H Predictive Control Design for Discrete-Time Networked Control Systems
Yiming ZHANG*. Vincent SIRCOULOMB.*
Nicolas LANGLOIS*
* Institut de Recherche en Systèmes Electroniques Embarqués, Avenue Galilée, BP 10024, 76801 Saint-Etienne du Rouvray,
France (Tel: 0033-023-291-5823; e-mail: [email protected]).
Abstract: This paper investigates H predictive control problem for a class of networked control system,
and presents a method for quantitatively selecting the control weighting parameter. Based on Lyapunov
function approach and H control theory, a generalized predictive state feedback controller is derived to
guarantee the closed-loop networked system stable. Then, an algebraic method for selecting a feasible
GPC control-weighting parameter is proposed. The contribution realizes the adaptive parameter tuning
for GPC with respect to response demands and performance. A simulation example demonstrates the
feasibility and practicability of the proposed method.
Keywords: Lag Networks, Generalized Predictive Control, Adaptive control, H-infinity control,
Parameter Optimization
1. INTRODUCTION
Networked Control Systems (NCS) (Wang et al., 2008, and
Hespanha et al., 2007) has been widely applied to many areas
such as vehicle control or industrial process, which can
realize information sharing and communication among all the
devices of a system. Meanwhile, along with its
advantages, network also induces new problems of time
delays, packet losses, and out-of-order (Luo and Chen, 2000),
which degrade the control performance of the controlled
system and may even cause the system to be unstable (Zhang
et al., 2001).
As we know, network-induced delays are stochastic,
disordering, unpredictable, and confusing. To overcome
unknown delay and dropout, generalized predictive control
(GPC) (Clarke et al., 1987) begins to be extended to
networked systems. Its control prediction generator is
designed to generate a set of future control laws in order to
cover the situation of no arrival packet (Hu, Bai, Shi and Wu,
2007, Lam, Gao and Wang, 2007, and Xiong and Lam,
2007). However, ordinary GPC lacks of stability analysis and
its parameters such as smooth one or control-weighting one
are always designed by experience. Furthermore, few papers
referring to GPC talk about the topic of disturbance problem.
Motivated by the above mentioned problem, in this paper, a
newly model of generalized predictive control is proposed,
and based on this model, a quantized H problem for
networked systems is studied. The first contribution is to
solve the stability problem for generalized predictive control,
and also achieve the prescribed H disturbance level. The
second contribution is to provide a method of quantitative
selection for the control-weighting parameter. Based on two
above results, we can get a relation table between the smooth
parameter and the control-weighting parameter. We assume
that: when the plant is running, the predictive controller
determines a proper smooth parameter by following a rule
(For example, estimation through a least square algorithm).
Then, it achieves an adaptive control-weighting parameter
guaranteeing the system stable and better performance. A
numerical example is given to illustrate the effectiveness of
the proposed GPC design method.
This paper is organized is as follows. Section 2 gives an
introduction to the modelling of the system. Modelling of
network-faced predictive control under bounded H norm is
presented in section 3. The proposed method is experimented
in Section 4 and concluding remarks are given in Section 5.
2. PROBLEM STATEMENT
Let us consider a noisy discrete-time model obtained from
the discretization of a continuous time-invariant plant with
sampling time Ts . For such a networked control system, the
state-space model of the physical discrete-time system is
assumed to satisfy:
­ xk 1 )xk *um EZk
°
,k t1,k Nu 1d m k
® zk Cxk
° y z X
¯ k k k
(1)
where at time kTs , xk ƒm is the state vector, zk ƒn is the
output vector, yk ƒn is the sensor’s feedback, umƒ p is the
control vector, and ) , * , C are constant matrices of
appropriate dimension. ^Zk ` and ^Xk ` are Gaussian,
uncorrelated, white, and zero mean processes.
