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Robust Stabilization of Interval Plants with Specified Damping Margins
Copyrighl © 1996IFAC 13th Trienllial Wo rM (;ollgre!;.
2d-086
ROBUST ST ABILIZATION OF INTERVAL PLANTS WITH SPECIFIED DAMPING MARGINS
Ahmed F. Sakr
Department of Electrical Engineering Cairo University, Cairo, EGYPT
Abstract: In this note, It is shown that a first order compensator robustly left·sector stabilizes an interval plant if and only if it left-sector stabilizes a finite number of speci~dly chosen veneX plants. The number of venex plants depends on either the system order or the damping angle that defines the left-sector. A simple procedure for designing this robust compensator is provided, and is illustrated via a numerical example. Keywords: Uncemin linear systems, Robust control, Compensators, Stabilization, Kharitonov theorem.
1. INTRODUCTION
The stability and feedback control of systems with unknown but bounded parametric unceminties (interval plants), is a problem of current interest in control theory. It has been stimulated by the fundamental work of Kharitonov (1979) who stated that an interval polynomial is Hurwitz invariant over the uncertainty bounding set if and only if four speeially construcled venex polynomials are stable. Kharitonov's original result was subsequently generalized to cover stability regions other than the left-half plane. Of practical importance is the generalization ~o left-sector stability (or synonymously relative stability) (Soh and Berger, 1988: Fao and Soh, 1989, 199Oa, 199Ob. 1992: Kathab and Jury, 1993) Those results arc useful in anaIyzing the stability robustness of a given feedback control system. However, there are relatively few efforts in extending those results from the analysis context to the design context. Among these
effons are the work of Chappellat and Bhanacharyya ( 1989), HallO! and Yong (19901, Barmish el al. (1990), Bernstein and Haddad (1992), and Chapellat el. 01. (1993). which give necessary and suffic ient conditions for robust stabilization of interval plants using either static or dynamic compensators. Tbese methods are concerned with the left-half plane stabilizatioll. Nonetheless, left· half plane stability does not guarante" proper system response. I! is more practical 10 consi:ler left-sector (relative) stabilization~ i.e. the entire family of interval plants will have only poles in a left-sectc'r defined bY adamping margin. The aim of this paper is to develop a method for constructing a robust controller that left-Sf'ctor stabilizes an interva1 plant. The controller is considerd in the general form of a first order compensator. This :'(Iml covers a variety of cvntrol actions such as P :'1, PD, lead. or lag compensators. Jt is proved he:dn that for an interval plant with a first o:..der compe'l';3tof, it is necessary and
3299
sufficient to left-sector stabilize only a finite number of specially chosen vertex plants in order to left-sector stabilize the entire family of interval plant. The number of these vertices depends on the degree of the plant or the angle of the damping margin that defines the left sector. It should be pointed out that this result is a generalization of the result reported in (Barmish et al.; 1992) for left-half plane stabilization.
,~
I..,
.,,'-........,
L1
Left Re
sector I...
2. PROBLEM FORMULATION
Fig. I. The left sector
In the standard closed loop feedback system, consider a strictly proper interval plant family !,(S) = N (S) I D(S)
polyoomial f(S) has only zeros within the left sector if and only if its polar plot encircles the origin n times (2mt phase advance) in the counter clockwise direction, as S traces the closed contour (L I'L ,L ) in a counter 2 3 clockwise direction. As S traces L , f(S) will undergo a 2 phase advance of 2n(" - ~). Hence and due to symmetry about the real axis, f(S) has only zeros within the left sector if and only if the pllaSe of f(S) increases monotonically through a net phase advance of n as S traces the upper segment LI . Noting that segment LI is defined as xei ; x El 0, 00) then the polar plot off(S) as S traverses L 1 is equivalent to the polar plot of
(I)
srn S + .....+ "n sn
N(S) = bo + bl S + .... + b m
(2)
O(S) = 30 + al
(3)
The numerator and denominator polynomials coefficients are uncertain but bounded by lower and upper limits; that is b.E[!!);'] for j=O,I, ... ,m and a.EI!!)i.] for i=O,I, ... ,n. The] fanlil~ of polyoomials of the !'orm't2\ is called an interval polyoomial; and the family of plants (1) is called an interval plant since its coefficients vary in a specified interval. The case where some coefficients are perturbation free is handled by equating their upper and lower limits. Consider also a fix.ed first order compensator of the fonn
n
f( xei+) = L Pk eik~ x k (7) k=0 as x increases frorn 0 to 00. The following observations and lemmas provide a machinery t>r the proof of the main result.
where Cl' C , C , and C are nonnegative real constants. 2 3 4 The closed-loop system is called r-stable if for all possible values of b.; j=O,l,,, .,m and a.; i=O,I, ... ,n the resulting closed,loopharacteristic polyno~al 8 (S) = (Cl S+C2) N(S) + (C3S+C4) D(S)
Observation i: f(xei; where R(x) and Q(x) are real polyoomials ofx.
(5)
Observation 2: f(xei+) ei6 for any real constant e, can be written as f(xej
has all its roots in a specified region r in the complex plane. If we consider r to be a left-sector, ther-stability means a minimum damping ratio margin.
Notice that if the phase off(xei+, is increasing through net phase advance of Rep as x. increases from 0 to 00 , then the
The problem is to determine the compensator C(S) that left-sector stabilizes the entire family of the interval plant I'(S). In this case, C(S) is called a robust left-sector stabilizer, and the system is robustly left-sector stable.
polar plot off(xei+) will interse't the real axis and the ei axis alternately in the counter clockwise direction. This is called an interlacing zero property; i.e. the roots ofR(x) interlace the roots ofQ(x).
