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Process controllability analysis using linear programming
ELSEVIER
J. Proc. Cont. Vol. 8, No. 3, pp. 165 174, 1998 ,d, 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0959-1524/98 $19.00 + 0.00
PII: S0959-1524(97)00044- 9
Process controllability analysis using linear programming S. D. Chenery and S. P. Walsh Centrefor ProcessSystemsEngineering,Imperial College, London, UK Controllability analysis is concerned with determining the limitations on achievable dynamic performance. This paper proposes the use of linear programming to determine the best linear controller and corresponding dynamic performance for problems of the form: min J(K, uo)s.t.c(K, u0, w) < 0Vw C W. That is, K,u0
a controller, K, and a reference operating point, u0, are selected to minimise a specified objective, J, while ensuring feasibility for all disturbances, w, within a specified set, 14". When K is a linear time invariant (LTI) controller and the objective function J and the constraints c can be expressed as linear functions then the above problem can be solved by linear programming. This formulation encompasses a wide range of problems ranging from minimising the maximum deviation in the regulated outputs subject to disturbances of magnitude less than one (the l~ optimal control problem) to optimising the expected value of a linear economic objective (the Optimal Linear Dynamic Economics (OLDE) problem). The relationship of this work to other approaches to controllability analysis is discussed. A highly flexible framework for addressing typical process performance requirements through appropriate selection of J, c and W is presented. The relative merits of alternative approaches for defining the achievable closed loop transfer functions based on the Q-parameterisation are carefully discussed. The feasibility of the proposed approach is demonstrated on an industrial reactor example. The needs for further work are discussed. 'u 1998 Elsevier Science Ltd. All rights reserved
Keywords: controllability; linear optimisation; It optimal control
Controllability has been defined in several different ways. In the 60s K a l m a n introduced the concept of state controllability which tells us whether the state vector can be taken from any initial state to any final state within a finite time. In the 70s Rosenbrock defined functional controllability which tells us whether there exists a set of input trajectories which can generate any output trajectory. He also introduced the importance of the notion of right half plane (RHP), i.e. 'unstable', zeros for multivariable systems. Skogestad I defines input-output controllability as:
affect the achievable control performance of both SISO and M I M O linear systems 2, e.g. location of R H P zeros, condition number, minimum time delay, etc. Morari 3 notes that "we often do not quite understand yet when and how these indicators should be used', but even so ~they are applied widely and indiscriminately and lead to erroneous conclusions about controllability'. The ambiguity involved in the interpretation of these measures is a major problem for controllability analysis. One reason for the ambiguity is that many controllability techniques based on linear models rely on frequency domain analysis. However process constraints and disturbances are typically defined in the time domain. Another reason is that each indicator typically only considers one of the above limitations in isolation. Therefore there is a need for an easily interpreted linear controllability analysis technique that deals directly with time domain performance requirements and quantifies the combined effect of as many fundamental limitations as possible. Another group of techniques are optimisation-based techniques 4.5 which may be based on linear or nonlinear models. These techniques differ from each other in the model they use and the degree of idealisation of the controller. Much recent work has fbcused on the direct
"The ability to achieve acceptable control performance, that is, to keep the outputs 0') within specified bounds from their setpoints (r), in spite of unknown variations in the plant (e.g., disturbances (d) and model perturbations) using available inputs (u) and available measurements (e.g. y , , or din)." It is this definition of controllability that this paper refers to. Achievable control performance is limited by: righthalf plane ( R H P ) zeros, time delays, input constraints, sensor noise and process-model mismatch (uncertainty). M a n y controllability indicators have been developed to give measures of how these fundamental limitations
165
166
Process controllability analysis: S. D. Chenery and S. P. Walsh
optimisation of PI controllers for both linear and nonlinear systems. This analysis gives a pessimistic bound on achievable performance in that the class of controllers considered is deliberately restricted. Other work has used various idealisations of controller performance to get optimistic bounds on achievable performance, e.g. perfect control, delay limited control. Therefore an optimisation based technique which gives a measure of achievable performance, i.e. controllability, for a very broad, yet realisable, set of controllers would be desirable. Any such method must be computationally tractable as well as conceptually attractive. In this paper we use linear programming techniques to address problems of the form: rain J(K, uo)s.t.c(K, Uo, w) <_ 0Vw c W
K,uo
That is, a controller, K, and a reference operating point, u0, are selected to minimise a specified objective, J, while ensuring feasibility for all disturbances, w, within a bounded set, W, with respect to the constraints c. The objective J is a measure of performance, therefore the solution of this problem gives a measure of controllability, the best achievable control performance despite disturbances. When K is a linear time invariant (LTI) controller and the objective function J and the constraints c can be expressed as linear functions then the above problem can be solved by linear programming. This formulation encompasses a wide range of problems ranging from minimising the maximum deviation in objective variables subject to disturbances of magnitude less than one (the ll optimal control problem) to optimising the expected value of a linear economic objective (the Optimal Linear Dynamic Economics (OLDE) problem). The achievable closed loop transfer functions for a generalised plant can be expressed as an affine function of a stable parameter Q. This allows the achievable discrete time closed loop impulse responses to be defined by an infinite set of linear constraints and approximated by a finite set of linear constraints. The methods for doing this are well established 6-s and are summarised here as a precursor to a careful comparison of the alternative methods within the overall analysis framework set out below. The disturbance set W can be expressed in a flexible manner and can include a mixture of magnitude bounded signals, step signals and steady-state signals. The allowable combinations of disturbances can be restricted if desired. This flexibility is important to allow specific process requirements to be captured with adequate precision. We show how the effect of this general set W can be captured in a linear program. Possible objective functions are then discussed and it is shown that many objectives of interest can be captured in a linear program, this controllability measure therefore covers a wide range of performance definitions.
Having defined the LP problems of interest we then discuss the properties of alternative formulations. We apply this approach to investigate the controllability of an industrial reactor system about a fixed operating point to see if the required performance can be met with a controller. This demonstrates the ability of the proposed technique to solve realistic problems. Finally we review the method proposed and the needs for further development.
Achievable closed loop transfer function The closed-loop system is defined by the relationships below, where P is the generalised plant, K is the controller, w are the disturbances, u are the control inputs, z are the regulated outputs and y are the measured outputs:
z = P l l w + Pi2u Y = P21 w + P22u u=Ky
We take P and K to be linear time invariant transfer functions, z may contain any process variable including elements of u and y. w may include measurement noise. The number of elements in each signal vector (nz, nw, nu, ny) are not required to satisfy any particular relationship. The generalised regulator problem is to choose K so as to optimise some property of the closed loop transfer function (mapping) from w to z while maintaining internal stability. The mapping from w to z is given by = PI1 + P I 2 K ( I - P22K)-lp21
(1)
The Q-parameterisation 9 is used so that the closed-loop map • is affine in a free stable parameter Q
• =H-UQV
(2)
If Q is selected from the set of stable (bounded input bounded output) LTI transfer functions (Q c l l ) then all stabilising linear time-invariant (LTI) controllers are parametrised by the above expression. If P and K are expressed as discrete time LTI systems and q~ is a discrete time impulse response matrix then it is possible to express the above parameterisation in terms of linear equality constraints. The first method for achieving this uses interpolation constraints 8. There are a set of constraints on qb called the interpolation conditions that ensure that the closedloop map • can be written in the form of (2) for some
QEll. There are two types of interpolation conditions, the zero interpolation conditions and the rank interpolation conditions. The first capture limitations on the achievable ~ due to fundamental process limitations like R H P zeros and time delays. The latter capture the limitations
Process controllability analysis." S. D. Chenery andS. P. Walsh on the achievable qb due to Q not having sufficient degrees of freedom, i.e. more disturbances than measurements n,. > ny and/or more regulated outputs than control inputs n: > n,. Any problem with insufficient degrees of freedom is known as multi-block and has an infinite number of constraints due to the rank interpolation conditions. Only one-block problems, with nw <_ny and n: _< n,, avoid these infinite constraints. Therefore if the problem is multi-block it is embedded in a one-block problem which has only zero interpolation conditions. In this paper the delay augmentation (DA) algorithm is used which involves U and V being augmented by N pure delays so that the problem is oneblock. As N increases the zero interpolation conditions of the DA problem approximate the original problem increasingly closely. The resulting constraints on qb are linear and in the time-domain. The interpolation conditions are given by a finite number of zero interpolation conditions of the kind shown in Equation (3).
