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Synthesis of decentralized dynamic quantizer within invariant set analysis framework
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Synthesis of decentralized dynamic quantizer within invariant set analysis framework Kenji Sawada ∗ Seiichi Shin ∗∗ ∗
The University of Electro-Communications, Chofu-city, Tokyo, Japan, (e-mail: [email protected]). ∗∗ The University of Electro-Communications, Chofu-city, Tokyo, Japan, (e-mail: [email protected]). Abstract: This paper focuses on the synthesis of decentralized dynamic quantizers for linear systems with discrete-valued signal in terms of invariant set analysis. We derive the synthesis condition that can be recast as parameter dependent linear matrix inequalities and has a closed-form solution under some circumstances. In the case of minimum phase systems, our method can provide an optimal decentralized dynamic quantizer in the sense that the quantizer gives an optimal output approximation property. Also, in the case of non-minimum phase systems, our method can provide a suboptimal stable decentralized dynamic quantizer. As a result, this paper gives a unified design method of the dynamic quantizer based on the invariant set analysis. Keywords: Discrete-valued signal; Invariant set; Decentralized dynamic quantizer; LMI. 1. INTRODUCTION One of the resent active control studies is the discrete-valued control problem (see Antsaklis [2004], IEEE [2007]), which covers various systems including discrete-valued signals such as networked systems, hybrid systems, embedded devices with D/A A/D converters and ON/OFF actuators. For example, when the signals within the networked system are quantized into the discrete-valued signals without considering the capacity of the communication channels, the control performance may seriously deteriorate and the system may be unstable at worst. Then there is an increasing need to consider discrete-valued control theory.
(a) Usual feedback system
have unstable zeros, their optimal dynamic quantizer becomes unstable. Also, their numerical design method in Azuma et al. [TAC 2008] provides a stable optimal dynamic quantizer, the order of which is basically higher than that of the given systems. Therefore, the authors have reconsider the dynamic quantizer design within the invariant set analysis framework in Shingin et al. [2004]. The framework can design a stable suboptimal dynamic quantizer such that the order of the quantizer is exactly the same as that of the given system for non-minimum phase systems (Sawada et. al. [2011]). In addition, it can also design a quantization interval such that the dynamic quantizer can achieve good approximation performance for feedforward-type networked systems with a small number of quantization levels (Okajima et. al. [2010]). On the other hand, from the perspective of the networked control feature, it is natural to synthesize multiple (decentralized) quantizers rather than a centralized quantizer. For example, in wireless sensor/actuator networks, the sensors and actuators that the quantizers are connected to are distributed (Stankovic [2008]). For such systems, it is difficult for the existing result of Minami et. al. [2009] to guarantee the stability of each quantizer. Thus it is important to consider the decentralized quantizer synthesis for applications.
(b) Control system with I/O quantizers
Fig. 1. Two control systems For the above problem, Azuma et. al. [Automatica 2008,C], Azuma et al. [TAC 2008], Minami et. al. [2009] have proposed the optimal dynamic quantizer framework that focuses on optimality of systems controlled by the discrete-valued signals. When a plant P (z) and a controller C(z) are given in the usual feedback system in Fig. 1, the framework provides a dynamic quantizer Qd such that the system in Fig. 1 (b) optimally approximates the system in Fig. 1 (a) in the sense of the inputoutput relation. Their optimal quantizer synthesis problems have closed-form solutions. However, when the given systems 978-3-902661-93-7/11/$20.00 © 2011 IFAC
(b) Usual feedback system (a) Quantized feedback system with decentralized quantizers
Fig. 2. Generalized quantized and unquantized systems
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Motivated by the above, this paper deals with the generalized quantized feedback system with the decentralized dynamic quantizers in Fig. 2 (a) that includes the various systems such as Fig. 1 (b). We recast the synthesis problem as parameter dependent linear matrix inequality optimization problem that has a closed-form solution under some circumstances. If the all transmission zeros of the given systems are stable, the proposed decentralized quantizer gives an optimal output approximation property. Otherwise, our method can design a stable suboptimal decentralized quantizer such that the order of each subquantizer is exactly the same as that of the given system and the infinite time control performance is always guaranteed. As a result, this paper gives a unified framework of the dynamic quantizer synthesis based on the invariant set analysis. Notation: The set of n × m (positive) real matrices is denoted n×m ). The set of n×m (positive) integer matrices by IRn×m (IR+ n×m is denoted by INn×m (IN+ ). 0n×m and Im (or for simplicity of notation, 0 and I) denote the n × m zero matrix and the m × m identity matrix, respectively. ⌊a⌋ denotes the floor of a ∈ IR+ . For a matrix M , M T , ρ(M ), σmax (M ) and abs(M ) denote its transpose, its spectrum radius, its maximum singular value and the matrix composed of the absolute values of its elements, respectively. For a vector x, xi is the i-th entry of x. For a symmetric matrix X, X > 0 (X ≥ 0) means that X is positive (semi) definite. For a full row rank matrix M , M † denotes its pseudo inverse matrix. For a matrix X, kXk2 denotes its 2-norm. For a vector x and a sequence of vectors X := {x1 , x2 , ...}, kxk and kXk denote their ∞norms, respectively. Finally, we use the “packed” notation for A B transfer functions: := C(zI − A)−1 B + D. C D 2. PRELIMINARIES Consider the linear time invariant (LTI) discrete-time system given by ξ(k + 1) = Aξ(k) + Bw(k)
(1)
ξ ∈ X, w ∈ W
(2)
n
m
where ξ ∈ IR and w ∈ IR denote the state vector and disturbance input, respectively. We define the invariant set. Definition 1. Define the invariant set of the system (1) to be a set X which satisfies m
where W := {w ∈ IR
⇒
T
Aξ + Bw ∈ X
: w w ≤ 1}.
The analysis condition can be expressed in terms of matrix inequalities as summarized in the following proposition. Proposition 2. (Shingin et al. [2004]) Consider the system (1). For a matrix 0 < P ∈ IRn×n , the ellipsoid E(P) := ξ ∈ IRn : ξ T Pξ ≤ 1 is an invariant set if and only if there exists a scalar α ∈ [0, 1 − ρ(A)2 ] satisfying T A PA − (1 − α)P AT PB ≤ 0. (3) B T PA B T PB − αIm The all ellipsoidal invariant sets are parameterized by Proposition 2. Also, the ellipsoidal invariant set allows us to approximate the reachable set from outside since the former covers the latter. Shingin et al. [2004] considers the criterion denoted by f (P) for the approximation of E(P) to the reachable set because the matrix P determines the ellipsoid. f (P) has the
monotonical decreasingness in the sense that its value for the set of inside is less than that of outside. When α is fixed in (3), Shingin et al. [2004] clarifies that the infimum of f (P) does not change even if P is restricted to P(α) given by P(α)−1 =
∞ X
k=0
1 Ak BB T (AT )k α(1 − α)k
(4)
where α ∈ (0, 1 − ρ(A)2 ). Thus the criterion f (P) can be replaced by f (P(α)) as well as the invariant sets in (3) can be parameterized by α ∈ (0, 1 − ρ(A)2 ). 3. PROBLEM FORMULATION Consider the control system with discrete-valued signal (quantized feedback system) depicted in Fig. 2 (a), which consists of the discrete-time LTI generalized plant G(z) and the dynamic quantizer Qd . The system G(z) is represented by # #" # " " x(k) A B1 B2 x(k + 1) r(k) zp (k) (5) = C1 D11 0 v(k) C2 D21 D22 y(k) where x ∈ IRng , zp ∈ IRq , r ∈ IRp , v ∈ IRm and y ∈ IRm denote the state vector, the controlled output, the exogenous input, the measured input and output, respectively. The system G(z) is stable, that is, the matrix A + B2 (I − D22 )−1 C2 is stable in the discrete domain. The system (5) covers the various systems. Consider the LTI plant P (z) with the state xp ∈ IRnp and controller C(z) with the state xc ∈ IRnc given by Ap Bp Ac Bc1 Bc2 zp v2 = Cp1 0 v1 , y1 = y2 r Cc Dc1 Dc2 Cp2 0
where y1 ∈ IRm1 , y2 ∈ IRm2 , v1 ∈ IRm1 , and v2 ∈ IRm2 denote the controller output, the plant measured output, the plant input, and the controller input, respectively. For example, by defining vectors: x := [ xTp xTc ]T ∈ IRng (ng := np + nc ), y := [ y1T y2T ]T , v := [ v1T v2T ]T and matrices: Bp 0 0 Ap 0 , , B2 := , B1 := A := 0 Bc1 Bc2 0 Ac 0 Cc C1 := [ Cp1 0 ] , D11 := 0, C2 := , Cp2 0 Dc2 0 Dc1 D21 := , D22 := , (6) 0 0 0 one gets the control system with I/O quantizers in Fig. 1 (b).
