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Application of the Finite Inclusions Theorem to Robust Multivariable Design
Copyright © 1996 IFAC 13th Triennia! World Congress, San Francisco, USA
2d-08 1
APPLICATION OF THE FINITE INCLUSIONS THEOREM TO ROBUST MULTIVARIABLE DESIGN Theodore E. Djaferis
Department of El
1
INTRODUCTION
tput (SISO) systems with pa.rameter uncertainty (see Djaferi. 1995). The techniques are based on the Nyquist Theorem and exploit properties of polynomials. A fundamental result, the Finite Inclusions Theorem (FIT), has been proposed as the vehicle for robust control analysis and synthesis. FIT addresses the V-stability of a polynomial family where 1> is a convex set in the complex plane and the interior of sc me Jordan curve r. This new test guarantees robust 1J-~tahility for a general polynomial family, if the polynomial value set at a finite number of points OD r lies in an appropriately chosen set of sectors in the complex plane. The number of such points is not large and can be as small &8 2n + 1, where n is the degree of the polynomial family. In the case of affine polynomial uncertainty l?IT provides a nece"..ary and sufficient condition for robust V-stability and for multiaffine uncertainty (using the Mapping Theorem) a sufficient condition. One can immediately take FIT and use it as the foundation for robust 'D-stabilization algorithms. These are iterative p:rocedures where at each iteration a finite set of linear inequalities in the controller parameters needs to be soIVI!d. Furthermore, one can suggest robust performance synthesis algorithms (e.g. robust asymptotic tracking, robust noise attenuation)
The problem of multivariable control for Linear Time Invariant (LTI) systems has received a great deal of attention in the control literature of the last thirty years (books in the area include Callier 1982, Francis 1986, Green and Limebeer 1995, Safonov 1980, Vidyasagar 1985). Justification for this effort comes from the fact that a multitude of dynamic systems have to be mer delled by descriptions that include several inputs and several outputs. This necessitates the development of analysis and synthesis iools for the investigation or multi input multi output (MIMO) systems. The experience with linear, time invariant, finite dimensional dynamical systems has shown that success in this endeavor depends on several faclon. High on the list are: 1) the development of analysis results that lead to synthesis techniques, 2) the consideration of robustness issues, and 3) when these techniques are inspired by SISO methods they must not place stringent requirements on the structure ofthe controller (e.g. diagonality). Perfecl examples of successful multivariable conbol methods are H~, p and LQG/LTR. Over the last five years powerful new robust control methods have been developed for single input single ou-
3270
ofthe Finite Inclusion. TheoreIll when V iz the len half complex plane. However, it is important 10 .tate this VtUiOD as in Section 4 it will facilitate the construction multivariable pedormance synthesis algorithms.
well as robult multiobjective performance synthesis algorithms. This paper shows how the Finite Inclusions Theorem does provide a framework. for multivariable design for systems with parameter uncertainty. It attacu the synthesu problem from a new direction and complements existing methodologies. After recalling the Finite Indusions Theorem in Section 2, a robust D-stabili.ation algorithm :is presented in Section 3. In Section 4 synthesis algorithm. are suggested for robust asymptopic tracking and noise attenuation. These are extensions of the SISO FIT .ynth..i. methodology where no reIItricive uaumptioM are placed on the .tructure of the controller. aB
Theorem 2 (Sim.llaneolU St4bilit, Finite Incl"';",.. Theorem, SSFIT) Let 'jI,(0, a) = Ei~o 0<,; (a)'; , a En., 1 ~ i ~ t, no ~ 0, ll,,:0. -+ C, Qi ... ,;:: Om. - R+ , Then, lor till a E 0 110 , 91,(0, a) ha.e all th.ir .ero. in Ihe ope" left MII plane il the ... e.ut mo ~ 1 intervals (eo,,, do.) c R, 1 ~ k ~ mo, I ~ i :5 l and real number. ""il :5 "',2 ~ ... :S LIlt"" IVC"
til.4t ""fli
2
V 1< i <, l -2 - .. < - "',
THE FINITE INCLUSION THEOREM
•
V I~,
Consider a polynomial family 'jI(.,a) with parameter uncertainty. The Finite Inclusions Theorem can be used In investigate robust V-stability. Specilically, let
'jI(., a)
V 1~ k
< mo,
t,
< mo, ., s. = {re' Ir > 0, B E V 1~ k
("'t,do.)}
Since the.. results address the stability of polynomial families with parameter uncertainty they can be applied
+ ... + 0<.(&)91.(_)
to SISO as well as MIMO sYlteml. Their application to di8CU5~ed fust.
robust 'D-stabilization is
at
ROBUST V-STABILIZATION 3
Consider a MIMO system with transfer function matrix P(.,a) described in a left polynomial matrix fraction reprelentation (Callier 1982):
..ch that int r u con.ez. Then for alia En., 'jI(., a) u of d.g .... n and lull all it.. root.. in int r if the ... ezUt.. m ~ 1 intervals (Ct, dt) C R and a countercloclc-wUe .equence 0/ painu 't Er, 1 ~ le ::; rn, .uclt. that
(1) where
N.,(_,a)
< m maz{d.+, -
< t,
91,(;""., n.) c
=
V 1 ~ l:
JI"",
1r71j
max{doHl - eol, do. - "'tH} ~ ..
where the ,p.(6), 0 :S i ~ u are given polynomials. Suppose that the parameter a takes values in the hypercube n. {a E R it la; ~ at ~ I I ~ i :5 k}, a; < 0, > 0, I :5 i ~ /e. The set R denotes the leal numbers. Further assume that Qi(a), I ~ i ~ u are polynomic in a and luch that ",(0) = o. Fo[ such a family the following result (see Kaminsky and Djaferis 1993, and Djaferis 1995) holds: Theorem 1 (The Finite Inclu.oiono Theorem, FIT) Let 91(0, a) Ei=o o<;(a)'; , a E n. n ~ 0, and Q; : n. - c. Fvrther, let r c C be a do.ed Jordan curve
=
...
