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Robust Control of LTI Square Mimo Plants Using Two Crone Control Design Approaches
Copyright ® IFAC Robust Control Design. Prague. Czech Republic. 2000
ROBUST CONTROL OF L TI SQUARE MIMO PLANTS USING TWO CRONE CONTROL DESIGN APPROACHES
P. Lanusse, A. Oustaloup and B. Mathieu Lahoratoire d'Automatique et de Productique - EP 2026 CNRS Bordeaux I University - ENSERB 351, Cours de la Liberation - 33405 Talence Cedex - FRANCE Tel: 33 556842417 - Fax: 33 556846644 - [email protected]
Abstract: A multi-SISO or a MIMO design approach is proposed for the robust control of uncertain LTI square MIMO plants, depending on the extent of the plant's diagonal nature. For the multi-SISO approach, new multi variable frequency uncertainty domains are defined using the Gershgorin and the small gains theorems. These domains are based both on the structured frequency uncertainty domains related to the diagonal elements of the plant transfer matrix, and on the Gershgorin circles related to the corresponding column elements. Each element of the diagonal controller can be thus designed independently of the others using the SISO Crone design method. The totally MIMO approach is based on the optimization of an open-loop diagonal transfer matrix to minimize the stability margin variations of the perturbed diagonal elements of the closed-loop transfer matrix. Copyright e 2000 IFAC Keywords: square MIMO plant, robustness, CRONE control, frequency domain design method
First, multi-SISO Crone control is defmed as the SISO Crone control of each diagonal element of an uncertain MIMO plant. The controller transfer matrix is diagonal and the coupling elements of plants are interpreted as further uncertainty on the plant diagonal elements.
1. INTRODUCTION Frequency domain design methods have already proved their efficiency for the robust control of LTI uncertain SI SO plants. Crone (the French acronym which means non-integer order robust control system) control design (Oustaloup, 1991; Oustaloup, et al., 1995a, 1999) is one of these frequency design methods and has already been compared with other methods, notably using a benchmark proposed by Landau (1995). This benchmark is based on robust digital control of a flexible transmission system. Of eight approaches, Crone control (Oustaloup, et al. , 1995b) came second with a satisfactory ratio of 98.6%, computed for all required specifications. A QFT approach by Nordin and Gutman came first, then other approaches based on: QFT ; a model-free tuning scheme; pole placement with sensitivity function shaping ; Hoo optimization with two DOF ; GPC ; and finally Hoo with one DOF. One can suppose that the first three approaches are the best because they use structured frequency uncertainty domains (called templates by QFT users), which is the least conservative way to model the plant uncertainty.
A totally multivariable approach may sometimes be necessary when the plant diagonal dominance degree (Arkun, et al., 1984) is weak. A multivariable CRONE control design using a non-diagonal controller is thus proposed. The parameters ofa diagonal (for the nominal plant) open-loop transfer matrix are optimized to minimize the variations of the stability margins of the perturbed diagonal elements of the closed-loop transfer matrix.
2. SI SO CRONE CONTROL DESIGN SISO Crone control design is a frequency approach for the robust control of plants under the common unityfeedback configuration. The controller or open-loop transfer function is defmed using integro-differentiation with non-integer (or fractional) order. The required robustness is that of both stability margins and performance, and particularly the robustness of the maximum Q of the complementary sensitivity function magnitude. Indeed, this magnitude (or resonant) peak is strongly correlated to the step response first overshoot. Three Crone control design methods have been developed, successively extending the application field.
LTI square MIMO uncertain plants are often, for the sake of simplicity, controlled using diagonal MIMO controllers. Each element of the controllers is usually designed taking into account only SISO uncertain behaviors of the MIMO plants. When these SISO controllers are designed to achieve rapid dynamics, a decrease of the stability margins of each closed-loop can often be seen. As in the case of QFT design (Horowitz, 1993), several frequency SISO control design methods have been extended to control MIMO plants. After a short overview of the SISO Crone control design, we present two different extensions for MIMO plants.
The first two Crone methods use rear non-integer integro-differentiation and permit the robustness of stability margins at the time of plant gain variation to be obtained. The first is based on a constant phase of the controller around the required open-loop gain crossover
379
(argAmG(jlO~IAmG{jlO~dB) ' As the uncertain open-
frequency lOcg' The controller is defmed by a real noninteger difJerentiator transfer function and the robustness is ensured only if lOcg is within a constant phase frequency range of the plant. The second is based on a constant phase of the open-loop defmed in the Nichols chart as a vertical frequency template. Around frequency lOcg' this template is defined by the transfer function of a real non-integer integrator:
loop transfer function f:X..s) is defined by: fJ...s) = G(s)C(.s) = G«S)L1mG(S)C(S) = fi(S)L1mG(.s) ,(5) where Ab) is the nominal open-loop transfer function, the uncertainty domains related to the Nichols locus of /30{j m) are also defmed by the possible values of the pair (arg Am G{jlO lAm G(jlO ~dB) ' By minimizing the variation of the magnitude peak, the optimal template positions the frequency uncertainty domains so that they overlap the low stability margin areas close to the critical point (Fig. 1) as little as possible. This minimizes the cost function J= (Qmax - Qr)2 , (6) where Qmax is the maximal value of magnitude peak Q.
t
(1)
where the real order n E IR determines the phase placement of the vertical template, namely -mt/2. The shape and a vertical sliding of the template due to gain variation ensure the constancy of the phase margin, of the magnitude peak Q, of the modulus margin (the minimal distance between the open-loop Nyquist locus and the critical point) and of the damping ratio of the closed-loop system (Lanusse, 1994). For more general cases, the vertical template does not slide on itself at the time of parameter variation of the plant. This template cannot therefore ensure the best robustness of the control system. It is more convenient to consider a template that is still defined as a straight line segment (around frequency lOcg) for the nominal parametric state of the plant, but with any direction, and called a generalized template. The open-loop transfer function is now defmed by the restriction in the operational plane Cj of the transfer function of a complex non-integer integrator:
P(')+'+ ;)f 91,,[(",: with n=a+ibEC j
and
n
Fig. 1. Generalized template Nichols loci and frequency uncertainty domains: (a) any given generalized template; (b) optimal template Designing the optimal template through the optimization of only two parameters (as a tangency is imposed) from the three high-level parameters cq,g, a and b, is the initial approach of third generation Crone control design (Oustaloup, 1991). In the version used (Oustaloup, et af. 1995a), the nominal open-loop transfer function takes into account: the accuracy specifications at low frequencies; the generalized template around frequency mcg; and the specifications on the control effort at high frequencies. This transfer function can be described as a transfer function based do frequency-limited complex non-integer integration :
(2)
S=O'+ jlOEC j . In the
Nichols chart, the real order a determines the phase placement of the template, and then the imaginary order b determines its angle to the vertical. From an infmity of generalized templates which tangent the magnitude contour related to a required magnitude peak Qr for the nominal parametric state of the plant, the optimal template can be defined as the generalized template which permits minimization of the variations of Q stemming from the uncertainty domains which in turn stem from the various parametric states of the plant.
