E L S E V I E R 0959-1524(95)00029-1

J Proc. Cont. Vol. 6, No. 1, pp. 37-48, 1996 Copyright (~) 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0959-1524/96 $15.00 + 0.00

An extension of singular value analysis for assessing manipulated variable constraints

Yi C a o 1 and D iane Biss 2 Centre for Systems and Control Engineering, University of Exeter, Exeter, EX4 4QF, UK

Received 27 December 1994; revised 4 July 1995

A new approach called feasible output radius analysis for linear or linearised models is introduced to address the problem of scaling dependency. This problem arises when assessing the effect of manipulated variable constraints (MVCs) on the closed-loop performance of chemical processes prior to carrying out control designs. The new indicators, R and R can be used to rank alternative control schemes on the basis that the larger _R and R, the better the closed-loop performance in the presence of control constraints. These indicators are determined from extending the concept of the 'feasible output amplitude region' and are independent of the input scaling chosen. Theoretical analysis shows that this method is an extension of the more traditional singular value analysis approach and is more flexible in dealing with various kinds of manipulated variable constraints. A case study, i.e. a two-CSTR process, is investigated using the new method. Via the case study, some superior characteristics of the new technique are demonstrated, such as ease of calculation, and flexibility in coping with different kinds of constraints.

Keywords: input-output controllability analysis, control constraints, control structure selection

Of interest is how to assess the effect of manipulated variable constraints (MVCs) on the closed-loop perfor- mance of chemical processes prior to carrying out con- trol designs. Previous work for linear systems has used singular value (SV) analysis, 1-4 as part of so-called input- output controllability analysis, to rank alternative process designs on the basis of the relative magnitudes of each configuration's SVs and their Euclidean condition num- ber. Although these indicators are easy to calculate, 5 the SV analysis suffers from two drawbacks: firstly, the anal- ysis is only applicable to linear systems, or a linearised model of the system if the process is nonlinear; and sec- ondly, the results are strongly dependent on the selected scalings for the process' inputs and outputs.

Previous work by the authors 2 has looked at an alter- native optimisation-based approach for assessing nonlin- ear systems, removing the necessity to linearise the model. This has proved successful in predicting the closed-loop performance of a simple case study. However, the com- putational load is significantly greater than for the SV analysis.

In this paper, a new approach for linear, or linearised, models is introduced to address the problem of scaling dependency. Perkins 6 showed that the achievable output range can be quantified by analysing the achievable out- put amplitudes when the outputs respond to sinusoidal inputs. In this case the 'feasible output amplitude region' can be described by the SVs when the Euclidean norm of the inputs is less than one. However in most physical plant, the manipulated variable constraints cannot ex- actly be expressed by bounded Euclidean norms. In such

I Supported by the ORS Award of CVCP and the Research Selectivity Fund of the University of Exeter. z To whom correspondence should be addressed.

cases, using SVs will generally lead to a conservative es- timation of the process' controllability. To overcome this deficiency, the concept of the 'feasible output amplitude region' has been extended to any form of MVCs. In par- ticular, for the case of linear MVCs, some new indicators which describe the 'feasible output amplitude region' are derived. The new indicators are independent of the input scaling and have some other superior features as well.

The paper is organised as follows: some rigorous math- ematical definitions are given; next algorithms for calcu- lating the new indicators with linear constraints are dis- cussed; followed by a case study to show the usage of the new indicators; and finally some conclusions are pro- vided.

Feasible output radius analysis

For an input-constrained linear system, the feasible out- put radius is to be defined. The associated maximum and minimum values are also to be defined and a theorem relating these concepts to the maximum and minimum singular values proven.

Feasible output radius

Consider an m-input, m-output input-constrained linear system, G(s). Its frequency response matrix is G(iw). In the time-domain, the system's inputs, u ~ IR m, are bounded by k constraints,

h(u) = [h i (u ) , hE(u) . . . . . hk(u) ] T _< 0 (1)

where here, and in the rest of this paper, a vector inequal- ity means element by element inequalities. The aim of the

37

38

new method is to describe the feasible output range when the system is driven by a set of feasible sinusoidal inputs. These inputs can be defined as follows:

Assessing manipulated variable constraints: Y. Cao and D. Biss

-R ( to ) = ~ (#T(-~

i(to) = m a x f T e - i * P ( t o ) e i* f ii,*

s.t. f E the edge of U Definition 1. A set of sinusoidal inputs, ui = ~i sin(tot + qbi), i = 1 . . . . . m, where tTi >_ 0 and qbi E R, are feasible iff the vector u = [ul . . . . . urn] T satisfies the input con- straints, Equation (1).

and

Definition 2. A subset U c •m is a feasible input ampli- tude region iff for any f = [tTt . . . . . /~m] r ~ U and qb = [qb 1 . . . . . qbm] T ~ R m, the composed sinusoidal input vec- tor u is feasible.

According to linear frequency response theory, 7 in the steady-state, the output responses to the feasible sinu- soidal inputs are also sinusoidal signals with the same frequency as the inputs. The sinusoidal output ampli- tudes ~(to) = [)71 (to) . . . . . ym(to)] T are feasible and can be analysed by Equation (2) when the sinusoidal inputs are feasible.

~T(t,O)~(tD) = vHGH(ito)G(ito)V

= fiT e-i* p(to) ei* f (2)

where the superscript H represents complex conjugate transpose, P = GHG, v = e i* fi and ~I, is a diagonal matrix with the (i, i)th element being qbi.

