Robust H2 Control of Linear Systems with ...

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Robust H2 Control of Linear Systems with Mismatched Quantization Mouquan Shen, Sing Kiong Nguang, Senior Member, IEEE, Choon Ki Ahn, Senior Member, IEEE, and Qing-Guo Wang

Abstract—This paper is concerned with the H2 control of linear systems with multiple quantization channels. The quantization parameters of each channel are not required to be identical. The resultant mismatches are represented by polytopic uncertainties. A composite controller composed of linear and nonlinear parts is designed to meet the required H2 performance and offset the quantization error. Resorting to a vertex separation technique and Finsler lemma instead of matrix inverse operations, new synthesis conditions for the desired linear part are derived in terms of linear matrix inequalities (LMIs), which are further extended to treat systems with norm-bounded uncertainties. A comparison of conservativeness between the proposed methods and the existing ones is demonstrated by two numerical examples. Index Terms—Mismatched quantization, robust control, linear matrix inequalities.

I. I NTRODUCTION With the development of digital technology, networked control systems (NCSs) have been widely applied in the fields of intelligent vehicles, robotics, industrial processes, and smart buildings due to their advantages of simple installation, low cost, and easy maintenance [1], [2]. However, owing to limited bandwidth or security constraints on communication, inevitable signal quantization renders the analysis and synthesis of NCSs complex [3], [4]. It is known that many hybrid phenomena caused by the quantization can deteriorate system performance as well as stability. Consequently, many research efforts have been expended on this topic. Some interesting results are developed in [5], [6] for static quantizers and [7], [8] for dynamic quantizers. To be specific, [5] puts the quantized state feedback Manuscript received July 9, 2017. (Corresponding author: Choon Ki Ahn) This work was supported partially by the National Natural Science Foundation of China under Grants (61773200, 61403189), the peak of six talents in Jiangsu Province under Grant 2015XXRJ-011, the Doctoral Foundation of Ministry of Education of China under Grant 20133221120012, the Natural Science Foundation of Jiangsu Province of China under Grant BK20130949, partially by National Research Foundation of Korea funded by the Ministry of Science, ICT and Future Planning under Grant NRF2017R1A1A1A05001325, and partially by the Brain Korea 21 Plus Project in 2018. Mouquan Shen is with the College of Electrical Engineering and Control Science, Nanjing Technology University, Nanjing, 211816, China (Email: [email protected]). Sing Kiong Nguang is with the Department of Electrical and Computer Engineering, The University of Auckland, Auckland 1142, New Zealand (Email: [email protected]). Choon Ki Ahn is with the School of Electrical Engineering, Korea University, Seoul, 136-701, Republic of Korea (Email: [email protected]). Qing-Guo Wang is with the Institute for Intelligent Systems, University of Johannesburg, South Africa (Email: [email protected]).

control of single-input-single-output discrete linear systems into the framework of the linear quadratic regulator problem and points out that the logarithmic quantizer is optimal to achieve quadratic stabilization. Additionally, this work discovers a relationship between minimum quantization information and unstable plant poles. Inspired by this work, [6] generalizes it to the multiple-input-multiple-output case and discovers that a sector-bound method is non-conservative to meet certain performance criteria with the involved quantization density. In [7], [8], under the supposition that the quantized sensitivity could be tuned by the quantized measurements, a zoom-in/zoom-out strategy is employed to dynamically adjust the quantization parameters. Based on this hybrid strategy, global asymptotic stability is established without exogenous disturbances. To integrate the influence of external disturbances, [9] adopts an input-to-state stability approach to study the quantized stabilization of nonlinear systems with this dynamic strategy. The above dynamic strategy with matched disturbances is also investigated in [10], where a composite controller is designed to ensure the required H2 performance and offset the problems induced by quantization and matched disturbances. Noticeably, in the above results, a standard implicit assumption is made that the quantizer parameters in the encoder and decoder are the same. Due to hardware limitations, this assumption is hard to realize in practical engineering. To relax it, the allowable bound on the mismatch between the encoder and decoder to system stability is investigated by [11] where the mismatch is a constant. Along this line, a changeable case is studied in [12], which discovers the mutual restraint between the mismatch and the quantizer density. However, the ratio to reflect the mismatch is confined to be less than 1. To remove this technique constraint, a general scenario is investigated by [13] where the ratio can vary in an interval without any extra constraint. Moreover, a robust controller design method is established in terms of linear matrix inequalities (LMIs). Pertinent to this idea, sliding mode control of linear systems with mismatched quantizers is discussed in [14]. Notably, most of them suppose that signals in different channels are quantized by the same quantizer. Practically, in NCSs, different channels usually employ different quantizers due to the asynchronism of different nodes, which is also a possible solution to release network load [15]. Therefore, [15] utilizes the logarithmic approach to study the H∞ control of NCSs with multiple quantizers and an iterative algorithm is developed to obtain the controller gain. Unfortunately, the employed logarithmic quantizer requires infinite quantization levels around the equilibrium and this result has not considered

