State Estimation Using Dependent Evidence Fusion: Application to Acoustic Resonance-Based Liquid Level Measurement.

Estimating the state of a dynamic system via noisy sensor measurement is a common problem in sensor methods and applications. Most state estimation methods assume that measurement noise and state perturbations can be modeled as random variables with known statistical properties. However in some practical applications, engineers can only get the range of noises, instead of the precise statistical distributions. Hence, in the framework of Dempster-Shafer (DS) evidence theory, a novel state estimatation method by fusing dependent evidence generated from state equation, observation equation and the actual observations of the system states considering bounded noises is presented. It can be iteratively implemented to provide state estimation values calculated from fusion results at every time step. Finally, the proposed method is applied to a low-frequency acoustic resonance level gauge to obtain high-accuracy measurement results.

1. Introduction

Estimating the state of a dynamic system based on noisy sensor measurements is a common problem in sensor methods and applications [1,2]. Mainstream estimation methods all assume that both the system state noise and measurement noise can be modeled as random variables with known statistical properties. The Kalman filter, which supposes both of noises obey Gaussian distributions, is, by far, the most popular method [3]. The basic Kalman filter is only applicable to linear systems. In order to deal with nonlinear cases Bucy and Sunahara proposed the extended Kalman filter (EKF) [4,5]. The EKF uses the first order Taylor expansion technique to linearize state and observation equations, and then obtains state estimations by the Kalman filter. On the other hand, approximation to a state probability distribution of a nonlinear system is, to a great extent, easier and more feasible than a linear approximation to a nonlinear function [6]. Based on this idea, Gordon and Salmond proposed the particle filter (PF) [6]. The performance of the PF is commonly superior to the EKF because it can usually provide more precise information about state posterior probability distribution than does the EKF, especially when it takes a multimodal shape or noise distributions are non-Gaussian [6,7].

The precondition of the above methods is that the noise statistical properties must be known. However, in some practical applications, what engineers can obtain are not precise statistical distributions [8], but ranges of noises. Hence, a group of state estimation methods considering bounded noises, also known as the bounded-error methods, appeared [9,10,11,12]. Assuming that all variables belong to known compact sets, these methods build simple sets, such as ellipsoids or boxes, guaranteed to contain all state vectors consistent with given constraints. For linear systems, some scholars began to study such state estimation methods in the 1960s [9,10,11]. For nonlinear systems, the corresponding studies are relatively rare. Khemane et al. and Jaulin proposed bounded-error state and parameter estimations for nonlinear systems [13,14]. Gning proposed a relatively simple and fast bounded-error method based on interval analysis and constraint propagation, which was successfully applied to dynamic vehicle localization [12], but when the noise bounds cannot be precisely determined, its robustness will unavoidably decline [7]. That is to say, if the bounds are too tight, then the data may become inconsistent with the system equations, and in this case, this method fails to provide a solution. On the contrary, if the bounds are overestimated, then the estimated state becomes very imprecise, and this method becomes overly pessimistic [7].

In order to deal with this problem, Nassreddine proposed an improved method by integrating interval analysis with DS evidence theory. Its key idea is to replace the set-based representation of uncertainty by a more general formalism, namely, mass functions in evidence theory [7]. It introduces possibility distributions to model bounded noises, and then uses mass functions, i.e., evidence composed of interval focal elements and their masses to approximate these distributions. Essentially, such mass functions can be regarded as “generalized boxes” composed of a collection of boxes with associated weights. These mass functions can be propagated in the system equations using interval arithmetic and constraint-satisfaction techniques to get the mass function of system state at each time step. Pignistic expectation of this mass function is calculated as the state estimation value. Therefore, this approach extends the pure interval approach, making it more robust and accurate.

Nassreddine’s research showed the powerful ability of DS evidence theory to deal with the uncertainty of dynamic systems. Hence, this paper further presents a new state estimation method, which uses not only evidential description of uncertainty, but also dependent evidence fusion. Here, the state equation and the observation equation of a dynamic system and the actual observations of system states are regarded as three information sources. The random set description of evidence and extension principle of random set is used to obtain state evidence and observation evidence from these three information sources and to propagate them in the system equations. There are correlation among these evidence, so the proposed combination rule of dependent evidence is used to fuse the propagated evidence and Pignistic expectation of fusion results is calculated as state estimation value at each time step. Compared with Nassreddine’s method, it is shown that the proposed approach generates more accurate estimation results by combining dependent evidence. An industrial liquid level detection apparatus was employed to show the better performance of the approach.

