Numerical Optimization Design of Dynamic Quantizer via Matrix Uncertainty Approach

In networked control systems, continuous-valued signals are compressed to discrete-valued signals via quantizers and then transmitted/received through communication channels. Such quantization often degrades the control performance; a quantizer must be designed that minimizes the output difference between before and after the quantizer is inserted. In terms of the broadbandization and the robustness of the networked control systems, we consider the continuous-time quantizer design problem. In particular, this paper describes a numerical optimization method for a continuous-time dynamic quantizer considering the switching speed. Using a matrix uncertainty approach of sampled-data control, we clarify that both the temporal and spatial resolution constraints can be considered in analysis and synthesis, simultaneously. Finally, for the slow switching, we compare the proposed and the existing methods through numerical examples. From the examples, a new insight is presented for the two-step design of the existing continuous-time optimal quantizer.


Introduction
With the rapid network technology development, the networked control systems (NCSs) have been widely studied [1][2][3][4][5][6][7][8][9][10].One of the challenges in NCSs is quantized control.In NCSs, the continuous-valued signals are compressed and quantized to the discrete-valued signals via the quantizer of the communication channel, and such quantization often degrades the control performance.Hence, a desirable quantizer minimizes the performance error between before and after the quantizer insertion.
Motivated by this, researchers [11][12][13][14] have provided optimal dynamic quantizers for the following problem formulation in the discrete-time domain.For a given plant , synthesize a "dynamic" quantizer   such that the system Σ  composed of  and   in Figure 1(a) "optimally" approximates the plant  in Figure 1(b) in the sense of the input-output relation.The obtained quantizer allows us to design various controllers for the plant  on the basis of the conventional control theories.Also, this framework is helpful in not only the NCS problem but also various control problems such as hybrid control, embedded system control, and on-off actuator control.
When we consider controlling a mechanical system with an on-off actuator, first the controlled object and its uncertainties are usually modeled in the continuoustime domain.Second, the model and its uncertainties are discretized to apply the above dynamic quantizer.However, the discretization sometimes results in uncertainties more complicated than those in the original model and creates undesirable complexity in robust control.The continuoustime setting quantizer is more suitable for the robust control of the quantized system than discrete-time one.Thus, our previous works [15,16] have considered the continuous-time setting, while a number of the discrete-time settings have been studied by others [11][12][13][14].In these works, it is assumed that the switching process of discretizing the continuousvalued signal is sufficiently quick relative to the control frequency and only the spatial determination (quantized accuracy) is considered as the quantization effect.This is Mathematical Problems in Engineering because the switching speed of the continuous-time deltasigma modulator for wireless broadband network systems is from 1 MHz to 100 MHz [17,18].On the other hand, the above assumption is essentially weak in the case of the slow switching such as the mechanical systems with on-off actuators [19].For the slow switching, we need to consider the quantization effect on both the switching speed and the spatial constraints in continuous time.For example, Ishikawa et al. proposed a two-step design of a feedback modulator [20]: (i) the control performance of the modulator is considered under only the spatial constraint, and (ii) the modulator is tuned in terms of the switching speed constraint.However, the structure of the modulator is more restricted than that of the dynamic quantizer and the obtained modulator is not always optimal.Therefore, the dynamic quantizer under temporal resolution (switching speed) and spatial resolution constraints has still to be optimally designed.The simultaneous consideration of the two constraints is the particular challenge we address in this paper.
