Minimum Variance Benchmark for Performance Assessment of

Article pubs.acs.org/IECR

Minimum Variance Benchmark for Performance Assessment of Decentralized Controllers† Antonius Yudi Sendjaja‡ and Vinay Kariwala*,§ School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore ABSTRACT: The available minimum variance (MV) benchmarks do not take the controller structure into account and thus can lead to incorrect conclusions regarding the performance of decentralized controllers. In this paper, we present a method for computing a lower bound on the least output variance achievable using decentralized controllers, where the nonconvexity of the optimization problem is handled using sums of squares programming. Though tight, the computation of this lower bound requires that the process model be known. As a more practical approach, we derive an MV benchmark for performance assessment of decentralized controllers on a loop by loop basis. The MV benchmark for loop by loop analysis can be estimated from the closed loop operating data with the knowledge of time delays between all the inputs and outputs. The usefulness of the proposed benchmarks is demonstrated using simulation examples.

1. INTRODUCTION The performance of a well-designed control system can degrade over time due to changes in operating conditions and disturbance dynamics. Controller performance assessment is useful for identifying the opportunities for performance improvement of industrial controllers. Among the various available methods,2−4 minimum variance (MV) benchmarking is one of the most widely used methods for controller performance assessment. In this approach, the controller is deemed to provide satisfactory performance, if the observed variance of the outputs is close to their least achievable value, i.e., the MV benchmark. On the other hand, reduction in the output variance is considered to be feasible through controller retuning, when the observed output variance is significantly higher than the MV benchmark. The origin of the MV benchmark can be traced back to Åström5 and Box et al.,6 who demonstrated that the achievable output variance for a single-input single-output (SISO) process under feedback control depends on the first few impulse response coefficients of the disturbance model. Harris7 showed that, with a priori knowledge of time delay, the MV benchmark can be estimated from closed loop operating data and established it as a tool for performance assessment of SISO systems. Using the concept of interactor matrices, Harris et al.8 and Huang et al.9 proposed the MV benchmark for multipleinput multiple-output (MIMO) systems. This methodology has also been extended to different controller architectures including feedback/feedforward controllers,10,11 cascade controllers12,13 and constrained model predictive controllers,14 and process types such as linear time varying processes15,16 and nonlinear processes.17−19 The success of the MV benchmark for controller performance assessment is highlighted by its incorporation into a number of industrial software that are used routinely in process industries.2,4 This paper focuses on the performance assessment of decentralized controllers used for the feedback control of linear time invariant processes. Owing to their simplicity in design and maintenance, decentralized controllers are widely used for regulation purposes in process industries.20 Similar to time © 2012 American Chemical Society

delay, the diagonal structure of the decentralized controllers can pose fundamental limitations on the least achievable output variance.21 The available MV benchmarks for SISO and MIMO processes do not take the controller structure into account and thus can lead to incorrect conclusions regarding the performance assessment of decentralized controllers. For example, the MV benchmark for SISO processes7 assumes that the loop is operating in isolation from the rest of the process and thus implicitly ignores the interactions between the loops of the decentralized controller; see Figure 1. Due to this assumption,

Figure 1. continued

it may be possible to improve the performance of the existing controller further than indicated by the MV benchmark for SISO processes. On the other hand, the MV benchmark for MIMO processes8,9 ignores the diagonal structure of the decentralized controller and thus has more degrees of freedom for variance minimization than are available in the actual controller. Using the example of a 2 × 2 process, we demonstrate in section 3 that the least output variance, which can be achieved using a diagonal controller, can be 4 times higher than that which can be achieved using a full multivariable controller. Except for some limiting cases, such as systems which are diagonal at high frequencies22 and sequentially minimum-phase systems,23 the characterization of the least achievable output variance is difficult, when all the loops of the decentralized Received: Revised: Accepted: Published: 4288

June 24, 2011 December 7, 2011 February 15, 2012 February 15, 2012 dx.doi.org/10.1021/ie201361d | Ind. Eng. Chem. Res. 2012, 51, 4288−4298

Industrial & Engineering Chemistry Research

Article

controller are tuned simultaneously. This happens as the optimization problem involving minimization of output variance becomes nonconvex, once the diagonal structure is imposed on the controller.24,25 With this difficulty, one may alternately look for tight upper and lower bounds on the MV benchmark for decentralized controllers. Some approaches for finding upper bounds on the MV benchmark for decentralized controllers have been reported using nonconvex optimization26,27 or by utilizing the structure of the optimization problem.28,29 Recently, Kariwala21 derived an explicit lower bound on the MV benchmark by considering those impulse response coefficients of the closed loop transfer function between disturbances and outputs, which depend linearly on the impulse response coefficients of the decentralized controller. In general, however, the lower bound proposed by Kariwala21 can be conservative due to the neglected impulse response coefficients. In this paper, we show that, though nonlinear, the impulse response coefficients of the closed loop transfer function between disturbances and outputs can be represented as polynomials in unknown impulse response coefficients of the controller. The resulting polynomial minimization problem is solved using sums of squares (SOS) programming,30−32 which guarantees a lower bound on the achievable output variance for processes under decentralized control. The proposed lower bound is useful for analyzing the effect of the structure of the decentralized controller on the achievable output performance. Based on this lower bound, however, assessing the performance of the decentralized controller using closed loop operating data is difficult, as it requires that the process model be known. We note that although theoretically the least achievable output variance is seen, when all the subcontrollers of the decentralized controller are allowed to be tuned simultaneously, control loops are mostly tuned one at a time in practice. With this view, we propose an MV benchmark for loop by loop analysis of decentralized controllers, which provides a measure of reducing the variance of the ith output through retuning of the ith controller. This result requires a modification of the available MV benchmark for SISO processes to account for the loop interactions. It is further shown that the modified MV benchmark can be directly estimated from the closed loop data with a priori knowledge of the delays between all the inputs and outputs. Several examples taken from the literature are used to demonstrate the usefulness of the proposed approaches.