Hereby, we consider a class of NCS with short networkinduced delays in the forward channel (Fig. 1). Timestamps
(Nilsson, 1998, and Nilsson et al., 1998) keep the same
clock. Any out-of-order arrival of packets can be reordered
within one buffer span.
The model (1) is equivalent to the following incremental
state-space model:
­ xk 1 ) I xk )xk 1 *'um E 'Zk
°
,k t1,k Nu 1d m k (2)
® zk Cxk
° y z X
¯ k k k
G
where 'um um um 1 , 'Zk Zk Zk 1 , and I is the identity
matrix.
In practice, the priority principle of CAN bus is set to keep
smooth communication in the forward channel. We are
interested in designing a one-step ahead predictive controller
being robust against disturbance.
3. MODELLING OF GPC
For the sake of compensation for network-induced delay
existing in the forward channel, the controller is designed to
generate an incremental vector of predictive control laws:
'U k
^'u
`
T
T
,'ukT1 k ,...,'ukT N 1 k
kk
u
(3)
where 'ui j is the incremental control low for time iTs based
on the states at time jTs .
The first control signal will be sent to the actuator via the
network as usual. We know it is impossible to avoid one
sampling period delay. The system indeed becomes a timedelay system.
7k
ª C*
0
«
C
I
C
)
*
*
« «
#
«
Nu 2
« Nu 1 i
i
«C ¦ ) * C ¦ ) *
0
0
i
i
«
«
#
#
«
N 1
N 2
« C ¦ )i* C ¦ )i*
«¬ i 0
i 0
°­ N
E ® ¦ yk j k J k j
°¯ j N1
2½
Nu
Hence, the optimal future outputs can be simplified by the
vector form:
(7)
Yˆk G 'U k 7 k
Defining
j
¿
j
(4)
h
yk j k ¦ C) i 1xk 1 ¦ C ) i 1xk ¦ ¦ C ) h i *'uk j h k
i 1
i 2
j
h
¦ ¦ C)
h 2i 2
h i
h 2i 2
j
(5)
(6)
¦ ¦ C ) h i *'uk j h k
h 2i 2
The topic about how to compute the expectations E> xk @ and
E> xk 1 @ have been discussed (Zhang et al., 2013).
Let us define:
J k ,'U k ª ˆT
º
T
T
«¬ yk 2 k , yˆ k 3 k ,..., yˆ k N k »¼ ,
ª T
º
T
T
«¬ 'uk k ,'uk 1 k ,...,'uk Nu 1 k »¼ ,
T
`
T
(8)
Taking into account packet dropouts and delay in the
feedback channel, the future reference trajectory is specially
designed by the expectation of output, which is given by the
following rule:
J k 2
D y k 2 1D J k
J k j
D
j -1
y k j 1D j -1 J k , j t 3
So,
ªD 0
«
2
«0 D
Rk «
# %
«
«¬ 0 "
$ Yˆ $
By
T
'U k
^
E >Yk Rk @ >Yk Rk @'U kT O'U k
ª¬Yˆk Rk ¼º ¬ªYˆk Rk ¼º 'U kT O'U k
b k
T
Yˆk
ª T
º
T
T
«¬ yk 2 k , yk 3 k ,..., yk N k »¼ ,
where D is the smooth parameter, 0dD d1 .