167
The effect of the choice of N and Ni/will be discussed later when the problem definition is complete. There is another technique which avoids calculating these interpolation conditions at all. For this technique the discrete impulse response of Q is approximated as 6 L
1
Q = Z q(k)z-k
(6)
k=O
with finite length L. The assumption that Q is a finite impulse response of length L means that this technique is an approximation and it is necessary to select L large enough to include the optimal Q. Using this approximation means that the discrete elements of qb can be expressed as follows -min(k, L - 1 )
¢~(k) +
= hi/(k) p=l
n = l rn=l
(7) tti,z. (H ,
-
a,)x/Xo)/~)(Xo)
= 0
(3) where
where 3,.~" and x~° are given by the left and right nullchains of U and V, respectively, and k0 is an 'unstable' zero of U or Vs. These conditions allow the elements of any feasible ap to be expressed as follows:
Z a~qx°'k(l)~)pq
(4)
(l) = b ij)'°'k
p = l q= I /=0
where n:
tl~
OC
p=l q=l
/=0
and
a )°k(tl =
t- slx t=0 s=0
(sl(Zl
] .1 ),=X 0
these elements of aO'qX°'k(1), qbpq(l) and b#'x°'k can be stacked and arranged into a constant real matrix Azcro, a variable vector ¢ and a constant real vector bzero, respectively, so that the zero interpolation conditions expressed by Equation (4) can be stated as Azero¢ = bzero
(5)
4~ is infinite dimensional. The method used to get a finite dimensional problem is to truncate ¢ for each zi and ~,) to a finite length N#. The elements of Azero associated with ¢ beyond this horizon are removed from the problem. This implies that any ¢ satisfying the truncated constraints also satisfies the infinite constraints for ¢/j(k) = 0Vk _> U#.
k O(k)(p) : ~ ~.~
u,.(r - p),mi(k - ,'1
r=p
Much as in the zero interpolation case these elements of C~(mk)(P), q,m(P), eli(k) and hi/(k) can be stacked and arranged into a real constant matrix Aq, variable vectors q and ¢ and a real constant vector h, respectively. In this way expression (7) can be stated as
Note that each SISO impulse response qnm(k) could be assigned a distinct length Lnm, but this would normally introduce an unhelpful increase in complexity. As with the zero interpolation constraints ¢ is truncated. In this case this implies nothing about the values of ¢~(k)k >_ N#. Either of these methods, the zero interpolation conditions or the Q-approximation, provides linear constraints defining the achievable ¢.
Disturbance description In typical process applications the disturbances will not be precisely characterised. However the following questions can usually be answered for the key disturbances. What range of values does the disturbance take? How quickly can the value change? Do changes occur frequently? Based on answers to these questions, an adequately precise disturbance description can be developed. Very slowly changing disturbances should be treated as steady-state. Disturbances which change significantly
Process controllability analysis."S. D. Chenery and S. P. Walsh
168
over the process transient response time can be modelled as instantaneous changes filtered through a transfer function W1. If the maximum rate of change is 10% of maximum variation per sample a finite impulse response (FIR) filter Wl,
infinite sum. However, the steady-state gain of Q can be evaluated as a finite sum and the corresponding steadystate gain of ¢ computed given the steady-state values of H, U, and V. Hence the closed loop steady-state gains, G~S.d zw , and V0, can be computed using finite dimensional linear constraints in both cases.
9
WI = 0 . 1 E q
-i
Step disturbances
i=0
would convert a step change between the limits to a ramp at the maximum rate, thereby excluding the unrealistic disturbances. Such filters can be included in the process model as shown in Figure 1. If changes occur infrequently compared to the process response time then a step disturbance description is more appropriate than a persistently varying signal which is only constrained to be magnitude bounded. Typical time-domain disturbance descriptions can therefore be represented in terms of three types of disturbance signals: persistently varying (persistent) disturbances, step disturbances and steady-state disturbances. Given a finite impulse response description of ¢ we can compute the worst-case response for each of these signal types. For convenience (and without loss of generality) we consider all disturbances relative to a reference point corresponding to Wcentre = (wh+ wt)/2 and scaled so that a unit variation corresponds to (w h - w t ) / 2 . The disturbance deviations about this point will be symmetric with a maximum deviation of unity (upper b o u n d = - l o w e r bound = 1) and the corresponding output deviations will also be symmetric. Steady-state disturbances
The maximum output deviation on zi in response to wj will be given by -v O
< vO
(9)
k=O
zo(t ) = - ~ j + 2
= - ~
¢0(k
Co(k) + 2
k=0
Co(k) k=0
The maximum deviation is therefore given by
- Vo <- - E
¢ij(k) + 2
Co(k)
_< vij
k=O
j
t>0
for
(10)
= 1. . . . . nz "= 1. . . . . nw
The steady-state component can be computed as above. With the interpolation constraints the maximum deviation for t < N 0 will be the maximum deviation for the infinite horizon. With the Q-approximation the peak step response calculated in this way will be a lower bound on the infinite horizon step response deviation as the later values of ¢ are unconstrained. Persistent disturbances
However we will only have access to ¢0 up to a finite horizon N o. With the interpolation constraints the later values of ¢0 have been set to zero and therefore the finite horizon sum equals the infinite horizon sum. With the Q-approximation the finite sum is not equal to the r--
w
We consider steps between any two levels within the lower and upper bounds on the disturbance value. The worst-case will be given by a step between the upper and lower bound or vice versa. The effect on output zi of a step in disturbance wj from the lower bound ( - 1 ) to the upper bound (1) would be,
..........
~
.
.
.
Wl
.
I
i . . . . .
SZ_ ..... J-l_
d
For persistently varying disturbances we have
vo = ~ ICu(k)l k=O The absolute value in the above expression requires a change of variable to give a suitable form for a LP. We define ~ = q ~ + - ~ - , where @+ and • have only nonnegative elements. We can then write OO
vU = ~
¢/~ (k) 4- ¢+ (k)
(11)
k=O
Figure 1
Block d i a g r a m for a u g m e n t e d system
Replacing the sum to infinity with the sum to N O• has the same implications as for the step disturbances.
169
Process controllability analysis: S. D. Chenery and S. P. Walsh Combining disturbances
For each disturbance type we can calculate v O. F o r a purely worst-case analysis we get the worst deviation on Zi as tIW
vi = Z
(12)
vii
i-I
If such an analysis is felt to be excessively conservative then we may define vi based on the m a x i m u m deviation over specified subsets of disturbances, e.g. any n step disturbances combined with all the steady-state and persistent disturbances. The constraints for each allowable combination could be appended to the LP. The values of v~ couple the disturbance description with the overall performance requirements.
Capturing performance requirements To complete the LP formulation we need to define the closed-loop performance requirements. That is, for the problem min J(K, uo)s.t.c(K, uo, w) < 0Vw c W K, ut,
we need to define J and c which specify what controllability problem we solve. The most basic requirement is feasibility, i.e. that constraints are not violated. We assume that the model used has been linearised about a steady-state with regulated output, measured output, disturbance and control input levels given by Zlin, Ylin, Wlin, Ulin and that all constraint limits are in terms of the original values. We denote the closed loop steady-state gains as G ss'ct and the open-loop steady-state gains as G ~s'°z. The requirement for feasibility can be expressed as below where each constrained variable is assumed, without loss of generality, to be an element of z. ~_l i < --
---i < __
Rather than imposing the feasibility requirement directly we may wish to determine the fraction of the disturbance deviations for which feasibility is achievable. This can only be expressed as a linear program if Zr~r is independent of K and u0, i.e. for a fixed operating point u0. In this case we can scale the deviations z~ by the minimum distance to a constraint (min(z) - : ~ e f E ~ e f Z~)) SO that a deviation less than 1 corresponds to constraint feasibility. We can then make the objective of the LP. v0 = min max vi K
The values of z and y corresponding to Wcentre and a corresponding input value u0 are given by
(J(K) = max vi) i
o(~')1. : u0 + K ( s ) ( y - y ' % 0 ref -- G':;:;,cI(~,T'-
Wcentre )
(17) (18)
So our optimisation becomes •"~SS, c l I - -
m i n o ref %- O-ow /~1 - Wcenlre )
_ref ,~ss,~ l/ .~ss,ol / .~ = Uzw ~14centre -- Wlin) %- Lrzu ~,U0 -- Ulin) %- Elin
(16)
I,n:
If the disturbances are scaled down relative to Wcentre by a factor ±~,0then feasibility will be maintained. ±~0therefore gives the fraction of the disturbance set which can be tolerated. The use of the 'disturbance fraction' as a controllability measure is discussed more generally in Walsh and Perkins 5. Note that if all the disturbances are persistent then v0 is the ll norm of the mapping from w to z and the problem being solved is an Ii optimal control problem. If we have a reasonable basis for penalising the maximum deviation on each objective z~ then it could be appropriate to minimise a weighted sum of the deviations vi rather than the m a x i m u m deviation. The weighted deviations could be minimised subject to feasibility constraints. This type of objective is highly flexible. Its key limitation is the difficulty of defining appropriate weights for the deviations. If we have reasonable information about the economics of the process then a more systematic approach can be taken to defining the objective for the LP. Given a linear(ised) economic objective we can determine the optimal expected value of this objective over the disturbance set. If the objective function is expressed as o(w,u) then by virtue of closed-loop linearity, the expected value of the objective, ~, is simply the objective for the expected value of the disturbance, ~7'.
0(~') :
z)Vw c Wi = 1 n:
i-l,n:
(19)
K.uo
(13) yref
,,~ss,ol /
~
.~ss,oD
= Uyw 0 ~centre -- Wlin) -F Oy u tU0 -- Ulin) -F Ylin
(14) with the controller given by u = uo + K ( s ) ( y - yref). The feasibility requirement can therefore be expressed as . l + v, __ "i < Z~ef < zhi - vi
i= 1,nz
(l 5)
Note that if o is included in : then G)";'I;~t can be constructed as discussed previously. Alternatively, if o is not in z and all inputs appear in : then G~;]',fl can be obtained as Go .,~,s,,,lG,,SSd + GS~,,,l
If we linearise around the steady-state optimum for i~' = Wcentre and the active set of constraints is assumed to remain constant over the disturbance set then this
170
Process controllability analysis: S. D. Chenery and S. P. Walsh n
problem reduces to minimising J = J o - ~ ~'iPi in the i--1
absence of constraints. J is the objective function at the optimal operating point, Jo is the steady state optimum and ~-i is the Lagrange multiplier of the constraint on zi. Young et al. 1° present a conceptually similar approach to evaluating operating economics based on the Q-approximation. Their approach diverges significantly in detail. They do not consider the use of interpolation constraints and consider only step disturbances and nominal economics. Their disturbance set is defined by r = {(ab, a d ) l d = de + a a ; d - < d < d +}
for which the base value db, the magnitude and direction of the step Aa are uncertain. They reduce the set of disturbances to be considered when testing for feasibility to just the critical (vertex) points ( d - , d + - d - ) and (d-, - ( d + - d - ) ) . For a large number of disturbances this set would still become large, 2 "w. By choosing the reference disturbance as dcentre we avoid the need to consider the above critical points individually. They allow convex objective functions which means that their problem size is limited by the capacity of convex programming algorithms rather than linear programming algorithms which can handle larger problems. They apply their method successfully to a simple problem. Despite the differences noted, their work is very much in the same spirit as ours.