points, respectively. Throughout this paper, the following assumptions are considered to hold: 1. G(q−1) with dimensions ny × ny is a stable, causal transfer function matrix with no zeros on or outside the unit circle except at infinity (due to time delays). 2. H(q−1) with dimensions ny × nd is a stable, causal transfer function matrix with no zeros on or outside the unit circle. 3. The input−output pairings are selected on the diagonal elements of G(q−1). 4. a(t) is a random noise sequence with the covariance being the identity matrix. 5. ysp(t) is zero for all t > 0. When H(q−1) contains zeros outside the unit circle, H(q−1) can be factored as H(q−1) = H̅ (q−1) DG−1(q−1), where H̅ (q−1) is invertible and DG(q) is the generalized unitary interaction matrix containing all the zeros of H(q−1) located outside the unit circle (including at infinity).33,34 Now, the performance can be assessed by treating H̅ (q−1) as the disturbance model and DG−1(q−1) a(t) as the noise sequence, as the spectra of a(t) and DG−1(q−1) a(t) are the same. Further, there is no loss of generality in assuming that the system is affected by noise, whose covariance is the identity matrix.21 When E[a(t) aT(t)] ≠ I, the disturbance model can be scaled by the square root of the covariance matrix of a(t) to satisfy this assumption. The set point ysp(t) is assumed to be zero for simplicity and the results presented in this paper can be easily generalized to the case where ysp(t) changes with time; see Remark 1 for details. For notational simplicity, we drop the arguments q−1 and t in the subsequent discussion. Our objective is to find the least achievable value of variance of y with respect to the controller K, i.e.

2. PRELIMINARIES We consider the closed loop system shown in Figure 2. For this system, we denote G(q−1) and H(q−1) as the process and disturbance models, respectively, such that

(3)

Jdecen = min Var(y) = min E[tr(y y T )] K

K

(2)

where K is assumed to have a diagonal structure, i.e. ⎡ K1 ⎢ ⎢0 K=⎢ ⎢⋮ ⎢0 ⎣

⎤ ⎥ K2 ··· 0 ⎥ ⎥ ⋮ ⋱ ⋮ ⎥ ··· ··· K ny ⎥ ⎦ ··· 0

0

In eq 2, E[·] and tr(·) denote the expectation and trace operators, respectively. Based on eq 2, the performance index for decentralized controller can be defined as ηdecen =

Jdecen E[tr(y y T )]

(4)

T

where E[tr(y y )] is the observed output variance. A related problem involves finding the least achievable variance of the ith output through tuning of ith controller, i.e. Ji ,decen = min E[(yi yiT )] Ki

Figure 2. Block diagram of closed loop system.

y(t ) = G(q−1) u(t ) + H(q−1) a(t )

(5)

Similar to eq 4, the performance index for the ith output can be defined as

(1)

Here, y(t) ∈ ny , u(t) ∈ ny , a(t) ∈ nd , and ysp(t) ∈ ny are controlled outputs, manipulated variables, disturbances, and set

ηi ,decen = 4289

Ji ,decen E[(yi yiT )]

(6)

dx.doi.org/10.1021/ie201361d | Ind. Eng. Chem. Res. 2012, 51, 4288−4298


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where E[(yi yiT)] is the observed variance of the ith output. In the rest of this paper, we present approaches to determine ηdecen in eq 4 (simultaneous analysis) and ηi,decen in eq 6 (loop by loop analysis). Remark 1. The focus of this paper is on regulatory control, and thus ysp is assumed to be zero for all t. When ysp changes with time, the MV benchmark can be defined based on control error, e = ysp − y.33 In particular, Jdecen in eq 2 can be defined as minK E[tr(e eT)] and Ji,decen in eq 5 can be defined as minK E[(ei eiT)] for i = 1, 2, ..., ny to assess the performance of decentralized controllers.

for any finite n. Thus, a lower bound on Jdecen can be found by n minimizing ∑i=0 tr(SiT Si). To relate Si with K, we note that when the closed loop system is stable, eq 10 can be expanded using Taylor Series expansion to get ⎡∞ ⎤ i i⎥ ⎢ S = ∑ ( −1) (GK ) H ⎢ ⎥ ⎣i = 0 ⎦

We define the following (n + 1)ny × (n + 1)ny dimensional Hankel matrices:

3. SIMULTANEOUS ANALYSIS In this section, we derive an MV benchmark for decentralized controllers, where all the loops are allowed to be tuned simultaneously. For easier comparison, the available MV benchmark for MIMO processes8,9 is presented next. 3.1. Conventional Approach. The MIMO process can be represented as G = D−1G̅

(7)

such that G̅ and D−1 contain the invertible and noninvertible parts of G, respectively. We consider that D(q) is a unitary interactor matrix, i.e., DT(q−1) D(q) = I.33 Let the filtered disturbance model be decomposed using the Diophantine identity as q−dDH = F + q−dR

(8)

S = (I + GK )−1H

Hv = [ H0T H1T ··· HnT ]T

(17)

Sv = [ S0T S1T ··· SnT ]T

(18)

(19)

