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I. INTRODUCTION
OPTIMAL control refers to finding a control input to the
system that optimizes a given performance index while
moving the system’s state from an initial value to a final value.
For linear time-invariant (LTI) systems, the linear-quadratic
regulator (LQR) problem is widely studied in the optimal
control literature [1], [2], [3]. The LQR consists of designing
a static full-state feedback optimal gain matrix that optimizes
a given quadratic cost. However, measuring all the system
states is not always possible or feasible. In such scenarios, the
system is controlled either by a static output-feedback control
[4] or state observer-based full-state feedback, i.e., dynamic
output-feedback control [2].
Static output-feedback is comparatively straightforward and
easy to implement than dynamic output-feedback. However,
dynamic output-feedback is preferred due to its inherent
robustness properties, especially when the system has
high-order dynamics and multiple outputs [3]. Additionally,
verifying the necessary and sufficient conditions for static
output-feedback stabilizability is a complex task [5], [6]. In
Manuscript received 21 January 2024; revised 27 March 2024;
accepted 14 April 2024. Date of publication 30 April 2024; date of
current version 15 May 2024. The work of Yogesh Kumar was supported
by the TCS Research, India, under TCS Research Scholar Program
under Grant FP-140. The work of Prasad Vilas Chanekar was supported
by the SERB Start-Up Research under Grant SRG/2023/001636.
Recommended by Senior Editor L. Menini. (Corresponding author:
Yogesh Kumar.)
The authors are with the Department of Electronics and
Communication Engineering, Indraprastha Institute of Information
Technology, New Delhi 110020, India (e-mail: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>;
<PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>; <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS>).
Digital Object Identifier 10.1109/LCSYS.2024.3395357
contrast, the stabilizability of the dynamic output-feedback
can be verified by evaluating the system’s stabilizability
and detectability. Thus, dynamic output-feedback is more
advantageous than static output-feedback if one has enough
computational resources to implement the observer in a
sampled-data setting [2].
The standard process of designing dynamic output feedback
control within the optimal control framework is as
follows. First, the full state feedback optimal gain is designed,
followed by a separate design of the observer gain [2],
[7], [8], [9], [10]. This is done by leveraging the so-called
separation principle. For deterministic systems, the observer
gain is selected such that for a fixed optimal control gain,
the performance of the observer-based system is close to
the optimal state-feedback setting. In contrast, for stochastic
systems, the observer gains are selected based on minimizing
yet another cost function to tune the variance of
the steady state [11], [12]. The weight matrices are decided
based on the process and measurement noise characteristics.
For deterministic systems, a recent analytical study [12] has
provided valuable insights into unifying the dynamic output
feedback problem for discrete dynamical systems. This letter
highlights the challenges and optimization landscape inherent
in the combined optimal controller-observer design problem.
A related approach is presented in the continuous-time domain
in [13], [14], using a block pulse function-based nonlinear
programming approach with a modified cost function. This
is computationally expensive and performs satisfactorily in
simulations, depending on discrete approximation parameters.
Motivated by the limited existing work in this domain, this letter
aims to delve deeper into the intricate problem of designing
optimal observer-controller gains within the LQR framework,
specifically focusing on continuous-time LTI systems.
In this letter, we formulate and solve the optimal observerbased
control design problem for a continuous-time LTI
system corresponding to the standard LQR performance index.
We leverage the extended-state linear system representation
to pose this problem as a static output feedback problem.
The static output-feedback gain matrix for the extended-state
