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Minimal Root Sensitivity in Linear Systems
Copyright ([) IFAC Applications of Nonlinear Programming Optimization and Control. Palo Alto. California. USA 1983
to
MINIMAL ROOT SENSITIVITY IN LINEAR SYSTEMS R. E. Skelton and D. A. Wagie School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, USA
Abstract. Nominal perfor~ance, root (eigenvalue) sensitivity, and stability are i~portant control design considerations. This paper deals only with the first two of these concerns. A lower bound is derived for root sensitivity and necessary and sufficient conditions are given for achievinq this minimum. This is the ~ain result of the paper. In addition, an optimal output feedhack control proble~ is discussed which penalizes an index related to root sensitivity. TI"/O hazards are clarified concerninq root sensitivity designs . First, we illustrate that root sensitivity has nothino to do with stability. Secondly, we show that normality of the plant ~atrix is the necessary and sufficient condition for minimal root sensitivity, but that normality measures can be convex even when root sensitivity measures are not. Hence, the opti~i zation problems are ouite different. Keywords. 5ensitivity analysis; linear systems; stability; root (eigenvalue) sensitivity; normal matrices. INTRODUCTION
substituted in the literature. To understand the second pitfall, we note that the necessary and sufficient condition for minimal root sensitivity is normality of the plant ~atrix . This fact has led researchers to presu~e that normality is a good design goal (t1acFarlane and Hung, 1982), but in fact the nor~ality measure can be a tractable convex function even when root sensitivity is not. The second nitfa 11, then, is that even if sf'1all ~oot sensitivity were a desirable desiqn ooal, the substitution of normality measures for root sensitivity ~easures is not alw~ys appropriate, as our exa~ple will illustrate .
The modal data for physical systems is rarely well known . This can make performance predictions unreliable in both open-loop and feedback cont~ol proble~s. This paper documents the smallest possible sensitivity of eiqenvalues A. with respect to the independent plant patameters in linear systems of the form (1) x = Ax , XE: Rn That is, the norf'l of the root sensitivity rlatrix ,n. ;n.1 _ en.1 1 aA aJl.ln
The norm of a
a7\l1
11 [. ] 11 2
~atrix
shall be denoted by
= t r [. ] * [ . J, t r [. ] ~ t race [ • ]
( 2) (3a)
dA i
dA .
~l
dAnn
1
and the norm of a vector shall be denoted by 11 (.) 11
and the lo~!er bound of its nor~ are of interest. Secondly, a J:letric related to root sensitivity is added to the opti~al output feedback proble~ in an attempt to achieve a compromise between perfor~ance and root sensitivity.
2
= (.) *(.)
(3b)
where * denotes comple x conjuqate transpose . Results herein are limited to the case of distinct eigenvalues for A. CONSTRUCTION OF A ROOT 5ENSITIVITY METRIC
This approach to system control desiqn is illustrated by a si~ple example to point out certain pitfalls in sensitivity analysis and desiqn which have not been previously clarified. The first pitfall is the fact that sensitivity has nothinq to do with "stabil ity" and yet these t\'!O objectives are often
The sensitivity of the ith eioenvalue dA ./ dA /', 1 is an nxn matrix denoted by S.1 = dA 1./ dA. The norm of Si fro~ (3) is
17
R. E. Skelton and D. A. Wagie
18
lis ,.11 2 ~
n
, ,
tr S. *S.
n
L L
a=l 13=1
aAi
(37\)
2
(4)
a13
The complete root sensitivity metric of i nterest is n 6 (5) s = 11 Si 112 ~ L II:~ i 11 2 i =1 i =1 Thus, from the point of vie~ of root sensitivity, a system desiqn with a larqe value of s miqht be considered less desirable than a system desiqn with a small value of s. This would be true if the analyst is s pec i f i cally concerned that root locations remain fixed in the presence of parameter uncertainties.
I
In order to compute the root sensitivity metric, we will assume that A has a linearly independent set of eiqenvectors e i Ae i = e i Ai The by
':';';:::'
,i = 1, ... , n
.b':i:";:'r:~[:J';
(6)
are defined -1 ,
E
also a more compact generalization of the eigenvalue sensitivity used in Van Ness (196~. The norm (4) may now be written liS,. 112 6 tr S.*S . = tr(l .e .T)*(l.e . T)
(7)
Multiplying (6) from the left by l.*, using (7), yields the eigenvalues in ter~s of A, its eigenvectors, and its reciprocal basis vectors, (8)
Differentiation of the scalar (8) with respect to A provides the required sensitivity aA. / aA. To derive this result two identities fr6m linear algebra are required tr AB = tr BA
a
a
(9)
T
aA(tr AB) = aA(tr BA) = B
(10)
where (9), (10) hold for real or cormlex matrices B and A, and (10) holds if the elements of A are independent. Hence from (8), usino (9), (10), a T ~ = aA[tr A(e i li*)] = (e i li*) aA i
liei
T
,,
( 13) where the last equal ity requires use of identity (9) a~ain, and where 2 2 Il l. 11 = l.*l. , Il e. 11 = e.*e .
,
,
"
"
(14)
One may note the similarity between (11) and the weak differential of A. qiven by Eq. (6.2.3) of Slum (1972). Aiso note the similarity between (13) and the upper bound of the weak derivative of Ai provided on the top of p. 235 in Blum (1972). This paper seeks lowe r bounds of root sensitivity rather than the upper bounds. Otherwise, the nature of the results are similar (see theorem 6.2.4 of Blum (1972)). Also note that the sensitivity model (11), (13) is different from the one used in Wilkinson (1965). Wilkinson, and later on, others (Postlethwaite, 1982) use the inner product of normali zed left and right eigenvectors: si = ri*e i , with Il r .11 = 1 = Ile ·ll . Our result (11) involves the outer product of the reciprocal basis vector and the (right) eiqenvector without requirin~ any special normalization. The two points of view use left eioenvectors, but with different normalizations: left eiqenvectors normalized to unit length, as opposed to that choice of left eiqenvectors correspondinq to a reciprocal basis of the (right) eigenvectors. He choose the latter. (Note, however, that if we used Wilkinson's model with normalized eiqenvectors and normalized recipro cal basis vectors, his sensitivity index would merely be the reciprocal of (13)). Since reciprocal basis vectors are uniquely determined from the eiqenvectors, only one normalization is required (the initial one on the eiqenvectors) in our metric, as opposed to two normalizations required by Wilkinson (1965) and Postleth~laite (1982). 1·le also would like the results to be independent of the particular normalization chosen for the eigenvectors; our approach has this property.
