Optimal Control of Networked Systems with Limited
Communication: a Combined Heuristic Search and
Convex Optimization Approach
Lilei Lu
Lihua Xie
BLK S2, School of Electrical and Electronic Engineering
Nanyang Technological University, Singapore 639798
Email: [email protected]
Minyue Fu
School of Electrical and Computer Engineering
The University of Newcastle, Australia
Abstract
This paper discusses the optimal H2 and H∞ control problems for networked systems with
limited communication constraint. The limited communication constraint in control networks is
taken into consideration in controller design by employing the notion of communication sequence.
Our objective is to find an optimal communication sequence and the corresponding optimal
controller for the given plant and communication resource under the H∞ or H2 performance
index. For a given communication sequence, the problem is formulated into a periodic control
problem for which a direct LMI design method is developed. We also propose a heuristic
search method for seeking a sub-optimal communication sequence, which in conjunction with the
convex optimization gives a solution to the optimal limited communication control problem. As
compared with the exhaustive search of communication sequence, our approach greatly reduces
the computational cost. Several examples are used to illustrate the design method. They
clearly indicate that the solution of the heuristic search converges to the optimal communication
sequence.
Keywords: Control networks, Periodic discrete-time system, H∞ and H2 control, Linear
matrix inequality, Communication sequence.
1
Introduction
Since networks may greatly decrease the hardwiring, the cost of installation and implementation,
it is popular to use networks in many complicated systems such as manufacturing plants, platoon
1
vehicles and robotic systems. In addition, the more modular and more flexible structure of
networked systems makes it much easier to remove, exchange, and add parts. However there
are also drawbacks in employing serial communication network to exchange information between
different system components. The main drawbacks are network-induced delay and the limitation
on bandwidth both of which can affect system performance. Time-delay may be induced by
networks when exchanging data among devices connected to the shared communication medium.
The characteristics of this kind of time delay in different control networks can be found in [11].
On the other hand, the effects of limited bandwidth on control system performance has attracted
a lot of interest recently; see, for example, [1]-[8].
Control networks are different from data networks in that in the former data are continuously
transmitted at relatively constant rates while in the latter, large data packets are sent out
occasionally at high data rates. Furthermore, control networks need to meet time-critical requirement, that is to say, message should be sent out successfully within a prespecified time.
The primary objective of control networks is to efficiently use the finite communication resources
while maintaining good system performance. Thus some standard protocols for control networks
have been developed such as Controlnet, Ethernet, Devicenet (see [12]), BACNet and Lonworks,
which have been used in many practical applications.
In classic models, it may simplify problems and still work well to separate the communication
aspects from the dynamics of a system. However when system performance is limited and degraded because of propagation delay and limited bandwidth, it is not proper without considering
the effects brought by networks. Conventional control theories such as synchronized control and
non delayed sensing and actuation must be reevaluated prior to application to networked control
systems [1, 2]. This brings forth a lot of efforts to investigate the effects of time-delay and finite
communication rates in control problems.
The problem of stabilizing an LTI system when only some elements of the outputs and/or some
of the control actions can be transmitted at one time has been investigated in [1]-[3]. The
LQ control with communication constraints is studied in [4] which indicates that an optimal
communication sequence is typically such that the sampling resources should be focused on
where they are needed most. In [5] and [6], stabilization of infinite-dimensional time-varying
ARMA model under limited data rates is considered. In [7] and [8] coding and state estimation
in limited communication channels are taken into account. [10] has shown that information
exchange between local controllers through a network can enlarge the class of plants to be
stabilized. An explicit stability condition dependent on the maximum time delay induced by
networks is given in [13] and [15]. Recently, [16] shows that by using the estimated values instead
of true values of system states at other nodes, a significant saving in bandwidth is achieved to
allow more resources to utilize the network.
On the other hand, the standard H∞ and H2 control problems are tackled by assuming that
all the outputs are available to the controller at any time instant. Obviously, this is impossible
due to the serial communication in control networks. In this paper, we want to find an optimal
2
communication sequence and a controller to obtain the minimum H∞ or H2 cost for networked
control systems with limited bandwidth. The problem is first formulated as a periodic control
problem by employing the notion of “communication sequence”. It is shown that under a given
communication sequence, the design of an optimal periodic controller can be converted to a convex optimization. We then propose a heuristic search approach for a suboptimal communication
sequence, which together with the convex optimization of controller, gives a solution to the joint
optimization problem. The approach is known to be convergent. As compared to the exhaustive search in [4], our approach greatly reduces the computational cost. Several examples are
included to demonstrate that the heuristic search in fact converges to a global optimal solution
although no theoretical proof is given. It is worth noting that stabilization problem under limited communication constraint has been considered in [1, 2] and [10] using the lifting technique
which will deal with a much higher dimensional system. Our proposed direct approach has the
advantage that it can avoid this and can be extended easily to deal with uncertain systems.
2
Problem Formulation
Consider a networked control system (NCS) shown in Fig 1. The plant, the sensors and the
controller are spatially distributed and connected together through a control network, whereas
conventional point-to-point link may be used. Now suppose that the spatially distributed plant
is a linear time-invariant system described by
x(k + 1) = Ax(k) + B1 w(k) + B2 u(k)
(2.1)
z(k) = C1 x(k) + D11 w(k) + D12 u(k)
(2.2)
y(k) = C2 x(k) + D21 w(k)
(2.3)
where x(k) ∈ Rn is the state vector, w(k) ∈ Rp the disturbance input, u(k) ∈ Rm the control
input, y(k) ∈ Rr the output, and z(k) ∈ Rq the controlled output. All the system matrices
are with appropriate dimensions. The initial state x0 is considered to be known and without
loss of generality it is set to be zero. The system is assumed to be interconnected with spatially
distributed subsystems. The output of each subsystem can only be sent to the controller through
the network at a given time. Here we consider a simple case in which the control u is not
transmitted by the network but in a way that it is transmitted to the plant directly.
We assume that the control network adopts scheduled releasing policy to transmit data. Under
this policy the start-sending time is scheduled to occur for each node and the signal is transmitted
periodically. Hence, as pointed out in [12], time delay is little and the possibility of message
collision is much lower. However, it is clear that under this policy the controller can’t have
simultaneous access to all outputs of the plant, but in a way that the multiple outputs are
sequentially multiplexed from the sensors to the controller at every step periodically. The way
of multiplexing can be described by the r switches
σ1 = [ 1 0 · · ·
0 ] , σ2 = [ 0
1 ···
3
0 ] , · · · , σr = [ 0 0
···
1].
z
Controller
u
y1
Plant
switches
Sensors
yr
ys
Control
network
w
Figure 1: Networked Control System
The switch σi , i = 1, · · · , r, is a 1 × r matrix with the ith element being 1 and all other entries
being zero. It determines the controller to communicate with which element of the outputs,
because σi y(k) = yi (k), where yi (k) is the ith element of the outputs.
We employ the idea of “communication sequence” which was originally introduced in [9] to jointly
formulate the control and communication problem. Here the kth element of a communication
sequence, sk , is defined as an arbitrary element of the above switches. Hence a communication
sequence leads the controller to read which of the output signals at each time instant. It
is reasonable to assume that the controller communicates with the plant following a periodic
pattern, which can be specified by an N -periodic communication sequence sk , where sk+N =
sk , ∀k ∈ Z.
Definition 2.1 An N -periodic communication sequence sk , k = 0, · · · , N − 1, where sk ∈
{σ1 , σ2 , · · · , σr }, is admissible if the following condition is satisfied




