Design of a decoupling controller structure for first order hyperbolic PDEs with distributed control action

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1 21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 ThA6.2 Design of a decoupling controller structure for first order hyperbolic PDEs with distributed control action Franz J. Winkler* and Boris Lohmann Abstract A new decoupling controller structure by attaching the sensors to the moving goods is developed. This approach is numerically efficient and thus suitable for online application, in contrast to most existing control strategies. Using the method of characteristics, the system of partial differential equations PDEs is transformed into ordinary differential equations for every measuring point on the transported good. A nonlinear flatness-based tracking controller is designed, based on a new decoupling strategy. The approach is applied to a continuous furnace. I. INTRODUCTION Transport processes which occur in a variety of industrial applications, for example in soldering, painting, baking, printing, shaping processes, in heat exchangers and tube reactors, are described by first order hyperbolic partial differential equation PDE systems. Source terms or control actions which are distributed in space significantly complicate the controller design. In this article especially transport processes with multiple control actions, which are arranged along the process, and sensors attached to the transported unit load is being analyzed. In most control approaches for distributed parameter systems DPS, the case of boundary control is considered. The control variable is usually locally fixed in space often located at the end of the process and the controller design is based on the approximation by a finitedimensional system using Galerkins e.g. modal and pseudomodal method or the approximation by power series. For the feedback control design two classical approaches are common: early lumping [5], [8] and late lumping [6]. Mainly parabolic systems of Sturm-Liouville type are considered [8]. Newer works on the design of trajectory tracking controller for DPS are largely based on the extension of the flatness property on DPS, using the power series method for the approximation of the DPS [1], [11], [13], [14], [22]. This approach is applied in [1], [22] to hyperbolic systems. In [1] a method for a process output feedback control with distributed control action is presented, which transforms the process into a system with boundary control. In general no parameterization of the distributed control variable can be achieved with a flat output variable and the approach is limited so far to linear DPS. In common, the application of approximation methods result in high order models, especially in the case of convection dominated systems. Moreover, the so-called observation spillover, which results from the neglected dynamics, can lead to This work was supported by the German National Academic Foundation. All authors are with the Institute of Automatic Control, Technische Universität München, Garching, Germany winkler@tum.de instability of the closed-loop system [1]. Consequently, the stability proof is another problem caused by approximation methods. Both, the power series method and the Galerkins method establish connections between neighboring points of the goods which are transported through the process and force the system to a behavior, which in fact does not exist. In transport processes a sudden change of the boundary conditions can occur and is transported through the process [18], [19]. This cannot be modeled by an approximation method, since the basis functions are continuously differentiable. Also finite element, finite volume and finite difference methods are not useful for modeling and the controller design of convection dominated systems [12]. For solving, as well as for simulation and controller design of first order hyperbolic PDEs the method of characteristics is better suited for application, because it solves the PDE exactly. In [21] a control approach, based on this method for a continuous furnace with separated heating zones, is presented. A control approach based on the combination of the method of characteristics and sliding mode techniques for processes modeled by a first order quasi-linear hyperbolic PDE is introduced in [17]. A nonlinear distributed output feedback controller for this class of systems is proposed in [2], [3]. In [16] an output feedback control method for the output of the process and in [15] the model predictive control is used, based on the method of characteristics. Demetriou introduces in [4] a control approach for DPS based on a moving collocated actuator and sensor. The approach is used in this work for fixed goods. In