,
hence l ,. *e.J = 0 ,..J
,,
"
,
Gilbert (1983) has computed the sufficient conditions for minimality for a variety of seAsitivity indices " :~A i 111,W::Ail/, and
(11 )
where
I
W:A1 /1.
denotes complelCC conju(late .
Result (11) can also be obtained from Jacobi's formula for small root perturbations (Jacobi, 1846)
, ,
,
61.. . = l .*6Ae.
(12 )
where 6Ai is the chanoe (to first order approximation) of Ai in the presence of a perturbation of A to A+6A. ECl ua t ion (11) i s
In addition, he has proved the differ~ntiability which we merely assumed in (2) and (11). We seek necessary and sufficient conditions for minimality of (13) . Now consider (13) again. The Schwartz inequality (Blum, 1972) holds for any two vectors l i' e i
, ,
,
Il .*e· 1 -< Il l · 1I
lie,. 11
(15 )
Hinimal Root Sensitivity
Since the pOI'ticuZOI' vectors .ti' e i are rel ated by (8), (16 )
.t.1 *e.1 = 1 Equations (15) and (16) lead
i~mediately
to
( 17)
Squarinq both sides of (17), and usinf! (13), leads to 11 S1·11
2
> -
( 18)
1
The equality in (17), and hence in (18), holds if and only if .ti and e i are colinear (.t. = e.) (Blu~, 1972). Fro~ linear alqebra (C~llen; 1967), t i = e i if and only if A is nor~al (AA* = A*A). Thus, the main theoretical results of th e paper are summarized as fo 11 ows. Theorem 1. Let (A., e., .t .) be the i th eigenvaZue , eigenvJcto} ana its ~ecip~ocaZ basis vecto~ associated with the ~eaZ mat~ix A. If A has a Zinea~Zy independent set of eigenvecto~s e , i=l, . .. , n, then i
i =1, ... , n
the
whe~e
Zowe~
11~112 d A
( 19)
bound
=1
i=l,
..• , n
is achieved if and onZy if AA* = A*A. The sensitivity met~ic (5) i s bounded f~om beZow by
s
>
n
~n
Linear Systems
J9
sensitivity terms l: l f~y 112 (where p.,i=l, ... r i
0
Pi
1
represent the uncertain parameters). The resulting computational burdens are very qreat indeed, since the dimension of the constraint (state) equation becomes n(l+r). Also, minimizinq output sensitivity does not necessarily keep root sensitivity small. Previously, we showed that root sensitivity is minimized when A is normal. It has also been shown (Patel and Toda, 1980) that the robustness bound for a certain class of parameter errors is maximized when the plant matrix, A, is normal. has often been assumed to be a good desiqn aoal . MacFarlane and Hunq (1982) recently stated," ... an approximation to normality is somethinq which one strives to achieve in the feedback design process. Thus, normal systems and their properties play a key role in the formulation and implementation of what we call the auasi-classical approach to feedback systems." The sugqestion here, of course, is that normal matrices A are desirable, and that app~oximation s of normal A are desirable. Therefore, with this backqround, to~ether with the motivation of Theorem 1, we shall desi(Jn a controller to yield a "more normal" closed loop plant matrix, rather than attemptin~ to minimize root sensitivity dire.ctly. lie shall later return, by way of example, to the question of how normality relates to root sensitivity outside the neiqhborhood of the global minimum of root sensitivity. This comparison should provide considerable insiqht toward the understanding of "normal" desiqn objectives, such as in MacFarlane and Hung (1982) and Postlethwaite (1982) . Nor~ality
(20)
and, the minimum sensitivity s = n is achieved if and onZy if A is normaZ (AA* = A*A).
The theorem provides necessary and sufficient conditions for minimum root sensitivity. If one wishes to keep roots relatively fixed in the presen ce of parameter variations, Theorem 1 indi cates that normality of A is a necessary and sufficient condition for a globally ~inimal value of dA i Gilbert (1983) has
3A shown that symmetry of A is a sufficient condition for a glob all y mini~al value of 1/: :e Aill. Since a symmetric A is also normal, and only has real eiqenvalues, this is in a~reement with The orem 1. The ne xt section sun~ests a means to incorporate this information into the output feedback control desian problem.
Consider a new performance index for optimization that includes an "abnormal ity" penal ty V
= ~: E( II Y I I~ + Il u ll ~) + BII(A+BGM)(A+BGM)T - (A+BGM)T(A+GBM) 11 2 CTQC
( 21)
subject to the state equations
.
x
Ax + Bu
y
Cx
+
Dw
E(w) = 0 , v
E(~~~I)(wT( r ),
[~
~]
vT(r ))
o(t- r )
z = Mx + v u
Gz
E(w(t) ) xT(O) v(t)
0, t
>
0
(22) OUTPUT FEEDBACK DESIGN Parameter sensitivity has long been a concern in optimal control. Some authors (Yedavalli and Skelton, 19 82; Byrne and Burke, 1976) have sugaested modifying a quadratic performance index by the addition of trajectory
When B is much smaller than the norms of Q and R the solution tends toward the standard optimal control result (Levine, Johnson, and Athans, 1971; Kosut, 1970). On the other extre~e, when B is chosen much larqer than the norms of Q and R, the closed-loop system
R. E. Skelton and D. A. Wagie
20
approaches the smallest possible root sensitivity. (From Theorem 1 note that root sensitivity is minimized if and only if A+BGM is nomal, in which case the latter term in (21) is zero.) Other choices of weights on the matrix norm may be chosen besides cTqC. This choice is su~~erted only to make the sensitivity weight C OC the same as the state weight in yTQy = xT[CTQC]x. Using the same matrix norm ~ as in previous sections, and defining A = A+BGM, V becomes
Theorem 3. The minimum s ensi tivity s=n can be guaranteed by outpu t feedba ck contr ol fo r an arbitrary A i f and on ly if rank B = r ank M = n. Fur t hermore, the contro l gain in t his cas e is not unique . TWo gain s that provide mini mum sensitivity are _B-1AM- l
(28)
G
B- 1AT M- 1
(29)
Proof: Substitute (28), (29) into the normality condition for minimum sensitivity
V = tr P(CTQC + MTGTRGM) + tr VGTRG
(23)
G
(A+RGM)(A+BGM)T - (A+BGM)T(A+~GM) = 0 ( 30) to see that the condition (30) holds.
