Proceedings of the 6th World Congress on Intelligent Control and Automation, June 21 - 23, 2006, Dalian, China
Suboptimal Guaranteed Cost Controller Design for Networked Control Systems Chen Peng, Dong Yue Center for Information & Control Engineering Technology Nanjing Normal University 78 Bancang Street Nanjing, 210042, China
[email protected],
[email protected]
Abstract-This paper is concerned with the guaranteed cost controller design of networked control systems. The continuous time plant with state-delay is studied. A new model of the networked control systems is proposed under consideration of the network-induced delay and data packet dropout. In terms of the given model, a suboptimal guaranteed cost controller design method is proposed based on linear matrix inequalities (LMI) approach. Suboptimal Allowable Equivalent Delay Bound (SAEDB) and the feedback gain of a memoryless controller can be derived by solving a set of LMI for the networked control system based on Lyapunov functional method. An example is given to show the effectiveness of our method. Index Terms- Suboptimal Allowable Equivalent Delay Bound, Guaranteed Cost, NCS I. I NTRODUCTION An Networked Control Systems (NCS) involves communication patterns in which both informational and physical control loops are closed through a real-time network. Recently, much attention has been paid to the study of stability analysis and control design of the NCS [1], [8], [10], [11], since their low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. In an NCS, one of the important issues to treat is the effect of the networkinduced delay on the system performance. Performance of the feedback control in the NCS is directly dependent upon the network-induced delay. Time-varying characteristics of the network-induced delay not only degrade control performance but also introduce distortion of the controller signal [4]. When designing a robust controller, in addition to the simple stabilization, they have been various efforts to assign certain performance criteria, such as quadratic cost minimization, pole placement, K2 @K4 norm minimization etc. Among them, the guaranteed cost control aims at stabilizing the systems while maintaining an adequate level of performance represented by the quadratic cost. The existing guaranteed cost controller design methods can be classified into two types: first, delay independent stabilization [3], [7], [9], in which a controller which can stabilize a system is provided irrespective of the size of the delay. Second, delay-dependent stabilization [5], [6] , in which the controller design is concerned with the size of the
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delay. To the best of our knowledge, it seems that there are no previous results on the state-delay guaranteed cost control for NCS, but it is very important to NCS. This has great motivated our research. This paper is structured as follows: Modelling of NCS is given and the definition of SAEDB is presented in Section 2. The SAEDB and guaranteed cost controller design for NCS with state-delay system is considered and an optimal problem is formulated in Section 3. Suboptimal minimum of the quadratic performance index is obtained for NCS in Section 4. A numerical example is given to show the effectiveness of our method and conclusions follow in Section 5 and 6. II. S YSTEM DESCRIPTION AND PRELIMINARIES In this paper we consider the following system with statedelay. {(w) ˙ = D{(w) + D1 {(w g) + Ex(w) {(w) = *(w)> w 5 [g> 0]
(1)