Equation (2) gives the relation between the output am- plitude ~ and the input amplitude f. Based on the feasi- ble input amplitude region, U, the Euclidean norm of the vector ~ can be used to measure the feasible output am- plitude range. The largest range corresponds to inputs at the edge of the feasible region.

Definition 3. The feasible output radius (f.o.r.), R, is the Euclidean norm of ~(w).

R(~, to) = 4~r(w)~(to) (3)

when fi is at the edge of U.

Using Equation (2), the f.o.r., R, can be represented w.r.t. f and el,:

R ( f , ~ , 03) = ff~ T e - i* P(to) e i* f , f E the edge o f U

(4)

Maximum and minimum feasible output radii

Definition 4. The maximum and minimum f.o.r, are de- fined as follows.

R(w) = max R(f, q~, to) (5)

__R(to) = min R(fi, q,, to) (6)

According to Equation (4), R(to) and R(to) can be ob- tained by solving the following optimisation problems:

(7)

_R(co) = ~/r(to)

r(w) = min r iTe- i* P(to) e i* f fi,*

s.t. f ~ the edge of U

(8)

D

Remark 5. R(to) and R(to) are independent of input scaling.

Generally, these optimisation problems (7) and (8) have to be solved using numerical techniques. However, for some special types of constraint, the solutions of Equations (7) and (8) can be rigorously derived. The following theorem gives the solutions for the constraints of Ilul12 --< 1, while the linear constraint problems are solved in the next sec- tion.

If a special case of UTU _< 1 is considered, then The- orem 6 shows that the maximum SV and minimum SV are equivalent to the maximum f.o.r, and minimum f.o.r. Hence feasible output radii can be considered as an ex- tension of SVs for any kind of input bound.

Theorem 6. If the system's inputs are constrained by uTu _< 1 then R(to) = ~ ( w ) and __R(w) = o-(to), where

and o- are the maximum and minimum singular values of G respectively.

PROOE Since uZu _ ~Tf, the amplitude constraint iiTfi _< 1 ensures the sinusoidal inputs are feasible. Thus the edge of U is firfi _ 1 = 0. A necessary condition for both problems (7) and (8) is that

(e - i # P e i* -M)f i = 0

where A is the Lagrange multiplier. Note that e i* is a unitary matrix. Thus 2, is an eigenvalue of the matrix P, i.e. the square singular value of G, and

= m a x f T ~ f = max A = ~ 2 fi

Therefore, R(to) = ~( to) is derived. Following the same arguments, R_(to) = o-(w) can also be obtained. []

Algorithms for linear constraints

In most circumstances, the MVCs are just linear con- straints, i.e. the sinusoidal inputs are bounded by k linear constraints: hiru _d i , i = 1 . . . . . k, where hi ~ R", or in vector form:

h(u) = Hu - d _< 0 (9)

where d = [dl . . . dk] T and k × m matrix, H = [h i . . . hk] T. To ensure the region defined by the constraints (9) in- cludes the steady state operating point, u = 0, it is as-

Assessing manipulated variable constraints: Y Cao and D. Biss 39

sumed that di > 0, i = 1 . . . . . k. For this case, the solu- tions of R(w) and R(w) can be derived.

Feasible input amplitude region

Before solving the optimisation problems (7) and (8), it is necessary to determine the feasible input amplitude region.

Lem ma 7. A set of sinusoidal inputs, u, is feasible under a single constraint hTu -< di, iff the input amplitude vector, fi, satisfies the following inequality:

fTfi _< di (1 O)

where fi is a vector whose elements are the absolute values of the corresponding elements of h.

P R O O E For any 4) E R m, the following inequality is true:

m m

h~u = 2 h i j f j s i n ( w t + dpj) < 2 Ihvl~J j=l j=l

Thus expression (10) ensures the inequality h/ru _ di, i.e. u is feasible. Furthermore, if the constraint (10) is not satisfied then the 4bs would be chosen such that hij sin(wt + q~j) : Ihijl, j = 1 . . . . . m, giving hTu = fiTfi. This leads to u being infeasible. []

Lemma 7 states that changing from an input constraint to an input amplitude constraint just requires hi to change to hi = ]hil. Thus, if a whole set of constraints is consid- ered, the feasible input amplitude region can be expressed using Theorem 8.

Theorem 8. Let matrix 17t = [h l . . . ilk] T E R kxm. Then the feasible input amplitude region w.r.t, the constraints (9) is

V = {~ >__ 0 1 lZlfi- d_< O} (11)

Theorem 8 states that the feasible input amplitude region is also bounded by a set of linear constraints if the si- nusoidal inputs are constrained linearly. This makes the optimisation problems (7) and (8) easier to solve.

Lemma 10. If G is invertible, so that P is invertible as well, the minimum costs of Equation (12) are

ri(w) = min d2 fiTi e _ i , p _ l ( w ) e i ~ f i i , i = 1 . . . . . k

(13)

P R O O E Using the Lagrangian method, s define

J = riTe-i* P(to) ei*u + A(hTfi - di)

Since P is invertible, the necessary condition, M/Off = 0 gives:

fi = ~ p - l h i A

Combining with the edge equation, fiTfi = di, yields that

26 A---- - - fiTP-I

Thus the solution (14) of the minimisation problem (12) is

die-i,t, p - I ei,~ hi fi = fiT e-i•, p - I ei~, l-li (14)

This solution leads to Lemma 10. []

Remark 11. l fG is not invertible, the generalised inverse, 9 P+ could be used to replace p-1. In this case, the solu- tion, fi, is the minimum Euclidean norm solution of the problem

rain 0J

The results of Lemma 10 are related to the phase shifts, 4). Generally, to obtain the minimum value with respect to 4) involves using numerical methods. However the following lemma determines the upper and lower bounds for R_.

