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Control System Analysis Cycle

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? (zeta)=damping ratio. ?n=natural frequency. Transfer ... ? (zeta)=damping ratio , ?n=natural frequency ... Step Responses with different damping ratio (zeta) ... – PowerPoint PPT presentation

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Title: Control System Analysis Cycle


1
Control System Analysis Cycle
CH 5-System Performance Analysis
2
CH. 5 Performance of Feedback Control Systems
  • This chapter describes how the input and the
    poles and zeros of the system will affect system
    performance, i.e. system output (or response).

3
CH. 5 Performance of Feedback Control Systems
  • This chapter describes how the input and the
    poles and zeros of the system will affect system
    performance, i.e. system output or response.
  • Topics to be covered
  • - Input Signals for testing performance (5.2)
  • - Performance of 1st-order Systems
  • - Performance of 2nd-order Systems (5.3)
  • - Root Location and the Transient Response
    (5.5)
  • - Steady-State Error of Feedback Systems (5.6)
  • - Design Example (5.9)
  • - Performance Analysis using Matlab and
    Simulink
  • (5.10,5.11)

4
System Performance
  • System Responses
  • 1. Time (or Transient) Response lt
  • 2. Frequency (or Steady-State) Response (Ch.
    8)
  • Key Concepts
  • 1. System Characteristic Equation
  • 2. Poles and Zeros

5
Impulse Response A Special Case of Time Response
  • The impulse response is the response for control
    systems to an input d(t) where
  • d(t) 1 (Table 2.3, p. 51).
  • So, if the input is the impulse, the output
  • is the inverse Laplace transform of the
    transfer function G(S)
  • R(s)1 Y(s)G(S)
  • or
    y(t)g(t)

G(s)
6
Matlab Command for Impulse Response impulse(sys)
  • So, if the input is the impulse, the output is
    the inverse
  • Laplace transform of the transfer function
    G(S)
  • R(s)1
    Y(s)G(S)
  • Matlab command for the unit impulse response of
    LTI systems is
  • gtgtimpulse(sys) Plots the impulse response of
    an arbitrary LTI model sys. This model can be
    SISO or MIMO, i.e. state models. Zero initial
    state is assumed for state models.

G(s)
7
Example
  • So, if the input is the impulse, the output is
    the inverse
  • Laplace transform of the transfer function
    G(S)
  • R(s)1
    Y(s)G(S)
  • Ex. G(s)1/(sst) gt g(t)e-st, tgt0.

G(s)
8
Time (Transient) Response
  • Let the system transfer function be
  • G(s)p(s)/q(s). Eq.(2.45), p. 58
  • Then,
  • 1. The (System) Characteristic Eqn. is q(s)0.
  • 2. Poles are the roots of q(s)o.
  • 3. Zeros are the roots of p(s)0.

9
Example
  • Consider the transfer function
  • G(s)p(s)/q(s)(2s1)/(s23s2)
  • gt
  • Poles q(s) s23s20 gt s-1,-2
  • Zeros p(s)2s10 gt s-1/2

10
Poles and System Stability
  • Location of Poles will determine the system
    stability (Ch. 6)

11
Finding system poles and zeros
  • Example pole and zero (Ch. 2, p. 107)

12
Commonly Used Input Signals(Table 5.1, Fig. 5.2,
p. 279)
  • (a) Step (b) Ramp
    (c) Parabolic

r(t)A, tgt0
r(t)At, tgt0
r(t)At2, tgt0
13
Performance of First-Order Systems
  • Ex. Transfer function of the RC circuit below
  • (p. 53, Ch. 2)
  • V1(s)(R1/Cs)I(s) and
  • V2(s)I(s)(1/Cs)
  • ? G(s)V2(s)/V1(s)1/(RCs1)
  • where RC is called the time constant ?.

14
Transfer Function of First-Order Systems
  • Ex. G(s)V2(s)/V1(s)1/(RCs1) where RC is called
  • the time constant ?.
  • ?In general,
  • G(s) K/(?s1), where Ksystem DC gain
  • (i.e.,
    KG(0)).
  • R(s)
    Y(s)

G(s)
15
Performance of First-Order Systems
  • G(s) K/(?s1), where Ksystem DC gain (G(0)),
  • and ? time
    constant
  • Step response R(s)1/s Y(s)
  • gt
  • Y(s)K/s K/(s1/?)
  • gt
  • y(t) K(1-e t/? )

G(s)
?
16
Performance of First-Order Systems
  • G(s) K/(? s1), where Ksystem DC gain (i.e.,
    KG(0)).
  • 1. Step response U(s)1/s gt
  • Y(s)K/s K/(s1/? gt y(t)
    K(1-e -t/? )
  • 2. Ramp response R(s)1/s2 gt
  • Y(s) K/s2 K? /s K? /(s1/?)
  • gt
  • y(t) Kt - K? K? e t/?

