Fixed linear systems with generalized performance criteri

Aggarwal, J. K.

Mc Clamroch, N. H.

Rhoten, R. P.

English

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ENGINEERING MECHANICS RESEARCH LABORATORY
THE CN1\'FI?SI'I'Y OF TEXAS
	 AUSTIN. TEXAS
N69-37017
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MABA CR OR TM% OA AD AWMBCRI 	 ICAYEOORYI
ON FIXED LINEAR SYSTEMS WITH A GENERALIZED
PERFORMANCE CRITERIA
by
R.P. Rhotenj J.K. Aggarwall N.H. McClamroch**
EMRL RM 1042
	
November, 1967
*	 Department of Electrical Engineering, The University of
Texas at Austin, Austin, Texas.
** Information and Control Engineering, University of
Michigan, Ann Arbor, Michigan
The research reported herein was supported by NASA Contract
Number NAS8-18120, NSF Grant Number GK-261 and Joint Services
Electronics Program Grant Number AF-AFOSR-766-66.
ON FD(ED LINEAR SYSTEMS WITH A
GENERALIZED PERFORMANCE CRITERIA
ABSTRACT
This paper considers optimization and sensitivity problems as
related to a fixed linear system with a generalized performance criteria.
A technique is used which solves for the performance index as a poly-
nomial in the system initial conditions. The method is not restricted
to quadratic form loss functions, -but applies to any index whose
inte% and is the product of a time function and a homogeneous polynomial
in the states and controls.
1
I. 114TRODUCT10N
Considerable research is currently in progress concerning pro-
blems that come under the broad headings of optimization and sensitivity
studies. These problems, however, are difficult in their complete
generality while significant results have been obtained by considering
restricted systems, in particular, systems linear in the states and
controls. Methods have been found to design optimally linear plants
with quadratic loss functions, with some solutions [1,21 assuming the
states to be continuously measurable and others [31 optimizing with
respect to a set of control release coordinates, the controller having
been synthesized by classical means. Controls that are in some sense
both optimal and insensitive were generated in [41 by adding a weighted
sensitivity function to the performance index. This paper utilizes the
results of (31 and [41 and extends them in that the integrand of the per-
formance index is not restricted to a quadratic form but may be a time
function multiplying a homogeneous polynomial in the states and controls.
The technique is simple in that it consists mainly of a gradient operation
and a matrix inversion, but may be tedious, since the matrices involved
may be non-numerical in nature.
2
3II. STATEMENT OF THE PROBLEM
The problem may be stated as follows. A plant and controller
are given which may be described by the linear state equations
it = Ax+Bu	 (1)
u - Cx + Du	 (2)
where x and x are n-vectors, u and t1 are m-vectors and A, B, C and D
are n x n, n x m, m x n and m x m time invariant matrices respectively.
The optimization problem is to find the control initial condition u(0) so
as to minimize a performance measure of the form
J'=  f(t)Q(x, u)dt	 (3)
0
j =
where f(t) is a Laplace transformable function of time and Q(x, u) is a
homogeneous polynomial in x and u. The "optimally-insensitive"
solution will be that initial condition u(0) which makes a measure of
the form of (3) small yet relatively insensitive to variations of para-
meters in the matrices of (1) and (2). The importance of the value of
the index relative to its sensitivity to parameter variations will be
decided by a sensitivity weighting matrix, much as the coefficients
of the polynomial Q(x, u) determine the importance of the states vs.
controls.
4III. PROBLEM SOLUTION
MacFarlane [51 has devised a technique to calculate functionals
of the form
J = 1 0
 
f(t)R(x)dt	 (4)
for a system
*= Fx	 (5)
where f(t) is Laplace transformable, R(x) is a homogeneous polynomial
in x, and the system of (5) is such as to insure the existence of the
integral of (4) . Application to the problem of (1), (2) and (3) is immediate,
simply considering the augmented system
A	 B
Y-	 ... ••	 y
C D
= HY
	 (6)
where y and t are m+n vectors and H is an (m+n) x (m+n) matrix. It is
assumed that the augmented matrix H assures the existence of the integral
of (3) and is stable. This assumption of the existence of the integ --11 may
be more restrictive than the requirement that H be stable, since the addi-
tion of an explicit dependence on time in the integrand may cause diver-
gene of the integral even for an asymptotically stable system. Conversely,
certain time-weighting functions may cause the integral to exist even for
unstable systems..
5Using the results of (51 directly, it may be shown that
f m Q(x, u)dt = -P(x(0), u(0))	 (7)0
where P(x, u) satisfies
Q(x, u) - P ,u [Ax + Bu1 + P(x ' u [Cx + Du1,
ax	 BU
or
Q(x, u) _ (T Hy,	 (8)
superscript T denoting transpose, and that P(x, u) will be a homogeneous
polynomial in x and u of the same degree as Q(x, u) . Defining the vector
z so that each of its components is of the form of a term of Q(x, u), then
for appropriate constant vectors p and q,
Q(x, u) = q T z
	 (8)
and
P(x, u) = p T z .	 (14)
Using the identity (8), equating the coefficients of tike terms of z yields
a matrix M such that
q = MTp.	 0 1)
For H a stable matrix, Lyapunov (see [61) has shown that (8) has a unique
solution for P(x, u), and hence the transformation M is non-singular giving
p- (MT)-1q.
	