Lemma 12. Let z E I[~ m, D E C '' 'x'' and • be an m x m real diagonal matrix, then

zTDz -< max zTe-i'l 'Dei'l 'z < [zTI . IDI. Izl

• (15)

where [ • [ represents the matrix (vector) with the modulus taken for each of its elements.

Minimum feasible output radius

L emma 9. The minimum f.o.r, problem (8) with U de- fined by Equation (11) is equivalent to the following op- timisation problem.

R(w) = min @i(eo) l

f.i(O.)) ---- min fiT e-i,l, P(co) e i'~ fi (12)

s.t. fiTfi = di, i = 1 . . . . . k

P R O O E 5 ~ the edge of U means that fi should at least satisfy one of the k equalities hifi = di, i = 1 . . . . . k. Hence the optimisation problem (8) is equivalent to the new optimisation problem (12). []

P R O O E The lower bound of expression (15) comes from the fact that, for any function, f ( . ), f(~I, = 0) _< max , f(q~). Conversely:

zTe-i*Dei' t 'z < [zVl " IDI. Izl

leads to the upper bound. []

For a special case, when z has no more than two non-zero elements, the following lemma can be stated.

Lemma 13. Let D be a positive semidefinite Hermitian matrix, and z E I~ m, having no more than two non-zero elements. Then

max z T e-i~' D e i~ z = [z y] • I D I - I zl

40 Assessing manipulated variable constraints: Y Cao and D. Biss

P R O O E Let the non-zero elements of z be zi and zt and J = z T e -i~ D e i~ z. Then

J = z2dii + z iz l (d i le -i(cb'-q'') +dliei(4',-q't)) + z~dll

Since D is a positive semidefinite Hermitian matrix, the submatrix composed ofdu, da, dti and dtt is also a positive semidefinite Hermitian matrix. Therefore, dii >- O, d, >_ 0 and dit = dr*, where the superscript * represents a com- plex conjugate number. Assume zizlda = I zillzt I I ditl e i°", then

J = Izi[ 2 Idiil + Izil Iztl Id, I (e -~l~'-~t-° ' l + eil~'-¢'-°"))

+ Izzl2ldttl

Hence the solution, qb~ - qbz = Oil, leads to the maximum value

m a x J = [ l z i l ]zll]. [Idiil [dill] [ Izi[] ~,,,¢,, Idtil Idttl " Iztl

Thus proving the lemma. []

Combining Lemmas 10 and 12 leads to the following theorem.

Theorem 14. The minimum f.o.r, is bounded as follows.

where

Rmin( to ) < R ( t o ) < Rmax(tO) (16)

Rmax( to ) = m i n (17) - - i ~ T p - 1 (U~)~ i

dt Rmin( to ) = min , _ _ (18)

i ~/h~. I P - n ( w ) l • fii

~i<di, i= 1 . . . . . m

i.e. hi = el, where ei is the ith column of an m × m unit matrix. Using Theorem 15 leads to Equation (20). []

Maximum feasible output radius

In this section, the solution of Equation (7) for linear constraints is discussed. Since the function (2) is positive and convex, its maximum value always locates at an ex- treme point of U. An extreme point is composed of m equality constraints of Equation (11), which are referred to as active constraints. The remaining k - m constraints are referred to as inactive constraints. Suppose there are l~ extreme points in U. At each extreme point, the ma- trix I:I can be partitioned into two submatrices: HAt and HIi, i = 1 . . . . . lc, where the m × m active matrix HAt is non-singular and corresponds to the m active constraints of the ith extreme point, and the (k - m) × m inactive matrix Hig corresponds to the inactive constraints at the same extreme point. The vector d can correspondingly be partitioned into dAi and dli, i = 1 . . . . . lc. Therefore the constraints in the l~ extreme points, fiEPi, i = 1 . . . . . l~, can be represented as:

UEPi ---- HA~dAt, HliUEPi -< dIi, i = 1 . . . . . lc (21)

Lemma 17. The maximum f.o.r, problem (7) with U de- fined by Equation (11) and lc extreme points defined by Equation (21) is equivalent to the following optimisation problem:

R(to) = m a x ~ i ( w )

PROOF. The minimisation problem of (13) is equivalent to maximising the denominator of its right-hand side. Thus using Lemma 12, the upper and lower bounds of the denominator can be obtained. These bounds lead to Theorem 14. []

Since P is a positive semidefinite Hermitian matrix, using Lemma 13, the following theorem can be stated.

Theorem 15. If each of the k input constraints in Equa- tion (9) is not related to more than two input variables, then the minimum f.o.r, is

di __R(w) = rain (19)

i ~/fi/r. ip_~(to)l . f i .

Corollary 16. If the inputs are constrained as - d i <_ ui <- di, i = 1 . . . . . m, and di= min(di, dr), then the minimum f.o.r, is

di _R_(w) = m i n - (20)

t 4 p . ( w )

where/~, is the (i, i)th element of p - l .

P R O O E The feasible input amplitude region of such sys- tems is determined by the following m input amplitude constraints:

-ii(w) = m a x uTpie l~P(to)ei~fiEPi, i = 1 . . . . . Ic cp

(22)

Combining Lemmas 12 and 17 gives the following theo- rem.

Theorem 18.

where

The maximum f.o.r, is bounded as follows.

Rmin( to ) -< R ( t o ) _ Rmax(~O) (23)

Rmin(O3) = m a x 4UTpiP(oO)UEPi (24) i

~T Rmax( to) -- m a x x/IuEPil • I P ( t o ) l • [UEPil

i (25)

Using Lemmas 13 and 17 leads to the following theorem.