where P is the solution of
APPLICATION OF CLOSED-LOOP ROOT SENSITIVITY DESIGN
P(A+BGM)T + (A+8GM)P + BGVGTB T
o
(24) The necessary conditions for the optimum G are obtained by augmenting the constraint equation (24) to (23) via Lagrange multiplier matrix K and differentiating the aunmented V with respect to P, K, and G. The result is that K and G must satisfy
o
K( A+BGI1) + (A+BGM) \
+ I1T GTRGM (25)
o=
RGMPf1 T + RGV + BTKPt1 T + BTKBGV
+ ~
(26)
Example 1: The pitch motion of a rigid aircraft is governed by (Bryson and Ho, 1975)
where a is the anqle of attack, q is the pitch rate, u is the elevator anrjle, T is the liftinq time constant, w is the undamped pitch natural frequency, °and Q is the elevator effe ct ivenes s. In the open-loop case, (u+w)=O we are interested in the root sensitivity properties of (31). Note from Theorem 1 that minimum sensitivity
where
~ ~ 8BT{[(AA T_A TA)C TQC + CTQC(AAT_ATA)]A _ A[(AA T _ ATA)CT()C + CTQC(AA T_A TA)]}t1 T ( 27)
s =
It has been shown (Kosut, 1970; Levine, Johnson, and Athans, 1971), that the necessary conditions for the output feedback solution for the system (22) to minimize the standard quadratic cost function (eq. (21) with 6=0) are given by (24)-(26) with ~ =O. Various suboptimal strate9ies for approximating the solution of (24)-(26) may be found in the 1iterature (Kosut, 1970). The following conclusion is a very special case of the sensitivity minimization problem and not usually practical, but the results are easily given. For an ar bitr ar y A, the matrix A+BGM can be made normal (by choice of G) only if rank B = rank M = n. Hence, we have the following:
a\ . 2
If-;;-i- II = 2 is achieved if and on l y if i =1 aA
A is normal. Computinq the "abnormality" matrix (AA*-A*A) yields 1_wo4
These results are summarized as follows: Theorem 2. The necessary conditions for mini mizing (21) subjec t to the cons t r aints (22) are given by (24)-(26) and (27).
2
L
AA*-A*A = 1 2 [ T( 1+w0 )
~( 1+wo2~ 4
I Wo - 1
(32)
Thu s, root sensitivityltakes on its absolute minimum when wo = 1, -T = O. This, of course, is not a practical possibility for the aircraft. Now consider using the output feed back desiqn of the previous section. Example 2. For the aircraft in Example 1, l et the an~le of attack measurement be made z = a+V where E[v] = 0, E[V(t)V( T)] = 5(t - T) describes the white measurement noise v and E[w] = 0, E[W(t)W( T)] = 6(t- T) describes the white actuator noise w. For the system (31) desi~n a measurement feedback control law for reQulatinrj u such that V ~: {E(q2+ pu2 ) + sll(A+BGM)(A+RGM)T -
(A+BGM)T(A+GRM) !I~TC }
(33)
21
Minimal Root Sensitivity in Linear Systems
is minimized. (We have assumed 0=1, R=p, and C = [0, lJ; with these values, equation (21) yi e 1ds (33).) The solution is provided by (26), where P is obtained froM (24),
P
and K is obtained from (25)
IIAA* - A*AII
VA =
T ,and the control effort. C QC For both figures w 2 = 1, T = .7, 6 and p o vary. In the standard output feedback desiejn ( 6=0), the output performance is improved with an increase in control effort (Fiq. 1) whereas the abnormality index qreatly increases with control effort (Fig. 2 with 6 = 0). Note also that 6 > 1 is not desired, since larqer values of 6 do not yield substantially larqer abnormality reductions (Fiq. 2) but do accelerate the dearadation of the nominal -output performance (Fiej. 1). o o o
6
en
2
o
o o
(35)
For the aircraft exaMple, (26) yields a fifth order equation in G as a function of 6 , p , W 2 and T. For given values of p , 6 , w2 and TO this equation will yield five candidate values of G. The optimal G is that root that mini~izes the cost function. In the present case G < 1 is required for stability. If none of the real candidate values of G are < 1, then analysis of this problem shol'/s that a decrease in 6 will drive G toward more stable values. This means that normality and stability are diametrically opposed in this circumstance.
6
C\J N
>,
8
w
.- 6
.1
E 8 o. - - p - - .,.... t .....-. ~+' 6 =0
Increasing
p
o
o
o~
________
.0000
~
________+-_____
.1000
Fi g. 1. Assume wo2=1, F.7 (this corresponds to darlping ratio = .71). Settinq e= 0, p = 1 yields the standard optimal measurement feedback control
.2000
_ +-__ .3000
Vu /: ; lim Ellu ll ~ t-+= Output cost Vy vs. input cost Vu [1 = .7, Wo = 1.0]
0
~
C\J
( 36)
G=-0.118
<- - - - - - -
s
0
Ll
er
and settinn e = sensitive desiqn
00 ,
p«
00
G = 2.0
yields the optimally ( 37)
This choice of G in (37) forces the closedloop system matrix to be symmetric (hence, norma 1) -l /T l]=[l/T lJ A+RGM [ -w 2+w 2G O . 1 0 o
0
I-
LlD N
,.« « ,.