/3o{s} = K(~+ S
[
It is easy to show that the multiplicative uncertainty 4J3(s) of the open-loop frequency response, which defines the frequency uncertainty domains (Lanusse, 1994) in the Nichols chart, is invariant and equal to that of the plant. Let an uncertain LTI SISO plant be: G(s} = Go {s }Am G{s} = Go {s}+ Aa G{s} , (3)
wh.,e
m.[(c iJ
l)nl(1l+s/lO) + S/lOh )Q
1 + sf £Oh °1+S/ lO)
Jib lJ-Sjgn(b)
1
(7)
(l+S/lOh}nb '
C, =[(1+ ~q/(I+ :DJ"
(8)
The order n) of f:X..jlO) at low frequencies fixes the steady state behavior of the closed-loop system. The value of nh has to be chosen as equal to or greater than the low frequency order of the plant.
where Go(s) is the nominal plant transfer function and where L1",G(s) and LtaG(s) are respectively multiplicative and additive uncertainty models. The relation between LtaG(s) and L1",G(s) is:
A nonlinear optimization algorithm searches for a vector of independent parameters which minimizes the cost function (6). As the nominal Nichols locus of /3o(s) tangents the Qr M. contour for a given resonance frequency ~ (close to the required closed-loop bandwidth frequency), and has integer orders n) and nh fixed by the designer, only 3 independent high-level
AmG{s} = 1 + Aa G{s )Go 1(s). (4) The uncertainty domains related to the Nichols locus of Go{jlO) are defined by all the possible values of the pair 380
parameters have to be optirnized. The minimization is carried out under a set of shaping constraints on the four usual sensitivity functions. These frequency constraints have to take into account as many specifications as possible. When the optimal open-loop transfer is detennined, the controller is designed from the frequency response of the non-integer transfer function Cni(s), which is deduced from the ratio of /3o(s) by Go(s). The design of the achievable controller consists in fitting frequency response Cni(jm) using a low integerorder transfer function Cin(s).
between the application on the one hand of the small gains theorem to uncertain LTI SISO systems, and on the other of the Gershgorin theorem to certain LTI MIMO systems. Certain LT! MIMO systems and Gershgorin 's theorem (Rosenbrock, et al., 1970; Rosenbrock, 1974; Maciejowski 1989). The Gershgorin theorem can be used for complex matrices of rational functions and, in particular, of transfer functions. The stability of closedloop MIMO systems can thus be studied using Gershgorin circles.
The interest of using Crone control design lies in the fact that it permits not only the genuine uncertainty domains of the plant (owing to the absence of norms) being taken into account, but also all types of uncertainty, whether structured or not. In addition, complex non-integer integration is introduced, which makes it possible to optirnize the open-loop transfer function only using few high-level parameters. The rational Crone controller is then only designed for the optimal open-loop transfer function, thus avoiding the iterative design of a controller with many parameters. Crone control design has already been extended to unstable or nonminimum-phase plants, to plants with bending modes, and to discrete-time control problems (Oustaloup, et aI., 1995b). From a fmite set of generalized templates, a curvilinear template has been defmed to improve the optimality of Crone control (Mathieu, et al., 1996).
Let Q(s) be an LTI and certain n*n open-loop transfer matrix. For all integer i between 1 and n, and for all mof [0, +(0), the Gershgorin's circles are defmed by the centers qji(jm) and radii rj (m) , with: rj(m) =
min(
L IS] Sn and I.,i
Iqij(jm~,
L IS)Sn and j~i
,qjj(jm~l ·
(9)
The n hulls drawn from the union of these circles are called Gershgorin bands, and they contain the Nyquist loci of the n eigenvalue frequency responses of Q(s). The closed-loop system is stable if: - the number of times the point (-1,0) is encircled by the n Gershgorin bands, satisfies the Nyquist stability condition generally used for SISO systems ; - all the n Gershgorin bands exclude the point (-1 ,0). So, the closed-loop system with an open-loop transfer function [Q(s)] can be stable if: rj (m)~II+qii(jm~ 'VI ~ i ~ nand 'Vm E [0,+(0) .(to)
3. MULTI-SISO CRONE CONTROL DESIGN 3.1 Introduction
Uncertain LTI SISO sy stems and small gains theorem (Zames, 1966,· Vidyasagar and Viswanadham , 1982). Let Q(s) be an uncertain LTI open-loop transfer function: (11)
The Gershgorin theorem has already been extended to deal with uncertain multivariable systems. As shown by Arkun, et al. (1984), this extension results in overestimation which can lead to excessive conservatism We here defme multivariable uncertainty domains based both on the structured parametric uncertainty of SISO systems as taken into account by the SISO Crone robust control approach, and on unstructured uncertainty computed from the coupling elements of the plant transfer matrix (Lanusse, et aI. , 1996). This stems from a comparison of the sufficient stability conditions given by the small gains theorem for uncertain SI SO systems, and the Gershgorin theorem for certain MIMO systems. Because these conditions are similar, for an uncertain MIMO system it is possible to defme a closed-loop stability condition, which takes into account both the uncertain off-diagonal elements, and the uncertainty on the diagonal elements, of the openloop transfer matrix. This definition permits an analysis of the robust stability which is as little conservative as possible. These results can be used by any initiallySI SO design approach using structured frequency uncertainty domains.
where Qo(s) is the nominal transfer function and, L1aQ(s) the additive form of the uncertainty. The small gains theorem states that the closed-loop system with an openloop transfer function Q(s) can be stable if: r(m)~II+Qo(jm) 'Vm E [0,+00) , (12) where r( m) is defmed by: r(m)~maxIQ(jm)-Qo(jm)
.