2-r:sci1Jl,;:
V 1~i ~
= 'jIo(.) + O
at
r",
< do, < -2 - 2 +
Ct, dt
-
CtH} ~ ..
Dp.(O,Il)
maz{d". - (Cl + Zm), (d, + Z..n) - c".} ~ .. VI ~ k ~ m9l(0.,n.)c S. = {re' Ir > O, BE (c. , d.)}
••
= =
diag(,··-').Vph.(Il) + N,... (.,a) diag( ...· )D••, + D.... (o,a) (2)
The plant family denominator matrix D,.(o, a) is an m x m polynomial matrix in $, with row degreel J.4i and where Pl ~ P2 ~ .. . ~ J.'.m are positive integers, J.'i == n.. The highest row coefficient matrix Dphc is a constant invertible matrix which does not involve the parameters. The polynomuJ matrix DplOC(II, a) contains the lower order terms in " and its entries have
The Finite Inclusions Theorem can also provide a characterization of robust (Hurwih) .tability for a finite number of polynomial families t;6.(8, a), I ~ i::; i, each oflhem as in (1). III th. context ofanalysis the following result, called the Simultaneous Stability Finite Inclusions Theorem (SS FIT) is in .... nce a [estatement
:>::::1
3271
coefficienll that are polynomic in a
En..
For this plant family the polynomial .p(., a) has coefficients that in general are polynomic in a and multiaffi... in X. Therefore, the problem of V-.tabilisation of multivariable systems with parameter uncertainty is expr....d in lerms of the V-stabilisation of a single polynomial family. It is evidem Ihat lhis problem can be attacked using the Finite Inclusions Theorem in a manner .inillar to that presented in Djaleris 1995 for SISO systems. Firsl assume thal by introducing additional uncertain parameters (and consequently ovcrhcr unding), the polynomial coefficienls are made mw/tia/fine in a. This allows one to work with value set verticel as a consequence of the Mapping Theorem (Zadeh and Desoer 1963) and leads to computationally efficient synthelia algorithms.
The plant
nnmerator matrix Np' (6, a) is an m x l polynomial matrix in ., with row degrees P-i - I, 1 ~ i ~ m. The highest column coefficient matrix NphiC(a) may include parameter uncertainty and the polynomial matrix Np lo.l{6,a) contain. lower order terms in.s, which have coefficients that are polynomic in a. Assume that the representation is left coprime for all values of a EO". f'a.mi)y
By construction the plant f&miIy is strictly proper. Let the plant family be in the standard unity feedback configuration of Figure 1:
~L_e
______ c_(._)________P_(_')__
~~
The basic idea behind one robust V-stabililation algorithm is the following: Pick an initial controller X(I), and an appropriate set of points and sectors that satisfy the conditions of the Finite Inclusions Theorem for oil) COli.' Expand the uncertainty range and solve for a new controller that fits the new value sets in the sectors. Center the value sets (i.e. pick new frequencies) and repeat the pl'ocess. The only "structural" difference with the SISO case is Ihe fact that the coefficients of a general ~(8', a) depend on controller coefficients in a. multiaffine manner rather than in an affine manner. Consequently, the inequalities that need to be solved in a FIT bas«:d procedure ate muUiaffine in the controller parameters. The SISO FIT based 'V-stabilization algorithm must therefore be modified to address this issue. However, it is important to point out Ihat if all bu lone (.ay the ,"), of the colnmns of X take specific values then the coefficients of .p(., a) become 4JJine in the remaining ,:ontroller parameters Xi and 10 do the corresponding inequalities. A natural suggestion for solving the required: inequalities would then be to go through an additional iteration where all but one of the columns of X are lept fixed. A FIT based algorithm for multivanable robust V-stabilisation can be structured all follows:
Figure 1: Feedback System with Tracking and Noi.e Requirements
The controller is in the right matrix fraction representation:
where
N«(.) D«(.)
=
1
X'pH'" + X2.... - + ... X.+1 I ... + X p ... - l + ... Xl (3)
The matrix D~(.) is m x m and N~(.) is l x m. The controller is proper I order q = mp and the column degrees of D~(.) and N~(.) are all taken to be equal to p for notational convenience. The (pm+ (p+ l)l) x m matrix
FIT m Synthelis Algorithm STEP 1 Let X(l) E Rd and n~')
c n.
be such Ihat
.p(., n~l), X(l») is V-stable and set j:=1. STEP 2: Determine m(j) 2: 1 sectors S~), 1 :s; 10 :s;
contains the d = m(pm + (p + 1)l) controller parameters. Let Xi,l :5 i ::; m be the itA. column of X. The closed loop chaIa.cteristic polynomial has degree n = n + q and is given by:
mU), and points .~) along the boundary of V such thal
sf).