3. PRELIMINARY RESULTS
Lemma i: Let g(xei+ ,fi)= f(xei O ) ei6+ fi(a x ei~ + P)v(x), where f(s)is a left-sector stable polyoomial, 6 is a real constant, a ~ p are nonnegati\'e real constants, v(x) is polynomial in x with nonnegative coefficients, and fiE[O,I] 1S a scalar, then g(xd+ ,fi) has interlacing zero
Let f(S) be an nth order polynomial of the form n
f(S) = POTPI S+ ... +p S (6) and consider the left-sector defined b;the angle <1>, where 1tI 2:0; ~ <" as illustrated by Fig. I. It is well known that the
3300
property for all l'e[O.I) if gexei+. 0) and g(xei'. I) have interlacing zero property.
proof in view of observation 2 'We have g(xei$ .1") = (R(x) + I'Jl vex»~ + (Q(x) + I"a xv(x» ei$ Rt(x) + QI(x)ei$ At fI = 0 the roots of RI(X) and QI(x)are Iho,e of R(x) and Q(x) respectively. Since v(x) is positive increasing function. then as '" increases from 0 to I. each individual root of either RICX) or QI(x) moves in one direction only. The roots ofRl(X) are detennined from RCx ) = - I" Jl v (x); and the gradient of their change with respect to I" i, given by Z,.(x) = dxldl" = 4dx1dR(x)XdRCx)l dfI) = -Jlv (x)fR' (x) where R'Cx) denotes dR(x)ldx . Similarly. the roots of QICX) are determined from Q(x) = - I"a xv(x); and the gradient of their change with respect to fI is given by Zq
=
We proceed by contradiction. Assume for some I'e (0.1) the roots of RI(x) and Ol(x) do not alternate. Since at fI = 0 and fI= 1 Rj(x) and Ql(x) have interlacing zero property. and the roots depend continuously on 1'. and each individual root moves only in one direction as ~ increases: then the following rentaIks are obvious: a) There exists at least two values of fI. say fll and 1'2. and two value, of x. say x I and X2. at which R(x I) = -fll llV(xll. Q(xiJ = -l'laxlV(xll. R(x2) =-fl2 llv(x2). and Q(X2)= -1'2'" x2v(x2)· b)For xelx I. x2) either R' (x) and O' (x) are both negative or both positive. cl Ifz(xl » 1 then 7.(x2)<1; el,e ifz(xj)<1 then z(x2) > I. Or in other words. by using the definition of 7.(x) and remark Ca) if (Q'Cxl)/(R' (xI») > (Q(xl)IRCxl» Ihen (Q'ex2) I (R' (x2)) < (Q(x2)lR(x2)) else if (Q'(xl)/(R' (XI»< (Q(x 1)IR(x 1» then (Q'(x2) I (R' (x2)) > (Q(x2)IR(x2))'
Consider now an interval polynomial oflhe form (J). It has been shown in (Foo and Sob. 1mb. 1992) that the interval polynomial will have only roots in a left-sector if and only if a finite number of spedally chosen venex polynomials have only roots in "he left-sector. The vertex polynomial, are given in (Foo ",1d Soh. I 990b. 1992) for special cases of the damping margins (angle ~). However. we introduce here the following lemma which provide procedure 10 identify Ihe ven"x polynomials. for any general value of $. First let uS define +I< = k+ - 2i,,; where i is selected such that 0$+1«2" for k=O.I ..... n: alld +k+n+l=k±lt such that OS+k+n+ <2n for k=O.I .... . n. Lel also 60- 61 .... .. 6m; m !>2n+l 10 be the values of the angles +1<. 0,;kSln+ I arranged in ascending order; i.e. ()=90<91 <......6 m<21<. We note thal m=2n+ I if the values oflhe angles fk are distinct for all k; and m<2n+ I if fkl = +k2 for some inlegers kl and k2. For exarn~·le if , =2"'J. lhen m=S regardless or the polynomial orde r. In this case 00 = ~O = J = ..... = O. 91 = o+J = ~n+6 = .... = "'3 92 = ~l = <1>4 = ..... = 2"'3. 93 = <1>0+ I = $n+4 = .... = n. 94 = h = <1>5 = ..... = 4"'. 65 = '~+2 =n+S = .... = 5"'J Notice that m is always an odd number. Let q4m-I)/2 and
no = {90. 6\. °2 ......... Bq}
01 = {Ol .02. 6 J .. ....... 6q+1}
11m = {9 m • 90. 9 1 ......... 6q.1}
(8)
lemma 2: The roots oflhe inte"a1 polynomial D(S) oflhe form (3) will lie only in a left.·sector defined by if and only if Ihe m+ I vertex polynomials Or CS): r = O.I ..... m have only roots in that left-sector. where
+
Or (5) = OCS): "i = 3; if +; E llr and a; =..l!i otherwise (9) for all i = O.I .....n; and r = O.I ..... m proof The necessity pan is ob> ious since Ihe m+ I venex polynomial. are members oflhe family of polynomials (3). We now prove sufficiency by contradiction. Assume that O,.(S); r=O,l ..... m have only roots in the left-sector. Assume also that D(S) has root" outside the left-sector for some values of a.~ i=O.l ....•n. Hence, the continuous root dependence on th~ coefficients of O(S) dictates that Ihere exists some values of a; =
Notice that R(xl). Q('I). R(x2). and Q(x2) are all negative. Since f(,) is left- sector stable then the polar plot of f(xi~) ei9 = R(x) + Q(x) ei has a monotonically increasing phase. This necessitates that: ifboth R'(x) and Q'(x) are negative for xelxl • x21 then (Q' Cxl)/(R' CXI» > (Q(x 1)fR(xl» and CQ'(x2)1CR' CX2» > (Q(x2)fRCx2)) else if both R'(x) and Q'(x) are positive for xelx I • x2) then (Q'Cx 1)/ (R' (x 1))< CQ(x I )fRCx I)) and (0' (x2) / (R' (x2» < (Q(x2)fR(x2»' The above inequalities are in contradiction with those of remark (c); and consequently with Ih. main assumption. This completes the proof.
for this case. the polar plot ofD·Cxei') will pass through the origin as illustrat by fig. 2. The tangent to the plot al the origin separates the complex plane into two regions; say region-l and region-H. From (8) it can be readily 'hown that one can always seh:ct one sel ~ that has all its members contained in region-L
3301
I= ReQ.i. on
"-
is left-sector stable for a1ll1e[O,I] if and only ifH(5,0) and H(S, I) are left-sector stable.
