Disturbance Decoupling of Coupled Tanks: From Theory to Practice *

4th IFAC Symposium on System, Structure and Control Università Politecnica delle Marche Ancona, Italy, Sept 15-17, 2010

Disturbance Decoupling of Coupled Tanks: From Theory to Practice ? ˇ Vladim´ır Zilka and Miroslav Hal´ as Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology Ilkoviˇcova 3, 812 19 Bratislava, Slovakia e-mail: [vladimir.zilka,miroslav.halas]@stuba.sk Abstract: Mathematical technicalities, which are involved in the modern theory of non-linear control systems, many times prevents a wider use of the impressive theoretical results in practice. Attempts to overlap this gap between theory and practice are usually more than welcome and form the main scope of our interest in this work. An important control problem given by the disturbance decoupling is studied for a real laboratory model of coupled tanks. It is shown that the theoretical solution to the disturbance decoupling problem does not satisfy practical control requirememts. Accordingly, the solution is modified and yields a nonlinear controller with the distrubance decoupling. Experiments on the real plant, verifying and comparing the results with and without the disturbance decoupling, are included as well. It is shown that the disturbances practically do not affect the system output. Keywords: nonlinear systems, applications, algebraic methods, disturbance decoupling, coupled tanks 1. INTRODUCTION The modern theory of nonlinear control systems owes a large part of its succes to the systematic use of differential geometric/algebraic methods, forming the scope of interest of many authors. To refer to a few see for instance Isidori (1989); Fliess et al. (1995); Conte et al. (2007) and references therein. Nowadays, such methods offer solutions to a wide range of nonlinear control problems. For instance, feedback linearization (Aranda-Bricaire et al., 1995), model matching and disturbance decoupling (Huijberts, 1992), realization problem, non-interacting control, observer design and others as summarized in Conte et al. (2007). Many were carried not only to the case of discrete-time nonlinear systems (Aranda-Bricaire et al., 1996; Kotta et al., 2001) but also to the case of nonlinear time-delay systems (Xia et al., 2002). However, a price one has to pay for such impressive and elegant solutions is given by a necessity of involving many mathematical technicalities. Of course, this prevents a wider use of the results in practice, making the big gap between control theory and control practice even bigger in this case. It is generally known that in practice the way of dealing with nonlinear control systems is many times based just on the linearization in a fixed operating point and then methods of linear control systems are applied. Therefore, attempts to overlap the gap are usually more than welcome which forms the main scope of our interest ? The work has been partially supported by the Slovak Grant Agency grants No. VG-1/0656/09 and VG-1/0369/10 and by grant No. NIL-I-007-d from Iceland, Liechtenstein and Norway through the EEA Financial Mechanism and the Norwegian Financial Mechanism. This project is also co-financed from the state budget of the Slovakia.

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in this work. In particular, an important control problem given by the disturbance decoupling, which is quite frequent control problem in practice, is studied. We begin with the theoretical solutions of Conte et al. (2007) and apply them to the laboratory model of coupled tanks, which is a demonstrative and well know system having contact points to many real control processes, for instance from chemical engineering. It is shown that the theoretical solutions cannot be directly applied and additional problems, related for instance to the difference between model and real system, constrained controller output and some others, have to be considered as well. 2. DISTURBANCE DECOUPLING PROBLEM We begin with an introduction to the disturbance decoupling problem of nonlinear control systems as discussed in Conte et al. (2007) to which the reader is referred for additional details and references. In the disturbance decoupling our task is to design, if possible, a control law such that the disturbances do not affect the output of a system. Technically speaking, a solution consits of finding for instance a state feedback under which a subspace of the state space, influenced by the disturbances, becomes unobservable in the compensated system. The situation can be explained by the following introductory system from Conte et al. (2007) x˙ 1 = x2 + u x˙ 2 = w y = x1 10.3182/20100915-3-IT-2017.00016