.p(.~), Eztrll/) , X(;)) c By FIT, .p(., nl/), X(;») is "D-stable. Each .~) should roughly center (angularly) the sel .p( .~), Eztrll/l, X(;») in ~il. STEP 3: Choose a slightly larger .et nl/+1) ::> nl/). STEP 4: Compute a new vector of controller parame-
3272
D.,
ters X<;+') such that 9I(0~), E"t 0~+1), X<;+')) C S~) for all k. This is equivalent to Bolving a system of multiafline inequalities in X(;+1). Fix all columns ofX(;+') hut one at cnrrent values. Solve the corresponding set of linear inequalities. Iterate over the choice of column. If no solutions are found to this system of inequalities, returD to Step 3 and choose a smaller O~+l). STEP 5: Let i := i + 1, and if O~) ::> 0., stop; otherwise, go to Step 2.
with N.,(o,a) and (0, a) as in Section 3. The plant family is in the uuity feed back configuration of Fignre 1. The controller is described in the right matrix fraction representation:
with N~(.) and D~(o) as in Section 3. The performance specification in addition to robust stability is robust asymptotic tracking or noise attenuation. The sensitivity transfer function malt;;. S(o,a) (0;;(0, a)) is given by:
=
If this algorithm terminates with O~) ::> O. then a multivariable controller would have been constructed that guarantees V-siabWation. No simplifying assumptions about the structure of the conboller have been made (like diagonality), that would restrict its applicability. However, this is an iterative algorithm which may not be able to find a robustly V-stabililing controller if one exists. Furthermore, the results will depend on the initial controller, and it is therefore prudent to use available information and insight to guide the choice. One can certainly employ the H~ methodology, LQG/LTR,
S(o,a) =
= (I + Pto, .)C(o))-'
=D~(o)(DpI(o, a)D~(o) + N.,(o, a)N~(.))-' DpI(o, a) =D~(o)~(o,at'D.,(o,a) where 91(0,.) = det~(o,a) is the closed loop characteristic polynomial. This polynomial family must have roots in the left half complex plane to guarantee robust stability. Each entry 0;;(0, a) cf S(o,a) can be thought of as a rational function with denominator ;(5, a). At this point there are several options for imposing the robust asymptotic tracking requirement. One ill to place a bound on the largest singular value of the weighted sensitivity. In other words suggest appropriate transfer function matrices W ll (o), W 12 (o) such that:
Pole Placement or any other multivariable design technique for this purpose. It is shown how this can he successfully done in the SISO case (see Djaferis 1995) and this is expected to be true for multivariable systems.
4
ROBUST PERFORMANCE SYNTHESIS
u(Wll (iw)S(iw, a)W12 (iw)) < 1 V a E
[1.
and w E [0,(0)
Another option is to place a requirement on each entry oUhe sensitivity transfer function mahix (&ee Horowib 1979, Van Diggelen and Glovel 1991):
Robust performance synthesis for multivariable systems has been at center stage over the last two decades. Very successful techniques for robust multivariable design such as H ~, LQG /LTR and p synthesis have been established for which MATLAB software is currently readily available. In addition to these methodologies, others (Horowih: 1979 and Machiejowski 1989) have been proposed which are generalizations of SISO design techniques to muItivariable systems. This Section demonstrates that for systems with parameter uncertainty FIT synthesis does provide a viable alternative method for robust multivariable system design. It can he seen to be a very natural extension of the design techniques for SISO systems. The focus is on the robust design for asymptotic hacking and noise attenuation, but it will be dear that the methods can be used so solve a wider class of controller synthesis problems. Consider a multivariable plant family Pto, a) with parameter uncertainty a E na which is described in the left matrix fraction representation:
IWH;(iw)o;;(iw, a))1 < 1 V w E [0, 00), a E O. and all i, j where WH;(O) is the (i, i)" dement of some matrix W.(o) which is strictly proper and has stable and 00nimum phase entries. Since th,e plant transfer function matrix is m x t there will be n12 such conditions. The complementary sensitivity transfer function matrix T(o,a) = (t;;(o,a)) is giyen by:
T(o, a)
=I
-
8(0, a)
It. diagonal entries aret;;(o,.) = 1 - 0;;(0,.) and the offdlagonalentriesaretij(.s,a) = -"ij(6,a). They too can be thought of as rational fundions with denominator ~(",a). In a similar manm:r the robust asymptotic noise attenuation requirement ,:an either be imposed as
(5)
3273
a bound on the largest singular value of the weighted complementary sensitivity transfer function matriJ: or elementwise:
all I < 1 V wE [0,(0), a E
I....;(jw)t;j(jw,
where
'W2ij(S-)
(la
aooalli,j
n",;;(.)n,;j(jw, a) d.,;;( ')1'1(_, a)
I
I<
1
~'
w E [0,00).
a
E
(la
(6)
This i. equivalent to:
is the (i,j)th. element of some matrix
W,(.) which is proper and ha. .table and minimum phase entries. There will be m 2 such conditions. One can immediately see that the weight. W , (.) and W.(.) cannot be chosen independently because the diagonal entries 6ii(jW, a) and 1 - 6&i(jw, a) are very related. In particular, both cannot be made "small" at the same frequency. Fortunately, in many cases tracking and noise attenuation requirements are placed at sufficiently disjoint frequency ranges. Lemma 1 The e"tM" of S(" a) and T(., a) are rational fundion. in , who6e numerator and denominator polynomial. are multiaffine ezpreuioru of tAe controller pam.meter•. Proof: It is known that S(" a) D""(,)~(,, a)-1 D.I(" a). Write ~(.,a)-1 = ·'~~~~i" where atij(.) denot.. adjoint of a matrix. This implies that:
e(J'w) n",;;(jw)n,;;(jw, a) I'~ 1 V [ ) (l d.li;(jW)I'I(jw, a) w E 0,00 , a E • (7)
l
where 0(.) is any stable proper transfer function with 19(jw)1 :'5: 1 for all w E [0, (0). In other words, for no plant in the family and no frequency can one have:
