Simultaneous Design of Controllers and Instrumentation:
ILQR/ILQG
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Citation
ShiNung Ching, P.T. Kabamba, and S.M. Meerkov.
“Simultaneous Design of Controllers and Instrumentation:
ILQR/ILQG.” IEEE Transactions on Automatic Control 55.1
(2010): 217–221. Web. 4 Apr. 2012. © 2010 Institute of Electrical
and Electronics Engineers
As Published
http://dx.doi.org/10.1109/tac.2009.2033879
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Institute of Electrical and Electronics Engineers (IEEE)
Version
Final published version
Accessed
Wed May 25 18:32:01 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/69928
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Detailed Terms
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 1, JANUARY 2010
and similarly
2
@L
@L
=
@ q_kc
@ q_2k 01
0 | @@q_L
2k
;
2
@L
@L
@L
=
+|
:
@ q_kc3
@ q_2k 01
@ q_2k
This provides the following complex formulation of the real Euler-Lagrange (24)
d
dt
2
d
dt
@L
@ q_kc3
@L
@ q_kr
=2
=
@L
+ Skc ;
@qkc3
@L
+ Skr ;
@qkr
k = 1; . . . ; nc
k = 1; . . . ; nr :
(25)
d
@H
+ Sk ;
pk = 0
dt
@qk
[8] W. Leonhard, Control of Electrical Drives. Amsterdam, The Netherlands: Elsevier, 1985.
[9] R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez, “PassivityBased Control of Euler-Lagrange Systems,” in Communications and
Control Engineering. Berlin, Germany: Springer-Verlag, 1998.
[10] Y. Yourgrau and S. Mandelstam, Variational Principles in Dynamics
and Quantum Theory, third ed. New York: Dover, 1979.
Simultaneous Design of Controllers and
Instrumentation: ILQR/ILQG
(26)
ShiNung Ching, Pierre T. Kabamba, and Semyon M. Meerkov
In the usual complexification procedure ([4, page 87]) the coefficient
2 appearing in the above equations is not present. This is due to our
qkc = q2k01 + |q2k instead of the usual choice qkc =
special choice p
q2k01 + |q2k = 2. This special choice preserves the correspondence,
commonly used in electrical engineering, between complex and real
electrical quantities
Let us assume that, for each q , the mapping q_ 7! @ L=@ q_ is a smooth
bijection. Then the Hamiltonian formulation of (24) reads
@H
d
;
qk =
dt
@pk
217
k = 1; . . . ; n
(27)
with H = @ L=@ q_ 1 q_ 0 L and p = @ L=@ q_ . Let us decompose p into
2 n and pr 2 n . Then pr = @ L=@ q_r and pc = 2@ L=@ q_c3 .
Simple computations yield another derivation of the Hamiltonian from
L directly
Abstract—The instrumentation, i.e., sensors and actuators, in feedback
control systems often contain nonlinearities, such as saturation, deadzone,
quantization, etc. Standard synthesis techniques, however, assume that the
actuators and sensors are linear. This technical note is intended to modify
the LQR/LQG methodology into the so-called Instrumented LQR/LQG
(referred to as ILQR/ILQG), which allows for simultaneous synthesis of
optimal controllers and instrumentation.
Index Terms—Linear plant/nonlinear instrumentation (LPNI).
I. INTRODUCTION
pc
H = @@qL_c q_c + @@q_Lc3 q_c3 + @@qL_r q_r 0 L
(28)
where H denotes the Hamiltonian H when is a considered as a function
of (q c ; q c3 ; q r ; pc ; pc3 ; pr ). Then (27) becomes
d c
q
dt k
d c
p
dt k
d r
q
dt k
d r
p
dt k
@H
;
@pkc3
@H
= 0 2 c3 + Skc ; k = 1; . . . ; nc
@qk
@H
= r;
@pk
@H
= 0 r + Skr ; k = 1; . . . ; nr :
@qk
=2
(29)
(30)
REFERENCES
[1] I. Boldea and S. A. Nasar, The Induction Machine Handbook. Orlando, FL: CRC Press, 2002.
[2] C. Canudas, A. Youssef, J. P. Barbot, Ph. Martin, and F. Malrait, “Observability conditions of induction motors at low frequencies. In,” in
Proc. 39th IEEE Conf. Decision Control, 2000, pp. 2044–2049.