rank 
s0
s1
..
.



 = r.

(2.4)
sN −1
The above definition has a more direct expression and is more flexible in converting a NCS to
a periodic system compared with the similar definition given in [1]. This condition requires
that no more than one of the outputs be measured by the controller at each time instant
and the controller communicate with each of the plant output at least once within a period
[2]. In this way, the bandwidth limitation in NCS is modeled in a manner that the controller
can communicate with only one of the sensors at a discrete-time constant according to the
communication sequence.
Remark 2.1 If more than one, say l, output measurements of the sensors can be transmitted
4
in one-packet, then sk ∈ {σ̄0 , σ̄1 , · · · , σ̄N −1 }, where



σ̄i = 


σ̃1
σ̃2
..
.



,


σ̃i ∈ {σ1 , σ2 , · · · , σr },
i = 0, 1, · · · , N − 1
σ̃l
Note that sk needs to satisfy (2.4). Here we mainly consider the case in which no element of the
outputs is lumped into one packet. But the results can be extended to the general case easily.
Since for a given periodic communication sequence, the NCS is in fact a periodic system, we
introduce a periodic controller of the form of (2.5)-(2.6) whose period is equal to that of the
communication sequence
x̂(k + 1) = Âk x̂(k) + B̂k ys (k)
(2.5)
u(k) = Ĉk x̂(k) + D̂k ys (k)
(2.6)
where x̂(k) ∈ Rn is the state of the controller, Âk ∈ Rn×n , B̂k ∈ Rn×1 , Ĉk ∈ Rm×n , D̂k ∈ Rm×1
are the controller matrices which are N -periodic, i.e.,
Âk+N = Âk , B̂k+N = B̂k , Ĉk+N = Ĉk , D̂k+N = D̂k , ∀k ∈ Z.
For convenience, we gather all the controller parameters into the following compact form
Θk =
Âk
Ĉk
B̂k
.
D̂k
(2.7)
And ys (k) is the information which is transmitted from the plant and fed into the controller.
Notice that in general ys (k) is not the same as y(k) because not all elements of y(k) are communicated to the controller at time instant k. If the transmission delay of the data from the plant
to the controller is negligible, then ys (k) can be expressed in the following way:
ys (k) = sk y(k)
(2.8)
where sk is the communication sequence, y(k) is the output of the system.
By defining the augmented state vector ξ(k) = [ xTk
following closed-loop system (Sc ) :
ξ(k + 1) = Āsk ξ(k) + B̄sk w(k)
(2.9)
z(k) = C̄sk ξ(k) + D̄sk w(k)
(2.10)
where
Āsk
x̂Tk ]T , then from (2.1)-(2.8) we have the
A + B2 D̂k sk C2
=
B̂k sk C2
C̄sk = [ C1 + D12 D̂k sk C2
B2 Ĉk
B1 + B2 D̂k sk D21
, B̄sk =
,
Âk
B̂k sk D21
(2.11)
D12 Ĉk ] , D̄sk = D11 + D12 D̂k sk D21 .
(2.12)
It is clear that the closed-loop system (2.9)-(2.10) is periodic in k.
5
The H∞ or H2 control problem can be stated as follows: find a communication sequence sk and
a periodic controller Θk of the form of (2.5)-(2.6) such that the closed-loop system (2.9)-(2.10)
is asymptotically stable and has an optimal H∞ or H2 performance under scheduled releasing
policy.
3
H∞ Control Problem
With the introduction of communication sequence, the optimal H∞ control problem under scheduled releasing policy will be formulated as that of periodic systems. Thus we first give the direct
approach for periodic systems analysis and design which is in comparison with the traditional
lifting technique. Now consider a linear discrete-time N -periodic system (Sk ) described by the
