1,

0 10 20 SF

Figure 7.4. (a) Operating diagram for Example 1 with the cell growth following Monod kinetics (fim = 1.0 h~l, K$ = 0.05 gL-1, and Kj —> oo in Table 3, D in h~l and SF in gL-1). p(SF) = D on the curve ABCDE. Forced periodic operation with variations in D and SF is superior to steady-state operation (i) at all frequencies in region I [(SFq,Dq) lying below the curve ABFG], (ii) for 0 < u) < wi and U2 < u> < oo (u>2 > u>i) in region II [(5Vo, D0) lying between the curves BFG and BCDE], and (iii) for all uj except uj = w* for (SFq,Dq) lying on the curve BFG =

Pi 1/922 at u> = a;»), (b) For a particular SFo, forced periodic operation involving variations in D and SF is superior to steady-state operation for (z, D0)(z = to2) lying outside the curve ABCDEF [f^ax = pnp22 on the curve ABCDEF, f2nax < P11P22 for (z, Do) enclosed by the curve ABCDEF, and fmax > P11P22 for (2, D0) lying outside the curve ABCDEF]. Regions I and II in (a) correspond to Do < D» and Do > D„, respectively. D„ corresponds to asterisk [449].

Figure 7.4. (a) Operating diagram for Example 1 with the cell growth following Monod kinetics (fim = 1.0 h~l, K$ = 0.05 gL-1, and Kj —> oo in Table 3, D in h~l and SF in gL-1). p(SF) = D on the curve ABCDE. Forced periodic operation with variations in D and SF is superior to steady-state operation (i) at all frequencies in region I [(SFq,Dq) lying below the curve ABFG], (ii) for 0 < u) < wi and U2 < u> < oo (u>2 > u>i) in region II [(5Vo, D0) lying between the curves BFG and BCDE], and (iii) for all uj except uj = w* for (SFq,Dq) lying on the curve BFG =

Pi 1/922 at u> = a;»), (b) For a particular SFo, forced periodic operation involving variations in D and SF is superior to steady-state operation for (z, D0)(z = to2) lying outside the curve ABCDEF [f^ax = pnp22 on the curve ABCDEF, f2nax < P11P22 for (z, Do) enclosed by the curve ABCDEF, and fmax > P11P22 for (2, D0) lying outside the curve ABCDEF]. Regions I and II in (a) correspond to Do < D» and Do > D„, respectively. D„ corresponds to asterisk [449].

Do < /ii (5'fo) [under the curve ABCEFG in Figure 7.5(a)], a unique locally, asymptotically stable non-trivial steady state is admissible, the global asymptotic stability of which is also assured since the washout state is unstable. Two non-trivial steady states are admissible in a portion of the Sf — D space where D0 > /-¿1 (5>o) [(Sfo, Do) lying inside the envelope CEFHC in Figure 7.5(a)]. One of the non-trivial steady states and the washout state are locally stable in this portion. On the curve CEF [excluding points C and F, ni(Spo) = D0], a unique non-trivial steady state, which is globally, asymptotically stable, is admissible.

Since positive J is of interest, it follows that (u + c) must be positive (Eq. 7.97). In the entire region of the Sf — D space where a stable non-trivial steady state is admissible [below the curve ABCHFG in Figure 7.5(a)], periodic control with variation in D alone is not proper. When u = Sp, Ptz(u) is positive for some w in region I [(Sfo, Do) lying to the right of the curve FIJ and below the curve FG, Figure 7.5(a)] and negative for all uj in region II [{Spo, D0) lying below the curve ABCHFIJ] and for {Sfo, D0) lying on the curve FIJ excluding point F. In region I, forced periodic operations subject to weak variations in SF will yield superior performance vis-a-vis steady-state operation.

The effect of simultaneous periodic variations in D and Sp on the bioreactor performance was examined for v = w — 0 (Eq. 7.95). In these operations, 5J is positive at all frequencies in region I and for uj\ < u> < oo {ui\ / 0, wi depends on D0 and Sfo) in region II. On the interface between the two regions [('SVo- D0) lying on the curve FIJ excluding point F], 5J is positive for uj > 0.

