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Chapter 8 Some Further Recent Contributions
Chapter
8
SOME FURTHER RECENT CONTRIBUTIONS
In the present chapter we continue the treatment of absolute stability of the indirect control system of (VII, Ql), and notably still assume y > 0. Let P(a,fl,w) and V ( x , o ) be the Popov and Liapunov functions of (VII, 4.6, 4.7). By strengthening a theorem of Yacubovich [3] (our main lemma below) Kalman [2] succeeded in completing essentially Popov’s second theorem to a n.a.s.c., his “sufficiency” part however requiring a further strong restriction. Our proposed more modest task is to prove, following largely Kalman, that P > 0 plus a very simple restriction yield n.a.s.c. to have V and - 3 both positive definite and also absolute stability. While our treatment is thus relatively simple, it is only fair to say that most of the difficulties in Kalman’s treatment were caused by weakening “>” to “ 2 ” ( P 2 0, 3 5 0). However our simpler attack will suffice to give the reader an idea of that of Kalman. In the last section we discuss the effect of restricting the admissible class of characteristic functions by an inequality 0 icp(a)/o 5 K. While we shall lean almost entirely upon the work of Kalman, the notations are those of the previous chapters. For convenience the reader may use the following transfer table :
$1. CONTROLLABILITY A N D OBSERVABILITY Notations ofthetext: A
h
c
5
7
p ~
Notations oFKalman: F
g
h
-<
y
p
$1. Controllability and Observability While these concepts have appeared here and there in the literature before Kalman, he has had the great merit of giving them explicit form, content and application. See notably Kalman [2] and Kalman and associates [11. Roughly speaking a controlled system is completely controllable (c.c.) if one cannot decompose it into two systems with the control operating in only one of the systems; otherwise it is merely partially controllable or simply noncontrollable. Controllability operates through the vector b. Something similar takes place regarding observability and the vector c appearing in the expression of 0. The latter is the feedback or "visible" effect of the control, hence observable. Without going into further detail we adopt the following explicit definition (Kalman): The pair (A, b) is completely controllable (c.c.) whenever the vectors b, Ab, A'b, ..., A"- ' b are linearly independent. The pair (c', A ) is completely observable (c.o.) whenever ( A ' ;c) is C.C.As a matter of fact the important property of independence of the vectors Akb has been considered by many authors. See for example LaSalle [3, Introduction and p. 151. Our main purpose in the present chapter is the proof'of the fundamental theorem of $6. Now from a very general theory developed by Kalman [2, No. 121 one may infer: (1.1) There is no loss ofgenerality in assuming that the pair ( A , b) is completely controllable and the pair (c',A ) is completely observable.
The reduction of our system to this type and also its justification will be carried out in the next section. Recall this well-known property : if the characteristic equation of the matrix A is (14
2
+ a,l"-' + ... + u , = o
then A satisfies the matrix equation (1.3)
A"
+ a,An-' + ... + a , E = 0. 107
8.
SOME FURTHER RECENT CONTRIBUTIONS
This property is usually expressed as: the matrix satis3es its own characteristic equation. If ( A ,b) is c.c., (1.2) is the equation of lowest degree (up to a constant factor) satisfied by A : it is also its minimal equation. In fact if A satisfied an equation of degree p < n, e.g., AP + a,'AP-
' + ... + al'E = 0
then the p + 1 vectors b, Ab,..., APb, would not be independent, contradicting the complete controllability of ( A ,b). (1.4) The property of complete controllability of ( A ,b) is independent of the choice of coordinates.
For the change of coordinates x -+ Px yields A -+ P- ' A P , b -+ P - 'b. Hence the set of vectors S = {Akb}, 0 k n goes into S1 = {P-'Akb}. Since P and P - are nonsingular, linear independence of one of the sets implies that of the other, hence (1.4) follows. We shall also require this property :
-=
'
(1.5) A n.a.s.c. for complete controllability of the pair ( A ,b) is that
vector x is such that x'eA'b = 0 for all t then x
=
0.
if a
It will be convenient to utilize orthogonality. Two vectors u,u are orthogonal if u'u = u'u = 0. As is well known and readily proved, a n.a.s.c. for the existence of a vector u # 0 orthogonal to n vectors u l ,..., u, is that the uh be linearly dependent. Observe also that (1.3) yields Ak+"b+ a,Ak+"-lb
+ ..- + Akb = 0
whatever k. Hence the vectors Akb all depend linearly upon the set S defined above. We come now to the actual proof of (1.5). NECESSITY. Let ( A ,b) be C.C. and let x'eArb= 0 for all t. Upon differentiating k times and setting t = 0 there follows in particular x'Akb = 0, 0 5 k < n. Since x is orthogonal to the linearly independent vectors of S , necessarily x = 0. This proves necessity. SUFFICIENCY. Let the property of (1.5) hold. As a consequence if x'Akh = 0 whatever k then x' = 0. Since these Akb depend linearly upon
108
52.
REDUCTION OF THE SYSTEM
the vectors of S , if x is orthogonal to all the vectors of S alone then x = 0. Hence S consists of linearly independent vectors. Hence (A, b) is c.c.: sufficiency is proved and so is (13.
$2. Reduction of the System to One with a Completely Controllable Pair ( A , b) and Completely Observable Pair (c', A ) The above title states our objective. However to describe the reduction process a convenient notation is needed. Let A = diag(A,, ..., A,) and let nh be the order of Ah so that Xnh = n. Iff is any vector let f h denote the vector whose components of order n, + n2 + ... + nh-' + s, 1 5 s g nh, are the same as those off and the rest zero. Thus f = Cf h . Method of reduction. The only manner in which the pair ( A , b) will affect later arguments is through expressions of type f ' A - 'b, where under a transformation of coordinates x + Px, f behaves like c : f +,''P so that f 'Az- ' b is unchanged. Note that this implies freedom in changing coordinates. Now the following operation does not modify f 'Az- ' b : If A
then
=
diag(A,, ..., A,)
f 'Az- 'b
=
1f h'A,'bh.
Hence if bh = 0 the summand f h ' A i ' bh = 0 whatever f. Hence Ah and related coordinates will not affect any later argument and so they may be suppressed. That is, one may freely replace A, as far as the sequel goes, by diag(A, ,..., Ah- Ah+ ,..., A,). This may also be justified in the following manner. Since bh = 0 the vector xh satisfies 2 = A h X h which is a system with constant matrix and no control: control-neutral. Since A is stable so is A h . Hence the solutions xh(t)all + 0 naturally as t -+ +a and so one may as well disregard Ah and related coordinates. An analogous process may be applied in the following case. Let A h be a block matrix : Ah = c h ( n h ) , such as.occurs in the Jordan normal form (IX, $1). Let bh be such that its coordinate of order v = n , + ... + nh is zero. In view of the form of Ah the componentsf,, b, off, b enter inf 'A; ' b
,, ,
109
8.
SOME FURTHER RECENT CONTRIBUTIONS
solely through the expression
As above, then, one may suppress the coordinate x , and components without affecting anything. In particular C(A) will merely be replaced similar block but of order one unit less. This operation may of course be repeated. Our reduction consists, then, in the suppression of certain submatrices and terms and allowable coordinate transformations, i.e. in which real points are always represented by certain conjugate pairs of coordinates. These operations will not affect the nature of the Jordan normal form. A final remark referring to the submatrices Ah in the diagonal form. The statement : (A,,,bh) is C.C.merely means that the vectors A,,%', 0 =< s c nh are linearly independent. We now proceed with our process of reduction. As we may assume that A is in the Jordan normal form it is convenient to consider first a single block matrix. Matrix A = CG), 3, # 0. Since one may assume that the preceding reductions have already been applied one may assume that b # 0 and also that its component b, # 0. We prove: Property a. Under the preceding assumptions the pair (C(A),b) is completely controllable.
The proof is a consequence of the following two properties: Property fl. One may choose coordinates such that C(A) is unchanged but b' becomes the vector (0,..., 0,l). Property y. If b' = (0,..., 0, 1) the pair (C(A),b) is completely controllable.
PROOFOF /?.The transformation y,
= ClXl
y,
= ClX,
+ ... + c,x, + ... + C.-IX,
. . .
Yn = C ~ x n
is nonsingular provided that c1 # 0 and it preserves the block property 110
$2.
REDUCTION OF THE SYSTEM
of A. Choose it so that b has the prescribed form. This yields the system in the C h : C1bh -k CZbh+l -k
**'
-k Cn-h+lbn = 0,
clbn
=
h < n,
1.
The determinant is +b," # 0, hence there is a unique solution. Since c1 = l/b,,, the transformation of coordinates x + y is legitimate, and as it does not modify the form of A, property B follows. PROOFOFY. We will show that if x'eArb= 0 for all t then x Thus y will be a consequence of (1.5). Let eAr= (&). Then x'eArb=
=
0.
1
Xh/?hn.
