Numerical Optimization Design of Dynamic Quantizer via Matrix Uncertainty Approach

Abstract

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.

1. Introduction

With the rapid network technology development, the networked control systems (NCSs) have been widely studied [1–10]. 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 [11–14] have provided optimal dynamic quantizers for the following problem formulation in the discrete-time domain. For a given plant 𝑃, synthesize a “dynamic” quantizer 𝑄_𝑑 such that the system Σ_𝑄 composed of 𝑃 and 𝑄_𝑑 in Figure 1(a) “optimally” approximates the plant 𝑃 in Figure 1(b) in the sense of the input-output relation. The obtained quantizer allows us to design various controllers for the plant 𝑃 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.

(a) Discrete-valued input system

(b) Usual system

Figure 1 

Two systems.

When we consider controlling a mechanical system with an on-off actuator, first the controlled object and its uncertainties are usually modeled in the continuous-time 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 continuous-time 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 [11–14]. In these works, it is assumed that the switching process of discretizing the continuous-valued signal is sufficiently quick relative to the control frequency and only the spatial determination (quantized accuracy) is considered as the quantization effect. This is because the switching speed of the continuous-time delta-sigma 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 [7–9]. 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 sampled-data 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 𝑛×𝑚 (positive) real matrices is denoted by ℝ^𝑛×𝑚 (ℝ₊^𝑛×𝑚). The set of 𝑛×𝑚 (positive) integer matrices is denoted by ℕ^𝑛×𝑚 (ℕ₊^𝑛×𝑚). We denote by ℒ^𝑝_∞ the set of piecewise-continuous functions of 𝑝-dimensional finite vectors such that ∞-norm of its functions is finite. 0_𝑛×𝑚 and 𝐼_𝑚 (or for simplicity of notation, 0 and 𝐼) denote the 𝑛×𝑚 zero matrix and the 𝑚×𝑚 identity matrix, respectively. For a matrix 𝑀, 𝑀^𝑇, 𝜌(𝑀) and 𝜎_max(𝑀) denote its transpose, its spectrum radius, and its maximum singular value, respectively. For a vector 𝑥, 𝑥_𝑖 is the 𝑖th entry of 𝑥. For a symmetric matrix 𝑋, 𝑋>0  (𝑋≥0) means that 𝑋 is positive (semi) definite. For a vector 𝑥 and a sequence of vectors 𝑋∶={𝑥₁,𝑥₂,…}, ‖𝑥‖ and ‖𝑋‖ denote their ∞-norms, respectively. Finally, we use the “packed” notation ^𝐴𝐵_𝐶𝐷∶=𝐶(𝑠𝐼−𝐴)⁻¹𝐵+𝐷.

2. Problem Formulation

Consider the discrete-valued input system Σ_𝑄 in Figure 1(a), which consists of the linear time invariant (LTI) continuous-time plant 𝑃 and the quantizer 𝑣=𝑄_𝑑(𝑢). The system 𝑃 is given by 𝑥̇𝑧=𝑥𝑣𝑃∶𝐴𝐵𝐶0, (1) where 𝑥∈ℝ^𝑛, 𝑧∈ℝ^𝑞, 𝑢∈ℝ^𝑚, and 𝑣∈ℝ^𝑚 denote the state vector, the measured output, the exogenous input, and the quantizer output, respectively. The continuous-valued signal 𝑢 is quantized into the discrete-valued signal 𝑣 via the quantizer 𝑄_𝑑. We assume that the matrix 𝐴 is Hurwitz; that is, the usual system in Figure 1(b) is stable in the continuous-time domain. The initial state is given as 𝑥(0)=𝑥₀.

For the system 𝑃, consider the continuous-time dynamic quantizer 𝑣=𝑄_𝑑(𝑢) with the state vector 𝑥_𝑄∈ℝ^𝑛𝑄 as shown in Figure 2. Its switching speed ℎ∈ℝ₊ (or its temporal resolution) is determined by the operator 𝐻𝑆, which converts the continuous-time signal 𝑔 into the low temporal resolution signal 𝑔 as follows: 𝑔()[𝑘],𝑔[𝑘]()[).𝐻𝑆∶𝑔⟶𝑔∶𝑘ℎ+𝜃=𝑔=𝑔𝑘ℎ,𝑘=0,1,2,3,…,𝜃∈0,ℎ (2) That is, 𝑢_𝑄=𝐻𝑆𝑢_𝑄. 𝑆 is the ideal sampler with the sampling period ℎ and 𝐻 is the zero-order hold operator. The spatial resolution of the quantizer 𝑄_𝑑 is expressed by the static quantizer 𝑞∶ℝ^𝑚→𝑑ℕ^𝑚 with the quantization interval 𝑑∈ℝ₊, that is, 𝑣=𝑞𝐻𝑆𝑢_𝑄,𝑢_𝑄=𝑢+𝑣_𝑄 (3) and the continuous-time LTI filter 𝑄 is given by 𝑥̇_𝑄𝑣_𝑄=𝐴_𝑄𝐵_𝑄𝐶_𝑄0𝑥_𝑄𝑒_𝑄,𝑒_𝑄∶=𝑣−𝑢. (4) Note that 𝑞 is of the nearest-neighbor type toward −∞ such as the midtread quantizer in Figure 3 (‖𝑞_𝑖(𝑢_𝑄𝑖)−𝑢_𝑄𝑖‖≤𝑑/2 where 𝑞_𝑖 and 𝑢_𝑄𝑖 are the 𝑖th row of 𝑞 and 𝑢_𝑄) and the initial state is given by 𝑥_𝑄(0)=0 for the drift free of 𝑄_𝑑 [11, 12].

Figure 2 

Continuous-time dynamic quantizer with switching speed ℎ.

Figure 3 

Midtread quantization.

Remark 1. In synthesis, our previous works [15, 16] ignored the operator 𝐻𝑆. 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 𝐻𝑆 in synthesis.