Properties of the LP formulations We have shown that many performance requirements of practical interest can be formulated in terms of a LP which can be solved to determine the best achievable performance for an optimal choice of K and, if appropriate, u0, thus providing a measure of controllability. The LPs involve a number of approximations so it is important to consider how the results of the LPs relate to the solution of the original infinite horizon discrete time problem. In the interpolation constraint method the finite horizon response is an upper bound on the infinite horizon response due to the additional requirement that the zero impulse response beyond the horizon is achievable. If delay augmentation is used with finite N then the resulting solution is a lower bound on the finite horizon problem. For a single block ll problem it is possible to calculate the values for N o• such that the constraints (bo(k) = 0Vk > N 0. are inactive 8 and the finite horizon optimum becomes equal to the infinite horizon optimum. For a multi-block 11 problem this calculation can be carried out for each value of N and the finite horizons N O can be chosen accordingly. The solutions as N increases will then provide a sequence of lower bounds which converge to the infinite horizon optimum. For problems other than the ll problem the minimum value of N O needed to make the finite horizon optimum equal
to the infinite horizon optimum is not known. The solutions generated are therefore lower bounds on an upper bound and hence simply approximations to the infinite horizon problem. In the Q-approximation method the optimal finite horizon response is a lower bound on the optimal infinite horizon response (with the same parameterisation of Q) since the finite horizon problem disregards potential deviations beyond the horizon. If L is less than any horizon N 0 ÷ 1 the resulting solution is an upper bound on the finite horizon problem, with arbitrary Q, as elements of Q affecting the finite horizon response are restricted to a value of zero. For L <_ N O + 1Vi, j the solution for the Q-parameterisation is a lower bound on the infinite horizon problem provided the finite horizon objective and constraints are not affected by qmn(k) being zero for k > L - 1. This condition is not guaranteed for step or steady-state disturbances (due to the steady-state effect being ~ ¢0(k)) and the Q-approxk=0
imation gives just an approximate result for these types of disturbances. With both methods, if the solution for varying L or N takes an almost constant value as L or N vary over most of the horizon then the behaviour beyond the end of the horizon can be presumed not to be significant and the result can be taken as a true estimate of the achievable performance. This convergence can be tested by reconstructing the controller K and calculating the infinite horizon response to within a desired precision (Dahleh and Diaz-Bobillo, 1995) to give an upper bound on the infinite horizon solution. In Ajbar et al.~l the interpolation constraint approach is criticised for giving only an approximate solution to the Ii problem, in contrast to the Q-approximation. However, both methods can be used to construct a sequence of lower bounds to the infinite horizon optimum or to construct upper bounds based on a specific solution. Also, both methods exhibit the qualitative property that consistent results as N or L increase indicates convergence to the true optimum. At least for the ll problem both methods seem to be equally well founded. If the problem is one-block then it is more efficient and exact to solve the zero interpolation condition problem. There are also differences in the structure of the associated linear programs. The zero interpolation conditions generate nz
nw
X06Auv i = l j = l
(~ru,(,k0), av,(Z0) = structural indices of U, V) constraints and t/z
V/w
ZZN~+
1
(N o = FIR length of q~0)
i--1 j = l
variables within the LP.
171
Process controllability analysis. S. D. Chenery andS. P. Walsh The Q-approximation method generates
Industrial reactor problem
ZZN +I
The case study used to demonstrate the LP controllability analysis is a linear model of an industrial reactor example presented by Walsh and Malik t3. The linear model of the exothermic plug flow reactor in Figure 2 is in the form of a transfer matrix of gain~delay lag elements. Note that the analysis could equally be applied to a more general linear model. The perlbrmance requirements are given as a set of target steady-state values and constraints for the variables Tin and Tou t (Tabh, 1). Note that the desired close loop response is otherwise unspecified. There are no steady-state degrees of freedom so uo is fixed. In a previous control study serious difficulties had been encountered in meeting the constraints above, even assuming rate limits on the disturbances. The controllability analysis below therefore focuses on feasibility. There are only two possible manipulated variables S~ the split between the product and the recycle and $2 the split between the heat exchanger and the bypass (& = 1 means 100% recycled and $2 - 1 means 100% bypassed). Fin is determined by the requirements of another plant. There are four measured variables Ain (a delayed composition measurement of the key reactant), Fr~cyd~, Tin and Tom The disturbance d is a well defined change between two operating regimes, A and B, which involves three feed component flowrates. The operators wish to make this change as fast as possible, but there is some flexibility available to reduce the rate of change if a step change cannot be accommodated. The objective is therefore to minimise the switching time while meeting the performance requirements. Since the disturbance only occurs in a single direction with respect to these flowrates it is represented as a single disturbance. The temperature measurements, disturbance and manipulated variables are all normalised by their maximum acceptable deviation relative to steady state with a disturbance d~em~. This gives the transfer function matrix
iljl constraints and
I~ Z
.... Ni/ +
ki=] / 1 (Lnm = FIR
l] + [nn~l"'mLnm~j + ]1
length
of
qnm)
variables. On the whole the number of constraints in the zero interpolation conditions will be less than the number of variables, in which case the interpolation condition will produce a smaller LP problem. However the Q-approximation problem on the whole will be more sparse which produces an LP which is faster to solve. Also if the DA algorithm must be used to produce the one-block problem then the number of constraints produced by the zero interpolation conditions will grow as the number of delays augmenting the problem are increased in the algorithm. Therefore which method is preferable in terms of LP complexity is highly problem specific. The Q-approximation is by far the simpler and more straightforward of the two techniques to implement. There are examples of various problems solved by each technique in the literature, (e.g. Refs 10-12) for the Q-approximation and in Dahleh and Diaz-Bobillo 8 for the zero interpolation method. We solved the very simple SISO problem presented in example 1 of Ajbar et al.I 1 using both techniques. Several iterations of the Q-approximation technique were required to check if a good choice of L had been made. The interpolation constraint technique solved immediately and exactly since the problem is one-block. In other examples, including the case study below we have successfully used interpolation constraints while the Q-approximation LP has not solved successfully. It should be noted that both approaches have a variant in which finite step responses replace the finite impulse response. For step and steady-state disturbances this will give a sparser LP and may have better solution characteristics. It is clear from the above discussion that there is an element of approximation in the LP formulation. We believe that provided convergence of estimates is obtained for varying L or N this approximation is of little consequence. The approximation of a nonlinear process by a linear model is of primary importance followed by the approximation of the continuous time linear model by a discrete time model. Variable transformations should be used where appropriate to improve the quality of the linear model and sampling should be faster than both open-loop transients in response to disturbances and the fastest expected closed loop response time. Some iteration might be necessary to achieve this.
, Freer
Recycle '
Fin
]
i
Feed {disturbance, d)
'
-
'
"tSll
Tm ' Exchanger] i bypass l
i
Atn Plug flow [e;lcl(lf
[ Figure 2
Prodtlcl
Process schematic
: Tout:
Process controllability analysis: S. D. Chenery and S. P. Walsh
172
Table 1
Performance
requirements
Steady-state
Variable
value
Upper
limit
c
300°C None
(251 $)r
to
(AinFrecycre Tout$n)r
0.065 2.1 5e~“.z5s 1+0.6s
shown
5
z
+
+
+
60
100
120
140
160
160
200
+ 1.96-
2 1.94-
0
g,.,,.
6.61
0
l19-
-4iTJle&45
t
1 86
+
t+
_ 1.77
_
0.49em04s 1+0.6s
1.66’
0
20
40
60
No.
