Znd.Eng. Chem. Res. 1992,31,1920-1927
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PROCESS ENGINEERING AND DESIGN Comparison of Internal Model Control and Linear Quadratic Optimal Control for SISO Systems Claudio Scali* and Daniele Semino Dipartimento di Ingegneria Chimica, Universitd di Pisa, Via Diotkalvi, 2, 1-56100 Pisa, Italy
Manfred Morari Chemical Engineering 210-41, California Institute of Technology, Pasadena, California 91125
In this paper internal model control (IMC) is formulated for general inputs and compared with linear quadratic (Hz) optimal control (LQOC) with a general (dynamic) input penalty weight. Analogies and differences between the two methods for robustness study are clearly pointed out. Both controllers can be expressed in terms of the same nominal controller and a robustness filter. The main difference is that the LQOC filter depends also on process and disturbance parameters. As a consequence, performance may be different in general, even if in many cases quite similar results are obtained. The design for robustness is much more straightforward in IMC. 1. Introduction A high controller gain implies generally tighter control, i.e., a larger bandwidth and better disturbance rejection and setpoint tracking. On the other hand, it leads to increased measurement noise amplification, more severe action of the manipulated variable, and sensitivity to plant changes, i.e., robustness problems. In multivariable systems there are also trade-offs among the different manipulated variables and among the different outputs. Different controller design procedures aim at achieving all these trade-offs in various ways. In the single input-single output (SISO)case discussed in this paper the degrees of freedom are inherently limited and it is to be expected that the different techniques can lead to very similar results, when properly applied. Some techniques, however, are easier to use because they address the pertinent design issues in a more direct manner. We will compare linear optimal control (LQOC) with internal model quadratic (Hz) control (IMC). Palmor and Shinnar (1979)and more recently Bergh and MacGregor (1986)have analyzed constrained minimum variance controllers (the stochastic version of LQOC) for sampled-data systems and have put them into the IMC framework. They have also investigated how robustness is affected by the disturbance models and the control weight. Kozub et al. (1987)have compared Wiener-Hopf controller design with IMC on a pilot plant tubular reactor. Harris and MacGregor (1987)studied the design of LQOC via transfer functions. Grimble et al. (1989)have analyzed LQG controllers from the IMC viewpoint. This paper uses a completely deterministic approach in continuous time. Therefore the form of presentation should be appealing to a wider audience. A control action weight transfer function is introduced which adds some freedom to the LQOC approach. This freedom can be valuable for improving the performance when robustness ‘Author to whom correspondence should be addressed. Phone: +39 (50) 511-241. Fax: +39 (50) 511-266. BITNET: S@ 1CNUCEVM.BITNET.
constraints are present. Ale0 an explicit expression for the LQOC for arbitrary inputs is presented which gives some insight into the structure of the optimal controller and is a convenient starting point for the IMC design. We will concentrate on the disturbance rejection problem. The tracking problem can be addressed analogously. Convenience of design (can the design problem be stated in relevant engineering terms?) and tuning (are the adjustable parameters directly related to performance and robustness?) w i l l be investigated rather than the specific numerical techniques used to obtain the controller parameters. 2. Internal Model Control The SISO system to be studied is shown in Figure 1. Here P is the plant, u the manipulated variable, y the output to be controlled, and d the effect of a disturbance on the output. The disturbance transfer function W,is chosen such that d ’ = 1;i.e., d’is an impulse. For the IMC design procedure (Garcia and Morari, 1982)the controller is parametrized by Q as K = Q(l - PQ)-l (1) and Q is selected as follows. Step 1. Determine 8 to minimize the integral square error (ISE): A r n e 2dt = A m y f 2dt for the particular input d. Assume that P is stable with the exception of 1 poles at the origin. Factor P into an all-pass portion PA and a minimum-phase portion PM: P = PAP, (3) 90 that PAincludes all the zeros belonging to the right half plane (RHP) and delays of P and IPAI = 1, v o. In general PA has the form -5
+ s;.
PA = e-aoni s+Ti
0888-5885/ 9212631-1920$03.Oo/O 0 1992 American Chemical Society
Re
(li), > 0
(4)
Ind. Eng. Chem. Res., Vol. 31, No. 8,1992 1921
T
~
rd
Figure 2. Block diagram used for design of LQOC.