Now an elementary calculation yields +n-h
Hence
The assumption implies that the polynomial of the sum is identically zero. Hence every xk = 0, x = 0 and y, hence also a, follows. General pair (A, b). We may take A in the Jordan normal form and without blocks whose bh = 0 or last component b, = 0. Take again a definite characteristic root A with two blocks @(A), C"(A) of orders p,q with p least for such blocks so that p S q. Let x,,', h = 1,2, ..., p and xi, k = 1, 2,..., q denote the associated coordinates. One may assume that their bJ, e.g. b' and b" are both of type (0,..., 0, 1). Apply the coordinate transformation x i - , + + xi-,+ - xl',..., xf + XI; - xp'. As a consequence the types of C'(A) and C"(A) will be unchanged but b" will be replaced by zero. Hence C"(A) may be suppressed. Upon repeating this operation as many times as necessary, A will still be in normal form but with distinct characteristic roots for distinct blocks and all the bh of type (0,...,0, 1). We must now show that when this happens (A, b) is C.C. Write simply A = diag(A,, ..., A,) where Ah = c&). Referring again to (1.5) one merely needs to show that if (2.1)
x'eArb= 0 111
8.
for all t then x
=
SOME FURTHER RECENT CONTRIBUTIONS
0. Now (2.1) is equivalent to this:
for all t implies that every X” = 0. As this property is a consequence of the complete controllability of (Ah, bh), (2.1) is proved. Hence finally ( A , b) is C.C. This completes the reduction. Complete observability of (c’, A ) . Let generally T denote the preceding operations on A which lower its order n and alone affect C.C.of ( A , b) or C.O.of (c’, A). Now starting with the initial A, to achieve C.C.of ( A , b ) may have required to apply operations T from A to A , of order n, < n and associated b’, c1 with ( A , , b’) C.C.If (c”, A , ) is not C.O.an analogous (dual) procedure will yield A , of order n, < n, and (c”, A,) c.o., etc. The process must clearly stop, e.g. with an A , = 0: final system control neutral, which is not realistic, and therefore ruled out, or with A , # 0, hence (in evident notations) with (A,, bo) C.C. and (c”, A,) C.O. Thus the reduction of (1.1) will have been achieved. We will say briefly that (A,,bo,co) is C.C.and C.O. Comparison of initial and reduced systems. Let all the designations of (VII, $1) be reserved for the reduced system. In view of the reduction process the initial system, conveniently assumed of dimension n p , has the general form
+
Here y is a p vector, the matrices A , , A , are p x p and n x p matrices and the triple (A, b, c) is C.C. and C.O. We are now faced with two distinct problems-mathematical and practical. As a mathematical problem one must deal with the system (2.2) as it stands and not suppress any coordinates : no reduction may be made. Practically however the situation is quite different. Let b, denote the vector like b corresponding to (2.2). This vector and the analog c, are design elements. If b,, c, have been chosen so that the parts b, c alone are #O, it means that one has considered the role of the vector y as immaterial as regards control. At this point one must recall that the vectors x , y merely represent deviations from certain initial system coordinate vectors 112
$3.
A SPECIAL FORM FOR SYSTEMS WITH COMPLETELY CONTROLLABLE PAIR
(see the Introduction). Thus the deviation y has been considered, by design, as immaterial. Hence it may reasonably be neglected. That is, one may replace y by zero and what is then left of the system (2.2) is really the system (VII, 1.1) but with ( A , b, c) C.C.and C.O. This assumption will be made throughout the rest of the chapter.
93. A Special Form for Systems with Completely Controllable Pair ( A , b) When the pair ( A , b) is completely controllable the following vectors en = b
en-,
+ a,E)b = ( A 2 + a,A + a,-,E)b =
(A
+ a,An-* + ... + a,E)b,
el = (An-'
where the a, are as in (1.2), are linearly independent and hence they constitute a base e. The effect of A on this base is given by Ae,
=
,
en- - anen
A e , - , = e n - 2 - a,-,e,
. . . Ae, =
-
ale,.
Hence if we adopt e as a base for coordinates, A will become
and b will be represented by (3.4)
b'
=
(0,0,..., 0,1). 113
8. SOME FURTHER
RECENT CONTRIBUTIONS
It will turn out later that this type of matrix A and vector b will alone need to concern us. Given the importance in the sequel and also, e.g., in Popov's relation of the expression c'A;'b, it is convenient to calculate it for the above pair (A, b). One must first calculate the vector A;'b = u. If A;' = (ap), the components of the vector u are aln,a2,,,...,a,,,,, that is, the last column of A2-'. This column consists of the cofactors of the last row of
divided by /A,(.The cofactors are readily found to be 1, z,...,z"-l. Hence if c,, are the components of c we have CIA; ' b =
(3.5)
C'
+ czz + -..+ c,zn-' lAzl
In particular
Since no particular properties of the vector c have been utilized in deriving (3.5) we may state: (3.7) Iff is any vector with componentsf, then (3.7a)
f ' A - ' b = f'
+ f z z + ... + f , z n lAzl
& Main I. Lemma (Yacubovich and Kalman) This lemma is at the root of the fundamental theorem to follow ($6). It comes closest to a result of Yacubovich [3, Theorem 31, and the necessity proof below differs very little from that of Yacubovich. However, the sufficiency proof, which is the more difficult part, is essentially inspired 114
§4.
MAIN LEMMA (YACUBOVICH AND KALMAN)
by the same part of the proof of Kalman's main lemma [2], perhaps the most original feature of his treatment. (4.1) Main 1emma.Given the stable matrix A, a symmetric matrix D > 0, vectors b # 0 and k, and scalars z 2 0, E > 0, then a n.a.s.c. for the existence of a solution as a matrix B (necessarily > O ) and vector q of the system
(a) A'B
(44
E
-
ED
B b - k = $9
(b)
is that
+ B A = -qq'
be small enough and that the Kalman relation z
(4.3)
+ 2Rek'AG'b
>0
be satisfied for all real o. As in (VII, $5) set m(io) = A i ' b . Thus m(io) is a complex vector function of o.With this notation one may also write (4.3) in the form z
(4.4)
+ k'rn + m*k > 0.
Notice now the identity ALB
+ BAi,
=
- (A'B
+ BA).
If one multiplies the right-hand side by AG'b and the left-hand side by b'A&-', then take account of (4.2) there follows (4.5)
rn*Bb
+ b'Bm = m*qq'rn + Ern*Dm.
This relation will be used at once. We come now to the proof of the lemma proper.
PROOFOF NECESSITY. In (4.5) replace Bb from (4.2). As a consequence (4.6)
2Re krn = Iq'rnl' - 2$
Re q'm
+ Em*Dm.
Moreover if one considers D as a hermitian matrix then D , = A~-'DA,<' is the hermitian matrix deduced from D by the change of coordinates x + A,<'x. Hence D , > 0 like D, and so since b # 0
6 = Eb'D,b
=
Em*Dm > 0.
Upon applying (4.6) there follows (4.7)
2 Re k'rn.= Iq'rnl' - 2& Re q'm
+ 6. 115
If q'm
=
8.
SOME FURTHER RECENT CONTRIBUTIONS
7
+ 2Rek'm = (1- &)' + pz + 6 > 0
1 + ip, (4.7) yields
which is (4.3). This proves necessity.
PROOFOF
SUFFICIENCY.
We first establish a preliminary result.
(4.8) If u is a real constant uector such that Re u'm (iw) = 0 whatever w then u = 0. Suppose that u # 0 and let we have I)~(Z)
=
u'm(z) =
uh
u1
be its components. Referring to (3.7)
+ u2z + ... + u,zn-l lAzl
This function has the following properties : (a) It is rational in z and not identically zero. (b) Its poles are among the characteristic roots of A and hence, since A is stable, they are all to the left of the complex axis. (c) Since the numerator of I),&) is of smaller degree than the denominator there is at least one such pole. (d) $,,(z) takes only complex values on the complex axis. It follows that +(z) = ii,b0(iz) is a rational function of z which takes only real values on the real axis and hence it is real. Moreover it has one or more poles and they are all to one side of the real axis. Now if a is such a pole so is d and the two are separated by the real axis. This contradiction shows that u = 0. Passing now to the sufficiency proof proper since both (4.9)
~ ( w=) m*k
+ km,
n(w) = m*Dm
are real rational functions of w with numerators of degree 5 n - 1 and denominator of degree n, both -,0 as w + + 00. Furthermore they are continuous for w finite. Hence they have finite upper and lower bounds. Let p be the upper bound of n(w) and v the lower bound of ~ ( w )Since . n(w) 0 for all finite w, we have p > 0. Hence
=-
+ m*k + k'm - Em*Dm 2 z + v - ep. Moreover owing to (4.3) 7 + v > 0. Hence if one chooses E < *[(z + v)/p], 7
we have (4.10)
7
116
+ m*k + k'm - Em*Dm =- 0.
$4.
