Steady-state operability of multi-variable non-square systems: application to Model Predictive Control (MPC) of the Shell Heavy Oil Fractionator (SHOF) L.R.E. Shead, C.G. Anastassakis, J.A. Rossiter
Abstract— This paper focuses on the application of steadystate operability to Model Predictive Control (MPC) of the Shell Heavy Oil Fractionator (SHOF) control problem. Operability for non-square systems is a relatively new area in the literature, and investigation into its applicability for higher order dimensions has not yet been addressed. This was done primarily to assess steady-state operability as a tool for evaluating the steady-state performance of high-order non-square systems. New theory and techniques have, however, been developed in the process. For non-square systems there are a number of alternatives in selecting the type (e.g. set-point, zone/interval and funnel) and number of controlled variables. This paper aims to investigate how operability can help the control designer to select these CVs whilst ensuring robustness to expected disturbances. Different options are assessed for the SHOF, and conclusions and recommendations are made regarding how the concept of operability might be improved. Keywords: non-square systems, operability, MPC, model predictive control.
I. I NTRODUCTION When designing a controller, one main concern is to ensure that system controlled variables (CVs) can be driven to desired values given the available system manipulated variables (MVs). Ideally, plant and controller design can be undertaken concurrently, with sufficient MV number/range incorporated to achieve the required CV behaviour. However, it is often the case that a controller has to be designed for an existing plant, and together with cost and physical constraints imposed on the selection of MVs, it is possible that control of the CVs has to be compromised in some way. A. Non-square systems When the number of MVs (m) is equal to the number of CVs (p) the system is called square, and is non-square otherwise. Non-square systems may be partitioned into the following categories ([3], [2]): •
•
Tall and thin, where the number of CVs exceeds the number of MVs. These systems are over-determined and have insufficient degrees of freedom (d.o.fs). Short and fat, where the number of CVs is less than the number of MVs. In these systems there are spare degrees of freedom and are characterised as underdetermined.
This work is funded by the EPSRC Department of Automatic Systems and Control Engineering, Mappin Street, University of Sheffield, S1 3JD, UK.
[email protected]
Systems are usually designed to be fat or square, generally allowing set-point control of each CV (termed SCVs). However, a system can become thin as a result of hardware failure or valve saturation ([2]) for example. For fat systems, their under-determined nature means that different combinations of MVs will all give desired output performance, such that the particular choice of MV combination is made from further optimisation. For thin systems the specification on the CVs has to be relaxed in some way to be achievable. There are a number of ways to relax the specification on CVs ([3], [2]). For example, the system can be squared down, such that only a subset of the original CVs are controlled. Alternatively, CVs can be controlled to be within a certain range or interval (ICVs) in a regulation problem, or within an interval that is a function of the desired set-point in a tracking problem (i.e. a funnel) ([11], [2]. Indeed, even for fat/square systems, it may be preferable to relinquish control of SCVs in favour of controlling a greater relative number of ICVs. A natural question arises regarding the ability of the controller to keep SCVs/ICVs at/within their desired values/intervals, at steady-state, given bounded disturbances. Appropriate selection of SCVs and ICV intervals for maximum plant operability can be tackled through ”operability” analysis. B. Operability The operability framework is a tool that has been developed to measure the ability of a system to reach its desired outputs for a given set of inputs and subject to bounded disturbances and constraints. The tool was introduced in [10] as a quantitative and potentially visual plant design and evaluation tool of process steady-state performance, and can be applied to the process without regard for controller structure selection. In [8], [9] a method for determining interval operability in the input space was developed. When excess inputs are left to control ICVs, there is normally a choice of input combinations: an optimisation program is required to choose the optimal input combination given the desired set-point and disturbance values. It was then recommended in [8], [9] that rather than exploring the entire set-point/disturbance space, it is more tractable to determine the minimum and maximum optimal inputs in each input dimension, and compare this (hyper-)rectangular space to the available input operating range. This area has been considered further in [5], introducing an operability index dedicated to ICV variables only. Output priorities are prespecified in [5], according to the shape of the Available
Operating Interval Space (AOIS), bounded by the mapped input space corresponding to the worst case disturbance values.