∑ tr(SiT Si) = min tr(SvT Sv)

K 0 , K1,..., K n i = 0

KH

(20)

Note that in general integrating poles may be present in H (due to the presence of nonstationary disturbances) and K (due to presence of integral action). In such cases, the sequences of impulse response coefficients of H and K, i.e., Hi and Ki, may not converge with i. The preceding derivation still holds, provided the sequence Si converges with i, which is guaranteed due to the stability of the closed loop system. On the basis of eq 19, we note that Sv and thus tr(SvT Sv) depends polynomially on the impulse response coefficients of the controller, Ki, i = 0, 1, ..., n. Thus, the optimization problem in eq 20 can be solved by finding the global minimal value of a polynomial. For this purpose, we use sums of squares (SOS) programming in this paper, which requires solving30,31,35 max p (21)

(10)

(11)

∞ (12)

s.t.

where Si represents the ith impulse response coefficient matrix of S. As tr(SiT Si) ≥ 0 for any i, it follows that S 22 ≥

(16)

0

n

∑ tr(SiT Si) i=0

··· 0 ⎤ ⎥ K 0 ··· 0 ⎥ ⎥ ⋮ ⋱ ⋮ ⎥ ··· K1 K 0 ⎥⎦

min

When a diagonal structure is imposed on K, the optimization problem in eq 11 becomes nonconvex,24,25 which makes characterization of Jdecen difficult. To derive a lower bound on Jdecen, using Parseval’s theorem, we have S 22 =

⎡ K0 ⎢ ⎢ K1 KH = ⎢ ⎢⋮ ⎢⎣ K n

Now, a lower bound on Jdecen can be found by solving

Since E[aa ] = I K

(15)

⎡ n ⎤ Sv = ⎢ ∑ ( −1)i (GH KH )i ⎥Hv ⎢ ⎥ ⎣i = 0 ⎦

T

Jdecen = min S 22

··· 0 ⎤ ⎥ G0 ··· 0 ⎥ ⎥ ⋮ ⋱ ⋮ ⎥ ··· G1 G0 ⎥⎦ 0

Based on eqs 16−18, Sv can be compactly written as

(9)

where Jfull is the MV benchmark for fully centralized controllers and ∥·∥2 denotes the /2 norm. The MV benchmark in eq 9, however, does not take the diagonal structure of the controller into account and thus may classify a well-performing decentralized controller as poorly performing. In the subsequent discussion, we present a lower bound on MV benchmark for processes under decentralized control. 3.2. MV Benchmark for Decentralized Controllers. For regulatory control, we have u = −Ky. Thus, the closed loop transfer function between a and y can be written as y = Sa;

⎡G0 ⎢ ⎢G1 GH = ⎢ ⎢⋮ ⎢⎣Gn

and the following (n + 1)ny × nd-dimensional matrices

where d denotes the order of the interactor matrix. Then8,9

Jfull = F 22

(14)

tr(SvT Sv) − p = zQz T

(22)

n

∑ tr(SiT Si) i=0

where z = z(K0, K1, ..., Kn) is a vector containing monomials of Ki until order n. By comparing the coefficients of the

(13) 4290



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controllers using the MV benchmark for full multivariable controllers may incorrectly classify a well-performing controller as poorly performing. As demonstrated using Example 1, SOS programming can be used to find a tight lower bound on Jdecen. For performance assessment of industrial controllers, it is preferred that the MV benchmark be estimated from the closed loop data. The estimation of Jdecen directly from the closed loop data, however, is difficult. When G is known (or has been identified using open or closed loop identification experiments), H can be estimated by prewhitening the pseudosignal (y − Gu).39 Then, SOS programming can be used to identify a lower bound on Jdecen and ηdecen based on the identified disturbance model. We point out that the knowledge of G is also required by other available approaches for performance assessment of decentralized controllers.26,27 In practice, the task of identifying G should be undertaken only when Jfull differs significantly from the observed output variance.

polynomials on both sides, eq 22 can be expressed as a set of linear equality constraints.30,31,35 Thus, the optimization problem in eqs 21−23 can be seen as a semidefinite program, which is convex. For any controller K n+1

∑

tr(SiT Si) ≥

n

∑ tr(SiT Si) i=0

i=0

(24)

Thus, a lower bound on Jdecen can be found by solving a series of convex programs for increasing values of n until convergence. In comparison with Kariwala,21 where only the first (2d − 1) impulse response coefficients of S are used to find a lower bound on Jdecen, the use of SOS programming provides a tighter lower bound, whenever n > (2d − 1). Note that SOS programming does not necessarily provide the minimum value of the polynomial, but guarantees a global lower bound.30,31,36 On the other hand, the available methods based on nonconvex optimization26,27 or structure of the optimization problem28,29 provide an upper bound on Jdecen. As the determination of the exact value of Jdecen is difficult, the knowledge of both upper and lower bounds is useful for performance assessment purposes. Numerical experiments suggest that the lower bound on Jdecen found using SOS programming is usually tight in the sense that the lower bound is close to the upper bound on Jdecen found using the Matlab command fmincon with multiple randomized initial guesses. Example 1. To illustrate the usefulness of the SOS programming approach, we consider the following 2 × 2 process, where21

4. LOOP BY LOOP ANALYSIS In this section, we consider an alternate method, where the performance of each loop of the decentralized controller is analyzed individually. For clarity of presentation, we first consider 2 × 2 systems under decentralized control. The result is generalized to ny × ny systems in section 4.3. For a 2 × 2 system, eq 1 can be written as y1 = G11u1 + G12u2 + H1a (26) y2 = G21u1 + G22u2 + H2a

