Parameters Design for Logarithmic Quantizer Based on Zoom Strategy

This paper is concerned with the problem of designing suitable parameters for logarithmic quantizer such that the closed-loop system is asymptotic convergent. Based on zoom strategy, we propose two methods for quantizer parameters design, under which it ensures that the state of the closed-loop system can load in the invariant sets after some certain moments.Then we obtain that the quantizer is unsaturated, and thus the quantization errors are bounded under the time-varying logarithm quantization strategy. On that basis, we obtain that the closed-loop system is asymptotic convergent. A benchmark example is given to show the usefulness of the proposed methods, and the comparison results are illustrated.


Introduction
With the continuous improvement of the communication capability and reliability of the network, the data in the control system is transmitted through the network increasingly.The network bandwidth, although it may be very large, is always limited, so the data quantization is inevitable.At present, zoom strategy proposed in [1,2] is a popular method in quantized control system.
Based on zoom strategy, literature [3] analyzes the stabilization problem for linear systems with multidimensional state and one-dimensional input.The main contribution of that paper is the trade-off between the quantized controller complexity and the system performance.Literature [4] proposes a unified framework to describe both the network conditions and the state quantization of linear systems.A model describing the nonideal network conditions and input/output state quantization is given by [5], and the problem of the quantized output feedback controller is designed there to asymptotically stabilize the closed-loop system.Hereafter, zoom strategy is widely used in quantized feedback stabilization problem.
As for the system affected by data quantization and timedelay, literature [6] proves that under some conditions, the closed-loop system can be global asymptotically stabilized via a dynamic quantization strategy by integral mean value theorem.Literature [7] designs an optimal dynamic quantizer which is able to minimize the maximum output error between the quantized system and unquantized systems.Supposing that the quantizer is to be saturated, two types of quantizer are discussed in [8] such that the state of the closedloop system starting from a neighborhood of the origin exponentially converges to a bounded region.If the closed-loop system is affected by data quantization and packet dropout, we study the quantized stabilization of linear discrete-time systems and discrete-time and continuous-time fuzzy systems, respectively, in [9] and [10].Assuming that the system is relating to quantization and disturbance, the related results can be shown in [11][12][13].In literature [11], the authors propose two specific control strategies that yield the input-to-state stability of the closed-loop system based on quantized state.When the sampled state/output can only be obtained by the controller, literature [12] designs the full state feedback controller and the output feedback controller to stabilize the system.We generalize the results of [12] to the case that the system is also affected by packet dropout in [13].For the stabilization problem of the closed-loop system affected by quantization and saturation, literature [8,14] gets some results.
In summary, zoom strategy is a popular method to adjust the quantizer parameters, especially for uniform quantizer, such that the quantized control system is stable.As we know, logarithmic quantizer is another kind of commonly used quantizer [30,31].Compared with uniform quantizer, logarithmic quantizer has better performance near the origin.Consider this advantage, some papers adopt logarithmic quantization method to quantize the system state or output.Recent results include literature [32][33][34][35][36].However, most of the existing papers assume that the parameters of the logarithmic quantizer are given previously.Few articles study the parameters design for logarithmic quantizer, which is the main research of this paper.
In this paper, we will propose two design methods for the parameters of the logarithmic quantizer.By zoom strategy, we design the logarithmic quantizer parameters including V 0 , , and quantization density , which determine the saturation boundary, dead zone, and the quantization intervals of the quantizer.Under the parameters designed, we guarantee the unsaturation of the quantizer and the asymptotic convergence of the closed-loop system.
The rest of this paper is organized as follows.We describe the problem discussed here in Section 2. Two design methods for the parameters of the logarithmic quantizer are illustrated in Sections 3 and 4, respectively.A well-known benchmark example is adopted in Section 5 to show the effectiveness of the design methods, and their comparison is also illustrated there.Finally, some conclusions are given in Section 6.
Notation.R  denotes the -dimensional Euclidean space.R + and N denote the set of positive real numbers and positive integers, respectively.We denote by ‖ ⋅ ‖ the standard Euclidean norm in R  and the corresponding induced matrix norm in R × . max () and  min () denote the maximum and minimum eigenvalue of matrix , respectively. ⊤ ∈ R × denotes the transposed of matrix  ∈ R × .The signal ⌈⌉ indicates the least integer not less than .0 ×1 denotes zero vector with dimension  × 1.

Problem Formulation
This paper considers the following linear time-invariant discrete systems: where  ∈ R  is the state vector,  ∈ R  is the control input, and  and  are constant matrices with proper dimensions.
In the following, we assume that the open-loop system is unstable; that is, ‖‖ > 1.
Assuming that the network is located at the sensor side, thus the controller can only receive the quantized values of the system states due to the limited bandwidth.Hence the controller can be illustrated as where  is the feedback matrix to be designed and (⋅) is a logarithmic quantizer defined as Suppose that the quantization level of the quantizer is equal to 2 + 1,  ∈ N.For any  ∈ {1, 2, . . ., }, if  ∈ [  ,  +1 ),  ∈ N ∪ {0}, where   will be designed below, we define with  ∈ (0, 1) and V +1 = V  , V 0 > 0, in which the quantization density  is given by  = (1 − )/(1 + ) ∈ (0, 1).Obviously, , , and V 0 are the critical parameters of the logarithmic quantizer, which are assumed to be undetermined.The aim of this paper is to design the quantizer parameters , , and V 0 based on zoom strategy to ensure the unsaturation of the quantizer; that is, From which we can see that the quantization errors are bounded and the system state tends to zero when  tends to infinity according to the definition of V  .The main innovation of this paper is that the zoom strategy, which is always adopted to discuss the properties of the uniform quantizer, is used here to determine the parameters of the logarithmic quantizer.
In what follows, we will propose two methods for quantizer parameters design and compare them in simulation example.

