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ECE5530: Multivariable Control Systems II. 8–1
H1 FULL-INFORMATION CONTROL
AND ESTIMATION
8.1: Differential games
■ H2 control optimizes average-case performance. Assumes complete
and perfect plant knowledge.
■ H1 control optimizes worst-case performance (gain). Also assumes
perfect and complete plant knowledge.
! H1 does not guarantee robustness directly.
! We will see more next chapter re. !-synthesis to guarantee
robustness.
■ LQR, LQE and LQG are all posed as H2 optimization problems. The
cost functions may be modified easily to pose them as H1
optimization problems.
■ So, for H1 control, we will have a similar set of problems to solve, but
different cost functions to optimize.
■ The H1 problem seeks to minimize the maximum gain of some
system over the set of disturbance inputs.
! “Disturbance” inputs W are true disturbance, reference inputs, etc.
! Performance “outputs” Y comprise the system state x.t/, input
u.t/, etc.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–2
■ The H1 problem may be posed as a two-player game: The designer
seeks a control to minimize the system gain; Nature seeks a
disturbance input to maximize the gain.
■ Dynamics specified by differential equations ➠ “Differential Game.”
■ The state dynamics are modeled by
Px.t/ D Ax.t/ C Buu.t/ C Bww.t/
■ The objective function is a real function of the state, control, and
disturbance: J fx.t/; u.t/; w.t/g:
■ The solution consists of the optimal control trajectory u".t/ and the
worst-case disturbance input w".t/.
J fx.u"; w/; u"; wg # J fx.u"; w"/; u"; w"g # J fx.u; w"/; u; w"g:
■ The control solution is then of the mini-max problem
u" D arg minu
.maxw
J fx.u; w/; u; wg/:
■ Lagrange multipliers may be used to convert a constrained mini-max
problem into an unconstrained mini-max problem of higher order
Ja.x; u; w; "/ D J.x; u; w/C
Z tf
0
"T .t/fAx.t/CBuu.t/CBww.t/$ Px.t/g dt:
■ A necessary condition for a saddle point is that the variation of Ja
must be zero.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–3
8.2: Full-information control
■ Our first control design will assume “full information:”
! Plant state known,
! Input “disturbance” known.
■ The plant is modeled as
Px.t/ D Ax.t/ C Buu.t/ C Bww.t/
y.t/ D Cyx.t/ C Dyuu.t/;
where DTyuCy D 0 and DT
yuDyu D I .
■ The output consists of two parts: Linear combinations of the state
x.t/ and linear combinations of the control u.t/.
! The conditions DTyuCy D 0 and DT
yuDyu D I enforce separation
between these two parts, and normalize the contribution of u.t/ to
the output.
! For example, Cy Dh
In 0iT
and Dyu Dh
0 Im
iT
.
■ These restrictions may be relaxed at the expense of more difficult
mathematics to follow (we won’t relax them!)
■ The H1 full-information control problem
is to find a feedback controller, using the
state and disturbance, that minimizes
the closed-loop 1-norm
J D!!Gyw
!!
1;Œ0;tf #D sup
kw.t/k2;Œ0;tf #¤0
ky.t/k2;Œ0;tf #
kw.t/k2;Œ0;tf #
:
w.t/
u.t/ x.t/
y.t/
ControllerK.%/
Plant
■ The controller is a linear system (not necessarily time-invariant),
denoted K.%/.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–4
■ This objective function may not be used directly as a differential game
since it is a function of only the controller, and not of any particular
single disturbance signal.
! The sup.%/ makes the cost J independent of any particular
disturbance input.
! We can drop the sup.%/ but do not end up with a tractable
optimization problem.
■ We modify the function in a few steps to get a more useful
representation that does remove the sup.%/.
■ First, consider that the optimum controller must be better than some
suboptimal controller having cost $ . So,
!!Gyw
!!
1;Œ0;tf #D sup
kw.t/k2;Œ0;tf #¤0
ky.t/k2;Œ0;tf #
kw.t/k2;Œ0;tf #
< $:
■ Controllers that satisfy this bound are suboptimal solutions to the H1
full-information control problem (or simply suboptimal controllers).
■ The same result is obtained by squaring both sides of the inequality
!!Gyw
!!
