Performance Assessment of distributed control systems with communication delays

Yousef Alipouri, Biao Huang

Department of Chemical and Materials Engineering, University of Alberta, Canada T6G 2V4

{alipouri; bhuang }@ualberta.ca

Abstract:

In distributed control systems, a local regulator does not have complete knowledge of the whole system. Therefore, some information is transmitted between local regulators through communication lines. The delays in communication lines are not negligible and they can make the closed loop system unstable or impose a limit on the achievable control performance. Due to the process and communication delays, controllers cannot reject all disturbances. Although there has been a plethora of literature available for assessing control performance in presence of process time delay, no similar study has been conducted for distributed control systems in presence of communication delays. In order to address this problem, a new solution for performance assessment of distributed control systems, which takes communication delays into account, is proposed in this paper. Further, the corresponding minimum variance controller is also derived under the distributed control framework.

Keywords:

Distributed Control Performance Assessment; Distributed Minimum Variance Controller; Communication Delay; Unitary Transfer Function Matrix.

Introduction

Control performance assessment (CPA) is an important asset-management technology used to monitor controllers and maintain the performance of automation systems in production plants. Since decreasing output variance has implications such as improving product quality and reducing wear of the actuators, minimum variance-based performance index, which compares the implemented controller performance with the minimum variance (MV) benchmark, is one of the most important indices for controller performance assessment (CPA) 1 in the literature as well as in industrial practice. The MV benchmark is feedback invariant and it depends on the process delays and stochastic disturbances under the centralized control framework.

The field of CPA began to blossom around 30 years ago with the study by Harris (1989). Since 1989, the research activities on performance assessment have mainly consisted of (i) control performance assessment for univariate controllers 2,3, (ii) control performance assessment for multivariate controllers 4,5, (iii) control performance assessment with linear quadratic Gaussian (LQG) benchmark 2,6, (iv) economic analysis of advanced control systems such as model predictive control (MPC) systems 7,8, (v) monitoring of control actuator nonlinearity 9,10, (vi) isolation and diagnosis of the underlying root causes responsible for performance degradation 11 and so on.

All of these CPA methods rest on the common assumption of univariate control loop or centralized control in the case of multivariate processes, which means either 1) all the information available about the system, and the calculations based upon the information are centralized, 2) control action calculations take place at a single location and communication delays do not exist. Communication delays, similar to the process delay, limits the achievable control performance, and thus alters the achievable performance (such as MV benchmark). Though there is a plethora of literature available for assessing control performance, the following two fundamental questions have not been addressed in the literature: 1) What is impact of communication delays on the MV benchmark? 2) How is the optimal MV controller designed considering that controller structure is restricted by distributed framework? Answering these questions will allow us to calculate the CPA index for distributed control systems.

In 12, a k-step ahead prediction along with the optimal prediction error variance under the distributed framework without considering exogenous inputs has been derived. According to 13, the k-step ahead prediction error variance is same as the variance under minimum variance control when the k is taken as the process delay. In this paper, we extend 12 to performance assessment of distributed control systems involving both input and output communication delays by estimating distributed MV benchmark as well as designing the corresponding distributed MV controller. Two kinds of communication delays are considered and they include 1) input communication delay which is the delay in transmitting control signals between local regulators; 2) output communication delay which is the delay in transmitting process outputs to local regulators. In order to calculate the distributed minimum variance, unitary transfer function matrices are introduced, which converts centralized control performance indices to that of distributed control. In this paper, it is assumed that each local regulator has access to all the information of the system and other local regulators but with delays. The solution approach is discussed in two sections: In the first section, the delays in all communication lines are assumed to be equal (namely equal communication delay). In the next section, utilizing the method proposed in the first section, the distributed MV benchmark is extended to the general case in which each communication line can have different delays (namely unequal communication delays). The main contribution of the manuscript is the determination of the MV benchmark along with the design of the MV controller by taking into account the communication delays and control structure restriction.