Theorem 19. If a system has no more than two inputs, i.e. m -< 2, then

R(w) = max4lli~pil . I P ( w ) l . IfiEPil (26)

If the inputs are bounded as -~ . _< ui < di, the feasi- ble input amplitudes are constrained as ~7i -< min(di, di).

Assessing manipulated variable constraints: Y Cao and D. Biss

There is then only one extreme point in the feasible input amplitude region:

fiEP = d (27)

where d is a vector with its ith element, di = min(d~, d~). Therefore the following theorem can be stated.

B Theorem 20. If the input's constraints a r e d i <_~ u i ~ di, i = I . . . . . m, then the maximum f.o.r, is bounded as

Rmin(tO) N R(U)) _< Rmax(tA)) (28)

41

~ F e e d 1 ~ F e e d 2

'l H oW:; '1 FTo:: °' Water In ~ W a t e ~ ~ ~ o d u c t

Figure 1. A two-CSTR process

where

Rmin(O)) = ~fiTpP(to)fiEp

R--max(~) = N/I~TpI • IP(w)I- liiEpl

with fizp being defined in Equation (27).

Corollary 21. If a system has no more than two inputs which are bounded as d i <_ ui <- di, i = 1, 2, then the maximum f.o.r, is

R(w) = ~/dTIP(co)ld (29)

where

d = [ min(dl, all) min(d2, d2) ]T

Remark 22. If co = 0, that is the system is at steady state, then qb = 0, e i~ = I, u = u and I7t = H, and the expres- sions for _R_(0) and R(0) simplify to:

dt _R_(0) = min (30)

i ~h~P-l(O)hi

R ( 0 ) = m a x f f ~ T p i P ( 0 ) ~ E P i (31 ) i

Input-output controllability analys& using f o.r.

The f.o.r, represents the maximum allowed output range when the inputs are bounded. Assume the output vari- ables, y, are scaled such that the maximum change of the output satisfies 11YI[2 < 1. Then the maximum and min- imum f.o.r., R and R__, can be used as input-output con- trollability indicators in the following ways. (1) At steady state, if the maximum f.o.r., R(0) _< 1,

then the corresponding control scheme is not input- output controllable. This control structure should be eliminated.

(2) At steady state, if the minimum f.o.r., R_(0) >__ 1, then the corresponding control scheme is input-output controllable and the control structure could be ac- cepted.

(3) The frequency, WR, defined as

R(w) >_ 1, m _ ~R (32)

describes the largest bandwidth within which perfect control could be achieved.

(4) The frequency, w__ R, defined as

R(oJ) > 1, w < w___ R (33)

describes the least bandwidth within which perfect control could be achieved.

(5) In the frequency range, co n < w _< ~R, the achie- veability of perfect control depends on the input di- rection.

(6) Within the bandwidths WR and ~R, the larger the values of R ( w ) and R(w), the better the expected closed-loop performance of the corresponding con- trol structure.

Case study

Modelling o f a t w o - C S T R process

A process comprising two CSTRs in series with an inter- mediate mixer introducing a second feed l° is investigated (see Figure 1). A first-order irreversible exothermic reac- tion:

A - - B (34)

is carried out in the process. Water at ambient tempera- ture (Tcwl and Tcw2) is used to cool the reactors. The densities and heat capacities are assumed to be constant and independent of temperature and concentration. Thus the process can be modelled in terms of the concentration of the raw material (A) by the following nonlinear differ- ential and algebraic equations (see the Nomenclature):

dxl dt = Qn - K v l ~ (35)

d(xlx2) dt

- - - KlXlX2 + QIICII - K v l x 2 v / ~ (36)

d(xlx3)

dt - - - A H K l x l x 2 + QIITII - Kvlx3v/~

- U,1 (x3 - x4) (37)

dx4 Vj1---~- = Qcwl (Tcwl - x4) + U a l ( X 3 - x4)

(38)

dx5 dt = QI2 - Kv2-fx--5 (39)

d(x5x6) d---~ = -K2x5x6 -F / V l X 2 x / ~ + QIzG2 - Kv2x6v/~

(40)

42 Assessing manipulated variable constraints: Y Cao and 19. Biss

Table 1. Physical and process constants

Variable Value Unit KVI 0.16 m3/2s -1 KV2 0.256 m3/2 s-1 Ual, Ua2 0.35 m3s - t E / R 6000 K A H 5 m 3 • K mol -I K0 2.7 x 108 s - t VJ1 , VJ2 1 m 3

Table 2. Operating point

Variable Value Unit xl 4.489 m 3 x2 0.084 mol m -3 x3 362.995 K x4 327.560 K x5 5.493 m 3 x6 0.053 mol m -3 x7 362.995 K x8 335.447 K QI1 0.339 m3s -1

7]1, 7]2, TCWl, Tcw2, 300 K QI2 0.261 m 3s-I Cn, Q2 20 mol m -3 Qcwl 0.45 m3s -1 Qcw2 0.272 m 3s-!

d ( x 5 x 7 ) - - -- A H K 2 x s x 6 + gVlX3 x/'~" + Q I 2 ~ 2

dt - - / v 2 X 7 ~ t ' ~ -- Ua2(x 7 - x8) (41)

~j dx8 2 - - ~ - = Q c w 2 ( T c w 2 - x 8 ) + Uaz(X7 - x 8 )

(42)

where K1 = Koexp(-E/Rx3) and K2 = Koexp(-E/Rx7). The states are Xl = VI, volume of CSTR1; x2 = Col, out- let concentration ofCSTR1; x3 = Tol, outlet temperature of CSTR 1; x4 = Tcwol, cooling water outlet temperature of CSTR1; xs = V2, volume of CSTR2; x6 = Co2, outlet concentration of CSTR2; x7 = TOE, outlet temperature of CSTR2; and x8 = Tcwo2, cooling water outlet tem- perature of CSTR2. The physical process constants are given in Table 1. The process is assumed to be operated at the point given in Table 2.

