Robust control for a multi-stage evaporation plant in the presence of uncertainties
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1 Preprint 11th IFAC Symposium on Dynamics and Control of Process Systems including Biosystems June NTNU Trondheim Norway Robust control for a multi-stage evaporation plant in the presence of uncertainties Philipp Nguyen and Robert Tenno Abstract A Multi Stage Flash evaporation plant is investigated as a partly unknown process represented by structured uncertainties concerning several model parameters. Robust control designs in form of H loop-shaping and s are applied to the plant. The control objective has been obtained from previous work where the optimal operating point has been shown to be in form of a reference liquid level profile throughout all the tanks of the plant. The incorporation of uncertainties and the controller into the generalized plant is done via Linear Fractional Transformations. The nominal as well as robust stability and performance are investigated for the controllers. I. INTRODUCTION One of the most popular and established sea water desalination technology is the multi-stage flash evaporator plant MSF for conversion of sea water into potable water. Lots of work has been invested in developing models which are utilized in optimization of the plant design and control for instance. In [1 and [ steady-state and dynamic mathematical models of a Multi Stage Flash Evaporator plant MSF have been reported accounting for known stage geometries and considering the physico-chemical properties of salty sea water by the means of correlations. In [3 a comprehensive overview is provided on the numerous control tasks and industrial control methods for desalination processes in MSF plants where the global control task has been divided into numerous sub-problems and investigated independently. In [4 a nonlinear 18-stage MSF in once-through configuration is investigated based on deterministic models. The optimal operating point regarding total water production and thermal efficiency of the plant has been obtained by backwards motion-planning resulting in a reference liquid height profile throughout the stages of the plant. Different controllers - Feedback linearization LQR and PI - are implemented for the nonlinear process in order to control the entire height profile of the plant. Based on defined evaluation criteria the performance of the controllers is elaborated. P. Nguyen is with School of Electrical Engineering Department of Automation and Systems Technology Aalto University 15 Espoo Finland philipp.nguyen@aalto.fi R. Tenno is with School of Electrical Engineering Department of Automation and Systems Technology Aalto University 15 Espoo Finland robert.tenno@aalto.fi Fig. 1: MSF desalination plant in once-through configuration modified from [3. Few works have been dealing with uncertain MSF processes. In [5 a conventional PI controller is applied for the top brine temperature control of the nominal plant for uncertain time delay and time constants due to operation point changes. It has been shown that the fixed PI controller was able to deal with the parametric deviations from the nominal value and provides good results for temperature stabilization. This paper is a further development of [4; here the cascaded MSF plant is regarded as an uncertain process due to the existence of such as incomplete physical models or unknown model parameters. The parameter uncertainties are incorporated into the plant via Linear Fractional Transformations LFTs and are assumed to be structured. Three robust controllers - H loop-shaping and µ synthesis controller - are designed and applied for the liquid profile reference control throughout the MSF plant; validation is done with respect to the nonlinear model. Further order reduction is carried out for the obtained controllers and nominal as well as robust stability and performance of the closed-loop system are investigated. II. PROCESS DESCRIPTION A MSF desalination plant is shown in Fig. 1 in a oncethrough configuration where the brine is fed through the plant only once before being discharged back to the sea. In a series of flashing chambers the evaporation and distillate formation process take place when the saturated seawater undergoes a reduction in pressure and is evaporated at lower temperatures due to the design vacuum conditions. In this paper the main focus Copyright 16 IFAC 54
