A Variational Approach Based on the

Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer 245 system model, both classical numeric and novel interval arithmetic solvers for sets of differential algebraic equations DAEs have been implemented to compute desired trajectories and control inputs in such a way that the output temperature of the system at a specific position matches a predefined time response. Moreover, the interval-based DAE solver ValEncIA-IVP has been used in [18], [19] to verify the estimation quality of classical estimator concepts which can be employed for online identification of internal system states which are not measured directly or which are not directly accessible for measurements. The algorithmic details of ValEncIA- IVP have been presented in [18]. Interval tools for verified sensitivity analysis as well as verified reachability analysis and observability analysis are summarized in [20]. One of the drawbacks of the early lumping approach is that it is rather difficult to know the connection between the original distributed parameter model and its discretized version a priori. However, this connection can be quantified by the explicit error estimates following directly from the method of integrodifferential relations MIDR formulation for inverse problems as shown in [21], [22], [23]. These estimates allow one to verify the quality of the finite-dimensional modelling, to refine numerical solutions and to make corresponding corrections of the control laws online. Among various control strategies, it is worth to note two basic directions, namely, feedforward and feedback. Feedforward control can eliminate, in the ideal situation, or at least reduce the effect of measured disturbances on the dynamical process if accurate system models are known and their initial states are consistent with the desired trajectories. A feedback control system is required to suppress undesirable measured as well as unmeasured disturbances that are always present in any real process. The combination of feedforward and feedback control can significantly improve system performance. Powerful approaches to system analysis, trajectory planning, and feedforward as well as feedback control have been derived after extending the method of flatness- based control from finite to infinite-dimensional systems. The combination of backstepping-based state-feedback control and flatness-based trajectory planning and feedforward control is considered in [24] for the design of an exponentially stabilizing tracking controller for a linear diffusion-convection-reaction system with parameters varying in space and time and a nonlinear boundary input. In [25], two new flatness-based tracking control strategies are developed, which are numerically efficient and applicable online. In these approaches, the PDE is transformed into an ODE for every measuring point using the method of characteristics. The flatness-based solution procedures proposed in [26]-[28] for systems with distributed and boundary control inputs are based on a mathematical discretization of the PDE by an ansatz function separating the dependencies on time and spatial coordinates. Adaptive control is becoming popular in many fields of engineering and science and faces several important challenges, especially in real-time applications for distributed parameter systems, which do not have precise models applicable to control design. In [29], some common and efficient adaptive control approaches, including model reference adaptive control, adaptive pole placement control, and adaptive backstepping control are presented and analyzed. The book [30] introduces 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. In Section II, the statement of an initial-boundary value problem for parabolic PDEs is given and the variational approach based on the MIDR is discussed. After that, the finite element algorithm is proposed in the frame of this approach in Section III. In the next section, after the formulation of the control problem for tracking of a desired temperature profile, both pure feedforward and adaptive control strategies are developed. The actual control structure is described in Section V. The robustness of the adaptive control strategy proposed is demonstrated and numerically verified in Section VI. Finally, the paper is concluded with an outlook on future research in Section VII.

