state variable description of lti systems · eytan modiano slide 3 the state of a system •the...

Eytan ModianoSlide 1

State Variable Description of LTI systems

Eytan Modiano


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


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


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


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


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


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


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


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)


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)

∫


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


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


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


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


Example, continued

!v1= !v

1

1

C1R1

+1

C1R2

"

#$%

&'+ v

2

1

C1R2

+ u(t)1

C1R1

!v2= !v

2

1

C2R2

+ v1

1

C2R2

!V =

!1

C1R1

!1

C1R2

1

C1R2

1

C2R2

!1

C2R2

(

)

****

+

,

----

V +

1

C1R1

0

(

)

***

+

,

---u(t)

y(t) =u(t) ! v

1

R1

.Y =!1R1

0(

)*

+

,-V +

1

R1

(

)*

+

,-u(t)

state variable description of lti systems · eytan modiano slide 3 the state of a system •the...

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