~
Figure 1. Block diagram d e f i i structure of eyetern under study.
where the overbar denotes complex conjugate. Factor the input d similarly: d = dAdM (5) Assume that d has at least 1 poles at the origin. The controller 8 which minimizes the ISE is given by
g
(p~d~)-'{fi'd~]* (6) where the operator { .] denotes that after a partial fraction of the operand all terms involving the poles of PA-^ are omitted. Morari and Zafiriou (1989) derived a more general result for unstable systems. It appears that the case of multiple poles at the origin had not been covered in the literature correctly before. The analytic form (6) is convenient for the simple models common in process control, because it clearly points out the performance limitations imposed by nonminimum phase elements. It also provides some insight into the structure of the controller which is often lost in numerical schemes. For complex models a numerical method as outlined for the LQOC below might be preferable. Step 2. Augment 8 by a low-p-ass filter F, which "detunes" the aggressive controller Q, to form Q = OF (7) Choose F for robustness. (Increased robustness goes hand in hand with smaller bandwidth and decreased control action.) The ISE criterion for the rejection of a particular disturbance has direct engineering significance. F is convenient for tuning because it appears in an affine (Le., linear) manner in all the closed loop transfer functions of interest y = PeFr (8) e = (1 - PgF)(d - r ) (9) Here r is the reference or setpoint. The choice of F determines directly closed loop response time, bandwidth, roll-off characteristic, etc. The following simple low-pass filters with the tuning parameter X were found to be effective. for asymptotically step-like inputs
order n is selected to make Q proper and to achieve the desired roll-off for robustness purposes.. In the absence of modeling error closed loop stability is guaranteed for all X 1 0. Q is ISE optimal for X = 0. For minimumphase (MP) systems X is the closed loop time constant. For nonminimum phase (NMP) systems the maximum achievable bandwidth is limited by the RHP zeros and time delays. X is the dominant closed loop time constant when 1/X is chosen to be smaller than this maximum bandwidth. In summary, in IMC the trade-off between performance and robustness/control action severity is affected by a single parameter X which has direct physical significance. There is a minimum order n of the filter required for properness of the controller. Generally, higher values of n tend to make the system more sluggish, but might be required for robustness reasons depending on the type of model uncertainty. Similar controller design methods have been suggested repeatedly. For a historical review see Morari and Zafiiou (1989). In particular Dahlin's algorithm is cited often. Note, however, that the well-known problems (rippling) associated with Dahlin's algorithm are not relevant here because they are specific to sampled-data systems. They can also be dealt with in the IMC framework (Zafiiou and Morari, 19851, however. 3. Linear Quadratic Optimal Control The block diagram of the configuration to be studied is shown in Figure 2. Here W, is a fictitious weight (transfer function) introduced to penalize the control action u (u' = W,u).The controller design objective is to minimize:
for asymptotically ramp-like inputs
for a particular input d . Objective 12 does not have much physical significance. Its main advantage is that well-established methods exist for finding the optimal controller. For any reasonable control problem there will be a trade-off between the two terms in (121, the output error y' and the weighted control action u ! By adjusting the weight W,, small control action or a small error can be favored. Thus minimizing (12) is a convenient, albeit indirect, technique to effect the performance/robustness trade-off. Via Parseval's theorem, objective (12) can be expressed in the frequency domain as
These are the simplest filters leading to controllers which compensate step (ramp-) like inputs offset-free as t m. More general filters are discussed by Morari and Zafiiou (1989). They may lead to improved performance if properly tuned. With an increased number of tuning parameters, this is not straightforward, however. The
where Cabdenotes the closed loop transfer function between input b and output a. The optimal controller minimizes the "average" (in the sense of the integral) s u m of transfer function magnitudes. In the frequency range where W,is small, much control action is allowed and the closed loop performance measured by the magnitude of
F=
-
1
(As + l)n'
n l l
(10)