[3] J. Chiasson, Modeling and High Performance Control of Electric Machines. New York: Wiley-IEEE Press, 2005.
[4] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and
Atoms: Introduction to Quantum Electrodynamics. New York: Wiley,
1989.
[5] H. R. Fudeh and C. M. Ong, “Modeling and analysis of induction machines containing space harmonics. Part-I: Modeling and transformation,” IEEE Trans. Power Apparatus Syst., vol. PAS-102, no. 8, pp.
2608–2615, Aug. 1983.
[6] J. Holtz, “Sensorless control of induction motor drives,” Proc. IEEE,
vol. 90, no. 8, pp. 1359–1394, Aug. 2002.
[7] S. Ibarra-Rojas, J. Moreno, and G. Espinosa-Pérez, “Global observability analysis of sensorless induction motors,” Automatica, vol. 40,
pp. 1079–1085, 2004.
LQR/LQG is a widely used methodology for designing linear controllers for linear plants. Within this methodology, the instrumentation,
i.e., actuators and sensors, are also assumed to be linear. In reality,
however, the instrumentation is often nonlinear, e.g., having saturation, deadzones, quantization, etc. This leads to the so-called Linear
Plant/Nonlinear Instrumentation (LPNI) system. Is it possible to extend
LQR/LQG to such systems? A positive answer to this question was provided in [1], where systems with saturating actuators were considered
and a methodology, referred to as SLQR/SLQG (with S standing for
‘saturating’), has been developed.
The results of [1] have been obtained using the method of stochastic
linearization [2], which is a global quasilinearization technique that reduces an LPNI system to a linear one with the instrumentation gains
being functions of all systems parameters, including functional blocks
and exogenous signals. The results of [1] have been extended in [3] to
LPNI systems with nonlinearities in actuators and sensors simultaneously.
In [1] and [3] the instrumentation was assumed to be given prior to
the controller design. The goal of this Technical Note is to develop a
method for simultaneous design of controllers and instrumentation. To
accomplish that, we parameterize the instrumentation by the severity
of its nonlinearities, e.g., levels of saturations, steps of quantization,
etc. Then, we introduce a performance index, which includes both the
system behavior and the parameters of the instrumentation. Assuming
that this performance index is quadratic, we derive synthesis equations
for designing optimal controllers and instrumentation simultaneously.
Manuscript received November 25, 2008; revised April 09, 2009, July 21,
2009, and September 24, 2009. First published November 24, 2009; current
version published January 13, 2010. Recommended by Associate Editor D.
Liberzon.
S. Ching is with the Department of Brain and Cognitive Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA (e-mail:
shinung@mit.edu; shinung@gmail.com).
P. T. Kabamba and S. M. Meerkov are with the Department of Electrical
Engineering and Computer Science, University of Michigan, Ann Arbor, MI
48109 USA (e-mail: kabamba@umich.edu; smm@eecs.umich.edu).
Digital Object Identifier 10.1109/TAC.2009.2033879
0018-9286/$26.00 © 2009 IEEE
218
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 1, JANUARY 2010
The resulting technique is referred to as ILQR/ILQG, where I stands
for ‘instrumented’.
As for the prior results in this area, to the best of our knowledge no
methods for simultaneous design of controllers and nonlinear instrumentation are available in the literature. The closest to our work is [4],
where the locations of linear instrumentation and its precision are optimized along with the controller design.
The outline of this Note is as follows: In Sections II and III, the ILQR
and ILQG problems, respectively, are formulated and solved for SISO
LPNI systems with saturating actuators and sensors. Generalizations
to other types of nonlinearities and to MIMO systems are outlined in
Section IV. An application to a ship roll damping problem is described
in Section V. The conclusions are formulated in Section VI. With some
modifications, the proofs of most theorems included in this Note follow
the pattern of [1]; therefore, and due to the space limitations, only comments on the proofs are included in the Appendix. Complete proofs and
other details can be found in [5].
where the minimization is over all pairs (K; ) such that A + B2 NK
is Hurwitz.