rei°
°
n"li;(jw)n,;,(jw,a) =-1 d.lij(jW)I'I(iw, a)
(8)
wh.r. :'5: r :'5: 1 and 0 :'5: a < 2... This is equivalent to saying (sine. 4li;(') and 1'1('" a) are stable) that the polynomial:
=
S(" a)
D,,"(.)adj(~(.,
allD,1 (" a) 1'1(" a)
a)) (",,;(., 1'1(" a) = I _ S(
) _
',a -
should not be zero for any 0 ~; r :5 1, Q E [0,27), • E (l., wE [0, (0). Clearly if the polynomial:
'I/J.ij(.,a,r,Ct)
(""i(.,a)) 1'1("a)
The elements of the ith column of
o Consider now anyone of the m 2 conditions for robust tracking. If Wli,·(.) n._.. u,1i ~ •.)) it can be equivalently stated as:
3274
=
rei°n,.,li; (_ )n,;;(., a) + 4li;( .)1'1(" a)
is stable for all Cl< E [0,2,,), r E [0,1] and a E {l., then the robust performance rt~quirement corresponding to the (i, j)th sensitivity tram fer function element will have been met. One can express the performance requirements for all the other sensitivity transfer function element! in a similar manner. The robust tracking requirement can therefore be expressed as a simultaneous polynomial family IItabilisation requirement. In t"..J:actly the same manner one can express the robust noise attenuation requirement as a simultaneous polynumial family stabilization problem where polynomial fanilies are of the form:
1/1,ij(".' r, et) = rei°n,.,2i;( ')'''';('' a) + 4,,;(.)1'1(., a) In order to d.velop an efficient SSFIT b ....d performance synthesis algorithm there is a need to overbound the disc uncertainty rei a a5scciated with performance by some polygonal overbound l.nd in the following algorithm the square overbound rib. is chosen. Now, each such polynomial family will in general have coefficients that are polynomic in the pla.nt uncertain parameters a and multiaffine in the cont.roller parameters X. In order to employ the FIT based algorithms for simultaneous stabilization developed for SISO systems, one
must do two things: Firstly, by introducing additional uncertain parameters one makes the plant uncertain parameter dependence mv.ltiaffi~ whenever it is required. Specifically, & term like 0 2 can be replaced by the poduct a1 a2. This of course will be a source of conservatism in the design. Secondly, one modifies the emting FIT algorithms, just as was done for multivanable robust stabilisation, to deal with the multiaffine controller parameter dependence. One easy solution is to inhoduce an iteration when solving the multiaffine inequalities, where controller pa.rameters are kept fixed for all but one column of x. The following algorithm is suggested where for notational convenience polynomial families are relabeled ,p'i;(6, a,r,a),'I.j1tij(6,8,r,a)
as
4>.(6, all:
I. Let Xl') E Rd and o~~) CO •• be such
that 4>.(., O~~), Xl'») are stable and set j:=l. STEP 2. Determine 2: 1 sectors
mY)
sn),
1 :s k :s ml;), 1 :s i :s l and frequencies along the jw axis such that 4>.(jwl~),Ezto?),XU») C By FIT, 4>.(., O?~, XU») are stable. Each wl!) should roughly center (angularly) the set 4>.(jwl~), E"'tO?~, X U») in U) . 5 i1 STEP s. Choose a slightly larger set O?+l) ::) O~. First this should affect the b parameters. When (if)
wW
sW
(}~) :J Ob", the enlargement in the b-diredion terminates and the enlargement in the a-direction commences. STEP 4: Compute a new vector of controller parameters XIH ') such that 'Pc ~·(J·wl;) Ezt OIH') XU+t») C "~I 06 J
S~') for all le and i. This is equivalent to solving a system of multiaffine inequalities in X(.i+l). Fix. all C~ lumns of X(i+l) but one at current values. Solve the corresponding set of linear inequalities. Iterate over the choice of column. If no solutions are found to this system of inequalities, return to Step 3 and choose a smaller O~+'). STEP 5. Let j := j otherwise, go to Step 2.
CONCI~USION
A powerful framework for rob"st control design of multivariable systems with parameter uncertainty hu heen presented. The approach is based on the Finite Inclusions Theorem and leads to iterative algorithms for design
in terms of IOlutions of linear inequalities in conholler parameters. The methods complement emtmg tools for multivariable design. Even though it would be greatly desirable, efficient software algorithms that implement the algorithms are not available. OUI hope is that this will change in the near future.
REFERENCES
PSSFITm Synthesis Algorithm
STEP
5
+ 1,
and if O?~ ::) 0 •• , stop;
It .hould be clear that this algorithm can be employed for both single objective and multiobjedive rohust performance synthesis. This formulation and algorithms can also be used to solve a number of other robust multivariable synthesis problems. One such is the problem of Robu6t Decoupling.
3275
Callier F.M. and Desoer C.A. (1982). Multi.ambk Feedback S1ldem6, SpringerM Verlag, New York. Djaferis T.E. (1995). Ro/nut Control D..ign: A Polynomial Approach, Kluwer Academic Publishers. Francis B.A. (1986). A Cou", in H~ Control Theory, Springer-VerIag. Green M. and Lirnebeer D.J.N. (1995). Linear Rob...t ContTO~ Prentice Hall. Horowih 1. M. (1979). Quantitative Synthesis of uncertain Multiple Input-Output Feedback Systems, Int. Jour. ContTO~ Vol. 30, pp. 81-106. Kaminsky R.D. and Djaferis T.E. (1993). The Finite Inclusions Theorem, Proceeding6 IEEE ODC, San Antoruo TX, 1993, pp. 508.518, (modified ve"ion IEEE 7"",•. on AC, Vol. 40, No. 3, 1995, pp. 549551). Machiejowslri J .M. (1989). Multi.ariable Feedback Design, Addison· Wesley. Van Diggelen F. and Glover K. (1991). Element-byElement Weighted Hoo-Frobenius and H2 Norm Problems, Proceeding$ IEEl? ODC, Brighton, England, pp. 923-924. Safonov M.G. (1980). Stability and RobU6tn... of Multivariable Feedback SY$te'IfU, MIT Press. Vidyasagar M. (1985). Sy.tem Control Synth ..is: A Factorization Approach, MIT Press. Zadeh L.A. and Desoer C.A. (1963) Linear Syltem Theory, McGr&w-Hill, New York.
