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Санкт-Петербургский государственный электротехнический университет "ЛЭТИ"

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Электротехника

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Jack H.Dynamic system modeling and control.2004.pdf

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state space control - 21.5

The state equations and the desired response of the system are,

·													·
·	0		1	x +	0	u							·		=		1	x
x =	–2 –3			x +	4	u							y		=		0	x
	–2 –3				4												0

Φ d( s) = s2 + s( 2) + ( 5)
The Markov (?) parameter can be calculated first.


Mc =		0		0	1		0			=		0	4
Mc =		0		0	1		0			=		4 –12
		4		–2 –3			4					4 –12

					–1							2

				0 4				0		1					0 1
																		( 2) + (			5) I
K =	0 1												+					( 2) + (			5) I
				4 –12				–2 –3							–2 –3
				4 –12				–2 –3							–2 –3

				–12 –4
				–4	0
				–4	0			–2			–3				0	2			5 0
K =	0 1			----------------------				–2			–3	+			0	2		+	5 0
				0 – 16					6	7				–4 –6					0 5
				0 – 16					6	7				–4 –6					0 5

K =				–0.75 –0.25							3 –1		=							3 –1		=
	0 1			–0.75 –0.25							3 –1					–0.25 0				3 –1			–0.75 0.25
	0 1			–0.25			0				2	6				–0.25 0				2	6		–0.75 0.25
				–0.25			0				2	6								2	6

NOTE: DOESN’T MATCH, TRACK DOWN ALGEBRA PROB

21.3OBSERVERS

•Observers are used to estimate the next system state. They can also be used to predict future state values.

•As an example, an observer might be used for a targeting system that is tracking a moving object. The estimator can be used to direct a missle to an estimated location so that the missle and target arrive at the same time.

•An observer is shown in block diagram form

state space control - 21.6

	Actual Plant
U	(and assumed state equations)
U	d
	d	Y
	----X = AX + BU	Y
	dt
	Y = CX

Observer
d ˆ	ˆ ˆ	ˆ	ˆ	ˆ
----X =	AX + BU + K( Y – Y)			Y
dt

ˆ	ˆ	ˆ
Y = CX		X

Figure 21.2 Block diagram of an observer

state space control - 21.7

Aside: For an open loop estimator a simulation is run, while the system is running. The simulation values are compared to the actual system responses, and the model is adjusted. It is assumed that the initial value of the state is unknown. If A is stable then the estimation will eventually converge (this can be checked using the Eigenvalues). However, there is no guarantee that it will converge quickly.

The actual system
d			Ax + Bu	y = Cx			C ≠ I
----x =			Ax + Bu	y = Cx			C ≠ I
dt
The state estimator model
d	ˆ		ˆ
----x =			Ax + Bu
dt
The estimation error
˜			ˆ
x	= x – x
d			ˆ	ˆ	d	˜	˜
----	( x – x) = A( x – x)				----x =		Ax
dt					dt
˜	= e	At ˜
x( t)	= e		x( 0)

• A closed loop estimatior is shown below. It uses a learning matrix L to adjust the convergence rate. A larger L value will result in faster convergence, but the exponent must

			state space control - 21.8
now be checked to ensure observability.
The system and estimator
y = Cx			ˆ	ˆ
y = Cx			y = Cx
The estimator error
˜	ˆ		˜
y = y – y = Cx
The estimator
d ˆ		ˆ	+ Bu( t)	+ L( y( t)	˜
----x	= Ax( t)		+ Bu( t)	+ L( y( t)	– y( t) )
dt		ˆ
ˆ		ˆ
y( t)	= Cx( t)
where,
		L =	a gain matrix
The convergence rate,
d ˜			˜
----x = ( A – LC) x
dt
˜	= e	( A – LC) t ˜
x( t)	= e		x( 0)

• Estimator gain selection

If a system is controllable the Eigenvalues of A-BK can be placed anywhere,

rank( A) = rank B AB … An – 1B = n

If a system is observable the Eigenvalues of A-LC can be placed anywhere,

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

• Consider the example below, for a second order system. The learning rate values are selected to determine how quickly the model converges.

state space control - 21.9

Consider the state equations,

–a

y =

----

1 0

–a

Find the convergence rate and then the Eigenvalues,

A – LC =

–a1 1

–

– a1 – l1 1

1 0

–a2 0

– a2 – l2 0

sI – ( A – LC)

s + a1 + l1

–1

a2 + l2

= s( s + a1 + l1) – ( –1) ( a2 + l2)

=s2( 1) + s( a1 + l1) + ( a2 + l2)

•This can also be done with Ackermann’s formula,

					0
T	= Φ		–1	…
L = Ke	= Φ	e	( s) Mo	0
				0
					1

state space control - 21.10

Consider the state equations for a mass spring damper system (y is the position).

y =

1 0

----

–Ks

–Kd

--------

---------

----

B =

A =

–Ks

–Kd

C =

D =

1 0

--------

---------

----

First test for observability, (note: it is observable)


					1 0
					1 0

								1 0		1 0
rank		C	= rank				= rank		= rank		= 2
		C			0	1
	CA				0	1
	CA			1 0	–Ks	–Kd		0 1		0 1
				1 0	–Ks	–Kd		0 1		0 1
					--------	---------
					--------	---------
					M	M

Consider the open loop estimator

d	ˆ			ˆ
----x =			Ax + Bu
dt

					0		1
d		ˆ			0		1
d		x	=
----		ˆ	=		–Ks		–Kd
dt		ˆ			–Ks		–Kd
		v			-------- ---------
						M	M
						M	M

			0
ˆ			0
x	+	1		F
ˆ		----
v		M
		M

ˆ	ˆ
y = Cx

ˆ		ˆ
ˆ		x
y =	1 0	ˆ
		v

• The open loop estimator will normally ignore the response to initial conditions and will have long term errors resulting from modeling errors and random disturbances in the system.

Consider the closed loop estimator with a learning rate.

----x =

Ax( t) + Bu( t)

+ Ly( t)

–K

F +

----

--------

---------

ˆ		ˆ
y( t)	= Cx( t)

ˆ			ˆ
ˆ	=		x
y( t)	=	1 0	ˆ
			ˆ
			v

These are then simulated numerically using initial conditions of zero.

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