⎡ −0.1q−2 −0.25q−1(1 − 0.3q−1) ⎤⎥ ⎢ − 1 − 1 ⎢ (1 − 0.1q )(1 − 0.2q ) (1 − 0.1q−1)(1 − 0.2q−1) ⎥ ⎥ G=⎢ ⎥ ⎢ 0.5q−1(1 + 0.9q−1) −0.1q−2 ⎥ ⎢ ⎢⎣ (1 − 0.1q−1)(1 − 0.2q−1) (1 − 0.1q−1)(1 − 0.2q−1) ⎥⎦

(27)

where Hi = [Hi1 Hi2 ··· Hind] for i = 1, 2. We assume that Gij(i,j = 2) does not contain unstable zeros. The time delay associated with Gij is denoted as dij, i.e. Gij = q−dijGij̅

(28)

where G̅ ij denotes the invertible part of Gij. Without loss of generality, we consider that the objective is to characterize the MV benchmark for y1. 4.1. Conventional Approach. A common approach for loop by loop analysis involves using the MV benchmark for SISO systems. Here, H1 is decomposed using the Diophantine identity as

(25) −1

and H = 1/(1 − q )I. For this process, the interactor matrix is D = qI. Then, F = I and the least output variance achievable using full multivariable controllers is Jfull = ∥F∥22 = 2. To find a lower bound on the least output variance achievable using diagonal controllers, Kariwala21 considered the contribution of the first (2d − 1) impulse response coefficients of S toward ∥S∥22 and found that Jdecen ≥ 4. This lower bound is, however, not very tight due to the neglected impulse response coefficients of S. Using nonconvex optimization with multiple randomized initial guesses, Kariwala21 showed that the value of Jdecen is approximately 8.16, which differs significantly from the reported lower bound. To find a tighter lower bound on Jdecen, we use SOS programming in this paper. The optimization problem in eqs 21−23 is solved using SeDuMi37 interfaced with Matlab through Yalmip.38 Using this approach, we find that Jdecen ≥ 8.023, which is close to the upper bound found by Kariwala.21 This example sufficiently demonstrates the advantages of the SOS programming technique for finding a tighter lower bound on Jdecen, as compared to the approach presented by Kariwala.21 In addition, for this process, the MV benchmark for a full multivariable controller is approximately 4 times lower than the MV benchmark found by accounting for the controller structure, which shows that the controller structure can significantly limit the achievable performance. Thus, the conventional approach for performance assessment of decentralized

H1 = F1 + q−d11R1

(29) 5,7

Then, the least achievable variance of y1 is taken as J1,siso = min E[(y1 y1T )] = F1 22 K1

(30)

The performance index for individual outputs is defined similar to eq 6. An inherent assumption in the use of eq 30 for loop by loop analysis is that u2 = 0 at all times or, in other words, the first loop is being operated in isolation from the rest of the process. We next demonstrate that, when the presence of other loops is accounted for, the first d11 impulse response coefficients of H1 are not necessarily feedback invariant and thus the least achievable variance of y1 can be lower than J1,siso in eq 30. 4.2. Modified MV Benchmark. Consider that the second loop is closed with u2 = −K2y2. Under partially closed loop conditions, we have20 y1 = P11u1 + Pd1a 4291

(31)



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Both these factors can lead to incorrect conclusions regarding the performance assessment of decentralized controllers using the MV benchmark for SISO processes. Though the result in eq 39 may seem entirely mathematical, a physical reasoning with this result can be associated by considering the block diagram of the closed loop system for a 2 × 2 system shown in Figure 3. Here, u1 can affect y1 directly

where P11 = G11 − Pd1 = H1 −

G12K2G21 1 + G22K2

(32)

G12K2H2 1 + G22K2

(33)

In subsequent analysis, we assume that P11 does not contain unstable poles and zeros. Some discussion on relaxing this assumption is provided in section 4.4. Let Pd1,m denote the minimum phase version of Pd1 obtained by mirroring all unstable zeros of Pd1 across the unit circle (including at infinity) and ( be the all-pass factor containing the unstable zeros of Pd1. In particular, consider that Pd1 with dimensions 1 × nd contains zeros at z = ci, i = 1, 2, ..., nz. Then, ( takes the form nz

(=

∏ (i i=1

where

34

⎧ (1 − 1/ci*q−1)(1 − ci) ⎪ ⎪ −ci* (i = ⎨ (1 − ciq−1)(1 − ci*) ⎪ ⎪q ⎩

Figure 3. Presence of parallel path (shown with thick line) from u1 to y1 for 2 × 2 systems.

if |ci| < ∞

through G11, but also through a parallel path involving G12 and G21 (shown with a thick line in Figure 3). Thus, d1′ = min(d11, d12 + d21) represents the delay of the fastest path through which u1 can affect y1. In this sense, loop interaction can sometimes be beneficial for reducing output variance, often referred to as positive interactions.40 Example 2. To illustrate the findings of this section, we consider the example of the Wood−Berry (WB) distillation column.41 For this process, the overhead (y1) and bottom (y2) product compositions need to be controlled using the reflux (u1) and steam (u2) flow rates, while the feed flow rate (a) is considered to be the disturbance. The continuous-time model is discretized with a sampling time of Ts = 1 min to get

if |ci| = ∞ (34)

