
with
E = B R - 1 B ' (: symmetric).
(5)
Since the exact determination of the true optimal control law is generally impossible for nonlinear systems, the present aim is to obtain a suboptimal feedback control for the nonlinear system (1). For a sufficiently small value of e, it may be reasonable to assume that the optimal feedback control be analytic in 5 for all t~[z, T]. Therefore we try to find a suboptimal control in a power-series form in 5. The Hamiltonian of the problem is given by
H=½x'Qx+½u'Ru+p'(Ax+ef+Bu)
(6)
where p(x, t) is the n-dimensional costate vector satisfying the differential equation
[~: _ t ~ H : _ Q x _ A, p_e(~___f~'p, PROBLEM STATEMENT Consider dynamical systems governed by the equation
Yc= A( t)x + ef(x, t) + B( t)u
(1)
where x is the n-dimensional state vector, u the m-dimensional control vector; A and B are n x nand n x m-matrices, respectively, continuous in t. The n-vector functionf(x, t), continuous in t, is an analytic function in x satisfying f(0, t ) = 0 for any t. 5 is a small scalar parameter. Here and throughout the paper a dot over a quantity denotes differentiation with respect to t. The problem is to find a feedback control law u(x, t) for which the quadratic index of performance
~
T
d(x, z) = ½
[x'Q(t)x + u'R(t)u]dt
(2)
is minimized. In equation (2), Q and R are symmetric positive definite matrices, both continuous
~x
\~x} p(x, T ) = 0 .
(7)
The matrix (dJ'/dx) is defined in such a way that the (i, j)-element of (t3f/dx) is (dfdt3xj). Due to the minimum principle, a necessary condition for the optimality is that the Hamiltonian be minimum with respect to u. Hence the optimal control is given by
u= - R - 1 B ' p .
(8)
Here we develop the costate vector p(x, t) into a power series with respect to e:
p(x, t)=P(t)x+g(x, t; e) =P(t)x+ ~ ~gC~)(x,t).
(9)
k=l
Differentiation of equation (9) with respect to t and use of equations (1) and (8) gives
A method for suboptimal design of nonlinear feedback systems
dO(~) Og(~) Ot +'-~x :ix + S' g(~)
= ekOO(t,) 1~=px+ k~=l := Ot +[p+
oo
705
O,,(k)-I
k~=tek2~x J(Ax+~f-Ep).
_
(10)
Substituting equations (9) and (10) into (7) and equating the coefficients of the like powers of e separately yields a sequence of equations:
X ~1: 63,,0) ~ + C%, ~ (1).Sx+S'90)= - OO_~x'Pf), (11-1) Ot Ox
Ol-
k-lg(O,Eg(k_O]. (13-k) _1
Hence, in each step of the successive calculation, it is required to solve the vector-matrix equation of the form:
~tg(x, t)+[}g(x, t)]S(t)x +S'(t)O(x, t)=fff-h(x, t) (14) Ox
~k :
O~(k) + c),-,(k) ~Ox Sx+S'o(')=-(Of~o('-t)
where #(x, t) is an unknown n-vector and h(x, t) is a given scalar. The function h is assumed to be, for the time being, a homogeneous polynomial of the degree, say, r + 1 in x, and is written as
\Ox]
Og(k-1) f
Ox k - 1 ,qrf(k-i)
+ ~ ~'~ i= 1
Eg(O
OX
(ll-k) (k=2, 3 . . . . ) with the boundary condition g(k)(x, (k = 1, 2 . . . . ). S is the matrix defined by
S=A-EP.
T) = 0 (12)
The zero-order equation is identical with equation (4). Successive solutions of equations (l l-k) with increasing k may determine the functions g(k)(x, t). Though equations (11) are linear in #(~), their solutions are difficult given a general form of the perturbation f. However, if f is a polynomial in x, g(h) are also polynomials in x and their coefficients are exactly determined under appropriate conditions. First the following theorem is established:
Theorem 1 If f (x, t) is a polynomial of the degree q in x, the coefficients being continuous in t, g(k)(X, t) given
h(x, t)=Hjl h.../,+ l(t)xjlxj2.., xi, +1" (15) Here and in what follows we use the summation convention that a twice repeated index is to be interpreted as a summation over that index from 1 to n. The coefficients Hjlj2...j,+l(t ) are assumed to be, without loss of generality, symmetric with respect to any pair of indices.* Hence there are (n + r) !/(n- 1) !(r + 1) ! different coefficients. The i-th component of is given by
Oh/Ox, the gradient of h,
a h = (r + 1)Hut .. " j r x j , . . . axi
xj.