The remainder of the paper is outlined as follows: In the next section, centralized minimum variance benchmark and challenges in assessing control performance under the distributed control framework are explained. In section III, the distributed MV benchmark is derived and the corresponding distributed MV controller is obtained. Section IV presents a simulation example. Conclusions are given in Section V.

We use bold capital letters for transfer function matrices, capital letters for constant matrices, small bold letters for vectors, and small letters for scalars. For ease of representation, the backshift operator z-1 will be omitted.

Preliminaries and Problem Statement

In this section, an introduction to the classic MV benchmark estimation, the concerned distributed control structure and challenges in estimating MV benchmark under the distributed control framework is provided.

II.1) Centralized MV Benchmark and Corresponding MV Controller

In determining the MV benchmark, the control objective is to minimize the process output variance. The achievable lower bound of output variance is named as the MV benchmark. This benchmark provides us useful information such as the improvement potential of the current control system. We consider the linear discrete Multi-Input Multi-Output (MIMO) process:

(1)

where is the back shift operator, is the n×m transfer function matrix of the process; is the n×n transfer function matrix of the disturbances; is the vector of process outputs ; is the vector of process inputs ; represents a vector of disturbances which are white noise with zero mean and covariance ?d.

Assume is a full rank transfer function matrix. For simplicity, we assume that and contain no non-minimum phase zeros. For this process, there exists an n×n unitary interactor matrix 14. The interactor matrix is a unitary transfer function matrix (i.e. ). Multiplying it to the system transfer matrix () will make it delay free. i.e.:

(2)

where is the delay-free transfer function matrix of , and K is a full rank constant matrix. The interactor matrix can be considered as the generalization of univariate system time delay to the multivariate system. In equation (2), contains the infinite zeros of , and is an invertible transfer function matrix which only contains the finite zeros of . The important property of a unitary matrix is that it does not change the spectral properties of a filtered signal which is useful in MV benchmark estimation 14. The interactor matrix can be written in the Markov parameter representation as

(3)

where d denotes the order of the interactor matrix and is unique for a given transfer function matrix, is the relative degree of the interactor matrix and Di are the coefficient matrices. The interactor matrix can be calculated using the given transfer function matrix or by using data of the system according to the algorithm proposed in 14,15.

Using Eq. (2), the process model in Eq. (1) can be rewritten as

(4)

Multiplying interactor matrix to both sides of Eq. (4) and defining the interactor-filtered output , Eq. (4) can be transferred to a process with a single common time delay d, as

(5)

where is a proper transfer function matrix. Now, the transfer function matrix can be further split into past and future terms by the following Diophantine equation (also known as the polynomial division identity),

(6)

where F is a polynomial of z-1 and R is the remaining proper and rational transfer function matrix in z-1.

Substituting Eq. (6), in Eq. (5), gives:

(7)

The last term in the above equation cannot be affected by the control action; hence it can be shown that 16:

(8)

where cov stands for the covariance operator. Therefore, the lower bound of the following cost function is

(9)

where J is known as the MV cost function, and is its lower bound and also known as MV benchmark. The MV benchmark is invariant regardless of the control structure. From equations (6) and (9) it can be seen that the MV benchmark is solely determined by the process time delay and the dynamics of unmeasured disturbances.

Considering that the interactor matrix is unitary, we have

(10)

where stands for centralized MV controller. Eq. (10) shows that the MV benchmark is not altered by filtering the output with interactor matrix. Thus, minimization of the interactor-filtered output variance is equivalent to minimization of the real output variance.

The MVC (or MV benchmark control) is achieved when the sum of the first two terms on the right-hand side of Eq. (7) is set to zero, i.e 16

(11)

After some manipulations, the explicit expression of the control law can be obtained 16:

(12)

or

(13)

where. Considering that and in the above equation are delay free, the controller is realizable.

It is clear from the above explanation that there are two main steps in MV benchmark determination or MV controller design, namely: 1) calculating the interactor matrix Eq. (4), which is a result of extracting process delay from the process model (); 2) splitting noise sequence to past and future terms using the Diophantine Equation, Eq. (6), which is a result of applying process delay to the disturbance model ().

