Robust H-infinity Filter Design for Networked Control System with Random Time Delays

This paper investigates a convex optimization approach to the problem of robustH1 ltering for uncertain linear systems connected over a common digital communication network. We consider the case where quantizers are static and the parameter uncertainties are norm bounded. Firstly, we propose a new model to investigate the eect of both the output quantization levels and the network conditions. Secondly, by introducing a descriptor technique, using Lyapunov-Krasovskii functional and a suitable change of variables, new required sucient conditions are established in terms of delay-dependent linear matrix inequalities (LMIs) for the existence of the desired network-based quantized lters with simultaneous consideration of network induced delays and measurement quantization. The explicit expression of the lters is derived to satisfy both asymptotic stability and a prescribed level of disturbance attenuation for all admissible norm bounded uncertainties.


Introduction
Networked control systems (NCS) in which control and communication issues are combined together, and all the delays and limitations of the communication channels between sensors, actuators, and controllers are taken into account has become an enabling technology for many military, commercial and industrial applications.In practice, due to the finite switching speed of amplifiers or finite speed of information processing, time delays including delays in the state or in the derivative of the state are often encountered in hardware implementation, which may be a source of oscillation, divergence, and instability in system Gao and network-induced delays.Recently, the robust H ∞ control problem for a class of networked systems with random communication packet losses has been studied by Z. Wang and Liu (2007).
However, due to network bandwidth restriction, the insertion of communication network in the feedback control loop inevitably leads to communication delays and makes the analysis and design of NCSs complex.Communication delays can deteriorate the performance of NCSs and even can destabilize the systems when they are not considered in the design of NCSs.So far, a variety of efforts have been devoted to analyzing NCSs with communication delays (see, e.g., Branicky et al. (2000); Gao and Chen (2008); Gao et al. (2008); Hu and Zhu (2003); Kim et al. (2003); Matveev and Savkin (2001); Montestruque and Antsaklis (2003), Nilsson et al. (1998); Wong and Brockett (1999); Yu et al. (2003); Yue et al. (2004); Yue et al. (2005), Yue and Han (2006); Zhang et al. (2001) and the references therein).Specifically, Branicky et al. (2000) and Zhang et al. (2001) analyzed the stability of NCSs and obtained stability regions using a hybrid systems technique.Kim et al. (2003) presented linear matrix inequality (LMI) conditions for obtaining maximum allowable delay bounds, which guarantee the stability of NCSs.Based on Lyapunov-Razumikhin function method, Yu et al. (2003) presented conditions on the admissible bounds of data packet loss and delays for NCSs in terms of LMIs.Based on stochastic control theory, optimal controller design of NCSs with stochastic network delays was investigated in Nilsson et al. (1998); Matveev and Savkin (2001); Hu and Zhu (2003).For other control schemes, we refer readers to the survey Tipsuwan and Chow (2003).To reduce the network traffic load, Montestruque and Antsaklis (2003), Montestruque and Antsaklis (2004) proposed a model-based control scheme for NCSs without/with delays.Necessary and sufficient conditions for the exponential stability of discrete-time and continuoustime NCSs without/with communication delays were established in both cases of state feedback and output feedback.However, they did not present any method for controller design when communication delays are considered.Moreover, it is in general not an easy task to design the controller based on their conditions.Mu et al. (2004) proposed an improved model based control scheme for NCS without/with delays and presented conditions for exponential stability together with controller design procedures.Particularly, an impulsive model based control scheme for discrete-time NCSs without communication delays was discussed.Recently, the problem of output feedback control for networked control systems (NCSs) with limited communication capacity was studied by Tian et al. (2008).In this paper, we contribute to the further development of a convex optimization approach to the problem of robust H ∞ filtering for uncertain linear systems connected over a common digital communication network.The network can be considered as depicted in Fig. 1.Here, we consider the case where quantizers are static and the parameter uncertainties are norm bounded.Firstly, we propose a new model to investigate the effect of both the output quantization levels and the network conditions.Secondly, by introducing a descriptor technique, using Lyapunov-Krasovskii functional and a suitable change of variables, new required sufficient conditions are established in terms of delay-dependent linear matrix inequalities (LMIs) for the existence of the desired network-based quantized filters with simultaneous consideration of network induced delays and measurement quantization.The explicit expression of the filters is derived to satisfy both asymptotic stability and a prescribed level of disturbance attenuation for all admissible norm bounded uncertainties.