II. BACKGROUND A. Model Predictive Control A state space model and resultant prediction equations are as follows: xk+1 = Axk + Buk + Edk , yk = Cxk
C. Contribution
n
m
xk ∈ R , uk ∈ R , yk ∈ R zk = F xk zk ∈ Rp
In this paper, it is postulated that for non-square systems the process operability does in fact rely on the structure of the MPC controller. This is essentially because an MPC controller can determine optimal inputs that can minimise input constraint violations, and therefore the level of suboperability as measured in [8], [9]. Previous non-square operability work has been in the input space: perhaps a more useful way of approaching this concept is in the output space: given hard input constraints (a normal requirement), an MPC controller can minimise output constraint violations according to priority, including set-point equality constraints (through the use of feasible MPC strategies [15]). A number of ideas will be pursued regarding non-square operability in the output space. Another issue to be addressed is the physical meaning of the operability index. If some particular set-points are more important than others, or disturbances more likely to occur, then the volume of space becomes less useful for quantifying operability. It has been noted in [9] that one might weight different parts of the DOS or EDS depending on importance. This paper suggests that a new version of operability index could be used which is more directly linked to the proportion of situations for which a feasible solution exists. Stochastic disturbances can then be considered more easily in the framework. In order to set the discussion in context, this shall be attempted in this paper for the Shell Heavy Oil Fractionator control problem ([1], [11]), which is higher order in terms of numbers of inputs/outputs than previously published operability case studies ([6], [7], [8]. Operability can give a quantitative result for higher order systems, but is more difficult to visualise. Investigation of the use of operability where the number of MVs is much less than the number of CVs (as with the SHOF) is interesting generally because the control objectives can not be easily satisfied ([3], [2]).
(1)
l
z = diag(F )− x x = Px xk + Hx − u + Dx dk , − → − → → − → →
(2)
where (·) = [(·)Tk+1 (·)Tk+2 . . .]T , and dk is an input − → disturbance vector 1. A typical MPC dynamic optimisation program is as follows ([13], [14]): min Jk = → x − xs 2Q + − u − us 2R − → u
(3)
→ −
2 (4) (+Δu −→S ) (+ρ1|2|∞) s. t. zmin ≤ − z + ≤ zmax , umin ≤ → u ≤ umax , → − Δumin ≤ Δu (5) −→ ≤ Δumax
where Q, R, S are semi-positive definite weighting matrices (S, ρ are optional). For ICVs, mutually consistent steadystate variables xs , us need to be found satisfying the following equations (assuming no plant integrators): zs = F xs ,
xs = Axs + Bus + Eds −1
⇒ zs = F (A − I) G
(6) −1
B us + F (A − I) Gd
E ds
s.t. zmin ≤ zs + ≤ zmax , umin ≤ us ≤ umax
(7) (8)
B. Existing operability framework Operability is essentially concerned with determining families of solutions to equation (7) subject to constraints (8). For square systems, CVs within the Achievable Output Space (AOS) can be driven to exact set-points. The set of possible desired set-points is termed the Desired Operating Space (DOS). The Servo Operability Index for square systems is as follows ([5]):. s − IOy =
μ(AOSu (dN ) ∩ DOS) μ(DOS)
(9)
where μ represents a function that calculates the size of the space. For two dimensions it will be the area while for three dimensions it will be the volume ([5]). The servooperability index can take values between 0 and 1. If the servo operability index is equal to 1 then all the CVs of the square system inside the DOS may be controlled to exact set-points. Theory for non-square systems was developed in [8], [9] (and has been revisited more recently in [5]). When ICVs are employed, flexibility is introduced into the control scheme, that allows reaction to disturbances and set-point changes
D. Summary This paper attempts to design aspects of an MPC controller for the SHOF case study by applying the operability framework. Initially, the necessary mathematical background is presented in section 2, both for MPC and operability, and theory is presented for tackling non-square systems. Then in section 3 the case study is introduced, and operability applied to investigate robustness to disturbances and expected regulatory performance. Finally, conclusions are made, indicating where future research efforts are to be directed.