From equation (14) it easily follows that the i-th component of g is a homogeneous polynomial of the degree r in x. Hence the i-th component of g may be written as ~I(X, t)-~G}, ...j,(t)xj,
(17)
. . . xj,
where the coefficients G are symmetric with respect to the lower indices. Substituting equations (16) and (17) into (14) leads to . . . j , d- r S k j l Gkj2 . . . j,"l- S k i G j l , . . . 1,
by equation (1 l-k) is the polynomial of the degree
- (r + 1)Hij , . . j . ] x y , . . , x j, = 0
k(q-1)+ 1 in x whose coefficients are continuous
( i = 1, 2 . . . . .
in t. In particular, g(1)(x, t) is the polynomial of the same degree as f(x, t). Secondly the stepwise solution of equations (11) will be discussed. The right side of equation (11-1) is integrable with respect to x. If (Og(°/Ox) are symmetric for i = I, 2 . . . . . k - 1 , then the right side of equation (l l-k) is also integrable with respect to x, and the equation is rewritten into
(16)
n).
(18)
Equations (18) hold for arbitrary values of x if and
* H are said to be symmetric with respect to any pair of indices, or simply with respect to indices, if the values of H are invariant with any interchange of indices. For example, H j 1 = i , j z = k , j 3 . . . j r + l = H J l = k . h = i , J3 . . . J r + t "
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
706
only ifthe coefficients o f x i , x j , . . , x~, vanish for any combinations of the values of indices. Hence we have n(n+r-1)!/(n-1)!r! simultaneous equations for G~..... j :
Theorem 2 guarantees the symmetry of {.3g(k)/63X) for every k > l . Then the successive solution of equations (11) is completed. By virtue of symmetry of (Og/Ox), equation (14) is equivalent to
i d}, ... j. + Skj,G~j,... j. + S ~ j f i m j , . . . j.
+
-~9( g x, t)+ O=-[xS'o'(Xv,x
k j~-~k+SktGjl...j,
• • • +Skj.G}~...
t)-h(x, t)]=O. (22)
-(r+l)Hij,...j =O (i, j~ . . . . . j , = 1, 2 . . . . .
n).
(19)
By virtue of Lemma in Appendix, a Lemma for an extended Liapunov equation, equations (19) are rewritten into the vector-matrix form:
If S and q in equation (20) are time-invariant, and if all the eigenvalues of S have negative real parts, ~(t) is uniformly asymptotically stable relative to 4o as t ~ - 0% where 40 is the solution of the linear algebraic equation S (' + I)~o= q.
~(t) + ~ {' + l)(t)~(t) = r/(t)
(20)
where S ('+x) is the r + 1 repeated Kronecker sum of S' [10]; ~ and r/are the n'+~-vectors such that the m-th components are G~°... j, and (,+ t)Hjoj,.., i,, respectively, m being
(Jk-- 1)n'-k+ 1. k=0
Now the following theorem can be established.
(23)
Due to Corollary L1 in Appendix, equation (23) is uniquely solvable under the given condition. Thus the following theorem is obtained:
Theorem 3 In equation (14), if S reduces to a stable constant matrix and h reduces to a form free of explicit dependence on t as t ~ - ~ , then the function #(x, t) is uniformly asymptotically stable as t ~ - 0% relative to the polynomial go(x) which satisfies the equation:
x'S'Oo(x)=h(x).
Theorem 2 The matrix (Oglc~x) given by equation (14) is symmetric, and consequently there exists a scalar function v(x, t) such that 0/3
g=-0x
for
te[z, T ] .
(21)
The coefficients of the polynomial 9 are determined by solving equation (20). Proof Since the coefficients H are symmetric with respect to all the indices, the coefficients G are also symmetric with respect to all the indices, including the upper and the lower ones. This is due to Corollary L2 in Appendix. It implies that (Ogi/Oxj)=(Ogj/Oxi) for any pair of i and j. When (O0/Ox) is symmetric, there exists a scalar function v satisfying equation (21) [11].
