state space analysis - 20.13

20.2OBSERVABILITY

-a system is observable iff the system state x(t) can be found by observing the input u and output y over a period of time from x(t) to x(t+h).

-If an input is to be observable it must be detectable in the output. For example consider the following state equations.

d

x

=

a b

x

+

e

u

----

dt

v

c d

v

f

y

=

x

+

j

u

g h

v

k

if the parameters c, d or f were not in the final expression for y, the second state equation would be said to be not observable. This will happen when an output parameter is zero in the C matrix and an equation is decoupled in the A matrix, i.e. the same variable column is zero.

For example,
d
	x				=				1			0					x				+				3					u
----
dt	v									0 –a							v								b
y =													x		+								u
					5 0														6
													v																																													–1
G = C( sI – A) –1B + D =																																												s 0								1 0							3
																																			5 0														–													+		6
																																												0 s								0 –a							b
																																																										s + a				0
																																	–1																									0 s – 1
														s – 1														0									3																																	3
G =						5 0																																				+		6			=					5 0				--------------------------------																	+	6
																																																								( s – 1) ( s + a)
															0								s + a														b																																	b
															1
																																																				3
													----------								0																															----------											15
G =													s – 1																	3					+								=									s – 1						+												+ 6
						5 0																																	6									5 0											6
																																																															= ----------
																					1									b																						b													s – 1
															0						-----------																															-----------
																					s + a																															s + a

Note: the variables ’a’ and ’b’ both dissapeared because this system is not observable.

state space analysis - 20.14

•This often happens when a system has elements that are decoupled, or when a pole and zero cancel each other.

•Observability can be verified formally for an LTI system with the following rela-

tionship.

The system is unobservable if the equation below is true for any state,

CeAtx* = 0	t ≥ 0		1				0				0
	x* =	0			,	1			, …	0
		…				…				…
			0				0				1

One example,

A =

1 2

C =

5 6

3 4

et e2t

Ce

At

x

*

=

1

,

0

1

,

5 6

=

t

3t

2t

4t

e

3t

e

4t

0

1

( 5e

+ 6e

) ( 5e

+ 6e

)

0

CeAtx*

1

,

( 5et + 6e3t) ( 5e2t

+ 6e4t)

( 5et + 6e3t) ( 5e2t + 6e4t)

=

0

CeAtx* = {( 5et + 6e3t) , ( 5e2t + 6e4t) }

Both values are non-zero, so both states are observable.

One example,
A =			1		0				C =
														3 0
			0		2
												et
Ce	At	x	*		=									0					1			,	0										1			,	0				t	, 0	}
								3 0
																										=	3e		t	0										=	{3e
													0 e	2t					0				1										0				1

The second state is not observable.

• Another theorem for testing observability is given below. If any of the states satisfies the equation it is unobservable.

state space analysis - 20.15

	C
	CA		x* = 0
	…
CAn – 1

• Yet another test for observability is,

		C
rank( Mo) = rank		CA		= n
		…
	CAn – 1

• If a system in unobservable, it is possible to make it observable by changing the

model.

•A pole-zero cancellation is often the cause of the loss of observability.

•If all unstable modes are observable, the system is detectable.

20.3CONTROLLABILITY

-a system is controllable iff there is an input u(t) that will cause the system to go from any initial state to any final state in a finite time.

-stabilizable if it is controllable or if the uncontrollable nodes are stable.

-If an input is to be observable it must be detectable in the output. For example consider the following state equations.

state space analysis - 20.16

d

x

=

a b

x

+

e

u

----

dt

v

c d

v

f

y =

x

+

j

u

g h

v

k

if the parameters c, d or f were not in the final expression for y, the second state equation would be said to be not uncontrollable. This will happen when an output parameter is zero in the C matrix and an equation is decoupled in the A matrix, i.e. the same variable column is zero.

For example,
d
	x				=				1				0					x				+			2			u
----
dt	v									a –b								v							0
y =														x	+									u
					3 4															5
														v																																																–1
G = C( sI – A) –1B + D =																																													s 0								1 0													2
																																		3 4																	–																		+					5
																																													0 s									a –b												0
																																																									s + B 0
																											–1																														0								s – 1
														s – 1								0										2																																										2
G =						3 4																														+					5						=				3 4							--------------------------------																						+		5
														0								s + b										0																								( s – 1) ( s + b)																		0
														1
														----------								0																																																											6
G =														s – 1														2				+												=				3								4							2			+								=												+ 5
						3			4																												5																																	5
																																																----------								-----------																								----------
																						1						0																				s – 1 s + b															0																	s – 1
														0								-----------
																						s + b

Note: the variables ’a’ and ’b’ both dissapeared because this system is not controllable.

• This can be verified with the

state space analysis - 20.17

The basic relationship is,

	*			T			At				B = 0																								t ≥			0
( x )				e
An example is,
A =							1 2																	B =							5
							3 4																								6
												T			1 2			t
																																									1t		2t
	1			,			0								3 4						5										,								e			e				5
												e												=			1 0							0 1
		0					1														6																		e3t e4t							6
							,										5e				1t			+ 6e	2t				= {5e1t + 6e2t, 5e3t + 6e4t }
	1 0										0 1
																	5e3t + 6e4t
The results are both non-zero, so both states are controllable.
An example is,
A =							1 0																	B =							0
							0 2																								3
												T			1 0			t																							1t
	1			,			0								0 2						0										,								e			0				0
												e												=			1 0							0 1
		0							1												3																		0 e2t							3
							,												0					= {0, 3e2t }
	1 0										0 1
																	3e2t

The first state has a zero, so it in not controllable.

• Another test for controllability is,

	( x	*) T			n – 1
			B AB … A			B

• Yet another test for controllability is,