Stabilization of uncertain systems using quantized andlossy observations and uncertain control inputs\thanksreffootnoteinfo

# Stabilization of uncertain systems using quantized and lossy observations and uncertain control inputs\thanksreffootnoteinfo

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###### Abstract

In this paper, we consider a stabilization problem of an uncertain system in a networked control setting. Due to the network, the measurements are quantized to finite-bit signals and may be randomly lost in the communication. We study uncertain autoregressive systems whose state and input parameters vary within given intervals. We derive conditions for making the plant output to be mean square stable, characterizing limitations on data rate, packet loss probabilities, and magnitudes of uncertainty. It is shown that a specific class of nonuniform quantizers can achieve stability with a lower data rate compared with the common uniform one.

Okano]Kunihisa Okano, Ishii]Hideaki Ishii\corauthrefcor

Department of Intelligent Mechanical Systems, Okayama University, Okayama, 700-8530, Japan

Department of Computer Science, Tokyo Institute of Technology, Yokohama, 226-8502, Japan

11footnotetext: This work was supported in part by the JST-CREST Program, by JSPS Postdoctoral Fellowship for Research Abroad, and by JSPS KAKENHI Grant Number JP16H07234.

## 1 Introduction

This paper studies stabilization of a linear system in which the plant outputs are transmitted to the controller through a bandwidth limited lossy channel and the exact plant model is unavailable. For control over finite data rate channels, it is well known [1, 2] that there exists a tight bound on the data rate for stabilization of linear systems, which is expressed simply by the product of the unstable poles of the plant. Such data rate limitations have been developed under a variety of networked control problems. For general nonlinear systems, it has been pointed out that the limitation is related to topological entropy [3, 4, 5]. For an overview on the topic, we refer to [6]; for more recent works, see, e.g., [7, 8]. On the other hand, control over packet dropping channels has also been studied actively (see, e.g., [9, 10]). Interestingly, by modeling the behavior of the losses as i.i.d. random processes, the maximum packet loss probability for achieving stabilization can also be characterized solely by the product of the unstable poles of the plant. Recent works have extended such results to the case of Markovian packet losses. In particular, [11] and [12] have derived the minimum data rates for the static and time-varying rate cases, respectively. We note that the works mentioned above assume perfect knowledge of the plant models.

Despite the active research in the area of networked control, uncertainties in plant models have received limited attention and are thus the focus of this work. In general, it is difficult to deal with the combination of uncertainties in the systems and incompleteness in the communication. In particular, in data rate limited control problems, the state evolution must be estimated through quantized information, but in the uncertain case, this task becomes complicated and often conservative. In [13, 14], linear time-invariant systems with norm bounded uncertainties are considered, and controllers to robustly stabilize the systems are proposed. In [15], scalar nonlinear systems with stochastic uncertainties and disturbances are studied, and a data rate bound sufficient for the moment stability is derived. Moreover, related stabilization problems are studied from the viewpoints of adaptive control [16] and switching control [17] as well. In these results, however, only sufficient conditions on data rates have been obtained, and they are not concerned with characterizing the minimum. On the other hand, observation problems of nonlinear time-varying uncertain systems are studied in [4]. Both required and sufficient data rates for observability are characterized by using the notion of topological entropy.

More specifically, we consider the stabilization of a parametrically uncertain plant over a Markovian lossy channel. The plant is represented as an autoregressive system whose parameters vary within given intervals. We develop bounds on the data rate, the packet loss probability, and plant uncertainty for stabilizability. The results become tight for the scalar plants case. In the course of our analysis, we demonstrate that the data rate can be minimized by employing a class of nonuniform quantizers, which is constructed in an explicit form. These quantizers have an interesting property that the cells are coarser around the origin and finer further away from the origin; such a structure is in contrast to the well-known logarithmic quantizer [18].

For the case of uncertain state parameters, the authors have studied the minimum data rate under model uncertainties and packet losses for the i.i.d. [19] and the Markovian [20] lossy channels. It is also noted that, in [21], for a similar class of uncertain plants, stabilization techniques have been developed based on the logarithmic quantizers [18]. This paper aims at further studying the more realistic situation where uncertainty is also present in the actuator of the plant. That is, the parameter of the control input may also be uncertain.

