State Variable Description of LTI systems
Eytan Modiano
Eytan Modiano
Slide 1
Learning Objectives
•
Understand concept of a state
•
Develop state-space model for simple LTI systems
–
–
–
RLC circuits
Simple 1st or 2nd order mechanical systems
Input output relationship
•
Develop block diagram representation of LTI systems
•
Understand the concept of state transformation
–
Eytan Modiano
Slide 2
Given a state transformation matrix, develop model for the
transformed system
The State of a System
•
The “state” of a system is the minimum information needed about
the system in order to determine its future behavior
–
Given the state at time t0, and input up to time t > t0; can determine the
output for time t.
• State Variables
–
Set of variables of smallest possible size that together with any input
to the system is sufficient to determine the future behavior (I.e.,
output) of the system.
–
Each state variable has “memory”
E.g., voltage in capacitor
–
Each state variable has an “initial condition”
E.g., its state at time t0
–
State variables are typically associated with energy storage
• State vector: vector of state variables
Eytan Modiano
Slide 3
Example: RLC circuit
e1
R
C
ic
+
V(t)
-
dv "
!!ic = C
V (t)
$ dv
!
=!
dt # &
dt
RC
!ic = !V / R $%
V (t) = V (0)e!t / RC
•
V(t) is the state of the system at time t
–
•
Initial condition - V(0) is the voltage
across the capacitor at time 0
V(t)
V(0)
If we know v(t) at any time t, we know it
for all future time
–
No input in this case
t
Eytan Modiano
Slide 4
State Variables
•
In electric circuits, the energy storage devices are the capacitors and
inductors
–
–
They contain all of the state information or “memory” in the system
State variables:
Voltage across capacitors
Current through inductors
•
In mechanical systems, energy is stored in springs and masses
–
State variables
Spring displacement
Mass position and velocity
•
Example” Single mass M, moving in one dimension (x), under force F
–
State variables (x1, x2)
x1 = mass position
x2 = velocity
F
Eytan Modiano
Slide 5
M
x1 = x !
x!1 = x2
"$
x2 = x! #
x!2 = !!
x=F/M
F = M!!
x
Position (x)
State Equations
•
State equations
in matrix form:
! ! x1 $
x = # & ,!!input :!u = F / M
" x2 %
State!equations :! x" = Ax + Bu
! x"1 $ ! 0 1 $ ! x1 $ ! 0 $
# x" & = # 0 0 & # x & + # 1 & F / M
" 2% "
%" 2% " %
•
Output equations:
–
suppose output is y=x1 (position)
! x1 $
y = [1 0 ] # &
" x2 %
x! = Ax + Bu
y = Cx + Du
•
General form of state equations:
•
We will focus (to start) mainly on homogeneous case:
Eytan Modiano
Slide 6
x! = Ax
General form of state equations
•
In general a system can have
–
n states, state vector X(t), m inputs, u(t), and l outputs, y(t)
! x1 (t) $
! u1 (t) $
! y1 (t) $
!
!
!
#
&
#
&
X(t) = # " & ,!!!U(t) = # " & ,!!Y (t) = ## " &&
#" xn (t) &%
#"un (t) &%
#" yn (t) &%
•
The state and output equations can be written as:
•
A,B,C,D are constant matrices
"
"
"
X! = AX + BU
"
"
"
Y = CX + DU
–
–
–
–
Eytan Modiano
Slide 7
A is an nxn “system dynamics” matrix
B is an nxm “input matrix”
C is an lxn matrix relating states to outputs
D is an lxm matrix relating inputs to outputs
Why the state-space approach?
•
Very general approach to describe Linear time-invariant (LTI)
systems
–
–
–
•
Rich theory describing the solutions
Simplifies analysis of complex systems with multiple inputs and
outputs
Approach is central to “modern” control
History of state-space approach
–
State-space approach to control system design was introduced in the
1950’s
Up to that point “classical” control used root-locus or frequency response
methods (more in 16.060)
Eytan Modiano
Slide 8
–
“New” approach was named “modern” control and still have that name
–
Related state space approach to describing differential equations is
over 100 years old
Block Diagram Representation
Integrator block diagram
$
∫
x(t)
"#
dx(t)
= ax(t) + bu(t)
dt
x(t) =
$
t
"#
t
ax(! ) + bu(! )d!