For the system Σ_𝑄 in Figure 1(a) with the initial state 𝑥₀ and the exogenous input 𝑢∈ℒ^𝑚_∞, 𝑧(𝑡,𝑥₀,𝑄_𝑑(𝑢)) denotes the output of 𝑧 at the time 𝑡. Also, for the system in Figure 1(b) without 𝑄_𝑑, 𝑧^∗(𝑡,𝑥₀,𝑢) denotes its output at the time 𝑡. Consider the following cost function: 𝐽𝑄_𝑑∶=sup_{(𝑥0,𝑢)∈ℝ𝑛×ℒ𝑚∞}sup_{𝑡∈ℝ+∪{0}}𝑧𝑡,𝑥₀,𝑄_𝑑(𝑢)−𝑧^∗𝑡,𝑥₀,𝑢. (5) If the quantizer minimizes 𝐽(𝑄_𝑑), the system Σ_𝑄 “optimally” approximates the usual system 𝑃 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 𝐽(𝑄_𝑑) 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 𝐻𝑆, this paper modifies the cost function as follows: 𝐽_𝐻𝑆𝑄_𝑑∶=sup_{(𝑥0,𝑢)∈ℝ𝑛×ℒ𝑚∞}sup_{𝜃∈[0,ℎ)}sup_{𝑘∈ℕ+∪{0}}𝑧𝑘ℎ+𝜃,𝑥₀,𝑄_𝑑(𝑢)−𝑧^∗𝑘ℎ+𝜃,𝑥₀,𝑢. (6) Fixing 𝜃=0 ignores the output error between before and after the quantizer is inserted over the 𝑘th sampling interval and leads to the cost function setting that is utilized for the discrete-time optimal dynamic quantizers [11–14]. Therefore, the optimal quantizer for 𝐽_𝐻𝑆(𝑄_𝑑) 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_𝑄 composed of 𝑃 and 𝑄_𝑑 with the initial state 𝑥₀∈ℝ^𝑛 and the exogenous input 𝑢∈ℓ^𝑚_∞, suppose that the quantization interval 𝑑∈ℝ₊, the switching speed ℎ∈ℝ₊, and the performance level 𝛾∈ℝ₊ are given. Characterize a continuous-time dynamic quantizer 𝑄_𝑑  (i.e., find parameters (𝑛_𝑄,𝐴_𝑄,𝐵_𝑄,𝐶_𝑄)) achieving 𝐽_𝐻𝑆(𝑄_𝑑)≤𝛾.

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 discrete-time cases [11–16], so this paper considers the basic feedforward system composed of 𝑃 and 𝑄_𝑑 similar to the previous works on optimal dynamic quantizer design [11, 12].

Remark 3. The plant 𝑃 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 [13–16]. This is our future task.

3. Main Result

3.1. System Expression

In this subsection, we consider the system expression for the quantizer analysis. Define the quantization error 𝑒 as 𝑒∶=𝑞𝑢_𝑄−𝑢_𝑄=𝑣−𝑢_𝑄. (7) From the properties of the quantizer 𝑞 and the operator 𝐻𝑆, 𝑒()[𝑘]∈−𝑑𝑘ℎ+𝜃=𝑒2,𝑑2^𝑚,𝑘=0,1,2,…,𝜃∈[0,ℎ), (8) holds where 𝑒[𝑘]=𝑒(𝑘ℎ). Then, one obtains 𝑣(𝑘ℎ+𝜃)=𝑣_𝑄[𝑘]+𝑢[𝑘]+𝑒[𝑘], (9) where 𝑣_𝑄[𝑘]=𝑣_𝑄(𝑘ℎ) and 𝑢[𝑘]=𝑢(𝑘ℎ) for 𝑘=0,1,2,…, 𝜃∈[0,ℎ). In this case, by using the sampled-data control technique, the following lemma holds.

Lemma 4. Denote by 𝑥 the state vector of the usual system in Figure 1(b) and define the signals as follows: 𝜉∶=𝑥^𝑇−𝑥^𝑇𝑥^𝑇_𝑄^𝑇,𝑧_𝑝∶=𝑧−𝑧^∗. (10) For the cost function 𝐽_𝐻𝑆(𝑄_𝑑), the difference between 𝑧(𝑘ℎ+𝜃,𝑥₀,𝑄_𝑑(𝑢)) and 𝑧^∗(𝑘ℎ+𝜃,𝑥₀,𝑢) for 𝑘=0,1,2,…, 𝜃∈[0,ℎ) is given by the following system: 𝜉[][𝑘][𝑘]+Σ∶𝑘+1=𝓐𝜉+𝓑𝑒_ℎ0𝑒^{𝐴(ℎ−𝜏)}𝐵𝑢(𝑘ℎ+𝜏)d𝜏_ℎ0𝑒^{𝐴𝑄(ℎ−𝜏)}𝐵_𝑄𝑢()𝑧𝑘ℎ+𝜏d𝜏_𝑝(𝑘ℎ+𝜃)=𝓒(𝜃)𝜉[𝑘]+𝓓(𝜃)𝑒[𝑘]+𝐶^𝜃₀𝑒^{𝐴(ℎ−𝜏)}𝐵𝑢(𝑘ℎ+𝜏)d𝜏, (11) where 𝜉[0]=0, 𝑢(𝑘ℎ+𝜏)∶=𝑢[𝑘]−𝑢(𝑘ℎ+𝜏); the matrices 𝓐, 𝓑, 𝓒(𝜃), and 𝓓(𝜃) are defined as follows: 𝑒𝓐∶=^𝐴ℎℎ0𝑒^𝐴𝜏d𝜏𝐵𝐶_𝑄0𝑒^𝐴𝑄ℎ+^ℎ₀𝑒^𝐴𝑄𝜏d𝜏𝐵_𝑄𝐶_𝑄,𝓑∶=^ℎ0𝑒^𝐴𝜏d𝜏𝐵_ℎ0𝑒^𝐴𝑄𝜏d𝜏𝐵_𝑄,𝓒(𝜃)∶=𝐶𝑒^𝐴𝜃𝐶^𝜃₀𝑒^𝐴𝜏d𝜏𝐵𝐶_𝑄,𝓓(𝜃)∶=𝐶^𝜃₀𝑒^𝐴𝜏d𝜏. (12)

Proof. See Appendix A.

