BOND-GRAPH BASED CONTROLLER DESIGN OF A TWO-INPUT TWO-OUTPUT FOUR-TANK SYSTEM

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1 BOND-GAPH BASED CONTOLLE DESIGN OF A TWO-INPUT TWO-OUTPUT FOU-TANK SYSTEM Nacsse, Matías A. (a) and Jnco, Sergio J. (b) LAC, Laboratorio de Atomatización y Control, Departamento de Control, Facltad de Ciencias Exactas, Ingeniería y Agrimensra, Universidad Nacional de osario, íobamba 245 Bis S2EKE osario Argentina. (a) nacsse@fceia.nr.ed.ar, (b) sjnco@fceia.nr.ed.ar ABSTACT The qadrple-tank process has been proposed as a benchmark for mltivariable control system design. This paper addresses the design in the bond graph domain of a robst controller having the volmetric flows of two pmps as maniplated variables and the level of the two lower tanks as the reglated otpts. The basic control objectives, expressed in terms of desired closed-loop energy and power-dissipation fnctions, are captred in the bond graph domain means a so-called Target Bond Graph, and the controller design is performed via Bond-Graph prototyping. The basic control law is frther robstified against parameter ncertainties, measrement deviations and falts sing the diagnostic bond graph concept, what leads to an additional loop consisting in a PI-law also being expressed with a physically meaningfl bond graph sbsystem. Some casal maniplations on the for-tank bond graph allow to extend to this mltivariable case the techniqe developed to solve the simpler monovariable two-tank control problem. Keywords qadrple-tank system, bond graph prototyping, robst falt-tolerant control, non-linear energy-based control.. INTODUCTION The qadrple-tank system proposed in (Johansson 2) trned into a very poplar benchmark allowing to test different control algorithms for mltivariable processes. It is a Two-Inpt Two-Otpt or TITO nonlinear plant consisting of for interconnected water tanks fed by two pmps, whose linearized model has a mltivariable zero, which can be made minimm or non-minimm phase by simply changing the position of two distribtion valves. This system has been sed to test and design several control schemes. In (Abdllah and Zribi 22) a bibliographical review and three different control schemes are presented; gain schedling, a linear parameter varying controller and inpt-otpt feedback linearization have been compared, measring the pressre of the for tanks. In (Johnsen and Allgöwer 27) an interconnection and damping assignment pls passivity based control (IDA- PBC) algorithm for the minimm phase configration of the for-tank system is presented showing simlation and experimental reslts. Limon, et al. 2 present a robst model predictive control (MPC) algorithm for level tracking. In both cases the level of the for tanks are measred. This paper addresses the design in the Bond Graph (BG) domain of an energy-based nonlinear control law which only measres the pressres of the two bottom tanks. The control system design objectives are to track the levels (or pressres) of the two bottom tanks, to reject distrbances originated in model ncertainties and measrement errors, and to be tolerant to some interconnection falts. Falt tolerant control (FTC) can be classified in two main categories, Passive Falt Tolerant Control (PFTC) and Active Falt Tolerant Control (AFTC). Both approaches are sally complemented in the praxis to improve the performance and stability of the falt tolerant system (Blanke, et al. 26). efer to (Zhang and Jiang 28) for a bibliographical and historical review on FTC. The passive approach defines a niqe control law to achieve the control objectives even in the presence of a falt. Generally speaking, the passive approach ensres stability and confers robstness nder falts to the control system, bt there exists a trade-off between performance and robstness (Isermann 26). Althogh this paper follows a PFTCapproach, some concepts commonly fond in the design of AFTC controllers are sed. The active approach modifies the control law according to the falts occrred, so that in this approach a falt detection and isolation (FDI) phase is mandatory before making a decision on how to reconfigre the control law. Analytical redndancy relationships (A), which cont among the many soltions sed to generate residal signals for FDI, have been implemented in the BG domain via the Diagnostic Bond Graphs (DBG) techniqe presented in (Samantaray, et al. 26). Using the plant inpts and plant measrements, residal signal are generated that depend on the model parameters and the real plant parameters. Nacsse and Jnco (2) address the PFTCproblem of a two-tank system in the BG domain sing 87

