Systematic design of active constraint switching using selectors

Selector logic is a simple and eﬀective tool to switch between diﬀerent controlled variables associated with change in active constraints. Selector blocks have been extensively used in the process control industry for decades, but their design has been based on engineering intuition and experience. Currently, there is a lack of systematic procedure to design selectors for active constraint switching. In this paper, we address this gap and provide a systematic procedure, which can be applied without the need for detailed process models. Illustrative examples are used to demonstrate the proposed framework.


A.1. Simulator Model
We consider a two-product distillation column with N T stages as shown in The total mass balance and the mass balance for the light component on stage i, except in the condenser (i = N T ), feed stage (i = N f ) and reboiler (i = 1) is given by: The mass balance on the feed stage (i = N f ) is given by, The mass balance on the reboiler (i = 1) is given by, where B is the bottom flow rate and V is the boilup as shown in Fig. 1. The mass balance on the condenser (i = N T ) is given by, where D is the distillate flow rate and L is the reflux as shown in Fig. 1.
From this, we get the expression for the rate of change of liquid mole fraction The model therefore has 2N T differential states denoted by [{x The liquid flows depend on the liquid holdup on the stage above and the vapor flow as follows where L0 i ( in kmol/min) and M 0 i (in kmol) are the nominal values for the liquid flow and holdup on stage i. The effect of vapor flow on the liquid flow is captured by λ.
The vapor composition can then be computed from the vapor-liquid equilibrium where α is the constant relative volatility.

A.2. Controller design
We assume that the overhead vapour V D is used to maintain a constant pressure. Stable operation of the column requires the levels M B and M D to be controlled. In this model, the column is stabilized using the LV-configuration where we use D to control M D , and B to control M B as shown in Fig. 1. We use a P-controllers for each level control loop, with the controller gain K P = 10 for both the loops.
As mentioned in the manuscript, the purity constraint on x D will always be active, since this is the most valuable product. The x D composition is controlled using the reflux L using a PI controller that is tuned using the SIMC tuning rules. For a desired closed loop time constant of τ c = 10, this results in the proportional gain K P = 7.8947 and K I = 0.2193.
The composition control for the bottom product x B is also achieved using a PI controller that is tuned using the SIMC rules. For a desired closed loop time constant of τ c = 10, this results in the proportional gain K P = 2.2140 and K I = 0.123.
The MATLAB scripts for the distillation column example is given below or can be found in https://github.com/dinesh-krishnamoorthy/Feedback-based-RTO/ tree/master/Selectors/ColA.

B.1. Simulator model
The benchmark Williams-Otto reactor converts the raw materials A and B into useful products P and E along with a byproduct G via a series of reactions, A + B → C k 1 = 1.6599 × 10 6 e −6666.7/Tr B + C → P + E k 2 = 7.2177 × 10 8 e −8333.3/Tr C + P → G k 3 = 2.6745 × 10 12 e −11111/Tr The reactor is modeled as, where the mass holdup W = 2105 kg. The reactor is controlled using the reactor temperature M V 1 := T r and the feed rate M V 2 := F B with pure B component. Feed rate F A with pure A component is a disturbance and we assume that it is expected to vary between 1kg/s and 2kg/s. We assume perfect level control such that the outflow F = F A + F B .

B.2. Controller design
The objective is to maximize the production of useful products P and E. In addition, there are purity constraints on G and A on the product stream, and a minimum outflow rate F out . The steady-state optimization problem is formulated as, x G x Gmax where x G,max = 0.08, x A,max = 0.12, and F min = 4.4 kg/s.
Since the purity constraint on x G is very low, it will always be active for the assumed disturbance range of F A ∈ [1, 2]kg/s. Therefore the relevant active constraint combinations are • only x G active (R-I) • x G and x A active (R-II) • x G and F active (R-III) Since we have two MVs, we first pair the reactor temperature u 1 = T r to tightly control x G to its limit of x G,max = 0.08 using a PI control. This leaves us with one degree of freedom, namely u 2 = F B , which will be used to control either the self-optimizing CV y 0 to a desired setpoint in region R-I, or control x A to its limit of x A,max = 0.12 in region R-II, or control F to its minimum limit of F min = 4.4 kg/s in region R-III. In this case, and since we have only Y − , a single maximum selector block can be used to switch between the different active constraint regions ( cf. Remark ??).
Therefore, we use four SISO controllers to control the CVs in each active constraint region, namely, 1. Composition controller denoted by CC G that uses T r to control x G to x G,max (in regions R-I, RII, and R-III).
2. Composition controller denoted by CC A that uses F B to control x A to x A,max (in region R-II). 3. Flow controller denoted by F C that uses F B to control F to F min (in region R-III). 4. Self-optimizing controller denoted by JC that uses F B to control the selfoptimizing CV y 0 to y 0,sp (in region R-I).
In region R-I, we have used a linear gradient combination as the self-optimizing CV y 0 = N T ∇ u J controlled to a constant setpoint of y 0,sp = 0. In this example, the linear gradient combination is given by y 0 = 0.9959∇ F B J + 0.0906∇ Tr J. The control structure design is shown in Fig. 2. The controller tuning parameters are shown in Table 1. Fig. 3 shows the simulation results using the proposed control structure design. It can be clearly seen that as the disturbance changes, the max selector block is able to automatically handle the CV-CV switching between R-I, R-II and R-III. The MATLAB scripts for the distillation column example is given below or can be found in https://github.com/dinesh-krishnamoorthy/Feedback-based-RTO/ tree/master/Selectors/WilliamsOtto.