For the system G(z), we define v := [ v1T , ..., vsT ]T ∈ IRm and y := [ y1T , ..., ysT ]T ∈ IRm , respectively, and consider the dynamic quantizer v = Qd (y) which is the decentralized structure composed of sub-dynamic quantizers vi = Qdi (yi ) (i = 1, ..., s) with the state vector xqi ∈ IRnqi . The sub-quantizer Qdi consists of the static quantizer qst : IRmi → dINmi with the quantization interval d ∈ IR+ , i.e., vi = qst (ui ), ui := uqi + yi
and the dynamic compensator Qi (z) Aqi Bqi xqi (k) xqi (k + 1) = eqi (k) uqi (k) Cqi 0
(7)
where vi ∈ IRmi , yi ∈ IRmi , uqi ∈ IRmi and eqi := vi − yi . For s = 2 and the system G(z) defined by (6), one gets the
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
i/o type decentralized dynamic quantizer in Fig. 1 (b). Then the quantizer Qd := diag(Qd1 , ..., Qds ) is realized by the static T T T ] : IRm → dINm , i.e., , .., qst quantizer Qst := [qst v = Qst (u), u := uq + y
represented by # # " #" " A B1 B2 ξ(k) ξ(k + 1) u(k) = C1 D11 D12 e(k) e = Qe (u), C2 0 D22 r(k) zp (k)
(8)
and the dynamic compensator Q(z) := diag(Q1 (z), ..., Qs (z)) xq (k + 1) Aq Bq xq (k) = , (9) uq (k) Cq 0 eq (k) Aq := diag(Aq1 , ..., Aqs ), Bq := diag(Bq1 , ..., Bqs ), Cq := diag(Cq1 , ..., Cqs )
(13)
where ξ := [ xT xTq ]T ∈ IRn (n := ng + nq ) and the system H(z) is the feedback connection of G(z) and Q(z).
where xq := [ xTq1 , ..., xTqs ]T ∈ IRnq , uq := [ uTq1 , ..., uTqs ]T ∈ IRm , u := [ uT1 , ..., uTs ]T ∈ IRm and eq := [ eTq1 , ..., eTqs ]T ∈ IRm . Note that qst is of the nearest-neighbor type toward −∞ and the initial state is given by xq (0) = 0 for the drift-free of Qd (see Azuma et. al. [Automatica 2008]). One such static quantizer is the midtread type quantizer in Fig. 3. Also, the quantizer Qd is said to be stable if its matrix Aq + Bq Cq is stable in the discrete domain (Azuma et. al. [CDC 2008]).
Fig. 4. Feedback system with quantization error Let T ∈ IN+ ∪ {∞} be the period over which we consider the quantizer performance. For the system in Fig. 2 (a) with the exogenous signal sequence RT := {r(0), r(1), ..., r(T − p 1)} ∈ l∞ , zp (k, x0 , RT ) denotes the output of zp at the kth time for the initial state x0 = x(0). Also, for the system in Fig. 2 (b) without the quantizer, zp∗ (k, x0 , RT ) denotes its output at the k-th time for the initial state x0 . Zp (x0 , RT ) and Zp∗ (x0 , RT ) denote the vector sequence of zp (k, x0 , RT ) and zp∗ (k, x0 , RT ) for k = 1, ..., T , respectively. This paper considers the following cost function:
Fig. 3. Midtread type quantization (mi = 1) We define the following matrices: D := (I − D22 )−1 , C2 := DC2 , D21 := DD21 , D22 := DD22 , A := A + B2 C2 , B1 := B1 + B2 D21 , B A B2 Cq , B1 := 2 , B2 := B2 D, A := Bq 0 Aq + Bq Cq B1 B2 := , C1 := [ C2 DCq ] , D11 := D22 , 0
D12 := D21 , C2 := [ C1 0 ] , D22 := D11 .
ET (Qd ) :=
(10)
For the system in Fig. 2 (a), the system G(z) with the static quantizer Qst seen by the linear compensator Q(z) can be recast as the linear fractional transformation (LFT) of a generalized plant G(z): x(k) A B1 B2 B2 x(k + 1) u(k) C2 D21 D22 D r(k) (11) z (k) = C D 0 0 e(k) 1 11 p uq (k) 0 0 I I eq (k)
and the quantization error Qe :
e = Qe (u), Qe (u) := Qst (u) − u
(12)
m
where the signal e ∈ [−d/2, d/2] . For the matrix B2 ∈ IRng ×m and i = 1, ..., s, B2i ∈ IRng ×mi denotes the i-th block column of B2 , i.e., B := [ B21 , ..., B2s ]. Assumption 1. For every i = 1, ..., s, the matrix C1 Aτi B2i is full row rank where τi ∈ {0} ∪ IN+ is the smallest integer satisfying C1 Aτi B2i 6= 0. The control system in Fig. 2 (a) can be described as a LFT (Fig. 4) of the quantization error Qe and a LTI system H(z)
sup
p (x0 ,RT )∈IRn ×l∞
kZp∗ (x0 , RT ) − Zp (x0 , RT )k.