2 an energy and power shaping method (Jnco 24). This method first expresses the control system specifications in terms of desired closed-loop energy and power dissipation fnctions, proceeds frther captring both fnctions in a so called Target Bond Graph (TBG) that represents the desired closed-loop behavior, and concldes constrcting the controller via Bond Graph prototyping. This prototyping is sch that the copling of the reslting controller-bg and the plant-bg renders the whole eqivalent to the TBG. In Nacsse and Jnco (2) the basic control law obtained in this way is robstified with additional terms derived considering a Diagnostic Bond Graph (DBG) of the closed-loop the nominal control system represented by the TBG (originally proposed nder ideal assmptions) is fed with the actal reference signals and measred plant otpts. Ths, the residal signal obtained from the closed loop DBG (CL-DBG) is a measre of the error between the desired and the actal dynamics of the control system. So, the control law aims at making the residal signal vanish in time, making the closed-loop system to behave asymptotically like the original TBG. The rest of the paper is organized as follows. Section 2 reviews some backgrond reslts and smmarizes the main reslts presented in Nacsse and Jnco (2). Section 3, revisiting this previos reslt on the two-tank system, addresses some BG maniplations that simplifies the controller and, simltaneosly, provides a way to extend the design method employed in the simpler system to the mltivariable for-tank process which in shown in Section 4. Next, Section 5 presents some simlation responses that prove the good dynamic response of the control system and, finally, Section 6 addresses some conclsions. 2. BACKGOUND AND PEVIOUS ESULTS This section briefly smmarizes the main ideas on performing energy shaping and damping assignment directly in the BG domain throgh BG prototyping and recalls their application to solve a control problem on a two-tank system as presented in Nacsse and Jnco (2). This reslt will be revised in the next section as a prelde to the development of the main reslt in this paper, the design of a controller for the qadrple-tank benchmark process. 2.. Energy- based control in the BG domain The power and energy shaping control techniqe defines the control problem as a stabilization one, choosing desired closed-loop energy- and powerdissipation fnctions, and obtaining the control law throgh eqations that match the control open-loop energy fnction (a kind of control Lyapnov fnction, see Sontag 998) and the desired closed-loop fnctions. Passivity-based control on port-hamiltonian models cont among the most sccessfl (Ortega, et al., 22, Ortega, et al., 28). In the BG domain, the closed-loop specifications are expressed by a so-called Target Bond Graph (TBG) representing the eqivalent closed-loop behavior. In order to obtain the control law, the controlled sorces which provide the maniplated variables in the BG model of the plant are prototyped (meaning that their behavior is expressed means BG components) in sch a way that their power-interconnection with the rest of the plant BG which is called a Virtal BG (VBG) matches the TBG.The control law is obtained from the VBG by simply reading the otpts of the prototyped sorces with the help of the casal assignment in the VBG. This method is exemplarily performed below on a two-tank system. For more details refer to (Jnco 24) Control via BG-prototyping the 2-tank system As shown in Figre, the tanks are located one above the other, the pper tank discharging into the lower tank. Both tanks are fed with a niqe inpt flow splitted between them throgh a distribtion valve whose parameter, determinates how the inpt flow is distribted to the tanks as indicated by the BG in Figre. The control objectives imposed on Tank are Qi= Tracking of constant reference levels ejection of constant distrbances. obstness regarding parametric ncertainties and falts. ( ) Qi pmp Qi Tank2 Tank Q2 Q Figre Two-tank system and its BG model. Measred plant otpts encircled in red. The state eqations can be read from the BG of Figre sing the standard procedre, giving as state variables the stored liqid volmes. Here the gage pressres at the bottom of the tanks are chosen as state variables (instead the liqid levels sed in Nacsse and Jnco 2) and presented in () = + + = + x2 x C C C 2 Q2 C a P a 2 88

