An Integrodifferential Approach to Adaptive Control Design for Heat Transfer Systems with Uncertainties

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1 An Integrodifferential Approach to Adaptive Control Design for Heat Transfer Systems with Uncertainties Vasily Saurin Georgy Kostin Andreas Rauh Luise Senkel Harald Aschemann Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow, Russia {saurin, Chair of Mechatronics, University of Rostock, Rostock, Germany {Andreas.Rauh, Luise.Senkel, Abstract: Open-loop and closed-loop control problems for distributed parameter systems, described by parabolic partial differential equations, are considered in this contribution. The goal of the study is the development of strategies for control and estimation of states, disturbances, and parameters. These strategies are based on the method of integrodifferential relations, a projection approach, and a suitable finite element technique. A real-time applicable control algorithm is proposed and its specific features are discussed. A verification of the control laws derived in this contribution is performed taking into account explicit error estimates resulting directly from the integrodifferential approach. The parameters, geometry, and actuation principles of a heat transfer system available at the Chair of Mechatronics, University of Rostock, are used for the numerical simulation and experimental validation. The test setup consists of a metallic rod equipped with a finite number of Peltier elements which are used as distributed control inputs allowing for active cooling and heating. Keywords: Heat transfer, adaptive control, optimal control, distributed parameter systems. 1. INTRODUCTION The design of adaptive and optimal control strategies for dynamic systems with distributed parameters has been actively studied in recent years. Processes such as heat transfer, diffusion, and convection are part of a large variety of applications in science and engineering. The theoretical foundation for optimal control problems with linear partial differential equations PDEs) and convex functionals was established in Lions 1971). In Tao ), some common and efficient adaptive control approaches, including model reference adaptive control, adaptive pole placement, and adaptive backstepping were presented and analyzed. The book Krstic and Smyshlyaev 1) introduced a comprehensive methodology for adaptive control design of parabolic PDEs with unknown functional parameters, including reaction-convection-diffusion systems ubiquitous in chemical, thermal, biomedical, aerospace, and energy systems. Different approaches to the discretization of dynamical models with distributed parameters have been developed in the past few years to reduce the original initialboundary value problem to a system of ordinary differential equations ODEs). Among these, variational and projection methods are of special importance to solve control problems for distributed parameter systems. The method of integrodifferential relations MIDR) is proposed in Kostin and Saurin 6) for the optimal control design of elastic beam motions. The variational principle was applied in Aschemann et al. 1) on the basis of a MIDR formulation of a parabolic PDE system. This system describes an application from the field of heat transfer for which tracking control strategies are to be designed. Moreover, a projection approach, which is also based on the MIDR, has been developed in Rauh et al. 1) for the same application. In this paper, the above-mentioned projection approach and a novel FEM technique are applied to design adaptive and optimal control strategies. The basic theoretic background of these techniques has been described by the authors in Saurin et al. 11b,a). A brief summary of the corresponding procedures is given in Sections and. The adaptive control approach summarized in Section 4 is capable of including an online parameter identification procedure as well as predictive techniques for tracking of desired output trajectories. The efficiency of this approach is demonstrated on the basis of numerical simulations. To improve the robustness of the control procedure with respect to measurement noise, an online applicable combined state and disturbance estimation approach is described in Section 5. This online applicable approach can be easily combined with the tracking control procedure in Section 4 and with an optimal feedforward control design that has been performed in Rauh et al. 1) with the help of Pontryagin s maximum principle. Experimental results for the latter approach extended by a feedback control structure for compensation of non-modeled disturbances