2d-08 1
APPLICATION OF THE FINITE INCLUSIONS THEOREM TO ROBUST MULTIVARIABLE DESIGN Theodore E. Djaferis
Department of El
1
INTRODUCTION
tput (SISO) systems with pa.rameter uncertainty (see Djaferi. 1995). The techniques are based on the Nyquist Theorem and exploit properties of polynomials. A fundamental result, the Finite Inclusions Theorem (FIT), has been proposed as the vehicle for robust control analysis and synthesis. FIT addresses the V-stability of a polynomial family where 1> is a convex set in the complex plane and the interior of sc me Jordan curve r. This new test guarantees robust 1J-~tahility for a general polynomial family, if the polynomial value set at a finite number of points OD r lies in an appropriately chosen set of sectors in the complex plane. The number of such points is not large and can be as small &8 2n + 1, where n is the degree of the polynomial family. In the case of affine polynomial uncertainty l?IT provides a nece"..ary and sufficient condition for robust V-stability and for multiaffine uncertainty (using the Mapping Theorem) a sufficient condition. One can immediately take FIT and use it as the foundation for robust 'D-stabilization algorithms. These are iterative p:rocedures where at each iteration a finite set of linear inequalities in the controller parameters needs to be soIVI!d. Furthermore, one can suggest robust performance synthesis algorithms (e.g. robust asymptotic tracking, robust noise attenuation)
The problem of multivariable control for Linear Time Invariant (LTI) systems has received a great deal of attention in the control literature of the last thirty years (books in the area include Callier 1982, Francis 1986, Green and Limebeer 1995, Safonov 1980, Vidyasagar 1985). Justification for this effort comes from the fact that a multitude of dynamic systems have to be mer delled by descriptions that include several inputs and several outputs. This necessitates the development of analysis and synthesis iools for the investigation or multi input multi output (MIMO) systems. The experience with linear, time invariant, finite dimensional dynamical systems has shown that success in this endeavor depends on several faclon. High on the list are: 1) the development of analysis results that lead to synthesis techniques, 2) the consideration of robustness issues, and 3) when these techniques are inspired by SISO methods they must not place stringent requirements on the structure ofthe controller (e.g. diagonality). Perfecl examples of successful multivariable conbol methods are H~, p and LQG/LTR. Over the last five years powerful new robust control methods have been developed for single input single ou-
3270
ofthe Finite Inclusion. TheoreIll when V iz the len half complex plane. However, it is important 10 .tate this VtUiOD as in Section 4 it will facilitate the construction multivariable pedormance synthesis algorithms.
well as robult multiobjective performance synthesis algorithms. This paper shows how the Finite Inclusions Theorem does provide a framework. for multivariable design for systems with parameter uncertainty. It attacu the synthesu problem from a new direction and complements existing methodologies. After recalling the Finite Indusions Theorem in Section 2, a robust D-stabili.ation algorithm :is presented in Section 3. In Section 4 synthesis algorithm. are suggested for robust asymptopic tracking and noise attenuation. These are extensions of the SISO FIT .ynth..i. methodology where no reIItricive uaumptioM are placed on the .tructure of the controller. aB
Theorem 2 (Sim.llaneolU St4bilit, Finite Incl"';",.. Theorem, SSFIT) Let 'jI,(0, a) = Ei~o 0<,; (a)'; , a En., 1 ~ i ~ t, no ~ 0, ll,,:0. -+ C, Qi ... ,;:: Om. - R+ , Then, lor till a E 0 110 , 91,(0, a) ha.e all th.ir .ero. in Ihe ope" left MII plane il the ... e.ut mo ~ 1 intervals (eo,,, do.) c R, 1 ~ k ~ mo, I ~ i :5 l and real number. ""il :5 "',2 ~ ... :S LIlt"" IVC"
til.4t ""fli
2
V 1< i <, l -2 - .. < - "',
THE FINITE INCLUSION THEOREM
•
V I~,
Consider a polynomial family 'jI(.,a) with parameter uncertainty. The Finite Inclusions Theorem can be used In investigate robust V-stability. Specilically, let
'jI(., a)
V 1~ k
< mo,
t,
< mo, ., s. = {re' Ir > 0, B E V 1~ k
("'t,do.)}
Since the.. results address the stability of polynomial families with parameter uncertainty they can be applied
+ ... + 0<.(&)91.(_)
to SISO as well as MIMO sYlteml. Their application to di8CU5~ed fust.
robust 'D-stabilization is
at
ROBUST V-STABILIZATION 3
Consider a MIMO system with transfer function matrix P(.,a) described in a left polynomial matrix fraction reprelentation (Callier 1982):
..ch that int r u con.ez. Then for alia En., 'jI(., a) u of d.g .... n and lull all it.. root.. in int r if the ... ezUt.. m ~ 1 intervals (Ct, dt) C R and a countercloclc-wUe .equence 0/ painu 't Er, 1 ~ le ::; rn, .uclt. that
(1) where
N.,(_,a)
< m maz{d.+, -
< t,
91,(;""., n.) c
=
V 1 ~ l:
JI"",
1r71j
max{doHl - eol, do. - "'tH} ~ ..
where the ,p.(6), 0 :S i ~ u are given polynomials. Suppose that the parameter a takes values in the hypercube n. {a E R it la; ~ at ~ I I ~ i :5 k}, a; < 0, > 0, I :5 i ~ /e. The set R denotes the leal numbers. Further assume that Qi(a), I ~ i ~ u are polynomic in a and luch that ",(0) = o. Fo[ such a family the following result (see Kaminsky and Djaferis 1993, and Djaferis 1995) holds: Theorem 1 (The Finite Inclu.oiono Theorem, FIT) Let 91(0, a) Ei=o o<;(a)'; , a E n. n ~ 0, and Q; : n. - c. Fvrther, let r c C be a do.ed Jordan curve
=
...