following state space model:
x(k + 1) = Ak x(k) + Bk w(k)
(3.13)
z(k) = Ck x(k) + Dk w(k)
(3.14)
where x(k) ∈ Rn is the state vector, w(k) ∈ Rp is the disturbance input, z(k) ∈ Rq is the output
of the system, and Ak ∈ Rn×n , Bk ∈ Rn×p ,Ck ∈ Rq×n , Dk ∈ Rq×p are N -periodic matrices
satisfying
(3.15)
Ak+N = Ak , Bk+N = Bk , Ck+N = Ck , Dk+N = Dk , ∀k ∈ Z.
The transition matrix of the system (3.13)-(3.14) is defined as
Φ(k, l) =
Ak−1 Ak−2 · · · Al , k > l
I,
k = l.
(3.16)
The transition matrix Φ(k, l) is N -periodic in k and l, i.e., Φ(k + N, l + N ) = Φ(k, l), ∀k, l ∈ Z.
The eigenvalues of Φ(k + N, k) which are independent of k are referred to as characteristic
multipliers. Moreover the periodic system is stable if and only if all the characteristic multipliers
of Ak are inside the unit circle of the complex plane [18].
Definition 3.1 Given a scalar γ > 0, the periodic system (3.13)-(3.14) is said to have an H∞
performance γ if it is asymptotically stable and satisfies
sup
0=w∈2
z2
<γ
w2
(3.17)
under zero initial state condition.
In the following we present the Periodic Bounded Real Lemma which will play a key role for the
results in this section.
Lemma 3.1 [21] The N -periodic system (3.13)-(3.14) has an H∞ performance γ if and only if
either of the following equivalent conditions holds:
6
(a) There exists an N -periodic positive definite solution Pk to the periodic Riccati difference
inequality
ATk Pk+1 Ak −Pk +(ATk Pk+1 Bk +CkT Dk )[γ 2 I −(BkT Pk+1 Bk +DkT Dk )]−1 (ATk Pk+1 Bk +CkT Dk )T +CkT Ck < 0
(3.18)
such that γ 2 I − (BkT Pk+1 Bk + DkT Dk ) > 0.
(b) There exists an N -periodic positive definite matrix Xk satisfying the LM Is

−Xk+1
 AT Xk+1
 k
 B T Xk+1
k
0
Xk+1 Ak
−Xk
0
Ck

Xk+1 Bk
0
−γI
Dk
0
CkT 
 < 0,
DkT 
−γI
(3.19)
for k = 0, 1, · · · N − 1, simultaneously.
By Lemma 3.1, the closed-loop system (2.9)-(2.10) has an H∞ performance γ under a given
communication sk if and only if there exists an N -periodic positive definite matrix Xk such that

−Xk+1
 ĀTs Xk+1
 k
 B̄ T Xk+1
sk
0
Xk+1 Āsk
−Xk
0
C̄sk
Xk+1 B̄sk
0
−γI
D̄sk

0
C̄sTk 
 < 0, k = 0, 1, · · · N − 1.
D̄sTk 
−γI
(3.20)
Next we shall use the approach of change of variables as proposed in [20] to derive an explicit
expression for the controller parameters that solve the H∞ control problem.
Denote
Rk
Xk =
MkT
and
Φ2k
I
=
0
Mk
,
Uk
Xk−1
Yk
=
NkT
Yk
,
NkT
Φ1k
Rk
=
MkT
Nk
Vk
(3.21)
I
.
0
(3.22)
Note that the matrices Rk , Yk , Mk , Nk , Uk and Vk are N -periodic. Further it can be easily verified
Mk NkT
= I − Rk Yk ,
Xk Φ2k = Φ1k ,
Rk
ΦT2k Xk Φ2k =
I
I
.
Yk
(3.23)
(3.24)
(3.25)
Now introduce the new controller variables as below
Ck = D̂k sk C2 Yk + Ĉk NkT
(3.26)
Bk = Rk+1 B2 D̂k + Mk+1 B̂k
Ak = Rk+1 (A + B2 D̂k sk C2 )Yk +
(3.27)
Rk+1 B2 Ĉk NkT
+
Mk+1 B̂k sk C2 Yk + Mk+1 Âk NkT .(3.28)
Observe that given matrices Ak , Bk , Ck and D̂k and nonsingular matrices Mk and Nk , the controller parameters Âk , B̂k , Ĉk and D̂k are uniquely determined by (3.26)-(3.28).
7
Theorem 3.1 Given a scalar γ > 0, the H∞ control problem under a given communication
sequence sk for the system (2.1)-(2.3) is solvable if and only if there exist N -periodic matrices
Rk > 0, Yk > 0, Ak , Bk , Ck and D̂k satisfying the following set of LMIs