2 conclude this section. Finally, conclusions and an outlook on future research are given in Section 6.. MATHEMATICAL MODELING OF DISTRIBUTED HEATING SYSTEMS AND STATEMENT OF THE CONTROL PROBLEM Consider a one-dimensional controlled heating process in a metallic rod with the length l which is described by a parabolic PDE that is split up into the heat flux law Fourier s law) and the first law of thermodynamics. Fourier s law is given by ) ξ ϑz, t), qz, t) ϑz, t) := qz, t) + λ =, 1) while the first law of thermodynamics represents the dynamic balance of internal energy according to qz, t) + κ 1 ϑz, t) t + κ ϑz, t) = µz, t). ) Here, ϑz, t) and qz, t) are the state variables representing both the temperature distribution in the rod and the corresponding heat flux density. The parameter λ describes the heat conductivity. Moreover, the coefficients κ 1 = ρc p and κ = α h depend on the density ρ, the specific heat capacity c p, the convective heat transfer coefficient α and the height h of the rod. The function µz, t) is assumed to be given by two parts according to µz, t) = µ d z, t) + µ c z, t) ) with µ d = a d z)vt) and µ c = a c z)ut). Here, vt) is the function of external disturbances and ut) is the control input distributed along one quarter of the rod. If the ambient temperature is the source for the external disturbance, a d z) is given by a d z) = κ. Moreover, the spatial distribution of the control input is described by 4 for z l a c z) = bhl 4 for where b is the width of the rod., 4) l 4 < z l In terms of the heat flux density qz, t), the boundary conditions are q, t) = q t) and ql, t) = q l t), 5) where q t) = and q l t) = denote adiabatic insulation of both ends of the rod. Finally, the initial temperature distribution is specified by ϑz, ) = ϑ z). 6) It is assumed that z = z d, z d l, denotes the output position of the system. The goal of control design is to find a suitable control input ut) in such a way that the output yt) = ϑz d, t) coincides with a sufficiently smooth profile y d t). To solve the initial-boundary value problem, the MIDR is applied in which the local equality ξ = defined in 1) and the initial conditions 6) are replaced by integral relations, whereas the first law of thermodynamics ) and the boundary conditions 5) are satisfied exactly. The corresponding integral formulation of Fourier s law is given by Φ = ϕ dzdt = with ϕ = ξ ) ϑz, t), qz, t). 7) As an alternative to the design of tracking controllers, energy-optimal feedforward control laws are designed offline which are extended for online applications in real-life experiments by a combined state and output feedback. The feedback structure makes use of estimates provided by a model-based state and disturbance observer.. SPATIAL DISCRETIZATION ALGORITHM For the derivation of a spatially discretized system representation, the unknown heat flux density is eliminated in a first stage. This is done by taking into account the boundary conditions by means of integration of the first law of thermodynamics with respect to z according to z [ ] ϑx, t) qz, t) = µx, t) κ 1 κ ϑx, t) dx + q t). t The second boundary condition in 5) takes the form of a linear integrodifferential equation according to [ ] ϑ l κ 1 t + κ ϑ dx = µx, t)dx + q t) q l t). 9) Using the expression 8) for the heat flux density qz, t), the constitutive relation 1) can be rewritten as ξ = λ ϑ z [ ] + ϑ µx, t) κ 1 t κ ϑ dx + q t) =. 8) 1) For the purpose of numerical simulation and for the design of controllers as well as state and disturbance estimators, the rod is now discretized into N finite elements in the spatial variable z with the mesh nodes z i, i =,..., N, i.e., the finite elements are defined by z [z i 1 ; z i ], = z < z 1 <... < z N 1 < z N = l. The corresponding approximation ϑ h,i z, t) of the temperature profile in the i-th finite element is defined by M ϑ h,i z, t) = b i,k,m z)θ i,k,m t) 11) k= with z [z i 1 ; z i ], i = 1,..., N, where θ i,k,m t) are unknown time-dependent coefficients and M is the fixed polynomial degree of the functions b i,k,m z) with respect to the coordinate z. To simplify the inter-element conditions guaranteeing continuity of the temperature distribution at the common node z j between the neighboring finite elements j and j + 1, Bernstein polynomials ) ) k ) M k M z zi 1 zi z b i,k,m z) = 1) k z i z i 1 z i z i 1 of degree M are used to approximate the temperature distribution in each rod segment. The inter-element conditions then sum up to θ j,m,m t) = θ j+1,,m t). 1)