2 IFAC DYCOPS-CAB 16 June NTNU Trondheim Norway lies on the linearized liquid level dynamics presented in the next section. III. LINEARIZED LIQUID LEVEL DYNAMICS AND CONTROL TASK A linear approximation of the nonlinear liquid level dynamics ẋ = fx u from [4 see 1 has been obtained. The operation point x u has been determined by backwards motion-planning and the system is locally linearized utilizing the Jacobians defined as: A = f x x u and B = f u x u. Eventually the local linear state space model is obtained in standard form: ẋ = A x + B u 1 with x R 18 u R 1 y R 1 A R and B R The system matrix A is sparse due to the cascaded structure of the MSF plant: A 11 A Ak j τ = A jj 1 A jj A jj A 1817 A 1818 The input matrix Bτ has the following structure: x u Bτ 18 1 = [ B 11 x u 3 with the only non-zero entry of the input matrix B being B 11 = χ1 A. The following substitute term χ j is introduced: 1 cp τ T λ s1 for j = 1 χ j = j 4 1 cp T sj 1 T sj for j = λ j Note how the dependence on the uncertain parameters are introduced for the system matrix Ak j τ and input matrix Bτ; the single elements are presented in detail in Appendix I. IV. UNCERTAIN GENERALIZED MSF PLANT The main uncertainties occur at the MSF inlet in the form of an unknown control feed rate due to unknown discharge coefficient of the inlet orifice and the deviating top brine temperature of the entering fluid which is corrupted by the deteriorating performance of the heat exchangers in the pre-heating phase. Inside the plant the stages are connected via valves which are subject to analogous parametric uncertainties as the inlet orifice. A. Structured uncertainties Parametric uncertainties are incorporated by using Linear Fractional Transformations LFTs [6. The following variables are considered as uncertain: a discharge coefficient c of the inlet orifice b top brine temperature τ and c valve coefficients k j connecting the tanks and the following structured model is used: c = c + p c δ c τ = τ + p τ δ τ and k j = k j + p kj δ k 5 with p c p τ and p kj being the known range of the corrupted variables the accented indicates the uncertain variables the accented is the nominal value and the subscript j is the location of the stage. Parameter c causes the control variable to be uncertain defined as: ũ = u c + p c δ c 6 where u is the nominal computed control law by the respective controller. The components of the uncertainty blocks satisfy the following condition: 1 δ c δ τ δ k 1. The uncertain variables and parameters can be expressed as an upper LFT in. F U M Ω := [M + M 1 Ω I M 11 Ω 1 M 1 with [ [ 1 1 M c = M c τ = τ p c p τ B. General model of the first stage S 1 M kj = 7 [ 1 8 p kj kj The uncertain liquid level dynamics of the first stage - first row of - incorporate structured parameter uncertainties and are expressed by: x 1 = Ã11 x1 x + B11 ũ 9 where Ã1 = Ã11 holds and the uncertain coefficients are given as à 11 = k1 + p k δ k a g [A p 1 p + g x 1 1 x B 11 = 1 1 cp τ + pτ δ τ T s1 ũ = A λ 1 c + p cδ c u. The entries of the linearized system matrices A and B are evaluated in the operating point x u. Defining A jj = k A jj the generalized model of the first stage S 1 is presented in 1 where the variable x is temporarily treated as an exogenous input; it reads: ẋ 1 ka 11 Q13 p ka 11 ka x 11 Q1 1 Q11 z c M C 11 z τ 1 w c w τ M τ 11 M C 1 z k1 M C w k1 = z x 1 y 1p r 1 1 d y e }{{} u :=S 1 1 with A 11 = A11 k a1 = 1 cp a = a 3 = a T s1 A Aλ 1 Q 11 = a 1M C + a T s1m C MM τ C Q 13 = a M1 τ Q 1 = a 1M1 C + a T s1m1 C MM τ 1 C. Later the generalized first stage S 1 is connected to the remaining downstream stages which makes x an internal state as supposed to. 55
3 IFAC DYCOPS-CAB 16 June NTNU Trondheim Norway C. General model of the downstream stages DS The cascaded plant consists of a sequential arrangement of flashing chamber which requires the coupling of the downstream stages DS to the general model of the first stage S 1. All the uncertainties of the interconnecting orifices k j are accounted in the modelling. For any downstream stage j > 1 each uncertainty block M kj is modelled as: w kj = δ k z kj z kj = v kj ẋ j = p k w kj + k v j 11 where v kj = A jj 1 x j 1 + A jj x j + A jj+1 x j+1 and note that A jj+1 = for j = 18. D. Global general plant MSF The generalized global plant MSF is presented next using the following definitions: Ẋ = [ ẋ 1 ẋ ẋ j ẋ 18 Z = [ z c z τ z k1 z k z kj z k18 X = [ x 1 x x j x 18 W = [ w c w τ w k1 w k w kj w k18 Y = [ y 1p e and U = [ r d y u 1 where Ẋ X R18 are the internal states Z W R are the outputs and inputs of the structured uncertainty block Y R is the exogenous output vector consisting of the perturbed output y 1p and control error e. The exogenous input vector U R 3 consists of the reference signal r output disturbance d y and control signal u. Its state space representation is: Ẋ Z = A B 1 B C 1 D 11 D 1 X W 13 Y C D 1 D U }{{} :=MSF 4 41 where MSF is the generalized global plant displayed as an interconnection block in Fig.. Detailed description of the submatrices is presented in Appendix II. The uncertainty blocks is: = diag{δ c δ τ δ k1 δ k... δ kj... δ k18 }. 