2-r:sci1Jl,;:
V 1~i ~
= 'jIo(.) + O
at
r",
< do, < -2 - 2 +
Ct, dt
-
CtH} ~ ..
Dp.(O,Il)
maz{d". - (Cl + Zm), (d, + Z..n) - c".} ~ .. VI ~ k ~ m9l(0.,n.)c S. = {re' Ir > O, BE (c. , d.)}
••
= =
diag(,··-').Vph.(Il) + N,... (.,a) diag( ...· )D••, + D.... (o,a) (2)
The plant family denominator matrix D,.(o, a) is an m x m polynomial matrix in $, with row degreel J.4i and where Pl ~ P2 ~ .. . ~ J.'.m are positive integers, J.'i == n.. The highest row coefficient matrix Dphc is a constant invertible matrix which does not involve the parameters. The polynomuJ matrix DplOC(II, a) contains the lower order terms in " and its entries have
The Finite Inclusions Theorem can also provide a characterization of robust (Hurwih) .tability for a finite number of polynomial families t;6.(8, a), I ~ i::; i, each oflhem as in (1). III th. context ofanalysis the following result, called the Simultaneous Stability Finite Inclusions Theorem (SS FIT) is in .... nce a [estatement
:>::::1
3271
coefficienll that are polynomic in a
En..
For this plant family the polynomial .p(., a) has coefficients that in general are polynomic in a and multiaffi... in X. Therefore, the problem of V-.tabilisation of multivariable systems with parameter uncertainty is expr....d in lerms of the V-stabilisation of a single polynomial family. It is evidem Ihat lhis problem can be attacked using the Finite Inclusions Theorem in a manner .inillar to that presented in Djaleris 1995 for SISO systems. Firsl assume thal by introducing additional uncertain parameters (and consequently ovcrhcr unding), the polynomial coefficienls are made mw/tia/fine in a. This allows one to work with value set verticel as a consequence of the Mapping Theorem (Zadeh and Desoer 1963) and leads to computationally efficient synthelia algorithms.
The plant
nnmerator matrix Np' (6, a) is an m x l polynomial matrix in ., with row degrees P-i - I, 1 ~ i ~ m. The highest column coefficient matrix NphiC(a) may include parameter uncertainty and the polynomial matrix Np lo.l{6,a) contain. lower order terms in.s, which have coefficients that are polynomic in a. Assume that the representation is left coprime for all values of a EO". f'a.mi)y
By construction the plant f&miIy is strictly proper. Let the plant family be in the standard unity feedback configuration of Figure 1:
~L_e
______ c_(._)________P_(_')__
~~
The basic idea behind one robust V-stabililation algorithm is the following: Pick an initial controller X(I), and an appropriate set of points and sectors that satisfy the conditions of the Finite Inclusions Theorem for oil) COli.' Expand the uncertainty range and solve for a new controller that fits the new value sets in the sectors. Center the value sets (i.e. pick new frequencies) and repeat the pl'ocess. The only "structural" difference with the SISO case is Ihe fact that the coefficients of a general ~(8', a) depend on controller coefficients in a. multiaffine manner rather than in an affine manner. Consequently, the inequalities that need to be solved in a FIT bas«:d procedure ate muUiaffine in the controller parameters. The SISO FIT based 'V-stabilization algorithm must therefore be modified to address this issue. However, it is important to point out Ihat if all bu lone (.ay the ,"), of the colnmns of X take specific values then the coefficients of .p(., a) become 4JJine in the remaining ,:ontroller parameters Xi and 10 do the corresponding inequalities. A natural suggestion for solving the required: inequalities would then be to go through an additional iteration where all but one of the columns of X are lept fixed. A FIT based algorithm for multivanable robust V-stabilisation can be structured all follows:
Figure 1: Feedback System with Tracking and Noi.e Requirements
The controller is in the right matrix fraction representation:
where
N«(.) D«(.)
=
1
X'pH'" + X2.... - + ... X.+1 I ... + X p ... - l + ... Xl (3)
The matrix D~(.) is m x m and N~(.) is l x m. The controller is proper I order q = mp and the column degrees of D~(.) and N~(.) are all taken to be equal to p for notational convenience. The (pm+ (p+ l)l) x m matrix
FIT m Synthelis Algorithm STEP 1 Let X(l) E Rd and n~')
c n.
be such Ihat
.p(., n~l), X(l») is V-stable and set j:=1. STEP 2: Determine m(j) 2: 1 sectors S~), 1 :s; 10 :s;
contains the d = m(pm + (p + 1)l) controller parameters. Let Xi,l :5 i ::; m be the itA. column of X. The closed loop chaIa.cteristic polynomial has degree n = n + q and is given by:
mU), and points .~) along the boundary of V such thal
sf).
.p(.~), Eztrll/) , X(;)) c By FIT, .p(., nl/), X(;») is "D-stable. Each .~) should roughly center (angularly) the sel .p( .~), Eztrll/l, X(;») in ~il. STEP 3: Choose a slightly larger .et nl/+1) ::> nl/). STEP 4: Compute a new vector of controller parame-
3272
D.,
ters X<;+') such that 9I(0~), E"t 0~+1), X<;+')) C S~) for all k. This is equivalent to Bolving a system of multiafline inequalities in X(;+1). Fix all columns ofX(;+') hut one at cnrrent values. Solve the corresponding set of linear inequalities. Iterate over the choice of column. If no solutions are found to this system of inequalities, returD to Step 3 and choose a smaller O~+l). STEP 5: Let i := i + 1, and if O~) ::> 0., stop; otherwise, go to Step 2.