This is a constrained optimization problem, since (6) can be rewritten
[6] as
KRK T
+
2
A + B2 NK )R + R(A + B2 NK )T
+
B1 B1T
K;
min
(
KRK T
= 0
0
2
2
erf
01 (N ) 02 = 0:
N
BT Q
+ 2
p p
= erf 01 (N ) 2 KRK T
K
(1)
(8)
(9)
(10)
= 0
(11)
where (Q; R; N; ) is the unique solution of
N2
QB2 B2T Q + C1T C1 = 0
(12)
+
T
N2
N2
A0
B2 B2T Q R + R A 0
B BT Q
+
+ 2 2
T
+ B1 B1 = 0
(13)
(14)
= 0
0 N p
01 2
2erf (N ) exp erf (N ) 0 1
0
(15)
2 =0
0
1
2 erf
(N )
AT Q + QA 0
with linear state feedback
u = KxG :
(2)
Here, xG is the state vector, u is the control, z is the controlled (performance) output, w is a standard white noise process and sat (1) is
given by
;
u > +
u;
0 u 0;
u < 0.
(3)
Introduce the following:
Assumption 1: (a) (A; B2 ) is stabilizable; (b) (C1 ; A) is detectable;
p
T C1 = 0; (e) A has no eigenvalues
(c) D12 = [0 ]T , > 0; (d) D12
in the open right-half plane.
Remark 1: Assumptions (a)–(d) are standard in conventional LQR
theory. Assumption (e) is used to ensure stability of the resulting
closed-loop LPNI system.
From (1) and (2), the closed-loop system is described by
x_ G = AxG + B2 sat (KxG ) + B1 w
z = (C1 xG + D12 K )xG :
(4)
Applying stochasic linearization (see [1] and [5] for details), (4) reduces to the following quasilinear system:
xG = (A + B2 NK )x^G + B1 w
z^ = (C1 xG + D12 K )x^G
u^ = K x^G
N = erf p
2u
^
_
^
(5)
^.
where u^ is the standard deviation of u
The ILQR Problem is stated as follows: Find the value of the gain K
and parameter of the actuator, which ensure
2
2
K; z^ + ; > 0
min
(7)
with N defined by
Consider an LPNI system given by
(u) =
+
where R satisfies
A. ILQR Synthesis Equations
sat
C1 RC1T
Using the Lagrange multiplier method to find the minimum of (7),
we obtain:
Theorem 1: Under Assumption 1, the ILQR problem is solved by
II. ILQR THEORY
x_ G = AxG + B1 w + B2 sat (u)
z = C1 xG + D12 u
tr
(6)
while the optimal ILQR cost is
K;
min
z^2 + 2
= tr
C1 RC1T
+
N2
T
2 B2 QRQB2
( + )
T 01 (N )2 : (16)
+2KRK erf
The solution to (12)–(15) can be found from a standard bisection
algorithm. Specifically, substituting in (15) by its expression from
(14), yields
=0
p
h(N ) = N erf 01 (N )
01 (N )2
2 exp erf
h(N ) 0
(17)
0 2erf
01 (N )2 :
(18)
It is shown in the proof of Theorem 1 that h(N ) is continuous and
monotonically increasing for N 2 [0; 1). This leads to the following:
ILQR Solution Algorithm: For a given > 0, (i) Find an -precise
solution of (17) using bisection (with initial conditions N1 = 0, N2 =
1); (ii) Find from (14) or (15); (iii) Find Q from (12); (iv) Find R
from (13); (v) Compute K and from (10) and (11).
Remark 2: Note that the optimal equivalent gain N resulting from
the ILQR solution is independent of the plant parameters. This is to be
expected since N is simply the percentage of time that the actuator does
not saturate. Thus, it depends only on the ratio of the control penalty and the instrumentation penalty .
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 1, JANUARY 2010
Remark 3: ILQR is a proper generalization of conventional LQR.
Indeed, observe from (17) that as approaches 0, N tends to 1 and,
from (15), tends to 0. Hence, tends to (i.e., the actuator becomes
linear) and (10), (12) and (13) reduce to the standard LQR equations.