(24)
The coefficients Gji .... Jr of the polynomial #0(x) are given by equation (23). A direct consequence of Theorem 3 is:
Corollary 1 If the unperturbed linear system is completely controllable and time-invariant, all the eigenvalues of S = A - B R - 1 B ' P ha~e negative real parts as t ~ - ~ . Then, for every k > 1, 9(k)(x, t), the polynomial function in x, is uniformly asymptotically stable as t ~ - 0% relative to the polynomial 9(k)(x) which satisfies the equation: for k = l
x'S'o°)=-x'Pf,
for k ~ 2
x,S,g(k)= _f,g(k-1)+½ ~_~ g(1),Eg(k-i) "
(25-1) k-1
i=1
(25-k)
Q.E.D.
The polynomial function o~k)(x), satisfying equation (25-k), exists and is unique for every k > 1.
In the above discussion, the function h is assumed to be a homogeneous polynomial in x. If h is composed of terms of different degrees, the calculations should be done separately for different degree terms. Due to the linearity of equation (14), the total solution is obtained by simply summing up the resulting 9 polynomials.
Now we have comoleted the procedure for calculating a suboptimal control. The solution of equations (11) or equations (25) up to the order l in e gives the suboptimal control of the l-th order. It is noted that, when T--* oo, the complete controllability of the unperturbed system suffices to yield a suboptimal control of an arbitrary order.
A method for suboptimal design of nonlinear feedback systems
707
[Ovtl)\ x'S'~-~-x )=-x'Pf'
(32-I)
Further the following corollary is useful in the next section.
Corollary 2
~k :
If the equation
x'S' g=O
81-
=
or
g'Sx=O
\
)
0x
k-I
(26)
-½ i=l Z o(°'E0
holds for any value of x with a stable S, and if (OglOx) is known to be symmetric, then g is identically a zero vector.
k-lg(O'E(OV(k-O~ i=1 +Z
(k=2, 3 . . . . PERFORMANCE EVALUATION
Theorem 4 If g(x) is the optimal one, i.e. if g(k)=g *(k) for every k > 1, where g *(k) is the solution of equation (25-k), then J is given by
J=J*=kx'Px+ ~ ekv*(k)[x(~)]
(27)
where gs(x) is a suboptimal feedback function satisfying equations (25) up to a certain order. The value of the performance index J resulting from the feedback control of equation (27) would be a function of x(z) and be written as
Js= ½f ~ (x'Qx + u'sRus)dt =vs[x(z)] .
).
Then the following theorem can be established:
In this section the quality of a suboptimal control law is examined. For simplicity we consider only a completely controllable time-invariant system. As shown in Corollary 1, when T ~ ~ the suboptimal feedback control is free of explicit dependence on t and is given by
us(x ) = - R- 1B' gs(x)
(32-k)
\Ox)
(28)
Evidently vs(0)=0 if gs(0)=0. Now the task is to examine the function vs(x). Differentiation of equation (28) with respect to z and use of equations (1) and (27) gives:
(33)
k=l
where v*(k)(x) is the scalar function satisfying Or*(k)
Ox
= g *(k) for every k ~ 1.
(34)
Theorem 4 can readily be proved inductively by comparing equations (32) with equations (25). Further the following theorem of interest is obtained.
Theorem 5 If g(x) is optimal up to the order I in 8 and is truncated after that, then (i) J is equal to J* up to the order 2/+ 1 in 8.
x Qx+g
Eg+2l~xlk](Ax+ef-Eg)=0.
(29)
In equation (29) all the values are estimated at t = ~. The subscript s is omitted here and throughout the following part of this section, but should be understood. We develop g and v into power series in e:
g(x)= Px + ~ and
k=l oo
ekoCk)(x) 1 (30)
v(x) = ½x'Px + k~=l ekvtk)(x) where the matrix P is the solution of
PA + A ' P - P E P + Q = 0 .
