Model-Based Observer Design Considering Unequal Measurement Delays

: State observers are essential components of a modern control system. It is often designed based on a mathematical model of the process, thus requiring detailed process knowledge. However, in the existing state estimation methods, equal delays are commonly assumed for all communication lines, which is unrealistic and poses problems such as instability and a degraded performance of observers when unequal time delays exist. In this paper, a design of observers considering the measurement delays is presented. To deal with this problem, a chain-based observer has been proposed in which each chain deals with one output delay, performs prediction for the unavailable output value, and passes it to the next chain. Convergence of each chain observer as well as overall state estimation were proven. To illustrate the performance of the proposed scheme, simulation studies were performed on a benchmark continuous stirred tank heater (CSTH) process.


Introduction
State observers play a key role in process monitoring, control, and automation by providing estimates of internal state variables using hardware output measurements. Given a process model, observers can be designed by a large number of available methods such as high gain [1], Kalman filter [2], sliding mode [3], and so on. Various types of observers have also been used in the feedback control design in the presence of delays [4,5]. One of the profound challenges in state estimation is that sensor measurements are often time delayed from the actual process outputs. The control problem of dynamic systems with delays has received growing attention in recent years due to novel application areas such as control over networks [6]. This might also occur due to the slow dynamics of physical sensors as well as communication delays from measurements transmitted between different processing units. If these delays are not compensated, they can lead to the performance degradation of the observers and can have a strong impact on their stability and robustness properties [7,8]. When designing an observer or controller for a system affected by such delays, it is often expedient to lump the measurement delay together with any system and actuation delay, as they are indistinguishable in terms of the ideal closed-loop dynamics of the system [9]. However, in some cases, it can be advantageous to model the measurement delays separately from the other delays.
For linear systems, when the system model is available, there exists literature to deal with equal output delays in state estimation [10][11][12]. Assuming equal delays for all output channels is unrealistic due to different communication line lengths or different sensors and technology types. Hence, the involvement of the delay could be in the communication line, be multiple, and different for each channel, which, however, has rarely been studied in the literature. Supposing equal delays will simplify the observer designing step, but can cause similar problems to ignoring them [13]. Others have assumed the delays on outputs [9,14] or states [15], which again, is simpler than considering them in the communication line (out 1.
Designing an observer by considering equal measurement delays.

2.
Designing a chain observer to deal with unequal measurement delays. 3.
Proving the convergence of each chain as well as the overall observer.
The rest of this paper is organized as follows. In Section 2, the preliminaries are presented. The problem formulation is discussed in Section 3. Section 4 presents the theory of the proposed Luenberger-based observer for the case of equal and unequal measurement delays. In Section 5, a simulation through the continuous stirred tank heater (CSTH) process is used to illustrate the performance of the proposed scheme. The paper ends with concluding remarks in the final section.

Preliminaries of Model-Based (Luenberger) Observer Design
In this section, we present the preliminaries of designing a model-based observer approach following the work of [17]. Consider a discrete-time LTI (linear time invariant) system, which is described by where u(k) ∈ R l , y(k) ∈ R m , and x(k) ∈ R n represent process input, output, and state variable vectors, respectively. Suppose that the pair (C, A) with A ∈ R n × n , C ∈ R m × n is observable. The s (= n)-order observer with the following state and output equations is asymptotically stable, provided that G is stable (all of its eigenvalues lie within the circle one) and the observer parameters H, L, Q, W, V satisfy the following Luenberger equations: where z(k) is the observer states related to the process states through the transformation matrix T ∈ R s × n so that z(k) = Tx(k), wherex(k) is an estimation of true state x(k).