<-------
~
.1
(Xl
Fiq. 1 shows the tradeoff
between output
/: ; t1->
p
I
« « 0
~
«:i-
6
:::>
2
0
~
.0000
and by Theorem 1, the sensitivity is at its minirlum in this case. Note, however, that stability is lost by this minimum sensitivity desiejn, (G < 1 is required for stability). Thus, minimally sensitive desiqns mioht not be stable.
Increasing
.1000
/: ;
Fig. 2.
.2000
.3000
2
Vu = lim El lullR t-Abnormality cost VA vs. input
cost Vu [ T = .7, w0 = 1.OJ Now, if we were to assume that abnormality was a consistent metric for root sensitivity (both were convex functions), then we could use Figs. 1 and 2 as design tools to choose the feedback law that achieves the best compromise between performance and root sensitivity (represented by abnormality). Hawever, abnormality is not a consistent index of root
R. E. Skelton and D. A. Wagie
22
sensitivity, except in the local neiC]hborhood of the normal condition, as illustrated by Figs. 3 and 4. From these finures it is obvious that (for this example) abnormality and root sensitivity are only consistent indices above G = .49 ( S ~ 2) and are actually opposing indices below that point. From Fiaures 1 and 2, we stated that we would not choose 13 > 1 if desiqninC] for performance/ normality. Hence, for this problem, usinC] Figs. 1 and 2 would place us in the realm where abnormality is an inconsistent index for root sensitivity. Note also that the sensitivity l'ihen we have repeated roots (G = .49). ->CO
o
o
Legend:
Lf)
[;] 13
0
I\n "abnormality" term is therefore added to the traditional nuadratic performance metric of optimal control in an attempt to decrease root sensitivity. The necessary conditions are given for the solution of this problem and an example (jives some practical insights. The solution to this problem points out that while "normality" implies minimal root sensitivity, minimizing abnormality will not consistently decrease root sensitivity. In fact, decreasinCl abnormality may very well increase root sensitivi~y, especially when two roots are close toqether. The desinn cautions produced by the results include the fact that the necessary and sufficient condition (normality) for minimal root sensitivity is only a local result and also that it applies only for non-defective Dl ant matrices (those having 1inearly independent eiqenvectors).
Os 100
£::,.13
to the plant matrix may be readily computed. A necessary and sufficient condition for minimum root sensitivity is that the plant matrix of the state equations be normal.
I\CKNOWLEDGMENT Portions of this work were sponsored by AFOSR Grant No. 820209 and NSF Grant No. ECS-8119598.
o o
.000
1.000
2.000
3.000
G( S) Fig. 3 Root sensitivity s vs. feedback gain G(S). [.=.7, Wo lob, p=lJ o
Legend:
u
Blum, E.K. (1972). Numerical Analysis and Computation: Theory and Practice. Addison-Hesley, Reading, Massachusetts. Bryson, 1\. and V.C. Ho (1975). Applied Optimal Control. Hemisphere Publishing, Hashington, D.C.
o
Lf)
REFERENCES
EJ 13
0
8
13
1
h.
S
100
Byrne, P.C. and M. Burke (1976). Optimizabon with trajectory sensitivity considerations. IEEE Trans. Auto. Control, Vol. I\C-21, 232-283.
c-
l-
u
ClIllen, C.G. (1967). Transformations. t1assachusetts.
N_
-« -l< -«, -l<
~
0
~
~1atrices
and Linear Addison-\'/esley, Readin(1,
lJ)
Gilbert, E.G. (1983). Conditions for minimizing the norm sensitivity of characteristic roots. Submitted for publication in IEEE Trans. Auto. Control.
-«
:>
0
~
.000
1.000
2 . 000
3.000
G( S) Fig. 4 Abnormality cost VA vs. feedback gain G( S). [.=.7, wo=LO, p=lJ cmlCLUS IOrlS
Explicit expressions for a scalar metric of root sensitivity is (jiven in terms of the left and right ei(jenvectors of the system, so that sensitivity of each eiqenvalue with respect
Jacobi, C.G.J. (1846). Crelle's Journal fur die Reine und Angewandte Mathematik, Vol. 30. DeGruyter, Berlin. pp. 51-95. Kosut, R.L. (1970). Suboptimal control of linear time-invariant systems subject to control structure constraints. IEEE Trans. Auto Control, Vol. AC-15, 557-563. Levine, W.S., T.L. Johnson, and M. Athans (1971). Optimal limited state variable feedback controllers for linear systems. IEEE Trans. Auto. Control, Vol. AC-15, 557-563.
Minimal Root Sensitivity in Linear Systems
MacFarlane, A.G.J., and Y.S. Hung (1932). A quasi-classical approach to multivariable feedback systems design. Proceedings of the Second IFAC SymposiLiiilOnComputer Aided Design of Multivariable Technological Systems, West Lafayette, Indiana, USA. Se pt. , 1982, 39-48. Patel, R.V., and M. Toda (1980). Quantitative measures of robustness for multivariable systems. Proceedi ngs of the 1980 Joint Auto. Control Conf., San Francisco, California, USA. Aug., 1980. Postlethv,aite,1. (1982). Sensitivity of the characteristic gain loci. Proceedinqs of the Second IFAC Symposium on Computer Aided Desiqn of Multivariable Technological Systems, ~est Lafayette, Indiana, USA. Sept., 1982, 153-158. Van Ness, J.E., J.M. Boyle, and F.P. Imad (1965). Sensitivities of large, multipleloop control systems. IEEE Trans. Auto. Control, Vol. AC-10, 308-3J5. Yedavalli, R.K., and R.E. Skelton (1982). Controller desiun for parameter sensitivity reduction in linear regulators. Optimal Control Applications & Methods, Vol. 3, 221-240. Wilkinson, J.H. (1965). The Algebraic Eioenvalue Problem . Oxford University Press, London.