1 Outputs are normally used to reconstruct the state vector, and can be used directly as CVs, but the use of z and E allows CVs to be defined more generally (e.g. as a linear combination of measured and unmeasured states)
2
whilst keeping CVs within constraints. A constrained steadystate target optimisation program (SSTOP) can be employed to determine the optimal output and input set-points (z s∗ , u∗s ) for any given disturbance. An example of the SSTOP described in [8], [9] is as follows: J
∗
([zs∗
u∗s ]T )
= min ut − us Rs + ρ1|2|∞ [zs ,us ]T
s. t. Au ≤ bu + , Az zIs ≤ bz , ≥ 0 zs = GBus + GEds , zSP s = r
1) Minimum and maximum bound for each ICV: A SSTOP reflecting more clearly the SSTOP that will be used in the actual MPC controller can be used, including hard input constraints and soft output constraints: J ∗ ([zs∗ , u∗s , ∗ ]T ) =
(10)
max
r ∈ DOS ds ∈ EDS
u∗j (r, ds )
usj =
min
r ∈ DOS ds ∈ EDS
ut − us Rs + r − zSP s Qs
+ ρ1|2|∞ (14) s. t. Au u ≤ bu , Az zs ≤ bz + , ≥ 0 (15) (16) zs = GBus + GEds
(11) (12)
Where r and ut are output and input set-points, and z s is partitioned into SCVs and ICVs: z sT = [zSP s zIs ]T . Note that u∗ is effectively a function of r and d s . Nonsquare linear systems now attain a non-linear steady-state relationship. In principle, this SSTO should be repeatedly solved for all r ∈ DOS and d s ∈ EDS, but it may be more tractable in general to determine the bounding box of the DIS: i.e. the maximum and minimum values for each input usi : u ¯sj =
min
[zs , us , ]T
with Rs suitably revised to give output set-points priority over input set-points [12]. The bounding box approach could again be used to yield the Achievable Output Interval Space (AOIS), introduced in [5] as the minimum interval space that CVs can be kept within assuming worst case disturbance values, i.e.: z¯Isj =
u∗j (r, ds )
max
r ∈ DOS ds ∈ EDS
∗ zIsj (r, ds )
z Isj =
min
r ∈ DOS ds ∈ EDS
∗ zIsj (r, ds )
(17) The Interval Operability Index is then introduced in [5] as 2 :
(13)
OIIz =
This space defines the Desired Input Interval Space (DIIS), which can be used as the DIS for OI calculations. A direct search algorithm was used in [8] because the optimisation problem is not necessarily differentiable (or continuous). An alternative is to map the r − d s space, which is sensible for low-dimension problems, but may not be tractable for higher dimensions. This approach is set up to determine whether process input constraints are too stringent to achieve a particular required output, or whether they are over-specified. If the DIIS is too large, the DOS and EDS can be modified iteratively to be realistic. The OI calculated in this method will not necessarily give a good representation of the proportion of set-points that can actually be achieved. For large dimension problems, access to a direct search algorithm is required, which may have convergence problems. It should also be noted that if u ¯ sj = usj there is a problem with using volumes to calculate the OI. An alternative is to operate equivalently in the output space. This is considered in the next section, developing a new approach.
μ(DOS ∪ AOIS) μ(AOIS)
(18)
This method does not reflect properly the ability of the system to enforce SCVs. This method requires either gridding or a search algorithm. 2) Maximum constraint violations as an operability measure: For this method, Q s is set to zero, and set-point constraints are included as slackened inequality constraints. The slack variable vector is useful because it represents the absolute violation of both SCV and ICV constraints. ρ can be a weighting vector (or diagonal matrix for 2-norm) reflecting the priority of ICVs, SCVs, SCVs over ICVs etc. The result of the analysis is the maximum absolute constraint violations: ¯j =
max
r ∈ DOS ds ∈ EDS
∗j (r, ds )
(19)
This method does not lead to a space and therefore an OI, but it does show which output constraints are problematic, and the extent to which all constraints are violated, including SCVs. This method requires either gridding or a search algorithm. 3) Use of stochastic/gridding methods to determine an operability probability directly: It can be difficult to obtain a solution using search methods, and be sure that the worst case values are found. Instead, r ∈ DOS, w ∈ EDS can either be gridded or randomly quizzed, and solutions found to (14). z sn , usn can be recorded to determine maximum and minimum values for the whole space, but more importantly, an operability measure can be directly evaluated from these
C. Extension to the operability framework: SSTO in the output space It is difficult to consider SCVs and ICVs simultaneously in the output space in terms of OI, especially for suboperability. Three approaches have been considered: 1) Determine the minimum and maximum bound for each ICV (useful if there are no SCV requirements); 2) Determine the maximum constraint violations and use this as an operability measure; 3) Map out the output space (uniformly or stochastically), and determine an operability probability.