2. Foundations of Dempster-Shafer (DS) Evidence Theory

The DS theory is a mechanism formalized by Shafer for representing and reasoning with uncertain, imprecise, and incomplete information. It is initially based on Dempster’s original work on the modeling of uncertainty in terms of upper and lower probabilities induced by a multi-valued mapping rather than as a single probability value [15]. One of the specificities of this theory is that the objects of study are no more the universe, i.e., a set, defined as the frame of discernment hereinafter, but the power set of this universe. In this section we introduce some main concepts of this theory and some necessary notions that will be used in the proposed approach. A more detailed exposition and some background information can be found in [16].

A set is called a frame of discernment if it contains mutually exclusive and exhaustive possible hypotheses. This set is usually denoted as Θ. The power set of Θ is denoted as 2Θ.

A function m: 2Θ → [0, 1] is called a mass function on Θ if it satisfies the following two conditions: (1) ; (2) . This function is also named as a basic belief assignment (BBA). A subset A with a non-null mass is viewed as a focal element. Commonly, if an information source can provide a mass function on Θ, then this mass function is called a body of evidence, abbreviated to evidence (E).

If m

Note that the Dempster’s combination rule is meaningful only when , i.e., m1 and m2 are not totally conflicting. This rule can be used to synthesize uncertain, imprecise or incomplete information coming from different sources.

2.2. The Degree of Dependence and the Combination of Dependent Evidence

In DS evidence theory, Dempster’s combination rule is the most important tool for computing a new BBA from two BBAs based on two pieces of evidence. This rule requires that the two pieces of evidence must be independent, which is considered to be a very strong constraint and cannot always be met in practice. Wu, Yang and Liu [17] pointed that if there are two pieces of evidence which are partially derived from the same information source, then both of them are mutually dependent. This interpretation concentrates on the connotation of independence conception in evidence combination operation. In this case, Wu, Yang and Liu [17] proposed the energy of evidence concept, and then, deduced the degree of dependence and the dependency coefficient between the two from the energy of the intersection of the two. Based on these notions, the combination of dependent evidence can be realized.

The energy of evidence E, En(E) is defined as: where |A

Suppose that the BBAs of evidence E1 and E2 are m1 and m2, respectively, and their focal element sequences are A1 and E2 are induced by the same information source. In this case, E1 and E2 will be dependent, then the energy of the intersection of the two pieces of evidence can be described by: where D denotes dependent focal element, |{D1 and m2.

The relationship of En(E1), En(E2) and En(E1, E2) is illustrated in Figure 1; especially, En(E1, E2) = 0 implies the independence between E1 and E2. The value of En(E1, E2) measures the dependency of the two pieces of evidence.

Figure 1

The relationships of two pieces of dependent evidence.

En(E then two corresponding independent pieces of evidence can be generated from E1 and E2.

For E1, its final independent energy can be calculated as:

Similarly:

The dependency coefficient of E1 to E2 is defined as: and the dependency coefficient of E2 to E1 is defined as:

E1 and E2 can be modified by R12 and R21, respectively, to obtain their corresponding independent E1′ and E2′, their BBA functions are given by:

Consequently, the requirement of Dempster’s rule is met and the combination of E1′ and E2′ can be implemented according to Dempster’s rule in (1). Finally, the combination of E1 and E2 is indirectly realized by the combination of E1′ and E2′. Actually, reference [17] gives the decorrelation method to correct E1 and E2 by dependency coefficients such that the corrected E1′ and E2′ can be deemed as the independent evidence and combined using Dempster’s rule.

(Random set [18,19]). A finite support random set on Θ is a pair (ℱ,m) where ℱ is a finite family of distinct non-empty subsets of Θ and m is a mapping ℱ → [0, 1] and such that ∑

ℱ is called the support of the random set and m is called a basic belief assignment. Such a random set (ℱ,m) is equivalent to a mass function in the sense of Shafer.

