AN APPROACH TO LUMPED CONTROL

Appl. Comput. Math. 6 (2007), no.1, pp.69-79

AN APPROACH TO LUMPED CONTROL SYNTHESIS IN DISTRIBUTED SYSTEMS K.R. AIDA-ZADE †, A.V. HANDZEL †, § Abstract. In the paper we study a class of problems of lumped control synthesis for objects with distributed parameters with the use of the values of object’s current states at the given points of the object. We search control on the class of piecewise constant functions. The values of these functions are determined by belonging of current states of object at observable points to given subsets (zones) which partition the phase state. The formulas which allow for application of the finite-dimensional smooth optimization methods for the search of optimal values of zone control are proposed. Key words: object with distributed parameters, lumped control, zone control, feedback, object state, initial conditions, boundary conditions, functional gradient.

Introduction In recent years with the development of technical and measurement tools of control systems more attention is paid to the problems of synthesis of control laws for objects with lumped parameters, governed by ordinary differential equations, as well as for objects with distributed parameters, described by partial differential equations [1-5]. In this paper we develop further the ideas of [1] and study the problem of synthesis of lumped sources control for the objects with distributed parameters on the basis of continuous observation of phase state at given points of object. In the proposed approach the phase state space (phase space) is beforehand somehow partitioned at observable points into given subsets (zones). The synthesizing control actions therewith are taken from the class of piecewise constant functions. The current values of control actions are determined by the subset of phase space that contains the aggregate of current states of object at the observable points (in these states control actions take constant values). In the paper such synthesized control actions are called zone control actions. A technique to obtain optimal values of zone control actions with the use of smooth optimization methods is given. With this aim, the formulas of objective functional gradient in the space of zone control actions are obtained. 1. The formulation of the problem of lumped control synthesis As an illustration of the proposed approach to the study of the problem of control synthesis in distributed systems, we consider the problem of control of plate heating process by means of lumped point sources, that can be described by the following equation: 2

ut = a ∆u +

l X

¡ ¢ ϑj (t) δ x − x ¯j ,

(x, t) ∈ Ω × (t0 , T ] ,

(1)

j=1

Here Ω is the two dimensional domain, occupied by the plate; the sources of heat are placed ³ ´ j j j at the plate’s points x ¯ = x ¯1 , x ¯2 ; the intensities of the sources ϑj (t) , j = 1, ..., l are to †Institute of Cybernetics of National Academy of Sciences of Azerbaijan, e-mail: .kamil [email protected] §Manuscript received 12 July 2005. 69

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APPL. COMPUT. MATH., VOL. 6, NO.1, 2007

be optimized; lis given number of sources; ∆is two dimensional Laplace operator; δ (.) is two dimensional generalized Dirac function; a2 is the thermal conductivity coefficient. Let us suppose, that there are N sensors mounted on the plate at the points x ˜s = (˜ xs1 , x ˜s2 ) ∈ Ω, s = 1, ..., N . These sensors perform on-line monitoring and reading-in of information about the state of heating process at these points into the process control system. This information is defined by the vector: ¡ 1 ¢ ¡ ¡ 1 ¢ ¡ N ¢¢ u ˜ (t) = u ˜ (t) , ..., u ˜N (t) = u x ˜ , t , ..., u x ˜ ,t , Initial-boundary conditions may be as follows: u (x, t0 ) = g0 (x) , u (x, t)|x∈Γ1 = g1 (t) ,

t ∈ [t0 , T ] .

x ∈ Ω,

(2)

t ∈ (t0 , T ] ,

(3)

¯ du ¯¯ = g2 (t), t ∈ (t0 , T ], dn ¯x∈Γ2

(4)

here g0 (x) , g1 (t) , g2 (t) are given initial-boundary functions, Γ = Γ1 ∪ Γ2 is the boundary of plate domain, Γ1 ∩ Γ2 = ∅. In more general case, the exact values of these functions are unknown, instead, the sets of values, attained by these functions are given, i.e., in place of (2)-(4) we have: u (x, t0 ) ∈ G0 (x) ,

x ∈ Ω,

(5)

u(x, t)|x∈Γ1 ∈ G1 (t), t ∈ (t0 , T ],

(6)

¯ du ¯¯ ∈ G2 (t) , dn ¯x∈Γ2

(7)

t ∈ (t0 , T ] .

