BASE VECTORS FOR SOLVING OF PARTIAL DIFFERENTIAL EQUATIONS

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1 BASE VECTORS FOR SOLVING OF PARTIAL DIFFERENTIAL EQUATIONS J. Roubal, V. Havlena Department of Control Engineering, Facult of Electrical Engineering, Czech Technical Universit in Prague Abstract The distributed parameters sstems can be described b linear two-dimensional (dependent on two spatial directions) parabolic partial differential equations. Using the finite difference method a distributed parameters sstem can be transformed to a linear discrete state space model. The controller design based on this description is complicated because of the large dimension of the model. Therefore, a model reduction method has to be used. We transform the state space model to the balanced realization of the sstem and show that the state vector of the model can be epressed as the series of columns of the transformation matri. These columns can be imaged as base vectors of the state space. 1 Introduction There are man industrial processes that have distributed parameters behaviour. Consequentl, these processes cannot be modelled b lumped inputs and/or lumped outputs models for correct representation. This paper deals with two-dimensional dnamic processes (sstems with parameters dependent on two spatial directions) that can be described b lumped inputs and distributed output models. These models can be mathematicall described b partial differential equations (PDE) [5]. Unlike ordinar differential equations, the PDEs contain, in addition, derivatives with respect to spatial directions. Consequentl, the partial differential equations lead to more accurate models but their compleit is larger. The dnamic behaviour of the distributed parameters sstem, which is described b the PDE, can be approimatel described b a finite-dimensional model, for eample, b using the finite difference method [1]. Then the ordinar differential equation model with large dimension is obtained and can be used for a finite-dimensional controller design. Unfortunatel, for online solving of an optimization problem, e.g. the model predictive control approach [6, 2], the large model dimension introduces a problem for the control design. Therefore, a model reduction method has to be used. Partial differential equations are usuall solved b base functions. This approach is based on that the solution of the PDEs can be epressed as series of the base functions. This paper presents similar methodolog. The discrete state space description of a distributed parameters sstem is transformed to the balanced realization [8]. Then the sstem state can be epressed as the product of the transformation matri and the balanced state. We can imagine the columns of the transformation matri as base vectors to which the sstem state can be epanded. Then using onl the first few base vectors corresponds to the truncation of states with smaller influence on the input/output behaviour of the sstem. The paper is organized as follows. In section 2, the distributed parameters model for the finite controller design is developed. In section 3, the balanced truncation method is described. In section 4, the relationship between the balanced realization and the sstem state epansion to an orthonormal base vectors is developed. In section 5, this approach is demonstrated on a heat transfer process.

2 2 Distributed Parameter Process Description In this section, the model of a heat transfer process described b a linear two-dimensional parabolic PDE [5] is developed for a finite-dimensional controller design. At first, the stationar PDE is transformed to a linear equation sstem using the finite difference approimation [1]. Then the implicit scheme [1] and this equation sstem are used for the transformation of the evolutionar PDE to a linear dnamic discrete sstem. 2.1 Stationar Partial Differential Equation For the surface thermal conductivit λ [W/K] independent on the temperature [K] and a surface heat source f [W/m 2 ], the heat transfer process in the stationar case can be described b a parabolic PDE ( 2 ) (, ) λ (, ) 2 = λ (, ) = f(, ), (1) where is the Laplace operator. The unknown temperature must satisf equation (1) on an open set Ω = (, L 1 ) (, L 2 ) and boundar condition on Ω ( Ω means the boundar of set Ω). In this paper, the boundar condition which specifies the temperature gradient on the boundar Ω is described b the following statement (, ) λ = α ( (, ) s (, ) ), (2) n where n is the unit normal vector, α [W/(mK)] is an eternal heat transfer coefficient and s is the surrounding temperature. Note that equation (2) is known as Newton (or the third kind) boundar condition [5]. i=,j= δ i=,j=n2 δ i,j-1 i-1,j i,j Ω i,j+1 i+1,j i=n1,j= Ω i=n1,j=n2 Figure 1: The mesh on the set Ω For the transformation of PDE (1) with boundar condition (2) to the finite dimensional model, the set Ω is covered b an imaginar mesh so that the values of the mesh points satisf i,j = (i δ, j δ) on the closed set Ω, F i,j = f(i δ, j δ) on the open set Ω and F i,j = s (i δ, j δ) on the set Ω, where δ and δ are the grid sizes of the imaginar mesh and i, j are row and column indices, respectivel (see Figure 1). Matri is the matri of the temperature values in the mesh points and matri F represents the heat source f and the surrounding temperature s.

