ИЗВЕСТИЯ РАН. ТЕОРИЯ И СИСТЕМЫ УПРАВЛЕНИЯ, 2010, № 3, с. 40-51
= ОПТИМАЛЬНОЕ УПРАВЛЕНИЕ
УДК 517.977.56
APPROACHES TO CONTROL DESIGN AND OPTIMIZATION IN HEAT TRANSFER PROBLEMS* © 2010 г. H. Aschemann1, G. V. Kostin2, A. Rauh1, V. V. Saurin2
1 Germany, Rostock, Chair of Mechatronics, University of Rostock 2 Russia, Moscow, Institute for Problems in Mechanics of the Russian Academy of Sciences
Received January 11, 2010
Abstract. In this paper, boundary control problems are considered for a distributed heating system. The dynamical model of the heating system under consideration is given by a parabolic partial differential equation. In the first stage, the implementation of the Fourier method is discussed for the problem of heat convection and conduction. In the second stage, two alternative solutions to the design of tracking controllers are discussed. On the one hand, an optimal control problem is solved based on the method of integrodifferential relations. On the other hand, this procedure is used to verify the quality of a flatness-based control strategy. The results obtained by the integrodifferential approach are compared with finite-mode Fourier approximations. After derivation of suitable, general-purpose solution procedures for the design of open-loop as well as closed-loop boundary control strategies, experimental results are presented. These results highlight the applicability of these procedures in a real-world experiment. For the experimental validation, a test setup at the University of Rostock has been used.
Introduction. The conventional method for the analysis of linear distributed heating systems is known as the method of separation of variables [1]. In this method, the solution is represented as an infinite series in which the terms are given by products of two functions: one function depends on time only and the other one on the spatial coordinates. In this paper, we aim at the computation and experimental validation of control strategies for distributed parameter systems. Many publications have been devoted to control problems for such systems, for example [2—5].
The control method proposed in [6] enables one to construct a constrained distributed control in closed form and ensures that the heating system is brought to a given state in a finite time. This method is based on decomposition of the original system into simple subsystems by the Fourier approach.
Other numerical approaches, for example, the Petrov—Galerkin method [7, 8], are actively developed in continuum mechanics. For all these methods it is supposed that some of the constitutive relations are generalized and that the exact solution is approximated by a finite set of trial functions. Various a priori and a posteriori heuristic criteria have been applied to improve the solution quality [9]. However, the compu-
* The authors would like to thank their students Aleksandar Pavlov and Bofei Li for their support in the implementation of the experiments. This work was supported by the Russian Foundation for Basic Research, project nos. 08-01-00234, 09-01-00582 and the Leading Scientific Schools Grants NSh-4315.2008.1, NSh-169.2008.1. This project is furthermore supported by the German Research Foundation (Deutsche Forschungsgemeinschaft DFG) under the grant number AS 132/2-1.
tation of reliable error estimates for these methods requires advanced mathematical techniques.
Another approach developed in this paper for reliable modeling and boundary control of systems with heat convection and conduction as well as for verification of the alternative strategy flatness-based control, is based on the method of integrodifferential relations (MIDR). Originally, this method has been presented in [10] and [11] for the linear theory of elasticity. Its main idea is the specification of the constitutive relation (Hooke's law) by an integral equality instead of a local differential law. Afterwards, the modified boundary value problem is reduced to the minimization of a suitable non-negative functional over all admissible displacements and equilibrium stresses. This approach has been extended to initial-boundary value problems for distributed systems and applied to inverse dynamic problems [12]. In this contribution, a modification of the MIDR is derived for heat transfer problems and applied to the optimization of boundary control strategies. The approach is based on a spatial polynomial representation of the unknown temperature profile as well as the heat flux density function. In addition, it is combined with a variational method. The minimization algorithm that is used to solve this problem allows us to estimate the local and integral quality of numerical solutions explicitly.
In this paper, two alternative control laws are derived: the approaches based on the MIDR and differential flatness. A modification of the Fourier method is used to verify the accuracy of these control strategies.
The flatness-based solution procedure that is discussed in this contribution is based on a mathematical
discretization of the PDE by an ansatz function separating the dependencies on time and spatial coordinates [13—15]. The corresponding approximate solution of the PDE, which is transferred into a set of ODEs by the mathematical discretization procedure, describes the temperature profile within the heating system under consideration, a metallic rod. The control of the temperature profile of this rod is implemented by active cooling and heating using Peltier elements. Using suitable boundary control inputs corresponding to the temperature at one end of the rod, the output temperature at the other end is controlled in such a way that it follows sufficiently smooth prescribed trajectories as closely as possible.
The design of control strategies for the distributed heating system considered in this article has already been studied in previous publications. In [16, 17], a procedure for the numerical computation of a different feedforward control strategy has been presented which makes use of a finite-volume discretization of the heat transfer equation in order to replace this parabolic partial differential equation (PDE) by a set of ordinary differential equations (ODEs). For this spatially discretized 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 function of time.
Moreover, the above-mentioned interval-based DAE solver VAlEncIA-IVP has been used in [16, 17] 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 (e.g., the temperatures at internal positions of the heating system) or which are not directly accessible for measurements (such as uncertainties of the heat transfer coefficients for convection and conduction). 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 [19].
In Section 1, the statement of initial-boundary value problems for parabolic PDEs is given. The open-loop control problem for tracking of a desired temperature profile is formulated and the heating system under consideration is specified. Then, the implementation of the Fourier method for this control problem is discussed in Section 2. The integrodifferential formulation of optimal control problems is proposed and the variational algorithm solving the problem of heat convection and conduction is developed in Section 3. In Section 4, an alternative flatness-based control strategy is developed to design the open-loop control sequences for the heating system. In Section 5, the MIDR and Fourier approaches are applied to verify
both optimal and flatness-based control laws. Simulations and experimental results are summarized to assess the robustness of the control strategy. Finally, the paper is concluded with an outlook on future research in the last section.
1. Statement of the problem. To derive a general procedure for control of parabolic PDEs, the following equation
ki dp- Ц + кз$ = g (г, t )
dt dz
(1.1)
is considered. In this equation, the time-dependent and position-dependent distribution of the physical variable of the process (e.g., the temperature in the case of the heat equation) is denoted by 3( z, t ). The distributed input g(z, t) can — depending on the application under consideration — either represent the control input or a disturbance.
To obtain a unique solution for the parabolic PDE, it is necessary to specify spatial boundary conditions at the physical edges of the system together with consistent initial conditions. The positions for which spatial boundary conditions are formulated are denoted by z = 0 and z = L. To stay close to the heating system discussed in this paper, we consider the boundary conditions
3Q dz
&z,0(t) and &z=z = &z(t) (1.2)
z=0
with the initial conditions
Ц t=0 = So(z).
(1.3)
In (1.2) and (1.3), & 0(t) and &0(z) are given functions of the time t and the position variable z. For simulation purposes, the function &L( t) is given, while it has to be determined if open-loop and closed-loop control tasks are considered.
In the open-loop control problem for the system (1.1)—(1.3), we assume that z = zd, 0 < zd < L, denotes the output position of the system and that z = L is the location of the system's boundary input. The goal of the control strategies derived in the following is the computation of a boundary input
3l(0 = u(t) (1.4)
such that the output &(zd, t) coincides with a desired sufficiently smooth temperature profile yd(t) according to
Щ z = z, = ^ ).
(1.5)
The exclamation mark means that this relation is the
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