Input-output specifications

For the analysis, x3 and x7 are selected as the measured variables. These were chosen since economic analysis of the above case study for control structure selection 11 shows that the optimal operation point with respect to economics is very sensitive to variation in the outlet temperatures of the reactors, hence these need to be controlled. For this output specification, two possible two-input, two-output configurations, named S1 and $2 (see Table 3) are considered. In Table 3, Qn and QI2 are the inlet flowrates of CSTR 1 and CSTR2 respectively, and Qcwl and Qcw2 are the cooling water flowrates of CSTR 1 and CSTR2 respectively.

The MVCs for S1 and $2 are given in expressions (43) and (44) respectively.

Table 3. Input specifications

Name u I u 2 S1 Qn QI2 $2 Qcw! Qcw2

Qn + QI2 -< 0.8 (m3s -1) for s1 Oll > 0.05 (m3s -1) (43)

QI2 >- 0.05 (m3s -I)

for $2 ~ 0.05 (m3s -1) < Qcwl -< 0.8 (m3s -I) 0.05 (m3s -1) < Qcw2 < 0.8 (m3s -1) ( (44)

L&earised models

Using the Control Design Interface of SpeedUp,12 the iin- earised models of the two control schemes are obtained. The state-input equations of the linearised models for S 1 and $2 with state variables being scaled by their steady state values are given by equations (45) and (46) respec- tively. For S1,

~¢ = Ax + BIn (45)

and for $2,

where

A =

k = A x + B2u (46)

-0.038 0 0 0 -17.900 -17.975 295.869 0

0.021 0 . 0 2 1 0.189 0.070 0 0 0.388 -0.800

0.031 0 0 0.018 0.098 0

0 0 0.062 0 0 0

B1 =

8 2 =

0.2228 52.8018 -0.0387

0 0 0 0 0

0 0 0

-0.0841 0 0 0 0

0 0 0 0

0 -0.055 0 -17.900 0 0.013 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

-18.009 295.865 0 0.013 0.043 0.059

0 0.379 -0.622

0 0 0 0

O. 1820 68.4427 -0.0316

0

0 0 0 0 0 0 0

-0.1057

Assessing manipulated variable constraints: Y. Cao and D. Biss 43

By specifying the maximum outlet temperature change to be 1.5 (K), the output scaling factor can be determined from dividing the steady state value, 362.995, by 1.5. This forces the scaled output-state equation to be the same for both schemes:

where

y = C x (47)

0 0 2 4 2 0 0 0 0 0 ] C = 0 0 0 0 0 0 2 4 2 0

Using the matrices A, Bl, B2 and C, the transfer matrices of S1 and $2 can be derived as:

G1 (s) = C(sl - A)-IBi (48)

G2 (s) = C (sl - A) - 1 B2 (49)

The Hermitian matrices of S1 and $2, PI (to) and P2(oo), which are needed in the f.o.r, analysis are defined using Gl (iw) and G2(iw) respectively, i.e.

P1 (to) = GH(iw)G1 (ito) (50)

P2(w) = GH(it-o)G2(iw) (51)

Steady-state fo.r. analysis

The steady state f.o.r, analysis is carried out using Re- mark 22, i.e. Equations (30) and (31). Firstly, the input constraints of S 1 and $2 are determined from the MVCs of expressions (43) and (44) to give expressions (52) and (53). These are determined from the fact that the variables of the linearised models are the offsets of the physical vari- ables from their steady state values, e.g. ul = Ql] -0 .339, where 0.339 is the steady state value of Qn (see Table 2).

[,1] l S1 -1 0 u_< 0.289 (52) 0 -1 0.211

$2

1 0 -1 0

0 1 0 - 1

0.35 0.4

u <_ 0.528 0.222

(53)

0.5

0.4

0.3

0.2

o.1

0

-0.1

-0.2

I - , , , ' ' ' 19 f , ' '

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 ul

Figure 2. The contour lines of the output radius of SI

0.5- ' - - - - ' . . . . ' . . . . . . ? ' ~ ' ~ - ' - - - ' - - - '- ~ - 12.9

,, 0.4" ~ "~10 I, _

0 .2 - i" -0.1-

-0.2 2 ~ ~ ~ - - - . - 6.3

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 ul

Figure 3. The contour lines of the output radius of $2

Table 4. The steady state f.o.r.

Next, the steady state gain matrices of SI and $2 are calculated as:

r 46.5965 0 ] Gt(O) = L 21.0153 32.7704

G2(0) = -6.7706 -17.8289

After calculating PI(0) = Glr(0)Gl(0) and P2(0) = G2r(0)G2(0), the contour lines of the output radii, ~/ul"p1(0)u of SI and ~/uTP2(0)U of S2 are considered and shown in Figures 2 and 3 respectively. The dashed line triangle in Figure 2 and the rectangle in Figure 3 represent the feasible input regions of S 1 and $2 from ex- pressions (52) and (53), respectively. Figure 2 shows that

Scheme R(0) __R(0) S1 19.2285 6.3031 $2 12.9263 3.6149

the minimum f.o.r, of SI is 6.3 on the line, u2 = -0.211, and the maximum f.o.r, of $1 is 19.2 at the extreme point: (ul, u2) = (0.411, -0.211). Whereas Figure 3 indicates that minimum f.o.r, of $2 is 3.6 on the line u2 = 0.222, and the maximum f.o.r, of $2 is 12.9 at the extreme point: (ul, u2) = (0.35, 0.528). More accurate results are calculated using Equations (30) and (31) and are shown in Table 4. These results show that both S 1 and $2 have f.o.r, larger than one, thus both schemes are controllable. However, S1 has larger f.o.r, values than $2, hence S1 is expected to have better closed-loop performance than $2.