Then

y1 = P11u1 + Pd1, ma ̃

(35)

where a ̃ = (a with E[ã ãT] = E[a aT] = I. In eqs 32 and 33, since 1/(1 + G22K2) and K2 are rational and invertible, it follows that the delay associated with P11 or the effective delay of the first loop is d1′ = min(d11, d12 + d21)

(36)

Now, let Pd1,m be decomposed using the Diophantine identity as Pd1, m = F1′ + q−d1′R1′

⎡ ⎢ ⎡ y1 ⎤ ⎢ 1 ⎢ ⎥=⎢ ⎣ y2 ⎦ ⎢ ⎢ ⎢⎣ 1

(37)

Using eqs 35 and 37, it follows that y1 = F1′a ̃ + q−d1′(P11 ̅ u1 + R1′a)̃

(38)

0.942q−1

⎡ 0.247q−5 ⎤ ⎥ ⎢ ⎢ 1 − 0.935q−1 ⎥ +⎢ ⎥a 0.358 ⎥ ⎢ ⎢ −1 ⎥ q 1 0.927 − ⎦ ⎣

−d1′

where P11 = q P̅11 and P̅11 denotes the invertible part of P11. Since the first term in eq 38 cannot be affected by u1 (invariant of K1), it follows that J1,decen = F1′ 22

− 0.879q−4 ⎤⎥ 1 − 0.954q−1 ⎥⎡ u1 ⎤ − ⎥⎢ ⎥ u 0.579q−8 − 1.302q−4 ⎥⎣ 2 ⎦ ⎥ − 0.912q−1 1 − 0.933q−1 ⎥⎦ 0.744q−2

(40)

(39)

For this process, the diagonal and off-diagonal terms of the relative gain array (RGA)42 are positive and negative, respectively. Thus, diagonal pairings are selected to ensure integrity against loop failure.43 The individual controllers of the decentralized control system are designed to be proportional− integral (PI) controllers, whose parameters are tuned using the internal model control (IMC) method.44 In this method, for

Here, ∥F1′ ∥22 represents the least achievable variance of y1, when the presence of the second loop is accounted for. In comparison with eq 30, we note the following: 1. For decentralized control, one needs to consider the impulse response coefficients of the filtered P d1 (disturbance model for partially closed loop system) and not H1 (open loop disturbance model). 2. The number of impulse response coefficients to be considered is d1′ (effective delay) and not d11 (open loop delay).

g (s ) = 4292

Kp τs + 1

e−θs

(41)



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and modified approaches is seen, as the former approach uses the impulse response coefficients of the open loop disturbance model (H1) instead of the disturbance model for the partially closed loop system (Pd1). We point out that, for this process, the first d1′ impulse response coefficients of Pd1,m and thus J1,decen depend on K2. For example, when the gain of K2 is decreased by a factor of 0.25, J1,decen increases to 0.1. With this controller tuning, the variance of y1 is 0.151. Thus, we have η1,decen = 0.665, which indicates that the variance of y1 can still be reduced by approximately 50%. Example 2 demonstrates that the MV benchmark for SISO processes can overestimate the performance index for decentralized controllers, as the change in the disturbance model due to loop interactions is not accounted for. Furthermore, for Example 2, the effective delays for both loops are the same as open loop delays, i.e., d1′ = d11 and d2′ = d22. This is not true in general, as shown next. Example 3. We consider the following process model:45

the tuning parameters are chosen as Kc =

1 τ ; K p τc + θ̃

τI = min(τ1, 4(τc + θ̃))

(42)

where Kc, τI, and τc denote the controller gain, integral time, and desired closed loop time constant, respectively, and θ̃ = θ + Ts/2. The recommended value for τc is θ̃.44 Using the continuous-time model of the process and the recommended value of τc for both loops, the following discrete-time decentralized PI controller is obtained: ⎡ 0.652 − 0.571q−1 ⎤ ⎢ ⎥ 0 ⎢ ⎥ 1 − q−1 ⎥ K=⎢ ⎢ −0.124 + 0.115q−1 ⎥ ⎢ ⎥ 0 ⎢⎣ ⎥⎦ 1 − q−1

(43)

which provides Var(y1) = 0.122 and Var(y2) = 0.759. The least achievable variances of y1 and y2 according to the conventional approach discussed in section 4.1 are J1,siso = 0.114 and J2,siso = 0.413, respectively. Accordingly, the performance indices for loops 1 and 2 are η1,siso = 0.932 and η2,siso = 0.544, respectively, which indicate that the variance of y2 can be reduced significantly by retuning K2, but tuning K1 will not reduce the variance of y1 significantly. Using the modified MV benchmark, we next show that this conclusion is not entirely correct. We have d11 = 2, d12 = 4, d21 = 8, and d22 = 4. Thus, the effective delays for the two loops are d1′ = min(d11, d12 + d21) = 2 and d2′ = min(d22, d12 + d21) = 4, respectively, which are the same as the corresponding open loop delays d11 and d22, respectively. Although Pd2 ≠ H2, the first d2′ = d22 impulse response coefficients of Pd2 and H2 are the same; see Figure 4.