14 Further it is assumed that < 1. The obtained generalized model MSF is of large dimensions: Fig. : Generalized plant S 1 coupled with the general model of the downstream stages MSF along with the structured uncertainty blocks in each stage. V. REQUIREMENTS ON THE CONTROL SYSTEM A. Performance criteria The controller to be designed shall stabilize the system internally accounting for the given parameter uncertainties in 5 while meeting defined performance specifications. In this case study a mixed sensitivity optimization task is formulated [ Wp I + GK 1 1 < 1 15 W u KI + GK where S = I + GK 1 is the sensitivity function based on the nominal plant G and K is the linear feedback controller yet to be found. The performance criteria is known as the S over KS design and it presents a tradeoff solution between reference tracking disturbance rejection and utilized control effort. B. Weighting functions In order to meet the mixed-sensitivity performance critera 15 the weighting functions W p s and W u s are to be designed appropriately. It is known that the sensitivity Ss is desired to be small for low frequencies and 1 for frequencies larger than the bandwidth which ensures reasonable tracking capability and disturbance rejection of the system. Avoiding saturation or damage of the actuator a constant upper limit is set over all frequencies achieved by W u s. The following weighting functions have been chosen in this case study: W p s =.65 s + 8.8s + 1.8s + 8s +.1 and W us = 1 3. VI. CONTROL DESIGN 16 For the reference control of the liquid profile throughout all the stages of the MSF plant three controllers have been applied: H loop-shaping and. All have been designed based on the linearized dynamics; and 3 where the control signal u is the inflow towards the first stage. A. H controller Usually the optimization task of 15 is relaxed by setting a finite positive number γ for which the following must hold: [ Wp I + GK 1 W u K I + GK 1 < γ This is known as the suboptimal H problem which minimizes the infinite-norm of the performance criteria over all stabilizing controllers K. The resulting controller is a dynamic system itself and is of th order and has the same number of states as the general plant including all the states from the performance weights. 56
4 IFAC DYCOPS-CAB 16 June NTNU Trondheim Norway B. H loop-shaping controller LSHC [7 The LSHC approach is formulated by the following optimization problem: [ KLSHC I GK I LSHC 1 1 M = γ 17 where G = M 1 Ñ is represented by a normalized left coprime factorization: [ [Ñ A + HC B + HD H M := R 1/ C R 1/ D R 1/. The parameter γ is the lowest achievable value of the infinite norm above for all stabilizing controllers K LSHC. Further the LSHC design is similar to the conventional loop-shaping design which is well-known and is attractive in control applications. Here a pre-compensator s+1 W 1 s = 3.9s+.1 with quasi integral behaviour is utilized to alter the frequency response of the plant G. C. µ synthesis [6 Due to the known diagonal structure of parametric uncertainties the synthesis of the controller is done by the means of the structured singular value µ. Robust performance can be formulated as a robust stabilization problem with respect to the augmented block [ := : R P R 1 P which augments the parametric uncertainty block by the performance block P. The optimization problem is formulated in terms of the upper bound of the scaled structured singular value: µnk µ min D σ D NK µ D 1. The controller shall minimize the infinite-norm of the following performance cost over frequency of this upper bound: min min D NK µ D 1 K µ D 18 where D is a constant scaling matrix and NK µ is the lower LFT which incorporates the controller into the generalized plant. The algorithm alternates between minimizing 18 with respect to D and K µ while keeping the other respective parameter constant. For detailed explanations on the method and notations the