Network Medium
The rest of this paper is organized as follows.Section 2 is the problem formulation and related preliminaries.In section 3, we investigate the robust H ∞ performance analysis of the filtering error system.The robust H ∞ filter design problem with normbounded uncertainties is addressed in Section 4 and the result is obtained based on the notion of asymptotic stability and LMIs.A numerical example is provided to illustrate the effectiveness of the approach presented in this paper in Section 4. And, we conclude the paper in Section 5.
The notations used throughout the paper are fairly standard.I n and 0 n represent, respectively, n by n identity matrix and n by n zero matrix; the superscript T stands for matrix transposition; n denotes the ndimensional Euclidean space; n×m is the set of all real m by n matrices.The matrices Î and Ĩ are defined, respectively, as Î := [I, 0] and Ĩ := [0, I]. .refers to the Euclidean vector norm or the induced matrix 2-norm and diag {• • • } represents a block diagonal matrix.λ min (A) and λ max (A) denote, respectively, the smallest and largest eigenvalue of the square matrix A. The operator sym {A} denotes A + A T and [.] is the operation of round.The notation P > 0 means that P is real symmetric and positive definite and the symbol * denotes the elements below the main diagonal of a symmetric block matrix.In addition, L 2 [0, ∞] is the space of square-integrable vector functions over [0, ∞].Matrices, if the dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

System description
Consider the following continuous-time system with time-varying structured uncertainties: where x(t) ∈ n is the state vector, y(t) ∈ m is the measured output, considered as the control input; w(t) ∈ l and z(t) ∈ r are the disturbance and the signal to be estimated, respectively.The coefficient matrices A, B, C, G are real matrices with appropriate dimensions.The time-varying structured uncertainties ∆A(t) and ∆B(t) are said to be admissible if the following form holds where L a , L b are constant matrices with appropriate dimensions; and F (t) is an unknown, real, and possibly time-varying matrix with Lebesgue measurable elements, and its Euclidean norm satisfies We are interested in investigating the stability property of systems when the observer undergoes quantization and delays.This kind of problem arises in scenarios in which a finite bandwidth channel lies in the feedback loop and introduces a delay.
In this paper, a quantizer means a piecewise constant function q : p → Q, where Q is a finite subset of l .This leads to a partition of l into a finite number of quantization regions of the form z ∈ l : q(z) = i , i ∈ Q where z ∈ l is the variable to be quantized.When z does not belong to the union of quantization regions of finite size, the quantizer saturates.More precisely, it is assumed that there exist positive real numbers M and ∆ such that the following two conditions hold: We will refer to M and ∆ as the range of q(z) and the quantization error, respectively.Condition 1 in ( 6) gives a bound on the quantization error when the quantizer does not saturate.Condition 2 in (7) provides a way to detect the possibility of saturation.We also assume that q(z) = 0 for z in some neighbourhood of the origin, i.e., the origin lies in the interior of the set {z : q(z) = 0}, Liberzon (2003); Tian et al. (2008).
In addition, in this paper, we will use quantized measurements of the form where µ > 0 and the range of this quantizer is µM and the quantization error is µ∆, Tian et al. (2008).
The problem considered here is to estimate the signal z(t) in ( 1) by a network-based quantized filter, shown in Fig. 1, of a general structure described by where x f (t) is the filter state vector, µ 1k q 1 is the quantized plant output with i k h as the sampling instant of the sensor and h as the sampling period, z f (t) is the filter output, and A f , B f , C f are appropriately dimensioned filter matrices to be designed.η sf k denotes the transmission delay from sensor to the filter.When considering the network conditions from the filter to the plant output, the quantized output signal can be expressed as ) with a natural assumption on the network induced delays as follows where constants η im and η iM , i = 1, 2, denote the minimum and maximum delays, respectively.η f o k denotes the transmission delay from the filter to the plant output.Then, from ( 12)-( 13) we have where ηi := η iM − η im .It is assumed that the values in both sets {i 1 , i 2 , i 3 , • • • } and {j 1 , j 2 , j 3 , • • • } are ordered as follows i k+1 > i k and j k+1 > j k , which means that there is no wrong packet sequences in the network.Therefore, the following conditions are satisfied, respectively, Furthermore, it is worth noting that there are n − 1 continuous packets dropped or lost if Yue and Han (2006).
we obtain the filtering error system as and where Finally, the problem of robust H ∞ filtering for uncertain linear systems with both the output quantization levels and the network conditions can be expressed as below.Problem: Given system (1)-( 3), design the filter ( 9)-( 10) such that the filtering error system ( 22)-( 23) from w(t) to e(t) is asymptotically stable with a prescribed H ∞ performance γ, that is e(t) 2 2 < γ 2 w(t) 2 2 under zero initial conditions for all admissible uncertain parameters.
Remark 1.It can be easily seen that the model under consideration in this paper is different from existing results in Yue and Han (2006) and Yue et al. (2006) in the following perspective: in comparison with our case that the filtering error system in (22) also considers the network conditions from the filter to the plant output, i.e. the quantized controlled output signal in (11), the references Yue and Han (2006); Yue et al. (2006) do not center on this case, i.e., the results in Yue and Han (2006) and Yue et al. (2006) can not be directly applied to the system ( 22)-( 23).

