DNT 354 - Control Principle

State-Space Models
DNT 354 - CONTROL PRINCIPLE
Date: 24th July 2008
Prepared by: Megat Syahirul Amin bin Megat Ali
Email: [email protected]
CONTENTS
Introduction
 State-Space Model
 Signal Flow Graph

INTRODUCTION

The basic questions that will be addressed in state-space
approach include:
i.
What are state-space models?
ii.
Why should we use them?
iii. How are they related to the transfer function used in
classical control system?
iv. How do we develop a space-state model?
STATE-SPACE MODEL
A representation of the dynamics of Nth-order
system as a first-order equation in an N-vector,
which is called the state.
Convert the Nth-order differential equation that
governs the dynamics of the system into N firstorder differential equation.
STATE-SPACE MODEL

The state of a system is described by a set of first-order
differential equations written in terms of the state variable.
STATE-SPACE MODEL

In a matrix form, we have:

State vector:
STATE SPACE MODEL
Input equation
Output equation
REPRESENTATION OF DYNAMIC SYSTEM

Example: Springer-mass-damper system
Dynamic equation of the system:
d 2 y (t )
dy (t )
M

b
 ky(t )  r (t )
2
dt
dt
Therefore, we define variable x1 and x2.
x1  y
x2  y

The 2 first-order equations are:
REPRESENTATION OF DYNAMIC SYSTEM

Example: Springer-mass-damper system
In matrix form:
If the measured output of the system is
position, then we have:
General State-Space Model:
REPRESENTATION OF DYNAMIC SYSTEM

Example: Simple mechanical system
Dynamic equation of the system:
d 2 y (t )
dy (t )
M
b
 ky(t )  r (t )
2
dt
dt
Let us define variable x1 and x2.
x1  y
The we will obtain:
x2  y
REPRESENTATION OF DYNAMIC SYSTEM

Example: Simple mechanical system
In vector form:
The output equation:
y  x1
REPRESENTATION OF DYNAMIC SYSTEM

Problem: Find the space-state for the following mechanical
system.
REPRESENTATION OF DYNAMIC SYSTEM

Problem: Find the space-state for the following RLC circuit.
t
di(t )
1
L
 Ri (t )   i( )d  v(t )
dt
C0
SIGNAL FLOW GRAPH






A signal flow graph is a graphical representation of the
relationships between the variables of a set of linear algebraic
equations.
The basic element of a signal flow graph is a unidirectional
path segments called branch.
The input and output points or junctions are called nodes.
A path is a branch or continuous sequence or branches that
can be traversed from one signal node to another signal node.
A loop is a closed path that originates and terminates on the
same node, and along the path no node is met twice.
Two loops are said to be non-touching if they do not have a
same common node.
SIGNAL FLOW GRAPH

Signal flow graph of control systems
SIGNAL FLOW GRAPH

Signal flow graph of control systems
SIGNAL FLOW GRAPH

Mason’s Gain Formula for Signal Flow Graph
Tij 
P
ijk
 ijk
k

Where,
Pijk
∆
∆ijk
: kth path from variable xi to xj
: Determinant of the graph
: Cofactor of the path Pijk
Δ  1 - (sum of all different loop gains)
 (sum of the gain products of all combinatio ns of 2 nontouchin g loops)
- (sum of the gain products of all combinatio ns of 3 nontouchin g loops)
 ...
SIGNAL FLOW GRAPH MODEL

Example: Transfer function of interacting system
SIGNAL FLOW GRAPH MODEL

Example: Transfer function of interacting system
The paths connecting input R(s) to output Y(s) are:
P1 = G1G2G3G4
P2 = G5G6G7G8
There are four individual loops:
L1 = G1H1
L2 = G2H2
L3 = G3H3
L4 = G4H4
SIGNAL FLOW GRAPH MODEL

Example: Transfer function of interacting system
Loops L1 and L2 does not touch loops L3 and L4. Therefore, the
determinant is:
  1  ( L1  L2  L3  L4 )  ( L1L3  L1L4  L2 L3  L2 L4 )
The cofactor of the determinant along path 1 is evaluated by
removing the loops that touch path 1 from ∆. Therefore have:
L1  L2  0
and,
1  1  ( L3  L4 )
Similarly, the cofactor for path 2 is:
2  1  ( L1  L2 )
SIGNAL FLOW GRAPH MODEL

Example: Transfer function of interacting system
Therefore, the transfer function of the system is:
Y ( s)
P  P 
 T ( s)  1 1 2 2
R( s )

G1G2G3G4 (1  L3  L4 )  G5G6G7G8 (1  L1  L2 )

1  L1  L2  L3  L4  L1 L3  L1 L4  L2 L3  L2 L4
SIGNAL FLOW GRAPH MODEL

Problem: Obtain the closed-loop transfer function by use of
Mason’s Gain Formula
FURTHER READING…

Chapter 3
i.
ii.

Dorf R.C., Bishop R.H. (2001). Modern Control Systems (9th Ed),
Prentice Hall.
Nise N.S. (2004). Control System Engineering (4th Ed), John
Wiley & Sons.
Chapter 5
i.
Nise N.S. (2004). Control System Engineering (4th Ed), John
Wiley & Sons.
“A mystic is someone who wants to understand the universe, but
are too lazy to study physics…”
THE END…