this article a new control approach for the trajectory tracking problem along the process of the DPS, with distributed control action is introduced. The control approach is based on the method of characteristics in combination with sensors attached to the goods. The presented approach allows a decoupled controller design along the characteristic curves. As an example the flatness-based tracking control is used to solve the tracking problem. The remainder of the paper is organized as follows: At first, the control problem is discussed and the method of characteristics is described briefly. Afterwards the controller design based on this method and the strategy to decouple the system is introduced. Finally, the approach is applied to a continuous furnace and qualitative and quantitative simulation results are given. II. PROBLEM STATEMENT In this article, transport processes with constant velocity v are considered, which can be described by the following /1/$ AACC 2563

2 system of first order semi-linear PDEs t w + v z wz,t cw,ū,z,t =, 1 with initial and boundary conditions wz,t = = gz, wz =,t = w t, 2 in the region Ω = [,L] [,, with z,t Ω, the state vector functions wz,t L 2 Ω, R n, the piecewise continuous input ūz,t : Ω R, c : L 2 Ω, R n C pω, R Ω R n, g L 2 [,L], R n and the piecewise continuous function w C p[,, R n. The distributed input variable ūz,t is considered as being separable in a function in time and space ūz,t = b T zūt. 3 The sensors are moving with the continuous good through the process. In this article a trajectory tracking problem is considered, where each control output should follow a trajectory. III. METHOD OF CHARACTERISTICS In contrast to approximation methods, the method of characteristics solves the PDE exactly. A geometric interpretation of the method of characteristics for this class of systems is given in [2] and will only be outlined briefly. This method solves the PDE by solving ODE systems along the so-called characteristic curves. These characteristic curves are parameterized by the parameter µ, which corresponds in this case to the time t. The ODE system describing the characteristic curves of PDE 1 can be formulated with ξµ = z t T, xµ = wξµ, uξµ = ūz,t and boundary condition x = wξ as = d dµ ξ 1µ = v 4 ξ 2µ = 1 5 x µ = cxµ,uξµ. 6 These equations are called characteristic differential equations of the semi-linear PDE 1. Equation 4 and 5 describes the projection of the characteristic curves on the z,t- plane and are called characteristic base curves. These curves describe the transport of the points of the good through the process in the z,t-plane. Fig. 1 shows a possible solution surface of PDE 1, in the case of a scalar function w L 2 Ω, R, with the characteristic curves and their projections to the z,t-plane. The method of characteristic depicts, that no connection between neighboring points exists. IV. CHARACTERISTIC-BASED CONTROL As described in [2], each point of the incoming goods moves along the characteristic base curves. The change in state is described by the equations along the characteristic. For a profile control, as considered in this article, one element of the state vector x or a linear combination of them, respectively, along the characteristic will be the control output variable. Thus, the controller design can be developed along the characteristic curve and only a system of ODEs has to Fig. 1. Possible solution surface of PDE 1, in the case of a scalar function w, with the characteristic curves and their projections to the z, t-plane. be considered for the design [2]. The ODEs of the different good points are coupled by the distributed characteristic of the actuating variable ūz,t = b T zūt. Consequently, the whole process can be described by the following ODE system. Without loss of generality, the system equations along the characteristic curve can be parameterized by the time t consider 5. ẋ i = cx i t,b i t,ut,t, i = 1...p, 7 with ut = uξ 2 µ = ūt, b i ξ 1 µ = b i z b i t and p denotes the number of considered good points inside the process. To regard more points the average of the points close to the sensor can be used as control output, or the dimension of the whole system has to be increased. A. Strategy to Decouple the System The following idea to decouple system 7 is based on the approach in [2] for the feedforward control. For the controller design it is assumed, that the system can be controlled by the distributed variable ūz,t for all z,t Ω. Thus, 7 converts into a decoupled SISO-system, in the case of one control variable as considered in this article, and can be described by ẋ i = cx i t,u opt,i t,t. 8 Consequently, the controller design can be done separately for each of the p good points. The controllers determine at each time step t k, p control actions u opt,i t k at different locations z i i = 1...p. The control actions have to be realized by the characteristic b i z i of the real actuators [2]. The problem can be formulated as u opt t k = B ut k 9 with u opt = u opt,1,...,u opt,p T and b1 z 1 bm z 1 B =.. R p m. 1 b1 z p bm z p Equation 9 is solved by optimization of 1 min ut k 2 u optt k But k 2 2, subject to u min u j u max, j = 1...m, 11 at each time step t k. The controller structure is shown in Fig. 2. The distributed characteristic of the actuators permits 2564