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the multiple quantization mismatches. To make the obtained quantization control results consistent with the engineering requirement, a possible solution is to combine the results of [13] and [15]. However, this combination is nontrivial since the following three aspects should be reconsidered: (I) a solvable presentation to present the multiple mismatches; (II) an effective treatment to handle all mismatches; and (III) a simple controller design method without a complex iterative algorithm. To explore feasible solutions for the above issues, this paper is concerned with the quantized H2 control of linear systems with mismatched coders and decoders. To give a possible solution for (I), the multiple mismatches are represented by polytopic uncertainties, which fall within the robust framework. Resorting to the obtained polytopic presentation, a vertex separation technique is employed to introduce a slack matrix Θ to make full use of the mismatches, which is an effective treatment for (II). For (III), a structure separation controller design method based on the Finsler lemma is developed instead of pre- and left-multiplying an inverse matrix, which could be readily extended to handle the considered systems with time-delay. With the help of the technical analysis of the multiple mismatches and the controller design based on the developed structure separation method, a composite controller consisting of linear and nonlinear parts is designed to meet the required H2 performance and eliminate the negative influence of quantization and matched disturbances, which covers the existing result as a special case. Numerical examples are provided to show the validity of the proposed method. The paper has five parts. In Section 2, the mismatches of quantizers in different channels are modeled by polytopic uncertainties and the desired composite controller is presented. In Section 3, two theorems are established to give the explicit design conditions for the desired controller. Numerical examples are employed to demonstrate the effectiveness of the proposed method in Section 4. Lastly, a conclusion is given in Section 5. Notation: The positive definite matrix X is represented by X > 0. Its transpose is X T . The lower and upper bounds of a are briefly represented by a and a ¯, respectively. For a matrix X, He(X) stands for X + X T . The p-norm of the vector x is 1 denoted by |x|p = (|x1 |p +|x2 |p +· · ·+|xn |p ) p . Once p = ∞, |x|p = max1≤i≤n |xi |. The matrix p-norm for X ∈ Rm×n is |X x| represented by |X |p = supx̸=0 |x|pp . sign() is a sign function. The transpose of X is X T and ∗ denotes the symmetry. II. P ROBLEM S TATEMENT AND P RELIMINARIES Consider a linear time-invariant plant described as below x(t) ˙ = Ax(t) + BQ(u(t)) + Bw(t),

(1)

where x(t) ∈ Rn is the system state, u(t) ∈ Rm denotes the control input, w(t) is the exogenous disturbance, and Q(u(t)) is a quantized control input. A ∈ Rn×n and n×m B are known ]real constant matrices. Let B = [ ∈ R [ ] b1 · · · bi · · · bm , Bi = 0 · · · bi · · · 0 , [ T ]T u1 (t) · · · uTi (t) · · · uTm (t) . Addiand u(t) = tionally, some assumptions on system matrices and w(t) are given as follows:

Assumption 1. The pair (A,B) is controllable. Assumption 2. For a positive constant εw , ||w(t)||∞ ≤ εw . Similar to [13], a mathematical function round(·) is utilized to map the input to its nearest integer, which is also called the quantization function. In contrast to the single quantizer for all input channels in [13], this paper is concerned with the case in which each channel has a different one, which is shown in Fig. 1. u (t ) u1 (t ) Encoder 1:

P F 1 (t)

ui (t )

...

Encoder i:

um (t )

P F L (t)

...

Encoder m: P F P(t) Quantizers

Network

Decoder 1: P d 1 (t )

...

Decoder i: P di (t )

...

Decoder m: P dm (t )

§ u (t ) · ¸ © Pc(t ) ¹

Pd (t)q ¨

Fig. 1. A sketch of multiple quantizers

It is assumed that the communication channels here are perfect (i.e., there is no time delay, packet loss, channel noise, etc.). Subsequently, the control input ( ui (t))is treated by the ith encoder, which is denoted by q µucii (t) (i = 1, · · · , m). (t) Then, it is routed via a digital network to the corresponding decoder. After receiving it, the quantized output Q(ui (t)) of the ith quantizer is decoded as below: ( ) ( ) ui (t) ui (t) Q(ui (t)) = µdi (t)round , µdi (t)q , µci (t) µci (t) (2) where µci (t) and µdi (t) are quantizer parameters of the encoders and decoders with known bounds, respectively. Without loss generality, set µci (t) ∈ [µci , µ ¯ci ] and µdi (t) ∈ [µdi , µ ¯di ]. Define ri (t) = µdi (t)/µci (t). Then, one has ri (t) =∈ [ri , r¯i ], and Q(ui (t)) in (2) is also rewritten as ( ) ui (t) Q(ui (t)) = ri (t)µci (t)q . µci (t) As a result, the quantized Q(u(t)) is represented by ( ) −1 Q(u(t)) = r(t)µc (t)q µc (t) u(t) ,

(3)

where µd (t) = diag{µd1 (t), µd2 (t), · · · , µdm (t)}, µc (t) = diag{µc1 (t), µc2 (t), · · · , µcm (t)}, and r(t) = diag{r1 (t), r2 (t), · · · , rm (t)}. To facilitate the following derivation, an equivalent transformation to ri (t) is given as ri (t) = rˆi + δi (t),

(4)

where rˆi = (ri + r¯i )/2, and δi (t) ∈ [ri − rˆi , r¯i − rˆi ]. Accordingly, r could be rewritten as r(t) = rˆ + δ(t), where rˆ = diag{ˆ r1 , rˆ2 , · · · , rˆm }, diag{δ1 (t), δ2 (t), · · · , δm (t)}.

