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Robust absolute stability of Lur'e control systems in parameter space
Automatica, Vol. 27, No. 1, pp. 147-151. 1991 Printed in Great Britain.
0005-1098/91 $3.00 + 0.00 Pergamon Press pie © 1990 International Federation of Automatic Control
Brief Paper
Robust Absolute Stability of Lur'e Control Systems in Parameter Space* A. TESIt
and A. VICINOt:~
Key Words--Absolute stability; Lur'e control systems; Popov criterion; robust stability; parameter variations; uncertainty sets; positive realness.
Abstract--This paper deals with the problem of robust absolute stability analysis for nonlinear Lur'e control systems in the presence of system parameter variations. The well known Popov criterion for absolute stability is used in order to characterize the boundary of the region of absolute stability in the parameter plane when the coefficients of the transfer function of the linear plant are polynomial functions of the uncertain parameters. For a scalar parameter, a method is given to determine the maximal interval of variation around a fixed nominal value preserving absolute stability. This result is also used to derive a technique for checking absolute stability of Lur'e systems with parameters in given planar uncertainty sets. Numerical examples showing the application of the method are reported.
stated by the celebrated Popov criterion (Popov, 1962). This condition reduces the absolute stability problem to positive realness of a suitable function of frequency. Few contributions to the study of positive realness in the presence of parametric uncertainty can be found in the recent literature. Bose and Delansky (1989) give easy conditions to check positive realness of a rational functon whose numerator and denominator polynomials are interval polynomials. In Siljak (1989b) it is shown how convexity of the domain of positivity in polynomial coefficient space allows one to check positivity of a polytope of polynomials by considering only its vertices. In particular, in this work implications of this result on the study of robust absolute stability of certain classes of perturbed Lur'e systems are discussed. In this paper we consider Lur'e type control systems in which either the linear part is affected by parametric uncertainties or the feedback nonlinearity sector is unknown. For these cases, we give an analytical method to study the robustness of absolute stability in the face of perturbations, when one or two physical parameters causes uncertainty in the control system. More precisely, we solve the following problems:
1. Introduction THE RENEWED interest in the problem of linear systems stability analysis in parameter space in last years has brought into fashion much of the work done since several decades ago in linear and nonlinear systems robust stability analysis [see e.g. Neimark (1949) and in particular Siljak (1969, 1989a) for extensive lists of references on t:_, subject]. While a great number of contributions h a - . appeared recently on robust stability analysis of linear systems against parametric perturbations, there has not been a similar explosion in the area of nonlinear systems analysis and control. One of the main reasons is certainly the fact that while for linear systems necessary and sufficient conditions for asymptotic stability are well known and relatively simple, on the whole, in the nonlinear case usually only sufficient conditions for asymptotic stability are known. A practical implication of this fact is that in general it is not possible to characterize the exact region of absolute stability in parameter space. However, relatively less complicated is the description of subsets of the domain of absolute stability. In this case it is recognized as a widely open problem to ascertain to which extent these subsets approximate the true absolute stability domain, i.e. how far the available absolute stability sufficient conditions are from being also necessary. An important class of nonlinear control systems is that of Lur'e-Postnikov systems [see e.g. Siljak (1969)]. For these systems, several sufficient conditions for absolute stability have been given since the beginning of the sixties [see e.g. Narendra and Taylor (1973); also very recent contributions can be found in the literature, e.g. Voronov (1989)]. The most widely used sufficient condition is undoubtedly that
• Consider a Lur'e system with a feedback nonlinearity in a prescribed sector and with the linear part transfer function coefficients (or state equation matrix entries) depending on a scalar physical uncertain parameter according to polynomial functions. Given a nominal value of the parameter for which the Popov criterion is satisfied (and hence the system is absolutely stable), compute the maximum interval around this value for which the Popov criterion is satisfied. The solution of this problem provides a region of absolute stability which in general is not the true one, due to the inherent conservativeness of the Popov criterion. A case of special interest arises when the nonlinearity sector represents the uncertain parameter. For this problem, which can be given a graphical solution in terms of the well known Popov locus of the linear part, it is shown how the proposed general analytical procedure simplifies and how it can be interpreted in terms of the Popov locus in the complex plane. • Given a Lur'e system as in the preceding point, with the only difference being that two physical uncertain parameters and a planar uncertainty set are involved, check if the Popov criterion is satisfied for all parameter values in the uncertain set. The paper is organized as follows. Section 2 presents problem formulation and basic results necessary for the successive development. Sections 3 and 4 solve the problems addressed in the two points above respectively, while Section 5 reports two numerical examples showing the application of the proposed method.
* Received 31 July 1989; revised 7 February 1990; received in final form 9 April 1990. The preliminary version of this paper was presented at the l l t h IFAC World Congress, Tallinn, Estonia in 1990. This paper was recommended for publication in revised form by Associate Editor P. Dorato under the direction of Editor H. Kwakernaak. t Dipartimento di Sistemi e Informatica, Universith di Firenze, Via di Santa Marta, 3-50139 Firenze, Italy. :[:Author to whom all correspondence should be addressed: Prof. A. Vicino, Dipartimento di Automatica e lnformatica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy.
2. Problem formulation and basic results We consider Lur'e type control systems given by
i = Ax + bu =cTx = -f(y) 147
(1)
148
Brief Paper
where x E R”, A E R”,“, b, continuous function belonging follows
CER”, to the
f(.):R-*R is a class %k defined as
.Y~={~(~):f(0)=O,O~yf(y)~kyZ,O
the transfer
function
(2)
G(s)
which represents the input-output realization of the linear plant in system (1). We make the assumption that the linear plant (A, b, c) is controllable, observable and asymptotically stable. The Lur’e control system (1) is said to be Absolutely Stable (AS) if the equilibrium state x = 0 is asymptotically globally stable, for each f(.) E &. It is well known that the fundamental Popov’s theorem (Popov, 1962) provides one of the most widely used su#icient conditions for (1) to be AS in terms of the sector bound k and the transfer function G(s). The following theorem states the Popov criterion in terms of a polynomial function. Theorem. inequality P(o’)
If there exists a real 0 E R such that the following holds
= ]D(jw)]’
{km’ + Re[(l+
joO)G(jo)]}
>O,
Vo ~0 (4)
then system
(1) is absolutely
stable.
Unfortunately, it is rather difficult to transform the above definition in an analytical description easily usable for testing positivity of a given uncertainty set U,. However, if coefficients a,(p) are linear in p, then D, is convex. In fact, it follows from (8) that if p‘ and P”E D, then (1 - n)p’ + kp” E D,, V E [0, 11. This consideration allows to conclude that positivity, and hence AS, of polytopes in parameter space is implied by positivity of the polytope vertices. This important property for a polynomial can be verified by a Routh-like test (Siljak, 1971). Unfortunately, the linearity hypothesis restricts the applicability of the above result only to very special classes of problems, where parameters p, enter linearly the numerator coefficients of G&P). To investigate the polynomial dependence case, we use an analytical description of sets including dD+. Consider the envelope E(p) of the family of surfaces in parameter space generated by the equation P(d; p) = 0 for w 2 0. Such an envelope, if it exists, is defined by the ore parameter family of real solutions of the equation system (Stljak, 1969) P(d;p)
dP(d;p)/&J
F,(Q;p)
0
VX E R+.