Problem Formulation
Sensors and communication networks are an integral part of a data transmission system. One of the main issues regarding measurements and network transmission is the presence of delays in each output measurement (say delay τ i for output y i ). Figure 1 illustrates the communication network and corresponding delays. Due to these delays, output signals y(k) required for state estimation (2) are not available instantaneously. Therefore, the objective of this work was to design an observer that utilizes delayed measurements (i.e., y(k − τ) = [y 1 (k − τ 1 ) . . . y m (k − τ m )]) to estimate z(k). As a special case, these delays may be equal in all outputs (i.e., τ i = d, for i = 1, . . . , m.) First, we developed an efficient observer scheme in the presence of constant equal measurement delays. The proposed observer, which deals with equal measurement delays, will have the form of: where it utilizes available data until the (k − d) th instant to estimate x(k). The next section will also propose a novel method to deal with unequal measurement delays using a chainobserver structure.
is asymptotically stable, provided that is stable (all of its eigenvalues lie within the circle one) and the observer parameters , , , , satisfy the following Luenberger equations: where ( ) is the observer states related to the process states through the transformation matrix T ∈ R s×n so that ( ) =̂( ), where ̂( ) is an estimation of true state ( ).

Problem Formulation
Sensors and communication networks are an integral part of a data transmission system. One of the main issues regarding measurements and network transmission is the presence of delays in each output measurement (say delay for output ). Figure 1 illustrates the communication network and corresponding delays. Due to these delays, output signals ( ) required for state estimation (2) are not available instantaneously. Therefore, the objective of this work was to design an observer that utilizes delayed measurements (i.e., y(k − τ) = [y1(k − τ1) … ym(k − τm)]) to estimate ( ). As a special case, these delays may be equal in all outputs (i.e., = , for = 1, … , .) First, we developed an efficient observer scheme in the presence of constant equal measurement delays. The proposed observer, which deals with equal measurement delays, will have the form of: where it utilizes available data until the ( − ) ℎ instant to estimate ( ). The next sec-

Observer Design in the Case of Equal Measurement Delays
In this section, it was assumed that the delays were equal in all outputs (i.e., τ i = d, for i = 1, . . . , m). The following theorem designs an asymptotically stable observer (5) with the consideration of equal measurement delays when the system model is available. It is an extension of the Luenberger condition (4) for the case of equal measurement delays. Theorem 1. Suppose system model (1) is known with equal measurement delay d in all measured outputs. Furthermore, it is assumed that the pair (C,A) is observable. The observer with the following state and output equations is asymptotically stable, namely lim k→∞ (Tx(k) − z(k)) → 0 , and has unbiased estimation for y(k), that is lim k→∞ (ŷ(k) − y(k)) = 0, provided the observer parameters defined in Equations (6) and (7) satisfy the following delayed-Luenberger equations: State convergence condition: Output convergence condition: where G should be chosen so that it is stable (i.e., every eigenvalue of G lies within the circle one). In addition, the closed loop dynamics of estimation error for state and output is reduced to Proof. The state equation of Equation (1) can be rewritten as: Substituting x(k) from (11) in (1) yields Similarly, x(k − 1) can be replaced by Ax(k − 2) + Bu(k − 2), which yields Continuing the above procedure, after d step reaches or The state estimation error is defined as Substituting Equations (15) and (6) in Equation (16) gives Now by substitution of The estimation error (18) will converge to zero, provided G is a stable matrix. Performing similar steps forŷ(k) − y(k) gives Converging e(k) to zero proves relation (10). Thus, the proof of Theorem 1 is completed.

Remark 1.
The matrix G can be defined as follows: then, the designed observer poles are roots of polynomial function x s −g s x s−1 −···−g 1 = 0. Therefore, to have the desired performance, one can select proper poles and determine the corresponding polynomial function coefficients (g i for i = 1,..., s).