3 When we define vectors ( :, ) ( :, 1) θ =., f = ( :, N 2 1) ( :, N 2 ) with indices l = j(n 1 + 1) + i, F ( :, ) F ( :, 1). F ( :, N 2 1) F ( :, N 2 ), (3) where (:, ) means the zero column of the matri, (:,1) the first column and so on, partial differential equation (1) with Newton boundar condition equations (2) can be written in compact form P θ = f. (4) More details were described in [7]. 2.2 Evolutionar Partial Differential Equation In the non stationar case, PDE (1) can be written as ρ c d(,, t) dt λ (,, t) = f(,, t), (5) where ρ [kg/m 2 ] is the surface densit of the medium and c [Ws/kgK] is its thermal capacit. In this case, the unknown temperature profile (,, t), dependent on time t, must satisf, for an initial condition (,, t ) = init (, ), equation (5) on the open set Ω and boundar condition (2) on Ω for all time horizon t t, t end. Using equation (4) and the implicit discretization scheme [1] with a sampling period δt, evolutionar PDE (5) with Newton boundar condition (2) can be approimated as θ(k + 1) = Mθ(k)+Nf(k), θ(k ) = θ init, (6) ( M = I + δt ) 1 ( P, N = I + δt ) δt P 1, ρ c ρ c ρ c where I is the identit matri with the corresponding dimension. 3 Model Reduction Method The accurac of model (6) increases with decreasing grid sizes δ and δ. Unfortunatel, for the advanced controller design, for eample the predictive controller, a low dimension model is needed. In this section, one reduction method is shortl described. 3.1 Model Reduction b the Balanced Truncation Method There are infinitel man different state space realizations for a given transfer function. But some realizations are more useful in control design. One of these realizations is the balanced realization which gives balanced Gramians for controllabilit W c and observabilit W o [8]. In addition, these Gramians are equal to the diagonal matri Σ W c = W o = Σ = diag (σ 1, σ 2,..., σ n ).

4 Note that the decreasingl ordered numbers, σ 1 σ 2... σ n, are called the Hankel singular values of the sstem. We suppose σ r σ r+1 for some r 1; n). Then the balanced realization implies that the states corresponding to the singular values of σ r+1,..., σ n are less controllable and observable than the states corresponding to σ 1,..., σ r. The states corresponding to the singular values of σ r+1,..., σ n have smaller influence on the input/output behaviour of the sstem. Therefore, truncating those less controllable and observable states will not lose much information about the sstem input/output behaviour and the dimension of the model can be significantl reduced. 3.2 Transformation to the Balanced Realization Model (6) can be epressed as (k+1) = A(k) + Bu(k) + Ez(k), (k )=, (k) = C(k) + Du(k), (7) where = θ is a state vector of the model (temperature profile), is its output vector (temperature measured in several points on the set Ω), u is its input vector (manipulated variable), z represents the surrounding temperature profile (measurable disturbance) and A, B, C, D, E are state matrices. Gramians for controllabilit W c and observabilit W o [4] of model (7) can be epressed b W c = W o = A i [ B, E ] [ B, E ] T ( A T ) i, (8) i= ( A T ) i C T C A i. (9) i= Model (7) can be transformed, b a matri Q to another realization = Qv, (1) v(k+1) = Av(k) + Bu(k) + Ez(k), v(k )=v, (k) = Cv(k) + Du(k), (11) where the matrices satisf A = Q 1 AQ, B = Q 1 B, E = Q 1 E, C = CQ, D = D. Note that Gramians of the transformed realization (11) are equal to W c = Q 1 W c (Q 1 ) T, W o = Q T W o Q, (12) = W c W o = Q 1 W c W o Q. (13) Now we have to find transformation matri Q, such that realization (11) is the balanced realization of the sstem. Since Gramians are smmetric positive definite matrices, we can factor Gramian for observabilit as W o = P T P, where matri P is the Cholesk factor of matri W o. Then equation (13) can be written as Q 1 W c P T P Q = Σ 2. Using elementar rearrangement, the equation can be obtained as Q 1 P 1 P W c P T P Q= ( P Q ) 1 P Wc P T( P Q ) =Σ 2. (14)

