Recomendados
Más contenido relacionado
Similar a 14th_Class_19-03-2024 Control systems.pptx
Similar a 14th_Class_19-03-2024 Control systems.pptx (20)
Último
Último (20)
14th_Class_19-03-2024 Control systems.pptx
- 1. Updated New Syllabus UNIT III CLASS NO: 7 DATE:19-03-2024
- 2. First Order System & its Response Fig 1 : Block diagram of a 1st order system Input Output Relationship: 𝑪 𝒔 𝑹(𝒔) = 𝟏 𝑻𝒔 + 𝟏 Objective: To analyse the system responses to such inputs as the unit-step unit-ramp unit-impulse functions The initial conditions are assumed to be zero.
- 3. Unit Step Response of First Order System Input Output Relationship: 𝑪 𝒔 𝑹(𝒔) = 𝟏 𝑻𝒔 + 𝟏 𝑪(𝒔) = 𝟏 𝑻𝒔 + 𝟏 𝑹(𝒔) Laplace Transform of Input S Here it is 𝓛 𝒖 𝒕 = 𝟏 𝒔 𝑪 𝒔 = 𝟏 𝑻𝒔 + 𝟏 𝟏 𝒔 𝑪 𝒔 = 𝟏 𝑻𝒔 + 𝟏 𝟏 𝒔 Using Partial Fraction Technique: 𝑪 𝒔 = 𝟏 𝒔 − 𝑻 𝑻𝒔 + 𝟏 = 𝟏 𝒔 − 𝟏 𝒔 + 𝟏 𝑻 𝓛−𝟏 𝑪(𝒔) 𝒄 𝒕 = 𝟏 − 𝒆− 𝒕 𝑻, for 𝒕 ≥ 𝟎 Time (𝒕) 𝒄(𝒕) 𝑢 𝑡 = 1, 𝑡 ≥ 0 𝑢 𝑡 = 0, 𝑡 < 0
- 4. Interesting Observations: • 𝒄 𝒕 𝒕=𝟎 = 𝒄 𝟎 = 𝟎 and 𝒄 𝒕 𝒕=∞ = 𝒄 ∞ = 𝟏 • The curve behaviour is exponential • At 𝒕 = 𝑻, 𝒄 𝑻 = 𝟏 − 𝒆−𝟏 = 𝟎. 𝟔𝟑𝟐 • The smaller the Time-Constant 𝑻 , the faster the system response. • The Slope of the Tangent Line at 𝒕 = 𝟎 is 𝟏 𝑻 since 𝒅𝒄(𝒕) 𝒅𝒕 𝒕=𝟎 = 𝟏 𝑻 𝒆− 𝒕 𝑻 𝒕=𝟎 = 𝟏 𝑻 ?
- 5. Unit-Ramp Response of First Order System Input Output Relationship: 𝑪 𝒔 𝑹(𝒔) = 𝟏 𝑻𝒔 + 𝟏 𝑪(𝒔) = 𝟏 𝑻𝒔 + 𝟏 𝑹(𝒔) Laplace Transform of Input Si Here it is 𝓛 𝒓 𝒕 = 𝟏 𝒔𝟐 𝑪 𝒔 = 𝟏 𝑻𝒔 + 𝟏 𝟏 𝒔𝟐 𝑪 𝒔 = 𝟏 𝑻𝒔 + 𝟏 𝟏 𝒔𝟐 Using Partial Fraction Technique: 𝑪 𝒔 = 𝟏 𝒔𝟐 − 𝑻 𝒔 + 𝑻𝟐 𝑻𝒔 + 𝟏 𝓛−𝟏 𝑪(𝒔) 𝒄 𝒕 = 𝒕 − 𝑻 + 𝑻𝒆− 𝒕 𝑻, for 𝒕 ≥ 𝟎 Time (𝒕) 𝒄(𝒕)
- 6. Interesting Observations: The error signal 𝒆 𝒕 𝒆 𝒕 = 𝒓 𝒕 − 𝒄 𝒕 = 𝑻 𝟏 − 𝒆− 𝒕 𝑻 𝒆 ∞ = 𝑻 𝟏 − 𝒆− ∞ 𝑻 = 𝑻 *** The error signal 𝒆 𝒕 approaches to 𝑻 at steady state
- 7. Unit-Impulse Response of First Order System Input Output Relationship: 𝑪 𝒔 𝑹(𝒔) = 𝟏 𝑻𝒔 + 𝟏 𝑪(𝒔) = 𝟏 𝑻𝒔 + 𝟏 𝑹(𝒔) Laplace Transform of Input Si Here it is 𝓛 𝜹 𝒕 = 𝟏 𝑪 𝒔 = 𝟏 𝑻𝒔 + 𝟏 𝑪 𝒔 = 𝟏 𝑻𝒔 + 𝟏 𝓛−𝟏 𝑪(𝒔) 𝒄 𝒕 = 𝟏 𝑻 𝒆− 𝒕 𝑻 for 𝒕 ≥ 𝟎 Time (𝒕) 𝒄(𝒕) 𝟏 𝑻 = 𝟐