219
Applying stochastic linearization to (23)results in
1
x^_ G = A^
xG + B1 w + B2 Na K x^C
x^_ C = M x^C LNs C2 x^G LD21 w
z^ = C1 x^G + D12 K x^C
u
^ = K x^C
y^ = C2 x^G
y^m = Ns C2 x^G + D21 w
Na = erf
2u^
Ns = erf
2y^
B. ILQR Stability Verification
According to the method of stochastic linearization, the standard deviation of z in the LPNI system (4) with the ILQR controller (10) and
the saturation level (11) is close to the standard deviation of z^ in the
quasilinear system (5) if this closed loop LPNI system is stable (see
[1], [3], and [5] for details). Therefore, it is important to establish that
(4) with (10) and (11) is indeed stable in the appropriate sense. To accomplish this, consider the following undisturbed version of (4):
x_ G = AxG + B2 sat (KxG )
z = C1 xG + D12 u:
(19)
Assume that the pair (K; ) is obtained from (10) and (11), and
(Q; R; N; ) is the corresponding solution of (12)–(15).
Theorem 2: For the closed-loop system (19) with (10), (11):
i) xG = 0 is the unique equilibrium;
ii) this equilibrium is exponentially stable;
iii) a subset of its domain of attraction is given by
X=
xG
2 Rn
: xTG ("Q)xG
4
BT ("Q)B
2
2
2
; "=
N2
: (20)
+
Note that a similar result has been derived in [7] in the context of
semi-global stability.
A. ILQG Synthesis Equations
z^ = C~1 x^
~ x^
u
^= K
A 0
~ = Na
; N
A~ =
0 M
0
C~1 = [C1
(25)
0
B1
~1 =
; B
LD21
Ns
~2 = B2 0 ; C~2 = 0 K :
D12 K]; B
0
L
C2 0
0
0
(26)
The ILQG Problem is stated as follows: Find K , L, M , and ,
which ensure
min
z^2 + a 2 + s 2 ; a > 0; s > 0
K;L;M;;
(27)
(28)
(21)
~ C~2 )P~ + P~ (A~ + B
~1 B
~2 N
~ C~2 )T + B
~1T = 0
~2 N
(A~ + B
(29)
~ defined by
with N
x_ C = M xC
u = KxC
0 Lym
(22)
The signals u; y; ym
are, respectively, the control, plant output,
are independent white noise
and measured output, while w1 ; w2
.
processes. The controlled outputs are z1 ; z2
Assumption 2: (a) (A; B2 ) is stabilizable and (C2 ; A) is detectable; (b) (A; B1 ) is stabilizable and (C1 ; A) is detectable; (c)
T C1 = 0
D12 = [0 ]T , > 0 and D21 = [0 ], > 0; (d) D12
T
and B1 D21 = 0; (e) A has no eigenvalues in the open right-half plane.
Remark 4: Assumptions (a)–(d) are standard in conventional LQG
theory, while (e) is used to ensure stability of the closed-loop system.
From (21) and (22), the closed-loop LPNI system is
2R
2R
p
p
T
~ B
~2 N
~ C~2 )^
~1 w; x^ = x^TG x^TC
x^_ = (A+
x+B
where P~ satisfies
with the dynamic output feedback
2R
(24)
which can be rewritten as
min
tr C~1 P~ C~1T + a 2 + s 2
K;L;M;;
Consider the open loop LPNI system
= AxG + B1 w + B2 sat (u)
= C1 xG + D12 u
= C2 xG
= sat (y) + D21 w
p
p
where the minimization is over all (K; L; M; ; ) such that (A~ +
~ C~2 ) is Hurwitz.
~2 N
B
Similar to the ILQR case, this problem can be rewritten as
III. ILQG THEORY
x_ G
z
y
ym
0
0
x_ G = AxG + B1 w + B2 sat (KxC )
x_ C = M xC L (sat (C2 xG ) + D21 w)
z = C1 xG + D12 KxC :
0
2=
2
0
~)
erf 01 (N
02
=0
0
:
2
(30)
(31)
The Lagrange multiplier method is again used obtain a solution of
this optimization problem.