5 Equation (14) means that P W c P T is similar to Σ 2 and is positive definite. Therefore, there eists an orthonormal transformation matri U (U T U = I), such that P W c P T = UΣ 2 U T. (15) From equations (12)a and (15), we can derive the formula for the transformation matri Q = P 1 UΣ 1/2. (16) Then model (11) is the balanced realization of model (7) and its Gramians for controllabilit and observabilit are W c = W o = Σ [8]. So if model (7) is a minimal realization and matri A is stable, the transformation matri Q can be obtained through the following procedure: compute the controllabilit and observabilit Gramians of model (7) W c >, W o >, find matri P such that W o = P T P, diagonalize P W c P T = UΣ 2 U T b using the singular value decomposition [3], let the Q = P 1 UΣ 1/2. 4 Base Vectors Considering the Hankel singular values are decreasingl ordered, the first state of state vector v has the biggest influence on the input/output behaviour of the sstem, the second state has less influence and so on. As it has been said in section 3, we can truncate states corresponding to small singular values σ i and the model keeps enough information about the input/output behaviour of the sstem. In this section, we show that the sstem state vector can be epressed in an orthonormal base. Then using onl the first r orthonormal base vectors (for epression of the sstem state vector) corresponds to the truncation of the states with smaller influence on the input/output behaviour of the sstem. 4.1 Decomposition of the Sstem State Vector to the Balanced Base Vectors Equation (1) can be rewritten as = Qv = n v i q i, (17) i=1 where v i are the elements of state vector v in balanced realization (11) and vectors q i are the columns of matri Q = [ ] q 1, q 2,..., q n. Since the transformation matri Q is not singular, the vectors q i are linearl independent. So equation (17) means that state vector can be epressed as the series of base vectors q i that are multiplied b weighted coefficients v i. Note that if we want to truncate states corresponding to the singular values of σ r+1,..., σ n, we epress state vector onl b the first r base vectors q i = r v i q i. i=1

6 4.2 Orthonormal Base Vectors The base vectors q i can be transformed to an orthonormal base b using the Gram-Schmidt orthonormalization. We decompose the matri Q such that Q = OR, where R is an upper triangular matri and O is an orthonormal matri (O T O = I). Then equation (17) can be rewritten as n = Qv = ORv = Oṽ = ṽ i o i, (18) where ṽ = Rv and vectors o i are the columns of matri O = [ o 1, o 2,..., o n ]. So equation (18) means that state vector can be epressed b the orthonormal base vectors o i. i=1 5 DEMONSTRATION EXAMPLE Consider a heat transfer process in a furnace where L 1 = L 2 =.9 m described b equation (5) with constants λ = 51 W/K, ρ = 25 kg/m 2, c = 1259 Ws/(kg K) and α = 1.14 W/(mK). The grid sizes are δ = δ =.2 m and the sampling period is δt = 3 s. The sstem has five lumped inputs (manipulated variables, see Figure 2) and the surface temperature is measured in 64 points which are uniforml distributed over the area Ω, see Figure 3b. Source distribution Figure 2: The heat source distribution f(, ) Stead state temperature profile Sstem output end Figure 3: (a) The stead-state temperature distribution (, ); (b) The sstem output (the temperature in 64 measurement points)

7 Hankel Singular Sstem Values σ > : (129) σ >.1: (18) σ >.1: (13) σ > 1: (9) σ > 1: (5) σ > 1: (1) σ Figure 4: The Hankel singular values of the sstem Figure 3a presents the stead-state temperature distribution (sstem state) for the unit step as input signal (see Figure 2) and the surrounding temperature s = 34 K. Figure 3b shows the sstem output temperature in 64 measurement points. Figure 4 shows the Hankel singular values of the sstem. Figure 5a presents the sstem output based on the first sstem state (in balanced realization), Figure 5b shows the base function corresponding to the first base vector (17) and Figure 5c shows the orthonormal base function corresponding to the first orthonormal base vector (18). Figure 6a presents the sstem output based on the first two sstem states and Figures 6b and 6c show the base function corresponding to the second base vector and the orthonormal base function corresponding to the second orthonormal base vector, respectivel. From Figures 5-14, it follows that the sstem state vector can be epressed in the series of the base functions (the columns of the transformation matri) or the orthonormal base function, respectivel. This methodolog is similar to the analtical solution of PDEs based on the nondiscrete base functions that are usuall nonlinear even if the PDEs are linear. Sstem output Base vector q 1 Orthonormal base vector o Figure 5: (a) Stead-state sstem output based on the first state; function (q 1 ); (c) The first orthonormal base function (o 1 ) (b) The first base Sstem output Base vector q 2 Orthonormal base vector o Figure 6: (a) Stead-state sstem output based on the first two states; (b) The second base function (q 2 ; (c) The second orthonormal base function (o 2 )