(13)
At each frequency, the left term of (13) can be interpreted as the radius r( m) of a circle that localizes the uncertainty. Its center is given by the nominal openloop frequency response. The closed-loop system is always stable, if it is stable for the nominal parametric state, and if all the circles exclude the point (-1,0). Comparison and illustration of stability conditions. It is interesting to compare the stability condition ( 12), obtained through the small gains theorem for uncertain SISO systems, with (10), given by the Gershgorin approach for nominal MIMO systems. These two conditions are illustrated by Fig. 2 which shows that the two theorems lead to similar graphical stability conditions. Indeed, the two formalisms use the distance between: the edge of the circles depending on the
3.2 Modeling the uncertainty of MIMO systems
We demonstrate that the definition of multivariable uncertainty is obvious when a parallel is established
381
uncertain or multivariable nature of the open-loop system, and the point (-1,0). So, in the nominal MIMO case, the coupling elements can be considered as uncertainty on a SI SO system. This is very important if a multi-SI SO design approach is considered. Thus, the certain or uncertain coupling elements can be considered as further uncertainty on the uncertain diagonal elements of the open-loop transfer matrix. So, these diagonal elements are now a little more uncertain.
rj{w)=
Construction of multivariable uncertainty domains of the diagonal elements. Crone methodology uses uncertainty domains constructed in the Nichols chart. Nevertheless, in order to facilitate their presentation, the uncertainty domains are here constructed in the complex plane. The multivariable uncertainty domains related to the diagonal elements, are constructed both through the frequency responses. of the diagonal elements for all the parametric states 01' : the plant, and through the unstructured uncertainty' related to the off-diagonal elements of the plant. For example, Fig. 3 shows the frequency response of a diagonal element for its nominal parametric state (numbered 0), and for the three other parametric states considered (numbered 1,2 and 3). Gershgorin circles from further off-diagonal column element uncertainty are drawn around each frequency response.
Consideration of off-diagonal plant elements. The methodology now proposed aims not to bound the elements resulting from the uncertainty on the diagonal and off-diagonal elements of the open-loop transfer matrix.
.
Take an n*n uncertain MIMO plant G and a n*n multiSISO (diagonal) controller C in series. As the previous section shows, the row and column off-diagonal elements of G can be considered as uncertainty on the diagonal elements. For a given parametric state, the additive uncertainty on gilD w) is given by:
L
1:5,:5n
Igij (j W ~,
L
1:5,:5n andj",j
Ig jj (j W ~l '
l:5j:5n
Igij(~W~,I+ l:5j:5nL .lgAJw Igjj(~W~l. ~
.lgAJw ~
and '''''
rio(~~f4.)
ril(~)0giil(j~)
(14)
Fig. 3. Nominal and perturbed diagonal elements, and their Gershgorin circles
(15)
and '''''
The multiplicative uncertainty on f3ji (j w), a diagonal element of the open-loop ~(s), with
Cl (s )gll (s)
p(s) = G(s)c(s) =
: [
0~)
0ii,(j~) ri3(f4.)~ ri,(~) G.
This uncertainty can be written in a multiplicative form: min[l+ L
(18)
9t(.)
(a) (b) Fig. 2. Graphical illustration of: (a) small gains theorem; (b) Gershgorin's theorem
andj~j
Igjj(jW~.
It is true that if the row elements are smaller than the column elements, such a determination of uncertainty can be a little conservative. It is also very easy to show that only the column diagonal dominance remains true in the case of a control with a multi-SISO controller in series.
:3(.) -1
min[
L
l:5j~n .lgAJw ~ and,,,,,
... cn (s )gln (S)] ". : ,(16)
The construction of tpe multivariable uncertainty domains consists in the determination of the hull that takes into account both the uncertainty related to the diagonal element (SISO uncertainty domain) and the further multivariable uncertainty. In fact, the multivariable uncertainty domain is defmed by the convex hull that includes all four Gershgorin circles (Fig. 4). Of all known ways of construction. this is the least conservative because it is the most realistic.
cl(S)gnl(S) ... cn(s)gnn(s)
is given by:
. [
mm 1+
L
ICjGw)gijGw~ . . ,1+
l:5jSn IC;(Jw )gAJw ~
and,,,,,
~
L
IgjjGw~l . .(17)
ISj:5n.lgjiGw ~
and,,,,,
When the diagonal controller elements are designed, (15) and (17) show that only uncertainty related to the off-diagonal column elements remains unchanged and is independent of the controller. To permit independent design of each controller element, it is this column uncertainty which is used. So, the circles chosen to represent the multivariable uncertainty on Gji (s), each have their radius in the complex plane:
Fig. 4. Construction of multivariable uncertainty domains in the complex plane In the Nichols chart, the method of construction used is the same, except that the circles are replaced by ellipsoids if the plant is column-diagonal dominant.
382
margins are obtained when each generalized template defmed by f30,um) tangents its specific Qri M. contour. The order nbi of each ,8,u m) at low frequencies fixes the accuracy of each closed-loop. The orders nhi of the open-loop transfers at high frequencies permit the elements of a controller to be proper. The controller can be defmed by the non-integer transfer matrix:
The construction of the multi variable uncertainty domains for each diagonal element of a MIMO plant, and the robust control design for each new uncertain SISO plant, is the multi-SISO Crone control approach. If the multivariable uncertainty domains are nevertheless too wide, decoupling techniques can be used to decrease the effects of the off-diagonal elements on the size of the multivariable uncertainty domains.
Cni(s)=GO(stlpo(s). (24) The inverse of the transfer matrix of the nominal plant is written as:
4. MIMO CRONE CONTROL DESIGN
Go ( S ) -1 = [nij(s)] -_-
4.1 Introduction MIMO Crone control design is basically the design of decoupling open-loops. It is the most general MIMO Crone control design. Wang and Davison (1975) first proposed decoupling control systems using static output feedback. Then, Desoer, et a/. (1981; 1984) proposed a parameterization of the set of stabilizing diagonal closed-loop transfer matrices. Later many authors (Dickrnan and Sivan, 1985; Desoer and Giindes, 1986; Lin and Hsieh, 1993) extended the results of Youla, et al. (1976) to show the optimality and Hoo robustness of diagonal approaches. An advantage of the MIMO Crone approach is its ability to describe each diagonal open-loop with few parameters through the use of complex non-integer integration orders. Another advantage is its ability to take into account the uncertainty of the MIMO plant without over-estimation. 4.2
where dij(s)andnij(s) are Hurwitz. Then (24) becomes Cni (s) = [
nhj
and
SOi(S) =
•
,80i(S) 1+ ,80i(S) 1 1 + ,8oi
( )"
(26)
(27)
~ max(deg(nij(s))-deg(dij(s))) .