- 8. An Important Property of Linear Time- Invariant (LTI) Systems 𝒄 𝒕 = 𝒕 − 𝑻 + 𝑻𝒆− 𝒕 𝑻, for 𝒕 ≥ 𝟎 𝒄 𝒕 = 𝟏 − 𝒆− 𝒕 𝑻, for 𝒕 ≥ 𝟎 𝒄 𝒕 = 𝟏 𝑻 𝒆− 𝒕 𝑻 for 𝒕 ≥ 𝟎 Unit-Ramp Response Unit-Step Response Unit-Impulse Response Interesting Observations: d (Unit-Ramp Response) / dt = Unit-Step Response d (Unit-Step Response ) / dt = Unit-Impulse Response The response to the integral of the original signal can be obtained by integrating the response of the system to the original signal and by determining the integration constant from the zero-output initial condition. This is a property of LTI systems. Linear time-varying (LTV) systems and nonlinear systems do not possess this Remark
- 9. Second Order System and Its Response Fig 2 : Servo system Simplified Block Diagram Plant
- 10. Relevant Equations related to Servo system ||Transfer Function of Servo system The equation for the load elements is 𝑱𝒄 + 𝑩𝒄 = 𝑻 • 𝑱 & 𝑩 are inertia and viscous- friction elements • 𝑻 denotes the Torque produced by the proportional controller whose gain is 𝑲 Taking Laplace Transform of both side and considering initial conditions to be zero: 𝑱𝒔𝟐𝑪(𝒔) + 𝑩𝒔𝑪(𝒔) = 𝑻(𝒔) 𝑪 𝒔 𝑻(𝒔) = 𝟏 𝒔(𝑱𝒔 + 𝑩) 𝑪 𝒔 𝑹(𝒔) = 𝑲 𝑱𝒔𝟐 + 𝑩𝒔 + 𝑲 = 𝑲 𝑱 𝒔𝟐 + 𝑩 𝑱 𝒔 + 𝑲 𝑱 2nd Order system Closed-Loop Transfer Function ∃ 2 Closed-Loop Poles
- 11. Step Response of Second Order System 𝑪 𝒔 𝑹(𝒔) = 𝑲 𝑱𝒔𝟐 + 𝑩𝒔 + 𝑲 = 𝑲 𝑱 𝒔𝟐 + 𝑩 𝑱 𝒔 + 𝑲 𝑱 Closed-Loop TF = 𝑲 𝑱 𝒔 + 𝑩 𝟐𝑱 + 𝑩 𝟐𝑱 𝟐 − 𝑲 𝑱 𝒔 + 𝑩 𝟐𝑱 − 𝑩 𝟐𝑱 𝟐 − 𝑲 𝑱 Closed-Loop Poles will be COMPLEX CONJUGATES if 𝑩𝟐 − 𝟒𝑱𝑲 Closed-Loop Poles will be REAL if 𝑩𝟐 − 𝟒𝑱𝑲 ≥ 𝟎 Remark: For Transient Response analysis, it is convenient to write 𝑲 𝑱 = 𝒘𝒏 𝟐 𝑩 𝑱 = 𝟐𝜻𝒘𝒏 = 𝟐𝝈 Key Terms 𝝈 : Attenuation 𝒘𝒏 : Undamped Natural Frequency 𝜻 : Damping Ratio 𝑩 : Actual Damping 𝑩𝒄 : Critical Ratio 𝜻 = 𝑩 𝑩𝒄 = 𝑩 𝟐 𝑱𝑲
- 12. 𝑮 𝒔 = 𝒘𝒏 𝟐 𝒔(𝒔 + 𝟐𝜻𝒘𝒏) 𝐶(𝑠) 𝑅(𝑠) Fig 3 : Standard Second-order feedback system Open Loop Transfer Function 𝐸 𝑠 = 𝑅 𝑠 − 𝐶(𝑠) 𝑪 𝒔 𝑹(𝒔) = 𝒘𝒏 𝟐 𝒔𝟐 + 𝟐𝜻𝒘𝒏 + 𝒘𝒏 𝟐 3 Important Cases To Study Case 1: Underdamped || 𝟎 < 𝜻 < 𝟏 Case 2: Critically Damped || 𝜻 = 𝟏 Case 3: Overdamped || 𝜻 > 𝟏 𝑪 𝒔 𝑹(𝒔) = 𝒘𝒏 𝟐 (𝒔 + 𝜻𝒘𝒏 + 𝒋𝒘𝒅)(𝒔 + 𝜻𝒘𝒏 − 𝒋𝒘𝒅) Where, 𝑤𝑑 = 𝑤𝑛 1 − 𝜁2 𝑪(𝒔) = 𝒘𝒏 𝟐 𝒔 + 𝜻𝒘𝒏 + 𝒋𝒘𝒅 𝒔 + 𝜻𝒘𝒏 − 𝒋𝒘𝒅 𝟏 𝐬 𝑹 𝒔 = 𝟏 𝒔 Unit Step Function 𝑪 𝒔 𝑹(𝒔) = 𝒘𝒏 𝟐 𝒔 + 𝒘𝒏 𝟐 Where, 𝑤𝑑 = 𝑤𝑛 1 − 𝜁2 = 0 𝑪(𝒔) = 𝒘𝒏 𝟐 𝒔 + 𝒘𝒏 𝟐 𝟏 𝐬 𝑹 𝒔 = 𝟏 𝒔 Unit Step Function 𝑪 𝒔 𝑹(𝒔) = 𝒘𝒏 𝟐 (𝒔 + 𝜻𝒘𝒏 + 𝒘𝒏 𝜻𝟐 − 𝟏)(𝒔 + 𝜻𝒘𝒏 − 𝒘𝒏 𝜻𝟐 − 𝟏) 𝑹 𝒔 = 𝟏 𝒔 Unit Step Function 𝑪(𝒔) = 𝒘𝒏 𝟐 𝒔(𝒔 + 𝜻𝒘𝒏 + 𝒘𝒏 𝜻𝟐 − 𝟏)(𝒔 + 𝜻𝒘𝒏 − 𝒘𝒏 𝜻𝟐 − 𝟏)