3 reference trajectory x Fig. 2. controller u opt optimization u system Principle controller structure. an amount of realizable control actions along the process. However, depending on the permitted range of the actuating variables and the amount of goods, the desired control action will not be realizable exactly for all time instants. The deviation is considered as uncertainty of the actuator and the stability of the closed loop will be analyzed by using robust control methods, e.g. the Small Gain Theorem. Furthermore, 11 can be used to determine the required number and characteristic of the actuators in order to solve the desired control task satisfactory and to analyze which control tasks are realizable with the actual configuration respectively. B. Controller Design A flatness based tracking controller is considered to solve the trajectory tracking problem. Other control strategies as the two degrees of freedom structure lead to similar results. The principle of the flatness based method is to find an output, a so-called flat output, which can be used to describe all states and control inputs. If such an output is found, it is relatively easy to calculate a control for a desired trajectory of an output signal. For definition and more details on differential flatness see [7], [14] for instance. The controller design is based on the decoupling strategy introduced above. Hence, it is relatively easy to solve the tracking problem, which is extremely complicated in the coupled case. This will be demonstrated in the example in section V and VI, where the temperature of the goods should follow three different reference trajectories. The controller design is realized along the characteristic curve using 8. It will be assumed that system 8 is flat with the flat output y. Hence, the control input can be parameterized by the flat output u opt,i t = f y,ẏ,...,y n x, i = 1...p. 12 For the flatness-based tracking controller design a new input ν i t is defined by ν i = y n [7]. Thus, for the control law follows u opt,i t = f ν i,y,...,y n Using ν i as new control input, the system is described by ν i = y n 14 and the trajectory tracking control problem can be solved by the control law e y,i = y y ν i = y n + k n 1,i e n 1 y,i + + k,i e y,i + k I,i e y,i dt. 15 In the case of a non-flat control output y y i, the reference trajectory y can be determined by solving y i = h y,ẏ,...,y n r y. 16 reference trajectory ψ p ψ 1 ψ 1 y n f,p y n f,1 controller 1 f ν p,ψ p ν p u opt,1 u opt,p controller p k T p k I,p optimization u e y,p system Fig. 3. Controller structure using flatness based tracking control, with ψ i = y n 1 T,, y, i = 1... p. Eq. 16 describes the internal dynamics of system 8 with respect to the output y i of relative degree r [9]. The Control law 15 is implemented for each good point to be controlled inside the process, whereas the reference trajectory is defined by yi. The real actuating variable is determined by optimization 11. Fig. 3 shows the resulting controller structure. An observer as introduced in [2] can be used to determine the state vector and thus the vector ψ i = in order to realize the control law 15. C. Robust Stability y n 1 ψ p - y,,y T The controller design does not consider the deviation of the control input resulting from the optimization. This deviation will be regarded as uncertainty of the actuator and is modeled as a multiplicative uncertainty as shown in Fig. 4 right. The control approach leads to a linear system y n = ν i. The model of the uncertainty is established in respect to the virtual input ν i, in order to simplify the stability proof. The uncertainty is defined to be i 1, i = 1...p. 17 The transfer function P W,i s between e i and d i has to be considered for the robustness analysis. Regarding Fig. 4 right transfer function P W,i s = P i sw A,i s follows to P W,i s= k n 1,is n + +k,i s + k I,i s n+1 +k n 1,i s n + + k,i s + k I,i W A,i s, 18 with the weighting function W A,i s. The small gain theorem is used to analyze the robust stability [23]: Theorem 4.1: Small Gain Theorem Suppose P W RH. Then the interconnected system shown in Fig. 4 left is well-posed and internally stable for all s RH with 1 1/γ if and only if P W s < γ, 2 < 1/γ if and only if P W s γ. In this article γ is set to 1. The resulting system P W,i s is a SISO system. Thus, the H norm is equivalent to the maximum magnitude of P W,i jω. The magnitude of P i jω decreases at frequencies higher than the roll-off frequency ω R,i, corresponding to 18. In general it is required, that at higher frequencies an uncertainty notably above 1% is allowed. In order to fulfill this condition for transfer function P i s, the weighting function can e.g. be chosen to be 1/ω R,i > T 1,i > T 2,i W A,i s = k w,i 1 + T 1,i s 1 + T 2,i s

