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How Good is Your Controller? Application of Control Loop Performance Assessment Techniques to MIMO Processes
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easy to implement on MIMO processes. The key ingredient of this scheme is a method of filtering and subsequent cross correlation analysis (FCOR) to estimate the achievable multivariable minimum variance performance from routine operating data. A computationally simple algorithm is developed for performance assessment of multivariable control systems.
Definition 2 Instead of taking the lower triangular form, if a interact or matrix as per definition 1 also satisfies
2. FEEDBACK-INVARIANCE OF MINIMUM VARIANCE CONTROL PERFORMANCE MEASURE AND ITS SEPARATION FROM ROUTINE OPERATING PROCESSES
DT(q-l)D(q) = I then this interactor matrix is denoted as the unitary interactor matrix.
2.1 Unitary Interactor Matrix
For the sake of brevity and convenience, the backshift operator q-l will be omitted from hereon, unless circumstances necessitate its presence. For example, the transfer function matrix T(q-l) will be expressed simply as T. Now consider the MIMO process yt = TUt
+ Nat
(1)
where Tand N are proper (causal), rational transfer function matrices in the backshift operator q-l; yt, Ut, and at are output, input and noise vectors of appropriate dimensions. To solve the multi variable minimum variance control problem, Wolovich and Falb(1976), Goodwin and Sin (1984), and Shah et al.(1987) introduce a lower triangular interactor matrix D, which is the generalization of the SISO time delay for the MIMO case. Definition 1 For every n x m proper, rational polynomial transfer function matrix T, there is a unique, nonsingular, n x n lower left triangular' polynomial matrix, D, called the interactor matrix ofT, such that IDI = qr and lim DT =
q-l--+O
unitary interactor matrix, which is a special case of the nilpotent interactor matrix. Huang et al.(1996a) have also found important properties of the unitary interactor matrix in minimum variance or singular LQG control. In general, the interactor matrix can also be written as D = qdDo+qd-l Dl + .. '+qD d- 1 , where D 1 , D 2 , · · · , Dd are constant coefficient matrices of the general interactor matrix.
t q-l--+O lim
= K
(2)
which is a full rank constant matrix, where the integer r is defined as the number of infinite zeros of the transfer function matrix T, and is called as the delay-free transfer function matrix of T.
t
Wolovich and Falb(1976) and Goodwin and Sin(1984) have shown existence of a unique lower triangular form of the general interactor matrix. However, the interactor matrix can also take other forms. It can be a full matrix or an upper triangular matrix (Shah et al., 1987; Huang et al., 1996b). Rogozinski et aL(1987) have introduced an algorithm for the calculation of a nilpotent interactor matrix. Peng and Kinnaert (1992) have introduced the
Existence of the unitary interactor matrix is established in the following lemma due to Rogozinski et al.(1987) and Peng and Kinnaert(1992). Lemma 1 For a full rank (in the field of q-l) rational, proper transfer function matrix T, there exists a non-unique unitary interactor matrix. However, any two unitary interactor matrices D(q) and D(q) satisfy the following relation: D(q) where
r
Proof:
= rD(q)
is a n x n unitary real matrix.
See Peng and Kinnaert(1992) for the proof. _
The unitary interactor matrix can be shown to be an "optimal" factorization of the time delay for minimum variance or singular LQG type control(Huang et al., 1996b). A straightforward algorithm exists for the calculation of the unitary interactor matrix(Rogozinski et al., 1987; Peng and Kinnaert, 1992; Huang et al., 1996b).
