Challenges and Paradigms in Applied Robust Control Part 9 pps

Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System 23 0 1 23 4 5 −150 −100 −50 0 50 100 150 Time, s Displacement, μm 2 4 6 8 10 Speed, ×1000 rpm (a) Robust H ∞ controller 0 1 23 4 5 −150 −100 −50 0 50 100 150 Time, s Displacement, μm 2 4 6 8 10 Speed, ×1000 rpm (b) LPV controller Fig. 16. Rotor acceleration responses. point of 6500 rpm where the system crosses the first flexible mode. The second point where the system experiences oscillations is close to the maximum speed and it can be explained by the deceleration of the rotor. The LPV controller has a lower magnitude of oscillations around this point; the difference is 35 %. Such a behavior can be explained by an adaptive nature of an LPV controller. In each step, the gains are modified according to the rotational speed. During the acceleration process, the system does not have enough time to adapt. This results in a higher amplitude of oscillations. During the later deceleration phase, the coefficients do not change that fast and performance is better. The speed of the parameter variation is a significant problem for the LPV controllers, and usually the main point of conservatism in that approach (Leith & Leithead, 2000). The second simulation experiment in the steady state proves that LPV controller provides a better performance. In this experiment, a step disturbance to the x channel of the rotor A-end is applied at the maximum rotational speed. The simulation results are presented in Fig. 17. The magnitude of the disturbance response for an LPV controller is about three times smaller than that of a robust controller. Additionally, the LPV controller does not have coupling between different ends, so the disturbance does not propagate through the system. 6. Real-time operating conditions The AMB-based system requires hard-real time controllers. In the case of a robust control strategy, the control law is of higher complexity than other solutions. Therefore, the implementation of the control law must fulfill the requirements of the target control system such as finite precision of the arithmetic and number format and available computational 229 Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System 24 Will-be-set-by-IN-TECH 0 0.05 0.1 0.15 0.2 0.25 0.3 0 100 200 300 Time, s Displacement, μm LPV End A LPV End B Robust End A Robust End B Fig. 17. Step disturbance response for controllers in the x direction. power. The digital control realization requires a digital controller that matches the continuous form in the operating frequency range. The controllers for the radial suspension of the AMB rotor system are tested using a dSpace DS1005-09 digital control board and a DS4003 Digital Input/Output system board as a regulation platform. The Simulink and Real-time Workshop software are applied for automatic program code generation. The selected sampling rate is 10 kHz. The resolution of the applied ADCs is 16 bits. The control setup limits the maximum number of states of the implemented controllers to 28 states. 7. Conclusions The chapter discusses options and feasible control solutions when building uncertain AMB rotor models and when designing a robust control for the AMB rotor systems. The review of the AMB systems is presented. The recommendations for difficult weight selection in different weighting schemes are given. Design-specific problems and trade-offs for each controller are discussed. It is shown that the operating conditions of the selected real-time controllers satisfy the control quality requirements. The resulting order of the controller depends on the complexity of the applied weighting scheme, plant order, and applied uncertainties. The detailed interconnections lead to controllers, which are difficult to implement and are not transparent. However, the too simple weighting schemes cannot provide sufficient design flexibility with respect to the multi-objective specification. For the systems with considerably gyroscopic rotors and high rotational speeds, the LPV method provides a significantly better solution than nonadaptive robust control methods. 8. Acknowledgement This chapter was partially founded by AGH Research Grant no 11.11.120.768 9. References Apkarian, P. & Gahinet, P. (1995). A convex characterization of gain-scheduled H ∞ controllers, Automatic Control, IEEE Transactions on 40(5): 853–864. Apkarian, P., Gahinet, P. & Becker, G. (1995). Self-scheduled H ∞ control of linear parameter-varying systems: a design example, Automatica 31(9): 1251–1261. Battachatyya, S. P., Chapellat, H. & Keel, L. H. (1995). 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Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ. 