dynamics has a novel block-diagonal structure with the controller
and observer gains of the original system embedded in
two independent blocks. We solve this reformulated problem
using a computationally efficient second-order gradient-based
algorithm that uses the action of the Hessian of the objective
function for step-size computation. A novel feature of our
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KUMAR et al.: OPTIMAL DYNAMIC OUTPUT FEEDBACK DESIGN FOR CONTINUOUS-TIME LTI SYSTEMS 455
problem setup is that the gradient of the objective function
is also block-diagonal, which eliminates the use of projection
methods [12], [15]. Further, we discuss a sufficient condition
for the convergence of our proposed algorithm to a stationary
point. Additionally, we show that when the initial state is
known, the optimal controller gain for the original system is
the same as the standard optimal LQR full state-feedback gain.
Finally, we present simulation results to show the efficacy and
utility of our proposed algorithm.
This letter is organized as follows: Section II presents
the preliminaries, followed by the problem formulation and
methodology in Section III. Sections IV and V present the
results, conclusion, and future directions.
II. PRELIMINARIES
Consider1 a linear time-invariant (LTI) system with the
following dynamics
˙x = Ax + Bu, y = Cx, x(0) = x0, (1)
where A ∈ Rn×n, B ∈ Rn×m, and C ∈ Rr×n are the system
matrices. Whereas x(t) ∈ Rn, y(t) ∈ Rr, u(t) ∈ Rm and x0 ∈
Rn denote the state, measured output, control input and initial
state the system, respectively. We assume that the system
(A, B) is stabilizable and (A,C) is detectable. Since all the
states are not directly measurable, so by virtue of detectability,
we can estimate the complete state by using a Luenberger
observer for the system (1) [3]. The observer dynamics is
defined by
˙ˆ
x = Aˆx + Bu + Ly − Cˆx, (2)
where ˆx(t) is the estimate of x(t) and L ∈ Rn×r is the static
observer gain. We use the state estimate to define the control
input and the closed loop system as,
u = Kˆx, ˙x = Ax + BKˆx, (3)
where K ∈ Rm×n is the static controller gain. We quantify the
system performance by the following quadratic objective,
fc = 
∞
0
xQx + uRudt, (4)
where Q ∈ Rn×n, Q = Q  0 and R ∈ Rm×m, R = R  0
are weighing matrices. We assume that (A,Q
1
2 ) is observable.
Note that fc is a function of x and ˆx (through u).
We intend to solve the problem of computing K, L for
a given Q, R such that the performance objective fc is
minimized and the dynamics (1) is stabilized. Mathematically,
the problem is expressed as,
min
K, L
fc = 
∞
0
xQx + uRu dt
s.t. ˙x = Ax + Bu, x(0) = x0,
1Notation: Let R denote the set of real numbers. For a matrix X ∈ Rm×n
having m−rows and n−columns, we use X for its transpose, tr(X) for its
trace and X for its Frobenius norm. We use 0 and I to denote matrix of zeros
and identity matrix, respectively of appropriate sizes. For matrix X, whenever
a subscript is present, i.e., Xmn implies X has m−rows and n−columns. For
X ∈ Rm×m, X()  0 denotes X is positive (semi-)definite. For scalar function
f , notation ∇Xf denotes the gradient of f with respect to X.
˙ˆ
x = Aˆx + Bu + Ly − Cˆx, ˆx(0) = ˆx0, u = Kˆx,
ˆx → x and x →0 as t→∞ asymptotically. (5)
In (5), the controller gain K and the observer gain L are
simultaneously optimized. The constraint ˆx → x and x →
0 as t→∞ensures that the state estimates finally converge
to the true state with time and guarantee the stability of
the system. To eliminate dependence on x0 and ˆx0, we
consider them to be uncorrelated random variables distributed
uniformly on the surface of the n−dimensional unit sphere.
In the next section, we convert (5) in a computationally
viable format and propose a solution procedure.
III. PROBLEM FORMULATION AND SOLUTION METHOD
We first present the general controller-observer design
problem. Then based on Lyapunov stability theory [16] we
present an equivalent reformulation followed by a gradientbased
solution procedure of (5).
A. General Controller-Observer Design Problem
Consider the observer error e  x−ˆx. Using (1), (2) and (3)
the error dynamics is
˙e = (A − LC)e. (6)
We introduce a new extended state-variable z = [x e]
whose dynamics is derived by combining (1) and (6) as
˙z = A z + B u, y = C z u = K y, (7)
where,
A =  A 0nn
0nn A