1922 Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992
the sensitivity function Gy'd' will be good and vice versa. This discussion gives some guidelines for the selection of the weight W,,: ita magnitude should increase with frequency. When w d has m integrators, then for low frequencies W, has to behave like sm to guarantee no offset. An increase of the polezero difference of W,,increases the roll-off order for the closed loop system. The controller which minimizes (12) can be found through an extension (to the case W,, # 0) of the results given (for W,, = 0) by Newton et al. (1957), via the frequency domain Wiener-Hopf technique. The controller Q'in an open loop structure, for the case of disturbance rejection, can be obtained from
where
D(s) = d(s) d(-s) = D+(s)D-(s) A(s) = P(s) P(-S) + W,,(S)W,,(-S)= A+(s) A-(s)
(144 (14b)
A+(s) and D+(s)contain negative zeros and poles of A(s) and D(s),respectively. For W , = 0 formula 14 gives the controller 8 minimizing the ISE (2) and becomes similar to (6). Analytical solutionsderived from (14) will be used in the sequel for some simple systems. This point is discussed in more details in the Appendix, where a method of numerical solution is also presented. Comparing the two procedures, the first difference to be stressed is that in LQOC the design of the robust controller Q'is made in one step, taking simultaneously into account process, disturbance, and input weight. Contrary to the IMC filter F, the effect of the weight W , is not reflected transparently in the closed loop transfer functions of interest. The controller tuning, to meet robustness constraints by adjusting the weight parameters, has to be made largely by trial and error. 4. Robustness and Performance Measures Controllers w i l l be compared in terms of the closed loop frequency response. Performance is best judged by the sensitivity function E defined as follows:
e = (y - r ) = ( 1 + PK)-l(d - r ) = ( 1 - PQ)(d - r ) A e(d - r ) (15) The sensitivity function e should be small for low frequencies (i.e., good steady-state tracking) and approaches unity for high frequencies (i.e., the control system is incapable of dealing with inputs of very high frequency). To judge robustness, it is best to define a set of II of elants P in the neighborhood of the nominal plant (model)
y = PK(I
+ f%)-lr
= PQr A qr
(18)
(Doyle and Stein, 1981). Note that
z + 1 = ( 1 -PQ)+ PQ = 1
(19) At high frequencieswhen 1, is large, 1 has to be small k d therefore 7 = 1. Thus there is a basic trade-off between good nominal performance (Z small) and good robustness (8 small).
5. Comparison between the Two Methods A formal analogy between the two methods can be established by rewriting the LQOC controller in the form given by (7) as Q' = GF' (20) where 8 minimizes the ISE (2) and F', which can be obtained from (14)and (20), becomes a type of LQOC filter. The parallelism allows an easy comparison: main analogies and differences can then be pointed out starting from the simplest cases for which an analytical solution is possible. A system of general interest in process control is given by a first-order process P (gain K , time constant r ) , with with a step-through-lag disdelay (e) and RHP zero (l), turbance wd (gain Kd, time constant r d ) :
-p
+1
p = K-e-88 rs + 1
(21)
The controller which minimizes the ISE is
The filters to be added for robustness in the two designs, for step-like inputs, are now presented: IMC Design: For step-like input a type-1 filter is recommended (10): F = (As + l)-n n 1 1 LQOC Design: An input weight W,, = asrn,m 2 1, should be used. For m = 1, the fiiter F'from (15) and (20) is
with
n
n = (p:((P - P)PI II,(w)l
(16)
Here Z,(w) is the (frequency dependent) bound on the multiplicative uncertainty. Definition 16 can be interpreted in the following manner: at each frequency w the frequency response P(io)_canlie anywhere in a disk with radius PZ,(w) centered at P(i0). Generally 1, increases with frequency and reaches unity at a frequency when the phase becomes completely uncertain. A closed loop system is stable for the family of plants described by n if and only if 11, < 1 (17) where the complementary sensitivity function 1 is defined through
Comparing the two expressions (10) and (25), the main differencebetween the two methods becomes evident: the IMC filter (F)depends only on the coefficient X (and on the filter order n),while the LQOC filter (F?, in addition to the weight coefficient a (and to the weight order m) depends on process parameters ( K , r , 0) and on the disturbance time constant ( r d ) . Thus, in general, the two controllers may behave differently, also if in many cases almost identical results can be obtained.
Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992 1923 Table I. Values of the Coefficient a for the LQOC Filter W ,= 4 8 , as a Function of Process Parameters K and T To Have Roll-off at the Two Different Frequencies: wB1 = 1 rad/@and w a s = 0.1 rad/s process parameters WBl OB2 K=l,r=l a = l a = 20 K = 1, T = 100 a = 0.01 a = l K = 100, T = 1 a = 100 a s 2000 K = 100, T = 100 a = l a = 100 K = 10, T = 10 a=l a = 100 1
10
10
10
1.
10
lo
10
1.
frequency
B
1
-1
10
0.a
-2 10
0.6
-3
10
0.4
2 10
10
.
10
1.
10
10
frequency 0.2
1 0.
0.
20
10
30
0.8
time Figure 3. (A) Example 1: sensitivity (2, dashed line) and complementary sensitivity function (TI, solid line) for different IMC filter orders n and time conatanta A. Robustness constraint: roll-off at osl = 1 rad/s and %z = 0.1 rad/& At wl: h = 1e; (1)n = 1; (2) n = 2. At %,: A = 10 s; (3) n = 1; (4) n = 2. (B) Example 1: time reaponsee for IMC in the different cases of (A).
The complex dependence of the LQOC filter on the tuning parameter a,which‘appears in a highly nonlinear manner, makes the design for robustness much more difficult. A new design is required for different values of process or disturbance parameters. Practical implications of these assertions will be clear from the few selected examples which will be shown in the next section. 6. Examples The two design methods have been compared on a variety of situations including minimum and nonminimum phase systems, simple and more complicated “disturbance models”, and systems with and without delay. The three examples presented here allow the realization of how the different characteristics of process and disturbance affect the design for robustness in the two methods. For a fair comparison the same robustness specifications are required in each case. Example 1. For the simple w e of a fmt-order system with step disturbance, a control system is to be designed to give roll-off at two different frequencies (aB1 = 1rad/s and qz= 0.1 rad/s), corresponding qualitatively to lower and higher uncertainty in the model. P = k(7s + 1)-1, Wd = s-1
The ISE optimal controller is Q = P’= (7s
+ 1)k-1
IMC Design: The filter is, from (10) F = (As + l)-* n 11
0.6
0.4 0.2
0. 0.
10
time
20
30
Figure 4. (A) Example 1: sensitivity (5, dashed line) and complementary sensitivity function (q, solid line) for LQOC with different values of process parameters. Input weight W, = as. Robustness constraink roll-off at osl = 1rad/s and 0 8 ~= 0.1 rad/s. At os1: (1) = 1 s (K = 1, T = 1). At W B ~ : (2) (Y 20 (K= 1, T = 1). At W B ~ : (3) (Y = 1 s (K = 1, T = 100). (B) Example 1: time responses for LQOC in the different cases of (A).
The effect of the filter parameter X and of the filter order n on 7 (performance) and ii (robustness) is shown in Figure 3A. The corresponding time responses, which are independent of process parameters, are shown in Figure 3B. A filter order 2 would be required if the uncertainty increased “rapidly”with w. Values of X to have roll-off at the two frequencies (ql and ~ 2 are) promptly computed. For both filter orders: X = 1 s and X = 10 s. A secondorder fdter gives a more sluggish response than a firsborder one. LQOC Design: The filter F’is, m a particular case from (25) F’ = K(as2+ bs + c)-l In this case the filter F’depends on all the process parameters and therefore also the weight coefficient a,necessary to achieve certain roll-off characteristics, does. Table I clearly illustrates that even for this very simple example it is impossible to predict a priori which value of a is needed to achieve specified robustness characteristics for a particular problem. The proper value of the weight
1924 Ind. Eng. Chem. Res., Vol. 31, No. 8,1992 -3
A
'10
L'
I
I
I
1. frequency Figure 5. Example 2 multiplicative uncertainty 1, and complementary sensitivity q for IMC (lower curve) and LQOC (upper curve).
0.
20
10
time
40
30
50
~ o l i d :I W C
20
100
0.