It is clear that the study of robust absolute stability against possible variations of parameters in control systems (1) is equivalent to studying robust positivity of families of polynomials with perturbed coefficients. We denote by p E R4 a parameter vector and by P(w*;p) the corresponding Popov polynomial with coefficients depending on p. Several variables can be interpreted as parameters in P(w’;p). Typical components of vector p may be physical uncertain parameters entering the transfer function coefficients (an uncertain transfer function will be denoted by G(s;p)) or the linear plant state matrices, the sector bound k or the Popov parameter 0. Of course, for a fixed sector bound k, the Popov condition (4) can be easily extended to include parameter dependence as follows. For a given uncertainty set U, in parameter space and a fixed sector bound k, the Lur’e system is AS if the following conditions are verified Vp E Up G(s; p) is asymptotically 1 30~R:P(w’;p)>O,
stable
(5)
VwzO.
(6)
We consider the case in which the coefficients of G(s;p) depend polynomially on the parameters p. Under this assumption, the Popov polynomial to be tested for positivity can be expressed as P(w’;p)
= i ai(p)02’ i=”
i
where coefficients a,(p), i = 0, 1, , n are polynomial functions in p. We will assume that condition (5) is satisfied. In fact, this can be checked for several classes of assumption perturbations (see e.g. Bhattacharyya 1987; Tesi and Vicino, 1988; Vicino, 1989) for the case in which parameters enter linearly the coefficients of G(s;p) and Sideris and Petia, 1988; Genesio and Tesi, 1988; Vicino er al., 1988 for the more complicated case where these coefficients are polynomial or rational functions of p). The domain of positiuify D,, whose boundary is denoted by dD+, plays a key role (8)
= i a,(p)U = 0 ,=” n-1 F,(sZ;p) = C (i + l)ai+,(p)Q’ i=o
. Q>O.
(10)
= 0
The first step we take now is to give an implicit representation of the envelope E(p). This can be done by using a well known theorem on the resultant of two polynomials in an independent variable with indeterminate coefficients (see e.g. Jacobson, 1964). By considering the two polynomials in (lo), we define the resultant of F, and Fz with respect to the indeterminate 52 as follows R,(P) where
= det [H(P)]
H(p) E Rr’-‘.*“~’
the matrix
(II)
is given by
H(P)= %(P)
a,-,@)
0
O,(P)
0
i
‘.
a,(P)
%(P)
‘.
%(P)
O,(P)
0
..-l(P)
no,(p) (n-l)L,(P)
‘.
0
no,(p)
0
0
The following
‘.
%(P)
O,-,(P)
.(
0
”
0
”
0
”
0
O,(P)
0
”
0
0
”
(n-lb,(p)
allows one to solve system
I
q,(P)
0
q(P)
no,(p)
theorem
a,-z(P)
0
4P)
O,(P) (12)
(10).
Theorem [see (Jacobson, 1964) for a general statement]. R,(p) = 0 if and only if one of the conditions occurs: l
(7)
020
=O’
where w 2 0 is the parameter. Excluding the special case o = 0 (which can be accounted for separately) and setting 8 = cc’, (9) can be written as
The above theorem relates AS to posifiuity of the polynomial P(o’), which will be called Popou polynomial. We recall that a polynomial f(x) is said to be positive if f(x) > 0,
= 0
1
l
a,(p) = 0 The two polynomials extension field.
As a consequence described as
have
a common
of this theorem,
root
the envelope
E(P) = J%(P) u E”(P) u L(P)
in a suitable 0 E(p)
can be (13)
where
and3Q>O:F,(Q;p)=OandF,(Q;p)=O}
(14)
.5(p)
= (P E RY:aO(p) = 0)
(15)
E,(p)
= {p E RY:a,(p)
(16)
Observation 1. Observe
that p E E,(p)
= 0).
if and only if there
149
Brief Paper exists Q > 0 which solves simultaneously both equations in (IO). Since the second equation in (10) is the derivative of the first one with respect to Q, it follows that corresponding to p E E,(p), the polynomial F,(S&p) must necessarily have a zero of multiplicity at least 2. Moreover, if F,(Q;p) has, for a fixed p, r positive real zeros (Q,, . , $2,) of multiplicity (p,, . . . , n,) (~~22) respectively, then the resultant R,(p)
has at p a zero of multiplicity i$, pi - r.
3. Robust stability against scalar perturbations In this section we give a solution to the following problem. Let k be a fixed sector bound and p = [p,, pJT a vector with p, representing the Popov parameter 6 and pz a “physical” uncertain parameter entering polynomially in system matrix A and/or vectors b, c. Let a nominal value of the parameter p2 =pi be given (corresponding to the nominal linear plant) such that the Popov inequality (4) is satisfied. With reference to the perturbed Popov polynomial P(w’;p), we want to evaluate the maximal connected domain of positivity of the variable p2, containing the nominal parameter value p$ To do this, we consider equations (13)-(16) defining the envelope containing dD+. First of all, observe that the Popov function is linear in p1 and polynomial in pz, so that P(o*;p) can be written as follows P(w? P) =h(w’;
PJP,
+fo(d
Pz)
= 0
W(oZ;p)/ap,
= 0
i aP(wZ;p)/do
= 0.
R&P,,
(18)
fi(w’; Pz) = 0
S(P2) = UP2)
R,z(P~) = det
W(P~)I= 0
(20)
and choose only those which solve simultaneously both equations (19) for some positive value w2. Notice that H(.) is defined as in (12). Of course, the two special cases w = 0 and w = m must be considered separately, since in both cases degeneracies of the equation system (19) occur. The second critical set S2(p2) is obtained by computing critical points of R&p,, p2) = 0, i.e. self intersection points the curve of R&p,, p2) = 0. These points are obtained by imposing that the gradient of R&p,, p2) is null P,)/~P,
= 0
( ~RQ(P,, P,)IJP,
~R,(P,,
= 0.
(21)
Again, the resultant theorem allows one to solve the above system. The critical set &(p2) includes all solutions pi of (21), which solve also (10) for positive Q. Obseruation
2. The
solutions in &(pJ
above conditions for the existence of become of easy interpretation if the order
u UP*)
u UP*).
(24).
Let us now introduce the definition of an “extremal” point of the parameter plane. We say that a point p = (p,, p2) of the plane such that pz E S(pJ is “extremal” if it satisfies the two properties II, and II, II,:
P~JD+
i.e. P(w*;p)
is nonnegative.
II,: Vc > 0 arbitrarily small p- = (p, - E, p2) $ D,
i.e. P(w*;p-)
and
P+ = (P, + E, p2) $ D,
and
P(w’;p+)
are not positive.
Observe that these properties can be easily checked means of algebraic nonnegativity tests (Siljak, 1971). Let us define the “extremal” set S*(pJ as follows UP,)
by
= {PI E S(p2): (P,, p2) is extremal)
q(P2)
and S;(pJ
as
= {Pz E S*(P2) :Pz>P:)
ST(Pz) = {Pz E S*(P,) :Pz < P3
(25) (26)
and the following quantities py=
Observe that, once the above system has been solved for p2 and o, the last equation in (18) provides, if desired, the corresponding values of p,. System (19) can be solved by using the resultant theorem reported in Section 2. In particular, in this case the resultant of the system with respect to the indeterminate o* depends only on pz, so that we need to find real solutions of the equation
(23)
and selecting as elements of &(pJ only those solutions which solve both equations (10) simultaneously for positive values of Q. Define now the following sets
and the two subsets S:(pz)
(19)
(22)
R&P,, ~2) = 0 1 a,(p) = 0
By substituting (17) in (18) the elements pi belonging to the first critical set &(p2) can be obtained by solving the following polynomial equation system
i h(02; P2) = 0.
