]> 3.1.3 Example 3: Automatic control system with PID controller

3.1.3 Example 3: Automatic control system with PID controller

The dynamics of the system depicted in Figure 3.3


PICT

Figure 3.3: Automatic control system with PID controller

is governed by the equations

ɛ(t) = y(t r) + w T0(t) + qy(t) =K0 δ(t) + K1ɛ(t) + K2ɛ̇(t) + K30tɛ(τ)dτ ,t > 0

where δ denotes the Dirac pseudofunction and w, K0, K1, K3, T0, r > 0 and q are constants (usually q = 0 or q = 1).

If we assume that the system is asymptotically stable and until the appearance of the disturbance it remains in the equilibrium then for t < 0

ɛ = 0,y = w,0ɛ(t)dt = qw K0K3 .

Hence denoting

c = K0K2 T0 ,a = q T0,b = K0K1 T0 ,d = K0K3 T0 ,z0 = K0 T0 ,

and introducing the system variables

z1(t) = y(t) w,z2(t) = d0tɛ(τ)dτ + aw

we obtain the system of equations

d dt z1(t) cz1(t r) =az1(t) + z2(t) + bz1(t r) d dt z2(t) =dz1(t r) z1(θ) =0, r θ < 0 z1(0) =z0 z2(θ) =0, r θ < 0 (3.14)

The system (3.14) has the form (3.5) with

n = 2,A1 = a1 0 0 ,A2 = b0 d0 ,A0 = c 0 00 , v0 = z0 0 ,φ 0 .

The problem is to express explicitly the integral

J =0ɛ2(t)dt =0ɛ2(t + r)dt =0z 12(t)dt(ɛ 0 on [0,r)) (3.15)

by the parameters a, b, c, d.