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On the design and properties of multivariable dead time compensators
0005-1098/83/030255 10503.00/0 PergamonPressLtd. 1983InternationalFederationof AutomaticControl
Automatica, Vol. 19. No. 3, pp. 255-264, 1983 Printedin GreatBritain.
On the Design and Properties of Multivariable Dead Time Compensators* Z. J. P A L M O R f and Y. HALEVIf
The notion of practical stability for MIMO systems with multiple dead times controlled by dead time compensators, defined and used with diagonal dominance theory, provides analytical desion aids for these systems. Key Words--Time lag system; delays; multivariable control systems; dead time compensators; control system synthesis; stability; models; controllers.
Abstract--The design problem of multivariable dead time compensators for MIMO systems with multiple dead times is considered. Analytical design aids which take into account the stability and sensitivity properties of these systems are presented. Conditions for practical stability, estimates on allowable tolerances in process models and methods for determining controller gains are derived. Based upon sufficient conditions, these methods lead to conservative gains. The extent of conservation is related to the amount of computational effort involved in each method and to the available plant information. It is shown that the type of dead time compensator treated here cannot be applied to unstable processes. Diagonal dominance theory is used to derive some of the results. Illustrative examples demonstrate the main results.
1979, 1982) that some optimal stochastic control schemes for DT processes contain DTCs. The main role of the DTC is in converting the infinite spectrum design problem to that of a finite spectrum one through the removal of all DTs from the characteristic equation. This has two obvious advantages. It allows for tighter control, since higher gains may be used and, second, design techniques suitable for MIMO systems without DTs may be applied instead of the scarce design procedures for systems with DTs. It has been shown (Palmor, 1980), however, that in the cases of SISO systems a straightforward application of design techniques suitable for systems with no DTs may lead, under practical circumstances, to wrong conclusions regarding the properties of the system so designed. Therefore, special precautions must be added into the design. In extreme situations, for example, systems which appear to be stable with large stability margins may loose stability under infinitesimal parameter changes. The design of DTC controllers for MIMO systems is subject to similar though more complex difficulties. Design techniques which take into account the special properties of such systems are, therefore, desirable. This paper is a first step towards this goal. The results described herein provide analytical means for designing DTC controllers for MIMO systems with multiple DTs. Among these are included conditions for the socalled practical stability of the design, estimates for allowable tolerances in models used in the DTC and ways to determine suitable controller gains. Another related problem which seems to have been overlooked to some extent, is the application of the DTC controller to open-loop unstable processes. This question is resolved in a formal manner in Section 3, following a brief description of the DTC control scheme given in Section 2. A necessary
1. INTRODUCTION
This paper is concerned with the design of MIMO system with multiple dead times (DTs). The presence of considerable DTs in control loops of many industrial processes often prevents effective control from being achieved with these processes. A significant improvement in the control in such cases may be obtained by the incorporation of the so-called dead time compensator (DTC) in the control scheme. The DTC was first suggested by Smith (1957, 1959) for SISO systems with single DT. Later Alevisakis and Seborg (1973, 1974) extended the DTC to MIMO systems with a single DT, and recently Ogunnaike and Ray (1979) have further extended the DTC to cope with MIMO processes with multiple DTs. These authors have also demonstrated through simulation studies that the well known improvements offered by the DTC in SISO systems carry over to the MIMO cases as well. Although the DTC has been developed on purely deterministic grounds, it has been shown (Palmor, * Received 12 October 1981; revised 22 October 1982. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor P. Dorato under the direction of editor B. D. O. Anderson. t Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, Israel. Aoz 19/3-c
255
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Z.J. PALMORand Y. HALEVI
condition for practically stable designs is stated in Section 4. Estimates on allowable mismatches in models used in the DTC are derived in Section 5 with the aid of diagonal dominance theory. A way to determine controller gains is presented in Section 6. We summarize our conclusion in Section 7.
y_..
2. DTC CONTROLLERS FOR MIMO PROCESSES: DESCRIPTION AND NOTATION
Linear time-invariant processes with input dead times may be represented in both the time and the frequency domains. Their representation in the time domain usually takes the form
Xl = A x l + ~Biu(t - ~i)
(la)
i
=
(Ib)
+ y Oju(t J
where x l ~ R " is the state vector, u ~ R " is a vector of manipulated variables and y l e R " is the output vector. The real matrices, A, B, C and Dj are assumed to have appropriate dimensions. By taking the Laplace transform of (1) the corresponding (r x m) transfer function matrix Gp(s), relating y(s) to u(s) will have the following form
Gp(s) = C(sl - A)-' ~ Bi e-"s + E Dje-aJS. (2) •
j
The DTC contains models of the process and of other elements in the control loops. Variables and parameters associated with the model will be denoted by ( A ) above the corresponding symbols used for the true functions or parameter values. thus, for example, (Jp(S) denotes the model used for Gp(s) and dq, is the estimated value o f ~ and so forth. Although the general descriptions given in (1) and (2) will be used to derive the result in the next section, the form of the elements of Gp(s) encountered in engineering practice is slightly simpler, especially when modelling is carried out experimentally. In these cases, a typical element of Gp(s) is usually given by
(Gp(s))ij = 9o(s)e -°'is,
i = 1. . . . . r; j = 1. . . . . m
(3) where go(s) is a rational transfer function. A MIMO feedback control system, with a DTC controller, is shown in Fig. 1. H(s) represents the (r x r) diagonal transfer function matrix of the sensors or control elements in the feedback. Each element of H(s) may consist of a rational transfer function, h,(s), and of a DT. Thus, it may be written
FIG. 1. The multivariable DTC.
As noted above, (Jp(s) and/q(s) are models of Gp(s) and H(s), respectively, used in the dead time compensation scheme. Gpo(S) and Ho(s) are Gp(s) and H(s) respectively without DTs. Go(s) is the (mx r) transfer function of the primary controller. In process control applications, the elements Oco(s) of G~(s) are conventional P, PI or PID controllers and, in many applications, G~(s) is taken to be a diagonal matrix. In this situation, and because we are concerned in output feedback control m, the number of inputs and r, the number of outputs are equal. The role of the DTC can be readily understood if the overall closed-loop transfer function matrix, relating the set point vector, v(s), to y(s), is computed. This relation is given by (for convenience, all arguments in all matrices are dropped)
y = Gp(I + GcHoGpo )- XGcv.
(5)
Equation (5) has been derived under the assumption that models are perfect. It holds for the more general process descriptions, like the one in (2), as well as for the simplified ones. It is seen that all DTs have been removed from the characteristic equation. This is the basic advantage of the DTC scheme and the characteristic from which its potential improvements arise. The design of Go(s), however, cannot be based upon (5) since it does not provide a full description of the stability properties of systems with DTC under practical circumstances where models do not match exactly process units. Systems designed with large stability margins using (5) may loose stability when infinitesimal mismatches occur. To motivate our discussion, we illustrate this point with a simple example. Means by which this phenomenon may be detected are given in Section 4.
Example 1. Let Gc and (Jp be given by
ocE 1 o21
as
(H(s))i = h,~s)e -r's,
i = 1, 2 . . . . . r.
(4)
Fe
+ l) 1
G" = Le-°25'l(s + I) e-°75'l(s + I) :/~ = 12.
Multivariable dead time compensators When the model dp used in the DTC exactly matches Gp, the characteristic equation of the overall closed loop is given by
257
Y2 -~ CX2 + Z D j u J
-
-
(9)
~Dju(t - fir)" J
Therefore, the control u is
det(I + HoGpoQ ) = Is2(1 + K1)
u = - K ( y 1 + Y2) = - K C ( x l + x2) - K ~ D j u
+ s(2 + K2)(1 + K~) + 1 + K~ + K2]/(1 + s) 2 = 0. (6) The roots of (6) are clearly in the L H P for any positive K~ and K 2. On the other hand, if the DT, 0~1, changes slightly such that 0 1 1 - ~11 = A0tl # 0, the system loses stability even for moderate gains. To see this, let us choose K1 = K2 = 2. The characteristic equation under these circumstances becomes
J or
u = --(I + K 2 D j ) - I K C ( x 1 + x2). $
Upon substituting (10) into (la) and (8) and taking the Laplace transforms (assuming zero initial conditions), we obtain t
sxl(s) = (A - ~ Z i e-='~)xl(s)
~,Zi
-
i
1 - [2e-°'s~(1 - e-'X°'-"S)(s + 1)(s + 3)]/ (3s 2 + 12s + 5) = 0
(7)
(10)
i
e-="x2(s)
sx2(s) = ( - ~,Z, + ~ Z , e-=")xl(s) i
i
and from the Nyquist plot of (7), shown in Fig. 2, it is concluded that the system is in fact unstable for A01t = 0.05, only 0.1 of the nominal DT.
+ ( A - ~i Zi + ~Z,e-'")x2(s ) (11) where
l.rrn
Zi = Bi(I + K~.Dj)- 1KC. 3
-I-0
The modes of the overall system are thus the roots of
-0"5
sI -- A +
~Zte -*,"
det
~ Z , - ~ Z,e
TM
'~"Zie-=" 1 = 0.
sI- A + ~Z,-
(12)
~Z#-='~J
FIG. 2. N y q u i s t plot for Example 1 with A011 = 0.05.