If the quantizer minimizes ET (Qd ), the system in Fig. 2 (a) optimally approximates the usual system in Fig. 2 (b) in the sense of the input-output relation. Azuma et. al. [Automatica 2008], Azuma et al. [TAC 2008], Azuma et. al. [CDC 2008], Minami et. al. [2009] have proposed the optimal quantizers in the sense that ET (Qd ) is minimized. On the other hand, our objective is to solve the following decentralized dynamic quantizer synthesis problem (E): For the system (13) with the exogenous signal sequence RT := {r(0), r(1), ..., r(T − 1)} ∈ p l∞ , suppose that the quantization period T ∈ IN+ ∪ {∞}, the quantization interval d ∈ IR+ and the performance level γ ∈ IR+ are given. Characterize a stable decentralized dynamic quantizer Qd (i.e., find parameters (nqi , Aqi , Bqi , Cqi ) i = 1, ..., s) achieving ET (Qd ) ≤ γ based on Proposition 2. 4. MAIN RESULT 4.1 Quantizer analysis Suppose that the stable decentralized quantizer Qd is given. The usual feedback system (v = y) in Fig. 2 (b) is given by ∗ ∗ x (k + 1) x (k) A B1 = zp∗ (k) r(k) C1 D11
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where x∗ ∈ IRng and zp∗ ∈ IRq denote its state vector and controlled output respectively, and x∗ (0) = x(0). Define the signals as follows:
∞
1 d2 m X C2 Ak B1 BT1 (AT )k CT2 4α (1 − α)k k=0 √ d m ⇔ γ(α) = √ σα . 2 α ⇔ γ(α)2 Iq ≥
ξ := [ xT − x∗ T xTq ]T , z := zp − zp∗ . Then the difference between zp∗ (k, x0 , RT ) and zp (k, x0 , RT ) is generated by the following error system H(z): ξ(k + 1) A B1 ξ(k) = , ξ(0) = 0 z(k) C2 0 e(k) m
where e ∈ IR is given by (12) and the matrices A, B1 and C2 are defined by (10). For the output e of Qe , we define the set E := {e ∈ IRm : 2 e satisfies (12)}. In this case, the relation E ⊆ md 4 W clearly 2 holds since the set md 4 W is an independent bounded disturbance without the relation (12). That is, the reachable set of H(z) with the error e ∈ E is no larger than that of H(z) with the 2 disturbance e ∈ md 4 W. Considering the reachable set from the origin ξ(0) to estimate the influences of the quantization error, this paper utilizes the invariant set which covers the reachable set from outside. Define √ d m , C := C2 . (14) A := A, B := B1 2 In this case, the ellipsoidal invariant set E(P) can be parameterized by Proposition 2, while covering the reachable set from the outside for the system H(z). In addition, if there exists the set E(P), there exists a scalar γ ∈ IR+ satisfying P CT T max sup |ci ξ| = γ ⇐ ≥0 (15) C γ 2 Iq i ξ∈E(P) where cTi is the i-th entry of C (see Sawada et. al. [2011]). That is, the performance level γ in (15) evaluates the upperbound of the difference between zp∗ (k, x0 , RT ) and zp (k, x0 , RT ) , and E∞ (Qd ) ≤ γ
(16)
is satisfied. The quantization period T is set to be ∞ because of the definition of the invariant set. In analysis, it is appropriate to treat P as a variable and search for P minimizing the performance level γ. For Proposition 2 defined by (14), we have the optimization problem (Aop): min 2
P>0,1−ρ(A) >α>0,γ>0
γ s.t. (3) and (15).
Focusing on the left side of (15), we see that γ is corresponding to the criterion f (P(α)). From the parameterization P(α) in (4), the infimum of γ can be expressed by the following lemma. Lemma 3. For the feedback system (13), suppose that the quantization interval d ∈ IR+ is given. Consider the optimization problem (Aop) with (14). The infimum of γ is given by √ d m inf γ = inf √ σα , 0 < α < 1 − ρ(A)2 , (17) α 2 α P ∞ 1 T k T k T σα := σmax k=0 (1−α)k C2 A B1 B1 (A ) C2 . Proof. Define γ(α) which is obtained from the problem (Aop) for the fixed α. Considering (4) and (14), we obtain (15) ⇔ γ(α)2 Iq − CP(α)−1 C T ≥ 0
Hence, the infimum of γ is given by (17).
2
4.2 Quantizer synthesis The analysis problem (Aop) with (14) suggests that the synthesis problem (E) reduces to the following non-convex optimization problem (OP1): min
P>0,Aq ,Bq ,Cq ,0>α>1−ρ(A)2 ,γ>0
γ s.t. (3) and (15)
where the matrices A, B and C are defined by (14). Define the matrices: ˆ := diag(A, ..., A) ∈ IRs·ng ×s·ng , A ˆ := diag(B21 , ..., B2s ) ∈ IRs·ng ×m , B Cˆ := [ C1 . . . C1 ] ∈ IRq×s·ng .
(18)
In this case, one gets k A ˆ ˆ q ˆ BC B ˆ = C2 Ak B1 , C 0 Bq 0 Aq + Bq Cq ˆ ˆ q A BC ρ = ρ(A). 0 Aq + Bq Cq
Then the following lemma holds from Lemma 3. Lemma 4. For the feedback system (13), define the matrices (18) and consider the non-convex optimization problem (OP2): min
P>0,Aq ,Bq ,Cq ,0>α>1−ρ(A)2 ,γ>0
γ s.t. (3) and (15)
where the matrices A, B and C are defined by √ ˆ d m ˆ ˆ q BC , , B := B A := A Bq 0 Aq + Bq Cq 2 C := Cˆ 0 .
(19)
In this case, the solutions (γ, α, Aq , Bq , Cq ) to the problem (OP2) with (19) are the exactly same as those to the problem (OP1) with (14). From Lemma 4, the problem (OP1) is feasible if and only if the problem (OP2) is feasible. As a result, Lemma 4 clarifies that the synthesis problem (E) reduces to the problem (OP2) as summarized in the following theorem. Theorem 5. For the feedback system (13), suppose that the quantization interval d ∈ IR+ and the performance level γ ∈ IR+ are given. For a scalar α ∈ (0, 1), there exists a stable decentralized dynamic quantizer Qd achieving (16) if one of the following equivalent statements holds. (i) There exist a matrix 0 < P ∈ IRn×n and a decentralized dynamic quantizer Qd satisfying (3) and (15) with (14). (ii) There exist matrices 0 < Xi ∈ IRng ×ng , 0 < Yi ∈ IRng ×ng , Fi ∈ IRmi ×ng , Wi ∈ IRng ×ng , and Ui ∈ IRng ×mi satisfying (1 − α)ΞP 0 ΞTA ΞP ΞTC 4α T ≥ 0, ≥0 (20) 0 I Ξ md2 m B ΞC γ 2 I q ΞA ΞB ΞP
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where
. . . C1 (A + B2s−1 Cqs−1 )k B2s−1 C1 (A + B2s Cqs )k B2s
ˆ := diag(X1 , ..., Xs ), X Fˆ := diag(F1 , ..., Fs ), ˆ := diag(U1 , ..., Us ), U ˆ I X , ΞP := I Yˆ ˆ U ΞB := ˆ , B
Yˆ := diag(Y1 , ..., Ys ), ˆ := diag(W1 , ..., Ws ), W
where C1 (A + B2i Cqi )k B2i = 0 holds for k ≤ τ1 − 1 and C1 (A + B2i Cqi )k B2i = C1 Aτi B2i holds for k = τi . When Cqi is given by (22), C1 (A + B2i Cqi )k B2i = 0 holds for k ≥ τi + 1. Since the cost function E∞ (Qd ) is characterized by (17), then the parameters (22) leads to (23). 2
ˆ ˆA ˆ X W ˆ A ˆ Yˆ + B ˆ Fˆ , A ΞC := Cˆ Cˆ Yˆ .
ΞA :=
In this case, one such sub-quantizer (i = 1, ..., s) is given by Bqi = Zi−1 (Ui − Xi B2i ), Cqi = −Fi Yi−1 , Aqi = Zi−1 (Xi AYi + Ui Fi − Wi )Yi−1, nqi = ng
(21)
where Zi = Xi − Yi−1 .
For the decentralized dynamic quantizer synthesis problem minimizing γ of (16), we have the optimization problem (Sop): min
ˆ ˆ ,U ˆ ,1>α>0,γ>0 X>0, Yˆ >0,Fˆ ,W
γ s.t. (20).