II. A Variational Approach Based on the

Method of Integrodifferential Relations II.1. Statement of the Heat Transfer Problem and Existing Solution Approaches Consider a one-dimensional heat transfer process in a rod with length l . The heat flux law Fouriers law relates the heat flux density , q z t and the temperature gradient to each other according to: , := = 0 q q z ϑ ξ ϑ λ ∂ + ∂ 1 In this equation, the temperature is denoted by , z t ϑ and λ is the heat conductance. The first law of thermodynamics leads to: = , p q c z t t z ϑ ρ αϑ µ ∂ ∂ + + ∂ ∂ 2 where ρ is the density of the rod material, p c is the specific heat capacity, and α is the heat transfer coefficient. Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer 246 The function , z t µ represents both the distributed control and external disturbances. Let us reduce the heat transfer system 1, 2 to a dimensionless form by changing of variables: 2 f f p p f p z t q z , t , q , , l c l , , c c ϑ ϑ τ λϑ ϑ ρ τµ ατ µ α τ ρ ϑ ρ λ = = = = = = = 3 Here, f ϑ is some characteristic temperature. In all following equations, the tildes are omitted, and, then, Eqs. 1 and 2 take the form: , := = 0 q q z ϑ ξ ϑ ∂ + ∂ 4 = , q z t t z ϑ αϑ µ ∂ ∂ + + ∂ ∂ 5 In terms of the dimensionless heat flux density q , the boundary conditions are given by: 1 0, = and 1, = q t q t q t q t 6 To close the formulation of an initial-boundary value problem, specify the initial temperature distribution by: , 0 = z z ϑ ϑ 7 In Eqs. 6 and 7, q , 1 q , and z ϑ are given functions. For example, if = 0 q t holds, adiabatic insulation of the rod end at the position = 0 z is taken into account. In cases where the PDEs and space domains are simple, it is possible to obtain the solution in so-called closed form, often as an infinite series in which the terms are given by a product of one function only depending on time and others depending on the spatial coordinates. The conventional approaches for the analysis of such systems are known as the Fourier and Laplace methods [31]. For solving more complicated problems e.g., systems with non-homogeneous properties and irregular shapes, variational and projective approaches have been thoroughly developed and studied by scientists. It is rather usual that the PDEs describing different physical phenomena are stationary conditions of some variational problem. Among these formulations, the Hamilton principle in dynamics corresponding to the minimum principle for potential energy of static problems can be mentioned cf. [32]. Alternative variational principles for initial-boundary value problems were obtained on the basis of the Laplace transformation in [33]. Another approach is the MIDR which has been presented in [34] and applied to static problems regarding the linear theory of elasticity. One of the important features of the MIDR is that an original boundary or initial-boundary value problem in PDEs can be reduced to a variational problem: minimize a non- negative functional following from constitutive relations such as Hookes law in the case of linear elasticity or Fouriers law for heat transfer. During evaluation of the system model, the value of the functional can serve as an integral estimate for the quality of any approximate solution, whereas the integrand characterizes the local error distribution. Variational formulations in finite element methods are frequently used in scientific and engineering applications. The mathematical origin of the method can be traced back to a paper by Courant [35]. Other numerical approaches, e.g., the Petrov-Galerkin method [36], [37] or the least squares method [38], are being developed actively for numerical modeling of dynamical processes. Various a priori and a posteriori heuristic criteria have been applied to improve the solution quality [39]. The numerical algorithm based on the MIDR and variational techniques was worked out and applied to linear elasticity problems [40]-[42]. The FEM realization gives one the possibility to develop various strategies for adaptive mesh refinement by using a local error estimate [43]. Separated approximations including on the one hand an expansion of finite dimension over some coordinate components and on the