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Robust multivariable controller design methodology: stability and performance requirements
Chemrcal Ewineermg Science, Vol. 46, Nu. S/6, pp. 1299-1310, Printed in Great Britain.
1991.
ooE-250Y/91 53.00 + 0.00 .a 1991 Pergamon Press plc
ROBUST MULTIVARIABLE CONTROLLER DESIGN METHODOLOGY: STABILITY AND PERFORMANCE REQUIREMENTS J. L. FIGUEROA, Automatic
0.
E. AGAMENNONI,’
Control Group,
(First
PLAPIQUI
received 21 December
A. C. DESAGES
and J. A. ROMAGNOLIt 8000 Bahia Blanca, Argentina
(UNS-CONICET),
1989; accepted in revisedform
15 June 1990)
Abstract-The problem of the synthesizing controller which verifies joint specifications of robust stability and robust performancehas been formulated by the p-theory, but the problem has not yet been solved. In this paper an algorithm is presented which ailows the evaluation of a suboptimal solution of the unsolved problem stated above. This solution is carried out by using a simple parametrization of controllers and an algorithm to adjust the IIfl(M)ll, to a desired value. Application to a practical example is discussed by controlling a triple effect evaporator.
1. INTRODUCIION Frequency
response
techniques
have produced
effect-
ive compensator
designs for a wide variety of systems. Moreover, in the late seventies, singular values were found for the correct multivariable generalization of classical SISO concepts. Since their application in control theory, a new generation of the so-called “robust” design approaches has appeared, which produces controllers under the assumption of model uncertainty [see, for instance, Doyle and Stein (19X1), Hung and MacFarlane (1982) and Agamennoni et al. (1985, 1988 and references therein)]. Robust control theory is now able to deal with a large class of uncertainties, including unstructured and structured perturbations in frequency and time domains (Doyle, 1982; Yedavalli, 1985), and highly structured perturbations (Rotstein et al., 1989; Bhattacharyya, 1987). In the eighties, the design techniques which had previously treated problems of robust stability were extended to robust performance. Regarding this subject, see for example, Doyle et al. (1982), and the applications of this technique in practical examples (Skogestad and Morari, 1988; Arkun and Morgan, 1988). The problem of synthesizing a controller of a perturbed plant has been formulated by the p-theory. Referring to Fig. 1 the problem can be posed as follows: find K(s) such that K(s) stabilizes all possible plants GP and 11p(M) 11 co < 1, where M is an operator which includes the performance requirements. This problem has not yet been soIved. H, synthesis solves a closely related problem: find K(s) which stabilizes GP(s) and IIMII, < 1. In this paper a methodology is presented which allows the evaluation of a suboptimal solution of the unsolved problem stated above. This solution is car-
ried out by using a simple parametrization of controllers and a previous algorithm to adjust 11p(M) llm to a desired value (Agamennoni et al., 1989). A single controller parameter is used as adjustable parameter. Furthermore, a discussion, in terms of the control of a tripple effect evaporator, provides the appropriated framework for the analysis of the inherent trade-off between stability and performance. We believe that the methodology proposed in this paper is truly user friendly, because the designer may specify and adjust the performance to satisfy his robustness constraints. The resulting compensated system has stability and performance margins which may be fixed beforehand. This paper is organized as follows. In Section 2 some performance criteria are defmed. In Section 3 the necessary background is introduced. In Section 4 a method to adjust the robust performance requirement is described together with a step by step algorithm. In Section 5 an example is given, before concluding in Section 6. 2 SOME PERFORMANCE Let
us consider
the standard
CRITERIA feedback
configura-
tion illustrated in Fig. 1, where K(s) is a controller matrix (n x n) and G’(s) is a member of the family of feasible
plants
II = {Gp(s):
II and
where
GE(s) = Gii(s)
II is characterized
+ Aij(s)
as
with
IA&)I
G ~ij)
(1)
where G is the nominal matrix transfer funct!.,a of the plant. This family implies that the entry ij of the set of
fA. Desages and 0. Agamennoni are with the Department of Electrical Engineering and the Comisi6n de Investigaclones Cientificas de la provincia de Buenos Aires. *Author to whom correspondence should be addressed.
Fig. 1. Multivariable 1299
closed loop system.
J. L. FIGUEROA et d.
1300
feasible plants lies in the circle with centre in G(s) and radius Pij. The objective of using feedback control is to keep the controlled output, y(s), closer to its set point. The performance specifications define what is referred to as “closer”. Furthermore, these performance requirements should be satisfied in spite of the plant uncertainty. Consequently the ultimate goal of the controller design is to achieve robust performance. The implications of these requirements are easier to understand if we consider some subobjectives which have to be satisfied in order to achieve this goal. These subobjectives are defined as follows.
2.1. Nominal stability The controller K applied has to be stable.
to the nominal
plant G
‘41
’} 6 W,(w)
for all w
6’ “I
2.3. Nominal complementary sensitivity The system formed by the nominal plant G and controller K should satisfy the following condition: + GK)-‘GK)
_ ..
6 W,(w)
for all w
(3)
where W, is a scalar function of frequency. This requirement implies a bound over the gain between the reference input r(s) and the system output y(s).
ia,
. .
(6)
and
M=(I+KG)-'K.
(7)
2.6. Robust sensitivity The closed loop system of Fig. I should satisfy the subobjective 2.2 for all plants G, E lT. This is verified if the nominal system is stable, and if it satisfies the inequality (2) for the nominal plant and < 1
is computed
for all w
(8)
with respect to the structure
A = diag {A,, AD >
(9)
with o(A,)
(2)
where W, is a scalar function of frequency. This requirement implies a bound over the gain between the noise input d(s) and the error system e(s), i.e. this gives a measure of the disturbance rejection index of the control system.
1
6 1”
with IS,1 6 pij
p(M)
2.2. Nominal sensitivity The system formed by the nominal plant G and the controller K should satisfy the following condition:
c{(I
_. . _
A, =
where p(M)
a((1 + GK)-
..
6 l/w,
(10)
and M=
-(I+KG)-‘K
(I+KG)-‘K
-((I+GK)-’
[ (I + GK)-’ Furthermore,
the system is robustly
1. (11)
stable.
2.7. Robust complementary sensitivity The closed loop system of Fig. 1 should satisfy the subobjective 2.3 for all plants G, E lT. This is verified if the nominal system is stable, if it satisfies the inequality (3) for the nominal plant and
P(W -s 1 where p(M)
is computed
for all w
(12)
with respect to the structure
A = diag (A,, A,} 2.4. Nominal input bound To have practical significance the gains from inputs of the controlled system, r(s), to inputs of the plant, U(S), must be bounded. This is expressed as 5((I
+ GK)-‘K}
< W,(w)
for all w
2.5. Robust stability The closed loop system of Fig. 1 should remain stable for all possible plants GP E XI. This is verified if the system is stable for the nominal plant and if p(M)
< 1
for all w
c(A,)
(14)
-(I
+ KG)-lK
- (f + GK)-‘GK Furthermore,
the system is robustly
stable.
1(13 ’
2.8. Robust bounded input The closed loop system of Fig. 1 must satisfy the inequality (4) by all feasible plants Gpe lT. This is verified if the nominal system is stable, if it satisfies (4) for the nominal plant and P(M)
(5)
where p is the structured singular value (SSV) (Doyle, 1982), computed with respect to the structure
< l/W,
and
(4)
where W, is a scalar function weighting used to specify the frequency range over which the plant inputs must be bounded. This range may be determined by the actuators dynamics.
(13)
with
where p(M)
-= 1
is computed
for all w
(16)
with respect to the structure
A = diag {AI, AB}
(17)
Robust multivariablecontroller design methodology
1301
with @((I + GK)-‘K)
< LO/W,(w)
(18)
and
M _
(I + KG)-‘K
-(I
+ KG)-‘K
-[
(I + KG)-‘K
-(I
+ KG)-‘K
1.
(19)
Furthermore, the system is robustly stable. The fulfilment of robust sensitivity and robust complementary sensitivity implies robust mixed sensitivity (RMS). In the following lemma this is formally expressed. Lemma 1. Consider the system of Fig. 1, formed by the controlled K(s) and the plant GP~ II. If the closed loop system satisfies nominal stability, nominal sensitivity and nominal complementary sensitivity, then these characteristics are satisfied for al1 perturbed plants if and only if P(M) (
1
for all w
where p(M) is computed with respect
(20) to the
A = diag {AI, Aa, AR}
structure
Fig. 2. Multivariable closed loop system with virtual uncertainties. with three fewer A-blocks the upper bound becomes an equality. Furthermore, the corresponding optimization problem is always convex. Therefore for less than three blocks the optimization algorithm used computes p exactly. In the case of having more than three blocks inf c?(DMD-~) is an approximation to the exact p value. 3.
MULTIVARIABLE
CONTROL
DESIGN
a previous work (Agamennoni et al., 1988) a multivariable controller design methodology was presented, which leads to a controller formulation given by In
(21)
K’(jw)
=
G- ‘(jw)
diag {mi(jw)}
(23)
with Al, A,, and AX defined as before, and
(I + GK)-’
-
(Z + KG)-‘K
-
(Z + GK)-’
(Z + GK)-
(I + KG)-lK (I + GK)-’
’GK
Proof The proof is straightforward, from Doyle et al. (1982). The following remarks are in order.