(5) and

δ(t)

=

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Remark 1. The above multiple quantization scheme covers [11], [13] where each channel has the same quantizer. Due to µci and µdi , these quantizers are allowed to be dynamic.

III. M AIN R ESULTS Before presenting the main results, some abbreviations are made as below:

The main objective of this paper is to design a composite controller that consists of linear uk (t) = Kx(t) and nonlinear un (t) parts as

ϕ = max{¯ µci }, λ = min{ri }, Pt = P Br(t), [ ] He(X ) + S Y A(X , Y) = + GT ΘG. ∗ 0

u(t) = uk (t) + un (t).

In this section, four theorems are established to achieve the required H2 performance and eliminate the adverse effects induced by quantization error and external disturbance.

(6)

The designed linear part uk (t) = Kx(t) should meet the following defined system performance [10]: ∫ ∞ { T } x (v)Sx(v) + uTk (v)Ruk (v) dv, (7) J (t) , t

where S ∈ Rn×n and R ∈ Rm×m are given positive definite matrices. Remark 2. Since the nonlinear part un (t) to be designed later is employed to eliminate the adverse effects induced by the quantization and external disturbance, the above defined H2 performance only contains the linear part uk (t) = Kx(t). To explore the main results in the next part, some technical lemmas are listed as below. Lemma 1. [10] For α, β ∈ R , p ≥ 1, and q ≥ 1, one has n

|α β| ≤ |α|p |β|q , T

p

−1

+q

−1

= 1.

(8)

Lemma 2. [16] If there exists a symmetric matrix Θ with ] [ Θ11 Θ12 , Θ= ΘT12 Θ22

Theorem 1. For a given positive scalar γ, under Assumptions 1 and 2, if there exist matrices P (P > 0), Z, L, and Θ satisfying Θii 22 ≤ 0

(i = 1, · · · , m)

(11)

which is the (i, i)th block of Θ22 , and Θ11 + He(∆(δ)Θ12 ) + ∆(δ)Θ22 ∆(δ) ≥ 0(δ ∈ ∆δ ), [ ] Π1 Π2 Π= < 0, ∗ Π3

(12)

xT (0)P x(0) − γ < 0,

(14)

where



[

A(P A + Bˆ rL, M1 ) Π1 =  ∗  P Bˆ r − Bˆ rZ + τ0 LT  0m1 Π2 =

LT R 0 −R

(13)

]  ,

 Π22 diag{ τi LT }  , | {z } i=1,··· ,m

where Θ11 , Θ22 ∈ Rmn×mn , such that Θii 22 ≤ 0(i = 1, ..., m) n×n ∈ R is the (i, i)th block of Θ with Θii 22 , and for ρ ∈ ∆ρ 22

[ ] Π22 = P B1 − B1 Z · · · P Bm − Bm Z ,   −τ0 He(Z) 01m 0m1 diag{−τi He(Z)}  , Π3 =  | {z } i=1,··· ,m

Θ11 + He (∆(ρ)Θ12 ) + ∆(ρ)Θ22 ∆(ρ) ≥ 0, [ ] U E + GT ΘG < 0, ET 0

M1 = [B1 L · · · Bm L], (9)

then one has W (ρ) = U +

m ∑

ρi Ei + (

i

m ∑

ρi Ei )T < 0

(10)

i

for all ρi ∈ [ρi , ρ¯i ], where U = U T ∈ Rn×n , ∆ρ = {ρ = (ρ1 , · · · , ρm ) : ρi ∈ {ρi , ρ¯i }}, ∆(ρ) = n×n diag{ρ I · · · ρm I}, ),  1  E = [E1 E2 · · · Em ](Ei ∈ R In×n   ..   0   .   G= .   In×n 0 Imn×mn Lemma 3. [17] Let P = P ∈ R and B ∈ R be given matrices. The following statements are then equivalent: T (a) B⊥ PB ⊥ < 0; (b) ∃S ∈ Rn×m such that P + He(SB) < 0, where B ⊥ denotes the arbitrary base of the null space of B. T

n×n

m×n

then the system (1) with the controller (6) is asymptotically stable and meets the required H2 performance level γ. Moreover, K = Z −1 L, and un (t) = −{ϕ/2 + εw /λ}sign(B T P x(t)). Proof. Substituting (3) into (1) provides ( ) x(t) ˙ = Ax(t) + Br(t)µc (t)q µc (t)−1 u(t) + Bw(t). (15) ( ) Setting qµc (u(t)) = µc (t)q µc (t)−1 u(t) and eµc = qµc (u(t)) − u(t), (15) is rewritten as x(t) ˙ = Ax(t) + Br(t)(u(t) + eµc (t)) + Bw(t).

