RANGE CONTROL MPC APPROACH FOR TWO-DIMENSIONAL SYSTEM 1

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1 RANGE CONTROL MPC APPROACH FOR TWO-DIMENSIONAL SYSTEM Jirka Roubal Vladimír Havlena Department of Control Engineering, Facult of Electrical Engineering, Czech Technical Universit in Prague Karlovo náměstí 3, 35 Prague, Czech Republic Honewell Prague Laborator, Honewell Intl Pod vodárenskou věží, 8 8 Prague 8, Czech Republic havlena@htchonewellcz Abstract: This paper deals with model predictive control of a distributed parameters sstem which is described b a linear two-dimensional (dependent on two spatial directions) parabolic partial differential equation This partial differential equation is transformed to the discrete state space description using the finite difference approimation A model with a large dimension is obtained and has to be reduced for an advanced control design The balanced truncation method is used for the model dimension reduction For this low dimension model, the range control approach is applied Copright c 5 IFAC Kewords: Distributed Parameters, Partial Differential Equations, Reduced-order Models, Modelling, Predictive Control INTRODUCTION There are man industrial processes which 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) which can be described b lumped inputs and distributed output models These models can be mathematicall described b partial differential equations (PDE) (Long, C A, 999) Unlike ordinar differential The authors acknowledge financial support of Honewell Prague Laborator (9/957/335) and grants /3/H and /5/75 of the Grant Agenc of the Czech Republic 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 finitedimensional model, for eample, b using the finite difference method (Babuška, I et al, 9) 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, eg the model predictive control approach, the large model dimension introduces a problem for the control design Therefore a model reduction method has to be used

2 Variables of ever real process have certain limits given b laws of phsics Unlike the classical control law, the model predictive control (MPC) considers eplicitl the future implication of current control action This approach enables us to include the constraints on inputs/outputs to the control algorithm (Maciejowski, J M, ) In (Havlena, V and Findejs, J, 5), the range control approach is described for lumped inputs and lumped outputs sstems The main idea of the range control concept is to replace the set point (reference) b low and high limits This methodolog leads to ver stable and robust control because the manipulated inputs do not compensate the high-frequenc component of the noise In this paper, this concept is applied for the distributed parameters sstem The paper is organized as follows In section, the distributed parameters model for the finite controller design is developed In section 3, the balanced truncation method is shortl described In section, the basic idea of model predictive control with the range control approach is described In section 5, this methodolog is applied to a heat transfer process as a demonstration eample DISTRIBUTED PARAMETER PROCESS DESCRIPTION In this section, the model of a heat transfer process described b a linear two-dimensional parabolic PDE (Long, C A, 999) is developed for the finite-dimensional controller design At first, the stationar PDE is transformed to a linear equation sstem using the finite difference approimation (Babuška, I et al, 9) Then the implicit scheme (Babuška, I et al, 9) and this equation sstem are used for the transformation of the evolutionar PDE to a linear dnamic discrete sstem Stationar Partial Differential Equation For the surface thermal conductivit λ [W/K] independent on the temperature Θ [K] and a surface heat source f [W/m ], the heat transfer process in the stationar case can be described b a parabolic PDE ( ) Θ(, ) λ + Θ(, ) = () = λ Θ(, ) = f(, ), where is the Laplace operator The unknown temperature Θ must satisf equation () on an open set Ω =(, L ) (, L ) 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 Θ(, ) λ = α ( Θ(, ) Θ s (, ) ), () n where n is unit normal vector, α [W/(mK)] is an eternal heat transfer coefficient and Θ s is the surrounding temperature Note that equation () is known as Newton (or the third kind) boundar condition (Long, C A, 999) i=,j= i=n,j= δ δ i,j- i-,j i,j i+,j i,j+ Ω Ω Figure The mesh on the set Ω i=,j=n i=n,j=n For the transformation of PDE () with boundar condition () to the finite dimensional model, the set Ω is covered b an imaginar mesh so that the values of mesh points satisf Θ i,j = Θ(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 = Θ s (i δ, j δ) on the set Ω, where δ = L /N and δ = L /N are the grid sizes of the imaginar mesh and i, j are row and column indices, respectivel (see Figure ) Matri Θ is the matri of the temperature values in the mesh points and matri F represents the heat source f and the surrounding temperature Θ s Using the second order finite difference approimation (Babuška, I et al, 9), the distributed parameters sstem, equations () and (), can be obtained as a linear equation sstem Pθ = f, (3) where Θ(:, ) F(:, ) Θ(:, ) θ =, f = F(:, ), Θ(:, N ) F(:, N ) where Θ(:, ) means the zero column of the matri Θ, Θ(:, ) the first column and so on Note that the square matri P contains (N + ) (N + ) rows and its structure and derivation can be found in (Roubal, J et al, ) Evolutionar Partial Differential Equation In the non stationar case, PDE () can be written as dθ(,, t) ρ c λ Θ(,, t) = f(,, t), () dt