By premultiplying and postmultiplying (12) by the non-singular matrices F and R respectively, where 3. D T C W I T H O P E N - L O O P U N S T A B L E P R O C E S S E S
In this section, it is shown that the DTC presented in the previous section cannot be applied to openloop unstable processes. The general time domain representation of the process, given by (1), is used. For simplicity, but with no loss of generality, we shall assume that H(s) = I and that the control is static; that is Gc = K. The realization of the feedback loop around the primary controller (see Fig. 1) is given by £2 = Ax2 + ~,Biu -- ~,B~u(t -- cq) t
i
(8)
and "'I-i it is obtained that (12) is equivalent to
,,.ot[sz;.., sl
1
- A + I;Zi] = 0
(13)
or
det (sl - A) det (sI - A + ~ Z , ) = 0. i
(14)
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Z.J. PALMORand Y. HALEVI
Therefore we have proved the following:
Theorem 1. The closed-loop system with a DTC [equations (1) and (8)--(10)] has 2n modes, n of which are those of the open-loop process; and the other n are the modes of an equivalent closed-loop system without DTs. Remark 1. An immediate conclusion from the above is that the DTC discussed here cannot be used with unstable processes. This clearly applies to the well-known Smith DTC (Smith, 1957, 1959), as has been noted by Gawthrop (1977). Remark 2. While the servomechanism properties of conventional controllers are significantly enhanced by the addition of the DTC, the improvements in the regulatory properties do not appear to be as good. Theorem 1 clarifies the reasons for this. When set-point changes occur, the open-loop modes are not excited and the system behaves according to the n modes of the equivalent system without DTs. On the other hand, when input disturbances (denoted by d in Fig. 1) enter the system, at least some of the n open-loop modes are excited too. Any slow open-loop mode will, therefore, influence the response in these cases. A way to improve on this is to introduce a special type of feedforward control (Palmor and Powers, 1981). Remark 3. A different DTC, which utilizes statefeedback control and which may be applied to unstable processes, appears in a paper by Manitius and Olbrot (1979). Its origin may be traced to optimal control (Kleinman, 1969; Donoghue, 1977; Kwon and Pearson, 1981). Questions related to the performance and implementation of this DTC are, however, still open and are currently under investigation. State observer designs for systems with measurement delays have been recently suggested (Olbrot, 1981). 4. STABILITYOF SYSTEMSWITH DTC CONTROLLERS In this section we assume, as is always the case in practice, that models do not exactly match the process. We call these cases non-ideal as opposed to the ideal cases discussed in the previous section. It is assumed that the processes are adequately represented by (3), the measurement devices as well as other elements in the feedback loops by (4), and that Gc(s) is diagonal so that the number of inputs and outputs are the same (i.e. m = r; r will be used to denote both). Let/a be a vector consisting of all parameters contained in the transfer matrices G~(s) and H(s). The elements of # are thus the various coefficients of the rational transfer functions in both matrices as well as of the dead times. Similarly,/~
is a vector consisting of all parameters in the models Gp(s) and H(s). The dimensions of # and/J are taken to be equal. The vector ~ can be augmented by zeros if necessary. Such cases may arise, for example, when low-order models are used. In the ideal cases /~ =/J. In the non-ideal cases the differences # - / J is denoted A#. The closed-loop transfer-matrix from v(s) to y(s) is given by G(s, #, IJ). Initially we have:
Definition. A control system with a DTC is practically stable (p.s.) if (a) it is asymptotically stable in the ideal case [i.e. all poles of G(s, #, ~) are in the open LHP], and (b) there exists a real number 6 > 0, for which all poles of G(s, # + A/~,/J) are in the open L H P for all A/t in 11AktII < ~. Similarly, a control system with a DTC is practically unstable if it satisfies condition (a) but there is no 6 > 0 for which condition (b) can be satisifed. In certain circumstances, systems containing DTC may be practically unstable, as has been demonstrated in Example 1. Next, a necessary condition is derived for stability of systems with multiple input and measurement DTs and with a DTC. For conciseness of the results to follow, we define the (r x r) matrix S(s) S(s)
(i, +
1
(15)
which is the inverse of the return difference or the sensitivity matrix of the system with no DTs in the ideal case. Since det(S-l(s)) is the characteristic polynomial in the ideal case, it may be assumed that all its roots are in the LHP. The (i, j) element of S(s) is denoted by %. The difference between the products HGpGc and H(~pGc is denoted by matrix A(s) A(s) -& (HGp- ISld,)Gc.
(16)
With definitions (15) and (16) one has:
Theorem 2. For a system with a DTC to be practically stable (p.s.), it is necessary that lim Isjigc~hi~ij(s)l < ½, Yi, j.
(17)
8~c~
Proof: The characteristic equation of the overall system in the non-ideal cases is det S - 1 det (I + SA) = 0.
(18)
Since det(S-1) has been assumed to have all its roots in LHP, the proof continues with det (I + SA). Assume that (17) is not satisfied for some arbitrary i and j (say i = l, j = q), but that the closed-loop
Multivariable dead time compensators system is p.s. We shall establish a contradiction here. For this, consider the following case
air = Ou, Vi, j
(19)
~tq - &zq = A %
(20)
259
Example 2. Condition (17) is applied to the system in Example 1. For i = j = 1 we have (s + 1 + K2)(s + 1) Six = (s + 1 + K2)(s + 1)(1 + K1) - K I K 2 ' 9cl = K1,/~1 = 1, 91x = 1
but hence
~tij = &ij, Vi, j,
i v~ l, j v~ q
limlslloc1910xll = K1/(1 + K1) 8 ~
where 0to --4 y~ + 0 o.
(21)
With (19) and (20), (18) reduces to det (I + SA) = 1 + sqlocqhtgl~ e -~'~ (e -~''~ -- 1). (22) Given that sql, Oc~,/tl and 01~ are analytic in the RHP, it has been shown (Palmor, 1980) that (22) has R H P roots for all A~q¢ ¢ 0 when (17) is not satisfied. Since the parameter A~z¢ is a component of the vector A/~ it is clear that there is no 6 > 0 in this case. Consequently a hyperspherical region IIArt II < ~ does not exist and the system is practically unstable contradicting our assumption.
Remark 4. Throughout this section, it is assumed that Gc is a diagonal matrix. If, however, G~ is a full (m x r) matrix, then (17) is replaced by lim It~iOijpji(s)l < ½, Vi, j
(23)
S--b oO
where Pji is an element of the ( m x r) matrix P(s) given by
P(s) = (I + 6cB0d.o)- 1~.
0t3
and the necessary condition for p.s. implies that K1/(1 + Kx)<½ from which K 1 < 1. In Example 1 both gains were taken to be 2 for which Kx/(1 + K1) = ~ > ½, and the overall system is indeed practically unstable as has been confirmed in Example 1. It is thus seen that the p.s. condition limits drastically the gain space. These limits could not have been predicted through the ideal cases. It is worthwhile noting that, when the two inputs are interchanged, no restriction on the gains will be imposed by condition (17) (p.s. problems arise in these cases, of proportional controllers, if the single proper element in Gpo is on the diagonal). When the overall system is practically stable, a way to determine controller gains in the non-ideal cases is clearly desired. The next theorem provides conditions for stability in the non-ideal cases for any mismatches in the DTs. Determination of controller gains by this theorem is straightforward, though the resulting gains are very conservative. A less conservative determination of gains through the utilization of diagonal dominance theory is developed in Section 6. Since the non-ideal case is dealt with, it is assumed that 0u ¢: 9o, [ii# hi and A~ij # 0 for all i and j. Let o~ = Im (s) and define
(24)
Remark 5. Each of the products sjioJiiOu(s) must be at least proper to satisfy condition (17). This is the case when all products are strictly proper. On the other hand, when one of the products is improper, the system is practically unstable• When it is proper, stability limits in the gain space may be found from condition (17). Such cases may arise when, for example, G~ consists of P or PI controllers and one or more of the go are proper (see Example 2). When G~ contains PD controllers and at least one of the O~j is proper, condition (17) will be violated, In some cases, cancellations of the higher powers in the denominator polynomial of s~j may occur resulting in practical instabilities. Such situations are common in some optimal stochastic designs (Palmor, 1982)• The next example illustrates some of the above points.
p = m,j,ax [guh'(J°~)[ ~
(25)
Then we have:
Theorem 3. The overall system is stable in the nonideal cases for any Aau(Vi, j) in - ~tu < Acto < aij if (2 + p)kCk(jto ) < 1, Vo9 > 0
(26)
k=l
where Ck(s) denotes the sum of the absolute values of all the products in all the principal minors of order k of the matrix GcHo t~po(S).
Proof: Denote
6r(s) --- ~odpo -/-/alp + t-/6p.
(27)
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Z.J. PALMORand Y. HALEVI
With (27) the characteristic equation of the overall system in the non-ideal case is given by det (I + G~Gr) = 0.
(28)
Let
q'J(s) zxl-e-;'''~ --
1 ~gijhi --
- a~, .s'~ e
J
)
.