MAIN LEMMA (YACUBOVICH AND KALMAN)
Let now $(z) = IAZI. Thus $(z) is a real polynomial with leading coefficient unity. Now the left hand side of (4.10)may be written z
Him) + k'm(io) + m*(io)k - Em*Dm = -
Here q(z) is a polynomial of degree 2n with leading coefficient t. Since q(io) is real and > O (4.10),q(iw) = q l ( 0 2 ) , q l a real polynomial without real roots. Hence q1(w2)= O(io)O(- io),
where O(z) is a real polynomial. Since the leading coefficient of O(z)O(-z) is t , that of O(z) is and the degree of O(z) is n. By division and since the leading coefficient of $(z) is unity
&,
where v(z) is a polynomial of degree at most n - 1. If q l , q2,..., are its (real) coefficients, define q by q' = (-q1,
..., -4").
Once q is known one obtains the matrix B from (4.2)and as we know B > 0. The above leads to the relation (4.1 1) z
+ k'm + m*k - &m*Dm=
v(io)
io)
Referring on the other hand to (2.2) and recalling the meaning of m we have
117
8.
SOME FURTHER RECENT CONTRIBUTIONS
Hence the relation (4.1 1) yields for the chosen q
+ m*k - Em*Dm = (m*q - &)(q'm
k'm
- &)
= m*qq'm - Jz(q'm = -
- T
+ m*q)
(m*Bb + b'Bm) - Em*Dm - &(q'm
+ m*q),
the last step by (4.5). Hence whatever w : m*(Bb - k - &q) =
+ (Bb - k - &q)'m
2Re(Bb - k - &q)'m
=
0.
Since the vector in parentheses is real it must vanish, showing that (4.2b) is satisfied. That is, a solution ( B , q) has been found for the system (4.2). Thus sufficiency of (4.3) is proved. This completes the proof of (4.1).
$5. Liapunov-Popov Function and Popov Inequality Their connection has already been emphasized (VII, §§4,5) and we recall that the function V ( x ,a), the related and the Popov inequality are
v
+ a(a - c ' x ) ~+ B@(o); - 3 = x'Cx + Bpcp2(a)+ 2d0'xcp(cr)+ 2ayacp(o); do = Bb - (@A'c + UYC). P(a,b,w) = f l y + Re((2ay + iwfl)c'&'b} 2 0. V ( X ,a) = X'BX
(5.1) (5.2)
(5.3) We also state the following generalization of the Lurie problem resembling one due to Kalman: Generalized Lurie problem. To3nd n.a.s.c. to assure absolute stability by means of the function V ( x ,a) of (5.1), i.e., through Vand - both positive for all x, a not both zero, and all admissible characteristic cp(a).
v
This problem is solved by the fundamental theorem ($6). Reduction of Kalman's relation (4.3) to Popov's (5.3). At first glance, although quite similar, they seem to deal with two different problems. 118
$6.
FUNDAMENTAL THEOREM
Actually by a specialization of the constant k appearing in (4.3) one obtains (5.3). Referring to the expression (5.2) of
+ UYC relation let z = /?p = /?(y + c'b). As a consequence (4.3) k
and in Kalman's yields (5.4)
let
/?p
=
Bb - do = @A'c
+ 2 Re{(&'A + uyc')A,'b}
The bracket may be written
> 0.
+ uyc'AG'b + i/?c'ioA,'b + uyc'A,'b.
@c'(ioE - Ai,)A,'b = - &'b
Since p - c'b = y, (5.4) reduces to (4.3). That is, with the substitutions indicated for the constants (5.3) reduces to (4.3).
56. Fundamental Theorem With the lemma behind us we are in position to prove : (6.1) Theorem. N.a.s.c. for both V and - V of 45 to be positive dejinite for all ( x , ~ )and choice of an admissible cp are the Popov-Kalman inequality (4.3) together with (6.2)
(a) u 2 0 , p 2 0 , u + / ? > O ; (b) z > 0 or z
=
0, do = 0, u > 0.
When these properties are satisfed the system (VII, 1.1) is absolutely stable.
(6.3) REMARK.It is quite instructive to compare the above theorem and the apparently similar theorem (11, 2.1 1). The earlier theorem refers to the Lurie-Postnikov function V and in its conditions there enter the matrix C and the control parameters b, c, p. In the present theorem the V function is the more general Popov type and in its conditions there enter merely the scalars u, /? and the control parameters. The difference is due, of course, to the appearance of the powerful Popov condition.
PROOF OF NECESSITY. The necessity of (6.2a) has already been proved. Regarding (6.2b) let 7 # 0. Then the substitution x + E X , cp -,cp, 0 + E% 119
8.
SOME FURTHER RECENT CONTRIBUTIONS
yields - V A zcp’ and so one must have z > 0. Suppose now z = 0, do # 0. Then the same substitution yields - p A 2c(p(o)d 0 ’ x : the sign of pchanges with that of E hence one must have do = 0. Then however u # 0, hence u > 0, since otherwise V = 0 for x = 0, CT 0. Thus (6.3b) must hold.
+
Consider now separately z > 0 and z
=
0.
I. z > 0. One may then write - V = X’(C - qq’)x
(6.4)
+ (&q + q’x)2 + 2uyacp,
where q is defined by (4.2b). Choose llxll large, E small, oo # 0 and fixed q’x = 0 if and o = E ’ O ~ ,q(o)= p 2 0 0 with p such that E~,uo,& q’x # 0, and any p > 0 if q’x = 0. Then - V A x ’ ( C - qq’)x, hence C - qq’ = D > 0. Thus (4.2) holds, hence by the lemma (4.3) is satisfied. Thus necessity is proved in this case. 11. z = 0, do = 0, u > 0. Taking o = 0 and any x , we have - V A x ‘ C x , hence C > 0 and so by (6.2~)B > 0. Since do = 0, (4.2) holds with q = 0. E = 1. Therefore the Popov-Kalman inequality is satisfied, and necessity is completely proved.
+
E
PROOFOF SUFFICIENCY. Since (4.3) holds given D > 0 there exists > 0 such that the system (4.2) has a solution (B, q). Note that C = ED
+ 44’ > 0,
hence also B > 0. It is convenient now to deal separately with Vand V. Take first K Ifu = 0 then B > 0 and so V > 0 for x # 0 or i f x = 0 for o # 0. Hence Vis positive definite for all X , C Tand admissible cp. On the other hand if u # 0 the sum of the first two terms in (5.1) is a positive definite quadratic form in x , CT and the conclusion is the same. Consider now V. If z # 0 (5.2) becomes
+
- V = EX’DX (&cp(a)
+ q’x)’ + 2~yocp(~).
Hence the sum of the first two terms is positive definite for all x , o and admissible cp and so - V has the same property. If z = 0 then do = 0, u > 0. Hence (5.2) reduces to - V = X’CX
+ 2uyocp(o).
Since C > 0 and u > 0 this expression is likewise positive definite for all x , o and admissible cp. 120
$7.
A RECENT RESULT OF MOROZAN
Since both V and - are, always, positive definite for all x, t~ and admissible cp, sufficiency is proved. PROOFOF ABSOLUTE STABILITY. All that is now needed is to show (Barbashin-Krassovskii complement, IX, 4.7). that V + 00 with llxll + Owing to B > 0 and property I11 of (I, $1) for cp this is true if a = 0 since then B > 0. It holds also when a > 0 since the first two terms in the expression (5.1) of Vmake up a positive definite quadratic form in x and 0.
$7. A Recent Result of Morozan It is interesting to return to the inequality (Fi)of (11, $2) for the number
p. In our present notations and since c loc. cit. is here A'c, and hence k for a = 0, /l= 1 is fA'c, we have (Fi)
p > (Bb
- k)'C-'(Bb
-
k).
At a meeting in Kiev in September 1961, the author raised the question of finding the minimum of p for all choices of the basic matrix C > 0. This question has recently been solved by Morozan [l]. We have however all that is required to obtain an answer here. Namely when a = 0, taking B = 1, (4.3)yields p
+ 2Rek'A,;'b
> 0.
Here, however, k = $4'~. Hence p
+ Re(c'A
*
Ak'b) > 0.
From Aim= i o E - A there follows p - c'b
+ Re iwc'Ai,'b
> 0.
Since this last inequality must hold for all real o we find (7.1)
p > c'b
+ sup Im oc'Ai,'b. m
Owing to the n.a.s.c. of the fundamental theorem the right-hand side represents the true least value of p. Owing no doubt to differences in notations this result does not coincide with that of Morozan. 121
8.