other hand unknown functions over one remaining component can be used in the MIDR [44]. For this expansion, the original problem described by PDEs is reduced to a finite-dimensional system of ODEs. This approach has been applied to heat conduction problems [21], where the first law of thermodynamics and the corresponding initial and boundary conditions are exactly satisfied, while Fouriers law is given in a weak form. Various projective approaches which are based on the MIDR can also be developed and applied effectively to reliable numerical modeling of physical processes. In contrast to the variational technique, projections of constitutive relations on a functional space that is chosen in a special way are used to compose a consistent system of equations. A modification of the MIDR which is based on this projective technique and an ansatz representation of unknown functions is derived in [22]. A corresponding numerical algorithm has been developed to define the temperature profile and heat flux density for one- dimensional heat transfer problems. II.2. Integrodifferential Formulation of the Heat Transfer Problem To solve the initial-boundary value problem 4–7, Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer 247 the MIDR is applied in which the constitutive relation 4 is replaced by the corresponding integral relation defined as: 1 2 0 0 = = 0 with = t f dzdt ,q ϕ ϕ ξ ϑ Φ ∫∫ 8 In 8, the interval [0, ] f t denotes the time horizon over which the process is considered, f t is the terminal instant of the process. Thus, the initial-boundary value problem of the linear theory of heat conduction can be reformulated: find the admissible temperature , z t ϑ and heat flux , q z t that obey the boundary condition 6, the initial condition 7, as well as the PDE constraint 5 and satisfy the integral relation 8. In addition to computing an approximation to the true temperature distribution, this integrodifferential formulation gives us the possibility to estimate the solution quality. Note that the integrand ϕ in Eq. 8 is non-negative. Hence, for any admissible temperature , z t ϑ and heat flux density , q z t satisfying the constraints 5–7, the integral = ,q ϑ Φ Φ is non- negative and reaches its absolute minimum on the solution , z t ϑ and , q z t see [21]. The value Φ of this functional can serve as a measure for the integral quality of the approximate solution ϑ and q , whereas its integrand ϕ shows the distribution of the local error. The dimensionless ratio: 1 2 0 0 = with = , t f q z t dzdt Φ ∆ Ψ Ψ ∫∫ 9 can be used as the relative integral error of the admissible fields ϑ and q . II.3. Variational Formulation The integrodifferential problem 5–8 can be reduced to a minimization problem: find the unknown functions ϑ ∗ and q ∗ which minimize the functional Φ : min ,q ,q ,q ϑ ϑ ϑ ∗ ∗ Φ = Φ = 10 under the constraints 5–7. Denote actual and arbitrary admissible temperatures and heat flux densities by ϑ ∗ , q ∗ and ϑ , q , respectively, and specify ϑ ϑ δϑ ∗ = + and q q q δ ∗ = + . Then, 2 q ,q ϑ ϑ δ δ δ Φ = Φ + Φ + Φ holds according to Eq. 10. Here, ϑ δ Φ , q δ Φ are the first variations of the functional Φ with respect to ϑ , q and 2 δ Φ is its second variation. The second variation: 2 0 0 f t L , q dzdt , δ ϕ δϑ δ Φ = ≥ ∫ ∫ 11 which is quadratic with respect to δϑ and q δ , is non- negative, since the inequality , q ϕ δϑ δ ≥ holds for any variation of the unknown functions. After integrating the first variations by parts and taking into account 5, the first variations of Φ have the form: 1 1 0 0 1 0 0 2 2 2 f f f t t z z t q dzdt dt z qdzdt ϑ ξ δ δϑ ξδϑ δ ξδ = = ∂ Φ = − + ∂ Φ = ∫ ∫ ∫ ∫ ∫ 12 It can be seen from Eq. 12 that the sum of the first variations for the functional Φ is equal to zero for any admissible variations δϑ , q δ if Eq. 4 holds. The temperature ϑ and the heat flux density q as well as their variations δϑ and q δ are related to each other through Eq. 5 according to: q z t δ δϑ αδϑ ∂ ∂ = − − ∂ ∂ 13 If we introduce an auxiliary function χ such that: z χ ξ ∂ = ∂ 14 holds, the stationary condition for the functional Φ can be represented in the form: [ ] 1 2 2 0 0 1 1 1 2 f f f t t z t t z , dzdt t z dt dz z ϑ ϑ δ χ χ δ αχ δϑ χ δϑ χδϑ = = = Φ = ⎛ ⎞ ∂ ∂ Φ = − + − + ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ∂ ⎡ ⎤ + + ⎢ ⎥ ∂ ⎣ ⎦ ∫ ∫ ∫ ∫ 15 after integration by parts. It follows from Eq. 15 that the PDE: 2 2 t z χ χ αχ ∂ ∂ + − = ∂ ∂ 16 Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer 248 with the homogeneous conditions at the edges of the rod and at the terminal instant f t t = : 1 f t t z z , z z χ χ χ = = = ∂ ∂ = = = ∂ ∂ 17 must hold. It can be seen that the terminal-boundary value problem 16 and 17 has only a trivial solution χ = . Thus, the stationary condition 15 with reference to Eq. 14 is equivalent to the relation 4. Together with the constraints 5–7 it constitutes the complete mathematical representation of heat transfer equations. III. Finite Element Discretization In this section, a new numerical algorithm for the finite element discretization of linear heat transfer problems is considered. The algorithm is based on the weak formulation 5–7 and 10 introduced in Subsection II.3 and a piecewise polynomial approximation h ϑ of the temperature function ϑ . Let us introduce new functions p z,t and t ω such that: z t t t z p z ,t d z e t e dt α α ϑ ϑ ω ′ − − ′ ′ ′ ′ = + + ∫ ∫ 18 Then, relation 5 can be transformed into the form: z p z ,t q p z ,t dz z,t t z t α µ ω ′ ∂ ⎡ ⎤ ∂ ′ ′ = − + + − ⎢ ⎥ ∂ ∂ ⎣ ⎦ ∫ 19 Integrating Eq. 19 with respect to the coordinate z and taking into account the boundary conditions 6 leads to the explicit expression for the heat flux density: 0 0 , = z z z p z ,t q z t q t p z ,t dz dz t z ,t dz z t α µ ω ′ ′′ ∂ ⎡ ⎤ ′′ ′′ ′ − + + ⎢ ⎥ ∂ ⎣ ⎦ ′ ′ + − ∫∫ ∫ 20 where the function ω is defined as: 1 1 1 0 0 = z t q t q t z ,t dz p z ,t p z ,t dz dz t ω µ α ′ ′ ′ − + + ′′ ∂ ⎡ ⎤ ′′ ′′ ′ − + ⎢ ⎥ ∂ ⎣ ⎦ ∫ ∫∫ 21 As a result, the function ξ defined in Eq. 4 can be rewritten using Eqs. 18, 20 and 21 as: 0 0 1 1 = 1 = 0 with: 1 = z z z p z,t P z,t zP ,t F z,t p z ,t P z,t p z ,t dz dz t z F z,t z q t z zq t Q z,t Q z,t z ,t dz z z ,t dz ξ α ϑ µ µ ′ − + + ⎧ ′′ ∂ ⎡ ⎤ ′′ ′′ ′ = + ⎪ ⎢ ⎥ ∂ ⎪ ⎣ ⎦ ⎪ ∂ ⎪ = + − + ⎪ ∂ ⎨ ⎪ − + ⎪ ⎪ ⎪ ′ ′ ′ ′ − ⎪⎩ ∫∫ ∫ ∫ 22 As follows from Eqs. 7 and 18, the initial condition for the function p is homogeneous, that is: p z, = 23 Let the length of the rod [ ] 0,1 z ∈ be divided into a finite number N of interval elements: [ ] 1 1 1 , , 0 = = 1 i i N N z z z z z z z − − ∈ … 24 where i z , = 0, , i N … , are the nodal coordinates. The approximation h p of the unknown function p z,t is defined on the set of polynomial splines: [ ] 1 1 1 h h i M k M k h i i ik k i i p z,t : p p S g z g z p t , z z , z , i , , N − = − = = ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = = ⎨ ⎬ ⎪ ⎪ ⎪ ⎪ ∈ = … ⎩ ⎭ ∑ 25 where ik p t are unknown time-dependent coefficients and M is the fixed degree of the complete polynomial with respect to the spatial coordinate z . For any interval [ ] 1 , i i z z z − ∈ , = 1, , i N … , the linear functions 0i g and 1i g have the following form: 1 1 1 1 = , = i i i i i i i i z z z z g g z z z z − − − − − − − 26 Now, it is possible to compose the vector w t of independent unknown functions: { } { } 1 1 = , , = , , , = 1, , N i i iM w t w w , w p p i N ′ ′ ′ … … … 27 The dimension of the vector w t is = K MN . Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer 249 After substituting the expressions 25–27 for the corresponding arguments in the function ξ defined in Eq. 22, the following minimization problem with respect to the unknown vector function w t is formulated: find a solution w t ∗ such that the functional: 1 2 with f t w t ,w dt z,t ,w dz, z,t ,w ζ ζ ϕ ϕ ξ Φ = = = ∫ ∫ 28 reaches its minimum: min w w w ∗ Φ = Φ 29 under the constraint: w = 30 The necessary optimality condition w δ Φ = is equivalent to a system of ODEs consisting of Euler’s vector equations: d dt w w ζ ζ ∂ ∂ − = ∂ ∂ 31 with the corresponding natural terminal condition: f t t w ζ = ∂ = ∂ 32 Note that the Weierstrass–Erdmann conditions should be taken into account at the discontinuous points of the function ζ . Eq. 31 with the conditions 30 and 32 constitute the consistent boundary value problem. As shown in Subsection II.3, the solution w t ∗ of the problem 30– 32 minimizes the value of the functional Φ .

IV. Optimal Feedforward and Adaptive