Consider the level-control problem of the tanks in series system of Example 13.1. Sup- pose that,...

   

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Consider the level-control problem of the tanks in series system of Example 13.1. Sup- pose that, in addition to the level sensor for the second tank that measures the con- trolled variable, we also have a level sensor installed for the first tank, and we want to use this additional measurement to build a cascade control system for the tanks. The model equations for the tanks are given by A dh dt A = dh, dt h +k,u+ F R = h h R R and the parameter values are A = R = A = R = k = 1. The primary output of the system (controlled variable) is h, whereas h is a secondary output, and we assume that they are both measured with very fast level sensors (negligible time constants). The main disturbance of the system is the flowrate F, which can vary in size in an unpredictable manner. (a) Draw the block diagram of a cascade control system, where the inner (secondary) loop regulates h and the outer (primary) loop regulates h. (b) Build a model-based PI controller for the inner loop and a model-based PID con- troller with filter for the outer loop. (c) Calculate and plot the closed-loop response of the controlled output h, in deviation form, when F, undergoes a unit step increase. Use = 0.1 as the filter parameter for both controllers. (d) If single-loop feedback control is applied to the system, with the same PID control- ler as in the previous question, what will be the closed-loop response? In a common graph, compare it with the closed-loop response of the cascade control system of the previous question (c). Repeat the previous problem for the case where, instead of h, the primary output is the outlet flowrate F = h/R, and it is measured downstream with dead time 0= 2. Consider the level-control problem of the tanks in series system of Example 13.1. Sup- pose that, in addition to the level sensor for the second tank that measures the con- trolled variable, we also have a level sensor installed for the first tank, and we want to use this additional measurement to build a cascade control system for the tanks. The model equations for the tanks are given by A dh dt A = dh, dt h +k,u+ F R = h h R R and the parameter values are A = R = A = R = k = 1. The primary output of the system (controlled variable) is h, whereas h is a secondary output, and we assume that they are both measured with very fast level sensors (negligible time constants). The main disturbance of the system is the flowrate F, which can vary in size in an unpredictable manner. (a) Draw the block diagram of a cascade control system, where the inner (secondary) loop regulates h and the outer (primary) loop regulates h. (b) Build a model-based PI controller for the inner loop and a model-based PID con- troller with filter for the outer loop. (c) Calculate and plot the closed-loop response of the controlled output h, in deviation form, when F, undergoes a unit step increase. Use = 0.1 as the filter parameter for both controllers. (d) If single-loop feedback control is applied to the system, with the same PID control- ler as in the previous question, what will be the closed-loop response? In a common graph, compare it with the closed-loop response of the cascade control system of the previous question (c). Repeat the previous problem for the case where, instead of h, the primary output is the outlet flowrate F = h/R, and it is measured downstream with dead time 0= 2.