, B =  B 0nn
0nm Inn

,
K =  K 0mr
0nn L

, C = Inn −Inn
0rn −C

.
To quantify the performance of the extended state dynamics
(7), we have the quadratic objective function
¯J
= 
∞
0
zQ z + uR u dt, (8)
where
Q =  Q 0nn
0nn 0nn

, R =  R 0mn
0mn 0nn

,
To optimally determine K, we state the general controllerobserver
design problem as
min
K
¯J
= 
∞
0
zQ z + uR u dt
s.t. ˙z = A z + B u, y = C z, u = Ky z(0) = z0,
z →0 as t→∞ asymptotically. (9)
The constraint z → 0 as t → ∞ ensures stability of the
closed-loop system. Note that K is the static output-feedback
gain and (9) is an optimal static output-feedback gain synthesis
problem. The relation between problem (5) and problem (9)
is illustrated next.
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456 IEEE CONTROL SYSTEMS LETTERS, VOL. 8, 2024
Proposition 1 (Relation Between (5) and (9)): The problems
(5) and (9) are identical provided z(0) = [x(0)

, x(0)
−
ˆx(0)
].
Proof: Using u = Ky, KC =  K −K
0nn −LC
, we have from(8)
¯J
= 
∞
0
x
e


Q + KRK −KRK
−KRK KRK
x
e
 dt.
Now using ˆx = x−e in (4) and expanding we get fc = ¯J as well
as identical dynamics (5). The constraint z →0 as t→∞
ensures ˆx → x and x →0 as t→∞. Hence proved.
Next, we reformulate (9) into a solvable format.
B. Reformulated Controller-Observer Design Problem
We use Lyapunov stability theory [16] to derive an equivalent
reformulation of (9). First, we define the following sets
of matrices,
K := 	K 



K =  K 0mr
0nn L
; A + B K C Hurwitz
,
KO := K 



K ∈ K; (A − LC,K) observable
 (10)
Note that K ∈ R(m+n)×(n+r) has a block-diagonal structure and
KO ⊆ K. Now assumptions (A, B) to be stabilizable, and
(A, C) to be detectable guarantees the existence of K ∈ K
such that A + B K C is Hurwitz. Next we reformulate the
objective function.
Theorem 1 (Reformulation of Objective Function):
Consider the dynamics (1), (7) and the performance objective
defined in (8). Then for K ∈ K there exists a P = P
 ∈
R2n×2n, P  0 such that
A + B K C

P + PA + B K C + Q + C

K

R K C = 0
and ¯J = tr(P Z0), where Z0 = z0z
0 . Furthermore, if K ∈ KO
then P  0.
Proof: Consider V = zPz with P  0. Differentiating with
respect to time and using (7) we have
˙V
= ˙z P z + z P ˙z = zA + B K C

P + PA + B K Cz.
Integrating both sides with respect to t with limits 0 to ∞ and
adding ¯J = 
∞
0 [zQz+uRu] dt on both sides with u = K y
with y = Cz we have

∞
0
˙V
dt + ¯J = 
∞
0
zA + B K C

P + PA + B K C
+Q + C

K

R K C z dt
Now 
∞
0
˙V
dt = V(∞) − V(0). As A + B K C is Hurwitz so
z → 0 as t →∞ which implies V(∞) = 0 using definition
of V. As (A+B K C) is stable and Q+C

K

R K C  0 then
from [17, Lemma 3.18] we must have P = P
  0 such that
A + B K C

P + PA + B K C + Q + C

K

R K C = 0.
Thus ¯J = V(0) = z
0 P z0 = tr(P z0z
0 ) = tr(P Z0). Now
when K ∈ KO, let DD = Q + C

K

R K C ≥ 0. Then for
stable Ac = (A + B K C), the existence of a ¯P  0 requires
(Ac,D) to be observable [17, Lemma 3.18]. The matrix D ∈
Rn+m×2n can be given as
 Q
12
0
−R
12
K R
12
K

