Ind. Eng. Chem. Res. 1992, 31, 1418-1421
1418
Time Delay Structure of a Multiple Input-Multiple Output System and Its Factorization and Compensation The concept of a time delay structure matrix, which can only be determined from the structural information of the system, is introduced. The time delay factorization and compensation are discussed concerning the weakening of the requirement of the prior information for a system in MIMO adaptive controller design and concerning also the design of a dynamic decoupled internal model controller. The equivalent relationship of time delay factorizations and compensations for the two purposes are proved. The conclusion is derived that any time delay unbalanced system can be transformed to a time delay balanced one by using an appropriate diagonal proper precompensator, and its design procedure is given. It is shown that an internal model controller design can be simplified by changing the time delay unbalanced system t o a time delay balanced one, if the system time delays are unbalanced and a decoupled closed loop response is required. 1. Introduction
Many chemical processes are multiple input-multiple output (MIMO) systems with multiple time delays. When a system has no time delay or equal time delays in all elements of the transfer matrix, it behaves essentially the same as a single input-single output (SISO) system. With multiple time delays, however, the system will have quite different properties from a SISO system. This makes the system much more difficult to control and design the controller. The concept of interador matrix presented by Wolovich and Falb (1976) has played an important role in MIMO adaptive control. This interactor matrix, at least its diagonal elements, is usually assumed known. Clearly, prior knowledge of the interactor matrix, except diagonal, cannot be derived from the structural information of the system only. This contradicts the proposition of adaptive control in principle, where system parameters are absolutely unknown. For solving this problem, Singh and Narendra (1984) proved that any MIMO system, whatever its time delay structure is, can be transformed to a system which has a diagonal interactor matrix by using an appropriate diagonal proper precompensator. Furthermore, both the precompensator and the interactor matrix can be determined only from the structural information of the system, without any parametric information of the system. Later, Ichikawa (1987) discussed this problem from another way and gave a simpler algorithm for the determination of the precompensator. Therefore, by using an appropriate diagonal proper precompensator, MIMO adaptive control can be made as an extension of SISO adaptive control and the requirement of prior information for the MIMO adaptive controller can be weakened. The internal model control (IMC) structure was introduced by Garcia and Morari (l982,1985a,b). Because of its simplicity, transparency, and better performance, the IMC has become an effective procedure for the design of complex MIMO control systems. The model predictive control law formulation suggested by Garcia and Morari (1985b) for the computation and implementation of the internal model controller has advantages of calculating controller parameters explicitly without inversion or factorization of polynomial matrices and obtaining the perfect internal model controller as a special case. However, when time delays in the controlled system are unbalanced and a decoupled closed loop response is sought, the formulation will be extremely tedious. In this paper, the equivalence between time delay compensations for the MIMO adaptive controller design suggested by Singh and Narendra (1984) and the internal model controller design by Garcia and Morari (1985a,b) will be proved. It will be shown that any time delay unbalanced system can be transformed into a time delay
balanced one and the design procedure of a dynamic decoupled internal model controller given by Garcia and Morari (1985b) for a MIMO time delay unbalanced system can be much simplified by using an appropriate diagonal proper precompensator. 2. Time Delay S t r u c t u r e Formulation An rn-input and m-output system transfer matrix is
assumed as 4" h l l ( Q )
*
-. q4L"h,,(q)
i
Gtq'=['
i
q4m1hmlCq)
***
q4mmhmm(q)
]
(1)
where dij is the time delay from input j to output i, and hij(q)are semiproper rational functions of q. Definition 1. The following matrix is defined as a time delay structure matrix: D =
(2) Dm1
Dmm
where
Example 1. For a system of
the time delay structure matrix is
[:: :]
D = 11 0
12
The time delay structure matrix describes completely the time delay relationship between each input and output, and it can be determined from the structural information of the system only, without requiring any parametric information about the system. 3. Adaptive Control
In many MIMO adaptive control schemes, the interador matrix is required to be known a priori. Unfortunately, this requirement appears too strong. In practice, it is difficult to know a priori the interador matrix of a system of unknown parameters. Singh and Narendra (1984) presented an ingenious idea for design of a dynamic precompensator which transforms the interactor matrix to a diagonal one.