(29)
The (i, j) element of GcGr is (GcGr)ij = gci~lgijqij
H = ] 2. Assume that the nominal values for process gain and time constants are: ~1 = k22 = 1, ~12 = k21 = 0.5 and t l l -- T22 = 1, t 1 2 --- "~21 = 2. F u r t h e r assume that the estimation of process gains may be off by + 10% and that the time constants by + 20%. Based on this data, we wish to estimate the control gain K which guarantees stability for any AOo. Note that in the ideal case, the system is stable for any positive K. From the given data, the following is found
(30) P =
from (25) and (29) we obtain Iqu(J09)l < 2 + p
(31)
2K C1(j09) - (co2 + 1)½ c
hence
[(GcGr(j09))ul < (2 + p)lgciO,jli,(j09)l.
(32)
Equation (28) can be written in the following form (Gantmacher, 1959)
kijlj:cij09 + 11 7 - - - - - - 7 = 1.375 i,j,co -~ijlJZij09+ 1 1
max
K 2 009) = - - - ¢ - f
092
K2/4 + - - .
4092 + 1
Substituting these into (26) we obtain K < 0.12. If #u = gu, it is found from Corollary 4 that the allowable gain may be slightly increased to K = 0.133.
[]
Remark 6. The determination of Ck(j09 ) in Theorem 3 and Corollary 4 is straightforward since it consists only of known functions like Gc(s), t~po(S) and Ho(s). The evaluation of p in (25) is somewhat more involved, though the information needed is similar to the one used frequently in classical control, namely, open-loop frequency response. If frequency response test between each input and each output, with all loops open, can be performed, then the determination of p is quite simple. In cases where the differences between giih~ and the corresponding models ~ijrai arise only from small time constants negligible in the course of mathematical modelling, then p is unity and (26) reduces to (36). When the actual functions guh~ are unavailable and frequency response test cannot be done, then the best that can be done to estimate p is to perturb the gains and time-constants from their nominal values in a fashion similar to Example 3, and compute the ratios IgijhJ~ijtli(j09)l. Such calculation will be usually carried out on a computer especially for high-order systems. The amount of parameter variations taken depends then on the accuracy of the models and possible process variations during operation. Having determined p one can then apply in a straightforward manner criterion (26) [or (36)] to check the stability of a system for a given set of control gains.
Example 3. Let Gp(s) be a (2 x 2) matrix with gu(s) = k u e-°'Js/(tos + 1), Go(s) = diag (K) and
While the determination of controller gains by
det (I + GcGr) = 1 + ~ Crk(S) = 0 k=l
(33)
where C Tk is the sum of all principal minors of order k in GcGr. Z~,=1Crk(s) is analytic in the RHP. All zeros of (33) will be in the LHP, according to Rouche's theorem, if r
k~l Crk(J09) < 1, V09 > 0.
(34)
From (32) and the definitions of CTk and Ck we obtain
~=1Crk(J09) < k
(2 + p)kCk(j09 )
(35)
k=l
and condition (26), then follows from (34). [] If mismatches are assumed to be only in the DTs, condition (26) reduces to:
Corollary 4. The overall system is stable for any mismatches Actu in the DTs ( - &u < A~ij < cqj) if ~. 3kck(j09) < 1, V09 > 0.
(36)
k=l
Proof: When guhi = gifii, P in (25) is 1.
5. ESTIMATION OF ALLOWABLE MISMATCHES
Multivariable dead time compensators Theorem 3 is simple and straightforward, the resulting gains are often very conservative since stability is guaranteed for any DT values and for any mismatches in them. With the aid of diagonal dominance theory (Rosenbrock, 1974), we may derive results which are helpful in checking the stability of a given system in the non-ideal cases and in estimating the allowable tolerances in the mismatches within which the stability of the system is maintained. In the next section, diagonal dominance will be used to determine controller gains. Let D be the Nyquist path in the s-plance consisting of the imaginary axis and a semi-circle of a large radius in the RHP. Further, define the transfer functions Mi(s) and IGi(s ) as follows: Mi(s) = gc,~,~i/(1 + g~i~ifli),
261
Q = diag [(1 + ga~it~,)-i].
(40)
If L is row dominant (column dominant) on D, so is QL (LQ). Observe that the elements on the diagonal of QL may be expressed in the following form (QL)u = (LQ)ii = 1 - Mi e-~"SlGi •
(41)
We proceed with the row dominance noting that the case of column dominance is completely analogous. Let det (QL) map D into FQL,and similarly let (QL), map D into Fi. Let FQL and F i encircle the origin NQ,. and Ni times, respectively. Clearly, if QL is diagonally dominant on D, then
i= 1,...,r
(42)
lq i = NQL. i=1
and IGi(s) = 1 - gi~hi e-a'"s/(~,/t~),
i = 1. . . . . r. (37)
The function M~(s) is the transfer function seen between the ith input and the corresponding ith measured output in the ideal case with no DTs and with all other loops open. In practice, it is often required that stability is maintained in the ith loop when all the other loops open. It is, therefore, assumed that the Mi(s) is analytic in the RHP. The function IGi(s) measures the amount of mismatching or ignorance in g,, hi as well as in a,. When the model exactly matches the process then IG~(s) = 0. Since it is required that 0,h~ be analytic in the R H P it follows that MJGi(s) (i = 1..... r) is analytic in the RHP. Recall that the characteristic equation in the non-ideal cases is given by (28) and denote the matrix (I + GcGT) by L. In the sequel we apply the definitions of diagonal dominancy suggested by Rosenbrock (1974, p. 142). Theorem 5. Let L be diagonally dominant on D. Let - M~e- ~,,sIG~ map D into F,(i = 1,2 ..... r) which encircles the point ( - 1, 0) N~ times. Then the overall closed-loop system is asymptotically stable iff
N~ = 0.
(38)
i=1
Proof: Let det (L) map D into FL which encircles the origin N~_ times. It is well known from stability theory that when all poles of GcGr are in the LHP, all roots of det (L) = 0 are in the LHP if and only if NL = 0. Now, let the (r x r) matrix Q be defined as
(39)
Since all poles and zeros of det (Q) are in the LHP, it follows that NL = NaL = ~ Ni.
(43)
1=1
Furthermore, notice that F~ encircles the origin as often as Fi encircles the point ( - 1, 0). Hence I~ i = ~ N i i=1
(44)
i=1
and (38) follows at once from (44), (43), and (39). [] A direct consequence of Theorem 5, which will be more useful for subsequent results, is the following: Corollary 6. Let L be diagonally dominant on D. Then the overall closed-loop system is asymptotically stable if N i=O,
i = 1. . . . . r.
(45)
From Corollary 6, several results which allow us to estimate the tolerances in the mismatches may be derived. Recall that the mismatches between the model and process are contained in the matrix Gr. different mismatches lead to different GT matrices. In the following, we assume that there exist some set of Gr matrices (denoted {Gr}), including Gr of the ideal case, for which L is diagonally dominant on D. Clearly, we assume that the design is p.s. First, we have: Theorem 7. Let L be diagonally dominant on D for some {Gr}. Then the overall closed-loop system is asymptotically stable for all GT in {Gr} for which IMi(jto)lGi(jo~)l < 1; Va~ > 0;
i = 1.... , r.
(46)
262
Z.J. PALMORand Y. HALEV!
Proof: From (46) and the analyticity of Mi(s)IGi(s) in the RHP, it follows with the aid of the PhragmenLindelof theorem that IMi(s)IG~(s)l <1 in the RHP. Hence, from Rouche's theorem, (QL(s))ii, (i = l, ..., r) in (41) is analytic in the RHP. This implies that N~ = 0 (i = 1, ..., r) and the result follows at once from Corollary 6. [] Example 4. Consider the process Gp(s) in Example 3 with all DTs equal to unity. Both controller gains are taken to be K = 3. It is required to estimate the allowable tolerances in the time constants z~j within which stability is maintained. In this case, IG,(s) = Az,s/[(1 + Az,s) + 1] and Mii(s) = 3/(s + 4) and condition (46) is clearly satisfied for at least all Az;~ in the range: A~, > -0.7. However, L is found to be diagonally dominant for Azii > - 0 . 8 and AzI2 = Az2x > - 1 . It is, therefore, concluded from Theorem 7 that stability is guaranteed for Az, > - 0 . 7 and Azl2 = A~21 > -- 1. Theorem 7 is now exploited to obtain estimates for allowable mismatches in two particular cases. First consider the case where mismatches in cq~ are likely to exist while all other parameters in the diagonal elements of the HGp are assumed to be perfectly known and fixed. However, mismatches in the other parameters in the off diagonal elements of the HGp are allowed. Under these circumstances, IGi(s ) takes the form 1Gi(s) = 1 - e -a'''~,
i = 1. . . . . r.