−Rk+1
 ∗

 ∗

 ∗

∗
∗
−I
−Yk+1
∗
∗
∗
∗
Rk+1 A + Bk sk C2
A + B2 D̂k sk C2
−Rk
∗
∗
∗
Ak
AYk + B2 Ck
−I
−Yk
∗
∗
Rk+1 B1 + Bk sk D21
B1 + B2 D̂k sk D21
0
0
−γI
∗
0
0
(C1 + D12 D̂k sk C2 )T
(C1 Yk + D12 Ck )T
(D11 + D12 D̂k sk D21 )T
−γI




<0


(3.29)
for k = 0, 1, · · · , N − 1, simultaneously, where ∗ denotes the entries which can be known from
the symmetry of the matrix.
Proof First, using (3.23)-(3.25), and the notations in (3.26)-(3.28), the following identities can
be easily derived:
ΦT2k C̄sTk
ΦT2(k+1) Xk+1 B̄sk
ΦT2(k+1) Xk+1 Āsk Φ2k
(C1 + D12 D̂k sk C2 )T
=
,
(C1 Yk + D12 Ck )T
Rk+1 B1 + Bk sk D21
=
,
B1 + B2 D̂k sk D21
Rk+1 A + Bk sk C2
Ak
=
.
A + B2 D̂k sk C2 AYk + B2 Ck
(3.30)
(3.31)
(3.32)
Then, the LMI (3.29) can be obtained easily by pre-multiplying and post-multiplying (3.20) by
diag{ΦT2(k+1) , ΦT2k , I, I} and diag{Φ2(k+1) , Φ2k , I, I}, respectively.
Further, note that if the set of LMIs (3.29) admits N -periodic solutions Rk > 0, Yk > 0, Ak , Bk , Ck
and D̂k , then it follows from (3.29) that
Rk
I
I
> 0.
Yk
This implies that the matrix I − Rk Yk is nonsingular and thus nonsingular matrices Mk and Nk
that satisfy (3.23) are guaranteed to exist. Therefore, the controller parameters can be obtained
from (3.26)-(3.28).
Remark 3.1 It is clear that under different communication sequences, the optimal achievable
H∞ performance will be different. Note that the optimal H∞ performance relies on both the
communication sequence and the periodic controller Θk . Let γo (sk , Θk ) be the optimal H∞
performance under a given communication sequence sk , then the optimal H∞ performance under
scheduled releasing policy can be obtained by the following optimization
γo (sk , Θk )
min
sk ,Θk
subject to (3.29).
Remark 3.2 It should be noted that the above problems are in fact non-convex optimization ones
with integer and rank constraints, and are very difficult to settle directly. It may be solved by
8
combining an exhaustive search [4] with the LMI optimization (3.29). However when the number
of the measurements increases, the size of the search tree will grow quickly. To avoid combinatoric
explosion, in the following, we shall propose a heuristic search method for a communication
sequence which, in conjunction with a convex optimization approach for controller parameters,
gives a simple solution to the H∞ control problems.
Heuristic Search Method
• Form so0 = {σ1 , σ2 , · · · , σr }. If the optimal H∞ performance γ0o under this communication
sequence satisfies γ0o − γ opt < , where is a pre-specified tolerance and γ opt is the optimal
H∞ performance for the system without communication constraint, i.e., ys (k) = y(k), then
so0 is the optimal communication sequence and the period is r. Otherwise, proceed to the
next step.
• Step i (1 ≤ i < Nm − r, where Nm is the maximum period which the period of the desired
sequence cannot exceed): Assume that the optimal communication sequence obtained in
o . Add an additional sampling σ , j = 1, 2, · · · , r,
step i−1 is soi−1 and the optimal cost is γi−1
j
to soi−1 to form a set of r new switching sequences sij = {soi−1 , σj }, j = 1, 2, · · · , r and
the period is increased by one. Calculate the H∞ optimal performance under these r
communication sequences by using Theorem 3.1. Assume that the optimal communication
sequence among sij , j = 1, 2, · · · , r, is soi and the optimal cost is γio .
o | < or i = N − 1− r, where is the pre-specified
• Step i+ 1: If γio − γ opt < or |γio − γi−1
1
m
1
o
tolerance, then stop and record the optimal sequence si and the optimal controller Θok .
Otherwise, let i = i + 1 and go back to step i.
Remark 3.3 From the heuristic search method, we can see that for each period we can just
consider r switching sequences. This can avoid the combinatoric explosion and thus reduces the
computation cost greatly compared to the exhaustive search method, especially when r and the
period of the optimal sequence are large. The examples in section 5 will show that the heuristic
search method is convergent with respect to the period of the communication sequence.
4
H2 Control Problem
In this section, we will first characterize the H2 norm of a periodic system by means of LMIs to
analyze the optimal H2 control problem under scheduled releasing policy. For the N -periodic
system (3.13)-(3.14) with p inputs and q outputs, ej is the j-th column vector of the p×p identity
matrix, δk is the Dirac delta function, ej δk is an impulse applied to the j-th input and gkj is the
corresponding output of the system under the zero initial condition. For each k = 0, 1, · · · , N −1,
it is easy to verify that gkj = {0, · · · , 0, Dk ej , Ck+1 Bk ej , Ck+2 Ak+1 Bk ej , · · ·}.
k−1
9
Definition 4.1 [17] For a stable system (Sk ) of (3.13)-(3.14), its H2 norm is defined as
Sk 2
N −1 p
1 = gkj 22
N
k=0 j=1
N −1 p ∞
1 = eTj DkT Dk ej +
eTj W T (k, l)W (k, l)ej
N
k=0 j=1
(4.33)
l=1
where W (k, l) = Ck+1 Ak+l−1 · · · Ak+1 Bk .
This is a commonly used definition of H2 norm for periodic systems, and is an extension of the
well known H2 norm for linear time-invariant systems. Such a norm can be calculated by the
following lemma.
Lemma 4.1 [17] If the N -periodic system (3.13)-(3.14) is stable, then its H2 norm can be
obtained in either of the following two ways:
(a)
N −1
1 trace(DkT Dk + BkT Qk+1 Bk )
Sk 2 = N
(4.34)
k=0
where Qk = QTk and Qk+N = Qk is the periodic solution of the following Lyapunov equation
ATk Qk+1 Ak − Qk + CkT Ck = 0, k = 0, 1, · · · , N − 1.
(4.35)
(b)
Sk 22 =
subject to
−1
1 N
trace(Wk+1 )
(Pk+1 ,Wk+1 ) N
k=0
min