2
1;Œ0;tf #D sup
kw.t/k2;Œ0;tf #¤0
(
ky.t/k22;Œ0;tf #
kw.t/k22;Œ0;tf #
)
< $2:
■ To satisfy this strict inequality, the term in f%g must be bounded away
from $2.ky.t/k2
2;Œ0;tf #
kw.t/k22;Œ0;tf #
# $2 $ "2:
■ Regrouping,
ky.t/k22;Œ0;tf # $ $2 kw.t/k2
2;Œ0;tf # # $"2 kw.t/k22;Œ0;tf # :
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–5
■ Satisfying this inequality means that the closed-loop infinity norm is
bounded by $ for all disturbance inputs and for some ".
■ We can then use the expression on the left as an objective function
for differential game theory
J$.x; u; w/ D ky.t/k22;Œ0;tf # $ $2 kw.t/k2
2;Œ0;tf # :
Bisection Algorithm
■ If $ is initially set high enough (we will see how to test for this later),
then a solution exists for J$ (to be developed in the next sections).
■ This solution is a suboptimal solution to the original H1 optimization
problem.
■ The goal is to find the smallest $ and approximate the H1 optimal
solution as closely as possible.
■ The following algorithm may be used to search for a good suboptimal
controller.
Bisection search algorithm steps
1. Select $u, $l such that $l #!!Gyw
!!
1;Œ0;tf ## $u.
2. Test .$u $ $l /=$l < tol.
■ If true, stop.!!Gyw
!!
1;Œ0;tf #&
1
2.$u C $l /.
■ If false, continue.
3. With $ D1
2.$l C $u/, test if
!!Gyw
!!
1;Œ0;tf #< $ using a test TBD.
■ If true, set $u D $ (test value too high) and go to step 2.
■ If false, set $l D $ (test value too low) and go to step 2.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–6
8.3: The Hamiltonian equations
■ We now solve the suboptimal H1 control problem.
■ We form the augmented cost function
Ja;$.u; w; "/ D
Z tf
0
yT .t/y.t/ $ $2wT .t/w.t/
C2"T .t/fAx.t/ C Buu.t/ C Bww.t/ $ Px.t/g dt:
■ Note that, by our assumptions on Cy and Dyu,
yT .t/y.t/ D xT .t/C Ty Cyx.t/ C xT .t/C T
y Dyuu.t/„ ƒ‚ …
0
C uT .t/DTyuCyx.t/
„ ƒ‚ …
0
CuT .t/ DTyuDyu
„ ƒ‚ …
I
u.t/:
■ Next, we form the increment of the cost function
%Ja;$.u; w; "; ıu; ıw; ı"/
D Ja;$.u C ıu; w C ıw; " C ı"/ $ Ja;$.u; w; "/
D
Z tf
0
.x C ıx/T C Ty Cy.x C ıx/ C .u C ıu/T .u C ıu/
$$2.w C ıw/T .w C ıw/
C2." C ı"/T fA.x C ıx/ C Bu.u C ıu/ C Bw.w C ıw/ $ . Px C ı Px/g dt
$
Z tf
0
xT C Ty Cyx C uT u $ $2wT w C 2"T fAx C Buu C Bww $ Pxg dt:
■ Expanding and grouping,
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–7
%Ja;$ D
Z tf
0
ıxT C Ty Cyıx C ıuT ıu $ $2ıwT ıw
C2ı"T fAıx C Buıu C Bwıw $ ı Pxg C 2uT ıu
C2xT C Ty Cyıx $ 2$2wT ıw C 2ı"T fAx C Buu C Bww $ Pxg
C2"T fAıx C Buıu C Bwıw $ ı Pxg dt:
■ The variation is (set to zero to find optimum)
ıJa;$ D
Z tf
0
2xT C Ty Cyıx C 2uT ıu $ 2$2wT ıw
C2ı"T fAx C Buu C Bww $ Pxg
C2"T fAıx C Buıu C Bwıw $ ı Pxg dt D 0:
■ Integration by parts yieldsZ tf
0
"T .t/ı Px.t/ dt D "T .tf /ıx.tf / $ "T .0/ıx.0/ $
Z tf
0
P"T .t/ıx.t/ dt:
■ We assume that the initial state is fixed so ıx.0/ D 0. Substituting:
ıJa;$ D $2"T .tf /ıx.tf / C
Z tf
0
.2 P"T C 2xT C Ty Cy C 2"T A/ıx
C.2"T Bw $ 2$2wT /ıw C .2uT C 2"T Bu/ıu
C2ı"T .Ax C Buu C Bww $ Px/ dt D 0:
■ The conditions for optimality is then
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–8
".tf / D 0I
P".t/ D $C Ty Cyx.t/ $ AT ".t/I
u.t/ D $BTu ".t/I
w.t/ D $$2BTw ".t/I
Px.t/ D Ax.t/ C Buu.t/ C Bww.t/:
■ Combining, we get a Hamiltonian equation
"
Px.t/
P".t/
#
D
"
A $BuBTu C $$2BwBT
w
$C Ty Cy $AT
#
„ ƒ‚ …
H1 Hamiltonian Matrix
"
x.t/
".t/
#
D Z1
"
x.t/
".t/
#
with initial condition x.0/ D 0 and final condition ".tf / D 0.