0888-588519212631-1418$03.00/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31, No. 5,1992 1419 6. Time Delay Structure Analysis Two factorizations of transfer matrix G(q),eqs 3 and 4, are for different purposes. The former is for simplification of adaptive controller design, and the latter is for design of a realizllble and decoupled internal model controller. For example 1 they have following interrelationship. Figure 1. Basic IMC structure.
From theorem 3.1 given by Singh and Narendra (1984), transfer matrix 1 can be factored as (3) G ( q ) = T(q-') G(q) K ( q ) where
T(q-') = diag
( q t l
q-t2
K ( q ) = diag (qkl qk2
...
...
Q-~,),
qkm),
ti > 1
min kj = 0 I
G(q)is nonsingular and ti and ki can be determined only from the time delays dip It can be known from eq 3 that the system described by transfer matrix G(q)will have a diagonal interactor matrix T(q) when a precompensator K(4-l) is cascaded into the system. Consider example 1. T(q-9 = diag (q+ 1 q-4)
T(q-') G+(q-l) This interrelationship will be proved that it is not only true for example 1, but also true for any system. First consider the transfer matrix described by eq 1. It can always be rearranged so that the sum of all diagonal elements of its time delay structure matrix is the least among all arrangements. For example 1, it has six arrangements, whose time delay structure matrices are 6 [8
lm+-
K ( q ) = diag (1 1 q4)
1imq4- G ( q ) is nonsingular. So, when a precompensator K(q-9 = diag ( 1 1 q4) is cascaded into the system, the new system will have following diagonal interactor matrix: T(q) = diag (q6 1 q4) 4. Internal Model Control The internal model control (IMC) structure was introduced by Garcia and Morari (1982,1985a). The basic IMC structure is shown in Figure 1. Ideally it is preferred to have regulatory behavior, i.e. Y(t)= s(t) at all times. In the IMC structure, perfect control can be achieved by choosing
D4
where G+(q-') is a diagonal matrix of only time delays and G-(q) is realizable with smallest time delays. Consider example 1: G+(q-l)= diag (q4 1 q")
]
11 2 6 2 6 11 2 11 6 = 0 12 11 , D5 = 12 11 0 , D6 = 12 O 11 [0 8 131 [0 13 8 [13 0 8
]
]
1
D1 has the least sum of diagonal elements. Lemma 1. If the sum of diagonal elements of the time delay structure matrix of a transfer matrix G(q)is the least among all arrangements, then the time predictor of each diagonal element of G-'(q) is equal to the time delay of the corresponding diagonal element of G ( q ) ,Le. dii = T ~ ~i =, 1, 2, ..., m Proof. The inverse of transfer matrix G ( q ) is G W = A(q)/lG(q)l lG(q)1can be described as IG(d1 = q - W q )
where g(q) is semiproper rational function of q, and i=l
Considering the adjoint matrix of G(q), it can be described as q-T11611(q)
A(q) =
G c ( d = 6-W
However, this perfect controller cannot be implemented mainly for the following two reasons. (i) If G ( q ) contains time delays, its inverse involves predictive terms which are unrealizable. (ii) G(q) has transmissioy outside the unit circle, which become unstaple poles in G-'(q). Assuming G(q) = G ( q ) ,for the design of a realizable MIMO internal model controller giving decoupled closed loop response, Garcia and Morari (1985a,b) factor the transfer matrix G ( q ) as G ( q ) = G+(q-l)G-(q) (4)
11 2 6 2 11 11 6 2 12 , Dz = 11 12 0 , D3 = 0 11 12 [8 0 131 [13 8 0 13 0
D1 = 11 0
: [ q-Tmla * a,l
**
...