. Now, to prove
that (Ac,D) is observable, we need to show that one can assign
the arbitrary eigenvalues of (Ac−LDD) using some gain LD =
LD11 LD12
LD21 LD22
. Evaluating the expression for (Ac − LDD) by
setting LD12
= −BR−1
2 , we get
Ac − LDD =
 A − LD11Q
12
0
−LD21Q
1
2 + L22R
1
2K A− LC − LD22R
12
K

From (4) and (10), (A,Q
1
2 ) and (A − LC,K) are observable,
therefore (Ac,D) is observable.
Now as x0 and ˆx0 are assumed random variables
distributed uniformly on the surface of the n−dimensional
unit sphere [18], we have the mean E[z0] = 0. From the
expression of Z0, we have Z0 =  x0
x0 − ˆx0
 x0
x0 − ˆx0


=
 x0xT
0 x0xT
0
− x0 ˆxT
0
x0xT
0
− ˆx0xT
0 x0xT
0
+ ˆx0 ˆxT
0
− 2ˆx0xT
0
. Thus, by taking
expectation of Z0 and using E[x0xT
0 ] = E[ˆx0 ˆxT
0 ] = I,
E[x0 ˆx
0 ] = 0 we have
V0 = E[Z0] = I I
I 2I

.
Using Theorem 1, we reformulate (9) using the mean value
of ¯J , i.e., E[ tr(P Z0)] = tr(P E[Z0]) = tr(P V0), as
min
K, P0
J = trP V0
s.t. A

c P + P Ac + Q + C

K

R K C = 0, K ∈ K. (11)
We now propose an iterative solution procedure to
solve (11).
C. Solution Procedure and Analysis
To solve (11), we use the gradient descent
method [19], [20]. Next we derive the expression for the
gradient of J with respect to K. First, we define some sparse
matrices that will be used in the theorem.
1 = Imm
0nm

, 2 = Inn
0
nr


, 3 = 0mn
Inn

, 4 = 0
rn
Irr


.
Theorem 2 (Gradient of J With Respect to K): Consider J
defined in (11). The gradient of J with respect to K is
∇
KJ = ∇KJ 0
0 ∇LJ
 ∈ R(m+n)×(n+r)
where
∇KJ = 2

1 R K C + B

PW C



2 ,
∇LJ = 2

3 R K C + B

PW C



4 ,
with P = P
  0 and W = W
 ∈ R2n×2n are the solutions of
A

c P + P Ac + Q + C

K

R K C = 0, (12a)
AcW + W A

c
+ V0 = 0, (12b)
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KUMAR et al.: OPTIMAL DYNAMIC OUTPUT FEEDBACK DESIGN FOR CONTINUOUS-TIME LTI SYSTEMS 457
respectively.
Proof: To derive our results, we use known matrix properties
[2] such as X, Y, tr(X Y) = tr(Y X), tr(X) = tr(X
)
provided the matrices are compatible for multiplication. From
matrix calculus we have
∇
KJ = ∇KJ 0
0 ∇LJ

.
We write the gain matrix K as
K = 1K2 + 3L4. (13)
First, we derive the expression for ∇KJ, substituting equation
(13) in (12a) and differentiating it with respect to K, we
have
A + B K C

PKdK + PKdK A + B K C
B 1dK2 C

P + P B 1dK2 C
+ C

(1dK2)
R K C + C

K

R (1dK2) C = 0, (14)
where dK is a matrix and is the differential of K. PK is
the derivative of the matrix P with respect to the matrix K
and is a fourth-order tensor. Pre-multiplying (14) by W, postmultiplying
(12b) by Pk dK, taking trace and doing matrix
manipulations we have
tr22C WP B + C