2.2 Feedback Invariance and Separation of MV Control Performance Measure from Routine Processes
Unlike previous work on multivariable minimum variance control by Borison(1979), Goodwin and Sin(1984), Harris and MacGregor(1987), and Walgama et al.(1989), the main focus of this study is the property of feedback invariance of the minimum variance term and the method to separate this term from the output variance. In the following section the terms offset or tracking error is considered in the mean sense, i.e., zero offset implies that E(yt) = ~sp. The minimum variance control (or the benchmark control) is defined as a linear control which minimizes the output LQ objective function J
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= E(yt _ ~SP)T(yt _ ~BP)
= E(ft - E(ft))T(ft - E(ft))!min = E(ei)(et) = tr(Var(q-d Mat)) where et = q-d M at, M and S satisfy the identity:
or the weighted LQ objective function J
= E(Yi -
-y;sPfW(Yi - -y;sP)
without control action constraints (Harris and MacGregor, 1987). This is also regarded as Singular LQ control (Peng and Kinnaert, 1992). Minimum variance or singular LQ control takes the disturbance model or spectrum into account, and performance is measured by variance or mean square error. It is therefore different from the minimum time or minimum ISE control law in (Tsiligiannis and Svoronos, 1988), which assumes a step-type disturbance or set point change.
DN = MOqd ,
+ ... + M d- 1q+S "
(5)
or
M
and S is a rational proper transfer function matrix. (2) If closed-loop routine operating data under feedback control is modelled by a multivariate Moving Average process:
Now for a multivariable process Yi
= TUt + Nat
+ LOat-d ~
using the notation of multi variable minimum variance control due to Goodwin and Sin(1984), the minimum variance control law can be designed to make the variance of the output DYi or equivalently ft = q-d DYi minimum, where the positive integer d is the maximum order (highest power of q) of all the elements of the interactor matrix, D. This filter, q-d D, removes infinite zeros from the transfer function matrix. If D is a lower triangular interactor, the minimum variance control law and performance measure of the interactor-filtered variable ft are different from that of the original variable Yi. However if D is a unitary interactor matrix, this filtering makes no difference for minimum variance control and the performance measure based on it and Yi is the same. The relation between Yi and ft is discussed in Lemma 2. A general, yet implicit, minimum variance control law has been developed by Goodwin and Sin(1984). In the following theorem, the feedback-invariant property and the method to separate the minimum variance term from output variance for the general multi variable process is developed.
and with the linear quadratic objective function (Singular LQ) defined by
J where
ft
= E(ft -
ytSP)T(ft - ytsP)
(4)
= q-d DYi, ytsp = q-d DY;sP then
(1) The optimal quadratic performance measure or minimum variance control performance measure is given by - sp T - sp E(Yi - Y; ) (Yi - Y; )!min
(6)
#
Wt-d
then et = q-d M at is independent of feedback control. Under minimum variance control, Wt-d will vanish, and therefore et represents the process output under minimum VrLriance control. et depends only on the first d terms of the Moving Average process and is therefore feedback-invariant. The minimum variance control performance can then be estimated from this term.
Proof: Proof is omitted due to space limits. See (Huang et al., 1995a) for details. _ Lemma 2 If D is a unitary interactor matrix, then the optimal control law which minimizes the following objective function of the intemctor-filtered variable ft: J1
= E(Yt -
ytSP) T (ft
-
ytsP)
(7)
also minimizes the objective function of the original variable Yi (8)
Theorem 1 For a multivariable process with a general interactor matrix
(3)
+ L 1at-d-1 + ...
and J 1 = J2 • Thus the performance measure of the original variable Yi can be obtained via the performance measure of the interactor-filtered variable ft. In fact, it can be shown (Huang and Shah, 1996a) that this control law is the same as the singular LQG control law solved from spectral factorization, and is unique.
Proof: The proof is straightforward and obtained by substituting ft = q-dDYi and ytsp = q-dDY;sP into (4) and using the fact that DT D = I. _ Corollary 1 With the general interact or matrix substituted by a weighted unitary interactor matrix (Huang
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4. CONCLUSION The main contributions of this paper are: • Development of a computationally simple algorithm to estimate performance measure of MIMO control loops. The use of this measure for preliminary process diagnosis and monitoring of multi variable processes under multiloop control has been illustrated by application to an industrial process. This later topic is bound to be the subject of considerable industrial interest for pre- and post-audit of advanced control applications. • The derivation of this algorithm is based on the idea of a minimum variance benchmark standard that has been extended from the SISO to the MIMO case.