(4.36)

−Pk+1
 AT Pk+1
k
0
Pk+1 Ak
−Pk
Ck
0
CkT  < 0
−I
(4.37)
−Pk+1
 B T Pk+1
k
0
Pk+1 Bk
−Wk+1
Dk
0
DkT  < 0
−I
(4.38)


simultaneously for k = 0, 1, · · · , N − 1, where PkT = Pk satisfying PN = P0 .
By Lemma 4.1, the square of the H2 norm of the closed-loop system (2.9)-(2.10) under a given
communication sk can be calculated by the optimization
−1
1 N
trace(Wk+1 )
(Xk+1 ,Wk+1 ) N
k=0
min
subject to

−Xk+1
 ĀT Xk+1
sk
0
Xk+1 Āsk
−Xk
C̄sk
10
(4.39)

0
C̄sTk  < 0
−I
(4.40)

−Xk+1
 ĀT Xk+1
sk
0
Xk+1 B̄sk
−Wk+1
D̄sk

0
D̄sTk  < 0
−I
(4.41)
simultaneously for k = 0, 1, · · · , N − 1, where XkT = Xk satisfying XN = X0 .
Obviously, the above matrices are nonlinear in unknown variables. Thus similar to the H∞ case,
we again use the definition of the new set of variables in (3.22) which will convert the problem
under consideration to a convex feasibility problem expressed in terms of LMIs. To this end,
we also use the partition forms of Xk and its inverse given by (3.21) and the variables Φ2k , Φ1k
introduced in (3.22). In this way, we will get the following theorem.
Theorem 4.1 The optimal H2 control problem under a given communication sequence sk for
the system (2.1)-(2.3) can be solved by the optimization
−1
1 N
trace(Wk+1 )
(Rk ,Yk ,Ak ,Bk ,Ck ,D̂k ) N k=0
(4.42)
min
subject to the following LMIs

−Rk+1

∗


∗



∗
∗
−I
−Yk+1
∗
∗
∗

−Rk+1

∗


∗
∗
Rk+1 A + Bk sk C2
A + B2 D̂k sk C2
−Rk
∗
∗
−I
−Yk+1
∗
∗
Ak
AYk + B2 Ck
−I
−Yk
∗
Rk+1 B1 + Bk sk D21
B1 + B2 D̂k sk D21
−Wk+1
∗
0
0
(C1 + D12 D̂k sk C2 )T
(C1 Yk + D12 Ck )T
−I




<0 ,


(4.43)

0

0
<0
T
(D11 + D12 D̂k sk D21 ) 
−I
(4.44)
simultaneously for k = 0, 1, · · · , N − 1, RN = R0 and YN = Y0 . The controller parameters are
given by (3.26)-(3.28).
Proof Using the notations in (3.26)-(3.28), the LMI (4.43) can be obtained by pre-multiplying
and post-multiplying (4.40) by diag{ΦT2(k+1) , ΦT2k , I} and diag{Φ2(k+1) , Φ2k , I} respectively. Similarly, pre-multiplying and post-multiplying (4.41) by diag{ΦT2(k+1) , I, I} and diag{Φ2(k+1) , I, I}
respectively, then using the notations in (3.26)-(3.28), the LMI (4.44) can be obtained.
Remark 4.1 Let Sc (sk , Θk )2 denote the optimal achievable H2 norm for a given communication sequence sk , then the optimal H2 control performance under scheduled releasing policy can
be obtained by the following optimization
min
sk ,Θk
Sc (sk , Θk )2
subject to (4.43) − (4.44).
11
Note that the H2 performance Sc (sk , Θk )2 is a nonlinear function of sk and the controller Θk .
Similar to the H∞ case, the H2 optimal control problem is also a non-convex optimization with
integer and rank constraint, and is very difficult to solve using any direct optimization method.
Hence we can use the heuristic search method used in the previous section again to deal with
this problem.
5
5.1
Illustrative Examples
Example 1
Consider a discrete-time linear system described by




x(k + 1) = 
z(k) =
y(k) =
1.45
0
1
0

0.2 0
0
0.4 0
0.2
0.2 1.1 0.75
0.4
−1 0





 x(k) + 


1 0 0 1
1 1 0 0
0 0 1 0
0.6
0
0.6
0
x(k) + w(k) + u(k)
x(k) +
1
1
w(k)






 w(k) + 


0
1
0
1



 u(k)