(I + GK)
it follows
2.9. Remarks (i) The inverse of the p-function of (5) is naturally interpreted as a stability margin, because it determines the minimun value by which the uncertainty bound p must be multiplied to make the system unstable for some GPEII. (ii) From the analogy of(i), in a more general case, the inverse of the p-function of eqs (8), (12), (16) and (20) can be interpreted as a robust performance margin because it determines the minimun value by which the desired bound of the performance must be multiplied to satisfy it, or to make the system unstable for some GP~ lT. (iii) The p-function of expressions (8), (12), (16) and (20) must be calculated considering the perturbations associated with A*, A, and AD as unstructured blocks of dimension (n x n). These blocks may be interpreted as virtual uncertainty appropriately placed, see for example Fig. 2 (Doyle et al., 1982). We will return to this consideration later. (iv) It is known that p(M) < inf b(DMDD
’);
1
.
(22)
‘GK]
where a,(s), i = 1,2, _ . , n, are complex scalar functions. A suitable choice of the functions ai may be used to introduce closed loop specifications. The controller described in (23) is called ideal, because if it were physically realizable the closed loop system would exactly satisfy the design specifications. The ideal controller is approximated with a controller of desired order C(s). This design step will be discussed in the followjng subsection. Let xi(s) be as follows:
pi exp ( - SOi) aiW
=
1 + &(l
- exp ( -
sOi))
i = 1,2,.
. . .n (24)
where fii(s) are complex functions and 0; are delayed terms. The ai of eq. (24) allows us to state a multivariable predictor control scheme. In the nominal case the following equalities are verified:
(25) SOi)
1 + pi(s) (1 -
exp ( -
sOi))
(26) (27)
1302
J. L. FIGUEROA
where H,(s) is the transfer matrix from the reference input r(s) to the output y(s), and HI(s) is the transfer matrix from the disturbance input d(s) to the output
With a similar reasoning for a robust complementary sensitivity problem, we obtain the following bounds:
Y(S). The evaluation of Oi is beyond the scope of this work. However, the designer may interactivefy assume different values of Oi searching for the set which produces a closer approximation between the ideal controller and C(s) (Agamennoni et al., 1989). In those problems in which the delay compensator scheme is not necessary (i.e. Oi = 0, i = 1,2, . . . , n), eqs (24j(26) may be written as %cs) = Bits)
Hd(4 = diag
(24’)
1
Pi(')
Hr(s) = dk
1
pics)~
+
(25’)
l
et al.
I1+iitsl}.
In the next section we will consider the concept of robust performance in the context of this methodo-
{(I
+ GK’)-‘GK’},
@r(w)
{H,}iil (1 + E,(W)) 1,
if
i = j
E,(W)
if
i#j
r(Z +
KG)-
’K’
(I + GK’)-’
1
(Z + GK’)-’
From the previous section the function H, of eq. (25) is a natural choice of IV, of (2) in the nominal sensitivity problem. When we introduce plant uncertainties, it is obvious that these specifications break down for some plant GP~ II. For this reason we consider
l{H,tiil
W, = (1 + dw))max
dw)~R+
(28)
for the analysis of the robust sensitivity problem, where cd is a positive real scalar which defines the tolerance band of the sensitivity bound. Moreover, from (25) we obtain the bound over the norm of U, and the structure of H,; for this reason the analysis of section 2 will, in general, be conservative. To avoid this problem, we consider the inequality (2) element by element, i.e. {(Z + GK’)-‘)ij
G
I{
(33)
and in (10) I
D,lij
In an analogous shown below.
<
‘/(
l-C
wrIij.
(34)
form, lemma 1 may be rewritten as
Lewvna 2. Consider the system of Fig. 1, constituted by the controller K’(s) and the plant G,EII. If the closed loop system satisfies nominal stability, nominal sensitivity and nominal complementary sensitivity, then these characteristics are satisfied for al1 perturbed plants if and only if P(M)
M =
(32)
where
logy. 3.1. Robust performance in the melhodology context In Section 2 we introduced some performance objectives for the design, but we did not explain how IV, and W, functions were established.
< @‘r(w)}ijl
where p(M)
for all w
G 1
is computed
(35)
with respect to the structure
A = diag I&,
AD. A,}
(36)
with AI, AD and A, defined as in (31) and (34) respectively, and - (I + KG)-’
-(I
K’
- (I + GK’)-’ (I
+
+ K'G)-lK (I + GK’)_
1
.
(I + GK’) - i GK’
GK’)-‘GK’
(37)
I
In this case, the virtual uncertainty AD, AR of Fig. 2 must be considered as structured blocks. Figure 3 shows the block diagram of the resulting control scheme, where C(s) is an approximation to K’ of a given order, calculated as in the following section. 3.2. Controller approximation The approximation procedure in an order reduction framework may be formulated as an optimization problem. Assume that the frequency response matrix K’(jw) = [k,,(jw)] of eq. (23) has been calculated and let the contr&ler C(s) be c(s)
a,jo = Ccij(s)l= b,jcsI
where aij(s) and b&)
are polynomials
(38)
and b,(s)
is
@‘d(W>}ijI
where
@d(W.,t1 + Ed(W)) [ E*(W)
(Hd}ij
and in (10)
if if
i=j
i#j
(30)
Fig. 3. Multivariable closed loop system with the proposed delay compensator.
Robust multivariablecontroller design methodology manic and Hurwitz. These polynomials should be chosen in order to minimize a norm II-11 of the distance between K’(s) and C(s): min
+LbW
IIK’(jw) -
C(W)
tions. This technique allows us to satisfy the following design criterion: max C&M)1 =
II.
w
(3%
min .Z *.b
(40)
is a continuous function of M.
Lemma 3. p(M)
where
sn
I k,, (jw) - c&v)
I2 Wii(w) dw
Lemma 4. Let M(s) be a matrix transfer function without poles on the imaginary axis. Then p(M(s)) is a continuous function of the frequency.
(41)
in which a positive weighting matrix W(w) = [ Wij(w)] has been included. In the practical implementation, the integral of the right was replaced by a summation over a set of suitably spaced frequencies a. Agamennoni et al. (1989) proposed a weighting matrix of the form Wij(W) =
l
2,
Pr& See Agamennoni et al. (1989). Suppose now that the functions Bi(s) of eq. (24) are modified as follows: qER
Pi(s) = q/7is
O>q21
(45)
giving
(42)
I kf,(iw) I Wi
(4)
PMS
where p is calculated with the structure of (5) and ,uMs is the maximun specified value of the p-function for all w. Immediately, we extend this idea to the adjustment of robust performance margin, but first remember two important properties of the p-function.
For computational convenience, the Frobenius norm II - IIF may be used. This norm leads to n2 scalar optimization problems with objective functions
J=
1303
K”(jw)
normalizing the error between cij and kij and enhancing approximation at low frequencies. The evaluation of C(s) by the minimization of expression (41) with weighting function (42) gives a good open loop controller. However, the approximation errors may have serious closed loop consequences. This problem can be avoided by means of the
= qK’(jw).
(46)
Expression (45) relates the time constant of the closed loop system (H,(s)) to the controller gains. We wish to explore the effect of varing q over p. The following result is needed. Lemma 5. Consider M as in eq. (37) of lemma 2, then M as a function of g is continuously differentiable. Proof: Consider M as in lemma 2, now M function of q in the form
q(I+ M =
-
qK’G)-‘K’
q(1 + qGK’)-
(I + qGK’)-’
= Cl + s((Z
+ GC)-’
1
q(Z + qK’G)-‘K’
(I + qGK’)q(Z + qGK’)-
1GK’
GCjl$&
1
1,
(43)
M,
=
M=M,M,
(48)
(1 + qK’G)-’ 0
0 (1 + qGK’)-’
0 0 (I + qGK’)-
0
and This function enhances the low frequency range and the critical frequencies where the closed loop has the worst performance.
qK’ 1
M,= [
PERFORMANCE
(47)
’ GK’
where
0
4. ROBUST
1
Furthermore, M may be expressed as
following weighting function: wij(w)
-
q(Z + qK’G)-‘K
- (I + qGK’)-
(I + qGK’)-’
is a
ADJUSTMENT
In a previous work, Agamennoni et al. (1989) proposed a simple controller design procedure for an MIMO plant under additively structured perturba-
1
-
qX’
qK’
-1
1
qGK’
qGK’
1
1 WV (49)
1
Now, as the system is nominally stable, it verifies p(K’G) p(GK’)
< 1 < 1.
J. L. FKXJEROA et al.
1304
Then, we may write (I + qGK’)-’
= I -
qGK’ f
(I + qK’G) - ’ = I - qK’G
q’(GK’)‘-
__. . . . . .
+ q*(K’G)*
-
. ..
. ..