5. REFERENCES Astrom, K.J. (1970). Introduction to Stochastic Control Theory. Academic Press. New York. Astrom, K.J. and B. Wittenmark (1973). On self-tuning regulators. Automatica 9, 457-476. Astrom, K.J. and B. Wittenmark (1990). ComputerControlled Systems, Theory and Design. second ed .. Prentice-Hall. Borison, U. (1979). Self-tuning regulators for a class of multivariable systems. Automatica Vol.15, 209215. Desborough, L. and T. Harris (1992). Performance assessment measure for univariate feedback control. The Canadian Journal of Chemical Engineering Vo1.70, 1186-1197. Eriksson, P.G. and A ..J. Isaksson (1994). Some aspects of control performance monitoring. In: Proceedings of the third IEEE conference on control applications. pp. 1029-1034. Goodwin, G.C. and K.S. Sin (1984). Adaptive Filtering Prediction and Control. Pr-entice-Hall. Harris, T. (1989). Assessment of closed loop performance. Canadian Journal of Chemical Engineering Vo1.67, 856-86l. Harris, T.J. and J.F. MacGregor (1987). Design of multivariable linear quadratic controllers using transfer functions. AIChE J. 33, 1481-1495. Huang, B. and S.L. Shah (1996a). Unitary interactor and explicit singular LQG using input-output transfer functions. Internal Report, Department of Chemical Engineering, University of Alberta. Huang, B., S.L. Shah and H. Fujii (1996b). Identification of the time delay /interactor matrix for MIMO systems using closed-loop data. to be presentation at the 1996 IFAC world congress.
Huang, B., S.L. Shah and K.Y. Kwok (1995a). Good, bad or optimal? performance assessment of MIMO processes. Submitted for publication. Huang, B., S.L. Shah and K.Y. Kwok (1995b). Online control performance monitoring of MIMO processes. In: Proceedings of American Control Conference. American Control Conference. Seattle, Washington, U.S.A .. pp. 1250 - 1254. Kozub, D.J. and C.E. Garcia (1993). Monitoring and diagnosis of automated controllers in the chemical process industries. In: AIChE Meeting. St. Louis. Lynch, C.B. and G.A. Dumont (1993). Closed loop performance monitoring. In: Proceedings of the Second IEEE Conference on Control Applications. Vancouver, B.C. Peng, Y. and M. Kinnaert (1992). Explicit solution to the singular LQ requlation problem. IEEE 1'rans AC 37, 633-636. Rhinehart (1995). A watch dog for controller performance monitoring. In: Proceedings of the 1995 American Control Conference. Seattle, Washington, U.S.A .. pp. 2239 - 2240. Rogozinski, M.W., A.P. Paplinski and M.J. Gibbard (1987). An algorithm for calculation of a nilpotent interactor matrix for linear multi variable systems. IEEE Trans. AC 32(3), 234-237. Shah, S.L., C. Mohtadi and D.W. Clarke (1987). Multivariable adaptive control without a prior knowledge of the delay matrix. Systems and Control Letters pp. 295-306. Stanfelj, N., T.E. Marlin and J.F. MacGregor (1993). Monitoring and diagnosing process control performance: the single-loop case. Ind. Eng. Chem. Res. Tsiligiannis, C.A. and S.A. Svoronos (1988). Dynamic interactors in multivariable process control-I, the general time delay case. Chemical Engineering Science 43(2), 339-347. Tyler, M.L. and M. Morari (1995a). Performance assessment for unstable and nonminimum-phase systems. In: IfA-Report No 95-03. Tyler, M.L. and M. Morari (1995b). Performance monitoring of control systems using likelihood methods. In: Proceedings of the American Control Conference. Seattle, Washington, U.S.A .. pp. 1245 - 1249. Walgama, K.S., D.G. Fisher and S.L. Shah (1989). Control of processes with noise and time-delays. AIChE Journal 35, 213-222. Wolovich, W.A. and P.L. Falb (1976). Invariants and canonical forms under dynamic compensation. SIAM J. Control Vol.14, 996-1008.