232 Challenges and Paradigms in Applied Robust Control Part 3 Distillation Process Control and Food Industry Applications 11 Reactive Distillation: Control Structure and Process Design for Robustness V. Pavan Kumar Malladi 1 and Nitin Kaistha 2 1 Department of Chemical Engineering, National Institute of Technology Calicut, Kozhikode, 2 Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur, India 1. Introduction Reactive Distillation (RD) is the combination of reaction and distillation in a single vessel (Backhaus, 1921). Over the past two decades, it has emerged as a promising alternative to conventional “reaction followed by separation” processes (Towler & Frey, 2002). The technology is attractive when the reactant-product component relative volatilities allow recycle of reactants into the reactive zone via rectification/stripping and sufficiently high reaction rates can be achieved at tray bubble temperature. For equilibrium limited reactions, the continuous removal of products drives the reaction to near completion (Taylor & Krishna, 2000). The reaction can also significantly simplify the separation task by reacting away azeotropes (Huss et al., 2003). The Eastman methyl acetate RD process that replaced a reactor plus nine column conventional process with a single column is a classic commercial success story (Agreda et al., 1990). The capital and energy costs of the RD process are reported to be a fifth of the conventional process (Siirola, 1995). Not withstanding the potentially significant economic advantages of RD technology, the process integration results in reduced number of valves for regulating both reaction and separation with high non-linearity due to the reaction-separation interaction (Engell & Fernholtz, 2003). Multiple steady states have been reported for several RD systems (Jacobs & Krishna, 1993; Ciric & Miao 1994; Mohl et al., 1999). The existence of multiple steady states in an RD column can significantly compromise column controllability and the design of a robust control system that effectively rejects large disturbances is a principal consideration in the successful implementation of the technology (Sneesby et al., 1997). In this Chapter, through case studies on a generic double feed two-reactant two-product ideal RD system (Luyben, 2000) and the methyl acetate RD system (Al-Arfaj & Luyben, 2002), the implications of the non-linear effects, specifically input and output multiplicity, on open and closed loop column operation is studied. Specifically, steady state transitions under open and closed loop operation are demonstrated for the two example systems. Input multiplicity, in particular, is shown to significantly compromise control system robustness with the possibility of “wrong” control action or a steady state transition under closed loop operation for sufficiently large disturbances. Challenges and Paradigms in Applied Robust Control 236 Temperature inferential control system design is considered here due to its practicality in an industrial setting. The design of an effective (robust) temperature inferential control system requires that the input-output pairings be carefully chosen to avoid multiplicity in the vicinity of the nominal steady state. A quantitative measure is developed to quantify the severity of the multiplicity in the steady-state input output relations. In cases where an appropriate tray temperature location with mild non-linearity cannot be found, it may be possible to “design” a measurement that combines different tray temperatures for a well- behaved input-output relation and consequently robust closed loop control performance. Sometimes temperature inferential control (including temperature combinations) may not be effective and one or more composition measurements may be necessary for acceptable closed loop control performance. In extreme cases, the RD column design itself may require alteration for a controllable column. RD column design modification, specifically the balance between fractionation and reaction capacity, for reduced non-linearity and better controllability is demonstrated for the ideal RD system. The Chapter comprehensively treats the role of non-linear effects in RD control and its mitigation via appropriate selection/design of the measurement and appropriate process design. 2. Steady state multiplicity and its control implications Proper regulation of an RD column requires a control system that maintains the product purities and reaction conversion in the presence of large disturbances such as a throughput change or changes in the feed composition etc. This is usually accomplished by adjusting the column inputs (e.g. boil-up or reflux or a column feed) to maintain appropriate output variables (e.g. a tray temperature or composition) so that the purities and reaction conversion are maintained close to their nominal values regardless of disturbances. The steady state variation in an output variable to a change in the control input is referred to as its open loop steady state input-output (IO) relation. Due to high non-linearity in RD systems, the IO relation may not be well behaved exhibiting gain sign reversal with consequent steady state multiplicity. From the control point of view, the multiplicity can be classified into two types, namely, input multiplicity and output multiplicity as shown in Figure 1. In case of output multiplicity, multiple output values are possible at a given input value (Figure 1(a)). Input multiplicity is implied when multiple input values result in the same output value (Figure 1(b)). To understand the implications of input/output multiplicity on control, let us consider a SISO system. Let the open loop IO relation exhibit output multiplicity with the nominal operating point denoted by ‘*‘(Figure 1(a)). Under open loop operation, a large step decrease in the control input from u 0 to u 1 would cause the output to decrease from y 0 to y 1 . Upon increasing the input back to u 0 , the output would reach a different value y 0 ‘ on the lower solution branch. For large changes in the control input (or alternatively large disturbances), the SISO system may exhibit a steady state transition under open loop operation. For RD systems, this transition may correspond to a transition from the high conversion steady state to a low conversion steady