1
q-Th,DOlm(q) q-Tmma
amm(q)
where a&) are semiproper rational functions of q. It can be easily proved that qjQis calculated by m
qi(I = E d j j j=l j#i
i = 1, 2, ..., m
then pi
= 'i - 7..Q ii dii
Q.E.D. Lemma 2. The largest time predictor element in each column (row) of G-'(q) will be on the diagonal if and only if the rows and columns of transfer matrix G ( q ) can be rearranged so that the smallest time delay of G(q) in each row (column) is on the diagonal. Furthermore, if this is true, then the time predictor of each diagonal element of G-'(q) will be equal to the time delay of corresponding diagonal element of G ( q ) ,i.e. qi = dii, i = 1, 2, ..., m
1420 Ind. Eng. Chem. Res., Vol. 31, No. 5 , 1992
Proof. See Holt and Morari (19851, lemma 3. Lemma 3. If the s u m of diagonal elements of the time delay structure matrix of a transfer matrix G(q)is the least among all arrangements, and G(q)is factored as eq 4,i.e. G ( q ) = G+(q-')G-(q) then the smallest time delay element or the largest time predictor element in each column of G ( q )will be on the diagonal. Proof. See the Appendix. Theorem 1. After transfer matrix G ( q )is factored as given in eq 4,G J q ) can further be factored as G-(q) = G-i(q) G-2(q) so that the factorization equation (5) G(q) = G+(q-')G-i(q)G-2(4) is equivalent to eq 3, i.e. G+(q-') = T(q-') G-i(q) = G ( q ) G - ~ ( Q=) K(q) (6) Proof. It has been discussed that any transfer matrix can be rearranged so that the sum of diagonal elements of its time delay structure matrix is least among all arrangements. According to lemma 3, let G-2(q)= diag (q-dl" q42"
...
It is easy to find that the time delays of the above transfer matrix are balanced. Considering eq 4,if the internal model controller is selected as Gc(q)= G--l(q),a dynamic decoupled response can be obtained. However, if G-(q-') has zeros outside the unit circle, the G--'(q) would be unstable. According to the IMC strategy, it is desirable that the internal model controller is stable and approximately equals the inverse of 6 ( q )with the exact inverse of G J q ) as a special case. The model predictive method suggested by Garcia and Morari (1985b) for the computation and implementation of the internal model controller has the advantages that the controller parameters can be calculated explicitly without inversion or factorization of polynomial matrices and that the perfect internal model controller, Le., Gc(q) = G--'(q),is obtained as a special case. However, when the time delay of the system is unbalanced and a decoupled system response is sought, the method suggested by Garcia and Morari (1985b) will be extremely tedious, in which the Gaussian elimination algorithm has to be used. The following example is a time delay unbalanced system given by Garcia and Morari (1985b). It is used to illustrate the role of precompensator in the internal model control law computation. Example 2.
g4mm)
where djm = min di;,
j = 1, 2,
1
..., m where
then 1imq4- G-l(q)is nonsingular. Comparing eq 3 with eq 5, eq 6 can be obtained. Q.E.D. 6 . The Role of Time Delay Precompensator in Internal Model Control The following definition is given by Garcia and Morari (1985b). Definition 2. Let diminbe the earliest output i can be affected by one of the inputs, i.e. dimin= min dij .I
Time delays are called balanced if the system can settle as soon as the outputs are affected by the inputs, i.e. dimin= di+, i = 1, 2, ..., m Otherwise the time delays are unbalanced, and the measure of inbalance is provided by do = m p (di+ - dimin) (7)
So let the precompensator be and let then
the time delay of which is balanced. Now consider the following cost function.