Theorem 3: Under Assumption 2, the ILQG problem (27) is solved
by
0 N+a BT Q
Ns
L = 0 P CT
M = A + B Na K + LNs C
pp
= erf 0 (Na ) 2 KRK T
p
= erf 0 (Ns ) 2 C (P + R)C T
K=
2
1
2
1
(23)
0 21 2
diag C~2 P~ C~2T
1
2
2
2
2
(32)
(33)
(34)
(35)
(36)
220
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 1, JANUARY 2010
where (P; Q; R; S; Na ; Ns ; 1 ; 2 ) is a solution of
AP + P A
T
For example, if g (y ) is a symmetric deadzone of width then
Ns2 P C T C P + B B T = 0
1 1
2 2
Na2 QB B T Q + C T C
2 2
1 1
+ 1
0
A Q + QA 0
T
T
+ 2 C2
C2 = 0
T
T
(A + B2 Na K )R + R(A + B2 Na K ) + LL = 0
T
T
(A + LNs C2 ) S + S (A + LNs C2 ) + K K = 0
= 0;
1 0 N p
01
exp
erf (Na )2 0 1
2erf (N )
p 2
C2 P SP C2T NsT 0 2 erf 01 (Ns )03
4
2 exp erf 01 (Ns )2 = 0
1
a 0
2 =0
01
2 erf (Na )
2
s 0
2 =0
0
1
2 erf (Ns )
G (y^ ) = 1 0 erf p=2 :
(37)
2
=0
2
(38)
(39)
m
k=1
[sgn (2y + (2k
(50)
KRK T 0
(42)
(43)
(44)
diag
C~2 P~ C~2T
Ns = G (v^ ) =
01
1
01
g0 (x)
y^
p1
2
exp
exp
F 01 (N ) 02 = 0
B. Multivariable Systems
(45)
(46)
(52)
=0
(53)
R
Consider the MIMO version of (23), where u 2 p and y; ym 2
p; q > 1, where , are understood as [1 . . . p ]T ,
[1 . . . q ]T , and sat (u) [sat (u1 ) . . . sat (up )]T ,
sat (y) [sat (y1 ) . . . sat (yq )]T . As before, the quasilinearization of this system is given by (24)–(26) with the equivalent gains specified by
R
q,
Na = diag Na ; . . . ; Na
Na = erf p k
; k = 1; . . . ; p
2u
^
Ns = diag Ns ; . . . ; Na
Ns = erf p l
; l = 1; . . . ; q:
2y^
(54)
(55)
The ILQG problem (27) now becomes
min
K;L;M;;
z^2 + T Wa + T Ws (56)
where Wa ; Ws are diagonal and positive definite. Clearly, this can be
rewritten as
min
K;L;M;;
tr
C~1 P~ C~1T
T
+
Wa + T Ws (57)
subject to the constraints (29) and (30), with 2 in (30) becoming
T
If f (u) and g (y ) are the actuator and sensor nonlinearities, the
general expressions for the quasilinear gains are
2
(51)
0diag F 01 (Na ) 02 ; G 01 (Ns ) 02
A. Arbitrary Nonlinearities
p1
:
similarly, for ILQG, (30) becomes
2=
u^
0
(2k 1) (z )
where F and G are assumed to be invertible. Existence and uniqueness
of the ILQR/ILQG solution will, of course, depend on the specific form
of these functions.
IV. GENERALIZATIONS
f0 (x)
k=1
e0
For the ILQR problem (7), the constraint (9) now becomes
(44), (37), (38), (39) and (40). Hence, the left hand side of (42) can
be expressed as function of Ns ; (iv) Find all Ns 2 [0; 1] that satisfy
(42) by using a root-finding technique such as numerical continuation
or generalized bisection; (v) For each Ns found in the previous step,
compute K; L; M; ; from (32)–(36); (vi) Using (32)–(36), find the
quintuple (K; L; M; ; ), which minimizes JILQG from (45).
Remark 5: In contrast to conventional LQG, due to the interdependence of (37)–(44) on both Na and Ns , the separation principle does
not hold for ILQG.
Na = F (u^ ) =
m
G (y^ ) = Qm p2 ; Qm (z) := p2z
y^
(41)
using bisection (with initial conditions Na;1 = 0, Na;2 = 1); (ii) Find
1 from (41) or (43); (iii) For any Ns , the left hand side of (42) can now
be determined by finding 2 , P , Q, R, and S , by solving, in sequence,
1
0 1)) 2 sgn (2y 0 (2k 0 1))]
then
(40)
ILQG Solution Algorithm: For a given > 0, (i) With h(1) defined
in (18), find an -precise solution Na of the equation
h(Na ) 0 a
Similarly, if g (y ) is the mid-tread quantizer
g (y) = which minimizes the ILQG cost
2
JILQG = tr C1 (P + R)C1T + N 2 B2T QRQB2
( + )
T
0
1
2
+ 2a KRK erf (Na )
T
01
2
+ 2s C2 (P + R)C erf (Ns ) :
(49)
2y^
0 2x2 dx
(47)
0 2x2 dx:
(48)
u^
y^
diag( )
0
0
T
diag( )
:
(58)
The optimization is carried out in a manner analagous to Theorem 3,
and the necessary conditions for minimality are obtained in terms of
the Lagrange multiplier 3 = [1 ; . . . ; (p+q) ].