I
"-
Proof Necessity is obvious. To prove sufficiency. we start by rewriting H(S, 11) as
""""-
Regi.on
H(S,Il) = HI (5) + 11(<<5 +
"IX
Re
"-
n Dp (S) = D*(S)+d +d S+ ... +dnS O l Dp(xei
(10) (11 )
H(xei.,I1)= HI(xei.) + 11 (0 xei. + Il) v(x)clcr
g(xei+,JI)= H(xei+. JJ)e-i"= =H I(xeit) dcr+ !l(,.xei++ll) v(x)
'1>
D'(xei~) lies in region-I. Dp(xei~) will also lie in region-I.
Thus Dp(xeiq,) will not increase monotonicaly through a net phase advance of nq, by sweepi ng x from 0 to 00. That is to say there exists one vertex polynomial DP
It is worth noting that Kharitonov theorem (Kharitonov. 1979) IS a special case of the above lemma for the case q,= 7tl2. In this case. m = 3. Q o= {O. 7tl2}, QI = {7tl2. ltl, Q2 =1". 37t12 ). and Q3 = (37t12. 0 ). It can be also shown t hat if the above lemma is specialized to the cases =27t13 and + =37t14 we get the same results as in (1'00 and Soh, 199Ob. 1992). ObservatIOn 3: For each x e[O. 00). O(xei+) is bounded by a convex polygon in the complex plane with vertices Drlxei~): r=O.I .. ... m. The edges of the polygon are of fixed orientation for all x and they are at angles 9 0 , 91 •.... ,8 01 .
+
lemma 3: Let Ho(S) be a polynomial, et, Il are real constants. and
HI(S ) = Ho(S) + CctS+1l )Dr(S)
(12)
H2CS ) = HoCS) + (as+Il)Dr+I(S)
(13)
where DrlS) and Dr+ I(S) are two successive vertex polynomials of 0(5). Then the polynomial H(S,Il) = ( 1- Il)H I(S) + !iH2(5 )
(14)
(16)
where v(x) is a polynomial in x with nonnegativ. coefficients and crE{8",9 1• ....• Srn}. We proceed by contradiction. We assume that for so:ne IlE(O,J). H(S,Il) is not left-sector stable. Then continuous root dependence on ~ necessitates that there exists some ~*and x· such that H(x·eif, ~. )=0. Let
or From (9) it can be seen that dt<>O if +k e equivalently if the vector e)kxk lies in region-l. SiIrularly, dk
(IS)
Thus, in view of observation 3 an,1 the definition (9)
"fig. 2 Polar plot ofD(S) with roots on the sector boundries
Note also that we can Mite Dp (S) and Dp(xei+) as
Jl:'[Dr+ I (S) - D"
Since H(S,O) and H(S, I) are left-sector stable,
(1 7) then
g(xei>!>.O) and g(xeiq., I) have interlacing zero property.
x'
Since H(x' ei>!>, ,,')=0 then g( ei+, ~ *) = 0 which is not possible in view of lemma I. Thi, contradiction completes the proof
4. MAIN RESULTS Theorem 1: A first order compen,;ator C(S) in the form (4) robustly left·sector stabilizes the entire family of the interval plant (I) if and only iLt left-sectorstabilizes the following vertex plants
Pqr (S) = Nq(S)I D"
cFl.1 .. ·..ml
• r=O.I •... ,mZ (18)
where Nq(S): q=O.I .... ,UlI andD"
5q"
( 19)
are all contained in a left-sector defined by damping margin 4>. Let N*(S) denote th" polynomial N(S) with arbitrary but fixed coefficients which are taken from the bounding ,et. Thus. in view 0:.; observation 3. for each x e[O. oO)
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follows that S",,(S}=(CI S+C2)N+(S)+(C3S+C4)DO(S) is left-sector stable. is bounded by a convex polygon in the compl.. plane with vertices
The application of theorem I requires a compotationaly intensive search for a compensator that left sector stabilizes m I m2 vertex plants. However, with the aid of the following remarks, a computrtionaly feasible design procedure may be invoked.
Or*(xei+C2)N*(xei+c4)Dr Cxei
Remark 1: Although theorem I considers, in principle, a first order compensator C(S) that involves four variable parameters, yet for practical applications, C(S) involves no more than two variable parar""ters. For example, for PI controller CI=K I, C2=K2' C3~~ 1 and C4= 0; for PO controller CI=KI' C2=K2' ('3=0 and C4=1: for lag compensator CI=KI' C2=KIK2, C3=1 and C4=aK2; where et >5 is a preselected constant; while for lead compensator CI=KI' C2=K I K2 , C3=1 and C4=K2/et. Thus what is only required is gridding two parameters K 1>0 and K2>O and evaJuatin;~ the root locations for the vertex polynomials for each grid point.
r *(xei
Remark 2: The effort associated with identifying the root locations of an nth order polynomial may be reduced tu checking only the Hurwitz stability (using the classical Routh criterion) for a 2nth order polynomial via tile following theorem.
we have (\-I1)o/(xei
To this end, we get the following fact Fact I: For arbitrary but fixed coefficient polynontial N*(S), if the vertex polynontials or *(S); FO,l, ... ,m2 are left-sector stable, then the interval polynomial O"(S) is left-sector stable. Defining 80 (x
eir=
Theorem 2: Let T(S) = t o+ tl S+ .. + InS n : and ~ where Q(Sl=CJo+qIS+. +Q2n S2n i qi = L tr ti-r cos ((i- 2rX.-,,}2)); i=O,I, ... ,2n (24) r=O
(Clxei+C2)N(xei+) + (C3xei+C4)Do(xei
then T(S) has only roots in the left-sector defined by ~ if and only ifQ(S) has only roots ir, the left-half plane.