(46) or (48) is denoted by {Gr}', then Theorem 7 (Corollary 8) guarantees stability only for {Gr}'n
Remark 8. If IMi(jw)l < ½, Wo > 0 then no restriction is imposed by (48) on A~,. Remark 9. Notice that while the corollary discusses mismatches in the DTs of the diagonal elements in HGp, the mismatches allowed in the offdiagonal elements are not restricted to be in the DTs only. Example 5. Consider the system of Example 4. We wish to estimate the DT tolerances within which the system will remain stable. It has been found that L in this case is diagonally dominant for those { Gr} for which IAOolOijl < 0.5. Variations in the other parameters in the off-diagonal elements of Gp were not considered. M 1 and M 2 are both 3/(s + 4). Therefore, (Mi)m = 0.75 and toOl = x/20. From (48) we obtain: AO, < 0.327 (i = 1, 2). Following Remark 7, it is concluded that stability is retained for {Gr}, for which AOii < 0.327 (i = 1, 2) and A012 (= A021) <0.5. The second particular case arises when mismatches in process and sensor gains are considered. g~j is written now as ko#'o where ko is the steady state gain of #~j. The difference k~j - l~ij is denoted by Ak w IG~ now takes the form
(47)
I Gi(s) = - Akii/fqi. Denote maxlMi(joJ)l by (Mi)m and let O9oi be the frequency above which IMi(jog)l < ½. Then from Theorem 7 we have:
Corollary 8. Let L be diagonally dominant on D for some {GT}. Let the mismatches in the diagonal elements of HGp be only in the DTs, ~,. Then the overall closed-loop system is asymptotically stable for all Gr in {Gr} for which Aaii < (2/O9oi) sin- 1 [2(Mi),.]- 1, i = 1, ..., r. (48) Proof: Condition (46) is clearly satisfied for the IGi in (47) for co > O~o~.Now, as [MilGi(jco)[ < (Mi)m2 sin (Actiiog/2), Vco > 0 it is required that (Mi)m2 sin (A~ilog/2) < 1, Vo9 < Ogol. Substituting O9o~for ~o in the left-hand side of the last inequality completes the proof. []
Remark 7. If the set of all G r matrices arising from
(49)
Corollary 9. Let L be diagonally dominant on D for some (Gr}. Let mismatches in the diagonal elements of Gp(s) be only in the gains k,. Then the overall closed-loop system is asymptotically stable for all Gr in { Gr } for which
IAk./~.l < (Mi)~ 1,
i = 1. . . . . r
(50)
Proof: From (46) we have IM(jog)llAk,/l¢iil < 1, Vo9 > 0 whence (50).
[]
Remark 10. Notice that mismatches in the offdiagonal elements of Gp are not restricted to gains only. Example 6. Let the process and controller be as in Example 5. Using Corollary 9, we shall estimate the gain tolerances Ak,. It is easy to verify that L is diagonally dominant for at least all GT consisting of Gp with IAko/fqj[ < 0.5. From (50) we obtain [Ak,[ < ~. Stability, however, may be guaranteed only for IAk~j/l~ijl < 0.5.
Multivariable dead time compensators hence
6. C O N T R O L L E R S G A I N D E T E R M I N A T I O N
While the estimation of model tolerances through the utilization of Theorem 7 and Corollaries 8 and 9 is quite simple, the search for {Gr}, for which L is diagonally dominant on D is quite tedious, especially due to the transcendental terms involved. This search may be avoided, in some cases, by suitable choices of controller gains which a priori guarantee diagonal dominance. The resulting gains will, in general, be conservative; but it turns out, in many cases, that they are much less conservative than the ones derived by means of Theorem 2. Once a suitable set of gains has been found, the results of the previous section may be effectively utilized. Since mismatches in DTs are usually the most destructive ones to stability, we shall base our derivation on them. It will be assumed that G~Gt,o is strictly proper and that none of its elements contain pure imaginary poles. As controller gains are sought, the transfer function Oct(s) is rewritten in a more explicit form: ga(s) = Kffa(s). Let fl denote the maximum mismatch in all uo, i.e. fl = m a x
i,j
263
I(L(jw)),ll <
T(~)lKio'a[~,Oo(jw)l, Vo~ > 0, i, j, i ~ j.
(53)
and I(L(j~o)),l > 1 - T(o~)lgff~[i~i(joJ)l, Va~>0, i = l . . . . . r.
(54)
For row dominance, it is required that I(L(jco)),l > ~ I(L(j~o))ol, j=l
Wo > 0, i = 1. . . . . r.
(55)
Substituting (53) and (54) into (55) we obtain
K~T(o~) ~ Ig'ag~0ij(jo~)l < 1, i = 1. . . . . r. jft
Now if K~ is selected such that
mO~lj
and define the function T(~o) as follows:
Ki < 1/lT(co) ~ Ig'ag~#~g(j~o)l], j=l
r(~o)
j'l + 2 sin (flco/2) 3
co < n/fl o~ > n / # '
(56)
then dominance is assured. Finite K~ can be always found since GcGr is assumed to be strictly proper. Equation (52) is obtained by replacing the product
We have:
Theorem 10. Let controller gains K~ be given by K i < 1/max[T(og) ~ Ig',il~i~o(jog)l], a}
V~o>0, i = l , . . . , r
(51)
by its maximum value in ~o > 0.
j=l i =
r(co)j ~= l Ig'Jzi~o(jco)l
1 ....
, r.
(52)
Then L is diagonally dominant on D.
Proof: The elements of the matrix L(s) are given by
Lii = 1 + Kig'ci[~igii~ii Lii = Kig'at~lgij~ij, i :~ j
[]
Remark 11. The theorem is developed for row dominance. For column dominance one considers the matrix L ' = I + GrQ. Condition (52) is then obtained with i and j interchanged. Example 7. Consider the G~(s) in Example 3 with 0 o = 1. Controllers gains are to be determined by (52). It is easy to obtain that 2
max T(r~) ~ Ih,#ij(jco)l co
j=l
where ~,s = 1 - e-='~s(1 - e-A='Js). Clearly I~o(jog)l ~ T(oJ)
is approximately 1.594. From (52) we get Ki < 0.63. Since in this case (Mt)m < ½, no restrictions on A~i arise from Corollary 8. Note that gains obtained here are larger than the corresponding ones derived in Example 3 (by Theorem 2) by approximately a factor of 5, but are considerably less than those used
264
Z.J. PALMOR and Y. HALEVI
in Example 5. This is the price we pay for avoiding the search for {GT}. 7. CONCLUSIONS
On one hand the cancellation of all DTs by the DTC scheme offers significant enhancements to the control of stable MIMO processes with multiple DTs, but, on the other hand, introduces considerable difficulties in the design problem. Designs based on the ideal cases are inadequate. For this reason, an attempt has been made in this paper to develop analytical aids for designing the DTC, which take into account the stability and sensitivity properties of these systems. A necessary condition for practical stability and further, means for estimating the allowable tolerances in process models were stated and proven. Also, methods for determining appropriate controller gains were presented. In addition, it was shown that the type of DTC treated in this paper cannot be applied to open-loop unstable processes. Some of the results rely rather heavily on the assumption that the characteristic equation of overall closed-loop is diagonally dominant in the ideal case. No attempt, however, to achieve diagonal dominance other than by choice of controller gains, has been made. This might be a fruitful area for further research. Acknowledgements--This work was supported by a grant from the Engineering Division of Taylor Instrument Company, Rochester, New York. Valuable discussions with Mr D. V. Powers of Taylor are greatly appreciated.
REFERENCES Alevisakis, G. and D. E. Seborg (1973). An extension of the Smith
predictor method to multivariable linear systems containing time delays. Int. J. Control, 3, 541. Alevisakis, G. and D. E. Seborg (1974). Control of multivariable system containing time delays using a multivariable Smith predictor. Chem. Engng Sci., 29, 373. Donoghue, J. F. (1977). A comparison of the Smith predictor and optimal design approaches for systems with delay in control. IEEE Trans. Ind. Electron. & Control Instrum., IECI24, 109. Gantmacher, F. R. (1959). The Theory of Matrices, Vol. I. Chelsea, New York, p. 70. Gawthrop, P. J. (1977). Some interpretation of self-tuning controllers, Proc. lEE, 124, 889. Kleinman, D. L. (1969). Optimal control of linear systems with time-delay and observation noise. IEEE Trans Aut. Control, AC-13, 524. Kwon, N. H. and A. E. Pearson (1980). Feedback stabilization of linear systems with delayed control. IEEE Trans Aut. Control, AC-25, 266. Manitius, A. Z. and A. W. Olbrot (1979). Finite spectrum assignment problem for systems having multiple time-delays. IEEE Trans Aut. Control, AC-24, 541. Ogunnaike, B. A. and N. H. Ray (1979). Multivariable controller design for linear systems having multiple time-delays. J. Am. Inst. Chem. Engrs, 25, 1043. Olbrot, A. W. (1981). Observability and observers for a class of linear systems with delays. IEEE Trans Aut. Control, AC-26, 513. Palmor, Z. J. (1979). Structural, stability and sensitivity properties of optimal stochastic control systems for dead time processes. TME-368, Technion, Israel Institute of Technology, Mechanical Engineering. Palmor, Z. J. (1980). Stability properties of Smith dead-time compensator controllers. Int. J. Control, 32, 937. Palmor, Z. J. (1982). Properties of optimal stochastic control systems with dead-time. Automatica, 18, 107. Palmor, Z. J. and D. V. Powers (1981). Feedforward properties of dead time compensator controllers. JACC, Charlottesville, Virginia. Rosenbrock, H. H. (1974). Computer-aided Control System Design. Academic Press. Smith, O. J. M. (1957). Closed control of loops with dead time. Chem. Engn 9 Pro9., 53, 217. Smith, O. J. M. (1959). A controller to overcome dead time. ISA J., 6, 28.