(Random relation [18,19]). Let Θ = Θ1 × Θ2 × …× Θ

The projections of a random relation on Cartesian product Θ1 × Θ2 ×…× Θ

For , A = C1 × C2 ×…× C1(C1) × m2(C2) ×…× m1,m1 ), (ℱ2,m2),…, (ℱ

2.3. The Random Set Description of Evidence

2.3.2. Extension Principles

Let ξ = (ξ1, ξ2,…, ξ) be a variable on Θ = Θ1 × Θ2 × …× Θ, ζ = f(ξ), , f : Θ→Φ is the function of ξ. The random set (ℛ,ρ) of ζ, which is the image of random relation (ℱ,m) of ξ through f, is given by extension principles [20,21,22]: where: M is the number of element of ℱ. The summation in Equation (14) accounts for the fact that more than one focal element A may yield the same image R through f.

The key of constructing (ℛ,ρ) is to calculate the image of A through f. If ξ is a continuous variable on Θ, then Θ = ℝ, ℱ becomes a finite family of distinct non-empty sub-intervals on Θ. In this case, the process of constructing (ℛ,ρ) is given as follows:

For each ξ in ξ, let its marginal random set be (ℱ,m) and the focal element of (ℱ,m) be a interval [a,a], then the focal element of (ℱ,m) can be given as:

The image of A can be calculated by using the methods of Interval Analysis [19,20,21]; if A is a convex set, then A has 2 vertices, denoted as v (j = 1,2,…, 2). If function f has certain properties, the Vertex Method can help reduce the calculation time considerably [22]:

, if ζ = f(ξ) is continuous in A and also no extreme point exists in this region (including its boundaries), then the value of interval function can be obtained by:

Thus, function f has to be evaluated 2

If f is continuous, if its partial derivatives are also continuous and if f is a strictly monotonic function with respect to each ξ

There is a case in point. Let ξ = (ξ1,ξ2,ξ3), A = [a11 × [a22 × [a33. Assume f and its partial derivatives are all continuous. If f is increasing with respect to ξ1 and ξ2, decreasing with respect to ξ3 respectively, then f has to be calculated only twice for each focal element A, namely, f(A) = [f(v, v123 Totally, 2M evaluations of f are needed to obtain complete (ℛ,ρ).

Furthermore, the expectation of (ℛ,ρ) is given by [ where R

3. State Estimation Based on Dependent Evidence Fusion

3.1. Dynamic System Model under Bounded Noises

The dynamic systems mode constructed by the state and observation equations is as follows: where the relationship between state x1 at time k + 1 and state x at time k is described as function f. The relationship between observation z1 at time k + 1 and state x1 at time k + 1 is described as function g. v and w are bounded additive state noise variable and observation noise variable respectively, which are independent of each other. These two noises can be approximated to triangle possibility distributions [7], noted as π and π, respectively, (the noise distributions are identical at each time step), as shown in Figure 2.

Figure 2

(a) Triangle possibility distributions of state noises, (b) Triangle possibility distributions of observation noises.

where [v] is the support interval of the state noise, v is the mode of state noise, similarly:

3.2. Recursive Algorithm of State Estimation Based on Extension Principles and Dependent Evidence Fusion

Figure 3 shows the flow of the proposed recursive algorithm. The following steps will be introduced in detail.

Figure 3

Flowchart of state estimation iterative algorithm.

Step 1: Construct noise evidence to approximate π. Initially, we construct evidence to approximate the possibility distribution π of state noise variable v. For any α ∈ (0, 1], α cut set of π is [9]:

If there exist α0, α1,⋯, α−1 which satisfy 0 = α0 < α1 <⋯< α−1 < 1, then their corresponding α-cut sets will satisfy where p is a positive integer. Take these p α-cut sets as focal elements with nested closed interval forms, then their corresponding BBAs are:

Figure 4 gives an example that when p = 3, and m can be constructed by uniformly cutting α three times. Distinctly, m corresponds to a possibility distribution π that approximates π. Certainly, a better approximation of the continuous possibility distribution can be obtained by increasing the number p of cut sets, at the expense of higher computational complexity.

Figure 4

The possibility distribution of state noise and its evidence construction.