Following conditions must be satisfied at the optimal steady-state (D = D*. Sr = S'F)

For v — w — 0 and parameters in Eq. 7.101, the optimal steady-state lies in region II [Figure 7.5(a)]. The performance of the optimal steady-state operation cannot therefore be improved by weak periodic variations in D or Sp alone. Weak periodic variations in both D and Sp (around D* and Sp, respectively) will lead to improved performance for u> > 4.165 cycles h~l.

An analytical expression for the optimum frequency (wo) for maximal improvement in performance of a steady-state operation via weak periodic variations in Sf alone (in region I) is provided in [448]. For Do = 0.3045 /i-1, a comparison between the maximum improvement attainable (vis-a-vis steady-state operation) in forced periodic operations involving weak variations in Sp alone and in both D and Sp (SJ as in Eq.

Figure 7.5. Results for Example 2. (a) Operating diagram for Example 2 (D in h~] and Sp in gL'1). ^i(Sf) = D on the curve ABCEFG. Non-trivial steady-states are admissible for (Sp, D) lying below the curve ABCHFG. The number of non-trivial steady-states is (i) one for (Sp, D) lying below the curve ABCEFG and on the curve CEF (excluding points C and F) and (ii) two for {Sp, D) lying inside the envelope CEFHC. Periodic operations involving weak variations in Sp are superior to steady-state operation only in region I [(Sfo, Do) lying to the right of the curve FIJ and below the curve FG]. For v = w = 0, periodic operations involving weak variations in Sp and D are superior to periodic operations involving weak variations in Sf and steady-state operation (i) at all frequencies in region I and for (Sfo,Do) lying on the curve FIJ (excluding point F) and (ii) for loi < cj < oo (wi > 0) in region II [(SF0,D0) lying below the curve ABCHFIJ]. The asterisk denotes the optimal steady-state for v = w = 0. (b) Portraits of Q [curve 1: Q — £22(^0), curve 2: Q = G2U(^o)] and Sfo and ¿>1 {uja) and SF0 (dashed curve) for D0 = 0.3045 h'1. (c) Portraits of G2 for < = 0.0417 and u> (dashed curve) and G2u and u> (solid curve) for Do = 0.3045 h~l and Sfo = 101.5 gL~x. C, = 0.0417 is the optimum amplitude ratio (G2 = G2u) at u = 1.904 cycles h'1. G2 = p22 + 2/C + pu(2, G2u = p22 - f^ax/Pn [449].

7.90 with / = fmax) is provided in Figure 7.5(b) for various Spo's in region I (u> = ujo in both types of operations). The benefit of simultaneous variation in D and Sf over variation in Sp alone is self-evident. The differential in the maximum improvement attainable in the two forced periodic operations is significantly sensitive to SFo■ For certain sets of operating parameters, (Sfo, Do) therefore, periodic operations involving variation exclusively in Sf can be substantially inferior to those involving variations in both D and Sf- In the narrow range of Spo considered in Figure 7.5(b), there is substantial variation in the optimum phase difference (0 = \) that leads to maximum positive interaction between D and Sp (/ = fmax)- The optimum frequency cjq for forced periodic operation involving variation in Sf alone decreases with increasing Sfo (profile not shown).

For D0 = 0.3045 hand SFo = 101.5 gL~l, variations in G2 (G2 = P22 — f2/pn, Pu < 0, Eq. 7.90) for the optimal amplitude ratio (G2 for / = fmax) and G2 for a fixed amplitude ratio (( = 0.0417) are presented in Figure 7.5(c). The amplitude ratio in the latter case is the optimal amplitude ratio only at u) — 1.904 cycles h~l. Periodic operations employing this amplitude ratio are suboptimal at other frequencies [Figure 7.5(c)], The difference between the performance of periodic operation employing optimal amplitude ratio and that of the periodic operation employing a fixed amplitude ratio increases as the deviation of u> from the frequency for which the fixed amplitude ratio is the optimal one [to = 1.904 cycles h~l in Figure 7.5(c)] increases.

Example 3.

The expressions for p, a and e are provided in Eq. 7.68, with the parameter values being [190, 600]

K's = 0.666 g/L, Pm = 87 g/L, P"m = 114 g/L, Kr = 203.49 g/L,

A unique non-trivial steady state is admissible in that portion of the Sf~D space where Pi{Spo) > D0 [(SfOj D0) lying below the curve ABCDEF in Figure 7.6]. The non-trivial steady state does not undergo any Hopf bifurcations and since the washout state is unstable when Pi{Sfo) > Do, the non-trivial steady state is globally, asymptotically stable.