(28)
1:S}:Sn
4.3 Optimization of the open-loop behavior
Perturbed transfer matrices T(s) and 8(s) are nondiagonal and their frequency responses are: T(jm) = [I + G(jm)cni(jm)tIG(jm)cni (jm) (29) l . (30) and S(jm) = [I + G(jm )coi (jm Nevertheless, it is possible to defme n perturbed openloop frequency responses: T·Gm) 1- S··Gm) ,8i (jm) = II • = ". for 1 ~ i ~ n ,(31)
)t
I-1jiGm)
SiiGm)
where T;,{jw) and Si,{jm) are the diagonal elements of T(jw) and S(jm). The open-loop frequency multiplicative uncertainty Lli(jm) is defined by:
L1i(jm) = ,8oiGm tl ,8iGm). (32) A major difference with the SISO case is that the openloop uncertainty is now different to the plant uncertainty. The open-loop uncertainty domains are not only computed from the plant, but also from the controller. The nominal open-loop behavior is defined as a set of n generalized templates, each tangenting a required Qri M. contour and ensuring a gain crossover frequencym.,gi' From the infinite number of sets within this definition, the optimal set of n templates is that which minimizes the peak magnitude variations. The cost function that the optimal set minimizes is:
1I0{s}= diag (fJOi{S)1sisn' (19) The nominal complementary sensitivity function and sensitivity function transfer matrices are
To'(s) =
. diag[,Boi(s) Si:Sn '
IS.Sn
From (27) the minimal value of each nhi is:
The decoupling objective with an output feedback leads to a nominal diagonal open-loop transfer matrix with n elements:
with
~ ()
nij(s) ] Cni(s)= [~(),80i(S) . . dij s IS,Sn l:SjSIl
or
For minimal-phase square n*n plants, our aim is to parameterize a set of open-loop matrices that reach the four following objectives: - perfect decoupling for the nominal plant - the accuracy specifications at low frequencies - the required nominal stability margin of the control system (behaviors around the required cut-off frequencies) - the specifications on the n control efforts at high frequencies.
and
1
n.. (s)] dij s
l:Sj:S1l
Parameterization of the nominal open-loop transfer matrix
To(s) = [J + Po(s )tlpo(s)= diag[Toi(s )t,isn l So(s) = [J + Po(s )t = diag[Soi(s )1sisn '
(25)
,
dij (s) ISi:Sn ISjSn
(20)
(21) (22)
J =
(23)
~(Qmaxi _Qri)2 + (Qmioi _Qri)2,
1=1
S
The three other objectives are obtained when each openloop transfer function /3o.{s) is defmed by relation (7) of the SI SO Crone approach. The n nominal stability
Qri
Qri
(33)
where Qmin i and Qmax i are the minimal and maximal values of the peak magnitude of the perturbed transfer function T;,(s). 383
Simultaneously optimizing the elements of the nominal open-loop transfer function does not permit the use of varying uncertainty domains for easy modeling of uncertainty. However, it does bring about both optimal placement and shaping of the varying uncertainty domains. The vector to be optimized is the set of the n vectors of independent parameters of each generalized template. Thus the number independent parameters is 3n. The optimization procedure is a constrained nonlinear optimization which minimizes (33) and respects the constraints imposed by the following requirements: - shaping the diagonal elements of the four sensitivity function matrices - bounding the off-diagonal elements of sensitivity function matrices to decrease the coupling effect resulting from plant uncertainty - stabilizing all the elements of sensitivity function matrices by respecting the multivariable Nyquist theorem for all possible plants. When the optimization is complete, each rational integer element of the controller transfer matrix is identified from the frequency response of(24).
5. CONCLUSION The multi-SISO and MIMO Crone control designs proposed here are extensions of SISO Crone control design to the multi variable domain. This study only deals with the case of square minimum-phase plants. The brief review of SISO Crone control in section 2 shows how the uncertainty must be modeled to provide simplicity of use in SISO Crone design. Section 3 presents a multi-SISO approach based on the successive determination of the elements of a diagonal controller using SISO Crone control design. The Gershgorin results related to closed-loop stability of the nominal MIMO systems are reviewed. A comparison is then made between the Gershgorin approach, and the approach using the small gains theorem for analysis of the closed-loop stability of uncertain SISO systems. It is shown that the coupling effects on closed-loop stability can be studied, if the off-diagonal elements of the plant are taken into account as further uncertainty on the diagonal uncertain elements of the plant. Thus new multivariable frequency uncertainty domains have been constructed through both the structured representation of the SISO uncertainty, and the Gersgorin circles related to all the parametric states of the plant. This new type of uncertainty representation permits the design of less conservative robust multi-SISO controls of uncertain MIMO plants. Section 4 presents a complete extension of SI SO Crone control design to MIMO plants. MIMO Crone control design is based on a complex non-integer parameterization of each element of a diagonal openloop transfer function matrix. These elements have to ensure nominal performances on transient and steady states. A simultaneous optimization of these elements minimizes the variations of each stability margin resulting from the plant uncertainty, while respecting constraints on the sensitivity function matrices. By using the parameterization of the set of all achievable 384
decoupled transfer functions proposed by Lin and Hsieh (1991), the MIMO Crone approach can now be used for nonminimum-phase or unstable plants (Oustaloup and Mathieu, 1999). The design of multi-SISO and MIMO Crone control relies on complementary approaches. Multi-SISO Crone control design is easy to use for MIMO plants, but is still a diagonal design. MIMO Crone control design is an entirely multi variable tool which is more efficient, but also more difficult to use. One or the other has to be chosen, depending on the multi variable degree of the plant, to optimize the ease-of-use/perforrnance trade-off.