We focus on 𝑢(𝑘ℎ+𝜏) of Σ. For the operator 𝐻𝑆 and the signal 𝑢∈ℒ^𝑚_∞, lim_ℎ→0‖[(𝐼−𝐻𝑆)𝑢](𝑡)‖≠0 (13) holds. This implies that we cannot ignore the temporal resolution constraint on the cost function 𝐽_𝐻𝑆(𝑄_𝑑) even if ℎ→0. On the other hand, low-pass prefiltering rectifies this situation [25]. In fact, for the stable LTI system 𝐹, lim_ℎ→0)𝑡𝐴‖[(𝐼−𝐻𝑆𝐹𝑢]()‖=0,𝐹∶=^𝐹𝐵_𝐹𝐶_𝐹0 (14) holds. For the evaluation of the cost function 𝐽_𝐻𝑆(𝑄_𝑑), then this paper utilizes 𝑢=𝐻𝑆𝐹𝑟,𝑟∈ℒ^𝑚_∞ (15) as the exogenous input. Note that 𝑢∈ℒ^𝑚_∞ if stable 𝐹 is strictly proper and 𝑟∈ℒ^𝑚_∞. For the signal (15), 𝑢[𝑘]=𝑢(𝑘ℎ+𝜃)(𝑘=0,1,2,…,𝜃∈[0,ℎ)) holds, so the terms of 𝑢(𝑘ℎ+𝜏) in (11) are eliminated. Then, Σ is rewritten as Σ_𝐻𝑆∶𝜉[][𝑘][𝑘],𝑧𝑘+1=𝓐𝜉+𝓑𝑒_𝑝(𝑘ℎ+𝜃)=𝓒(𝜃)𝜉[𝑘]+𝓓(𝜃)𝑒[𝑘]. (16) Also, this paper solves the following synthesis problem (𝐄^′): for the system  Σ_𝑄 composed of 𝑃 and 𝑄_𝑑  with the initial state  𝑥₀∈ℝ^𝑛  and the exogenous input  𝑢∈ℒ^𝑚_∞  in (15), suppose that the quantization interval  𝑑∈ℝ₊, the switching speed  ℎ∈ℝ₊, and the performance level  𝛾∈ℝ₊  are given. Characterize a continuous-time dynamic quantizer  𝑄_𝑑 (i.e., find parameters  (𝑛_𝑄,𝐴_𝑄,𝐵_𝑄,𝐶_𝑄)) achieving  𝐽_𝐻𝑆(𝑄_𝑑)≤𝛾.

3.2. Quantizer Analysis

The quantization error 𝑒 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 𝜉̃[𝑘+1]=𝒜𝜉̃[𝑘]+ℬ𝑤[𝑘], (17) where 𝜉̃∈ℝ^𝑛𝜉 and 𝑤∈ℝ^𝑚 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 ℝ_∞∶=𝜉̃∈ℝ^𝑛𝜉∃𝑘∈ℕ₊[⋅]𝜉̃[𝑘]=∃𝑤∈𝕎,_𝑘−1𝑖∑₌₀𝒜^{𝑘−1−𝑖}[𝑖],ℬ𝑤𝕎∶=𝑤∈ℝ^𝑚∶𝑤^𝑇𝑤≤1. (18)

Definition 6. Define the invariant set of the system (17) to be a set 𝓍 which satisfies 𝜉̃∈𝓍,𝑤∈𝕎⟹𝒜𝜉̃+ℬ𝑤∈𝓍. (19)

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^𝑇∈ℝ^{𝑛𝜉×𝑛𝜉}, the ellipsoid 𝔼(𝒫)∶={𝜉̃∈ℝ^𝑛𝜉∶𝜉̃^𝑇𝒫𝜉̃≤1} is an invariant set if and only if there exists a scalar 𝛼∈[0,1−𝜌(𝒜)²] satisfying 𝒜^𝑇𝒫𝒜−(1−𝛼)𝒫𝒜^𝑇ℬ𝒫ℬ_𝑇𝒫𝒜ℬ^𝑇𝒫ℬ−𝛼𝐼_𝑚≤0. (20)

Note that the ellipsoidal set 𝔼(𝒫) covers the reachable set ℝ_∞ from outside. Define the set ℰ∶={𝑤∈ℝ^𝑚∶𝑒=(√𝑚𝑑/2)𝑤satisfies(7)} and rewrite the system (16) as Σ_𝐻𝑆∶𝜉̃[]𝜉̃[𝑘]𝑤[𝑘],𝑧𝑘+1=𝓐+𝓑_𝑝(𝑘ℎ+𝜃)=𝓒(𝜃)𝜉̃[𝑘]+𝓓(𝜃)𝑤[𝑘], (21) where 𝜉=(√𝑚𝑑/2)𝜉̃, 𝑧_𝑝=(√𝑚𝑑/2)𝑧_𝑝, 𝑒=(√𝑚𝑑/2)𝑤, and 𝑤∈ℰ. The left multiplication of Σ_𝐻𝑆 with √𝑚𝑑/2 leads to Σ_𝐻𝑆. The relation ℰ⊆𝕎 clearly holds because 𝑒^𝑇𝑒≤𝑚𝑑²/4 and the set 𝕎 is an independent bounded disturbance without the relation (7). That is, the reachable set of Σ_𝐻𝑆 with 𝑤∈ℰ is no larger than that of Σ_𝐻𝑆 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 𝐽_𝐻𝑆(𝑄_𝑑) by substituting 𝒜=𝓐 and ℬ=𝓑 into (20). Move on to the matrix exponential 𝑒^𝐴𝜃 of 𝓒(𝜃) and 𝓓(𝜃) in (21), which is rewritten as 𝑒^𝐴𝜃=𝐼+Ω(𝜃)𝐴,Ω(𝜃)∶=^𝜃₀𝑒^𝐴𝜏d𝜏. (22) Along with this, 𝑧_𝑝 of (21) is also rewritten as 𝑧_𝑝()=(𝜃)(𝜃)𝑘ℎ+𝜃𝐶+𝐶Ω𝐴𝐶Ω𝐵𝐶_𝑄𝜉̃[𝑘](𝜃)𝑤[𝑘]=(𝜃)𝓓𝜉̃[𝑘](𝜃)𝑤[𝑘],,+𝐶Ω𝓒+𝐶Ω+𝐶Ω𝓒∶=𝐶0𝓓∶=𝐴𝐵𝐶_𝑄. (23) In addition, from the properties of ℝ_∞ and 𝔼(𝒫), 𝐽_𝐻𝑆𝑄_𝑑≤sup_{[)𝜃∈0,ℎ}𝜉̃sup_∈ℝ∞,𝑤_∈ℰ𝓒(𝜃)̃(𝜃)𝑤√𝜉+𝓓𝑚𝑑2≤sup_{[)𝜃∈0,ℎ}𝜉̃sup_∈ℝ∞,𝑤_∈𝕎𝓒(𝜃)̃(𝜃)𝑤√𝜉+𝓓𝑚𝑑2(∵ℰ⊆𝕎)≤sup_{[)𝜃∈0,ℎ}𝜉̃sup_{(𝒫),∈𝔼}𝑤_∈𝕎𝓒(𝜃)̃(𝜃)𝑤√𝜉+𝓓𝑚𝑑2∵ℝ_∞⊆𝔼(𝒫)≤sup_{[)𝜃∈0,ℎ}𝜉̃sup_{∈𝔼(𝒫)}(𝓒+𝐶Ω(𝜃)𝓓)_𝛾1𝜉̃√𝑚𝑑2+sup_{[)𝜃∈0,ℎ}𝑤sup_∈𝕎‖𝐶Ω_𝛾2(𝜃)𝑤‖√𝑚𝑑2 (24) holds. Similarly to our previous papers [13, 15, 16], by using the ℒ₁ control technique in [21], we provide the sufficient conditions for computing 𝛾₁∈ℝ₊ and 𝛾₂∈ℝ₊ of (24) as follows: 𝒫𝓒^𝑇+𝓓^𝑇Ω(𝜃)^𝑇𝐶^𝑇𝓒+𝐶Ω(𝜃)𝓓𝛾²₁𝐼_𝑞Ω(𝜃)≥0,_𝑇𝐶^𝑇𝐶Ω(𝜃)≤𝛾²₂𝐼_𝑛,∀𝜃∈[0,ℎ). (25)