3 To simplify the notation in the following, the vectors b i,m z) = [b i,,m... b i,m,m ] T 14) and B M z) = [b 1,M z)... b N,M z)] T 15) are introduced to denote all Bernstein polynomials of order M for either one rod segment i or for the union of all segments, respectively. Accordingly, the coefficient vectors θ i,m t) = [θ i,,m t),..., θ i,m,m t)] T 16) and Θ M t) = [θ 1,M t),..., θ N,M t)] T 17) are defined. A set of ODEs for the coefficient vector Θ M := Θ M t) can be obtained by projecting ξ in 1) onto the set of basis functions { bi,k,m 1 z), z [z i 1 ; z i ] η i,k z) = 18), z / [z i 1 ; z i ] according to ) ξ ϑz, t), qz, t) η i,k z)dz = 19) with i = 1,..., N and k =,..., M 1. These projections correspond to a system of M N ODEs for the vector of unknown coefficients Θ M. Finally, the boundary condition 9) is expressed as an ODE by [ ] ϑ h x, Θ M ) κ 1 + κ ϑ h x, Θ M ) dx = fl, t) q l t) t ) which has to be solved together with the ODEs 19). The corresponding initial conditions of this system of ODEs are computed from a least squares approximation of the initial temperature distribution ϑ z) according to Θ [ M ) = arg min ϑh z, Θ M )) ϑ z) ] dz. 1) Θ M ) In addition to computing an approximation to the true temperature distribution, this integrodifferential formulation provides the possibility to estimate the solution quality if approximations ϑz, t) to the exact solution ϑ z, t) are used. For any admissible temperature distribution ϑz, t) satisfying the constraints 6) and 9), the integral Φ = Φ ϑ) is non-negative and reaches its absolute minimum on the exact solution ϑ z, t) see Aschemann et al. 1)). The value Φ of this functional, therefore, serves as a measure for the integral quality of the approximate solution ϑ, whereas its integrand ϕ shows the distribution of the local error in both space and time. The dimensionless ratio = Φ Ψ with Ψ = λ ϑz, t) ) dzdt ) can be used as the relative integral error of any admissible temperature field ϑz, t). 4. ADAPTIVE CONTROL STRATEGY 4.1 Formulation of the Tracking Control Task The general scheme of the adaptive control procedure is depicted in Fig. 1. The proposed adaptive control strategy takes into account a sequence of time steps t [t k 1, t k ], t k = kt c, k, with the length t c of each piecewise constant control interval. Trajectory planner y d Adaptive controller u Data storage Plant Fig. 1. Adaptive control scheme. y ŷ û Parameter identifier The control signal ut) can be found by means of the adaptive controller by using the current output y = {y 1,..., y Ny } of measurements y i = θt, z y i ) gathered at discrete positions z y i [; l], i = 1,..., N y, the identified function ˆvt) of external disturbances and the desired temperature profile y d t) generated by the trajectory planner. The corresponding values are written in the data storage as û. These values are then applied to the plant at the beginning of the time step t = t k. At the end of this time step, the vector y is measured and again used together with the previously saved values ŷ and ˆv in the parameter identifier to produce a new estimate for the function ˆvt). After that, the current vector y and the identified function ˆvt) are put into the adaptive controller, saved in the data storage as ˆv, and then the control cycle is repeated in the following time step. Using the vector yt k 1 ) of system outputs, the function v k t) = ˆv = const, t t k, of external disturbances and the desired profile y d t), the control u k t) is determined for the k-th step by minimizing the output error according to t k +t p u k = arg min y dt ) u k with y = θt, z d, u k, v k ) y d t) 4) over the prediction horizon [t k ; t k + t p ]. At the end of the control step, the output vector yt k ) is measured and used together with the values yt k 1 ) and u k to produce a new disturbance estimate v k+1 according to N y vk+1 = arg min θt v k k, z y i, u k, v k ) y i t k )). i=1 5) After that, the current vector yt k ) and the identified function vk+1 are used as the initial data for the next step. The ratio of both the control and prediction horizons t c and t p, respectively, has to be chosen in such a way as to guarantee the stability of the overall control process. t k ˆv