H ∞ performance analysis
In this section, we first investigate the problem of H ∞ performance analysis for nominal system (1)-(3) with no uncertainties and exactly known filter matrices.Specifically, we will be concerned with the conditions under which the filtering error system with finite delay components is asymptotically stable from w(t) to e(t) with an H ∞ performance γ.The following theorem shows that the H ∞ performance of the filtering error system can be guaranteed if one can find certain matrices so that some LMIs are satisfied.
Theorem 1.Given the positive constants γ, ∆ i and the matrices A f , B f , C f , if there exist positive-definite matrices Π 11,11 = −I and other elements Π i,j for j ≥ i are equal to zero.Then, system ( 22)-( 23) is asymptotically stable with the H ∞ performance level γ > 0.
Proof.Firstly, we represent ( 22) in an equivalent descriptor model form as Define the Lyapunov-Krasovskii functional where ξ(s) ds By Lemma 1 (in Appendix) and from ( 25), it is clear that Differentiating other Lyapunov terms in (27) give and, using Jensen's Inequality in Lemma 2 (in Appendix), one gets ξ(s) ds) and, similarly, ξ(s) ds) Moreover, from the Leibniz-Newton formula, the following equations hold for any matrices {N i } 10 i=1 with appropriate dimensions: is an augmented state vector.According to the property of the quantizers q i (.) and using the Leibniz-Newton formula, we readily obtain and, similarly, Now, to establish the H ∞ performance measure for the system (1)-( 3), assume zero initial condition, then we have ] dt , then along the solution of (1) for any nonzero w(t) the following equation holds From ( 19), ( 28)-( 32) and adding the left and right sides of equations ( 33)-( 36) and ( 37)-( 38), respectively, into V (t), we get where means that the L 2 -gain from the disturbance w(t) to the filtering error e(t) is less than γ .By applying Schur complements, we find that Σ < 0 is equivalent to (24).This completes the proof.