Remark 8. For the inequalities (25) and any vectors 𝜉̃∈ℝ^𝑛 and 𝑤∈ℝ^𝑚, we have 𝜉̃^𝑇𝒫𝜉̃𝜉̃^𝑇𝓒^𝑇+𝓓^𝑇Ω(𝜃)^𝑇𝐶^𝑇𝓒+𝐶Ω(𝜃)𝓓𝜉̃𝛾²₁𝐼_𝑞⟺𝜉̃≥0_𝑇𝓒+𝐶Ω(𝜃)𝓓^𝑇𝓒+𝐶Ω(𝜃)𝓓𝜉̃≤𝛾²₁𝜉̃^𝑇𝒫𝜉̃∵Schurcomplement (26) and 𝑤^𝑇Ω(𝜃)^𝑇𝐶^𝑇𝐶Ω(𝜃)𝑤≤𝛾²₂𝑤^𝑇𝑤. Then, we see that (24) holds if 𝜉̃∈𝔼(𝒫) and 𝑤∈𝕎.

The inequalities (25) are difficult to test since we need to find 𝒫, 𝛾₁, and 𝛾₂ satisfying (20) and (25) for infinitely many values of 𝜃∈[0,ℎ). Then, using the matrix uncertainty technique [23, 24], we consider their sufficient conditions, which are easy to compute. Considering Ω(𝜃) in (22) as a matrix uncertainty, we introduce the following lemma regarding the matrix exponential [26, 27].

Lemma 9. For the matrix Ω(𝜃) in (22), 𝜎_max(Ω(𝜃))≤𝛿(𝜃)≤𝛿(ℎ),∀𝜃∈[0,ℎ), (27) holds where 𝛿(𝜃)𝑒∶=^{𝜇(𝐴)𝜃}−1𝜇(𝐴)(𝐴)(𝐴)𝜇(𝐴),𝜇≠0,𝜃,𝜇=0,∶=max𝜆∶𝜆∈eig𝐴+𝐴^𝑇2. (28)

Proof. Since 𝜎_max(𝑒^𝐴𝜃)≤𝑒^{𝜇(𝐴)𝜃} (see [26]), 𝜎_max(Ω(𝜃))≤^𝜃₀𝜎_max𝑒^𝐴𝛽𝑑𝛽≤^𝜃₀𝑒^{𝜇(𝐴)𝛽}𝑑𝛽 (29) holds.

By using Lemma 9 and the 𝑆-procedure [23, 28, 29], the sufficient condition analyzing the cost function 𝐽_𝐻𝑆(𝑄_𝑑) of the system Σ_𝑄 can be expressed in terms of matrix inequality as summarized in the following theorem.

Theorem 10. Consider the system Σ_𝑄 composed of 𝑃 and 𝑄_𝑑 with the initial state 𝑥₀∈ℝ^𝑛 and the exogenous input 𝑢∈ℒ^𝑚_∞ in (15). For the quantization interval 𝑑∈ℝ₊ and the switching speed ℎ∈ℝ₊, the upper bound of the cost function 𝐽_𝐻𝑆(𝑄_𝑑) is given by 𝐽_𝐻𝑆𝑄_𝑑(ℎ)(ℎ)≤𝛾𝑠.𝑡.𝛾∶=𝛾+𝜎_max(𝐶)𝛿(ℎ)√𝑚𝑑2 (30) if there exist 0<𝒬=𝒬^𝑇∈ℝ^{(𝑛+𝑛𝑄)×(𝑛+𝑛𝑄)}, 0<𝑆=𝑆^𝑇∈ℝ^𝑛×𝑛, 𝛼_ℎ∈[0,1/2ℎ−𝜌(𝓐)²/2ℎ] and 𝛾∈ℝ₊ satisfying Φ_ℎ𝒬+𝒬Φ^𝑇_ℎ+2𝛼_ℎ𝒬Γ_ℎ√ℎ𝒬Φ^𝑇_ℎΓ^𝑇_ℎ−2𝛼_ℎ𝐼_𝑚√ℎΓ^𝑇_ℎ√ℎΦ_ℎ𝒬√ℎΓ_ℎ𝓒−𝒬≤0,𝒬𝒬^𝑇𝛿(ℎ)𝒬𝓓^𝑇𝓒𝒬𝛾²𝐼_𝑞−𝛿(ℎ)𝐶𝑆𝐶^𝑇0𝛿(ℎ)(𝜃)(𝜃)[),𝓓𝒬0𝑆≥0,𝑆Ω=Ω𝑆,∀𝜃∈0,ℎ (31)(32) where the matrices Φ_ℎ and Γ_ℎ are defined by Φ_ℎ1∶=ℎ^ℎ₀𝑒^𝐴𝜏1d𝜏𝐴ℎ^ℎ₀𝑒^𝐴𝜏d𝜏𝐵𝐶_𝑄01ℎ^ℎ₀𝑒^𝐴𝑄𝜏d𝜏𝐴_𝑄+𝐵_𝑄𝐶_𝑄,Γ_ℎ1∶=ℎ^ℎ₀𝑒^𝐴𝜏1d𝜏𝐵ℎ^ℎ₀𝑒^𝐴𝑄𝜏d𝜏𝐵_𝑄. (33)

Proof. See Appendix A.