reader is referred to [6. D. Order reduction of controllers The resulting controllers are a separate dynamic system itself of high order which is inconvenient in implementation for instance. A significant order reduction of the controllers is achieved by means of the Hankel singular values and the results are listed in Table I. For the frequency range of [ rad s it has been verified that the reduced controllers accurately replicate the behaviour over the original high-order controllers as depicted in Fig. 3. Therefore the low-order approximations are applied in the following. Log Magnitude Phase degrees TABLE I: Order reduction of the initial controllers. Order # Initial design Order # Reduction H controller 1 5 Loop-shaping H Infinity controller Frequency radians/sec full: 1rst order reduced: 5th order Frequency radians/sec a H controller Log Magnitude Phase degrees Loop shaping controller Frequency radians/sec full: nd order reduced: 6th order Frequency radians/sec b Loop-shaping Phase degrees Log Magnitude Frequency radians/sec full: nd order reduced: 6th order Frequency radians/sec c Fig. 3: Comparison of the reduced red against the full blue order controllers over a frequency band of [ rad s. A. Robust stability VII. NUMERICAL RESULTS Robust stability analysis is carried out in terms of structured µ values. The upper and lower bound of the µ values are shown for all three controllers in Fig. 4. Robust stability is achieved for all the investigated controllers as the maximum value of µ is around.. Furthermore the maximum singular value is depicted in black which characterizes the robust stability with respect to unstructured perturbations; for ω <.4 rad s robust stability cannot be preserved. It is emphasized how further knowledge of the uncertainty structure results in a improved and less conservative design. B. Nominal and robust performance For all three controllers the nominal and robust performance criteria is investigated [6; the nominal performance condition is given in terms of a lower LFT: FL MSF K < 1 19 where where MSF is provided in 13 and K is one of the designed controllers. The robust performance criteria is mu 1 Upper and lower bounds of µ values unstructured uncertainty H infinity Loop shaping Frequency rad/s Fig. 4: Upper continuous and lower dashed bounds for the structured µ values in blue green and red. The unstructured uncertainty bound is displayed in black. 57
5 IFAC DYCOPS-CAB 16 June NTNU Trondheim Norway Nominal and robust performance.84.8 nominal upper bound.8 lower bound Frequency rad/s a H controller Nominal and robust performance nominal upper bound.6 lower bound Frequency rad/s b Loop-shaping Nominal and robust performance.8.8 nominal upper bound.78 lower bound height m height m Reference tracking reference H infinity Loop shaping time s Disturbance rejection 1 disturbance H infinity Loop shaping time s Fig. 6: Top: Reference tracking for x set 1 = 3 m and d y =. Bottom: disturbance rejection for x set 1 = m and d y = 1 for t 1 seconds Frequency rad/s c Fig. 5: Nominal and robust performance. m reference profile FBL profile LQR profile PI profile Loop shaping profile µ profile TABLE II: Results of the nominal left column and robust right column performance criteria in terms of the infintenorm. σ FL MSF K µ Njω H controller Loop-shaping formulated as: F U NK < 1 µ Njω < 1 < 1 The results of the nominal and robust performance in terms of µ values are presented in Fig. 5 and Table II where the left column represents the nominal performance and the right column the robust performance. C. Reference tracking and disturbance rejection The obtained controllers are tested for a given reference profile for the liquid level x in all the stages and against a unit disturbance separately. The respective other variable was set to zero during the test scenario. An exemplary test result is shown in Fig. 6 for the first tank of the MSF plant. D. Nonlinear validation of the reduced controllers The reduced controllers are implemented for the nonlinear model of the plant presented in [4 in order to evaluate the liquid level profile control throughout the 18 stages of the plant. Moreover the characteristic numbers of the operation are evaluated over a simulation time of T = 4 seconds. The total distillate production is denoted with w[t the plant s efficiency is the ratio between the total throughput of seawater and the amount of produced distillate from stage no. Fig. 7: Achieved liquid level profile throughout the MSF