]> 3.1.1 Example 1: Nuclear reactor temperature control I

3.1.1 Example 1: Nuclear reactor temperature control I

Consider the nuclear reactor temperature control system depicted in Figure 3.1.


PICT

Figure 3.1: The nuclear reactor temperature control system

The dynamics equations of the system described in [36] are

T(t) + y(t) =p(t r) 1l(t) + f(t) =p(t) K1ɛ(t) + K20tɛ(τ)dτ =f(t) w y(t) =ɛ(t) ,t > 0 (3.1)

where K1, K2 are parameters, 1l denotes the Heaviside step function and r, T are fixed positive constants. If we assume that the system is asymptotically stable and until the moment of the appearance of a disturbance it remains in equilibrium, then for t < 0

ɛ = 0,f = K20ɛ(t)dt = w,p = w,y = w (3.2)

From (3.1) and (3.2) we get

ɛ̈(t) = 1 Tɛ̇(t) 1 Tδ(t r) K1 T ɛ̇(t r) K2 T ɛ(t r)

where δ denotes Dirac’s pseudofunction, together with the initial conditions

ɛ(θ) = 0,ɛ̇(θ) = 0forθ [r, 0] .

Hence, introducing the state variables z1(t) = ɛ(t + r), z2(t) = ɛ̇(t + r) and the notation

z20 = a0,a = 1 T,b = K1 T = 5K1,d = K2 T = 5K2

we obtain the final version of the dynamics equations

ż1(t) =z2(t) ż2(t) =az2(t) + bz2(t r) + dz1(t r) z1(θ) =0, r θ 0 z2(θ) =0, r θ < 0 z2(0) =z20 (3.3)

The problem is to determine a pair (b,d) minimizing the integral performance index

J =0ɛ2(t)dt =0ɛ2(t + r)dt =0z 12(t)dt(ɛ 0on[r, 0]) (3.4)

The system (3.3) is a special case of the neutral system

v̇(t) =A1v(t) + (A1A0 + A2)z(t r), t 0 v(t) =z(t) A0z(t r), t 0 v(0) =v0 z(θ) =φ(θ)for almost every θ [r, 0] (3.5)

where A1, A2, A0 L(n), r > 0, v0 n, φ is a function defined on (r, 0) with values in n. This can be seen by taking

n = 2,A1 = 01 0a ,A2 = 00 db ,A0 = 0 L(2),(v = z) , v0 = 0 z20 ,φ 0 .