(16)

Moreover, according to the definition of round(), one has |eµci | ≤ µci /2. Choose the Lyapunov function as V (x(t)) = xT (t)P x(t),

(17)

where P > 0. Computing the time derivative of V (t) in (17) yields V˙ (t) = xT (t)He(P A + Pt K)x(t) + 2xT (t)P Bw(t) + 2xT (t)P Br(t)(un (t) + eµc ).

(18)

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From Lemma 1, Assumption 2, and the fact that |eµc |∞ ≤ ϕ/2, the term 2xT (t)Pt (un (t) + eµc ) + 2xT (t)P Bw(t) in (18) is tackled as

where



K B =  0m1

2xT (t)Pt (un (t) + eµc ) + 2xT (t)P Bw(t) ≤ 2xT (t)Pt un (t)



−1

+ 2|x (t)Pt |1 |eµc |∞ + 2|x (t)Pt |1 |r(t) |∞ |w(t)|∞ ϕ εw ≤ 2xT (t)Pt un (t) + 2|xT (t)Pt |1 ( + ). (19) 2 λ T

T

According to un (t) = −{ϕ/2 + εw /λ}sign(B T P x(t)), one obtains ϕ εw ) = 0, (20) 2xT (t)Pt un (t) + 2|xT (t)Pt |1 ( + 2 λ which also means 2xT (t)Pt (un (t) + eµc ) + 2xT (t)P Bw(t) ≤ 0.

(21)

Combining (18) and (21), one has V˙ (t) ≤ xT (t)He(P A + Pt K)x(t). If the following inequality holds He(P A + Pt K) + S + K T RK < 0,

(22)

then one has V˙ (t) < −xT (t)(S + K T RK)x(t) = J˙ (t).

(23)

On the basis of V (∞) = J (∞) = 0, from (23), J (0) < V (0). On the other hand, by recalling (4), (22) is rewritten as T

He(P A + P Bˆ rK + P Bδ(t)K) + S + K RK < 0. (24) ∑m Resorting to Lemma 2, with P Bδ(t)K = i=1 δi (t)P Bi K, (24) holds once the following inequalities hold [ T ] K RK 0 A(P A + P Bˆ rK, E1 ) + < 0, (25) ∗ 0 [ ]T [ ] I I Θ ≥ 0, (26) ∆(δ(t)) ∆(δ(t))

   S=   

01m −I diag{|{z} K } 0m1 m

 01m diag{ −I }  , |{z} m 

Bˆ rZ − P Bˆ r X 0m1 0mm RZ − R 01m τ0 Z 01m 0m1 diag{ τi Z } |{z}

   ,   

m

X = [B1 Z − P B1 · · · Bm Z − P Bm ], which is just (13). According to the above derivation, one obtains J (0) < V (0) when (11)-(14) hold. Consequently, the required stability and H2 performance are ensured. Remark 3. The polytopic uncertainties caused by the mismatched quantization is handled by the vertex separation technique, where a slack matrix Θ is introduced. Therefore, the proposed method could render less conservative H2 performance. Remark 4. By employing the Finsler lemma, a new controller design method is performed without left- and right- multiplying a matrix, which is also called the direct method. Remark 5. Since sign(B T P x(t)) may cause the chattering T x(t) problem, it will be replaced with |BBT PPx(t)|+ξ as [18] in the simulation part. As all the channels have the same quantizer in [11], [13], the proposed method in Theorem 1 reduces to the following corollary. Corollary 1. For a given positive scalar γ, under Assumptions 1 and 2, if there exist matrices P (P > 0), Z, L, and Θ satisfying

where E1 = [P Bi K · · · P Bm K]. Applying the Schur complement to (25) leads to  [ T ]  K R A(P A + P Bˆ r K, E ) 1   < 0. 0 (27) ∗ −R

Θ11 + He(∆(δ)Θ12 ) + ∆(δ)Θ22 ∆(δ) ≥ 0(δ ∈ ∆δ ), [ ] Ξ1 Ξ2 Ξ= < 0, ∗ Ξ3

Thus, (27) is also rewritten as

xT (0)P x(0) − γ < 0,

⊥T

PB ⊥ < 0, (28) [ ] P1 P2 where P = , P1 =  [ T ∗ ] P3 K R rK, E1 )  A(P A + P Bˆ , 0 P2 = ∗ −R   [ ] [ ⊥ ] 0 01m 0 01m B1 ⊥  0m1 0mm , P3 = , B = , ∗ 0mm B2⊥ 0 01m   K 01m 0 B1⊥ = diag{I, |{z} · · · , I}, B2⊥ =  0m1 diag{|{z} K } 0m1 . B

m

m

Applying Lemma 3 to (28) gives P + He(SB) < 0,

(29)