j
h
T
ªJ kT 2 ,J kT3 ,...,J kT N º ,
¬
¼
We get the vector form of the cost function:
The optimal predictor of yk j k is:
j
`
diag O1 ,O2 ,...,ONu ,
Rk
E 'Zk 1 j h Xk j
yˆ k j k E ª yk j k º ¦ C )i 1E> xk 1 @ ¦ C ) i 1E> xk @
¬
¼ i 1
i 2
^
O
Yk
where J k is the future reference trajectory, N is the maximal
predictive horizon, N1 is the minimal predictive horizon
( N1 2 for this situation, jt 2 , and N N1 t N u ), N u is the
control horizon and O j is the weighting sequence, and where
yk j k is the j-step future output computed as follows:
j
01 xˆ k 1 0 2 xˆ k
T
j¦1O j 'uk j 1 k °¾°
2
0
#
0
ª 1 iº
ª 1 iº
«C ¦ ) »
«C¦) »
« i 0 »
« i 1 »
« 2 i»
« 2 i»
«C ¦ ) »
«C¦) »
ª
º
E
x
« i 0 » ¬ k 1 ¼ « i 1 » E ª¬ xk º¼
«
»
«
»
#
#
«
»
«
»
« N 1 )i »
« N 1 ) i »
«C ¦
»
«C ¦
»
¬ i 0 ¼
¬ i 1 ¼
We can choose the cost function as the following form:
J k ,'U k º
»
»
»
»
»
C*
"
»
»
»
"
#
»
N Nu
" C ¦ )i* »
»¼
i 0
N N1 *Nu
"
%
%
0 º
ª 1D º
ª 1D º
»
«
«
2 »
2 »
%
# »
« 1D »
« 1D »
ˆ
E Yk «
J k D Yk «
Jk
%
# »
# »
0 »
»
«
»
«
»
«¬1D N -1 »¼
«¬1D N -1 »¼
0 D N -1 »¼
"
aJ k
minimizing
the
wJ k ,'U k w'U k 0 , then
'U k
cost
ª I $b GT I $b G O I º
¬
¼
function
1
following
I $b G T » $aJ k
I $b 01xˆk 1 I $b 0 2 xˆk º¼
(9)
For the sake of simplicity, let us introduce the symbol Ei for
1
) I xk
xk 1
denoting every row of + ª¬ I $b GT I $b G O I º¼ :
Ei
ª0
1 01* Nu i 1 º
¬ 1i
¼
T
ª
º
¬) I *E1+ I Ab G I Ab 01 ¼ xk
The component in the control horizon computed at time
k d Ts is then:
T
E d + I $b G ª¬ $aJ k d I $b 01xˆk 1 d
I $b 0 2 xˆk d º¼
(10)
ª¬)*E1+ I Ab GT I Ab 0 2 º¼ xk 1 E 'Z k 4[ k
where
4 ª*E1+ I Ab GT I Ab 01 *E1+ I Ab GT I Ab 0 2
¬
*E1+ I Ab GT $a º
¼
T
[ k ª xkT
xkT1 J kT1 º .
¼
¬
where 1d d d Nu 1 .
According to this control law, in fact, generalized predictive
control is equivalent to state feedback control. The interest of
this paper is to stabilize the system (1) by a proper gain. The
functional relationship between the gain and the controlweighting parameter O provides a method of quantitative
selection.
For the time-delay system (2), we design a one-step ahead
predictive control law as H controller, which should satisfy:
Given a positive constant r and zero initial condition, if
zk 2 d r 'Zk 2 is satisfied for 'Zk A 2 [0,f ) , the closed-loop
system (2) is H norm stabilizable.
Lemma 1(The Schur complement lemma (Zhang, 2005, and
Haynsworth, 1968)): Given constant matrices s1 , s2 , s3 of
appropriate dimensions where s1 and s2 are symmetric, then
s2 !0 and s1 s3T s21s3 0 if and only if
Hereby, the separation principle is guaranteed to be
applicable and allows for reducing the study of the closedloop dynamic to a simplified stabilization problem.