where {(w) 5 Rq and x(w) 5 Rp are the state vector and the control input vector respectively. D> D1 and E are constant matrices with appropriate dimensions. g is a scalar representing the delay in the system state. Suppose the sensor is clock-driven and controller and actuator are event-driven, the data are transmitted with a single-packet and the full state variables are available for measurements, the real input x(w) realized through zero-order hold in (1) is a piecewise constant function. In addition, if we consider the effect of the nonideal Quality of Services (QoS), such as network-induced delay and network packet dropout on the NCS, the real control system can be modeled as (2) {(w) ˙ = D{(w) + D1 {(w g) + Ex(w), ¢ £ ln k + ln > ln+1 k + ln+1 w 5 + x(w ) = N{(w ln ), w 5 {ln k + ln , n = 1> · · · } (3) where k is the sampling period, ln (n = 1> 2> 3> ===) are some integers and {l1 > l2 > l3 > ======} {0> 1> 2> 3> ===}. ln is the network-induced delay, which denotes the time from the instant ln k when sensor nodes sample from plant to the instant when actuator transmit data to the plant. Obviously, ^4 n=1 [ ln k + ln > ln+1 k + ln+1 ] = [ w0 > 4 )> w0 0=
In this paper, we assume that x(w) = 0 before the first control signal reaches the plant. Assume that the sensor to actuator delay is ln ( ln = vf fd vf fd ln + ln ), where ln is the sensor to controller delay, ln is controller to actuator delay and compute and overhead delay is included in vf ln . The system (2) can be rewritten as {(w) ˙ = D{(w) + D1 {(w g) + EN{(ln k)> £ ¢ ln k + ln > ln+1 k + ln+1 w 5
(4)
Remark 1: In (4), {l1 > l2 > l3 > ======} is a subset of {0> 1> 2> 3> ===} = Moreover, it is not required that ln+1 A ln = If | (ln+1 ln ) |= 1 it means no data packet dropout in the transmission. If ln+1 A ln + 1> it means that some data packet dropout and no non-order sequence occurs. If ln+1 ? ln + 1, it means non-order sequence occurs, which includes n = 0 and n ? k as the special cases. Therefore, (4) can be viewed as a general form of NCS, where the effect of the networked-induced delay, data packet dropout and non-order sequence are simultaneously considered. Given positive definite symmetric matrices T1 and U1 , we will consider the cost function Z 4 [{W (w)T1 {(w) + xW (w)U1 x(w)]gw (5) M= 0
Associated with the cost function (5), the guaranteed cost controller is defined as follows: Definition 1: Consider the uncertain time-delay system of NCS (4), if there exists a control law x(w) and a positive scalar , such that, for all admissible uncertainties, the closed-loop system is stable and the closed-loop value of the cost function (5) satisfies M , then is said to be a guaranteed cost and x(w) is said to be a guaranteed cost controller for the system (4). To facilitate developments, we first introduce the following definition. Definition 2: Suboptimal Allowable Equivalent Delay Bound (SAEDB) of NCS, denoted by > which satisfies (ln+1 ln )k + ln+1 > n = 1> 2> 3 · ·· III. S UBOPTIMAL GUARANTEED COST CONTROLLER DESIGN FOR NCS In this section, we assume that the full state variables are available for measurement and present a SAEDB calculation method for system (4) based on a Lyapunov functional method. Theorem 1: For given scalars and l (l = 2> 3> 4), if e A 0, [> \ and P fl (l = there exist matrices Se , Ve and U 1> 2> 3> 4) with appropriate dimensions such that 6 5 e 11 f1 e 12 e 13 e 14 P [ 0 9 f2 e 23 e 24 P e 22 0 0 : : 9 : 9 f3 e 34 P e 33 0 \W : 9 : 9 e 44 P f4 ?0 (6) 9 0 0 : : 9 9 e U 0 0 : : 9 7 -T1 0 8
1
-U11
((ln+1 ln )k + ln+1 ) , n=1> 2 · · ·
are satisfied, e 11 = Ve + D[ W + [DW + P f1 + P f1W W W f1 + P f e 13 = E\ + 3 [D P 3 W e e 22 = V + 2 D1 [ + 2 [DW1 e 24 = 2 [ W + 4 [DW1 f4W e = 3 [ W + 4 \ W E W P where e 34 , then the fW 12 = D1 [ W + 2 [DW + P 2 e 14 = Se + 4 [DW [ W + P f4W f2 e 23 = 2 E\ + 3 [DW1 P W W f3 P f3W e 33 = 3 E\ + 3 \ E P W e 4 [ 4 [ e 44 = U system (4) is asymptotically stable with the feedback gain N = \ [ 1 and the cost function M in (5) satisfies the following bound: (8) M *W (0)[ 1 Se[ W *(0) Z 0 e W *(w)gw *W (w)[ 1 V[ + g Z 0 Z 0 e W *(y)gygv *˙ W (y)[ 1 U[ ˙ +
v
Proof: We set |(w) = {(w) ˙ and construct a Lyapunov functional candidate as (9) Y (w) = Y3 (w) + Y2 (w) + Y1 (w) Rw Rw W | (y)U|(y)gygv, Y2 (w) = where Y3 (w) = w v Rw W { (y)V{(y)gy, Y1 (w) = {W (w)S {(w), and S A 0, V A 0, wg Rw U A 0. Using the formula {(w) {(ln k) ln k |(v)gv = 0 and (4), we can show that, for arbitrary matrices Ql and Pl (l = 1> 2> 3> 4) of appropriate dimensions, ¸ · Z w ¤ £ W |(v)gv = 0 (10) {(w) {(ln k) (w)P ln k
