Soft constraint

Soft constraints. In Figure 19.2b, there was a situation in which there was some flexibility to change the temperature at which a filtration takes place. These are termed soft constraints. There is not complete freedom to choose the conditions under which the operation takes place, but there is some flexibility to change the conditions. Another example of a soft constraint is product storage temperature. There is sometimes flexibility to choose the temperature at which material is stored. How should such soft constraints be directed to benefit the overall heat integration problem ... 

Soft constraints should exploit the plus-minus principle to improve the targets. 

The idea of the new approach is to, first, define only one model that includes all constraints - be it shift models, personnel, batch sizes, maximum perishabilities or soft constraints such as costs and values, characteristic numbers or feasibility and optimum of the plan. The case studies will show that the model follows an intuitive representation of the relevant items. Second, the operator based approach makes the overall solution procedure extensible. If one operator gets stuck in a local optimum another operator may help out and could be added at any time. Third, the whole approach works on understandable objects such that at any time during the solution procedure an easy check can be made what happens. 

The demands are given as orders which are partially movable or have a fixed assignment to a resource with dearly defined setup, production and deaning times. There are also anonymous demands that were calculated from forecasts. The target inventory is a soft constraint that is used to model dynamic safety stocks. Most quants must fulfill integer batch sizes and often minimum lot sizes. 

Another important feature of the case study scenario and the resulting cost model is inventory control. High seasonality effects and long campaign durations necessitate considerable build-up of stocks. To avoid an unbalanced build-up of stock, soft constraints for safety stock and maximum stock levels are used. To achieve an even better inventory leveling across products and locations, piece-wise linear cost functions for falling below safety stock as well as for exceeding maximum stock levels are employed. 

Process simulators contain the model of the process and thus contain the bulk of the constraints in an optimization problem. The equality constraints ( hard constraints ) include all the mathematical relations that constitute the material and energy balances, the rate equations, the phase relations, the controls, connecting variables, and methods of computing the physical properties used in any of the relations in the model. The inequality constraints ( soft constraints ) include material flow limits maximum heat exchanger areas pressure, temperature, and concentration upper and lower bounds environmental stipulations vessel hold-ups safety constraints and so on. A module is a model of an individual element in a flowsheet (e.g., a reactor) that can be coded, analyzed, debugged, and interpreted by itself. Examine Figure 15.3a and b. 

A reasonable choice for bo is the spectrum of the analyte of interest because that is the solution for b in the absence of noise and interferents. Another choice is the net analyte signal41 calculated using all of the known pure analyte spectra. Such flexibility in the selection of b0 is owing to the manner in which the constraint is incorporated into the calibration algorithm. For CR, the spectral constraint is included in a nonlinear fashion through minimization of , and is thus termed a soft constraint. On the other hand, there is little flexibility for methods such as HLA, in which the spectral constraint is algebraically subtracted from each sample spectrum before performing PCA. We term this type of constraint a hard constraint. 

The steam turbines require one-third the energy of An electric motor. Each refrigeration unit has a different horsepower/ton characteristic, which also depends upon ambient conditions. There are hard constraints on compressor loads and cost penalties (soft constraints) on electrical load. Steam, refrigeration, compressed air, and electrical loads to the plant vary continually. While the author does not suggest that the optimum operating and control strategy is known, he does imply that a computer control system is the only way to operate the plant in an optimum manner. 

A linear model predictive control law is retained in both cases because of its attracting characteristics such as its multivariable aspects and the possibility of taking into account hard constraints on inputs and inputs variations as well as soft constraints on outputs (constraint violation is authorized during a short period of time). To practise model predictive control, first a linear model of the process must be obtained off-line before applying the optimization strategy to calculate on-line the manipulated inputs. The model of the SMB is described in [8] with its parameters. It is based on the partial differential equation for the mass balance and a mass transfer equation between the liquid and the solid phase, plus an equilibrium law. The PDE equation is discretized as an equivalent system of mixers in series. A typical SMB is divided in four zones, each zone includes two columns and each column is composed of twenty mixers. A nonlinear Langmuir isotherm describes the binary equilibrium for each component between the adsorbent and the liquid phase. 