8 Sstem output Base vector q 3 Orthonormal base vector o Figure 7: (a) Stead-state sstem output based on the first three states; (b) The third base function (q 3 ); (c) The third orthonormal base function (o 3 ) Sstem output Base vector q 4 Orthonormal base vector o Figure 8: (a) Stead-state sstem output based on the first four states; (b) The fourth base function (q 4 ); (c) The fourth orthonormal base function (o 4 ) Sstem output Base vector q 5 Orthonormal base vector o Figure 9: (a) Stead-state sstem output based on the first five states; (b) The fifth base function (q 5 ); (c) The fifth orthonormal base function (o 5 ) Sstem output Base vector q 6 Orthonormal base vector o Figure 1: (a) Stead-state sstem output based on the first si states; (b) The sith base function (q 6 ); (c) The sith orthonormal base function (o 6 )

9 Sstem output Base vector q 7 Orthonormal base vector o Figure 11: (a) Stead-state sstem output based on the first seven states; (b) The seventh base function (q 7 ); (c) The seventh orthonormal base function (o 7 ) Sstem output Base vector q 8 Orthonormal base vector o Figure 12: (a) Stead-state sstem output based on the first eight states; (b) The eighth base function (q 8 ); (c) The eighth orthonormal base function (o 8 ) Sstem output Base vector q 9 Orthonormal base vector o Figure 13: (a) Stead-state sstem output based on the first nine states; (b) The ninth base function (q 9 ); (c) The ninth orthonormal base function (o 9 ) Sstem output Base vector q 1 Orthonormal base vector o Figure 14: (a) Stead-state sstem output based on the first five states; (b) The tenth base function (q 1 ); (c) The tenth orthonormal base function (o 1 )

10 6 CONCLUSION The state space model of the distributed parameters sstem described b the linear two-dimensional parabolic partial differential equation and the model reduction b the balanced truncation method are described. The connection of the model reduction b the balanced truncation method and the solution of the partial differential equations based on the base functions is developed. The result is demonstrated on the heat transfer process eample. You can observe from the figures that the base functions q i and the orthonormal base functions o i, respectivel, are similar to the Fourier decomposition of a signal (the first harmonic, the second harmonic, etc.) Acknowledgements This work was partl supported b the Ministr of Industr and Trade of the Czech Republic under project 1H-PK/22. References [1] Babuška, I., Práger, M., and Vitásek, E. Numerical Processes in Differential Equations. London New York Sdne, Interscience Publishers, [2] Havlena, V. and Findejs, J. Application of Model Predictive Control to Advanced Combustion Control. Control Engineering Practice, 13 (No.6):671 68, 25. [3] Horn, R. A. and Johnson, Ch. R. Matri Analsis. Cambridge Universit Press, United Kingdom, [4] Kailath, T. Linear Sstems. Prentice Hall, New Jerse, 198. [5] Long, C. A. Essential Heat Transfer. Longman Group, England, [6] Roubal, J. and Havlena, V. Range Control MPC Approach for Two-Dimensional Sstem. In Proceedings of IFAC 16th World Congress, Prague, Czech Republic, 25. [7] Roubal, J., Trnka, P., and Havlena, V. Simulation of Two-Dimensional Distributed Parameter Sstems. In Matlab Conference, Prague, Czech Republic, [8] Zhou, K., Dole, J. C., and Glover, K. Robust and Optimal Control. Prentice Hall, New Jerse, Jirka Roubal Department of Control Engineering, Facult of Electrical Engineering, Czech Technical Universit in Prague, Karlovo náměstí 13, Prague 2, Czech Republic, roubalj@control.felk.cvut.cz Vladimír Havlena Department of Control Engineering, Facult of Electrical Engineering, Czech Technical Universit in Prague, Karlovo náměstí 13, Prague 2, Czech Republic, havlena@control.felk.cvut.cz