with N.,(o,a) and (0, a) as in Section 3. The plant family is in the uuity feed back configuration of Fignre 1. The controller is described in the right matrix fraction representation:
with N~(.) and D~(o) as in Section 3. The performance specification in addition to robust stability is robust asymptotic tracking or noise attenuation. The sensitivity transfer function malt;;. S(o,a) (0;;(0, a)) is given by:
=
If this algorithm terminates with O~) ::> O. then a multivariable controller would have been constructed that guarantees V-siabWation. No simplifying assumptions about the structure of the conboller have been made (like diagonality), that would restrict its applicability. However, this is an iterative algorithm which may not be able to find a robustly V-stabililing controller if one exists. Furthermore, the results will depend on the initial controller, and it is therefore prudent to use available information and insight to guide the choice. One can certainly employ the H~ methodology, LQG/LTR,
S(o,a) =
= (I + Pto, .)C(o))-'
=D~(o)(DpI(o, a)D~(o) + N.,(o, a)N~(.))-' DpI(o, a) =D~(o)~(o,at'D.,(o,a) where 91(0,.) = det~(o,a) is the closed loop characteristic polynomial. This polynomial family must have roots in the left half complex plane to guarantee robust stability. Each entry 0;;(0, a) cf S(o,a) can be thought of as a rational function with denominator ;(5, a). At this point there are several options for imposing the robust asymptotic tracking requirement. One ill to place a bound on the largest singular value of the weighted sensitivity. In other words suggest appropriate transfer function matrices W ll (o), W 12 (o) such that:
Pole Placement or any other multivariable design technique for this purpose. It is shown how this can he successfully done in the SISO case (see Djaferis 1995) and this is expected to be true for multivariable systems.
4
ROBUST PERFORMANCE SYNTHESIS
u(Wll (iw)S(iw, a)W12 (iw)) < 1 V a E
[1.
and w E [0,(0)
Another option is to place a requirement on each entry oUhe sensitivity transfer function mahix (&ee Horowib 1979, Van Diggelen and Glovel 1991):
Robust performance synthesis for multivariable systems has been at center stage over the last two decades. Very successful techniques for robust multivariable design such as H ~, LQG /LTR and p synthesis have been established for which MATLAB software is currently readily available. In addition to these methodologies, others (Horowih: 1979 and Machiejowski 1989) have been proposed which are generalizations of SISO design techniques to muItivariable systems. This Section demonstrates that for systems with parameter uncertainty FIT synthesis does provide a viable alternative method for robust multivariable system design. It can he seen to be a very natural extension of the design techniques for SISO systems. The focus is on the robust design for asymptotic hacking and noise attenuation, but it will be dear that the methods can be used so solve a wider class of controller synthesis problems. Consider a multivariable plant family Pto, a) with parameter uncertainty a E na which is described in the left matrix fraction representation:
IWH;(iw)o;;(iw, a))1 < 1 V w E [0, 00), a E O. and all i, j where WH;(O) is the (i, i)" dement of some matrix W.(o) which is strictly proper and has stable and 00nimum phase entries. Since th,e plant transfer function matrix is m x t there will be n12 such conditions. The complementary sensitivity transfer function matrix T(o,a) = (t;;(o,a)) is giyen by:
T(o, a)
=I
-
8(0, a)
It. diagonal entries aret;;(o,.) = 1 - 0;;(0,.) and the offdlagonalentriesaretij(.s,a) = -"ij(6,a). They too can be thought of as rational fundions with denominator ~(",a). In a similar manm:r the robust asymptotic noise attenuation requirement ,:an either be imposed as
(5)
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a bound on the largest singular value of the weighted complementary sensitivity transfer function matriJ: or elementwise:
all I < 1 V wE [0,(0), a E
I....;(jw)t;j(jw,
where
'W2ij(S-)
(la
aooalli,j
n",;;(.)n,;j(jw, a) d.,;;( ')1'1(_, a)
I
I<
1
~'
w E [0,00).
a
E
(la
(6)
This i. equivalent to:
is the (i,j)th. element of some matrix
W,(.) which is proper and ha. .table and minimum phase entries. There will be m 2 such conditions. One can immediately see that the weight. W , (.) and W.(.) cannot be chosen independently because the diagonal entries 6ii(jW, a) and 1 - 6&i(jw, a) are very related. In particular, both cannot be made "small" at the same frequency. Fortunately, in many cases tracking and noise attenuation requirements are placed at sufficiently disjoint frequency ranges. Lemma 1 The e"tM" of S(" a) and T(., a) are rational fundion. in , who6e numerator and denominator polynomial. are multiaffine ezpreuioru of tAe controller pam.meter•. Proof: It is known that S(" a) D""(,)~(,, a)-1 D.I(" a). Write ~(.,a)-1 = ·'~~~~i" where atij(.) denot.. adjoint of a matrix. This implies that:
e(J'w) n",;;(jw)n,;;(jw, a) I'~ 1 V [ ) (l d.li;(jW)I'I(jw, a) w E 0,00 , a E • (7)
l
where 0(.) is any stable proper transfer function with 19(jw)1 :'5: 1 for all w E [0, (0). In other words, for no plant in the family and no frequency can one have:
rei°
°
n"li;(jw)n,;,(jw,a) =-1 d.lij(jW)I'I(iw, a)
(8)
wh.r. :'5: r :'5: 1 and 0 :'5: a < 2... This is equivalent to saying (sine. 4li;(') and 1'1('" a) are stable) that the polynomial:
=
S(" a)
D,,"(.)adj(~(.,
allD,1 (" a) 1'1(" a)
a)) (",,;(., 1'1(" a) = I _ S(
) _
',a -