1
Obviously do 3 0
From theorem 1, the system with a diagonal interactor matrix will be a time delay balanced system, and any time delay unbalanced system can be transformed to a time delay balanced system by using a precompensator. Considering example 1, let the precompensator be G-2(q-1)= diag (1 1 q4) then &q)
=
G(q)
G&)
= q-"
Selecting r l = I,BI = 0, P = 1 and minimizing the cost function (€0,the same control law as given by Garcia and Morari (1985b) can be obtained. Therefore, by using an appropriate precompensator, the procedure of computation control law is much simpler than that given by Garcia and Morari (1985131, because of the avoidance of the Gaussian elimination algorithm. 7. Conclusions
Two procedures of time delay factorization, one for weakening the requirement of the prior information about the system in MIMO adaptive controller design and the other for design of a realizable dynamic decoupled internal model controller, are given. It is proved that, in spite of
Ind. Eng. Chem. Res., Vol. 31,No. 5, 1992 1421 the two procedures for different purposes, the results of time delay factorization are the same for any MIMO systems. It is demonstrated that any time delay unbalanced system can be transformed to a time delay balanced one by using an appropriate diagonal proper precompensator, whose design procedure is also suggested. It is shown that, when the time delay of the system is unbalanced and a decoupled system response is sought, the design procedure of the internal model controller by the model predictive method is much simplified if it is first transformed to a time delay balanced one. Although the role of precompensator is only shown in the design decoupled internal model controller by the model predictive method, the results concerning time delays have an important practical sense in designs of decoupled adaptive controllers, internal model controllers, model predictive controllers, and other controllers based on an input-output model.
Appendix. Proof of Lemma 3 For simplicity, this proof is carried out for a three-dimensional transfer matrix, for higher dimensions the proof is similar except more complex. Express GJq) as
1
hll(q)
4d12&2(q) 4d13&(q)
G l q ) = qdz1h21(q) 4&) qd31h31(q) qdSh32(q)
qdZ3&3(q)
1
qd33h33(q)
assuming di; 2 0, i = 1, 2, 3, j = 1, 2. It is obvious that GJq) has the same property as G(q), that the sum of diagonal elements of the time delay structure matrix is the least among all arrangements. By using lemma 1, G--'(q) can be expressed as
1
h11(q)
=
G-l(q)
q+i,
,&q)
q413'+d33'h 13(q)
qd31&1(q)
1
4d32h32(4) h33(q)
gll(q)
GT:(q) = 4-T21821(q)
1
qd23-+d33h 23h)
qdZ1h21(q) h22(q)
q"12gi2(q)
q-"38i3(S)
g22(4)
q-Q3823(4)
q-r31'-d33g31 (4) q-T32-d33g32(g)
g&)
By contradiction: Assume d33-> dlc,i.e. -d13d33- > 0 According to lemma 2, we have -731- - d33- or -732- - d3< > 0 But if -731- - d33-> 0, then
+
h11(s) g l l ( q )
+ q-d12--r21-h12(q)g21(q) + g31(q)# 1
q-d13~+d33~-r31--d33~~13(q)
and if
-732-
- d33-> 0, then
q-r12-hl(q) g 1 2 k )
+ qd"h12(4)
g22M
+
q-d~3-+dss--rs~--dss-h13(4) g32(q)#
0
Further G-i(q) G-i-l(q) #
I
Q.E.D. Literature Cited Garcia, C. E.; Morari, M. Internal Model Control. 1. A Unifying Review and Some New Results. Znd. Eng. Chem. Process Des. Deu. 1982,21, 308. Garcia, C. E.; Morari, M. Internal Model Control. 2. Design Procedure for Multivariable. Znd. Ena. - Chem. Process Des. Dev. 1985a, 24,472. Garcia, C. E.; Morari, M. Internal Model Control. 3. Multivariable Control Law ComDutation and Tuning Guidelines. Znd. Ena. Chem. Process Des. Deu. 198513,24,481. Holt, B. R;Morari, M. Design of Resilient Processing Plants-V The Effect of Dead Time on Dynamic Resilience. Chem. Eng. Sci. 1985,40,1229. Ichikawa, K. Multivariable Adaptive Control As A Natural Extension of SISO Adaptive Control. Znt. J. Control 1987, 46, 2056. Singh, R. P.; Narendra, K. S. Prior Information in The Design of Multivariable Adaptive Controllen. ZEEE Trans. Autom. Control 1984, AC-29, 1108. Wolovich, W. A.; Falb, P. L. Invariants and Canonical Forma Under Dynamic Compensation. SZAM J. Control Optim. 1976,14,996.
Xiaoming Luo,* Youxian Sun, Chunhui Zhou
and factor GJq) into then
Institute of Industrial Process Control, Zhejang University 310027 Hangzhou, People's Republic of China Received for review July 15,1991 Revised manuscript received January 15,1992 Accepted January 30,1992