(31)
Substitution of equations (30) into (29) results in a sequence of equations:
(ii) v(2t+2), the (2/+2)th term in the expansion of J, is given by xtS,[ -0t)(21+2)
L
g,(21+2)1_ -½a*°+x)'Eg
Proof (i) Again the theorem is proved by induction. First, corresponding to l=0, assume that g(x)=Px. From equations (25-1) and (32-1), it is readily observed that v(1)= v*(~). Second, assume that, if gti)=g.(O for i=l, 2, . . . . l - 1, v(°= v*(° for i= 1, 2 . . . . , 2 1 - 1. Besides if g(t)=g.(t), comparison of equation (32-k) with (25-k) for k = 21leads to x'S'[(Ov(Zt)/Ox) - g,(2t)] _- 0. Due to Corollary 2, it implies that v(2t)= v*(2/). Further, examination of equations (25-k) and (32-k) for k = 21+ 1 leads to v(2t+ l) = v,(2t+ 1). (ii) Comparison of equation (32-k) with (25-k) for k = 2 / + 2 yields equation (35). Q.E.D.
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
708
Theorem 5 is an extension of the theorem obtained by KOKOTOVIt~and CRUZ [6, 7] for linear systems. A similar result has been reported by WERNER and CRUZ [8] regarding an approximate construction of the optimal feedback control for systems with unknown parameters; where the control is expanded in a Taylor series with respect to the unknown parameters. However the present proof is given in a much simpler way by use of induction. It is noted that Theorems 4 and 5 apply also to a time-varying system with a finite T.
Example 1 Consider the third-order system governed by:
~IX2X3
~
~2X3XI
L ~3XIX2
+
u2
iI ul
(38)
u3
with the index of performance APPLICATION
TO A LARGE-SCALE
SYSTEM
The method developed so far has a wide variety of applications. One of the straightforward applications is the synthesis of a suboptimal control for a large-scale system as composed of several subsystems of lower dimensions. Consider the problem of optimizing the system
J = ½ y f (x'Qx+u'Ru)dt, where
(39)
Q=diag(ql, q2, q3), R=diag(rx, r2, r3), and q~, r~>0.
£i=Aixi+efi+Biu~
(i=1, 2 . . . . .
N)
(36)
with respect to the performance index
J = ½ i=t ~
f: (x[Qix~+u~Rtu~)dt
(37)
where x~ is the ncvector of the i-th substate, and u~ is the m~-vector of the i-th subcontrol. An analytic function f~, representing the coupling of the subsystems, may depend on all the substates xi(i= 1, 2 . . . . , N). At, B~, Qi and Ri are matrices of the appropriate dimensions. A particular problem in which all the couplings f~ are linear has been discussed by KOKOTOVI(~et al. [6] and the present authors [12], independently. The method of the preceding sections has much enlarged the range of applications. The first step of the solution involves the solution of lower-order Riccati equation of each subsystem. In the subsequent steps one has only to solve linear equations. This fact greatly relaxes the complexity of computation. Evidently the foregoing theorems apply to the composite system of equations (36). Especially, the complete controllability of all the subsystems is sufficient to guarantee the uniqueness of an improved solution.
ILLUSTRATIVE
It is well known that, if ~l + ~2 + ~3 =0, equation (38) describes the rotational motion of a rigid body. First, the suboptimal control law is calculated by the present procedure up to the third order:
U - - IJ (0) .4- p.lJ (1) 4- ~2~t(2) -1- ~3sn(3) i----i _ _ v _ i __v --i - - v Hi
where
,,I°'= -,/ q,lrlx,, U~ 1) =
C(1)XjXk/ri,
(2)_
Ut
(2)
2
(2) 2
--Xi[C k Xj~I-Cj Xk]/r i,
U(3) __ ~. ~ 1"1.,(3),,2 -1- , . ( 3 ) y 2 +
i
--'~j~'ktJ~i "vi "~'j ~j
c~3)x2]/ri
3
c(')= Z ct,x/q,r,/a, l=l
ctl)[o~i+c(1)/2ri]
(2)
Ci
-
-
tr--4qi/r i
c~3)_ [¢xj+ c (')1r i]c~2) + [gk + C~t)/rk]C~2) a + 2x/qi/rl
EXAMPLES
Three examples are presented to illustrate the application of the present method. The suboptimal controls are compared with those obtained by other methods.
3
a= Z
1=1
x/q,/r, (i,J, k = l , 2, 3; i # j # k # i ) .
(40)
A method for suboptimal design of nonlinear feedback systems It follows from equations (40) that, if e ° ) = 0 , u(k)=0
(41)
709
Example 2
In order to examine the influence of truncation in the power-series solution, consider the secondorder system:
for every k_->l. This implies that u = u (°) is the exact optimal control for this case. KUMAR [13] previously discussed a special case of c ° ) = 0 , i.e.