44

EP1

l u2

0.211

0.2

Assessing manipulated variable constraints: Y. Cao and D. Biss

EP2 0.289

Figure 4. Feasible input amplitude region of S1

Dynamic fo.r. analysis

When w :~ 0, the feasible input amplitude region is used instead of the feasible input region defined for the steady- state analysis. Using Lemma 7, i.e. Equation (10), the constraints of the input amplitudes, fi, for S 1 and $2 can be obtained from expressions (52) and (53), respectively, as; [,!] o-.1

for S1 1 fi _< 0.289 (54) 0 L 0 . 2 1 1 J

~_<

0.35 ] 0.4 0. 528 / 0.222 J

(55)

10 10

for $2 01 01

For S 1, the second and third constraints can be omitted because the first constraint is tighter than the other two (see Figure 4). This gives the feasible input amplitude region for S1 as:

U1 = {U = [/~I t~2 ]T I Ul -F t~ 2 ~ 0.2}

The shadowed area in Figure 4 represents the feasible region. The edge of U1 is:

~l + t72 _ 0.2 (56)

which is represented by the solid line, h, in Figure 4. There are two extreme points for this edge:

HEPI = [ 0 0.2 ]T

UEP2 = [ 0.2 0 ]T (57)

In Figure 4, the extreme points are marked as EP1 and EP2.

The same analysis can be done for $2 where the second and third constraints on ~ in expression (55) are elimi- nated, since these are automatically satisfied if the first and fourth constraints are satisfied. Therefore the feasible input amplitude region is determined as (the shadowed area in Figure 5):

U2 = {fi = [ Ul fi2 ]T [ fil --<- 0.35, ~72 -< 0.222}

Therefore, the edge of IU 2 is composed of two constraints:

hl: ~j _< 0.35

h2:if2 _< 0.222 (58)

0.528

0.222

u2

h2 EP ~ ~<~i~.~i~f~ I ~/~

0.35 0.4

Figure 5. Feasible input amplitude region of $2

hl

ul

These correspond to the solid lines h 1 and h2 in Figure 5. There is only one extreme point:

uEp = [ 0.35 0.222 IT (59)

Next the frequency based f.o.r, can be calculated using P1 and P2 of equations (50) and (51). Note that in this case study m = 2. According to Theorem 15, R = Rmi n and according to Theorem 19, R = Rmax. Therefore, us- ing Equation (19) and expression (56), the frequency de- pendent _R_ of SI is derived as:

0.2 _Rl (co) = [']

Whilst using Equation (26) and the extreme points of Equations (57), the R of SI is yielded as:

R l ( w ) = m a x ( l [ 0 0 ' 2 ] l P , ( w ) l [ 0 0 2 ],

Similarly, the _R_ of $2 is determined by calculating Equation (17) using hl and h2 of expression (58), while the R of $2 is directly calculated for the extreme point ~EP of Equation (59) using Equation (26), i.e.

/

R2(f-O) = min [ 0.35 ,

/

and

0.222

i[01 liP [ °

E °" ] R2(m) = 0.35 0.222] IP(w)I 0.222

Assessing manipulated variable constraints: Y Cao andD. Biss 4 5

10 ~

10 0

10 -1

c~10 ~

10 -3

10 4

~ ~ . . . - . - $1 Rma)

"'-'.::, ......

10 -s . . . . . . . . i , . . . . . . 10 -1 10 0 101 10 2

Frequency (rad/s)

Figure 6. The feasible output radii of SI and $2

Table 5. The frequency bands of f.o.r.

~ 10 °

._ ._ . " - .

~ . ~ " " . " " ~ [ - - $1 SVmin ~ ' " - " " . | - - S2 SVmin

~ ' " - " : l - - sl SVr.~, "- / s svm ,

x \ \

x \ \

\

10 -~ 10 0 F requency (rad/s)

Figure 7. The maximumandminimum singular values of S1 and $2

Scheme ~R (rad/s) w R (rad/s) S1 1.1 0.62 $2 0.62 0.41

The calculated curves of RI, R2, R__ 1 and R 2 a re shown in Figure 6. According to expressions (32) and (33), the frequency bands are determined from the curves in Fig- ure 6 and given in Table 5. The larger frequency bands of S1 (see Table 5) indicate that S1 could be perfectly controlled within a wider frequency range than $2. The curves in Figure 6 also show that input scheme S1 has larger R and R than input scheme $2 for all frequencies. Thus, better closed-loop performance subject to MVCs is expected with scheme S 1.