0
10
20
time
30
40
10
30
20
40
300
200
Figure 7. (A) Example 3 output response for the IMC controller. Robustness constraint roll-off at wel = 1 rad/s. Parameter valuea: V K,V 7 ; K d = 1, 7 d = 100. (1) Type-1 filter (A = 1); (2) type-2 filter (A = 2). (B)Example 3 output response for the IMC controller. Robustness constraint roll-off at wez = 0.1 rad/s. Parameter valuea: V K, V 7;Kd = 1, rd = 100. (1) Type-1 filter (A = 10); (2) type-2 flter (A = 20).
50
B
0
time
50
t ime
Figure 6. Example 2: responses of the closed loop system with IMC/LQOC. (A) Nominal system P = P = e-151. (B)Uncertain system P = e+'.
a has to be determined by trial and error. Furthermore, any adjustment in a requires all the controller parameters to be recomputed, which is rather complicated when an analytical solution is available, as in the example here, and is involved when a numerical solution has to be used, as in the general case. It is also noted that the shape of the F'(w) = ?(a)curves may vary at intermediate frequencies, depending on the process parameters. For this reason, in the cases indicated with z, the value of a should be computed more exactly for the specific robustness problem, starting from the indicated value. This is shown in Figure 4A, while Figure 4B shows the corresponding time responses. It is clear that for appropriately chosen a ' s the performance and robustness characteristics of the two schemes can be made very similar. A comparison of performance
-
between the two methods (with n = 2 for IMC and m = 1for LQOC, to have the same slope at w m) shows that in some cases almost completely coincident results can be obtained (e.g., IMC curve 2 in Figure 3 and LQOC curve 1 in Figure 4), while in other cases results cannot match exactly. Adopting an input weight of order m = 2 (see the Appendix), analogous conclusions would be reached about variations of the required weight coefficient a with process parameters and about the comparison with IMC. Example 2. In this example a specific problem of robustness is faced. A control system is to be designed for a pure dead-time process so that it rejects ramplike inputs asymptotically without error. Closed loop stability is to be pzeserved for variations in the delay. Assume 0 = 15, le - el = 2. Z, e e-1&, W,= s-2
P = e-Bs, 13 I 0 I17 An upper bound on the multiplicative uncertainty caused by the time delay error can be easily constructed (Figure 5): 1, =
- 11,
1, = 2,
0
w
L a/2
Ia / 2
Note that this uncertainty description is Yconservativen in that it dowa other types of modeling errors apart from delay errors. Therefore condition 17 is only sufficient for
Ind. Eng. Chem. Res., Vol. 31,No. 8, 1992 1925 -3
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time time Figure 8. (A)Example 3: output response for the LQOC controller. Robustness contraint: roll-off at W B ~= 1 rad/s. (Al)Parameter values: K = 1, T = 1; k d 1, Td = 100. (1)w,= as, 10. (2)w,= as2, = 2. (A2) Parameter Values: K 1, 7 100; Kd 1, Td 100. (1)w, = as, a = 0.02. (2)W , = as2, a = 0.02. (B)Example 3: output response for the LQOC controller. Robustness constraint: roll-off at aa = 0.1 rad/s. (Bl)Parameter values: K = 1 , =~1;& = 1, Td = 100. (1)W, = as, a = 50. (2) W, = as2,a = 200. (B2)Parameter values: K = 1, T = 100, Kd 1, Td = 100. (1)w, (YS, = 2. (2) w,= 05*, CY = 20. (Y
robust stability in the presence of a pure time delay error of f 2 . The ISE optimal controller is = 8s + 1 = 15s + 1
e
IMC Design: For ramp-like inputs a type-2 filter has to be used:
F=
(nAs + 1) n12 (As 1)"
+
LQOC Design: An input weight W,, = asm,rn 1 2, should be used. From (14) the corresponding filter becomes
F' =
dls
+1
1
+ bs + c 8s + 1 with a = a,b = (2a)'I2, c = 1, dl = 8 + CY)'/^. as2