We propose a numerical optimization method for the continuous-time dynamic quantizer under switching speed and quantized accuracy constraints.To achieve the method, this paper solves the design problem via sampled-data control framework that has so far provided various results for networked control problems [7][8][9].We refer to the previous work on optimal dynamic quantizer design [11,12] and consider the basic feedforward system in Figure 1(a).In addition to the invariant set analysis [21,22] similarly to our previous works [13,15,16], this paper utilizes a matrix uncertainty approach [23,24] that is proposed in a sampleddata control framework.Although the obtained results can be more conservative than those in the previous works on continuous-time dynamic quantizer [15,16] from the viewpoint of the class of the exogenous input and the applicable plants, both temporal and spatial resolution constraints can be addressed in analysis and synthesis, simultaneously.For the fast switching case, the proposed conditions converge to the corresponding conditions of our previous works.Finally, for the slow switching, we compare the proposed and existing methods [15,16] through numerical examples.In particular, a new insight is presented for the two-step design of the existing continuous-time optimal quantizer.Notation.The set of  ×  (positive) real matrices is denoted by R × (R × + ).The set of  ×  (positive) integer matrices is denoted by N × (N × + ).We denote by L  ∞ the set of piecewise-continuous functions of -dimensional finite vectors such that ∞-norm of its functions is finite.0 × and   (or for simplicity of notation, 0 and ) denote the  ×  zero matrix and the  ×  identity matrix, respectively.For a matrix ,   , () and  max () denote its transpose, its spectrum radius, and its maximum singular value, respectively.For a vector ,   is the th entry of .For a symmetric matrix ,  > 0 ( ≥ 0) means that  is positive (semi) definite.For a vector  and a sequence of vectors  := { 1 ,  2 , . ..}, ‖‖ and ‖‖ denote their ∞-norms, respectively.Finally, we use the "packed" notation (     ) := ( − ) −1  + .

Problem Formulation
Consider the discrete-valued input system Σ  in Figure 1(a), which consists of the linear time invariant (LTI) continuoustime plant  and the quantizer V =   ().The system  is given by where  ∈ R  ,  ∈ R  ,  ∈ R  , and V ∈ R  denote the state vector, the measured output, the exogenous input, and the quantizer output, respectively.The continuous-valued signal  is quantized into the discrete-valued signal V via the quantizer   .We assume that the matrix  is Hurwitz; that is, the usual system in Figure 1(b) is stable in the continuoustime domain.The initial state is given as (0) =  0 .
For the system , consider the continuous-time dynamic quantizer V =   () with the state vector   ∈ R   as shown in Figure 2. Its switching speed ℎ ∈ R + (or its temporal resolution) is determined by the operator , which converts the continuous-time signal  into the low temporal resolution signal ĝ as follows: [] =  (ℎ) ,  = 0, 1, 2, 3, . . .,  ∈ [0, ℎ) . ( That is, û =   . is the ideal sampler with the sampling period ℎ and  is the zero-order hold operator.The spatial resolution of the quantizer   is expressed by the static quantizer  : R  → N  with the quantization interval  ∈ R + , that is, and the continuous-time LTI filter  is given by Note that  is of the nearest-neighbor type toward −∞ such as the midtread quantizer in Figure 3 (‖  (û  ) − û ‖ ≤ /2 where   and û are the th row of  and û ) and the initial state is given by   (0) = 0 for the drift free of   [11,12].
Remark 1.In synthesis, our previous works [15,16] ignored the operator .In implementation, however, the continuous-time quantizer needs the switching process discretizing the continuous-valued signal.Of course, the applicable interval of switching depends on controlled objects such as narrowband or broadband networked systems and mechanical systems with on-off actuators.Therefore, it is important to consider the operator  in synthesis.