Oq 0 (x eir={Clxei+C 2 )N q(xei+C4)Do(xei
Proof: Let T I (S)=to+t 1ei(+-"'2) S+ ... +lnei nC.-"'2) sn and T2(S)=to+tle -j(-"'2) S+ ... +tn" -j(.-"'2) Sn Notice that the roots of TI(S) and T2(S) are those of T(S) rotated by angle (.-"'2) in the c\ockw.se and anticlockwise directions, respectively. Utili:,;ng the fundamental
(23)
and using a similar line of proof, we get the following fact Fact 2: For arbitrary but fixed coefficient polynomial OO(S), if the vertex polynomials Oq ,,(S); q=O: 1,... ,m are L left-sector stable, then the interval polynomIal 80 (~) IS left-sector stable.
identities: ei a =cosa + jsina, and cos(a - 11 }=cosa cosp + sinasin p: while noting that t·= 0 for i > n, it can be shown that I . Q(S) = TI(S) T2(S) Thus the roots of the real polynomIal Q(S) are the union of the rcots of the two complex polynomials T I (5) and T 2(S) ,md should form conjugate pairs. Suppose that 2r conjugate roots ofT(S) are lying outside the left sector. Then by rotating the roots ofT(S) by angle (<1>-"'2) in the clockwise direction, exactly r roots will be located to the right of the imaginary axis. Thus, T 1(S) will have exactly r right-half side roots. Sintilarly, it is clear that T 2(S) will have exactly r right-half side roots .
The proof of the theorem is completed by combining facts I and 2. Let N"CS) and O,,(S) be any polynontials with arbitrary but fixed coefficients which are taken from their bounding sets. Since the polynomials 0qr(S); q=O,I, ... ,m I, FO,I, ... ,m2 are left-sector stable, then applying fact I, m I times while replacing N*(S) by Nq(S); q=O,I, ... ,mb' it follows that Oq O(Sp (CIS+C2)NqCS)+(C3S+C4)D (S), q=O, I, ... ,m I are left-sector stable. Then applying fact 2, it
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Consequently, Q(S) will have exactly 2r right-half side conjugate roots .The proof is then clear by taking _
6. CONCLUSION
5. EXAMPLE
Necessary and sufficient conditions have been derived for robust stabilization of interval plants, under the restriction of damping margins, using first order compensator. Based on these conditions a procedure for constructing the robust compensator is presented. Extensions to more general
Consider the following interval plant family
classes of compensators is an open research area
with the following parameter uncertainties: bl E[40,50], bo E[90,11O], a3E[14,16], a2E[85,95), al E [130,150], ao E[8,14). It is required to find a robust PI compensator of the form C(S)=XI+K2/S that left-sector stabilizes the entire family of the interval plant P(S) for the following three cases'
case I: oF 7tl2 (damping margin9l: i.e. Hurwitz stability) case 2: oF 27t13 (damping margrn9l.5) case 3: oF 37t14 (damping margin9l.7) KI and K2 are grided between 0 and 3 with a 0.1 grid 5tep. In view of theorems I and 2, for each grid point, only 16 routh tables are checked for case I: 24 for case 2; and 32 for case 3. Fig.3 shows the set of stabilizing parameters for the three cases. If it is required to stabilize the interval plant with a minimum damping ratio of 0.7. then any member of the set of stabilizing parameters for case 3 may be selected~ e.g. KI=I and K2= 0.2
. _ .... -_ ..... -_ .... . . . . . . . - - .... - ..... · ........... - . - .... . · ..... - - .... . ...... - ...... - - ..... - . - .. . . - - ........ - .... - - .. . . . - - .. - ... - ..... - - - - . .. .... - - .. - - ........ . . . -- ........ - - - ....... . - . - .. - - .......... - - - .. . - . - . . - ........... - .... . . - - - ....... - ... - - - .. - .. . - ....... - - ...... - ...... -
3
2
• - . . . . . . . . . . . . . . . -++ . . - • _. - - _. _ • . . - + + + + _ . - -
K2
• . . • . • • _ • . . • • • . +++++-1"' • . .
. . . . . - • • . . . • • ++++++++ . . . . . - . . • • - . . . _.++++++++++_._. • . . . . . . . _ . . +++++++++++_ . . . . . . . . • • . . . -++++++++++++- . _ . . . _ . . • - . • -+++++++++++++- . . • . . . • . . . . -++++++++++++++- . . • _ . . . . . . -++++++++++++++++_ . . . . . . . . . . +++++++++++++++++ . . • . . . . . . . +++-++++++++++++++ . . . . . . . • . +++_ • • +++++++++++++ . . . .
o
~
easel
+
ca.se2
..
Fig.3 .Set of PI parameters for robust stabilization
ea,se3
REFERENCES
Barmish B.R.,Hollol CV.,Krans F.J and Tempo R. (1992). Extreme point results for robust stabilization of interval plants with first order compensators, IEEE Trans. Automat. Contr., vol.37, 707-714. Bernstein D.S. and Haddad \V.M. (1992). Robust controller synthesis nsing Kharitonovs theorem, IEEE Trans. A utomat. Cono·.,vol.37, 129-132. Chapellat H. and Bhattacharyya S.B. (1989). A generalization of Kharitonovs theorem: Robust stability of interval plants, IEEE Trans. Automat. Contr., vol,34, 306-311 Chapellat H., Dahleh M. and Bhattacharyya S.P. (199:;). Robust stability manifolds for multilinear inte,,'a! systemsJEEE Trans AutomdContr.,voI38, 314-:1 '! Foo YK and Soh Y.C (1989). Rllot clustering ofinter,'a] polynomials in the left sector, Syst. Contr. Lell.. vol.!3,239-245. Foo YK.and .Soh Y.C (1990a). Damping margins of interval polynomials, IEEE Trans. Automat. Contr.. voI.35,417-479 . Foo Y.K. and Soh Y.C.(1990b). Generalization of strong Kharitonov theorems to the left sector, IEEE Trans. Automat. Contr.. yo1.35. 137S-1382 . Foo YK and Soh YC (1992). Strong Kharitonov theorems for low-order po:ynomials. IEEE Trans. Automat. Contr.• vol.37,1816-1820 . Hollot C V. and Yang F. (1990:,. Robust stabilization of interval plants using lead or lag compensators, Syst. Contr. Lell., Yol,14, 9-12 Katbab A. and JUlY E.!. (1993) A note on two methods related to stability robnstnes:; of polynomials in a sector (relative stability), IEEE Trans.Automat . Contr.• vol,38,380-383 . Kharitonov V. L. (1979). Asynptotic stability of an equilibrium position of a family of systems of linear differential equations, Differ(~ntial Equations, vo1.14~ 1483-1485. Soh CB. and Berger CS (1988,. Damping margins of polynomials with perturbE,d coefficients: IEEE Trans. Automat. Contr.,vol.~;3, 509~511.