4 P W y n W A,i controller i d i e i system i ν i y... k T i k I,i Fig. 4. Small Gain Theorem left and the resulting system by flatnessbased tracking control right with multiplicative uncertainty ψ i = T y n 1,, y, i = 1... p. - ψ i - y view factors z m y m y 3 i+1 y 3 i+2 y3 i z m Fig. 6. Two-dimensional view factor of twelve radiators left and reference trajectory for good i right with i N. y z K V. EXAMPLE: CONTINUOUS FURNACE As an example a continuous furnace for heating homogeneous goods is considered. This process is similar to different of the mentioned transport processes. The process is shown schematically in Fig. 5. The goods are transported at constant velocity. The process consists of twelve radiators. The mathematical model of the process is described by the semi-linear hyperbolic first order PDE as proposed in [21] for z L Tz,t t + v Tz,t z } {{ } convection =c α T Γ z,t Tz,t }{{} boundary heat transfer m + c rad F j z Trad,jt 4 T 4 z,t, j=1 2 } {{ } heat radiation T,t = T t, Tz, = T z z, 21 with c α = α κ c ρ and c rad = κ rad σ B ǫ F c ρ. The temperature around the good is denoted by T Γ and the temperature of radiator j by T rad,j. The parameter κ and κ rad denotes geometric factors, σ B the Stefan-Boltzmann-constant, α the heat transfer coefficient and ǫ the emission coefficient, which includes the absorption and reflection properties of the radiator and the good. The view factor F j z describes the energy distribution of radiator j along the process. Fig. 6 shows the two-dimensional view factors in the case of twelve radiators as indicate in Fig. 5. Table I depicts the values of the process parameters. Here, PVC plates are considered as goods. The actuators are controlled by linear PI-controllers in a cascade structure, i.e. the resulting dynamic as well as the Fig. 5. v z good sensor Tz,t T Γz,t radiator Schematic assembly of the considered furnace. TABLE I PROCESS PARAMETER description variable value unit m velocity v.5 sec heat transfer constant c α sec heat radiation constant c rad K 3 sec length of the furnace L 1 m L dynamic of the sensors can be neglected in comparison to the dynamics of the process. Using the method of characteristics the change in state for p good points along the characteristic base curves is described by ẋ i t = c α u O t x i t m + c rad F i,j t u 4 jt x 4 it, i = 1...p, j=1 22 whereas the characteristic base curves are defined by ξt = v 1 T and x i t = T i ξt, u j t = T rad,j t, u O t = T Γ ξt and F i,j t = F j ξ 1 t for good i. The decoupled system description results, corresponding to 8, in ẋ i t=c α u O t x i t+c rad qopt,i t F i tx 4 it, 23 with F i t = m j=1 F i,jt. The temperatures of the good points are the desired controlled output variables. These variables correspond to the states along the characteristics. The temperature around the goods u O will be considered as ascertainable disturbance. In order to demonstrate the capabilities of the controller design using the decoupling approach the goods should follow three different reference trajectories s-shaped, sine curve and Gaussian, as shown in Fig. 6, subsequently. At first lumped goods and then expanded goods are considered. A. Control Design System 23 is flat with the flat output y = x i, where x i corresponds to the measurement output y i. Hence, for the control input follows y = y i, i = 1...p q opt,i t= ẏit c α u O t y i t+c rad F i ty 4 i t c rad. 24 For the flatness-based tracking controller a new input is defined to ν i = ẏ = y i. Hence, the control laws with e y,i = yi y i are k,i := k, k I,i := k I q opt,i = 1 νi c α u O y i + c rad F i ty 4 i, 25 c rad ν i = ẏi + k e y,i + k I e y,i dt. 26 B. Robust Stability The transfer function from e i to d i is P W,i =P W s=psw A s = k s + k I s 2 + k s + k I W A s