REFERENCES Arkun, Y., B. Manousio.uthakis and P. Putz (1984). Robust Nyquist array methodology: 11 riliw theoretical framework for analysis and design of robust feedback system, Int. J. Control, 40, 603-629. Desoer, C .A. and MJ. Chen (1'981). Design of multi variable feedback systems with stable plant, IEEE Trons. on A.C., 26, 408-415. Desoer, C.A. and C.L Gustafson (1984). Design of multivariable feedback systems with simple unstable plant", IEEE Trons. on A.C., 29, 901-908. Oesoer, C .A. and A.N. Giindes (1986). Decoupling linear multiinput multi output plants by dynamic output feedback: an algebraic theory, IEEE Trons. on A.C. , 31,744-750. Oickman, A. and R. Sivan (1985). On the robusmess of multi variable linear feedback systems, lEEE Trans. on A.C., 30, 401-404. Horowitz, LM. (1994). Quantitative feedback design theory - QFT, QFT Publications, Boulder, Colorado. Landau, 1.0., D. Rey, A. Karimi, A. Voda and A. Franco (1995). A flexible transmission system as a benchmark for digital control, European Journal a/Control, 1,77-96. Lanusse, P. (1994). De la commande CRONE de premiere generation it la commande CRONE de troisieme generation, PhD Thesis, Bordeaux I University, France. Lanusse, P., A. Oustaloup and O. Sutter (1996). Multi-scalar CRONE control of multivariable plants; Proc. WAC'96-ISIAC Symphosia , Montpellier, France. Lin, C.A. and T.F. Hsieh (1991). Decoupling controller design for linear multivariable plants, IEEE Trans. on A.C., 36, 485-489. Linneman, A. and Q.G. Wang (1993). Decoupling by precompensation while maitaining stabilizability, IEEE Trans. on I A.C., 38, 735-744. Maciejowski, J. M. (1989). Multivariable feedback design, Addison Wesley, England. Mathieu , B., A. Oustaloup and P. Lanusse (1996). Third generation CRONE control: generalized template and curvilinear template; Proc. CESA '96, Lille,' France. Oustaloup, A. (1991). La commande CRONE, Ed. Hermes, Paris. Oustaloup, A., P. Lanusse and B. Mathieu (I 995a). Robust control of SISO plants: the Crone control, Proc. ECC'95, Roma. Oustaloup, A., B. Mathieu and P. Lanusse (1995b). The Crone control of reason ant plants: application to a flexible transmission, European Journal a/Control, 1, 113-121. Oustaloup, A. and B. Mathieu (1999). La commande Crone: du scalaire au multi variable, Ed. Hermes, Paris. Rosenbrock, H.H. and C. Storey (1970). Mathematics of dynamical systems, Nelson, London. Rosenbrock, H.H. (1974). Computer Aided System Design, Academic Press, New York. Vidyasagar, M. and N. Viswanadham (1982). Algebraic design techniques for realizable stabilisation, IEEE Trans. on A.C., 27, n° 5. Wang, S.H. and E.J. Davison (1975). Design of decoupled control systems: a frequency domain approach, Int. J. Control, 21 . You la, D.C., HA Jabr and J.J. Bongiomo (1976). Modem WienerHopf design of optimal controllers (part II),IEEE Trans. on A. c., 21, 319-338. lames, G. (1966). On the input-output stability of time varying feedback systems; Part I: conditions using concepts of loop gain, conicity and positivity, IEEE Trans. on A.C., 11, 228-238.
ROBUST CONTROL OF L TI SQUARE MIMO PLANTS USING TWO CRONE CONTROL DESIGN APPROACHES
P. Lanusse, A. Oustaloup and B. Mathieu Lahoratoire d'Automatique et de Productique - EP 2026 CNRS Bordeaux I University - ENSERB 351, Cours de la Liberation - 33405 Talence Cedex - FRANCE Tel: 33 556842417 - Fax: 33 556846644 - [email protected]
Abstract: A multi-SISO or a MIMO design approach is proposed for the robust control of uncertain LTI square MIMO plants, depending on the extent of the plant's diagonal nature. For the multi-SISO approach, new multi variable frequency uncertainty domains are defined using the Gershgorin and the small gains theorems. These domains are based both on the structured frequency uncertainty domains related to the diagonal elements of the plant transfer matrix, and on the Gershgorin circles related to the corresponding column elements. Each element of the diagonal controller can be thus designed independently of the others using the SISO Crone design method. The totally MIMO approach is based on the optimization of an open-loop diagonal transfer matrix to minimize the stability margin variations of the perturbed diagonal elements of the closed-loop transfer matrix. Copyright e 2000 IFAC Keywords: square MIMO plant, robustness, CRONE control, frequency domain design method
First, multi-SISO Crone control is defmed as the SISO Crone control of each diagonal element of an uncertain MIMO plant. The controller transfer matrix is diagonal and the coupling elements of plants are interpreted as further uncertainty on the plant diagonal elements.
1. INTRODUCTION Frequency domain design methods have already proved their efficiency for the robust control of LTI uncertain SI SO plants. Crone (the French acronym which means non-integer order robust control system) control design (Oustaloup, 1991; Oustaloup, et al., 1995a, 1999) is one of these frequency design methods and has already been compared with other methods, notably using a benchmark proposed by Landau (1995). This benchmark is based on robust digital control of a flexible transmission system. Of eight approaches, Crone control (Oustaloup, et al. , 1995b) came second with a satisfactory ratio of 98.6%, computed for all required specifications. A QFT approach by Nordin and Gutman came first, then other approaches based on: QFT ; a model-free tuning scheme; pole placement with sensitivity function shaping ; Hoo optimization with two DOF ; GPC ; and finally Hoo with one DOF. One can suppose that the first three approaches are the best because they use structured frequency uncertainty domains (called templates by QFT users), which is the least conservative way to model the plant uncertainty.
A totally multivariable approach may sometimes be necessary when the plant diagonal dominance degree (Arkun, et al., 1984) is weak. A multivariable CRONE control design using a non-diagonal controller is thus proposed. The parameters ofa diagonal (for the nominal plant) open-loop transfer matrix are optimized to minimize the variations of the stability margins of the perturbed diagonal elements of the closed-loop transfer matrix.
2. SI SO CRONE CONTROL DESIGN SISO Crone control design is a frequency approach for the robust control of plants under the common unityfeedback configuration. The controller or open-loop transfer function is defmed using integro-differentiation with non-integer (or fractional) order. The required robustness is that of both stability margins and performance, and particularly the robustness of the maximum Q of the complementary sensitivity function magnitude. Indeed, this magnitude (or resonant) peak is strongly correlated to the step response first overshoot. Three Crone control design methods have been developed, successively extending the application field.
LTI square MIMO uncertain plants are often, for the sake of simplicity, controlled using diagonal MIMO controllers. Each element of the controllers is usually designed taking into account only SISO uncertain behaviors of the MIMO plants. When these SISO controllers are designed to achieve rapid dynamics, a decrease of the stability margins of each closed-loop can often be seen. As in the case of QFT design (Horowitz, 1993), several frequency SISO control design methods have been extended to control MIMO plants. After a short overview of the SISO Crone control design, we present two different extensions for MIMO plants.
The first two Crone methods use rear non-integer integro-differentiation and permit the robustness of stability margins at the time of plant gain variation to be obtained. The first is based on a constant phase of the controller around the required open-loop gain crossover
379
(argAmG(jlO~IAmG{jlO~dB) ' As the uncertain open-
frequency lOcg' The controller is defmed by a real noninteger difJerentiator transfer function and the robustness is ensured only if lOcg is within a constant phase frequency range of the plant. The second is based on a constant phase of the open-loop defmed in the Nichols chart as a vertical frequency template. Around frequency lOcg' this template is defined by the transfer function of a real non-integer integrator:
loop transfer function f:X..s) is defined by: fJ...s) = G(s)C(.s) = G«S)L1mG(S)C(S) = fi(S)L1mG(.s) ,(5) where Ab) is the nominal open-loop transfer function, the uncertainty domains related to the Nichols locus of /30{j m) are also defmed by the possible values of the pair (arg Am G{jlO lAm G(jlO ~dB) ' By minimizing the variation of the magnitude peak, the optimal template positions the frequency uncertainty domains so that they overlap the low stability margin areas close to the critical point (Fig. 1) as little as possible. This minimizes the cost function J= (Qmax - Qr)2 , (6) where Qmax is the maximal value of magnitude peak Q.