Singular value analysis

To perform singular value analysis, the inputs need to be properly scaled. TM 14 From the input constraints (52) and (53), the largest feasible symmetric input regions of S I and $2 can be determined as the following input con- straints:

f luzl ~ 0.1 (60) for S1 lu2l -< 0.1

t tull -< 0.35 (61) for $2 l u21 -< 0.222

For S1, the choice of the bounds is not unique and those given are selected on the basis that they give the same bound for ul and u2. For $2, the selections are more straightforward, since the minimum values are chosen from the two possible options. Thus the input scaling matrices of SI and $2 are chosen to be:

0.1 0 ] SIl = 0 O. 1

and

Table 6. PI controller parameters

Controller 1 Controller 2 Name Kc r l K~ T/ SI 0.0144 1.6535 0.0204 1.4629 $2 -0.0441 1.7739 -0.0375 1.6600

] 0.222

The maximum and minimum singular values of the scaled frequency response matrices of S1 and $2, Gl(iw)Sn and G2(io3)812, are calculated and shown in Figure 7. By specifying the lower frequency band, w__ B, and upper frequency band, wB, as the lowest frequencies at which the minimum and maximum singular values are equal to one respectively, i.e.

~ (w) >_ I, 09 _< w___B (62)

~(~o) > 1, co _< ~ (63)

Then, according to Postlethwaite and Skogestad,t5 within the frequency range lower than the frequency bands, per- fect control could be approximately achieved. Figure 7 shows that the frequency bands of S1 and $2 are almost the same. Considering the behaviour below wB and o3_ B, $2 has larger singular values than S1. These results sug- gest that $2 is a better control structure than S 1. However the simulation results indicate that the prediction from the singular value analysis is wrong. This is due to spec- ifying the symmetric bounds for S1 in expression (60) which are too conservative (see Figure 2).

Simulation results

SIMULINK 16 was used to model the process. The PI controller parameters were tuned using the process reac- tion curve method with the one-quarter decay ratio cri- terion 17 (see Table 6).

In Figures 8(a) and (b) and 9(a) and (b), the outputs of S1 reach steady state more quickly than those of $2 when-

46 Assessing manipulated variable constraints: Y. Cao and D. Biss

(a) First Output Response 364

~. 363.5

363

-362.5

I- 362

361.5 0

0.45

I I

50 100 150 Time (s)

(c) Input Response of Sl

200

0.4

E v 0.35

O ~. 0.3

/ \

I "\ / ~ ' ~ . ~ " . . . . . . . . . . . . .

(b) Second Output Response 363"41 !~

~" 363"21 ;/~ t"

® i i

, ,

~- 362.6lV 362 4 [

0.5

0.4

E 2 0.3 E 0 ~. 0.2

50 100 150 Time (s)

(d) Input Response of $2

0.25 - - 0.1 0 50 100 150 200 0

Time (s)

200

/ ' \ r / \ / \ , / " ~ . , . . . ' ~ . _ ~ ' ~ . ~

~./"

50 100 150 200 Time (s)

Figure 8. Closed-loop response to a -12 (K) cooling water temperature disturbance

364.5

~" 364

363.5 Q_ E I- 363

362.5 0

(a) First Output Response

; I

i

0.35

50 100 150 Time (s)

(c) Input Response of $1

200

0.3

E 0.25

O ~. 0.2

t

" l / . \ . ~- .

\ .1" . ]

~" 364 [

"~'363.5 '-1

Ill Q. E 363 I-

362.5 0

0.8

~0 .6 E G)

0.4 14.

0.15 0.2 0 50 100 150 200 0

Time (s)

Figure 9. Closed-loop response to a 12 (K) cooling water temperature disturbance

(b) Second Output Response

I -

i . \

i . .

\ . / "

50 100 150 Time (s)

(d) Input Response of $2

200

f , /" \.

\ /" .\ . / ' ~ x . ~ . - - - . _ . . . .

/ . . J

50 100 150 Time (s)

200

Assessing manipulated variable constraints: Y. Cao and D. Biss 47

ever cooling water temperature disturbances of - 1 2 (K) or 12 (K) are applied to the process. Figure 9(d) shows that the constraint ul -< 0.35, corresponding to Qcwl -< 0. 8, of $2 is the most active constraint. These results con- firm the prediction made by the f.o.r, method.

Conclusions

A new technique, called feasible output radius analysis, for assessing the effect of MVCs on a process' input- output controllability has been proposed. Theoretical analysis shows that this method is an extension of the more traditional SV analysis method and is more flexible in dealing with various kinds of MVC.

A couple of new indicators, R and R, are introduced for ranking alternative control schemes on the basis that the larger R and R, the better the closed-loop performance in the presence of control constraints.

The new indicators can be calculated by solving optimi- sation problems. For linear MVCs, some simple bounds on R and R can be derived. For the case co = 0, or when the number of related variables in the constraints is less than 3, the indicators can be analytically expressed.

At the steady state, the f.o.r, can be exactly calculated for unsymmetric input constraints. This leads to a more accurate prediction of the expected closed-loop perfor- mance of a particular scheme and is less conservative than the traditional SV analysis. In the dynamic f.o.r, analy- sis, the feasible input amplitude region has to be used in- stead of the unsymmetric feasible input region. Therefore the dynamic controllability prediction is more conserva- tive than that of the steady state f.o.r, analysis. However, the results are still less conservative than those of the SV analysis. This is because the f.o.r, analysis can deal with any linear input constraints without needing to change them into Euclidean norm bounds as is required with SV analysis.

A case study, i.e. a two-CSTR process was investigated using the new method. Via the case study, some superior characteristics of the new technique were demonstrated, such as ease of calculation, flexibility in coping with dif- ferent kinds of constraint and less conservative results when assessing a process' controllability.

Acknowledgements

This paper also includes results of the Network in Chem- ical Process Control which is supported by the Human and Capital Mobility Programme of the European Com- munity, contract CHRX-CT94-0672.