~2) = 0
1 a&p) = 0
(17)
where fa(.) and f,(.) are suitable polynomial functions. We select the extremal values of p2 considering points in the plane belonging to certain sets, called “critical” sets. For simplicity of notation, only the second components of the corresponding points will be included in these sets, denoted by Si(p2), i = 1, 2, 3. The first critical set S,(p2) is obtained by collecting points in the plane (pl, p2) belonging to E(p) and satisfying necessary conditions for the existence of a horizontal line (parallel to the axis pl) tangent to E(p), i.e. such that P(l.J;p)
of the linear plant is n = 2 or n = 3. In fact, from Observation 1 of Section 2 it follows that for n = 2, S,(p,) must necessarily be empty; for n = 3, S2(p2) is nonempty if and only if there exists a p such that the polynomial equation F,(Q;p) = 0 in the indeterminate 8 has one only positive root of multiplicity 3. The last critical set S3(p2) is obtained by looking for real solutions of the following two equation systems (compare (13, 15, 16))
PY=
.max pi +S;oJZ) &~~~*)p~.
(27) (28)
In the above equations pT(pp)
is set to -m(+m) if the set is empty. Assuming that G(s;p) is asymptottcally stable for pz E (p?, pr), the following theorem follows readily from the previous considerations.
UP*W,3P,))
Theorem. The parameter uncertainty domain (~7, p,“) is the maximal connected domain of absolute stability, according to the Popov criterion, containing the nominal 0 parameter for a given Lur’e control system. Notice that maximality of the domain of absolute stability computed above does not necessarily mean that it is not possible to extend the interval (~7, py) still preserving AS, because the Popov condition is sufficient but not necessary for AS. 3.1. Maximal sector of absolute stability. In this subsection, we briefly show how the procedure given before simplifies when we consider the special case in which the parameter p2 represents the reciprocal of the sector bound k. In this case, we assume for coherence with (2) that the allowed minimum k is 0 and accordingly p: = m. We want to estimate the maximal k for which AS is preserved according to the Popov criterion. The Popov polynomial turns out to be linear in both parameters p, = 0 and pz = k-l. The envelope equations
150
Brief Paper
(10) become f,(J)Pi +f2(J)p2 +fo(w*) = 0 I (llo)[f;(W2)p, +f;(o*)p* +&(o2)]
=0’
liJ‘O
(29)
where f,!(.), i = 0, 1, 2 denote first derivatives with respect to O. Polynomial functions fa, f,, f2 have an easy interpretation in terms of the frequency response G(jo). In particular, setting G(jw) = K(jw) + jN;(jo) (30) 4(jw) + jQ(jw) it is easy to show that fo(w’) = X(jo)Q(jw) fi(w’) = +$(jw)Q(jo)
+ N,(jw)Q(jw) -
sides of the rectangle, i.e. setting alternatively pi =p: (or i = 1, 2, possible intersections can be computed by applying the procedure given in Section 3 for each side. Step 2. To check for absence of envelope points inside U,, we have to verify that the polynomial equation systems (22) and (23) have no solutions belonging to U,,. Moreover, it must be checked that the following polynomial equation system
p, =py),
NWPAw)l
(31)
R,(P,,P,)=O %(~,,~z)la~i =0 i (or SR~P,, P,)/~P, = 0)
has no real solution in U,. Notice that if we denote by S,(p) the set of solutions of (35). only the following subset must be considered
f2(0*) = lO(jw
C(P)
The above equations allow one to give a graphical interpretation of the criticalgoints in terms of the Popov locus in the complex plane (Siljak, 1969). In the parameter plane each curve of the envelope is a straight line. As a consequence, the set S,(p,) may be readily obtained by computing the real positive solutions of of the polynomial equation f,(wZ) = 0 (32) and then computing the corresponding solutions for pr P; = -f;,(w~)/f2(w2).
(33)
Notice that from the second equation in (31) solution frequencies wi correspond to points where the Popov locus crosses the real axis. Computation of the solutions belonging to the critical set S2(p2) does not simplify significantly in the special case p2 = k-‘. These solutions correspond to intersections of the real axis with straight lines admitting possible multiple tangency points with the Popov plot. In fact, these solutions represent critical points of the curve R&p,, pz) = 0, i.e. points such that there exist at least two different values of frequency w satisfying the envelope equations (29). The set S,(p,) can be readily computed because the last equations of (22, 23) are linear in p,, p2, so that the solution of each of the two systems requires to solve one polynomial equation. From a graphical point of view, these points correspond to lines passing through one of the end points of the Popov plot and tangent to it at that point and/or some other point(s). Observation 3. The feedback gain interval [0, kH) for which the linear system G(s) is closed loop asymptotically stable is usually called Hurwitz secfor for system (1). It can be easily checked that if pyeS,(p2), then py= kjj’, i.e. the Popov sector coincides with the Hurwitz sector. Hence, the Popov criterion allows one to conclude that the well known Aizerman conjecture is true. An example of computation of the maximum k for which the Lur’e system (1) is AS according to the Popov criterion for a classical example taken from the literature is given in Section 5.
4. Robust absolute stability for planar uncertainty sets In this section we assume that the sector bound k in (2) is given. We consider the case in which the components of vector p E R2 are uncertain physical parameters affecting polynomially the coefficients of G(s;p) (or the entries of (A, b, c)). Let the uncertainty set be defined as a rectangle in the parameter plane Up = {p E R*:p:sp,
‘p:,
i = 1, 2).
(34)
Assuming that Vp E Up the linear part of (1) is asymptotically stable and that Cl’ contains at least one point p =p” for which G(s;p) satisfies the Popov criterion, absolute stability can be studied in two successive steps. In a first step we ascertain if the boundary au, intersects the envelope E(p). In the second step it is checked if there exist points of the envelope in the interior of VP. Step 1. We look for possible intersections of the rectangle sides with the envelope E(p) given in (13-16). Considering
(35)
= {K(P)
fl E(P)).
(36)
From a practical point of view, system (35) can be solved, as shown for other cases in Section 3, by applying the resultant theorem. The set S:(p) can be computed immediately selecting solutions p ES,(~) for which there exists some positive real value w solving simultaneously both equations (9). As a last observation on Step 2, it is worth noting that if for a fixed value of the Popov parameter the test fails, i.e. the set S:(p) is not empty, we have to perform the test for different values of 0. Hence, in general it may happen that Step 2 must be repeated for all real values 0, meaning that the solution of a family of problems like that solved above may be needed. As a final comment, we notice that since the Popov criterion is only sufficient for AS, the fact that the test proposed fails in assessing AS of an uncertainty set Cl, in general does not allow to conclude that system (1) is not AS for p E Cl,. A negative answer of the test would mean only that the Popov criterion does not allow one to assess robust absolute stability, so that other alternative criteria should be employed. 5. Numerical examples
In this section we present two examples showing applications of the results presented in previous sections. In the first example, a well known system is considered and the maximum Popov sector is computed. In the second example, we determine the maximum allowable uncertainty domain for a scalar parameter affecting the linear part of a Lur’e system, for a prescribed class of nonlinear functions gc*. Example 1 (Narendra and Taylor, 1973, p. 173; Safonov and
Weytzner, 1987). Consider a Lur’e system where the linear part is described after a pole-shifting by G(s) =
3(s + 1) s4+s3+25s2+3s+3’
From standard arguments it follows that the Munoitz sector is [0,7). Thus, by defining (p,, p2) t (0, k-l), we observe that we need only consider values p2 in the interval (0.14285,ml. Moreover, it can be easily checked from the structure of P(o*;p) that the point (p,, p2) = (0,~) satisfies the Popov condition (4), so that we can assume as nominal value pt = 00. The functions f;(.), i = 0, 1, 2 in (31) are given by f0(uJ2) = -6602 + 9 f,(w*) = -3w6 + 72w4 (38) [ f2(02) = o8 - 490~ + 625~~ - 1410~ + 9. From equations (32) and (33) we obtain the set S,(pJ
= I-1,
0.14285).
(39)
The envelope equations are F,(Q;p)
=p@
- (49~~ + 3p,)Q3 + (625~~ + 72p,)Q*
- (141p1 + 66)Q + 9p, + 9 = 0 F,(P;p)
= 4p2Q3 - 3(49p2 + 3p,)Q* + 2(625p, + 72pJQ +(141p,+66)=0. (40)
Brief Paper
Also the set $3(P2 ) turns out to be empty because the coefficients ao(p) and a3(p) never vanish. Consequently, the set S(P2) is equal to $1(P2) and S . ( p 2 ) = {1}. Hence, we obtain, p~' = 1 and p2M = m, obtaining that the considered Lur'e system is AS for P2 • (1, o0). The corresponding values of 0 and to for p2 = 1 are 0 = 0 . 2 and to =3.162.
wlmIG(Jw)l
-1.5 -1.5
-1
-0.5
0
0.5
l
1.5
2
2,5
FiG. 1. Popov line for maximal absolute stability sector of Example 1. Since the degree of polynomial F~(2;p) in 2 is 4, from Observation 2 of Section 3, we can expect that self intersection points of the envelope may exist. In fact, computing the resultant Ru(pl,P2 ) and solving (21), we obtain the set $2(P2 ) = {-1.443, 0.37}. (41) The set $3(p2 ) obtained by solving equations (22) and (23) is given by $3(P2) :- { - 1 , 0}. (42) From (39), (41) and (42) we obtain S(p2) = { - 1.443, - 1, 0, 0.14285, 0.37}.
(43)
The extremal set S.(p2) is obtained by checking properties H1 and H 2 S.(pz ) = {0.37}. Hence, we select p~' = 0.37 and p2M = ~ obtaining that the considered system is AS for k • [0, 2.7). The corresponding values of 0 and to for k = 2.7 are found to be 0 = 0.781 and to1 = 1.54, to2 = 5.24. These two values of to correspond to the tangency points of the straight line defined by k = 2.7 and 0 = 0.781 with the Popov locus (see Fig. 1).
Example 2. Consider the system described by the transfer function 11 G(s;p2) = (s + 1)(s 2 + p2 s + 9) (44) where P2 is an uncertain parameter with nominal value pO= 10 and the nonlinearity f ( - ) • ~k with k = 1. From standard arguments, we obtain that the closed loop system with a linear constant feedback with gain k = l is asymptotically stable for P2 • (1, oo). As in Section 3, we set Pl = 0. The functions fi(-), i = 0 , 1 in (17) are given by fo( to2, P 2 ) = to6 + (p2 _ 17)o) 4 + (52 - llp2 + p 2 ) t o 2 + 1 8 0 fl(to2,'p2) = -llto
4 + 11(9 + p 2 ) t o 2.
From equations (19) we obtain the set $1(P2) = {0, 1}.
(45) (46)
The envelope equations are
;
F~(fl;p) = 2 3 + ( p 2 _ 11pl - 17)2 z F2(2;p)
151
+ (p2 + 11p~p2 _ 11p2 + 99pl + 52)2 + 180 = 0 3 2 2 + 2 ( p 2 - l l p ~ - 17)2 + (p2 + llp~p2 _ llp2 + 99p~ + 52) = 0. (47)
From Observation 2 in Section 3, we notice that since the degree of the polynomial F~(2;p) in 2 is 3, the only possible points belonging to the set S2(p2 ) must necessarily yield a triple positive solution for 2 in F~(2;p). This cannot happen because ao(p) = 180 > 0. Therefore, the set S2(P2 ) is empty. This can be also verified by computing the resultant RQ(Pl, P2) and solving (21).
6. Conclusions In this paper an analytical method is proposed for robust absolute stability analysis of Lur'e control systems subject to parameter variations. The case where system perturbations are due to one or two uncertain parameters has been considered and a method is given which allows one to estimate maximal domains of absolute stability in parameter space, based on the Popov criterion. Further research is needed in the direction of studying robust absolute stability by means of different criteria, which may take into account more information about the feedback nonlinear function than the sector condition. Moreover, effective methods of analysis for problems involving several physical parameters appear to be of primary interest for future work. Acknowledgements--This work was partially supported by funds of Ministero della Universit~ e della Ricerca Scientifica e Tecnologia. References Bhattacharyya, S. P. (1987), Robust Stabilization Against Structured Perturbations. Lecture Notes in Control and Information Sciences 99, Springer, Berlin. Bose, N. K. and J. F. Delansky (1989). Boundary implications for interval positive rational functions. IEEE Trans. Circ. Syst. CAS-36, 454-458. Genesio, R. and A. Tesi (1988). Results on the stability robustness of systems with state space perturbations. Syst. Control Lett. 11, 39-47. Jacobson, N. (1964). Lectures in Abstract Algebra, Vol. III. Von Nostrand, Princeton, NJ. Narendra, K. S. and J. H. Taylor (1973). Frequency Domain Criteria for Absolute Stability. Academic Press, New York. Neimark, Yu. I. (1949). Stability of Linearized Systems (in Russian). LKVVIA, Leningrad. Popov, V. M. (1962). Absolute stability of nonlinear systems of automatic control. Aut. Remote Control, 22, 857-875. (Russian original published in 1961). Safonov, M. G. and G. Weytzner (1987). Computer-aided stability analysis renders Popov criterion obsolete. IEEE Trans. Aut. Control, AC-32, 1128-1131. Sideris, A. and R. S. S. Pefia (1988). Fast computation of the multivariable stability margin for real interrelated uncertain parameters. Proc. ACC. Atlanta, CA. Siljak, D. D. (1969). Nonlinear Systems: The Parameter Analysis and Design. Wiley, New York. Siljak, D. D. (1971). New algebraic criteria for positive realness. J. Franklin Inst., 291, 109-120. Siljak, D. D. (1989a). Parameter space methods for robust control design: A guided tour. IEEE Trans. Aut. Control, AC-34, 674-688. Siljak, D. D. (1989b). Polytopes of nonnegative polynomials. Proc. ACC, Pittsburgh, PA. Tesi, A. and A. Vicino (1988). Robustness analysis for uncertain dynamical systems with structured perturbations. Proc. 27th Conf. on Decision and Control. Austin, TX, 519-525. Also IEEE Trans. Aut. Control 1990, AC-35, 191-195. Vicino, A., A. Tesi and M. Milanese (1988). An algorithm for nonconservative stability bounds computation for systems with nonlinearly correlated parametric uncertainties. Proc. 27th IEEE Conf. on Decision and Control, Austin, TX, 1761-1766. Also IEEE Trans. Aut. Control 1990, AC-35, 835-841. Vicino, A. (1989). Maximal polytopic stability domains in parameter space for uncertain systems. Int. J. Control, 49, 351-361. Voronov, A. A. (1989). On improving absolute stability criteria for systems with monotonic nonlinearities and the method of absolute stability regions construction. Preprints IFAC Syrup. Nonlinear Control Systems Design, Capri, Italy, pp. 231-235.
