Discrete-Time Quantized H Filtering with Quantizer Ranges Consideration

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1 009 American Control Conference Hyatt Regency Riverfront St. Louis MO USA June FrC17.1 Discrete-Time Quantized H Filtering with Quantizer Ranges Consideration Wei-Wei Che and Guang-Hong Yang Abstract This paper is concerned with the quantized H filtering problem for discrete-time systems. The quantizer considered here is dynamic and composed of a dynamic scaling and a static quantizer. Motivated by practical transmission channels requirements the static quantizer ranges are fully considered in this paper. A quantized H filter design strategy is proposed with taking quantizer errors into account a convex optimization method is developed to minimize static quantizer ranges with meeting H performance requirement for quantized augmented systems. A numerical example is given to illustrate the effectiveness of the proposed filter design method. I. INTRODUCTION The recent interest in quantized systems is the involvement of communication channels of limited bandwidth in data transmission. In this setting the signal is quantized and then coded for transmission. From the controller or filter design point of view a fundamental problem is how to design a control strategy composed of a controller and a quantizer (or a filtering strategy composed of a filter and a quantizer ) with the minimum information rate in order to guarantee a given control objective or filtering objective. Filters are the most essential building blocks of signals processing. And the problem of filtering with quantized signals has been considered by several researchers 17 1 ect. Many contributions have appeared in the literature examining the impact of quantization on linear systems. These quantization policies can be mainly categorized depending on whether the quantizer is static or dynamic. A static quantizer is a memoryless nonlinear function as a dynamic quantizer uses memory and thus can be much more complex and potentially more powerful. There has been many quantized results using static quantizers such as and. The attraction of static quantizers is the simplicity of their coding/decoding schemes. However the main drawback of static policies is that they require an infinite number of quantization bits to ensure asymptotic stability 5. In contrast quantization policies with memory (dynamic quantization policies) have the advantages to scale the quantization levels dynamically so that the region of attraction is increased and the steady state limit cycle is reduced. This is indeed the basic idea behind In fact it is shown in 16 that stabilization of an SISO LTI system (in some stochastic sense) can be achieved This work was supported in part by the Funds for Creative Research Groups of China (No ) the State Key Program of National Natural Science of China (Grant No ) National 973 Program of China (Grant No. 009CB30604) the Funds of National Science of China (Grant No ) the 111 Project (B08015) and the Funds of PhD program of MOE China (Grant No ). Wei-Wei Che is with the College of Information Science and Engineering Northeastern University Shenyang P.R. China. Tel: (86) ; Fax: (86) ; She is also with the School of Electrical and Electronic Engineering Nanyang Technological University Singapore cww 1980@16.com Guang-Hong Yang is with the College of Information Science and Engineering Northeastern University Shenyang China Corresponding author. Tel: (86) ; Fax: (86) ; yangguanghong@ise.neu.edu.cn using only a finite number of quantization levels. In this setting the dynamic quantizer effectively consists of two parts: an encoder at the output end and a decoder at the input end 14. Another type of dynamic quantizers uses dynamic scaling in conjunction with a static quantizer. Noticeable work along this line includes Note that dynamic scaling is a very popular idea in signal processing for reducing quantization errors 15 but the key difference here is that the system stability and performance are also considered. Noting that the works