Eytan ModianoSlide 2
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– Given a state transformation matrix, develop model for the
transformed system
Eytan ModianoSlide 3
The State of a System
• The “state” of a system is the minimum information needed aboutthe system in order to determine its future behavior
– Given the state at time t0, and input up to time t > t0; can determine theoutput 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 ModianoSlide 4
Example: RLC circuit
• V(t) is the state of the system at time t– Initial condition - V(0) is the voltage
across the capacitor at time 0
• If we know v(t) at any time t, we know itfor all future time
– No input in this case
iicc +V(t)
-
R
C
ee11
!!!i
c= C
dv
dt
!ic= !V / R
"
#$
%$&
dv
dt= !
V (t)
RC
V (t) = V (0)e! t /RC
V(t)
V(0)
t
Eytan ModianoSlide 5
State Variables
• In electric circuits, the energy storage devices are the capacitors andinductors
– They contain all of the state information or “memory” in the system– State variables:
Voltage across capacitors Current through inductors
• In mechanical systems, energy is stored in springs and masses– State variables
Spring displacement Mass position and velocity
• Example” Single mass M, moving in one dimension (x), under force F– State variables (x1, x2)
x1 = mass position x2 = velocity
MPosition (x)
F F = M!!x
x1= x
x2= !x
!"#$
!x1= x
2
!x2= !!x = F /M
Eytan ModianoSlide 6
State Equations
• State equations in matrix form:
• Output equations:– suppose output is y=x1 (position)
• General form of state equations:
• We will focus (to start) mainly on homogeneous case:
!x =
x1
x2
!
"#
$
%&,!!input :!u = F /M
State!equations :! "x = Ax + Bu
"x1
"x2
!
"#
$
%& =
0 1
0 0
!
"#
$
%&x1
x2
!
"#
$
%& +
0
1
!
"#
$
%&F /M
y = 1 0[ ]x1
x2
!
"#
$
%&
!x = Ax
!x = Ax + Bu
y = Cx + Du
Eytan ModianoSlide 7
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)
• The state and output equations can be written as:
• A,B,C,D are constant matrices– 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
!X(t) =
x1(t)
"
xn(t)
!
"
###
$
%
&&&
,!!!!U(t) =
u1(t)
"
un(t)
!
"
###
$
%
&&&
,!!!Y (t) =
y1(t)
"
yn(t)
!
"
###
$
%
&&&
!"
X = A"X + B
"U
"Y = C
"X + D
"U
Eytan ModianoSlide 8
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)– “New” approach was named “modern” control and still have that name
– Related state space approach to describing differential equations isover 100 years old
Eytan ModianoSlide 9
Block Diagram Representation
• x(t) - state variables• u(t) - inputs
∫x(t) x(! )"#
t
$ d!
Integrator block diagram
dx(t)
dt= ax(t) + bu(t)
x(t) = ax(! ) + bu(! )d!"#
t
$
u(t) b
a
+ ∫ x(t)x(t) !x(t)
State is “fed-back” into the system
System block diagram
x(t) ! integrator!output
!x(t) ! integrator!input
!x(t) = ax(t) + bu(t)
Eytan ModianoSlide 10
General system block diagram
!"
X = A"X + B
"U
"Y = C
"X + D
"U
u(t) b
a
+ ∫
!y(t)
!x(t)
!"x(t) c +
d
x1= x
x2= !x
!"#$
!x1= x
2
!x2= !!x = !x
1= F /M
Force-mass example:
F 1/M ∫x1(t)
x2(t) = !x
1 !x2(t)
∫
Eytan ModianoSlide 11
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)
• State transformation can simplify system description and analysis
Eytan ModianoSlide 12
State Transformation, continued
• In general, we can transform to a new state vector by,
• Where T is the state transformation matrix
• Relationship between and must be one-to-one (i.e., themapping must be invertible)⇒ T must be non-singular⇒ T-1 must exist
!x
!"x
!"x = T
"x
!x
!"x
original !system :!!!!! !x = Ax + Bu
Transformed !system :! "x = Tx! "!x = T!x = TAx + TBu
! "!x = TAT"1x + TBu !!(recall :!!x = T
"1"x)
!! "A = TAT"1,!!! "B = TB
!! "!x = "Ax + "Bu
Eytan ModianoSlide 13
State Transformation (example)
• Try to write down the stateequations explicitly
• Notice that the statetransformation is a linearcombination of the originalsystem states
• Notice that the newtransformed system has amuch simpler (tounderstand) structure
– Two Decoupled first orderdifferential equations
– The system is still thesame - just the descriptionis simpler
Original!system:! !! !x = Ax + Bu
A =!1 4
4 1
"
#$
%
&',!!!B =
2
4
"
#$
%
&'
State!transformation:
"x1 =!x1 + x22
,! "x2 =x1 + x2
2
T =1
2
!1 1
1 1
"
#$
%
&',!!T
!1=
!1 1
1 1
"
#$
%
&'
"A = TAT!1=
!5 0
0 3
"
#$
%
&',!!"B = TB =
1
3
"
#$%
&'
"!x1 = !5 "x1 + u
"!x2 = 3 "x2 + 3u
Eytan ModianoSlide 14
State variable description of RLC circuits
• Input: voltage source ~ u(t)• System state: capacitor voltages ~ v1(t), v2(t)• Output: current through resistor R1 ~ y(t)• Node equations:
ee11 ee22
ii22 +v2-
ii11+v1-
R1 R2
C1 C2 + -
y(t)
u(t)i1(t) = C
1
d
dtv1(t)
i2(t) = C
2
d
dtv2(t)
e1:!!v1! u(t)
R1
+ i1+v1! v
2
R2
= 0"dv
1
dt=v2! v
1
C1R2
!u(t) ! v
1
C1R1
e2:!!v2! v
1
R2
+ i1= 0"
dv2
dt= !
v2! v
1
C2R2
=v1! v
2
C2R2