Θ22 ≤ 0,

where



Ξ1 =  [

∗

LT R 0 −R

(32)

]  ,

P Bˆ r − Bˆ rZ + τ0 LT 0 0 P B − BZ + τ1 LT [ ] −τ0 He(Z) 0 Ξ3 = , 0 −τ1 He(Z) Ξ2 =

(31)

(33) [

A(P A + Bˆ rL, BL)

(30)

] ,

then the system (1) with the controller (6) is asymptotically stable and meets the required H2 performance level γ. Moreover, K = Z −1 L, and un (t) = −{ϕ/2 + εw /λ}sign(B T P x(t)). Paralleling to the structure of K = Z −1 L, an alternative result is established below with the dual structure of K = N Y −1 .

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Theorem 2. For a given positive scalar γ, under Assumptions 1 and 2, if there exist matrices Y (Y > 0), N , and Θ satisfying

Corollary 2. For a given positive scalar γ, under Assumptions 1 and 2, if there exist matrices Y (Y > 0), N , and Θ satisfying

Θii 22 ≤ 0, (i = 1, · · · , m)

Θ22 ≤ 0,

(34) and

where Θii 22 is the (i, i)th block of Θ22 , and Θ11 + He(∆(δ)Θ12 ) + ∆(δ)Θ22 ∆(δ) ≥ 0(δ ∈ ∆δ ),  [ ]  Y S NT R rN, M2 )  A(AY + Bˆ 0 ]    < 0, [ 0   −Q 0 ∗ ∗ −R [ ] −γ xT (0) < 0, ∗ −Y

(35) (36)

(37)

where M2 = [B1 N · · · Bm N ], then the system (1) with the controller (6) is asymptotically stable and meets the required H2 performance level γ. Moreover, K = N Y −1 , and un (t) = −{ϕ/2 + εw /λ}sign(B T Y −1 x(t)). Proof. Choose a candidate Lyapunov function as V (x(t)) = xT (t)Y −1 x(t),

(38)

where Y > 0. Defining θ(t) = Y −1 x(t) and calculating the differential of V (x(t)) in (38) gives V˙ (t) = x˙ T (t)Y −1 x(t) + xT (t)Y −1 x(t) ˙ (39)

Along the same lines as Theorem 1, V˙ (t) < J˙ (t) could be guaranteed once the following inequality holds: He(AY + Bˆ rKY + Bδ(t)KY ) + Y SY + Y K T RKY < 0 (40) T −1 with un (t) = −{ϕ/2 + εw /λ}sign{B Y x(t)}. Resorting ∑m to Lemma 2, with BδKY = i=1 δi Bi KY , (40) holds if [ ] He(AY + Bˆ rKY ) + Y SY + Y K T RKY E2 0 E2T T

+ G ΘG < 0, [ ]T [ ] I I Θ ≥ 0, ∆(δ) ∆(δ)

Θ11 + He(∆(δ)Θ12 ) + ∆(δ)Θ22 ∆(δ) ≥ 0(δ ∈ ∆δ ),  [ ]  Y S NT R A(AY + r ˆ BN, BN )  0 ]    < 0, [ 0   −S 0 ∗ ∗ −R [ ] −γ xT (0) < 0, ∗ −Y

(45) (46)

(47)

then the system (1) with the controller (6) is asymptotically stable and meets the required H2 performance level γ. Moreover, K = N Y −1 , and un (t) = −{ϕ/2+εw /λ}sign(B T Y −1 x(t)). Remark 7. Owing to the introduction of Θ, the vertex separation technique is employed to make full use of the bounds of mismatches. Consequently, the conditions of Corollary 2 may be less conservative than those of Theorem 1 in [13], which is also verified via numerical examples in the next section. Subsequently, the uncertain system in [13] is given below: x(t) ˙ = (A + ∆A(t))x(t) + B(I + ∆B(t))Q(u∆ (t)) + Bω(t), (48)

= θT (t)He(AY + Br(t)KY )θ(t) + 2θT (t)Br(t)(un (t) + eµc ) + 2θT (t)Bw(t).

