### Control System Analysis#### Block Diagram of Control SystemThe image depicts a control system with a feedback loop. The main components include:- **Summer (Σ):** Combines inputs \( R(s) \) and feedback to produce error \( E(s) = R(s) - Y(s) \).- **Gain \( K \):** Amplifies the error signal.- **Plant Transfer Function \( G_p(s) = \frac{100}{1 + 0.2s} \):** Represents the system dynamic response.- **Integrator \( \frac{1}{20s} \):** Part of the feedback loop affecting system output \( Y(s) \).- **Feedback Component \( K_t \):** Influences the feedback path.#### Problem Statement(a) **Error Constants:**Determine the step, ramp, and parabolic-error constants (\( K_p, K_v, \) and \( K_a \)). The error signal is defined as \( e(t) \).(b) **Steady-State Errors:**Calculate steady-state errors in terms of \( K \) and \( K_t \) for the cases below. Assume system stability.- (b.1) For input \( r(t) = u_s(t) \).- (b.2) For input \( r(t) = t \cdot u_s(t) \).#### Explanation- **Error Signal \( e(t) \):** The difference between the reference input and the output.- **\( K_p, K_v, K_a \):** Constants related to system steady-state errors for different input types: - \( K_p \) for step inputs. - \( K_v \) for ramp inputs. - \( K_a \) for parabolic inputs. Understanding these concepts and analyzing this control system is essential for designing systems with minimal steady-state error and desired dynamic response.

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The block diagram of a control system is shown in the figure. (a) Find the step-, ramp-, and parabolic- error constants (K₂, K₂, and Ka). The error signal is defined as e(t). (b) Find the steady-state errors in terms of K and K, when the following inputs are applied. Assume that the system is stable. (b.1) r(t) = us(t) (b.2) r(t) = tus(t) R(s) + E(s) K Com 10 G₂(s)=1 +0.2s 100 K₁ 20s Y(s)

The block diagram of a control system is shown in the figure. (a) Find the step-, ramp-, and parabolic- error constants (K₂, K₂, and Ka). The error signal is defined as e(t). (b) Find the steady-state errors in terms of K and K, when the following inputs are applied. Assume that the system is stable. (b.1) r(t) = us(t) (b.2) r(t) = tus(t) R(s) + E(s) K Com 10 G₂(s)=1 +0.2s 100 K₁ 20s Y(s)

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Question

### Control System Analysis

#### Block Diagram of Control System

The image depicts a control system with a feedback loop. The main components include:

- **Summer (Σ):** Combines inputs \( R(s) \) and feedback to produce error \( E(s) = R(s) - Y(s) \).
- **Gain \( K \):** Amplifies the error signal.
- **Plant Transfer Function \( G_p(s) = \frac{100}{1 + 0.2s} \):** Represents the system dynamic response.
- **Integrator \( \frac{1}{20s} \):** Part of the feedback loop affecting system output \( Y(s) \).
- **Feedback Component \( K_t \):** Influences the feedback path.

#### Problem Statement

(a) **Error Constants:**
Determine the step, ramp, and parabolic-error constants (\( K_p, K_v, \) and \( K_a \)). The error signal is defined as \( e(t) \).

(b) **Steady-State Errors:**
Calculate steady-state errors in terms of \( K \) and \( K_t \) for the cases below. Assume system stability.

- (b.1) For input \( r(t) = u_s(t) \).
- (b.2) For input \( r(t) = t \cdot u_s(t) \).

#### Explanation

- **Error Signal \( e(t) \):** The difference between the reference input and the output.
- **\( K_p, K_v, K_a \):** Constants related to system steady-state errors for different input types:
- \( K_p \) for step inputs.
- \( K_v \) for ramp inputs.
- \( K_a \) for parabolic inputs.

Understanding these concepts and analyzing this control system is essential for designing systems with minimal steady-state error and desired dynamic response.

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