(12)
In this manner, the coefficients of P(x, u) may be determined from those
Of Q(x, U) .
Since d(P(x, u))/dt - Q(x, u),
dt (p Z) - qTz
_ (MTp)Tz
- pTMz,
and (13) is satisfied by
I - Mz
giving
z = eMtz(0) .
Using (9) and (15) in (3),
j - 1 10 ONTeMtz(0)dt
0
- gTF(M)z(0)
where
F(M) = r® f(t)eMtdt.
J0
As is discussed in [51, F(M) is easily evaluated by noting that
F(-M) - J : f(t)e Mtdt,0
which may be thought of as the matrix equivnient of the Laplace trans-
form. Hence, examples would be
f(t) - 1	 F(M) _ -M-1
f(t) - eat	 F(M) - -[M-all=1
f(t) = tr 	F(M) = rl [-M-llr+i
f(t) a sin (at)	 F(M) - a(M2+a21]-1
6
(13)
(14)
(15)
(15)
(17)
7where I denotes the identity matrix.
With the index given in the form of (16), the solution of the
system of algebraic equations
.Ai—) = 0	 (18)
will yield necessary conditions for the existence of a minimum of J.
If information concerning the sensitivity of the index with respect
to certain variable parameters is desired, evaluation of j is accomplished
as above, except that the matrix to be inverted will now be non-numerical.
Thus, the index will be given by (16) as ration of polynomials in para-
meters of the matrices of (1) and (2) multiplying control initial conditions.
At this point, considerable information concerning the sensitivity of the
system is readily available. Specifically, the infinitesimal performance
index sensitivity is given by
of I f nominal
	 (19)
where f is any one of the parameters. Unfortunately, this information
is often of limited value since it only serves as some indication of the
effects of a finite parameter change. These finite changes can be in-
vestigated, however, with the index given in essentially polynomial
form. For the variable f and some maximum permissible index value
I max, an inequality solution of (16) will yield constants fl and f2 such
that
8j < j max	 (20)
if
fl < f 
-S f2 . (21)
Additional information might be desired concerning the effects of varia-
tions in state and control initial conditions. This is again simple and,
for example, for a single state - single control system with a homogeneous
polynomial of order 4 in the loss function, (16) will yield constants
0 0 , $ 1 , ..., 04 such that
Al E Oi[x(0)iu(0)4-i x(Q)nominalu(d) nominal]	 (22)1-0
where
Aj = j - 7nominal	 (23)
Although the above information ; doubtless of value, it is
basically of an analysis nature yielding littler information concerning
the deign of a more acceptable solution. This latter problem was
investigated in [41 by the addition of a sensitivity term to the performance
index. Although the controls were restricted to a fa m
u - Kx (24)
with a quadratic loss function, the results are here extended to include
controls of the form of (2) with loss functions of the form of (3).
It is assumed that the matrices of (1) and (2) contain an 4-vector,
w, of variable parameters with nominal value wo. Defies the sensitivity
9of the performance index with respect to changes in the parameter
vector as
AL _^ W—
	
A T
	
(25)dw .
AL-
wl awl	 Bwi
a new index is formed
I = ^' °° f(t)Q(x, u)dt +Qw/Y Go 	 QL'"^
T	 (26)
where G is a positive semi-definite t x.t weighting matrix and evaluation
is made at w = wo . The G matrix determines the importance of the
sensitivity of the performance index relative to its actual value. That
is, as the norm of G approaches infinity, the problem is concerned com-
pletely with sensitivity while if the norm approaches zero, the problem
is again a pure optimization.
To find the initial conditions u(0) minimizing I, the integral
portion of (26) is evaluated as a function of w and u(0). The gradient
operation is performed, evaluation is made at w = wo, and there results
I as a function only of the control initial conditions. Hence, the
solution of
a_I 
= 0	 (27)Bu(0)
will yield necessary conditions for a minimum of I.
10
N, EXAMPLE PROBLEM
As an example of a non-quadratic time-weighted performance
index, consider the system
is = -x+ u
--x- u
X(0) = 1
with the index defined by
1 = f ine t(x4+u4)dt.0
Proceeding with the solution,
qT=[10001],
4
x
3
x u
2 2Z =	 x u
3
xu
4u
-4	 -1	 0 0	 0
4	 -4	 -2 0	 0
MT = 0	 3	 -4 -3	 0
0	 0	 2 -4	 -4
0	 0	 0 1	 -4
and
0,176 0.118 0.059 0.020 0.004
-0.029 0.147 0.073 0.025 0.005
F(M)= 	 0.098 -0.049 0.141 0.049 0,010
-0.005 0.025 -0.073 0.147 0.029
0.004' -0.024 0.€159 -0.118 0.176
11
From (16) ,
I = 0.180u(0)4- 0.098u(0) 3+ 0.118u(0) 2+ 0.098u(0) + 0.180,
and (18) yields a minimum of j at u(0) = -0.267. This gives an index
value of j = 0.166, compared with that of j = 0.180 for starting the
control at the origin.
Now from the sensitivity viewpoint, the previous problem will
be reconsidered including the effects of a variable parameter. That is,
given the system
}C=-x+wu
u = -x- u
X(0) = 1
wo = 1
with original index
j = f CO e t(x4 + u4)dt,
0
it is desired to minimize the new index
I = j1 W= } 10 Q!Lw I,,.
Evaluation of the integral portion of I by (16) yields
j =
	