Denote by Σ the system Σ without operator 𝐻𝑆. In this case, the system Σ is given by Σ∶𝜉̇(𝑡)=𝓐𝜉(𝑡)+𝓑𝑒(𝑡),𝑧_𝑝(𝑡)=𝓒𝜉(𝑡), (34) where 𝓐∶=𝐴𝐵𝐶_𝑄0𝐴_𝑄+𝐵_𝑄𝐶_𝑄,𝐵𝐵𝓑∶=_𝑄. (35) Regarding the definition of 𝜉(𝑡), 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)∫^ℎ₀𝑒^𝐴ℎd𝜏→𝐼 and (1/ℎ)∫^ℎ₀𝑒^𝐴𝑄ℎd𝜏→𝐼 hold, so Φ_ℎ→𝓐 and Γ_ℎ→𝓑 hold. In the same limit, from 𝛿(ℎ)→0, conditions (31) and (32) converge to the analysis conditions of the continuous-time dynamic quantizer for the system Σ in [15, 16]. On the other hand, for a small ℎ, 𝑒^𝐴ℎ→𝐼, 𝑒^𝐴𝑄ℎ→𝐼, ∫^ℎ₀𝑒^𝐴ℎd𝜏→0 and ∫^ℎ₀𝑒^𝐴𝑄ℎd𝜏→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 𝑆 such that 𝑆Ω(𝜃)=Ω(𝜃)𝑆 holds. For example, we can set 𝑆=𝑠_𝜃𝐼_𝑛, 𝑠_𝜃∈ℝ₊ and this setting leads to the following optimization problem (Aop):min_{𝒬=𝒬𝑇>0,𝑆=𝑠𝜃𝐼𝑛(𝓐)>0,1/2ℎ−𝜌2/2ℎ≥𝛼ℎ≥0,𝛾>0}𝛾²s.t.(31)and(32). (36)When scalar 𝛼_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 𝛼_ℎ, we can obtain an upper bound of 𝐽_𝐻𝑆(𝑄_𝑑).

3.3. Quantizer Synthesis

The problem (Aop) suggests that the quantizer synthesis problem (𝐄^′) is reduced to the following nonconvex optimization problem (OP): min_{𝒬=𝒬𝑇>0,𝑆=𝑠𝜃𝐼𝑛>0,𝐴𝑄,𝐵𝑄,𝐶𝑄(𝓐),1/2ℎ−𝜌2/2ℎ≥𝛼ℎ≥0,𝛾>0}𝛾²s.t.(31)and(32). (37) That is, if (OP) is feasible, (𝐄^′) is feasible.

From the matrix product such as (1/ℎ)∫^ℎ₀𝑒^𝐴𝑄ℎd𝜏 and 𝐴_𝑄+𝐵_𝑄𝐶_𝑄 in (31), the synthesis condition is difficult to derive from Theorem 10 unlike the continuous-time case without the operator 𝐻𝑆 in [15, 16]. Thus, we fixed the parameters as follows: 𝑛_𝑄=𝑛,𝐴_𝑄=𝐴,𝐵_𝑄=𝐵. (38) The structure (38) does not severely limit the synthesis because 𝐴_𝑄 and 𝐵_𝑄 of the continuous-time dynamic quantizer for the system Σ in [15, 16] are also (38). See Appendix B. In other words, 𝑥̇_𝑄=𝐴𝑥_𝑄+𝐵𝑒_𝑄 estimates the quantization influence on the system 𝑃. Along with this, we fix 𝒬 of (31) as follows: 𝒬=𝑌𝑉𝑉𝑉,𝑌=𝑌^𝑇>0,𝑉=𝑉^𝑇>0. (39) 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 𝑛_𝑄=𝑛 [30].

Under some circumstances (38) and (39), we obtain the following synthesis condition.

Theorem 11. Consider the system Σ_𝑄 composed of 𝑃 and 𝑄_𝑑 with the initial state 𝑥₀∈ℝ^𝑛 and the exogenous input 𝑢∈ℒ^𝑚_∞ in (15). Suppose that the quantization interval 𝑑∈ℝ₊, the switching speed ℎ∈ℝ₊, and the performance level 𝛾∈ℝ₊ are given. For a scalar 𝛼_ℎ∈[0,1/2ℎ], there exist a continuous-time dynamic quantizer 𝑄_𝑑 achieving (30) if one of the following equivalent statements holds.(i)There exist matrices 0<𝒬=𝒬^𝑇∈ℝ^{(𝑛+𝑛𝑄)×(𝑛+𝑛𝑄)}, 0<𝑆=𝑆^𝑇∈ℝ^𝑛×𝑛 and a dynamic quantizer 𝑄_𝑑 satisfying (31), (32), and (38).(ii)There exist matrices 0<𝑌=𝑌^𝑇∈ℝ^𝑛×𝑛, 0<𝑉=𝑉^𝑇∈ℝ^𝑛×𝑛, 𝑊∈ℝ^𝑚×𝑛, 0<𝑆=𝑆^𝑇∈ℝ^𝑛×𝑛 satisfying Θ_𝐴ℎ+Θ^𝑇_𝐴ℎ+2𝛼_ℎΘ_𝑃Θ_𝐵ℎ√ℎΘ^𝑇_𝐴ℎΘ^𝑇_𝐵ℎ−2𝛼_ℎ𝐼_𝑚√ℎΘ^𝑇_𝐵ℎ√ℎΘ_𝐴ℎ√ℎΘ_𝐵ℎ−Θ_𝑃Θ≤0,_𝑃Θ^𝑇_𝐶Θ^𝑇_𝐷ℎΘ_𝐶𝛾²𝐼_𝑞−𝛿(ℎ)𝐶𝑆𝐶^𝑇0Θ_𝐷ℎ(𝜃)(𝜃)[),0𝑆≥0,𝑆Ω=Ω𝑆,∀𝜃∈0,ℎ (40)(41)where Θ_𝑃∶=𝑌𝑉𝑉𝑉,Ψ_ℎ1∶=ℎ^ℎ₀𝑒^𝐴𝜏Θd𝜏,_𝐴ℎΨ∶=^ℎ(𝐴𝑌+𝐵𝑊)Ψ_ℎ()Ψ𝐴𝑉+𝐵𝑊_ℎ(𝐴𝑉+𝐵𝑊)Ψ_ℎ(),Θ𝐴𝑉+𝐵𝑊_𝐵ℎΨ∶=^ℎ𝐵Ψ_ℎ𝐵,Θ_𝐶,Θ∶=𝐶𝑌𝐶𝑉_𝐷ℎ∶=𝛿(ℎ)(𝐴𝑌+𝐵𝑊)𝛿(ℎ)(𝐴𝑉+𝐵𝑊). (42) In this case, such a quantizer parameter is given by 𝑛_𝑄=𝑛,𝐴_𝑄=𝐴,𝐵_𝑄=𝐵,𝐶_𝑄=𝑊𝑌⁻¹. (43)