3 Where, with i=,2, represents the gage pressre at the bottom of Tanki; are the tanks hydralic capacities and are coefficients depending on the cross sections of the otlet holes of the tanks. The proposed TBG for the closed loop system is shown in Figre 2 where the desired stored energy and power dissipation are expressed in terms of the tracking error state variable in (2) and (3). Figre 2 TBG for pressre control of lower tank. = = = C 2 3 The tracking error = is the state variable of the TBG and is the Tank reference pressre. To enforce the desired closed-loop dynamics specified by the TGB, the Virtal BG (VBG) of Figre 3 is constrcted. It shows how to proceed in order to obtain the control law. The left half of the figre is obtained prototyping the controlled power sorce in sch a way that access is gained to the chosen otpt, the pressre x, and an overall eqivalent behavior to the TBG is achieved. The first objective is achieved via the exact compensation of the Tank2 pressre on the pmp over the distribtion valve and of the discharge of Tank2 on Tank. The second objective is reached first adding the virtal elements with negative gains that cancel the own dynamics of Tank and later bilding the incremental dynamics arond the reference pressre for Tank via the insertion of the virtal elements and (shapes the closed-loop energy) and (damping assignment). a 2 P ref C C a VBG P C C C H Figre 3 Virtal and plant BG. Using the standard casality reading procedre the control law (2)can be read directly from the VBG as C C 2 Q2 C PlantBG a P a 2 = + 4 Assming exact model knowledge and perfect measrements, this control law yields a closed-loop behavior eqivalent to the TBG of Figre 2, i.e., the closed-loop dynamics satisfies (3). emark the rated control law (4) performs a partial energy shaping and damping assignment, since only the dynamics of Tank is captred in the TBG. As no objectives are imposed on Tank2 and its dynamics is hidden in closed-loop, its stability mst be analyzed after the controller has been designed, property that can be easily verified in this case. Pertrbed closed-loop dynamics. Becase of parameter dispersion, falts, modeling errors, sensor limited precision, noise, etc., neither the model nor the measrements are exact. To deal with this it is convenient to think the control inpt as composed by two terms as in (5), where is the rated part of, i.e., the control inpt part that performs the power and energy shaping nder ideal plant and measrement conditions. In the same expression, is the nknown controller part de to modeling errors, parametric dispersion, falts, etc. The BG of Figre 4 reflects this sitation. = + δ P e Figre 4 Pertrbed TBG. 5 Under this sitation the closed-loop dynamics no longer satisfies (3) bt (6). = + C obstifying the control law The CL-DBG is defined injecting the pressre tracking error (as measred on the real control system) into the TBG throgh modlated sorces and collecting a residal signal as shown in Figre 5. x x ref x e Figre 5 Proposed CL-DBG. Measrements to be fed encircled in red. The CL-DBG yields to the new error dynamics in (7) which is driven by the residal signal. The residal signal, which is the power co-variable of the error C res C H P e C H 89

4 injected into the CL-DBG, is a measre of the difference between the actal and the ideally expected closed-loop dynamics, i.e., when =, responds as previosly defined in the TBG of Figre 2. This sggests that the control objectives cold be reached extending the previosly compted control law to a new one =,,, incorporating the residal signal in sch a way that tends to zero with growing time. = + (7) The residal expression (8) obtained reading the CL-DBG clearly shows that choosing as in (4) (rated control law) yields =, in absence of falts and modeling errors. = ( ) (8) To improve the control system robstness, the extra term 4 shown in (9) is added to the expression (4) for. = + + (9) Choosing = yields the residal dynamics () +=( ) () Ths, with constant, goes asymptotically to zero with time constant /. As already anticipated, this forces to approach asymptotically the desired error dynamics defined in the TBG of Figre 2. epresenting in terms of yields (), expression showing that, in this case, the residal signal defined in the CL-DBG has a PI strctre. Note however that this does not necessarily generalize, since the reslting strctre depends on the TBG. = () 3. THE TWO-TANK POBLEM EVISITED In this Section the two-tank problem is revisited. First, a casal maniplation of the original system BG is shown which leads to a simplification of the control law and will be advantageosly sed in the process of deriving the control law for the qadrple-tank system. Second, a BG interpretation of the robstifying additional term is provided. 