23
to
MINIMAL ROOT SENSITIVITY IN LINEAR SYSTEMS R. E. Skelton and D. A. Wagie School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana, USA
Abstract. Nominal perfor~ance, root (eigenvalue) sensitivity, and stability are i~portant control design considerations. This paper deals only with the first two of these concerns. A lower bound is derived for root sensitivity and necessary and sufficient conditions are given for achievinq this minimum. This is the ~ain result of the paper. In addition, an optimal output feedhack control proble~ is discussed which penalizes an index related to root sensitivity. TI"/O hazards are clarified concerninq root sensitivity designs . First, we illustrate that root sensitivity has nothino to do with stability. Secondly, we show that normality of the plant ~atrix is the necessary and sufficient condition for minimal root sensitivity, but that normality measures can be convex even when root sensitivity measures are not. Hence, the opti~i zation problems are ouite different. Keywords. 5ensitivity analysis; linear systems; stability; root (eigenvalue) sensitivity; normal matrices. INTRODUCTION
substituted in the literature. To understand the second pitfall, we note that the necessary and sufficient condition for minimal root sensitivity is normality of the plant ~atrix . This fact has led researchers to presu~e that normality is a good design goal (t1acFarlane and Hung, 1982), but in fact the nor~ality measure can be a tractable convex function even when root sensitivity is not. The second nitfa 11, then, is that even if sf'1all ~oot sensitivity were a desirable desiqn ooal, the substitution of normality measures for root sensitivity ~easures is not alw~ys appropriate, as our exa~ple will illustrate .
The modal data for physical systems is rarely well known . This can make performance predictions unreliable in both open-loop and feedback cont~ol proble~s. This paper documents the smallest possible sensitivity of eiqenvalues A. with respect to the independent plant patameters in linear systems of the form (1) x = Ax , XE: Rn That is, the norf'l of the root sensitivity rlatrix ,n. ;n.1 _ en.1 1 aA aJl.ln
The norm of a
a7\l1
11 [. ] 11 2
~atrix
shall be denoted by
= t r [. ] * [ . J, t r [. ] ~ t race [ • ]
( 2) (3a)
dA i
dA .
~l
dAnn
1
and the norm of a vector shall be denoted by 11 (.) 11
and the lo~!er bound of its nor~ are of interest. Secondly, a J:letric related to root sensitivity is added to the opti~al output feedback proble~ in an attempt to achieve a compromise between perfor~ance and root sensitivity.
2
= (.) *(.)
(3b)
where * denotes comple x conjuqate transpose . Results herein are limited to the case of distinct eigenvalues for A. CONSTRUCTION OF A ROOT 5ENSITIVITY METRIC
This approach to system control desiqn is illustrated by a si~ple example to point out certain pitfalls in sensitivity analysis and desiqn which have not been previously clarified. The first pitfall is the fact that sensitivity has nothinq to do with "stabil ity" and yet these t\'!O objectives are often
The sensitivity of the ith eioenvalue dA ./ dA /', 1 is an nxn matrix denoted by S.1 = dA 1./ dA. The norm of Si fro~ (3) is
17
R. E. Skelton and D. A. Wagie
18
lis ,.11 2 ~
n
, ,
tr S. *S.
n
L L
a=l 13=1
aAi
(37\)
2
(4)
a13
The complete root sensitivity metric of i nterest is n 6 (5) s = 11 Si 112 ~ L II:~ i 11 2 i =1 i =1 Thus, from the point of vie~ of root sensitivity, a system desiqn with a larqe value of s miqht be considered less desirable than a system desiqn with a small value of s. This would be true if the analyst is s pec i f i cally concerned that root locations remain fixed in the presence of parameter uncertainties.
I
In order to compute the root sensitivity metric, we will assume that A has a linearly independent set of eiqenvectors e i Ae i = e i Ai The by
':';';:::'
,i = 1, ... , n
.b':i:";:'r:~[:J';
(6)
are defined -1 ,
E
also a more compact generalization of the eigenvalue sensitivity used in Van Ness (196~. The norm (4) may now be written liS,. 112 6 tr S.*S . = tr(l .e .T)*(l.e . T)
(7)
Multiplying (6) from the left by l.*, using (7), yields the eigenvalues in ter~s of A, its eigenvectors, and its reciprocal basis vectors, (8)
Differentiation of the scalar (8) with respect to A provides the required sensitivity aA. / aA. To derive this result two identities fr6m linear algebra are required tr AB = tr BA
a
a
(9)
T
aA(tr AB) = aA(tr BA) = B
(10)
where (9), (10) hold for real or cormlex matrices B and A, and (10) holds if the elements of A are independent. Hence from (8), usino (9), (10), a T ~ = aA[tr A(e i li*)] = (e i li*) aA i
liei
T
,,
( 13) where the last equal ity requires use of identity (9) a~ain, and where 2 2 Il l. 11 = l.*l. , Il e. 11 = e.*e .
,
,
"
"
(14)
One may note the similarity between (11) and the weak differential of A. qiven by Eq. (6.2.3) of Slum (1972). Aiso note the similarity between (13) and the upper bound of the weak derivative of Ai provided on the top of p. 235 in Blum (1972). This paper seeks lowe r bounds of root sensitivity rather than the upper bounds. Otherwise, the nature of the results are similar (see theorem 6.2.4 of Blum (1972)). Also note that the sensitivity model (11), (13) is different from the one used in Wilkinson (1965). Wilkinson, and later on, others (Postlethwaite, 1982) use the inner product of normali zed left and right eigenvectors: si = ri*e i , with Il r .11 = 1 = Ile ·ll . Our result (11) involves the outer product of the reciprocal basis vector and the (right) eiqenvector without requirin~ any special normalization. The two points of view use left eioenvectors, but with different normalizations: left eiqenvectors normalized to unit length, as opposed to that choice of left eiqenvectors correspondinq to a reciprocal basis of the (right) eigenvectors. He choose the latter. (Note, however, that if we used Wilkinson's model with normalized eiqenvectors and normalized recipro cal basis vectors, his sensitivity index would merely be the reciprocal of (13)). Since reciprocal basis vectors are uniquely determined from the eiqenvectors, only one normalization is required (the initial one on the eiqenvectors) in our metric, as opposed to two normalizations required by Wilkinson (1965) and Postleth~laite (1982). 1·le also would like the results to be independent of the particular normalization chosen for the eigenvectors; our approach has this property.