outlined above with dynamic quantizers deal with only stability/stabilization problems as the H control problem has not been addressed in them. As pointed out in 4 that the control strategies of updating the quantizer s parameter are dependent on time in the existing works and 13 and such control strategies cannot be applied for the case of H control systems since we do not know the value of the disturbance inputs and thus cannot drive the state into an invariant region as done in 13. In contrast in4 a state or output dependent control strategy is proposed and by using which the quantized continuoustime H control problem is solved. Similarly the latest work 4 studies the networked-based H control problem with dynamic quantizers. But these works do not consider the minimum number of quantization levels required to assure the H performance requirements for quantized systems. However a major question about the quantized systems concerns the minimum number of quantization levels required to assure closed loop system stability and performance which may have many benefits that include lower cost higher reliability and easier maintenance. Motivated by the above reasons this paper is concerned with the quantized H filtering problem for discrete-time linear systems with a type of dynamic quantizers which are conjuncted with static quantizers via dynamic scalings. By taking quantizer errors into account a quantized H filter design strategy is proposed a convex optimization method is developed to minimize static quantizer ranges. The resulting design guarantees that the quantized augmented system is asymptotically stable and with a prescribed H performance bound. The effectiveness of the proposed filter design method is demonstrated by a numerical example. The organization of this paper is as follows. Section II presents the problem under consideration and some preliminaries. Section III gives design methods of quantized H filtering strategies. In Section IV an example is presented to illustrate the effectiveness of the proposed methods. Finally Section V gives some concluding remarks. Notation: Given a matrix E E T and E 1 denote its transpose and inverse when it exists respectively. The symbol within a matrix represents the symmetric entries. For a vector x R k the -norm of x is defined as x := (x T x) 1 and for a matrix Q R m n Q λ max(q) and λ min(q) is defined as the largest singular value the maximum eigenvalue and the minimum eigenvalue of matrix Q respectively. For an appropriate square matrix Y denote He(Y ) = Y + Y T. II. PROBLEM STATEMENT AND PRELIMINARIES A. Quantizer Firstly we introduce the quantizer considered in this paper. The definition of a quantizer is given with general form as in 13. Let /09/$ AACC 5659

2 z R l be the variable being quantized. By a quantizer we mean a piecewise constant function q : R l D D is a finite subset of R l. This leads to a partition of R l into a finite number of quantization regions of the form {z R l : q(z) = i} i D. These quantization regions are not assumed to have any particular shapes. When z does not belong to the union of quantization regions of finite size the quantizer saturates. More precisely we assume that there exist positive real numbers M and such that the following two conditions hold: q(z) z if z M. (1) q(z) z > if z > M. () Condition (1) gives a bound on the quantization error when the quantizer does not saturate. Condition () provides a way to detect the possibility of saturation. M and represent the range and the quantization error bound of the quantizer q( ) respectively. Assume that q(x) = 0 for x in some neighborhood of the origin i.e. the origin lies in the interior of the set {x : q(x) = 0}. In the filtering strategy to be developed below we consider the oneparameter family of quantizers q µ(z) = µq( z ) µ > 0 (3) µ µ is the quantizer s parameter. The range and the error of this quantizer is Mµ and µ respectively. µ can be seen as the zoom variable: increasing µ corresponds to zooming out and essentially obtaining a new quantizer with larger range and larger