Here Gi (.) are set-valued mappings, in which every value of argument is related to a closed bounded set. The corresponding functions of distribution of the values of initial-boundary conditions Φ0 (g0 ) , Φ1 (g1 ) , Φ2 (g2 ) are given. The presented problem of plate heating consists in the choice of the values of sources’ intensities ϑj (t) ∈ V j , j = 1, ..., l, with respect to the state values at observable plate points u ˜s (t) = u(˜ xs , t), s = 1, ..., N : ¡ 1 ¢ ϑj (t) = ϑj u ˜ (t) , ..., u ˜N (t) ,

j = 1, ..., l, t ∈ [t0 , T ] ,

(8) ¡ 1 ¢ in order to minimize the given functional. Here V the set of admissible values V = V , ..., V l . In the case of initial-boundary conditions (2) - (4), the functional may be given as follows: j is

Z

l Z X £ j ¤2 [u (x, T, ϑ) − u (x)] dx + ϑ (t) dt, T

∗

J (v) = Ω

2

j=1 t

(9)

0

while in the case of non-fixed initial-boundary conditions (5) - (7), the functional may become: R R R R J(v) = u[(x, T ; ϑ, g0 , g1 , g2 )− G0 G1 G2 Ω

−u∗ (x)]2 dxdΦ2 (g2 )dΦ1 (g1 )dΦ0 (g0 ) +

l RT P

[ϑj (t)]2 dt,

(10)

j=1 t0

here u (x, T ; ϑ, g0 , g1 , g2 ) is the solution of the problem (1), (5) - (7), which corresponds to specific initial-boundary functions g0 (x) ∈ G0 , g1 (t) ∈ G1 , g2 (t) ∈ G2 and to admissible values

K.R. AIDA-ZADE,A.V. HANDZEL: AN APPROACH TO LUMPED ...

71

of control actions ϑ (t) ⊂ V ; u∗ (x) is given function, describing desired temperature distribution at the terminal time instance of heating process. We shall study the control problem with boundary conditions (2) - (4) (called problem A) and the control problem with conditions (5) - (7) (called problem B). It is clear, that the problem B is a general case of the problem A, that is why the following mathematics will be performed for the problem B, since similar results are easy to obtain for the problem A. We shall assume that the functions involved in the formulations satisfy all the necessary conditions of existence and uniqueness of the solution of the considered problems. We shall find control functions (8) on the following class of piecewise constant functions. Let the values of plate phase states satisfy the following inequality: u ≤ u (x, t; ϑ, g0 , g1 , g2 ) ≤ u, (x, t) ∈ Ω, under all admissible control actions and initial-boundary conditions: ϑ (t) ∈ V, g0 (x) ∈ G0 (x) , g1 (t) ∈ G1 (t) , g2 (t) ∈ G2 (t) . Hence it follows, that we can consider N -dimensional phase space of process states. This space is formed by the states at observable points of plate: u (˜ xs , t) , s = 1, ..., N . The admissible set of states under all admissible values of control actions and initial-boundary conditions constitutes N -dimensional parallelepiped in this space: u ≤ u (˜ xs , t; ϑ, g0 , g1 , g2 ) ≤ u ¯, s = 1, ..., N. Let us partition the set of all possible values of temperature [u, u] by points uk , k = 0, ..., m, such, that uk < uk+1 , into m temperature intervals: [u, u] =

m−2 [

[uk , uk+1 )

[

[um−1 , um ],

u0 = u,

um = u.

k=0

We shall choose piecewise constant values of control actions with respect to plate state at observable points (more strictly, with respect to belonging of the temperature to one or other temperature interval). Let control actions satisfy the following conditions of piecewise constancy: ϑj (t) = ϑji1 i2 ,...,iN = const, is = 1, ..., m, s = 1, ..., N, j = 1, ..., l (11) in the cases when the values of current states at observable points satisfy the following inequality conditions: ¡ s˙ ¢ (12) uis −1 ≤ u x ˜ , t; ϑ (t) , g0 , g1 , g2 < uis , is = 1, ..., m, s = 1, .., N.. The sets (12) constitute N - dimensional parallelepipeds in the N - dimensional phase space u ˜s (t) = u (˜ xs , t) , s = 1, ..., N . The total number of these parallelepipeds is mN . It is clear, that the presence of feedback is assumed for control (11) as well as for (8), in case (11) the values of intensity of controllable sources therewith vary in the process of plate heating only at that time instances, when the set of states at observable points moves from one phase parallelepiped (12) to another one. The number of different values of intensity of each source is equal to the number of phase parallelepipeds, defined by inequalities (12), i.e., mN . The total number of optimized parameters ϑji1 i2 ,...,iN , is = 1, ..., m, s = 1, .., N, j = 1.., l. They govern the behavior of the sources at every possible state of plate at observable points, that may occur at various admissible initial-boundary conditions and control actions, determining the belonging of current state u˜ (t) to one or other phase parallelepiped (12). So, the considered problem of plate heating control with the use of feedback at the class of piecewise constant functions consists in optimization of lmN - dimensional vector:

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³ ´ ϑ = ϑ11,...1 , ..., ϑlm,...,m ,

(13)

which directly determines the course of plate heating process and, consequently, the value of objective functional (10). 2. Solving problem of zone control synthesis Original problem of optimal control of distributed object without condition (11) on the class of control actions was studied in many papers, especially, for the case of formulation A [4]. For studying and solving the problem B, we can use, for instance, the results of [5]. The assumed in this work condition of piecewise constancy of control (11) leads to parametric problem of optimal control of distributed system, in which finite-dimensional vector of parameters (13) is to be optimized. On the other hand, the obtained problem may be considered as finitedimensional optimization problem. Its objective function depends on optimized vector ν and is determined by solving boundary value problem (1)-(4) or (1),(5)-(7) and by computation of functional (9) or (10). Taking into consideration the aforesaid, for numerical solving zone control synthesis problem, we can use the methods of finite-dimensional smooth optimization, in particular, iterative method of gradient projection type: ³ ´ ϑ(q+1) (αq ) = PV ϑ(q) − αq gradJ (ϑq ) , q = 0, 1, ..., (14) ³ ´ αq = argmin J ϑ(q+1) (α) . α>0

ϑ0

Here: is some given value of initial guess vector of zone control actions ϑ, PV (z) is the operator of projection of element z on admissible set V . With this aim, we shall present the obtained formulas of functional gradient in the space of optimized parameters. ˜ (t) beLet us denote by Πi1 ,...,iN (ϑ, g0 , g1 , g2 ) ⊂ [t0 , T ] the time interval when phase state u longs to the (i1 , ..., iN )-th phase parallelepiped at chosen control ϑ (t) and at functions g0 (x) , g1 (t) , g2 (t), involved in initial-boundary conditions. It is clear, that m [

...

i1 =1

m [

Πi1 ,...,iN (ϑ, g0 , g1 , g2 ) = [t0 , T ] .

iN =1

The interval Πi1 ,...,iN (ϑ, g0 , g1 , g2 ) in general case can be multilinked, since the process can enter repeatedly into some phase parallelepiped during time interval [t0 , T ], i.e.: p(i1 ,...,iN ;ϑ,g0 ,g1 ,g2 )

Πi1 ,...,iN =

[

Πµi1 ,...,iN (ϑ, g0 , g1 , g2 ) ,

µ=1

here p (i1 , ..., iN ; ϑ, g0 , g1 , g2 )is the number of entering of values of plate phase states at the observable points into the (i1 , ..., iN ) - th phase parallelepiped at chosen control ϑ (t) and at initial-boundary functions g0 (x), g1 (t), g2 (t). Let us consider the following boundary value problem, adjoint to (1), (5) - (7), (10): ψt = −a2 ψxx ,

(x, t) ∈ Ω × [t0 , T ) ,

ψ (x, T ) = −2 [u (x, T ; ϑ, g0 , g1 , g2 ) − u∗ (x)] , ψ (x, T )|x∈Γ1 = 0,

¯ dψ (x, T ) ¯¯ = 0. dn ¯x∈Γ2

(15) (16) (17)


73

Here the adjoint variable ψ (x, t) = ψ (x, t; ϑ, g0 , g1 , g2 ) is the solution of the problem (15) (17), under corresponding specific control actions ϑ and initial-boundary functions g0 (x) , g1 (t) , g2 (t) , which determine corresponding solution u (x, t; ϑ, g0 , g1 , g2 ) of boundary value problem (1), (5) (7). The value of this solution at t = T is involved in initial condition (16) of adjoint boundary value problem. In particular, in the case of problem A, ψ = ψ (x, t; ϑ) is the solution of (15) (17), with the solution u(x, T ; ϑ) of initial-boundary problem (1) - (4) involved in (16). The following theorem holds. Theorem 1. The components of functional gradient in the problem (1), (5) - (7) in the space of piecewise constant control actions (11), (12) for arbitrary control ϑ ∈ V under appropriate normalization are defined by the formula: ∂J(ϑ) ∂ϑji ,...,i 1