SOME FURTHER RECENT CONTRIBUTIONS
$8. Return to the Standard Example In the preceding chapter we have already made a comparison between the two types of function V ( x ,0 ) : (VII, 4.6), form of Lurie-Postnikov (a = 0), and Popov form (VII, 4.Q (a > 0). We return to the same question here and arrive at comparisons based upon rather simple estimates obtained from Popov's inequality. For simplicity the discussion will be restricted to the case /3 > 0. Thus Popov's inequality may be written
y
(8.1)
+ Re(6 + iw)c'A,'b
>0
where 6 = 2ay//3. In our earlier notation m(iw) = c'AL'b one may write (8.1) as
y > os,
(8.2)
-
=
Sl(o) + iS,(w)
6S1,
which is to hold for some 6 2 0 and all real o. It is evident that one may take /3 = 1. Then 6 = 0 corresponds to the Lurie-Postnikov type of function V ( x , a ) and 6 > 0 to the Popov generalization. Since (8.1) is independent of the choice of coordinates (a fact readily established) we may assume that A = diag(Al,..., An). As a consequence
and therefore
The relation (9.1) will only yield rather simple estimates when all the & are real. We confine our attention to this case. Thus = -ph < 0. Hence C'AL'b
Hence
122
=
1-ph + iw' bhch
$9.
DIRECT CONTROL
Taking into account the relation p p >
(8.3)
c
+ c‘b we obtain from (8.1):
=y
Ph(Ph Ph2
+
6)bhCh
oz
.
It is now necessary to distinguish between the signs of the products bhCh. Let bh’Ch‘ denote the positive products and Ph‘ their P h , and bit;, p i the negative products and their p,,. Suppose also that PI’
5 Pz‘
s ... 5 PP”
P;
s Pi s ... s P;.
Now it is clear that (8.3) will hold if one merely preserves the bh’Ch‘, chooses 6 = p l ’ and o = 0. Let
Similarly set
It is evident that if p p is the least of the numbers pp’, p: then p p is a suitable lower bound for the number p. Now let us see what one obtains as lower bound for p from our inequality (Fi). Referring to (II,5) its vector c, now written c,,, has for components -p,,c,,. Hence the inequality (Fi)yields here p >
1bh‘ch‘ = Pm.
It is clear that if the products bhch are not all negative p p < pm, hence the Popov type of Liapunov function, i.e. with a suitable u > 0, is then more advantageous than the Lurie-Postnikov type with u = 0 (our earlier type).
99. Direct Control The most interesting direct control of order n is the one which reduces to an indirect control of order n - 1 and is fully discussed in (IV, §§6,7). The state matrix A of the direct control has zero as simple characteristic root. All that we propose to do here is to adapt the theorem of $6 to that case.
123
8. SOME FURTHER
RECENT CONTRIBUTIONS
In the notations loc. cit. the system is Xo = A O X O
c i = g’x,
- boV(0) - pCp(0)
where A, is a stable ( n - 1) x (n - 1) matrix. One may apply directly the fundamental theorem of $6 under the following identifications : A, corresponds to A ;x, to x , bo to b, g’ = co’A, to c’A,co to c. Here also y = p - c,’A,b,. Finally p and q(a) have the same meaning as in $6.
$10. RQumC (Indirect Control: y > 0) The variety of results on the Popov expression P ( a , p , o ) and the Liapunov function V ( x , 6)of (5.1) as related to absolute stability, may be summarized as follows :
I. Popou’s first theorem. A sufficient condition for absolute stability is (10.1) P(cr,p,o) 2 0 for some a,B 2 0, a B > 0, and all real o.
+
11. Popou’s second theorem. (10.1) is a necessary condition to have absolute stability via Vand - p positive definite, with a and /?the same in P and K 111. Kalman’s theorem. (10.1) plus another (complicated) condition is a n.a.s.c. to have absolute stability through V > 0, p 4 0, with same a, fi in P and K
IV. Theorem of $6.(10.1) with P > 0 plus Bp > 0 or Bp = 0, do = 0: a > 0 are n.a.s.c. to have absolute stability secured through Vand -V both positive definite. V. However, in 111 and IV the pair (A, b) is assumed completely controllable. In both also a certain theorem of Yacubovich plays a major role. Suppose now that in the initial system (VII, 1.1) the pair ( A ,b) is not completely controllable. As we have shown in $2, one may choose coordinates, and select a reasonable vector cz such that the initial system is replaced by (2.2) together with (2.3) where now (Al, b) is completely controllable. Popov’s first theorem provides a sufficient condition for the 124
$11.
COMPLEMENT ON THE FINITENESS OF THE RATIO cp(0)/0
absolute stability of the full system (2.2, 2.3). It would evidently be most desirable to prove that P 2 0, supplemented, perhaps, by some simple inequality is also a necessary condition for absolute stability. Since (2.3) already has this property, it might suffice to obtain this result for a completely controllable pair ( A , b). Up to the present, however, this remains an open question.
$11. Complement on the Finiteness of the Ratio cp(a)/a Two recent publications led to this complement : (a) a noteworthy paper by Yacubovich [4] in which he deals not only with the restriction in the title but even with a possible isolated function cp(a); (b) an extensive monograph by Aizerman and Gantmacher [l] where the restriction in question is accepted throughout. This has induced the author to examine the possible modifications in the results of the chapter presented by the added condition (11.1)
0
# 0:
to our admissible class. As indirect and direct controls proceed along entirely distinct lines, the two cases are separated. Indirect control. Take I/=
and modify
X’BX+ U(O - c ‘ x ) ~+ /3@(~)
v by adding and subtracting A(0) = 2ay
Thus A(o)> 0 for
0
# 0, or cp(o)
- V = X’CX
A’B
(.q) -
cp(0).
-= KO and A(0)= 0 if q(a) = KO. Then + 2dO’xcp(0)+ A ( 0 )
+ z0cp2(0)
+ B A = -c,
do = Bb - ~ B A ‘ c- UYC. Replace now Popov’s initial expression by the K-Popov expression :
P(a, B, w, K)
=
P(a, p, w )
+ 2UKY ~
=
By
+ 2UY + Re((2a + iw/?)c’A,;’b}. K -
125
8.
SOME FURTHER RECENT CONTRIBUTIONS
Under the same modifications as before in $5 (expression of k ) it is identical with the K-Kalman expression
K(a, b, w,
K) =
zo
+ 2 Re k'AZ'b.
The new fundamental theorem is : (11.2) &Theorem for indirect control. N.a.s.c., in order that, with V as above, both V and - V be positive definite for all x, o and all K-admissible functions p (p restricted by 11.1) is that the K-Popov-Kalman inequality:
P(a, b, w, 4 > 0 hold for all real w together with
(1 1.3)
zo
> 0.
When these properties are satisfied the system is absolutely stable in the sense that cp is restricted by (11.1).
PROOFOF
NECESSITY. To
-V
prove (11.3) take cp(a) = K O so that
= X'CX
+ zocp2 + 2d0'px.
For x = 0 and zo # 0 then - = Z ~ K ~ hence O ~ , zo > 0. On the other hand zo = 0 is ruled out since then - V cannot be positive definite in x, O. Thus (1 1.3) holds. Write now - V = x'(C - qq')x
+ (&p + q'x)2.
Take any x # 0 and determine a by J z o ~ a = -q'x. As a consequence - V = x'(C - qq')x > 0 for all x # 0. Hence C - qq' = D > 0. Thus all the necessary conditions of the main lemma are fulfilled with z = zo. Hence the K-Popov-Kalman inequality holds and necessity is proved. PROOFOF SUFFICIENCY.It is practically the same as in (6.3) save that one need not consider zo = 0. The proof of absolute stability with the K restriction added is the same as in $6, with the modification
+
- V = EX'DX (&&a)
which does not affect the proof. 126
+ q'x)2 + 2ay
(o
- q(a)
?:p
$1 1.
COMPLEMENT O N THE FINITENESS OF THE RATIO cp(O)/U
Direct control. This time the system is (11.4) X’ = A X - bq(a), CT = C’X.