0005-1098/91 $3.00 + 0.00 Pergamon Press pie © 1990 International Federation of Automatic Control
Brief Paper
Robust Absolute Stability of Lur'e Control Systems in Parameter Space* A. TESIt
and A. VICINOt:~
Key Words--Absolute stability; Lur'e control systems; Popov criterion; robust stability; parameter variations; uncertainty sets; positive realness.
Abstract--This paper deals with the problem of robust absolute stability analysis for nonlinear Lur'e control systems in the presence of system parameter variations. The well known Popov criterion for absolute stability is used in order to characterize the boundary of the region of absolute stability in the parameter plane when the coefficients of the transfer function of the linear plant are polynomial functions of the uncertain parameters. For a scalar parameter, a method is given to determine the maximal interval of variation around a fixed nominal value preserving absolute stability. This result is also used to derive a technique for checking absolute stability of Lur'e systems with parameters in given planar uncertainty sets. Numerical examples showing the application of the method are reported.
stated by the celebrated Popov criterion (Popov, 1962). This condition reduces the absolute stability problem to positive realness of a suitable function of frequency. Few contributions to the study of positive realness in the presence of parametric uncertainty can be found in the recent literature. Bose and Delansky (1989) give easy conditions to check positive realness of a rational functon whose numerator and denominator polynomials are interval polynomials. In Siljak (1989b) it is shown how convexity of the domain of positivity in polynomial coefficient space allows one to check positivity of a polytope of polynomials by considering only its vertices. In particular, in this work implications of this result on the study of robust absolute stability of certain classes of perturbed Lur'e systems are discussed. In this paper we consider Lur'e type control systems in which either the linear part is affected by parametric uncertainties or the feedback nonlinearity sector is unknown. For these cases, we give an analytical method to study the robustness of absolute stability in the face of perturbations, when one or two physical parameters causes uncertainty in the control system. More precisely, we solve the following problems:
1. Introduction THE RENEWED interest in the problem of linear systems stability analysis in parameter space in last years has brought into fashion much of the work done since several decades ago in linear and nonlinear systems robust stability analysis [see e.g. Neimark (1949) and in particular Siljak (1969, 1989a) for extensive lists of references on t:_, subject]. While a great number of contributions h a - . appeared recently on robust stability analysis of linear systems against parametric perturbations, there has not been a similar explosion in the area of nonlinear systems analysis and control. One of the main reasons is certainly the fact that while for linear systems necessary and sufficient conditions for asymptotic stability are well known and relatively simple, on the whole, in the nonlinear case usually only sufficient conditions for asymptotic stability are known. A practical implication of this fact is that in general it is not possible to characterize the exact region of absolute stability in parameter space. However, relatively less complicated is the description of subsets of the domain of absolute stability. In this case it is recognized as a widely open problem to ascertain to which extent these subsets approximate the true absolute stability domain, i.e. how far the available absolute stability sufficient conditions are from being also necessary. An important class of nonlinear control systems is that of Lur'e-Postnikov systems [see e.g. Siljak (1969)]. For these systems, several sufficient conditions for absolute stability have been given since the beginning of the sixties [see e.g. Narendra and Taylor (1973); also very recent contributions can be found in the literature, e.g. Voronov (1989)]. The most widely used sufficient condition is undoubtedly that
• Consider a Lur'e system with a feedback nonlinearity in a prescribed sector and with the linear part transfer function coefficients (or state equation matrix entries) depending on a scalar physical uncertain parameter according to polynomial functions. Given a nominal value of the parameter for which the Popov criterion is satisfied (and hence the system is absolutely stable), compute the maximum interval around this value for which the Popov criterion is satisfied. The solution of this problem provides a region of absolute stability which in general is not the true one, due to the inherent conservativeness of the Popov criterion. A case of special interest arises when the nonlinearity sector represents the uncertain parameter. For this problem, which can be given a graphical solution in terms of the well known Popov locus of the linear part, it is shown how the proposed general analytical procedure simplifies and how it can be interpreted in terms of the Popov locus in the complex plane. • Given a Lur'e system as in the preceding point, with the only difference being that two physical uncertain parameters and a planar uncertainty set are involved, check if the Popov criterion is satisfied for all parameter values in the uncertain set. The paper is organized as follows. Section 2 presents problem formulation and basic results necessary for the successive development. Sections 3 and 4 solve the problems addressed in the two points above respectively, while Section 5 reports two numerical examples showing the application of the proposed method.
* Received 31 July 1989; revised 7 February 1990; received in final form 9 April 1990. The preliminary version of this paper was presented at the l l t h IFAC World Congress, Tallinn, Estonia in 1990. This paper was recommended for publication in revised form by Associate Editor P. Dorato under the direction of Editor H. Kwakernaak. t Dipartimento di Sistemi e Informatica, Universith di Firenze, Via di Santa Marta, 3-50139 Firenze, Italy. :[:Author to whom all correspondence should be addressed: Prof. A. Vicino, Dipartimento di Automatica e lnformatica, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy.
2. Problem formulation and basic results We consider Lur'e type control systems given by
i = Ax + bu =cTx = -f(y) 147
(1)
148
Brief Paper
where x E R”, A E R”,“, b, continuous function belonging follows
CER”, to the
f(.):R-*R is a class %k defined as
.Y~={~(~):f(0)=O,O~yf(y)~kyZ,O
the transfer
function
(2)
G(s)
which represents the input-output realization of the linear plant in system (1). We make the assumption that the linear plant (A, b, c) is controllable, observable and asymptotically stable. The Lur’e control system (1) is said to be Absolutely Stable (AS) if the equilibrium state x = 0 is asymptotically globally stable, for each f(.) E &. It is well known that the fundamental Popov’s theorem (Popov, 1962) provides one of the most widely used su#icient conditions for (1) to be AS in terms of the sector bound k and the transfer function G(s). The following theorem states the Popov criterion in terms of a polynomial function. Theorem. inequality P(o’)
If there exists a real 0 E R such that the following holds
= ]D(jw)]’
{km’ + Re[(l+
joO)G(jo)]}
>O,
Vo ~0 (4)
then system
(1) is absolutely
stable.
Unfortunately, it is rather difficult to transform the above definition in an analytical description easily usable for testing positivity of a given uncertainty set U,. However, if coefficients a,(p) are linear in p, then D, is convex. In fact, it follows from (8) that if p‘ and P”E D, then (1 - n)p’ + kp” E D,, V E [0, 11. This consideration allows to conclude that positivity, and hence AS, of polytopes in parameter space is implied by positivity of the polytope vertices. This important property for a polynomial can be verified by a Routh-like test (Siljak, 1971). Unfortunately, the linearity hypothesis restricts the applicability of the above result only to very special classes of problems, where parameters p, enter linearly the numerator coefficients of G&P). To investigate the polynomial dependence case, we use an analytical description of sets including dD+. Consider the envelope E(p) of the family of surfaces in parameter space generated by the equation P(d; p) = 0 for w 2 0. Such an envelope, if it exists, is defined by the ore parameter family of real solutions of the equation system (Stljak, 1969) P(d;p)
dP(d;p)/&J
F,(Q;p)
0
VX E R+.