K

R1dK = trV0Pk dK. (15)
Similar to ∇KJ the expression for ∇LJ is
tr24C WP B + C

K

R3dL = trV0PL dL, (16)
where PL is the derivative of P with respect to L and dL is
the differential of L. With differential dJ = tr(∇
K
J dK) =
tr(∇
K J dK)+tr(∇
L J dL) = tr(V0PL dL)+tr(V0Pk dK), we get
the required result from comparison.
Note that the gradient of J also has the same block-diagonal
structure as K. With the gradient expression in Theorem 2,
a Vanilla gradient descent algorithm for solving (11) can be
straightforwardly derived [15, Algorithm 1]. However, being
a first-order method, its convergence would be slow due to the
linear rate of convergence[15]. To alleviate this, we propose a
second-order optimization procedure motivated by the results
in [21].
Theorem 3 (Action of Hessian of J With Respect to K): The
action of the Hessian of J applied to a direction dK is
∇2
KJdK, dK = trdK

F1 + trdK

F2
where,
F1 =
2

1 M1W C



2 0
0 2

3 M1W C



4

,
F2 =
2

1 M2W


C



2 0
0 2

3 M2W


C



4

,
M1 = (R dK C + B

P


), M2 = (R K C + B

P). P

 =
PKdK, W

 = WKdK are the solutions of
A

c P

 + P


Ac + M
2 dK C + M
2 dK C

= 0, (17a)
Algorithm 1 Switched Gradient Descent Algorithm
Input: A, B, C, K0
, Q, R, V0, , α, β, T > 0
Output: K
∗
1: Set j ← 0, Kj ← K0
2: while ∇
KJ(Kj
) >  do
3: Solve (12a) and (12b) to obtain Pj
, Wj
4: Compute ∇
KJ(Kj
) using Theorem 2
5: dKj ←∇
KJ(Kj
)
6: Solve (17a) and (17b) to obtain P
j
, W
j
7: Compute ∇2
K
J[dKj
, dKj] using Theorem 3
8: sj ← min

T,

∇
KJ(Kj
)
2
∇2
K
J[∇
KJ(Kj
),∇
KJ(Kj
)]

9: if sj > 0 then
10: Compute step size sj using Armijo rule with initial
sj as given in step: 8  See Algorithm 2
11: else
12: Compute step size sj using Armijo rule with initial
sj = 1  See Algorithm 2
13: end if
14:
15: Kj+1 ← Kj − sj∇
KJ(Kj
)
16: j ← j + 1
17: end while
18: K
∗ ← Kj
19: return K
∗
AcW

 + W


A

c
+ BdKC W + BdKC W
 = 0. (17b)
Proof: Differentiating (12a) and (12b) with respect to K
yield (17a) and (17b), respectively. Here PK and WK are
fourth-order tensors and are the derivatives of matrices P and
W with respect to the matrix K. P