A state space realisation of the transfer function was found and discretised using the bilinear Tustin method. The sampling rate was chosen to be substantially faster than the frequencies up to which disturbance attenuation was required. The measured outputs y and regulated outputs z are given by
(20)
whilst the exogenous input w and the control inputs u are given by
w=d,u=
+
1.96-
I +0.02s
-3,5geA=” I f0.6~
+
2-
_ , ,8ge-0.W8”
-13.98e-035’ I +0.6s
“f&6;
+
2.02
2
D
T=
+
+
.P
is
T relating below:
+
2 04
Lower limit
None 650°C
400°C 600°C
i-ii, Tout
2.06
for Tin and T,,,
0s1 s2
The scaled manipulated inputs 51 and $2 have been included in the regulated outputs to incorporate the input constraints on the splits. Zero steady-state offsets for Tin and To,, must be enforced. This involves adding two linear time domain constraints onto the resulting 1, LP. u. should be less than 1 for both the performance specifications and input constraints to be met. The problem is a two-block problem, with more regulated outputs than control inputs, i.e. n, > n,, which requires the use of the DA algorithm. This was solved using several different LP solvers (based on MINOS, CPLEX and OSL subroutines) which all gave the same results as would be expected. Overall we have found CPLEX to be the most robust of these routines. The convergence of the solution is shown in Figure 3. This gives gN for N = 200 as 2.0551 which means that the performance specifications can only be met using an LTI controller for about half the disturbance range. To investigate how restrictive the input constraints on the control inputs u are on the achievable performance the Ii analysis can be repeated excluding the control inputs from the regulated output z. Now the problem is one block nZ = n, and n, > n, which means that the problem is good rank, i.e. there are no rank interpolation conditions. This means that only zero interpolation conditions will be produce and the-DA algorithm is not
Figure 3
Convergence
of DA delays, N
of DA algorithm
required. Constraints are still added to the LP to enforce zero steady-state offset. When the resulting LP was solved the solution was u. = 1.8287. The performance is improved only slightly by ignoring input constraints. As stated previously the disturbance can be rate limited through modifications of the upstream operation. Introducing a limit on the rate of change of the disturbance we found that the constraints could be satisfied (with no input constraints) if the disturbance varied only by 14.3% of its maximum variation per min. With input constraints a variation of just 10% per min could be tolerated. The disturbance value was expected to change only infrequently so a step description is arguably more appropriate. The problem in this case is still multiblock, since input constraints are still included as are zero steady state conditions, and will require the DA algorithm. The convergence of the DA for this problem is shown in Figure 4 giving z,,, for N = 200 of 1.1428. Although this is a considerable reduction on the value for the Ii problem it still does not quite satisfy the performance
115
++
r
+
+
60
60
+
+
+
+
+
+
+ 091
0
20
40
100
No. of DA Figure 4
Convergence
120
140
delays, N
of DA algorithm
160
160
200
Process controllability analysis."S. D. Chenery and S. P. Walsh requirements and would need a further restriction of the disturbance set to do so. This could be done as for the 11 analysis by rate limiting the disturbance. This highlights the importance of appropriate selection of the class of disturbances considered. Overall this LP based controllability analysis provides strong confirmation of the difficulty of meeting the performance requirements, particularly with rapidly varying disturbances. We were unable to solve the problem in this case study using the Q-approximation due to numerical problems.
Review of method Our aim was to develop a method which: • • • •
captures typical performance requirements directly; finds the best performance with a broad but realisable class of controllers; takes account of as many of the fundamental limitations on controllability as possible; is computationally tractable.
This LP based method manages to achieve quite well on most of these four categories. It captures a wide range of typical performance requirements depending on its formulation. It optimises the performance over a very general set of controllers, the set of LTI controllers. It takes account of most of the fundamental limitations on controllability: time delays, R H P zeros, input constraints, noise. These last two points mean that the bound that it provides on achievable performance is likely to be tighter than either the pessimistic or optimistic bounds mentioned in the introduction. It is computationally tractable in as far as it has been successfully applied to industrial examples, one of which is presented here. However this method has some shortcomings in meeting these four criteria. How serious these are and possible techniques for tackling them are discussed. Many typical primary performance requirements are captured in a very flexible manner without need for approximation. Little attention has been paid to classical control criteria such as damping and integral squared error. This reflects their secondary importance, the difficulty of weighing such issues objectively against economic objectives, and the difficulty of including them within a linear programming framework. The controllers obtained from the linear program may therefore be lightly damped or otherwise "poor' controllers. We prefer to let a control engineer make the trade-off between "nice" responses and economics/feasibility, with the knowledge of the best that can be achieved ignoring such issues. However there are certain well defined economic criteria that are not well approximated by linear objectives and constraints, such as the cost associated with deviations in quality variables.
173
Typically quality is defined in terms of discrete thresholds separating product grades with each grade having an associated value. The maximum deviation gives some indication of the impact on quality, but the relationship is not a simple one. This relationship will not be linear and will depend on the probability distribution of the disturbance set on which there is typically little information. One possible approach is to define a set of disturbance scenarios with associated weights to approximate the probability distribution. If these disturbances cause violations then an appropriate penalty would be added to the objective to reflect the expected cost of lost product implied. This would require solution of a MILP problem to accommodate the discrete decision as to what grade of product has been produced. An alternative is to use a simple linear penalty on the maximum deviation, as for the back-otis. This would necessarily involve a large element of judgement and iteration would be needed to confirm the validity of the weights, e.g. by MonteCarlo simulation. Further work is clearly needed on estimating the economic performance associated with quality. The class of controllers considered is a broad and meaningful one, but it is not complete. The most common conventional and advanced controllers in the process industry--PI with output limiting and constrained model predictive control--lie outside the set of LTI controllers as their characteristics change when constraints become active. This is an important limitation and requires radically different approaches. One possibility is to solve an optimal control problem for critical disturbance scenarios identified from the above analysis, using either a linear or nonlinear model '4. If the difference in performance between the LTI controller and the optimal control is small then the gap between the performance of LTI control and controllers taking a more active approach to constraints will also be small. If the gap is large then no conclusion can be drawn as the optimal control policy does not capture limitations associated with measurement availability and characteristics. The techniques discussed above, while very powerful, are only one element in the ideal control engineers' toolbox. The constraints on the achievable closed loop transfer functions incorporate the effect of time delays and ~unstable' zeroes on the achievable closed loop response. Input constraints and noise can be included through z and w, respectively. This framework therefore allows four of the five fundamental limitations on controllability to be considered simultaneously. The only missing controllability limit is uncertainty. As discussed in Dahleh and Diaz-Bobillo s, it is straightforward to add unstructured robust stability conditions to the above problems at the expense of increasing problem size. Such descriptions are however notoriously conservative. Dealing with robust performance or structured uncertainty in a non-conservative manner requires further research. One possibility would be to use the methods above as a guide to developing a controller and
174
Process controllability analysis: S. D. Chenery and S. P. Walsh
subsequently address robustness directly on the nonlinear model. The computational tractability has been verified to some extent by successful performance on industrial problems. In addition to the example above, the O L D E analysis technique was applied successfully to a large industrial problem (two exothermic P F R s in series) in Walsh e t a l . 14 in which the objective was to minimise the production of one component of the output. Nonetheless there are limitations on the size of problem which can be analysed at present. A problem with n z = n u = n y = n w = n = 5 and a finite horizon of 100 would give several thousand constraints and variables, close on a million non-zero elements and take several hours to solve on a S P A R C 10 workstation. Simple linear p r o g r a m m i n g algorithms will fail on a problem of this size. The number of variables and constraints increases with n 2 and the number of non-zero terms in the constraints increases with n a. Solution time increases with about n 5. Problems with n > 10 are likely to be out of reach for some time! In the short-term, careful formulation and the use of high quality LP software is necessary to allow realistic problems to be tackled.
Conclusions The techniques presented above substantially meet the objectives set out in the introduction. Some needs for further work have been identified.
Acknowledgement The work described in this paper has been carried out with the support of the EPSRC and ICI.
References 1. Skogestad, S., A procedure for SISO controllability analysis-with application to design of pH processes. In IFAC Workshop on Integration of Process Design and Control, Baltimore, MD, 1994, pp. 23-28. 2. Skogestad, S. and Postlethwaite, I., Multivariable Feedback Control. John Wiley & Sons, New York, 1996. 3. Morari, M., Effect of design on the controllability of chemical plants. In IFAC Workshop on Integration of Process Design and Process Control, London, UK., 1992. 4. Perkins, J. D. and Walsh, S. P. K., Optimisation as a tool for design/control integration. Computers and Chemical Engineering, 1996, 20, 315 323. 5. Walsh, S. and Perkins, J., Operability and control in process synthesis and design. In Advances in Chemical Engineering 2: Process Synthesis, ed. J. L. Anderson, Academic Press, New York, 1996, 301~402. 6. Boyd, S. P., Balakrishnan, V., Barratt, C. H., Khraishi, N. M., Li, X., Meyer, D. G. and Norman, S. A., A new CAD method and associated architectures for linear controllers. 1EEE Transactions on Automatic' Control, 1988, 33(3), 268-282. 7. Boyd, S. P. and Barratt, C. H., Linear Controller Design: Limits of Performance. Prentice Hall, New York, 1991. 8. Dahleh, M. A. and Diaz-Bobillo, I. J., Control of Uncertain Systems." A Linear Programming Approach. Prentice Hall, New York, 1995. 9. Youla, D. C., Jabr, H. A. and Bongiorno, J. J,, Modern Wiener Hopf design of optimal controllers--Part II: The multivariable case. IEEE Transactions on Automatic Control, 1976, AC21(3), 319-338. 10. Young, J. C. C., Swartz, C. L. E. and Ross, R., On the effects of constraints, economics and uncertain disturbances on dynamic operability assessment. Computers and Chemical Engineering, 1996, 20S, $677. 11. Ajbar, A., Keenan, M. R. and Kantor, J. C., Optimal linear regulation with hard constraints. AIChEJournal, 1995, 41(11), 24392450. 12. Swartz, C. L. E., A computational framework for dynamic operability assessment. In IFAC Workshop on Integration of Process Design and Control, Baltimore, MD, 1994, pp. 153-158 13. Walsh, S. P. and Malik, T. I., Controllability analysis and modelling requirements: an industrial example. Computers and Chemical Engineering, 1995, 19S, $369. 14. Walsh, S. P., Chenery, S. D., Owen, P. and Malik, T. I., A case study in control structure selection for a chemical reactor. Computers and Chemical Engineering, 21S, 1997, $391 396.