4 4. Numerical Simulation Results Consider the heating system which has been built up at the Chair of Mechatronics, University of Rostock, Aschemann et al. 1); Rauh et al. 1). The following system data are given: h =.1 m, b =.4 m, l =. m, z n = n l, n = 1,, 4 z y i = i 1)l/8, i = 1,..., 4, z d = z y 4 λ = 11 W/m K), α = 5 W/m K) ρ = 78 kg/m, c p = 4 J/kg K) y d t) = ϑ + 5 K 1 + tanh[σt t f /)] coth[σt f /]) σ =.15, t f = 6 s vt) = ϑ + t t f, ϑ = K. The control input of this system is provided by a Peltier element located beneath the first quarter of the rod. In Fig., three control signals are displayed for 1) optimal polynomial feedforward control under the assumption v k = ϑ, ) adaptive control with identification of disturbances v k const), and ) adaptive control without identification of disturbances v k = ϑ ). The corresponding temperatures at the output position z = z d and at the middle of the control segment z = z y 1 are shown in Fig.. As expected, the adaptive control strategy with identification of external disturbances leads to the smallest deviation from the desired profile y d, see Fig. in which the control quality of the approaches ) and ) is compared. ut) in W Fig.. Control functions ut). It can be seen from Figs. and that the control quality of the pure feedforward strategy can be improved significantly by an adaptive control procedure which exploits estimates for non-measured external disturbances. However, it has to be noted that such estimates can be quite sensitive with respect to measurement noise, especially in cases in which only a small number of previous measurements here: only one previous step) is used to estimate slowly varying disturbances. To improve the robustness of the disturbance estimate shown in Fig. 4 which determines the external disturbance over the complete time horizon [; t f ] with errors smaller than.8 K, a model-based state and disturbance observer derived also on the basis of the MIDR is introduced in the following section. 1, t) in K ϑzd, t), ϑz y ϑzd, t) ydt) in K Fig.. Temperature trajectories at the input and output positions left) as well as temperature deviations from the desired profile right). ˆvt) in K Fig. 4. Identified ambient temperature ˆv. One of the most important properties of the MIDR procedure is the fact that both local and global error estimates can be determined directly. The local error distribution ϕz, t) introduced in 7) is displayed in Fig. 5 for the adaptive control strategy with simultaneous identification of the external disturbance in the case that Bernstein polynomials of the order M = are used in each finite element to describe the temperature distribution. The corresponding relative integral error defined in ) is already quite small: = A further increase of the approximation order M leads to a notable decrease of the integral error, which becomes equal to = for M = and = for M = 4. In addition, the function ϕz, t) allows for identifying regions in space and time for which the finite-dimensional system model exhibits its largest errors compared to the original distributed parameter system. 5. EXPERIMENTAL VALIDATION OF OPTIMAL CONTROL AND MODEL-BASED OBSERVER DESIGN Assume that the linear state equations resulting from the projection approach with M = are abbreviated by ẋt) = Axt) + but) = fxt), ut)), 6) where the vector x contains all time-dependent coefficients of the temperature distribution in the rod according to 17). The goal of the control and observer design is the implementation of an offline computed optimal heating-up