The main difficulty in the current setup can be described as follows. Evaluation of the estimation error and its evolution is the key to derive the minimum data rate. Due to plant instability, the estimation error grows over time, but it can be reduced based on state observations. In our previous work [20], we have assumed uncertainty only in the state coefficients. In the presence of uncertainty in the actuator, the results there are not applicable. In particular, when the control input is large, the estimation error will grow further, making the analysis more involved. We show that uncertainty in the actuator side introduces additional nonuniformity in the quantizer structure when compared to our previous results.

This paper is organized as follows. In the next section, we formulate the stabilization problem for the networked control system. In Section 3, we consider the fundamental case for the scalar plant systems. The general order plants case is considered and a sufficient condition for the stability is shown in Section 4. In [20], a necessary condition for the general order plants case is also provided. However, in this paper, we do not include the corresponding result since in general it contains some conservativeness and its significance may be limited. Finally, concluding remarks are given in Section 5. The material of this paper was presented in [22] in a preliminary form, but this version contains updated results with their full proofs.

Notations: is the set of nonnegative integers and stands for the set of natural numbers. is simply written as . For a given interval on , denote its infimum, supremum, and midpoint by , , and , respectively; its width is given by .

## 2 Problem setup

We consider the networked system depicted in Fig. 1, where the plant is connected with the controller by the communication channel. At time , the encoder observes the plant output and quantizes it to a discrete value. The quantized signal is transmitted to the decoder through the channel. Here, the set represents all possible outputs of the encoder and contains symbols. Thus, the data rate of the channel is given as ; thus, the term refers to the bit rate of the communication, which is consistent with the literature [11, 12]. The decoder receives the symbol and decodes it into the interval , which is an estimate of . The transmitted signal may be lost in the channel. The result of the communication is notified to the encoder by the acknowledgment signals before the next communication starts. Finally, using the past and current estimates, the controller provides the control input .

The plant is the following autoregressive system with uncertain parameters:

 yk+1=a1,kyk+a2,kyk−1+⋯+an,kyk−n+1+bkuk. (1)

The parameters are bounded and may be time varying as

 ai,k∈Ai:=[a∗i−ϵi,a∗i+ϵi],i=1,2,…,n, bk∈B:=[b∗−δ,b∗+δ], (2)

where and . The initial values , , are in the known intervals as , where . To ensure controllability at all times, we introduce the following assumption: For every time , the input parameter is nonzero. That is,

 |b∗|−δ>0. (3)

In the communication channel, transmitted signals may be randomly lost. Denote the channel state at time by the Markovian random variable . This state represents whether the packet is received () or lost (). The loss probability at time depends on the previous state and is denoted by the failure probability and the recovery probability as follows: . To make the process ergodic, assume . Moreover, without loss of generality, assume that at the initial time the packet is successfully transmitted, i.e., .

The communicated signal is the quantized value of the plant output and is generated by the encoder as . Here, the quantizer is a time-invariant map from to with a scaling parameter . The quantizer divides its input range into cells and its output is the index of the cell into which the input falls. We assume the boundaries of the quantization cells to be symmetric about the origin. When is even, we denote the boundary points of nonnegative quantization cells as , , where

 h0=0,h⌈N/2⌉=12,hl

If is odd, the nonnegative boundaries can be written by , , where . For notational simplicity, we add and use the same notation for this case also.

Based on the channel output , the decoder determines the interval , which is the estimation set of . When the packet arrives successfully, i.e., , is the quantization cell which falls in. Otherwise, is taken as the entire input range of the quantizer.