It is clear that the study of robust absolute stability against possible variations of parameters in control systems (1) is equivalent to studying robust positivity of families of polynomials with perturbed coefficients. We denote by p E R4 a parameter vector and by P(w*;p) the corresponding Popov polynomial with coefficients depending on p. Several variables can be interpreted as parameters in P(w’;p). Typical components of vector p may be physical uncertain parameters entering the transfer function coefficients (an uncertain transfer function will be denoted by G(s;p)) or the linear plant state matrices, the sector bound k or the Popov parameter 0. Of course, for a fixed sector bound k, the Popov condition (4) can be easily extended to include parameter dependence as follows. For a given uncertainty set U, in parameter space and a fixed sector bound k, the Lur’e system is AS if the following conditions are verified Vp E Up G(s; p) is asymptotically 1 30~R:P(w’;p)>O,
stable
(5)
VwzO.
(6)
We consider the case in which the coefficients of G(s;p) depend polynomially on the parameters p. Under this assumption, the Popov polynomial to be tested for positivity can be expressed as P(w’;p)
= i ai(p)02’ i=”
i
where coefficients a,(p), i = 0, 1, , n are polynomial functions in p. We will assume that condition (5) is satisfied. In fact, this can be checked for several classes of assumption perturbations (see e.g. Bhattacharyya 1987; Tesi and Vicino, 1988; Vicino, 1989) for the case in which parameters enter linearly the coefficients of G(s;p) and Sideris and Petia, 1988; Genesio and Tesi, 1988; Vicino er al., 1988 for the more complicated case where these coefficients are polynomial or rational functions of p). The domain of positiuify D,, whose boundary is denoted by dD+, plays a key role (8)
= i a,(p)U = 0 ,=” n-1 F,(sZ;p) = C (i + l)ai+,(p)Q’ i=o
. Q>O.
(10)
= 0
The first step we take now is to give an implicit representation of the envelope E(p). This can be done by using a well known theorem on the resultant of two polynomials in an independent variable with indeterminate coefficients (see e.g. Jacobson, 1964). By considering the two polynomials in (lo), we define the resultant of F, and Fz with respect to the indeterminate 52 as follows R,(P) where
= det [H(P)]
H(p) E Rr’-‘.*“~’
the matrix
(II)
is given by
H(P)= %(P)
a,-,@)
0
O,(P)
0
i
‘.
a,(P)
%(P)
‘.
%(P)
O,(P)
0
..-l(P)
no,(p) (n-l)L,(P)
‘.
0
no,(p)
0
0
The following
‘.
%(P)
O,-,(P)
.(
0
”
0
”
0
”
0
O,(P)
0
”
0
0
”
(n-lb,(p)
allows one to solve system
I
q,(P)
0
q(P)
no,(p)
theorem
a,-z(P)
0
4P)
O,(P) (12)
(10).
Theorem [see (Jacobson, 1964) for a general statement]. R,(p) = 0 if and only if one of the conditions occurs: l
(7)
020
=O’
where w 2 0 is the parameter. Excluding the special case o = 0 (which can be accounted for separately) and setting 8 = cc’, (9) can be written as
The above theorem relates AS to posifiuity of the polynomial P(o’), which will be called Popou polynomial. We recall that a polynomial f(x) is said to be positive if f(x) > 0,
= 0
1
l
a,(p) = 0 The two polynomials extension field.
As a consequence described as
have
a common
of this theorem,
root
the envelope
E(P) = J%(P) u E”(P) u L(P)
in a suitable 0 E(p)
can be (13)
where
and3Q>O:F,(Q;p)=OandF,(Q;p)=O}
(14)
.5(p)
= (P E RY:aO(p) = 0)
(15)
E,(p)
= {p E RY:a,(p)
(16)
Observation 1. Observe
that p E E,(p)
= 0).
if and only if there
149
Brief Paper exists Q > 0 which solves simultaneously both equations in (IO). Since the second equation in (10) is the derivative of the first one with respect to Q, it follows that corresponding to p E E,(p), the polynomial F,(S&p) must necessarily have a zero of multiplicity at least 2. Moreover, if F,(Q;p) has, for a fixed p, r positive real zeros (Q,, . , $2,) of multiplicity (p,, . . . , n,) (~~22) respectively, then the resultant R,(p)
has at p a zero of multiplicity i$, pi - r.
3. Robust stability against scalar perturbations In this section we give a solution to the following problem. Let k be a fixed sector bound and p = [p,, pJT a vector with p, representing the Popov parameter 6 and pz a “physical” uncertain parameter entering polynomially in system matrix A and/or vectors b, c. Let a nominal value of the parameter p2 =pi be given (corresponding to the nominal linear plant) such that the Popov inequality (4) is satisfied. With reference to the perturbed Popov polynomial P(w’;p), we want to evaluate the maximal connected domain of positivity of the variable p2, containing the nominal parameter value p$ To do this, we consider equations (13)-(16) defining the envelope containing dD+. First of all, observe that the Popov function is linear in p1 and polynomial in pz, so that P(o*;p) can be written as follows P(w? P) =h(w’;
PJP,
+fo(d
Pz)
= 0
W(oZ;p)/ap,
= 0
i aP(wZ;p)/do
= 0.
R&P,,
(18)
fi(w’; Pz) = 0
S(P2) = UP2)
R,z(P~) = det
W(P~)I= 0
(20)
and choose only those which solve simultaneously both equations (19) for some positive value w2. Notice that H(.) is defined as in (12). Of course, the two special cases w = 0 and w = m must be considered separately, since in both cases degeneracies of the equation system (19) occur. The second critical set S2(p2) is obtained by computing critical points of R&p,, p2) = 0, i.e. self intersection points the curve of R&p,, p2) = 0. These points are obtained by imposing that the gradient of R&p,, p2) is null P,)/~P,
= 0
( ~RQ(P,, P,)IJP,
~R,(P,,
= 0.
(21)
Again, the resultant theorem allows one to solve the above system. The critical set &(p2) includes all solutions pi of (21), which solve also (10) for positive Q. Obseruation
2. The
solutions in &(pJ
above conditions for the existence of become of easy interpretation if the order
u UP*)
u UP*).
(24).
Let us now introduce the definition of an “extremal” point of the parameter plane. We say that a point p = (p,, p2) of the plane such that pz E S(pJ is “extremal” if it satisfies the two properties II, and II, II,:
P~JD+
i.e. P(w*;p)
is nonnegative.
II,: Vc > 0 arbitrarily small p- = (p, - E, p2) $ D,
i.e. P(w*;p-)
and
P+ = (P, + E, p2) $ D,
and
P(w’;p+)
are not positive.
Observe that these properties can be easily checked means of algebraic nonnegativity tests (Siljak, 1971). Let us define the “extremal” set S*(pJ as follows UP,)
by
= {PI E S(p2): (P,, p2) is extremal)
q(P2)
and S;(pJ
as
= {Pz E S*(P2) :Pz>P:)
ST(Pz) = {Pz E S*(P,) :Pz < P3
(25) (26)
and the following quantities py=
Observe that, once the above system has been solved for p2 and o, the last equation in (18) provides, if desired, the corresponding values of p,. System (19) can be solved by using the resultant theorem reported in Section 2. In particular, in this case the resultant of the system with respect to the indeterminate o* depends only on pz, so that we need to find real solutions of the equation
(23)
and selecting as elements of &(pJ only those solutions which solve both equations (10) simultaneously for positive values of Q. Define now the following sets
and the two subsets S:(pz)
(19)
(22)
R&P,, ~2) = 0 1 a,(p) = 0
By substituting (17) in (18) the elements pi belonging to the first critical set &(p2) can be obtained by solving the following polynomial equation system
i h(02; P2) = 0.