 = PKdK∈ R2n×2n, W

 =
WKdK∈ R2n×2n are matrices. dK∈ R(m+n)×(n+r) is a matrix
and is the differential of K. Then, using the gradient expression
in Theorem 2, we obtain the required result.
The expression of the action of Hessian derived in
Theorem 3 enables us a step-size for the gradient descent
method, which ensures faster convergence. Next, We present
a suitably modified version of the algorithm given in [21].
Note that the expression ∇2
K
J[∇
KJ(Kj
),∇
KJ(Kj
)] can take
any values in R. Hence, it was necessary to introduce both an
upper bound T on the step size sj and the switching between
Vanilla and Hessian-based step size selection in Algorithm 1
to ensure a bounded sj > 0. The algorithm for Armijo rule [19]
to determine the step size sj is given next.
Notably, there are many ways to choose the parameters α, β,
, and T [19], [21] and the discussion along these lines deserve
a separate consideration. However, as a general guideline, one
can select the parameters from the typical range given as
follows: α ∈ [0.2, 0.5], β ∈ [0.4, 0.7], T ∈ [1, 10] and
 ∈ [10−4, 10−2] for faster convergence. Next, we analyze
some properties of Algorithm 1. Since our controller-observer
design problem (9) is a static output-feedback problem, we
follow the results from [21] to prove the convergence.
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458 IEEE CONTROL SYSTEMS LETTERS, VOL. 8, 2024
Algorithm 2 Armijo Rule [19]
Input: α, β, Kj
, sj
Output: sj
1: Kj
1
= Kj − sj∇
KJKj
2: while J(Kj
1) ≥ JKj − αsj
∇
KJ(Kj
)
2 or (A + B Kj
1 C)
Hurwitz do
3: sj = βsj,
4: end while
5: return sj
Theorem 4 (Properties of Algorithm 1): The following
statements are true for the Algorithm 1
(i) If K0 ∈ K, then Kj ∈ K is guaranteed for all iterations
of Algorithm 1.
(ii) If J(¯K) is L−smooth with constant Ls and 0 < sj <
2
Ls
, ∀j, then Algorithm 1 converges to a stationary point.
Proof:
(i) For any K ∈ K, we need to ensure (A + B K C))
is Hurwitz as well as the structural constraints of K,
having K and L as block-diagonal elements. The former
is ensured in Armijo rule step of the both the Algorithms,
whereas for the latter, each iteration Kj+1 ← Kj −
sj∇
KJ(Kj
) with Armijo results in Kj+1 ∈ K due to the
specially structured gradient given in Theorem 2 for an
initial K0 ∈ K.
(ii) From [21, Th. 4.6], if the step size 0 < sj < 2
Ls
,
then the gradient-descent sequence given in step:15
of Algorithm 2 leads to a monotonically decreasing
sequence of objective function values as iterations j
progresses [21, Th. 4.6]. Also, the objective function is
lower-bounded by zero, so the algorithm converges to a
stationary point.
Thereom 4 (ii) gives a sufficient condition for the convergence
of the Algorithm 1. However, the condition depends
on the existence and computation of Ls, which is a hard
problem, in general, [21]. For our case, due to the positive
semidefinite nature of matrices ¯Q and ¯ R, we cannot compute
Ls using [21, Th. 3.15]. However, our extensive numerical
simulations demonstrate the convergence of the algorithm.
Further analysis in this matter is part of our future work.
We now show that when the initial state x0 is known, then
the solution to the problem (11) is the same as the solution to
the full-state feedback linear-quadratic regulator problem.
Proposition 2 (Equivalence of Problem (11) to Linear-
Quadratic Regulator Problem for Known Initial State):
Consider the dynamics (1) and its extended form (7) with the
control design optimization problem (11). If the initial state
x(0) = x0 is known, then with ˆx(0) = x(0), the controller K in
the solution K to the problem (11) is unique and is the same as
the solution to the full-state feedback linear-quadratic regulator
problem for the system (A, B) with weighing matrices Q
and R.
Proof: For known x(0) we have ˆx(0) = x(0) yields e(0) = 0
and z0 = [x
0 0
n1]. Now
V0 = E[Z0] =  I 0nn
0nn 0nn

.
As Ac is Hurwitz then, the solution to (12b) has the form
W = W11 0nn
0nn 0nn
, where W11  0 is the unique solution of
(A + BK)W11 + W11(A + BK)
 + I = 0. (18)
Further, let P =
P11 P12
P