Observer Design in the Case of Unequal Delays
This section considers the extension of the proposed observers (6) and (7) to deal with unequal measurement delay. The proposed observer consists of m chain-observers, each one being in the form of (6) and (7). Without loss of generality, let τ 1 ≤ τ 2 ≤ · · · ≤ τ m and define Before explaining the proposed approach of designing the observer considering unequal delay, a theorem that provides stability conditions for a system with multiple delays is presented. This theorem is an extension to the single delay case, which was presented in [20]. The result of this theorem enables us to design an observer with the convergence property.
Theorem 2. Suppose the system model with the assumption A 0 2 = 0, is given by where N is the value of time delays. Then, it is asymptotically stables if there exists a real symmetric matrix P > 0 so that: Remark 2. When A 0 2 = 0, system model (21) can be rewritten as Then, the augmented model becomes Then, Theorem 2 can be applied for stability analysis of (25).
Suppose that due to different sensors or measurement techniques or communication topologies, each output measurement is received with a delay that may be different from delays in other output channels. Therefore, in order to estimate the state x(k) at sample time k, the output y i is only available up to k − τ i for i = 1, . . . , m. The observer gains can be parameterized by the set of the available measurements. In order to handle the unequal measurement delays, we propose constructing a new observer scheme. The proposed observer scheme is shown in Figure 2, in which y i (1 : s) is the output i from sample 1 to s. The proposed observer gains can be calculated in a composition of m chain-observers, each of which predicts the unavailable output due to measurement delay as well as updates the values of the estimated states and then passes them to the next chain. The following theorem provides the design of the asymptotically stable chain-observer with the consideration of unequal measurement delays provided that the system model is available.
in Equation (1) are known, and τ = [ τ1 … τm] represents unequal measurement delays, and define 1 = 1 , = − −1 , = 2, … , . The design of -chain-observer is achieved by solving the chain-Luenberger equations for = 1, … , chainobservers. The observer is designed using the estimated output ̂+ 1 ( − ) for = , … , − 1 provided by the previous chain-observers from 1 to − 1 as Each chain-observer satisfies the following convergence conditions: where for = 1, … , should be chosen so that it is stable and should satisfy the stability condition Theorem 2 as where > 0, = − ∏ = +1 . It follows then, that the constructed observer is asymptotically convergent, and the observer output equation delivers unbiased estimation for ( ), that is, Proof. See Appendix B. □ . . , m. The design of m-chain-observer is achieved by solving the chain-Luenberger equations for i = 1, . . . , m chain-observers. The observer i is designed using the estimated outputŷ j+1 (k − τ i ) for j = i, . . . , m − 1 provided by the previous chain-observers from 1 to i − 1 as Each chain-observer satisfies the following convergence conditions: where G i for i = 1, . . . , m should be chosen so that it is stable and should satisfy the stability condition Theorem 2 as It follows then, that the constructed observer is asymptotically convergent, and the observer output equation delivers unbiased estimation for y(k), that is, To design an observer with unequal output delay, Theorem 3 can be utilized to construct each chain observer provided that g i (i.e., G i ) in each chain must be selected in such a way that the stability condition (29) is satisfied. Therefore, the chain observer can be designed using the following algorithm (Algorithm 1):

Algorithm 1. Chain observer design algorithm
Step 1: set i = m Step 2: Select a proper G i that satisfies stability conditions (29).
Step 3: Design the chain-observer i using Theorem 1 with d = d i .
Step 4: Produce the complementary output y(k − τ i−1 ) using available output and input, then estimate output data by Equation (27).
Step 5: Go to step 2 and repeat till i = 1.

Simulation
In this section, the effectiveness of the proposed approach was demonstrated on a simulation setup of a continuous stirred tank heater (CSTH). The system is written as a linear state-space model to generate input-output data with the following parameters [16]: V out ≡ u 1 , P h ≡ u 2 are two input variables, and the water level h T , the temperature of the water in the tank T T as well as in the heating jacket T hj are output variables. The physical meanings of the process variables and parameters used above are listed in Table 1. The output variables had the delay vector of τ = [2 5 7]. The order of the observer was s = 3. Set g 1 = [10 −2 , −3 × 10 −8 , 3 × 10 −4 ] so that all three poles were located in 10 −4 at each chain. Then, we designed a chain-observer using Algorithm 1 when m = 3.
The following is the design procedure:  The designed chain-observer (Method 1) was compared with the classic observer (Method 3), which ignores the measurement delays, and the observer (Method 2) [21], which supposes (one) equal measurement delay (delay = 2 in this test). The method proposed in [21] deals with both state and output delays; in this test, state delay is supposed to be equal to zero. Figures 3-5 show the estimated values for the system states x 1 , x 2 , and x 3 by the above-mentioned observers, respectively. These figures show that the proposed observer estimation error converges to zero in 10 s. Observer #2 (considered equal delays) had a bias in estimation and was slow in convergence, and observer #3 (ignored delays) diverged. The input signal to the system is depicted in Figure 6.