3.. Casal maniplation and simplified control law In Figre the flow from Tank2 into Tank is compted by the -element representing the discharge orifice of the former. Ptting in derivative casality the pper C-element (models the potential energy stored in the pper tank) yields the BG of Figre 6, where the discharging flow is compted as the sm of the incoming flow from the sorce and the otpt flow to the pper C element, i.e., =. This new comptation jstifies the eqivalent BG of Figre 7. C C 2 Figre 6 BG model of the two tanks system with the pper C in derivative casality. C C Figre 7 Eqivalent plant model after maniplation of the original BG. Frthermore, as the control objectives are placed only on Tank, for control system design prposes the eqivalent plant model shown in Figre 8 can be considered, where the effect of Tank2 enters as a distrbance. C Figre 8 Eqivalent control system design BG. Q 2 C C C Q a 2 P a a 2 c2 C 2 P a C C 2 9

5 Following similar steps as those detailed in Section 2.2, the VBG of Figre 9 can be constrcted to enforce the desired closed-loop dynamics specified by the TGB. C C VBG ref P H P. 2 C C The control law (2) can be read directly from the VBG sing the standard casality reading procedre = + + (2) emark Eqations (4) and (2) are flly eqivalent and both show the need to measre not only the pressre of Tank bt also that of Tank2 to implement the control law. However, eqation (2) which reslts from the above maniplation performed on the BG has the advantage of showing that its last term can be eliminated from the control law as it is an evanescent pertrbation, i. e., it vanishes in steady state, and as sch it does not affect the eqilibrim point. Doing so yields the control law (3) which can be implemented with the sole measrement of. However, it mst be realized that this simplification amonts to modifying the TBG as indicated in Figre. = + Figre New TBG withot measring. a P C C PlantBG (3) 3.2. BG interpretation of the robstifying term This sbsection revisits the reslt of (Nacsse and Jnco 2) smmarized in Section 2.3 of this paper in order to provide a BG implementation of the additional term of the control law given in (). The TBG defines the closed loop dynamics of the system. To robstify its behavior it is necessary to inject an additional control action. In order to analyze this problem in the BG domain a concise word BG version of Figre 9 is presented in Figre. Departing from Figre, Figre 2 shows the word BG of a power interconnection proposed as a means to provide the additional control action. There, the word BG block named mdbg mst be capable of rejecting all of the above-mentioned distrbances. C a P Figre 9 Virtal and plant BG. P2 C 2 C Pe C. C P H 2 Virtal BG With or C 2 P Plant Word BG (Withot) TBG (New TBG Figre ) Figre Power copling between plant BG and VBG to obtain the TBG. Virtal BG (ated Controller) VBG Figre 2 Proposed power interconnection In the approach presented in Nacsse and Jnco (2) a residal signal is generated throgh a CL-DBG as indicated in Figre 3 (cf. Figre 5) (see Appendix B for a brief description of the DBG for FDI prposes). This residal measres the error between the PTBG and the TBG and is sed to generate an additional control action forcing this error to vanish in steady-state (as well as the residal itself). Virtal BG CL-DBG - res res ref mdbg P Figre 3 Signal copling between PTBG and TBG throgh closed loop DBG. P mdbg VBG com P Plant Word BG Pertrbed TBG The additional control action (a hydralic flow) given in () in terms of the tracking error (a pressre) can be interpreted as prodced by an effort-sharing set of a hydralic resistance and inertance, i.e., an I- and a -element. Figre 4 shows the reslting closed loop BG with power copling between the PTBG and the mdbg, where = and =. Pertrbed TBG δ C Figre 4 Power copling between the PTBG and the mdbg. com C Plant Word BG Pertrbed TBG otpts P e H mdbg I I P e P2 C 2 9