(5.45)
(5.46)
(5.47)
where x(k), y(k), z(k), u(k), w(k) are the same as those in (2.1)-(2.3). It can be easily seen
that the eigenvalues of the first subsystem are {1.45, 0.4} and the eigenvalues of the second
subsystem are {1.1, 0.4}. The feedback from the sensor to the controller is connected by a
network with scheduled releasing policy, which can transmit only one measurement within one
sampling period. From the previous section, we know that the switches for this system are
σ1 = [ 1
0],
σ2 = [ 0 1 ] .
In the following we will discuss the problem of optimal H∞ and H2 control under scheduled
releasing policy for the system (5.45)-(5.47). First it can be known that the optimal H∞ performance and H2 performance without communication limitations are 3.9368 and 2.7254. We
start from the sequence {σ1 , σ2 } and obtain the optimal H∞ performance 4.8842 and the H2
performance 3.3383. Then by adding the switch σ1 and the switch σ2 to the sequence {σ1 , σ2 }
respectively, we arrive at the sequences {σ1 , σ2 , σ1 } and {σ1 , σ2 , σ2 }. By Theorem 3.1, we can
obtain the H∞ performance 4.3947 under the sequence {σ1 , σ2 , σ1 } which is smaller than that
under the sequence {σ1 , σ2 , σ2 } as shown in Table 1. So the sequence {σ1 , σ2 , σ1 } is the resulted
sequence at this step. Proceeding the heuristic search we have the following results shown in
Table 1, in which the sequence in boldface stands for the resulted sequence at each step.
From the results shown above, we can see that under different communication sequences with
the same period the optimal H∞ performance or H2 performance are different and the more
sampling we allocate for y1 (k) the better performance is achieved. This may be explained by
12
step
i=1
i=2
i=3
i=4
period
N=3
N=4
N=5
N=6
sequence
{σ1 , σ2 , σ1 }
{σ1 , σ2 , σ1 , σ1 }
{σ1 , σ2 , σ1 , σ1 , σ1 }
{σ1 , σ2 , σ1 , σ1 , σ1 , σ1 }
H∞ norm
4.3947
4.1390
3.9647
3.9368
H2 norm
2.9906
2.8313
2.7387
2.7254
sequence
{σ1 , σ2 , σ2 }
{σ1 , σ2 , σ1 , σ2 }
{σ1 , σ2 , σ1 , σ1 , σ2 }
{σ1 , σ2 , σ1 , σ1 , σ1 , σ2 }
H∞ norm
5.6995
5.1950
4.8717
4.6338
H2 norm
3.7493
3.3506
3.1360
3.0006
Table 1: the results of using the heuristic search method
the fact that the first subsystem is less stable than the second one. As a result, communicating
more often with y1 will lead to better performance. Clearly, when we choose the communication
sequence s1opt = {σ1 , σ2 , σ1 , σ1 , σ1 , σ1 }, we can achieve the optimal H∞ or H2 performance which
are in fact the same as those without limited communication. However it should be noted that
even though the measurement of the second subsystem seems to play a less important role, it
cannot be ignored. In fact, there does not exist any controller to stabilize the system by only
using the measurement of the first subsystem.
We can also obtain the optimal sequence and the corresponding performance value by using the
exhaustive search, which will need 651691 flops in H∞ control problem and 735597 flops in H2
control problem. However, in the above heuristic search, only 355796 flops and 392651 flops are
needed respectively to get the desired sequence. When the number of the outputs of the original
system and the periods of the desired sequence become larger, more time will be saved by using
the heuristic search method. The convergence of the heuristic search method with respect to
the period of the communication sequence for this example is shown in Figure 2.
5
4.5
Performance
4
3.5
3
2.5
2
2.5
3
3.5
4
4.5
5
Period of the sequence
5.5
6
6.5
7
Figure 2: Convergence of the heuristic search method: the solid is for H∞ norm and the dashed
is for H2 norm
In the simulation, we also tried to get different systems by changing the value of the element
A(1, 1) from 1.3 to 1.45 and use the heuristic search method to deal with these systems. Here
we give four typical cases and the results are shown in the following table. It should be noted
13
A(1, 1) optimal sequence
1.43
{σ1 , σ2 , σ1 , σ1 , σ1 }
1.4
{σ1 , σ2 , σ1 , σ1 }
1.38
{σ1 , σ2 , σ1 }
1.3
{σ1 , σ2 }
H∞ norm f lopse f lopsh
3.8559
381642 248268
3.7336
189656 151772
3.6508
75254 75254
3.3106
13924 13924
H2 norm f lopse f lopsh
2.6335
403492 258743
2.4947
197050 156421
2.4014
75151 75151
2.0202
17941 17941
Table 2: The optimal performance and sequence for different systems: f lopse stands for the
case of exhaustive search and f lopsh stands for the case of heuristic search
that the value of H∞ norm or H2 norm for each case is exactly the same as that obtained when
there is no communication constraint. It can be clearly seen from Table 2 that the period of the
optimal communication sequence varies with the characteristic of the system.
5.2
Example 2 [22]
It can be known that the optimal H∞ and H2 performance without communication constraint
are 12.4846 and 2.6660 respectively. Using the heuristic search method with one step search, we
can easily obtain the desired sequence s2opt = {σ1 , σ2 } which gives the value of 12.5878 for H∞
performance and 2.6678 for H2 performance. The H∞ controller under the sequence of s2opt is
given as follows