The initial value of the scaling parameter and its update law are shared between the encoder and the decoder. The parameter is updated as follows. At time , the encoder and the decoder predict the next plant output based on the past estimates , . As time progresses from to , the past outputs are multiplied by . Let be the set of predicted values for in the form of and is given by

 Y−k+1:={ a1y′k+⋯+any′k−n+1:a1∈A1,…,an∈An, y′k∈Yk,…,y′k−n+1∈Yk−n+1}. (5)

Moreover, since the applied input is , the set is large enough to include . Note that the above prediction set in (5) is computable on both sides of the channel by the acknowledgment signal regarding from the decoder to the encoder. Finally, this set must be covered by the quantizer to avoid saturation. Hence, the scaling parameter is a function of and must be large enough that

 σk+1≥2supy−∈Y−k+1, b∈B|y−+buk|. (6)

The controller provides the control input based on the past and current estimates as

 uk=n∑i=1fi,k(Yk−i+1), (7)

where are maps from an interval in to a real number, which determine the input based on the estimates.

This paper investigates stabilization of the uncertain networked system in Fig. 1 by designing the encoder, the decoder, and the controller under the constraints (4)–(7).

###### Definition 1.

The feedback system depicted in Fig. 1 is stabilizable if there exists a pair of an encoder with the scaling parameter satisfying (6) and a controller (7) such that the worst case output over all deterministic perturbations is mean square stable (MSS): as . Here, is the decoder output at time and the expectation is taken with respect to the packet losses , , .

###### Remark 2.

We note that the results and the proofs in this paper have been polished compared with our previous paper [20] thanks to the help of the anonymous reviewers. In [20], the definition of stabilizability and the proof of the necessary condition for the scalar plants case should be updated. This however does not change the bounds on the data rate and the loss probabilities shown in the theorem.

###### Remark 3.

To simplify the analysis in dealing with uncertainties, we have introduced some structures in the encoder and the controller. While they are closely related to those employed in, e.g., [13, 23, 15] for obtaining sufficient conditions, there is some conservatism. In Definition 1, the supremum is taken over , which contains all possible over , , , , , and , . If we do not limit the controller class to (7), there may exist one which can compute an estimation set tighter than , though it is difficult to describe the tightest estimation set analytically. Furthermore, for Definition 1, it is important that the quantizer does not saturate. This is guaranteed by (6), and there is always a quantization cell containing .

## 3 Scalar plants case

We first analyze the simple setup with the scalar plant (a first-order autoregressive process):

 yk+1=akyk+bkuk, ak∈A:=[a∗−ϵ,a∗+ϵ],bk∈B=[b∗−δ,b∗+δ], (8)

where and . We assume that its dynamics is always unstable in the sense that the parameter has magnitude greater than 1 at all times, i.e.,

 |a∗|−ϵ>1. (9)

To express the main result of this section, let

 ra:=|a∗|−ϵ|a∗|+ϵ,rb:=|b∗|−δ|b∗|+δ,Δ:=ϵ+δ|a∗||b∗|, ν:=√1+p{(|a∗|+ϵ)2−1}1−(1−q)(|a∗|+ϵ)2. (10)

Here, , , and reflect the magnitudes of the uncertainties and represents the effect of packet losses in the required data rate as we will see in (11). We show later, in the proof of the next theorem, that the radicand of is positive when the feedback system is stabilizable. If is a constant, i.e., for all time , it has been shown in [11] that mean-square stability implies that . Taking account of the case that , we have . The following theorem shows a condition on the data rate , the loss probabilities , and the magnitude of uncertainty for stabilizability.

###### Theorem 4.

Consider the feedback system in Fig. 1 with the scalar plant in (8). If the system is stabilizable, then the following inequalities hold:

 R>Rnec:=⎧⎨⎩loglog{(1−Δν)2}log(rarb)if ϵ>0 or δ>0,log|a∗|+logνif ϵ=δ=0, (11) q>qnec:=1−1(|a∗|+ϵ)2+Δ2p2{1−(|a∗|+ϵ)−2}1−Δ2, (12) 0≤Δ<1. (13)

Furthermore, if these inequalities are satisfied with an even , it is possible to construct a stabilizing controller.

We note that when (12) and (13) hold and or , we have that and hence is well defined; see the last part of the proof of Theorem 4 for details. This theorem provides limitations for stabilization on the data rate, the packet loss probabilities, and the plant uncertainty. The required data rate and the recovery probability are monotonically increasing with respect to the uncertainty bounds and . This means that more plant uncertainty requires better communication with higher data rate and recovery probability.