quantization error such that any signals can be adequately measured while decreasing µ corresponds to zooming in and obtaining a quantizer with smaller range but also smaller quantization error such that the signals can be driven to 0. B. Problem statement Consider an LTI model described by x(k + 1) z(k) y(k) = Ax(k) + B 1ω(k) = C 1x(k) = C x(k) x(k) R n is the state y(k) R p is the measured output ω(k) R m is the disturbance and z(k) R q is the regulated output respectively. A B 1 C 1 and C are known constant matrices of appropriate dimensions. With the quantizer defined by (1)-(3) the quantized filter is given as ξ(k + 1) = A Fξ + B Fq µ1 (y) = A Fξ + B Fµ 1q 1( y µ 1 ) (5) z F(k) = C Fξ + D Fq µ1 (y) = C Fξ + D Fµ 1q 1( y µ 1 ) ξ(k) R n is the filter state z F(k) is the estimation of z(k) and the constant matrices A FB FC F and D F are filter matrices to be designed. q µ1 ( ) is a dynamic quantizer defined by (3) and is composed of dynamic scaling µ 1 and static quantizer q 1( ) defined by (1) with range M 1 and error 1. The main technical difficulty is that there is no separate communication channel to communicate the gain value. One approach is that both sides of the communication channel compute the same µ 1 independently. This is possible only when the gain µ 1 can be computed using only the quantized signal because this signal is available to both sides of the communication channel. Applying (5) to (4) the following quantized augmented system is obtained as x e(k + 1) = A ex e(k) + B eω(k) + B 1ē z e(k) = C ex e(k) + D eē (4) (6) x e(k) = x(k) T ξ(k) T T z e(k) = z(k) z F(k) is the estimation error and A 0 B1 A e = B B FC A e = F 0 C e = ( ) 0 C 1 D FC C F B1 = B F and D e = D F ē = µ 1e with e = q 1( y µ 1 ) y µ 1. Due to the effect of quantization errors the problem addressed in this paper is as follows: Quantized H Filtering Problem: Find a method to optimize the static quantizer range M 1 and then design a quantized H filtering strategy with the minimized quantizer range such that the augmented system (6) is asymptotically stable and with a prescribed H performance bound. C. Preliminaries The following lemma presented will be used in this paper. Lemma 1: Let γ > 0 be a given constant. Then the following statements are equivalent: (i) there exists a symmetric positive matrix X > 0 such that the following LMI holds He(Ω 1) < 0; (7) X 0 A T e X C T e 0 0 γ I B e T X 0 Be T X Ω 1 = 0 0 X 0 0 I 0. 0 X (ii) there exist a nonsingular matrix T and a symmetric matrix P > 0 with Y N P = (8) N N such that the following LMI holds with and He(Ω ) < 0 (9) P 0 A T eap C T ea 0 0 γ I B eap T 0 B T eap Ω = 0 0 P 0 0 I 0 0 P A ea = A 0 B B FaC A ea = Fa B1 0 C ea = C 1 D FaC C Fa (10) A Fa = T 1 A FT B Fa = T 1 B F C Fa = C FT. (11) Proof: Due to the limit of the space it is omitted. III. QUANTIZED H FILTERING STRATEGY DESIGN In this section firstly taking quantizer errors into account a quantized H filtering strategy design method is presented in subsection A without the consideration of optimizing static quantizer ranges. Then in subsection B a convex optimization method is developed to optimize static quantizer ranges and a quantized H filtering strategy is proposed to solve the Quantized H Filtering Problem. 5660

3 A. Quantized H filter design without considering static quantizer ranges Firstly taking quantizer errors into account the following lemma is presented to design filters and matrix variables P Q which will be used in the sequel. Lemma : For plant (6) γ > 0 is a given scalar assume that there exist matrices F A F B C FD F and S > 0 N < 0 such that the following LMI holds with Denote He(Ω 3) < 0 (1) S 0 S 0 0 γ I 0 0 Ω 3 = SA SA SB 1 S 0 Γ 1 Γ 3 (S N)B 1 S Γ Γ SB (S N)B I S 0 0 S Γ 1 = (S N)A + F BC + F A Γ = C 1 D FC C FΓ 4 = C 1 D FC Γ 3 = (S N)A + F BC. (13) A F = N 1 F A B F = N 1 F B C F = C F D F = D F. (14) Then there exist two positive definite matrices P > 0 and Q11 Q Q = 1 > 0 (15) Q such that the following LMI holds Θ 1 + Q 11 A T e PB e + Q 1 Be T PA e + Q T < 0 (16) 1 Θ + Q Θ 1 = A T e PA e P + Ce T