t
(1)
where the real order n E IR determines the phase placement of the vertical template, namely -mt/2. The shape and a vertical sliding of the template due to gain variation ensure the constancy of the phase margin, of the magnitude peak Q, of the modulus margin (the minimal distance between the open-loop Nyquist locus and the critical point) and of the damping ratio of the closed-loop system (Lanusse, 1994). For more general cases, the vertical template does not slide on itself at the time of parameter variation of the plant. This template cannot therefore ensure the best robustness of the control system. It is more convenient to consider a template that is still defined as a straight line segment (around frequency lOcg) for the nominal parametric state of the plant, but with any direction, and called a generalized template. The open-loop transfer function is now defmed by the restriction in the operational plane Cj of the transfer function of a complex non-integer integrator:
P(')+'+ ;)f 91,,[(",: with n=a+ibEC j
and
n
Fig. 1. Generalized template Nichols loci and frequency uncertainty domains: (a) any given generalized template; (b) optimal template Designing the optimal template through the optimization of only two parameters (as a tangency is imposed) from the three high-level parameters cq,g, a and b, is the initial approach of third generation Crone control design (Oustaloup, 1991). In the version used (Oustaloup, et af. 1995a), the nominal open-loop transfer function takes into account: the accuracy specifications at low frequencies; the generalized template around frequency mcg; and the specifications on the control effort at high frequencies. This transfer function can be described as a transfer function based do frequency-limited complex non-integer integration :
(2)
S=O'+ jlOEC j . In the
Nichols chart, the real order a determines the phase placement of the template, and then the imaginary order b determines its angle to the vertical. From an infmity of generalized templates which tangent the magnitude contour related to a required magnitude peak Qr for the nominal parametric state of the plant, the optimal template can be defined as the generalized template which permits minimization of the variations of Q stemming from the uncertainty domains which in turn stem from the various parametric states of the plant.
/3o{s} = K(~+ S
[
It is easy to show that the multiplicative uncertainty 4J3(s) of the open-loop frequency response, which defines the frequency uncertainty domains (Lanusse, 1994) in the Nichols chart, is invariant and equal to that of the plant. Let an uncertain LTI SISO plant be: G(s} = Go {s }Am G{s} = Go {s}+ Aa G{s} , (3)
wh.,e
m.[(c iJ
l)nl(1l+s/lO) + S/lOh )Q
1 + sf £Oh °1+S/ lO)
Jib lJ-Sjgn(b)
1
(7)
(l+S/lOh}nb '
C, =[(1+ ~q/(I+ :DJ"
(8)
The order n) of f:X..jlO) at low frequencies fixes the steady state behavior of the closed-loop system. The value of nh has to be chosen as equal to or greater than the low frequency order of the plant.
where Go(s) is the nominal plant transfer function and where L1",G(s) and LtaG(s) are respectively multiplicative and additive uncertainty models. The relation between LtaG(s) and L1",G(s) is:
A nonlinear optimization algorithm searches for a vector of independent parameters which minimizes the cost function (6). As the nominal Nichols locus of /3o(s) tangents the Qr M. contour for a given resonance frequency ~ (close to the required closed-loop bandwidth frequency), and has integer orders n) and nh fixed by the designer, only 3 independent high-level
AmG{s} = 1 + Aa G{s )Go 1(s). (4) The uncertainty domains related to the Nichols locus of Go{jlO) are defined by all the possible values of the pair 380
parameters have to be optirnized. The minimization is carried out under a set of shaping constraints on the four usual sensitivity functions. These frequency constraints have to take into account as many specifications as possible. When the optimal open-loop transfer is detennined, the controller is designed from the frequency response of the non-integer transfer function Cni(s), which is deduced from the ratio of /3o(s) by Go(s). The design of the achievable controller consists in fitting frequency response Cni(jm) using a low integerorder transfer function Cin(s).
between the application on the one hand of the small gains theorem to uncertain LTI SISO systems, and on the other of the Gershgorin theorem to certain LTI MIMO systems. Certain LT! MIMO systems and Gershgorin 's theorem (Rosenbrock, et al., 1970; Rosenbrock, 1974; Maciejowski 1989). The Gershgorin theorem can be used for complex matrices of rational functions and, in particular, of transfer functions. The stability of closedloop MIMO systems can thus be studied using Gershgorin circles.
The interest of using Crone control design lies in the fact that it permits not only the genuine uncertainty domains of the plant (owing to the absence of norms) being taken into account, but also all types of uncertainty, whether structured or not. In addition, complex non-integer integration is introduced, which makes it possible to optirnize the open-loop transfer function only using few high-level parameters. The rational Crone controller is then only designed for the optimal open-loop transfer function, thus avoiding the iterative design of a controller with many parameters. Crone control design has already been extended to unstable or nonminimum-phase plants, to plants with bending modes, and to discrete-time control problems (Oustaloup, et aI., 1995b). From a fmite set of generalized templates, a curvilinear template has been defmed to improve the optimality of Crone control (Mathieu, et al., 1996).
Let Q(s) be an LTI and certain n*n open-loop transfer matrix. For all integer i between 1 and n, and for all mof [0, +(0), the Gershgorin's circles are defmed by the centers qji(jm) and radii rj (m) , with: rj(m) =
min(
L IS] Sn and I.,i
Iqij(jm~,
L IS)Sn and j~i
,qjj(jm~l ·
(9)
The n hulls drawn from the union of these circles are called Gershgorin bands, and they contain the Nyquist loci of the n eigenvalue frequency responses of Q(s). The closed-loop system is stable if: - the number of times the point (-1,0) is encircled by the n Gershgorin bands, satisfies the Nyquist stability condition generally used for SISO systems ; - all the n Gershgorin bands exclude the point (-1 ,0). So, the closed-loop system with an open-loop transfer function [Q(s)] can be stable if: rj (m)~II+qii(jm~ 'VI ~ i ~ nand 'Vm E [0,+(0) .(to)
3. MULTI-SISO CRONE CONTROL DESIGN 3.1 Introduction
Uncertain LTI SISO sy stems and small gains theorem (Zames, 1966,· Vidyasagar and Viswanadham , 1982). Let Q(s) be an uncertain LTI open-loop transfer function: (11)
The Gershgorin theorem has already been extended to deal with uncertain multivariable systems. As shown by Arkun, et al. (1984), this extension results in overestimation which can lead to excessive conservatism We here defme multivariable uncertainty domains based both on the structured parametric uncertainty of SISO systems as taken into account by the SISO Crone robust control approach, and on unstructured uncertainty computed from the coupling elements of the plant transfer matrix (Lanusse, et aI. , 1996). This stems from a comparison of the sufficient stability conditions given by the small gains theorem for uncertain SI SO systems, and the Gershgorin theorem for certain MIMO systems. Because these conditions are similar, for an uncertain MIMO system it is possible to defme a closed-loop stability condition, which takes into account both the uncertain off-diagonal elements, and the uncertainty on the diagonal elements, of the openloop transfer matrix. This definition permits an analysis of the robust stability which is as little conservative as possible. These results can be used by any initiallySI SO design approach using structured frequency uncertainty domains.
where Qo(s) is the nominal transfer function and, L1aQ(s) the additive form of the uncertainty. The small gains theorem states that the closed-loop system with an openloop transfer function Q(s) can be stable if: r(m)~II+Qo(jm) 'Vm E [0,+00) , (12) where r( m) is defmed by: r(m)~maxIQ(jm)-Qo(jm)
.