ess Steady-state error
17
Transfer Function of a Second-Order Linear System
  • Ex. A spring-mass-damper system
  • From Ch. 2, Md2y/dt2bdy/dtkyu(t)
  • gt
  • G(s) 1/(Ms2 bs k)

18
Transfer Function of a 2nd-Order System
  • Standard Form
  • Y(s)G(s)/(1G(s))U(s)
  • ?n2/(s22? ?ns ?n2)R(s) (5.7)
  • where,
  • ? (zeta)damping ratio
  • ?nnatural frequency

19
Transfer Function of a 2nd-Order System
  • Standard Form
  • Y(s)G(s)/(1G(s))U(s)?n2/(s22? ?ns
    ?n2)R(s) (5.7)
  • where,
  • ? (zeta)damping ratio , ?nnatural
    frequency
  • Notes Damping ratio is a real number between 0
    and 1, and defines the damping properties of the
    system, i.e.
  • the smaller ? is, the bigger system
    oscillation becomes.
  • Natural frequency or natural mode of a system
    determines system frequency response with no
    forcing function, i.e. u(t)0.

20
Fig. 5.5 Step Response of y(t) 1 (1/ß)e - ?
?nt sin(?n ß t ?)
? 0.1 0.2 0.7 1.0 2.0
21
Performance of a 2nd-Order System
  • Unit Step Responce
  • Y(s)G(s)R(s), R(s)1/s
  • ?n2/s(s22? ?ns ?n2) (5.8)
  • gt
  • y(t) 1 (1/ß)e - ? ?nt sin(?n ß t ?)
    (5.9)
  • where
  • ß ?2

22
Ex. Effect of ?n on the step response (Figure
5.10)
23
Step Response of a 2nd-Order System
  • Standard form G(s) ?n2/s(s2? ?ns ?n2)
  • For complex poles, the unit step response
    is
  • y(t) 1 (1/ß)e - ? ?nt sin(?n ß t
    ?)
  • Key Response Parameters (used in Design)
  • -Tr (Rise time)
  • -Tp (Peak time)
  • -Mpt(Peak Value)
  • -P.O.( Overshoot)
  • ((Mpt-fv)/fv)x100
  • where fvthe steady-state,
  • or final, value of y(t)

24
Response Parameters of a 2nd-Order System
  • Key Response Parameters
  • -Tr (Rise time) The time it takes for the
    system to reach the vicinity of its target value
    fv.
  • -Ts (Settling Time) The time it takes to
    settle within a certain percentage of the input.
  • -Tp (Peak time) The time to take to reach the
    maximum overshoot point.
  • -Mpt(Peak Value) The output value at tTp
  • -Mp(Overshoot) (Mpt-fv)/fv)
  • The maximum amount the system
  • overshoots its final value divided
  • by its final value ( and often
  • expressed as a percentage).

25
Step Response
  • 2nd-Order System Response (Figure 5.7, p. 284)

Overshoot
Peak Time
Input
Settling Time
Rise Time
26
Response Parameters of a 2nd-Order System
  • Key Response Parameters
  • -Tr (Rise time) The time it takes for the
    system to reach the vicinity of its target value
    fv.
  • -Ts (Settling Time) The time it takes to
    settle within a certain percentage of the input.
  • -Tp (Peak time) The time to take to reach the
    maximum overshoot point.
  • -Mpt(Peak Value) The output value at tTp
  • -Mp(Overshoot) (Mpt-fv)/fv)
  • The maximum amount the system
  • overshoots its final value divided
  • by its final value ( and often
  • expressed as a percentage).

27
Response Parameters of a 2nd-Order System
  • Key Response Parameters
  • -Tr (Rise time) The time it takes for the
    system to reach the vicinity of its target value
    fv.
  • -Ts (Settling Time) The time it takes to
    settle within a certain percentage of the input.
    Normally Ts4/??n (5.13)
  • -Tp (Peak time) The time to take to reach the
    maximum overshoot point.
  • -Mpt(Peak Value) The output value at tTp
  • -Mp(Overshoot) (Mpt-fv)/fv)
  • The maximum amount the system
  • overshoots its final value divided
  • by its final value ( and often
  • expressed as a percentage).

28
Overshoot and Damping Ratio
  • Key Response Parameters (used in Design)
  • Tp (Peak time)
  • pi/(?n ?2) (5.14)
  • gt
  • Percent Overshoot
  • 100exp(-?pi/ ?2 ) f(?)
  • (5.16)
  • and
  • ?n Tp pi( ?2)
  • gt
  • Figure 5.8, p. 285

29
Step Responses with different damping ratio (zeta)
  • This script will plot step responses for the
    system
  • transfer function G(s)1/(s2 2zetas 1) for
  • zeta0.2, 0.4,1 and 2
  • zet0.2 0.4 1 2
  • for k14
  • zetazet(k)
  • Gtf(1,1 2zeta 1)
  • step(G)
  • hold on
  • end
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