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where w is the disturbance. The main idea is to use a static state feedback to cancel those states which affect the output and which are affected by the disturbance. As can be seen, through x2 the disturbance w affects the output y y¨ = u˙ + w However, the state feedback u = v − x2 makes x2 unobservable and thus decouples the disturbance w from the output y in the closed-loop system y˙ = v The general solution follows the same idea. That is, if possible, it makes unobservable the subspace of the state space affected by the disturbance. Problem statement. Consider the system x˙ = f (x) + g(x)u + p(x)w y = h(x) where the state x ∈ Rn , the input u ∈ Rm , the output y ∈ Rp , the disturbance w ∈ Rq and the entries of f , g, p and h are elements of the field of meromorphic functions K. Find, if possible, a static state feedback u = α(x, v) such that dy (i) ∈ spanK {dx, dv, . . . , dv (i) } for any i ∈ N. The existence of such a static state feedback is a special case of a more general class of the so-called quasi-static state feedbacks and is ensured by fulfilling the conditions of the following theorem. Theorem 1. (Conte et al. (2007)). Let X = spanK {dx} and Y = spanK {dy (i) ; i ≥ 0}. The disturbance decoupling problem is solvable if and only if p(x) is orthogonal to the subspace X ∩ Y. The proof, technical details and additional references can be found in Conte et al. (2007). Remark 2. Note that for the case of SISO systems, considered later in this paper, conditions for the existence of a solution in terms of static and quasi-static state feedback coincide. 3. COUPLED TANKS In this section, we proceed with the solution of the disturbance decoupling problem for a laboratory model of coupled tanks. The plant is depicted in Fig. 1. It consists of three tanks, five valves and two pumps. Each of three tanks is equipped by a valve itself and remaining two valves couple tanks 1 and 2 and, respectively, tanks 2 and 3. Thus, the system can be used as one-, two- or three-tank system, SISO or even MIMO, depending on a proper combination of active valves and pumps. In addition, the system is easy to use, for the communication is established via USB interface. There is no need of additional data acquisition cards. Hence, it allows us to control the system for instance in Matlab/SIMULINK and even change the structure of the system directly from Matlab/SIMULINK by switching on and off corresponding valves and pumps. This offers a 173

Fig. 1. Coupled tanks: front and back view. wide variety of possible experiments and even makes the system ideal for use in teleexperiments with web camera visual feedback. In what follows, we restrict our attention to a standard coupled two-tank system, however, with all three valves and both two pupms active for each of the tanks. Structure of such a system is depicted in Fig. 2. The aim is to control the level in the first tank coupled with the second tank by a valve with the flow coeficient c12 . Each of the tanks is equipped by a pump and a valve itself, having the flow coeficients of valves c1 and c2 respectively. However, the valve c2 and the second pump are considered as disturbances w1 and, respectively, w2 coupled with the first tank by the valve c12 . Thus, we deal here with a SISO system with disturbances which can be modelled by the following state-space equations p √ 1 u − c1 x1 − c12 sign(x1 − x2 ) |x1 − x2 | A1 p √ 1 x˙ 2 = w1 − w2 c2 x2 + c12 sign(x1 − x2 ) |x1 − x2 | A2 y = x1 (1) x˙ 1 =

where x1 and x2 are levels in tank 1 and tank 2 respectively and A1 and A2 respectively are cross-sections of the tanks, see Fig. 2. Note that the disturbance w1 ∈ h0, u2 max i, where u2 max is assummed to be an upper limit for the

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Fig. 3. Closed-loop structure with disturbance decoupling and a controller. Fig. 2. Coupled tanks. capacity of the second pump, and w2 ∈ {0, 1}, depending on whether the valve c2 is switched off or on respectively. 3.1 Disturbance decoupling problem One can show that the standard theoretical solution of the disturbance decoupling problem of the system (1) does not meet basic practical control requirements and thus it is necessary to modify it accordingly. Following the lines of Conte et al. (2007) recalled in Section 2 we compute p √ 1 y˙ = u − c1 x1 − c12 sign(x1 − x2 ) |x1 − x2 | A1 Since y˙ directly depends on the input u, that is the relative degree of the system is 1, and it is not affected by any of the distrubances w1 or w2 respectively, both can be decoupled. Clearly, the conditions of Theorem 1 are satisfied and since (1) is SISO system there exists a static state feedback which solves the problem. By solving for u the equation y˙ = v one gets p √ u = A1 v − c1 A1 x1 + c12 A1 sign(x1 − x2 ) |x1 − x2 | (2) where v represents input to the compensated system which is reduced to the first order linear system y˙ = v with the transfer function 1 F (s) = (3) s To complete the solution we design a controller for the compensated system. Since we have a linear system with the transfer function (3) this is a trivial task. A straightforward solution is given by a P -controller, within the standard closed-loop structure, with 1 P = T where T represents a time constant of the closed-loop system with the transfer function P F (s) 1 G(s) = = 1 + P F (s) Ts + 1 The whole structure is shown in Fig. 3. However, it is important to say that the state feedback (2), which achieves the disturbance decoupling, is not a controller actually. It only achieves the disturbance decoupling from the output without any information about how successful the decoupling is. It is thus easy to conclude 174