(13)
At each frequency, the left term of (13) can be interpreted as the radius r( m) of a circle that localizes the uncertainty. Its center is given by the nominal openloop frequency response. The closed-loop system is always stable, if it is stable for the nominal parametric state, and if all the circles exclude the point (-1,0). Comparison and illustration of stability conditions. It is interesting to compare the stability condition ( 12), obtained through the small gains theorem for uncertain SISO systems, with (10), given by the Gershgorin approach for nominal MIMO systems. These two conditions are illustrated by Fig. 2 which shows that the two theorems lead to similar graphical stability conditions. Indeed, the two formalisms use the distance between: the edge of the circles depending on the
3.2 Modeling the uncertainty of MIMO systems
We demonstrate that the definition of multivariable uncertainty is obvious when a parallel is established
381
uncertain or multivariable nature of the open-loop system, and the point (-1,0). So, in the nominal MIMO case, the coupling elements can be considered as uncertainty on a SI SO system. This is very important if a multi-SI SO design approach is considered. Thus, the certain or uncertain coupling elements can be considered as further uncertainty on the uncertain diagonal elements of the open-loop transfer matrix. So, these diagonal elements are now a little more uncertain.
rj{w)=
Construction of multivariable uncertainty domains of the diagonal elements. Crone methodology uses uncertainty domains constructed in the Nichols chart. Nevertheless, in order to facilitate their presentation, the uncertainty domains are here constructed in the complex plane. The multivariable uncertainty domains related to the diagonal elements, are constructed both through the frequency responses. of the diagonal elements for all the parametric states 01' : the plant, and through the unstructured uncertainty' related to the off-diagonal elements of the plant. For example, Fig. 3 shows the frequency response of a diagonal element for its nominal parametric state (numbered 0), and for the three other parametric states considered (numbered 1,2 and 3). Gershgorin circles from further off-diagonal column element uncertainty are drawn around each frequency response.
Consideration of off-diagonal plant elements. The methodology now proposed aims not to bound the elements resulting from the uncertainty on the diagonal and off-diagonal elements of the open-loop transfer matrix.
.
Take an n*n uncertain MIMO plant G and a n*n multiSISO (diagonal) controller C in series. As the previous section shows, the row and column off-diagonal elements of G can be considered as uncertainty on the diagonal elements. For a given parametric state, the additive uncertainty on gilD w) is given by:
L
1:5,:5n
Igij (j W ~,
L
1:5,:5n andj",j
Ig jj (j W ~l '
l:5j:5n
Igij(~W~,I+ l:5j:5nL .lgAJw Igjj(~W~l. ~
.lgAJw ~
and '''''
rio(~~f4.)
ril(~)0giil(j~)
(14)
Fig. 3. Nominal and perturbed diagonal elements, and their Gershgorin circles
(15)
and '''''
The multiplicative uncertainty on f3ji (j w), a diagonal element of the open-loop ~(s), with
Cl (s )gll (s)
p(s) = G(s)c(s) =
: [
0~)
0ii,(j~) ri3(f4.)~ ri,(~) G.
This uncertainty can be written in a multiplicative form: min[l+ L
(18)
9t(.)
(a) (b) Fig. 2. Graphical illustration of: (a) small gains theorem; (b) Gershgorin's theorem
andj~j
Igjj(jW~.
It is true that if the row elements are smaller than the column elements, such a determination of uncertainty can be a little conservative. It is also very easy to show that only the column diagonal dominance remains true in the case of a control with a multi-SISO controller in series.
:3(.) -1
min[
L
l:5j~n .lgAJw ~ and,,,,,
... cn (s )gln (S)] ". : ,(16)
The construction of tpe multivariable uncertainty domains consists in the determination of the hull that takes into account both the uncertainty related to the diagonal element (SISO uncertainty domain) and the further multivariable uncertainty. In fact, the multivariable uncertainty domain is defmed by the convex hull that includes all four Gershgorin circles (Fig. 4). Of all known ways of construction. this is the least conservative because it is the most realistic.
cl(S)gnl(S) ... cn(s)gnn(s)
is given by:
. [
mm 1+
L
ICjGw)gijGw~ . . ,1+
l:5jSn IC;(Jw )gAJw ~
and,,,,,
~
L
IgjjGw~l . .(17)
ISj:5n.lgjiGw ~
and,,,,,
When the diagonal controller elements are designed, (15) and (17) show that only uncertainty related to the off-diagonal column elements remains unchanged and is independent of the controller. To permit independent design of each controller element, it is this column uncertainty which is used. So, the circles chosen to represent the multivariable uncertainty on Gji (s), each have their radius in the complex plane:
Fig. 4. Construction of multivariable uncertainty domains in the complex plane In the Nichols chart, the method of construction used is the same, except that the circles are replaced by ellipsoids if the plant is column-diagonal dominant.
382
margins are obtained when each generalized template defmed by f30,um) tangents its specific Qri M. contour. The order nbi of each ,8,u m) at low frequencies fixes the accuracy of each closed-loop. The orders nhi of the open-loop transfers at high frequencies permit the elements of a controller to be proper. The controller can be defmed by the non-integer transfer matrix:
The construction of the multi variable uncertainty domains for each diagonal element of a MIMO plant, and the robust control design for each new uncertain SISO plant, is the multi-SISO Crone control approach. If the multivariable uncertainty domains are nevertheless too wide, decoupling techniques can be used to decrease the effects of the off-diagonal elements on the size of the multivariable uncertainty domains.