¤ £ (w)Q W [D{(w) + D1 {(w g) + EN{(ln k) |(w)] = 0 (11) where (w) = [{(w) {(w g) {(l k) |(w)], P n £ W ¤ £ W W W W= ¤ P1 P2W P3W P4W , Q = Q1 Q2 Q3 Q4 . Taking the time derivative of Y (w) for w 5 [ ln k + ln > ln+1 k + ln+1 ). Then using (10) and £ (11), the whole time ¢derivative of Y (w) for w 5 ln k + ln > ln+1 k + ln+1 yields: Y˙ (w) = {W (w)[11 ]{(w)+2{W (w)[12 ]{(w g) +2{W (w)[13 ]{(w g)+2{W (w g)[23 ]{(ln k) +{W (w g)[22 ]{(w g)+2{W (w)[14 ]|(w) +2{W (w g)[24 ]|(w)+{W (ln k)[33 ]{(ln k) +2{W (ln k)[34 ]|(w)+|(w)[44 ]|(w) Z w Z w | W (y)U|(y)gy-2[(w)P W ] |(v)gv w
ln k
From (7), it can be ¢seen that, when w 5 £ ln k + ln > ln+1 k + ln+1 , Z w Z w W | (v)U|(v)gv | W (y)U|(y)gy (12) w
225
(7)
ln k
Z -2(w)P
W
Z
w
w
|(v)gv ln k
| W (y)U|(y)gy
(13)
can obtain
Z
ln k
+ W (w)P W U1 P (w)
Combining (10)-(13) and considering the performance index (5), we obtain Y˙ (w) W (w)z(w) + W (w)P W U1 P (w)
(14)
6 13 14 11 + T1 12 9 22 23 24 : :, where z = 9 7 33 + N W U1 N 34 8 44 11 = V + Q1 D + DW Q1W + P1 + P1W 13 = Q1 EN + DW Q3W P1 + P3W 22 = V + Q2 D1 + DW1 Q2W 24 = Q2 + DW1 Q4W 34 = Q3 + N W E W Q4W P4W . 12 = Q1 D1 + DW Q2W + P2W W W W 14 = S + D Q4 Q1 + P4 23 = Q2 EN + DW1 Q3W P2 33 = Q3 EN + N W E W Q3W P3 P3W 44 = U Q4 Q4W By Suchr complements, we can show that the solvability of (6) implies that of (15).
9 9 9 9 9 7
¯ e 12 e 22
e 13 e 23 ¯
e 14 e 24 e 34 e 44
f1 P f2 P f3 P f4 P e U
6
9 9 9 9 7
11 +T1
12 22
13 23 33 +N W U1 N
14 24 34 44
[{W (w)T1 {(w) + xW (w)U1 x(w)]gw
(18)
W g W
Z +
Z
W v Z 0 Z 0
g W
| W (y)U|(y)gygv
| W (y)U|(y)gygv
v
As the closed-loop system (4)) is asymptotically stable, we can obtain Z 4 [{W (w)T1 {(w) + xW (w)U1 x(w)]gw (19) 0 Z 0 *W (y)V*(y)gy *W (0)S *(0) + g Z 0 Z 0 + *˙ W (y)U*(y)gygv ˙
v
e = From the definition of (16) to (15), Se = [S [ W > U W W 1 e W e [U[ > V = [V[ > we get S = [ S [ , U = e W , V = [ 1 V[ e W and putting them in (19), we [ 1 U[ can obtain (8). This completes the proof. IV. S UBOPTIMAL MINIMUM OF THE QUADRATIC
: : : :?0 : 8
PERFORMANCE
(15)
¯ = e 11 + [T1 [ W > ¯ = e 33 + \ W U1 \= Defining where Q2 = 2 Q1 , Q3 = 3 Q1 , Q4 = 4 Q1 and \ = N[ W , e = [U[ W , Ve = defining [ = Q 1 , Se = [S [ W , U W W f [V[ and Pl = [Pl [ (l = 1, 2, 3, 4) then pre, postmultiplying both sides of (16) with gldj([ [ [ [ [) and its transpose, we can also obtain (15) from (16). Therefore, (16) is equivalence to (15). 5
0
{W (W )S {(W ) {W (0)S {(0) Z W Z 0 W + { (y)V{(y)gy {W (y)V{(y)gy
5
5
W
P1 P2 P3 P4 U
6
P lq M1 + M2 + M3 vxemhfw wr : (6) and (7)
: : :?0 : 8 (16)
By Suchr complements, we can obtain Y˙ (w) -{W (w)T1 {(w)-{W (ln k)N W U1 N{(ln k)?0
In terms of Theorem 1, the upper bound (8) of guaranteed e Hence, cost is apparently not a convex function in Se, Ve and U. finding the minimum of this upper bound can not be considered as an LMI generalized eigenvalue minimization problem. In this Section, a method to find a suboptimal value for this upper bound and corresponding feedback gain N = \ [ 1 is presented. In order to obtain a controller feedback gain N = \ [ 1 , which achieves the least guaranteed cost value among all e U, e [> \ and P fl (l = 1> 2> 3> 4 possible choices of Se, V, and given scalars and l (l = 2> 3> 4)> ), we have to solve the following minimization problem:
(17)
Therefore by using Lyapunov theorem, the closed-loop system (4) is asymptotically stable. Furthermore, by integrating both sides of (17) from 0 to W and using the initial condition, we
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(20)
M1 = *W (0)[ 1 Se[ W *(0) R0 e W *(y)gy where M2 = g *W (y)[ 1 V[ = Let us R0 R0 W e W *(y)gygv M3 = v *˙ (y)[ 1 U[ ˙ derive the upper bound on the cost function (5). Assume that there exist new variables 1 = W1 , 2 = W2 , 3 = W3 which satisfies e W ? 2 , [ 1 U[ e W ? 3 (21) [ 1 Se[ W ? 1 , [ 1 V[ By introducing new variables O, W , Z and K, defining O = e1 . The condition (21) [ 1 , W = Se1 ,Z = Ve1 and K = U