Xl=X2'
t
(48)
X2=SX~+U
qx =q2 =q3 = 1,
r 1 =r2=ra=K,
and ~x + ~2 '[- 0~3~ O.
(42)
subject to the performance index
Secondly, the method of instantaneous linearization by PEARSON [3] is applied to equation (38). According to the method, the nonlinear system (1) is rewritten into a linearized one such as
d=
q
oo
0
(x 2 + x 2 + u2)dt.
(49)
The suboptimal control of the second order is ~=~(x, t)x+B(t)u
with
4(0,
t)=0 U~ U(0)"1-gU(1)"1-g2U(2)
where .4 is an n x n state-dependent matrix. The state-dependent matrix ,4 for the system (38) is here assumed in the form:
where
U(0) = - x 1-1"7321x2, I
.~=~
0
¢tlVlX3
~2(1 --V2)X3
0
O~3V3X2
gl(1 -- Vl)X2 1 O~2V2X1
~t3(1 -- V3)X1
U(1)= - x ~ - l ' 3 2 3 5 X ~ X z - O ' 6 9 2 3 x l x ~
(50) -0"1323x~,
0 (43)
where v~(i= 1, 2, 3) are arbitrary constants. Then the suboptimal control is given by
//(2)= - 0 " 5 x { - O ' 7 3 9 2 x ~ x 2 - O ' 3 3 2 0 x ~ x ~ +O,O190x~x~+O'O523xlx~
+0"0112x~. (44)
u = - R - 1Px
where P is the positive definite solution of the following quadratic equation: A'P+PA-PR-
1p+ Q = 0 .
(45)
In order for equation (45) to have the exact solution with the parameters of equations (42), it is required that
iv1 iv 0iill 1 -vx
0
v3
~2
0
v2
1 - v3
a3
----0. (46)
Equation (46) holds for nontrivial values of the parameters st, if and only if arbitrary constants v~ satisfy the following relation: V1p2 -~t-p2p3 -~--k'aVl--[-1 = P1 -~t-V2-]--V3.
(47)
It is noted that the form of ~ should skillfully be chosen in order to obtain a good solution.
Now the stabilization property of the controller is examined. With u = u t°), the origin of equations (48) is globally asymptotically stable with e<0, while it is only locally asymptotically stable with 8 > 0. In fact, for e > 0, there exist unstable singular points at (+x/l/e, 0), and trajectories passing through either of these points give the stability boundaries. With u = u ( ° ) + e u (1), the origin becomes locally stable for e < 0, so that the first-order approximation gives a better result than the zeroorder one only in a limited region around the origin. On the other hand, with e > 0, the first-order controller stabilizes the origin in the large. The second-order controller is of a locally stabilizing nature for both signs of e. Figure 1, showing the loci of the initial states for which J = 5, reflects the stabilization property mentioned above. The curves are shown only on the upper-half plane, because they are symmetrical about the origin. Within the regions illustrated, the higher-order approximations improve the performance with e= 1, but it is not necessarily with e=-l.
Y. NISHIKAWA,N. SANNOMIYAand H. ITAKURA
710
Xl =X2,
I ..... -
-
-
-
-
~
:
by the zero-order control
:
by the first-order control
:
by the second-order control
/t,,t<
X2 = g'CX4 + gX4X 6 + U 1 -{" 13X3 U 3,
!
"~3= X4, ~~'4 =
3
--
(51)
~X2X6 + 112 -- ~X1123,
X6 = EX2X4 "F 123 + ~X 1 ~12 •
1 ,
0 -3
-2
\',,
\\
-1
0 X1
The quantities xl, x 3 and x 5 represent the roll, yaw and pitch motion, respectively, of the body about its principal axes. c is a constant associated with the orbiting angular velocity. The performance index is taken to be
,
1
2
=
(a) e= 1
'
-2
-I
-- 0
i
When e = 0 , the system of equations (51) is decoupled into three subsystems, each of them being a double-integral system. The method for composite systems can be applied then. Although equations (51) contain the product terms of x and u, the method still works effectively. In fact, if an additional term eB(')(x, t)u exists on the right side of equation (1), there appear terms containing B (~t and 9 (t) on the right side of equation (ll-k) ( / < k ; k = 1, 2 , . . . ); while its left side is kept unchanged. Then the recursive calculation proceeds in much the same way.
2
X I
(b) e - - : FIG. 1.