C e Θ = Be T PB e γ I and A e B e C e are defined in (6). Proof: Due to the limit of the space it is omitted. Remark 1: It is well-known that the standard H filter design for discrete-time systems is reduced to solving the following inequality Θ 1 A T e PB e Be T PA e γ I + Be T < 0. (17) PB e Compare (16) with (17) except the added matrix variable Q the block ( ) in the left of inequality (16) is with an additional term B T e PB e which reflects the effectiveness of the quantization errors. To facilitate the presentation of Theorem 1 the following algorithm is given first based on Lemma. Algorithm 1: Step 1. By using Lemma design the filter gains A FB FC FD F and matrix variables P Q satisfying (16). η C Step. Compute the value of 1 η = φ + φ + ϕλ min(q) with φ = A T e P B 1 + Ce T D e and ϕ = B 1 T P B 1 + De T D e. Now the following result for H filtering is proposed based on Algorithm 1 without the consideration of optimizing static quantizer ranges which guarantees that system (6) is global asymptotic stability and with the H performance attenuation level γ. Theorem 1: Consider system (4) assume that M 1 is chosen large enough such that M 1 > η C 1 λ min(q). (18) Then filtering strategy (5) with the designed filter gains A FB FC FD F and with the dynamic scaling µ 1 = y M 1 + η 1 C (19) renders the augmented system (6) asymptotically stable and with the H disturbance attenuation level γ. Proof: By using the properties of (1) for the quantizer q 1( ) it is easy to check that whenever y M 1µ 1 ē = µ 1 q 1( y µ ) y µ 1 µ 1 1. (0) Then consider the Lyapunov function candidate V (k) = x T e (k)px e(k) for the quantized augmented system (6) and by using (16) the difference of V (k) along solutions of (6) is computed as V (k) = V (k + 1) V (k) = (A ex e + B eω + B 1ē) T P(A ex e + B eω + B 1ē) x T e Px e γ ω T ω ze T z e + x T e (A T e PA e P + Ce T C e)x e + x T e A T e PB eω + x T e (A T e P B 1 + Ce T D e)ē + ω T Be T PB eω + ē T ( B 1 T P B 1 + De T D e)ē γ ω T ω T ze T z e + γ ω T xe xe ω Q ω ω + x e φ ē + ē ϕ z T e z e + γ ω T ω λ min(q) x e + x e φ ē + ē ϕ z T e z e + γ ω T ω λ min(q)( x e ( x e (φ φ +ϕ) ē ) η ē ) (1) According to (18) there always exists a sufficiently small scalar ε (0 1) such that M 1 > η C 1 λ min(q)(1 ε). () Combining () with (19) we obtain that there always exists a sufficiently small scalar ε (0 1) such that the following holds η C 1 µ1 y λ min(q)(1 ε) = (M 1 + η C 1 )µ1/ M1µ1. (3) λ min(q) In other words if we always choose µ 1 satisfying (3) then (1) holds and thus V (k) ze T z e + γ ω T ω ε λmin(q) xe y. (4) C By setting ω = 0 obviously V (k) < 0 i.e. the system is asymptotically stable. 5661

4 In addition for any k > 0 we can obtain n V (k) V (0) (ze T (k)z e(k) γ ω T (k)ω(k)). (5) k=0 Using V (k) 0 V (k) < 0 and zero initial condition the following is obtained z e(k) < γ ω(k) (6) which implies that the H disturbance attention level γ is achieved. Remark : Theorem 1 offers a quantized H filtering strategy for discrete-time systems such that the required H performance can be guaranteed. A similar result for continuous-time state feedback systems has been obtained in 4. The difference between them is that in 4 the controller gain used in the quantized control strategy is firstly designed by the standard state-feedback H controller design as in this paper the filter gains used in the quantized filtering strategy of Theorem 1 is firstly designed by Lemma with the consideration of the effect of the quantization errors. In fact the additional term Be T PB e in the block ( ) of inequality (16) compared to (17) which has been mentioned in Remark 1 is resulted from the term ω T Be T P B 1e in (1) by using ω T Be T P B 1e ω T Be T PB eω + e T BT 1 P B 1e which renders the effect of the quantization errors involved in the proposed design conditions. Remark 3: In Theorem 1 the static quantizer range M 1 for the existence condition of the quantized H filtering strategy is given based on Algorithm 1. But the static quantizer range achieved by this method may be very large and does not accord with