J. Proc. Cont. Vol. 8, No. 3, pp. 165 174, 1998 ,d, 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0959-1524/98 $19.00 + 0.00
PII: S0959-1524(97)00044- 9
Process controllability analysis using linear programming S. D. Chenery and S. P. Walsh Centrefor ProcessSystemsEngineering,Imperial College, London, UK Controllability analysis is concerned with determining the limitations on achievable dynamic performance. This paper proposes the use of linear programming to determine the best linear controller and corresponding dynamic performance for problems of the form: min J(K, uo)s.t.c(K, u0, w) < 0Vw C W. That is, K,u0
a controller, K, and a reference operating point, u0, are selected to minimise a specified objective, J, while ensuring feasibility for all disturbances, w, within a specified set, 14". When K is a linear time invariant (LTI) controller and the objective function J and the constraints c can be expressed as linear functions then the above problem can be solved by linear programming. This formulation encompasses a wide range of problems ranging from minimising the maximum deviation in the regulated outputs subject to disturbances of magnitude less than one (the l~ optimal control problem) to optimising the expected value of a linear economic objective (the Optimal Linear Dynamic Economics (OLDE) problem). The relationship of this work to other approaches to controllability analysis is discussed. A highly flexible framework for addressing typical process performance requirements through appropriate selection of J, c and W is presented. The relative merits of alternative approaches for defining the achievable closed loop transfer functions based on the Q-parameterisation are carefully discussed. The feasibility of the proposed approach is demonstrated on an industrial reactor example. The needs for further work are discussed. 'u 1998 Elsevier Science Ltd. All rights reserved
Keywords: controllability; linear optimisation; It optimal control
Controllability has been defined in several different ways. In the 60s K a l m a n introduced the concept of state controllability which tells us whether the state vector can be taken from any initial state to any final state within a finite time. In the 70s Rosenbrock defined functional controllability which tells us whether there exists a set of input trajectories which can generate any output trajectory. He also introduced the importance of the notion of right half plane (RHP), i.e. 'unstable', zeros for multivariable systems. Skogestad I defines input-output controllability as:
affect the achievable control performance of both SISO and M I M O linear systems 2, e.g. location of R H P zeros, condition number, minimum time delay, etc. Morari 3 notes that "we often do not quite understand yet when and how these indicators should be used', but even so ~they are applied widely and indiscriminately and lead to erroneous conclusions about controllability'. The ambiguity involved in the interpretation of these measures is a major problem for controllability analysis. One reason for the ambiguity is that many controllability techniques based on linear models rely on frequency domain analysis. However process constraints and disturbances are typically defined in the time domain. Another reason is that each indicator typically only considers one of the above limitations in isolation. Therefore there is a need for an easily interpreted linear controllability analysis technique that deals directly with time domain performance requirements and quantifies the combined effect of as many fundamental limitations as possible. Another group of techniques are optimisation-based techniques 4.5 which may be based on linear or nonlinear models. These techniques differ from each other in the model they use and the degree of idealisation of the controller. Much recent work has fbcused on the direct
"The ability to achieve acceptable control performance, that is, to keep the outputs 0') within specified bounds from their setpoints (r), in spite of unknown variations in the plant (e.g., disturbances (d) and model perturbations) using available inputs (u) and available measurements (e.g. y , , or din)." It is this definition of controllability that this paper refers to. Achievable control performance is limited by: righthalf plane ( R H P ) zeros, time delays, input constraints, sensor noise and process-model mismatch (uncertainty). M a n y controllability indicators have been developed to give measures of how these fundamental limitations
165
166
Process controllability analysis: S. D. Chenery and S. P. Walsh
optimisation of PI controllers for both linear and nonlinear systems. This analysis gives a pessimistic bound on achievable performance in that the class of controllers considered is deliberately restricted. Other work has used various idealisations of controller performance to get optimistic bounds on achievable performance, e.g. perfect control, delay limited control. Therefore an optimisation based technique which gives a measure of achievable performance, i.e. controllability, for a very broad, yet realisable, set of controllers would be desirable. Any such method must be computationally tractable as well as conceptually attractive. In this paper we use linear programming techniques to address problems of the form: rain J(K, uo)s.t.c(K, Uo, w) <_ 0Vw c W
K,uo
That is, a controller, K, and a reference operating point, u0, are selected to minimise a specified objective, J, while ensuring feasibility for all disturbances, w, within a bounded set, W, with respect to the constraints c. The objective J is a measure of performance, therefore the solution of this problem gives a measure of controllability, the best achievable control performance despite disturbances. When K is a linear time invariant (LTI) controller and the objective function J and the constraints c can be expressed as linear functions then the above problem can be solved by linear programming. This formulation encompasses a wide range of problems ranging from minimising the maximum deviation in objective variables subject to disturbances of magnitude less than one (the ll optimal control problem) to optimising the expected value of a linear economic objective (the Optimal Linear Dynamic Economics (OLDE) problem). The achievable closed loop transfer functions for a generalised plant can be expressed as an affine function of a stable parameter Q. This allows the achievable discrete time closed loop impulse responses to be defined by an infinite set of linear constraints and approximated by a finite set of linear constraints. The methods for doing this are well established 6-s and are summarised here as a precursor to a careful comparison of the alternative methods within the overall analysis framework set out below. The disturbance set W can be expressed in a flexible manner and can include a mixture of magnitude bounded signals, step signals and steady-state signals. The allowable combinations of disturbances can be restricted if desired. This flexibility is important to allow specific process requirements to be captured with adequate precision. We show how the effect of this general set W can be captured in a linear program. Possible objective functions are then discussed and it is shown that many objectives of interest can be captured in a linear program, this controllability measure therefore covers a wide range of performance definitions.
Having defined the LP problems of interest we then discuss the properties of alternative formulations. We apply this approach to investigate the controllability of an industrial reactor system about a fixed operating point to see if the required performance can be met with a controller. This demonstrates the ability of the proposed technique to solve realistic problems. Finally we review the method proposed and the needs for further development.
Achievable closed loop transfer function The closed-loop system is defined by the relationships below, where P is the generalised plant, K is the controller, w are the disturbances, u are the control inputs, z are the regulated outputs and y are the measured outputs:
z = P l l w + Pi2u Y = P21 w + P22u u=Ky
We take P and K to be linear time invariant transfer functions, z may contain any process variable including elements of u and y. w may include measurement noise. The number of elements in each signal vector (nz, nw, nu, ny) are not required to satisfy any particular relationship. The generalised regulator problem is to choose K so as to optimise some property of the closed loop transfer function (mapping) from w to z while maintaining internal stability. The mapping from w to z is given by = PI1 + P I 2 K ( I - P22K)-lp21
(1)
The Q-parameterisation 9 is used so that the closed-loop map • is affine in a free stable parameter Q
• =H-UQV
(2)
If Q is selected from the set of stable (bounded input bounded output) LTI transfer functions (Q c l l ) then all stabilising linear time-invariant (LTI) controllers are parametrised by the above expression. If P and K are expressed as discrete time LTI systems and q~ is a discrete time impulse response matrix then it is possible to express the above parameterisation in terms of linear equality constraints. The first method for achieving this uses interpolation constraints 8. There are a set of constraints on qb called the interpolation conditions that ensure that the closedloop map • can be written in the form of (2) for some
QEll. There are two types of interpolation conditions, the zero interpolation conditions and the rank interpolation conditions. The first capture limitations on the achievable ~ due to fundamental process limitations like R H P zeros and time delays. The latter capture the limitations
Process controllability analysis." S. D. Chenery andS. P. Walsh on the achievable qb due to Q not having sufficient degrees of freedom, i.e. more disturbances than measurements n,. > ny and/or more regulated outputs than control inputs n: > n,. Any problem with insufficient degrees of freedom is known as multi-block and has an infinite number of constraints due to the rank interpolation conditions. Only one-block problems, with nw <_ny and n: _< n,, avoid these infinite constraints. Therefore if the problem is multi-block it is embedded in a one-block problem which has only zero interpolation conditions. In this paper the delay augmentation (DA) algorithm is used which involves U and V being augmented by N pure delays so that the problem is oneblock. As N increases the zero interpolation conditions of the DA problem approximate the original problem increasingly closely. The resulting constraints on qb are linear and in the time-domain. The interpolation conditions are given by a finite number of zero interpolation conditions of the kind shown in Equation (3).