,
hence l ,. *e.J = 0 ,..J
,,
"
,
Gilbert (1983) has computed the sufficient conditions for minimality for a variety of seAsitivity indices " :~A i 111,W::Ail/, and
(11 )
where
I
W:A1 /1.
denotes complelCC conju(late .
Result (11) can also be obtained from Jacobi's formula for small root perturbations (Jacobi, 1846)
, ,
,
61.. . = l .*6Ae.
(12 )
where 6Ai is the chanoe (to first order approximation) of Ai in the presence of a perturbation of A to A+6A. ECl ua t ion (11) i s
In addition, he has proved the differ~ntiability which we merely assumed in (2) and (11). We seek necessary and sufficient conditions for minimality of (13) . Now consider (13) again. The Schwartz inequality (Blum, 1972) holds for any two vectors l i' e i
, ,
,
Il .*e· 1 -< Il l · 1I
lie,. 11
(15 )
Hinimal Root Sensitivity
Since the pOI'ticuZOI' vectors .ti' e i are rel ated by (8), (16 )
.t.1 *e.1 = 1 Equations (15) and (16) lead
i~mediately
to
( 17)
Squarinq both sides of (17), and usinf! (13), leads to 11 S1·11
2
> -
( 18)
1
The equality in (17), and hence in (18), holds if and only if .ti and e i are colinear (.t. = e.) (Blu~, 1972). Fro~ linear alqebra (C~llen; 1967), t i = e i if and only if A is nor~al (AA* = A*A). Thus, the main theoretical results of th e paper are summarized as fo 11 ows. Theorem 1. Let (A., e., .t .) be the i th eigenvaZue , eigenvJcto} ana its ~ecip~ocaZ basis vecto~ associated with the ~eaZ mat~ix A. If A has a Zinea~Zy independent set of eigenvecto~s e , i=l, . .. , n, then i
i =1, ... , n
the
whe~e
Zowe~
11~112 d A
( 19)
bound
=1
i=l,
..• , n
is achieved if and onZy if AA* = A*A. The sensitivity met~ic (5) i s bounded f~om beZow by
s
>
n
~n
Linear Systems
J9
sensitivity terms l: l f~y 112 (where p.,i=l, ... r i
0
Pi
1
represent the uncertain parameters). The resulting computational burdens are very qreat indeed, since the dimension of the constraint (state) equation becomes n(l+r). Also, minimizinq output sensitivity does not necessarily keep root sensitivity small. Previously, we showed that root sensitivity is minimized when A is normal. It has also been shown (Patel and Toda, 1980) that the robustness bound for a certain class of parameter errors is maximized when the plant matrix, A, is normal. has often been assumed to be a good desiqn aoal . MacFarlane and Hunq (1982) recently stated," ... an approximation to normality is somethinq which one strives to achieve in the feedback design process. Thus, normal systems and their properties play a key role in the formulation and implementation of what we call the auasi-classical approach to feedback systems." The sugqestion here, of course, is that normal matrices A are desirable, and that app~oximation s of normal A are desirable. Therefore, with this backqround, to~ether with the motivation of Theorem 1, we shall desi(Jn a controller to yield a "more normal" closed loop plant matrix, rather than attemptin~ to minimize root sensitivity dire.ctly. lie shall later return, by way of example, to the question of how normality relates to root sensitivity outside the neiqhborhood of the global minimum of root sensitivity. This comparison should provide considerable insiqht toward the understanding of "normal" desiqn objectives, such as in MacFarlane and Hung (1982) and Postlethwaite (1982) . Nor~ality
(20)
and, the minimum sensitivity s = n is achieved if and onZy if A is normaZ (AA* = A*A).
The theorem provides necessary and sufficient conditions for minimum root sensitivity. If one wishes to keep roots relatively fixed in the presen ce of parameter variations, Theorem 1 indi cates that normality of A is a necessary and sufficient condition for a globally ~inimal value of dA i Gilbert (1983) has
3A shown that symmetry of A is a sufficient condition for a glob all y mini~al value of 1/: :e Aill. Since a symmetric A is also normal, and only has real eiqenvalues, this is in a~reement with The orem 1. The ne xt section sun~ests a means to incorporate this information into the output feedback control desian problem.
Consider a new performance index for optimization that includes an "abnormal ity" penal ty V
= ~: E( II Y I I~ + Il u ll ~) + BII(A+BGM)(A+BGM)T - (A+BGM)T(A+GBM) 11 2 CTQC
( 21)
subject to the state equations
.
x
Ax + Bu
y
Cx
+
Dw
E(w) = 0 , v
E(~~~I)(wT( r ),
[~
~]
vT(r ))
o(t- r )
z = Mx + v u
Gz
E(w(t) ) xT(O) v(t)
0, t
>
0
(22) OUTPUT FEEDBACK DESIGN Parameter sensitivity has long been a concern in optimal control. Some authors (Yedavalli and Skelton, 19 82; Byrne and Burke, 1976) have sugaested modifying a quadratic performance index by the addition of trajectory
When B is much smaller than the norms of Q and R the solution tends toward the standard optimal control result (Levine, Johnson, and Athans, 1971; Kosut, 1970). On the other extre~e, when B is chosen much larqer than the norms of Q and R, the closed-loop system
R. E. Skelton and D. A. Wagie
20
approaches the smallest possible root sensitivity. (From Theorem 1 note that root sensitivity is minimized if and only if A+BGM is nomal, in which case the latter term in (21) is zero.) Other choices of weights on the matrix norm may be chosen besides cTqC. This choice is su~~erted only to make the sensitivity weight C OC the same as the state weight in yTQy = xT[CTQC]x. Using the same matrix norm ~ as in previous sections, and defining A = A+BGM, V becomes
Theorem 3. The minimum s ensi tivity s=n can be guaranteed by outpu t feedba ck contr ol fo r an arbitrary A i f and on ly if rank B = r ank M = n. Fur t hermore, the contro l gain in t his cas e is not unique . TWo gain s that provide mini mum sensitivity are _B-1AM- l
(28)
G
B- 1AT M- 1
(29)
Proof: Substitute (28), (29) into the normality condition for minimum sensitivity
V = tr P(CTQC + MTGTRGM) + tr VGTRG
(23)
G
(A+RGM)(A+BGM)T - (A+BGM)T(A+~GM) = 0 ( 30) to see that the condition (30) holds.