N

R R R

=

¡ j ¢ ψ x ¯ , τ ; ϑ, g0 , g1 , g2 ×dΦ2 (g2 ) dΦ1 (g1 ) dΦ0 (g0 ) +

R

G0 G1 G2 Πi1 ,...,iN +2Πi1 ,...,iN ϑji1 ,...,iN ,

j = 1, ...l,

is = 1, ...m,

(18)

s = 1, ..., N,

here ψ (x, τ ; ϑ, g0 , g1 , g2 ) is the solution of adjoint problem (15) - (17), that corresponds to current zone control. For more special problem (1) - (4), (9), i.e., in the case of fixed initial-boundary conditions, the components of functional gradient are determined by the formula: ∂J(ϑ) ∂ϑji ,...,i

=

R

¡ j ¢ ψ x ¯ , τ ; ϑ dτ + 2Πi1 ,...,iN ϑji1 ,...,iN ,

(19) j = 1, ..., l, is = 1, ..., m, s = 1, ..., N. The formulas (18), (19) allow us to compute the functional gradient at current value of control actions and to use it further in iterative procedures of smooth optimization like (15). We must note, that actual dimensions of optimized parameters vector are rather large. So, a lot of calculations and computer time may be needed to solve control synthesis problem, presented in the paper. Taking into consideration, on the one hand, the importance and complexity of control synthesis problem, and, on the other hand, the absence of demand of these problems’ real-time solving, we think that the proposed approach may be applied when designing a number of automatic and automated systems of control of technical objects and processes. 1

N

Πi1 ,...,iN

3. Numerical experiments. Let us consider one more application of the proposed approach, namely, numerical solving of zonal control synthesis for heat-exchanger process [3]. Let a liquid goes through the heatexchanger. The temperature of steam in the jacket is ν(t). The temperature of the liquid u(x,t) at the point x ∈ (0, l) at the time instance t is defined by boundary problem, namely, by differential equation with partial derivatives of the 1st order: ut (x, t) = −a1 ux (x, t) − a0 u (x, t) + av (t) , (20) u (x, 0) = ψ0 (x) , u (0, t) = ψ1 (t) . Here l is the length of heat-exchanger, ψ0 (x) is initial temperature of liquid in the heatexchanger, ψ1 (t) is the temperature of liquid at the entry point of heatexchanger, a0 , a1 , a2 are given parameters of the process. The stem temperature ν (t) is the control parameter, the value ν (t) is to minimize the functional: ZT

Zl [u (x, t) − u∗ (x)]2 dx,

2

J (v) = α

v (t) dt + 0

(21)

0

and the following condition must fulfill: v ≤ v (t) ≤ v,

t ∈ [0, T ],

(22)

74


Here u∗ (x) is given function of desired temperature distribution in the heat-exchanger, α is given small positive value. The proposed method of the considered problem solving was applied to some test problems (20)-(22) with variation of problem data and the parameters of numerical calculations. When executing numerical experiments with a beforehand known solution of the problem (20)-(22), poor convergence to the solution or termination of the optimum search far away from the solution have been revealed. Analysis and further numerical experiments have shown that this fact is linked with the possibility of abrupt variation of ν (t) when trajectory transits from one phase parallelepiped to another if the values of piece-wise constant control actions assigned to these parallelepipeds are essentially different. To overcome this, the calculations were performed in two stages. At the first stage we used control function the value of which for a given fixed trajectory is equal to weighted linear combination of all values of control actions for all phase parallelepipeds. The weighted coefficients for the control νi1 ... iL were inversely proportional to the distance from the current trajectory to the i1 ...iL -th phase parallelepiped, i.e.: νsm (t) =

m X i1 =1

−

...

m X

λi1 ... iL νi1 ... iL ,

(23)