5 Using both controller layouts, which are presented in section VI-B, the magnitude of the allowable weighting function W A jω can be calculated to.87 for frequencies lower than the roll-off frequency of P W jω, which is located at approximately.15 rad/sec and.5 rad/sec respectively. At higher frequencies the allowable uncertainty increases linear with the frequency. Consequently, the demand on robustness at higher frequency is fulfilled. By a set of simulations it was found, that the deviation for the considered process is much smaller than 1%. Thus, the control of the process is robustly stable. A. Applied Disturbances VI. SIMULATION This paper considers three disturbances for the continuous furnace. One of them is Gaussian measurement noise with variance σ 2 y = 1, which has been added to the measured temperature of the good. The second disturbance is that the starting temperature of the good differs from the starting temperature of the reference trajectory T =1 K. When the good s initial temperature is higher than the initial value of the reference trajectory, the controller cannot change it because it would need to cool the heater. To prevent this situation, the reference trajectory starts above the ambient temperature. This gives the controller a chance to eliminate the disturbance. As third disturbance a failure of the sixth radiator is considered. Changes of system parameters have only few effects on the controller performance. Thus, these uncertainties are not considered in this article. B. Controller Layout The controller parameter are chosen as k =.25 1/sec., k I =.15 1/sec. 2 for the lumped goods and k =.1 1/sec., k I =.26 1/sec. 2 for the expanded goods, in order to achieve a good transient behavior. The actuating variable is limited between 293 K and 1 K. C. Quality Criterion The quadratic arithmetic mean of the error is used as a measure for the performance of the controller. For an expanded good an average of the quadratic arithmetic means of all the measured points along the good is calculated ē i = 1 N N y[k] y [k] 2, ē = 1 M k=1 M ē i, 28 i=1 with N representing the number of measurement values along the trajectory and M the number of measuring points along the good. The measurement points can be obtained by several sensors along the expanded good or by approximation. To compare the performance of the designed controller with a feedforward control, a relative quality criterion ē rel is calculated as ē rel = ēno controller ē controller. 29 TABLE II QUALITY CRITERION traject. 1 traject. 2 traject. 3 ē rel lumped ē rel expanded D. Results of the Simulations In simulations twelve actuators as shown in Fig. 5 and the dynamics of the actuators and sensors are considered. The actuators are controlled separately by PI-controllers. The tracking errors are comparable for the three different trajectories. Thus, only the result for one good is presented. Fig. 7a shows the temperature profile of one good, which has to follow the trajectory y 1t in Fig. 6 in presence of the discussed disturbances. As it can be seen, the control action delivers satisfying results, which are much better than the feedforward control, as Table II points out. The performance of the controller is visualized in Fig. 7b, which shows the tracking errors. Moreover, expanded goods can be handled with this approach and will be considered as second control task. The goods are 1 mm long and are located as illustrated in Fig. 8. Again the goods have to follow the trajectories depicted in Fig. 6, subsequently. The first, middle and last point of each good are regarded as controlled output variables. The results for one controlled output variable of three different goods are shown in Fig. 9. Fig. 1 shows the result in a 3-dimensional plot for one good along the furnace. The results for the other points of the goods and the other goods respectively are comparable and are omitted. The control approach can compensate the applied disturbances relatively fast and the results are much better than using single feedforward control, as depicted in Table II. Temperature T K Temperature error K Fig. 7. Fig a 5 Ref. trajectory No controller Track. controller No controller Track. controller t sec. b Simulation result a and error profile b of one lumped good. v z control variable z g Considered assembly of the furnace in the case of expanded goods. L 2567

6 Temperature error K 4 2 No controller good 1 good 2 good t sec. Fig. 9. Simulation result for one point of three expanded goods following different reference trajectories. Fig. 1. Error profile of one expanded good. VII. CONCLUSION AND FUTURE WORKS In this article a control strategy to solve the trajectory tracking problem for transport processes with distributed control action has been introduced. It has been shown, that this complex control task can be solved by using an optimization to decouple the system equations in combination with sensors attached to the goods. The controller design is based on the method of characteristics and has been demonstrated for a flatness-based trajectory tracking controller. Finally, a continuous furnace has been considered to realize different temperature profiles of lumped and expanded goods. This approach improves