Remark 3. Based on the results presented above, we can conclude that
• By changing the observer poles, the convergence speed can be tuned.

•
Each chain is an observer in the form of (6).

•
Increasing the communication delays may increase the convergence time and transient estimation error magnitude, but according to theorem 3, it will not make the observer unstable.

Conclusions
Delays are an inherent part of any system. Ignoring it could cause instability or degradation in the performances of the control or estimation blocks. Measurement and communication lines produce some delays, which may be unequal due to different line

Remark 3. Based on the results presented above, we can conclude that
• By changing the observer poles, the convergence speed can be tuned.

•
Each chain is an observer in the form of (6).

•
Increasing the communication delays may increase the convergence time and transient estimation error magnitude, but according to theorem 3, it will not make the observer unstable.

Conclusions
Delays are an inherent part of any system. Ignoring it could cause instability or degradation in the performances of the control or estimation blocks. Measurement and communication lines produce some delays, which may be unequal due to different line

Remark 3.
Based on the results presented above, we can conclude that

•
By changing the observer poles, the convergence speed can be tuned.

•
Each chain is an observer in the form of (6).

•
Increasing the communication delays may increase the convergence time and transient estimation error magnitude, but according to theorem 3, it will not make the observer unstable.

Conclusions
Delays are an inherent part of any system. Ignoring it could cause instability or degradation in the performances of the control or estimation blocks. Measurement and communication lines produce some delays, which may be unequal due to different line distances or sensor types and technologies. Supposing equal measurement/communication delays causes a similar problem to ignoring them. In this paper, we proposed a new approach to constructing an observer, consisting of a bank of chain-observers to deal with equal and unequal measurement delays. Analytical results for the construction of the observers are provided, and convergence of the chain-observers is shown. Simulations demonstrate the efficacy of the proposed design scheme through a CSTH system in the presence of measurement delays. Having a chain-based observer design facilitates the extension of the proposed approach to a data-driven method. Therefore, the advantage of the proposed approach is that it can be used for designing data-driven (subspace-based) observers [17] in the presence of equal and unequal communication delays. Designing such a data-driven observer is in our future research scope. Data Availability Statement: Data sharing is not applicable (no new data were created or analyzed in this study, therefore, data sharing was not applicable to this article).

Acknowledgments:
The authors gratefully acknowledge the financial support from Alberta Innovates.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Proof of Theorem 2
Before proving Theorem 2, the following lemma is presented.
Lemma A1. For any two matrices F and G of dimension n × m and for any square matrix P = P T > 0 of dimension n, the following matrix inequality holds, where ε is a positive constant.
Now, to prove Theorem 2, consider the following Lyapunov function: Taking forward the difference from the above equation gives: Applying Lemma A1 on inequality (A3) yields Now, based on Cauchy-Schwarz inequality [22], namely for any real vector v i , the Taking two factors x T k − d j and x k − d j out of the left-and right-hand side of two last terms in Equation (A5) yields Since matrices A T 0 PA 0 and A T j PA j are symmetric and positive semi-definite, based on [23], we have The minimum of the function g can be found by taking its derivative with respect to the parameter ε, which yields Therefore, the optimum value for ε is We can conclude A j T PA j − P < 0 is satisfied, then considering Equation (A8), we have The proof is completed.