Application of 7r-criterion was considered for v ~= 0 (Eq. 7.95). Periodic control was found not to be proper when u = D in the entire region where fit (Sj.'o) > Do- When u — Sf, />22(>') is positive for some uj in region I [(<SVo, Do) lying to the right of the curve DGH and below the curve DEF] and negative for all w in region II [(5^0, -Do) lying below the curve ABCDGH] and for (Sfo, D0) lying on the curve DGH (excluding point D)

Figure 7.6. Operating diagram for Example 3 (D in h~l and Sy in gL_1). Hi (Sp) = D on the curve ABCDEF. A unique non-trivial steady-state is admissible for (Sf, D) lying below the curve ABCDEF. For v = 0, periodic operations involving weak variations in SF are superior to steady-state operation only in region I [(Sfo, do) lying to the right of the curve DGH and below the curve DEF]. For v = w = 0, periodic operations involving weak variations in Sf and D are superior to periodic operations involving weak variations in Sf and steady-state operation (i) at all frequencies in region I and for (Sfo, -Do) lying on the curve DGH (excluding point D) and (ii) for <x>i < uj < oo (wi > 0) in region II [(Sfo, do) lying below the curve ABCDGH], The asterisk denotes the optimal steady-state for v = w = 0 [449].

Figure 7.6. Operating diagram for Example 3 (D in h~l and Sy in gL_1). Hi (Sp) = D on the curve ABCDEF. A unique non-trivial steady-state is admissible for (Sf, D) lying below the curve ABCDEF. For v = 0, periodic operations involving weak variations in SF are superior to steady-state operation only in region I [(Sfo, do) lying to the right of the curve DGH and below the curve DEF]. For v = w = 0, periodic operations involving weak variations in Sf and D are superior to periodic operations involving weak variations in Sf and steady-state operation (i) at all frequencies in region I and for (Sfo, -Do) lying on the curve DGH (excluding point D) and (ii) for <x>i < uj < oo (wi > 0) in region II [(Sfo, do) lying below the curve ABCDGH], The asterisk denotes the optimal steady-state for v = w = 0 [449].

(Figure 7.6). The performance of steady-state operation can be improved via periodic variations in Sp in region I. The effect of simultaneous periodic variations in D and Sp on the bioreactor performance was examined for v = w = 0 (Eq. 7.103). In such operations, 5J is positive (i) at all frequencies in region I and for (Spo, Do) lying on the curve DGH and (ii) for cji < to < oo (loi / 0, depends on Dq and Sfo) in region II. For v = w = 0 and the parameters in Eq. 7.103, the optimal steady-state (subject to Eq. 7.102) lies in region II (Figure 7.6). The performance of the optimal steady-state operation cannot therefore be improved by weak periodic variations in D or Sp alone. Weak periodic variations in both D and SF around D* and Sp, respectively, will lead to improved performance only for large to [oj > 2921 cycles h~l, r < 1.23 s). The rapid cycling required may need to be restricted to weak perturbations since large and very rapid perturbations in the extracellular environment may not be suitable for cellular metabolism.

Additional examples are discussed in [448, 449, 450]. Unstructured mod els, such as those considered in this section, predict a faster response to changes in operating parameters, such as D and Sf, than that observed experimentally [192]. This is presumably due to the inherent assumption in these models of no time lag between changes in the extracellular environment (abiotic phase) and adjustment of cellular metabolism. This assumption may be relaxed by considering that the specific rates (such as /i, <7, and e) are functions not only of the current substrate concentration but also of previous substrate concentrations. Delay models for cell growth that account for this have been used previously [5, 440, 448]. For Example 1, periodic forcing in D or Sf does not lead to any improvement in bioprocess performance. However, accounting for the lag between changes in extracellular environment and alteration in cell growth rate revealed that forced periodic operations involving variations in D or Sf alone provide superior performance vis-a-vis steady-state operation [pn(w) > 0 for < u> < oo (u>* > 0) and p22(t^) > 0 for u' < u) < oo (J ^ 0)] [448]. The generalized 7r-criterion, being based on weak variations around a steady state, provides a sufficient condition (and not a necessary one) for superiority of a forced periodic operation over a steady-state operation. Satisfaction of the criterion guarantees superiority of periodic operations involving both weak and strong variations in process inputs. It is anticipated that stronger input variations will lead to higher enhancement in performance under these conditions. Violation of the criterion does not necessarily rule out such superiority when strong variations in process inputs are considered. When strong variations in one or more of the bioreactor inputs, viz., D and Sf, are considered, the performance of the bioreactor subject to periodic forcing must be evaluated via solution of Eqs. 7.29-7.31 subject to the periodic boundary conditions in Eq. 7.76 and the objective function in Eq. 7.95 for particular forms of variations in the input(s). Application of the generalized 7r-criterion allows one to identify the regions in the multidimensional operating parameter space (Sfo — Do space for this case study) where periodic forcing may lead to improved process (bioreactor in the present case) performance. One may anticipate enlargement of these regions in bioprocess operations involving strong variations in the feed conditions (D and Sf)-