Again the effect of the LQOC -weight coefficient a changes with the process parameter 0, while the IMC filter coefficient A does not. For the specific example under study essentially identical results are found; a filter order n = 3 (IMC) and an input weight order m = 2 (LQOC) were chosen for comparison to achieve the same slope of the ~ ( w curves ) for w a. The robust stability condition (17)is satisfied in the two cases for A = 9.25 and a = 50. The do) plots are shown in Figure 5, while the-corresponding time responses in the nominal case (6 = B = 15) and for the worst uncertain case (0 = 17) are reported in Figure 6A and Figure 6B, respectively. Example 3. For the case of a first-order process with step-through-lag disturbance, a control system is to be
-
designed to give roll-off at the same frequencies wB1 and as in example 1. P = K / ( m + 1)
wB2,
w d
= Kd/S(TdS
+ 1)
The ISE optimal controller is = (7s
+ 1)/K
IMC Design: Because the filter F does not depend on process and disturbance parameters, the output response Y ( d ) = (1- F ) w d depends only on the disturbance. For small values of the disturbance lag 7 d , no differences are found with respect to example 1. For large values of T d , by adopting a type-1 filter F = (As + l)-",n 1 1, the response exhibits a very slow return to the steady state (Figure 7). The peak is very low, but it can reach higher values for disturbance gain K d >> 1. This can be undesirable in some cases of practical relevance, for example when the disturbance acts at the process input P = Wd. Adopting a higher order type-1 filter does not help in this case, while significant improvements can be obtained by using a type-2 filter as shown in the same figures. This is not surprising, as for large values of Td the examined disturbance becomes similar to a ramp. LQOC Design: The filter F' depends on the process parameters K,T and on the disturbance lag Td (25), (in addition on the weight coefficient a). Thus, the output response Y(d)= (1 - F?Wd depends on both process and disturbance. Again the LQOC filter must be recomputed for different valuea of the process parameters and it may not be poseible
1926 Ind. Eng. Chem. Res., Vol. 31, No. 8, 1992
to obtain completely coincident results when process parameters change, owing to the different slopes of the q ( w ) curves. Some sample time responses are reported in Figure 8, for different values of process and disturbance parameters. For the case i d = 100 and T = 1 (slow disturbance and fast process), the same sluggish return to steady state is shown when an input weight of order m = 1 is used (Figure 8A1,Bl). Analogously to IMC, responses can be improved by using a higher order input weight m = 2. For the case i d = 100 and i = 100 (process and disturbance equally slow), the use of a higher order filter is unnecessary (Figure 8A2,B2). In this case, especially for roll-off at the lower frequency, performance obtained with a first-order input weight is superior to the one given by both IMC filters. 7. Summary and Conclusions In the proposed general formulations for IMC and LQOC, ‘detuning” to meet robustness constraints can be obtained via the IMC filter parameter X and the LQOC weight parameter a. The formal analogy introduced between the two methods, which expresses the gobust controller in terms of the same ISE controller (&) with different filters, clearly points at similarities and differences for robustness studies. The main difference is that the LQOC filter depends also on the process and disturbance parameters, while the IMC filter does not. As a general consequence, results given by the two methods are not completely equivalent. Some differences may be shown for specific control problems, though, in the large majority of cases, the quality of performance is very similar. The design for robustness is much more straightforward in IMC, because the filter coefficient X can be chosen on the basis of user requirement, while in LQOC a changes with system parameters and has to be chosen by trial and error.
Acknowledgment Partial support from the National Science Foundation (U.S.A.)and the Consiglio Nazionale delle Ricerche (Italy) is gratefully acknowledged. This paper was originally submitted to Industrial & Engineering Chemistry Research on June 6, 1991.