can be replaced by. · ¸ · 1 O 2 A0, OW W OW
O Z
¸
· A0,
3 OW
O K
¸ A0 (22)
Assuming (22) is satisfied, we can conclude that the following inequality holds: *W (0)[ 1 Se[ W *(0) ? wu(1 1 ) (23) Z 0 e W *(y)gy ? wu(2 2 ) *W (y)[ 1 V[ g Z 0 Z 0 e W *(y)gygv *˙ W (y)[ 1 U[ ˙ ? wu(3 3 )
4. If (21) is all satisfied, then return to Step 2 after decreasing to some extent. If (21) isn’t all satisfied within a specified number of iterations, then exit. Otherwise, set n = n + 1 and go to Step 3. 5. Output suboptimal guaranteed cost vxe and corresponding guaranteed cost feedback gain N = \ [ 1 = The above Algorithm gives the controller feedback gain N = \ [ 1 under the constraint of guaranteed cost vxe , and corresponding suboptimal guaranteed cost vxe . In the following Section, we will illustrate the effectiveness of the above Algorithm.
v
R0 where 1 , *W (0)*(0), 2 , g *W (y)*(y)gy,3 , R0 R0 W *˙ (y)*(y)gygv. ˙ For some constant A 0> assume v wu(1 1 ) + wu(2 2 ) + wu(3 3 ) ?
(24)
(22) includes nonlinear conditions, it can not be solved directly by LMI. However, in terms of the idea in a cone complementary linearization algorithm [2], [5], the above feasibility problem can be solved iteratively. So we present the following nonlinear minimization problem instead of the original nonconvex minimization in (20). e + UK) e P lq wu([O + SeW + VZ vxemhfw wr : Se A 0> Ve A 0> e A 0> W A 0> Z A 0> K A 0 U dqg (6), (7), (22), (24), · ¸ · ¸ [ L Se L A 0, A 0, L O L W · ¸ · ¸ e L Ve L U A 0, A 0= L Z L K
(25)
Although it is only a suboptimal solution of (20), it is easier to solve than the original non-convex minimization problem. We can easily find a suboptimal minimum of the guaranteed cost based on cone complementary linearization method. For finding solution of (25), we present the following iterative Algorithm. Here, (21) is used as a stopping criterion in the Algorithm. Algorithm: 1. Choose different l (l = 2> 3> 4) to obtain the suboptimal based on Theorem 1, such that there exists a feasible solution to LMI conditions in (6) and (7). 2. Choose a sufficiently large initial , search a feasible set f , V, e U, e [, \ , P fl (l = 1, 2, 3, 4), 1 , 2 , 3 , (S O, W , Z , K) satisfying LMI in (25). Set n = 1> vxe = . 3. Solve the following LMI problem for the variables e K> U) e (O> [> W> Se> Z> V> P lq wu([n O + On [ + Sen W + Wn Se en K + Kn U) e + Zn Ve + Ven Z + U
(26)
vxemhfw wr : OP L lq (25)= Set [n+1 = [, On+1 = O, Wn+1 = W , Zn+1 = Z , Kn+1 = e U en+1 = U e K, S˜+1 = Se, Ven+1 = V,
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V. N UMERICAL EXAMPLE In this Section, an example is provided to illustrate the proposed approach. Consider the state-delay system given in [5]. (27) {(w) ˙ = D{(w) + D1 {(w g) + Ex(w) where · ¸ · ¸ · ¸ 0 1 0=1 0 0 D= , D1 = ,E= = 0 0=1 0 0=1 0=1 Assume that the initial conditions are {1 (w) = 0=02hw+1 and {2 (w) = 0 for w 5 [ max(> g)> 0]. First, we consider the stabilization problem to derive the SAEDB . When the controller is implemented through a network, it can be written in the form of (4). Suppose that the full state variables are available for measurement. It has been found that the SAEDB that guarantees the stability of the system (2) is 4=95 and the corresponding feedback gain N = [0=1478 1=4779] based on Theorem 1 with 2 = 1=1, 3 = 0=3, 4 = 8=4. It means that, if k = 1 and the data packet dropout can be neglected in the transmission, the maximum allowable delay ln 3=95= The designed controller can stabilize the system (2) as long as the upper bound of the network-induced delay is less than 3=95. Second, we consider above system and performance criteria of guaranteed cost under network. Weighting matrices are chose as T1 = 0=1L2×2 and U1 = 0=1. Table1. Suboptimal guaranteed cost and feedback gain