If we replace this in eq. (49) it is clear that MI is a polynomial function of 4, and from eq. (48) M is also a polynomial function of q. Now, it is well known that the polynomial function is a continuously differentiable function. From these considerations we conclude that M is a continuously differentiable function of 4. Lemma 6. Consider M as in eq. (37) of lemma 2, then JJas a Function of 4 is continuously differentiable. ProofI
By definition
of the p-function
p = s;p
C(DMPD-‘)
(51)
we defined ti=DMPD-’
(52)
then, it verifies A*n?u
= &J
(53)
Of
Fu = p*u
(54)
where u is the right singular vector associated with the p-function. By differentiating (54) with respect to 4, we obtain dF au --u+F-=p*_++_
&
and multiplying
8~
a4
aq
a$ a4
cm
on the left by u* gives
Now, from lemma 5, it is easily verifiable that F is a continuously differentiable function of 4, which implies that, from (56),
w
(57)
&t is a continuously differentiable function of q. We are ready to state the main result.
4.1. Design algorithm Finally, we will summarize the results of the present paper in a step by step design algorithm to satisfy the following criterion: (58) where p(Ms is the maximum specified value of the pfunction for all w E n. Expression (45) is a good starting point since it relates the time constant of the closed loop system (H(S)) to the controller gains. A variation in the gain factor q E R may be taken as a variation in the closed loop time constant of the nominal system. Hence, we have a single parameter for adjusting the ideal controller K’ [see (46)] with a proportional shifting of all the time constants specified for every single loop. Besides, it may be easily shown that the frequency response of the normalized synthesis relative error of (41) will remain constant for all values of q. This is an important property of K’ which allows the evaluation of q with any initial controller C(s). Remark. It is important to note that since GP(s) is open loop stable, then there always exists a sufficiently small q for which the closed loop system is stable. The design specifications are:
(1) dominant time constants of each loop, (2) time delay for each loop,
(3) slow disturbance loop, (4) the performance systems.
modes to reject in the different margin of overall
closed loop
First, the designer may try to achieve the desired dominant time constant in each single loop (nominal performance objective). Then, if the performance margin is smaller than the one specified, the algorithm will provide the minimum proportional reduction of loop gains so that the robust performance specifications will be satisfied (robust performance objectives). A step by step procedure for the computation of the controller that achieves the design goals described above could read as follows:
(1) Specify the desired dominant time constants zi,
Theorem 2. Let us suppose that ,~(jw) attains its maximum in a unique isolated frequency w,. Then if the derivative of p at w, is not equal to zero, the overall performance margin may be decreased by a suitable variation of 4. Proof: This is evident (1989). Later, in the example formance trade-off will be effect of variation on q to robust performance in the triple effect evaporator.
from
Agamennoni
et al.
section, the stability-peranalysed by discussing the adjust robust stability and context of the control of a
i
1,2, . . ,n for each single loop, the desired pMs and 6 > 0. (The formulation and evaluation of the set of fii from the desired time constants are introduced in Appendix A.) (21 Evaluate the frequency response of K’ = G- ’ diag (aI (jw)}. (3) Synthesize K’ with the desired controller structure C(s) according to the discussion of Section 3. (4) Set 4 = qO. (a) Set C,(s) = C(s)+ =
(b) If the compensated system GC, is unstable, reduce q and return to 4a.
Robust multivariable controller design methodology STEAM
PRESSURE
1305
CONTROL TAP ‘.‘.T.~.“-._.-
r_____id?____
STEAM
_A
f
WATER
1
PRODUCT
-.2 i
I
-____-. -.-I-
JUICE PREHEATING JUlCE CONCENTRATION STEAM CONOENSATE TAP WATER VAPOR
Fig. 4. Schematic diagram
of
the
industrial scale triple effect evaporator study in the example.
Table 1. Evaporator G(s)
(5) Evaluate max (~1 over w, and note with wM the frequency of the maximum. (6) If P(W,Y 4) > PMSI modify q in order to satisfy the design criterion (58) by using the following iterative
I-_
PRODUCT COOLING
Entry
Evaporator 0.83 exp ( -
procedure:
s 0.077)
(s 0.0032 + 1) (S 0.0687 + 1)
i=O
0.00
I PL(w, a) - pMS 1 > 6
do while
-
1.3 exp ( - s 0.082)
(s 0.0042 + 1) (s 0.1422 + 1) -
22
i=i+
1.86 exp ( -
s 0.027)
(SO.165 + 1)
1
end do go to 5. (7) End. 0.96
Some implementation aspects of the routines of steps 4 and 5 may be found in Agamennoni et al. (1989). 5. EXAMPLE OF APPLICATION The feasibility of applying the theory presented in this paper is illustrated in the detailed multivariable control design of an apple juice evaporator. A triple effect industrial scale evaporator is considered with counter current pre-heating and parameters tuned to fit real parameter data. This evaporator is shown schematically in Fig. 4. The nominal transfer function at the desired operation point is a matrix of dimension 2 x 2 given in Table 1, with the concentration and level in the third effect as output, and inlet steam pressure and output flows as input. The model reflects the typical nonlinear, time-variant characteristics of these types of equipment: asymmetric behavior (i.e. different responses to positive and negative disturbances and set point changes), non-linear dependence with the amplitude of perturbations, changes in the heat transfer
0.0
1 0.01
L 0.
I
I
L
I.0
10.
I 100.
I 1000.
Fig. 5. Frequency response of the relative uncertainty mattix p,.
coefficients due to filthiness, etc. In view of these considerations structured additive uncertainty is applied to each Gi,(s). It is modelled as
P,,l(s) is shown in Fig. 5. As a design objective we choose a closed response of the system with a time constant of where
zL = 0.1 (h)
i = 1,2.
loop
J. L. FKXJEROA
1306
et al.
1." f
1.0
".80 _
0.8
-
0.60 -
0-6
-
0.4" -
0.4
".2" -
07 I cl.,
0.01
1 1."
Icl.
LOO.
0.01
1.
1".
Fig. 6. Frequency responseof the entries(1, 1) and (2, 2) of the complementarysensitjvity matrix. (a) Ideal controller K’. (b) Synthesizedcontroller C. (c) Adjusted controller C,,.
Table 2. Controller C(s) Entry
Controller 11.14 + 948.3 s + 60.67 s2
1,1
s(1.0 + 122.1 s) 0.00
L2 I
0. .01
.I
I 1.
I 1”.
2J
i 100.
Fig. 7. Frequency response of the p-function as a measure of the complementary sensitivity.(a) Synthesizedcontroller C. (b) Adjusted controller C,,. First, the frequency response to the ideal controller K’(jw) Ceq. (23)) is approximated by a PID controller, say C(s). This controller is included in Table 2. A weighting function such as eq. (43) for the synthesis of C(s) was used. The frequency range of this approximation is [0.01,100]. In Fig. 6, we show the diagonal entries of the complementary sensitivity matrix (i.e. {(I + GK)-i GK},,), for the ideal controller K’(s) [see curve a], and the synthesized controller C(s) (see curve b). This plot shows that the controller C(s) is a very good approximation to K’(s). Moreover, curve b presents a more abrupt slope in the crossover frequency than curve a. This characteristic may be desirable for several well known reasons. The specification For robust complementary sensitrvtty is max (p(w)) < 1 W where SSV are calculated as in Section 2.5 with *&l
= diag i(l + ~~(~11 I(H,)iil}
(61)
en(w) = 0.01 IWO’.
(62)
where
-
7.67 -
590.0 s -
54.83 s=
s (1.0 + x7.1 s) ~ 5.00 - 270.34 s -
292
46.0 s2
s (1.0 + 64.4 s)
Table 3. Controller C&s) Entry I,1
Controller 7.45 + 634.3 s + 40.58 s2 s(1.0 + 122.1 s) 0.00
t,2 2,1 X2
- 5.1 - 395.3 s - 36.68 s2 s (1.0 + 87.1 s) - 3.3 -
180.8 s - 30.81 s2
s (1.0 + 64.6 s)
computed controller is called C,,(s) and it is included in Table 3. The diagonal entries of the complementary sensitivity matrix using this controller are shown in Fig. 6c. Next, the mixed sensitivity requirements using this controller are analysed. These requirements are defined as (63) where the SSV are calculated as in lemma 3 with
of p as a function of w is shown in Fig. 7a. Here ,U presents an overshoot of 1.3 at a frequency of near 10 (l/h). The maximum of this function is reduced to 1 in four iterations of the algorithm described in Section 4. The resulting plot of p(w) is shown in Fig. 7. The The plot
@d(w) = diag {(l + ~~(~11 l{Hdlril)
(44)
@r(w) = diag {Cl + err(w)) I {ff,jiilj
(45)
ED(W) = 0.05 + 0.01 W’.O’
(66)
ER(w) = 0.05 + 100 W-1.o’
(67)
where
Robust multivariable controller design methodology
which are larger at the frequency band where the functions (sensitivity and complementary sensitivity) are small. We allow for those frequencies greater relative violations of the nominal requirements. It can be seen from Fig. 8a that the requirement of expression (63) is not verified. AppIying the algorithm of Section 4 to adjust the mixed sensitivity margin, it is reduced in two iterations as shown in Fig. 8b. The computed controller is called C,,, and it is included in Table 4. In Fig. 9(a) the (i, i) entries of the sensitivity function for the ideal controller (curve a), the controller C,,(s) (curve b), and the controller C,.(s) (curve c) are shown. Similar plots are shown in Fig. 9(b) for the complementary sensitivity. In these plots it is shown that the nominal characteristics are poor for controller C,,. The bound over p(w) equal to 1 for the mixed sensitivity requirement was then considered. This re-
1307
quirement may be strong if we do not consider the analysis of robust stability, because some feasible plants would produce oscillations. Figure 10 shows the p-function for robust stability analysis. it is clear that if the maximum of the p-function is lower than 0.2 then the closed loop system is robustly stable. The limit introduced by the actuators (control valves of steam and output flows) in the gain from the reference input to the process input was considered. In this case the maximum of this gain was fixed to be less than 1.5. Figure 11 shows that these specifications are verified. Figure 12 shows the time simulation of the nominal plant compensated with the controllers C (curve a), C, (curve b) and C,, (curve c). C,, provides a less interactive closed loop system, among the different than the other controllers. Figure 13 alterations,
Table 4. 1.2 _s
0.96
-
0.72
-
Q.4S
-
0.24
i
--
---
a ..___C
v
r*-
I Controller C,,
Entry
5.92 + 504.6 s + 32.28 s=
171
s (1.0 + 122.1 s)
132
0.00 -
2s
0.72
-
0.48
-
0.24
-
314.4 s -
29.1 s2
s (1.0 + 87.1 s) -
292
Fig. 8. Frequency response of the p-function as a measure of the mixed sensitivity. (a) Controller C,,. (b) Controller C,,.