5 ϕz, t) Fig. 5. Local error distribution ϕz, t)..1 z in m strategy in an experimental setup. The offline computed control input and the corresponding output trajectory are used as a feedforward control sequence and a reference trajectory in the experimental implementation. To compensate disturbances, the online implementation extends the offline computed feedforward control by a state and output feedback which makes use of estimates for the nonmeasured components of the vector x and the external disturbances v. In the experiment, these disturbances mainly stem from the fact that free convection on the top of the metallic rod is prohibited by an air canal which is used in further research work for active cooling of this system. Here, this air canal is deactivated and therefore represents an additional, a-priori unmodeled disturbances. 5.1 Energy-Optimal Heating-Up Strategy The goal of the optimal feedforward control synthesis is to transfer the temperature ϑ h,n z d, t) at a given position z d in given time t f to a desired final value ϑ d with a vanishing final variation rate ϑ h,n z = z d, t f ) =. This goal is taken into account by sufficiently large weighting factors ν 1 and ν in the terminal cost function f T := ν 1 ϑ h,n z d, t f ) ϑ d ) + ν ϑh,n z d, t f )) 7) of the performance criterion J C = f T + u t)dt = f T + f dt! = min. 8) The minimization of J C is performed numerically with the help of the maximum principle of Pontryagin. For that purpose, the Hamiltonian H := f +p T fxt), ut)) with the adjoint states p is maximized by the control u fulfilling the condition H u u=u =. Defining H := Hu = u ), the set of canonical equations [ẋ ] [ fxt), u ] t)) = ṗ H 9) x is obtained. For 9), the boundary value problem with the initial states x) = v and the terminal conditions p t f ) = f T ) x x=xtf ) is solved numerically by the boundary value problem solver bvp4c in Matlab. To guarantee its convergence, the problem is firstly solved for ν 1 = 1 5, ν =. Secondly, this solution is used to re-initialize bvp4c with ν 1 = ν = 1 5. For the online application, the control input is defined as θ 1,M t) v 1 t) ut) = u t) + u P I t) k T.., 1) θ 4,M t) v 4 t) where all non-measured values are replaced by the observer outputs described in the following subsection. The control part u P I t) represents an additional PI output feedback determined by the transfer function 1 + K R T I s + 1 T I s ) S v. ) U P I s) Y d s) ϑ h,4 7l 8, s) = The feedback and prefilter gains k and S v are chosen by a linear quadratic regulator design exploiting the condition for steady-state accuracy. Moreover, T I is set to compensate the largest time constant of the approximating system model 6) with K R =. 5. State and Disturbance Observer Design To reconstruct non-measured variables, the ODEs obtained from the projection approach in the MIDR are used to design the observer ˆ xt) = È xt) + but) + L y ŷ) ) with the estimates ˆ x for the extended state vector x. This extended state vector consists of the states x introduced in the previous subsection as well as values for the ambient temperature which are assumed to be piecewise constant for each rod segment, cf. Rauh et al. 1). Using the vector y = [ ϑ h,1 l 8, t) ϑ h,4 7l 8, t) ϑ h,1, t) ϑ h,4l, t) ] T = Cxt) 4) of temperature measurements in the midpoints of two rod segments as well as the information about adiabatic insulation of the rod edges according to ϑ h,1, t) := = and ϑ h,4 l, t) := ϑ h,4z,t) =, the z= z=l ϑ h,1 z,t) observer gain matrix L = P C T R 1 5) can be calculated by minimizing the error measure J O = 1 xt) T Q xt) + yt) T R yt) ) dt 6) with Q = Q T and R = R T >, where the errors with respect to the extended state vector are denoted by xt) and yt) corresponds to the deviations between the true and estimated system outputs. The solution to this optimization problem leads to the algebraic Riccati equation P C T R 1 CP AP P A T Q =, 7) for which the positive definite, symmetric matrix P = P T > has to be determined. Both matrices Q and R are set to identity matrices of appropriate dimensions in the experiment. It can be shown experimentally, that this observer approach is more robust with respect to measurement noise than the pure adaptive disturbance estimator introduced in the previous section. Moreover, the observer is also able to cope with a smaller number of measurement points here only two temperature measurements instead of four measurements that have been assumed before).