plant at terminal time instant T. that denoted with η[%. A RMS value is defined for each single stage and subsequently summed up over the entire simulation as an indicator for the accuracy of the profile control. The transition time t [s is defined as the time instant when all the liquid levels have been driven to their respective reference value with a tolerance of 5%. At terminal time T the achieved liquid profiles of the robust controllers are visualized in Fig. 7 along the reference profile and previously tested controllers such as Feedback linearization LQR and PI controllers. The evaluation of the characteristic numbers are summarized for all the robust controllers in Table III along with the previously tested controllers for the sake of comparison. VIII. CONCLUSION In this paper an uncertain model has been derived for the linearized liquid level dynamics of a cascaded MSF plant by using Linear Fractional Transformations in order to incorporate multiple parametric uncertainties into a deterministic system which has been derived in earlier work. Based on the given reference profile throughout the stages and defined mixed-sensitivity performance criteria three robust controllers have been designed: H loop-shaping and µ synthesis controllers. The high order of the obtained controllers could be reduced by using Hankel singular values without compromising the performance. The controllers have been tested and successfully verified with respect to nominal and robust stability and performance criteria. Moreover the controllers have been implemented in the original 58
6 IFAC DYCOPS-CAB 16 June NTNU Trondheim Norway TABLE III: Elaboration of the characteristic numbers for the FBL LQR PI loop-shaping and s. FBL LQR PI Loop-shaping w[t η[% RMS t [s nonlinear MSF model in order to validate their performance based on defined characteristic numbers such as total distillate production thermal efficiency of the plant control accuracy by means of RMS and transition time. The main outcome is that the robust controllers perform very similar along with the previously tested controllers such as feedback linearization LQR and PI controllers while accounting for the model uncertainties which has been inherited by design. APPENDIX I ELEMENTS OF SYSTEM MATRIX Ak j τ The elements of the system matrix Ak j τ from are presented next. First stage j = 1: A 11 = k a g A 1 = A 11 Intermediate stages j 17: [ A A jj 1 = k a g χ j [A [ A jj+1 = k a g A A jj = A jj 1 + A jj+1 Last stage j = 18: p 1 p + g x 1 1 x p j 1 p j + g x 1 j 1 x j p j p j+1 + g x 1 j x j+1 A 1817 = k a g χ 18 [A p 17 p 18 + g x 1 17 x 18 [ A 1818 = k a g A p 1 18 p amb + gx 18 A 1817 APPENDIX II SUBMATRICES OF GENERALIZED PLANT MSF with the submatrices defined as: A 11 A 11 A 1 A A 3. A = k A jj 1 A jj A jj A 1716 A 1717 A 1718 A 1817 A 1818 B 1 = Q 1 Q 13 p k1 A 11 O 1 17 O 17 3 p k I [ O 18 B = Q O 18 C 1 = 1 1 k 1 E [ 1 C = where E R is a submatrix of A by eliminating its first row. M11 C M11 τ D 11 = M 1 C M11 τ O 3 17 [ 1 D = 1 1 O 17 3 O T O 3 3 O 3 17 D 1 = M1 C M 1 τ M C O D 1 = O REFERENCES 3 [1 M. Rosso A. Beltramini M. Mazzotti and M. Morbidelli Modeling multistage flash desalination plants Desalination vol. 18 no. 1 pp [ M. Mazzotti M. Rosso A. Beltramini and M. Morbidelli Dynamic modeling of multistage flash desalination plants Desalination vol. 17 no. 3 pp [3 I. Alatiqi H. Ettouney and H. El-Dessouky Process control in water desalination industry: an overview Desalination vol. 16 no. 1 pp [4 P. Nguyen and R. Tenno Modelling optimization and control of a multi-stage flash evaporator by means of motion-planning in Conference on Decision and Control CDC 15. IEEE 15. [5 M. Chidambaram B. C. Reddy and D. M. Al-Gobaisi Design of robust siso pi controllers for a msf desalination plant Desalination vol. 19 no. pp [6 S. Skogestad and I. Postlethwaite Multivariable feedback control: analysis and design. Wiley New York 7 vol.. [7 D. McFarlane and K. Glover A loop-shaping design procedure using h synthesis Automatic Control IEEE Transactions on vol. 37 no. 6 pp The composition of the submatrices of the general global system MSF 13 are presented next. The general plant MSF consists of known variables making it deterministic the uncertainty block and controller K can be incorporated by upper and lower LFTs respectively. It is recalled that the MSF has the following structure: Ẋ Z = Y A B 1 B C 1 D 11 D 1 C D 1 D } {{ } :=MSF 4 41 X W U. 59
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