state. The transition can be easily prevented by installing a feedback controller with its setpoint as y 0 . Since the output values at the three possible steady states corresponding to u 0 are distinct, it is theoretically possible to drive the system to the desired steady state with the appropriate setpoint (Kienle & Marquardt, 2003). Note Reactive Distillation: Control Structure and Process Design for Robustness 237 that for the IO relation in Figure 1(a), the feedback controller would be reverse acting for y 0 /y 0 ‘ and direct acting for y 0 “ as the nominal steady state. The implications of input multiplicity in an IO relation are much more severe. To understand the same, consider a SISO system with the IO relation in Figure 1(b) and the point marked ‘*‘ as the nominal steady state. Assume a feedback PI controller that manipulates u to maintain y at y 0 . Around the nominal steady state, the controller is direct acting. Let us consider three initial steady states marked a, b and c on the IO relation, from where the controller must drive the output to its nominal steady state. At a, the initial error (y SP -y) is positive and the controller would decrease u to bring y to the desired steady state. At b, the error is again positive and the system gets driven to the desired steady state with the controller reducing u. At c, due to the y SP crossover in the IO relation, the error signal is negative and the direct acting controller would increase u, which is the wrong control action. Since the IO relation turns back, the system would settle down at the steady state marked ‘**’. For large disturbances, a SISO system with input multiplicity can succumb to wrong control action with the control input saturating or a steady state transition if the IO relation exhibits another branch with the same slope sign as the nominal steady state. Input multiplicity or more specifically, multiple crossovers of y SP in the IO relationship thus severely compromise control system robustness. Fig. 1. Steady state multiplicity, (a) Output multiplicity, (b) Input multiplicity The suitability of an input-output (IO) pairing for RD column regulation can be assessed by the steady state IO relation. Candidate output variables should exhibit good sensitivity (local slope in IO relation at nominal operating point) for adequate muscle to the control system where a small change in the input drives the deviating output back to its setpoint. Of these candidate sensitive (high open loop gain) outputs, those exhibiting output multiplicity may be acceptable for control while those exhibiting input multiplicity may compromise control system robustness due to the possibility of wrong control action. The design of a robust control system for an RD column then requires further evaluation of the IO relations of the sensitive (high gain) output variables to select the one(s) that are monotonic for large changes in the input around the nominal steady state and avoid multiple y SP crossovers. If Challenges and Paradigms in Applied Robust Control 238 such a variable is not found, the variable with a y SP crossover point (input multiplicity), that is the furthest from the nominal operating point should be selected. It may also be possible to combine different outputs to design one that avoids crossover (input multiplicity). The magnitude |u 0 -u c |, where u c is the input value at the nearest y SP crossover can be used as a criterion to screen out candidate outputs. For robustness, Kumar & Kaistha (2008) define the rangeability, r, of an IO relation as r = |u 0 – u c ’ | where u c ’ is obtained for y = y SP – y offset as shown in Figure 1(b). The offset from the actual crossover point ensures robustness to disturbances such as a bias in the measurement. In extreme cases, where a suitable output variable is not found that can effectively reject large disturbances, the RD column design may require alteration for improving controllability. Each of these aspects is demonstrated in the following example case studies on a hypothetical two-reactant two-product ideal RD column and an industrial scale methyl acetate RD column. 3. RD control case studies To demonstrate the impact of steady state multiplicity on RD control, two double feed two-reactant two-product RD columns with stoichiometric feeds (neat operation) are considered in this work. The first one is an ideal RD column with the equilibrium reaction A + B ↔ C + D. The component relative volatilities are in the order  C >  A >  B >  D so that the reactants are intermediate boiling. The RD column consists of a reactive section with rectifying and stripping trays respectively above and below it. Light fresh A is fed immediately below and heavy fresh B is fed immediately above the reactive zone. Product C is recovered as the distillate while product D is recovered as the bottoms. The rectifying and stripping trays recycle the reactants escaping the reactive zone and prevent their exit in the product streams. This hypothetical ideal RD column was originally proposed by Luyben (2000) as a test-bed for studying various control structures (Al-Arfaj & Luyben, 2000). In terms of its design configuration, the methyl acetate column is similar to the ideal RD column with light methanol being fed immediately below and heavy acetic acid being fed immediately above the reactive section. The esterification reaction CH 3 COOH + CH 3 OH ↔ CH 3 COOCH 3 + H 2 O occurs in the reactive zone with nearly pure methyl acetate recovered as the distillate and nearly pure water recovered as the bottoms. Figure 2 shows a schematic of the two RD columns. The ideal RD column is designed to process 12.6 mol s -1 of stoichiometric fresh feeds to produce 95% pure C as the distillate product and 95% pure D as the bottoms product. Alternative column designs with 7 rectifying, 6 reactive and 7 stripping trays or 5 rectifying, 10 reactive and 5 stripping trays are considered in this work. For brevity, these designs are referred to as 7/6/7 and 5/10/5 respectively. The methyl acetate RD column is designed to produce 95% pure methyl acetate distillate. The 7/18/10 design configuration reported by Singh et al. (2005) is studied here. Both the columns are operated neat with stoichiometric feeds. The reaction and vapor liquid equilibrium model parameters for the two systems are provided in Table 1. [...]