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2d-086
ROBUST ST ABILIZATION OF INTERVAL PLANTS WITH SPECIFIED DAMPING MARGINS
Ahmed F. Sakr
Department of Electrical Engineering Cairo University, Cairo, EGYPT
Abstract: In this note, It is shown that a first order compensator robustly left·sector stabilizes an interval plant if and only if it left-sector stabilizes a finite number of speci~dly chosen veneX plants. The number of venex plants depends on either the system order or the damping angle that defines the left-sector. A simple procedure for designing this robust compensator is provided, and is illustrated via a numerical example. Keywords: Uncemin linear systems, Robust control, Compensators, Stabilization, Kharitonov theorem.
1. INTRODUCTION
The stability and feedback control of systems with unknown but bounded parametric unceminties (interval plants), is a problem of current interest in control theory. It has been stimulated by the fundamental work of Kharitonov (1979) who stated that an interval polynomial is Hurwitz invariant over the uncertainty bounding set if and only if four speeially construcled venex polynomials are stable. Kharitonov's original result was subsequently generalized to cover stability regions other than the left-half plane. Of practical importance is the generalization ~o left-sector stability (or synonymously relative stability) (Soh and Berger, 1988: Fao and Soh, 1989, 199Oa, 199Ob. 1992: Kathab and Jury, 1993) Those results arc useful in anaIyzing the stability robustness of a given feedback control system. However, there are relatively few efforts in extending those results from the analysis context to the design context. Among these
effons are the work of Chappellat and Bhanacharyya ( 1989), HallO! and Yong (19901, Barmish el al. (1990), Bernstein and Haddad (1992), and Chapellat el. 01. (1993). which give necessary and suffic ient conditions for robust stabilization of interval plants using either static or dynamic compensators. Tbese methods are concerned with the left-half plane stabilizatioll. Nonetheless, left· half plane stability does not guarante" proper system response. I! is more practical 10 consi:ler left-sector (relative) stabilization~ i.e. the entire family of interval plants will have only poles in a left-sectc'r defined bY adamping margin. The aim of this paper is to develop a method for constructing a robust controller that left-Sf'ctor stabilizes an interva1 plant. The controller is considerd in the general form of a first order compensator. This :'(Iml covers a variety of cvntrol actions such as P :'1, PD, lead. or lag compensators. Jt is proved he:dn that for an interval plant with a first o:..der compe'l';3tof, it is necessary and
3299
sufficient to left-sector stabilize only a finite number of specially chosen vertex plants in order to left-sector stabilize the entire family of interval plant. The number of these vertices depends on the degree of the plant or the angle of the damping margin that defines the left sector. It should be pointed out that this result is a generalization of the result reported in (Barmish et al.; 1992) for left-half plane stabilization.
,~
I..,
.,,'-........,
L1
Left Re
sector I...
2. PROBLEM FORMULATION
Fig. I. The left sector
In the standard closed loop feedback system, consider a strictly proper interval plant family !,(S) = N (S) I D(S)
polyoomial f(S) has only zeros within the left sector if and only if its polar plot encircles the origin n times (2mt phase advance) in the counter clockwise direction, as S traces the closed contour (L I'L ,L ) in a counter 2 3 clockwise direction. As S traces L , f(S) will undergo a 2 phase advance of 2n(" - ~). Hence and due to symmetry about the real axis, f(S) has only zeros within the left sector if and only if the pllaSe of f(S) increases monotonically through a net phase advance of n
(I)
srn S + .....+ "n sn
N(S) = bo + bl S + .... + b m
(2)
O(S) = 30 + al
(3)
The numerator and denominator polynomials coefficients are uncertain but bounded by lower and upper limits; that is b.E[!!);'] for j=O,I, ... ,m and a.EI!!)i.] for i=O,I, ... ,n. The] fanlil~ of polyoomials of the !'orm't2\ is called an interval polyoomial; and the family of plants (1) is called an interval plant since its coefficients vary in a specified interval. The case where some coefficients are perturbation free is handled by equating their upper and lower limits. Consider also a fix.ed first order compensator of the fonn
n
f( xei+) = L Pk eik~ x k (7) k=0 as x increases frorn 0 to 00. The following observations and lemmas provide a machinery t>r the proof of the main result.
where Cl' C , C , and C are nonnegative real constants. 2 3 4 The closed-loop system is called r-stable if for all possible values of b.; j=O,l,,, .,m and a.; i=O,I, ... ,n the resulting closed,loopharacteristic polyno~al 8 (S) = (Cl S+C2) N(S) + (C3S+C4) D(S)
Observation i: f(xei
(5)
Observation 2: f(xei+) ei6 for any real constant e, can be written as f(xej
has all its roots in a specified region r in the complex plane. If we consider r to be a left-sector, ther-stability means a minimum damping ratio margin.