⎡ ⎤ −6 −7 1.106q−1 ⎥ ⎢ 0.0347q + 0.0287q − 1 − 2 − 1 ⎢ 1 − 1.542q + 0.5674q 1 − 0.7788q ⎥ ⎥ G=⎢ ⎢ − 0.1951q−7 ⎥ 0.2835q−1 ⎢ ⎥ ⎢⎣ 1 − 0.7165q−1 1 − 0.9512q−1 ⎥⎦

(44)

where the continuous-time model has been discretized with a sampling time of Ts = 1 unit. We assume the disturbance model to be an integrator, i.e., H = 1/(1 − q−1)I. The steady state RGA analysis42 suggests the use of diagonal pairing. As before, the following decentralized PI controller is designed using the IMC method using the continuous-time model and the recommended value of τc for both loops:44 ⎡ 0.533 − 0.5q−1 ⎤ ⎢ ⎥ 0 ⎢ ⎥ 1 − q−1 ⎥ K=⎢ − 1 ⎢ −0.417 + 0.396q ⎥ ⎢ ⎥ 0 ⎢⎣ ⎥⎦ 1 − q−1

(45)

which provides Var(y1) = 30.713 and Var(y2) = 4.346. As the disturbance model is a pure integrator, the least achievable output variance for the MV benchmark for SISO processes is the same as the time delay. For this process, we have d11 = 6 and d22 = 7. Thus, according to the conventional approach, the least achievable variances of y1 and y2 are J1,siso = 6 and J2,siso = 7, respectively. This is clearly incorrect, as the actual variance of y2 is less than the indicated least achievable output variance leading to a performance index larger than 1. Next, we show that this discrepancy can be resolved by taking the loop interactions into account. Using the proposed method, we have d1′ = min(d11, d12 + d21) = 2 and d2′ = min(d22, d12 + d21) = 2, which shows that the effective delays for the two loops are lower than the corresponding open loop delays. In addition, the impulse response coefficients of the disturbance models for the partially closed loop system (Pd1 and Pd2) differ from the open loop disturbance models (H1 and H2); see Figure 5. For loop 2, the impulse response coefficients of Pd2 are much smaller than the impulse response coefficient of H2, indicating the presence of positive interactions, i.e., performance enhancement due to the interactions from other loop. Based on eq 39, the least achievable variances are calculated to be J1,decen = 3.134 and

Figure 4. Impulse response coefficients of disturbance models of open loop and partially closed loop systems for Example 2.

Thus, we find that J2,decen = J2,siso = 0.413 and η2,decen = η2,siso = 0.544. Pd1 contains two unstable zeros, which are reflected across the unit circle to obtain Pd1,m. From Figure 4, we note that the impulse response coefficients of Pd1,m decay much faster than the impulse response coefficients of H1. Based on the first d1′ = d11 impulse response coefficients of Pd1,m, we find that J1,decen = 0.062 and η1,decen = 0.507. Thus, the modified MV benchmark identifies that the variance of y1 can be reduced by half through tuning of K1. This observation is in direct contrast to the findings of the conventional approach, which is based on the MV benchmark for SISO processes. The large difference between the performance indices computed using conventional 4293



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⎡ H21 H22 ··· H2n ⎤ d ⎥ ⎢ ⎥ ⎢ ⋮ ··· ⋮ H̃2 = ⋮ ⎥ ⎢ ⎢⎣ Hny1 Hny 2 ··· Hnynd ⎥⎦

(50)

⎡ K2 ⎢ ⎢0 K2̃ = ⎢ ⎢⋮ ⎢0 ⎣

(51)

⎤ ⎥ K3 ··· 0 ⎥ ⎥ ⋮ ⋱ ⋮ ⎥ ··· ··· K ny ⎥ ⎦ 0

··· 0

When all other loops are closed, y1 is still given by eq 31, but the expressions for P11 and Pd1 change as

Figure 5. Impulse response coefficients of disturbance models of open loop and partially closed loop systems for Example 3.

J2,decen = 1.721, and thus η1,decen = 0.102 and η2,decen = 0.396. This result indicates that the variance of the second loop can still be reduced by a factor of 2.5 through controller retuning, whereas the conventional approach significantly overestimates the performance index. 4.3. Generalization to ny × ny Systems. To generalize the modified MV benchmark for loop by loop analysis to ny × ny systems, let us define

̃ K2̃ (I + G̃ 22K2̃ )−1G̃ 21 P11 = G11 − G12

(52)

̃ K2̃ (I + G̃ 22K2̃ )−1H̃2 Pd1 = H1̃ − G12

(53)

Thus, J1,decen can be found by using the Diophantine identity for the minimum phase version of Pd1. Here, the effective delay d1′ needs to be calculated as d1′ = min(d11, min d1k + min dl1) k ,k≠1

l ,l≠1

(54)

which represents the delay of the fastest among (ny − 1) + 1 paths through which u1 can affect y1. Example 4. We consider the energy balance control for interacting sidestream column/stripper distillation column example taken from Ogunnaike et al.46 For this process, the outputs are overhead ethanol mole fraction (y1), sidestream ethanol mole fraction (y2), and temperature on the 19th tray (y3). The available inputs are reflux flow rate (u1), sidestream product flow rate (u2), and reboiler steam pressure (u3), and the disturbances are feed flow rate (a1) and feed temperature (a2). The discrete time transfer models of the process and disturbance are obtained with a sampling time of Ts = 1 min46 and are given as 2

̃ = [G12 G13 ··· G1ny ] G12

(46)

G̃ 21 = [G21 G31 ··· Gny1]T

(47)

⎡ G22 G23 ··· G2n ⎤ y⎥ ⎢ ⎢ ̃ ⋮ ··· ⋮ ⎥ G22 = ⋮ ⎥ ⎢ ⎢⎣Gny 2 Gny3 ··· Gnyny ⎥⎦

(48)

H1̃ = [ H11 H12 ··· H1nd ]

(49)