For the system Σ  in Figure 1(a) with the initial state  0 and the exogenous input  ∈ L  ∞ , (,  0 ,   ()) denotes the output of  at the time .Also, for the system in Figure 1(b) without   ,  * (,  0 , ) denotes its output at the time .Consider the following cost function: (,  0 ,   ()) −  * (,  0 , )     . ( If the quantizer minimizes (  ), the system Σ  "optimally" approximates the usual system  in the sense of the input-output relation.In this case, we can use the existing continuous-time controller design methods for the system in Figure 1(b) without considering the quantization effect.When the controlled object and its uncertainties are modeled in the continuous-time domain, therefore, the continuous-time quantizer can introduce robust control of the continuous-time setting directly, while the discrete-time quantizer requires discretization of the whole control system.Our previous works [15,16] proposed an optimal dynamic quantizer for the cost function (  ) for the fast switching case ℎ = 0.That is, only the spatial deterioration has been considered.On the other hand, the simultaneous consideration of the temporal and spatial resolution constraints is the problem we address in this paper.To consider the temporal resolution constraint caused by the operator , this paper modifies the cost function as follows: Fixing  = 0 ignores the output error between before and after the quantizer is inserted over the th sampling interval and leads to the cost function setting that is utilized for the discrete-time optimal dynamic quantizers [11][12][13][14].Therefore, the optimal quantizer for   (  ) minimizes the output error between the systems in Figures 1(a) and 1(b) in terms of the input-output relation under the temporal and spatial resolution constraints.Motivated by the above, our objective is to solve the following continuous-time dynamic quantizer synthesis problem (E): for the system Σ  composed of  and   with the initial state  0 ∈ R  and the exogenous input  ∈ ℓ  ∞ , suppose that the quantization interval  ∈ R + , the switching speed ℎ ∈ R + , and the performance level  ∈ R + are given.Characterize a continuous-time dynamic quantizer   (i.e., find parameters (  ,   ,   ,   )) achieving   (  ) ≤ .
This paper proposes continuous-time quantizers in terms of solving the problem (E) on the basis of invariant set analysis and the sampled-data control technique, while other researchers [11,12,14] have proposed the discrete-time optimal ones.
Remark 2. The cost function setting of this paper is more complicated than the existing continuous-time and discretetime cases [11][12][13][14][15][16], so this paper considers the basic feedforward system composed of  and   similar to the previous works on optimal dynamic quantizer design [11,12].Remark 3. The plant  is restricted to be stable because of the feedforward structure, while the existing results can address unstable plants.To remove this restriction, we need to consider a feedback system structure similar to existing ones [13][14][15][16].This is our future task.

System Expression.
In this subsection, we consider the system expression for the quantizer analysis.Define the quantization error  as From the properties of the quantizer  and the operator , holds where [] = (ℎ).Then, one obtains where V  [] = V  (ℎ) and [] = (ℎ) for  = 0, 1, 2, . ..,  ∈ [0, ℎ).In this case, by using the sampled-data control technique, the following lemma holds.
We focus on ũ(ℎ + ) of Σ.For the operator  and the holds.This implies that we cannot ignore the temporal resolution constraint on the cost function   (  ) even if ℎ → 0. On the other hand, low-pass prefiltering rectifies this situation [25].In fact, for the stable LTI system , holds.For the evaluation of the cost function   (  ), then this paper utilizes as the exogenous input.Note that  ∈ L  ∞ if stable  is strictly proper and  ∈ L  ∞ .For the signal (15), [] = (ℎ+) ( = 0, 1, 2, . . .,  ∈ [0, ℎ)) holds, so the terms of ũ(ℎ + ) in (11) are eliminated.Then, Σ is rewritten as Also, this paper solves the following synthesis problem (E  ): for the system Σ  composed of  and   with the initial state  0 ∈ R  and the exogenous input  ∈ L  ∞ in (15), suppose that the quantization interval  ∈ R + , the switching speed ℎ ∈ R + , and the performance level  ∈ R + are given.Characterize a continuous-time dynamic quantizer   (i.e., find parameters (  ,   ,   ,   )) achieving   (  ) ≤ .

Quantizer Analysis.
The quantization error  of ( 16) is bounded as mentioned earlier.The reachable set and the invariant set characterize such a system with bounded input.Consider the LTI discrete-time system given by where ξ ∈ R   and  ∈ R  denote the state vector and disturbance input, respectively.We define the reachable set and the invariant set.