■ The solution to this system, at final time, is
"
x.tf /
".tf /
#
D eZ1.tf $t /
"
x.t/
".t/
#
D
"
ˆ11.tf $ t/ ˆ12.tf $ t/
ˆ21.tf $ t/ ˆ22.tf $ t/
# "
x.t/
".t/
#
:
■ Substitute the final condition ".tf / D 0,
"
x.tf /
0
#
D
"
ˆ11.tf $ t/ ˆ12.tf $ t/
ˆ21.tf $ t/ ˆ22.tf $ t/
# "
x.t/
".t/
#
■ Solving the lower-triangular block
".t/ D $fˆ22.tf $ t/g$1ˆ21.tf $ t/x.t/ D P.t/x.t/:
■ Concluding,
u.t/ D $BTu P.t/x.t/ D BT
u fˆ22.tf $ t/g$1ˆ21.tf $ t/x.t/ D $K.t/x.t/:
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–9
EXAMPLE 9.1: We wish to control a plant that is modeled as
Px.t/ D x.t/ C u.t/ C 2w.t/
y.t/ D
"
10
0
#
x.t/ C
"
0
1
#
u.t/
where the objective function is
J$fx.t/; u.t/; w.t/g D ky.t/k22 $ $2 kw.t/k2
2 :
■ Note: State and control weightings are incorporated into the “output”
equation, which is scaled to yield a unity-weight on the control.
■ The Hamiltonian system is:"
Px.t/P".t/
#
D
"
1 $1 C 4$$2
$100 $1
# "
x.t/
".t/
#
;
which has state-transition matrix
eZ1t D L$1Œ.sI $ Z1/$1#
D L$1
"
1
s2 $ 101 C 400$$2
"
s C 1 $1 C 4$$2
$100 s $ 1
##
■ An H1 full-information suboptimal controller exists if ˆ22 is invertible
throughout the desired time interval.
■ This element of the state-transition matrix is:
ˆ22.t/ D L$1
"
s $ 1
s2 $ 101 C 400$$2
#
D
8
ˆˆˆˆˆˆ<
ˆˆˆˆˆˆ:
a $ 1
2aeat C
a C 1
2ae$at ; $ >
r
400
101
1 $ t; $ D
r
400
101r
!2 C 1
!2sin.!t C &/; $ <
r
400
101
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–10
where a Dp
101 $ 400$$2, ! Dp
$101 C 400$$2, and & D $ tan$1.!/.
■ Note that as $ gets smaller, a gets smaller, ! gets bigger, and & gets
more negative.
■ When $ <p
400=101 the eigenvalues of the Hamiltonian matrix are
imaginary, and the gain does not exist for long time intervals since
ˆ22.t/ periodically crosses zero.
■ The lack of solution for feedback gain means there is no solution of
the differential game for the given bound.
■ For the suboptimal bound
$ D 2:03 (a D 2/ and the final
time tf D 3, the H1 suboptimal
feedback gain is
K.t/ D100e2.3$t / $ 100e$2.3$t /
e2.3$t / C 3e$2.3$t /:
0 0.5 1 1.5 2 2.5 3
0
20
40
60
80
100
Time (s)
Feed
back
gai
n
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–11
8.4: Riccati equations to find state feedback
The Riccati equation
■ Solving the Hamiltonian for the time-varying state feedback may be
done via a Riccati equation.
■ Given that ".t/ D P.t/x.t/, differentiate to get
P".t/ D PP .t/x.t/ C P.t/ Px.t/:
■ Substitute for P".t/ and Px.t/ from the Hamiltonian system
$C Ty Cyx.t/$AT ".t/ D PP .t/x.t/CP.t/fAx.t/$.BuBT
u $$$2BwBTw /".t/g:
■ Then, substituting for ".t/ D P.t/x.t/,
f PP .t/CP.t/ACAT P.t/CC Ty Cy$P.t/.BuBT
u $$$2BwBTw /P.t/gx.t/ D 0:
■ Since this is valid for any x.t/
PP .t/ D $P.t/A $ AT P.t/ $ C Ty Cy C P.t/.BuBT
u $ $$2BwBTw /P.t/;
which is the Riccati equation for the H1 suboptimal control problem,
initialized with P.tf / D 0.