⎡ 0.0383q−3 + 0.0533q−4 ⎤ − 0.0343q−4 − 0.0324q−5 −(5.12 × 10−4)q−2 ⎢ ⎥ ⎢ ⎥ 1 − 0.8614q−1 1 − 0.8907q−1 1 − 0.8955q−1 ⎢ ⎥ ⎢ 0.1583q−7 + 0.1357q−8 − 0.4278q−4 − (1.28 × 10−3)q−2 − (2.98 × 10−4)q−3 ⎥ ⎥ G=⎢ ⎢ ⎥ 1 − 0.7351q−1 1 − 0.8187q−1 1 − 0.8665q−1 ⎢ ⎥ ⎢ − 3.2420q−10 − 0.7621q−11 2.4740q−10 + 1.5760q−11 ⎥ 0.1238q−2 − 0.1136q−3 ⎢ ⎥ − 1 − 1 − 1 − 2 ⎢⎣ ⎥⎦ 1 − 0.8845q 1 − 0.9123q 1 − 1.7220q + 0.7333q

(55)

⎡ −3.73 × 10−4 + (3.10 × 10−4)q−1 + (6.45 × 10−4)q−2 ⎤⎥ 0.0093q−11 ⎢ ⎢ ⎥ 1 − 0.851q−1 1 − 1.814q−1 + 0.8228q−2 ⎢ ⎥ ⎢ 0.0166q−9 + 0.0154q−10 −(2.28 × 10−3)q−1 + (5.45 × 10−4)q−2 + (1.54 × 10−3)q−3 ⎥ ⎥ H=⎢ ⎢ ⎥ 1 − 0.0.8651q−1 1 − 1.7660q−1 + 0.7796q−2 ⎢ ⎥ ⎢ −0.2545 − 0.3584q−1 ⎥ 0.0719 + 0.1011q−1 ⎢ ⎥ − 1 − 1 ⎢⎣ ⎥⎦ 1 − 0.8812q 1 − 0.8791q

(56)

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the use of recommended values of τc for the three loops results in a very aggressive controller. Using trial and error, τc is chosen as ̃ , 18θ22 ̃ , and 6θ33 ̃ for loops 1, 2, and 3, respectively, where θiĩ = 55θ11 θii with θii being the time delay of the first-order approximation of gii(s). The resulting decentralized PI controller is given as

The decentralized PI controller is designed using the IMC method44 based on the diagonal elements of the process. For this process, g33(s) has second order and is thus approximated as a firstorder transfer function using the model reduction techniques discussed by Skogestad44 prior to controller design. Furthermore,

⎡ 0.0697 − 0.0593q−1 ⎤ ⎢ ⎥ 0 0 ⎢ ⎥ 1 − q−1 ⎢ ⎥ ⎢ ⎥ −0.0372 + 0.0297q−1 ⎥ K=⎢ 0 0 ⎢ ⎥ 1 − q−1 ⎢ ⎥ ⎢ 1.0340 − 0.7685q−1 ⎥ 0 0 ⎢ ⎥ ⎢⎣ ⎥⎦ 1 − q−1

For diagonal pairing, we have d1′ = d1 = 3; d2′ = d2 = 4, and d3′ = d3 = 2. As the expected magnitudes of the two disturbances are 0.2 and 20, respectively,46 H is scaled by diag(0.21/2,201/2) to satisfy the assumption of disturbances having unit variance. For this process, every element of Pd contains finite zeros outside the unit circle. Thus, prior to the analysis, these unstable zeros are reflected across the unit circle to obtain Pd,m. The impulse response coefficients of Pd,m and H are shown in Figure 6, and the MV benchmarks obtained using the conventional and modified approaches are shown in Table 1.

(57)

of decentralized controllers. In comparison, the performance indices obtained using the modified MV benchmark for these loops are closer but still less than 1. Similar to earlier examples, this happens due to the use of the open loop disturbance model (H), instead of the disturbance model for the partially closed loop system (Pd), in the conventional approach. Furthermore, unlike Example 2, η3,siso obtained using the conventional approach is lower than η3,decen obtained using the modified approach. This shows that, in general, the MV benchmark for SISO processes can also underestimate the performance of decentralized controllers. 4.4. Estimation from Data. In this section, we show that with the knowledge of time delays between all the inputs and outputs, the modified MV benchmark can be estimated from the closed loop data. When all the loops are closed under feedback control, i.e., u = −Ky, we have y = (I + GK )−1Ha

(58)

or equivalently −1 ⎡1 + G K ̃ K2̃ ⎤ ⎡ H1̃ ⎤ G12 11 1 ⎢ ⎥ ⎢ ⎥a y= ⎢⎣ G̃ 21K1 I + G̃ 22K2̃ ⎥⎦ ⎢⎣ H̃2 ⎥⎦

(59)

where G̃ 12, G̃ 21, G̃ 22, H̃ 2, and K̃ 2 are given in eqs 46−eq 51. Using the formula for inversion of partitioned matrices, we have47 ∼ ∼ y1 = X −1(H͠ 1 − G12K͠ 2(I + G22K͠ 2)−1H͠ 2)a (60) Figure 6. Impulse response coefficients of disturbance models of open loop and partially closed loop systems for Example 4.

where ∼ ∼ Pd1 = H͠ 1 − G12K͠ 2(I + G22K͠ 2)−1H͠ 2

Table 1. Performance Assessment of Sidestream Column/ Stripper Distillation Column Using Loop by Loop Analysis

and

conventional

modified

loop

Var(y)