167
The effect of the choice of N and Ni/will be discussed later when the problem definition is complete. There is another technique which avoids calculating these interpolation conditions at all. For this technique the discrete impulse response of Q is approximated as 6 L
1
Q = Z q(k)z-k
(6)
k=O
with finite length L. The assumption that Q is a finite impulse response of length L means that this technique is an approximation and it is necessary to select L large enough to include the optimal Q. Using this approximation means that the discrete elements of qb can be expressed as follows -min(k, L - 1 )
¢~(k) +
= hi/(k) p=l
n = l rn=l
(7) tti,z. (H ,
-
a,)x/Xo)/~)(Xo)
= 0
(3) where
where 3,.~" and x~° are given by the left and right nullchains of U and V, respectively, and k0 is an 'unstable' zero of U or Vs. These conditions allow the elements of any feasible ap to be expressed as follows:
Z a~qx°'k(l)~)pq
(4)
(l) = b ij)'°'k
p = l q= I /=0
where n:
tl~
OC
p=l q=l
/=0
and
a )°k(tl =
t- slx t=0 s=0
(sl(Zl
] .1 ),=X 0
these elements of aO'qX°'k(1), qbpq(l) and b#'x°'k can be stacked and arranged into a constant real matrix Azcro, a variable vector ¢ and a constant real vector bzero, respectively, so that the zero interpolation conditions expressed by Equation (4) can be stated as Azero¢ = bzero
(5)
4~ is infinite dimensional. The method used to get a finite dimensional problem is to truncate ¢ for each zi and ~,) to a finite length N#. The elements of Azero associated with ¢ beyond this horizon are removed from the problem. This implies that any ¢ satisfying the truncated constraints also satisfies the infinite constraints for ¢/j(k) = 0Vk _> U#.
k O(k)(p) : ~ ~.~
u,.(r - p),mi(k - ,'1
r=p
Much as in the zero interpolation case these elements of C~(mk)(P), q,m(P), eli(k) and hi/(k) can be stacked and arranged into a real constant matrix Aq, variable vectors q and ¢ and a real constant vector h, respectively. In this way expression (7) can be stated as
Note that each SISO impulse response qnm(k) could be assigned a distinct length Lnm, but this would normally introduce an unhelpful increase in complexity. As with the zero interpolation constraints ¢ is truncated. In this case this implies nothing about the values of ¢~(k)k >_ N#. Either of these methods, the zero interpolation conditions or the Q-approximation, provides linear constraints defining the achievable ¢.
Disturbance description In typical process applications the disturbances will not be precisely characterised. However the following questions can usually be answered for the key disturbances. What range of values does the disturbance take? How quickly can the value change? Do changes occur frequently? Based on answers to these questions, an adequately precise disturbance description can be developed. Very slowly changing disturbances should be treated as steady-state. Disturbances which change significantly
Process controllability analysis."S. D. Chenery and S. P. Walsh
168
over the process transient response time can be modelled as instantaneous changes filtered through a transfer function W1. If the maximum rate of change is 10% of maximum variation per sample a finite impulse response (FIR) filter Wl,
infinite sum. However, the steady-state gain of Q can be evaluated as a finite sum and the corresponding steadystate gain of ¢ computed given the steady-state values of H, U, and V. Hence the closed loop steady-state gains, G~S.d zw , and V0, can be computed using finite dimensional linear constraints in both cases.
9
WI = 0 . 1 E q
-i
Step disturbances
i=0
would convert a step change between the limits to a ramp at the maximum rate, thereby excluding the unrealistic disturbances. Such filters can be included in the process model as shown in Figure 1. If changes occur infrequently compared to the process response time then a step disturbance description is more appropriate than a persistently varying signal which is only constrained to be magnitude bounded. Typical time-domain disturbance descriptions can therefore be represented in terms of three types of disturbance signals: persistently varying (persistent) disturbances, step disturbances and steady-state disturbances. Given a finite impulse response description of ¢ we can compute the worst-case response for each of these signal types. For convenience (and without loss of generality) we consider all disturbances relative to a reference point corresponding to Wcentre = (wh+ wt)/2 and scaled so that a unit variation corresponds to (w h - w t ) / 2 . The disturbance deviations about this point will be symmetric with a maximum deviation of unity (upper b o u n d = - l o w e r bound = 1) and the corresponding output deviations will also be symmetric. Steady-state disturbances
The maximum output deviation on zi in response to wj will be given by -v O
< vO
(9)
k=O
zo(t ) = - ~ j + 2
= - ~
¢0(k
Co(k) + 2
k=0
Co(k) k=0
The maximum deviation is therefore given by
- Vo <- - E
¢ij(k) + 2
Co(k)
_< vij
k=O
j
t>0
for
(10)
= 1. . . . . nz "= 1. . . . . nw
The steady-state component can be computed as above. With the interpolation constraints the maximum deviation for t < N 0 will be the maximum deviation for the infinite horizon. With the Q-approximation the peak step response calculated in this way will be a lower bound on the infinite horizon step response deviation as the later values of ¢ are unconstrained. Persistent disturbances
However we will only have access to ¢0 up to a finite horizon N o. With the interpolation constraints the later values of ¢0 have been set to zero and therefore the finite horizon sum equals the infinite horizon sum. With the Q-approximation the finite sum is not equal to the r--
w
We consider steps between any two levels within the lower and upper bounds on the disturbance value. The worst-case will be given by a step between the upper and lower bound or vice versa. The effect on output zi of a step in disturbance wj from the lower bound ( - 1 ) to the upper bound (1) would be,
..........
~
.
.
.
Wl
.
I
i . . . . .
SZ_ ..... J-l_
d
For persistently varying disturbances we have
vo = ~ ICu(k)l k=O The absolute value in the above expression requires a change of variable to give a suitable form for a LP. We define ~ = q ~ + - ~ - , where @+ and • have only nonnegative elements. We can then write OO
vU = ~
¢/~ (k) 4- ¢+ (k)
(11)
k=O
Figure 1
Block d i a g r a m for a u g m e n t e d system
Replacing the sum to infinity with the sum to N O• has the same implications as for the step disturbances.
169
Process controllability analysis: S. D. Chenery and S. P. Walsh Combining disturbances
For each disturbance type we can calculate v O. F o r a purely worst-case analysis we get the worst deviation on Zi as tIW
vi = Z
(12)
vii
i-I
If such an analysis is felt to be excessively conservative then we may define vi based on the m a x i m u m deviation over specified subsets of disturbances, e.g. any n step disturbances combined with all the steady-state and persistent disturbances. The constraints for each allowable combination could be appended to the LP. The values of v~ couple the disturbance description with the overall performance requirements.
Capturing performance requirements To complete the LP formulation we need to define the closed-loop performance requirements. That is, for the problem min J(K, uo)s.t.c(K, uo, w) < 0Vw c W K, ut,
we need to define J and c which specify what controllability problem we solve. The most basic requirement is feasibility, i.e. that constraints are not violated. We assume that the model used has been linearised about a steady-state with regulated output, measured output, disturbance and control input levels given by Zlin, Ylin, Wlin, Ulin and that all constraint limits are in terms of the original values. We denote the closed loop steady-state gains as G ss'ct and the open-loop steady-state gains as G ~s'°z. The requirement for feasibility can be expressed as below where each constrained variable is assumed, without loss of generality, to be an element of z. ~_l i < --
---i < __
Rather than imposing the feasibility requirement directly we may wish to determine the fraction of the disturbance deviations for which feasibility is achievable. This can only be expressed as a linear program if Zr~r is independent of K and u0, i.e. for a fixed operating point u0. In this case we can scale the deviations z~ by the minimum distance to a constraint (min(z) - : ~ e f E ~ e f Z~)) SO that a deviation less than 1 corresponds to constraint feasibility. We can then make the objective of the LP. v0 = min max vi K
The values of z and y corresponding to Wcentre and a corresponding input value u0 are given by
(J(K) = max vi) i
o(~')1. : u0 + K ( s ) ( y - y ' % 0 ref -- G':;:;,cI(~,T'-
Wcentre )
(17) (18)
So our optimisation becomes •"~SS, c l I - -
m i n o ref %- O-ow /~1 - Wcenlre )
_ref ,~ss,~ l/ .~ss,ol / .~ = Uzw ~14centre -- Wlin) %- Lrzu ~,U0 -- Ulin) %- Elin
(16)
I,n:
If the disturbances are scaled down relative to Wcentre by a factor ±~,0then feasibility will be maintained. ±~0therefore gives the fraction of the disturbance set which can be tolerated. The use of the 'disturbance fraction' as a controllability measure is discussed more generally in Walsh and Perkins 5. Note that if all the disturbances are persistent then v0 is the ll norm of the mapping from w to z and the problem being solved is an Ii optimal control problem. If we have a reasonable basis for penalising the maximum deviation on each objective z~ then it could be appropriate to minimise a weighted sum of the deviations vi rather than the m a x i m u m deviation. The weighted deviations could be minimised subject to feasibility constraints. This type of objective is highly flexible. Its key limitation is the difficulty of defining appropriate weights for the deviations. If we have reasonable information about the economics of the process then a more systematic approach can be taken to defining the objective for the LP. Given a linear(ised) economic objective we can determine the optimal expected value of this objective over the disturbance set. If the objective function is expressed as o(w,u) then by virtue of closed-loop linearity, the expected value of the objective, ~, is simply the objective for the expected value of the disturbance, ~7'.
0(~') :
z)Vw c Wi = 1 n:
i-l,n:
(19)
K.uo
(13) yref
,,~ss,ol /
~
.~ss,oD
= Uyw 0 ~centre -- Wlin) -F Oy u tU0 -- Ulin) -F Ylin
(14) with the controller given by u = uo + K ( s ) ( y - yref). The feasibility requirement can therefore be expressed as . l + v, __ "i < Z~ef < zhi - vi
i= 1,nz
(l 5)
Note that if o is included in : then G)";'I;~t can be constructed as discussed previously. Alternatively, if o is not in z and all inputs appear in : then G~;]',fl can be obtained as Go .,~,s,,,lG,,SSd + GS~,,,l
If we linearise around the steady-state optimum for i~' = Wcentre and the active set of constraints is assumed to remain constant over the disturbance set then this
170
Process controllability analysis: S. D. Chenery and S. P. Walsh n
problem reduces to minimising J = J o - ~ ~'iPi in the i--1
absence of constraints. J is the objective function at the optimal operating point, Jo is the steady state optimum and ~-i is the Lagrange multiplier of the constraint on zi. Young et al. 1° present a conceptually similar approach to evaluating operating economics based on the Q-approximation. Their approach diverges significantly in detail. They do not consider the use of interpolation constraints and consider only step disturbances and nominal economics. Their disturbance set is defined by r = {(ab, a d ) l d = de + a a ; d - < d < d +}
for which the base value db, the magnitude and direction of the step Aa are uncertain. They reduce the set of disturbances to be considered when testing for feasibility to just the critical (vertex) points ( d - , d + - d - ) and (d-, - ( d + - d - ) ) . For a large number of disturbances this set would still become large, 2 "w. By choosing the reference disturbance as dcentre we avoid the need to consider the above critical points individually. They allow convex objective functions which means that their problem size is limited by the capacity of convex programming algorithms rather than linear programming algorithms which can handle larger problems. They apply their method successfully to a simple problem. Despite the differences noted, their work is very much in the same spirit as ours.