In synthesis, the parameters (Aq , Bq , Cq ) to be designed lead to α ∈ (0, 1). When scalar α is fixed, the conditions in Theorem 5 are linear matrix inequalities (LMIs) in terms of the other variables. Using standard LMI software in combination with the line search of α for (Sop), we can obtain a stable decentralized dynamic quantizer, numerically. Figure 5 illustrates the relation between the problems (OP1), (OP2) and (Sop). Problem (E) (Lemma 3) ⇑∗ Problem (OP1): (nq , Aq , Bq , Cq ) (Lemma 4) m‡ Problem (OP2): (nq , Aq , Bq , Cq ) (Theorem 5) m⋆ ˆ Yˆ , Fˆ , W ˆ ,U ˆ) Problem (Sop) : (X,
Under some circumstances, Proposition 2 gives a dynamic quantizer which is expressed by the given generalized plant parameters. The following theorem denotes this fact. Theorem 6. Consider the optimization problem (OP1) for the matrices (14). Suppose that Assumption 1 holds. An optimal solution of (Aq , Bq , Cq ) to the problem (OP1) is given by Aqi = A, Bqi = B2i , Cqi = −(C1 Aτi B2i )† C1 Aτi +1 (22)
for every i = 1, ..., s and its infimum of γ ∈ IR+ is √ d m p inf γ = σρ , ρ = min{ρ(A), ρ(Aq + Bq Cq )}, (23) 2 1 − ρ2
1 σρ := τ1 C1 Aτ1 B21 1τ2 C1 Aτ2 B22 . . . 1τs C1 Aτs B2s ρ
2
if the matrix Aqi + Bqi Cqi defined in (22) is stable in the discrete domain for every i = 1, ..., s. ˆ and Bq = B, ˆ one gets Proof. For the case where Aq = A C2 Ak B1 = C1 (A + B21 Cq1 )k B21 C1 (A + B22 Cq2 )k B22
d τs τ2 τ1 E∞ (Qop d ) = kabs(C1 [ A B21 A B22 . . . A B2s ])k . 2 It is striking that the structure of their optimal quantizer is equivalent to our proposed one based on Proposition 2 even if the former performance evaluation is less conservative than the latter one. Theorem 6 points out that the proposed quantizer is also optimal in the sense that the quantizer gives an optimal output approximation property for minimum phase systems. When the system G(z) is non-minimum phase, the optimal quantizer (22) is unstable. In this case, Azuma et al. [TAC 2008] has provided the numerical design method in which the stable and optimal decentralized quantizer synthesis problem is recast as the following optimization problem:
−1
d
TX
k abs(C2 A B1 ) (= ET (Qd )) min
2 Aq ,Bq ,Cq ,P k=0
Fig. 5. Solutions to Problem (E)
ρ
Theorem 6 presents a closed-form solution to the problem (E). In the case of m = p, the stable Aq + Bq Cq in (22) implies that the all transmission zeros of the generalized plant G(z) are stable (see Azuma et. al. [CDC 2008]). When the matrix C2 is full row rank, the system G(z) is minimum phase and Assumption 1 holds, on the other hand, Minami et. al. [2009] has presented an optimal decentralized dynamic quantizer Qop d given by (22) and its performance given by
s.t. (Aqi + Bqi Cqi )T Pi (Aqi + Bqi Cqi ) < Pi , ∀i = 1, ..., s and P := diag(P1 , ..., Ps ) > 0
∗: (E) is feasible if (OP1) is feasible. ‡: Equivalent for (nq , Aq , Bq , Cq ). ⋆: Equivalent for the fixed α ∈ (0, 1).
ρ
for the given T ∈ IN+ . The order of each sub-quantizer in their method is given by ⌊T /2⌋+1 or (⌊T /2⌋+1)T . When T is set be large, we need the reduction technique. On the other hand, even if the system G(z) has unstable zeros, our method provides a suboptimal dynamic decentralized quantizer with nqi = ng in the sense that the upper bound of E∞ (Qd ) is minimized. This suboptimality is the conservativeness that the ellipsoidal approximation in Proposition 2 leads to. Even if there exists an optimal quantizer achieving a certain performance for a non-minimum phase system, it is not necessarily that such a quantizer is obtained from (OP1). The results of Theorems 5 and 6 are corresponding to the centralized case when we set s = 1. Then, this paper gives a unified framework of the dynamic quantizer synthesis based on the invariant set analysis. 5. NUMERICAL EXAMPLE Consider the system in Fig. 1 (b). The plant P (z) is the discretized system of the unstable non-minimum phase continuoustime LTI system 0 1 0 xp (t) x˙ p (t) , zp (t) = y2 (t) = 1 −1 1 v1 (t) y2 (t) −2 1 0 with the sampling time h = 0.1 and zero-order hold. Its eigenvalues are {1.064, 0.857} and its unstable zero is {1.224}. The stabilizing controller C(z) is given by
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
20 v (kh)
v1(kh)
20
0
1
0 −20 0
5
10 time kh [s]
15
−20 0
20
20
20
10
10
2
0 −10 0
5
10 time kh [s]
15
−10 0
20
20
5
10 time kh [s]
15
20
15
20
(b) v2 (the input of C(z))
10 z (kh)
10
5
p
5
p
z (kh)
15
0
(b) v2 (the input of C(z))
0
0
0
5
10 time kh [s]
15
20
0
(c) zp (the controlled output of P (z))
xc (k + 1) y1 (k)
=
0.741 0.086 −1 1
5
10 time kh [s]
(c) zp (the controlled output of P (z))
Fig. 6. Time responses for centralized quantizer (s=1)
10 time kh [s]
(a) v1 (the input of P (z))
v (kh)
v2(kh)
(a) v1 (the input of P (z))
5
Fig. 7. Time responses for decentralized quantizer (s=2) REFERENCES
xc (k) . v2 (k)
Consider d = 10 and s = 2. From (Sop), the stable suboptimal decentralized quantizers with γ = 2.09 are obtained as follows: Aqi = A, Bqi = B2i , and Cqi = [ 23.353 − 11.738 − 1.000 ] (s = 1, 2). As a result, the matrices Cqi are modified from the optimal one such that the quantizers are stable. To compare the decentralized quantizer with the centralized quantizer, the stable suboptimal centralized quantizer with γ = 2.09 is also obtained from (Sop) with s = 1. Figures 6 and 7 illustrate the time responses of v1 (kh), v2 (kh) and zp (kh) for the centralized and decentralized quantizers with the initial state x(0) = [ − 5 0 − 1 ]T , respectively. In Figs. 6 and 7, the thin lines and the thick lines illustrate the time responses of the usual feedback system in Fig. 2 (b) and the quantized feedback system in Fig. 2 (a), respectively. We see that the discrete-valued signals v1 and v2 in Fig. 7 switch more quickly compared with those in Fig. 6. However, we see that the decentralized quantizer attenuates the excess performance deterioration between both quantizers. In fact, both quantizer performances are nearly equal in terms of the performance level γ. 6. CONCLUSION We have proposed a unified framework of the decentralized dynamic quantizer synthesis based on the invariant analysis. This paper has proposed the quantizer synthesis condition that has a closed-form solution under some circumstances. Also, we have clarified the effectiveness and the limitation of the proposed condition.
P. Antsaklis and J. Baillieul (eds.). Special issue on networked control systems. IEEE Trans. Auto. Contr.. Vol. 49, No. 9, pages 1421–1423, 2004. Special Section on Complex Networked Control Systems. IEEE Control Systems Magazine. Vol. 27, No. 4, 2007. S. Azuma and T. Sugie. Optimal dynamic quantizer for discrete-valued input control. Automatica, Vol. 44, No. 12, pages 396–406, 2008. S. Azuma and T. Sugie. Stability analysis of quantized feedback systems including optimal dynamic quantizers. Proc. of 47th IEEE CDC, pages 3392–3397, 2008. S. Azuma and T. Sugie. Synthesis of optimal dynamic quantizers for discrete-valued input control. IEEE Trans. Auto. Contr., Vol. 53, No. 9, pages 2064–2075, 2008. Y. Minami, S. Azuma, and T. Sugie. Optimal Decentralized Dynamic Quantizers for Discrete-Valued Input Control: A Closed form solution and Experimental Evaluation. Proc. of ACC, pages 4367–4372, 2009. H. Shingin and Y. Ohta. Optimal invariant sets for discrete-time systems: Approximation of reachable sets for bounded inputs. 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems, Theory and Applications (LSS), pages 401– 406, 2004. K. Sawada and S. Shin. Synthesis of dynamic quantizers for quantized feedback systems within invariant set analysis framework. ACC, to appear, 2011. H. Okajima, N. Matsunaga, and K. Sawada. Optimal quantization interval design of dynamic quantizers which satisfy the communication rate constraints. Proc. of 49th IEEE CDC, pages 4733–4739, 2010. J. A. Stankovic. When sensor and actuator networks cover the world. ETRI Journal, Vol. 30, No. 5, pages 627–633, 2008.