As in the case of steady-state continuous bioprocesses, it is possible that in the inherently transient batch and fed-batch bioprocesses, periodic perturbations in one or more inputs around their optimal trajectories may lead to improvement in bioprocess performance. Identification of optimal trajectories for which such improvement occurs will however be a numerically challenging task, since unlike the continuous steady-state cultures where the variations in inputs are around a fixed point in the multidimensional space of inputs u, the systematic variations in inputs are around trajectories in this space in the case of batch and fed-batch operations.

7.4 Feedback Control 7.4.1 State-Space Representation

The majority of techniques for design of controllers for multivariable systems apply to linear systems. The system represented by Eqs. 7.1 and 7.2 is a nonlinear multivariable system. The behavior of such a system in a close neighborhood of a reference state, (xr,ur,dr), can be represented by linearizing Eqs. 7.1 and 7.2 using the approach discussed in Section 4.7.2. Following Eqs. 4.90-4.95, r/x

at where x = x — xr, u = u — ur, d = d — dr, and

The output is

A(t), B(t), C(i) and E(i) are the appropriately dimensioned system matrices with the respective multiplying vectors, the elements of which are partial derivatives evaluated at the reference state (xr, ur, dr). If the reference state happens to be a steady state (admissible only in a continuous bioreactor operation), then the system matrices are time-invariant. In that case, the state-space representation in Eqs. 7.104 and 7.106 can be transformed into transfer function representation by applying Laplace transform to Eqs. 7.104 and 7.106.

with the transfer functions having the form

As mentioned previously, the process models for biological reactors are inherently nonlinear due to large number of chemical reactions occurring in a typical cell. Where kinetic descriptions are available, the values of model

parameters (kinetic and equilibrium coefficients) may be subject to substantial uncertainty due to complexity of the reaction scheme and difficulty to account for all reactions. Further, batch and fed-batch operations, which are inherently transient operations, are more common than continuous operations which permit steady-state operation. Taking cognizance of these, the transfer function representation is considered here since it can still provide valuable guidelines on control of transient multivariable processes.

In multivariable systems, all the output variables (y) are measured and the information is sent to the controllers assigned to the task of regulating each output. The control decisions by the controllers are implemented as appropriate changes (manipulations) in certain process inputs (u). The transfer functions for a multivariable feedback-controlled (closed-loop) process can be obtained from the concise representation of the block diagram for the process in Figure 7.7 as follows. By substituting the following input-output relations for the measuring devices and the controllers ym(s) = Gm(s)y(s), y(s) = Gc(s)e(s), e(s) = yd(s) - ym(s), (7.109)

in Eqs. 7.107 and 7.108, one can obtain

In Eqs. 7.109 and 7.110, Gc(s) and G m(s) represent the transfer function matrices for the controllers and measuring devices, respectively, and ym(s) and yd(s) the Laplace transforms of the vector of measured outputs and the vector of set-points for the measured outputs, respectively. In view of relation 7.110, one obtains the following relations among the outputs and inputs for the feedback-controlled process y{s) = G!(s)y „(«) + G2(s)d(s), (7.111)

Gi(s) = (I + GGcGm)-1GGc, G 2(s) = (I + GGcGm)_1Gd.

For an uncontrolled process with p outputs, m manipulated inputs and md disturbances, the dimensions of G, Gm, Gc, and Gd are (p x m), {p x p), (m x p), and (p x m<j), respectively. It is evident from the dimension of Gc that a maximum of mp controllers will be needed to control p outputs by manipulating m inputs. Such controller configuration will be the most complex one for the given number of process inputs and outputs.