3 where ρ [kg/m ] is the surface densit of the medium and c [Ws/kgK] is its thermal capacit In this case, the unknown temperature profile Θ(,, t), dependent on time t, must satisf, for an initial condition Θ(,, t ) = Θ init (, ), equation () on the open set Ω and boundar condition () on Ω for all time horizon t t, t end Using equation (3) and the implicit discretization scheme (Babuška, I et al, 9) with a sampling period δt, evolutionar PDE () with Newton boundar condition () can be approimated as θ(k+) = Mθ(k)+Nf(k), θ(k ) = θ init, (5) ( M= I+ δt ) ( P, N= I+ δt ) δt P, ρ c ρ c ρ c where I is the identit matri with the corresponding dimension More details can be found in (Roubal, J et al, ) 3 MODEL REDUCTION METHOD The accurac of model (5) increases with decreasing grid sizes δ and δ Unfortunatel, for the advanced controller design such as the predictive controller, a low dimension model is needed In this section, one reduction method is shortl described 3 Model Reduction b Balanced Truncation 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 (Zhou, K et al, 99) In addition, these Gramians are equal to the diagonal matri Σ W c = W o = Σ = diag (σ, σ,, σ n ) Note that the decreasingl ordered numbers, σ σ σ n, are called the Hankel singular values of the sstem We suppose σ r σ r+ for some r ; n) Then the balanced realization implies that the states corresponding to the singular values of σ r+,, σ n are less controllable and observable than the states corresponding to σ,, σ r The states corresponding to the singular values of σ r+,, σ n have smaller influence on the input/output behaviour of the sstem Therefore, truncating the 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 Reduced Model for the Control Design The reduced model for control of evolution partial differential equation () with Newton boundar condition () can be written as (k+)=a(k) + Bu(k) + Ez(k), (k )=, (k)=c(k) + Du(k), () where is a state of the model, is its output (temperature in several points on the set Ω), u is its input (manipulated variable), z represents the surrounding temperature profile (measurable disturbance) and A, B, C, D, E are state matrices MODEL PREDICTIVE CONTROL Output Prediction For model (), the output prediction for the prediction horizon T p can be written in a compact form ŷ k = V(k) + Tẑ k + Sû k = ỹ k + Sû k, (7) where ỹ k = V(k) + Tẑ k and ŷ, û and ẑ are the output, inputs and disturbance prediction, respectivel ŷ k = (k) (k+) (k+t p ), û k = u(k) u(k+) u(k+t p ) and the prediction matrices are V = C CẠ CA T p, S= D CB CAB CA T p B, ẑ k = D CB D z(k) z(k+) z(k+t p ) CB D Figure Control variables specification in time domain the funnel for a controlled variable and the manipulated variable with reduced degrees of freedom and a constant value at the end of the correction horizon Note that matri T is similar to matri S with matri D replaced b zero matri and matri B replaced b matri E

4 Quadratic Program Problem Formulation Consider process (7) and constraints on the controlled variables and their rates of change ŷ L ŷ k ŷ H, ŷ L ŷ k ŷ H (8) and constraints on the manipulated variables and their rates of change û L û k û H, û L û k û H (9) The basic idea of the range control approach (Havlena, V and Findejs, J, 5) is to replace the setpoint for the controlled variable b a set range which is defined b the sequence of low and high limits ŷ L and ŷ H, see Figure Then the optimalit criterion can be epressed as a quadratic programming problem } min { ỹ k + Sû k ŵ k Q + û Qu u,w and constraints (9) () subject to ŷ L ŵ k ŷ H () For the penalization of control increments in criterion (), the increments of control can be obtained as û k = Φû k ũ k, () Φ = I I I, ũ k = I I u(k ) where I is the identit matri with the corresponding dimension Then the optimization criterion () can be written as min u,w [ ] [ S I ûk Φ ŵ k ] +, [ ỹk ũ k ] [ Q Q u ], (3) ŷ L ŵ k ŷ H, () û L û k û H, (5) û L + ũ k Φû k û H + ũ k () To reduce the dimension and computational requirements of the QP problem, the number of independent moves of manipulated variables ma be reduced (Havlena, V and Findejs, J, 5), see Figure The receding horizon approach is implemented The idea of this strateg is to find the optimal control sequence on the prediction horizon T p Then for feedback control, onl the first element of the optimal control sequence is applied to the plant and the optimal problem is recalculated for a new measured data (Maciejowski, J M, ) 5 DEMONSTRATION EXAMPLE Consider a heat transfer process in a furnace where L = L = 9 m described b equation () with constants λ = 5 W/K, ρ = 5 kg/m, c = 59 Ws/(kg K) and α = W/(mK) The grid sizes are δ = δ = m and the sampling period is δt = 3 s Source distribution Figure 3 Heat source distribution f(, ) The heat source distribution f(, ) is shown in Figure 3 We consider that the sample sstem has five lumped inputs (manipulated variables) and the surface temperature is measured in points which are uniforml distributed over the area Ω The inputs can take the values u ; 5 Figure a presents the stead-state temperature distribution (sstem state) for the unit step as inputs signal (see Figure 3) and the surrounding temperature Θ s = 3 K Figure b shows the sstem output temperature in measurement points Figure 5a shows the Hankel singular values of our sstem From this figure it follows that the sstem contains one singular value which is greater than (red point in Figure 5a), five singular values which are greater than (red and green points in Figure 5a), nine singular values which are greater than (red, green and blue points in Figure 5a) etc In this eample, the balanced truncation is used and the number of states of the reduced order model is chosen as r = 3 Figure 5b shows time response of Frobenius norm (Horn, R A and Johnson, Ch R, 985) of the model output error From figure it follows that the Frobenius norm reaches a stead state value Note that the input signal of the sstem is unit step The balanced truncation of this model for other numbers of states are compared in (Roubal, J et al, )

