Chapter 3: Dynamic Response

1. Chapter 3: Dynamic Response Part B: Dynamic Response versus the Pole and Zero Locations

2. B1. Introduction • The most important characteristic of the dynamic response is absolute stability, that is, whether the system is stable or unstable. • A system is: • stable if its transient response decays & • unstable if it does not decay.

3. Stable and Unstable Systems Unstable t = 0 t → ∞ Stable

4. Dynamic Response of a Stable System • It consists in two parts: the transient response & the steady-state response (also called final state or final value) • By transient response, we mean that which goes from the initial state to the final state. • By steady-state response,we mean the manner in which the system output behaves as t approaches infinity.

5. Initial value can be determined from the Initial Value Theorem Final value can be determined from the Final Value Theorem Dynamic Response Analysis Transient response Initial state Unstable If Final state exits Stable It can be analyzed using the poles and zeros of the output in the s-domain

6. Pole and Zero Definitions • In an input-output system from x(t) to y(t) whose transform function has the form: • The poles are the values of s for which a(s) = 0, and the zerosare the values of s for which b(s) = 0.

7. Why do we need to solve for the poles since we know how to solve in the t-domain? Outlines • B1: Introduction • B2: Effects of pole locations • B3: Transient response specifications • B4: Effects of zero and additional poles

8. Transient Response Analysis There are two ways to approach solving the dynamic model: • The linear analysis techniques, based on the pole and zero values of Y(s),which provide aqualitative characteristic behavior of the response y(t). • The numerical techniques, in order to solve the system equation, which provide a precise picture of the system. The linear analysis techniques allow a quick analysis, which provide a insight into why the solutions has certain features and how the system might be changed to modify the response in the desired direction

9. What is the significance of a pole? Outlines • B1: Introduction • B2: Effects of pole locations • B3: Transient response specifications • B4: Effects of zero and additional poles Why does the pole location matter? How do we define the pole location from its value? Where do additional poles come from?

10. Connection between the t- and s-domains • Consider an input-output system from x(t) from y(t) whose the dynamic model has the form: • The transfer function is defined by: → Characteristic Equation Poles of G(s) → Roots of the Characteristic Eq

11. Connection between the t- and s-domains • Consider an input system from x(t) from y(t) whose the dynamic model has the form: • The transfer function is defined by: Characteristic Eq Input → Additional poles

12. Connection between the t- and s-domains • Consider an input system from x(t) from y(t) whose the dynamic model has the form: • The transfer function is defined by: Characteristic Eq Natural response is given by an impulse input

13. Significance of poles Each of the pole values can be identified with a particular type of response.For instance: • A simple real polep = 2for Y(s) [i.e.Y(s)=1/(s - 2) + …] produces a growingy(t) = C e2t +…[see HW#4] • A simple pair of imaginary polesp = ± 4jproduces asinusoidal response y(t) = K1 cos(4t) + K2 sin (4t) [see midterm] • A simple pair of complex polesp = - ζω0 ± jω0 (1 - ζ2)1/2 [that is p= - σ ± jωdwithσ = ζω0andωd = ω0(1 - ζ2)1/2]produces adecayingresponse of the form: [see Appendix A]

14. Significance of poles (cont’d) The nature and value of the poles determine whether the system is stable or instable, and the type of response The nature and value of any pole is classified as a function of its location in the plan defined by: the Real part and Imaginary part of the pole [or in other words the s-domain] Im(s) Re(s)

15. Representation of a pole piin s-domain. Consider a linear system having n/2 poles: In rectangular coordinates, the poles are at(-σ, ±ωd)

16. Representation of , d ,  and 0in s-domain. In rectangular coordinates, the poles are at(-σ, ±ωd) In polar coordinates, the poles are at (ω0, sin-1ζ)

17. Outlines • B1: Introduction • B2: Effects of pole locations • B3: Transient response specifications & application to problem • B4: Effects of zero and additional poles What are the transient response specifications? Where does a zero come from?

18. Transient-Response Specifications Actual transient response specifications are well defined features characterizing: • The amplitudes and frequencies of the oscillations, • The time response of the system to reach its steady-state, • The slope at origin • etc… Or in other words, The performance of the system

19. Significance of a zero • Consider an input system from x(t) from y(t) whose the dynamic model has the form: • The transfer function is defined by: Input defined by a first order diff eq Zero modifies the coefficients of the exponential terms whose shape is governed by the poles.

20. B2. Dynamic Response versus Pole Locations • The dynamic response of a system to an impulse input is the natural response • Consider a linear system whose the transfer function has n poles: • The general form of the natural response varies as a function of the pole locations (i.e., the values of σi and ωdi)

21. The natural response is of the form: K is given by the initial condition Assuming K > 0: Location in s-domain: Simple Real Pole p = - Note: since  = 0, y(t) is located on the real axis.

22. Assuming K >0, if  >0: Location in s-domain: Note: Since  = 0, y(t) are located on the real axis. Simple Real Pole y(t) decays stable stable • Effect of the value of  >0: y(t) decay

23. Assuming K >0, if  <0: Location in s-domain: Note: Since  = 0, y(t) are located on the real axis. Simple Real Pole y(t) grows unstable unstable • Effect of the value of  <0: y(t) grow

24. The natural response is of the form: B is given by the initial condition Assuming B > 0: Location in s-domain: Simple Real Pole p = 0 unstable Note: since  = 0 and  = 0, y(t) is located on the origin.