We see that the sum of the uncertainties appears in the limitations. It is interesting that there is no explicit limitation on or , but the sum of these uncertainties must be smaller than 1. This indicates some tradeoff in the tolerable uncertainties for and . In particular, the product implies that for more unstable plants, the bound on the input parameter has more effect on the stability conditions in the theorem. We remark that when the input parameter is known and is constant as , i.e., , then the limitations and coincide with those shown in [20], where uncertainty is present only in . Moreover, if is also known, i.e., , then the limitations are equal to those in [11], where the exact plant model is assumed to be available.

We provide an example to illustrate the limitations in Theorem 4. Consider a plant with and and a channel with the loss probabilities and . Fig. 2 shows the bound on the data rate versus the uncertainties in and in . When the sum of the uncertainties is large as (12) is not satisfied, in (11) is nonpositive and hence the required data rate for stabilizability becomes infinite.

###### Remark 5.

The works of [13] and [15] have shown sufficient conditions for stabilization of uncertain plants via finite data rate and lossless channels. We remark that those conditions contain conservatism even for the scalar plants case. Consider the scalar plant (8) when the input coefficient is known () and the channel is lossless. For such systems, the sufficient bound on the data rate in [13] and the one from [15], respectively, become

 Rsuf:=log|a∗|−ϵ(|a∗|+ϵ)1−ϵ(2|a∗|+2ϵ+1)  and  R′suf:=log|a∗|1−ϵ.

It can be verified that our result is tighter than these bounds as and . For general order plants, however, it is difficult to compare these results since the types of uncertainties are different: In [13], unstructured uncertainties are considered, and it is hard to describe the data rate limitation in an explicit form, while [15] deals with nonlinear plants but only scalar ones.

We now present the proof of Theorem 4. The key idea lies in evaluating the expansion rate of the state estimation sets due to plant instability. The proof consists of two steps, which are presented in Sections 3.1 and 3.2, respectively.

###### Remark 6.

Compared with our previous work [20], the main difficulty is that we have to take account of the expansion in the state estimation sets by control inputs. If we know the exact control input applied to the plant, then the width of the estimation set is not affected by the input since we can track the variation of the state precisely. However, in the current setup, the estimation set may expand by the control input due to the uncertainty in . Hence, the scaling parameter must be selected to cover this expansion in addition to that by plant instability.

### 3.1 The quantizer minimizing the expansion rate

In this subsection, we introduce the expansion rate for a given quantizer. Then we show the optimal quantizer which minimizes the rate in the worst case.

For a given quantizer whose boundary points are , let

 (14)

for . We see later in Section 3.2 that this characterizes the expansion rate of the volume of the estimation set for one sampling period due to the uncertain parameters and . The rate varies depending on , which represents the cell which observed output falls into and we have to consider the worst case to guarantee stability against uncertainties.

The tight lower bound on the worst-case expansion rate can be derived as shown in Lemma 7 below. Let us define the boundary points dividing , where , as

 h∗l:=⎧⎪⎨⎪⎩hm1−t(rarb)l1−t(rarb)mif ϵ>0 or δ>0,hml−t′m−t′if ϵ=δ=0. (15)

Furthermore, define , which is used in the following lemma to represent the worst-case expansion rate as

 w∗m :=⎧⎪⎨⎪⎩hm(|a∗|+ϵ)(1+δ|b∗|)1−rarb1−t(rarb)m% if ϵ>0 or δ>0,hm|a∗|m−t′if ϵ=δ=0, t :={1+δ/|b∗|1−ϵ/|a∗|if N is odd,1if N is even, t′ :={12if N is odd,0if N is even. (16)
###### Lemma 7.