... ISBN 97 8-1560328254 256 Challenges and Paradigms in Applied Robust Control Tyreus, B.D., Luyben, W.L., 199 2 Tuning PI controllers for integrator/deadtime processes, Industrial and Engineering Chemistry Research, 31, pp.2625-2628 12 Robust Multivariable Control of Ill-Conditioned Plants – A Case Study for High-Purity Distillation Kiyanoosh Razzaghi and Farhad Shahraki Department of Chemical Engineering,... and Lundström, 199 0; Sriniwas et al., 199 5; Christen et al., 199 7; Shin et al., 2000; Razzaghi & Shahraki, 2005, 2007; Biswas et al., 20 09) Some possible improvements for linear multivariable predictive control of high-purity distillation columns are proposed by Trentacapilli et al ( 199 7) and a simple way of inserting a local model that contains part of the process nonlinearity into the controller is... errors, and if 258 Challenges and Paradigms in Applied Robust Control determinant of the gain matrix of the model and that of the plant have different signs, no controller with integral action exists that can stabilize both the model and plant (Grosdidier et al., 198 5) Many control design techniques have been applied to the high-purity distillation columns (e.g Georgiou et al., 198 8; Skogestad and Lundström,... series of step changes in FB when (a) T18 (b) T12 is controlled variable 248 Challenges and Paradigms in Applied Robust Control flow rate from 8.82 to 12.6 mol s-1 brings the column operation in the vicinity of point A in Figure 9 (relatively low FA) with the consequent wrong control action causing a valve shutdown For the second series of step changes (+20%, +20% and -40%), a stable and well behaved response... manipulation handle T18/T12 is therefore likely to be controlled tightly without significant deviations from its nominal setpoint Larger deviations in T2 (controlled using FA) can result in wrong control action due to input multiplicity corresponding to higher FA feed into the Reactive Distillation: Control Structure and Process Design for Robustness 247 column (Figure 8 and Figure 9) In the T2-FA IO... the T12-VS (T2 fixed) and T2-FA (T12 fixed) IO relations are also obtained 246 Challenges and Paradigms in Applied Robust Control These are shown in Figure 9 The nominal steady state is marked O and the corresponding crossover points are marked A, B etc A non-nominal steady state on a solution branch is stable if the local slope in the IO relation has the same sign as for the nominal steady state O,... ( 199 4) Steady state multiplicities in an ethylene glycol reactive distillation column, Industrial and Engineering Chemistry Research, 33, pp.2738-2748 Reactive Distillation: Control Structure and Process Design for Robustness 255 Dorn, C., Guttinger, T.E., Wells, G.J., Morari, M., Kienle, A., Klein, E., Gilles, E.-D ( 199 8) Stabilization of an unstable distillation column, Industrial and Engineering... single matrix The structured singular value (SSV) approach provides necessary and sufficient conditions for robust stability and performance for the situation in which uncertainty occurs simultaneously and independently in various parts of the overall control system (e.g input and output uncertainty) but the perturbation matrix is still norm-bounded One of the most difficult steps in analysing the robust. .. gain, Niederlinski Index and relative gain, the proposed rangeability metric is a useful tool for selecting robust controlled variables and rejecting poor choices that may potentially succumb to non-linear dynamic phenomena The steady state IO relations (bifurcation analysis) can also help in arriving at an inherently more controllable and economical RD process design 6 References Agreda, V.H., Partin,... temperatures and (b) ΔT = T20 – T8 with FHAc and Qr 2 49 250 Challenges and Paradigms in Applied Robust Control crossover in the IO relations for high rangeability, we also consider a combination of tray temperatures The difference between two reactive tray temperatures (ΔT = T20 - T8) was found to avoid input multiplicity with respect to FHAc and Qr with the corresponding IO relations in Figure 12(b) In the . of Robust and Nonlinear Control 6 (9- 10): 98 3 99 8. Zhou, K. ( 199 8). Essentials of Robust Control, Prentice-Hall, Upper Saddle River, NJ. Zhou, K., Doyle, J. & Glover, K. ( 199 6). Robust and. Approach, Prentice Hall. 230 Challenges and Paradigms in Applied Robust Control Discussion on Robust Control Applied to Active Magnetic Bearing Rotor System 25 Becker, G. & Packard, A. ( 199 4). Robust performance. are also obtained. Challenges and Paradigms in Applied Robust Control 246 These are shown in Figure 9. The nominal steady state is marked O and the corresponding crossover points are marked