Cni(s)=GO(stlpo(s). (24) The inverse of the transfer matrix of the nominal plant is written as:
4. MIMO CRONE CONTROL DESIGN
Go ( S ) -1 = [nij(s)] -_-
4.1 Introduction MIMO Crone control design is basically the design of decoupling open-loops. It is the most general MIMO Crone control design. Wang and Davison (1975) first proposed decoupling control systems using static output feedback. Then, Desoer, et a/. (1981; 1984) proposed a parameterization of the set of stabilizing diagonal closed-loop transfer matrices. Later many authors (Dickrnan and Sivan, 1985; Desoer and Giindes, 1986; Lin and Hsieh, 1993) extended the results of Youla, et al. (1976) to show the optimality and Hoo robustness of diagonal approaches. An advantage of the MIMO Crone approach is its ability to describe each diagonal open-loop with few parameters through the use of complex non-integer integration orders. Another advantage is its ability to take into account the uncertainty of the MIMO plant without over-estimation. 4.2
where dij(s)andnij(s) are Hurwitz. Then (24) becomes Cni (s) = [
nhj
and
SOi(S) =
•
,80i(S) 1+ ,80i(S) 1 1 + ,8oi
( )"
(26)
(27)
~ max(deg(nij(s))-deg(dij(s))) .
(28)
1:S}:Sn
4.3 Optimization of the open-loop behavior
Perturbed transfer matrices T(s) and 8(s) are nondiagonal and their frequency responses are: T(jm) = [I + G(jm)cni(jm)tIG(jm)cni (jm) (29) l . (30) and S(jm) = [I + G(jm )coi (jm Nevertheless, it is possible to defme n perturbed openloop frequency responses: T·Gm) 1- S··Gm) ,8i (jm) = II • = ". for 1 ~ i ~ n ,(31)
)t
I-1jiGm)
SiiGm)
where T;,{jw) and Si,{jm) are the diagonal elements of T(jw) and S(jm). The open-loop frequency multiplicative uncertainty Lli(jm) is defined by:
L1i(jm) = ,8oiGm tl ,8iGm). (32) A major difference with the SISO case is that the openloop uncertainty is now different to the plant uncertainty. The open-loop uncertainty domains are not only computed from the plant, but also from the controller. The nominal open-loop behavior is defined as a set of n generalized templates, each tangenting a required Qri M. contour and ensuring a gain crossover frequencym.,gi' From the infinite number of sets within this definition, the optimal set of n templates is that which minimizes the peak magnitude variations. The cost function that the optimal set minimizes is:
1I0{s}= diag (fJOi{S)1sisn' (19) The nominal complementary sensitivity function and sensitivity function transfer matrices are
To'(s) =
. diag[,Boi(s) Si:Sn '
IS.Sn
From (27) the minimal value of each nhi is:
The decoupling objective with an output feedback leads to a nominal diagonal open-loop transfer matrix with n elements:
with
~ ()
nij(s) ] Cni(s)= [~(),80i(S) . . dij s IS,Sn l:SjSIl
or
For minimal-phase square n*n plants, our aim is to parameterize a set of open-loop matrices that reach the four following objectives: - perfect decoupling for the nominal plant - the accuracy specifications at low frequencies - the required nominal stability margin of the control system (behaviors around the required cut-off frequencies) - the specifications on the n control efforts at high frequencies.
and
1
n.. (s)] dij s
l:Sj:S1l
Parameterization of the nominal open-loop transfer matrix
To(s) = [J + Po(s )tlpo(s)= diag[Toi(s )t,isn l So(s) = [J + Po(s )t = diag[Soi(s )1sisn '
(25)
,
dij (s) ISi:Sn ISjSn
(20)
(21) (22)
J =
(23)
~(Qmaxi _Qri)2 + (Qmioi _Qri)2,
1=1
S
The three other objectives are obtained when each openloop transfer function /3o.{s) is defmed by relation (7) of the SI SO Crone approach. The n nominal stability
Qri
Qri
(33)
where Qmin i and Qmax i are the minimal and maximal values of the peak magnitude of the perturbed transfer function T;,(s). 383
Simultaneously optimizing the elements of the nominal open-loop transfer function does not permit the use of varying uncertainty domains for easy modeling of uncertainty. However, it does bring about both optimal placement and shaping of the varying uncertainty domains. The vector to be optimized is the set of the n vectors of independent parameters of each generalized template. Thus the number independent parameters is 3n. The optimization procedure is a constrained nonlinear optimization which minimizes (33) and respects the constraints imposed by the following requirements: - shaping the diagonal elements of the four sensitivity function matrices - bounding the off-diagonal elements of sensitivity function matrices to decrease the coupling effect resulting from plant uncertainty - stabilizing all the elements of sensitivity function matrices by respecting the multivariable Nyquist theorem for all possible plants. When the optimization is complete, each rational integer element of the controller transfer matrix is identified from the frequency response of(24).
5. CONCLUSION The multi-SISO and MIMO Crone control designs proposed here are extensions of SISO Crone control design to the multi variable domain. This study only deals with the case of square minimum-phase plants. The brief review of SISO Crone control in section 2 shows how the uncertainty must be modeled to provide simplicity of use in SISO Crone design. Section 3 presents a multi-SISO approach based on the successive determination of the elements of a diagonal controller using SISO Crone control design. The Gershgorin results related to closed-loop stability of the nominal MIMO systems are reviewed. A comparison is then made between the Gershgorin approach, and the approach using the small gains theorem for analysis of the closed-loop stability of uncertain SISO systems. It is shown that the coupling effects on closed-loop stability can be studied, if the off-diagonal elements of the plant are taken into account as further uncertainty on the diagonal uncertain elements of the plant. Thus new multivariable frequency uncertainty domains have been constructed through both the structured representation of the SISO uncertainty, and the Gersgorin circles related to all the parametric states of the plant. This new type of uncertainty representation permits the design of less conservative robust multi-SISO controls of uncertain MIMO plants. Section 4 presents a complete extension of SI SO Crone control design to MIMO plants. MIMO Crone control design is based on a complex non-integer parameterization of each element of a diagonal openloop transfer function matrix. These elements have to ensure nominal performances on transient and steady states. A simultaneous optimization of these elements minimizes the variations of each stability margin resulting from the plant uncertainty, while respecting constraints on the sensitivity function matrices. By using the parameterization of the set of all achievable 384
decoupled transfer functions proposed by Lin and Hsieh (1991), the MIMO Crone approach can now be used for nonminimum-phase or unstable plants (Oustaloup and Mathieu, 1999). The design of multi-SISO and MIMO Crone control relies on complementary approaches. Multi-SISO Crone control design is easy to use for MIMO plants, but is still a diagonal design. MIMO Crone control design is an entirely multi variable tool which is more efficient, but also more difficult to use. One or the other has to be chosen, depending on the multi variable degree of the plant, to optimize the ease-of-use/perforrnance trade-off.
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