Control Design


Block Diagram and signal flow graph(Mason)



partial fraction expansion



Transfer funciton of poles and zeros


TF of poles and zeros



Periodic signal to Aperiodic signal

periodic to aperiodic


Fourier Transform (Continuous -Time)

Fourier transform c



Laplace Transform

s=σ + jω

Laplace transform



Input signal:

unit step function,unit ramp function, unit acceleration funciton,unit impluse function;


unit step response,unit ramp response, unit acceleration response,unit impluse response;






Typical Blocks:







First order system












Time delay








PID controller





Charactertistic equation (transfer funciton)


quadratic formula






Time domain analysis

Step response

(Figure come from Mordern Control Systems,Richard C and Robert H)


  rising time Tr

  peak time Tp

  settling timeTs

  percent overshoot)σ%

Routh-Hurwitz criterion



The Root Locus Method(linear system)

not directly solve the characteristic equation

On the s-plane (complex plane), where the x-axis is σ(real) ,and y-axis is jω (imaginary)

gain K






Frequency response Methods

Frequency response G(ω)

Frequency response



Bode plot




Nyquist Criterion




Non-linear control

Lyapunov stability analysis



State Variable Models


Signal flow gain formula


Signal flow gain


Pijk= gain of kth path from variable xi to variable xj,


Δ= determinant of the graph


Δijk=cofactor of the path Pijk




The state variables of a dynamic system


State var of dynamic sys


State differential equation

State differential equation


Output equation

Output equation



State space



Four first order differential equations


Four first order DE


TF numerator is b0


TF numerator is b0


TF numerator is a polynomial


TF numerator is a polynomial



Phase variable canonical form


Phase variable canonical form



Input feedforward canonical form


Input feedforward canonical form




Parallel manipulator: PD controller,augmented PD controller,PC torque controller.