Proof. We fix 𝒬 as shown in (39) and introduce the change of variables 𝑊=𝐶_𝑄𝑌. Hence, (31) and (32) result in (40) and (41). Also, designing 𝐶_𝑄 yields 𝛼_ℎ∈[0,1/2ℎ] because 𝜌(𝓐) is determined by 𝐶_𝑄 and [0,1/2ℎ−𝜌(𝓐)²/2ℎ]⊆[0,1/2ℎ].

In the limit of ℎ→0; Ψ_ℎ converges to 𝐼 and 𝛿(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 𝑆=𝑠_𝜃𝐼_𝑛 for Theorem 11, the quantizer synthesis problem (𝐄^′) is reduced to the following optimization problem (Sop):min_{𝑌=𝑌>0𝑇,𝑉=𝑉𝑇>0,𝑆=𝑠𝜃𝐼𝑛>0,𝑊,1/2ℎ≥𝛼ℎ≥0,𝛾>0}𝛾²s.t.(40)and(41). (44)If (Sop) is feasible, (𝐄^′) is feasible. Therefore, a continuous-time dynamic quantizer considering both spatial and temporal resolution constraints is obtained from Theorem 11.

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.

4. Discussion

For the slow switching, we compare the proposed method and existing continuous-time quantizer [15, 16]. Consider the system Σ_𝑄. The plant 𝑃 is the stable minimum phase LTI system: 𝑥̇(𝑡)𝑧(𝑡)=𝑥(𝑡)𝑣(𝑡)−3300−22110. (45) In the case without the operator 𝐻𝑆, an optimal form of the continuous-time quantizer 𝑄_𝑑^𝑜𝑝 [15, 16] is given by (B.2). See Appendix B. The continuous-time quantizer 𝑄_𝑑^𝑜𝑝 and its performance are parameterized by the free parameter 𝑓∈ℝ₊. For the simulation, we consider a two-step design for 𝑄_𝑑^𝑜𝑝; we first set 𝑓 and second insert the operator 𝐻𝑆 in the obtained 𝑄_𝑑^𝑜𝑝. Also, the achievable performance of 𝐽_𝐻𝑆(𝑄_𝑑^𝑜𝑝) is calculated by (Aop).

For the comparison, we set the switching speed ℎ=0.01 [s] and the quantization interval 𝑑=2. First, we set 𝑓=50 and then obtain 𝑄_𝑑^𝑜𝑝 with 𝛾_𝑐=0.102. Also, 𝛾(ℎ)=0.219 for 𝐽_𝐻𝑆(𝑄_𝑑^𝑜𝑝) is obtained from (Aop). Second, we solve the problem (Sop) and obtain 𝛾(ℎ)=0.219 and the matrix 𝐶_𝑄=−19.26−22.72. In this case, both quantizers can approximate well. Figures 4 and 5 illustrate the simulation results of the time responses of Σ_𝑄 with the proposed quantizer and the quantizers 𝑄_𝑑^𝑜𝑝 in (B.2). The initial state 𝑥₀=[00]^𝑇 and the input 𝑢(𝑡)=sin𝜋𝑡+cos0.7𝜋𝑡 are given. In Figures 4 and 5, the thin lines and the thick lines are for the conventional system in Figure 1(b) and the system Σ_𝑄 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.

(a)

(b)

Figure 4 

Time responses of Σ𝑄 with the proposed quantizer for ℎ=0.01.

(a)

(b)

Figure 5 

Time responses of Σ𝑄 with (B.2) for ℎ=0.01.

Next, we consider the case ℎ=0.1. In this case, the two controlled outputs approximated by the two quantizers differ. (Aop) for 𝑄_𝑑^𝑜𝑝 is infeasible. From (Sop), on the other hand, we obtain 𝛾(ℎ)=0.949 and the matrix 𝐶_𝑄=−1.1321−3.097. Figures 6 and 7 illustrate the simulation results on the time responses of Σ_𝑄 with (B.2) and the proposed quantizer in the same fashion. We see that 𝑧(𝑡) of the usual plant 𝑃 is approximated by 𝑧(𝑡) of the system Σ_𝑄 with the proposed quantizer, while 𝑧(𝑡) of the system Σ_𝑄 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 𝑄_𝑑^𝑜𝑝 is applicable to the given switching speed setting.

(a)

(b)

Figure 6 

Time responses of Σ𝑄 with (B.2) for ℎ=0.1.

(a)

(b)

Figure 7 

Time responses of Σ𝑄 with the proposed quantizer for ℎ=0.1.

Remark 13. In the above numerical experiments, the proposed quantizer is designed and the quantizer 𝑄_𝑑^𝑜𝑝 is analyzed for 𝑢=𝐻𝑆𝐹(sin𝜋𝑡+cos0.7𝜋𝑡), while the time responses of the quantizers are simulated for 𝑢=sin𝜋𝑡+cos0.7𝜋𝑡. 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.