Fig. 4. The results obtained in simulation (dashed line) and from the real plant (solid line): level in the first tank, the disturbances w1 and w2 and level in the second tank. that this will be sensitive not only to noise and additional disturbances that had not been considered, but also to the model inaccuracies. Clearly, the difference betwen the model (1) and the real plant causes that the compensated system does not reduce strictly to the linear system (3) in which case the P -controller might not be enought for controlling the system. This can be seen in Fig. 4 which compares the results obtained from both a simulation in Matlab/SIMULINK and an experiment on the real plant. Notice that the disturbance decoupling (2) with the P controller do not satisfy the basic control requirements for the real plant, for the output does not achieve required value, even though it does in the simulation.

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3.2 Nonlinear controller with the disturbance decoupling For this reason, the next step is to modify the both disturbance decoupling and controller design. Since our task is to control the level in the first tank, it is clear that a controller should be able to control satisfactorily a one-tank system while assumming the whole second tank and all related events just as a disturbance. Then the disturbance decoupling can be employed as an improvement of the controller behaviour. The latter can be easily achieved by modifying the state feedback (2) to the form p u = A1 v + c12 A1 sign(x1 − x2 ) |x1 − x2 | (4) Then the compensated system, i.e. the system (1) with the state feedback (4), reduces to a one-tank system of the form √ 1 v − c1 x1 A1 y = x1

x˙ 1 =

(5)

Note that under the state feedback (4) the state x2 in the system (1) is unobservable. That is, it is, together with both w1 and w2 , decoupled from the first tank. The difference here is that now we do not require the compensated system takes the form of a first order linear system with the transfer function (3) but rather takes the form of a standard one-tank system for which a plenty of controllers have been designed. This is even assummed as a trivial task and rather more complicated problems were studied for instance in Huba (2003); Pan et al. (2005); ˇ Hal´ as (2006); Almutairi and Zribi (2006); Zilka et al. (2009). Perhaps, the easiest way is to design a P I-controller based on system linearization in a fixed operating point. However, from a practical point of view this might result in a control structure having, due to the controller output constraint, wind-up effect in which case an anti-reset-wind (ARW) solution should be employed. Hence, better results can be obtained by using more sophisticated methods. Therefore, we will consider here the controller designed ˇ in Zilka et al. (2009) which adopts the control structure from Huba (2003) for linear systems and modifies to the nonlinear case, see Fig. 5. Such a structure has the following properties. It • satisfies a linearity of the closed loop, • eliminates an input disturbance δ, • deals with a control signal constraint. In particular, the structure has the properties of a P Icontroller except that no wind-up effect might occur, in contrast to the classical P I-controller with the control signal constraint, for now there is no direct integration in the controller. For more details see Huba (2003). The requirement of the closed loop linearity is satisfied easily by a feedback linearization. If we want the closed loop dynamics to be determined by a linear first order system with a time constant T we obtain regular static state feedback √ v 0 A1 x 1 A1 v= + A1 c1 x1 − (6) T T 175