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Synthesis of decentralized dynamic quantizer within invariant set analysis framework Kenji Sawada ∗ Seiichi Shin ∗∗ ∗
The University of Electro-Communications, Chofu-city, Tokyo, Japan, (e-mail: [email protected]). ∗∗ The University of Electro-Communications, Chofu-city, Tokyo, Japan, (e-mail: [email protected]). Abstract: This paper focuses on the synthesis of decentralized dynamic quantizers for linear systems with discrete-valued signal in terms of invariant set analysis. We derive the synthesis condition that can be recast as parameter dependent linear matrix inequalities and has a closed-form solution under some circumstances. In the case of minimum phase systems, our method can provide an optimal decentralized dynamic quantizer in the sense that the quantizer gives an optimal output approximation property. Also, in the case of non-minimum phase systems, our method can provide a suboptimal stable decentralized dynamic quantizer. As a result, this paper gives a unified design method of the dynamic quantizer based on the invariant set analysis. Keywords: Discrete-valued signal; Invariant set; Decentralized dynamic quantizer; LMI. 1. INTRODUCTION One of the resent active control studies is the discrete-valued control problem (see Antsaklis [2004], IEEE [2007]), which covers various systems including discrete-valued signals such as networked systems, hybrid systems, embedded devices with D/A A/D converters and ON/OFF actuators. For example, when the signals within the networked system are quantized into the discrete-valued signals without considering the capacity of the communication channels, the control performance may seriously deteriorate and the system may be unstable at worst. Then there is an increasing need to consider discrete-valued control theory.
(a) Usual feedback system
have unstable zeros, their optimal dynamic quantizer becomes unstable. Also, their numerical design method in Azuma et al. [TAC 2008] provides a stable optimal dynamic quantizer, the order of which is basically higher than that of the given systems. Therefore, the authors have reconsider the dynamic quantizer design within the invariant set analysis framework in Shingin et al. [2004]. The framework can design a stable suboptimal dynamic quantizer such that the order of the quantizer is exactly the same as that of the given system for non-minimum phase systems (Sawada et. al. [2011]). In addition, it can also design a quantization interval such that the dynamic quantizer can achieve good approximation performance for feedforward-type networked systems with a small number of quantization levels (Okajima et. al. [2010]). On the other hand, from the perspective of the networked control feature, it is natural to synthesize multiple (decentralized) quantizers rather than a centralized quantizer. For example, in wireless sensor/actuator networks, the sensors and actuators that the quantizers are connected to are distributed (Stankovic [2008]). For such systems, it is difficult for the existing result of Minami et. al. [2009] to guarantee the stability of each quantizer. Thus it is important to consider the decentralized quantizer synthesis for applications.
(b) Control system with I/O quantizers
Fig. 1. Two control systems For the above problem, Azuma et. al. [Automatica 2008,C], Azuma et al. [TAC 2008], Minami et. al. [2009] have proposed the optimal dynamic quantizer framework that focuses on optimality of systems controlled by the discrete-valued signals. When a plant P (z) and a controller C(z) are given in the usual feedback system in Fig. 1, the framework provides a dynamic quantizer Qd such that the system in Fig. 1 (b) optimally approximates the system in Fig. 1 (a) in the sense of the inputoutput relation. Their optimal quantizer synthesis problems have closed-form solutions. However, when the given systems 978-3-902661-93-7/11/$20.00 © 2011 IFAC
(b) Usual feedback system (a) Quantized feedback system with decentralized quantizers
Fig. 2. Generalized quantized and unquantized systems
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10.3182/20110828-6-IT-1002.01963
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Motivated by the above, this paper deals with the generalized quantized feedback system with the decentralized dynamic quantizers in Fig. 2 (a) that includes the various systems such as Fig. 1 (b). We recast the synthesis problem as parameter dependent linear matrix inequality optimization problem that has a closed-form solution under some circumstances. If the all transmission zeros of the given systems are stable, the proposed decentralized quantizer gives an optimal output approximation property. Otherwise, our method can design a stable suboptimal decentralized quantizer such that the order of each subquantizer is exactly the same as that of the given system and the infinite time control performance is always guaranteed. As a result, this paper gives a unified framework of the dynamic quantizer synthesis based on the invariant set analysis. Notation: The set of n × m (positive) real matrices is denoted n×m ). The set of n×m (positive) integer matrices by IRn×m (IR+ n×m is denoted by INn×m (IN+ ). 0n×m and Im (or for simplicity of notation, 0 and I) denote the n × m zero matrix and the m × m identity matrix, respectively. ⌊a⌋ denotes the floor of a ∈ IR+ . For a matrix M , M T , ρ(M ), σmax (M ) and abs(M ) denote its transpose, its spectrum radius, its maximum singular value and the matrix composed of the absolute values of its elements, respectively. For a vector x, xi is the i-th entry of x. For a symmetric matrix X, X > 0 (X ≥ 0) means that X is positive (semi) definite. For a full row rank matrix M , M † denotes its pseudo inverse matrix. For a matrix X, kXk2 denotes its 2-norm. For a vector x and a sequence of vectors X := {x1 , x2 , ...}, kxk and kXk denote their ∞norms, respectively. Finally, we use the “packed” notation for A B transfer functions: := C(zI − A)−1 B + D. C D 2. PRELIMINARIES Consider the linear time invariant (LTI) discrete-time system given by ξ(k + 1) = Aξ(k) + Bw(k)
(1)
ξ ∈ X, w ∈ W
(2)
n
m
where ξ ∈ IR and w ∈ IR denote the state vector and disturbance input, respectively. We define the invariant set. Definition 1. Define the invariant set of the system (1) to be a set X which satisfies m
where W := {w ∈ IR
⇒
T
Aξ + Bw ∈ X
: w w ≤ 1}.
The analysis condition can be expressed in terms of matrix inequalities as summarized in the following proposition. Proposition 2. (Shingin et al. [2004]) Consider the system (1). For a matrix 0 < P ∈ IRn×n , the ellipsoid E(P) := ξ ∈ IRn : ξ T Pξ ≤ 1 is an invariant set if and only if there exists a scalar α ∈ [0, 1 − ρ(A)2 ] satisfying T A PA − (1 − α)P AT PB ≤ 0. (3) B T PA B T PB − αIm The all ellipsoidal invariant sets are parameterized by Proposition 2. Also, the ellipsoidal invariant set allows us to approximate the reachable set from outside since the former covers the latter. Shingin et al. [2004] considers the criterion denoted by f (P) for the approximation of E(P) to the reachable set because the matrix P determines the ellipsoid. f (P) has the
monotonical decreasingness in the sense that its value for the set of inside is less than that of outside. When α is fixed in (3), Shingin et al. [2004] clarifies that the infimum of f (P) does not change even if P is restricted to P(α) given by P(α)−1 =
∞ X
k=0
1 Ak BB T (AT )k α(1 − α)k
(4)
where α ∈ (0, 1 − ρ(A)2 ). Thus the criterion f (P) can be replaced by f (P(α)) as well as the invariant sets in (3) can be parameterized by α ∈ (0, 1 − ρ(A)2 ). 3. PROBLEM FORMULATION Consider the control system with discrete-valued signal (quantized feedback system) depicted in Fig. 2 (a), which consists of the discrete-time LTI generalized plant G(z) and the dynamic quantizer Qd . The system G(z) is represented by # #" # " " x(k) A B1 B2 x(k + 1) r(k) zp (k) (5) = C1 D11 0 v(k) C2 D21 D22 y(k) where x ∈ IRng , zp ∈ IRq , r ∈ IRp , v ∈ IRm and y ∈ IRm denote the state vector, the controlled output, the exogenous input, the measured input and output, respectively. The system G(z) is stable, that is, the matrix A + B2 (I − D22 )−1 C2 is stable in the discrete domain. The system (5) covers the various systems. Consider the LTI plant P (z) with the state xp ∈ IRnp and controller C(z) with the state xc ∈ IRnc given by Ap Bp Ac Bc1 Bc2 zp v2 = Cp1 0 v1 , y1 = y2 r Cc Dc1 Dc2 Cp2 0
where y1 ∈ IRm1 , y2 ∈ IRm2 , v1 ∈ IRm1 , and v2 ∈ IRm2 denote the controller output, the plant measured output, the plant input, and the controller input, respectively. For example, by defining vectors: x := [ xTp xTc ]T ∈ IRng (ng := np + nc ), y := [ y1T y2T ]T , v := [ v1T v2T ]T and matrices: Bp 0 0 Ap 0 , , B2 := , B1 := A := 0 Bc1 Bc2 0 Ac 0 Cc C1 := [ Cp1 0 ] , D11 := 0, C2 := , Cp2 0 Dc2 0 Dc1 D21 := , D22 := , (6) 0 0 0 one gets the control system with I/O quantizers in Fig. 1 (b).