4.0 -
2.6 -
143.8 s -
24.5 sz
~(1.0 + 64.6 s)
Fig. 9(a) Frequency response of the entries (1, 1) and (2,2) of the sensitivity matrix. (a) Ideal controller K’. (b) Controthr C,,. (c) Controller C,,.
L.0
I.0
0.8
-
0.8
_
0.6
-
0.4
-
0.4
-
0.2
_
0.2
_
0.0
i 0 cl1
0.6
0.0 0.1
I.0
LO.
100.
0.01
0.1
Fig. 9(b) Frequency response of the entries (1, 1) and (2, 2) of the complementary controller K’. (b) Controller C,.. (c) Controller C,.
1".
sensitivity matrix. (a) Ideal
10".
1308
J. L. FIGUEROA et al.
shows the time simulation of the perturbated plant for a 30% increase in the stady state gain, with controllers C (curve a) and C,, (curve b). The robustness property of C,, is clearIy exposed since! C,, retains a nearly similar response, with respect to those obtained with the nominal plant, while C has a considerable overshoot. Finally, the stability-performance trade-off will be
discussed in the framework of the present work. At first, the effects of q variations on robust stability and performance will be studied separately. Then both effects will be gathered to give a clear view of the inherent trade-off. For the example under consideration, Fig. 14 shows the three-dimensional plot of p(M, AI) as a function of q and w when the ideal controller is used. It is clear that the stability margins
0.36
Fig.
10. Frequency
response of the p-function stability analysis.
for robust
Fig. 12. Time simulation of the nominal plant compensated with (a) controller C, (b) controller C,, and(c) controllerC,.
1 l.i,
.a
1.04
.6
0.68
-L
0.32
.2 0
L .o 1
-.o; I
I
1
1
1
10
100.
Fig. 11. Frequency response of the F-function as a measure of input bound.
Fig. 13. Time simulation of the pcrturbated plan1 compensated with (a) controller C and (b) controller C,,.
Fig. 14. ,u(M, A,) as a function of w and 4.
1309
Robust multivariable controller design methodology
Fig. 15. 11 a(1 +
qGK’)-
‘) -
11as
%(I + GKml
a function of w and 4.
etical framework and some implementation aspects were discussed. The following aspects are noted. (1) A simple parametrization with a gain factor 4 is stated for the ideal and approximated controller. (2) This parametrization allows us to have direct control on the closed loop poles of the system. (3) A simple algorithm is used to adjust 4 in order to verify the specified robust properties. (4) Practical results and the stability-performance trade-off were presented in terms of controlling a triple effect evaporator. s 11, parametrized in 4. Fig. 16. II 104, A,) IL vs 11 of the closed loop system reduce as q increases. It can p also be nuted that as y increases the maximum occurs at a higher frequency. We define the following quantity s = 6(1-t
qGK’)-
1 - @(I + GK’)-’
as the difference between the desired sensitivity function and the actual value when the ideal controller is used. Figure 15 shows the plot of s as a function of q and w. We can see that as 4 increases the peak of s decreases, and the actual closed loop system tends to satisfy the performance requirements. Finally, Fig. 16 shows the plot of 11&%f, A,) 11 m vs 11 s Ilrnparametrized in q. As q increases (as expected) the performance is improved (s decreases) and the stability margin is decreased. 6. CONCLusIONS A simple step by step methodology for the design of multivariable compensators was extended to evaluate a controller which verifies joint specifications of robust stability and performance. The necessary theor-
NOTATION
G(s)
or G
G’(s) K(s) or K H,(s) H, (4
or K’ C(s) or C
K’(jw) Bits)
or
Pi
dR,;j-)
max{-} diag { . >
I-I
nominal transfer matrix of the plant perturbed transfer matrix of the plant controller closed loop transfer matrix (reference input-plant output) disturbance transfer matrix (disturbance input-plant output) ideal controfler of the nominal plant a particular realization of K’ rational complex function set of positive real scalars determinant maximum diagonal matrix absolute value of a scalar function absolute values maximum singular value field of n x n complex matrices
{h
(wf/w~Ylog
initial frequency final frequency
(w,>l>
J. L. FIGUEROA et al.
1310 REFERENCES
Agamennoni, O., Desages, A. and Romagnoli, J. A., 1985, Multivariable controller design in frequency domain. Proc. Am. Control Canj: (ACC). Agamennoni, 0. E., Desages, A. and Romagnoli, J., 1988, Robust controller design methodology for multivariable chemical processes. Chem. Engng Sci. 43, 2937-2950. Agamennoni, O., Rotstein, H., Desages, A. and Romagnoli, J., 1989, Robust controller design methodology for multivariable chemical processes: structured perturbations. Chem. Engng Sci. 44, 2597-2605. Arkun, Y. and Morgan, C. 0. III, 1988, On the use of the structured singular value for robustness analysis of distillation column. Comput. Chem. Engng 12, 302-306. Bhattacharyya, S. P., 1987, Robust stabilization against structured perturbations. Lecture Notes in Control and Znjbrmation Sciences. Springer, Berlin. Doyle, J. and Stein, G., 1981, Multivariable feedback design: concepts for a classical/modern synthesis. ZEEE Trans. AC-26, Nl. autom. Control Doyle, J. C., I982, Analysis of feedback systems with structured uncertainties. Proc. IEE, Part D 129, 248-250. Doyle, J. C., Wail, J. E. and Stein, G., 1982, Performance and robustness analysis for structured uncertainty. PTOC. 20th Conf on Decision and Control, pp. 629-636. Hung_ Y. S. and MacFarlane, A. G. J., 1982, Multivariable feedback: a quasi-classical approach. Lecture Notes in Control and Information Sciences. Springer, Berlin. Rotstein, H. P., Desages, A. C. and Romagnoli, J. A., 1989, Calculation of highly structurated stability margins. Znt. J. Control49, 1079-1092. Skogestad, S. and Morari, M., 1988, LV-control of high purity distillation column. Chem. Engng Sci. 43, 3348. Yedavalli, R. K., 1985, Improved measures of stability robustness for linear state space models. ZEEE Trans. autom. Control 30, 577-579.
where Zi and Pi are polynomials with real coefficients in the variable s. The singular values of the return difference operator (i.e. F = Z + GK’) are then given by
I -Xi4 I I ZAiw) + (Pdjw) - ZiWN
Oi{ F’} =
and the closed loop reference H,(s)
Z,(s) -
= diag
&(s)
= diag
Pi(S)
Z,(s)
The transfer matrix Hd from system output, y(s), is
exp ( -&+@A I
transfer matrix exp(
[eq. (25)] by
- sBi)
disturbance
_
signal,
Z,(s) + (P,(s) - Z,(s)) exp ( - soi) Z,(s)
w(s), to
D(s)
where D(s) is the model of the dynamics of signal disturbance d(s). Then Z,(s) is the characteristic polynomial of the ith nominal reference and disturbance loop and is assumed to be defined by the designer. The polynomials Pi may be evaluated to have pole-zero cancellation between zeros of the numerator of H,(s) and the slow modes of D (see Fig. 1). This is achieved by the following condition: lim [(Z,(s) 5-S&J
+ (P;(s) - Z,(s)) exp ( -
se,)]
= 0
j = 1,
. . . , ndi
(Al)
where srij are the slow poles of the ith loop of D to be cancelled. The polynomials Z,(s) and P,(s) may be defined as Z,(s)=
fJ j=*
(s +
l/Tij) =
$ j=O
Zi,S'
I-L
(s + l/Pij) =
Pi(S) = s n j=L
2
PiiS’
j=1
where APPENDIX
A: FORMULATION
AND
EVALUATION
OF
t = gri + ndi
THE SETi S&,1
This appendix summarizes the formulation of the #$(s) functions in order to introduce the performance specifications into the nominal closed loop system. Let j&(s) be expressed as follows:
t%(s)=
1
ZiId
- piw Pi(S) >
and pij = zi,
vj
> ndi
where gri are the number of poles of the nominal closed loop system. If gri are equal to or greater than the maximum number of poles to infinity of the ith column of G(s)-‘, then the resulting ideal controller will be proper.