ª) I *E1+ I $b GT I $b 01 º x
¬
¼ k
xk 1
ª¬)*E1+ I $b GT I $b 0 2 º¼ xk 1 E 'Zk
We assume that there exist positive symmetric matrices P
and Q . Then, we can choose the following Lyapunov
function:
'Vk
Vk 1 Vk
T
T
T
T
xk 1 Pxk 1 xk Qxk xk Pxk xk 1Qxk 1
xk 1 Pxk 1 xk Q P xk xk 1Qxk 1
T
ª xk º
«
»
« xk 1 »
« 'Z »
¬ k¼
s3 º
» 0 .
s1 ¼»
º
»
»
» d 0 (11)
r2 »
»
PE P¼»
0
0
where X P*E1+ and the stars means its corresponding
transpose matrix:
ZT º ª
» «
¼» ¬ Z
(12)
T
xk 1Qxk 1
Its forward differential equation is given by:
Theorem 1: For a given r !0 , if there exist symmetric
positive definite matrices P and Q such that LMI (11) is
solvable, then the closed-loop system (2) is H norm
stabilizable.
ª
«
¬« Z
k 1
i k d
T
xk Pxk
or equivalently
ª
QPCTC
0
«
0
Q
«
«
0
0
«
«
T
T
¬«P)PX I Ab G I Ab 01 X I Ab G I Ab02P)
T
xk Pxk ¦ xiT Qxi
Vk
ª s1 s3T º
«
» 0 ,
«¬ s3 s2 »¼
ª s2
« T
¬« s3
T
I Ab 01xˆk I Ab 0 2 xˆk 1 º¼ E 'Zk
where 0nm is the null matrix with n lines and m columns.
'u k k d
)xk 1 *E1+ I Ab G ª¬ $aJ k 1
º
»
¼
Proof: substituting the first control signal at time k 1Ts
into the system and considering xk xk xˆk ,
T
T
ªQ P
«« 0
«¬ 0
T
Tº
§ª
T
¨ « ¬ª ) I *E1+ I $b G I $b 01 ¼º »
¨«
T »
¨« ª
T
º »
¨ « ¬ *E1+ I $ b G I $ b 0 2 ) ¼ » P>u@
¨«
»
ET
¨«
»
¨
¼
©¬
0 0 º · ª xk º
¸«
»
Q 0 »» ¸ « xk 1 »
0 0 »¼ ¸¹ «¬ 'Zk »¼
where for any matrix Z: ZP>u@ ZPZ T .
The H cost function H is:
H
f
f
¦ ª zTk zk r 2 'ZkT 'Zk º d ¦ ª zkT zk r 2 'ZkT 'Zk º Vk
¼ k 0¬
¼
k 0¬
f
¦ ª zkT zk r 2 'ZkT 'Zk 'Vk º
¼
k 0¬
Tº
­
§ ªª
) I *E1+ I $b G T I $b 01 º »
°
T ¨ «¬
¼
° ª xk º ¨ «
f °
T »
¦ ® «« xk 1 »» ¨ « ª *E1+ I $b G T I $b 0 2 ) º » P>u@
¼ »
¨« ¬
k 0°
« 'Z » ¨
»
T
°¬ k ¼ ¨ «
E
«
»
¨
°
¼
©¬
¯
Table1. Results of H norm-based parameter tuning
½
0 º · ª xk º °
» ¸«
°
0 » ¸ « xk 1 »» ¾
°
»¸
r 2 ¼ ¸¹ «¬ 'Zk »¼ °¿
Hereby, we consider a rule that a lower smooth parameter is
selected online when a larger difference of the reference is
detected. We expect to lower the overshoot. Then, we
introduce the following time-varying reference (Fig. 1) to test
the control performance.