6030
easy to implement on MIMO processes. The key ingredient of this scheme is a method of filtering and subsequent cross correlation analysis (FCOR) to estimate the achievable multivariable minimum variance performance from routine operating data. A computationally simple algorithm is developed for performance assessment of multivariable control systems.
Definition 2 Instead of taking the lower triangular form, if a interact or matrix as per definition 1 also satisfies
2. FEEDBACK-INVARIANCE OF MINIMUM VARIANCE CONTROL PERFORMANCE MEASURE AND ITS SEPARATION FROM ROUTINE OPERATING PROCESSES
DT(q-l)D(q) = I then this interactor matrix is denoted as the unitary interactor matrix.
2.1 Unitary Interactor Matrix
For the sake of brevity and convenience, the backshift operator q-l will be omitted from hereon, unless circumstances necessitate its presence. For example, the transfer function matrix T(q-l) will be expressed simply as T. Now consider the MIMO process yt = TUt
+ Nat
(1)
where Tand N are proper (causal), rational transfer function matrices in the backshift operator q-l; yt, Ut, and at are output, input and noise vectors of appropriate dimensions. To solve the multi variable minimum variance control problem, Wolovich and Falb(1976), Goodwin and Sin (1984), and Shah et al.(1987) introduce a lower triangular interactor matrix D, which is the generalization of the SISO time delay for the MIMO case. Definition 1 For every n x m proper, rational polynomial transfer function matrix T, there is a unique, nonsingular, n x n lower left triangular' polynomial matrix, D, called the interactor matrix ofT, such that IDI = qr and lim DT =
q-l--+O
unitary interactor matrix, which is a special case of the nilpotent interactor matrix. Huang et al.(1996a) have also found important properties of the unitary interactor matrix in minimum variance or singular LQG control. In general, the interactor matrix can also be written as D = qdDo+qd-l Dl + .. '+qD d- 1 , where D 1 , D 2 , · · · , Dd are constant coefficient matrices of the general interactor matrix.
t q-l--+O lim
= K
(2)
which is a full rank constant matrix, where the integer r is defined as the number of infinite zeros of the transfer function matrix T, and is called as the delay-free transfer function matrix of T.
t
Wolovich and Falb(1976) and Goodwin and Sin(1984) have shown existence of a unique lower triangular form of the general interactor matrix. However, the interactor matrix can also take other forms. It can be a full matrix or an upper triangular matrix (Shah et al., 1987; Huang et al., 1996b). Rogozinski et aL(1987) have introduced an algorithm for the calculation of a nilpotent interactor matrix. Peng and Kinnaert (1992) have introduced the
Existence of the unitary interactor matrix is established in the following lemma due to Rogozinski et al.(1987) and Peng and Kinnaert(1992). Lemma 1 For a full rank (in the field of q-l) rational, proper transfer function matrix T, there exists a non-unique unitary interactor matrix. However, any two unitary interactor matrices D(q) and D(q) satisfy the following relation: D(q) where
r
Proof:
= rD(q)
is a n x n unitary real matrix.
See Peng and Kinnaert(1992) for the proof. _
The unitary interactor matrix can be shown to be an "optimal" factorization of the time delay for minimum variance or singular LQG type control(Huang et al., 1996b). A straightforward algorithm exists for the calculation of the unitary interactor matrix(Rogozinski et al., 1987; Peng and Kinnaert, 1992; Huang et al., 1996b).