Definition 5. Define the reachable set of the system (17) to be a set R ∞ which satisfies Definition 6. Define the invariant set of the system (17) to be a set x which satisfies The analysis condition can be expressed in terms of matrix inequalities as summarized in the following proposition [22].Proposition 7. Consider the system (17).For a matrix 0 < P = P  ∈ R   ×  , the ellipsoid E(P) := { ξ ∈ R   : ξ P ξ ≤ 1} is an invariant set if and only if there exists a scalar Note that the ellipsoidal set E(P) covers the reachable set R ∞ from outside.Define the set E := {w ∈ R  :  = (√/2)w satisfies (7)} and rewrite the system (16) as where  = (√/2) ξ,   = (√/2)z  ,  = (√/2)w, and w ∈ E. The left multiplication of Σ with √/2 leads to Σ  .The relation E ⊆ W clearly holds because    ≤  2 /4 and the set W is an independent bounded disturbance without the relation (7).That is, the reachable set of Σ with w ∈ E is no larger than that of Σ with the disturbance w ∈ W.
Then, this paper utilizes the reachable set to estimate the influences of the quantization error and the invariant set to characterize the cost function   (  ) by substituting A = A and B = B into (20).Move on to the matrix exponential   of C() and D() in (21), which is rewritten as Along with this, z of ( 21 In addition, from the properties of R ∞ and E(P), holds.Similarly to our previous papers [13,15,16], by using the L 1 control technique in [21], we provide the sufficient conditions for computing  1 ∈ R + and  2 ∈ R + of (24) as follows: Remark 8.For the inequalities (25) and any vectors ξ ∈ R  and w ∈ R  , we have and w Ω()    Ω()w ≤  2 2 w w.Then, we see that (24) holds if ξ ∈ E(P) and w ∈ W.
The inequalities (25) are difficult to test since we need to find P,  1 , and  2 satisfying (20) and (25) for infinitely many values of  ∈ [0, ℎ).Then, using the matrix uncertainty technique [23,24], we consider their sufficient conditions, which are easy to compute.Considering Ω() in ( 22) as a matrix uncertainty, we introduce the following lemma regarding the matrix exponential [26,27].Lemma 9.For the matrix Ω() in (22), holds where Proof.Since  max (  ) ≤  () (see [26]), By using Lemma 9 and the -procedure [23,28,29], the sufficient condition analyzing the cost function   (  ) of the system Σ  can be expressed in terms of matrix inequality as summarized in the following theorem.
Theorem 10.Consider the system Σ  composed of  and   with the initial state  0 ∈ R  and the exogenous input  ∈ L  ∞ in (15).For the quantization interval  ∈ R + and the switching speed ℎ ∈ R + , the upper bound of the cost function   (  ) is given by where the matrices Φ ℎ and Γ ℎ are defined by Proof.See Appendix A.
Denote by Σ the system Σ without operator .In this case, the system Σ is given by where Regarding the definition of (), see Lemma 4.An advantage the condition (31) over conditions (20) is that it can be used for a small ℎ without numerical difficulty.This idea comes from [23,24].In the limit of ℎ → 0, (1/ℎ) ∫ ℎ 0  ℎ d →  and (1/ℎ) ∫ ℎ 0    ℎ d →  hold, so Φ ℎ → Â and Γ ℎ → B hold.In the same limit, from (ℎ) → 0, conditions ( 31) and ( 32) converge to the analysis conditions of the continuoustime dynamic quantizer for the system Σ in [15,16].On the other hand, for a small ℎ,  ℎ → ,    ℎ → , ∫ ℎ 0  ℎ d → 0 and ∫ ℎ 0    ℎ d → 0 hold, so A and B (A and B) are close to identity and zero matrices, respectively, and the left side of ( 20) is close to zero.