should not be zero for any 0 ~; r :5 1, Q E [0,27), • E (l., wE [0, (0). Clearly if the polynomial:
'I/J.ij(.,a,r,Ct)
(""i(.,a)) 1'1("a)
The elements of the ith column of
o Consider now anyone of the m 2 conditions for robust tracking. If Wli,·(.) n._.. u,1i ~ •.)) it can be equivalently stated as:
3274
=
rei°n,.,li; (_ )n,;;(., a) + 4li;( .)1'1(" a)
is stable for all Cl< E [0,2,,), r E [0,1] and a E {l., then the robust performance rt~quirement corresponding to the (i, j)th sensitivity tram fer function element will have been met. One can express the performance requirements for all the other sensitivity transfer function element! in a similar manner. The robust tracking requirement can therefore be expressed as a simultaneous polynomial family IItabilisation requirement. In t"..J:actly the same manner one can express the robust noise attenuation requirement as a simultaneous polynumial family stabilization problem where polynomial fanilies are of the form:
1/1,ij(".' r, et) = rei°n,.,2i;( ')'''';('' a) + 4,,;(.)1'1(., a) In order to d.velop an efficient SSFIT b ....d performance synthesis algorithm there is a need to overbound the disc uncertainty rei a a5scciated with performance by some polygonal overbound l.nd in the following algorithm the square overbound rib. is chosen. Now, each such polynomial family will in general have coefficients that are polynomic in the pla.nt uncertain parameters a and multiaffine in the cont.roller parameters X. In order to employ the FIT based algorithms for simultaneous stabilization developed for SISO systems, one
must do two things: Firstly, by introducing additional uncertain parameters one makes the plant uncertain parameter dependence mv.ltiaffi~ whenever it is required. Specifically, & term like 0 2 can be replaced by the poduct a1 a2. This of course will be a source of conservatism in the design. Secondly, one modifies the emting FIT algorithms, just as was done for multivanable robust stabilisation, to deal with the multiaffine controller parameter dependence. One easy solution is to inhoduce an iteration when solving the multiaffine inequalities, where controller pa.rameters are kept fixed for all but one column of x. The following algorithm is suggested where for notational convenience polynomial families are relabeled ,p'i;(6, a,r,a),'I.j1tij(6,8,r,a)
as
4>.(6, all:
I. Let Xl') E Rd and o~~) CO •• be such
that 4>.(., O~~), Xl'») are stable and set j:=l. STEP 2. Determine 2: 1 sectors
mY)
sn),
1 :s k :s ml;), 1 :s i :s l and frequencies along the jw axis such that 4>.(jwl~),Ezto?),XU») C By FIT, 4>.(., O?~, XU») are stable. Each wl!) should roughly center (angularly) the set 4>.(jwl~), E"'tO?~, X U») in U) . 5 i1 STEP s. Choose a slightly larger set O?+l) ::) O~. First this should affect the b parameters. When (if)
wW
sW
(}~) :J Ob", the enlargement in the b-diredion terminates and the enlargement in the a-direction commences. STEP 4: Compute a new vector of controller parameters XIH ') such that 'Pc ~·(J·wl;) Ezt OIH') XU+t») C "~I 06 J
S~') for all le and i. This is equivalent to solving a system of multiaffine inequalities in X(.i+l). Fix. all C~ lumns of X(i+l) but one at current values. Solve the corresponding set of linear inequalities. Iterate over the choice of column. If no solutions are found to this system of inequalities, return to Step 3 and choose a smaller O~+'). STEP 5. Let j := j otherwise, go to Step 2.
CONCI~USION
A powerful framework for rob"st control design of multivariable systems with parameter uncertainty hu heen presented. The approach is based on the Finite Inclusions Theorem and leads to iterative algorithms for design
in terms of IOlutions of linear inequalities in conholler parameters. The methods complement emtmg tools for multivariable design. Even though it would be greatly desirable, efficient software algorithms that implement the algorithms are not available. OUI hope is that this will change in the near future.
REFERENCES
PSSFITm Synthesis Algorithm
STEP
5
+ 1,
and if O?~ ::) 0 •• , stop;
It .hould be clear that this algorithm can be employed for both single objective and multiobjedive rohust performance synthesis. This formulation and algorithms can also be used to solve a number of other robust multivariable synthesis problems. One such is the problem of Robu6t Decoupling.
3275
Callier F.M. and Desoer C.A. (1982). Multi.ambk Feedback S1ldem6, SpringerM Verlag, New York. Djaferis T.E. (1995). Ro/nut Control D..ign: A Polynomial Approach, Kluwer Academic Publishers. Francis B.A. (1986). A Cou", in H~ Control Theory, Springer-VerIag. Green M. and Lirnebeer D.J.N. (1995). Linear Rob...t ContTO~ Prentice Hall. Horowih 1. M. (1979). Quantitative Synthesis of uncertain Multiple Input-Output Feedback Systems, Int. Jour. ContTO~ Vol. 30, pp. 81-106. Kaminsky R.D. and Djaferis T.E. (1993). The Finite Inclusions Theorem, Proceeding6 IEEE ODC, San Antoruo TX, 1993, pp. 508.518, (modified ve"ion IEEE 7"",•. on AC, Vol. 40, No. 3, 1995, pp. 549551). Machiejowslri J .M. (1989). Multi.ariable Feedback Design, Addison· Wesley. Van Diggelen F. and Glover K. (1991). Element-byElement Weighted Hoo-Frobenius and H2 Norm Problems, Proceeding$ IEEl? ODC, Brighton, England, pp. 923-924. Safonov M.G. (1980). Stability and RobU6tn... of Multivariable Feedback SY$te'IfU, MIT Press. Vidyasagar M. (1985). Sy.tem Control Synth ..is: A Factorization Approach, MIT Press. Zadeh L.A. and Desoer C.A. (1963) Linear Syltem Theory, McGr&w-Hill, New York.