iL =1

ρ2 i1 ... iL 2σ 2

λi1 ... iL = e , here ρi1 ... iL is the distance from trajectory u(x,t) to the i 1 ,, i L -th phase parallelepiped. Control in the form (16) we will call ”smoothed control”, since its value changes smoothly when trajectory moves from one phase parallelepiped to another. The less is σ, the less is the influence of control assigned to the phase parallelepipeds that do not contain trajectory at the current time instance in aggregate control (23). That is why in the course of optimization search we can decrease σ, thus approaching to the problem in its source formulation. At the end of optimization we can use optimal functions νsm (t) determined at the previous steps as initial points and try to solve the problem with piece-wise constant ν (t). Numerical experiments were carried out for three problems with different terminal function u*(x). Common data for all three problems are: a 1 =1, a 0 =0.5, a=1.5, L=1, T =1. Regularization coefficient: α = 4.0 ∗ 10−4 . The points of observation: x 1 =0.1, x 2 =0.4, x 3 =0.6, x 4 =0.9. _ _ The points that determine the intervals of space partition by u are: u ˆ 0 = −99999 , u ˆ1 = _ _ 1.7 , u ˆ 2 = 2.2 , u ˆ 3 = 99999 . Implicit grid method was used for solving boundary value problem (20). The following problems I, II, III were solved. Problem I. Boundary and initial conditions: Ψ0 (x) = 1, Ψ1 (t) = 1. The terminal function u*(x) is the solution u (x,t) of (20)-(22) when t=T under some given (t). Then we ”forget” ν optimal (t), and start solving the problem as if we do not know the solution. If we will be able to find ν optimal (t), then it can testify for correct algorithm of the problem solving. The results of numerical experiments are shown in figures 1,2. The figures show the values of liquid temperature u(x,T) at the terminal time instance T and the values of function ν(t) when t changes from 0 till 1. Notation ”without smoothing, without regularization” denotes the solution obtained without application of (23) and when α = 0. ν optimal

Problem II.


75

The case of linear u*(x): u*(x)=1+2.5 x. Here again Ψ0 (x) = 1, Ψ1 (t) = 1. The results of the problem solving are given at the figures 3,4. In numerical experiments for the problem II we studied the influence of increase of the number of discretization along x, t on the control behavior and on trajectory approaching to the given terminal trajectory u*(x). Problem III. The case of constant u*(x): u*(x)= 2. Here Ψ0 (x) = 1, Ψ1 (t) = 1. The results of the problem solving are presented at the figures 5,6. Notation ”smoothing and regularization” means application of (23) and that the regularization coefficient α is not equal zero. In numerical experiments for this problem one of the best solutions is the solution obtained from initial point that was optimal for the problem with decreased number of discretization points along x and t (”The best Nu obtained from the best Nu when nx=nt=11”). 4. Conclusion Let us emphasize important advantage of the proposed in the paper synthesized zone control as compared to synthesized classical control of ϑ (u (x, t)) type. It is caused by technical complexity of on-line acquirement of information about current states of objects at the observable points and construction of control actions on the basis of this information. Whereas, the construction of zone control is performed during the time, when the states of object lie in the definite zones of phase space. This argues for robust features of zone control, proposed in the paper [4]. Also it is of practical interest to apply the proposed approach to constructing control actions from the results of observation of object current state values only at certain points of object. In many cases, this corresponds to actual capacities of measurement systems at objects. And, finally, it is clear, that one can easy extend the study of illustrative problems of plate heating and the problem of heat-exchanger control to a number of other problems of control of distributed objects, governed by functional equations of another type.

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Figure 1

Figure 2


Figure 3

Figure 4

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Figure 5

Figure 6


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References [1] Aida-zade, K.R. The problem of control in the mean with respect to regional control actions. Transactions of the National Academy of Sciences of Azerbaijan, 4, 2003. [2] Harmon Ray, W. , Advanced Process Control. McGraw-Hill, 1981. [3] Ivanova, A.P. Feedback control for stochastic heat equation. Journal of Computer and System Sciences International. Vol. 42, No. 5, 2003 pp.683-691. [4] Lions, I.L., Control optimal de systemes gouvernes par des equations aux derivees partielles . Dunod Gauthier - Villars, Paris, 1968. [5] Ovsyannikov, D.A. ,Mathematical methods of clusters’ control. Leningrad, Leningrad State University Press, 1980. [6] Polyak, B.T., Scherbakov P.S., Robust stability and control. Moscow, ”Nauka”, 2002. [7] Systems and control encyclopedia . Ed. M. G. Singh. v.1-8. Pergamon Press,1987. [8] The control handbook. Ed. W.S. Levine. CDC. Press. IEEE Press,1996.

K.R. Aida-zade, for a photograph and biography, see Apl. Compt. Math. (2002),vol.4., no.2., p.177. A.V. Handzel -Alexander V. Handzel - was born in 1961, in Baku. He graduated from Applied Mathematics department in Azerbaijan State University in 1983. He received Ph.D. degree of technical sciences on a speciality ”Application of computer facilities, mathematical modelling and mathematical methods in scientific investigations” in 1998 at Baku State University. Currently he is a researcher in North Caucasus State Technical University, Stavropol, Russia. His research interests include numerical methods, optimization, optimal control and distributed systems.