Formula Notes Control Systems

Formula Notes (Control Systems)

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Control Systems Open Loop Control System: •

In this system the output is not feedback for comparison with the input.

•

Open loop system faithfulness depends upon the accuracy of input calibration.

When a designer designs, he simply design open loop system. system. Closed Loop Control System: It is also termed as feedback control system. Here the output has an effect on control action through a feedback. Ex. Human being Transfer Function:

Transfer function =

C (s ) R (s (s)

=

G(s) 1 + G(s)H (s (s)

Comparison of Open Loop and Closed Loop control systems: Open Loop: 1. Accuracy Accuracy of an open open loop syste system m is defined defined by the the calibration calibration of input. input. 2. Open loop system system is simple simple to construct construct and cheap. cheap. 3. Open Open loop loop system systemss are gene general rally ly stabl stable. e. 4. Operation Operation of this system system is is affected affected due to presenc presencee of non-lineari non-linearity ty in its its elements. Closed Loop: 1. As the error error between between the the reference reference input input and the the output output is cont continuou inuously sly measur measured ed throu through gh feedb feedback ack.. The close closed d system system work workss more more accurat accurately ely.. 2. Closed loop system systemss is complicat complicated ed to construc constructt and it is is costly. costly. 3. It becomes becomes unstab unstable le under under certain certain condition conditions. s. 4. In terms terms of performance performance the closed closed loop loop system system adjusts adjusts to the the effects effects of nonnonlinearity present. system may be defined as the Transfer Function: The transfer function of an LTI system ratio of Laplace transform of output to Laplace transform of input under the assumption G(s)= •

Y(s) X(s)

The transfer function is completely specified in terms of its poles and zeros and the gain factor.

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Formula Notes (Control Systems)

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•

The T.F. function of a system depends on its elements, assuming initial conditions as zero and is independent of the input f unction.

•

To find a gain of system through transfer function put  s = 0

Example:

G(s) =

s

2

4

s

+

+

6s

+

9

Gain =

4 9

If a step, ramp or parabolic parabolic response of of T.F. is given, then then we can find Impulse Impulse Response directly through differentiation of that T.F. d  dt  d  dt  d  dt 

(Parabolic Response) = Ramp Response (Ramp Response) = Step Response (Step Response) = Impulse Response

Block Diagram Reduction: Rule Original Diagram X 1G1G2 1. Combin Combining ing X 1 G1 X1 blocks in cascade G1 G2

Equivalent Diagram X 1G 1G2 X1 G1G 2

2.. oving a summing point after a block 

3. Moving Moving a summing point ahead of block 

4. Movi Moving ng a take take off point after a block 

X1

X1

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G

X1G

X1

G

X1

1/G

X 1G

Formula Notes (Control Systems) X1

5. Movin Moving g a tak takee off point ahead of  a block 

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X 1G

G

X1 X 1G

X 1G

6. Elim Elimin inat atin ing g a feedback loop

X 1G

G

G

X1

X2

G 1GH

(GX (GX1  ± X2 ) Signal Flow Graphs: It is a graphical representation of control system. • •

Signal Flow Graph of Block Diagram:

Mason’s Gain Formula:

 pk 

Transfer function =

Σ

pk

∆ k 

∆

th

gain of  k  forward path

→ Path

∆=

1 – [Sum of all individual loops] loops] + [Sum of gain products of of two non-touching loops] – [Sum of gain products of 3 non-touching loops] + ……….. th

of  ∆ obta obtain ined ed by by remo removi ving ng all all the the loo loops ps tou touch chin ing g k  forw forwar ard d path path as as well as non-touching to each other ∆ k  → Value

Some Laplace and

Z   Transforms

F ( F (s)

f ( f (nT ) nT )

F ( F (z )

1 s

1(nT  1(nT ))

z z−1

1 s2

nT 

Tz (z − 1)2

1 s + a  + a

e−anT 

z z − e−aT 

a s(s + a  + a))

1 − e−anT 

z (1 − e−aT  ) (z − 1)(z 1)(z − e−aT  )

a s2 (s + a  + a))

(anT  − 1 + e −anT  )/a

z [z (aT  − 1 + e −aT  ) + (1 − (1 + aT )e  aT )e−aT  )] a(z − 1)2 (z − e−aT  )