Automatica, Vol. 19. No. 3, pp. 255-264, 1983 Printedin GreatBritain.
On the Design and Properties of Multivariable Dead Time Compensators* Z. J. P A L M O R f and Y. HALEVIf
The notion of practical stability for MIMO systems with multiple dead times controlled by dead time compensators, defined and used with diagonal dominance theory, provides analytical desion aids for these systems. Key Words--Time lag system; delays; multivariable control systems; dead time compensators; control system synthesis; stability; models; controllers.
Abstract--The design problem of multivariable dead time compensators for MIMO systems with multiple dead times is considered. Analytical design aids which take into account the stability and sensitivity properties of these systems are presented. Conditions for practical stability, estimates on allowable tolerances in process models and methods for determining controller gains are derived. Based upon sufficient conditions, these methods lead to conservative gains. The extent of conservation is related to the amount of computational effort involved in each method and to the available plant information. It is shown that the type of dead time compensator treated here cannot be applied to unstable processes. Diagonal dominance theory is used to derive some of the results. Illustrative examples demonstrate the main results.
1979, 1982) that some optimal stochastic control schemes for DT processes contain DTCs. The main role of the DTC is in converting the infinite spectrum design problem to that of a finite spectrum one through the removal of all DTs from the characteristic equation. This has two obvious advantages. It allows for tighter control, since higher gains may be used and, second, design techniques suitable for MIMO systems without DTs may be applied instead of the scarce design procedures for systems with DTs. It has been shown (Palmor, 1980), however, that in the cases of SISO systems a straightforward application of design techniques suitable for systems with no DTs may lead, under practical circumstances, to wrong conclusions regarding the properties of the system so designed. Therefore, special precautions must be added into the design. In extreme situations, for example, systems which appear to be stable with large stability margins may loose stability under infinitesimal parameter changes. The design of DTC controllers for MIMO systems is subject to similar though more complex difficulties. Design techniques which take into account the special properties of such systems are, therefore, desirable. This paper is a first step towards this goal. The results described herein provide analytical means for designing DTC controllers for MIMO systems with multiple DTs. Among these are included conditions for the socalled practical stability of the design, estimates for allowable tolerances in models used in the DTC and ways to determine suitable controller gains. Another related problem which seems to have been overlooked to some extent, is the application of the DTC controller to open-loop unstable processes. This question is resolved in a formal manner in Section 3, following a brief description of the DTC control scheme given in Section 2. A necessary
1. INTRODUCTION
This paper is concerned with the design of MIMO system with multiple dead times (DTs). The presence of considerable DTs in control loops of many industrial processes often prevents effective control from being achieved with these processes. A significant improvement in the control in such cases may be obtained by the incorporation of the so-called dead time compensator (DTC) in the control scheme. The DTC was first suggested by Smith (1957, 1959) for SISO systems with single DT. Later Alevisakis and Seborg (1973, 1974) extended the DTC to MIMO systems with a single DT, and recently Ogunnaike and Ray (1979) have further extended the DTC to cope with MIMO processes with multiple DTs. These authors have also demonstrated through simulation studies that the well known improvements offered by the DTC in SISO systems carry over to the MIMO cases as well. Although the DTC has been developed on purely deterministic grounds, it has been shown (Palmor, * Received 12 October 1981; revised 22 October 1982. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor P. Dorato under the direction of editor B. D. O. Anderson. t Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, Israel. Aoz 19/3-c
255
256
Z.J. PALMORand Y. HALEVI
condition for practically stable designs is stated in Section 4. Estimates on allowable mismatches in models used in the DTC are derived in Section 5 with the aid of diagonal dominance theory. A way to determine controller gains is presented in Section 6. We summarize our conclusion in Section 7.
y_..
2. DTC CONTROLLERS FOR MIMO PROCESSES: DESCRIPTION AND NOTATION
Linear time-invariant processes with input dead times may be represented in both the time and the frequency domains. Their representation in the time domain usually takes the form
Xl = A x l + ~Biu(t - ~i)
(la)
i
=
(Ib)
+ y Oju(t J
where x l ~ R " is the state vector, u ~ R " is a vector of manipulated variables and y l e R " is the output vector. The real matrices, A, B, C and Dj are assumed to have appropriate dimensions. By taking the Laplace transform of (1) the corresponding (r x m) transfer function matrix Gp(s), relating y(s) to u(s) will have the following form
Gp(s) = C(sl - A)-' ~ Bi e-"s + E Dje-aJS. (2) •
j
The DTC contains models of the process and of other elements in the control loops. Variables and parameters associated with the model will be denoted by ( A ) above the corresponding symbols used for the true functions or parameter values. thus, for example, (Jp(S) denotes the model used for Gp(s) and dq, is the estimated value o f ~ and so forth. Although the general descriptions given in (1) and (2) will be used to derive the result in the next section, the form of the elements of Gp(s) encountered in engineering practice is slightly simpler, especially when modelling is carried out experimentally. In these cases, a typical element of Gp(s) is usually given by
(Gp(s))ij = 9o(s)e -°'is,
i = 1. . . . . r; j = 1. . . . . m
(3) where go(s) is a rational transfer function. A MIMO feedback control system, with a DTC controller, is shown in Fig. 1. H(s) represents the (r x r) diagonal transfer function matrix of the sensors or control elements in the feedback. Each element of H(s) may consist of a rational transfer function, h,(s), and of a DT. Thus, it may be written
FIG. 1. The multivariable DTC.
As noted above, (Jp(s) and/q(s) are models of Gp(s) and H(s), respectively, used in the dead time compensation scheme. Gpo(S) and Ho(s) are Gp(s) and H(s) respectively without DTs. Go(s) is the (mx r) transfer function of the primary controller. In process control applications, the elements Oco(s) of G~(s) are conventional P, PI or PID controllers and, in many applications, G~(s) is taken to be a diagonal matrix. In this situation, and because we are concerned in output feedback control m, the number of inputs and r, the number of outputs are equal. The role of the DTC can be readily understood if the overall closed-loop transfer function matrix, relating the set point vector, v(s), to y(s), is computed. This relation is given by (for convenience, all arguments in all matrices are dropped)
y = Gp(I + GcHoGpo )- XGcv.
(5)
Equation (5) has been derived under the assumption that models are perfect. It holds for the more general process descriptions, like the one in (2), as well as for the simplified ones. It is seen that all DTs have been removed from the characteristic equation. This is the basic advantage of the DTC scheme and the characteristic from which its potential improvements arise. The design of Go(s), however, cannot be based upon (5) since it does not provide a full description of the stability properties of systems with DTC under practical circumstances where models do not match exactly process units. Systems designed with large stability margins using (5) may loose stability when infinitesimal mismatches occur. To motivate our discussion, we illustrate this point with a simple example. Means by which this phenomenon may be detected are given in Section 4.
Example 1. Let Gc and (Jp be given by
ocE 1 o21
as
(H(s))i = h,~s)e -r's,
i = 1, 2 . . . . . r.
(4)
Fe
+ l) 1
G" = Le-°25'l(s + I) e-°75'l(s + I) :/~ = 12.
Multivariable dead time compensators When the model dp used in the DTC exactly matches Gp, the characteristic equation of the overall closed loop is given by
257
Y2 -~ CX2 + Z D j u J
-
-
(9)
~Dju(t - fir)" J
Therefore, the control u is
det(I + HoGpoQ ) = Is2(1 + K1)
u = - K ( y 1 + Y2) = - K C ( x l + x2) - K ~ D j u
+ s(2 + K2)(1 + K~) + 1 + K~ + K2]/(1 + s) 2 = 0. (6) The roots of (6) are clearly in the L H P for any positive K~ and K 2. On the other hand, if the DT, 0~1, changes slightly such that 0 1 1 - ~11 = A0tl # 0, the system loses stability even for moderate gains. To see this, let us choose K1 = K2 = 2. The characteristic equation under these circumstances becomes
J or
u = --(I + K 2 D j ) - I K C ( x 1 + x2). $
Upon substituting (10) into (la) and (8) and taking the Laplace transforms (assuming zero initial conditions), we obtain t
sxl(s) = (A - ~ Z i e-='~)xl(s)
~,Zi
-
i
1 - [2e-°'s~(1 - e-'X°'-"S)(s + 1)(s + 3)]/ (3s 2 + 12s + 5) = 0
(7)
(10)
i
e-="x2(s)
sx2(s) = ( - ~,Z, + ~ Z , e-=")xl(s) i
i
and from the Nyquist plot of (7), shown in Fig. 2, it is concluded that the system is in fact unstable for A01t = 0.05, only 0.1 of the nominal DT.
+ ( A - ~i Zi + ~Z,e-'")x2(s ) (11) where
l.rrn
Zi = Bi(I + K~.Dj)- 1KC. 3
-I-0
The modes of the overall system are thus the roots of
-0"5
sI -- A +
~Zte -*,"
det
~ Z , - ~ Z,e
TM
'~"Zie-=" 1 = 0.
sI- A + ~Z,-
(12)
~Z#-='~J
FIG. 2. N y q u i s t plot for Example 1 with A011 = 0.05.
