Many physical and biological systems involve time delays. A pure time delay has its output the...
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Many physical and biological systems involve time delays. A pure time delay has its output the same as its input but shifted in time. A more common type of delay is pooling delay. An example of such a feedback system is shown in Figure 5.3 Here the level of fluid in tank B determines the rate at which fluid enters tank A. Suppose this rate is given by R₁ (t) = a[V-V₂(1)], where a and V are positive constants and V₂() is the volume of fluid in tank B at time t. (a) If the outflow rate R3 from tank B is constant and the flow rate R₂ from tank A into B is R₂(t) = KV₁(1), where K is a positive constant and V₁ (1) is the vol- ume of fluid in tank A at time t, then show that this feedback system is governed by the system dV₁ dV₂ = a(V-V₂(t)) - KV₂ (t). = KV₁ (1)-R₂. Pump Tank A V₂(1) R₂ Tank B V₂(1) Power R3 Figure 5.3 Feedback system with pooling delay (b) Find a general solution for the system in part (a) when a = 5 (min)-¹, V = 20 L, K = 2 (min)-¹, and R3 = 10 L/min. (c) Using the general solution obtained in part (b), what can be said about the volume of fluid in each of the tanks as 1→→→ +∞? Many physical and biological systems involve time delays. A pure time delay has its output the same as its input but shifted in time. A more common type of delay is pooling delay. An example of such a feedback system is shown in Figure 5.3 Here the level of fluid in tank B determines the rate at which fluid enters tank A. Suppose this rate is given by R₁ (t) = a[V-V₂(1)], where a and V are positive constants and V₂() is the volume of fluid in tank B at time t. (a) If the outflow rate R3 from tank B is constant and the flow rate R₂ from tank A into B is R₂(t) = KV₁(1), where K is a positive constant and V₁ (1) is the vol- ume of fluid in tank A at time t, then show that this feedback system is governed by the system dV₁ dV₂ = a(V-V₂(t)) - KV₂ (t). = KV₁ (1)-R₂. Pump Tank A V₂(1) R₂ Tank B V₂(1) Power R3 Figure 5.3 Feedback system with pooling delay (b) Find a general solution for the system in part (a) when a = 5 (min)-¹, V = 20 L, K = 2 (min)-¹, and R3 = 10 L/min. (c) Using the general solution obtained in part (b), what can be said about the volume of fluid in each of the tanks as 1→→→ +∞?
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Related Book For
Principles of Information Systems
ISBN: 978-0324665284
9th edition
Authors: Ralph M. Stair, George W. Reynolds
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