Chapter 5 Dynamics and Regulation of Low-order Systems § 5.1 General Effects of Feedback § 5.2 Dynamics and Regulation of 1st-order System § 5.3 Dynamics and Regulation of 2nd-order System § 5.4 Time-domain Specifications of System Performance

§ 5.1 General Effects of Feedback (1) Closed-loop System: Negative Feedback Control:

§ 5.1 General Effects of Feedback (2) Proportional Regulator: Under Regulation Out of Regulation

§ 5.1 General Effects of Feedback (3) Proportional Controller: Mechanical and Hydraulic Controller Electronic Controller Governing of mechanical plant is liberated by using OP to realize electronic controller.

§ 5.1 General Effects of Feedback (4) Effectiveness of Feedback: (1) Set point regulation Error Response Following the first law of Newtonian Mechanics: At rest Zero error in position regulation At constant velocity Zero error in speed regulation

§ 5.1 General Effects of Feedback (5) Dynamic Response Poles of Closed-loop Transfer Function: The dynamic response is governed by new poles as D1(s)D2(s)+N1(s)N2(s)=0 For unity feedback system, feedback has no effect on system zeros.

§ 5.1 General Effects of Feedback (6) (2) Parameter Variations I/O transmission is dominated by sensor for high loop gain. System variations The effect of large variation is investigated by using robust analysis.

§ 5.1 General Effects of Feedback (7) Sensitivity analysis (small variation) Variation of G Loop gain The Sensitivity of I/O variation is reduced by employing high loop gain. Variation of H Feedback sensor directly and constantly affect the I/O transmission. In general, is higher than . High loop gain reduces the I/O variation due to system parameters.

§ 5.1 General Effects of Feedback (8) (3) Disturbance Rejection Disturbance Transmission

§ 5.1 General Effects of Feedback (9) Disturbance Error Note: steady state = set point - |offset error| - |disturbance error| |parameter uncertain error| Overall Effects of Feedback Advantages : Command following Disturbance rejection Improve system robustness Disadvantages : Reduce gain Increase system complexity Introduce instability

§ 5.2 Dynamics and Regulation of 1st-order Systems (1) Definition – A system that can store energy in only one form and location. Physical Examples: System pole-zero diagram t=0+ , ON Input signal Output signal Input signal Output signal t=0+ Open gate Input signal

§ 5.2 Dynamics and Regulation of 1st-order Systems (2) System and Input Model Differential Eq. Model: Transfer Function Model: Standard Form of Pure Dynamics

§ 5.2 Dynamics and Regulation of 1st-order Systems (3) System Dynamics: (1) Step response from differential equation

§ 5.2 Dynamics and Regulation of 1st-order Systems (4) Response of initial relaxed system

§ 5.2 Dynamics and Regulation of 1st-order Systems (5) (2) Step response from pole-zero diagram Steady state (Pure static gain) Transient (Pure dynamics)

§ 5.2 Dynamics and Regulation of 1st-order Systems (6) Overall response

§ 5.2 Dynamics and Regulation of 1st-order Systems (7) Closed-loop Regulation Ex: Liquid-level regulation

§ 5.2 Dynamics and Regulation of 1st-order Systems (8) Unity Feedback Regulator Set Point Regulation Equivalent I/O transmission Proportional regulator (Kp) changes the static amplification (K’) and response speed ( ).

§ 5.2 Dynamics and Regulation of 1st-order Systems (9) (1) Effect of Kp on Pure Dynamics Pole-zero diagram Response is faster when the proportional gain is increased.

§ 5.2 Dynamics and Regulation of 1st-order Systems (10) (2) Effect of Kp on Steady State Response Steady state error (offset) is reduced when the proportional gain is increased.

§ 5.2 Dynamics and Regulation of 1st-order Systems (11) Set Point and Response Regulation Response speed is a pure dynamic behavior, therefore the comparisons of response speeds are referred to the steady state responses at different Kp.

§ 5.2 Dynamics and Regulation of 1st-order Systems (12) Disturbance Rejection

§ 5.2 Dynamics and Regulation of 1st-order Systems (13) High Gain Regulation For infinite high gain Kp, the unity feedback regulator approaches ideal static system. Finite gain regulation will improve the speed of dynamic response but it has offset error in steady state.

§ 5.3 Dynamics and Regulation of 2nd-order Systems (1) Definition – A system having two separate energy-storage elements. Physical Examples: System pole-zero distribution

§ 5.3 Dynamics and Regulation of 2nd-order Systems (2) System and Input Model Differential Eq. Model: Transfer Function Model: Standard Form of Pure Dynamics

§ 5.3 Dynamics and Regulation of 2nd-order Systems (3) Step Response From Differential Equation For underdamping and initial relaxation system

§ 5.3 Dynamics and Regulation of 2nd-order Systems (4)

§ 5.3 Dynamics and Regulation of 2nd-order Systems (5) Step Response From Pole-zero distribution Pole-zero distribution Poles distribution Response: Steady state Transient

§ 5.3 Dynamics and Regulation of 2nd-order Systems (6) Overall response

§ 5.3 Dynamics and Regulation of 2nd-order Systems (7) Closed-loop Regulation Ex: DC Servomechanism Servo motor Motion and power transmission by reduction gear (Gear Ratio; N) Position sensor by potentiometer

§ 5.3 Dynamics and Regulation of 2nd-order Systems (8) Command response Disturbance response

§ 5.3 Dynamics and Regulation of 2nd-order Systems (9) Unity Feedback Regulator Set Point Regulation Equivalent I/O transmission Proportional regulator (Kp) changes the static amplification (K’) and dynamic response( ) through increasing the stiffness of a system.

§ 5.3 Dynamics and Regulation of 2nd-order Systems (10) (1) Regulation of Pure Dynamics Pole-zero distribution (2) Steady-state response The same result as that of 1st-order system with offset error

§ 5.3 Dynamics and Regulation of 2nd-order Systems (11) Disturbance Rejection

§ 5.4 Time-domain Specifications of System Performance (1) Step Testing of Black-Box System (2) Dynamic test (Within linear range) (1) Static test

§ 5.4 Time-domain Specifications of System Performance (2) Step Testing and Identification of Low-order Systems (1) Performance testing

§ 5.4 Time-domain Specifications of System Performance (3) (2) System Identification 1st-order system: 2nd-order system:

§ 5.4 Time-domain Specifications of System Performance (4) Effects of Additional Poles an Zeros Effect of real pole The effect of the real pole is to make the response more sluggish. Effect of real LHP zero The effect of the real zero is to make the response more oscillatory.

§ 5.4 Time-domain Specifications of System Performance (5) Effect of real RHP zero RHP zero: Nonminimum-phase zero The effect of nonminimum-phase zero is to cause initial reversal motion in step response.

§ 5.4 Time-domain Specifications of System Performance (6) Dynamic Model Simplification Order reduction: High order system Low order model Ignore real pole (zero) in oscillatory system modes: Pole-zero cancellation: Ignorance of far away poles and zeros: Poles and zeros only have s.s. effects