~CX2
X5 = X 6 ,
2
0-3
--
The loci of the initial states resulting J =
5.
Figure 2 illustrates examples of variation of J with e for various approximations.
Example 3
Table 1 compares values of J obtained by the zero- and the first-order control for various values of e when c = l . The first-order control affords better results even for moderately large e.
The following equations approximately describe a three-axis attitude-control system of an orbiting space body [14]:
,.8
/
I
1"4
....
:
by the zero-order control
J--J
i
by the f i r s t - o r d e r
:
by the second-order
I I t /
control
~o.trol
I
I
///
,/,
. ,/./
js p sI
1,0
0.E -i'0
1 -0-5
I 0"5
2. Variation of Jwith e for various approximations of the control (Initial condition: x 1= 1, x2=0).
FIG.
1.0
A m e t h o d for suboptimal design o f nonlinear feedback systems TABLE 1. COMPARISON OF VALUES OF PERFORMANCE INDEX jr. J0 : J BY THE ZERO-ORDER CONTROL ; J l : J BY THE FIRST-ORDER CONTROL
(a) Initial condition: xt =x3 =x5=2, X 2 : X 4 : X 6 : 0 0"1 0"2 0'5 1"0 2"0 5'0 J0
10.26
10"14
J1
10"23 10"06
9 " 9 6 11"28 17"84 9"78 10-26 11'61
14"58
(b) Initial condition: xl=xa=xs=O, X2=X4=X6:2 e 0-1 0-2 0"5 1"0 2'0 5'0 J0
10"53 10"61 10'79
J1
10"51 10"54 10"43
10"82 10'44 9"92
8'95
11-28 7"74
CONCLUSION A systematic procedure has been developed for the suboptimal design of a nonlinear state regulator. When the nonlinearity is characterized by a polynomial function in the state, the definite way to determine a suboptimal feedback control o f a power-series form is established by Theorems l and 2. The stability property o f the suboptimal feedback function is discussed in T h e o r e m 3. The contents o f T h e o r e m 3 as well as o f L e m m a in Appendix could also be used to construct a desired polynomial function, e.g. a desired Liapunov function. Theorems 4 and 5 clarify the approximation property o f the suboptimal policy qualitatively. However the quantitative estimation o f the performance degradation is generally difficult. When the nonlinearity is large, the convergence o f the series-form solution is not assured, and higher-order approximations do not necessarily give better results. The m e t h o d developed provides a promising tool for a variety o f problems, including linear or nonlinear, deterministic or stochastic systems optimization. A n application to a composite system, whether it is linear or non linear, is straightforward. The effectiveness o f the method is illustrated by examining typical examples. Since the procedure is completely systematic, it can efficiently be used in a c o m p u t e r - p r o g r a m m e d computation.
711
[7] P. V. KOKOTOVIt~and J. B. CRUZ, JR. : An approximation theorem for linear optimal regulators. J. Math. Anal. Applic. 27, 249-252 (1969). [8] R. A. WERNERand J. B. CRUZ, JR.: Feedback control which preserves optimality for systems with unknown parameters. 1EEE Trans. Aut. Control AC-13, 621-629 (1968). [9] R. E. KALMAN:Contributions to the theory of optimal control. Bol. Soc. Math. Mex. 5, 102-119 (1960). [10] R. BELLMAN: Introduction to Matrix Analysis, pp. 227-231. McGraw-Hill, New York (1960). [11] P. M. MORSEand H. FESHBACH: Methods of Theoretical Physics, pp. 14-15. McGraw-Hill, New York (1953). [121 Y. NISHIKAWA, N. SANNOMIYA, H. ITAKURA and T. FUKUDA" Suboptimal feedback control of weakly coupled systems. Preprints of Symp. on Systems Engng., Soc. of Instrument and Control Eng. of