For the system G(z), we define v := [ v1T , ..., vsT ]T ∈ IRm and y := [ y1T , ..., ysT ]T ∈ IRm , respectively, and consider the dynamic quantizer v = Qd (y) which is the decentralized structure composed of sub-dynamic quantizers vi = Qdi (yi ) (i = 1, ..., s) with the state vector xqi ∈ IRnqi . The sub-quantizer Qdi consists of the static quantizer qst : IRmi → dINmi with the quantization interval d ∈ IR+ , i.e., vi = qst (ui ), ui := uqi + yi
and the dynamic compensator Qi (z) Aqi Bqi xqi (k) xqi (k + 1) = eqi (k) uqi (k) Cqi 0
(7)
where vi ∈ IRmi , yi ∈ IRmi , uqi ∈ IRmi and eqi := vi − yi . For s = 2 and the system G(z) defined by (6), one gets the
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
i/o type decentralized dynamic quantizer in Fig. 1 (b). Then the quantizer Qd := diag(Qd1 , ..., Qds ) is realized by the static T T T ] : IRm → dINm , i.e., , .., qst quantizer Qst := [qst v = Qst (u), u := uq + y
represented by # # " #" " A B1 B2 ξ(k) ξ(k + 1) u(k) = C1 D11 D12 e(k) e = Qe (u), C2 0 D22 r(k) zp (k)
(8)
and the dynamic compensator Q(z) := diag(Q1 (z), ..., Qs (z)) xq (k + 1) Aq Bq xq (k) = , (9) uq (k) Cq 0 eq (k) Aq := diag(Aq1 , ..., Aqs ), Bq := diag(Bq1 , ..., Bqs ), Cq := diag(Cq1 , ..., Cqs )
(13)
where ξ := [ xT xTq ]T ∈ IRn (n := ng + nq ) and the system H(z) is the feedback connection of G(z) and Q(z).
where xq := [ xTq1 , ..., xTqs ]T ∈ IRnq , uq := [ uTq1 , ..., uTqs ]T ∈ IRm , u := [ uT1 , ..., uTs ]T ∈ IRm and eq := [ eTq1 , ..., eTqs ]T ∈ IRm . Note that qst is of the nearest-neighbor type toward −∞ and the initial state is given by xq (0) = 0 for the drift-free of Qd (see Azuma et. al. [Automatica 2008]). One such static quantizer is the midtread type quantizer in Fig. 3. Also, the quantizer Qd is said to be stable if its matrix Aq + Bq Cq is stable in the discrete domain (Azuma et. al. [CDC 2008]).
Fig. 4. Feedback system with quantization error Let T ∈ IN+ ∪ {∞} be the period over which we consider the quantizer performance. For the system in Fig. 2 (a) with the exogenous signal sequence RT := {r(0), r(1), ..., r(T − p 1)} ∈ l∞ , zp (k, x0 , RT ) denotes the output of zp at the kth time for the initial state x0 = x(0). Also, for the system in Fig. 2 (b) without the quantizer, zp∗ (k, x0 , RT ) denotes its output at the k-th time for the initial state x0 . Zp (x0 , RT ) and Zp∗ (x0 , RT ) denote the vector sequence of zp (k, x0 , RT ) and zp∗ (k, x0 , RT ) for k = 1, ..., T , respectively. This paper considers the following cost function:
Fig. 3. Midtread type quantization (mi = 1) We define the following matrices: D := (I − D22 )−1 , C2 := DC2 , D21 := DD21 , D22 := DD22 , A := A + B2 C2 , B1 := B1 + B2 D21 , B A B2 Cq , B1 := 2 , B2 := B2 D, A := Bq 0 Aq + Bq Cq B1 B2 := , C1 := [ C2 DCq ] , D11 := D22 , 0
D12 := D21 , C2 := [ C1 0 ] , D22 := D11 .
ET (Qd ) :=
(10)
For the system in Fig. 2 (a), the system G(z) with the static quantizer Qst seen by the linear compensator Q(z) can be recast as the linear fractional transformation (LFT) of a generalized plant G(z): x(k) A B1 B2 B2 x(k + 1) u(k) C2 D21 D22 D r(k) (11) z (k) = C D 0 0 e(k) 1 11 p uq (k) 0 0 I I eq (k)
and the quantization error Qe :
e = Qe (u), Qe (u) := Qst (u) − u
(12)
m
where the signal e ∈ [−d/2, d/2] . For the matrix B2 ∈ IRng ×m and i = 1, ..., s, B2i ∈ IRng ×mi denotes the i-th block column of B2 , i.e., B := [ B21 , ..., B2s ]. Assumption 1. For every i = 1, ..., s, the matrix C1 Aτi B2i is full row rank where τi ∈ {0} ∪ IN+ is the smallest integer satisfying C1 Aτi B2i 6= 0. The control system in Fig. 2 (a) can be described as a LFT (Fig. 4) of the quantization error Qe and a LTI system H(z)
sup
p (x0 ,RT )∈IRn ×l∞
kZp∗ (x0 , RT ) − Zp (x0 , RT )k.