In numerical computation, it is appropriate to fix the structure of  such that Ω() = Ω() holds.For example, we can set  =     ,   ∈ R + and this setting leads to the following optimization problem (Aop):    From the matrix product such as (1/ℎ) ∫ ℎ 0    ℎ d and   +    in (31), the synthesis condition is difficult to derive from Theorem 10 unlike the continuous-time case without the operator  in [15,16].Thus, we fixed the parameters as follows: The structure (38) does not severely limit the synthesis because   and   of the continuous-time dynamic quantizer for the system Σ in [15,16] are also (38).See Appendix B.
In other words, ẋ  =   +   estimates the quantization influence on the system .Along with this, we fix Q of (31) as follows: The structure (39) also does not impose a severe limitation on the synthesis because an appropriate choice of the quantizer state coordinates allows us to assume that Q has the special structure for the full order case   =  [30].Under some circumstances ( 38) and (39), we obtain the following synthesis condition.Theorem 11.Consider the system Σ  composed of  and   with the initial state  0 ∈ R  and the exogenous input  ∈ L  ∞ in (15).Suppose that the quantization interval  ∈ R + , the switching speed ℎ ∈ R + , and the performance level  ∈ R + are given.For a scalar  ℎ ∈ [0, 1/2ℎ], there exist a continuous-time dynamic quantizer   achieving (30) if one of the following equivalent statements holds.
(ii) There exist matrices where In this case, such a quantizer parameter is given by Proof.We fix Q as shown in (39) and introduce the change of variables  =   .Hence, (31) and (32) result in ( 40) and (41).Also, designing In the limit of ℎ → 0; Ψ ℎ converges to  and (ℎ) converges to 0; then conditions (40) and (41) also converge to the synthesis condition of the continuous-time dynamic quantizer for the system Σ in (34).Also, by setting  =     for Theorem 11, the quantizer synthesis problem (E  ) is reduced to the following optimization problem (Sop):  Remark 12.To consider numerical optimization analysis or synthesis of a quantizer as shown in (Aop) and (Sop), we need the signal assumption (15) in Theorems 10 and 11.On the other hand, for the high speed switching such that ℎ is very small, the assumption (15) ensures that solutions to the problem (E  ) converge to our previous results [15,16].Therefore, the results of this paper partly include our previous results [15,16] although each class of exogenous signals and plants is restricted.

Discussion
For the slow switching, we compare the proposed method and existing continuous-time quantizer [15,16].Consider the system Σ  .The plant  is the stable minimum phase LTI system: In the case without the operator , an optimal form of the continuous-time quantizer    [15,16] is given by (B.2).  4 and 5, the thin lines and the thick lines are for the conventional system in Figure 1(b) and the system Σ  in Figure 1(a), respectively.We see that the controlled outputs of the discrete-valued input systems with the dynamic quantizers approximate those of the usual systems even if the quantized outputs are applied.Also, the two controlled outputs approximated by both quantizers are exactly the same.
Next, we consider the case ℎ = 0.1.In this case, the two controlled outputs approximated by the two quantizers differ.(Aop) for    is infeasible.From (Sop), on the other hand, we obtain (ℎ) = 0.949 and the matrix   = [−1.1321−3.097].Figures 6 and 7 illustrate the simulation results on the time responses of Σ  with (B.2) and the proposed quantizer in the same fashion.We see that () of the usual plant  is approximated by () of the system Σ  with the proposed quantizer, while () of the system Σ  with (B.2) diverges.From this example, we see that the proposed method can address the spatial resolution and the temporal resolution issues, simultaneously.Also, Theorem 10 verifies whether the quantizer    is applicable to the given switching speed setting.
Remark 13.In the above numerical experiments, the proposed quantizer is designed and the quantizer    is analyzed for  = (sin  + cos 0.7), while the time responses of the quantizers are simulated for  = sin  + cos 0.7.That is, this is the conservativeness caused by the signal assumption (15).However, we see that the above results verify the effectiveness of the proposed method even if the signal conservativeness exists.