References

1. Biss D. and Perkins, J.D. 'ECC-93 Proceedings' (Groningen, The Netherlands), 1993, 1056.

2. Cao, Y., Biss, D. and Perkins, J.D. Computers and Chem. Engng 1996, 20.

3. Johnston, R.D. and Barton, G.W. Industrial and Engng

Chem. Res. 1987, 26, 830. 4. Morari, M. Chem. Engng Sci., 1983, 38, 1881. 5. Fararooy, S., Perkins, J.D., Malik, T.I., Oglesby, M.J. and

Williams, S.,Computers and Chem. Engng, 1993, 17, 617. 6. Perkins, J.D. Preprints IFAC Symposium DYCORD

(Maastricht, The Netherlands), 1989. 7. Owens, D.H. 'Feedback and Multivariable Systems' Peter

Peregrinus, 1978. 8. Fletcher, R. 'Practical Methods of Optimization, Vol. 2:

Constrained Optimization' John Wiley, New York, 1981. 9. Noble, B. and Daniel, J.W. 'Applied Linear Algebra'

Prentice-Hall, New Jersey, 1977. 10. de Hennin, S., Perkins, J.D. and Barton, G.W Fifth Int.

Symp. on Process Systems Engineering (Kyongju, South Korea) 1994, 297.

11. Heath, J.A. and Perkins, J.D. ICTwmE Research Event (London, UK) 1994, 841.

12. Prosys Technology Ltd, Cambridge, UK. 'SpeedUp User Manuals: Vols 1 and 2, Issue 5.Y2.', version 1.0 edition, July 1991.

13. Morari, M and Zafiriou, E. 'Robust Process Control' Prentice-Hall, New Jersey, 1989.

14. Skogestad, S. and Morari, M. Industrial and Engng Chem. Res. 1987, 26, 2029.

15. Postlethwaite, 1. and Skogestad, S. in 'Essays on Control: Perspectives in the Theory and its Applications' (Eds H. L. Trentelman and J. C. Willems) Birkhfiuser, Boston, 1993, 269.

16. MathWorks, South Natick. 'SIMULINK User's Guide' 1992.

17. Stephanopoulos, G., 'Chemical Process Control: An Introduction to Theory and Practice' Prentice-Hall, New Jersey, 1984.

Nomenclature

Symbols []

I1" 112 I ' I

A B A, B and C Cli

Co~

C C m X m

d di dA~

dIi

E / R

ei e G, (Gi or G2)

end of proof Euclidean norm absolute value, for vector and matrix it represents element by element absolute values superscript, represents the complex con- jugate number raw material product matrices of the system' state equation inlet concentration of the ith CSTR (mol/m 3) outlet concentration of the ith CSTR (mol/m 3) set of all complex numbers set of all m × m complex matrices input constraint vector, see (9) ith element of vector d a subset of d, corresponding to the active constraints at the ith extreme point a subset of d, corresponding to the inac- tive constraints at the ith extreme point activation energy divided by perfect gas constant (K) ith column of a unit matrix transcendental number transfer matrix (of S I or $2)

48

H

H I:l

HA/

Hli

hi

I i K0 K, gvi

P, (P1 or P2)

p+

Qii Qcwi R ~m

R R, (Rl or R2)

_R, (R 1 or R2)

m

Rmax Rmin Rmax Rmin Sl, and $2 S. (SI1) T

Tcwi

Tcwo~

Assessing manipulated variable constraints: Y. Cao and 19. Biss

superscript, the complex conjugate trans- t pose Uai input constraint matrix, see (9) input amplitude constraint matrix with elements being the absolute values of the U elements of H u and dui a subset of 171, corresponding to the active fi an zTi constraints at the ith extreme point a subset of I7I, corresponding to the inac- fiEP tive constraints at the ith extreme point ith column of H Vji ith column of IZl unit matrix x v/S-1 xi Arrhenius constant (s -l) y reaction rate of the ith CSTR (s -l) outlet valve constant of the ith CSTR (m 3/2 s- l) Greek symbols Hermitian matrix (of S1 or $2), equal to AH GHG A generalised inverse of P inlet flowrate of the ith CSTR o- cooling water flowrate of the ith CSTR 4) set of all real numbers qbi set of all m dimension real vectors feasible output radius maximum feasible output radius (of S1 w or $2) wB minimum feasible output radius (of S1 co B or $2) WR upper bound of R COR lower bound of R upper bound of R_ Abbreviations lower bound of_R_ CSTR first and second control schemes CW input scaling matrix of S1 ($2) EP superscript, represents transpose f.o.r. cooling water temperature of the ith iff CSTR MVC cooling water output temperature of the PI ith CSTR s.t. inlet temperature of the ith CSTR (K) SV outlet temperature of the ith CSTR (K) w.r.t.

time variable heat transfer coefficient multiplied by the heat transfer area of the ith CSTR (m3s -1) feasible input amplitude region input vector and its ith element input amplitude vector and its ith ele- ment extreme point of U volume of the ith CSTR (m 3) cooling jacket volume of the ith CSTR (m 3) vector of state variables ith state variable output vector output amplitude vector

reaction heat coefficient (m 3 K mo1-1) Lagrange multiplier maximum singular value minimum singular value phase shift vector of a sinusoidal input phase shift of the ith sinusoidal input diagonal matrix with the iith element be- ing (hi frequency variable maximum frequency band of maximum frequency band of cr maximum frequency band of minimum frequency band of R__

continuously stirred tank reactor cooling water extreme point feasible output radius if and only if manipulated variable constraint proportional plus integral (controller) subject to singular value with respect to