Ji,siso

ηi,siso

Ji,decen

η

1 2 3

1.87 × 10−4 2.31 × 10−3 1.1887

1.97 × 10−4 2.35 × 10−3 0.4364

1.0586 1.0158 0.3671

1.56 × 10−4 1.88 × 10−3 0.4898

0.8332 0.8144 0.4120

̃ K2̃ (I + G̃ 22K2̃ )−1G̃ 21K1 X = 1 + G11K1 − G12

(61)

∼ ∼ ∼ X = 1 + (G11 − G12K͠ 2(I + G22K͠ 2)−1G21)K1

(62)

with ∼ ∼ ∼ P11 = G11 − G12K͠ 2(I + G22K͠ 2)−1G21

It can be noted that similar to Example 3, for the conventional approach, Ji,siso exceeds the observed output variance for loops 1 and 2, which signifies a limitation of using the conventional MV benchmark for performance assessment

Thus, we have y1 = (1 + P11K1)−1Pd1a 4295

(63)


Industrial & Engineering Chemistry Research y1 = (1 + P11K1)−1Pd1, ma ̃

Article

(64)

Let M1 = K1/(1 + P11K1) (Youla parametrization), which implies that K1 = M1/(1 − P11 M1). Then, using eq 37 y1 = (1 − P11M1)Pd1, ma ̃

(65)

y1 = F1′a ̃ − q−d1′(P11 ̅ M1Pd1, m − R1′)a ̃

(66)

which shows that the first d1′ impulse response coefficients of the closed loop and open loop transfer functions between y1 and ã are the same. Thus, F1′ and ηi,decen can be estimated using the closed loop data, e.g., using the filtering and correlation (FCOR) algorithm,33 with a priori knowledge of the delays of gij for all i, j = 1, 2, ..., ny. It is interesting to note from eq 66 that the estimation of the MV benchmark using the closed loop data leads to estimation of impulse response coefficients of Pd1,m and not H1, as is traditionally believed. This observation highlights that, although the application of the MV benchmark derived for the SISO controller to decentralized controllers is theoretically not justified, the estimation using closed loop data fortunately still leads to the correct impulse response coefficients. The issue regarding the number of the impulse response coefficients to be considered for estimation of MV benchmark, however, still remains and requires the knowledge of the effective delays. Remark 2. In general, P11 can contain finite poles or zeros outside the unit circle, even though all the elements of G are stable and minimum phase. For example, the unstable poles may arise when the pairings are selected on negative elements of the RGA.42 Similar to the MV benchmark for SISO systems,33 the modified MV benchmark can be obtained by mirroring the unstable poles and zeros of P11 across the unit circle using all-pass factors. In general, however, the locations of these unstable poles and zeros depend on the controller settings for other loops48 and thus the estimation of modified MV benchmark from operating data can be difficult. Example 5. We revisit Example 4 with the PI controller settings given in eq 57. The closed loop system is simulated, where the two disturbances are assumed to be random noise sequences with variances of 0.2 and 20, respectively. The resulting output and input responses are shown in Figure 7. The observed output variances for the three loops are 1.90 × 10−4, 2.31 × 10−3, and 1.2227, respectively. For each loop, the output signals are prewhitened to obtain the following estimates of Fi′, i.e., F̂i′: F1̂ ′ = 0.0090 + 0.0070q−1 + 0.0051q−2

Figure 7. Input and output responses for Example 5.

5. CONCLUSIONS The use of decentralized or multiloop controllers is common in process industries. In this paper, we have shown that the use of existing minimum variance (MV) benchmarks for single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems for performance assessment of decentralized controllers may lead to incorrect conclusions regarding the opportunities for variance reduction through controller retuning. To find a lower bound on the achievable output variance using decentralized controllers, where individual loops are tuned simultaneously, we proposed the use of sums of squares (SOS) programming. Furthermore, a modified MV benchmark is proposed for loop by loop analysis of the decentralized controller, which can be directly estimated from the closed loop data. In summary, this paper takes a major step toward systematic performance assessment of decentralized controllers. Future work will focus on finding methods for estimation of the lower bound on the achievable output variance with simultaneous tuning of loops from the closed loop data. For loop by loop analysis, approaches to account for the effect of tuning an individual loop on the other loops of the decentralized controller due to interactions will be pursued. Extension of the results presented in this paper for fully decentralized controllers

(67)

F2̂ ′ = 0.0165 + 0.0286q−1 + 0.0228q−2 + 0.0178q−3 (68)

F3̂ ′ = 0.3773 + 0.5852q−1

(69)

The estimates of the performance indices η̂i are computed as η̂i = ∥F̂i′∥22/Var(yi), which are found to be 0.8343, 0.8285, and 0.3965 for the three loops. We note that the estimated values of performance indices are within ±5% of the theoretical values shown in Table 1. The same analysis is repeated 100 times, for which η̂i = 0.8519 ± 0.0130, 0.8234 ± 0.0201, and 0.4178 ± 0.0136 for i = 1, 2, and 3, respectively, which shows that the modified MV benchmark can be reliably estimated from the closed loop data. 4296



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to block decentralized controllers49,50 is also an issue for future research.

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AUTHOR INFORMATION

Corresponding Author

**Tel.: +91 8042069950. E-mail: [email protected]. Present Address ‡

Advanced Environmental Biotechnology Center (AEBC− NEWRI), Nanyang Technological University, Singapore. § ABB Global Industries & Services, Ltd., Mahadevpura, Bangalore 560048, India. Notes †

A preliminary version of this work was presented at the International Symposium on Advanced Control of Chemical Processes held in Istanbul, Turkey.1

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