It is worth noticing that m is constructed on the condition that all values outside the support interval [v] are completely impossible. However, in practice, the bounds va and vb are commonly given based on available measurement knowledge or real data, so they may not be precise and the values outside [v] may appear. To account for the imprecision of the support interval [7], constructs by discounting m with a small discount rate ε, in which, is defined as [9]: where , Θ = ℝ, accordingly, . In the course of implementing the proposed algorithm, Θ can be replaced by the closed interval [v′,v′], here v′ >> v and v′ >> v such that the following interval operations can be done easily.

In the same way, we can construct evidence using the possibility distribution πw of observation noise variable w.

Step 2: Obtain state prediction evidence. at time . Suppose the estimation result at time k is . When k = 1, is initialized as real observation z1. Considering the influence of noise to the state, we construct the state evidence of by adding noise to :

Thus, taking and as the inputs of state equation , we can get the state prediction evidence by mapping from the inputs to the outputs based on the extension principles in Equations (13) and (14).

Step 3: Obtain observation prediction evidence. at time k + 1 from observation equation.

Taking the state prediction evidence in Step 2 as the input of the observation equation , we can get based on the extension principles in (13) and (14).

Step 4: Obtain fusion evidence. at time k + 1 in observation domain.

Firstly, in Step 1, we get evidence using the possibility distribution πw of w:

After getting observation zk+1 at time k + 1, considering the influence of noise to the observation, we construct the evidence of z+1 through adding noise to z+1:

Secondly, using Dempster′s combination rule, we fuse and to get the fusion evidence in observation domain at time k + 1. As for the relationship between and . The former is obtained by propagating from state equation to observation equation ; the latter is constructed by adding noise to z+1. It can be seen that the former completely comes from the state information at past time step k which does not use the observation noise (), but uses the state noise (). Because () and () are independent of each other, so it is believed that the former and the latter are also independent of each other. Hence both of them can be directly fused using Dempster′s combination rule.

Step 5: Get new evidence. at time k + 1 in state domain.

Taking attained in the Step 4 as the input of inverse function , we can get by using the extension principles in Equations (13) and (14).

Step 6: Get state estimation evidence. and state estimate at time k + 1.

Using Dempster′s combination rule, we can fuse attained in Step 5 and attained in Step 2. That is to say, we utilize the former to revise the latter to get state estimation evidence . is obtained by inverse mapping of fusion evidence in observation domain. is obtained by the fusion of observation evidence and observation prediction evidence . In Step 3, it is noted that is related to , so and are certainly mutually dependent. Therefore, the combination of dependent evidence must be used for fusing both of them. For the focal elements of and are the closed intervals on real numbers, here we extend the combination of dependent evidence in the discrete frame of discernment introduced in Section 2.2 to that in the continuous frame of discernment (see the corresponding proposition and example in Appendix A). and can be fused using the extended combination of dependent evidence to get state estimation evidence at time k + 1.

Finally, Pignistic expectation of is calculated as state estimation value by Equation (20). Using state estimation at time k + 1 to do next iteration, we can estimate state at every time step.

In conclusion, as shown in Figure 3, the whole recursive algorithm is actualized under the framework of DS evidence theory. The corresponding evidence in state and observation domains are not only propagated and transformed by the extension principle, but also fused by the Dempster′s combination rule and the proposed combination rule for dependent evidence. Especially, fusion procedure can make that the masses focus to those interval focal elements that contain the system state, so as to get the accurate estimation results, which is the main difference from Nassreddine’s method under the framework of the interval analysis. In next section, our approach will be applied to liquid level estimation using an industrial level apparatus to show its better performance than possible with Nassreddine’s method.

4. Application to Liquid Level Measurement

Level measurement methods based on sound reflection phenomena have been successfully applied in some areas of process industry (chemical, waste water treatment, petroleum, etc.) because the level is the main monitored process variable used in industrial alarm systems. Ultrasonic measurement methods, with good directivity, convenient operation and so on, have become some of the most commonly used techniques [24]. Their measuring principle is to emit an ultrasound toward a liquid surface and receive the echos, then to calculate the distance from the surface to the acoustic receiver device by multiplying the sound velocity by the round-trip time [25]. However, this method is susceptible to the quality of the instrument itself and environmental noise, which will deteriorate the measurement accuracy. Besides, if the ultrasound encounters foams, residues, deposits, etc., in the measurement process, it is also prone to parasitic reflection, thereby the ultrasound propagation path is changed, which seriously affects the measurement accuracy [26].