As Liapunov function take V ( X )= X’BX (1 1.5) hence
+ P@(D),
+ 2 d ’ x ~ ( o+) 7cp2(a)
- V(X)= X‘CX
(11.6)
+ B A = - C , d = Bb - &?A’c, T = fic‘b. Actually the role of v is really played by the function W ( x )= - v - s 0 ( A’B
9
- cp(0)
=
X‘CX
+ 2(d - ~SC)’X(P +
( +3 7
-
q2
where S > 0. In the presence of the restriction ( 1 1.1) the adequate theorem here is : (11.7) K-Theorem f o r a direct control. Suficient conditions for V positive definite as a function of x for all admissible functions cp satisfying (11.1) and W as a quadratic form in x and cp (unrestricted) is the K - P O ~ Oinequality. V ( 1 1.8)
S
P(S, /3, o,K) = K
+ Re{(S + io/?)c’A,’b}
>0
for some # 0, some positive 6, and all real w. When these conditions are fuljilled both V and - V are positive definite and we have absolute stability. REMARK. This theorem does not really differ in substance from a theorem of Aizerman and Gantmacher [2, p. 781. They give conditions referring to a Liapunov function V with the property that if V is positive definite under the restriction (11.1) then W is positive definite in x, cp without restriction. The proof of sufficiency can be carried out by a slight modification of the argument of the sufficiency proof of our fundamental theorem ($6). It is also obvious that when the given conditions hold
- v = w+s
(
0--
V:))
cp(0)
is positive definite in x (arbitrary) and cp (restricted by 11.1). Absolute stability is then established as in $6. 127
8
SOME FURTHER RECENT CONTRIBUTIONS
In the present chapter we continue the treatment of absolute stability of the indirect control system of (VII, Ql), and notably still assume y > 0. Let P(a,fl,w) and V ( x , o ) be the Popov and Liapunov functions of (VII, 4.6, 4.7). By strengthening a theorem of Yacubovich [3] (our main lemma below) Kalman [2] succeeded in completing essentially Popov’s second theorem to a n.a.s.c., his “sufficiency” part however requiring a further strong restriction. Our proposed more modest task is to prove, following largely Kalman, that P > 0 plus a very simple restriction yield n.a.s.c. to have V and - 3 both positive definite and also absolute stability. While our treatment is thus relatively simple, it is only fair to say that most of the difficulties in Kalman’s treatment were caused by weakening “>” to “ 2 ” ( P 2 0, 3 5 0). However our simpler attack will suffice to give the reader an idea of that of Kalman. In the last section we discuss the effect of restricting the admissible class of characteristic functions by an inequality 0 icp(a)/o 5 K. While we shall lean almost entirely upon the work of Kalman, the notations are those of the previous chapters. For convenience the reader may use the following transfer table :
$1. CONTROLLABILITY A N D OBSERVABILITY Notations ofthetext: A
h
c
5
7
p ~
Notations oFKalman: F
g
h
-<
y
p
$1. Controllability and Observability While these concepts have appeared here and there in the literature before Kalman, he has had the great merit of giving them explicit form, content and application. See notably Kalman [2] and Kalman and associates [11. Roughly speaking a controlled system is completely controllable (c.c.) if one cannot decompose it into two systems with the control operating in only one of the systems; otherwise it is merely partially controllable or simply noncontrollable. Controllability operates through the vector b. Something similar takes place regarding observability and the vector c appearing in the expression of 0. The latter is the feedback or "visible" effect of the control, hence observable. Without going into further detail we adopt the following explicit definition (Kalman): The pair (A, b) is completely controllable (c.c.) whenever the vectors b, Ab, A'b, ..., A"- ' b are linearly independent. The pair (c', A ) is completely observable (c.o.) whenever ( A ' ;c) is C.C.As a matter of fact the important property of independence of the vectors Akb has been considered by many authors. See for example LaSalle [3, Introduction and p. 151. Our main purpose in the present chapter is the proof'of the fundamental theorem of $6. Now from a very general theory developed by Kalman [2, No. 121 one may infer: (1.1) There is no loss ofgenerality in assuming that the pair ( A , b) is completely controllable and the pair (c',A ) is completely observable.
The reduction of our system to this type and also its justification will be carried out in the next section. Recall this well-known property : if the characteristic equation of the matrix A is (14
2
+ a,l"-' + ... + u , = o
then A satisfies the matrix equation (1.3)
A"
+ a,An-' + ... + a , E = 0. 107
8.
SOME FURTHER RECENT CONTRIBUTIONS
This property is usually expressed as: the matrix satis3es its own characteristic equation. If ( A ,b) is c.c., (1.2) is the equation of lowest degree (up to a constant factor) satisfied by A : it is also its minimal equation. In fact if A satisfied an equation of degree p < n, e.g., AP + a,'AP-
' + ... + al'E = 0
then the p + 1 vectors b, Ab,..., APb, would not be independent, contradicting the complete controllability of ( A ,b). (1.4) The property of complete controllability of ( A ,b) is independent of the choice of coordinates.
For the change of coordinates x -+ Px yields A -+ P- ' A P , b -+ P - 'b. Hence the set of vectors S = {Akb}, 0 k n goes into S1 = {P-'Akb}. Since P and P - are nonsingular, linear independence of one of the sets implies that of the other, hence (1.4) follows. We shall also require this property :
-=
'
(1.5) A n.a.s.c. for complete controllability of the pair ( A ,b) is that
vector x is such that x'eA'b = 0 for all t then x
=
0.
if a
It will be convenient to utilize orthogonality. Two vectors u,u are orthogonal if u'u = u'u = 0. As is well known and readily proved, a n.a.s.c. for the existence of a vector u # 0 orthogonal to n vectors u l ,..., u, is that the uh be linearly dependent. Observe also that (1.3) yields Ak+"b+ a,Ak+"-lb
+ ..- + Akb = 0
whatever k. Hence the vectors Akb all depend linearly upon the set S defined above. We come now to the actual proof of (1.5). NECESSITY. Let ( A ,b) be C.C. and let x'eArb= 0 for all t. Upon differentiating k times and setting t = 0 there follows in particular x'Akb = 0, 0 5 k < n. Since x is orthogonal to the linearly independent vectors of S , necessarily x = 0. This proves necessity. SUFFICIENCY. Let the property of (1.5) hold. As a consequence if x'Akh = 0 whatever k then x' = 0. Since these Akb depend linearly upon
108
52.
REDUCTION OF THE SYSTEM
the vectors of S , if x is orthogonal to all the vectors of S alone then x = 0. Hence S consists of linearly independent vectors. Hence (A, b) is c.c.: sufficiency is proved and so is (13.
$2. Reduction of the System to One with a Completely Controllable Pair ( A , b) and Completely Observable Pair (c', A ) The above title states our objective. However to describe the reduction process a convenient notation is needed. Let A = diag(A,, ..., A,) and let nh be the order of Ah so that Xnh = n. Iff is any vector let f h denote the vector whose components of order n, + n2 + ... + nh-' + s, 1 5 s g nh, are the same as those off and the rest zero. Thus f = Cf h . Method of reduction. The only manner in which the pair ( A , b) will affect later arguments is through expressions of type f ' A - 'b, where under a transformation of coordinates x + Px, f behaves like c : f +,''P so that f 'Az- ' b is unchanged. Note that this implies freedom in changing coordinates. Now the following operation does not modify f 'Az- ' b : If A
then
=
diag(A,, ..., A,)
f 'Az- 'b
=
1f h'A,'bh.
Hence if bh = 0 the summand f h ' A i ' bh = 0 whatever f. Hence Ah and related coordinates will not affect any later argument and so they may be suppressed. That is, one may freely replace A, as far as the sequel goes, by diag(A, ,..., Ah- Ah+ ,..., A,). This may also be justified in the following manner. Since bh = 0 the vector xh satisfies 2 = A h X h which is a system with constant matrix and no control: control-neutral. Since A is stable so is A h . Hence the solutions xh(t)all + 0 naturally as t -+ +a and so one may as well disregard Ah and related coordinates. An analogous process may be applied in the following case. Let A h be a block matrix : Ah = c h ( n h ) , such as.occurs in the Jordan normal form (IX, $1). Let bh be such that its coordinate of order v = n , + ... + nh is zero. In view of the form of Ah the componentsf,, b, off, b enter inf 'A; ' b
,, ,
109
8.
SOME FURTHER RECENT CONTRIBUTIONS
solely through the expression
As above, then, one may suppress the coordinate x , and components without affecting anything. In particular C(A) will merely be replaced similar block but of order one unit less. This operation may of course be repeated. Our reduction consists, then, in the suppression of certain submatrices and terms and allowable coordinate transformations, i.e. in which real points are always represented by certain conjugate pairs of coordinates. These operations will not affect the nature of the Jordan normal form. A final remark referring to the submatrices Ah in the diagonal form. The statement : (A,,,bh) is C.C.merely means that the vectors A,,%', 0 =< s c nh are linearly independent. We now proceed with our process of reduction. As we may assume that A is in the Jordan normal form it is convenient to consider first a single block matrix. Matrix A = CG), 3, # 0. Since one may assume that the preceding reductions have already been applied one may assume that b # 0 and also that its component b, # 0. We prove: Property a. Under the preceding assumptions the pair (C(A),b) is completely controllable.
The proof is a consequence of the following two properties: Property fl. One may choose coordinates such that C(A) is unchanged but b' becomes the vector (0,..., 0,l). Property y. If b' = (0,..., 0, 1) the pair (C(A),b) is completely controllable.
PROOFOF /?.The transformation y,
= ClXl
y,
= ClX,
+ ... + c,x, + ... + C.-IX,
. . .
Yn = C ~ x n
is nonsingular provided that c1 # 0 and it preserves the block property 110
$2.
REDUCTION OF THE SYSTEM
of A. Choose it so that b has the prescribed form. This yields the system in the C h : C1bh -k CZbh+l -k
**'
-k Cn-h+lbn = 0,
clbn
=
h < n,
1.
The determinant is +b," # 0, hence there is a unique solution. Since c1 = l/b,,, the transformation of coordinates x + y is legitimate, and as it does not modify the form of A, property B follows. PROOFOFY. We will show that if x'eArb= 0 for all t then x Thus y will be a consequence of (1.5). Let eAr= (&). Then x'eArb=
=
0.
1
Xh/?hn.