1991.
ooE-250Y/91 53.00 + 0.00 .a 1991 Pergamon Press plc
ROBUST MULTIVARIABLE CONTROLLER DESIGN METHODOLOGY: STABILITY AND PERFORMANCE REQUIREMENTS J. L. FIGUEROA, Automatic
0.
E. AGAMENNONI,’
Control Group,
(First
PLAPIQUI
received 21 December
A. C. DESAGES
and J. A. ROMAGNOLIt 8000 Bahia Blanca, Argentina
(UNS-CONICET),
1989; accepted in revisedform
15 June 1990)
Abstract-The problem of the synthesizing controller which verifies joint specifications of robust stability and robust performancehas been formulated by the p-theory, but the problem has not yet been solved. In this paper an algorithm is presented which ailows the evaluation of a suboptimal solution of the unsolved problem stated above. This solution is carried out by using a simple parametrization of controllers and an algorithm to adjust the IIfl(M)ll, to a desired value. Application to a practical example is discussed by controlling a triple effect evaporator.
1. INTRODUCIION Frequency
response
techniques
have produced
effect-
ive compensator
designs for a wide variety of systems. Moreover, in the late seventies, singular values were found for the correct multivariable generalization of classical SISO concepts. Since their application in control theory, a new generation of the so-called “robust” design approaches has appeared, which produces controllers under the assumption of model uncertainty [see, for instance, Doyle and Stein (19X1), Hung and MacFarlane (1982) and Agamennoni et al. (1985, 1988 and references therein)]. Robust control theory is now able to deal with a large class of uncertainties, including unstructured and structured perturbations in frequency and time domains (Doyle, 1982; Yedavalli, 1985), and highly structured perturbations (Rotstein et al., 1989; Bhattacharyya, 1987). In the eighties, the design techniques which had previously treated problems of robust stability were extended to robust performance. Regarding this subject, see for example, Doyle et al. (1982), and the applications of this technique in practical examples (Skogestad and Morari, 1988; Arkun and Morgan, 1988). The problem of synthesizing a controller of a perturbed plant has been formulated by the p-theory. Referring to Fig. 1 the problem can be posed as follows: find K(s) such that K(s) stabilizes all possible plants GP and 11p(M) 11 co < 1, where M is an operator which includes the performance requirements. This problem has not yet been soIved. H, synthesis solves a closely related problem: find K(s) which stabilizes GP(s) and IIMII, < 1. In this paper a methodology is presented which allows the evaluation of a suboptimal solution of the unsolved problem stated above. This solution is car-
ried out by using a simple parametrization of controllers and a previous algorithm to adjust 11p(M) llm to a desired value (Agamennoni et al., 1989). A single controller parameter is used as adjustable parameter. Furthermore, a discussion, in terms of the control of a tripple effect evaporator, provides the appropriated framework for the analysis of the inherent trade-off between stability and performance. We believe that the methodology proposed in this paper is truly user friendly, because the designer may specify and adjust the performance to satisfy his robustness constraints. The resulting compensated system has stability and performance margins which may be fixed beforehand. This paper is organized as follows. In Section 2 some performance criteria are defmed. In Section 3 the necessary background is introduced. In Section 4 a method to adjust the robust performance requirement is described together with a step by step algorithm. In Section 5 an example is given, before concluding in Section 6. 2 SOME PERFORMANCE Let
us consider
the standard
CRITERIA feedback
configura-
tion illustrated in Fig. 1, where K(s) is a controller matrix (n x n) and G’(s) is a member of the family of feasible
plants
II = {Gp(s):
II and
where
GE(s) = Gii(s)
II is characterized
+ Aij(s)
as
with
IA&)I
G ~ij)
(1)
where G is the nominal matrix transfer funct!.,a of the plant. This family implies that the entry ij of the set of
fA. Desages and 0. Agamennoni are with the Department of Electrical Engineering and the Comisi6n de Investigaclones Cientificas de la provincia de Buenos Aires. *Author to whom correspondence should be addressed.
Fig. 1. Multivariable 1299
closed loop system.
J. L. FIGUEROA et d.
1300
feasible plants lies in the circle with centre in G(s) and radius Pij. The objective of using feedback control is to keep the controlled output, y(s), closer to its set point. The performance specifications define what is referred to as “closer”. Furthermore, these performance requirements should be satisfied in spite of the plant uncertainty. Consequently the ultimate goal of the controller design is to achieve robust performance. The implications of these requirements are easier to understand if we consider some subobjectives which have to be satisfied in order to achieve this goal. These subobjectives are defined as follows.
2.1. Nominal stability The controller K applied has to be stable.
to the nominal
plant G
‘41
’} 6 W,(w)
for all w
6’ “I
2.3. Nominal complementary sensitivity The system formed by the nominal plant G and controller K should satisfy the following condition: + GK)-‘GK)
_ ..
6 W,(w)
for all w
(3)
where W, is a scalar function of frequency. This requirement implies a bound over the gain between the reference input r(s) and the system output y(s).
ia,
. .
(6)
and
M=(I+KG)-'K.
(7)
2.6. Robust sensitivity The closed loop system of Fig. I should satisfy the subobjective 2.2 for all plants G, E lT. This is verified if the nominal system is stable, and if it satisfies the inequality (2) for the nominal plant and < 1
is computed
for all w
(8)
with respect to the structure
A = diag {A,, AD >
(9)
with o(A,)
(2)
where W, is a scalar function of frequency. This requirement implies a bound over the gain between the noise input d(s) and the error system e(s), i.e. this gives a measure of the disturbance rejection index of the control system.
1
6 1”
with IS,1 6 pij
p(M)
2.2. Nominal sensitivity The system formed by the nominal plant G and the controller K should satisfy the following condition:
c{(I
_. . _
A, =
where p(M)
a((1 + GK)-
..
6 l/w,
(10)
and M=
-(I+KG)-‘K
(I+KG)-‘K
-((I+GK)-’
[ (I + GK)-’ Furthermore,
the system is robustly
1. (11)
stable.
2.7. Robust complementary sensitivity The closed loop system of Fig. 1 should satisfy the subobjective 2.3 for all plants G, E lT. This is verified if the nominal system is stable, if it satisfies the inequality (3) for the nominal plant and
P(W -s 1 where p(M)
is computed
for all w
(12)
with respect to the structure
A = diag (A,, A,} 2.4. Nominal input bound To have practical significance the gains from inputs of the controlled system, r(s), to inputs of the plant, U(S), must be bounded. This is expressed as 5((I
+ GK)-‘K}
< W,(w)
for all w
2.5. Robust stability The closed loop system of Fig. 1 should remain stable for all possible plants GP E XI. This is verified if the system is stable for the nominal plant and if p(M)
< 1
for all w
c(A,)
(14)
-(I
+ KG)-lK
- (f + GK)-‘GK Furthermore,
the system is robustly
stable.
1(13 ’
2.8. Robust bounded input The closed loop system of Fig. 1 must satisfy the inequality (4) by all feasible plants Gpe lT. This is verified if the nominal system is stable, if it satisfies (4) for the nominal plant and P(M)
(5)
where p is the structured singular value (SSV) (Doyle, 1982), computed with respect to the structure
< l/W,
and
(4)
where W, is a scalar function weighting used to specify the frequency range over which the plant inputs must be bounded. This range may be determined by the actuators dynamics.
(13)
with
where p(M)
-= 1
is computed
for all w
(16)
with respect to the structure
A = diag {AI, AB}
(17)
Robust multivariablecontroller design methodology
1301
with @((I + GK)-‘K)
< LO/W,(w)
(18)
and
M _
(I + KG)-‘K
-(I
+ KG)-‘K
-[
(I + KG)-‘K
-(I
+ KG)-‘K
1.
(19)
Furthermore, the system is robustly stable. The fulfilment of robust sensitivity and robust complementary sensitivity implies robust mixed sensitivity (RMS). In the following lemma this is formally expressed. Lemma 1. Consider the system of Fig. 1, formed by the controlled K(s) and the plant GP~ II. If the closed loop system satisfies nominal stability, nominal sensitivity and nominal complementary sensitivity, then these characteristics are satisfied for al1 perturbed plants if and only if P(M) (
1
for all w
where p(M) is computed with respect
(20) to the
A = diag {AI, Aa, AR}
structure
Fig. 2. Multivariable closed loop system with virtual uncertainties. with three fewer A-blocks the upper bound becomes an equality. Furthermore, the corresponding optimization problem is always convex. Therefore for less than three blocks the optimization algorithm used computes p exactly. In the case of having more than three blocks inf c?(DMD-~) is an approximation to the exact p value. 3.
MULTIVARIABLE
CONTROL
DESIGN
a previous work (Agamennoni et al., 1988) a multivariable controller design methodology was presented, which leads to a controller formulation given by In
(21)
K’(jw)
=
G- ‘(jw)
diag {mi(jw)}
(23)
with Al, A,, and AX defined as before, and
(I + GK)-’
-
(Z + KG)-‘K
-
(Z + GK)-’
(Z + GK)-
(I + KG)-lK (I + GK)-’
’GK
Proof The proof is straightforward, from Doyle et al. (1982). The following remarks are in order.