0.2073
 −0.0106
Â1 = 
 −0.0021
0.0003
0.7859
Ĉ1 =
0.3474

4.0209
0.6716
0.1849
−0.0013
−3.8346
−1.4969
0.2728
 −0.0020
Â2 = 
 −0.0045
−0.0000
0.7515
Ĉ2 =
0.3320
8.9205
0.5125
0.2368
0.0000
−10.8991
−4.6346

10.2234
−0.6384
1.0837
−0.0048
129.3361
−3.3639 
,
14.2996 
4.7535


4.5883
 0.0419 

B̂1 = 
 0.0238  ,
−0.0027
−13.6105
−5.3457
−333.6673
,
−90.4704
8.0140
−0.6734
1.1833
0.0000
−66.1137
−0.2445 
,
−0.6370 
0.0001
4.8192
 −0.0331 

B̂2 = 
 0.0483  ,
0.0000
94.9859
,
41.9224
−7.1183
D̂2 =
.
−3.1472

−10.6223
−3.9920
−6.8397
D̂1 =
.
−3.0257




And the H2 controller under the sequence of s2opt is given as follows

0.8103
 −0.1743
Â1 = 
 0.0539
0.0000
0.0065
Ĉ1 =
0.0041
0.1946
0.9469
0.3295
0.0000
−0.0043
0.0013

−320.3214
23.3889 
,
−39.2444 
0.0474
0.0901
−0.2903
0.8547
−0.0000
−0.0067
−0.0005
14
38.0811
,
24.5208
−5.2085
 9.8974 

B̂1 = 
 −2.9110  ,
−0.0006
−0.4838
D̂1 =
.
−0.3041

0.8542
 −0.1262
Â2 = 
 0.0514
0.0000
Ĉ2 =
5.3
0.0062
0.0039
0.2426
0.7680
0.2320
−0.0000
−0.0092
−0.0025
−0.0778
0.0302
0.9305
0.0000
−0.0013
0.0036

−311.7734
28.3163 
,
−55.5182 
0.0458
37.5368
,
24.1451


−3.9938
 7.4985 

B̂2 = 
 −2.8762  ,
−0.0006
D̂2 =
−0.4878
.
−0.3066
Example 3
To further illustrate the effectiveness of the heuristic search method, in the following we will
consider another example with three outputs whose parameters are given by

1.5
 0


A =  0.1

 0
0.3
0.3 0.1
0.4
0
0.2 0.2
−0.1 0
−0.5 0.6

1
C2 =  0
0.1
0.5
0
0
0
0.2
−0.6
0.4
0.3
0
1
0.7







−0.4
0.6
0
0.2
 0 
 1 
 0 
−0.1 














0  , B1 =  0.6  , B2 =  0  , C1T =  0  ,







 0 
 1 
 0.8 
0.3 
−0.4
−0.3
0.6
0.3



0 0.3
1
0 0.5  , D21 =  0.6  , D11 = 0.4, D12 = −0.1.
0 0.6
0
The optimal H∞ norm without communication constraint for this example is 1.2128. By using the heuristic search method we can obtain the desired sequence s3opt = {σ1 , σ2 , σ3 , σ1 , σ1 }
under which the H∞ norm is known as 1.6450. This sequence can be viewed as the optimal
sequence since the simulation shows that no performance is improved when enlarge the period
of the sequence. The optimal H2 norm without communication constraint is 0.9596 and using the heuristic search method we arrive at the following results: 1.0914 under the sequence
{σ1 , σ2 , σ3 , σ1 , σ1 } and 1.0666 under the sequence {σ1 , σ2 , σ3 , σ1 , σ1 , σ1 }. If we choose the tolerance parameter to be 0.1, the sequence s3opt can also be viewed as the optimal sequence for
H2 performance. The convergence of the heuristic search method for this example is shown in
Figure 3.
5.4
Example 4: Inverted Pendulum System
The continuous inverted pendulum system model is given as follows [14]

0
1
0
0
−3m2 g/4M
−Bf /M
ẋp (t) = 
0
0
0
/2LM
3(m
+
m
0
3B
1
2 )g/2LM
f
1 0 0 0
x (t)
y(t) =
0 0 1 0 p



