Many industrial control systems have multiple "layered" control loops in what is known as a cascade configuration. For example, a drone's height controller requests a certain thrust, and the thrust controller requests a certain power output to the propeller actuator. A block diagram of such a configuration is shown in the figure below. R(s) E1(s) RU1(s) E₂(s) U₂(s) U₁(s) GC1 GC2 Gp2(S) GP1(S) Y(s) Nothing here is out of the ordinary, except that the output of Gc₁ is Ru₁(s), the set point for the inner loop, and the output from Gp2(s) is U₁(s), the input to process Gp₁(s). Since zero-offset action of inner loops is not something we typically need, the inner control loop is a P controller, while the outer loop is a PI controller. The transfer functions are: Gp1(S) 1 s + 1 Gp2(S) 1 10s + 1 Gci(s) = K₁ (1+) T GC2 = K2 Where K₁ and K₂ are the (selectable) gains for the controllers, and T = 1 is called the "integral gain" and is presented this way for the sake of easier algebra. It is our objective to design controllers for this process. Let's do it in steps. 2.1. Determine the inner closed-loop transfer function describing the relationship Simplify to obtain a transfer U₁(s) RU1(s) function of the form where N(s) and D(s) are polynomials of s. It will be a function of K₂. N(s) D(s)' 2.2. Using your result from above, determine the closed-loop transfer function . You do not need to simplify. R(s) Y(s) 2.3. We would like to analyze the stability of Y(s). Using the denominator of the transfer function, set up the R(S)* characteristic polynomial D(s) = 0 that can be used to form a Routh Array for this system. Collect like terms for each power of s. Your result should be a third-order polynomial that is a function of K₁, K₂, and T. 2.4. The first two rows of a slightly different (aka, not the answer to above but very close) characteristic polynomial are shown below. Using this (not your answer to [2.3]), determine terms b₂, bo, C₂, and co. ROW 3 ROW 2 ROW 1 ROW 0 10 (11 + K₂) b₂ C2 (2 + K₁ + K₁K₂) K₁K₂T bo Co 0 0 0 0 2.5. Assuming K₂ = 5 and K₁ > 0, use your results for b₂ and c₂ to explicitly write the conditions for parameter T that ensures closed-loop stability of this process.

Many industrial control systems have multiple "layered" control loops in what is known as a cascade configuration. For example, a drone's height controller requests a certain thrust, and the thrust controller requests a certain power output to the propeller actuator. A block diagram of such a configuration is shown in the figure below. R(s) E1(s) RU1(s) E₂(s) U₂(s) U₁(s) GC1 GC2 Gp2(S) GP1(S) Y(s) Nothing here is out of the ordinary, except that the output of Gc₁ is Ru₁(s), the set point for the inner loop, and the output from Gp2(s) is U₁(s), the input to process Gp₁(s). Since zero-offset action of inner loops is not something we typically need, the inner control loop is a P controller, while the outer loop is a PI controller. The transfer functions are: Gp1(S) 1 s + 1 Gp2(S) 1 10s + 1 Gci(s) = K₁ (1+) T GC2 = K2 Where K₁ and K₂ are the (selectable) gains for the controllers, and T = 1 is called the "integral gain" and is presented this way for the sake of easier algebra. It is our objective to design controllers for this process. Let's do it in steps. 2.1. Determine the inner closed-loop transfer function describing the relationship Simplify to obtain a transfer U₁(s) RU1(s) function of the form where N(s) and D(s) are polynomials of s. It will be a function of K₂. N(s) D(s)' 2.2. Using your result from above, determine the closed-loop transfer function . You do not need to simplify. R(s) Y(s) 2.3. We would like to analyze the stability of Y(s). Using the denominator of the transfer function, set up the R(S)* characteristic polynomial D(s) = 0 that can be used to form a Routh Array for this system. Collect like terms for each power of s. Your result should be a third-order polynomial that is a function of K₁, K₂, and T. 2.4. The first two rows of a slightly different (aka, not the answer to above but very close) characteristic polynomial are shown below. Using this (not your answer to [2.3]), determine terms b₂, bo, C₂, and co. ROW 3 ROW 2 ROW 1 ROW 0 10 (11 + K₂) b₂ C2 (2 + K₁ + K₁K₂) K₁K₂T bo Co 0 0 0 0 2.5. Assuming K₂ = 5 and K₁ > 0, use your results for b₂ and c₂ to explicitly write the conditions for parameter T that ensures closed-loop stability of this process.

Electric Motor Control

10th Edition

ISBN:9781133702818

Author:Herman

Publisher:Herman

Chapter40: Push-button Synchronizing

Section: Chapter Questions

Problem 5SQ

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