(I + GK)
it follows
2.9. Remarks (i) The inverse of the p-function of (5) is naturally interpreted as a stability margin, because it determines the minimun value by which the uncertainty bound p must be multiplied to make the system unstable for some GPEII. (ii) From the analogy of(i), in a more general case, the inverse of the p-function of eqs (8), (12), (16) and (20) can be interpreted as a robust performance margin because it determines the minimun value by which the desired bound of the performance must be multiplied to satisfy it, or to make the system unstable for some GP~ lT. (iii) The p-function of expressions (8), (12), (16) and (20) must be calculated considering the perturbations associated with A*, A, and AD as unstructured blocks of dimension (n x n). These blocks may be interpreted as virtual uncertainty appropriately placed, see for example Fig. 2 (Doyle et al., 1982). We will return to this consideration later. (iv) It is known that p(M) < inf b(DMDD
’);
1
.
(22)
‘GK]
where a,(s), i = 1,2, _ . , n, are complex scalar functions. A suitable choice of the functions ai may be used to introduce closed loop specifications. The controller described in (23) is called ideal, because if it were physically realizable the closed loop system would exactly satisfy the design specifications. The ideal controller is approximated with a controller of desired order C(s). This design step will be discussed in the followjng subsection. Let xi(s) be as follows:
pi exp ( - SOi) aiW
=
1 + &(l
- exp ( -
sOi))
i = 1,2,.
. . .n (24)
where fii(s) are complex functions and 0; are delayed terms. The ai of eq. (24) allows us to state a multivariable predictor control scheme. In the nominal case the following equalities are verified:
(25) SOi)
1 + pi(s) (1 -
exp ( -
sOi))
(26) (27)
1302
J. L. FIGUEROA
where H,(s) is the transfer matrix from the reference input r(s) to the output y(s), and HI(s) is the transfer matrix from the disturbance input d(s) to the output
With a similar reasoning for a robust complementary sensitivity problem, we obtain the following bounds:
Y(S). The evaluation of Oi is beyond the scope of this work. However, the designer may interactivefy assume different values of Oi searching for the set which produces a closer approximation between the ideal controller and C(s) (Agamennoni et al., 1989). In those problems in which the delay compensator scheme is not necessary (i.e. Oi = 0, i = 1,2, . . . , n), eqs (24j(26) may be written as %cs) = Bits)
Hd(4 = diag
(24’)
1
Pi(')
Hr(s) = dk
1
pics)~
+
(25’)
l
et al.
I1+iitsl}.
In the next section we will consider the concept of robust performance in the context of this methodo-
{(I
+ GK’)-‘GK’},
@r(w)
{H,}iil (1 + E,(W)) 1,
if
i = j
E,(W)
if
i#j
r(Z +
KG)-
’K’
(I + GK’)-’
1
(Z + GK’)-’
From the previous section the function H, of eq. (25) is a natural choice of IV, of (2) in the nominal sensitivity problem. When we introduce plant uncertainties, it is obvious that these specifications break down for some plant GP~ II. For this reason we consider
l{H,tiil
W, = (1 + dw))max
dw)~R+
(28)
for the analysis of the robust sensitivity problem, where cd is a positive real scalar which defines the tolerance band of the sensitivity bound. Moreover, from (25) we obtain the bound over the norm of U, and the structure of H,; for this reason the analysis of section 2 will, in general, be conservative. To avoid this problem, we consider the inequality (2) element by element, i.e. {(Z + GK’)-‘)ij
G
I{
(33)
and in (10) I
D,lij
In an analogous shown below.
<
‘/(
l-C
wrIij.
(34)
form, lemma 1 may be rewritten as
Lewvna 2. Consider the system of Fig. 1, constituted by the controller K’(s) and the plant G,EII. If the closed loop system satisfies nominal stability, nominal sensitivity and nominal complementary sensitivity, then these characteristics are satisfied for al1 perturbed plants if and only if P(M)
M =
(32)
where
logy. 3.1. Robust performance in the melhodology context In Section 2 we introduced some performance objectives for the design, but we did not explain how IV, and W, functions were established.
< @‘r(w)}ijl
where p(M)
for all w
G 1
is computed
(35)
with respect to the structure
A = diag I&,
AD. A,}
(36)
with AI, AD and A, defined as in (31) and (34) respectively, and - (I + KG)-’
-(I
K’
- (I + GK’)-’ (I
+
+ K'G)-lK (I + GK’)_
1
.
(I + GK’) - i GK’
GK’)-‘GK’
(37)
I
In this case, the virtual uncertainty AD, AR of Fig. 2 must be considered as structured blocks. Figure 3 shows the block diagram of the resulting control scheme, where C(s) is an approximation to K’ of a given order, calculated as in the following section. 3.2. Controller approximation The approximation procedure in an order reduction framework may be formulated as an optimization problem. Assume that the frequency response matrix K’(jw) = [k,,(jw)] of eq. (23) has been calculated and let the contr&ler C(s) be c(s)
a,jo = Ccij(s)l= b,jcsI
where aij(s) and b&)
are polynomials
(38)
and b,(s)
is
@‘d(W>}ijI
where
@d(W.,t1 + Ed(W)) [ E*(W)
(Hd}ij
and in (10)
if if
i=j
i#j
(30)
Fig. 3. Multivariable closed loop system with the proposed delay compensator.
Robust multivariablecontroller design methodology manic and Hurwitz. These polynomials should be chosen in order to minimize a norm II-11 of the distance between K’(s) and C(s): min
+LbW
IIK’(jw) -
C(W)
tions. This technique allows us to satisfy the following design criterion: max C&M)1 =
II.
w
(3%
min .Z *.b
(40)
is a continuous function of M.
Lemma 3. p(M)
where
sn
I k,, (jw) - c&v)
I2 Wii(w) dw
Lemma 4. Let M(s) be a matrix transfer function without poles on the imaginary axis. Then p(M(s)) is a continuous function of the frequency.
(41)
in which a positive weighting matrix W(w) = [ Wij(w)] has been included. In the practical implementation, the integral of the right was replaced by a summation over a set of suitably spaced frequencies a. Agamennoni et al. (1989) proposed a weighting matrix of the form Wij(W) =
l
2,
Pr& See Agamennoni et al. (1989). Suppose now that the functions Bi(s) of eq. (24) are modified as follows: qER
Pi(s) = q/7is
O>q21
(45)
giving
(42)
I kf,(iw) I Wi
(4)
PMS
where p is calculated with the structure of (5) and ,uMs is the maximun specified value of the p-function for all w. Immediately, we extend this idea to the adjustment of robust performance margin, but first remember two important properties of the p-function.
For computational convenience, the Frobenius norm II - IIF may be used. This norm leads to n2 scalar optimization problems with objective functions
J=
1303
K”(jw)
normalizing the error between cij and kij and enhancing approximation at low frequencies. The evaluation of C(s) by the minimization of expression (41) with weighting function (42) gives a good open loop controller. However, the approximation errors may have serious closed loop consequences. This problem can be avoided by means of the
= qK’(jw).
(46)
Expression (45) relates the time constant of the closed loop system (H,(s)) to the controller gains. We wish to explore the effect of varing q over p. The following result is needed. Lemma 5. Consider M as in eq. (37) of lemma 2, then M as a function of g is continuously differentiable. Proof: Consider M as in lemma 2, now M function of q in the form
q(I+ M =
-
qK’G)-‘K’
q(1 + qGK’)-
(I + qGK’)-’
= Cl + s((Z
+ GC)-’
1
q(Z + qK’G)-‘K’
(I + qGK’)q(Z + qGK’)-
1GK’
GCjl$&
1
1,
(43)
M,
=
M=M,M,
(48)
(1 + qK’G)-’ 0
0 (1 + qGK’)-’
0 0 (I + qGK’)-
0
and This function enhances the low frequency range and the critical frequencies where the closed loop has the worst performance.
qK’ 1
M,= [
PERFORMANCE
(47)
’ GK’
where
0
4. ROBUST
1
Furthermore, M may be expressed as
following weighting function: wij(w)
-
q(Z + qK’G)-‘K
- (I + qGK’)-
(I + qGK’)-’
is a
ADJUSTMENT
In a previous work, Agamennoni et al. (1989) proposed a simple controller design procedure for an MIMO plant under additively structured perturba-
1
-
qX’
qK’
-1
1
qGK’
qGK’
1
1 WV (49)
1
Now, as the system is nominally stable, it verifies p(K’G) p(GK’)
< 1 < 1.
J. L. FKXJEROA et al.
1304
Then, we may write (I + qGK’)-’
= I -
qGK’ f
(I + qK’G) - ’ = I - qK’G
q’(GK’)‘-
__. . . . . .
+ q*(K’G)*
-
. ..
. ..
If we replace this in eq. (49) it is clear that MI is a polynomial function of 4, and from eq. (48) M is also a polynomial function of q. Now, it is well known that the polynomial function is a continuously differentiable function. From these considerations we conclude that M is a continuously differentiable function of 4. Lemma 6. Consider M as in eq. (37) of lemma 2, then JJas a Function of 4 is continuously differentiable. ProofI
By definition
of the p-function
p = s;p
C(DMPD-‘)
(51)
we defined ti=DMPD-’
(52)
then, it verifies A*n?u
= &J
(53)
Of
Fu = p*u
(54)
where u is the right singular vector associated with the p-function. By differentiating (54) with respect to 4, we obtain dF au --u+F-=p*_++_
&
and multiplying
8~
a4
aq
a$ a4
cm
on the left by u* gives
Now, from lemma 5, it is easily verifiable that F is a continuously differentiable function of 4, which implies that, from (56),
w
(57)
&t is a continuously differentiable function of q. We are ready to state the main result.