25. The natural response is of the form: A and B are given by the initial conditions Assuming A > 0, B > 0: Location in s-domain: Simple Pair of Imaginary Poles p = ± jd Note: since  = 0, y(t) is located on the imaginary axis.

26. Assuming A > 0, B >0: Location in s-domain: Simple Pair of Imaginary Poles y(t) oscillates forever unstable unstable • Effect of the value of  < 0: unstable y(t) oscillate unstable Note: In the case  = 0, the system is said unstable or critically stable.

27. The natural response is of the form: A and B are given by the initial conditions Assuming A >0, B > 0: Location in s-domain: Simple Pair of Complex Poles p = - ±jd stable unstable stable unstable

28. Summary – Transient Response vs Simple Pole Locations

29. Stability Criterion vs Pole Locations Unstable Stable The locations of poles in s-domain determine whether the system is stable or unstable.

30. Stability Criterion based on the Pole Locations • A system is stable if all its poles have negative real parts (i.e., there are all strictly inside the left-side s-plane) • and unstable otherwise Note:This criterion is valid only if the system is linear time-invariant (i.e., has constant parameters).

31. Reminder: Representation of , d ,  and 0in s-domain. In rectangular coordinates, the poles are at(-σ, ±ωd) In polar coordinates, the poles are at (ω0, sin-1ζ)

32. Stability Criterion in terms of the rectangular components (-σ, ωd) • Given n poles the stability criterion of a such system is that all poles satisfy: • If any pole of the system is in the right-half s-plane (i.e., has a positive real part,  < 0), then the system is unstable.

33. If   0? • With any simple pole such as s = 0 (i.e.,  = 0, = 0), initial conditions persist. • With any simple pole on the imaginary axis ( = 0,   0), oscillatory motion persists.

34. Attenuation σ& Damped Frequency ωd Definition: •  is called attenuation. • d is called damped frequency.

35. Reminder: Representation of , d ,  and 0in s-domain. In rectangular coordinates, the poles are at(-σ, ±ωd) In polar coordinates, the poles are at (ω0, sin-1ζ)

36. Stability Criterion vs Pole Locations Unstable Stable The locations of poles in s-domain determine whether the system is stable or unstable.

37. Stability Criterion in terms of the polar components (ω0, sin-1ζ) • Given n poles the stability criterion of a such system is that all poles satisfy: • If any pole of the system is in the right-half s-plane (i.e., has a negative damping ratio, ζ≤ 0 ), then the system is unstable.

38. Damping Ratio ζ& Undamped Natural Frequency ω0 Consider a 2nd-order system whose the transfer function is: We wish to study the dynamic response to an unit-step input (that said an input x(t)=u(t) such as L{x(t)}=X(s)=1/s). Note: a response to an unit-step input is called unit-step response. a response to an impulse input is called impulse response, etc…

39. If0 <  <1 (complex poles): The system is underdamped. If  = 1(equal real poles): the system is critically damped If  > 1(negative real and unequalpoles) : the system isoverdamped. Damping Ratio ζ& Undamped Frequency ω0 (cont’d) The character of the transient unit-step response depends on the pole locations.

40. Unit-Step Response vs.  • Note that two 2nd-order systems having the same  but different 0 will exhibit the same overshoot and the same oscillatory pattern. • An underdamped system with 0.5 < ζ<0.8 gets close to the final value more rapidly than a critically damped or overdamped system.

41. A Few Comments on Transient Response Why does a physical control system exhibit a transient response before a steady-state can be reached? • Because a physical control system involves energy storage. Therefore, the output of the system, when subjected to an input, cannot follow the input immediately. What is the transient response used for in a control system? • To specify the performance characteristics of a system. Frequently, these are defined from the transient response to a unit-step input (with all initial conditions set to zero) because it is easy to generate and convenient for comparison.

42. B3. Transient-Response Specifications • In many practical cases, the desired performances of control systems are specified in terms of time domain quantities. • These quantities characterize the damped oscillations that a practical system exhibits before reaching steady-state, when subjected to an unit-step input.

43. Time-Domain Specificationsfor a first-order system Performances are commonly specified in terms of: Time constant τ.

44. Time Constant • The time constant is the time when the response is 1/e times the initial value: 1 Hence, τ is a measure of the rate of decay. 1/e τ

45. Time-Domain Specificationsfor second or higher-order systems Performances are commonly specified in terms of: • Delay time, • Rise time, • Peak time, • Maximum overshoot, • Settling time.

46. 1. Delay Time • The delay time is the time requiredfor the responseto reach half of the final valuethe very first time. 1 td 0.5 0

47. 2. Rise Time • The rise time is the time requiredfor the responseto rise from 0% to 100% of its final value. 1 td 0.5 0 tr Note: for overdamped systems, the 10% to 90% rise time is commonly used.

48. 3. Peak Time • The peak time is the time requiredfor the responseto reach the first peak of the overshoot. t p 1 t d 0.5 0 t r

49. 4. Maximum (percent) Overshoot • The maximum overshoot is the relative maximum peak value of the response curve measured from the final value. t p M p 1 t d 0.5 0 t r Note: the maximum overshoot directly indicates the relative stability of the system.

50. 5. Settling Time • The settling time is the time required for the response curve to reach and stay within a range about1% or 2% of the final steady-state value. t p M p ±1% 1 t d 0.5 0 t t r s Note: t is the time it takes the system transients to decay. s