Notice that if the phase off(xei+, is increasing through net phase advance of Rep as x. increases from 0 to 00 , then the
The problem is to determine the compensator C(S) that left-sector stabilizes the entire family of the interval plant I'(S). In this case, C(S) is called a robust left-sector stabilizer, and the system is robustly left-sector stable.
polar plot off(xei+) will interse't the real axis and the ei
3. PRELIMINARY RESULTS
Lemma i: Let g(xei+ ,fi)= f(xei O ) ei6+ fi(a x ei~ + P)v(x), where f(s)is a left-sector stable polyoomial, 6 is a real constant, a ~ p are nonnegati\'e real constants, v(x) is polynomial in x with nonnegative coefficients, and fiE[O,I] 1S a scalar, then g(xd+ ,fi) has interlacing zero
Let f(S) be an nth order polynomial of the form n
f(S) = POTPI S+ ... +p S (6) and consider the left-sector defined b;the angle <1>, where 1tI 2:0; ~ <" as illustrated by Fig. I. It is well known that the
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property for all l'e[O.I) if gexei+. 0) and g(xei'. I) have interlacing zero property.
proof in view of observation 2 'We have g(xei$ .1") = (R(x) + I'Jl vex»~ + (Q(x) + I"a xv(x» ei$ Rt(x) + QI(x)ei$ At fI = 0 the roots of RI(X) and QI(x)are Iho,e of R(x) and Q(x) respectively. Since v(x) is positive increasing function. then as '" increases from 0 to I. each individual root of either RICX) or QI(x) moves in one direction only. The roots ofRl(X) are detennined from RCx ) = - I" Jl v (x); and the gradient of their change with respect to I" i, given by Z,.(x) = dxldl" = 4dx1dR(x)XdRCx)l dfI) = -Jlv (x)fR' (x) where R'Cx) denotes dR(x)ldx . Similarly. the roots of QICX) are determined from Q(x) = - I"a xv(x); and the gradient of their change with respect to fI is given by Zq
=
We proceed by contradiction. Assume for some I'e (0.1) the roots of RI(x) and Ol(x) do not alternate. Since at fI = 0 and fI= 1 Rj(x) and Ql(x) have interlacing zero property. and the roots depend continuously on 1'. and each individual root moves only in one direction as ~ increases: then the following rentaIks are obvious: a) There exists at least two values of fI. say fll and 1'2. and two value, of x. say x I and X2. at which R(x I) = -fll llV(xll. Q(xiJ = -l'laxlV(xll. R(x2) =-fl2 llv(x2). and Q(X2)= -1'2'" x2v(x2)· b)For xelx I. x2) either R' (x) and O' (x) are both negative or both positive. cl Ifz(xl » 1 then 7.(x2)<1; el,e ifz(xj)<1 then z(x2) > I. Or in other words. by using the definition of 7.(x) and remark Ca) if (Q'Cxl)/(R' (xI») > (Q(xl)IRCxl» Ihen (Q'ex2) I (R' (x2)) < (Q(x2)lR(x2)) else if (Q'(xl)/(R' (XI»< (Q(x 1)IR(x 1» then (Q'(x2) I (R' (x2)) > (Q(x2)IR(x2))'
Consider now an interval polynomial oflhe form (J). It has been shown in (Foo and Sob. 1mb. 1992) that the interval polynomial will have only roots in a left-sector if and only if a finite number of spedally chosen venex polynomials have only roots in "he left-sector. The vertex polynomial, are given in (Foo ",1d Soh. I 990b. 1992) for special cases of the damping margins (angle ~). However. we introduce here the following lemma which provide procedure 10 identify Ihe ven"x polynomials. for any general value of $. First let uS define +I< = k+ - 2i,,; where i is selected such that 0$+1«2" for k=O.I ..... n: alld +k+n+l=k±lt such that OS+k+n+ <2n for k=O.I .... . n. Lel also 60- 61 .... .. 6m; m !>2n+l 10 be the values of the angles +1<. 0,;kSln+ I arranged in ascending order; i.e. ()=90<91 <......6 m<21<. We note thal m=2n+ I if the values oflhe angles fk are distinct for all k; and m<2n+ I if fkl = +k2 for some inlegers kl and k2. For exarn~·le if , =2"'J. lhen m=S regardless or the polynomial orde r. In this case 00 = ~O = J = ..... = O. 91 = o+J = ~n+6 = .... = "'3 92 = ~l = <1>4 = ..... = 2"'3. 93 = <1>0+ I = $n+4 = .... = n. 94 = h = <1>5 = ..... = 4"'. 65 = '~+2 =
no = {90. 6\. °2 ......... Bq}
01 = {Ol .02. 6 J .. ....... 6q+1}
11m = {9 m • 90. 9 1 ......... 6q.1}
(8)
lemma 2: The roots oflhe inte"a1 polynomial D(S) oflhe form (3) will lie only in a left.·sector defined by if and only if Ihe m+ I vertex polynomials Or CS): r = O.I ..... m have only roots in that left-sector. where
+
Or (5) = OCS): "i = 3; if +; E llr and a; =..l!i otherwise (9) for all i = O.I .....n; and r = O.I ..... m proof The necessity pan is ob> ious since Ihe m+ I venex polynomial. are members oflhe family of polynomials (3). We now prove sufficiency by contradiction. Assume that O,.(S); r=O,l ..... m have only roots in the left-sector. Assume also that D(S) has root" outside the left-sector for some values of a.~ i=O.l ....•n. Hence, the continuous root dependence on th~ coefficients of O(S) dictates that Ihere exists some values of a; =
Notice that R(xl). Q('I). R(x2). and Q(x2) are all negative. Since f(,) is left- sector stable then the polar plot of f(xi~) ei9 = R(x) + Q(x) ei
for this case. the polar plot ofD·Cxei') will pass through the origin as illustrat by fig. 2. The tangent to the plot al the origin separates the complex plane into two regions; say region-l and region-H. From (8) it can be readily 'hown that one can always seh:ct one sel ~ that has all its members contained in region-L
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I= ReQ.i. on
"-
is left-sector stable for a1ll1e[O,I] if and only ifH(5,0) and H(S, I) are left-sector stable.
I
"-
Proof Necessity is obvious. To prove sufficiency. we start by rewriting H(S, 11) as
""""-
Regi.on
H(S,Il) = HI (5) + 11(<<5 +
"IX
Re
"-
n Dp (S) = D*(S)+d +d S+ ... +dnS O l Dp(xei
(10) (11 )
H(xei.,I1)= HI(xei.) + 11 (0 xei. + Il) v(x)clcr
g(xei+,JI)= H(xei+. JJ)e-i"= =H I(xeit) dcr+ !l(,.xei++ll) v(x)
'1>
D'(xei~) lies in region-I. Dp(xei~) will also lie in region-I.