On the contrary, low-frequency sound waves have longer wavelength and it is easy to generate the diffraction phenomenon which can effectively overcome the problem of parasitic reflection due to foams, residues, deposits, etc. When a speaker emits sound waves with a uniform change from a frequency f to a higher frequency f toward the surface and a microphone receives the corresponding echoes, the generated standing wave signals extracted in the oscilloscope can be used to calculate the height of the liquid level. Kumperščak and Završnik [25,26] used this idea to measure liquid levels. However, they both directly used observations to calculate the liquid level. In practice, if the measurements obtained by using a speaker and a microphone are not precise enough and if the effect of environmental noise is inevitable, then the deviation of the final measurement results will be unacceptable, which is the most common shortcoming in the present level measurement methods.

In our earlier work [27], we have used the Evidential Reasoning(ER) rule to deal with liquid level estimation with bounded noises, but the ER-based method only provides an initial idea for state estimation under the framework of DS evidence theory and only gives precise estimated results when the level length is less than 1.6 m. In order to improve the evidence fusion-based state estimate method, this paper introduces a new information source, Dempster combination rule and evidence dependence conceptions. We construct the state equation and observation equation based on the principle of level measurement using acoustic standing waves, and then use the proposed algorithm to estimate the frequencies of the standing waves, which can be translated into the liquid level height (0 m–10 m). Compared with the direct measurement method and Nassreddine’s method, the estimation results verify that our algorithm has obvious advantages and improves the level estimation accuracy.

4.1. Acoustic Standing Wave Level Gauge

The structure of an acoustic standing wave level gauge is shown in Figure 5, and mainly consists of a waveguide (a tube), a speaker, a microphone, a thermometer and a controller. When sound waves in the frequency range [f,f] generated by a signal generator (audio card and speaker) vertically propagate to a surface and echoes appear, superposition of both waves will generate standing waves. Here, y1 denotes the sound wave generated by speaker and y2 denotes the echo reflected by the surface:

Figure 5

Structure of a level gauge.

The synthesis wave of y1 and y2 can be expressed as: where A is the amplitude of sound wave, P is the frequency of sound wave and L is the distance from the top of the tube to the surface of liquid as shown in Figure 5. From Equation (29), we know that when L and λ have the following relation, the amplitude of synthesis wave reaches the maximum:

In this case, this synthesis wave is defined as the standing wave and its wavelength is: where λ is the wavelength of kth standing wave, f is the frequency of kth standing wave (kth resonance frequency) in [f]. c is sound velocity, and T is temperature.

Substituting Equation (30) into Equation (31), we obtain: where, n is given as [28]: and:

Theoretically, in Equation (33), f+1 − f = f, f is the fundamental resonance frequency and f = n, n ∈ ℕ+ (the set of all positive integers) [26,28]. For example, if L = 9.6 m, T = 23.9 °C, and n = 1, then the fundamental resonance frequency can be calculated by (32):

If the frequency range [f] is [1000 Hz, 2500 Hz], then there are 82 resonance frequencies in this range, k = 1, 2,⋯, 82, and n = 56,57,⋯,137. Consequentially, f1 = 56 × 18 Hz, f2 = 57 × 18 Hz,⋯, f82 = 137 × 18 Hz.

4.2. System Model

Firstly, we consider the resonance frequency as the estimated state and construct the corresponding state equation. If we can continuously collect the resonance frequency f1, then we have the following equations:

From Equations (32) and (36), obviously, we can get:

Consequentially, we can establish the recursive linear state equation and observation equation, respectively: where x = f, z is the observation of f, w and v are independent noise sequences coming from speaker and microphone, respectively, satisfying the conditions:

The intervals [v] and [w] denote the boundaries of the state noise and observation noise, respectively. The state noise v and observation noise w can be expressed by possibility distributions π and π with the support intervals [v] and [w], respectively.

It should be noted that, in theory, n in (39) should be taken as a positive integer. However, in practice, it can be only calculated by observations z and z+1 according to Equation (33). Because of observation imprecision, the calculated n is commonly not a positive integer, so it should be approximated as: where “” denotes the operator that “round numbers to the nearest integer”.