From the above results, the existing continuous-quantizer in [15,16] may be suitable for a two-step design such that   (ℎ) is stable via the parameter .In terms of the upper where the parameters (  ,   ,   ) of Ψ ℎ and Γ ℎ are given by (B.2).This problem is LMI for the plane search of  and  ℎ .For ℎ = 0.1, its solution is  = 5.715 (  = 0.397) and then (ℎ) = 0.949 for   (   ) is obtained.This performance is about the same as that of the proposed quantizer.Therefore, we see that Theorem 10 is also helpful for the two-step design of the existing continuous-time quantizer [15,16] even if the tractable optimization method instead of the plane search remains an issue for future work.Such a method correlates the parameter  with the switching speed ℎ, so its insight is expected not only to result in a new two-step design but also to clarify the relationship between the discrete-time and continuous-time dynamic quantizers.Of course, important future topics also include considering the quantized feedback control system with unstable plants and generalizing the exogenous signal for the evaluation of the cost function.

Conclusion
Focusing on the broadbandization and the robustness of the networked control systems, this paper has dealt with the continuous-time quantized control.We have proposed numerical optimization methods analyzing and synthesizing the continuous-time dynamic quantizer on the basis of the invariant set analysis and the sampled-data control technique.The contributions of the proposed method can be summarized as follows.
(i) Both the temporal and spatial resolution constraints can be simultaneously considered, whereas Ishikawa et al. [20] considered the two constraints step-by-step and we [15,16] previously ignored the temporal constraint in synthesis.As a result, the proposed method is applicable to both the slow and fast switching cases.(ii) The maximum output difference for each sampling interval is proven to be evaluated numerically via the matrix uncertainty approach, while the existing results [11][12][13][14] evaluate that only for each sampling instance.(iii) The analysis and synthesis conditions are given in terms of BMIs.However, the quantizer analysis and synthesis problems are reduced to tractable optimization problems.(iv) The new insight is presented for the existing continuous-time quantizer design [15,16].
Also, this paper has clarified the following areas for future work.
(i) Because of the feedforward structure, the plant is restricted to be stable.To address unstable systems, we Mathematical Problems in Engineering need to propose design methods for feedback control systems.
(ii) The sensors and actuators are distributed in the networked control system [31], so it is necessary to design multiple (decentralized) quantizers rather than a centralized quantizer similar to the existing ones [14,16].
(iii) The class of exogenous signals evaluating the cost function is restricted.To avoid this conservativeness, it is necessary to propose the equivalent discretetime expression instead of (11) using adjoint operator similar to sampled-data control [25,32].
(iv) For networked control applications, it is important to consider the time-varying sampling period, time delay, packet loss, and so on similar to [6][7][8][9][10].For example, the works [6,8] using the LMI technique address time-varying sampling period and time delay, so it is expected that our method using the LMI technique also extends to such problems.

B. Continuous-Time Dynamic
Quantizer [15,16] For the system Σ in (34) without the operator , we consider the following non-convex optimization (OP  ): The dynamic quantizer without the operator  is obtained from (OP  ) and (  ) ≤   is achieved.Also, an optimal form of the continuous-time quantizer [15,16] is given by  where ]() := max{Re() :  ∈ eig()}.The continuoustime quantizer    and its performance are parameterized by the free parameter  ∈ R + .Note that the larger values of  not only provide the better approximation performance, but also switch the outputs V, more quickly.In other words, the quantizer from (OP  ) results in the switching that is too fast and is sometimes not applicable to the slow switching case.
scalar  ℎ is fixed, the conditions in Theorem 10 are linear matrix inequalities (LMIs) in terms of the other variables.Using standard LMI software and the line search of  ℎ , we can obtain an upper bound of   (  ).
) is feasible, (E  ) is feasible.Therefore, a continuoustime dynamic quantizer considering both spatial and temporal resolution constraints is obtained from Theorem 11.