2 A particular shape of the AOIS is assumed in [5], but that is not done here.
3
calculations. The new Mapped Operability Index measure is as follows: N ∗ n (r, ds ) = 0 j=1 (20) MOIz = N Where N is the number of simulations. A strength of the stochastic method is the ability to apply normal distributions for ds , representing the fact that extreme disturbances are unlikely. The gridding method will ensure that unlikely but important values for d s and w are analysed. D. Summary This section has laid out the mathematical framework for MPC and operability, and proposed some new ways for dealing with non-square systems. These new approaches will be used to investigate the SHOF with a view to selection of appropriate ICVs/SCVs in light of expected disturbances. III. A NON - SQUARE CASE STUDY IN HIGHER DIMENSIONS : THE S HELL H EAVY O IL F RACTIONATOR (S.H.O.F)
Fig. 1.
In this case study we will consider the SHOF, which behaves essentially as a distillation column [11], [4]. The presented fractionator is a linear process which does not represent an existing process, but has been created to represent the most significant control engineering features encountered on real oil fractionators. A schematic diagram of the fractionator is shown in figure 1. The process has 7 measured outputs, 3 inputs and 2 input disturbances. This section introduces the SHOF, and then evaluates operability for the system. Perturbations are then made to ICV and disturbance definitions to determine how much more disturbance can be introduced into the system before it becomes inoperable, and to what degree performance specifications could be tightened. Finally, stochastic disturbances are considered as a more realistic way of representing disturbance effects.
In order to have a challenging non-square example, a new requirement has been introduced for the purpose of this exercise: |zi | ≤ 0.5, i ∈ {3, 4, 5, 6}. Also, to simplify the analysis, the unmeasured disturbance d 2 was set to zero. B. Application of operability As the specification does not require set-point control it is ideally suited to the use of ICVs, and the Interval Operability Index is a useful tool for verifying performance satisfaction. The method summarised in II-C.1 can be used to determine the bounding box of the ICVs, with d 1 gridded. A problem with this method is that it is possible to have a collapsed hyper-rectangle with zero volume. This makes the polytope intersection operation in (18) more awkward to perform. An alternative for a hyper-rectangular DOS is as follows: ⎧ ∗ min(¯zIsi ,ymax,i )−max(,ymin,i ) ∗ ⎪ ⎪ z¯Isi
= z ∗Isi : ∗ −z ∗ ⎪ z¯Isi p Isi ⎧ ⎨ ∗ ≤ zIsi ≤ ymax,i : 1 y ⎨ min,i OIIz = ∗ ∗ ∗ ⎪ ≥ z y z ¯ = z : ⎪ min,i i=1 ⎪ Isi Isi Isi :0 ⎩ ⎩ ∗ ymax,i ≤ zIsi (21)
A. The SHOF control problem The inputs and outputs of the system are presented in I. The control structures must be designed in such a way to handle the following performance criteria: •
• •
•
•
Schematic of the SHOF.
The Top and Side Draw End Point products (z 1 , z2 ) should be kept within ±0.005 at steady state, and between -0.5 and 0.5 otherwise. Maximise steam make in the steam generators which translates to minimising the Bottoms Reflux Duty (u 3 ). The system should be able to reject the disturbances entering the column either by the Upper or the Intermediate Refluxes (d1 , d2 ) due to changes in the heat duty requirements from other columns. These disturbances could range between -0.5 and +0.5. The limits for all draws (u 1 , u2 ), and the Bottom Reflux Draw Temperature (u 3 ) should be between -0.5 and +0.5. The Intermediate Reflux Duty (d 1 ) is measurable and therefore may be used for feed-forward control.
