1 / 25

Chapter 3 Time-Domain Analysis of the linear systems

Chapter 3 Time-Domain Analysis of the linear systems. Introduction Stability and Algebraic criteria Analysis of stable error First-order system analysis Second-order system analysis High-order system analysis. Assume:. Zero initial condition; Typical input signal. 3.1 Introduction.

chaz
Download Presentation

Chapter 3 Time-Domain Analysis of the linear systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 3Time-Domain Analysis of the linear systems • Introduction • Stability and Algebraic criteria • Analysis of stable error • First-order system analysis • Second-order system analysis • High-order system analysis

  2. Assume: Zero initial condition; Typical input signal. 3.1 Introduction 3.1.1 Basic and macroscopically requirements to design a control system. 3.1.2 Basic assumption conditions for analyzing a linear control system The response of a system could be : zero-state zero-input response response transient steady-state portion portion 1) The system must be stable (stability) —First requirement. 2) The control should be accurate (accuracy). 3) The response should be quick-acting (rapidity).

  3. 3.1 Introduction 3.1.3 Typical test input signal 1. why to research the test input signal? • The actual input signal of the system is multifarious, normally a standard test input signal should be chosen→to analyze the system performance. • Allow the designer to compare several designs project. • Many input signals of the control systems are similar to the test signals. 2.Which types of the test input signal ? • Step input signal; • Ramp input signal; • Parabolic input signal; • Pulse input signal; • Sinusoidal input signal;

  4. r(t) A 0 t Such as: Such as: switch turn on; relay close train uniformity speed mo-tion; elevator uniformity speed raise. 3.1 Introduction 3. Typical test input signal 2) Ramp input signal 1) Step input signal R(s)=A/s A=1 — unity step input A=1— unity ramp function Application : test signal of the constant-input systems Application : some input-tracking systems

  5. 0 ε Unity impulse. Application: some impulse experiments such as: Servo-control system 3.1 Introduction 4) Pulse input signal 3) Parabolic input signal A = 1— unity parabolic signal Application : same as the Ramp signal Such as: Impactive disturbance

  6. Application: Frequency domain analysis of control systems such as:Communication system,Radar system … 3.1 Introduction 5) Sinusoidal signal

  7. 3.1 Introduction 3.1.4 Relationship between impulse response and other responses Theorem: 1. For the typical input signals proof: as above :

  8. 3.1 Introduction If: g(t) →impulse response of a linear system. g(t) = L-1[ G(s)] h(t) → step response of a linear system. ct(t) → ramp response of a linear system. ctt(t) →parabolic response of a linear system 2. For any signal r(t) we have the Convolution theorem : then: example

  9. A typical unity step response of a control system is shown in following figure 3.1 Introduction 3.1.5The transient performance specifications of a control system • For different system, the research aim is different. For example, the tracking servo control systems are different from the constant-regulating systems. • performance measure — in terms of the step responses of the systems.

  10. Performance specification definition: A Overshoot σ% = 100% B Rise Time tr A Setting time ts B Peak time tp

  11. (for theunder dampedsystems) (for the over damped systems) (for the under damped systems) 3.1 Introduction 1)Rise time tr 2)Peak time tp :

  12. (For the under damped systems) 3.1 Introduction 3)Percent overshoot : 4)Setting time ts : 5)Delay time td : σ% → smoothness of the response. tr、tp、td → rapidity of the response. ts → rapidity of the transient process of the response.

  13. Definition A system work at some original states and is effected by a disturbance (noise) . When the disturbance go off , the system can come back to the original states —— the system is stable. Or else the system is unstable. unstable system is of no practical value) assume: r(t)= δ(t) → R(s)=1 3.2 Stability analysis of the linear systems(Stability: the most important performance for a control system) 3.2.1 What is the Stability of a system ? 3.2.2 The sufficient and necessary conditions of the sta- bility for a linear system.

  14. Conclusion: The sufficient and necessary conditions of the stability for a linear system is: All poles of φ(s) are the poles with negative real part. or : All poles of φ(s) lie in the left-half of the s-plane . 3.2 Stability analysis of the linear systems

  15. Im S-plane Re The sufficient and necessary conditions of the stability for a linear system Graphic representation: Stable region Unstable region The relationship between the system’s stability and the position of poles in S-plane.

  16. R(s) C(s) G(s) - H(s) Assume the polynomial: 3.2 Stability analysis of the linear systems 3.2.3 Routh Criterion Then: for a control system : Characteristic equation of the system: 1+G(s)H (s)=0 Assume :

  17. 3.2 Stability analysis of the linear systems We have: we need: all roots of the characteristic equation lie in the left-half of s-plane for a stable system. For the equation : The question is: how could we know the roots all lie in the left-half of s-plane? Routh do it like this:

  18. Tabulate the → Routh table (array): 3.2 Stability analysis of the linear systems

  19. 3.2 Stability analysis of the linear systems 1) All elements of the first column of the Routh-table(array) are positive . — The system is and must is stable. conclusion(Routh Criterion): Necessary, sufficient 2) The number of roots with positive real parts is equal to the number of changes in sign of the first column of Routh-table .

  20. 3.2 Stability analysis of the linear systems Example 3.2.2 Example 3.2.1 Unstable . 2 roots With positive Real parts Stable

  21. transfer function: Estimate the stability of the system . 3.2 Stability analysis of the linear systems Example 3.2.3 A unity feedback system, Open-loop solution: 0<k<8, the system is stable

  22. 3.2 Stability analysis of the linear systems some or other element is equal to zero in the Routh-table. Example 3.2.4 s4 1 1 1 s3 2 2 s2 0 1 s1 ε>0 s0 1 Unstable. 2 roots with positive real parts. Note: Can use a infinitesimalε>0substituting the zero element in the first column.

  23. Example 3.2.5 There are all zero elements in some row of Routh-table. Make auxiliary polynomial : 0 0 multiplied by 1/2 no effect result Unstable,no root in the right-half of s-plane. but there are two pair of roots in the imaginary axis of s-plane:

  24. — unstable, be short of the item 3.2 Stability analysis of the linear systems The order of the auxiliary polynomial is always even indicating the number of symmetrical root pairs . Note: Solving the auxiliary equation, we have: Two inferences about Routh-criterion 1) The characteristic equation is short of one or more than one items —The system must be unstable . Example : 2) The coefficient of the characteristic equation are different in sign. The system must be unstable. Example : — unstable, The coefficient are different in sign.

  25. 3.3 steady-stable error Analysis of the linear system Connection to next part

More Related