Appendix Analytical Solution for LQOC. The availability of explicit expressions for the LQOC controller (Q’ = QF1 is valuable for realizing the dependence of the filter parameter F’ on process and disturbance parameters. For some of the simple cases for which the comparison between the two methods has been presented, they can be derived from (14), here rewritten:
with D ( s ) = d(s) d(-s) = D+(s)D-(s) A(s) = P(s)P(-s)
+ W,(S)W,(-S) = A+(s)
A-(s)
(A21
(A3)
Analogous relationships have been presented by Palmor (1982) in the case of stochastic control of systems with multiplicative delay. In his approach input disturbances are described through rational spectrum densities. He
derived explicit expressions for some simple examples in the case of unitary gain and zero- and first-order input weights. In solving (Al), the polynomial of highest degree to be factored is the numerator of A(s) of degree p = 2(n m), which can be reduced to p‘ = n + m;n and m are the degree of the denominator of P(s)and of the input weight Wu(s),respectively. In all the cases for which explicit expressions of Q’ have been presented in the examples,.p ’ I2. Even for the simple case of a fmborder process w t h second-order weight ( W , = as2, results shown in example 1) a sixth degree is obtained (p = 6, p’ = 3). In this case the filter F’becomes p +1 ds + 1 F’ = K (A4) (as2 + bs + c)(es + 1) 6s + 1
+
with the filter coefficients a, b, c, d, and e depending on the six roots of the polynomial to be factored. The increased order of the input weight m = 2 is reflected in a higher order of the filter denominator, which a. Apart from this considgives a higher slope for u eration, the dependence from the system parameters becomes too complex to be worth to be investigated and useless for robust design purposes. In these cams, and for higher order of process and input weight, a numerical solution, as outlined below, can be used advantageously. Numerical Solution for LQOC. A numerical solution to the controller Q’ minimizing (12) can be found via state-space techniques. Let G be the 2 X 2 system:
-
L
1
L
J
and let the quadruple (A,B,CP)be the state space realization of G,i.e. 3 = AX + B ~ u + Bid’ y‘ = C’X u’= C ~ + X DZ~U (A6) Here B, C, and D have been partitioned appropriately. This “singularl problem can be solved as follow. Because the measurementsare uncorrupted by noise, the states can be recovered through an observer essentially instantly. Thus we can formally design a state feedback controller to minimize (12). With the optimal state feedback controller (A6) becomes (Kwakernaak and Sivan, 1972)
+
3 = (A - B2D22-’B2TR)~ Bld’ y’ = C’X
u‘= C ~ +X D22u where R is the solution of the Riccati equation: ATR + RA + P C - RB2D22-2B2TR= 0
(A71
(A8) (A7) is the realization of the transfer function Gydt,which is according to Figure 2 Gytdt = (1 + PK)-’Wd (A91 Thus the optimal controller K in closed loop form is K = P’( WdGytdt-’ - 1) (AW
It can be easily shown that this controller is always proper. The operations implied by (A10) can be executed conveniently in state space with standard software, e.g., CONSM (Holt et aL, 1987). A Riccati equation solver is also part of most control system design packages. For the Riccati equation to have a solution DZ2# 0 and (A6) must
Ind. Eng. Chem. Res. 1992,31, 1927-1936
be stabilizable. For DZ2# 0 we require lime- W,(s) # 0; i.e., high-frequency inputs must have a finite penalty. For stabilizability W,must be stable. Though the proposed procedure for avoiding singular observer problems is straightforward, we are not aware that it has been reported in the literature. The procedure has been checked successfully for the simple cases for which an analytical solution is possible and then has been used to design the LQOC controller Q’ for higher order systems. Note that time delay systems can be included by using low-order Pad6 approximations. For the case presented in example 2, no practical differences were found by usinga second-order Pad6 approximation of the model delay 8 in the design of the controller.
Literature Cited Bergh, L. G.; MacGregor, J. F. “Constrained Minimum Variance Controllers: Internal Model Structure and Robustness Properties”. Process Control Laboratory, McMaster University: Hamilton, Ontario, Canada, 1986; Report No. 1001. Doyle, J. C.; Stein, G. Multivariable Feedback Design: Concepts for a Clessical/Modern Synthesis. ZEEE Trans Autom. Control 1981, AC-26, 4-16. Garcia, C. E.; Morari, M. Internal Model Control. 1. A Unifying Review and Some New Results. Znd. Eng. Chem. Process Des. Dev. 1982,21, 308-323; also presented at Annual AIChE Mtg., New Orleans, LA, 1981.
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Grimble, M. T.; DeLaSalle, S.; Ho,D. Relationship Between Internal Model Control and LQG Controller Structures. Automatica 1989, 25,41-53.
Harris, T. J.; MacGregor, J. F. Design of Multivariable LinearQuadratic Controllers Using Transfer Functions. AZChE J. 1987, 33,1481-1495.
