Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

Automatic Control Theory Excecies(12) 3 — 22, 23, 24, 25 3 — 26 （选做）

Review §3.5.1 The Concept of Stability §3.5.2 The Necessary and Sufficient Condition for System Stability §3.5.3 Stability Criterion （ 1 ） A necessary condition （ 2 ） Routh Criterion （ 3 ） Special cases of Routh’s array （ 4 ） The application of the Routh’s criterion Characteristic roots of the closed-loop transfer function have negative real parts; In other words, all the poles of the closed- loop transfer function lie in the left half s-plane

Automatic Control Theory （ Lecture 12 ） §3 Time-Domain Analysis and Adjustments of Linear Systems §3.1 Introduction §3.2 First-order System §3.3 Second-order System §3.4 Higher-order System §3.5 Stability Analysis of Linear Systems §3.6 The Steady-State Error of Linear Systems §3.7 Time-Domain Compensation

Automatic Control Theory §3.6 The Steady-State Error of Linear Systems （ Lecture 12 ）

§3.6 The Steady-State Error of Linear Systems (1) The steady-state error of control systems is a steady-state specification reflecting the system accuracy. In this section, we only discuss the system’s theoretical error, and neglect the error caused by the nonlinear factors. We obtain the steady-state error only for stable systems, since an unstable closed-loop system is generally of no practice value. A system is called non-error system if the steady-state error of its step-response is theoretically zero. Otherwise, it is an error system Introduction

§3.6 The Steady-State Error of Linear Systems (2) §3.6.1 Error and Steady-State Error §3.6.2 The General Approach to Obtain Steady State Errors （ 1 ） Determine the stability The system error with respect to the input. The system error with respect to the output Steady-state error Dynamic error: steady-state component in error Final value error : （ 2 ） Obtain the transfer function from the input or the disturbance to the error signal （ 3 ） Using the final value theorem to obtain the steady-state error

§3.6.2 The Formal Approach of Obtaining Steady State Errors (1) Example １ Consider the system shown in figure. Obtain the steady-state error when r(t) = n(t) = t. Solution ． depends on the structure and the parameters of the system depends on the input

§3.6.2 The Formal Approach of Obtaining Steady State Errors (2) Example 2 Consider the system shown in figure. Obtain the steady-state error when r(t) is A·1(t), At, At2/2 Solution ． depends on the structure and the parameters of the system e ss depends on the type of input （ step function, ramp function or parabolic function ） depends on the kind of input （ control input ， disturbance input and the input position ）

§3.6.3 Static Error Constant Method （ 1 ） Static Error Constant Method——The calculating rules of e ss when input is r(t) depends on the structure and the parameters (K, v) of the system depends on the type of the input

§3.6.3 Static Error Constant Method （ 2 ） The static position error constant The static velocity error constant The static acceleration error constant

§3.6.3 Static Error Constant Method （ 3 ）

§3.6.3 Static Error Constant Method （ 4 ） Example 3 Consider the system shown in figure. Obtain the steady-state error for the input signal Solution ．

§3.6.3 Static Error Constant Method （ 5 ）

§3.6.3 Static Error Constant Method （ 6 ） Example 4 Consider the system shown in figure. Determine the so that the steady-state error is zero when the input signal. Solution ． The Feedforward compensation (Composite compensation) can improve the steady state accuracy of control systems.

§3.6.3 Static Error Constant Method （ 6 ） Example 5 Consider the system shown in figure. Obtain the steady-state error when Solution. Increment of the gain or addition of integral factors between the summing points of the feedback and the disturbance can reduce or eliminate the steady state error due to the input or the disturbance.

Review §3.6.1 Error and Steady-State Error Definition of error: (1) The system error with respect to the input ； (2) The system error with respect to the output Steady-State Error ： (1) Static Error ； (2) Dynamic Error §3.6.2 The Formal Approach of Obtaining Steady State Errors (1) System’s Stability (2) Error Transfer Function (3) Solving Steady-State Error by Final Value Theorem §3.6.3 Static Error Constant Method (1) Static Error Constant Method : Kp, Kv, Ka (2)The Method to determine error (3) Stipulation §3.6.4 Steady-state Error Caused by Disturbances 1 ） System is stable 2 ） The system error with respect to the input 3 ） Input is r(t), and r(t) doesn't have other feedback path

Automatic Control Theory Excecies(12) 3 — 22, 23, 24, 25 3 — 26 （选做）