Properties of the LP formulations We have shown that many performance requirements of practical interest can be formulated in terms of a LP which can be solved to determine the best achievable performance for an optimal choice of K and, if appropriate, u0, thus providing a measure of controllability. The LPs involve a number of approximations so it is important to consider how the results of the LPs relate to the solution of the original infinite horizon discrete time problem. In the interpolation constraint method the finite horizon response is an upper bound on the infinite horizon response due to the additional requirement that the zero impulse response beyond the horizon is achievable. If delay augmentation is used with finite N then the resulting solution is a lower bound on the finite horizon problem. For a single block ll problem it is possible to calculate the values for N o• such that the constraints (bo(k) = 0Vk > N 0. are inactive 8 and the finite horizon optimum becomes equal to the infinite horizon optimum. For a multi-block 11 problem this calculation can be carried out for each value of N and the finite horizons N O can be chosen accordingly. The solutions as N increases will then provide a sequence of lower bounds which converge to the infinite horizon optimum. For problems other than the ll problem the minimum value of N O needed to make the finite horizon optimum equal
to the infinite horizon optimum is not known. The solutions generated are therefore lower bounds on an upper bound and hence simply approximations to the infinite horizon problem. In the Q-approximation method the optimal finite horizon response is a lower bound on the optimal infinite horizon response (with the same parameterisation of Q) since the finite horizon problem disregards potential deviations beyond the horizon. If L is less than any horizon N 0 ÷ 1 the resulting solution is an upper bound on the finite horizon problem, with arbitrary Q, as elements of Q affecting the finite horizon response are restricted to a value of zero. For L <_ N O + 1Vi, j the solution for the Q-parameterisation is a lower bound on the infinite horizon problem provided the finite horizon objective and constraints are not affected by qmn(k) being zero for k > L - 1. This condition is not guaranteed for step or steady-state disturbances (due to the steady-state effect being ~ ¢0(k)) and the Q-approxk=0
imation gives just an approximate result for these types of disturbances. With both methods, if the solution for varying L or N takes an almost constant value as L or N vary over most of the horizon then the behaviour beyond the end of the horizon can be presumed not to be significant and the result can be taken as a true estimate of the achievable performance. This convergence can be tested by reconstructing the controller K and calculating the infinite horizon response to within a desired precision (Dahleh and Diaz-Bobillo, 1995) to give an upper bound on the infinite horizon solution. In Ajbar et al.~l the interpolation constraint approach is criticised for giving only an approximate solution to the Ii problem, in contrast to the Q-approximation. However, both methods can be used to construct a sequence of lower bounds to the infinite horizon optimum or to construct upper bounds based on a specific solution. Also, both methods exhibit the qualitative property that consistent results as N or L increase indicates convergence to the true optimum. At least for the ll problem both methods seem to be equally well founded. If the problem is one-block then it is more efficient and exact to solve the zero interpolation condition problem. There are also differences in the structure of the associated linear programs. The zero interpolation conditions generate nz
nw
X06Auv i = l j = l
(~ru,(,k0), av,(Z0) = structural indices of U, V) constraints and t/z
V/w
ZZN~+
1
(N o = FIR length of q~0)
i--1 j = l
variables within the LP.
171
Process controllability analysis. S. D. Chenery andS. P. Walsh The Q-approximation method generates
Industrial reactor problem
ZZN +I
The case study used to demonstrate the LP controllability analysis is a linear model of an industrial reactor example presented by Walsh and Malik t3. The linear model of the exothermic plug flow reactor in Figure 2 is in the form of a transfer matrix of gain~delay lag elements. Note that the analysis could equally be applied to a more general linear model. The perlbrmance requirements are given as a set of target steady-state values and constraints for the variables Tin and Tou t (Tabh, 1). Note that the desired close loop response is otherwise unspecified. There are no steady-state degrees of freedom so uo is fixed. In a previous control study serious difficulties had been encountered in meeting the constraints above, even assuming rate limits on the disturbances. The controllability analysis below therefore focuses on feasibility. There are only two possible manipulated variables S~ the split between the product and the recycle and $2 the split between the heat exchanger and the bypass (& = 1 means 100% recycled and $2 - 1 means 100% bypassed). Fin is determined by the requirements of another plant. There are four measured variables Ain (a delayed composition measurement of the key reactant), Fr~cyd~, Tin and Tom The disturbance d is a well defined change between two operating regimes, A and B, which involves three feed component flowrates. The operators wish to make this change as fast as possible, but there is some flexibility available to reduce the rate of change if a step change cannot be accommodated. The objective is therefore to minimise the switching time while meeting the performance requirements. Since the disturbance only occurs in a single direction with respect to these flowrates it is represented as a single disturbance. The temperature measurements, disturbance and manipulated variables are all normalised by their maximum acceptable deviation relative to steady state with a disturbance d~em~. This gives the transfer function matrix
iljl constraints and
I~ Z
.... Ni/ +
ki=] / 1 (Lnm = FIR
l] + [nn~l"'mLnm~j + ]1
length
of
qnm)
variables. On the whole the number of constraints in the zero interpolation conditions will be less than the number of variables, in which case the interpolation condition will produce a smaller LP problem. However the Q-approximation problem on the whole will be more sparse which produces an LP which is faster to solve. Also if the DA algorithm must be used to produce the one-block problem then the number of constraints produced by the zero interpolation conditions will grow as the number of delays augmenting the problem are increased in the algorithm. Therefore which method is preferable in terms of LP complexity is highly problem specific. The Q-approximation is by far the simpler and more straightforward of the two techniques to implement. There are examples of various problems solved by each technique in the literature, (e.g. Refs 10-12) for the Q-approximation and in Dahleh and Diaz-Bobillo 8 for the zero interpolation method. We solved the very simple SISO problem presented in example 1 of Ajbar et al.I 1 using both techniques. Several iterations of the Q-approximation technique were required to check if a good choice of L had been made. The interpolation constraint technique solved immediately and exactly since the problem is one-block. In other examples, including the case study below we have successfully used interpolation constraints while the Q-approximation LP has not solved successfully. It should be noted that both approaches have a variant in which finite step responses replace the finite impulse response. For step and steady-state disturbances this will give a sparser LP and may have better solution characteristics. It is clear from the above discussion that there is an element of approximation in the LP formulation. We believe that provided convergence of estimates is obtained for varying L or N this approximation is of little consequence. The approximation of a nonlinear process by a linear model is of primary importance followed by the approximation of the continuous time linear model by a discrete time model. Variable transformations should be used where appropriate to improve the quality of the linear model and sampling should be faster than both open-loop transients in response to disturbances and the fastest expected closed loop response time. Some iteration might be necessary to achieve this.
, Freer
Recycle '
Fin
]
i
Feed {disturbance, d)
'
-
'
"tSll
Tm ' Exchanger] i bypass l
i
Atn Plug flow [e;lcl(lf
[ Figure 2
Prodtlcl
Process schematic
: Tout:
Process controllability analysis: S. D. Chenery and S. P. Walsh
172
Table 1
Performance
requirements
Steady-state
Variable
value
Upper
limit
c
300°C None
(251 $)r
to
(AinFrecycre Tout$n)r
0.065 2.1 5e~“.z5s 1+0.6s
shown
5
z
+
+
+
60
100
120
140
160
160
200
+ 1.96-
2 1.94-
0
g,.,,.
6.61
0
l19-
-4iTJle&45
t
1 86
+
t+
_ 1.77
_
0.49em04s 1+0.6s
1.66’
0
20
40
60
No.