- 13. ) STABILITY OF A MASS-SPRING-DAMPER System
- 14. B) STABILITY OF A MASS-SPRING System
- 15. Expression of 𝒄 𝒕 in all the cases Case 1: Underdamped || 𝟎 < 𝜻 < 𝟏 𝑪(𝒔) = 𝒘𝒏 𝟐 𝒔 + 𝜻𝒘𝒏 + 𝒋𝒘𝒅 𝒔 + 𝜻𝒘𝒏 − 𝒋𝒘𝒅 𝟏 𝐬 Using Partial Fraction 𝑪 𝒔 = 𝟏 𝒔 − 𝒔 + 𝟐𝜻𝒘𝒏 𝒔𝟐 + 𝟐𝜻𝒘𝒏𝒔 + 𝒘𝒏 𝟐 = 𝟏 𝐬 − 𝐬 + 𝜻𝒘𝒏 𝒔 + 𝜻𝒘𝒏 𝟐 + 𝒘𝒅 𝟐 − 𝜻𝒘𝒏 𝒔 + 𝜻𝒘𝒏 𝟐 + 𝒘𝒅 𝟐
- 16. Important Observations of Case 2 Case 2: Critically Damped || 𝜻 = 𝟏 𝑪(𝒔) = 𝒘𝒏 𝟐 𝒔 + 𝒘𝒏 𝟐 𝟏 𝐬 Case 3: Overdamped || 𝜻 > 𝟏 𝑪(𝒔) = 𝒘𝒏 𝟐 𝒔(𝒔 + 𝜻𝒘𝒏 + 𝒘𝒏 𝜻𝟐 − 𝟏)(𝒔 + 𝜻𝒘𝒏 − 𝒘𝒏 𝜻𝟐 − 𝟏)
- 17. Fig.: Unit-step response curves of the system A family of unit-step response curves 𝒄(𝒕) with various values of 𝜻 w.r.t. 𝒘𝒏𝒕
- 18. Second Order System and Its Transient Response Specifications Salient Features: Frequently, the performance characteristics of a control system are specified in terms of the transient response to a unit-step input, since it is easy to generate and is sufficiently drastic. The transient response of a system to a unit-step input depends on the initial conditions. For convenience in comparing transient responses of various systems, it is a common practice to use the standard initial condition that the system is at rest initially with the output and all-time derivatives thereof zero. Then the response characteristics of many systems can be easily compared. The transient response of a practical control system often exhibits damped oscillations before reaching steady state. In specifying the transient-response characteristics of a control system to a unit-step input, it is common to specify the following: 1) Delay Time 𝒕𝒅 2) Rise Time (𝒕𝒓) 3) Peak Time 𝒕𝒑 4) Maximum Overshoot 𝑴𝒑 5) Settling Time (𝒕𝒔)
- 19. Definitions: 1) Delay Time 𝒕𝒅 The delay time is the time required for the response to reach half the final value the very first time. 𝑐 𝑡𝑑 = 1 2 𝑐 ∞ = 1 2 1 2 = 1 − 𝑒−𝜁𝑤𝑛𝑡𝑑 1 − 𝜁2 sin 𝑤𝑑𝑡𝑑 + tan−1 1 − 𝜁2 𝜁 Example: For underdamped second-order system:
- 20. 2) Rise Time 𝒕𝒓 • The rise time is the time required for the response to rise from 10% to 90%, 5% to 95%, or 0% to 100% of its final value. • For underdamped second-order systems, the 0% to 100% rise time is normally used. • For overdamped systems, the 10% to 90% rise time is commonly used. 𝑐 𝑡𝑟 = 100% 𝑐 ∞ = 1 1 = 1 − 𝑒−𝜁𝑤𝑛𝑡𝑟 1 − 𝜁2 sin 𝑤𝑑𝑡𝑟 + tan−1 1 − 𝜁2 𝜁 Example: For underdamped second-order system: Where; 𝑤𝑑 = 𝑤𝑛 1 − 𝜁2, 𝜁𝑤𝑛 = 𝜎 And 𝛽 is shown here
- 21. 3) Peak Time 𝒕𝒑 The peak time is the time required for the response to reach the first peak of the overshoot. Example: For underdamped second- order system:
- 22. 4) Maximum % Overshoot 𝑴𝒑 The maximum overshoot is the maximum peak value of the response curve measured from unity. If the final steady-state value of the response differs from unity, then it is common to use the maximum percent overshoot. Example: For underdamped second- order system: (A)
- 23. 4) Settling Time (𝒕𝒔) The settling time is the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%). • The settling time is related to the largest time constant of the control system. Which percentage error criterion to use may be determined from the objectives of the system design in question. Example: For underdamped second- order system:
- 24. The relative dominance of closed-loop poles is determined by the ratio of the real parts of the closed-loop poles, as well as by the relative magnitudes of the residues evaluated at the closed-loop poles. The magnitudes of the residues depend on both the closed-loop poles and zeros. If the ratios of the real parts of the closed-loop poles exceed 5 and there are no zeros nearby, then the closed-loop poles nearest the 𝒋𝒘 axis will dominate in the transient-response behaviour because these poles correspond to transient-response terms that decay slowly. Those closed-loop poles that have dominant effects on the transient-response behaviour are called dominant closed-loop poles. Quite often the dominant closed-loop poles occur in the form of a complex- conjugate pair. The dominant closed-loop poles are most important among all closed-loop poles. Note that the gain of a higher-order system is often adjusted so that there will exist a pair of dominant complex-conjugate closed-loop poles. The presence of such poles in a stable system reduces the effects of such nonlinearities as dead zone, DOMINANT CLOSED LOOP POLES