By premultiplying and postmultiplying (12) by the non-singular matrices F and R respectively, where 3. D T C W I T H O P E N - L O O P U N S T A B L E P R O C E S S E S
In this section, it is shown that the DTC presented in the previous section cannot be applied to openloop unstable processes. The general time domain representation of the process, given by (1), is used. For simplicity, but with no loss of generality, we shall assume that H(s) = I and that the control is static; that is Gc = K. The realization of the feedback loop around the primary controller (see Fig. 1) is given by £2 = Ax2 + ~,Biu -- ~,B~u(t -- cq) t
i
(8)
and "'I-i it is obtained that (12) is equivalent to
,,.ot[sz;.., sl
1
- A + I;Zi] = 0
(13)
or
det (sl - A) det (sI - A + ~ Z , ) = 0. i
(14)
258
Z.J. PALMORand Y. HALEVI
Therefore we have proved the following:
Theorem 1. The closed-loop system with a DTC [equations (1) and (8)--(10)] has 2n modes, n of which are those of the open-loop process; and the other n are the modes of an equivalent closed-loop system without DTs. Remark 1. An immediate conclusion from the above is that the DTC discussed here cannot be used with unstable processes. This clearly applies to the well-known Smith DTC (Smith, 1957, 1959), as has been noted by Gawthrop (1977). Remark 2. While the servomechanism properties of conventional controllers are significantly enhanced by the addition of the DTC, the improvements in the regulatory properties do not appear to be as good. Theorem 1 clarifies the reasons for this. When set-point changes occur, the open-loop modes are not excited and the system behaves according to the n modes of the equivalent system without DTs. On the other hand, when input disturbances (denoted by d in Fig. 1) enter the system, at least some of the n open-loop modes are excited too. Any slow open-loop mode will, therefore, influence the response in these cases. A way to improve on this is to introduce a special type of feedforward control (Palmor and Powers, 1981). Remark 3. A different DTC, which utilizes statefeedback control and which may be applied to unstable processes, appears in a paper by Manitius and Olbrot (1979). Its origin may be traced to optimal control (Kleinman, 1969; Donoghue, 1977; Kwon and Pearson, 1981). Questions related to the performance and implementation of this DTC are, however, still open and are currently under investigation. State observer designs for systems with measurement delays have been recently suggested (Olbrot, 1981). 4. STABILITYOF SYSTEMSWITH DTC CONTROLLERS In this section we assume, as is always the case in practice, that models do not exactly match the process. We call these cases non-ideal as opposed to the ideal cases discussed in the previous section. It is assumed that the processes are adequately represented by (3), the measurement devices as well as other elements in the feedback loops by (4), and that Gc(s) is diagonal so that the number of inputs and outputs are the same (i.e. m = r; r will be used to denote both). Let/a be a vector consisting of all parameters contained in the transfer matrices G~(s) and H(s). The elements of # are thus the various coefficients of the rational transfer functions in both matrices as well as of the dead times. Similarly,/~
is a vector consisting of all parameters in the models Gp(s) and H(s). The dimensions of # and/J are taken to be equal. The vector ~ can be augmented by zeros if necessary. Such cases may arise, for example, when low-order models are used. In the ideal cases /~ =/J. In the non-ideal cases the differences # - / J is denoted A#. The closed-loop transfer-matrix from v(s) to y(s) is given by G(s, #, IJ). Initially we have:
Definition. A control system with a DTC is practically stable (p.s.) if (a) it is asymptotically stable in the ideal case [i.e. all poles of G(s, #, ~) are in the open LHP], and (b) there exists a real number 6 > 0, for which all poles of G(s, # + A/~,/J) are in the open L H P for all A/t in 11AktII < ~. Similarly, a control system with a DTC is practically unstable if it satisfies condition (a) but there is no 6 > 0 for which condition (b) can be satisifed. In certain circumstances, systems containing DTC may be practically unstable, as has been demonstrated in Example 1. Next, a necessary condition is derived for stability of systems with multiple input and measurement DTs and with a DTC. For conciseness of the results to follow, we define the (r x r) matrix S(s) S(s)
(i, +
1
(15)
which is the inverse of the return difference or the sensitivity matrix of the system with no DTs in the ideal case. Since det(S-l(s)) is the characteristic polynomial in the ideal case, it may be assumed that all its roots are in the LHP. The (i, j) element of S(s) is denoted by %. The difference between the products HGpGc and H(~pGc is denoted by matrix A(s) A(s) -& (HGp- ISld,)Gc.
(16)
With definitions (15) and (16) one has:
Theorem 2. For a system with a DTC to be practically stable (p.s.), it is necessary that lim Isjigc~hi~ij(s)l < ½, Yi, j.
(17)
8~c~
Proof: The characteristic equation of the overall system in the non-ideal cases is det S - 1 det (I + SA) = 0.
(18)
Since det(S-1) has been assumed to have all its roots in LHP, the proof continues with det (I + SA). Assume that (17) is not satisfied for some arbitrary i and j (say i = l, j = q), but that the closed-loop
Multivariable dead time compensators system is p.s. We shall establish a contradiction here. For this, consider the following case
air = Ou, Vi, j
(19)
~tq - &zq = A %
(20)
259
Example 2. Condition (17) is applied to the system in Example 1. For i = j = 1 we have (s + 1 + K2)(s + 1) Six = (s + 1 + K2)(s + 1)(1 + K1) - K I K 2 ' 9cl = K1,/~1 = 1, 91x = 1
but hence
~tij = &ij, Vi, j,
i v~ l, j v~ q
limlslloc1910xll = K1/(1 + K1) 8 ~
where 0to --4 y~ + 0 o.
(21)
With (19) and (20), (18) reduces to det (I + SA) = 1 + sqlocqhtgl~ e -~'~ (e -~''~ -- 1). (22) Given that sql, Oc~,/tl and 01~ are analytic in the RHP, it has been shown (Palmor, 1980) that (22) has R H P roots for all A~q¢ ¢ 0 when (17) is not satisfied. Since the parameter A~z¢ is a component of the vector A/~ it is clear that there is no 6 > 0 in this case. Consequently a hyperspherical region IIArt II < ~ does not exist and the system is practically unstable contradicting our assumption.
Remark 4. Throughout this section, it is assumed that Gc is a diagonal matrix. If, however, G~ is a full (m x r) matrix, then (17) is replaced by lim It~iOijpji(s)l < ½, Vi, j
(23)
S--b oO
where Pji is an element of the ( m x r) matrix P(s) given by
P(s) = (I + 6cB0d.o)- 1~.
0t3
and the necessary condition for p.s. implies that K1/(1 + Kx)<½ from which K 1 < 1. In Example 1 both gains were taken to be 2 for which Kx/(1 + K1) = ~ > ½, and the overall system is indeed practically unstable as has been confirmed in Example 1. It is thus seen that the p.s. condition limits drastically the gain space. These limits could not have been predicted through the ideal cases. It is worthwhile noting that, when the two inputs are interchanged, no restriction on the gains will be imposed by condition (17) (p.s. problems arise in these cases, of proportional controllers, if the single proper element in Gpo is on the diagonal). When the overall system is practically stable, a way to determine controller gains in the non-ideal cases is clearly desired. The next theorem provides conditions for stability in the non-ideal cases for any mismatches in the DTs. Determination of controller gains by this theorem is straightforward, though the resulting gains are very conservative. A less conservative determination of gains through the utilization of diagonal dominance theory is developed in Section 6. Since the non-ideal case is dealt with, it is assumed that 0u ¢: 9o, [ii# hi and A~ij # 0 for all i and j. Let o~ = Im (s) and define
(24)
Remark 5. Each of the products sjioJiiOu(s) must be at least proper to satisfy condition (17). This is the case when all products are strictly proper. On the other hand, when one of the products is improper, the system is practically unstable• When it is proper, stability limits in the gain space may be found from condition (17). Such cases may arise when, for example, G~ consists of P or PI controllers and one or more of the go are proper (see Example 2). When G~ contains PD controllers and at least one of the O~j is proper, condition (17) will be violated, In some cases, cancellations of the higher powers in the denominator polynomial of s~j may occur resulting in practical instabilities. Such situations are common in some optimal stochastic designs (Palmor, 1982)• The next example illustrates some of the above points.
p = m,j,ax [guh'(J°~)[ ~
(25)
Then we have:
Theorem 3. The overall system is stable in the nonideal cases for any Aau(Vi, j) in - ~tu < Acto < aij if (2 + p)kCk(jto ) < 1, Vo9 > 0
(26)
k=l
where Ck(s) denotes the sum of the absolute values of all the products in all the principal minors of order k of the matrix GcHo t~po(S).
Proof: Denote
6r(s) --- ~odpo -/-/alp + t-/6p.
(27)
260
Z.J. PALMORand Y. HALEVI
With (27) the characteristic equation of the overall system in the non-ideal case is given by det (I + G~Gr) = 0.
(28)
Let
q'J(s) zxl-e-;'''~ --
1 ~gijhi --
- a~, .s'~ e
J
)
.