2.2 Feedback Invariance and Separation of MV Control Performance Measure from Routine Processes
Unlike previous work on multivariable minimum variance control by Borison(1979), Goodwin and Sin(1984), Harris and MacGregor(1987), and Walgama et al.(1989), the main focus of this study is the property of feedback invariance of the minimum variance term and the method to separate this term from the output variance. In the following section the terms offset or tracking error is considered in the mean sense, i.e., zero offset implies that E(yt) = ~sp. The minimum variance control (or the benchmark control) is defined as a linear control which minimizes the output LQ objective function J
6026
= E(yt _ ~SP)T(yt _ ~BP)
= E(ft - E(ft))T(ft - E(ft))!min = E(ei)(et) = tr(Var(q-d Mat)) where et = q-d M at, M and S satisfy the identity:
or the weighted LQ objective function J
= E(Yi -
-y;sPfW(Yi - -y;sP)
without control action constraints (Harris and MacGregor, 1987). This is also regarded as Singular LQ control (Peng and Kinnaert, 1992). Minimum variance or singular LQ control takes the disturbance model or spectrum into account, and performance is measured by variance or mean square error. It is therefore different from the minimum time or minimum ISE control law in (Tsiligiannis and Svoronos, 1988), which assumes a step-type disturbance or set point change.
DN = MOqd ,
+ ... + M d- 1q+S "
(5)
or
M
and S is a rational proper transfer function matrix. (2) If closed-loop routine operating data under feedback control is modelled by a multivariate Moving Average process:
Now for a multivariable process Yi
= TUt + Nat
+ LOat-d ~
using the notation of multi variable minimum variance control due to Goodwin and Sin(1984), the minimum variance control law can be designed to make the variance of the output DYi or equivalently ft = q-d DYi minimum, where the positive integer d is the maximum order (highest power of q) of all the elements of the interactor matrix, D. This filter, q-d D, removes infinite zeros from the transfer function matrix. If D is a lower triangular interactor, the minimum variance control law and performance measure of the interactor-filtered variable ft are different from that of the original variable Yi. However if D is a unitary interactor matrix, this filtering makes no difference for minimum variance control and the performance measure based on it and Yi is the same. The relation between Yi and ft is discussed in Lemma 2. A general, yet implicit, minimum variance control law has been developed by Goodwin and Sin(1984). In the following theorem, the feedback-invariant property and the method to separate the minimum variance term from output variance for the general multi variable process is developed.
and with the linear quadratic objective function (Singular LQ) defined by
J where
ft
= E(ft -
ytSP)T(ft - ytsP)
(4)
= q-d DYi, ytsp = q-d DY;sP then
(1) The optimal quadratic performance measure or minimum variance control performance measure is given by - sp T - sp E(Yi - Y; ) (Yi - Y; )!min
(6)
#
Wt-d
then et = q-d M at is independent of feedback control. Under minimum variance control, Wt-d will vanish, and therefore et represents the process output under minimum VrLriance control. et depends only on the first d terms of the Moving Average process and is therefore feedback-invariant. The minimum variance control performance can then be estimated from this term.
Proof: Proof is omitted due to space limits. See (Huang et al., 1995a) for details. _ Lemma 2 If D is a unitary interactor matrix, then the optimal control law which minimizes the following objective function of the intemctor-filtered variable ft: J1
= E(Yt -
ytSP) T (ft
-
ytsP)
(7)
also minimizes the objective function of the original variable Yi (8)
Theorem 1 For a multivariable process with a general interactor matrix
(3)
+ L 1at-d-1 + ...
and J 1 = J2 • Thus the performance measure of the original variable Yi can be obtained via the performance measure of the interactor-filtered variable ft. In fact, it can be shown (Huang and Shah, 1996a) that this control law is the same as the singular LQG control law solved from spectral factorization, and is unique.
Proof: The proof is straightforward and obtained by substituting ft = q-dDYi and ytsp = q-dDY;sP into (4) and using the fact that DT D = I. _ Corollary 1 With the general interact or matrix substituted by a weighted unitary interactor matrix (Huang
6027
6028
6029
4. CONCLUSION The main contributions of this paper are: • Development of a computationally simple algorithm to estimate performance measure of MIMO control loops. The use of this measure for preliminary process diagnosis and monitoring of multi variable processes under multiloop control has been illustrated by application to an industrial process. This later topic is bound to be the subject of considerable industrial interest for pre- and post-audit of advanced control applications. • The derivation of this algorithm is based on the idea of a minimum variance benchmark standard that has been extended from the SISO to the MIMO case.