12 P22

 0, then using Theorem 2
yields
∇
KJ = 2RK + BP11W11 0
0 0

. (19)
From (19), it can be inferred that the stationary point of (11)
is K = −R−1BP11. Putting K = −R−1BP11 in (12a) gives
the equation in P11
A P11 + P11 A + Q − P11 B R−1 B P11 = 0. (20)
From [17, Corollary 13.8], for a controllable system
(A, B), (20) has a unique solution (P11  0) which minimizes
a cost J = tr(P11). Owing to the structure of Z0, problem (11)
is also reduced to minimizing J = tr(P11). From [2], K =
−R−1BP11 and (20) is the solution to the full-state feedback
linear quadratic regulator problem for the system (A, B) with
weighting matrices Q and R and hence the result follows.
Remark 1: Proposition 2 shows that if the initial system
state is measurable, the dynamic output feedback behaves like
a linear-quadratic regulator even if C = I. In contrast, for static
output-feedback problems, C = I is needed for equivalence to
the full-state feedback problem [21].
IV. RESULTS
We now demonstrate the utility of our proposed algorithms
through two examples. In the first example, we consider a
benchmark problem of a transport aircraft [22]. We apply
the Vanilla gradient-descent algorithm and Algorithm 1 to
compute the optimal dynamic output feedback controller gain
parameters K and L. We compare the performance of both
algorithms and present the comparison results. Since the
controller gains are obtained by minimizing the standard
quadratic objective as given in (4), we also compare the
optimal objective value with the static output feedback case.
In the second example, we take a set of benchmark examples
from the COMPleib library [23]. The following parameters
are used in the implementation of both the algorithms:  =
5 × 10−3(until specified otherwise), α = 0.3 β = 0.5, T =
10. For both examples, we start with an initial gain K0 ∈ K
by selecting the initial controller gain as the full-state feedback
LQR gain for system (A, B), and the initial observer gain is
selected by placing the poles of (A,C) at [−8 −7 −8 −5]
for Example 1 and randomly in [−30, 0) for Example 2.
A. Example 1
Consider the linearized dynamics of supersonic transport
aircraft at Mach 2.7 [22] with the weight matrices as Q = I,
R = I22. The initial value of J for K0 is 166.63. Applying
Algorithm 1, we compute the optimal K
∗
and the optimal cost
J = 156.82 (both cases) yielding an improvement of 5.88%.
The Vanilla gradient descent algorithm takes around 221.176 s
to converge to a stationary point, whereas Algorithm 1 takes
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KUMAR et al.: OPTIMAL DYNAMIC OUTPUT FEEDBACK DESIGN FOR CONTINUOUS-TIME LTI SYSTEMS 459
Table I
RESULTS WITH ALGORITHM 1 APPLIED TO COMPleIB EXAMPLES.
FOR ∗, AND ∗∗ MARKED EXAMPLES, WE USED  = 5 × 10−4, AND
 = 5 × 10−5, RESPECTIVELY
only 97.76 s, meaning that the Hessian-based algorithm
converges faster. In the static output feedback case, the final
cost is J = 159.06 [15].
B. Example 2
We analyze a set of practical benchmark examples from
the COMPleib library [23]. In this analysis, we considered the
weighing matrices as Q = CC and R = I, alongside an initial
gain K0 ∈ K. The results presented in Table I encompass
the initial cost J(K0
), the final cost upon convergence J(K
∗
),
the percentage improvements observed, and the cost with the
static output-feedback JSOF [5, Algorithm 3] across various
system configurations. Table I shows significant improvement
in the system performance exhibiting the effectiveness of
Algorithm 1 across diverse practical system configurations. In
general, we observe that the static output-feedback cost is
equal or greater than the optimal dynamic output-feedback
cost, and a rigorous analysis of this observation is part of our
future work.
V. CONCLUSION
This letter presents a novel framework for designing an
optimal observer-based controller for LTI systems. We posed
the optimal observer-based controller design problem as
an optimal static output-feedback synthesis problem on the
extended state dynamics. We analytically showed that the
controller of the extended state dynamics has a specific blockdiagonal
structure to embed the controller and observer gains
of the original system. We solved the optimal static outputfeedback
synthesis problem using a computationally efficient
second-order gradient-based method with a sufficient condition
for convergence to a stationary point. We also show the
equivalence of the dynamic output-feedback behavior to the
optimal full-state feedback when the initial system states are
known. Our numerical simulations show the efficacy and
utility of the proposed algorithm. Our future work involves
studying the nature of the stationary point, adding input
constraints in the formulation, and analyzing computational
aspects of the proposed algorithms and their application to
more practical examples.