4.1. Design algorithm Finally, we will summarize the results of the present paper in a step by step design algorithm to satisfy the following criterion: (58) where p(Ms is the maximum specified value of the pfunction for all w E n. Expression (45) is a good starting point since it relates the time constant of the closed loop system (H(S)) to the controller gains. A variation in the gain factor q E R may be taken as a variation in the closed loop time constant of the nominal system. Hence, we have a single parameter for adjusting the ideal controller K’ [see (46)] with a proportional shifting of all the time constants specified for every single loop. Besides, it may be easily shown that the frequency response of the normalized synthesis relative error of (41) will remain constant for all values of q. This is an important property of K’ which allows the evaluation of q with any initial controller C(s). Remark. It is important to note that since GP(s) is open loop stable, then there always exists a sufficiently small q for which the closed loop system is stable. The design specifications are:
(1) dominant time constants of each loop, (2) time delay for each loop,
(3) slow disturbance loop, (4) the performance systems.
modes to reject in the different margin of overall
closed loop
First, the designer may try to achieve the desired dominant time constant in each single loop (nominal performance objective). Then, if the performance margin is smaller than the one specified, the algorithm will provide the minimum proportional reduction of loop gains so that the robust performance specifications will be satisfied (robust performance objectives). A step by step procedure for the computation of the controller that achieves the design goals described above could read as follows:
(1) Specify the desired dominant time constants zi,
Theorem 2. Let us suppose that ,~(jw) attains its maximum in a unique isolated frequency w,. Then if the derivative of p at w, is not equal to zero, the overall performance margin may be decreased by a suitable variation of 4. Proof: This is evident (1989). Later, in the example formance trade-off will be effect of variation on q to robust performance in the triple effect evaporator.
from
Agamennoni
et al.
section, the stability-peranalysed by discussing the adjust robust stability and context of the control of a
i
1,2, . . ,n for each single loop, the desired pMs and 6 > 0. (The formulation and evaluation of the set of fii from the desired time constants are introduced in Appendix A.) (21 Evaluate the frequency response of K’ = G- ’ diag (aI (jw)}. (3) Synthesize K’ with the desired controller structure C(s) according to the discussion of Section 3. (4) Set 4 = qO. (a) Set C,(s) = C(s)+ =
(b) If the compensated system GC, is unstable, reduce q and return to 4a.
Robust multivariable controller design methodology STEAM
PRESSURE
1305
CONTROL TAP ‘.‘.T.~.“-._.-
r_____id?____
STEAM
_A
f
WATER
1
PRODUCT
-.2 i
I
-____-. -.-I-
JUICE PREHEATING JUlCE CONCENTRATION STEAM CONOENSATE TAP WATER VAPOR
Fig. 4. Schematic diagram
of
the
industrial scale triple effect evaporator study in the example.
Table 1. Evaporator G(s)
(5) Evaluate max (~1 over w, and note with wM the frequency of the maximum. (6) If P(W,Y 4) > PMSI modify q in order to satisfy the design criterion (58) by using the following iterative
I-_
PRODUCT COOLING
Entry
Evaporator 0.83 exp ( -
procedure:
s 0.077)
(s 0.0032 + 1) (S 0.0687 + 1)
i=O
0.00
I PL(w, a) - pMS 1 > 6
do while
-
1.3 exp ( - s 0.082)
(s 0.0042 + 1) (s 0.1422 + 1) -
22
i=i+
1.86 exp ( -
s 0.027)
(SO.165 + 1)
1
end do go to 5. (7) End. 0.96
Some implementation aspects of the routines of steps 4 and 5 may be found in Agamennoni et al. (1989). 5. EXAMPLE OF APPLICATION The feasibility of applying the theory presented in this paper is illustrated in the detailed multivariable control design of an apple juice evaporator. A triple effect industrial scale evaporator is considered with counter current pre-heating and parameters tuned to fit real parameter data. This evaporator is shown schematically in Fig. 4. The nominal transfer function at the desired operation point is a matrix of dimension 2 x 2 given in Table 1, with the concentration and level in the third effect as output, and inlet steam pressure and output flows as input. The model reflects the typical nonlinear, time-variant characteristics of these types of equipment: asymmetric behavior (i.e. different responses to positive and negative disturbances and set point changes), non-linear dependence with the amplitude of perturbations, changes in the heat transfer
0.0
1 0.01
L 0.
I
I
L
I.0
10.
I 100.
I 1000.
Fig. 5. Frequency response of the relative uncertainty mattix p,.
coefficients due to filthiness, etc. In view of these considerations structured additive uncertainty is applied to each Gi,(s). It is modelled as
P,,l(s) is shown in Fig. 5. As a design objective we choose a closed response of the system with a time constant of where
zL = 0.1 (h)
i = 1,2.
loop
J. L. FKXJEROA
1306
et al.
1." f
1.0
".80 _
0.8
-
0.60 -
0-6
-
0.4" -
0.4
".2" -
07 I cl.,
0.01
1 1."
Icl.
LOO.
0.01
1.
1".
Fig. 6. Frequency responseof the entries(1, 1) and (2, 2) of the complementarysensitjvity matrix. (a) Ideal controller K’. (b) Synthesizedcontroller C. (c) Adjusted controller C,,.
Table 2. Controller C(s) Entry
Controller 11.14 + 948.3 s + 60.67 s2
1,1
s(1.0 + 122.1 s) 0.00
L2 I
0. .01
.I
I 1.
I 1”.
2J
i 100.
Fig. 7. Frequency response of the p-function as a measure of the complementary sensitivity.(a) Synthesizedcontroller C. (b) Adjusted controller C,,. First, the frequency response to the ideal controller K’(jw) Ceq. (23)) is approximated by a PID controller, say C(s). This controller is included in Table 2. A weighting function such as eq. (43) for the synthesis of C(s) was used. The frequency range of this approximation is [0.01,100]. In Fig. 6, we show the diagonal entries of the complementary sensitivity matrix (i.e. {(I + GK)-i GK},,), for the ideal controller K’(s) [see curve a], and the synthesized controller C(s) (see curve b). This plot shows that the controller C(s) is a very good approximation to K’(s). Moreover, curve b presents a more abrupt slope in the crossover frequency than curve a. This characteristic may be desirable for several well known reasons. The specification For robust complementary sensitrvtty is max (p(w)) < 1 W where SSV are calculated as in Section 2.5 with *&l
= diag i(l + ~~(~11 I(H,)iil}
(61)
en(w) = 0.01 IWO’.
(62)
where
-
7.67 -
590.0 s -
54.83 s=
s (1.0 + x7.1 s) ~ 5.00 - 270.34 s -
292
46.0 s2
s (1.0 + 64.4 s)
Table 3. Controller C&s) Entry I,1
Controller 7.45 + 634.3 s + 40.58 s2 s(1.0 + 122.1 s) 0.00
t,2 2,1 X2
- 5.1 - 395.3 s - 36.68 s2 s (1.0 + 87.1 s) - 3.3 -
180.8 s - 30.81 s2
s (1.0 + 64.6 s)
computed controller is called C,,(s) and it is included in Table 3. The diagonal entries of the complementary sensitivity matrix using this controller are shown in Fig. 6c. Next, the mixed sensitivity requirements using this controller are analysed. These requirements are defined as (63) where the SSV are calculated as in lemma 3 with
of p as a function of w is shown in Fig. 7a. Here ,U presents an overshoot of 1.3 at a frequency of near 10 (l/h). The maximum of this function is reduced to 1 in four iterations of the algorithm described in Section 4. The resulting plot of p(w) is shown in Fig. 7. The The plot
@d(w) = diag {(l + ~~(~11 l{Hdlril)
(44)
@r(w) = diag {Cl + err(w)) I {ff,jiilj
(45)
ED(W) = 0.05 + 0.01 W’.O’
(66)
ER(w) = 0.05 + 100 W-1.o’
(67)
where
Robust multivariable controller design methodology
which are larger at the frequency band where the functions (sensitivity and complementary sensitivity) are small. We allow for those frequencies greater relative violations of the nominal requirements. It can be seen from Fig. 8a that the requirement of expression (63) is not verified. AppIying the algorithm of Section 4 to adjust the mixed sensitivity margin, it is reduced in two iterations as shown in Fig. 8b. The computed controller is called C,,, and it is included in Table 4. In Fig. 9(a) the (i, i) entries of the sensitivity function for the ideal controller (curve a), the controller C,,(s) (curve b), and the controller C,.(s) (curve c) are shown. Similar plots are shown in Fig. 9(b) for the complementary sensitivity. In these plots it is shown that the nominal characteristics are poor for controller C,,. The bound over p(w) equal to 1 for the mixed sensitivity requirement was then considered. This re-
1307
quirement may be strong if we do not consider the analysis of robust stability, because some feasible plants would produce oscillations. Figure 10 shows the p-function for robust stability analysis. it is clear that if the maximum of the p-function is lower than 0.2 then the closed loop system is robustly stable. The limit introduced by the actuators (control valves of steam and output flows) in the gain from the reference input to the process input was considered. In this case the maximum of this gain was fixed to be less than 1.5. Figure 11 shows that these specifications are verified. Figure 12 shows the time simulation of the nominal plant compensated with the controllers C (curve a), C, (curve b) and C,, (curve c). C,, provides a less interactive closed loop system, among the different than the other controllers. Figure 13 alterations,
Table 4. 1.2 _s
0.96
-
0.72
-
Q.4S
-
0.24
i
--
---
a ..___C
v
r*-
I Controller C,,
Entry
5.92 + 504.6 s + 32.28 s=
171
s (1.0 + 122.1 s)
132
0.00 -
2s
0.72
-
0.48
-
0.24
-
314.4 s -
29.1 s2
s (1.0 + 87.1 s) -
292
Fig. 8. Frequency response of the p-function as a measure of the mixed sensitivity. (a) Controller C,,. (b) Controller C,,.