Given a quantizer dividing into cells, consider a subset of the quantization region , where . The worst-case expansion rate of the cells in is bounded as

 maxl∈{0,1,…,m−1}wl≥w∗m. (17)

The equality in (17) holds if for . Furthermore, consider the subset of the quantization region , where . Then, it follows that

 maxl∈{m′,m′+1,…,m−1}wl ≥⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩{hm−(rarb)m−m′hm′}(|a∗|+ϵ)(1+δ|b∗|)1−rarb1−(rarb)m−m′if ϵ>0 or δ>0,(hm−hm′)|a∗|m−m′if ϵ=δ=0. (18)

Proof. To prove the inequalities (17) and (18), we first assume that there exists a set of boundary points , , such that , , and are the same for all , i.e., for a constant

 wl=ˆw,∀l∈{i,i+1,…,m−1}. (19)

We shall show that for all quantizers , it holds that

 maxl∈{i,i+1,…,m−1}wl(h)≥ˆw, (20)

where denotes the expansion rate in (14) with the quantization boundaries . This is done by contradiction. Suppose that . Then, from (19), it follows for all that

 wl(h)≤maxl′∈{i,i+1,…,m−1}wl′(h)<ˆw=wl(^h). (21)

We shall compare with for each using (21). For , we have . Substituting these into (14) yields

 wi(h) =⎧⎪ ⎪⎨⎪ ⎪⎩2(|a∗|+ϵ)hi+1if N is odd and i=0,(|a∗|+ϵ)(1+δ|b∗|)hi+1−(|a∗|−ϵ)(1−δ|b∗|)hielse, wi(^h) =⎧⎪ ⎪⎨⎪ ⎪⎩2(|a∗|+ϵ)^hi+1% if N is odd and i=0,(|a∗|+ϵ)(1+δ|b∗|)^hi+1−(|a∗|−ϵ)(1−δ|b∗|)^hielse.

Since from (21), it follows that

 hi+1<^hi+1. (22)

Moreover, by (14), we have for that

 wl(h) =(|a∗|+ϵ)(1+δ|b∗|)hl+1−(|a∗|−ϵ)(1−δ|b∗|)hl, wl(^h) =(|a∗|+ϵ)(1+δ|b∗|)^hl+1−(|a∗|−ϵ)(1−δ|b∗|)^hl.

By substituting these into (21), we obtain

 hl+1 ≤rarbhl+maxl′∈{i,i+1,…,m−1}wl′(h)(|a∗|+ϵ)(1+δ|b∗|), ^hl+1

By introducing the relation (22) to the above, we recursively obtain for every . This however is in contradiction with . Thus (20) holds.

To establish (17) and (18), we must show the existence of satisfying the assumptions made at the beginning of this proof. First, for (17), consider the quantizer in (15) dividing the subset of the quantization range. By a routine calculation and the relation in (4), one can confirm that satisfies , , and (19) with and . Hence, (17) follows with the equality condition.

Next, to show the case of (18) with , suppose that and consider the following sequence for :

 ^hm′+l=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(rarb)lhm′+{hm−(rarb)m−m′hm′}1−(rarb)l1−(rarb)m−m′if ϵ>0 or δ>0,hm′+(hm−hm′)lm−m′if ϵ=δ=0.

We then have that , , and (19) where is equal to the right-hand side of (18). This concludes the proof.

Let us denote by the quantizer consisting of the boundaries in (15). This quantizer is optimal in the sense that it minimizes , and the minimum is given as from (17). Quantization of is nonuniform and becomes more so when the plant has more uncertainty. To see this, consider the plant with , , and , and take . The boundaries of for are shown in Fig. 3 (a) and for in Fig. 3 (b).

The nonuniformity of is an outcome of minimizing the effect of the plant uncertainty on the state estimation. This characteristic can be explained as follows. Due to quantization, only the interval containing the true output is known to the controller. After one time step, because of the plant instability, the interval in which the output should be included will expand. When the plant model is known, the expansion ratio is constant and is equal to for any quantization cell. However, with plant uncertainties, the ratio depends on the location of the cell. In particular, cells further away from the origin expand more. Moreover, to bring such cells around the origin, larger control inputs are required compared with cells closer to the origin. Because of the uncertainty in the input parameter , larger control input will result in further expansion of the interval. This fact is illustrated in Fig. 3(a) when uniform quantization is employed. In contrast, when the proposed quantizer is used, the cells after one step have the same width (Fig. 3(b)). We note that when there is no uncertainty in the plant, i.e., , then becomes the uniform quantizer.