Following feedback control of SISO systems, the simplest controller configuration will involve control of one process output by manipulating only one process input. This one-to-one input-output pairing will require the least number of controllers, viz., min(mt, p), with mt {— m + m^) being the total number of process inputs (manipulated and non-manipulated).

When using minimum number of single-loop controllers, an important consideration is the input-output pairing. The decision on the input-output pairing is based on how a particular output that is to be controlled is affected by each of the inputs that are being manipulated. In the vicinity of a steady state, by invoking the final value theorem (s —* 0), one can relate the deviations in the outputs (y) to deviations in the manipulated inputs (u) as y(t) = Ku(i), K = G(0). (7.112)

The elements of K are referred to as the steady-state gains. The (i, j)th element of the gain matrix K represents the ratio of change in the output y, to change in the input Uj., i.e., (K)y = dyi/duj.

The most widely used measure of interaction has been the relative gain array (RGA) introduced by Bristol [81]. For q manipulated inputs [q = min(mt, p)], the array (denoted as A) is a square matrix of dimension q. The (i, j')th element of RGA, \lJ} is the ratio of gain between output yi and input uj (dyi/duj) when no control is implemented (the so-called open loop gain) and gain between output yi and input Uj when all control loops except the yi - Uj loop are functioning. Let R be the transpose of the inverse of the gain matrix K with elements rVj. The elements of the relative gain array (Ay) are then related to the elements of K and R as per the relation [81, 438]

The relative gain array (RGA) has some interesting properties, which are listed below.

1. RGA is a symmetric matrix.

2. The elements of RGA in any row or any column add up to unity.

4. The gain in the open loop pairing y% with Uj when all other loops are closed (operating), K*j, is related to the open-loop gain for this pair (Kij) as

At j

The open-loop gain is thus altered by a factor of 1/Ay when all controllers except that for the j/j- Uj loop are active. This alteration is due to action from other control loops, which may be complementary or retaliatory. The sign of Ay then assumes special significance.

5. If K is a diagonal, an upper triangular, or a lower triangular matrix and if not, can be arranged into one via appropriate switches of rows or columns, then RGA is an identity matrix. The process under consideration then is non-interactive.

Recommendations for Input-Output Pairings

Based on the magnitudes of Ay, the recommendations for pairing and implications for interactions among control loops are discussed briefly.

1. Ay = 1. The input Uj can control y, without interference from the other control loops. Pairing Uj with yt is therefore recommended. This always is the case for non-interactive processes (property 5 of RGA).

2. Ay = 0. Since Uj has no direct influence on y,;, pairing Uj with yt is absolutely not recommended.

3. 0 < Ay <1. In absolute values, the closed loop gain (all control loops except the y, — Uj loop closed) is larger than the open loop gain. The increase in gain is due to complementary effect from other active control loops. The complementary effect becomes increasingly pronounced as Ay is reduced. At the critical value of Ay = 0.5, the direct effect of Uj on is identical to the complementary effect of other control loops. As a result, the pairing Uj — is recommended when 0.5 < XtJ < 1 and should be avoided when 0 < \,j < 0.5.

4. Xij > 1. Here, the open-loop gain between yl and Uj exceeds the corresponding closed-loop gain. This is due to retaliatory effect of other control loops. The direct effect is still dominant. The retaliatory effect is enhanced as A,j is increased. The higher the Xij, the greater is the opposition Uj experiences from the other control loops in trying to control As a result, pair yi with Uj as long as Xij is not very large and where possible, avoid pairing yi with Uj if Ay is very large.

5. Xij < 0. When all loops except the Uj -yi loop are closed, a particular change in Uj will produce a change in yi in opposite direction to that when all loops are open (uncontrolled process). The retaliatory effect of the other control loops is in opposition to the direct effect of Uj on yi and is the dominant of the two effects. The y, - u} pairing is potentially unstable and should be avoided.