~2) = 0
1 a&p) = 0
(17)
where fa(.) and f,(.) are suitable polynomial functions. We select the extremal values of p2 considering points in the plane belonging to certain sets, called “critical” sets. For simplicity of notation, only the second components of the corresponding points will be included in these sets, denoted by Si(p2), i = 1, 2, 3. The first critical set S,(p2) is obtained by collecting points in the plane (pl, p2) belonging to E(p) and satisfying necessary conditions for the existence of a horizontal line (parallel to the axis pl) tangent to E(p), i.e. such that P(l.J;p)
of the linear plant is n = 2 or n = 3. In fact, from Observation 1 of Section 2 it follows that for n = 2, S,(p,) must necessarily be empty; for n = 3, S2(p2) is nonempty if and only if there exists a p such that the polynomial equation F,(Q;p) = 0 in the indeterminate 8 has one only positive root of multiplicity 3. The last critical set S3(p2) is obtained by looking for real solutions of the following two equation systems (compare (13, 15, 16))
PY=
.max pi +S;oJZ) &~~~*)p~.
(27) (28)
In the above equations pT(pp)
is set to -m(+m) if the set is empty. Assuming that G(s;p) is asymptottcally stable for pz E (p?, pr), the following theorem follows readily from the previous considerations.
UP*W,3P,))
Theorem. The parameter uncertainty domain (~7, p,“) is the maximal connected domain of absolute stability, according to the Popov criterion, containing the nominal 0 parameter for a given Lur’e control system. Notice that maximality of the domain of absolute stability computed above does not necessarily mean that it is not possible to extend the interval (~7, py) still preserving AS, because the Popov condition is sufficient but not necessary for AS. 3.1. Maximal sector of absolute stability. In this subsection, we briefly show how the procedure given before simplifies when we consider the special case in which the parameter p2 represents the reciprocal of the sector bound k. In this case, we assume for coherence with (2) that the allowed minimum k is 0 and accordingly p: = m. We want to estimate the maximal k for which AS is preserved according to the Popov criterion. The Popov polynomial turns out to be linear in both parameters p, = 0 and pz = k-l. The envelope equations
150
Brief Paper
(10) become f,(J)Pi +f2(J)p2 +fo(w*) = 0 I (llo)[f;(W2)p, +f;(o*)p* +&(o2)]
=0’
liJ‘O
(29)
where f,!(.), i = 0, 1, 2 denote first derivatives with respect to O. Polynomial functions fa, f,, f2 have an easy interpretation in terms of the frequency response G(jo). In particular, setting G(jw) = K(jw) + jN;(jo) (30) 4(jw) + jQ(jw) it is easy to show that fo(w’) = X(jo)Q(jw) fi(w’) = +$(jw)Q(jo)
+ N,(jw)Q(jw) -
sides of the rectangle, i.e. setting alternatively pi =p: (or i = 1, 2, possible intersections can be computed by applying the procedure given in Section 3 for each side. Step 2. To check for absence of envelope points inside U,, we have to verify that the polynomial equation systems (22) and (23) have no solutions belonging to U,,. Moreover, it must be checked that the following polynomial equation system
p, =py),
NWPAw)l
(31)
R,(P,,P,)=O %(~,,~z)la~i =0 i (or SR~P,, P,)/~P, = 0)
has no real solution in U,. Notice that if we denote by S,(p) the set of solutions of (35). only the following subset must be considered
f2(0*) = lO(jw
C(P)
The above equations allow one to give a graphical interpretation of the criticalgoints in terms of the Popov locus in the complex plane (Siljak, 1969). In the parameter plane each curve of the envelope is a straight line. As a consequence, the set S,(p,) may be readily obtained by computing the real positive solutions of of the polynomial equation f,(wZ) = 0 (32) and then computing the corresponding solutions for pr P; = -f;,(w~)/f2(w2).
(33)
Notice that from the second equation in (31) solution frequencies wi correspond to points where the Popov locus crosses the real axis. Computation of the solutions belonging to the critical set S2(p2) does not simplify significantly in the special case p2 = k-‘. These solutions correspond to intersections of the real axis with straight lines admitting possible multiple tangency points with the Popov plot. In fact, these solutions represent critical points of the curve R&p,, pz) = 0, i.e. points such that there exist at least two different values of frequency w satisfying the envelope equations (29). The set S,(p,) can be readily computed because the last equations of (22, 23) are linear in p,, p2, so that the solution of each of the two systems requires to solve one polynomial equation. From a graphical point of view, these points correspond to lines passing through one of the end points of the Popov plot and tangent to it at that point and/or some other point(s). Observation 3. The feedback gain interval [0, kH) for which the linear system G(s) is closed loop asymptotically stable is usually called Hurwitz secfor for system (1). It can be easily checked that if pyeS,(p2), then py= kjj’, i.e. the Popov sector coincides with the Hurwitz sector. Hence, the Popov criterion allows one to conclude that the well known Aizerman conjecture is true. An example of computation of the maximum k for which the Lur’e system (1) is AS according to the Popov criterion for a classical example taken from the literature is given in Section 5.
4. Robust absolute stability for planar uncertainty sets In this section we assume that the sector bound k in (2) is given. We consider the case in which the components of vector p E R2 are uncertain physical parameters affecting polynomially the coefficients of G(s;p) (or the entries of (A, b, c)). Let the uncertainty set be defined as a rectangle in the parameter plane Up = {p E R*:p:sp,
‘p:,
i = 1, 2).
(34)
Assuming that Vp E Up the linear part of (1) is asymptotically stable and that Cl’ contains at least one point p =p” for which G(s;p) satisfies the Popov criterion, absolute stability can be studied in two successive steps. In a first step we ascertain if the boundary au, intersects the envelope E(p). In the second step it is checked if there exist points of the envelope in the interior of VP. Step 1. We look for possible intersections of the rectangle sides with the envelope E(p) given in (13-16). Considering
(35)
= {K(P)
fl E(P)).
(36)
From a practical point of view, system (35) can be solved, as shown for other cases in Section 3, by applying the resultant theorem. The set S:(p) can be computed immediately selecting solutions p ES,(~) for which there exists some positive real value w solving simultaneously both equations (9). As a last observation on Step 2, it is worth noting that if for a fixed value of the Popov parameter the test fails, i.e. the set S:(p) is not empty, we have to perform the test for different values of 0. Hence, in general it may happen that Step 2 must be repeated for all real values 0, meaning that the solution of a family of problems like that solved above may be needed. As a final comment, we notice that since the Popov criterion is only sufficient for AS, the fact that the test proposed fails in assessing AS of an uncertainty set Cl, in general does not allow to conclude that system (1) is not AS for p E Cl,. A negative answer of the test would mean only that the Popov criterion does not allow one to assess robust absolute stability, so that other alternative criteria should be employed. 5. Numerical examples
In this section we present two examples showing applications of the results presented in previous sections. In the first example, a well known system is considered and the maximum Popov sector is computed. In the second example, we determine the maximum allowable uncertainty domain for a scalar parameter affecting the linear part of a Lur’e system, for a prescribed class of nonlinear functions gc*. Example 1 (Narendra and Taylor, 1973, p. 173; Safonov and
Weytzner, 1987). Consider a Lur’e system where the linear part is described after a pole-shifting by G(s) =
3(s + 1) s4+s3+25s2+3s+3’
From standard arguments it follows that the Munoitz sector is [0,7). Thus, by defining (p,, p2) t (0, k-l), we observe that we need only consider values p2 in the interval (0.14285,ml. Moreover, it can be easily checked from the structure of P(o*;p) that the point (p,, p2) = (0,~) satisfies the Popov condition (4), so that we can assume as nominal value pt = 00. The functions f;(.), i = 0, 1, 2 in (31) are given by f0(uJ2) = -6602 + 9 f,(w*) = -3w6 + 72w4 (38) [ f2(02) = o8 - 490~ + 625~~ - 1410~ + 9. From equations (32) and (33) we obtain the set S,(pJ
= I-1,
0.14285).