where P is the solution of
APPLICATION OF CLOSED-LOOP ROOT SENSITIVITY DESIGN
P(A+BGM)T + (A+8GM)P + BGVGTB T
o
(24) The necessary conditions for the optimum G are obtained by augmenting the constraint equation (24) to (23) via Lagrange multiplier matrix K and differentiating the aunmented V with respect to P, K, and G. The result is that K and G must satisfy
o
K( A+BGI1) + (A+BGM) \
+ I1T GTRGM (25)
o=
RGMPf1 T + RGV + BTKPt1 T + BTKBGV
+ ~
(26)
Example 1: The pitch motion of a rigid aircraft is governed by (Bryson and Ho, 1975)
where a is the anqle of attack, q is the pitch rate, u is the elevator anrjle, T is the liftinq time constant, w is the undamped pitch natural frequency, °and Q is the elevator effe ct ivenes s. In the open-loop case, (u+w)=O we are interested in the root sensitivity properties of (31). Note from Theorem 1 that minimum sensitivity
where
~ ~ 8BT{[(AA T_A TA)C TQC + CTQC(AAT_ATA)]A _ A[(AA T _ ATA)CT()C + CTQC(AA T_A TA)]}t1 T ( 27)
s =
It has been shown (Kosut, 1970; Levine, Johnson, and Athans, 1971), that the necessary conditions for the output feedback solution for the system (22) to minimize the standard quadratic cost function (eq. (21) with 6=0) are given by (24)-(26) with ~ =O. Various suboptimal strate9ies for approximating the solution of (24)-(26) may be found in the 1iterature (Kosut, 1970). The following conclusion is a very special case of the sensitivity minimization problem and not usually practical, but the results are easily given. For an ar bitr ar y A, the matrix A+BGM can be made normal (by choice of G) only if rank B = rank M = n. Hence, we have the following:
a\ . 2
If-;;-i- II = 2 is achieved if and on l y if i =1 aA
A is normal. Computinq the "abnormality" matrix (AA*-A*A) yields 1_wo4
These results are summarized as follows: Theorem 2. The necessary conditions for mini mizing (21) subjec t to the cons t r aints (22) are given by (24)-(26) and (27).
2
L
AA*-A*A = 1 2 [ T( 1+w0 )
~( 1+wo2~ 4
I Wo - 1
(32)
Thu s, root sensitivityltakes on its absolute minimum when wo = 1, -T = O. This, of course, is not a practical possibility for the aircraft. Now consider using the output feed back desiqn of the previous section. Example 2. For the aircraft in Example 1, l et the an~le of attack measurement be made z = a+V where E[v] = 0, E[V(t)V( T)] = 5(t - T) describes the white measurement noise v and E[w] = 0, E[W(t)W( T)] = 6(t- T) describes the white actuator noise w. For the system (31) desi~n a measurement feedback control law for reQulatinrj u such that V ~: {E(q2+ pu2 ) + sll(A+BGM)(A+RGM)T -
(A+BGM)T(A+GRM) !I~TC }
(33)
21
Minimal Root Sensitivity in Linear Systems
is minimized. (We have assumed 0=1, R=p, and C = [0, lJ; with these values, equation (21) yi e 1ds (33).) The solution is provided by (26), where P is obtained froM (24),
P
and K is obtained from (25)
IIAA* - A*AII
VA =
T ,and the control effort. C QC For both figures w 2 = 1, T = .7, 6 and p o vary. In the standard output feedback desiejn ( 6=0), the output performance is improved with an increase in control effort (Fiq. 1) whereas the abnormality index qreatly increases with control effort (Fig. 2 with 6 = 0). Note also that 6 > 1 is not desired, since larqer values of 6 do not yield substantially larqer abnormality reductions (Fiq. 2) but do accelerate the dearadation of the nominal -output performance (Fiej. 1). o o o
6
en
2
o
o o
(35)
For the aircraft exaMple, (26) yields a fifth order equation in G as a function of 6 , p , W 2 and T. For given values of p , 6 , w2 and TO this equation will yield five candidate values of G. The optimal G is that root that mini~izes the cost function. In the present case G < 1 is required for stability. If none of the real candidate values of G are < 1, then analysis of this problem shol'/s that a decrease in 6 will drive G toward more stable values. This means that normality and stability are diametrically opposed in this circumstance.
6
C\J N
>,
8
w
.- 6
.1
E 8 o. - - p - - .,.... t .....-. ~+' 6 =0
Increasing
p
o
o
o~
________
.0000
~
________+-_____
.1000
Fi g. 1. Assume wo2=1, F.7 (this corresponds to darlping ratio = .71). Settinq e= 0, p = 1 yields the standard optimal measurement feedback control
.2000
_ +-__ .3000
Vu /: ; lim Ellu ll ~ t-+= Output cost Vy vs. input cost Vu [1 = .7, Wo = 1.0]
0
~
C\J
( 36)
G=-0.118
<- - - - - - -
s
0
Ll
er
and settinn e = sensitive desiqn
00 ,
p«
00
G = 2.0
yields the optimally ( 37)
This choice of G in (37) forces the closedloop system matrix to be symmetric (hence, norma 1) -l /T l]=[l/T lJ A+RGM [ -w 2+w 2G O . 1 0 o
0
I-
LlD N
,.« « ,.