ªQ P C T C
«
«
0
«
0
¬
0
Q
0
According to lemma 1,
D
0.2
0.5
0.7
O
0.0043
0.0021
0.0011
40
T
ªª
T
º º
«¬)I *E1+ I $b G I $b 01¼ »
ªQPCTC 0
0 º
«
«
»
T »
« ª*E + I $ GT I $ 0 )º » P >u@ «
Q 0 » d 0
0
b
b
2
¼ »
«¬ 1
«
»
«
»
T
0
0 r2 ¼»
¬«
E
«
»
¬
¼
Reference
30
T
ªª
T
º º
«¬P)PP*E1+ I $b G I $b 01¼ »
ªQPCTC 0
0 º
«
T » 1
»
T
« ªP*E + I $ G I $ 0 P)º »P >u@««
Q
0
0 »d0
2
b
b
¼ »
« ¬ 1
»
«
«
»
0
0 r2 ¼
¬
PET
«
»
¬
¼
ª
QPCTC
0
0 º
«
»
Q
0
0 »
«
«
»d0
r2 »
0
0
«
«
»
T
T
¬«P)PX I $b G I $b 01 X I $b G I $b 02 P) PE P»¼
20
10
0
−10
0
P X
K
ª*T *E1K T **T K I $b GT I $b GK T *º
¼
* KK * ¬
T
1015
1525
D =0.7
D =0.7
D =0.7
Case 2
D =0.7
D =0.5
D =0.5
Case 3
D =0.7
D =0.5
D =0.2
80
Consider a discrete linear time-invariant plant like (1) with
sampling period Ts 0.1s , and:
ª1 º
« »,C
¬0 ¼
[0.5 0.1], E
ZkT ~ N 0,Q with variance Q
X kT ~ N 0, R with variance R
ª0.1º
« »;
¬0.1¼
20 ˜ I 3 ;
2.5 .
where I m denotes the identity matrix of dimension m .
Considering r 1 , Nu 3 and N 4 , this paper selects three
smooth parameters for samples. By means of Matlab LMI
toolbox, we compute corresponding feasible solutions based
on theorem 1.
case 1
case 2
case 3
60
Control signal
4. SPECIAL CASES AND EXAMPLES
ª1.2 0.7 º
«
»,*
0 ¼
¬1
010
Case 1
According to Tab. 2, we obtain the simulation results of
control signals and outputs:
Proof done
)
Simulation period
(13)
1
T
25
Table2. Three situations for parameter tuning
Ÿ*E1 K ª I $b GT I $b GO I º
¬
¼
Ÿ O1
20
Taking account of two sudden jumps at time 10s and 15s, the
controller would change the smooth parameter to improve the
performance. For comparison, we suppose three situations
(case 1 with a constant parameter; case 2 for a sudden jump;
case 3 for two sudden jumps) to illustrate the advantage of
the designed adaptive control.
If this LMI is solvable by a bounded r , the networked
control system is H norm stabilizable. O can be computed
by:
*E1+
10
15
Simulation time (s)
Fig. 1. Reference of the plant.
where X P*E1+ .
1
5
40
20
0
−20
−40
0
5
10
15
Simulation time (s)
Fig. 2. Control signal of the plant.
20
25
50
case 1
case 2
case 3
System output
40
30
20
10
0
−10
−20
0
5
10
15
Simulation time (s)
20
25
Fig. 3. Output of the plant.
case 1
case 2
case 3
Output Error
40
20
0
−20
0
5
10
15
Simulation time (s)
20
25
Fig. 4. Output errors.
Figure 4 demonstrates that case 3 achieves the smallest
difference. For three cases, it is clear that the presented selfadaptation approach can lower the system overshoot and
smooth the system response. The computation and selection
of parameters can be automatically finished online.
Meanwhile, a larger smooth parameter will lead to static error,
which refers to case 1 (Fig. 3). As a result, we can find that
the results keep the similar law with the rule from experience.
5. CONCLUSION
This paper provides a necessary and sufficient condition for
the stabilization problem of generalized predictive controller
under the bounded H norm. Based on LMI, Lyapunov and
H control theory, a discrete LMI-based theorem has been
presented and the control-weighting parameter can be
quantified by a derived algebraic method. It realizes the
adaptation control for GPC with the self-adjustable smooth
parameter and the online computed control-weighting
parameter, especially for the reference with sudden jumps.
As a result, simulation demonstrates that computed LMIbased feasible solutions can keep the system stable under H
norm and the environment of network-induced delay.
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