6 5. Experimental Results Experimental results have shown an accurate tracking of the energy-optimal output trajectory determined by means of the approach summarized in this section, see Fig. 6. In spite of the large disturbance Fig. 7, left) which acts on the rod like an additional convective heat input, estimated by the above-mentioned observer, the tracking errors remain in the range [.1;.1]K over the complete time horizon. Only at the end, a slightly larger deviation occurs which can be stabilized rapidly by the controller. The resulting control signal is depicted in Fig. 7 right). ϑzd, t), ydt) in K ydt) ϑzd, t) in K Fig. 6. Comparison of desired and actual outputs y d t) and ϑz d, t). v4t) in K ut), u t) in W Fig. 7. Disturbance estimate v 4 as well as control ut) closed-loop, solid line) and u t) offline computed optimal control, dashed line) CONCLUSIONS AND FUTURE WORK In this paper, adaptive and optimal control algorithms with online parameter identification as well as modelbased state and disturbance estimation have been derived for both robust trajectory tracking and optimal heatingup strategies of distributed heating systems. All control strategies are based on the MIDR, a projection approach, and a novel finite element technique. 5 An experimental validation has shown accurate tracking properties for an optimal desired output trajectory in spite of large disturbances acting on the system. Compared to other finite volume or finite element techniques for distributed heating systems, the MIDR procedure provides explicit estimates for both the local and integral quality of the resulting approximations of the temperature distribution which help to systematically improve the approximation quality by adaptation of the number of finite elements and the corresponding approximation orders. In future work, means for the improvement of the robustness of the adaptive control procedure with respect to measurement noise will be investigated. One possible approach is the combination of the observer which has already been applied successfully in the experiment with the adaptive control and estimation procedure Sec. 4). ACKNOWLEDGEMENTS This work was supported by the Russian Foundation for Basic Research, project nos , , the Leading Scientific Schools Grants NSh , NSh , as well as by the German Research Foundation Deutsche Forschungsgemeinschaft DFG) under the grant number AS 1/-1. REFERENCES Aschemann, H., Kostin, G.V., Rauh, A., and Saurin, V.V. 1). Approaches to Control Design and Optimization in Heat Transfer Problems. J. Comp. Sys. Sci. Int., 49), Kostin, G.V. and Saurin, V.V. 6). Modeling of Controlled Motions of an Elastic Rod by the Method of Integro-Differential Relations. J. Comp. Syst. Sci. Int., 451), Krstic, M. and Smyshlyaev, A. 1). Adaptive Control of Parabolic PDEs. Princeton University Press. Lions, J.L. 1971). Optimal Control of Systems Governed by Partial Differential Equations. Springer Verlag, New York. Rauh, A., Kostin, G.V., Aschemann, H., Saurin, V.V., and Naumov, V. 1). Verification and Experimental Validation of Flatness-Based Control for Distributed Heating Systems. Int. Rev. Mech. Eng., 4), 188. Rauh, A., Senkel, L., Aschemann, H., Kostin, G., and Saurin, V. 1). Reliable Finite-Dimensional Models with Guaranteed Approximation Quality for Control of Distributed Parameter Systems. In Proc. of American Control Conference ACC 1. Montreal, Canada. Under review. Saurin, V.V., Kostin, G.V., Rauh, A., and Aschemann, H. 11a). Adaptive Control Strategies in Heat Transfer Problems with Parameter Uncertainties Based on a Projective Approach. In A. Rauh and E. Auer eds.), Modeling, Design, and Simulation of Systems with Uncertainties, Mathematical Engineering, 9. Springer. Saurin, V.V., Kostin, G.V., Rauh, A., and Aschemann, H. 11b). Variational Approach to Adaptive Control Design for Distributed Heating Systems under Disturbances. Int. Rev. Mech. Eng., 5), Tao, G. ). Adaptive Control Design and Analysis. Wiley & Sons, Inc., Hoboken, New Jersey.