If the quantizer minimizes ET (Qd ), the system in Fig. 2 (a) optimally approximates the usual system in Fig. 2 (b) in the sense of the input-output relation. Azuma et. al. [Automatica 2008], Azuma et al. [TAC 2008], Azuma et. al. [CDC 2008], Minami et. al. [2009] have proposed the optimal quantizers in the sense that ET (Qd ) is minimized. On the other hand, our objective is to solve the following decentralized dynamic quantizer synthesis problem (E): For the system (13) with the exogenous signal sequence RT := {r(0), r(1), ..., r(T − 1)} ∈ p l∞ , suppose that the quantization period T ∈ IN+ ∪ {∞}, the quantization interval d ∈ IR+ and the performance level γ ∈ IR+ are given. Characterize a stable decentralized dynamic quantizer Qd (i.e., find parameters (nqi , Aqi , Bqi , Cqi ) i = 1, ..., s) achieving ET (Qd ) ≤ γ based on Proposition 2. 4. MAIN RESULT 4.1 Quantizer analysis Suppose that the stable decentralized quantizer Qd is given. The usual feedback system (v = y) in Fig. 2 (b) is given by ∗ ∗ x (k + 1) x (k) A B1 = zp∗ (k) r(k) C1 D11
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where x∗ ∈ IRng and zp∗ ∈ IRq denote its state vector and controlled output respectively, and x∗ (0) = x(0). Define the signals as follows:
∞
1 d2 m X C2 Ak B1 BT1 (AT )k CT2 4α (1 − α)k k=0 √ d m ⇔ γ(α) = √ σα . 2 α ⇔ γ(α)2 Iq ≥
ξ := [ xT − x∗ T xTq ]T , z := zp − zp∗ . Then the difference between zp∗ (k, x0 , RT ) and zp (k, x0 , RT ) is generated by the following error system H(z): ξ(k + 1) A B1 ξ(k) = , ξ(0) = 0 z(k) C2 0 e(k) m
where e ∈ IR is given by (12) and the matrices A, B1 and C2 are defined by (10). For the output e of Qe , we define the set E := {e ∈ IRm : 2 e satisfies (12)}. In this case, the relation E ⊆ md 4 W clearly 2 holds since the set md 4 W is an independent bounded disturbance without the relation (12). That is, the reachable set of H(z) with the error e ∈ E is no larger than that of H(z) with the 2 disturbance e ∈ md 4 W. Considering the reachable set from the origin ξ(0) to estimate the influences of the quantization error, this paper utilizes the invariant set which covers the reachable set from outside. Define √ d m , C := C2 . (14) A := A, B := B1 2 In this case, the ellipsoidal invariant set E(P) can be parameterized by Proposition 2, while covering the reachable set from the outside for the system H(z). In addition, if there exists the set E(P), there exists a scalar γ ∈ IR+ satisfying P CT T max sup |ci ξ| = γ ⇐ ≥0 (15) C γ 2 Iq i ξ∈E(P) where cTi is the i-th entry of C (see Sawada et. al. [2011]). That is, the performance level γ in (15) evaluates the upperbound of the difference between zp∗ (k, x0 , RT ) and zp (k, x0 , RT ) , and E∞ (Qd ) ≤ γ
(16)
is satisfied. The quantization period T is set to be ∞ because of the definition of the invariant set. In analysis, it is appropriate to treat P as a variable and search for P minimizing the performance level γ. For Proposition 2 defined by (14), we have the optimization problem (Aop): min 2
P>0,1−ρ(A) >α>0,γ>0
γ s.t. (3) and (15).
Focusing on the left side of (15), we see that γ is corresponding to the criterion f (P(α)). From the parameterization P(α) in (4), the infimum of γ can be expressed by the following lemma. Lemma 3. For the feedback system (13), suppose that the quantization interval d ∈ IR+ is given. Consider the optimization problem (Aop) with (14). The infimum of γ is given by √ d m inf γ = inf √ σα , 0 < α < 1 − ρ(A)2 , (17) α 2 α P ∞ 1 T k T k T σα := σmax k=0 (1−α)k C2 A B1 B1 (A ) C2 . Proof. Define γ(α) which is obtained from the problem (Aop) for the fixed α. Considering (4) and (14), we obtain (15) ⇔ γ(α)2 Iq − CP(α)−1 C T ≥ 0
Hence, the infimum of γ is given by (17).
2
4.2 Quantizer synthesis The analysis problem (Aop) with (14) suggests that the synthesis problem (E) reduces to the following non-convex optimization problem (OP1): min
P>0,Aq ,Bq ,Cq ,0>α>1−ρ(A)2 ,γ>0
γ s.t. (3) and (15)
where the matrices A, B and C are defined by (14). Define the matrices: ˆ := diag(A, ..., A) ∈ IRs·ng ×s·ng , A ˆ := diag(B21 , ..., B2s ) ∈ IRs·ng ×m , B Cˆ := [ C1 . . . C1 ] ∈ IRq×s·ng .
(18)
In this case, one gets k A ˆ ˆ q ˆ BC B ˆ = C2 Ak B1 , C 0 Bq 0 Aq + Bq Cq ˆ ˆ q A BC ρ = ρ(A). 0 Aq + Bq Cq
Then the following lemma holds from Lemma 3. Lemma 4. For the feedback system (13), define the matrices (18) and consider the non-convex optimization problem (OP2): min
P>0,Aq ,Bq ,Cq ,0>α>1−ρ(A)2 ,γ>0
γ s.t. (3) and (15)
where the matrices A, B and C are defined by √ ˆ d m ˆ ˆ q BC , , B := B A := A Bq 0 Aq + Bq Cq 2 C := Cˆ 0 .
(19)
In this case, the solutions (γ, α, Aq , Bq , Cq ) to the problem (OP2) with (19) are the exactly same as those to the problem (OP1) with (14). From Lemma 4, the problem (OP1) is feasible if and only if the problem (OP2) is feasible. As a result, Lemma 4 clarifies that the synthesis problem (E) reduces to the problem (OP2) as summarized in the following theorem. Theorem 5. For the feedback system (13), suppose that the quantization interval d ∈ IR+ and the performance level γ ∈ IR+ are given. For a scalar α ∈ (0, 1), there exists a stable decentralized dynamic quantizer Qd achieving (16) if one of the following equivalent statements holds. (i) There exist a matrix 0 < P ∈ IRn×n and a decentralized dynamic quantizer Qd satisfying (3) and (15) with (14). (ii) There exist matrices 0 < Xi ∈ IRng ×ng , 0 < Yi ∈ IRng ×ng , Fi ∈ IRmi ×ng , Wi ∈ IRng ×ng , and Ui ∈ IRng ×mi satisfying (1 − α)ΞP 0 ΞTA ΞP ΞTC 4α T ≥ 0, ≥0 (20) 0 I Ξ md2 m B ΞC γ 2 I q ΞA ΞB ΞP
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
where
. . . C1 (A + B2s−1 Cqs−1 )k B2s−1 C1 (A + B2s Cqs )k B2s
ˆ := diag(X1 , ..., Xs ), X Fˆ := diag(F1 , ..., Fs ), ˆ := diag(U1 , ..., Us ), U ˆ I X , ΞP := I Yˆ ˆ U ΞB := ˆ , B
Yˆ := diag(Y1 , ..., Ys ), ˆ := diag(W1 , ..., Ws ), W
where C1 (A + B2i Cqi )k B2i = 0 holds for k ≤ τ1 − 1 and C1 (A + B2i Cqi )k B2i = C1 Aτi B2i holds for k = τi . When Cqi is given by (22), C1 (A + B2i Cqi )k B2i = 0 holds for k ≥ τi + 1. Since the cost function E∞ (Qd ) is characterized by (17), then the parameters (22) leads to (23). 2
ˆ ˆA ˆ X W ˆ A ˆ Yˆ + B ˆ Fˆ , A ΞC := Cˆ Cˆ Yˆ .
ΞA :=
In this case, one such sub-quantizer (i = 1, ..., s) is given by Bqi = Zi−1 (Ui − Xi B2i ), Cqi = −Fi Yi−1 , Aqi = Zi−1 (Xi AYi + Ui Fi − Wi )Yi−1, nqi = ng
(21)
where Zi = Xi − Yi−1 .
For the decentralized dynamic quantizer synthesis problem minimizing γ of (16), we have the optimization problem (Sop): min
ˆ ˆ ,U ˆ ,1>α>0,γ>0 X>0, Yˆ >0,Fˆ ,W
γ s.t. (20).