Here, we focus on the eigenvalues of 𝓐 for the system Σ with 𝑄_𝑑^𝑜𝑝. The eigenvalues for 𝑓=50 and ℎ=0.1 are {0.741,0.819,0.550,−4.21} and then 𝓐 is unstable in the discrete-time domain. From Theorem 10, (Aop) is infeasible if 𝜌(𝓐) is bigger than 1 (in other words, 𝓐 is unstable). {0.741,0.819} are the eigenvalues of 𝑒^𝐴ℎ. That is, 𝑒^𝐴𝑄ℎ+∫^ℎ₀𝑒^𝐴𝑄𝜏d𝜏𝐵_𝑄𝐶_𝑄(=∶𝐴_𝑄(ℎ)) for 𝑄_𝑑^𝑜𝑝 is unstable. Then, we consider the case in which 𝑓=3 and ℎ=0.1 (𝛾_𝑐=0.707) such that 𝐴_𝑄(ℎ) is stable. The corresponding eigenvalues are {0.637,0.354}. In this case, 𝛾(ℎ)=1.085 for 𝐽_𝐻𝑆(𝑄_𝑑^𝑜𝑝) is obtained from (Aop).

From the above results, the existing continuous-quantizer in [15, 16] may be suitable for a two-step design such that 𝐴_𝑄(ℎ) is stable via the parameter 𝑓. In terms of the upper bound of cost function (B.3), first, let us consider the problem (P-1): maximize 𝑓 for 𝑄_𝑑^𝑜𝑝 such that ∃𝑋^𝑇=𝑋>0s.t.𝐴_𝑄(ℎ)^𝑇𝑋𝐴_𝑄(ℎ)−𝑋<0, (46) where the parameters (𝐴_𝑄,𝐵_𝑄,𝐶_𝑄) of 𝐴_𝑄(ℎ) are given by (B.2). This problem is LMI for the line search of 𝑓. For ℎ=0.1, its solution is 𝑓=18.9 (𝛾_𝑐=0.173). However, 𝛾(ℎ)=113.148 for 𝐽_𝐻𝑆(𝑄_𝑑^𝑜𝑝) is obtained from (Aop). By using Theorem 10, next, let us consider the problem (P-2): min_{𝒬=𝒬𝑇>0,𝑆=𝑠𝜃𝐼𝑛(𝓐)>0,1/2ℎ−(𝜌2)/2ℎ≥𝛼ℎ≥0,𝑓>0,𝛾>0}𝛾²s.t.(31)and(32), (47) where the parameters (𝐴_𝑄,𝐵_𝑄,𝐶_𝑄) of Ψ_ℎ and Γ_ℎ are given by (B.2). This problem is LMI for the plane search of 𝑓 and 𝛼_ℎ. For ℎ=0.1, its solution is 𝑓=5.715 (𝛾_𝑐=0.397) and then 𝛾(ℎ)=0.949 for 𝐽_𝐻𝑆(𝑄_𝑑^𝑜𝑝) 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 𝑓 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.

5. 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 [11–14] 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 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 discrete-time 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 [6–10]. 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.

Appendix

A. Proof

The proof of Lemma 4 is as follows.

Proof. 𝑧(𝑘ℎ+𝜃) for 𝑘=0,1,2,…, 𝜃∈[0,ℎ) (which is the behavior of 𝑧(𝑡) over the 𝑘th sampling interval) is given by the discretized system of 𝑃: 𝑥[𝑘+1]=𝑒^𝐴ℎ𝑥[𝑘]+^ℎ₀𝑒^{𝐴(ℎ−𝜏)}()𝑧()𝐵𝑣𝑘ℎ+𝜏d𝜏𝑘ℎ+𝜃=𝐶𝑒^𝐴𝜃𝑥[𝑘]+𝐶^𝜃₀𝑒^{𝐴(𝜃−𝜏)}𝐵𝑣(𝑘ℎ+𝜏)d𝜏, (A.1) where 𝑥[𝑘]=𝑥(𝑘ℎ). 𝑣(𝑡) is given by (9) and 𝑣_𝑄[𝑘] is given by the discretized system of 𝑄: 𝑥_𝑄[]=𝑒𝑘+1_𝐴𝑄ℎ+^ℎ₀𝑒^𝐴𝑄𝜏d𝜏𝐵_𝑄𝐶_𝑄𝑥_𝑄[𝑘]+^ℎ₀𝑒^𝐴𝑄𝜏d𝜏𝐵_𝑄𝑒[𝑘]+^ℎ₀𝑒^{𝐴𝑄(ℎ−𝜏)}𝐵_𝑄𝑢()𝑣𝑘ℎ+𝜏d𝜏,_𝑄[𝑘]=𝐶_𝑄𝑥_𝑄[𝑘], (A.2) where 𝑥_𝑄[𝑘]=𝑥_𝑄(𝑘ℎ). This is because 𝑒_𝑄 is given by 𝑒_𝑄=𝑣_𝑄̂+𝑒+𝑢−𝑢,𝑣_𝑄̂=𝐻𝑆𝑣_𝑄,𝑢=𝐻𝑆𝑢. (A.3) Then, 𝑧(𝑘ℎ+𝜃,𝑥₀,𝑄_𝑑(𝑢)) for 𝑘=0,1,2,…, 𝜃∈[0,ℎ) is expressed by the discretized system of Σ_𝑄: Σ^𝑑_𝑄∶𝑥[]𝑥𝑘+1_𝑄[]𝑥[𝑘]𝑥𝑘+1=𝓐_𝑄[𝑘][𝑘]++𝓑𝑒_ℎ0𝑒^{𝐴(ℎ−𝜏)}𝐵𝑢[𝑘]d𝜏_ℎ0𝑒^{𝐴𝑄(ℎ−𝜏)}𝐵_𝑄𝑢()𝑧()(𝜃𝑥[𝑘]𝑥𝑘ℎ+𝜏d𝜏𝑘ℎ+𝜃=𝓒)_𝑄[𝑘]+𝓓(𝜃)𝑒[𝑘]+𝐶^𝜃₀𝑒^{𝐴(ℎ−𝜏)}𝐵𝑢[𝑘]d𝜏. (A.4) Also, 𝑧^∗(𝑘ℎ+𝜃,𝑥₀,𝑢) for 𝑘=0,1,2,…, 𝜃∈[0,ℎ) without the quantizer is given by 𝑃^∗_𝑑∶𝑥[𝑘+1]=𝑒^𝐴ℎ𝑥[𝑘]+^ℎ₀𝑒^{𝐴(ℎ−𝜏)}()𝑧𝐵𝑢𝑘ℎ+𝜏d𝜏_∗(𝑘ℎ+𝜃)=𝐶𝑒^𝐴𝜃𝑥[𝑘]+𝐶^𝜃₀𝑒^{𝐴(ℎ−𝜏)}𝑢(𝑘ℎ+𝜏)d𝜏, (A.5) where 𝑥(0)=𝑥₀ and 𝑥[𝑘]=𝑥(𝑘ℎ). From Σ^𝑑_𝑄 and 𝑃^∗_𝑑, we obtain the system Σ in (11).

