3. Outline
• How to find mathematical model, called a state-
space representation, for a linear, time-invariant
system
• How to convert between transfer function and
state space models
4. State Variables
• Definition: State variables are variables that collectively
describe the internal state of a dynamic system. They
encapsulate all relevant information needed to predict the
future behavior of the system.
• Examples: In mechanical systems, state variables could
include position, velocity, and acceleration. In electrical
circuits, they might represent voltage and current across
different components.
• Significance: State variables are essential for modeling and
understanding the behavior of dynamic systems. They serve
as the foundation for state space representation.
5. State Equations:
• Definition: State equations are mathematical equations that
describe how the state variables of a system evolve over time.
They capture the system's dynamics in terms of its state
variables and inputs.
• Formulation: State equations are typically expressed as a set
of first-order ordinary differential equations for continuous-
time systems or difference equations for discrete-time
systems.
• Example: For a simple mechanical system, state equations
might describe how position, velocity, and acceleration
change over time in response to applied forces.
6. Components of State Space
Representation:
• State Vector (x): A column vector containing all the state variables
of the system. It represents the system's current state at any given
time.
• State Equation (x˙=f(x,u)): Describes the time rate of change of the
state vector. It's a function of the state vector x and the input
vector u.
• Input (u): Represents the external influences or control inputs
applied to the system. Inputs can affect the system's state
evolution.
• Output (y): The measurable quantities or signals produced by the
system. Outputs are often related to the state variables through
output equations.
7. Advantages of State Space
Representation:
• Unified Framework: State space provides a unified framework
for modeling diverse systems, including linear and nonlinear
systems, discrete-time and continuous-time systems, and SISO
and MIMO systems.
• Flexibility: It can handle complex systems with multiple inputs,
outputs, and state variables, making it suitable for a wide
range of engineering applications.
• System Analysis and Control Design: State space
representation facilitates system analysis, stability analysis, and
control design, enabling engineers to design advanced
controllers.
• Insight into System Dynamics: State space representation
offers insights into the dynamic behavior of systems, including
stability, controllability, observability, and transient response.
8. Applications of State Space
Representation:
• Control Systems: Widely used in designing and analyzing
feedback control systems, including PID controllers, state
feedback controllers, and observers.
• Robotics: Employed in modeling the dynamics of robotic
systems, such as robot arms, manipulators, and mobile
robots, enabling precise control and motion planning.
• Aerospace: Crucial for modeling and controlling aircraft,
spacecraft, and satellites, ensuring stable flight and
precise maneuvering.
• Electrical Engineering: Used in modeling electrical
circuits, power systems, and electronic control systems,
facilitating efficient energy management and control.
9. 9
State-Space Modeling
• Alternative method of modeling a system than
▫ Differential / difference equations
▫ Transfer functions
• Uses matrices and vectors to represent the system
parameters and variables
• In control engineering, a state space
representation is a mathematical model of a
physical system as a set of input, output and state
variables related by first-order differential equations.
To abstract from the number of inputs, outputs and
states, the variables are expressed as vectors.
10. 10
Motivation for State-Space Modeling
• Easier for computers to perform matrix algebra
▫ e.g. MATLAB does all computations as matrix math
• Handles multiple inputs and outputs
• Provides more information about the system
▫ Provides knowledge of internal variables (states)
Primarily used in complicated, large-scale
systems
11. Definitions
• State- The state of a dynamic system is the
smallest set of variables (called state variables)
such that knowledge of these variables at t=t0 ,
together with knowledge of the input for t ≥ t0 ,
completely determines the behavior of the
system for any time t to t0 .
• Note that the concept of state is by no means
limited to physical systems. It is applicable to
biological systems, economic systems, social
systems, and others.
12. State Variables or Phase Variables:
• The state variables of a dynamic system are the
variables making up the smallest set of variables
that determine the state of the dynamic system.
• If at least n variables x1, x2, …… , xn are needed
to completely describe the behavior of a
dynamic system (so that once the input is given
for t ≥ t0 and the initial state at t=t0 is specified,
the future state of the system is completely
determined), then such n variables are a set of
state variables.
13. State Vector:
• A vector whose elements are the state variables.
• If n state variables are needed to completely
describe the behavior of a given system, then
these n state variables can be considered the n
components of a vector x. Such a vector is called
a state vector.
• A state vector is thus a vector that determines
uniquely the system state x(t) for any time t≥ t0,
once the state at t=t0 is given and the input u(t)
for t ≥ t0 is specified.
14. State Space:
• The n-dimensional space whose coordinate axes
consist of the x1 axis, x2 axis, ….., xn axis, where
x1, x2,…… , xn are state variables, is called a
state space.
• "State space" refers to the space whose axes are
the state variables. The state of the system can
be represented as a vector within that space.
15. • State-Space Equations. In state-space analysis
we are concerned with three types of variables
that are involved in the modeling of dynamic
systems: input variables, output variables, and
state variables.
• The number of state variables to completely
define the dynamics of the system is equal to the
number of integrators involved in the system.
• Assume that a multiple-input, multiple-output
system involves n integrators. Assume also that
there are r inputs u1(t), u2(t),……. ur(t) and m
outputs y1(t), y2(t), …….. ym(t).