Fig. 5. Closed loop structure. where v 0 denotes input to the closed loop and T determines its dynamics. Under this feedback the input-output description of the closed loop represents a linear system with the transfer function 1 Ts + 1 The remaining requirements are satisfied as follows. The input disturbance δ is eliminated via the feedback compensator K2 which reconstructs δ and subtracts it from the controller output. Note that the disturbance δ is not a subject of the decoupling. It represents all other disturbances (noise, model inaccurancies, etc.) and events not having been modelled or considered. Finally, the compensator K1 only removes the impact of K2 while controlling the system (via the feedback linearization). To design the compensator K2 one, in linear case, simply computes the inverse transfer function which reconstructs δ. However, the transfer function formalism is already available also for nonlinear systems (Zheng and Cao, 1995; Hal´as, 2008) having contact points to the polynomial approach of Zheng et al. (2001) and to the module approach of Fliess (1994); Fliess et al. (1995). Here, we do not enter into details, only remark that a common feature of these works is given by the fact that differntials of the system variables are employed, which allows us to use linear methods, except that now the polynomial system description that relates the differentials is non-commutative. Nevertheless, such a transfer function formalism has many properties we expect, inlcuding the possibility to use the transfer function algebra when combining systems in series, parallel or feedback connection. Following the lines of Hal´as (2008) the transfer function of the system (5) can be computed from its input-output differential equation 1 √ y˙ = v − c1 y A1 as 1 1 dv − c1 √ dy A1 2 y c1 1 (s + √ )dy = dv 2 y A1 dy˙ =

and the transfer function is F (s) =

1/A1 s + 2c√1y

(7)

Once the transfer function is computed we can continue with the compensator design. Of course, the ideal com-

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pensator K2 (s) = F 1(s) is not realizable, even in linear case. Hence, we use K2 (s) =

s + 2c√1y 1 1 = (Tf s + 1) F (s) 1/A1 (Tf s + 1)

(8)

where Tf is a time constant which characterizes how fast the disturbance elimination will be. The transfer function (8) corresponds to the input-output differential equation as follows. dyK2 = K2 (s)duK2 A1 s + dyK2 =

A c √1 1 2 uK2

Tf s + 1

duK2

A1 c1 Tf dy˙ K2 + dyK2 = A1 du˙ K2 + √ duK2 2 uK2 √ Tf y˙ K2 + yK2 = A1 u˙ K2 + A1 c1 uK2

(9)

where uK2 and yK2 denote input and, respectively, output to the compensator K2 . Note that uK2 = y. The next step is to find, if possible, a state-space realization of the system (9). This is, of course, necessary when one wants to implement the compensator (for instance in Matlab/SIMULINK). To find a realization, we can follow the lines of Conte et al. (2007) or even Hal´ as and Kotta (2009) where the nonlinear realization problem were studied directly within the transfer function formalism. For the system (9) the following state-space realization can be found 1 1 1 √ xK2 + A1 c1 uK2 + A1 uK2 Tf Tf Tf 1 yK2 = xK2 + A1 uK2 Tf

x˙ K2 = −

Finally, the compensator K1 (s) is just a linear system with the transfer function 1 K1 (s) = Tf s + 1

Fig. 6. The results obtained for the real plant and the nonlinear controller with (solid line) and without (dashed line) the disturbance decoupling: level in the first tank, the disturbances w1 and w2 and level in the second tank. 4. CONCLUSION

3.3 Experiment on the real plant The nonlinear compensator K2 (s) together with the compensator K1 (s) and the feedback linearization (6) can be now used to control the real plant, modelled by the statespace equations (1), with the disturbance decoupling (4). The whole structure was depicted in Fig. 5. The closed loop responses are shown in Fig. 6. To compare the results the experiment was repeated also without the disturbance decoupling (4). As can be seen in the first case the disturbances w1 and w2 practically do not affect the system output y. Remark 3. Note that the same parameters and disturbances w1 and w2 were used in all experiments. The parameters A1 , c1 , c12 were identified as 10−3 m2 , 1.52×10−2 , 1.54 × 10−2 and the time constants T and Tf were chosen to be 10s and 4s respectively. The disturbance w1 was 4.6 × 10−6 m3 s−1 and the valve c2 , of which the switching off and on was considered as the disturbance w2 , had the flow coeficient c2 = 1.38 × 10−2 . 176

As mentioned, mathematical technicalities, which are involved in the modern theory of non-linear control systems, many times prevents a wider use of the impressive theoretical results in practice. In this work, an attempt to overlap such a gap between theory and practice was studied. An important practical control problem given by the disturbance decoupling problem were applied on coupled tanks. It was shown that the initial theoretical solution to the disturbance decoupling problem does not satisfy the basic control requirememts. In particular, the output of the system did not achieve the required value. For that reason, the solution was modified accordingly. For linear systems with uncertainties in the model, possible solution is based on a robust decoupling control in the sense of the H∞ control. But for nonlinear system it is not easy to say whether there exists a similar solution yet. So, we used the nonlinear controller design with the disturbance decoupling. To summarize the results, the whole control structure satisfied the closed loop linearity, eliminated other unmodelled disturbances and had the