For the proof of Theorem 10, we use the 𝑆-procedure [23, 28, 29].

Lemma A.1. For the real matrices 𝐿, 𝑀, and 𝑁 of appropriate size, the inequality 𝐿+𝑀Δ𝑁+𝑁^𝑇Δ^𝑇𝑀^𝑇≥0 (A.6) holds for any matrix Δ such that 𝜎_max(Δ)≤𝛿 if and only if there exists a matrix 𝑆=𝑆^𝑇>0 such that 𝐿−𝛿𝑀𝑆𝑀^𝑇√𝛿𝑁^𝑇√𝛿𝑁𝑆≥0,𝑆Δ=Δ𝑆. (A.7)

Then, the proof of Theorem 10 is as follows.

Proof. We use Lemmas 9 and A.1 with 𝒫𝓒𝐿=^𝑇𝓒𝛾²𝐼_𝑞0𝐶,(𝜃).,𝑀=𝑁=𝓓0,Δ=Ω (A.8) In this case, for the inequalities (25), we obtain their sufficient conditions as follows: 𝛾₁𝒫𝓒≤𝛾s.t.^𝑇𝛿(ℎ)𝓓^𝑇𝓒𝛾²𝐼_𝑞−𝛿(ℎ)𝐶𝑆𝐶^𝑇0𝛿(ℎ)(𝜃)(𝜃)[)𝛾𝓓0𝑆𝑆Ω=Ω𝑆,∀𝜃∈0,ℎ≥0,₂≤𝜎_max(𝐶)𝛿(ℎ)s.t.Ω(𝜃)^𝑇𝐶^𝑇𝐶Ω(𝜃)≤𝜎_max(𝐶)²𝛿(ℎ)²𝐼_𝑛,∀𝜃∈[0,ℎ). (A.9) Then, the upper bound of 𝐽_𝐻𝑆(𝑄_𝑑) is given by (30). By substituting 𝓐=𝐼+ℎΦ_ℎ and 𝓑=ℎΓ_ℎ into (20), we obtain 𝓐^𝑇𝒫𝓐−(1−𝛼)𝒫𝓐^𝑇𝓑𝒫𝓑_𝑇𝒫𝓐𝓑^𝑇𝒫𝓑−𝛼𝐼_𝑚=𝐼+ℎΦ_ℎ^𝑇𝒫𝐼+ℎΦ_ℎ−(1−𝛼)𝒫ℎ𝐼+ℎΦ_ℎ^𝑇𝒫Γ_ℎℎΓ^𝑇_ℎ𝒫𝐼+ℎΦ_ℎℎ²Γ^𝑇_ℎ𝒫Γ_ℎ−𝛼𝐼_𝑚Φ=ℎ^𝑇ℎ𝒫+𝒫Φ_ℎ+𝛼/ℎ𝒫𝒫Γ_ℎΓ^𝑇_ℎ𝒫−𝛼/ℎ𝐼_𝑚√+ℎℎΦ^𝑇_ℎ√ℎΓ^𝑇_ℎ𝒫√ℎΦ_ℎ√ℎΓ_ℎ≤0. (A.10) By Schur complement [29], (A.10) is equivalent to (31) where 𝒬=𝒫⁻¹ and 𝛼_ℎ=𝛼/2ℎ. Also, (A.9) is equivalent to (32) where 𝒬=𝒫⁻¹.

B. Continuous-Time Dynamic Quantizer [15, 16]

For the system Σ in (34) without the operator 𝐻𝑆, we consider the following non-convex optimization (𝐎𝐏^′):min_𝒫>0,𝜇(𝓐_{)≥𝛼≥0,𝐴𝑄,𝐵𝑄,𝐶𝑄,𝛾𝑐>0}𝛾_𝑐𝓐s.t.^𝑇𝓑𝓑𝒫+𝒫𝓐+2𝛼𝒫𝒫_𝑇𝒫−2𝛼𝐼_𝑚𝒫𝓒≤0,_𝑇𝓒𝛾²_𝑐𝐼_𝑞≥0. (B.1)The dynamic quantizer without the operator 𝐻𝑆 is obtained from (𝐎𝐏^′) and 𝐽(𝑄_𝑑)≤𝛾_𝑐 is achieved. Also, an optimal form of the continuous-time quantizer [15, 16] is given by 𝑄_𝑑^𝑜𝑝∶𝑥̇_𝑄=𝐴𝑥_𝑄()−()+𝐵𝑣−𝑢𝑣=𝑞𝐶𝐵⁻¹𝐶𝐴+𝑓𝐼𝑥_𝑄+𝑢, (B.2) where its achievable upperbound of 𝐽(𝑄^o𝑝_𝑑) is characterized by inf𝛾_𝑐=𝑑√𝑚4𝑔𝑔𝜎𝑓−^max(),𝜈(𝐴)()𝐶𝐵𝑔=max,𝜈𝐴−𝐵𝐶𝐵⁻¹𝐶𝐴+𝑓𝐼, (B.3) where 𝜈(𝐴)∶=max{Re(𝜆)∶𝜆∈eig(𝐴)}. The continuous-time quantizer 𝑄_𝑑^𝑜𝑝 and its performance are parameterized by the free parameter 𝑓∈ℝ₊. Note that the larger values of 𝑓 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.

Acknowledgments

The authors would like to thank the reviewers for their valuable comments. This work was partly supported by Grant-in-Aid for Young Scientists (B) no. 24760332 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.