In synthesis, the parameters (Aq , Bq , Cq ) to be designed lead to α ∈ (0, 1). When scalar α is fixed, the conditions in Theorem 5 are linear matrix inequalities (LMIs) in terms of the other variables. Using standard LMI software in combination with the line search of α for (Sop), we can obtain a stable decentralized dynamic quantizer, numerically. Figure 5 illustrates the relation between the problems (OP1), (OP2) and (Sop). Problem (E) (Lemma 3) ⇑∗ Problem (OP1): (nq , Aq , Bq , Cq ) (Lemma 4) m‡ Problem (OP2): (nq , Aq , Bq , Cq ) (Theorem 5) m⋆ ˆ Yˆ , Fˆ , W ˆ ,U ˆ) Problem (Sop) : (X,
Under some circumstances, Proposition 2 gives a dynamic quantizer which is expressed by the given generalized plant parameters. The following theorem denotes this fact. Theorem 6. Consider the optimization problem (OP1) for the matrices (14). Suppose that Assumption 1 holds. An optimal solution of (Aq , Bq , Cq ) to the problem (OP1) is given by Aqi = A, Bqi = B2i , Cqi = −(C1 Aτi B2i )† C1 Aτi +1 (22)
for every i = 1, ..., s and its infimum of γ ∈ IR+ is √ d m p inf γ = σρ , ρ = min{ρ(A), ρ(Aq + Bq Cq )}, (23) 2 1 − ρ2
1 σρ := τ1 C1 Aτ1 B21 1τ2 C1 Aτ2 B22 . . . 1τs C1 Aτs B2s ρ
2
if the matrix Aqi + Bqi Cqi defined in (22) is stable in the discrete domain for every i = 1, ..., s. ˆ and Bq = B, ˆ one gets Proof. For the case where Aq = A C2 Ak B1 = C1 (A + B21 Cq1 )k B21 C1 (A + B22 Cq2 )k B22
d τs τ2 τ1 E∞ (Qop d ) = kabs(C1 [ A B21 A B22 . . . A B2s ])k . 2 It is striking that the structure of their optimal quantizer is equivalent to our proposed one based on Proposition 2 even if the former performance evaluation is less conservative than the latter one. Theorem 6 points out that the proposed quantizer is also optimal in the sense that the quantizer gives an optimal output approximation property for minimum phase systems. When the system G(z) is non-minimum phase, the optimal quantizer (22) is unstable. In this case, Azuma et al. [TAC 2008] has provided the numerical design method in which the stable and optimal decentralized quantizer synthesis problem is recast as the following optimization problem:
−1
d
TX
k abs(C2 A B1 ) (= ET (Qd )) min
2 Aq ,Bq ,Cq ,P k=0
Fig. 5. Solutions to Problem (E)
ρ
Theorem 6 presents a closed-form solution to the problem (E). In the case of m = p, the stable Aq + Bq Cq in (22) implies that the all transmission zeros of the generalized plant G(z) are stable (see Azuma et. al. [CDC 2008]). When the matrix C2 is full row rank, the system G(z) is minimum phase and Assumption 1 holds, on the other hand, Minami et. al. [2009] has presented an optimal decentralized dynamic quantizer Qop d given by (22) and its performance given by
s.t. (Aqi + Bqi Cqi )T Pi (Aqi + Bqi Cqi ) < Pi , ∀i = 1, ..., s and P := diag(P1 , ..., Ps ) > 0
∗: (E) is feasible if (OP1) is feasible. ‡: Equivalent for (nq , Aq , Bq , Cq ). ⋆: Equivalent for the fixed α ∈ (0, 1).
ρ
for the given T ∈ IN+ . The order of each sub-quantizer in their method is given by ⌊T /2⌋+1 or (⌊T /2⌋+1)T . When T is set be large, we need the reduction technique. On the other hand, even if the system G(z) has unstable zeros, our method provides a suboptimal dynamic decentralized quantizer with nqi = ng in the sense that the upper bound of E∞ (Qd ) is minimized. This suboptimality is the conservativeness that the ellipsoidal approximation in Proposition 2 leads to. Even if there exists an optimal quantizer achieving a certain performance for a non-minimum phase system, it is not necessarily that such a quantizer is obtained from (OP1). The results of Theorems 5 and 6 are corresponding to the centralized case when we set s = 1. Then, this paper gives a unified framework of the dynamic quantizer synthesis based on the invariant set analysis. 5. NUMERICAL EXAMPLE Consider the system in Fig. 1 (b). The plant P (z) is the discretized system of the unstable non-minimum phase continuoustime LTI system 0 1 0 xp (t) x˙ p (t) , zp (t) = y2 (t) = 1 −1 1 v1 (t) y2 (t) −2 1 0 with the sampling time h = 0.1 and zero-order hold. Its eigenvalues are {1.064, 0.857} and its unstable zero is {1.224}. The stabilizing controller C(z) is given by
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
20 v (kh)
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Fig. 6. Time responses for centralized quantizer (s=1)
10 time kh [s]
(a) v1 (the input of P (z))
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(a) v1 (the input of P (z))
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Fig. 7. Time responses for decentralized quantizer (s=2) REFERENCES
xc (k) . v2 (k)
Consider d = 10 and s = 2. From (Sop), the stable suboptimal decentralized quantizers with γ = 2.09 are obtained as follows: Aqi = A, Bqi = B2i , and Cqi = [ 23.353 − 11.738 − 1.000 ] (s = 1, 2). As a result, the matrices Cqi are modified from the optimal one such that the quantizers are stable. To compare the decentralized quantizer with the centralized quantizer, the stable suboptimal centralized quantizer with γ = 2.09 is also obtained from (Sop) with s = 1. Figures 6 and 7 illustrate the time responses of v1 (kh), v2 (kh) and zp (kh) for the centralized and decentralized quantizers with the initial state x(0) = [ − 5 0 − 1 ]T , respectively. In Figs. 6 and 7, the thin lines and the thick lines illustrate the time responses of the usual feedback system in Fig. 2 (b) and the quantized feedback system in Fig. 2 (a), respectively. We see that the discrete-valued signals v1 and v2 in Fig. 7 switch more quickly compared with those in Fig. 6. However, we see that the decentralized quantizer attenuates the excess performance deterioration between both quantizers. In fact, both quantizer performances are nearly equal in terms of the performance level γ. 6. CONCLUSION We have proposed a unified framework of the decentralized dynamic quantizer synthesis based on the invariant analysis. This paper has proposed the quantizer synthesis condition that has a closed-form solution under some circumstances. Also, we have clarified the effectiveness and the limitation of the proposed condition.
P. Antsaklis and J. Baillieul (eds.). Special issue on networked control systems. IEEE Trans. Auto. Contr.. Vol. 49, No. 9, pages 1421–1423, 2004. Special Section on Complex Networked Control Systems. IEEE Control Systems Magazine. Vol. 27, No. 4, 2007. S. Azuma and T. Sugie. Optimal dynamic quantizer for discrete-valued input control. Automatica, Vol. 44, No. 12, pages 396–406, 2008. S. Azuma and T. Sugie. Stability analysis of quantized feedback systems including optimal dynamic quantizers. Proc. of 47th IEEE CDC, pages 3392–3397, 2008. S. Azuma and T. Sugie. Synthesis of optimal dynamic quantizers for discrete-valued input control. IEEE Trans. Auto. Contr., Vol. 53, No. 9, pages 2064–2075, 2008. Y. Minami, S. Azuma, and T. Sugie. Optimal Decentralized Dynamic Quantizers for Discrete-Valued Input Control: A Closed form solution and Experimental Evaluation. Proc. of ACC, pages 4367–4372, 2009. H. Shingin and Y. Ohta. Optimal invariant sets for discrete-time systems: Approximation of reachable sets for bounded inputs. 10th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems, Theory and Applications (LSS), pages 401– 406, 2004. K. Sawada and S. Shin. Synthesis of dynamic quantizers for quantized feedback systems within invariant set analysis framework. ACC, to appear, 2011. H. Okajima, N. Matsunaga, and K. Sawada. Optimal quantization interval design of dynamic quantizers which satisfy the communication rate constraints. Proc. of 49th IEEE CDC, pages 4733–4739, 2010. J. A. Stankovic. When sensor and actuator networks cover the world. ETRI Journal, Vol. 30, No. 5, pages 627–633, 2008.
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