Thus Dp(xeiq,) will not increase monotonicaly through a net phase advance of nq, by sweepi ng x from 0 to 00. That is to say there exists one vertex polynomial DP
It is worth noting that Kharitonov theorem (Kharitonov. 1979) IS a special case of the above lemma for the case q,= 7tl2. In this case. m = 3. Q o= {O. 7tl2}, QI = {7tl2. ltl, Q2 =1". 37t12 ). and Q3 = (37t12. 0 ). It can be also shown t hat if the above lemma is specialized to the cases =27t13 and + =37t14 we get the same results as in (1'00 and Soh, 199Ob. 1992). ObservatIOn 3: For each x e[O. 00). O(xei+) is bounded by a convex polygon in the complex plane with vertices Drlxei~): r=O.I .. ... m. The edges of the polygon are of fixed orientation for all x and they are at angles 9 0 , 91 •.... ,8 01 .
+
lemma 3: Let Ho(S) be a polynomial, et, Il are real constants. and
HI(S ) = Ho(S) + CctS+1l )Dr(S)
(12)
H2CS ) = HoCS) + (as+Il)Dr+I(S)
(13)
where DrlS) and Dr+ I(S) are two successive vertex polynomials of 0(5). Then the polynomial H(S,Il) = ( 1- Il)H I(S) + !iH2(5 )
(14)
(16)
where v(x) is a polynomial in x with nonnegativ. coefficients and crE{8",9 1• ....• Srn}. We proceed by contradiction. We assume that for so:ne IlE(O,J). H(S,Il) is not left-sector stable. Then continuous root dependence on ~ necessitates that there exists some ~*and x· such that H(x·eif, ~. )=0. Let
or From (9) it can be seen that dt<>O if +k e equivalently if the vector e)kxk lies in region-l. SiIrularly, dk
(IS)
Thus, in view of observation 3 an,1 the definition (9)
"fig. 2 Polar plot ofD(S) with roots on the sector boundries
Note also that we can Mite Dp (S) and Dp(xei+) as
Jl:'[Dr+ I (S) - D"
Since H(S,O) and H(S, I) are left-sector stable,
(1 7) then
g(xei>!>.O) and g(xeiq., I) have interlacing zero property.
x'
Since H(x' ei>!>, ,,')=0 then g( ei+, ~ *) = 0 which is not possible in view of lemma I. Thi, contradiction completes the proof
4. MAIN RESULTS Theorem 1: A first order compen,;ator C(S) in the form (4) robustly left·sector stabilizes the entire family of the interval plant (I) if and only iLt left-sectorstabilizes the following vertex plants
Pqr (S) = Nq(S)I D"
cFl.1 .. ·..ml
• r=O.I •... ,mZ (18)
where Nq(S): q=O.I .... ,UlI andD"
5q"
( 19)
are all contained in a left-sector defined by damping margin 4>. Let N*(S) denote th" polynomial N(S) with arbitrary but fixed coefficients which are taken from the bounding ,et. Thus. in view 0:.; observation 3. for each x e[O. oO)
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follows that S",,(S}=(CI S+C2)N+(S)+(C3S+C4)DO(S) is left-sector stable. is bounded by a convex polygon in the compl.. plane with vertices
The application of theorem I requires a compotationaly intensive search for a compensator that left sector stabilizes m I m2 vertex plants. However, with the aid of the following remarks, a computrtionaly feasible design procedure may be invoked.
Or*(xei
Remark 1: Although theorem I considers, in principle, a first order compensator C(S) that involves four variable parameters, yet for practical applications, C(S) involves no more than two variable parar""ters. For example, for PI controller CI=K I, C2=K2' C3~~ 1 and C4= 0; for PO controller CI=KI' C2=K2' ('3=0 and C4=1: for lag compensator CI=KI' C2=KIK2, C3=1 and C4=aK2; where et >5 is a preselected constant; while for lead compensator CI=KI' C2=K I K2 , C3=1 and C4=K2/et. Thus what is only required is gridding two parameters K 1>0 and K2>O and evaJuatin;~ the root locations for the vertex polynomials for each grid point.
r *(xei
Remark 2: The effort associated with identifying the root locations of an nth order polynomial may be reduced tu checking only the Hurwitz stability (using the classical Routh criterion) for a 2nth order polynomial via tile following theorem.
we have (\-I1)o/(xei
To this end, we get the following fact Fact I: For arbitrary but fixed coefficient polynontial N*(S), if the vertex polynontials or *(S); FO,l, ... ,m2 are left-sector stable, then the interval polynomial O"(S) is left-sector stable. Defining 80 (x
ei
Theorem 2: Let T(S) = t o+ tl S+ .. + InS n : and ~ where Q(Sl=CJo+qIS+. +Q2n S2n i qi = L tr ti-r cos ((i- 2rX.-,,}2)); i=O,I, ... ,2n (24) r=O
(Clxei+C2)N(xei+) + (C3xei
then T(S) has only roots in the left-sector defined by ~ if and only ifQ(S) has only roots ir, the left-half plane.
Oq 0 (x eir={Clxei+C 2 )N q(xei
Proof: Let T I (S)=to+t 1ei(+-"'2) S+ ... +lnei nC.-"'2) sn and T2(S)=to+tle -j(-"'2) S+ ... +tn" -j(.-"'2) Sn Notice that the roots of TI(S) and T2(S) are those of T(S) rotated by angle (.-"'2) in the c\ockw.se and anticlockwise directions, respectively. Utili:,;ng the fundamental
(23)
and using a similar line of proof, we get the following fact Fact 2: For arbitrary but fixed coefficient polynomial OO(S), if the vertex polynomials Oq ,,(S); q=O: 1,... ,m are L left-sector stable, then the interval polynomIal 80 (~) IS left-sector stable.
identities: ei a =cosa + jsina, and cos(a - 11 }=cosa cosp + sinasin p: while noting that t·= 0 for i > n, it can be shown that I . Q(S) = TI(S) T2(S) Thus the roots of the real polynomIal Q(S) are the union of the rcots of the two complex polynomials T I (5) and T 2(S) ,md should form conjugate pairs. Suppose that 2r conjugate roots ofT(S) are lying outside the left sector. Then by rotating the roots ofT(S) by angle (<1>-"'2) in the clockwise direction, exactly r roots will be located to the right of the imaginary axis. Thus, T 1(S) will have exactly r right-half side roots. Sintilarly, it is clear that T 2(S) will have exactly r right-half side roots .
