Hardware Reference

In-Depth Information

£

¤

£

¤

T
, C =

x
p

x
pi
x
vi

where x =

C
p
00

and

⎡

⎣

⎤

⎦

A
p
00

−C
p
10

A =

,

£

¤

0 −C
v

01

⎡

⎤

⎡

⎤

B
p

0

0

0

1

1

⎣

⎦

⎣

⎦

B =

,B
r
=

.

Let us consider the following quadratic performance index,

X

∞

¡

¢

,

J =
1

2

z(k)
T
R
1
z(k)+u
k
R
2
u(k)

(3.155)

k=0

£

¤

T
. Then the

design problem can be restated as:
fi
nd a proper controller for the generalized

system (equation 3.154) such that the closed-loop system is stable, and the

linear quadratic (LQ) performance J is minimized.

To obtain the dual-stage controller, one can separate the design into the

following two steps:

where R
1
≥ 0, R
2
> 0areweights,andz(k)=

x
pi
x
vi

1. In the first step, we assume that all the states of the generalized system

are available and design a static state feedback control law u(k)=Kx(k),

such that it solves the LQ optimal control problem. Since

©

ª

A, B

is

n

o

R
1
,A

stabilizable, if R
1
is selected such that

is detectable, then

it follows that this problem of static state feedback LQ optimal control

is solvable and the feedback gain can be obtained by:

K = −(R
2
+ B
T
PB)
−1
B
T
PA

(3.156)

where P>0 is the unique stabilizing solution of the following Riccati

equation:

A
T
PA−P + R
1
−A
T
PB(R
2
+ B
T
PB)
−1
B
T
PA=0.

(3.157)

2. Since not all the states of the system in our problem are measurable, we

need an observer to reconstruct the unmeasured states from the mea-

surements of inputs and outputs. Since the states of the integrator, x
pi

and x
vi
, can be directly derived from equation 3.153, only the states for

the model given by equation 3.151 need to be estimated. We can use the

following state observer:

x
o
(k +1) = A
p
x
o
(k)+B
p
u
p
(k),

+ L(C
p
x
o
(k) −y
p
(k)),

(3.158)