practical communication channel requirements. In the following subsection a method will be developed to minimize the static quantizer ranges. B. Quantized H filter design with considering static quantizer ranges In this subsection we will develop a convex optimization method to optimize M 1 and further give a quantized H filtering strategy to solve the Quantized H Filtering Problem. According to (18) we can minimize M 1 by minimizing the value of η C 1. For the fact that η C 1 is complicated because it depends on design parameters Ce T D e + A T e P B 1 λ min(q) and B 1 T P B 1 + De T D e. In the sequel we aim to optimize these parameters and consequently minimize M 1 indirectly. Let β 1 > 0 β > 0 α > 0 δ > 0 and ε > 0 be scalars and A T e P 1 < β1 (7) P 1 B1 < β (8) C T e < α (9) D e < δ (30) B T 1 P B 1 + D T e D e < ε. (31) Then according to (7)-(31) we can optimize the values of A T e P 1 1 P B1 Ce T D e and B 1 T P B 1 + De T D e by optimizing scalars β 1 β α δ and ε respectively. Obviously inequalities (7)-(31) are respectively equivalent to β 1 I A T e P < 0 PA e P (3) β I BT 1 P P B < 0 1 P (33) α I C e < 0 I (34) C T e δ I De T < 0 (35) I D e ε I Θ 3 < 0 (36) I Θ 3 Θ 3 = B 1 T P B 1 + De T D e and C e B 1 and D e are defined in (6). Now in order to solve the Quantized H Filtering Problem we need to solve inequalities (3)- (36) combined with (16). However inequalities (16) (3) (33) and (36) are not convex and cannot be solved directly. Thus the following lemma is presented to convert them to convex ones. Lemma 3: Let γ > 0 be a given scalar for scalars β 1 > 0 β > 0 α > 0 δ > 0 and ε > 0 matrix variables F A F B C FD F S > 0 N < 0 and Q with structure Q 11 Q 1 Q 13 Q = Q T 1 Q Q 3 > 0 (37) Q T 13 Q T 3 Q 33 the following statements hold: (i) (3) holds if and only if the following LMI holds He(Ω 4) < 0 (38) β 1 I Ω 4 = 0 β 1 I 0 0 SA 0 S 0 (S N)A + F BC F A S (ii) (33) holds if and only if the following LMI holds. β I 0 FB T 0 S S < 0. (39) F B S S + N (iii) (36) holds if the following LMI holds Ω 5 = He(Ω 5) < 0 (40) ε I 0 0 I 0 0 S 0 0 F B 0 S S + N 0 D F I F B 0 D F S 0 0 S S + N I (iv) (16) holds if and only if the following LMI holds. He(Ω 6) < 0 (41) 566

5 Ω 6 = Ξ 1 Ξ T Ξ Q T 13 + Q T 3 Q T γ I+Q SA SA SB 1 S Γ T 1 Γ T 3 (S N)B 1 S Γ T Γ T SB (S N)B I S S with Γ 1 Γ Γ 3 Γ 4 are defined by (13) and Ξ 1 = S + Q 11 + Q 1 + Q T 1 + Q Ξ = S + Q 11 + Q 1 Ξ 3 = S + N + Q 11. Proof: Due to the limit of the space it is omitted. Let ρ = c 0β1 + c 1β + c α + c 3δ + c 4ε c 0 c 1 c c 3 and c 4 are constants to be chosen. Then based on Lemma 3 by optimizing ρ the following algorithm is presented to give a convex optimization method to design the filter gain matrices and matrix variables P Q with the consideration of system performance and quantizer ranges at the same time. Denote Q11 ξ Q1 Q13 Q(ξ) = Q ξ Q 3 > 0 (4) Q 33 ξ Algorithm : Step 1. Solving the following optimization problem min ρ F A F BC FD FS N β 1 β α δ ε ξ Q 11 Q 1Q 13 Q Q 3 Q 33 subject to (34) (35) (38)-(4) output the optimal solutions as N = N opt S = S opt F A = F Aopt F B = F Bopt C F = C Fopt D F = D Fopt Q opt. Step. Compute A Fopt = NoptF 1 Aopt B Fopt = NoptF 1 Bopt C Fopt = C Fopt D Fopt = D Fopt Sopt N P opt = opt N opt N opt N B 0 1opt =. opt B Fopt (43) The resulting A Fopt B Fopt C Fopt and D Fopt will form the optimized filter gains and P opt Q opt are optimized matrices. η Step 3. Compute the value of opt C 1 λ min (Q opt ) η opt = φ opt + φ opt + ϕ optλ min(q opt) with φ opt = A eoptp 1 opt P 1 opt B1opt + Ceopt D T eopt ϕ opt = B 1optP T B 1opt + DeoptD T eopt. and Remark 4: Because λ min(q) has a significant effect on the value of η C 1 λ min condition (4) is introduced to restrict the value (Q) of λ min(q) such that λ min(q) ξ. Remark 5: Algorithm presents a convex optimization method to design the filter gains A FB FC FD F and matrix variables P Q with the consideration of optimizing the value of η C 1 (realized by conditions (34) (35) (38) (39) (40) (4)) as well as guaranteeing the H performance (realized by condition (41)). Now based on