5 Stead state temperature profile Sstem output 8 8 Θ end Θ Figure (a) Stead-state temperature distribution Θ(, ); (b) Sstem output 5 Hankel Singular Sstem Values σ > : (9) σ > : (8) σ > : (3) σ > : (9) σ > : (5) σ > : () Frobenius Norm of Error σ 5 Err F Figure 5 (a) Hankel singular values of the sstem; (b) Frobenius norm of the model output error For the reduced order model, the MPC controller with the range control strateg respecting the constraints u 5 and u is designed For the initial condition of temperature distribution as in Figure, the sstem in closed loop is controlled The input time responses and the Frobenius norm response of control error are shown in Figure 7a In Figure 7b, the temperature distribution, the Θ Initial temperature Distribution 8 Figure The initial temperature distribution 8 sstem output with the set range reference, the control error and the reference temperature profile, respectivel, are presented Note that the sstem is simulated with the measurement noise σ e During the control simulation, the temperature on the left side of the area was changed at time minutes The temperature was decreased b K, which is represented the addition of the material for melting into the furnace From Figure 7a it follows that the input constraints are not violated and the Frobenius norm of the control error reaches a stead state value The reference temperature profile is shown in Figure 7b Note that the low and high limits of the set range reference are set to ± K of this profile Figure 8 presents the simulation results as Figure 7 but in this case the predictive controller without the range control concept is used If we compare Figures 7 and 8 we will observe the result which we epected The input trajectories in Figure 7 are smoother than in Figure 8 because the manipulated inputs need not compensate the high-frequenc component of the measurement noise

6 Temperature Distribution Sstem Output u u u 3 u Err F u Frobenius Norm Error Θ 5 er 5 5 Control Error 5 5 m Reference Figure 7 Range control predictive control approach: (a) The inputs time responses and Frobenius norm of control error time response; (b) Stead state temperature distribution with its reference 5 5 Temperature Distribution Sstem Output u u u 3 u Err F u Frobenius Norm Error Θ er Control Error Reference Figure 8 Classical predictive control approach: (a) The inputs time responses and Frobenius norm of control error time response; (b) Stead state temperature distribution with its reference m 5 5 CONCLUSION The state space model of the distributed parameters sstem which is described b the linear twodimensional parabolic partial differential equation and the model reduction b the balanced truncation method are described The range control methodolog of the model predictive control is introduced and is applied to the distributed parameter model which can describe a heat transfer process Because of a large dimension of the model, the balanced truncation reduction method was used The predictive controller with the range control strateg was compared with the classical predictive control approach The epected results was obtained In the case of the range control approach the manipulated variables are smoother than in the classical predictive control approach because the manipulated variables do not compensate the high-frequenc component of the measurement noise or inaccurac of the model at high frequencies The predictive controller with the range control strateg leads to more calm control REFERENCES Babuška, I, Práger, M and Vitásek, E (9) Numerical Processes in Differential Equations London - New York - Sdne, Interscience Publishers Havlena, V and Findejs, J (5) Application of Model Predictive Control to Advanced Combustion Control Control Engineering Practice 3 (No), 7 8 Horn, R A and Johnson, Ch R (985) Matri Analsis Cambridge Universit Press, United Kingdom Long, C A (999) Essential Heat Transfer Longman Group, England Maciejowski, J M () Predictive Control with Constraints Prentice Hall, Harlow, England Roubal, J, Trnka, P and Havlena, V () Simulation of Two- Dimensional Distributed Parameter Sstems In: Matlab Conference Prague, Czech Republic Zhou, K, Dole, J C and Glover, K (99) Robust and Optimal Control Prentice Hall, New Jerse