The quantizer structure above is similar to that introduced in our previous work [20], which studied plant uncertainties only in . We also note that nonuniform quantizers have appeared in the networked control literature, e.g., in stabilization problems [18, 24] and identification problems [25].

### 3.2 Derivation of the limitations in Theorem 4

We are now ready to prove Theorem 4. The central problem in the proof is to evaluate the expansion rate of the estimation set during one sampling period. First, a tight bound on the expansion rate is shown, and then with the bound and the optimal quantizer , the limitations (11)–(13) for stabilizability are derived.

Proof of Theorem 4. (Necessity) Suppose that the feedback system is stable in the sense of Definition 1 with an encoder and a controller. We first show that as implies that . The estimation set available at the controller corresponds to a quantization cell. By (6), this set is guaranteed to contain . Let denote the smallest width of the quantization cells. Then, from the definition of the quantizer, we have that , and hence . Therefore, follows from the mean square stability of the system.

The rest of the proof consists of three steps. The first step is to prove the following inequality, which provides a bound on the expansion rate of :

 σk+1≥ηkσk. (23)

Here, is the random variable defined as

 ηk:={|a∗|+ϵ if γk=0,wlk if γk=1, (24)

where is defined in (14) and is the integer such that , which is the index of the quantization cell which falls into.

To establish (23), we recall (6) and claim that for any control input , we have that

 σk+1 ≥2supy−∈Y−k+1, b∈B|y−+buk| ≥2supy−∈Y−k+1, b∈B|y−+bu∗(Y−k+1)| =μ(Y−k+1)+2δ|b∗||c(Y−k+1)|=:σ∗k+1, (25)

where the input is defined as

 u∗(Y):=−c(Y)b∗ (26)

for an interval on . The input brings the midpoint of the prediction set into the origin when the parameter is equal to .

For the derivation of (25), we first consider the case that and . In this case, it is obvious that the input minimizing is nonpositive. For such inputs , we have that

 supy−∈Y−k+1, b∈B|y−+buk| =max{\lx@overaccentset\cc@style––Y−k+1+(b∗−δ)uk, −\lx@underaccentset\cc@style––Y−k+1−(b∗+δ)uk}. (27)

Take an arbitrary nonpositive input and denote it as . With this expression and (26), the right-hand side of (27) is equal to

 max{ μ(Y−k+1)2+δb∗c(Y−k+1)+(b∗−δ)Δuk, μ(Y−k+1)2+δb∗c(Y−k+1)−(b∗+δ)Δuk},

which takes its minimum when . This proves (25). The case of or can be reduced to the above by appropriately flipping signs of , , and .

The lower bound on in (25) is in fact equal to the right-hand side of (23), i.e.,

 σ∗k+1=ηkσk. (28)

Here, is present because the expansion rate of is affected by the packet losses . To derive (28), we consider three cases (i)–(iii) depending on the location of the estimation set as follows. For simplicity, assume .

(i) : This case occurs only when the packet arrives, i.e., . In addition, from (24), where is even or . From the basic results for products of intervals [26] and (9), for the interval , we obtain its supremum and infimum as and , respectively. Substitution of these into the left-hand side of (28) gives

 μ(Y−k+1)+2δ|b∗||c(Y−k+1)| ={(a∗+ϵ)(1+δb∗)hlk+1−(a∗−ϵ)(1−δb∗)hlk}σk.

Hence, by (14) the relation in (28) holds.

(ii) : In this case, we have

 \lx@overaccentset\cc@style––Y−k+1=(a∗+ϵ)\lx@overaccentset\cc@style––Yk,\lx@underaccentset\cc@style––Y−k+1=(a∗+ϵ)\lx@underaccentset\cc@style––Yk. (29)

Moreover, because of the symmetry in the quantization cells, it holds that and hence . Thus, . To compute this width , consider the following two cases: (ii-1) If , then . Hence, by (29), we have . (ii-2) Otherwise, must be odd and from the symmetry of the quantizer and the condition (ii). Thus, . Hence, (28) holds for this case also.

(iii)