the single feedforward control up to a factor of 54 when disturbances are applied. The approach can easily be extended to longer goods by considering the average of the good points as control variable or by increase the number of controlled output variables. The distributed characteristic of the actuators permits enough freedom to realize the required control actions. In further works, the optimization step will be improved, for example to regard the previous control action, in order to achieve small changes of the actuator variables. If it is taken into account for the controller design that the controlled points belong to an expanded good, better temperature homogeneity of the good can be achieved. Furthermore, the presented control approaches will be tested on a real continuous furnace available at the Institute of Automatic Control at the Technische Universität München. VIII. ACKNOWLEDGMENTS The authors thank the German National Academic Foundation for financial support. IX. APPENDIX Definition H is a closed subspace in L with functions that are analytic in the open right-half plane and bounded on the imaginary axis. The H norm is defined as F := sup σ [Fs] = sup σ [Fj ω]. Res> ω R The second equality can be regarded as a generalization of the maximum modulus theorem for matrix functions. The real rational subspace of H is denoted by RH which consists of all proper and real rational stable transfer matrices. REFERENCES [1] M. J. Balas. Active control of flexible systems. Journal of Optimization Theory and Applications, 253: , [2] P. D. Christofides and P. Daoutidis. Feedback control of hyperbolic PDE systems. AIChE Journal, 4211: , [3] P. D. Christofides and P. Daoutidis. Robust control of hyperbolic PDE systems. Chemical Engineering Science, 531:85 15, [4] M. A. Demetriou. Guidance of a moving collocated actuator/sensor for improved control of distributed parameter systems. In Proc. of the 47th IEEE Conference on Decision and Control, pages , Cancun, Mexico, 28. [5] J. Deutscher and C. Harkort. Parametric state feedback design of linear distributed-parameter systems. Int. J. Control, 826:16 169, 29. [6] H. O. Fattorini. Infinite-dimensional Optimization and Control Theory. Cambridge University Press, Cambridge, [7] M. Fliess, J. Lévine, P. Martin, and P. Rouchon. Flatness and defect of nonlinear systems: introductory theory and examples. International Journal of Control, 616: , [8] D. Franke. Systeme mit örtlich verteilten Parametern: Eine Einführung in die Modellbildung, Analyse und Regelung. Springer-Verlag, Berlin, Germany, [9] V. Hagenmeyer and M. Zeitz. Internal dynamics of flat nonlinear SISO systems with respect to a non-flat output. Systems & Control Letters, 52: , 24. [1] A. Kharitonov. Flachheitsbasierte Steuerungs- und Regelstrategien für Systeme mit verteilten Parametern bei Wärme- und Stoffübertragungsprozessen. PhD thesis, Universität Stuttgart, 27. [11] A. Kharitonov and O. Sawodny. Flatness-based feedforward and feedback control for heat and mass transfer processes. In Proc. of the IEEE Conference on Robotics, Automation and Mechatronics, pages 1 6, 26. [12] P. Knabner and L. Angermann. Numerik partieller Differentialgleichungen - Eine anwendungsorientierte Einführung. Springer Verlag, 2. [13] T. Meurer and M. Zeitz. Flatness-based tracking control for parabolic distributed-parameter systems with boundary input. at - Automatisierungstechnik, 548: , 26. [14] J. Rudolph. Flatness Based Control of Distributed Parameter Systems. Shaker Verlag, Aachen, Germany, 23. [15] H. Shang, J. F. Forbes, and M. Guay. Model predictive control for quasilinear hyperbolic distributed parameter systems. Ind. Eng. Chem. Res., 43: , 24. [16] H. Shang, J. F. Forbes, and M. Guay. Feedback control of hyperbolic distributed parameter systems. Chemical Engineering Science, 6:969 98, 25. [17] H. Sira-Ramirez. Distributed sliding mode control in systems described by quasilinear partial differential equations. Systems & Control Letters, 13: , [18] J. C. Strikwerda. Finite Difference Schemes and Partial Differential Equations. SIAM, 24. [19] G. B. Whitham. Linear and Nonlinear Waves. John Wiley & Sons, New York, London, [2] F. J. Winkler and B. Lohmann. Flachheitsbasierte Steuerung und Beobachtung von Transportprozessen mit verteiltem Eingriff. at - Automatisierungstechnik, 5711: , 29. [21] F. J. Winkler and B. Lohmann. Flatness-based control of a continuous furnace. In Proc. of the 3rd IEEE Multi-conference on Systems and Control MSC, pages , Saint Petersburg, RUSSIA, 29. [22] F. Woittennek and J. Rudolph. Motion planning for a class of boundary controlled linear hyperbolic PDE s involving finite distributed delays. ESAIM: COCV, 9: , 23. [23] K. Zhou. Robust and Optimal Control. Prentice Hall, Englewood Cliffs, New Jersey,