5. REFERENCES Astrom, K.J. (1970). Introduction to Stochastic Control Theory. Academic Press. New York. Astrom, K.J. and B. Wittenmark (1973). On self-tuning regulators. Automatica 9, 457-476. Astrom, K.J. and B. Wittenmark (1990). ComputerControlled Systems, Theory and Design. second ed .. Prentice-Hall. Borison, U. (1979). Self-tuning regulators for a class of multivariable systems. Automatica Vol.15, 209215. Desborough, L. and T. Harris (1992). Performance assessment measure for univariate feedback control. The Canadian Journal of Chemical Engineering Vo1.70, 1186-1197. Eriksson, P.G. and A ..J. Isaksson (1994). Some aspects of control performance monitoring. In: Proceedings of the third IEEE conference on control applications. pp. 1029-1034. Goodwin, G.C. and K.S. Sin (1984). Adaptive Filtering Prediction and Control. Pr-entice-Hall. Harris, T. (1989). Assessment of closed loop performance. Canadian Journal of Chemical Engineering Vo1.67, 856-86l. Harris, T.J. and J.F. MacGregor (1987). Design of multivariable linear quadratic controllers using transfer functions. AIChE J. 33, 1481-1495. Huang, B. and S.L. Shah (1996a). Unitary interactor and explicit singular LQG using input-output transfer functions. Internal Report, Department of Chemical Engineering, University of Alberta. Huang, B., S.L. Shah and H. Fujii (1996b). Identification of the time delay /interactor matrix for MIMO systems using closed-loop data. to be presentation at the 1996 IFAC world congress.
Huang, B., S.L. Shah and K.Y. Kwok (1995a). Good, bad or optimal? performance assessment of MIMO processes. Submitted for publication. Huang, B., S.L. Shah and K.Y. Kwok (1995b). Online control performance monitoring of MIMO processes. In: Proceedings of American Control Conference. American Control Conference. Seattle, Washington, U.S.A .. pp. 1250 - 1254. Kozub, D.J. and C.E. Garcia (1993). Monitoring and diagnosis of automated controllers in the chemical process industries. In: AIChE Meeting. St. Louis. Lynch, C.B. and G.A. Dumont (1993). Closed loop performance monitoring. In: Proceedings of the Second IEEE Conference on Control Applications. Vancouver, B.C. Peng, Y. and M. Kinnaert (1992). Explicit solution to the singular LQ requlation problem. IEEE 1'rans AC 37, 633-636. Rhinehart (1995). A watch dog for controller performance monitoring. In: Proceedings of the 1995 American Control Conference. Seattle, Washington, U.S.A .. pp. 2239 - 2240. Rogozinski, M.W., A.P. Paplinski and M.J. Gibbard (1987). An algorithm for calculation of a nilpotent interactor matrix for linear multi variable systems. IEEE Trans. AC 32(3), 234-237. Shah, S.L., C. Mohtadi and D.W. Clarke (1987). Multivariable adaptive control without a prior knowledge of the delay matrix. Systems and Control Letters pp. 295-306. Stanfelj, N., T.E. Marlin and J.F. MacGregor (1993). Monitoring and diagnosing process control performance: the single-loop case. Ind. Eng. Chem. Res. Tsiligiannis, C.A. and S.A. Svoronos (1988). Dynamic interactors in multivariable process control-I, the general time delay case. Chemical Engineering Science 43(2), 339-347. Tyler, M.L. and M. Morari (1995a). Performance assessment for unstable and nonminimum-phase systems. In: IfA-Report No 95-03. Tyler, M.L. and M. Morari (1995b). Performance monitoring of control systems using likelihood methods. In: Proceedings of the American Control Conference. Seattle, Washington, U.S.A .. pp. 1245 - 1249. Walgama, K.S., D.G. Fisher and S.L. Shah (1989). Control of processes with noise and time-delays. AIChE Journal 35, 213-222. Wolovich, W.A. and P.L. Falb (1976). Invariants and canonical forms under dynamic compensation. SIAM J. Control Vol.14, 996-1008.
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