4.3. Liquid Level Estimation Tests

In order to construct the level gauge in Figure 5, we use a low-precision microphone and speaker to emit and receive cosine sound waves, respectively, an electronic thermometer to collect temperature and a PVC tube with a diameter of d = 75 mm to transmit sounds. The estimated level L is the distance from the surface to the speaker platform. The controller transmits sine or cosine waves to drive the speaker to emit the signals vertically to the liquid surface. We use the software AUDIOSCSI (Brothers Studio, Shenzhen, China) which is based on an audio controller (82801HBM-ICH8M with sampling rate 44,100 Hz, Intel Corporation, Santa Clara, CA, USA) to generate sound waves. The frequencies of wave change with uniform speed from f = 1000 Hz to f = 2400 Hz in 5 s. Thus, the microphone receives the synthesis waves and sends them to the controller as shown in Figure 6 (L = 4.6 m). It can be seen that there are the frequencies of 39 adjacent standing waves collected by microphone in [1000 Hz, 2500 Hz]. Figure 7 shows the resonance frequency f (k = 1,2,⋯, 39) extracted from the spectrum of the synthesis waves by the fast detecting algorithm in [28]. In this experiment, set liquid level distance L = 4.6 m, the ambient temperature is 26.5 °C and sound velocity is 347.3 m/s. The state equation and observation equation of resonance frequency are shown in Equations (39) and (40).

Figure 6

Waveform graph (L = 4.6 m).

Figure 7

Resonance frequencies and amplitudes (L = 4.6 m).

For the state noise v, we use a high-precision oscillograph (TPS2024, Tektronix, Shanghai, China) to receive the cosine sound waves emitted by the audio controller and speaker in range [1000 Hz, 2500 Hz] and calculate errors about 100 frequency points uniformly selected from 1000 Hz to 2500 Hz. As in [7], the bounds of v, are taken to be plus or minus three times the standard deviation of errors. So the possibility distributions π of v can be constructed as in Figure 8. Where the expectation of π is 0, standard deviation σ is 0.1, then the support intervals [v] = [−0.3, 0.3], mode v= 0. Set α0 = 0, α1 = 1/3, α2 = 2/3, we can get three nested closed intervals and their BBAs to approximate π as in Figure 8.

Figure 8

Probability distribution π of state noise v.

Furthermore, in Step 1 of the proposed algorithm in Section 3.2, discounting m at rate ε = 0.05 and approximating as the closed interval [v 100σ 100σ] = [−10, 10], we can construct the evidence of v according to Equation (26) as shown in Table 1.

Table 1

Evidence of state noise.

ℱkv	[−0.1, 0.1]	[−0.2, 0.2]	[−0.3, 0.3]	[−10, 10]
mkv	0.3167	0.3167	0.3167	0.05

The observation noise w is mainly related to the microphone and the fast detecting algorithm. Firstly, we extract observation values of resonance frequencies in range [1000 Hz, 2500 Hz] about 30 level values uniformly selected from L = 1.3 m to L = 10.6 m by the fast detecting algorithm. Secondly, we calculate the errors between the theoretically correct values (true values) and observation values. In the same way, the possibility distributions π of w and the corresponding closed intervals and their BBAs can be constructed as in Figure 9, where, σ = 1.23, w = −6.9, so [w] = [−10.59, −3.21]. Furthermore, discounting m at rate ε = 0.05 and approximating as the closed interval [w − 100σ 100σ] = [−129.7, 115.7], we can construct the evidence of w shown in Table 2.

Figure 9

Probability distribution π of observation noise w.

Table 2

Evidence of observation noise.

ℱkw	[−8.13, −5.67]	[−9.36, −4.44]	[−10.59, −3.21]	[−129.7, 115.7]
mkw	0.3167	0.3167	0.3167	0.05

From Figure 7, it can be seen that the first observation value of resonance frequency z1 = 1023.3 Hz. According to Step 2) in Section 3.2, the first estimation result is initialized as the real observation z1. After obtaining and , our recursive algorithm presented in Section 3.2 can be used to estimate resonance frequencies at each step k. Figure 10a gives the estimation results of our method and Nassreddine’s method, together with the true values and observations (z). Figure 10b gives the absolute errors between true values and the estimated values of our method, the estimated values of Nassreddine’s method, and z respectively. It can be seen that the estimation accuracy and convergence of our method are better than those of Nassreddine’s method because of the focusing function of the proposed fusion procedure for dependent evidence.