4.0 -
2.6 -
143.8 s -
24.5 sz
~(1.0 + 64.6 s)
Fig. 9(a) Frequency response of the entries (1, 1) and (2,2) of the sensitivity matrix. (a) Ideal controller K’. (b) Controthr C,,. (c) Controller C,,.
L.0
I.0
0.8
-
0.8
_
0.6
-
0.4
-
0.4
-
0.2
_
0.2
_
0.0
i 0 cl1
0.6
0.0 0.1
I.0
LO.
100.
0.01
0.1
Fig. 9(b) Frequency response of the entries (1, 1) and (2, 2) of the complementary controller K’. (b) Controller C,.. (c) Controller C,.
1".
sensitivity matrix. (a) Ideal
10".
1308
J. L. FIGUEROA et al.
shows the time simulation of the perturbated plant for a 30% increase in the stady state gain, with controllers C (curve a) and C,, (curve b). The robustness property of C,, is clearIy exposed since! C,, retains a nearly similar response, with respect to those obtained with the nominal plant, while C has a considerable overshoot. Finally, the stability-performance trade-off will be
discussed in the framework of the present work. At first, the effects of q variations on robust stability and performance will be studied separately. Then both effects will be gathered to give a clear view of the inherent trade-off. For the example under consideration, Fig. 14 shows the three-dimensional plot of p(M, AI) as a function of q and w when the ideal controller is used. It is clear that the stability margins
0.36
Fig.
10. Frequency
response of the p-function stability analysis.
for robust
Fig. 12. Time simulation of the nominal plant compensated with (a) controller C, (b) controller C,, and(c) controllerC,.
1 l.i,
.a
1.04
.6
0.68
-L
0.32
.2 0
L .o 1
-.o; I
I
1
1
1
10
100.
Fig. 11. Frequency response of the F-function as a measure of input bound.
Fig. 13. Time simulation of the pcrturbated plan1 compensated with (a) controller C and (b) controller C,,.
Fig. 14. ,u(M, A,) as a function of w and 4.
1309
Robust multivariable controller design methodology
Fig. 15. 11 a(1 +
qGK’)-
‘) -
11as
%(I + GKml
a function of w and 4.
etical framework and some implementation aspects were discussed. The following aspects are noted. (1) A simple parametrization with a gain factor 4 is stated for the ideal and approximated controller. (2) This parametrization allows us to have direct control on the closed loop poles of the system. (3) A simple algorithm is used to adjust 4 in order to verify the specified robust properties. (4) Practical results and the stability-performance trade-off were presented in terms of controlling a triple effect evaporator. s 11, parametrized in 4. Fig. 16. II 104, A,) IL vs 11 of the closed loop system reduce as q increases. It can p also be nuted that as y increases the maximum occurs at a higher frequency. We define the following quantity s = 6(1-t
qGK’)-
1 - @(I + GK’)-’
as the difference between the desired sensitivity function and the actual value when the ideal controller is used. Figure 15 shows the plot of s as a function of q and w. We can see that as 4 increases the peak of s decreases, and the actual closed loop system tends to satisfy the performance requirements. Finally, Fig. 16 shows the plot of 11&%f, A,) 11 m vs 11 s Ilrnparametrized in q. As q increases (as expected) the performance is improved (s decreases) and the stability margin is decreased. 6. CONCLusIONS A simple step by step methodology for the design of multivariable compensators was extended to evaluate a controller which verifies joint specifications of robust stability and performance. The necessary theor-
NOTATION
G(s)
or G
G’(s) K(s) or K H,(s) H, (4
or K’ C(s) or C
K’(jw) Bits)
or
Pi
dR,;j-)
max{-} diag { . >
I-I
nominal transfer matrix of the plant perturbed transfer matrix of the plant controller closed loop transfer matrix (reference input-plant output) disturbance transfer matrix (disturbance input-plant output) ideal controfler of the nominal plant a particular realization of K’ rational complex function set of positive real scalars determinant maximum diagonal matrix absolute value of a scalar function absolute values maximum singular value field of n x n complex matrices
{h
(wf/w~Ylog
initial frequency final frequency
(w,>l>
J. L. FIGUEROA et al.
1310 REFERENCES
Agamennoni, O., Desages, A. and Romagnoli, J. A., 1985, Multivariable controller design in frequency domain. Proc. Am. Control Canj: (ACC). Agamennoni, 0. E., Desages, A. and Romagnoli, J., 1988, Robust controller design methodology for multivariable chemical processes. Chem. Engng Sci. 43, 2937-2950. Agamennoni, O., Rotstein, H., Desages, A. and Romagnoli, J., 1989, Robust controller design methodology for multivariable chemical processes: structured perturbations. Chem. Engng Sci. 44, 2597-2605. Arkun, Y. and Morgan, C. 0. III, 1988, On the use of the structured singular value for robustness analysis of distillation column. Comput. Chem. Engng 12, 302-306. Bhattacharyya, S. P., 1987, Robust stabilization against structured perturbations. Lecture Notes in Control and Znjbrmation Sciences. Springer, Berlin. Doyle, J. and Stein, G., 1981, Multivariable feedback design: concepts for a classical/modern synthesis. ZEEE Trans. AC-26, Nl. autom. Control Doyle, J. C., I982, Analysis of feedback systems with structured uncertainties. Proc. IEE, Part D 129, 248-250. Doyle, J. C., Wail, J. E. and Stein, G., 1982, Performance and robustness analysis for structured uncertainty. PTOC. 20th Conf on Decision and Control, pp. 629-636. Hung_ Y. S. and MacFarlane, A. G. J., 1982, Multivariable feedback: a quasi-classical approach. Lecture Notes in Control and Information Sciences. Springer, Berlin. Rotstein, H. P., Desages, A. C. and Romagnoli, J. A., 1989, Calculation of highly structurated stability margins. Znt. J. Control49, 1079-1092. Skogestad, S. and Morari, M., 1988, LV-control of high purity distillation column. Chem. Engng Sci. 43, 3348. Yedavalli, R. K., 1985, Improved measures of stability robustness for linear state space models. ZEEE Trans. autom. Control 30, 577-579.
where Zi and Pi are polynomials with real coefficients in the variable s. The singular values of the return difference operator (i.e. F = Z + GK’) are then given by
I -Xi4 I I ZAiw) + (Pdjw) - ZiWN
Oi{ F’} =
and the closed loop reference H,(s)
Z,(s) -
= diag
&(s)
= diag
Pi(S)
Z,(s)
The transfer matrix Hd from system output, y(s), is
exp ( -&+@A I
transfer matrix exp(
[eq. (25)] by
- sBi)
disturbance
_
signal,
Z,(s) + (P,(s) - Z,(s)) exp ( - soi) Z,(s)
w(s), to
D(s)
where D(s) is the model of the dynamics of signal disturbance d(s). Then Z,(s) is the characteristic polynomial of the ith nominal reference and disturbance loop and is assumed to be defined by the designer. The polynomials Pi may be evaluated to have pole-zero cancellation between zeros of the numerator of H,(s) and the slow modes of D (see Fig. 1). This is achieved by the following condition: lim [(Z,(s) 5-S&J
+ (P;(s) - Z,(s)) exp ( -
se,)]
= 0
j = 1,
. . . , ndi
(Al)
where srij are the slow poles of the ith loop of D to be cancelled. The polynomials Z,(s) and P,(s) may be defined as Z,(s)=
fJ j=*
(s +
l/Tij) =
$ j=O
Zi,S'
I-L
(s + l/Pij) =
Pi(S) = s n j=L
2
PiiS’
j=1
where APPENDIX
A: FORMULATION
AND
EVALUATION
OF
t = gri + ndi
THE SETi S&,1
This appendix summarizes the formulation of the #$(s) functions in order to introduce the performance specifications into the nominal closed loop system. Let j&(s) be expressed as follows:
t%(s)=
1
ZiId
- piw Pi(S) >
and pij = zi,
vj
> ndi
where gri are the number of poles of the nominal closed loop system. If gri are equal to or greater than the maximum number of poles to infinity of the ith column of G(s)-‘, then the resulting ideal controller will be proper.