■ A suboptimal H1 controller exists for performance bound $ iff there
exists a solution P ' 0 to
PA C AT P $ P.BuBTu $ $$2BwBT
w /P C C Ty Cy D 0;
where the matrix A $ .BuBTu $ $$2BwBT
w /P is stable.
! This may be used as the test in the bisection search algorithm for
existence of an H1 controller for performance bound $ .
EXAMPLE 9.1 CONTINUED: Here, we use the Riccati equation to solve
the same problem as in the prior example.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–12
■ The Riccati equation is (A D 1, Bu D 1, Bw D 2, Cy Dh
10 0iT
):
PP .t/ D $2P.t/ C .1 $ 4$$2/P 2.t/ $ 100;
with final condition P.3/ D 0.
■ The solution is
P.t/ D K.t/ D100e2.3$t / $ 100e$2.3$t /
e2.3$t / C 3e$2.3$t /;
which is the same as we found before.
■ (The solution can be verified by substituting P.t/ back into the Riccati
equation). Note, P.t/ D K.t/ because Bu D 1.
Steady-state result
■ The steady-state result may be found by setting PP .t/ D 0 and solving
the A.R.E.
■ An eigenvector method may again be used to solve for Pss
P D ‰21.‰11/$1;
where ‰ is the eigensystem of the Hamiltonian system, and ‰21 and
‰11 are formed from the eigenvectors associated with the stable
eigenvalues.
■ A suboptimal H1 controller exists for performance bound $ iff Z1
has no eigenvalues on the imaginary axis, ‰11 is invertible, and
P D ‰21.‰11/$1 ' 0.
! This may also be used in the test in the bisection search algorithm
for existence of an H1 controller for performance bound $ .
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–13
■ The suboptimal controller is then given as
u.t/ D $BTu ‰21.‰11/
$1x.t/ D $Kx.t/:
EXAMPLE 9.2: An antenna is required to remain pointed at a satellite in
the presence of disturbance torques caused by gravity, wind,
squirrels, etc. The state model is
Px.t/ D
"
0 1
0 $1
#
x.t/ C
"
0
1
#
u.t/ C
"
0
1
#
w.t/
e.t/ Dh
1 0i
x.t/:
■ As usual, u.t/ is the control torque, w.t/ is the disturbance torque,
and e.t/ is the tracking error.
■ A tracking error of less than 0:1 rad is desired, even in the presence of
disturbances up to 2 N-m.
■ The control magnitude is also required to be less than 10 N-m.
■ These specifications are appended to the plant as weighting functions
to yield a model in standard form:
Px.t/ D
"
0 1
0 $1
#
x.t/ C
"
0
10
#
u1.t/ C
"
0
2
#
w1.t/
y.t/ D
"
10 0
0 0
#
x.t/ C
"
0
1
#
u1.t/;
where w1.t/ and u1.t/ are the normalized disturbance and control
inputs.
■ In this form, w1.t/ is less than 1 and both elements of y.t/ are
required to be less than 1. The original control input can be recovered
from the normalized control as u.t/ D 10u1.t/.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–14
■ A full-information controller is generated for $ D 1:
u1.t/ D $h
10:2 1:36i
x.t/ or u.t/ D $h
102 13:6i
x.t/:
100 101 102−90
−80
−70
−60
−50
−40
Frequency (rad/s)
Gai
n (d
B)
Tracking error
γ = 1γ = 0.2
100 101 102−20
−15
−10
−5
0
5
Frequency (rad/s)G
ain
(dB)
Control effort
γ = 1γ = 0.2
■ The left plot shows frequency response of tracking error; the right plot
shows frequency response of control effort.
■ Maximum gain from sinusoidal disturbance to tracking error is
10$40=20 D 0:01, which is less than spec of 0:05.
■ Maximum gain from disturbance input to control is 1.26, which is less
than spec of 5.
■ A second full-information controller is generated for $ D 0:21.
u1.t/ D $h
32:8 7:39i
x.t/ or u.t/ D $h
328 73:9i
x.t/:
■ Performance is better, but gains are higher. Disturbance rejection is
higher, but control bandwidth is also higher.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–15
8.5: H1 Estimation
■ When investigating H2 control, we split the problem into a regulation
part (LQR) and an estimation part (LQE).