<-------
~
.1
(Xl
Fiq. 1 shows the tradeoff
between output
/: ; t1->
p
I
« « 0
~
«:i-
6
:::>
2
0
~
.0000
and by Theorem 1, the sensitivity is at its minirlum in this case. Note, however, that stability is lost by this minimum sensitivity desiejn, (G < 1 is required for stability). Thus, minimally sensitive desiqns mioht not be stable.
Increasing
.1000
/: ;
Fig. 2.
.2000
.3000
2
Vu = lim El lullR t-Abnormality cost VA vs. input
cost Vu [ T = .7, w0 = 1.OJ Now, if we were to assume that abnormality was a consistent metric for root sensitivity (both were convex functions), then we could use Figs. 1 and 2 as design tools to choose the feedback law that achieves the best compromise between performance and root sensitivity (represented by abnormality). Hawever, abnormality is not a consistent index of root
R. E. Skelton and D. A. Wagie
22
sensitivity, except in the local neiC]hborhood of the normal condition, as illustrated by Figs. 3 and 4. From these finures it is obvious that (for this example) abnormality and root sensitivity are only consistent indices above G = .49 ( S ~ 2) and are actually opposing indices below that point. From Fiaures 1 and 2, we stated that we would not choose 13 > 1 if desiqninC] for performance/ normality. Hence, for this problem, usinC] Figs. 1 and 2 would place us in the realm where abnormality is an inconsistent index for root sensitivity. Note also that the sensitivity l'ihen we have repeated roots (G = .49). ->CO
o
o
Legend:
Lf)
[;] 13
0
I\n "abnormality" term is therefore added to the traditional nuadratic performance metric of optimal control in an attempt to decrease root sensitivity. The necessary conditions are given for the solution of this problem and an example (jives some practical insights. The solution to this problem points out that while "normality" implies minimal root sensitivity, minimizing abnormality will not consistently decrease root sensitivity. In fact, decreasinCl abnormality may very well increase root sensitivi~y, especially when two roots are close toqether. The desinn cautions produced by the results include the fact that the necessary and sufficient condition (normality) for minimal root sensitivity is only a local result and also that it applies only for non-defective Dl ant matrices (those having 1inearly independent eiqenvectors).
Os 100
£::,.13
to the plant matrix may be readily computed. A necessary and sufficient condition for minimum root sensitivity is that the plant matrix of the state equations be normal.
I\CKNOWLEDGMENT Portions of this work were sponsored by AFOSR Grant No. 820209 and NSF Grant No. ECS-8119598.
o o
.000
1.000
2.000
3.000
G( S) Fig. 3 Root sensitivity s vs. feedback gain G(S). [.=.7, Wo lob, p=lJ o
Legend:
u
Blum, E.K. (1972). Numerical Analysis and Computation: Theory and Practice. Addison-Hesley, Reading, Massachusetts. Bryson, 1\. and V.C. Ho (1975). Applied Optimal Control. Hemisphere Publishing, Hashington, D.C.
o
Lf)
REFERENCES
EJ 13
0
8
13
1
h.
S
100
Byrne, P.C. and M. Burke (1976). Optimizabon with trajectory sensitivity considerations. IEEE Trans. Auto. Control, Vol. I\C-21, 232-283.
c-
l-
u
ClIllen, C.G. (1967). Transformations. t1assachusetts.
N_
-« -l< -«, -l<
~
0
~
~1atrices
and Linear Addison-\'/esley, Readin(1,
lJ)
Gilbert, E.G. (1983). Conditions for minimizing the norm sensitivity of characteristic roots. Submitted for publication in IEEE Trans. Auto. Control.
-«
:>
0
~
.000
1.000
2 . 000
3.000
G( S) Fig. 4 Abnormality cost VA vs. feedback gain G( S). [.=.7, wo=LO, p=lJ cmlCLUS IOrlS
Explicit expressions for a scalar metric of root sensitivity is (jiven in terms of the left and right ei(jenvectors of the system, so that sensitivity of each eiqenvalue with respect
Jacobi, C.G.J. (1846). Crelle's Journal fur die Reine und Angewandte Mathematik, Vol. 30. DeGruyter, Berlin. pp. 51-95. Kosut, R.L. (1970). Suboptimal control of linear time-invariant systems subject to control structure constraints. IEEE Trans. Auto Control, Vol. AC-15, 557-563. Levine, W.S., T.L. Johnson, and M. Athans (1971). Optimal limited state variable feedback controllers for linear systems. IEEE Trans. Auto. Control, Vol. AC-15, 557-563.
Minimal Root Sensitivity in Linear Systems
MacFarlane, A.G.J., and Y.S. Hung (1932). A quasi-classical approach to multivariable feedback systems design. Proceedings of the Second IFAC SymposiLiiilOnComputer Aided Design of Multivariable Technological Systems, West Lafayette, Indiana, USA. Se pt. , 1982, 39-48. Patel, R.V., and M. Toda (1980). Quantitative measures of robustness for multivariable systems. Proceedi ngs of the 1980 Joint Auto. Control Conf., San Francisco, California, USA. Aug., 1980. Postlethv,aite,1. (1982). Sensitivity of the characteristic gain loci. Proceedinqs of the Second IFAC Symposium on Computer Aided Desiqn of Multivariable Technological Systems, ~est Lafayette, Indiana, USA. Sept., 1982, 153-158. Van Ness, J.E., J.M. Boyle, and F.P. Imad (1965). Sensitivities of large, multipleloop control systems. IEEE Trans. Auto. Control, Vol. AC-10, 308-3J5. Yedavalli, R.K., and R.E. Skelton (1982). Controller desiun for parameter sensitivity reduction in linear regulators. Optimal Control Applications & Methods, Vol. 3, 221-240. Wilkinson, J.H. (1965). The Algebraic Eioenvalue Problem . Oxford University Press, London.
23