(44)

(41) (42) −1

where E2 = [B1 KY · · · Bm KY ]. By choosing K = N Y and using the Schur complement, it follows  [ ]  Y S NT R rN, M2 )  A(AY + Bˆ 0 ]    < 0, (43) [ 0   −S 0 ∗ −R which is just (51). Remark 6. The difference between Theorems 1 and 2 is that the former is a direct design method that does not involve performing an inverse operation. Once there is just one quantizer, the following corollary could be reduced from Theorem 2 directly.

where u∆ (t) is the control input, ∆A(t) and ∆B(t) are timevarying uncertainties and meet the following assumption Assumption 3. ∆A(t) = U1 ∆1 (t)V1 , ∆1 (t)∆T1 (t) ≤ I, ∆B(t) = U2 ∆2 (t)V2 , ∆2 (t)∆T2 (t) ≤ I, |∆B(t)|∞ ≤ φ < 1,

(49)

where Ui and Vi (i = 1, 2) are known real matrices, ∆i (t) is unknown, and φ is a known non-negative constant. The composite controller u∆ (t) is designed as u∆ (t) = K∆ x(t) + u∆n (t),

(50)

where K∆ and u∆n (t) could be obtained by the following theorem. Theorem 3. For a given positive scalar γ, under Assumptions 1-3, if there exist matrices P (P > 0), Z, L, and Θ satisfying Θii 22 ≤ 0, (i = 1, · · · , m) Θii 22

where

(51)

is the (i, i)th block of Θ22 , and

Θ11 + He(∆(δ)Θ12 ) + ∆(δ)Θ22 ∆(δ) ≥ 0(δ ∈ ∆δ ), (52) [ ] Λ1 Λ2 Λ= (53) < 0, ∗ Λ3 xT (0)P x(0) − γ < 0, where

[

Λ1 = [ Λ12 =

Λ11 ∗ U1T P 0

Λ12 Λ13

]

[

] Ξ Σ + GT ΘG, ∗ 0 ] [ ] LT R Ξ1 LT rˆV2T ,Ξ = , 0 ∗ −η2 I

, Λ11 =

U2T B T P 0

(54)

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

 T −η1−1 I 0 0 + η2 P BU2 U2T B T P + S + K∆ RK∆ . [ ] Λ13 =  ∗ −η2−1 I 0  , Σ = Σ1 . . . Σm , Applying the Schur complement to (58) yields ∗ ∗ −R [ ] ] [ T ]   [ T Bi L 0 ˜ Σ ˆ U1 P U2T B T P K∆ R Ξ T Σi = , Υ1 = [P B1 − B1 Z · · · P Bm − Bm Z], + G ΘG V2i L 0  0 0   ∗ 0    0 −1 −1 T   −η I 0 0 Ξ1 = He(P A + Bˆ rL) + η1 V1 V1 + S, 1     −1     T ∗ ∗ −η I 0 2 P Bˆ r − Bˆ r Z + τ0 L Υ1 ∗ ∗ −R V2 − V2 Z Υ2  , Υ3 = diag{ −τi Υ3i }, Λ2 =  | {z } < 0, (60) 02m×1 Υ3 i=1,··· ,m [ ] [ ] Υ2 = V21 − V21 Z · · · V2m − V2m Z , ˜ ˜ = Ξ1 V2 rˆK∆ and Ξ ˜ 1 = He(P A+P Bˆ   where Ξ rK∆ )+ [ ] −τ0 He(Z) 01×m ∗ −η2 I T τ L −1 i T 0m×1 diag{−τi He(Z)}  , Υ3i = Λ3 =  , η1 V1 V1 +S. With the help of Lemma 3, (53) is obtained. | {z } 0 i=1,··· ,m Theorem 4. For a given positive scalar γ, under Assumptions then the system (48) with the controller (50) is asymptoti- 1-3, if there exist matrices Y (Y > 0), N , and Θ satisfying cally stable and meets the required H2 performance level (61) Θii 22 ≤ 0, (i = 1, · · · , m) γ. Moreover, K∆ = Z −1 L, and u∆n (t) = −{ (1+φ)ϕ 2(1−φ) + εω T where Θii 22 is the (i, i)th block of Θ22 , and λ(1−φ) }sign{B P x(t)}. Proof. Taking an approach to the proof similar to that in Theorem 1, one obtains V˙ (t) = xT (t)He(P (A + ∆A(t)) + P B(I + ∆B(t))r(t)K∆ )x(t) + 2xT (t)P Bω(t) T

+ 2x (t)P B(I + ∆B(t))r(t)(u∆n (t) + eµc ).

(55)

Θ11 + He(∆(δ)Θ12 ) + ∆(δ)Θ22 ∆(δ) ≥ 0(δ ∈ ∆δ ), (62) ] [ ] [ ˆ1 Λ ˆ2 Λ −γ xT (0) ˆ Λ= < 0, (63) ˆ 3 < 0, ∗ −Y ∗ Λ where

From Lemma 1 and Assumptions 2 and 3, one has 2xT (t)P B(I + ∆B(t))r(t)(u∆n (t) + eµc ) + 2xT (t)P Bω(t)

[

ˆ1 = Λ [ ˆ= Ξ

] [ Y V1T Y S N T R ˆ , + G ΘG, Λ2 = 0 0 0   ] −η1 I 0 0 (V2 rˆN )T ˆ3 =  ∗ −S 0 , , Λ −η2 I ∗ ∗ −R [ ] ] ˆ i = Bi N 0 , ˆm , Σ ... Σ V2i N 0