Japan, Tokyo, pp. 9-12 (in Japanese, 1969). [13] K. S. P. KUMAR: On the optimum stabilization of a satellite. IEEE Trans. Aerospace and Electronic Systems AES-1, 82-83 (1965). [14] Y. NISmKAWA,C. HAYASHIand N. SANNOMIYA: Fuel and energy minimization in three dimensional attitude control of an orbiting satellite. Proc. IFAC Symp. on Peaceful Uses of Automation in Outer Space, Stavanger, Norway, pp. 287-298 (1965). [15] Y. NlSmKAWA, N. SANNOMIYAand H. ITAKURA: Suboptimal design of a nonlinear feedback system. Memoirs of the Faculty of Engineering, Kyoto University, Voi. XXXII, pp. 334-347 (1970). APPENDIX Lemma.for deriving equation (20) The following lemma regarding an extended L i a p u n o v equation provides a basis for rewriting equation (19) into equation (20): Lemma Consider the n ' algebraic equations for n ' quantities Xjt.,.jr: Ckj,Xkj2... jr + Ckj2Xj~k,,3... Jr + • • • + CkjrXj~...Jr-,k = Dy,...Jr
(A. 1)
where Cij and Dj~... jr are given quantities. All the indices run from 1 to n. Equations (A.1) can be rewritten into the vectormatrix f o r m : C~'J~ ~')= q(') (A.2) where t~(')=C ' ~ C ' ~ . . . ~ C ' . (A,3) r
REFERENCES [1] E. G. AL'BREKHT: On the optimal stabilization of nonlinear systems. J. Appl. Math. Mech. (PMM) 25, 1254-1266 (1961). [2] D.L.LuKES: Optimal regulation ofnonlinear dynamical systems. SIAMJ. Control 7, 75-100 (1969). [3] J. D. PEARSON: Approximation methods in optimal control, I. suboptimal control. J. Electron. Control 13, 453--469 (1962). [4] W. L. GARRARD,N. H. McCLAMROCHand L. G. CLARK: An approach to sub-optimal feedback control of nonlinear systems. Int. J. Control 5, 425-435 (1967). [5] W. L. GARRARD: Additional results on sub-optimal feedback control of non-linear systems. Int. J. Control 10, 657-663 (1969). [6] P. V. KOKOTOVIt~,W. R. PERKINS,J. B. CRUZ, JR. and G. D'ANS: e-coupling method for near-optimal design of large-scale linear systems. Proc. IEE 116, 889-892 (1969).
C is the n x n-matrix such that the (i, j)-element is Cij; ~ ' ) and r/(r) are the n'-vectors o f which the m-th components are X j , . . . j , and D j , . . . j r , respectively, rn being (jk--1)nr-k + l . k=l
The symbol @ denotes the Kronecker sum of matrices.
eroo/ Let us proceed inductively. equations (A.1) reduce to
First, for r = 2 ,
Ckjl Xkj2 "~ CkjlXjl k = Dj,j2.
(A,4)
712
Y. NISHIKAWA, N. SANNOMIYA a n d H. ITAKURA
By i n t r o d u c i n g the matrices X a n d D whose ( i , j ) elements are X~j a n d D~j, respectively, equations (A.4) are rewritten into
C ' X + X C = D.
(A.5)
In terms o f ¢(2) a n d ~/~2) as defined in L e m m a , e q u a t i o n (A.5) is equivalent to [10] (C,(~Ct)~(2)=
(A.6)
q(2}.
Secondly we show t h a t L e m m a is true for r + 1 if it is true for r. F o r r + 1, e q u a t i o n s (A.1) are written as [Ckj~Xkj~
• • . Jr+t + C k j z X j l R
+ CkjrXjt...
• • • Jr+~ + " " "
J r - ,kjr+ 1] + C k j . + I X j l . . .
jrk
=Dj . . . . ~,+~.
(A.7)
T h e terms in the b r a c k e t are identical, for a fixed value of.j,+ t, with those on the left side o f e q u a t i o n s (A.1). Hence, with use o f ~('+~) and r/("+t) equations (A.7) are rewritten into [1~") ® t ~ t ' ) + C ' ®I("')]~'+l)=q (~+~)
(A.8)
where the s y m b o l ® denotes the K r o n e c k e r p r o d u c t o f matrices [10], a n d I ~k) is the k x k identity matrix. D u e to the definition o f the K r o n e c k e r sum [10], e q u a t i o n (A.8) is equivalent to (~(r+ 1)~(r + 1)=/,](, + 1).
(A.9) Q.E.D.