■ When joined, the result turned out to be optimal as well (LQG).
■ One reason that this works is because H2 optimization of $K.t/x.t/
is equal to $K.t/ times the H2 optimization of x.t/.
! This same property does not hold for H1 estimation; we must be
more careful when generating an H1 output feedback controller.
! H1 estimation is still required, so we study it briefly here.
■ The plant model that is assumed looks like
Px.t/ D Ax.t/ C Buu.t/ C Bww.t/
m.t/ D Cmx.t/ C Dmww.t/
with constraints DmwBTw D 0 and DmwDT
mw D I .
■ Goal is to estimate a linear combination of the state, y.t/ D Cyx.t/.
! One example would be to estimate the state itself: Cy D I .
■ H1 estimation is the dual problem to H1 regulation. As you might
expect, there is a Hamiltonian system involved
2
4 Px Oh.t/
3
5 D
2
4$AT C T
m Cm $ $$2C Ty Cy
BwBTw A
3
5
2
4x Oh.t/
3
5 D Y1x Oh.t/:
■ The H1 suboptimal estimator gain may be found in terms of the
state-transition matrix of the Hamiltonian system:
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–16
L.t/ D Q.t/C Tm D ˆ21.t/fˆ11.t/g
$1C Tm
where the ˆij are terms of the state-transition matrix:
eY1t D
2
4ˆ11.t/ ˆ12.t/
ˆ21.t/ ˆ22.t/
3
5 :
■ A steady-state solution may be found from the eigensystem of Y1 as
Q D ‰22.‰12/$1;
where
2
4‰12
‰22
3
5
is a matrix whose columns are the eigenvectors of the estimator
Hamiltonian associated with the unstable eigenvalues.
■ A suboptimal H1 estimator exists if there is a solution to Y1 that has
no eigenvalues on the imaginary axis, ‰12 is invertible, and
Q D ‰22.‰12/$1 ' 0.
! This may be used in the bisection search algorithm for existence of
an H1 estimator for performance bound $ .
■ These result may also be obtained from a Riccati equation
PQ.t/ D Q.t/AT C AQ.t/ C BwBTw $ Q.t/fC T
m Cm $ $$2C Ty CygQ.t/;
with initial condition Q.0/ D 0.
■ A suboptimal H1 estimator exists if there is a solution to
AQ C QT A $ Q.C Tm Cm $ $$2C T
y Cy/Q C BwBTw D 0;
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–17
such that Q ' 0 and that the matrix A $ Q.C Tm Cm $ $$2C T
y Cy/ is
stable.
! This may also be used in the bisection search algorithm for
existence of an H1 estimator for performance bound $ .
EXAMPLE 9.3: An H1 estimator can be applied to the radar range
tracking of an aircraft. A model equations for the range r.t/ and radial
velocity of an aircraft is (range is measured):"
Pr.t/
Rr.t/
#
D
"
0 1
0 0
# "
r.t/
Pr.t/
#
C
"
0
1
#
w.t/
m.t/ Dh
1 0i
"
r.t/
Pr.t/
#
C v.t/:
■ The output to be estimated is the entire state:
y.t/ D
"
1 0
0 1
# "
r.t/
Pr.t/
#
■ The plant input and measurement error are assumed to be bounded:"
jw.t/j
jv.t/j
#
#
"
4 m=sec2
20 m
#
■ Model equations using normalized inputs are:"
Pr.t/
Rr.t/
#
D
"
0 1
0 0
# "
r.t/
Pr.t/
#
C
"
0 0
4 0
# "
w1.t/
v1.t/
#
m.t/ Dh
1 0i
"
r.t/
Pr.t/
#
Ch
0 20i
"
w1.t/
v1.t/
#
■ Normalizing this equation so that the input-output coupling matrix
satisfies DmwDTmw D I gives the measurement equation
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–18
m1.t/ D
"
1
200
#"
r.t/
Pr.t/
#
Ch
0 1i
"
w1.t/
v1.t/
#
where m.t/ D 20m1.t/.
■ The resulting H1 estimator is given as
POx.t/ D A Ox.t/ C L1.t/
"
m1.t/ $
"
1
200
#
Ox.t/
#
Oy.t/ D Cy Ox.t/:
■ We can write this in terms of the original measurements,
POx.t/ D A Ox.t/ C1
20L1.t/
"
20m1.t/ $ 20
"
1
200
#
Ox.t/
#
D A Ox.t/ C L.t/h
m.t/ $h
1 0i
Ox.t/i
;
where L.t/ D L1.t/=20:
■ Baseline results are plotted
below.