ˆ Σ ˆ Ξ ∗ 0 ˆ1 Ξ ∗

]

T

≤ 2xT (t)P Br(t)u∆n (t) + 2φ|xT (t)P Br(t)|1 |u∆n (t)|∞ [ (1 + φ)ϕ T ˆ= Σ ˆ1 + 2(1 − φ)xT (t)P Br(t)u∆n (t) + 2 |x (t)P Br(t)|1 Σ 2 εω (56) then the system (48) with the controller (50) is asymptoti+ 2 |xT (t)P Br(t)|1 . λ cally stable and meets the required H2 performance level Based on the designed controller u∆n (t) = −{ (1+φ)ϕ 2(1−φ) + γ. Moreover, K∆ = N Y −1 , and u∆n (t) = −{ (1+φ)ϕ 2(1−φ) + εω T T εω }sign{B P x(t)}, it leads to 2x (t)P Br(t)u (t) + T −1 ∆n λ(1−φ) x(t)}. λ(1−φ) }sign{B Y 2|xT (t)P Br(t)|1 ϕ2 + 2|xT (t)P Br(t)|1 ελω = 0, which also gives 2xT (t)P B(I + ∆B(t))r(t)(u∆ n(t) + eµc ) + Proof. The proof could be obtained based on Theorems 2 and 3, and thus, it is omitted here. 2xT (t)P Bω(t) ≤ 0. Therefore, one obtains Remark 8. Once all channels employ a common quantizer V˙ (t) ≤ xT (t)He(P A + P Br(t)K∆ + P U1 ∆1 (t)V1 (i.e., m = 1), the conditions proposed in Theorem 4 could be + P BU2 ∆2 (t)V2 r(t)K∆ ))x(t). (57) reduced to those of [13] by setting Θ = 0. Taking the approach to the proof similar to that in Theorem 1 once again, V˙ (t) < J˙ (t) could be guaranteed once the following inequality holds [ ] ˆ Σ ˆ Ξ (58) + GT ΘG < 0, ∗ 0 [ ]T [ ] I I (59) Θ ≥ 0, ∆(δ) ∆(δ) where

[

] [ ] ˆ 1 (V2 rˆK∆ )T Ξ ˆ i = P Bi K∆ 0 , , Σ V2i K∆ 0 ∗ −η2 I [ ] ˆ ˆ ˆ Σ = Σ1 . . . Σm , ˆ 1 = He(P A + P Bˆ Ξ rK∆ ) + η −1 V T V1 + η1 P U1 U T P

ˆ= Ξ

1

1

1

Remark 9. The computational burden is induced by the quantized channel magnitude m (i.e., all LMI conditions should be checked 2m + 1 times). To alleviate the burden, a feasible solution is to employ the structured Θ considered in [19]. IV. N UMERICAL EXAMPLES In this section, two numerical examples are supplied to illustrate the effectiveness of the proposed methods. Example 1. Consider system (1) with the following parameters: [ ] [ ] 0 1 2 0 A= , B= . 2 −3 1 0.5

IEEE TRANSACTIONS ON AUTOMATIC CONTROL

7

Set S = I, R = 0.01I, µc1 (t) ∈ [0.02, 0.1], µd1 (t) ∈ [0.01, 0.014], µc2 (t) ∈ [0.012, 0.02], and µd2 (t) ∈ [0.016, 0.024]. One then has r1 (t) ∈ [0.1, 0.7] and r2 (t) ∈ [0.8, 1.2]. Table I shows the H2 indices obtained by solving the conditions given in Theorem 1 and Corollary 1 (one quantizer).

2 x in [1] 1

x in [1] 2 1

x

2

1

x(t)

TABLE I γ

x

1.5

0.5

0

FOR DIFFERENT METHODS −0.5

Corollary 1 1.8730

γ

Theorem 1 0.9872

−1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

It is seen that the H2 performance level obtained by multiple quantizers is less conservative than that of the single case. On the other hand, applying the above system parameters to Theorem 1 in [13], Corollary 2 and Theorem 2 yield the corresponding H2 indices in Table II.

Fig. 2. Curves of x(t) 10 quantized u u

1

u

TABLE II

γ

Theorem 1 in [13] 2.030

1

(t) in [13]

(t)

(t)

0

FOR DIFFERENT METHODS

Corollary 2 1.8730

Theorem 2 0.9872

u 1 (t) and quantized u 1 (t)

γ

1

1

(t) in [13]

quantized u

5

-5

From this table, it is seen that the vertex separation technique is more effective than the classical polytopic approach. Additionally, since there is a dualistic relationship between Theorems 1 and 2, by tuning scalars τi , the H2 indices for these two theorems are equal. This similarity is also suitable for Corollary 1 and Corollary 2. Based on the above system matrices, consider the uncertain system parameters: [ ] (48) [with the following ] [ ] U1 = 0.2 0 0.1 0 0.1 0 , V1 = , U2 = , V2 = 0.2 0.5 0.1 0 [ 0 0.3 ] 0.1 0.3 . Applying Theorem 1 in [13] and Theorem 3 in 0 0.1 this paper, the solved H2 indices are obtained and shown in Table III.