Now an elementary calculation yields +n-h
Hence
The assumption implies that the polynomial of the sum is identically zero. Hence every xk = 0, x = 0 and y, hence also a, follows. General pair (A, b). We may take A in the Jordan normal form and without blocks whose bh = 0 or last component b, = 0. Take again a definite characteristic root A with two blocks @(A), C"(A) of orders p,q with p least for such blocks so that p S q. Let x,,', h = 1,2, ..., p and xi, k = 1, 2,..., q denote the associated coordinates. One may assume that their bJ, e.g. b' and b" are both of type (0,..., 0, 1). Apply the coordinate transformation x i - , + + xi-,+ - xl',..., xf + XI; - xp'. As a consequence the types of C'(A) and C"(A) will be unchanged but b" will be replaced by zero. Hence C"(A) may be suppressed. Upon repeating this operation as many times as necessary, A will still be in normal form but with distinct characteristic roots for distinct blocks and all the bh of type (0,...,0, 1). We must now show that when this happens (A, b) is C.C. Write simply A = diag(A,, ..., A,) where Ah = c&). Referring again to (1.5) one merely needs to show that if (2.1)
x'eArb= 0 111
8.
for all t then x
=
SOME FURTHER RECENT CONTRIBUTIONS
0. Now (2.1) is equivalent to this:
for all t implies that every X” = 0. As this property is a consequence of the complete controllability of (Ah, bh), (2.1) is proved. Hence finally ( A , b) is C.C. This completes the reduction. Complete observability of (c’, A ) . Let generally T denote the preceding operations on A which lower its order n and alone affect C.C.of ( A , b) or C.O.of (c’, A). Now starting with the initial A, to achieve C.C.of ( A , b ) may have required to apply operations T from A to A , of order n, < n and associated b’, c1 with ( A , , b’) C.C.If (c”, A , ) is not C.O.an analogous (dual) procedure will yield A , of order n, < n, and (c”, A,) c.o., etc. The process must clearly stop, e.g. with an A , = 0: final system control neutral, which is not realistic, and therefore ruled out, or with A , # 0, hence (in evident notations) with (A,, bo) C.C. and (c”, A,) C.O. Thus the reduction of (1.1) will have been achieved. We will say briefly that (A,,bo,co) is C.C.and C.O. Comparison of initial and reduced systems. Let all the designations of (VII, $1) be reserved for the reduced system. In view of the reduction process the initial system, conveniently assumed of dimension n p , has the general form
+
Here y is a p vector, the matrices A , , A , are p x p and n x p matrices and the triple (A, b, c) is C.C. and C.O. We are now faced with two distinct problems-mathematical and practical. As a mathematical problem one must deal with the system (2.2) as it stands and not suppress any coordinates : no reduction may be made. Practically however the situation is quite different. Let b, denote the vector like b corresponding to (2.2). This vector and the analog c, are design elements. If b,, c, have been chosen so that the parts b, c alone are #O, it means that one has considered the role of the vector y as immaterial as regards control. At this point one must recall that the vectors x , y merely represent deviations from certain initial system coordinate vectors 112
$3.
A SPECIAL FORM FOR SYSTEMS WITH COMPLETELY CONTROLLABLE PAIR
(see the Introduction). Thus the deviation y has been considered, by design, as immaterial. Hence it may reasonably be neglected. That is, one may replace y by zero and what is then left of the system (2.2) is really the system (VII, 1.1) but with ( A , b, c) C.C.and C.O. This assumption will be made throughout the rest of the chapter.
93. A Special Form for Systems with Completely Controllable Pair ( A , b) When the pair ( A , b) is completely controllable the following vectors en = b
en-,
+ a,E)b = ( A 2 + a,A + a,-,E)b =
(A
+ a,An-* + ... + a,E)b,
el = (An-'
where the a, are as in (1.2), are linearly independent and hence they constitute a base e. The effect of A on this base is given by Ae,
=
,
en- - anen
A e , - , = e n - 2 - a,-,e,
. . . Ae, =
-
ale,.
Hence if we adopt e as a base for coordinates, A will become
and b will be represented by (3.4)
b'
=
(0,0,..., 0,1). 113
8. SOME FURTHER
RECENT CONTRIBUTIONS
It will turn out later that this type of matrix A and vector b will alone need to concern us. Given the importance in the sequel and also, e.g., in Popov's relation of the expression c'A;'b, it is convenient to calculate it for the above pair (A, b). One must first calculate the vector A;'b = u. If A;' = (ap), the components of the vector u are aln,a2,,,...,a,,,,, that is, the last column of A2-'. This column consists of the cofactors of the last row of
divided by /A,(.The cofactors are readily found to be 1, z,...,z"-l. Hence if c,, are the components of c we have CIA; ' b =
(3.5)
C'
+ czz + -..+ c,zn-' lAzl
In particular
Since no particular properties of the vector c have been utilized in deriving (3.5) we may state: (3.7) Iff is any vector with componentsf, then (3.7a)
f ' A - ' b = f'
+ f z z + ... + f , z n lAzl
& Main I. Lemma (Yacubovich and Kalman) This lemma is at the root of the fundamental theorem to follow ($6). It comes closest to a result of Yacubovich [3, Theorem 31, and the necessity proof below differs very little from that of Yacubovich. However, the sufficiency proof, which is the more difficult part, is essentially inspired 114
§4.
MAIN LEMMA (YACUBOVICH AND KALMAN)
by the same part of the proof of Kalman's main lemma [2], perhaps the most original feature of his treatment. (4.1) Main 1emma.Given the stable matrix A, a symmetric matrix D > 0, vectors b # 0 and k, and scalars z 2 0, E > 0, then a n.a.s.c. for the existence of a solution as a matrix B (necessarily > O ) and vector q of the system
(a) A'B
(44
E
-
ED
B b - k = $9
(b)
is that
+ B A = -qq'
be small enough and that the Kalman relation z
(4.3)
+ 2Rek'AG'b
>0
be satisfied for all real o. As in (VII, $5) set m(io) = A i ' b . Thus m(io) is a complex vector function of o.With this notation one may also write (4.3) in the form z
(4.4)
+ k'rn + m*k > 0.
Notice now the identity ALB
+ BAi,
=
- (A'B
+ BA).
If one multiplies the right-hand side by AG'b and the left-hand side by b'A&-', then take account of (4.2) there follows (4.5)
rn*Bb
+ b'Bm = m*qq'rn + Ern*Dm.
This relation will be used at once. We come now to the proof of the lemma proper.
PROOFOF NECESSITY. In (4.5) replace Bb from (4.2). As a consequence (4.6)
2Re krn = Iq'rnl' - 2$
Re q'm
+ Em*Dm.
Moreover if one considers D as a hermitian matrix then D , = A~-'DA,<' is the hermitian matrix deduced from D by the change of coordinates x + A,<'x. Hence D , > 0 like D, and so since b # 0
6 = Eb'D,b
=
Em*Dm > 0.
Upon applying (4.6) there follows (4.7)
2 Re k'rn.= Iq'rnl' - 2& Re q'm
+ 6. 115
If q'm
=
8.
SOME FURTHER RECENT CONTRIBUTIONS
7
+ 2Rek'm = (1- &)' + pz + 6 > 0
1 + ip, (4.7) yields
which is (4.3). This proves necessity.
PROOFOF
SUFFICIENCY.
We first establish a preliminary result.
(4.8) If u is a real constant uector such that Re u'm (iw) = 0 whatever w then u = 0. Suppose that u # 0 and let we have I)~(Z)
=
u'm(z) =
uh
u1
be its components. Referring to (3.7)
+ u2z + ... + u,zn-l lAzl
This function has the following properties : (a) It is rational in z and not identically zero. (b) Its poles are among the characteristic roots of A and hence, since A is stable, they are all to the left of the complex axis. (c) Since the numerator of I),&) is of smaller degree than the denominator there is at least one such pole. (d) $,,(z) takes only complex values on the complex axis. It follows that +(z) = ii,b0(iz) is a rational function of z which takes only real values on the real axis and hence it is real. Moreover it has one or more poles and they are all to one side of the real axis. Now if a is such a pole so is d and the two are separated by the real axis. This contradiction shows that u = 0. Passing now to the sufficiency proof proper since both (4.9)
~ ( w=) m*k
+ km,
n(w) = m*Dm
are real rational functions of w with numerators of degree 5 n - 1 and denominator of degree n, both -,0 as w + + 00. Furthermore they are continuous for w finite. Hence they have finite upper and lower bounds. Let p be the upper bound of n(w) and v the lower bound of ~ ( w )Since . n(w) 0 for all finite w, we have p > 0. Hence
=-
+ m*k + k'm - Em*Dm 2 z + v - ep. Moreover owing to (4.3) 7 + v > 0. Hence if one chooses E < *[(z + v)/p], 7
we have (4.10)
7
116
+ m*k + k'm - Em*Dm =- 0.
$4.