0 5 10 15 20 250
1
2
3
4
5
Time (s)
Gai
ns
1
2L
L
0 5 10 15 20 250.8
0.9
1
1.1
1.2 x 104
Time (s)
Ran
ge
ActualEstimate
0 5 10 15 20 25−100
−50
0
50
100
Time (s)
Ran
ge ra
te
ActualEstimate
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–19
■ If meas. error decreases from 20
to 2, we get the these results.
■ Estimation gains increase;
estimates converge more quickly.0 5 10 15 20 250
5
10
15
Time (s)
Gai
ns
1
2L
L
0 5 10 15 20 250.8
0.9
1
1.1
1.2 x 104
Time (s)
Ran
ge
ActualEstimate
0 5 10 15 20 25−100
−50
0
50
100
Time (s)
Ran
ge ra
te
ActualEstimate
■ Finally, if reference output is only
range rate (we still estimate
range, but don’t care about its
accuracy), we get these results.
■ Gains change (slightly)! The H1
estimator depends on the output
of interest!0 5 10 15 20 250
1
2
3
4
5
Time (s)
Gai
ns
1
2L
L
0 5 10 15 20 250.8
0.9
1
1.1
1.2 x 104
Time (s)
Ran
ge
ActualEstimate
0 5 10 15 20 25−100
−50
0
50
100
Time (s)
Ran
ge ra
te
ActualEstimate
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–20
Summary
■ H1 controller and estimator design is an iterative procedure.
■ Perform the bisection algorithm to find the lowest value of the
performance bound $ that you can, within some tolerance.
! You will generally back off of this performance bound somewhat,
as designs that are nearly optimal can have numeric instabilities.
■ For controller design, the conditions for existence of some
suboptimal controller with bound $ are:
! There exists a solution P ' 0 to
PA C AT P $ P.BuBTu $ $$2BwBT
w /P C C Ty Cy D 0;
where the matrix A $ .BuBTu $ $$2BwBT
w /P is stable; or,
! Z1 has no eigenvalues on the imaginary axis, ‰11 is invertible,
and P D ‰21.‰11/$1 ' 0.
■ For estimator design, the conditions for existence of a suboptimal
controller with bound $ are:
! There exists a solution Q ' 0 to
AQ C QT A $ Q.C Tm Cm $ $$2C T
y Cy/Q C BwBTw D 0;
where the matrix A $ Q.C Tm Cm $ $$2C T
y Cy/ is stable; or,
! Y1 has no eigenvalues on the imaginary axis, ‰12 is invertible, and
Q D ‰22.‰12/$1 ' 0.
■ For controller implementation, with some $ , solve the differential
Riccati equation
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
ECE5530, H1 FULL-INFORMATION CONTROL AND ESTIMATION 8–21
PP .t/ D $P.t/A $ AT P.t/ $ C Ty Cy C P.t/.BuBT
u $ $$2BwBTw /P.t/;
backward in time, initialized with P.tf / D 0. Then,
u.t/ D $K.t/x.t/ D $BTu P.t/x.t/:
■ A steady-state suboptimal controller, with some $ , uses the
eigensystem ‰ of the Z1 matrix to find the steady-state gain matrix:
u.t/ D $Kx.t/ D $BTu ‰21.‰11/
$1x.t/;
where ‰21 and ‰11 are formed from the eigenvectors associated with
the stable eigenvalues of Z1.
■ For estimator implementation, with some $ , solve the differential
Riccati equation
PQ.t/ D Q.t/AT C AQ.t/ C BwBTw $ Q.t/fC T
m Cm $ $$2C Ty CygQ.t/;
forward in time, initialized with Q.0/ D 0. Then,
L.t/ D Q.t/C Tm :
■ A steady-state suboptimal estimator, with some $ , uses the
eigensystem ‰ of the Y1 matrix to find the steady-state gain:
L D QC Tm D ‰22.‰12/
$1C Tm ;
where ‰22 and ‰12 are formed from the eigenvectors associated with
the unstable eigenvalues of Y1.
Lecture notes prepared by Dr. Gregory L. Plett. Copyright © 2016, 2010, 2008, 2006, 2004, 2002, 2001, 1999, Gregory L. Plett
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