1.3. Digital Control A typical approach to computer control is based on the sample-and-hold technique, which can be described as follows. The values p(t)and (t)are measured only at discrete instants or sampling times The control law is updated by a program at each time t= ko on the basis of the sampled values (ko) and p(kd). The output of this program, a value Uk is then fed into the system as a control (held constant at that value) during the interval [ k6, k8 +d v4 a4+ Figure 1.3: Sampled control For simplicity we assume here that the computation of uk can be done quickly relative to the length o of the sampling intervals; otherwise, the model must be modified to account for the extra delay. To calculate the effect of applying the constant control u(t)≡ Uk if t∈[ko,k+6 (1.12) we solve the differential equation(1. 2)with this function u. By differentiation one can verify that the general solution is, for tE 6, ko+d p()=y(k60+5(0)+e-k6+(k6)-92(k6)+。-t+k6 vk,(1.13) (k6)+9(k6)+-k6-y(k6)- e-t+kd (1.14) hus, applying the constant control u gives rise to new values for p(k8 +8 and p(kd+d at the end of the interval via the formula (k6+6) p(k6+6) Buk,1.3. Digital Control 7 A typical approach to computer control is based on the sample-and-hold technique, which can be described as follows. The values ϕ(t) and ˙ϕ(t) are measured only at discrete instants or sampling times 0, δ, 2δ, 3δ,...,kδ,... The control law is updated by a program at each time t = kδ on the basis of the sampled values ϕ(kδ) and ˙ϕ(kδ). The output of this program, a value vk, is then fed into the system as a control (held constant at that value) during the interval [kδ,kδ + δ]. v v v v v 0 1 2 3 4 u(t) t u δ 2345 δδδδ Figure 1.3: Sampled control. For simplicity we assume here that the computation of vk can be done quickly relative to the length δ of the sampling intervals; otherwise, the model must be modified to account for the extra delay. To calculate the effect of applying the constant control u(t) ≡ vk if t ∈ [kδ,kδ + δ] (1.12) we solve the differential equation (1.2) with this function u. By differentiation one can verify that the general solution is, for t ∈ [kδ,kδ + δ], ϕ(t) = ϕ(kδ)+ ˙ϕ(kδ) + vk 2 et−kδ + ϕ(kδ) − ϕ˙(kδ) + vk 2 e−t+kδ − vk , (1.13) so ϕ˙(t) = ϕ(kδ)+ ˙ϕ(kδ) + vk 2 et−kδ − ϕ(kδ) − ϕ˙(kδ) + vk 2 e−t+kδ . (1.14) Thus, applying the constant control u gives rise to new values for ϕ(kδ +δ) and ϕ˙(kδ + δ) at the end of the interval via the formula  ϕ(kδ + δ) ϕ˙(kδ + δ)  = A  ϕ(kδ) ϕ˙(kδ)  + Bvk , (1.15)