•
•
x(! )d!
x(t) - state variables
u(t) - inputs
System block diagram
u(t)
b
+
!
x(t)
∫
x(t)
a
Eytan Modiano
Slide 9
State is “fed-back” into the system
x(t)
x(t) ! integrator!output
! ! integrator!input
x(t)
! = ax(t) + bu(t)
x(t)
General system block diagram
"
"
"
X! = AX + BU
"
"
"
Y = CX + DU
u(t)
b
+
"
!
x(t)
d
∫
!
x(t)
c
+
a
Force-mass example:
F
Eytan Modiano
Slide 10
1/M
x1 = x !
x!1 = x2
"$
x2 = x! #
x!2 = !!
x = x!1 = F / M
x!2 (t)
∫
x2 (t) = x!1
∫
x1 (t)
!
y(t)
State Transformation
•
The state variable description of a system is not unique
•
Different state variable descriptions are obtained by “state transformation”
–
–
•
New state variables are weighted sum of original state variables
Changes the form of the system equations, but not the behavior of the system
Some examples: original system ~ x1(t), x2(t)
–
Transformed systems ~ z1(t), z2(t)
(1) z1(t) = x2(t), z2(t)=x1(t)
(2) z1(t) = x1(t)+x2(t), z2(t)=x2(t)
(3) z1(t) = 2x1(t) - x2(t), z2(t) = x1(t)+2x2(t)
•
Eytan Modiano
Slide 11
State transformation can simplify system description and analysis
State Transformation, continued
!
• In general, we can transform x
"
"
x! = Tx
•
"
to a new state vector x!
by,
Where T is the state transformation matrix
!
• Relationship between x
"
and x!
must be one-to-one (i.e., the
mapping must be invertible)
⇒ T must be non-singular
⇒ T-1 must exist
original!system :!!!!! x! = Ax + Bu
Transformed!system :! x" = Tx ! x"! = Tx! = TAx + TBu
! x"! = TAT "1 x + TBu!!(recall :!!x = T "1 x" )
!! A" = TAT "1 ,!!! B" = TB
" + Bu
"
!! x"! = Ax
Eytan Modiano
Slide 12
State Transformation (example)
•
Try to write down the state
equations explicitly
•
Notice that the state
transformation is a linear
combination of the original
system states
•
Notice that the new
transformed system has a
much simpler (to
understand) structure
–
–
Eytan Modiano
Slide 13
Two Decoupled first order
differential equations
The system is still the
same - just the description
is simpler
Original!system:!!! x! = Ax + Bu
" !1 4 %
"2 %
A=$
,!!!B = $ '
'
# 4 1&
#4 &
State!transformation:
!x1 + x2
x1 + x2
x"1 =
,! x" 2 =
2
2
1 " !1 1%
" !1 1%
!1
T= $
,!!T = $
'
'
1
1
2 # 1 1&
#
&
" !5 0 % "
"1 %
!1
"
A = TAT = $
,!! B = TB = $ '
'
# 0 3&
# 3&
x"!1 = !5 x"1 + u
x"!2 = 3x" 2 + 3u
State variable description of RLC circuits
y(t)
u(t)
•
•
•
•
Eytan Modiano
Slide 14
+
-
R1
e1
i1
+
v1
-
e2
i2
R2
C1
C2
+
v2
-
d
i1 (t) = C1 v1 (t)
dt
d
i2 (t) = C2 v2 (t)
dt
Input: voltage source ~ u(t)
System state: capacitor voltages ~ v1(t), v2(t)
Output: current through resistor R1 ~ y(t)
Node equations:
v1 ! u(t)
v1 ! v2
dv1 v2 ! v1 u(t) ! v1
e1 :!!
+ i1 +
=0"
=
!
R1
R2
dt
C1 R2
C1 R1
v2 ! v1
dv2
v2 ! v1 v1 ! v2
e2 :!!
+ i1 = 0 "
=!
=
R2
dt
C2 R2
C2 R2
Example, continued
" 1
1 %
1
1
v!1 = !v1 $
+
+ v2
+ u(t)
'
C1 R2
C1 R1
# C1 R1 C1 R2 &
1
1
v!2 = !v2
+ v1
C2 R2
C2 R2
1
( 1
*! C R ! C R
1 1
1 2
V! = *
1
*
*
C2 R2
)
Eytan Modiano
Slide 15
1 +
( 1 +
C1 R2
- V + * C1 R1 - u(t)
*
1 !
*) 0 -,
C2 R2 ,
( !1
+
(1+
u(t) ! v1
y(t) =
.Y = *
0 - V + * - u(t)
R1
) R1
,
) R1 ,