vxe 50 28 26.3
Iterations 1 7 6
38.192 21.736 20.435
N [-0.1556 -1.5333] [-0.1553 -1.5323] [-0.1555 -1.5325]
According to Algorithm 1, we can obtain suboptimal guaranteed cost and corresponding controller feedback gain N. Table 1 shows the obtained suboptimal guaranteed cost and corresponding controller feedback gains when max = 4=75 and g = 0=3= It shows that under the guaranteed cost performance vxe , with corresponding feedback gain N, the designed controller can stabilize the system (2). A plot of the states of the above uncertain system with controller feedback gain n = [0=1555 1=5325] is shown in Figure 2. It appears that the system is asymptotically stable.
Value of state variable
0.05
[5] Y. S. Lee, Y. S. Moon, and W. H. Kwon. Delay-dependent guaranteed cost control for uncertain state-delayed stystems. In Proceedings of the American control conference, June 25-27, 2001. [6] X. Lie and C. E. De Souza. Criteria for robust stability and stabilization of uncertain linear systems with state delays. Automatic, 33(9):pp.1657– 1662, 1997. [7] S. O. R. Moheimani and I. R. Moheimani. Optimal quadratic guaranteed cost control of a class of uncertain time delay systems. IEE Proc.-control Theory Appl, 144:pp.1838–1845, 1997. [8] G. C. Walsh, Y. Hong, and G. B. Linda. Stability analysis of networked control systems. IEEE transactions on control systems technology, 10:pp.438–446, 2002. [9] L. Yu and F. R. Gao. Output feedback guaranteed cost control for uncertain discrete-time systems using linear matrix inequalities. Journal of optimization theory and applications, 113:pp.621–634, 2001. [10] D. Yue, Q. L. Han, and C. Peng. State feedback controller design of networked control systems. Circuits and Systems II: Analog and Digital Signal Processing, IEEE Transactions on, 51:640–644, 2004. [11] W. Zhang, M. S. Branicky, and S. M. Phillips. Stability of networked control systems. IEEE Control Systems Magazine, 21:pp.84–99, 2001.
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Fig. 1.
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VI. C ONCLUSION In this paper, we have presented a solution to the delaydependent guaranteed cost control problem via a memoryless state feedback for a class of state-delay systems of NCS. Firstly, the model of NCS is presented, which considers the effect of the nonideal network conditions, such as data packet dropout, non-order sequence etc. Secondly, the guaranteed cost control for state-delay system has been considered, and a cone complementary linearization algorithm is presented to solve non-convex problem of controller design. At last, It was shown that the SAEDB for the guaranteed cost of the NCS, the suboptimal guaranteed cost controller feedback gain and the corresponding cost upper bound can be acquired by numerical example. The multi-loops network scheduling algorithm, controller and scheduling co-design, the plant with nonlinearity and other performance criteria, such as K2 @K4 will be left for our future research. ACKNOWLEDGMENT This work is partially supported by National Natural Foundation of China (60474079), Ministry of Education of Jiangshu (111330B131). R EFERENCES [1] R. Asok and H. Yoram. Integrated communication and control systems: Part ii-design consideration. Journal of Dynamic Systems, Measurement, and Control, 110:pp.374–381, 1988. [2] L. E. Ghaoui, F. Oustry, and M. AitRami. A cone complementary linearization algorithm for static output-feedback and related problems. IEEE Trans. Automat.Contr., 42(8):pp.1171–1176, 1997. [3] X. Guan, Z. lin, and G. Guan. Robust guaranteed cost control for discrete time uncertain systems with delay. IEE Proc.-control Theory Appl, 146:pp.598–602, 1999. [4] S. H. Hong. Scheduling algorithm of data sampling times in the integrated communication and control systems. IEEE Transactions on Control System Technology, 3:pp.225–231, 1995.
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