Figure 10

(a) Estimation results of resonance frequencies, (b) Absolute values of frequency estimation errors.

Finally, we can calculate the estimate level L by Equation (32) according to the estimated resonance frequencies of our method, Nassreddine’s method, and z respectively as shown in Figure 11a,b gives the corresponding absolute values of length estimation errors. Obviously, the more accurate the estimations of resonance frequencies are, the more accurate the estimations of level L are. As our method always provides more accurate estimations of resonance frequencies, it is therefore always superior.

Figure 11

(a) Estimation results of level L, (b) Absolute values of length estimation errors.

More experiments are performed for different values of L to find the mean of absolute values of estimation errors and to show the efficacy of the proposed method as shown in Table 3. Here, for every values of L, from above to below, Table 3 gives in order the experiment results of our method, Nassreddine’s method, and direct measurement method (namely, substituting z into (32)).

Table 3

Experimental results for different values of L.

No	True L (m)	T (°C)	Runtime (s)	Mean Error (m)	No	True L (m)	T (°C)	Runtime (s)	Mean Error (m)
1	1.3	27	1.81	0.0126	6	5.6	26.5	20.11	0.018
			0.88	0.016				4.81	0.0374
			-	0.238				-	0.0661
2	2.1	26.5	2.49	0.0254	7	6.6	26.5	23.57	0.0238
			1.33	0.0364				5.61	0.0436
			-	0.0441				-	0.088
3	2.6	26.5	7.81	0.0144	8	7.6	26.5	27.94	0.0299
			2.05	0.0297				6.77	0.0530
			-	0.0591				-	0.1060
4	3.6	26.5	9.62	0.0141	9	8.6	23.9	31.23	0.0216
			2.34	0.0312				7.41	0.0456
			-	0.0468				-	0.1295
5	4.6	26.5	16.07	0.0160	10	9.6	23.9	35.15	0.0435
			3.81	0.0337				8.36	0.0732
			-	0.0552				-	0.1624

It should be noted that the calculation complexity of our algorithm is relatively high, and meanwhile, with the increase of the measured length of level, the corresponding synthetic wave contains more and more resonance frequency points, so the CPU time will increase, so it needs a longer time (the hardware in this test: CPU E8400, CPU Clock Speed 3.00 GHz, RAM 2 GB). But to the situation that liquid level change relatively slowly, our method is applicable. Certainly, the rapid development of data processing capability of computer hardware will make the complexity less of an issue.

5. Conclusions

The subject of Nassreddine’s method is still interval analysis, it introduces evidence, namely belief-function to elaborate bounded noises. In detail, it gives the improved form of noise bounds (the triangular possibility distribution), here the error interval is extended to the evidence construction, namely many interval focal elements with the corresponding belief assignments. Obviously, the latter has more information than the former. Then, it still uses interval arithmetic and constraint-satisfaction techniques to propagate not only interval focal elements, but also its belief assignments, hence its performance is slightly better than that of pure interval propagation. However, the subject of our method is DS evidence theory and random set theory and introduces Dempster combination rule and evidence dependence conceptions. Although we still use Nassreddine’s evidence construction technique, the random set description of evidence and extension principle of random set are used to obtain state evidence and observation evidence from the defined information sources and to propagate them in the system equations. The main contribution is to realize the fusion of the propagated evidence, in which the degree of dependence and the combination of dependent evidence are further considered.

As a whole, compared with Nassreddine’s method, our method increases the state estimation accuracy. The application in liquid level estimation using an industrial level apparatus shows the efficacy of the proposed method. Certainly, it is worth noting that, in a given application, there are some constraint conditions such as the continuous, monotonic and invertible properties of state and observation equations, and state observability. When they cannot be satisfied, the computational burden will inevitably be increased because of the use of additional complex interval operation algorithms or matrix operation algorithms (for multidimensional states) in [21,22]. Hence fast operation algorithms should be studied in the further. On the other hand, although the proposed procedure of evidence fusion can make the masses focus to those interval focal elements that contain the system state, so as to get the accurate estimation results, how to further evaluate the convergence of fusion using available theories is still a problem worthy studying which will promote the usage of evidence theory in state estimation.