	 [(24w2+400w+649) + (200w2+500w+120)u(0)
5(64w +500w}625)
• (4A• • 3+300w2+48w+300)u(0)2+(120w3-200w-500)u(8)3
• (24w4+24w2+400w+625)u(0)41.
t:
12
Taking the derivative with respect to w and evaluating at w - 1, I is
given by
I = 5(1189) [1073+580u( 0)+696u(0)2-580u(0)3+1073u(0)4]
+{ 1	 [1189(448+900u(0) +792u(C`)2+130u(0)3+.44u(0)4)
5(1189)2
- 628(1073+580u(0)+696u(0) 2-580u(0) 3+1073u(0)4)]}2 (10).
Finally, the minimum of I is found to be 0.180 for an initial condition
u(0) $ -0.144. A graph of I and J versus u(0) is given in Figure 1.
V. CONCLUSIONS AND DISCUSSION
A method has been presented which allows the evaluation of a
generalized performance index in terms of initial conditions and system
parameters. This evaluation allows sensitivity analysis and the genera-
tion of control initial conditions that are either optimal or "optimally-
insensitive". While the method is rather complex algebraically, some
recent papers have been concerned with the reduction of the complexity
of the problem. Specifically, [7] gives a method for evaluating M 1
which depends upon finding the eigenvalues and eigenvectors of F,
while [8] gives an algorithm for finding M efficiently and finding M - 1
by inverting matrices of smaller order than M.
Of course, the major disadvantage of this process is that the
control must be designed "a priori". With this in mind, an attempt
rformanoe Indices vs. Control Initial Condition
13
14
has been made to leave as free parameters the elements of the C and D
matrices of (2), evaluate the index as a function of these parameters,
and then optimize with respect to them. That is, an attempt was made
to find a control optimal over the space of all controls of the form of (2) .
A difficulty was immediately encountered, however, as is evidenced by
the following scalar example. Given a system
is = ax + bu	 (28)
A = cx + du
	
(29)
and a performance index
j = J, (x2 + u2)dt,	 (30)
0
the globally optimal control is known by standard techniques to be
u = kx
	
(31)
with control initial conditions
u(0) - kx(0).	 (32)
This control is obviously an element of the restricted control space since
u = kX
= kax + kbu .	 (33)
Consequently, the values ka, kb and kx(0) would be found for c, d and
u(0) by the process presented herein. Examination of the augmented
system, however, reveals that the augmented matrix is singular, an
obviously unacceptable solution.
BIBLIOGRAPHY
[11 Kalman, R. E . , "Contributions to the Theory of Optimal Control",
Bol. Soc. Mat. Mexican, 1960.
[21 Athans, Michael, "The Status of Optimal Control Theory and
Applications for Deterministic Systems", IEEE Trans. on Automatic
Control, vol. AC-11, no... 3, July 1966.
131	 Mueller, T. E. and Aggarwal, J. K., "Optimal Control with a Fixed
Controller Structure", Proceedings of the First Princeton Conference
on Information Sciences and Systems, Princeton University,
Princeton, New Jersey, March 1967.
[41 Cadzow, J. A., "Optimal Control of a System Subject to Parameter
Variation", Proceedings of 1966 Joint Automatic Control Conference,
University of Washington, Seattle, Washington, pp. 807-10, 1966.
[51 MacFarlane, A . G . J. , "A Method of Computing Performance Func-
tionals for Linear Dynamical Systems", Journal of Electronics and
Control, No. 4, October 1963.
[61 Lefshetz, Solomon, Differential Equations: Geometric The
Interscience Publishers, New York, p. 120.
[71 MacFarlane, A . G . J . , "Functional-matrix Theory for the General
Linear Electrical Network - Part 3", Proc . IEE (London), vol. 113,
July 1966.
181 Barnett, S. and Storey, C . , "On the General Functional Matrix for
a Linear System", IEEE Trans. on Automatic Control, vol. AC-12,
no. 4, August 1967.
15


On fixed linear systems with a generalized performance criteria

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