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Formula Notes (Control Systems) Laplace Transform:

Inverse Laplace Transform:

Fourier Transform:

Inverse Fourier Transform:

Star Transform:

Z Transform:

Inverse Z Transform:

Modified Z Transform:

Final Value Theorem:

Initial Value Theorem:

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Formula Notes (Control Systems)       

   

       

         

   

Unit Step Function:

Unit Ramp Function:

Unit Parabolic Function:

Closed-Loop Transfer Function:

Open-Loop Transfer Function:

Characteristic Characteristic Equation:

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Formula Notes (Control Systems)

Time Response of 2nd order system : Step i/P :

  (sin ω  1  ζ2 t ± tan− �− )  −   sin    ± tan− �− e(t) =  −   sin    ± tan− �− ess  = lim  → − → ζ → Damping ratio ; ζω → Damping factor •

C(t) = 1-

•

•

C(t) = 1

-=

c(t) = 1- cos

C(t) = 1

  Sin   ± tan− �−  −

ω

t

−(1 + ωt)

-e

      C(t) = 1 - 2  −− − −  −  T = −  −  −  Tundamped > Tovedamped > Tundedamped > Tciticaldamp Time Domain Specifications : •

•

• •

 − − tan ∅ �   =  π−∅   − π Peak time t   =   Max over shoot % M  = e− − × 100 Settling time t s = 3T 5% tolerance Rise time t   =

/

= 4T

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2% tolerance

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Formula Notes (Control Systems)

•

•

•

•

• •

•

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d +07  )   Dampingfactor Damping factor2 ζ2 = π  (lM + (lM2π) Time period of oscillations T =        No of oscillations oscillations = 2π  = 2π t  ≈ 1.5 t d t   = 2.2 T   ; ω = ω  1  2ζ2 Resonant peak M  = 2− Bandwidth ωb  = ω (1  2ζ2 +  4 4  4 2 + 2) 2 .

Delay time t   =

×

/

>  ω < ω < ωb >

/

Static error coefficients : •

•

•

()  ()  = lim ss → () = lim + →0 →0  ess  = +KP (positional error) K = lim →0 () ()  K v = lim   ( )() Ramp i/p (t) : ess = K →0 Parabolic i/p ( t 2 /2) : ess  = 1/ K  K   = lim s 2 ( )() →0 Step i/p :

e  = lim

Type < i/p Type = i/p Type > i/p •

•

→→ ss ∞ → ssss

e  = e  finite e  = 0

Sensitivity S =

∂A A sensitivity of A w.r.to K. ∂K K / /

Sensitivity of over all T/F w.r.t forward path T/F G(s) : Open loop: S =1 Closed loop :

S=

 +G( +G(s)H(s)

•

Minimum ‘S’ value preferable

•

Sensitivity of over all T/F w.r.t feedback T/F H(s) : S =

G(s)H(s) +G( +G(s)H(s)

Stability RH Criterion : • • •

Take characteristic equation 1+ G(s) H(s) = 0 All coefficients should have same sign There should not be missing ‘s’ term . Term missed means presence of at least one +ve real part root

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Formula Notes (Control Systems)

•

•

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If char. Equation contains either only odd/even terms indicates roots have no real part & posses only imag parts there fore sustained oscillations in response. Row of all zeroes occur if  (a) Equation has at least one pair of real roots with equal image but opposite sign (b) has one or more pair of imaginary roots (c) has pair of complex conjugate conjugate roots forming symmetry about origin.

Position Error Constant:

Velocity Error Constant:

Acceleration Error Constant:

General System Description:

Convolution Description:

Transfer Function Description:

State-Space Equations:

Transfer Matrix:

Transfer Matrix Description:

Mason's Rule:

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Formula Notes (Control Systems)

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State-Space Methods:

1.General 1.General State Equation Solution:

2. General Output Equation Solution:

3. Time-Variant Time-Variant General Solution:

4. Impulse Response Matrix:

Root Locus: The Magnitude Equation: Equation:

The Angle Equation: Equation:

 Number of Asymptotes: Asymptotes: Angle of Asymptotes: Asymptotes:

Breakaway Point Locations:

 or

Controllers and Compensators: Compensators: PID:

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