6 Figre 5 shows the reslting closed loop BG, where the power interconnection among the plant BG, the VBG and the mdbg is highlighted in dotted sqares. Notice that the redndancy in the which injects the can be eliminated by shifting the mdbg into the VBG as it is shown in Figre 6. H a P. C 2 ( ) Qi 2 Qi2 Tank3 Tank4 Q3 Q 42 Qi ( 2 )Qi 2 2 C C VBG ref P H P P ref C I C Figre 5 eslting closed loop BG. P a P C C Plant BG mdbg I P e a P. C 2 Qi = pmp Tank Q x x2 Q2 Tank2 pmp 2 Qi = 2 2 Figre 7 For-tank system with measrements encircled in red. 4.. Casal maniplation and controller design In this sbsection the control law for the for-tank system is designed. The control objectives are the same of that proposed for the two-tank system, in this case imposed on both of the bottom tanks. C C VBG mdbg I I ref P P P e C a P C C Plant BG Figre 6 Closed loop BG with mdbg copled into the VBG emark As already anticipated, with constant δ, the tracking error approaches zero asymptotically, as it can be easily verified throgh a casal and power analysis of the BG in Figre 4. Indeed, with constant distrbances at the otpts of both, the I-element integrates its inpt effort ntil it is driven to zero. At the same time, the I-element keeps at its otpt a constant flow-vale which exactly cancels the sm of the -flows and, ths, generates a zero-flow sitation at the inpt of the C-element, which in trn keeps in zero, which is precisely the control objective. The stability of this sitation depends on the elements being strictly dissipative (Jnco 2), which is of corse ensred by design. 4. APPLICATION TO THE FOU-TANK BENCHMAK POBLEM The for-tank system depicted in Figre 7 is a TITO nonlinear plant consisting of for interconnected water tanks fed by two pmps, whose linearized model has a mltivariable zero, which can be made minimm or non-minimm phase by simply changing the position of two distribtion valves. C P a 4 P 4 a 3 P 3 CC 4 CC 3 2 a P a 2 2 CC CC 2 Figre 8 BG model of the for-tank system. As the first step developing the mltivariable control law, a casal maniplation is performed on the original BG of Figre 8 as shown in Figre 9. Similarly as in the two-tank case, again the discharge flows of the pper tanks are expressed not as compted by their associated -elements bt as the difference between the flows of the sorces mins their net inpt flows ( and ). This maniplation permits to constrct the new BG given in Figre 2. This BG exhibits two internal variables and, which in the seqel are going to be treated as virtal control inpts. Seen from the bonds associated to these two axiliary flow variables the qadrple-tank problem appears as two decopled two-tank problems. Hence, the mltivariable problem is strongly simplified, as the previosly developed procedre can be first applied to each sb-problem and then the overall control law be recovered sing the casal relationships relating the 2 92

7 axiliary variables with the control inpts provided by the power-conserving strctre. 2 CC CC 2 Figre 9 BG of the for tanks system with pper tanks in derivative casality. The real and the virtal control inpts are related throgh a transformation matrix which is given in (4). Notice that when + = the transformation matrix is singlar. In this condition of the distribtion valves the mltivariable zeros (of the linearized system) are placed at the origin of the complex plane. = ( ) a 4 P 4 a 3 P 3 CC 4 CC 3 a P a 2 2 ( ) (4) The virtal control laws, obtained following the same procedre as in the two-tank example ignoring the vanishing term, are given in (5). 2 = + = + (5) 4.2. obstifying the control law The virtal control laws (5) are robstifyied, as in the two-tank example, via power copling of the mdbg or by simply adding a term like () to each eqation of (5). This yields the real control laws given in (6) and (7). = ( ) + x + x (6) = ( ) + + (7) Using (6) and (7) the stability of the hidden closed-loop dynamics of Tank_3 and Tank_4 can be verified. a 4 P 4 CC 4 Tank4 Tank CC. P 4C4 * * 2 a P a 2. C P CC 2 Tank2 Tank3 CC 3 Power Conserving a 3 P 3 Strctre Figre 2 BG model after maniplation for controller design. 93