MAIN LEMMA (YACUBOVICH AND KALMAN)
Let now $(z) = IAZI. Thus $(z) is a real polynomial with leading coefficient unity. Now the left hand side of (4.10)may be written z
Him) + k'm(io) + m*(io)k - Em*Dm = -
Here q(z) is a polynomial of degree 2n with leading coefficient t. Since q(io) is real and > O (4.10),q(iw) = q l ( 0 2 ) , q l a real polynomial without real roots. Hence q1(w2)= O(io)O(- io),
where O(z) is a real polynomial. Since the leading coefficient of O(z)O(-z) is t , that of O(z) is and the degree of O(z) is n. By division and since the leading coefficient of $(z) is unity
&,
where v(z) is a polynomial of degree at most n - 1. If q l , q2,..., are its (real) coefficients, define q by q' = (-q1,
..., -4").
Once q is known one obtains the matrix B from (4.2)and as we know B > 0. The above leads to the relation (4.1 1) z
+ k'm + m*k - &m*Dm=
v(io)
io)
Referring on the other hand to (2.2) and recalling the meaning of m we have
117
8.
SOME FURTHER RECENT CONTRIBUTIONS
Hence the relation (4.1 1) yields for the chosen q
+ m*k - Em*Dm = (m*q - &)(q'm
k'm
- &)
= m*qq'm - Jz(q'm = -
- T
+ m*q)
(m*Bb + b'Bm) - Em*Dm - &(q'm
+ m*q),
the last step by (4.5). Hence whatever w : m*(Bb - k - &q) =
+ (Bb - k - &q)'m
2Re(Bb - k - &q)'m
=
0.
Since the vector in parentheses is real it must vanish, showing that (4.2b) is satisfied. That is, a solution ( B , q) has been found for the system (4.2). Thus sufficiency of (4.3) is proved. This completes the proof of (4.1).
$5. Liapunov-Popov Function and Popov Inequality Their connection has already been emphasized (VII, §§4,5) and we recall that the function V ( x ,a), the related and the Popov inequality are
v
+ a(a - c ' x ) ~+ B@(o); - 3 = x'Cx + Bpcp2(a)+ 2d0'xcp(cr)+ 2ayacp(o); do = Bb - (@A'c + UYC). P(a,b,w) = f l y + Re((2ay + iwfl)c'&'b} 2 0. V ( X ,a) = X'BX
(5.1) (5.2)
(5.3) We also state the following generalization of the Lurie problem resembling one due to Kalman: Generalized Lurie problem. To3nd n.a.s.c. to assure absolute stability by means of the function V ( x ,a) of (5.1), i.e., through Vand - both positive for all x, a not both zero, and all admissible characteristic cp(a).
v
This problem is solved by the fundamental theorem ($6). Reduction of Kalman's relation (4.3) to Popov's (5.3). At first glance, although quite similar, they seem to deal with two different problems. 118
$6.
FUNDAMENTAL THEOREM
Actually by a specialization of the constant k appearing in (4.3) one obtains (5.3). Referring to the expression (5.2) of
+ UYC relation let z = /?p = /?(y + c'b). As a consequence (4.3) k
and in Kalman's yields (5.4)
let
/?p
=
Bb - do = @A'c
+ 2 Re{(&'A + uyc')A,'b}
The bracket may be written
> 0.
+ uyc'AG'b + i/?c'ioA,'b + uyc'A,'b.
@c'(ioE - Ai,)A,'b = - &'b
Since p - c'b = y, (5.4) reduces to (4.3). That is, with the substitutions indicated for the constants (5.3) reduces to (4.3).
56. Fundamental Theorem With the lemma behind us we are in position to prove : (6.1) Theorem. N.a.s.c. for both V and - V of 45 to be positive dejinite for all ( x , ~ )and choice of an admissible cp are the Popov-Kalman inequality (4.3) together with (6.2)
(a) u 2 0 , p 2 0 , u + / ? > O ; (b) z > 0 or z
=
0, do = 0, u > 0.
When these properties are satisfed the system (VII, 1.1) is absolutely stable.
(6.3) REMARK.It is quite instructive to compare the above theorem and the apparently similar theorem (11, 2.1 1). The earlier theorem refers to the Lurie-Postnikov function V and in its conditions there enter the matrix C and the control parameters b, c, p. In the present theorem the V function is the more general Popov type and in its conditions there enter merely the scalars u, /? and the control parameters. The difference is due, of course, to the appearance of the powerful Popov condition.
PROOF OF NECESSITY. The necessity of (6.2a) has already been proved. Regarding (6.2b) let 7 # 0. Then the substitution x + E X , cp -,cp, 0 + E% 119
8.
SOME FURTHER RECENT CONTRIBUTIONS
yields - V A zcp’ and so one must have z > 0. Suppose now z = 0, do # 0. Then the same substitution yields - p A 2c(p(o)d 0 ’ x : the sign of pchanges with that of E hence one must have do = 0. Then however u # 0, hence u > 0, since otherwise V = 0 for x = 0, CT 0. Thus (6.3b) must hold.
+
Consider now separately z > 0 and z
=
0.
I. z > 0. One may then write - V = X’(C - qq’)x
(6.4)
+ (&q + q’x)2 + 2uyacp,
where q is defined by (4.2b). Choose llxll large, E small, oo # 0 and fixed q’x = 0 if and o = E ’ O ~ ,q(o)= p 2 0 0 with p such that E~,uo,& q’x # 0, and any p > 0 if q’x = 0. Then - V A x ’ ( C - qq’)x, hence C - qq’ = D > 0. Thus (4.2) holds, hence by the lemma (4.3) is satisfied. Thus necessity is proved in this case. 11. z = 0, do = 0, u > 0. Taking o = 0 and any x , we have - V A x ‘ C x , hence C > 0 and so by (6.2~)B > 0. Since do = 0, (4.2) holds with q = 0. E = 1. Therefore the Popov-Kalman inequality is satisfied, and necessity is completely proved.
+
E
PROOFOF SUFFICIENCY. Since (4.3) holds given D > 0 there exists > 0 such that the system (4.2) has a solution (B, q). Note that C = ED
+ 44’ > 0,
hence also B > 0. It is convenient now to deal separately with Vand V. Take first K Ifu = 0 then B > 0 and so V > 0 for x # 0 or i f x = 0 for o # 0. Hence Vis positive definite for all X , C Tand admissible cp. On the other hand if u # 0 the sum of the first two terms in (5.1) is a positive definite quadratic form in x , CT and the conclusion is the same. Consider now V. If z # 0 (5.2) becomes
+
- V = EX’DX (&cp(a)
+ q’x)’ + 2~yocp(~).
Hence the sum of the first two terms is positive definite for all x , o and admissible cp and so - V has the same property. If z = 0 then do = 0, u > 0. Hence (5.2) reduces to - V = X’CX
+ 2uyocp(o).
Since C > 0 and u > 0 this expression is likewise positive definite for all x , o and admissible cp. 120
$7.
A RECENT RESULT OF MOROZAN
Since both V and - are, always, positive definite for all x, t~ and admissible cp, sufficiency is proved. PROOFOF ABSOLUTE STABILITY. All that is now needed is to show (Barbashin-Krassovskii complement, IX, 4.7). that V + 00 with llxll + Owing to B > 0 and property I11 of (I, $1) for cp this is true if a = 0 since then B > 0. It holds also when a > 0 since the first two terms in the expression (5.1) of Vmake up a positive definite quadratic form in x and 0.
$7. A Recent Result of Morozan It is interesting to return to the inequality (Fi)of (11, $2) for the number
p. In our present notations and since c loc. cit. is here A'c, and hence k for a = 0, /l= 1 is fA'c, we have (Fi)
p > (Bb
- k)'C-'(Bb
-
k).
At a meeting in Kiev in September 1961, the author raised the question of finding the minimum of p for all choices of the basic matrix C > 0. This question has recently been solved by Morozan [l]. We have however all that is required to obtain an answer here. Namely when a = 0, taking B = 1, (4.3)yields p
+ 2Rek'A,;'b
> 0.
Here, however, k = $4'~. Hence p
+ Re(c'A
*
Ak'b) > 0.
From Aim= i o E - A there follows p - c'b
+ Re iwc'Ai,'b
> 0.
Since this last inequality must hold for all real o we find (7.1)
p > c'b
+ sup Im oc'Ai,'b. m
Owing to the n.a.s.c. of the fundamental theorem the right-hand side represents the true least value of p. Owing no doubt to differences in notations this result does not coincide with that of Morozan. 121
8.
