Control system theory formulas, Cheat Sheet of Control Systems

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Introduction

Open Loop Control Systems:

• In this system the output is not feedback for comparison with the input.
• Open loop system faithfulness depends upon the accuracy of input calibration.

Transfer function, ( )

C s G s R s

When a designer designs, he simply design open loop 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 G s

R s 1 G s H s

Comparison of Open Loop and Closed Loop control systems:

Open Loop:

1. Accuracy of an open loop system is defined by the calibration of input.
2. Open loop system is simple to construct and cheap.
3. Open loop systems are generally stable.

CONTROL SYSTEMS (FORMULA NOTES/SHORT NOTES)

(GX 1 ± X 2 )

Signal Flow Graphs:

• It is a graphical representation of control system.
• Signal flow graph of Block diagram.

Mason’s Gain Formula:

Transfer function =

k k p 



pk → Path gain of k

th forward path.

∆ = 1 – [Sum of all individual loops] + [Sum of gain products of two non-touching loops] – [Sum of

gain products of 3 non-touching loops] + …

 → Value of ∆ obtained by removing all the loops touching k

th forward path as well as non-toughing

to each other.

Some Laplace and Z Transforms

Modified Z-Transform:

( )

 −

=−

  ^  ^ ^ 

n

n

X z,m Z x n ,m x n m 1 .z

Final Value Theorem:

( ) ( ) ( ) ( ) s 0 z 1

x lim sX s lim z 1 X z → →

Initial Value Theorem:

s

x 0 lim sX s →

Euler’s Formula:

( ) ( )

j

e cos jsin



Convolution:

( a * b) ( ) t a ( ) b t( )dT



−



Convolution Theorem:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

L f t * g t F s G s

L f t g t F s * G s

  =  

  =  

Characteristic Equation:

A I 0

Av v

wA lw

Decibels:

dB = 20 log(C)

Unit Step Function:

0, t 0 u t 1, t 0

Unit Ramp Function: r(t) = t.u(t)

Unit Parabolic function: ( ) ( )

p t t u t 2

Closed Loop Transfer Function:

p cl p

KG s H s 1 KG s Gb s

Open Loop Transfer Function:

Hol(s) = KGp(s)Gb(s)

Characteristics Equation:

F(s) = 1 + Hol

Time Response of 2 nd order System:

Step input:

n t 2 2 1 n 2

e 1 C t 1 sin 1 t tan

1

− −

nt 2 1 n 2

e^1 e t sin t tan

1

− −

n t^2 1 ss n t 2

e^1 e lim sin t tan

1

− −

→

= ^   ^ 

 ^ 

→  →^ Damping ratio;^ n →Damping factor

  1 (Under Damped):

n t^2 1 n 2

e 1 C t 1 sin t tan

1

− −

 = 0 (Un damped):

C(t) = 1 – cosωnt

 = 1 Critically damped :( )

nt n

C t 1 e 1 t

−

  1 Over damped :( )

2 (^1) nt

2 2

e C t 1

2 1 1

  − −  − (^)    = −

 −  −  −

→ → →

ss t s 0 s 0

sR s e lim e t lim E s lim 1 GH

ss P

e 1 K

(Positional error)

P (^ )^ (^ )

s 0

K lim G s H s →

• Ramp input (t) = ss

v

e K

P (^ )^ (^ )

s 0

K lim S G s H s →

• Parabolic input (t 2 /2):

→

2 ss a s 0 a

e , K lim s G s H s K

Type < input→ ess= 

Type = input ss →e finite

Type > input ss →e = 0

• Sensitivity,

A / A

S

K / K

sensitivity of A w.r.t. K.

• Sensitivity of overall T/F w.r.t. forward T/F G(s):

Open loop: S = 1

Closed loop:

S

1 G s H s

• Minimum ‘S’ value preferable
• Sensitivity of overall T/F w.r.t. feedback T/F H(s):

G s H s S 1 G s H s

Stability RH Criterion:

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

root

• If characteristic equations contains either only odd/even terms indicates roots have no real part

and possess only image parts therefore 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 roots forming symmetry about origin.

Position Error Constant:

p s 0

p z 1

K lim G s

K lim G s

→

→

Velocity Error Constant:

v (^ )^ (^ )^ (^ )

s 0 z 1

K lim sG s lim z 1 G s → →

Acceleration Error Constant:

2 2 a s 0 z 1

K lim s G s lim z 1 G s → →

General System Description:

y t( ) g t,r x r dr( ) ( )



−



Convolution Description:

y t( ) x t( ) h t( ) g ( ) h t( )dt



−



Transfer Function Description:

Y ( s) = H s X s( ) ( ) Y ( z) =H z X z( ) ( )

State Space Equations:

x  ( )t = Ax t( ) +Bu t( )

y t( ) = Cx t( ) +Du t( )

Transfer Matrix:

1 C sI A B D H s

− − + =

1 C zI A B D H z

− − + =

Transfer Matrix Description:

 KGH z ( )= 180 

Number of Asymptotes:

Na = P – Z

Angle of Asymptotes:

k (^ 2k^1 )

P Z

Breakaway Point Locations:

G s H s( ) ( ) GH z( )

0 or 0 ds dz

Controllers and Compensators:

PID:

i p d

K

D s K K s s

( ) p i a

T z 1 z 1 D z K K K 2 z 1 Tz

 +^   − 