8 5. SIMULATION ESULTS. The parameters sed in the simlations, shown in Table I, were obtained from (Johansson 2), where are the cross section areas of the tanks, related to the tanks hydralic capacities by the relation = where is the liqid (water) density and is the gravitational acceleration. The parameters of the mdbg are = = and = =.. Table I. Simlation parameters Parameter Vale, 28, 32,.7,.57 Using the control laws (6) and (7) with noisy measrements, with normal distribtion and amplitde =., of the bottom tanks pressres, the simlation scenarios involve abrpt falts in the system. To show the robstness of the control laws, the sed parameters are -% and +% for those related with Tank and Tank2 respectively. For illstration prposes the simlations show tanks levels instead of tanks pressres. The first simlation scenario, Scenario_, consists of a minimm phase configration with valves positions in =.3 and =.2. This are rated vales sed to parameterize the control laws. In this scenario two seqential falts occrred at time =2 the vale of the cross section area of otlet hole of Tank2 is increased by 5% (forcing the same increment in ); and at time =5 the vale of the valve position changes to = x x_ref x2 x2_ref x3 The simlation responses for Scenario_ are shown in Figre 2 and Figre 22, which show that despite the falts occrrence the control system behaves as expected and the bottom tanks levels follow their references. For both cases, the control inpts, and, force to zero the residal signals in order to reject the falts es_ es_ time {s} Figre 22 esidal signals and control inpts for seqential falts in simlation Scenario_. In simlation Scenario_2 a non-minimm phase configration is tested, where the valves position are placed at =.7, =.7. Again two seqential falts occrred, the first, at time =6 where the vale of the position valve changes to = which physically implies that pmp_2 injects all its flow to Tank_3; and the second, at time =5, where the vale of the cross section area of otlet hole of Tank ( ) is increased by 5%. In this scenario the bottom tank levels follow their references rejecting the distrbances originated by the falts occrrence as it is shown in Figre 23. Figre 24 shows the associated residal signals and the control inpts. Here again, the control inpts force the residal signals to remain at zero to reject the falts x time {s} Figre 2 Tanks levels responses for seqential falts in Scenario_. 94

9 x x_ref x2 x2_ref x3 feedback term was ignored in the (otherwise exact) control law in order to spare a measrement. The second stage proceeded to robstify the previos controller to which aim a closed-loop diagnostic bond graph was introdced. Finally, a casal maniplation was performed on the BG of the qadrple-tank that permitted handling the associated mltivariable problem as two monovariable decopled problems, each for a simple two-tank system. Simlation reslts demonstrate the good response and the falt tolerance of the control system. ACKNOWLEDGMENTS The athors wish to thank SeCyT-UN (the Secretary for Science and Technology of the National University of osario) for their financial spport time {s} Figre 23 Tanks levels responses for seqential falts in Scenario_ time {s} Figre 24 esidal signals and control inpts for seqential falts in simlation Scenario_2. 6. CONCLUSIONS This work addressed the design of a robst controller for a mltivariable for-tank system in three stages. A (partial) energy shaping and damping assignment control system design techniqe in the bond graph domain was first applied to obtain an almost-exact I/Ofeedback linearizing controller for a simpler two-tank problem. The controller is jst almost-exact becase a x4 es_ es_ APPENDIX A ESIDUAL SINKS The residal sink component injects the necessary effort or flow in order to make vanish the power conjgated variable into the sink. A residal sink element can be interpreted as an energy store where it parameter tend to zero. For example, an effort residal sink can be interpreted a C element in integral casality = If the parameter C tends to zero, then is determined by the algebraic eqation =. Figre 25 shows the graphical representation of the effort and flow residal sink sed in (Bortzky 29). rsf rse Figre 25flow and effort residal sink. APPENDIX B DIAGNOSTIC BOND GAPH The Diagnostic Bond Graph was first presented by (Samantaray, et al. 26) for nmerical evalation of analytical redndancy relationships (A). The As are calclated to perform FDI in an AFTC frame. Basically, the DBG is obtained from a BG model of the plant injecting the plant measrements and inpts throgh modlated sorces. The residal signal is obtained by measring the power co-variables of the modlated sorces, see Figre 26. eading directly from the BG the residals are = + + 2= + 8 As can be noted in (8), the residals depend on system parameters. If the model represents perfectly the controlled system, then the residal signals are zero. The differential casality is an advantage in FDI, becase no initial states are necessary to evalate the residals. 95

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