16. • Define n outputs of the integrators as state variables:
x1(t), x2(t), ……… xn(t). Then the system may be
described by
17. • The outputs y1(t), y2(t), ……… ym(t) of the
system may be given by
19. • then Equations (2–8) and (2–9) become
• where Equation (2–10) is the state equation and
Equation (2–11) is the output equation. If vector
functions f and/or g involve time t explicitly, then the
system is called a time varying system.
20. • If Equations (2–10) and (2–11) are linearized
about the operating state, then we have the
following linearized state equation and output
equation:
21. • A(t) is called the state matrix,
• B(t) the input matrix,
• C(t) the output matrix, and
• D(t) the direct transmission matrix.
• A block diagram representation of Equations (2–12) and (2–
13) is shown in Figure
23. State representation of a linear system
D
A
B
x
u
x
Bu
Ax
x
C
y
Du
Cx
y
A= System Matrix(n,n)
B= Input Matrix (n,m)
x= State Vector (n,1)
u= Input Vector (m,1)
C= Output Matrix (r,n)
D= Direct Transmission Matrix (r,m)
y= Output Vector (r,1)
24. • If vector functions f and g do not involve time t
explicitly then the system is called a time-
invariant system. In this case, Equations (2–12)
and (2–13) can be simplified to
•Equation (2–14) is the state equation of the
linear, time-invariant system and
•Equation (2–15) is the output equation for the
same system.
25. State-space Representation
• One advantage of the state-space representation
is that it can be used for the simulation of
physical systems on the digital computer,
25
Consider the differential
equation
32. Differential Equations
• Example: Springer-mass-damper system
• Assumption: Wall friction is a viscous force.
The time function of
r(t) sometimes
called forcing
function
Linearly proportional
to the velocity
)
(
)
( t
bv
t
f
33. Differential Equations
• Example: Springer-mass-damper system
• Newton’s 2nd
Law:
)
(
)
(
)
(
)
( t
Ma
t
r
t
ky
t
bv
)
(
)
(
)
(
)
(
2
2
t
r
t
ky
dt
t
dy
b
dt
t
y
d
M
34. Correlation Between Transfer
Functions and State-Space Equations
• The "transfer function" of a continuous time-
invariant linear state-space model can be derived
in the following way:
First, taking the Laplace transform of
Yields
37. Transition matrix Bu
Ax
x
Assuming that the system is continuous and linear
that A and B are time-invariant and
Using Laplace transform
)
s
(
BU
)
s
(
AX
)
0
(
x
)
s
(
sX
)
s
(
BU
)
0
(
x
)
s
(
X
)
A
sI
(
)]
s
(
BU
)
0
(
x
[
)
A
sI
(
)
s
(
X 1
Taking the inverse Laplace transform of resolvent matrix
]
)
A
sI
[(
L
)
t
( 1
1
Transition matrix
38. Transition matrix
The state vector will take the following form (convolution)
t
0
d
)
(
Bu
)
t
(
)
0
(
x
)
t
(
)
t
(
x
Or more generally
t
t
0
0
d
)
(
Bu
)
t
(
)
0
(
x
)
t
t
(
)
t
(
x
The output vector will take the following form
Assuming that C and D are time-invariant
)
t
(
Du
)
t
(
Cx
)
t
(
y
39. Transition matrix
Properties of the transition matrix
n
Akt
k
At
k
I
)
0
(
)
kt
(
e
)
e
(
)
t
(
]
)
A
sI
[(
L
)
t
( 1
1
nn
ni
2
n
1
n
in
ii
2
i
1
i
n
2
i
2
22
21
n
1
i
1
12
11
a
s
a
a
a
a
a
s
a
a
a
a
a
s
a
a
a
a
a
s
)
A
sI
(
The resolvent matrix
![Transition matrix Bu
Ax
x
Assuming that the system is continuous and linear
that A and B are time-invariant and
Using Laplace transform
)
s
(
BU
)
s
(
AX
)
0
(
x
)
s
(
sX
)
s
(
BU
)
0
(
x
)
s
(
X
)
A
sI
(
)]
s
(
BU
)
0
(
x
[
)
A
sI
(
)
s
(
X 1
Taking the inverse Laplace transform of resolvent matrix
]
)
A
sI
[(
L
)
t
( 1
1
Transition matrix](https://image.slidesharecdn.com/cmuadvancedcontrolsm60aclesson2-250601172915-9bdb9f5e/85/CMU-Advanced-Controls-M60AC-Lesson-2-ppt-37-320.jpg)
![Transition matrix
Properties of the transition matrix
n
Akt
k
At
k
I
)
0
(
)
kt
(
e
)
e
(
)
t
(
]
)
A
sI
[(
L
)
t
( 1
1
nn
ni
2
n
1
n
in
ii
2
i
1
i
n
2
i
2
22
21
n
1
i
1
12
11
a
s
a
a
a
a
a
s
a
a
a
a
a
s
a
a
a
a
a
s
)
A
sI
(
The resolvent matrix](https://image.slidesharecdn.com/cmuadvancedcontrolsm60aclesson2-250601172915-9bdb9f5e/85/CMU-Advanced-Controls-M60AC-Lesson-2-ppt-39-320.jpg)