V. APPLICATION TO SHIP ROLL DAMPING PROBLEM
A. Model and Problem
Ship roll oscillations caused by sea waves lead to passenger discomfort. To minimize this discomfort, the roll angle of the ship should be
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 55, NO. 1, JANUARY 2010
maintained at less than 3 degrees. One approach to reducing ship oscillations involves the use of two actively controlled stabilizing wings
attached to the stern. Clearly, the angular travel of these wings is constrained, leading to actuator saturation.
As demonstrated in [5] the above system can be modelled in statespace form as
0
:
x_ G =
0
:
0 985
1
0
0
0
0
0
0
1 125
0
0
0
0
:
0 286
0
:
0 311
1
xG
K; N; R; Q; ; )
0
+
(59)
Note that the system is normalized so that = 1 corresponds to an
angular travel of 18 degrees, which is the saturation authority given in
[8].
In [1], this problem was studied in the context of SLQR and SLQG.
In particular, when the instrumentation is fixed at = 1 and ym = y
(i.e., the sensor is linear), SLQR and SLQG are used to synthesize a
controller that achieves the performance specification z < 3 degrees,
where z is the standard deviation of the roll angle. Below, we demonstrate the ILQR and ILQG approaches to this problem.
B. ILQR Solution
0
:
5 641
0
:
7 565
0
:
3 672
:
;
= 0:78 ) 14 deg (60)
C. ILQG Solution
Using the tuned parameters a = 2:55 2 1003 , s = 10010 , 05 and = 1005 , the ILQG solution method results in
10
0
0
KRK T
+
2
A + B2 NK )R + R(A + B2 NK )T + B1 B1T Q
KRK T
0
2
2
01 (N )
erf
:
2
(64)
Proof of Theorem 2: Similar to that of Theorem 3 in [1] with the
same Lyapunov function.
Proof of Theorem 3: Similar to that of Theorem 1 but with the Lagrangian
~ ; ; ; )
K; L; M; Na ; Ns ; P~ ; Q;
1 2
9(
= tr
C~1 P~ C~1T
A~ + B~2 N~ C~2 P~
+ tr
P A
B NC T
+ ~ ( ~ + ~ 2 ~ ~2 )
3
diag
C~2 P~ C~2T
0
~
B BT Q
+ ~ 1 ~1
1
2
2
erf
01 (N~ ) 02
:
(65)
0 2058]
resulting in z^ = 2:72. Numerical simultation of the nonlinear system
with this controller and actuator reveals that z = 2:79, which verifies
the accuracy of the quasilinearization. Clearly, the design objectives
are met. Note that by simultaneously synthesizing the controller and
instrumentation, we find a solution that uses a saturation authority of
less than 18 degrees.
K = [02:029 0 2:798
L = [080:77 0 16:09
= 0:91; = 0:35
+
(
+ tr
Based on the previous subsection, the design objectives are z^ <
3 rad and 1. Using the ILQR solution method with the tuned
penalties = 3:5 2 1003 and = 1 2 1006 , we obtain
= [
C1 RC1T
+ tr
z1 = [0 0:109 0 0]xG
y = [0 1:248 0 0]xG
p
ym = sat (y) + w2 :
K
For complete proofs, see [5].
Proof of Theorem 1: Similar to that of Theorem 1 in [1] but using
the Lagrangian
sat (u)
0
0
APPENDIX
= tr
0
w1 +
1
based on the widely used LQR/LQG synthesis engine, ILQR/ILQG is
a promising design alternative for practicing control engineers.
9(
0
1
0
+
:
1 563
221
:
1 264
:
281 41
:
0 0709]
0 100:38]T
=
(61)
(62)
(63)
leading to z^ = 2:56. Simulation of the nonlinear system yields
= 2:77, which meets the performance specification.
z
VI. CONCLUSION
This technical note provides a method for simultaneous synthesis
of controllers and instrumentation for linear plants. It requires a computational effort comparable with standard LQR/LQG. Thus, being
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