First Method for Quantizer Parameters Design
If there exist a positive-definite matrix  and a feedback matrix  such that that is, the system can be stabilized by standard state feedback, then the asymptotic convergence of the closed-loop system can be guaranteed by the quantizer parameters designed in the following theorem.
Theorem 1.For any given matrices , , and Π defined by ( 5) and the quantization level 2 + 1, we select quantizer parameters  and  satisfying where  is an arbitrary given positive constant and Υ is defined by Υ = 2‖( + )‖/ min (Π).Given a constant  ∈ N, if  and  selected above satisfy the inequalities  ≥   and , where  0 and V 0 are determined by (9) below and  +1 fl   +  fl   + ⌈τ⌉ with Hence we have lim →∞ () = 0 ×1 ; that is, the closed-loop system is asymptotic convergent.
Remark 2. If the bandwidth of the network is large enough such that  ≥ ln / ln , then the condition  ≥   can always be guaranteed.Moreover, the inequality   < ‖‖ −1 must be ensured if  is selected large enough based on  < 1.
Comprehensively, if the bandwidth of the network and the constant  are set suitably, we can always find the quantizer parameters satisfying the conditions of the above theorem.
Stage 1 (zooming-out).In this stage, the system is as openmode; that is, ( + 1) = ().The aim of this stage is to determine the value of V 0 and a moment where the system state is unsaturated.
On the basis of that, we can ensure that Ṽ(), ∀ ∈ N ∪ {0}, is the quantized value of the logarithmic quantizer.

Stage 2 (zooming-in).
Let the zooming-out stage finish at the moment  0 and the system is transferred to closed-mode.The purpose of this stage is to design the quantizer parameters and   ,  ∈ N ∪ {0}, such that ‖()‖ ≤ (1/(1 − ))V  for any  ∈ [  ,  +1 ).To this end, we first illustrate that  0 is an invariant set.
Let Lyapunov function as () =  ⊤ ()(), then it is easy to get For an arbitrary given positive constant , if then we get Moreover, by (13) we know that the set B, defined by is an invariant set.Define B as and it is obvious that B ⊃ B. If  selected satisfies the following inequality: we can see that  0 ⊃ B ⊃ B, and thus  0 is an invariant set.

Second Method for Quantizer Parameters Design
If there are matrices  and  such that the following inequality holds then the asymptotic convergence of the closed-loop system can also be obtained by the quantizer parameters designed in Theorem 4.

Stage 1 (zooming-out).
The same as the proof of Theorem 1, we get a moment  0 and a positive constant V 0 such that Note that the positive-definite matrix  used here is defined by (31) rather than (5); thus the definitions of the sets  0 and  0 are different from each other.

Stage 2 (zooming-in).
If  and  selected satisfy  ≥   and () ∈  0 , ∀ ∈ N, then we know that ‖()‖ ≤ √(/(1 − ))V 0 holds, and thus If then we have Due to that the set B, defined by is an invariant one; if we select  small enough such that then we get  0 ⊃ B and thus  0 is an invariant set.Definition of  is as in (32) and  1 =  0 + ; we claim that In fact, if (40) does not hold, we know that and thus holds for any  ∈ [ 0 ,  1 ].Based on (37) we get for any  ∈ {1, 2, . . ., }.Hence, the following inequality can be given Journal of Control Science and Engineering But ( 34) and (41) give that which results in a contradiction.Thus the claim (40) holds, combined with the definition of  which results in Similar to the proof of Theorem 1, we obtain the unsaturation of the quantizer and the asymptotic convergence of the closed-loop system.This completes the proof.

Simulation
In this section, we adopt a well-known benchmark example to illustrate the effectiveness of the main results.Let discretization interval be 0.2; we convert the continuoustime linearized model of the benchmark to discrete-time one shown as   x 1 (k) x 4 (k) x 3 (k) 7, we know that the quantizer is unsaturated and the closedloop system is asymptotic convergent.
Comparison.From Case 1, we obtain that the first design method has a wider applicability than the second one.When  is large enough, two methods proposed here can guarantee the unsaturation of the quantizer and the asymptotic convergence of the closed-loop system according to the discussions of Case 2. Note that the decreasing period  of the second method is less than the one, that is, , of the first method.However, the decreasing rate  of the second method is x 1 (k) x 4 (k) x 3 (k) larger than the one of the first method.Comprehensively, the convergence speed of the closed-loop system under the second method is slower than the one under the first method.The reason of this is that the second method adopts the following inequality: +    ≤    +   , with positive-definite matrix  and matrices  and  to amplify the formula, but the first method avoids it.

Conclusion
This paper proposed two design methods for logarithmic quantizer parameters.Under both methods, we ensured the unsaturation of the quantizer and the asymptotic convergence of the closed-loop system.Further research includes designing the logarithmic quantizer parameters when the system is affected by network-induced imperfection, like time-delay, packet dropout, and so on.

Figure 5 :Figure 6 :
Figure 5: Asymptotic convergence of the system when  = 60 under method I.

Figure 7 :
Figure 7: Asymptotic convergence of the system when  = 60 under method II.