Using this method, in the instance of the SHOF, the OI Iz can be calculated to be 1, meaning that the system is fully operable. This result provides minimial information regarding the sensitivity of this figure. Examination of z¯Is ∗, z∗Is is helpful for determining how close CVs are to their thresholds. Also of interest is how both z¯Is ∗, z∗Is and OIIz varies when larger disturbances are introduced. C. Disturbance analysis Operability can now be used to give an indication of the conservatism in the design. This could be approached by considering how much more disturbance can be introduced into the system whilst maintaining OI Iz = 1. The threshold 4
TABLE I I NPUTS AND OUTPUTS FOR THE SHOF ( ADAPTED FROM [11]) Input Top Draw Side Draw Bottoms Reflux Duty Intermediate Reflux Duty Upper Reflux Duty Top End Point Side End Point Top Temperature Upper Reflux Temperature Side Draw Temperature Int. Reflux Temperature Bottoms Reflux Temperature
Role MV MV MV Measured DV Unmeasured DV ICV ICV ICV ICV ICV ICV ICV
Symbol u1 u2 u3 d1 d2 z1 z2 z3 z4 z5 z6 z7
Original Objectives
New Constraints
Constraints −0.5 ≤ u1 ≤ 0.5 −0.5 ≤ u2 ≤ 0.5 −0.5 ≤ u3 ≤ 0.5 −0.5 ≤ d1 ≤ 0.5 −0.5 ≤ d2 ≤ 0.5 −0.5 ≤ z1 ≤ 0.5 −0.5 ≤ z1 ≤ 0.5
Minimise −0.005 ≤ z1 ≤ 0.005 at ss −0.005 ≤ u2 ≤ 0.005 at ss
d2 = 0 −0.5 ≤ −0.5 ≤ −0.5 ≤ −0.5 ≤
−0.5 ≤ z7 ≤ 0.5
for the SHOF is |d1 | ≤ 2.489 (see Figure 4), so the factor of safety is 5. This result was verified by conducting simulations using the MPC Toolbox and a 54 state dynamic model3 . The factor of safety can be used to account for parameter uncertainty and dynamic effects. In further work, it would be more vigorous to make use of dynamic operability theory ([9], and robustness theory, which would require development for application to the non-square case. This result can be illustrated graphically through the use of a radar plot in figure 2 4 . Three different values of maximum disturbance are used. Projections of the AOIS onto individual ICV axes can be useful in illustrating the extent to which individual ICVs are within a hyper-rectangular bounding box of the DOS (which is hyper-rectangular itself in this case), but the concept of volume is not presented here: the operability index remains a numerical result.
0.8
z6
z5
z4
z3 z4 z5 z6
≤ 0.5 ≤ 0.5 ≤ 0.5 ≤ 0.5
z3
0.6 z2
0.4 z7
0.2
z1
0 −0.2
D. Use of SCVs As there is such a large factor of safety, it would be interesting to replace some ICVs with SCVs. The number of SCVs must be less than the number of MVs to retain sufficient d.o.f to control the ICVs adequately, so there can be up to 2. The natural choice is z 1 , z2 , with the steadystate constraints for each changed to [-0.5 0.5]. Use of the method described in II-C.2 with equal weighting on all slack variables resulted in values presented in figure 3. From figure 3(a) it can be seen that with set-point control z1 will have more severe constraint violations than z 2 and should be within 0.1 of r for all r ∈ DOS, d 1 inEDS, with constraint violations also expected in z 4 by about the same amount. Figure 3(b) shows that for equal weighting/probability of r ∈ DOS, d 1 inEDS, the set-point cannot be tracked about 12% of the time. Finally, 20 can be applied, giving an MOI of 0.77. The worst case disturbance values have play a significant role in the operability analysis. Assumption of a flat disturbance probability distribution could result in conservatism in
−0.4
DOS |wi|