SOME FURTHER RECENT CONTRIBUTIONS
$8. Return to the Standard Example In the preceding chapter we have already made a comparison between the two types of function V ( x ,0 ) : (VII, 4.6), form of Lurie-Postnikov (a = 0), and Popov form (VII, 4.Q (a > 0). We return to the same question here and arrive at comparisons based upon rather simple estimates obtained from Popov's inequality. For simplicity the discussion will be restricted to the case /3 > 0. Thus Popov's inequality may be written
y
(8.1)
+ Re(6 + iw)c'A,'b
>0
where 6 = 2ay//3. In our earlier notation m(iw) = c'AL'b one may write (8.1) as
y > os,
(8.2)
-
=
Sl(o) + iS,(w)
6S1,
which is to hold for some 6 2 0 and all real o. It is evident that one may take /3 = 1. Then 6 = 0 corresponds to the Lurie-Postnikov type of function V ( x , a ) and 6 > 0 to the Popov generalization. Since (8.1) is independent of the choice of coordinates (a fact readily established) we may assume that A = diag(Al,..., An). As a consequence
and therefore
The relation (9.1) will only yield rather simple estimates when all the & are real. We confine our attention to this case. Thus = -ph < 0. Hence C'AL'b
Hence
122
=
1-ph + iw' bhch
$9.
DIRECT CONTROL
Taking into account the relation p p >
(8.3)
c
+ c‘b we obtain from (8.1):
=y
Ph(Ph Ph2
+
6)bhCh
oz
.
It is now necessary to distinguish between the signs of the products bhCh. Let bh’Ch‘ denote the positive products and Ph‘ their P h , and bit;, p i the negative products and their p,,. Suppose also that PI’
5 Pz‘
s ... 5 PP”
P;
s Pi s ... s P;.
Now it is clear that (8.3) will hold if one merely preserves the bh’Ch‘, chooses 6 = p l ’ and o = 0. Let
Similarly set
It is evident that if p p is the least of the numbers pp’, p: then p p is a suitable lower bound for the number p. Now let us see what one obtains as lower bound for p from our inequality (Fi). Referring to (II,5) its vector c, now written c,,, has for components -p,,c,,. Hence the inequality (Fi)yields here p >
1bh‘ch‘ = Pm.
It is clear that if the products bhch are not all negative p p < pm, hence the Popov type of Liapunov function, i.e. with a suitable u > 0, is then more advantageous than the Lurie-Postnikov type with u = 0 (our earlier type).
99. Direct Control The most interesting direct control of order n is the one which reduces to an indirect control of order n - 1 and is fully discussed in (IV, §§6,7). The state matrix A of the direct control has zero as simple characteristic root. All that we propose to do here is to adapt the theorem of $6 to that case.
123
8. SOME FURTHER
RECENT CONTRIBUTIONS
In the notations loc. cit. the system is Xo = A O X O
c i = g’x,
- boV(0) - pCp(0)
where A, is a stable ( n - 1) x (n - 1) matrix. One may apply directly the fundamental theorem of $6 under the following identifications : A, corresponds to A ;x, to x , bo to b, g’ = co’A, to c’A,co to c. Here also y = p - c,’A,b,. Finally p and q(a) have the same meaning as in $6.
$10. RQumC (Indirect Control: y > 0) The variety of results on the Popov expression P ( a , p , o ) and the Liapunov function V ( x , 6)of (5.1) as related to absolute stability, may be summarized as follows :
I. Popou’s first theorem. A sufficient condition for absolute stability is (10.1) P(cr,p,o) 2 0 for some a,B 2 0, a B > 0, and all real o.
+
11. Popou’s second theorem. (10.1) is a necessary condition to have absolute stability via Vand - p positive definite, with a and /?the same in P and K 111. Kalman’s theorem. (10.1) plus another (complicated) condition is a n.a.s.c. to have absolute stability through V > 0, p 4 0, with same a, fi in P and K
IV. Theorem of $6.(10.1) with P > 0 plus Bp > 0 or Bp = 0, do = 0: a > 0 are n.a.s.c. to have absolute stability secured through Vand -V both positive definite. V. However, in 111 and IV the pair (A, b) is assumed completely controllable. In both also a certain theorem of Yacubovich plays a major role. Suppose now that in the initial system (VII, 1.1) the pair ( A ,b) is not completely controllable. As we have shown in $2, one may choose coordinates, and select a reasonable vector cz such that the initial system is replaced by (2.2) together with (2.3) where now (Al, b) is completely controllable. Popov’s first theorem provides a sufficient condition for the 124
$11.
COMPLEMENT ON THE FINITENESS OF THE RATIO cp(0)/0
absolute stability of the full system (2.2, 2.3). It would evidently be most desirable to prove that P 2 0, supplemented, perhaps, by some simple inequality is also a necessary condition for absolute stability. Since (2.3) already has this property, it might suffice to obtain this result for a completely controllable pair ( A , b). Up to the present, however, this remains an open question.
$11. Complement on the Finiteness of the Ratio cp(a)/a Two recent publications led to this complement : (a) a noteworthy paper by Yacubovich [4] in which he deals not only with the restriction in the title but even with a possible isolated function cp(a); (b) an extensive monograph by Aizerman and Gantmacher [l] where the restriction in question is accepted throughout. This has induced the author to examine the possible modifications in the results of the chapter presented by the added condition (11.1)
0
# 0:
to our admissible class. As indirect and direct controls proceed along entirely distinct lines, the two cases are separated. Indirect control. Take I/=
and modify
X’BX+ U(O - c ‘ x ) ~+ /3@(~)
v by adding and subtracting A(0) = 2ay
Thus A(o)> 0 for
0
# 0, or cp(o)
- V = X’CX
A’B
(.q) -
cp(0).
-= KO and A(0)= 0 if q(a) = KO. Then + 2dO’xcp(0)+ A ( 0 )
+ z0cp2(0)
+ B A = -c,
do = Bb - ~ B A ‘ c- UYC. Replace now Popov’s initial expression by the K-Popov expression :
P(a, B, w, K)
=
P(a, p, w )
+ 2UKY ~
=
By
+ 2UY + Re((2a + iw/?)c’A,;’b}. K -
125
8.
SOME FURTHER RECENT CONTRIBUTIONS
Under the same modifications as before in $5 (expression of k ) it is identical with the K-Kalman expression
K(a, b, w,
K) =
zo
+ 2 Re k'AZ'b.
The new fundamental theorem is : (11.2) &Theorem for indirect control. N.a.s.c., in order that, with V as above, both V and - V be positive definite for all x, o and all K-admissible functions p (p restricted by 11.1) is that the K-Popov-Kalman inequality:
P(a, b, w, 4 > 0 hold for all real w together with
(1 1.3)
zo
> 0.
When these properties are satisfied the system is absolutely stable in the sense that cp is restricted by (11.1).
PROOFOF
NECESSITY. To
-V
prove (11.3) take cp(a) = K O so that
= X'CX
+ zocp2 + 2d0'px.
For x = 0 and zo # 0 then - = Z ~ K ~ hence O ~ , zo > 0. On the other hand zo = 0 is ruled out since then - V cannot be positive definite in x, O. Thus (1 1.3) holds. Write now - V = x'(C - qq')x
+ (&p + q'x)2.
Take any x # 0 and determine a by J z o ~ a = -q'x. As a consequence - V = x'(C - qq')x > 0 for all x # 0. Hence C - qq' = D > 0. Thus all the necessary conditions of the main lemma are fulfilled with z = zo. Hence the K-Popov-Kalman inequality holds and necessity is proved. PROOFOF SUFFICIENCY.It is practically the same as in (6.3) save that one need not consider zo = 0. The proof of absolute stability with the K restriction added is the same as in $6, with the modification
+
- V = EX'DX (&&a)
which does not affect the proof. 126
+ q'x)2 + 2ay
(o
- q(a)
?:p
$1 1.
COMPLEMENT O N THE FINITENESS OF THE RATIO cp(O)/U
Direct control. This time the system is (11.4) X’ = A X - bq(a), CT = C’X.
As Liapunov function take V ( X )= X’BX (1 1.5) hence
+ P@(D),
+ 2 d ’ x ~ ( o+) 7cp2(a)
- V(X)= X‘CX
(11.6)
+ B A = - C , d = Bb - &?A’c, T = fic‘b. Actually the role of v is really played by the function W ( x )= - v - s 0 ( A’B
9
- cp(0)
=
X‘CX
+ 2(d - ~SC)’X(P +
( +3 7
-
q2
where S > 0. In the presence of the restriction ( 1 1.1) the adequate theorem here is : (11.7) K-Theorem f o r a direct control. Suficient conditions for V positive definite as a function of x for all admissible functions cp satisfying (11.1) and W as a quadratic form in x and cp (unrestricted) is the K - P O ~ Oinequality. V ( 1 1.8)
S
P(S, /3, o,K) = K
+ Re{(S + io/?)c’A,’b}
>0
for some # 0, some positive 6, and all real w. When these conditions are fuljilled both V and - V are positive definite and we have absolute stability. REMARK. This theorem does not really differ in substance from a theorem of Aizerman and Gantmacher [2, p. 781. They give conditions referring to a Liapunov function V with the property that if V is positive definite under the restriction (11.1) then W is positive definite in x, cp without restriction. The proof of sufficiency can be carried out by a slight modification of the argument of the sufficiency proof of our fundamental theorem ($6). It is also obvious that when the given conditions hold
- v = w+s
(
0--
V:))
cp(0)
is positive definite in x (arbitrary) and cp (restricted by 11.1). Absolute stability is then established as in $6. 127