(39)
The envelope equations are F,(Q;p)
=p@
- (49~~ + 3p,)Q3 + (625~~ + 72p,)Q*
- (141p1 + 66)Q + 9p, + 9 = 0 F,(P;p)
= 4p2Q3 - 3(49p2 + 3p,)Q* + 2(625p, + 72pJQ +(141p,+66)=0. (40)
Brief Paper
Also the set $3(P2 ) turns out to be empty because the coefficients ao(p) and a3(p) never vanish. Consequently, the set S(P2) is equal to $1(P2) and S . ( p 2 ) = {1}. Hence, we obtain, p~' = 1 and p2M = m, obtaining that the considered Lur'e system is AS for P2 • (1, o0). The corresponding values of 0 and to for p2 = 1 are 0 = 0 . 2 and to =3.162.
wlmIG(Jw)l
-1.5 -1.5
-1
-0.5
0
0.5
l
1.5
2
2,5
FiG. 1. Popov line for maximal absolute stability sector of Example 1. Since the degree of polynomial F~(2;p) in 2 is 4, from Observation 2 of Section 3, we can expect that self intersection points of the envelope may exist. In fact, computing the resultant Ru(pl,P2 ) and solving (21), we obtain the set $2(P2 ) = {-1.443, 0.37}. (41) The set $3(p2 ) obtained by solving equations (22) and (23) is given by $3(P2) :- { - 1 , 0}. (42) From (39), (41) and (42) we obtain S(p2) = { - 1.443, - 1, 0, 0.14285, 0.37}.
(43)
The extremal set S.(p2) is obtained by checking properties H1 and H 2 S.(pz ) = {0.37}. Hence, we select p~' = 0.37 and p2M = ~ obtaining that the considered system is AS for k • [0, 2.7). The corresponding values of 0 and to for k = 2.7 are found to be 0 = 0.781 and to1 = 1.54, to2 = 5.24. These two values of to correspond to the tangency points of the straight line defined by k = 2.7 and 0 = 0.781 with the Popov locus (see Fig. 1).
Example 2. Consider the system described by the transfer function 11 G(s;p2) = (s + 1)(s 2 + p2 s + 9) (44) where P2 is an uncertain parameter with nominal value pO= 10 and the nonlinearity f ( - ) • ~k with k = 1. From standard arguments, we obtain that the closed loop system with a linear constant feedback with gain k = l is asymptotically stable for P2 • (1, oo). As in Section 3, we set Pl = 0. The functions fi(-), i = 0 , 1 in (17) are given by fo( to2, P 2 ) = to6 + (p2 _ 17)o) 4 + (52 - llp2 + p 2 ) t o 2 + 1 8 0 fl(to2,'p2) = -llto
4 + 11(9 + p 2 ) t o 2.
From equations (19) we obtain the set $1(P2) = {0, 1}.
(45) (46)
The envelope equations are
;
F~(fl;p) = 2 3 + ( p 2 _ 11pl - 17)2 z F2(2;p)
151
+ (p2 + 11p~p2 _ 11p2 + 99pl + 52)2 + 180 = 0 3 2 2 + 2 ( p 2 - l l p ~ - 17)2 + (p2 + llp~p2 _ llp2 + 99p~ + 52) = 0. (47)
From Observation 2 in Section 3, we notice that since the degree of the polynomial F~(2;p) in 2 is 3, the only possible points belonging to the set S2(p2 ) must necessarily yield a triple positive solution for 2 in F~(2;p). This cannot happen because ao(p) = 180 > 0. Therefore, the set S2(P2 ) is empty. This can be also verified by computing the resultant RQ(Pl, P2) and solving (21).
6. Conclusions In this paper an analytical method is proposed for robust absolute stability analysis of Lur'e control systems subject to parameter variations. The case where system perturbations are due to one or two uncertain parameters has been considered and a method is given which allows one to estimate maximal domains of absolute stability in parameter space, based on the Popov criterion. Further research is needed in the direction of studying robust absolute stability by means of different criteria, which may take into account more information about the feedback nonlinear function than the sector condition. Moreover, effective methods of analysis for problems involving several physical parameters appear to be of primary interest for future work. Acknowledgements--This work was partially supported by funds of Ministero della Universit~ e della Ricerca Scientifica e Tecnologia. References Bhattacharyya, S. P. (1987), Robust Stabilization Against Structured Perturbations. Lecture Notes in Control and Information Sciences 99, Springer, Berlin. Bose, N. K. and J. F. Delansky (1989). Boundary implications for interval positive rational functions. IEEE Trans. Circ. Syst. CAS-36, 454-458. Genesio, R. and A. Tesi (1988). Results on the stability robustness of systems with state space perturbations. Syst. Control Lett. 11, 39-47. Jacobson, N. (1964). Lectures in Abstract Algebra, Vol. III. Von Nostrand, Princeton, NJ. Narendra, K. S. and J. H. Taylor (1973). Frequency Domain Criteria for Absolute Stability. Academic Press, New York. Neimark, Yu. I. (1949). Stability of Linearized Systems (in Russian). LKVVIA, Leningrad. Popov, V. M. (1962). Absolute stability of nonlinear systems of automatic control. Aut. Remote Control, 22, 857-875. (Russian original published in 1961). Safonov, M. G. and G. Weytzner (1987). Computer-aided stability analysis renders Popov criterion obsolete. IEEE Trans. Aut. Control, AC-32, 1128-1131. Sideris, A. and R. S. S. Pefia (1988). Fast computation of the multivariable stability margin for real interrelated uncertain parameters. Proc. ACC. Atlanta, CA. Siljak, D. D. (1969). Nonlinear Systems: The Parameter Analysis and Design. Wiley, New York. Siljak, D. D. (1971). New algebraic criteria for positive realness. J. Franklin Inst., 291, 109-120. Siljak, D. D. (1989a). Parameter space methods for robust control design: A guided tour. IEEE Trans. Aut. Control, AC-34, 674-688. Siljak, D. D. (1989b). Polytopes of nonnegative polynomials. Proc. ACC, Pittsburgh, PA. Tesi, A. and A. Vicino (1988). Robustness analysis for uncertain dynamical systems with structured perturbations. Proc. 27th Conf. on Decision and Control. Austin, TX, 519-525. Also IEEE Trans. Aut. Control 1990, AC-35, 191-195. Vicino, A., A. Tesi and M. Milanese (1988). An algorithm for nonconservative stability bounds computation for systems with nonlinearly correlated parametric uncertainties. Proc. 27th IEEE Conf. on Decision and Control, Austin, TX, 1761-1766. Also IEEE Trans. Aut. Control 1990, AC-35, 835-841. Vicino, A. (1989). Maximal polytopic stability domains in parameter space for uncertain systems. Int. J. Control, 49, 351-361. Voronov, A. A. (1989). On improving absolute stability criteria for systems with monotonic nonlinearities and the method of absolute stability regions construction. Preprints IFAC Syrup. Nonlinear Control Systems Design, Capri, Italy, pp. 231-235.