6. In summary, one should pair input and output variables that have positive RGA elements that are closest to unity.

The relative gain array is based on the gain matrix for a process under consideration. The kinetics of processes of interest here (bioprocesses) being highly nonlinear, the elements of a steady-state gain matrix are based on the linearized version of the nonlinear process model. As a result, the elements of the process gain matrix as well as the elements of RGA will be functions of steady-state operating conditions for the process. The input-output pairings based on RGA analysis will therefore be dependent on the process operating conditions and may be altered as the operating conditions are changed. The concept of the relative gain array can be extended, with appropriate caution, to dynamic processes [173]. For process operation in the vicinity of a steady-state, the system matrices A, B, C and E are considered to be time-invariant (Eqs. 7.104 and 7.106) since the reference state, (xr, ur, dr), is time-invariant. The linearized version of the description of a nonlinear process (Eqs. 7.1 and 7.2) is a reasonable approximation in a small interval of t (t — At < t' < t). The reference state, (xr, ur, dr), then must lie in this interval. The system matrices A, B, C and E (Eqs. 7.104 and 7.106) will then change from one time interval to another as the reference state, (xr, ur, dr), is altered. Obtaining information on dynamic process gains from these system matrices is not straightforward. Alternately, for each time interval, one can obtain an equivalent process gain matrix from the nonlinear process model (Eqs. 7.1 and 7.2), the individual gains, (K)^ (between y% and Uj), being obtained as

(K0-) « [yi(t) — yi(t — At)]/[uj(t) - uj(t - At)}. (7.115)

The choice of At is somewhat arbitrary. Witcher [657] has recommended At to be 20 to 100% of the dominant time constant in the process. The magnitude of At is reduced by the process time delay, if any, in effect of Uj on yit dij [173]. One can then proceed with obtaining RGA as described earlier (Eq. 7.114). This equivalent RGA has been referred to as the dynamic relative gain array. We will continue to refer to it as RGA. During the transient operation of a bioprocess from an initial state to a final state (this may be a steady state for continuous bioprocess operation) in a single operation (run or experiment), the elements of the process gain matrix and hence the elements of RGA may alter significantly. The input-output pairings therefore may not be the same throughout the operation and may have to be switched on one or more occasions.

It should be apparent from (Eq. 7.114) that even though the elements of RGA involve comparison of open-loop gain between an input Uj and an output j/i with the closed-loop gain for this pair (when all other control loops except the loop controlling yi by manipulating Uj are closed), RGA can be estimated solely from the open-loop gains. Although the discussion related to estimating the interaction among inputs and outputs thus far has been based on availability of a mathematical description of the process, the so-called process model, one should not be under the impression that availability of a model is essential for estimation of RGA and decision on input-output pairing (controller configuration). When process models are not available or when available are reliable only in a narrow region of operating conditions, it is still possible to obtain the RGAs from experimental data. In an uncontrolled process, one can implement changes in an input (one input at a time) and observe the changes in various output variables. The elements of the process gain matrix, K, can then be obtained, similar to Eq. 7.115 as

Generation of RGA then would follow as per Eq. 7.113.

A system where p > mt is an underdefined system since there are not enough input variables to control all output variables. Based on economic considerations, one must decide which mt of the p output variables are the most important. These will be paired with the mt inputs and the remaining outputs (p — mt in number) will have to be left uncontrolled. Multiple independent sets (subsystems) of input-output pairing are candidates in this case, the exact number of sets being pCmt[= pU{mt!(p — mt)!}j. The relative gain arrays for all sets must be obtained. Comparison of the RGAs for these subsystems will reveal which subsystem has RGA closest to the ideal situation (elements corresponding to particular input-output pairing as close to unity as possible) and therefore will provide the best possible control.

A system where mt > p is an overdefined system since there are not enough output variables to be controlled with the available input variables which can be manipulated (mt). The number of controllers in this situation is p and only p inputs can be manipulated. The remaining (mt — p) inputs would therefore not be manipulated and can be used for process optimization. If they cannot be regulated then they will be classified as disturbance. Multiple independent sets (subsystems) of input-output pairing are candidates in this case, the exact number of sets being mtCp[= mt!/{p!(mt—p)!}]. The relative gain arrays for all sets must be obtained. Comparison of the RGAs for these subsystems will reveal which subsystem has RGA closest to the ideal situation (elements corresponding to particular input-output pairing as close to unity as possible) and therefore will provide the best possible control.