The following c o r o l l a r i e s were used to establish T h e o r e m s 2 a n d 3:
Corollary L1--Uniqueness of the solution Let 2~ ( i = l, 2, . . . , n) be the eigenvalues o f C' o r equivalently o f C. T h e n if a n d only if a n y sum o f r eigenvalues o f C, 2 ~ + 2 ~ + ... +2~, (Jx, J2 . . . . rJ, = 1, 2 . . . . . n), is n o t zero, X s . . . . ~, o f e q u a t i o n s (A.1) are uniquely d e t e r m i n e d . In p a r t i c u l a r , if C is a stable matrix, X~, ...~, are unique.
Corollary L2--Symmetry with respect to the indices I f D ~ . . . ~ . are symmetric with respect to the indices, Xj . . . . j , are also symmetric. F o r m o r e details o f L e m m a a n d Corollaries, refer to Ref. [15]. E q u a t i o n (A.1) is a linear e q u a t i o n to determine the n-index quantities X given the two-index quantities C a n d the n-index quantities D. F o r the special case o f n = 2 , the so called L i a p u n o v e q u a t i o n , i. e., e q u a t i o n (A.5), is obtained. Hence equation ( A . I ) is considered an extended version o f the L i a p u n o v equation. In fact it can p r o v i d e a L i a p u n o v function o f the n-form for linear systems. Rrsumr--L'article prrsente une mrthode d'approximation pour construire un rrgulateur d'rtat optimal pour un systrme non-linraire avec indice de performance quadratique. La non-linrarit6 est considrrre comme une perturbation du systrme et un paramrtre s est introduit pour la reprrsenter.
En utilisant un drveloppement en srrie selon les puissances de ~, on obtient un systrme d'rquations diffelentielles aux d6rivbes partielles dont les solutions fournissent une Ioi de rraction sous-optimale. Etant donnre une non-linrarit6 polynomiale, les 6quations differentielles aux drrivres partielles sont reduites ~ des 6quations differentielles ordinaires par srparation des variables. Les termes d'ordre zrro fournissent une 6quation bien connue de Riccati. Les 6quations d'ordres sup6rieurs sont transformdes en 6quations linraires du type conventionnel grace /tun lemme se rapportant /t une 6quation grnrralisre de Liapunov. I1 est montr6 que l'approximation du l-irme ordre pour le loi de rraction correspond/t l'approxirnation du (2•+ 1)-i~me ordre pour la performance optimale. Le procrd6 developp6 possrde une large variet6 d'applications. A titre d'une des approximations directes, I'article discute la synthese d'une commande sous-optimale pour un systrme/t grande 6chelle considrr6 comme 6tant compos6 de plusieurs sous-syst~mes de dimensions infrrieures. Trois exemples inclus illustrent plusieurs caractrristiques de la mrthode. Zummmenfassmalg--Es wird eine Anniiherungsmethode zum Aufbau eines optimalen Zustandsreglers f'tir ein nicht lineares System mit quadratischem Leistungsindex dargestellt. Die Nichtlinearit~tt wird als StSrung des Systems angesehen, und es wird ein Parameter e dafilr eingef'tihrt. Unter Verwendung einer Potenzreihenentwicklung in ~ wird eine Reihe partieller Differentialgleichungen abgeleitet, deren Lrsungen ein suboptimales Rtickkoppelungsgesetz formen. Bei gegebener polynomischer Nonlinearit/~t werden die partiellen Differentialgleichungen auf gewrhnliche Differentialgleichungen durch Trennung von Veriinderlichen zurtickgefiihrt. Die Null-Grad Ausdrticke fiihren auf eine gut bekannte Riccati Gleichung. Gleichungen hSheren Grades werden infolge eines Hilfsatzes in Bezug auf eine erweiterte Liapunov Gleichung in lineare Gleichungen normaler Art umgewandelt. Es wird gezeigt, dass die Ann~iherung l-ten Grades ftir das Riickkoppelungsgesetz die Ann~iherung (21+ 1) ten Grades fiir die optimale Leistung ergibt. Das entwickelte Verfahren hat eine grosse Auswahl von Anwendungszecken. Die Synthese einer suboptimalen Regelung wird als einer der einfachen Anwendungszwecke fiJr ein System yon weiten Ausmassen besprochen, das sich aus mehreren Subsystemen kleinerer Ausmasse zusammensetzt. Drei beigeftigte Beispiele illustrie;en mehrere Merkmale der Methode. Ve31oMe---CTaTbR npemIaraer MeToJI npx6rmxceHaa mm KOHCTpyHpOBaHlifl peryJ~Topa OnTHMa.rmHOrO COCTOmma
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