(29)
The (i, j) element of GcGr is (GcGr)ij = gci~lgijqij
H = ] 2. Assume that the nominal values for process gain and time constants are: ~1 = k22 = 1, ~12 = k21 = 0.5 and t l l -- T22 = 1, t 1 2 --- "~21 = 2. F u r t h e r assume that the estimation of process gains may be off by + 10% and that the time constants by + 20%. Based on this data, we wish to estimate the control gain K which guarantees stability for any AOo. Note that in the ideal case, the system is stable for any positive K. From the given data, the following is found
(30) P =
from (25) and (29) we obtain Iqu(J09)l < 2 + p
(31)
2K C1(j09) - (co2 + 1)½ c
hence
[(GcGr(j09))ul < (2 + p)lgciO,jli,(j09)l.
(32)
Equation (28) can be written in the following form (Gantmacher, 1959)
kijlj:cij09 + 11 7 - - - - - - 7 = 1.375 i,j,co -~ijlJZij09+ 1 1
max
K 2 009) = - - - ¢ - f
092
K2/4 + - - .
4092 + 1
Substituting these into (26) we obtain K < 0.12. If #u = gu, it is found from Corollary 4 that the allowable gain may be slightly increased to K = 0.133.
[]
Remark 6. The determination of Ck(j09 ) in Theorem 3 and Corollary 4 is straightforward since it consists only of known functions like Gc(s), t~po(S) and Ho(s). The evaluation of p in (25) is somewhat more involved, though the information needed is similar to the one used frequently in classical control, namely, open-loop frequency response. If frequency response test between each input and each output, with all loops open, can be performed, then the determination of p is quite simple. In cases where the differences between giih~ and the corresponding models ~ijrai arise only from small time constants negligible in the course of mathematical modelling, then p is unity and (26) reduces to (36). When the actual functions guh~ are unavailable and frequency response test cannot be done, then the best that can be done to estimate p is to perturb the gains and time-constants from their nominal values in a fashion similar to Example 3, and compute the ratios IgijhJ~ijtli(j09)l. Such calculation will be usually carried out on a computer especially for high-order systems. The amount of parameter variations taken depends then on the accuracy of the models and possible process variations during operation. Having determined p one can then apply in a straightforward manner criterion (26) [or (36)] to check the stability of a system for a given set of control gains.
Example 3. Let Gp(s) be a (2 x 2) matrix with gu(s) = k u e-°'Js/(tos + 1), Go(s) = diag (K) and
While the determination of controller gains by
det (I + GcGr) = 1 + ~ Crk(S) = 0 k=l
(33)
where C Tk is the sum of all principal minors of order k in GcGr. Z~,=1Crk(s) is analytic in the RHP. All zeros of (33) will be in the LHP, according to Rouche's theorem, if r
k~l Crk(J09) < 1, V09 > 0.
(34)
From (32) and the definitions of CTk and Ck we obtain
~=1Crk(J09) < k
(2 + p)kCk(j09 )
(35)
k=l
and condition (26), then follows from (34). [] If mismatches are assumed to be only in the DTs, condition (26) reduces to:
Corollary 4. The overall system is stable for any mismatches Actu in the DTs ( - &u < A~ij < cqj) if ~. 3kck(j09) < 1, V09 > 0.
(36)
k=l
Proof: When guhi = gifii, P in (25) is 1.
5. ESTIMATION OF ALLOWABLE MISMATCHES
Multivariable dead time compensators Theorem 3 is simple and straightforward, the resulting gains are often very conservative since stability is guaranteed for any DT values and for any mismatches in them. With the aid of diagonal dominance theory (Rosenbrock, 1974), we may derive results which are helpful in checking the stability of a given system in the non-ideal cases and in estimating the allowable tolerances in the mismatches within which the stability of the system is maintained. In the next section, diagonal dominance will be used to determine controller gains. Let D be the Nyquist path in the s-plance consisting of the imaginary axis and a semi-circle of a large radius in the RHP. Further, define the transfer functions Mi(s) and IGi(s ) as follows: Mi(s) = gc,~,~i/(1 + g~i~ifli),
261
Q = diag [(1 + ga~it~,)-i].
(40)
If L is row dominant (column dominant) on D, so is QL (LQ). Observe that the elements on the diagonal of QL may be expressed in the following form (QL)u = (LQ)ii = 1 - Mi e-~"SlGi •
(41)
We proceed with the row dominance noting that the case of column dominance is completely analogous. Let det (QL) map D into FQL,and similarly let (QL), map D into Fi. Let FQL and F i encircle the origin NQ,. and Ni times, respectively. Clearly, if QL is diagonally dominant on D, then
i= 1,...,r
(42)
lq i = NQL. i=1
and IGi(s) = 1 - gi~hi e-a'"s/(~,/t~),
i = 1. . . . . r. (37)
The function M~(s) is the transfer function seen between the ith input and the corresponding ith measured output in the ideal case with no DTs and with all other loops open. In practice, it is often required that stability is maintained in the ith loop when all the other loops open. It is, therefore, assumed that the Mi(s) is analytic in the RHP. The function IGi(s) measures the amount of mismatching or ignorance in g,, hi as well as in a,. When the model exactly matches the process then IG~(s) = 0. Since it is required that 0,h~ be analytic in the R H P it follows that MJGi(s) (i = 1..... r) is analytic in the RHP. Recall that the characteristic equation in the non-ideal cases is given by (28) and denote the matrix (I + GcGT) by L. In the sequel we apply the definitions of diagonal dominancy suggested by Rosenbrock (1974, p. 142). Theorem 5. Let L be diagonally dominant on D. Let - M~e- ~,,sIG~ map D into F,(i = 1,2 ..... r) which encircles the point ( - 1, 0) N~ times. Then the overall closed-loop system is asymptotically stable iff
N~ = 0.
(38)
i=1
Proof: Let det (L) map D into FL which encircles the origin N~_ times. It is well known from stability theory that when all poles of GcGr are in the LHP, all roots of det (L) = 0 are in the LHP if and only if NL = 0. Now, let the (r x r) matrix Q be defined as
(39)
Since all poles and zeros of det (Q) are in the LHP, it follows that NL = NaL = ~ Ni.
(43)
1=1
Furthermore, notice that F~ encircles the origin as often as Fi encircles the point ( - 1, 0). Hence I~ i = ~ N i i=1
(44)
i=1
and (38) follows at once from (44), (43), and (39). [] A direct consequence of Theorem 5, which will be more useful for subsequent results, is the following: Corollary 6. Let L be diagonally dominant on D. Then the overall closed-loop system is asymptotically stable if N i=O,
i = 1. . . . . r.
(45)
From Corollary 6, several results which allow us to estimate the tolerances in the mismatches may be derived. Recall that the mismatches between the model and process are contained in the matrix Gr. different mismatches lead to different GT matrices. In the following, we assume that there exist some set of Gr matrices (denoted {Gr}), including Gr of the ideal case, for which L is diagonally dominant on D. Clearly, we assume that the design is p.s. First, we have: Theorem 7. Let L be diagonally dominant on D for some {Gr}. Then the overall closed-loop system is asymptotically stable for all GT in {Gr} for which IMi(jto)lGi(jo~)l < 1; Va~ > 0;
i = 1.... , r.
(46)
262
Z.J. PALMORand Y. HALEV!
Proof: From (46) and the analyticity of Mi(s)IGi(s) in the RHP, it follows with the aid of the PhragmenLindelof theorem that IMi(s)IG~(s)l <1 in the RHP. Hence, from Rouche's theorem, (QL(s))ii, (i = l, ..., r) in (41) is analytic in the RHP. This implies that N~ = 0 (i = 1, ..., r) and the result follows at once from Corollary 6. [] Example 4. Consider the process Gp(s) in Example 3 with all DTs equal to unity. Both controller gains are taken to be K = 3. It is required to estimate the allowable tolerances in the time constants z~j within which stability is maintained. In this case, IG,(s) = Az,s/[(1 + Az,s) + 1] and Mii(s) = 3/(s + 4) and condition (46) is clearly satisfied for at least all Az;~ in the range: A~, > -0.7. However, L is found to be diagonally dominant for Azii > - 0 . 8 and AzI2 = Az2x > - 1 . It is, therefore, concluded from Theorem 7 that stability is guaranteed for Az, > - 0 . 7 and Azl2 = A~21 > -- 1. Theorem 7 is now exploited to obtain estimates for allowable mismatches in two particular cases. First consider the case where mismatches in cq~ are likely to exist while all other parameters in the diagonal elements of the HGp are assumed to be perfectly known and fixed. However, mismatches in the other parameters in the off diagonal elements of the HGp are allowed. Under these circumstances, IGi(s ) takes the form 1Gi(s) = 1 - e -a'''~,
i = 1. . . . . r.