The idea behind the use of minimum number of controllers and RGA-based selection of the input-output pairings is to have the controller loops be essentially independent. This occurs only when the RGA elements corresponding to all input-output pairings are either unity or very close to unity. When the RGA element corresponding to an input-output pairing (Xij) is substantially farther off from unity, there will be significant interaction from other control loops while controlling yt. The interaction from the other control loops is due to process interactions. Consider as an example a process with two manipulated inputs ui and u2 and two controlled outputs yi and y2. Let the process transfer function matrix or the gain matrix be a full matrix. Let y\ be paired with U\ and y2 with u2. Controller for the y\ — u\ loop may change u\ subject to information feedback on y\. This change in ui would then lead to change not only in yi (the controlled variable for the Mi — 2/1 loop), but also in y2. The alteration in y2 due to action of controller 1 would then be fed back to controller 2. The action of controller 2 (manipulation of u2) is thus influenced by action of controller 1. The change in u2 will lead to change in not only y2 (the controlled variable for the u2 — y2 loop), but also in y\. The change in y\ would then result in change in ui. The action of controller 1 is thus influenced by action of controller 2. One can see that the two loops would be continually interacting.

The interaction among the minimum number of control loops can be minimized by use of appropriate decouplers. The decouplers, which are placed between the controllers and the process, try to compensate for the interaction in the process and therefore are also referred to as interaction compensators. The use of decouplers is intended to make the control loops independent. The controller-decoupler combination is also referred to as decoupling controller.

After RGA analysis, let the input-output pairings be such that Uj be paired with yj, j = 1,2,..., q, q = min(mt, p). If the RGA analysis suggests otherwise, then u or y and G(s) or K may have to be reconfigured. As an example of this reconfiguration, consider a process with three manipulated inputs and three controlled outputs. If the pairings based on RGA are ui — 2/3, «2 ~lh and «3— J/2) then (i) the output vector should be reconfigured as (y)new = [y3 Vi J/2]T and the third, first and second rows of G(s) or K should appear as the first, second and third rows, respectively, in the reconfigured G(s) or K; or (ii) the input vector should be reconfigured as (u)new = [tt2 «3 U\]T and the second, third, and first columns of G(s) or K should appear as the first, second and third columns in the reconfigured G(s) or K. Let the controller action (controller outputs) be denoted as v. These serve as the inputs to decouplers, the outputs from the decouplers being the manipulated process inputs u. The controlled outputs and the controller inputs can then be related as y (s) = G(s)G/(s)Gc(s)e(s) (7.117)

in the s domain and by analogy as

in the time domain in terms of steady-state or dynamic gains. In the above, Gi and Ki represent the transfer function matrix and gain matrix, respectively, for the interaction compensators (decouplers). The number of controlled outputs being equal to the number of manipulated inputs in the simple multi-loop controller configuration under consideration, considering the situation where no decoupler is employed [Gi(s) = Kj(i) = I], it can be deduced from Eqs. 7.117 and 7.118 that Gc(s) and Kc(i) are diagonal matrices. It should then be evident from Eqs. 7.117 and 7.118 that for the control loops to perform independently of one another, G(s)Gi(s) and K(t)Ki(i) must be diagonal matrices.

The elements of G(s)Gi(s) and K(t)Kj(i) can be selected in multiple ways. The more general way assigns the non-trivial elements of these matrices to be the diagonal elements of G(s) or K(t), as applicable. The interaction compensator may then be obtained as

Gj(s) = [GMl^diag G(s), [G(s)]_1 = adj G(s)/|G(s)| (7.119)

Ki(i) = [K(t)]_1diag K(t), [K(i)]_1 = adj K(t)/|K(t)| (7.120)

In Eqs. 7.119 and 7.120, adj M denotes the adjoint matrix of M. Some words of caution are in order here. Perfect decoupling is possible only if the process model is perfect and reliable. However, even with imperfect process models, decoupling can be applied with considerable success. The dynamic decouplers being based on model inverses (Eqs. 7.119 and 7.120), these can be implemented only if the inverses are causal and stable. For further discussion of this and other related issues, the reader should refer to Ogunnaike and Ray [438].