Robust H ∞ filter design
In this section we investigate the robust H ∞ filter design problem for system (1)-(3) with the norm bounded uncertainty parameters defined in (4)-( 5).
Given positive constants , γ and ∆ i , there exist a network-based quantized filter in the form of ( 9)-( 10) such that the filtering error system ( 22)-( 23) is asymptotically stable with an H ∞ disturbance attenuation level γ if there exist the scalar ρ > 0, positive-definite matrices with ,10 = ( ÎT + ĨT ) and other elements Πi,j are equal to their counterpart elements in the matrix Π.Moreover, if the above conditions are feasible, desired filter gain matrices in the form of ( 9)-( 10) are given by C f and Proof.If the state-space matrices Ā and B3 in (24) are replaced with Ā + M 1 F (t)L a and B3 + M 1 F (t)L b , respectively, and by considering P 3 = P 2 and introducing change of variables then the inequality ( 24) is equivalent to the following condition: By Lemma 3 (in Appendix), a necessary and sufficient condition for (44) is that there exists a scalar ρ such that then, applying Schur complements, we find that ( 45) is equivalent to (41).This completes the proof.Remark 2.
In Theorem 2, the results are expressed within the framework of LMIs, which can be easily computed by the interiorpint method.It is also observed that the LMIs (25) and ( 41) are linear in the set of scalar ρ, matrices P Then, the optimal solution to obtain the minimum disturbance attenuation level, i.e. γ, can be found by solving the following convex optimization problem min λ subject to LM Is (25) and (41) with λ := γ 2 Remark 3. The reduced conservatism of Theorems 1 and 2 benefits from the construction of the new Lyapunov function in (27), using a free weighting matrix technique, and no bounding technique is needed to estimate the inner product of the involved crossing terms Park (1999).It is also worth noting that, recently, the so-called 'delay fractioning' approach has been developed in Mou et al. (2008) that is shown to lead to much less conservative results than most existing literature.Of course, more detailed investigations using delay fractioning method would be of interest.

Numerical results
In this section, one example is provided to illustrate the effectiveness of the results obtained in the previous sections.
Consider the system (1)-( 3) with the following matrices: By applying Theorem 2 to the system under consideration with constants = 0.1, ) and exogenous disturbance inputs as below, which belongs to [0, ∞], w(t) = 1/(1 + 2t 2 ) 1/(1 + 0.5t 0.5 ) , t ≥ 0 then the filtering error signal e(t) is plotted in Fig. 2 which shows that the controlled output of the filter, z f (t), tracks the controlled output of the system, z(t), well.

Conclusion
In this paper we have developed the problem of robust H ∞ filtering for uncertain linear systems connected over a common digital communication network.We considered the case where quantizers are static and the parameter uncertainties are norm bounded.Firstly, we proposed a new model to investigate the effect of both the output quantization levels and the network conditions.Secondly, by introducing a descriptor technique, using Lyapunov-Krasovskii functional and a suitable change of variables, new required sufficient conditions were established in terms of delay-dependent linear matrix inequalities (LMIs) for the existence of the desired network-based quantized filters with simultaneous consideration of network induced delays and measurement quantization.The explicit expression of the filters was derived to satisfy both asymptotic stability and a prescribed level of disturbance attenuation for all admissible norm bounded uncertainties.Future work will investigate filter and control designs for a class of nonlinear systems over network with random communication packet losses by using the delay fractioning approach.

Appendix
Lemma 1 Han and Yu (2004): For any arbitrary column vectors a(t), b(t) , matrices Φ(t), H, U and W the following inequality holds:

Figure 1 :
Figure 1: A typical network-based filter

Lemma 2 :
Park (1999) (Jensen's Inequality) Given a positive-definite matrix P ∈ n×n and two scalars b > a ≥ 0 for any vector x(t) ∈ n , we haveKhargonekar et al. (1990): Given matrices Y = Y T , D, E and F of appropriate dimensions with F T F ≤ I , then the following matrix inequality Y + sym {DF E} < 0 holds for all F if and only if there exists a scalar > 0 such that Y + DD T + −1 E T E < 0.