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properties of a P I-controller, however, without any windup effect or a necessity to use an ARW solution. Together with the modified disturbance decoupling the disturbances practically did not affect the system output. More general situations, discussing the control structure with the input disturbance elimination, are discussed in Umeno and Hori (1991) and Ohishi et al. (1988). REFERENCES Almutairi, N.B. and Zribi, M. (2006). Sliding mode control of coupled tanks. Mechatronics, 16(7), 427 – 441. ¨ and Moog, C. (1996). Aranda-Bricaire, E., Kotta, U., Linearization of discrete-time systems. SIAM Journal of Control Optimization, 34, 1999–2023. Aranda-Bricaire, E., Moog, C., and Pomet, J. (1995). A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control, 40, 127–132. Conte, G., Moog, C., and Perdon, A. (2007). Algebraic Methods for Nonlinear Control Systems. Theory and Applications. Communications and Control Engineering. Springer-Verlag, London, 2nd edition. Fliess, M. (1994). Une interpr´etation alg´ebrique de la transformation de laplace et des matrices de transfert. Linear Algebra and its Applications, 203, 429–442. Fliess, M., L´evine, J., Martin, P., and Rouchon, P. (1995). Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Control, 61, 1327–1361. Hal´ as, M. (2006). Quotients of noncommutative polynomials in nonlinear control systems. In 18th European Meeting on Cyberntetics and Systems Research. Vienna, Austria. Hal´ as, M. (2008). An algebraic framework generalizing the concept of transfer functions to nonlinear systems. Automatica, 44, 1181–1190. ¨ (2009). Realization problem of Hal´ as, M. and Kotta, U. SISO nonlinear systems: a transfer function approach. In 7th IEEE International Conference on Control & Automation. Christchurch, New Zealand. Huba, M. (2003). Gain scheduled P I level control of a tank with variable cross section. In 2nd IFAC Conference on Control Systems Design. Bratislava, Slovakia. Huijberts, H. (1992). A nonregular solution of the nonlinear dynamic disturbance decoupling problem with an application to a complete solution of the nonlinear model matching problem. SIAM Journal of Control Optimization, 30, 350–366. Isidori, A. (1989). Nonlinear systems. Springer, New York, 2nd edition. ¨ Zinober, A., and Liu, P. (2001). Transfer equivKotta, U., alence and realization of nonlinear higher order inputoutput difference equations. Automatica, 37, 1771–1778. Ohishi, K., Ohnishi, K., and Miyachi, K. (1988). Adaptive dc servo drive control taking force disturbance suppression into account. Industry Applications, IEEE Transactions on, 24(1), 171 –176. Pan, H., Wong, H., Kapila, V., and de Queiroz, M.S. (2005). Experimental validation of a nonlinear backstepping liquid level controller for a state coupled two tank system. Control Engineering Practice, 13(1), 27 – 40. Umeno, T. and Hori, Y. (1991). Robust speed control of dc servomotors using modern two degrees-of-freedom 177

controller design. Industrial Electronics, IEEE Transactions on, 38(5), 363 – 368. ˇ Zilka, V., Hal´as, M., and Huba, M. (2009). Nonlinear controllers for a fluid tank system. In R. Moreno-D´ıaz, F. Pichler, A. Quesada-Arencibia (Eds.): Computer Aided Systems Theory - EUROCAST 2009, Lecture Notes in Computer Science, 618–625. Springer, Berlin, Germany. Xia, X., M´arquez-Mart´ınez, L., Zagalak, P., and Moog, C. (2002). Analysis of nonlinear time-delay systems using modules over non-commutative rings. Automatica, 38, 1549–1555. Zheng, Y. and Cao, L. (1995). Transfer function description for nonlinear systems. Journal of East China Normal University (Natural Science), 2, 15–26. Zheng, Y., Willems, J., and Zhang, C. (2001). A polynomial approach to nonlinear system controllability. IEEE Transactions on Automatic Control, 46, 1782–1788.