The /th objective component that corresponds to a hard constraint is assigned to the value of G(i) whenever the hard constraint has been satisfied. The underlying reason is that there is no ranking preference for any particular objective component that has the same value in an evolutionary optimization process, and thus the evolution will only be directed toward optimizing soft constraints and any unattained hard constraints. 

Again, as an example, let us imagine that the constraint on increasing feed rate to our case study heater is now a limit on maximum burner pressure. Unlike the fuel valve position, this is a soft constraint. Although violation is undesirable it is physically possible. Burner... 

Some MVC permit the definition of hard and soft constraints. In addition to their meaning as described above, often a soft constraint is used as a more conservative limit on a hard constraint. The controller will violate soft constraints if this is the only way that it can satisfy all the hard constraints. 

On our case study heater, the process operator will typically enter a SP of around 90 %. This effectively converts a hard constraint to a soft constraint. In order to maintain control of the outlet temperature during minor process disturbances some leeway is required. This means that the process capacity is not fuUy utilised. Conditioning the constraint, as described in Chapter 5, offers an alternative method of converting it to a soft constraint and would permit this leeway to be reduced. 

Some MVC packages permit hard and soft constraints. Soft constraints are adhered to if possible but will be violated if this is the only way of avoiding violating a hard constraint. Other packages permit weighting to be applied to constraints so that the engineer can specify which should be violated first if the MVC cannot identify a feasible solution. 

Figure 8.19 shows one of the detailed trends used in support of the exercise. It shows, for MV 1, the acmal value and the HI and LO constraints. If the MVC supports hard and soft constraints then both should be trended. The chart is useful in determining why the total number of constraining MVs has changed, and from the time of the constraint change, identifying who made the change and why. 

Figure 13.2 Benefit of improved control versus a soft constraint...

Bistarelli, S. Semirings for Soft Constraint Solving and Programming. LNCS, vol. 2962. Springer, Heidelberg (2004)... 

In the case of overrides with proportional-only action, we can visualize, as shown in Figure 9.18, a zone between hard and soft constraints. The width of this zone is determined by the override loop gain, which is limited by stability considerations. With a proportional control loop des ned fijr dead-beat response, there is a unique relationship between the value of the manipulated variable and the distance between the process variable and its hard constraint. The manipulated variable always readies its maximum (or minimum) value before the process variable exceeds its hard constraint. The soft constraint will correspond to Ae minimum (or maximum) value of the override output. The takeover point between normal and override controls will be at a variable position (depending on operating conditions) somewhere between the hard and soft constraints. 

It is usually necessary, and often desirable, to impose relations between the parameters of a model. The problem to be solved is then a minimization of Equation [6] while imposing subsidiary conditions. A constraint satisfies such a condition exactly. A restraint (soft constraint) is a condition accompanied by a measure of uncertainty or a weight, and accordingly satisfies the condition only approximately. 

To ensure that QM and MM molecules do not exit their regions, an infinitely steep constraint can be used. However, in practice, such a constraint would lead to divergent molecular dynamics. Therefore, a soft constraint is used, typically a simple half-harmonic potential that exerts a repulsive force on any MM molecules... 

Inequality constraints can be included in the control calculations in many different ways (Maciejowski, 2002 Qin and Badgwell, 2003). It is convenient to make a distinction between hard constraints and soft constraints. As the name implies, a hard constraint cannot be violated at any time. By contrast, a soft constraint can be violated, but the amount of violation is penalized by a modification of the objective function, as described below. This approach allows small constraint violations to be tolerated for short periods of time. 

Unfortunately, hard output constraints can result in infeasible solutions for the optimization problem, especially for large disturbances. Consequently, output constraints are usually expressed as soft constraints involving slack variables Sj (Qin and Badgwell, 2003) ... 

A time-varying Q matrix can also be used to implement soft constraints by real-time adjustment of Q, For example, if an output variable approaches an upper or lower limit, the corresponding elements of Q would be temporarily increased. 

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