1. STATE EQUATIONS
FOR PHYSICAL
SYSTEMS

2. State-Space
- referred to as the modern, or time-domain
approach.
- unified method for modelling, analyzing,
and designing a wide range of system.
- represents nonlinear (e.g. backlash,
saturation, and dead zone.
- represents time-varying systems (e.g.
Missiles with varying fluid levels or lift in an
aircraft flying through a wide range of
altitudes.

3. State-Space
1. Select a particular subset of all possible system
variables and call the variables in this subset state
variable.

4. State-Space
2) For an nth-order system, write n simultaneous, first
order differential equations in terms of the state
variables. We call this system of simultaneous
differential state equations.

5. State-Space
3) If we know all the initial condition of all of the state
variables at to as well as the system input for t≥to ,we
solve the simultaneous differential equations for the
state variables for t≥to

6. State-Space
4) We algebraically combine the state variables with the
systems input and find all of the other system
variables for t≥to. We call this algebraic equation the
output equation.

7. State-Space
5) We consider the state equations and the output
equations a viable representation of the system. We
call this representation of the system a state-space
representation

12. The General State-Space
Representation
Linear Combination
A linear combination of n variables, xi, for
i=1 to n, is given by the following sum:
S=Knxn + Kn-1xn-1 + ... +K1x1 NOTE: where K is constant

13. The General State-Space
Representation
Linear Independence
A set of variables is said to be linearly
independent, if none of the variables
can be written as a linear combination of
the others.

14. The General State-Space
Representation
Linear Combination
A linear combination of n variables, xi, for
i=1 to n, is given by the following sum:
S=Knxn + Kn-1xn-1 + ... +K1x1 NOTE: where K is constant

15. The General State-Space
Representation
System variable
Any variable that responds to an input or
initial conditions in a system

16. The General State-Space
Representation
State Variables
The smallest set of linearly independent
system variables such that the values of
the members of the set at time to along
with known forcing functions completely
determine the value of all system
variables for all t≥to

17. The General State-Space
Representation
State Space
The n-dimension space whose axes are the
state variables.
(e.g. vR and vC)

18. The General State-Space
Representation
State Equation
A set of n simultaneous, first order
differential equations with n variables,
where n variables to be solved are the
state variables.

19. The General State-Space
Representation
Output Equation
The algebraic equations that expresses the
output variables of a system as linear
combinations of the state variables and
the inputs.

20. The General State-Space
Representation
ẋ=Ax + Bu --- STATE EQUATION
y=Cx + Du ---OUTPUT EQUATION
for t≥to and initial conditions x(to), where
x=state vector
ẋ=derivative of the state vector with respect to time
y=output vector
u=input or control vector
A=system matrix
B=input matrix
C=output matrix
D= feedforward matrix

21. Representing an electrical
Network
Problem
Given the electrical network, find a state-space representation
if the output is the current through the resistor.
L
v(t) R C

22. Representing an electrical
Network
Step 1
Label all the branches currents in the network.
L
iL(t)
v(t) iR(t) R iC(t) C

23. Representing an electrical
Network
Step 2
Select the state variables by writing the derivative equation for
all energy storage elements, that is, the inductor and the
capacitor.

24. Representing an electrical
Network
Step 3
Apply network theory, such as Kirchhoffs voltage and current
laws

25. Representing an electrical
Network
Step 4
Substitute the results of Eqs. (3.24) and (3.25) into Eqs.
(3.22) and (3.23) to obtain the following state equations:

26. Representing an electrical
Network
Step 5
Find the output equation.

27. Representing a Translational
Mechanical System

28. Representing a Translational
Mechanical System

29. Representing a Translational
Mechanical System

30. Representing a Translational
Mechanical System

31. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Constant
Term in Numerator

32. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Constant
Term in Numerator

33. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Constant
Term in Numerator

34. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Constant
Term in Numerator

35. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Polynomial in
Numerator

36. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Polynomial in
Numerator

37. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Polynomial in
Numerator

38. Converting a Transfer Function to
State Space
 Converting a Transfer Function with Polynomial in
Numerator

39. State-Space Representation to
Transfer Function

40. State-Space Representation to
Transfer Function

41. State-Space Representation to
Transfer Function

42. State-Space Representation to
Transfer Function

43. Linearization

44. Linearization

45. Linearization

46. Linearization