A state space realisation of the transfer function was found and discretised using the bilinear Tustin method. The sampling rate was chosen to be substantially faster than the frequencies up to which disturbance attenuation was required. The measured outputs y and regulated outputs z are given by
(20)
whilst the exogenous input w and the control inputs u are given by
w=d,u=
+
1.96-
I +0.02s
-3,5geA=” I f0.6~
+
2-
_ , ,8ge-0.W8”
-13.98e-035’ I +0.6s
“f&6;
+
2.02
2
D
T=
+
+
.P
is
T relating below:
+
2 04
Lower limit
None 650°C
400°C 600°C
i-ii, Tout
2.06
for Tin and T,,,
0s1 s2
The scaled manipulated inputs 51 and $2 have been included in the regulated outputs to incorporate the input constraints on the splits. Zero steady-state offsets for Tin and To,, must be enforced. This involves adding two linear time domain constraints onto the resulting 1, LP. u. should be less than 1 for both the performance specifications and input constraints to be met. The problem is a two-block problem, with more regulated outputs than control inputs, i.e. n, > n,, which requires the use of the DA algorithm. This was solved using several different LP solvers (based on MINOS, CPLEX and OSL subroutines) which all gave the same results as would be expected. Overall we have found CPLEX to be the most robust of these routines. The convergence of the solution is shown in Figure 3. This gives gN for N = 200 as 2.0551 which means that the performance specifications can only be met using an LTI controller for about half the disturbance range. To investigate how restrictive the input constraints on the control inputs u are on the achievable performance the Ii analysis can be repeated excluding the control inputs from the regulated output z. Now the problem is one block nZ = n, and n, > n, which means that the problem is good rank, i.e. there are no rank interpolation conditions. This means that only zero interpolation conditions will be produce and the-DA algorithm is not
Figure 3
Convergence
of DA delays, N
of DA algorithm
required. Constraints are still added to the LP to enforce zero steady-state offset. When the resulting LP was solved the solution was u. = 1.8287. The performance is improved only slightly by ignoring input constraints. As stated previously the disturbance can be rate limited through modifications of the upstream operation. Introducing a limit on the rate of change of the disturbance we found that the constraints could be satisfied (with no input constraints) if the disturbance varied only by 14.3% of its maximum variation per min. With input constraints a variation of just 10% per min could be tolerated. The disturbance value was expected to change only infrequently so a step description is arguably more appropriate. The problem in this case is still multiblock, since input constraints are still included as are zero steady state conditions, and will require the DA algorithm. The convergence of the DA for this problem is shown in Figure 4 giving z,,, for N = 200 of 1.1428. Although this is a considerable reduction on the value for the Ii problem it still does not quite satisfy the performance
115
++
r
+
+
60
60
+
+
+
+
+
+
+ 091
0
20
40
100
No. of DA Figure 4
Convergence
120
140
delays, N
of DA algorithm
160
160
200
Process controllability analysis."S. D. Chenery and S. P. Walsh requirements and would need a further restriction of the disturbance set to do so. This could be done as for the 11 analysis by rate limiting the disturbance. This highlights the importance of appropriate selection of the class of disturbances considered. Overall this LP based controllability analysis provides strong confirmation of the difficulty of meeting the performance requirements, particularly with rapidly varying disturbances. We were unable to solve the problem in this case study using the Q-approximation due to numerical problems.
Review of method Our aim was to develop a method which: • • • •
captures typical performance requirements directly; finds the best performance with a broad but realisable class of controllers; takes account of as many of the fundamental limitations on controllability as possible; is computationally tractable.
This LP based method manages to achieve quite well on most of these four categories. It captures a wide range of typical performance requirements depending on its formulation. It optimises the performance over a very general set of controllers, the set of LTI controllers. It takes account of most of the fundamental limitations on controllability: time delays, R H P zeros, input constraints, noise. These last two points mean that the bound that it provides on achievable performance is likely to be tighter than either the pessimistic or optimistic bounds mentioned in the introduction. It is computationally tractable in as far as it has been successfully applied to industrial examples, one of which is presented here. However this method has some shortcomings in meeting these four criteria. How serious these are and possible techniques for tackling them are discussed. Many typical primary performance requirements are captured in a very flexible manner without need for approximation. Little attention has been paid to classical control criteria such as damping and integral squared error. This reflects their secondary importance, the difficulty of weighing such issues objectively against economic objectives, and the difficulty of including them within a linear programming framework. The controllers obtained from the linear program may therefore be lightly damped or otherwise "poor' controllers. We prefer to let a control engineer make the trade-off between "nice" responses and economics/feasibility, with the knowledge of the best that can be achieved ignoring such issues. However there are certain well defined economic criteria that are not well approximated by linear objectives and constraints, such as the cost associated with deviations in quality variables.
173
Typically quality is defined in terms of discrete thresholds separating product grades with each grade having an associated value. The maximum deviation gives some indication of the impact on quality, but the relationship is not a simple one. This relationship will not be linear and will depend on the probability distribution of the disturbance set on which there is typically little information. One possible approach is to define a set of disturbance scenarios with associated weights to approximate the probability distribution. If these disturbances cause violations then an appropriate penalty would be added to the objective to reflect the expected cost of lost product implied. This would require solution of a MILP problem to accommodate the discrete decision as to what grade of product has been produced. An alternative is to use a simple linear penalty on the maximum deviation, as for the back-otis. This would necessarily involve a large element of judgement and iteration would be needed to confirm the validity of the weights, e.g. by MonteCarlo simulation. Further work is clearly needed on estimating the economic performance associated with quality. The class of controllers considered is a broad and meaningful one, but it is not complete. The most common conventional and advanced controllers in the process industry--PI with output limiting and constrained model predictive control--lie outside the set of LTI controllers as their characteristics change when constraints become active. This is an important limitation and requires radically different approaches. One possibility is to solve an optimal control problem for critical disturbance scenarios identified from the above analysis, using either a linear or nonlinear model '4. If the difference in performance between the LTI controller and the optimal control is small then the gap between the performance of LTI control and controllers taking a more active approach to constraints will also be small. If the gap is large then no conclusion can be drawn as the optimal control policy does not capture limitations associated with measurement availability and characteristics. The techniques discussed above, while very powerful, are only one element in the ideal control engineers' toolbox. The constraints on the achievable closed loop transfer functions incorporate the effect of time delays and ~unstable' zeroes on the achievable closed loop response. Input constraints and noise can be included through z and w, respectively. This framework therefore allows four of the five fundamental limitations on controllability to be considered simultaneously. The only missing controllability limit is uncertainty. As discussed in Dahleh and Diaz-Bobillo s, it is straightforward to add unstructured robust stability conditions to the above problems at the expense of increasing problem size. Such descriptions are however notoriously conservative. Dealing with robust performance or structured uncertainty in a non-conservative manner requires further research. One possibility would be to use the methods above as a guide to developing a controller and
174
Process controllability analysis: S. D. Chenery and S. P. Walsh
subsequently address robustness directly on the nonlinear model. The computational tractability has been verified to some extent by successful performance on industrial problems. In addition to the example above, the O L D E analysis technique was applied successfully to a large industrial problem (two exothermic P F R s in series) in Walsh e t a l . 14 in which the objective was to minimise the production of one component of the output. Nonetheless there are limitations on the size of problem which can be analysed at present. A problem with n z = n u = n y = n w = n = 5 and a finite horizon of 100 would give several thousand constraints and variables, close on a million non-zero elements and take several hours to solve on a S P A R C 10 workstation. Simple linear p r o g r a m m i n g algorithms will fail on a problem of this size. The number of variables and constraints increases with n 2 and the number of non-zero terms in the constraints increases with n a. Solution time increases with about n 5. Problems with n > 10 are likely to be out of reach for some time! In the short-term, careful formulation and the use of high quality LP software is necessary to allow realistic problems to be tackled.
Conclusions The techniques presented above substantially meet the objectives set out in the introduction. Some needs for further work have been identified.
Acknowledgement The work described in this paper has been carried out with the support of the EPSRC and ICI.
References 1. Skogestad, S., A procedure for SISO controllability analysis-with application to design of pH processes. In IFAC Workshop on Integration of Process Design and Control, Baltimore, MD, 1994, pp. 23-28. 2. Skogestad, S. and Postlethwaite, I., Multivariable Feedback Control. John Wiley & Sons, New York, 1996. 3. Morari, M., Effect of design on the controllability of chemical plants. In IFAC Workshop on Integration of Process Design and Process Control, London, UK., 1992. 4. Perkins, J. D. and Walsh, S. P. K., Optimisation as a tool for design/control integration. Computers and Chemical Engineering, 1996, 20, 315 323. 5. Walsh, S. and Perkins, J., Operability and control in process synthesis and design. In Advances in Chemical Engineering 2: Process Synthesis, ed. J. L. Anderson, Academic Press, New York, 1996, 301~402. 6. Boyd, S. P., Balakrishnan, V., Barratt, C. H., Khraishi, N. M., Li, X., Meyer, D. G. and Norman, S. A., A new CAD method and associated architectures for linear controllers. 1EEE Transactions on Automatic' Control, 1988, 33(3), 268-282. 7. Boyd, S. P. and Barratt, C. H., Linear Controller Design: Limits of Performance. Prentice Hall, New York, 1991. 8. Dahleh, M. A. and Diaz-Bobillo, I. J., Control of Uncertain Systems." A Linear Programming Approach. Prentice Hall, New York, 1995. 9. Youla, D. C., Jabr, H. A. and Bongiorno, J. J,, Modern Wiener Hopf design of optimal controllers--Part II: The multivariable case. IEEE Transactions on Automatic Control, 1976, AC21(3), 319-338. 10. Young, J. C. C., Swartz, C. L. E. and Ross, R., On the effects of constraints, economics and uncertain disturbances on dynamic operability assessment. Computers and Chemical Engineering, 1996, 20S, $677. 11. Ajbar, A., Keenan, M. R. and Kantor, J. C., Optimal linear regulation with hard constraints. AIChEJournal, 1995, 41(11), 24392450. 12. Swartz, C. L. E., A computational framework for dynamic operability assessment. In IFAC Workshop on Integration of Process Design and Control, Baltimore, MD, 1994, pp. 153-158 13. Walsh, S. P. and Malik, T. I., Controllability analysis and modelling requirements: an industrial example. Computers and Chemical Engineering, 1995, 19S, $369. 14. Walsh, S. P., Chenery, S. D., Owen, P. and Malik, T. I., A case study in control structure selection for a chemical reactor. Computers and Chemical Engineering, 21S, 1997, $391 396.