(46) or (48) is denoted by {Gr}', then Theorem 7 (Corollary 8) guarantees stability only for {Gr}'n
Remark 8. If IMi(jw)l < ½, Wo > 0 then no restriction is imposed by (48) on A~,. Remark 9. Notice that while the corollary discusses mismatches in the DTs of the diagonal elements in HGp, the mismatches allowed in the offdiagonal elements are not restricted to be in the DTs only. Example 5. Consider the system of Example 4. We wish to estimate the DT tolerances within which the system will remain stable. It has been found that L in this case is diagonally dominant for those { Gr} for which IAOolOijl < 0.5. Variations in the other parameters in the off-diagonal elements of Gp were not considered. M 1 and M 2 are both 3/(s + 4). Therefore, (Mi)m = 0.75 and toOl = x/20. From (48) we obtain: AO, < 0.327 (i = 1, 2). Following Remark 7, it is concluded that stability is retained for {Gr}, for which AOii < 0.327 (i = 1, 2) and A012 (= A021) <0.5. The second particular case arises when mismatches in process and sensor gains are considered. g~j is written now as ko#'o where ko is the steady state gain of #~j. The difference k~j - l~ij is denoted by Ak w IG~ now takes the form
(47)
I Gi(s) = - Akii/fqi. Denote maxlMi(joJ)l by (Mi)m and let O9oi be the frequency above which IMi(jog)l < ½. Then from Theorem 7 we have:
Corollary 8. Let L be diagonally dominant on D for some {GT}. Let the mismatches in the diagonal elements of HGp be only in the DTs, ~,. Then the overall closed-loop system is asymptotically stable for all Gr in {Gr} for which Aaii < (2/O9oi) sin- 1 [2(Mi),.]- 1, i = 1, ..., r. (48) Proof: Condition (46) is clearly satisfied for the IGi in (47) for co > O~o~.Now, as [MilGi(jco)[ < (Mi)m2 sin (Actiiog/2), Vco > 0 it is required that (Mi)m2 sin (A~ilog/2) < 1, Vo9 < Ogol. Substituting O9o~for ~o in the left-hand side of the last inequality completes the proof. []
Remark 7. If the set of all G r matrices arising from
(49)
Corollary 9. Let L be diagonally dominant on D for some (Gr}. Let mismatches in the diagonal elements of Gp(s) be only in the gains k,. Then the overall closed-loop system is asymptotically stable for all Gr in { Gr } for which
IAk./~.l < (Mi)~ 1,
i = 1. . . . . r
(50)
Proof: From (46) we have IM(jog)llAk,/l¢iil < 1, Vo9 > 0 whence (50).
[]
Remark 10. Notice that mismatches in the offdiagonal elements of Gp are not restricted to gains only. Example 6. Let the process and controller be as in Example 5. Using Corollary 9, we shall estimate the gain tolerances Ak,. It is easy to verify that L is diagonally dominant for at least all GT consisting of Gp with IAko/fqj[ < 0.5. From (50) we obtain [Ak,[ < ~. Stability, however, may be guaranteed only for IAk~j/l~ijl < 0.5.
Multivariable dead time compensators hence
6. C O N T R O L L E R S G A I N D E T E R M I N A T I O N
While the estimation of model tolerances through the utilization of Theorem 7 and Corollaries 8 and 9 is quite simple, the search for {Gr}, for which L is diagonally dominant on D is quite tedious, especially due to the transcendental terms involved. This search may be avoided, in some cases, by suitable choices of controller gains which a priori guarantee diagonal dominance. The resulting gains will, in general, be conservative; but it turns out, in many cases, that they are much less conservative than the ones derived by means of Theorem 2. Once a suitable set of gains has been found, the results of the previous section may be effectively utilized. Since mismatches in DTs are usually the most destructive ones to stability, we shall base our derivation on them. It will be assumed that G~Gt,o is strictly proper and that none of its elements contain pure imaginary poles. As controller gains are sought, the transfer function Oct(s) is rewritten in a more explicit form: ga(s) = Kffa(s). Let fl denote the maximum mismatch in all uo, i.e. fl = m a x
i,j
263
I(L(jw)),ll <
T(~)lKio'a[~,Oo(jw)l, Vo~ > 0, i, j, i ~ j.
(53)
and I(L(j~o)),l > 1 - T(o~)lgff~[i~i(joJ)l, Va~>0, i = l . . . . . r.
(54)
For row dominance, it is required that I(L(jco)),l > ~ I(L(j~o))ol, j=l
Wo > 0, i = 1. . . . . r.
(55)
Substituting (53) and (54) into (55) we obtain
K~T(o~) ~ Ig'ag~0ij(jo~)l < 1, i = 1. . . . . r. jft
Now if K~ is selected such that
mO~lj
and define the function T(~o) as follows:
Ki < 1/lT(co) ~ Ig'ag~#~g(j~o)l], j=l
r(~o)
j'l + 2 sin (flco/2) 3
co < n/fl o~ > n / # '
(56)
then dominance is assured. Finite K~ can be always found since GcGr is assumed to be strictly proper. Equation (52) is obtained by replacing the product
We have:
Theorem 10. Let controller gains K~ be given by K i < 1/max[T(og) ~ Ig',il~i~o(jog)l], a}
V~o>0, i = l , . . . , r
(51)
by its maximum value in ~o > 0.
j=l i =
r(co)j ~= l Ig'Jzi~o(jco)l
1 ....
, r.
(52)
Then L is diagonally dominant on D.
Proof: The elements of the matrix L(s) are given by
Lii = 1 + Kig'ci[~igii~ii Lii = Kig'at~lgij~ij, i :~ j
[]
Remark 11. The theorem is developed for row dominance. For column dominance one considers the matrix L ' = I + GrQ. Condition (52) is then obtained with i and j interchanged. Example 7. Consider the G~(s) in Example 3 with 0 o = 1. Controllers gains are to be determined by (52). It is easy to obtain that 2
max T(r~) ~ Ih,#ij(jco)l co
j=l
where ~,s = 1 - e-='~s(1 - e-A='Js). Clearly I~o(jog)l ~ T(oJ)
is approximately 1.594. From (52) we get Ki < 0.63. Since in this case (Mt)m < ½, no restrictions on A~i arise from Corollary 8. Note that gains obtained here are larger than the corresponding ones derived in Example 3 (by Theorem 2) by approximately a factor of 5, but are considerably less than those used
264
Z.J. PALMOR and Y. HALEVI
in Example 5. This is the price we pay for avoiding the search for {GT}. 7. CONCLUSIONS
On one hand the cancellation of all DTs by the DTC scheme offers significant enhancements to the control of stable MIMO processes with multiple DTs, but, on the other hand, introduces considerable difficulties in the design problem. Designs based on the ideal cases are inadequate. For this reason, an attempt has been made in this paper to develop analytical aids for designing the DTC, which take into account the stability and sensitivity properties of these systems. A necessary condition for practical stability and further, means for estimating the allowable tolerances in process models were stated and proven. Also, methods for determining appropriate controller gains were presented. In addition, it was shown that the type of DTC treated in this paper cannot be applied to open-loop unstable processes. Some of the results rely rather heavily on the assumption that the characteristic equation of overall closed-loop is diagonally dominant in the ideal case. No attempt, however, to achieve diagonal dominance other than by choice of controller gains, has been made. This might be a fruitful area for further research. Acknowledgements--This work was supported by a grant from the Engineering Division of Taylor Instrument Company, Rochester, New York. Valuable discussions with Mr D. V. Powers of Taylor are greatly appreciated.
REFERENCES Alevisakis, G. and D. E. Seborg (1973). An extension of the Smith
predictor method to multivariable linear systems containing time delays. Int. J. Control, 3, 541. Alevisakis, G. and D. E. Seborg (1974). Control of multivariable system containing time delays using a multivariable Smith predictor. Chem. Engng Sci., 29, 373. Donoghue, J. F. (1977). A comparison of the Smith predictor and optimal design approaches for systems with delay in control. IEEE Trans. Ind. Electron. & Control Instrum., IECI24, 109. Gantmacher, F. R. (1959). The Theory of Matrices, Vol. I. Chelsea, New York, p. 70. Gawthrop, P. J. (1977). Some interpretation of self-tuning controllers, Proc. lEE, 124, 889. Kleinman, D. L. (1969). Optimal control of linear systems with time-delay and observation noise. IEEE Trans Aut. Control, AC-13, 524. Kwon, N. H. and A. E. Pearson (1980). Feedback stabilization of linear systems with delayed control. IEEE Trans Aut. Control, AC-25, 266. Manitius, A. Z. and A. W. Olbrot (1979). Finite spectrum assignment problem for systems having multiple time-delays. IEEE Trans Aut. Control, AC-24, 541. Ogunnaike, B. A. and N. H. Ray (1979). Multivariable controller design for linear systems having multiple time-delays. J. Am. Inst. Chem. Engrs, 25, 1043. Olbrot, A. W. (1981). Observability and observers for a class of linear systems with delays. IEEE Trans Aut. Control, AC-26, 513. Palmor, Z. J. (1979). Structural, stability and sensitivity properties of optimal stochastic control systems for dead time processes. TME-368, Technion, Israel Institute of Technology, Mechanical Engineering. Palmor, Z. J. (1980). Stability properties of Smith dead-time compensator controllers. Int. J. Control, 32, 937. Palmor, Z. J. (1982). Properties of optimal stochastic control systems with dead-time. Automatica, 18, 107. Palmor, Z. J. and D. V. Powers (1981). Feedforward properties of dead time compensator controllers. JACC, Charlottesville, Virginia. Rosenbrock, H. H. (1974). Computer-aided Control System Design. Academic Press. Smith, O. J. M. (1957). Closed control of loops with dead time. Chem. Engn 9 Pro9., 53, 217. Smith, O. J. M. (1959). A controller to overcome dead time. ISA J., 6, 28.