Limitations of Decouplers - Ill-Conditioned Processes

It should be evident from Eqs. 7.118 and 7.120 that if the determinant of the process gain matrix is very small, the system will be extremely sensitive to any errors in the process model and decoupling will be difficult to achieve. Small changes in e or v will lead to large changes in y. The process is said to be ill-conditioned when |K(i)| is very small. It is virtually impossible to achieve decoupling in an ill-conditioned process. There are situations where |K(t)| is not small, yet the process is poorly conditioned. Examples of these situations have been discussed in [438]. The most reliable indicator of the conditioning of a process is the condition number of the process gain matrix, k(K) , which is provided by the ratio of the largest singular value of this matrix to the smallest singular value. The singular values of the real-valued K(i) are the the square root of the eigenvalues of the matrix KT(£)K(t). Since KT(t)K(£) is a symmetric matrix with real elements, its eigenvalues and therefore the singular values of K(i) are non-negative. When the input-output pairings are based on RGA, all singular values are positive since K(t) is not singular. As |K(i)| is reduced in absolute value, so is the smallest singular value of K(t) and the condition number is increased. A very small |K(i) | is indicative of the degeneracy of the process. Such degeneracy is only a special case of ill-conditioning. If the condition number of the process gain matrix is quite large, then the process is said to be poorly conditioned. Use of a decoupler in such situations would do more harm than good and should be avoided.

The singular value decomposition (SVD) of the process gain matrix provides a much more general approach to decoupling. SVD allows for extension of matrix diagonalization to non-square process gain matrices, since KT(t)K(t) is a square matrix irrespective of whether or not K(i) is.

Let r(K) be the rank of K [r = r(K) < q, q = min(m, p)], which is an p x m matrix. Then only r singular values of K (denoted as aj, j = 1, 2,..., r) are non-trivial and the remaining (m — r) singular values are trivial. Let the non-trivial singular values be arranged as <j\ > 02 > . •. or . For any matrix such as the process gain matrix, there exist orthogonal (i.e., unitary) matrices W and V such that

with W, V and X being p x p, m x m and p x m matrices related to K as follows. The p columns of W, denoted as Wj (i = 1,2,... ,p), are the orthonormal eigenvectors of KKT. Thus,

Similarly, the m columns of V, denoted as Vj (i — 1,2,...,m), are the orthonormal eigenvectors of KTK. Thus,

Since V and W are composed of orthonormal vectors, these are orthogonal (or unitary) matrices, i.e., VTV = VVT = I (dimension of I = m) and WTW = WWT = I (dimension of I = p). It then follows that

In view of the above, upon pre-multiplication by W and post-multiplication by VT, Eq. 7.121 can be restated as

The eigenvectors vi of KTK and Wj of KKT are related to each other as per the following general pair of expressions.

With the singular values of K being arranged in descending order, the only non-trivial elements of the pxm matrix £ appear for i,j = 1,2,...,r, i.e.,

Eti = cri, i = 7 = L 2,..., r; S,;, = 0 otherwise. (7.127)

Figure 7.8. Block diagram of the multivariable controller based on SVD technique [438].

From BA Ogunnaike and WH Ray. Process Dynamics, Modeling, and Control. New York: Oxford University Press, Inc., 1994. Used by permission.

Figure 7.8. Block diagram of the multivariable controller based on SVD technique [438].

From BA Ogunnaike and WH Ray. Process Dynamics, Modeling, and Control. New York: Oxford University Press, Inc., 1994. Used by permission.

Decoupler Design

Substitution of the singular value decomposition of K, Eq. 7.125, into the input-output relations for the process leads to the following

Pre-multiplication of the above by WT and use of relation in Eq. 7.124 leads to the following restatement of Eq. 7.128

Ar](t) = SA p(t), A (J,(t) = VTAu(i), A v(t) = WTAy(i). (7.129a)

Since the non-trivial elements of S lie on a diagonal (Eq. 7.127), the system is totally decoupled, with ^ being paired with (i = 1,2,... ,r). The block diagram of the feedback-controlled multivariable process utilizing singular value decomposition is shown in Figure 7.8. The process outputs are mixed according to Eq. 7.129a to obtain T), the information on which is then fed to the comparators to obtain controller inputs. The manipulated inputs u are obtained from the controller outputs, /z, via the mixing rule in Eq. 7.129a, i.e.,

The application of the relative gain array method involves pairings among actual process inputs and outputs. For non-square process gain matrices (the number of inputs not being the same as the number of outputs), use of minimal controller configuration implies that either some inputs cannot be manipulated (overdefined system) or some outputs cannot be controlled (underdefined system). This problem does not arise when the controller configuration is based on SVD since all inputs and outputs are involved in the feedback control (Eq. 7.129).

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