10

5

5

0

0

-5

-5

-10

-10

-15

-15

-10

-15

10

-20

-20 -20

-25

-25 0

0.1

0.2

0.3

0

0.4

0.1

0.2

0.3

0.4

0.5

-25 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

Fig. 3. Curves of u∆1 (t) and Q(u∆1 (t))

Choosing S = I, R = 0.1I, µc1 (t) ∈ [0.02, 0.1], µd1 (t) ∈ [0.01, 0.014], µc2 (t) ∈ [0.012, 0.02], and µd2 (t) ∈ [0.016, 0.024], one has r1 (t) ∈ [0.1, 0.7] and r2 (t) ∈ [0.8, 1.2]. Then, the H2 indices computed by Theorem 1 in [13], Corollary 2, and Theorem 2 are given in Table IV. Moreover, the TABLE IV γ

TABLE III γ

γ

FOR DIFFERENT METHODS

Theorem 1 in [13] 2.1283

Theorem 1 1.4495

FOR DIFFERENT METHODS

Theorem 1 in [13] 4.4873

γ

Corollary 2 3.2543

Theorem 2 1.5638

controller gain and Lyapunov matrix for the H2 level 1.5638

10

quantized u u 5

2

2

2

(t) in [13]

(t) in [13]

quantized u u

2

(t)

(t)

0

u 2 (t) and quantized u 2 (t)

By choosing µd1 = 0.014, µc1 = 0.02, µd2 = 0.02, µc2 = 0.015, ∆1 (t) = 0.5 sin t, and ∆2 (t) = 0.8 cos t, under the initial condition x(0) = [2 − 1]T , simulation curves for the uncertain system with the controllers obtained by different methods are given in Figures 2-4. To be specific, the state response curves are depicted in Figure 2, and the quantized control inputs are displayed in Figures 3 and 4, respectively. From the above table and simulation curves, it is reasonable to conclude that the proposed method is more effective than the existing method. Example 2. An aircraft system borrowed from [16] is given below: [ ] [ ][ ] α(t) ˙ −1.175 0.9871 α(t) = q(t) ˙ −8.458 −0.8776 q(t) [ ][ ] −0.194 −0.03593 δE (t) + . −19.29 −3.803 δP T V (t)

2 0

0

-5

-2 -5 -4 -10 -10

-6 -8

-15

-15

-10 -12

-20

-14 -20 -25

-16 0

0.1

0.2

0.3

0

0.4

0.1

0.2

0.3

0.4

-25 0

0.5

1

1.5

2

2.5

Time(sec)

Fig. 4. Curves of u∆2 (t) and Q(u∆2 (t))

3

3.5

4

4.5

5

IEEE TRANSACTIONS ON AUTOMATIC CONTROL

8

are

V. C ONCLUSIONS [

1.3203 2.9122 0.8804 1.9525

K=

]

[

−0.5217 46.3354

2.1720 −0.5217

, Y =

] .

With the choice of x(0) = [2 − 0.5]T , given µd1 = 0.014, µc1 = 0.02, µd2 = 0.02, and µc2 = 0.015, system states and quantized control input curves are simulated in Figures 5 - 7, respectively.

2 x1 x2 1.5

The H2 control of linear systems with multiple mismatched quantizers was discussed in this paper. A polytopic representation was employed to present these mismatches. A vertex separation technique was utilized to make full use of uncertainty vertexes. Without imposing an inverse matrix operation, effective controller design methods were proposed in the form of LMIs. The validity of the proposed methods was verified by numerical examples. When the output is activated by an event-triggered scheme, how to extend the proposed method to multiple mismatched output quantization would be studied as a future work.

1

R EFERENCES x(t)

0.5

0

−0.5

−1

−1.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

Fig. 5. Curves of x(t)

1.5 quantization u1(t)

1

u1(t)

u1(t) and quantization u1(t)

0.5

0 0.01

−0.5

0 −1 −0.01 −1.5

−0.02 −0.03 2.7

−2

−2.5

0

0.5

1

2.8

1.5

2

2.9

3

2.5

3

3.1

3.5

4

4.5

5

Time(sec)

Fig. 6. Curves of u1 (t) and Q(u1 (t))

0.6 quantization u2(t) u2(t)

0.4

u2(t) and quantization u2(t)

0.2 0 −0.2 0.01 −0.4

0 −0.01

−0.6

−0.02 −0.8

−0.03 2.3

2.4

2.5

2.6

2.7

2.8

−1 −1.2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)

Fig. 7. Curves of u2 (t) and Q(u2 (t))

In terms of the above table and figures, it is observed that the H2 performance index yielded by the proposed method is less conservative than that of the existing result and the resultant controller could ensure the required stability.

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