Holt, B. R.; Buck, U.; Economou, C.; Grimm, W.; Grodidier, P.; Jerome, N. F.; Jordan, K.; Klewin, R.; Kraemer, W.; Kugueneko, G. V.; Mandler, J.; Ness, A.; Webb, C.; Zafiriou, E.; Morari, M.; Ray, W. H. CONSYD-Integrated Software for Computer Aided Control System Design and Analysis. Comput. Chem. Eng. 1987, 11, 187-203.
Kozub, D. J.; MacGregor, J. F.; Wright, J. D. Application of LQ and IMC Controllers to a Packed Bed Reactor. AZChE J. 1987,33, 1496-1506.
Kwakernaak, H.; Sivan,R. Linear Optimal Control Systems; Wiley Interscience: New York, 1972; p 260. Morari, M.; Zdiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. Newton, G. C.; Gould, L. A.; Kaiser, J. F. Anulytic Design of Linear Feedback Controls; Wiley: New York, 1957. Palmor, Z. J. Properties of Optimal Stochastic Control System with Dead-Time. Autornotica 1982,18, 107-116. Palmor, Z. J.; Shinnar, R. Design of Sampled Data Controllers. Znd. Eng. Chem. Process Des. Dev. 1979, 18,9. Zafiiiou, E.; Morari, M. Digital Controllers for SISO Systems. A Review and a New Algorithm. Znt. J. Control 1985,42,855-876.
Received for review March 24, 1992 Accepted May 1, 1992
Synthesizing Optimal Flowsheets: Applications to IGCC System Environmental Control Urmila M. Diwekar,* H. Christopher Frey, and Edward S. Rubin Department of Engineering and Public Policy and Center for Energy and Environmental Studies, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
A new process synthesis capability implemented in the public version of the ASPEN chemical process simulator is demonstrated via an illustrative case study of a complex flowsheet. The objective of the case study is to minimize the cost of an advanced integrated gasification combined-cycle (IGCC) plant design featuring hot-gas cleanup, subject to environmental constraints. The problem is formulated as a mixed integer nonlinear programming (MINLP) optimization problem, involving the selection of both an optimal process configuration and optimal design parameters for that configuration. Performance and cost models of the IGCC system developed for the ASPEN simulator, along with the newly developed process synthesis capability, are used. As a first step, alternative in situ and external desulfurization are considered as process alternatives. 1. Introduction There is a significant interest today in the ability of integrated coal gasification combined-cycle(IGCC) systems to provide electricity reliably and at lower cost relative to conventional fossil fuel alternatives. The ability of IGCC systems to meet stringent environmental emission standards is another attractive feature of this technology. Environmental control systems, however, account for a significant part of the cost and complexity of IGCC systems. Current systems require cooling of the gas stream prior to cleanup, thus generating a significant wastewater stream which must be treated in addition to the air pollutant and solid waste streams normally associated with coal-based electric power generation. Hobgas cleanup systems offer the potential for significantly simplifying and reducing the cost of environmental control for many IGCC systems. This is currently the subject of intensive research and development. In addition to the technical aspect of IGCC technology, there is also a strong need for ‘systems” research to
identity the best ways of configuring IGCC systems and of incorporating advanced cleanup and other technology to produce electricity at minimum cost. For example, the most common design for sulfur removal using hot-gas cleanup is through the use of solid sorbents. Sulfur capture occurs either through the addition of a solid reactant in the gasifier (i.e., in-bed desulfurization), or by external desulfurization of the flue gas (e.g., zinc ferrite process) or by a combination of these two methods. A number of computer models have been developed by the U.S. Department of Energy’s Morgantown Energy Technology Center (DOE/METC) to allow analysis of different configurations of IGCC systems using the ASPEN process simulator (e.g., Stone, 1985). Selection of the optimal flowsheet configuration and the optimal design parameters, however, requires additional capabilities in the modeling environment. A new process synthesis capability that has been implemented in ASPEN (Diwekar et al., 1992) provides a tool to obtain the optimal flowsheet configuration and design
0888-5885/92/2631-1927$03.00/00 1992 American Chemical Society