The proof of the theorem is completed by combining facts I and 2. Let N"CS) and O,,(S) be any polynontials with arbitrary but fixed coefficients which are taken from their bounding sets. Since the polynomials 0qr(S); q=O,I, ... ,m I, FO,I, ... ,m2 are left-sector stable, then applying fact I, m I times while replacing N*(S) by Nq(S); q=O,I, ... ,mb' it follows that Oq O(Sp (CIS+C2)NqCS)+(C3S+C4)D (S), q=O, I, ... ,m I are left-sector stable. Then applying fact 2, it
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Consequently, Q(S) will have exactly 2r right-half side conjugate roots .The proof is then clear by taking _
6. CONCLUSION
5. EXAMPLE
Necessary and sufficient conditions have been derived for robust stabilization of interval plants, under the restriction of damping margins, using first order compensator. Based on these conditions a procedure for constructing the robust compensator is presented. Extensions to more general
Consider the following interval plant family
classes of compensators is an open research area
with the following parameter uncertainties: bl E[40,50], bo E[90,11O], a3E[14,16], a2E[85,95), al E [130,150], ao E[8,14). It is required to find a robust PI compensator of the form C(S)=XI+K2/S that left-sector stabilizes the entire family of the interval plant P(S) for the following three cases'
case I: oF 7tl2 (damping margin9l: i.e. Hurwitz stability) case 2: oF 27t13 (damping margrn9l.5) case 3: oF 37t14 (damping margin9l.7) KI and K2 are grided between 0 and 3 with a 0.1 grid 5tep. In view of theorems I and 2, for each grid point, only 16 routh tables are checked for case I: 24 for case 2; and 32 for case 3. Fig.3 shows the set of stabilizing parameters for the three cases. If it is required to stabilize the interval plant with a minimum damping ratio of 0.7. then any member of the set of stabilizing parameters for case 3 may be selected~ e.g. KI=I and K2= 0.2
. _ .... -_ ..... -_ .... . . . . . . . - - .... - ..... · ........... - . - .... . · ..... - - .... . ...... - ...... - - ..... - . - .. . . - - ........ - .... - - .. . . . - - .. - ... - ..... - - - - . .. .... - - .. - - ........ . . . -- ........ - - - ....... . - . - .. - - .......... - - - .. . - . - . . - ........... - .... . . - - - ....... - ... - - - .. - .. . - ....... - - ...... - ...... -
3
2
• - . . . . . . . . . . . . . . . -++ . . - • _. - - _. _ • . . - + + + + _ . - -
K2
• . . • . • • _ • . . • • • . +++++-1"' • . .
. . . . . - • • . . . • • ++++++++ . . . . . - . . • • - . . . _.++++++++++_._. • . . . . . . . _ . . +++++++++++_ . . . . . . . . • • . . . -++++++++++++- . _ . . . _ . . • - . • -+++++++++++++- . . • . . . • . . . . -++++++++++++++- . . • _ . . . . . . -++++++++++++++++_ . . . . . . . . . . +++++++++++++++++ . . • . . . . . . . +++-++++++++++++++ . . . . . . . • . +++_ • • +++++++++++++ . . . .
o
~
easel
+
ca.se2
..
Fig.3 .Set of PI parameters for robust stabilization
ea,se3
REFERENCES
Barmish B.R.,Hollol CV.,Krans F.J and Tempo R. (1992). Extreme point results for robust stabilization of interval plants with first order compensators, IEEE Trans. Automat. Contr., vol.37, 707-714. Bernstein D.S. and Haddad \V.M. (1992). Robust controller synthesis nsing Kharitonovs theorem, IEEE Trans. A utomat. Cono·.,vol.37, 129-132. Chapellat H. and Bhattacharyya S.B. (1989). A generalization of Kharitonovs theorem: Robust stability of interval plants, IEEE Trans. Automat. Contr., vol,34, 306-311 Chapellat H., Dahleh M. and Bhattacharyya S.P. (199:;). Robust stability manifolds for multilinear inte,,'a! systemsJEEE Trans AutomdContr.,voI38, 314-:1 '! Foo YK and Soh Y.C (1989). Rllot clustering ofinter,'a] polynomials in the left sector, Syst. Contr. Lell.. vol.!3,239-245. Foo YK.and .Soh Y.C (1990a). Damping margins of interval polynomials, IEEE Trans. Automat. Contr.. voI.35,417-479 . Foo Y.K. and Soh Y.C.(1990b). Generalization of strong Kharitonov theorems to the left sector, IEEE Trans. Automat. Contr.. yo1.35. 137S-1382 . Foo YK and Soh YC (1992). Strong Kharitonov theorems for low-order po:ynomials. IEEE Trans. Automat. Contr.• vol.37,1816-1820 . Hollot C V. and Yang F. (1990:,. Robust stabilization of interval plants using lead or lag compensators, Syst. Contr. Lell., Yol,14, 9-12 Katbab A. and JUlY E.!. (1993) A note on two methods related to stability robnstnes:; of polynomials in a sector (relative stability), IEEE Trans.Automat . Contr.• vol,38,380-383 . Kharitonov V. L. (1979). Asynptotic stability of an equilibrium position of a family of systems of linear differential equations, Differ(~ntial Equations, vo1.14~ 1483-1485. Soh CB. and Berger CS (1988,. Damping margins of polynomials with perturbE,d coefficients: IEEE Trans. Automat. Contr.,vol.~;3, 509~511.
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