Algorithm the following corollary similar to Theorem 1 can be obtained: Corollary 1: Consider system (4) assume that M 1min is chosen large enough such that M 1min > ηopt C 1 λ min(q opt). (44) Then filtering strategy (5) with the designed filter gains A Fopt B Fopt C Fopt D Fopt and with the dynamic scaling µ 1 = y M 1min + η opt 1 C λ min (Q opt ) solves the Quantized H Filtering Problem. IV. EXAMPLE (45) To illustrate the effectiveness of the proposed optimized H filtering strategy using MATLAB via the LMI Control Toolbox 11 an example is given to provide a comparison between our design method with the consideration of optimizing M 1 and the design method without the consideration of optimizing M 1. Example 1: Consider a linearized model of an F-404 engine from 1 which is also studied in 7 and 3. Using the zeroorder hold equivalent method with a sample period T = 0.1s a discrete-time model of the system is obtained as x(k + 1) = x(k) (46) ω(k) 0. 0 z(k) = x(k) y(k) = x(k). A. Quantized filter design by Algorithm 1 In this part we design a quantized H filtering strategy based on Algorithm 1 without the consideration of optimizing M 1. By using Lemma we design a filter with the scalar γ ini = By Algorithm 1 with the designed filter parameters and ξ = = 0.01 for γ = and γ = respectively we obtain η C 1 λ min = 799. and η C 1 (Q) λ min = For (Q) these two cases by Theorem 1 let M 1 = > 799. and M 1 = > 796. respectively filtering strategy (5) guarantees the H filtering objective. B. Quantized filter design by Algorithm In this subsection we design a quantized H filtering strategy based on Algorithm with the consideration of optimizing M 1. Let c 0 = 1 c 1 = 0.1 c = 1 c 3 = c 4 = 0.1. For one case let γ = by Algorithm with ξ = and 1 = 0.01 we obtain the filter parameters and matrices P opt Q opt. It is easy to compute η opt C 1 λ min (Q opt = By Corollary 1 ) let M 1min = 91.5 > then filtering strategy (5) with the corresponding filter parameters and dynamic scaling (45) solves the Quantized H Filtering Problem. For the second case let γ = we obtain the filter parameters and matrices P opt Q opt. It is easy to compute η opt C 1 λ min (Q opt = ) For this case by Corollary 1 let M 1min = > then filtering strategy (5) with the corresponding filter parameters and dynamic scaling (45) solves the Quantized H Filtering Problem. 5663

6 C. Comparison Table 1 is given to compare the quantizer ranges obtained based on Algorithm 1 and the optimization method given in Algorithm. Algorithm 1 Algorithm M 1 (γ = ) M 1 (γ = ) TABLE I COMPARISON OF THE QUANTIZER RANGES From this table we can see that compared with the quantizer ranges obtained based on Algorithm 1 the optimized quantizer ranges obtained based on Algorithm are much more improved. This phenomenon shows the effectiveness of our optimization method. On the other hand for the example we see that the tighter the H performance bound to γ ini is the larger quantizer range is needed. D. Simulation Given initial condition x 0 = T ξ 0 = T for γ = let the disturbance ω(k) be ω(k) = { sin(k) 50 k 60 0 otherwise. Then Figure 1 shows the estimation error responses of system (6) with the quantized H filtering strategies designed with the consideration of optimizing M 1 (solid curve) and without the consideration of optimizing M 1 (deashed curve) respectively. z e Time (s) Fig. 1. Trajectories of estimation errors. This example shows the superiority and the necessity of our design method with the consideration of optimizing static quantizer ranges. V. CONCLUSION The quantized H filtering problem of discrete-time LTI systems has been investigated. In particular the static quantizer ranges are fully considered for their practical importance. By taking quantizer errors into account a quantized H filter design strategy is proposed a convex optimization method is developed to minimize static quantizer ranges. The resulting design guarantees that the quantized augmented system is asymptotically stable and with a prescribed H performance bound. 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