0
0


0
1/M 
 u(t)
xp (t) + 



1
0
0
−3/2LM
where xTp (t) = [ x ẋ θ θ̇ ]T and m1 = 1, Bf = 0, m2 = 0.5, L = 1, g = 9.8, M = m1 + m2 /4.
Assuming a zero-order sample-and-hold at the inputs with a sampling period equal to 0.1, we
15
2.8
2.6
2.4
Performance
2.2
2
1.8
1.6
1.4
1.2
1
3
3.5
4
4.5
5
5.5
Period of the sequence
6
6.5
7
Figure 3: Convergence of the heuristic search method: the solid is for H∞ norm and the dashed
is for H2 norm
can obtain the discrete-time system. To deal with H2 control problem we also assume that
D21 = 0, D11 = 0, D12 = 0 and
B1T = [ 0.2
−0.2 0.4
−0.0 ]T , C1 = [ 0.6
0
0 0.3 ] .
The optimal H2 performance without communication constraint for the discrete-time system
is 1.3267. We start from the sequence {σ1 , σ2 } under which the H2 performance is calculated
as 2.8962. The simulation results via the heuristic search method are given in Table 3 and the
convergence of this method with respect to the period of the sequence for the inverted pendulum
system is shown in Figure 4. From the results we can see that allocating more resources to the
step
i=1
i=2
i=3
i=4
i=5
i=6
period
sequence
N =3
{σ1 , σ2 , σ1 }
N =4
{σ1 , σ2 , σ2 , σ1 }
N =5
{σ1 , σ2 , σ2 , σ2 , σ1 }
N =6
{σ1 , σ2 , σ2 , σ2 , σ2 , σ1 }
N =7
{σ1 , σ2 , σ2 , σ2 , σ2 , σ2 , σ1 }
N = 8 {σ1 , σ2 , σ2 , σ2 , σ2 , σ2 , σ2 , σ1 }
H2 norm
3.7297
3.4148
3.2223
3.1142
3.0154
2.9397
sequence
H2 norm
{σ1 , σ2 , σ2 }
2.7552
{σ1 , σ2 , σ2 , σ2 }
2.6755
{σ1 , σ2 , σ2 , σ2 , σ2 }
2.6232
{σ1 , σ2 , σ2 , σ2 , σ2 , σ2 }
2.5859
{σ1 , σ2 , σ2 , σ2 , σ2 , σ2 , σ2 }
2.5579
{σ1 , σ2 , σ2 , σ2 , σ2 , σ2 , σ2 , σ2 } 2.5359
Table 3: the results of using the heuristic search method
angle between the stick and the vertical axis is essential to achieve better performance and when
the period of the sequence exceeds 5 the performance is improved very little. So the sequence
s4opt = {σ1 , σ2 , σ2 , σ2 , σ2 } may be regarded as the optimal sequence for this inverted pendulum
system. Under the sequence s4opt we can obtain the following controller

1.4295
 0.9741
Â1 = 
 0.1826
−0.0000
4.4826
4.4049
0.0038
−0.0000
14.3213
19.5476
−1.9211
−0.0000
16

30.6403
36.3217 
,
11.8135 
−0.1659


−64.2333
 −82.0351 

B̂1 = 
 −15.3770  ,
0.0000
Ĉ1 = [ −0.2436

−1.7950
−3.2983
−2.8520
0.3430
−0.0000
4.2817
 3.3749
Â2 = 
 −0.2440
0.0000
Ĉ2 = [ −1.2481

1.2679
−4.1886
−3.6711
0.2722
−0.0000
3.9554
 3.1990
Â3 = 
 −0.1218
0.0000
Ĉ3 = [ −1.0177

1.4563
−4.0289
−3.5826
0.1836
−0.0000
4.0177
 3.3581
Â4 = 
 −0.0894
0.0000
Ĉ4 = [ −0.9710

1.2993
−3.7294
1.4331
−0.2613
0.0000
3.8811
 −1.3586
Â5 = 
 0.1916
−0.0000
Ĉ5 = [ −0.9374
1.1950
−6.1463
−15.4440 ] ,

−2.0543
−2.4912
0.9059
0.0000
10.4318
9.4519 
,
−1.5025 
0.0008
0.7601
−3.9912 ] ,
−4.0229
−4.4284
1.0835
0.0000
10.7594
10.1028 
,
−1.3296 
−0.0040
1.3740
−3.7220 ] ,
−3.8659
−4.3284
1.0288
0.0000
10.4950
10.0302 
,
−1.1596 
−0.0046
1.2371
−3.3769 ] ,
−3.0390
2.2977
−1.1867
−0.0000
8.1365
−3.8885 
,
1.3691 
0.0180
0.9926
−2.6187 ] ,
D̂1 = [ 28.6484 ] ,

D̂2 = [ 112.7934 ] ,


D̂3 = [ 101.6118 ] ,


D̂4 = [ 101.3939 ] ,


D̂5 = [ 101.1956 ] .
Performance
3
2.5
2
3
3.5
4
4.5
5
Period of the sequence
5.5
6
6.5
7
Figure 4: Convergence of the heuristic search method
17

−301.4499
 132.8850 

B̂5 = 
 −18.7442  ,
0.0001
3.5
2.5

−302.5802
 −317.7872 

B̂4 = 
 8.4619  ,
−0.0000
4
2

−282.8581
 −289.5060 

B̂3 = 
 11.0252  ,
−0.0000
4.5
1.5

−282.4769
 −279.2660 

B̂2 = 
 20.1874  ,
−0.0000
6
Conclusion
This paper has investigated the optimal H∞ and H2 control problems for networked control
systems. Based on the notion of communication sequence and a direct controller design approach
for periodic systems, the solution to the problems is given in terms of a set of LMIs. Given a
sequence, an explicit expression for the controller is provided in terms of the solution of the
LMIs. It has been shown that communication sequences can affect the system performance and
a heuristic search method is also given to obtain the desired sequence under which the H∞ or
H2 performance is better than that under other sequence. More efficient algorithm to obtain
the optimal communication sequence will be investigated in the future. Moreover, we have
assumed that the control u is transmitted to the plant directly. In practice, the controller may
also transmit the control signals to the plant via a network. The approach of this paper can be
extended to deal with this situation.
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