II. 2) Problem Statement

Eq. (13) is a valid MV controller expression under the centralized framework in which there exist no communication delays. According to Eq. (13), in order to generate the MV control signal at sampling instant t, all input and output values up to t are required, i.e.

(14)

where

As explained in the introduction section, it is crucial to take communication network into consideration when studying distributed systems. In a communication network where signals are often transmitted among adjacent local subsystems, the problematic part is presence of communication delays. Hence, it is strongly recommended to account for communication delays in the procedure of designing distributed controllers.

As explained in the introduction section, communication network has become an essential part of data transmission for distributed systems. The main issue of a network transmission is the introduction of time delays (communication delays) when receiving signals from adjacent local subsystems. Therefore, communication delays must be considered when designing controllers for distributed systems. Due to communication delays, some of output and input signals from adjacent local subsystems which are required to produce the control signals in (14) are not available at sampling instant t. To explain it, consider the distributed configuration with communication delays as shown in Fig. 1. Under this framework, components located on networks communicate and coordinate their actions by passing messages to each other. The components interact with each other in their regular operation. The communication delays in input signals and output signals are du and dy, respectively. The input and output communication delay matrix du and dy are defined as:

(15)

where denotes delay value in transmitting input data from local regulator i to j, and denotes delay value in transmitting output data from system output j (yj) to local regulator i.

Problem – According to Eq. (13), in order to produce MV control signal , all system output and input signals up to sample time t are required; however, yj(s), s=t, …, t- for j=1,…n, i=1,…m and uj(s), s=t, …, t-for i,j=1,…m, due to output and input communication delays, respectively, are not available for the local regulator i, for i=1,…,m. The set of missing data giving set of their indices and the input and output communication delay matrices are defined as:

(16)

The achievable distributed MV controller in the presence of communication delays, with assumption (without loss of generality) of and , should have the following form (structure):

(17)

One trivial solution to obtain the achievable MV controller with missing data and is to simply set missing data coefficients in the optimal centralized MV controller (13) to zero, i.e.

This approach is not only non-optimal but may also introduce instability to the system. In order to determine the achievable optimal controller (determine optimal value for and ) the distributed MV controller is proposed in the following sections.

Fig. 1- Structure of distributed minimum variance controller considered in this paper

Distributed MV benchmark and controller

III. a) Case of equal communication delays:

Let us consider a simple case first in which all communication delays are equal, i.e , and , j=1,…n, i=1,…m. According to Eq. (14), when communication delays are equal, the achievable MV controller should have the form of:

(18)

Writing the above equation in vector form yields

(19)

where and are delay-free transfer function matrices, and are constant matrices whose elements are and , respectively. In this subsection, we shall determine optimal and .

Theorem 1 (Case of equal communication delay)-For the linear system which is described by model (1) and has the control structure shown in Fig. (1) with equal communication delays, the distributed MV benchmark () is

(20)

and the distributed MV control law with achievable structure (20) is

(21)

where Kdis, , Fdis, and Rdis will be defined in the following proof process.

Proof.

Similar to the centralized case, Theorem 1 can be shown in two steps: 1) considering the impact of the input communication delays on the process model; 2) calculating the impact of the output communication delays on the disturbance model.

Step-1: Considering impact of the system input communication delays on the process model- First, there exists a unitary transfer function matrix D1 (), which satisfies

(22)

or

(23)

where D1 is given by:

(24)

Appendix A demonstrates how the unitary transfer function matrix D1 which satisfies Eq. (23) can be constructed.

Now, considering Eq. (22) and multiplying D1 to both sides of Eq. (5), gives

(25)

where .

It is worth to note that due to the unitary nature of D1 and considering Eq. (10), we have

(26)

where stands for the MV controller whose structure is restricted by distributed framework.

Step 2: Considering impact of the output communication delay on the disturbance model – The term in Eq. (25) can be further split into unpredictable and predictable components using the Diophantine identity as:

(27)

Then, equation (25) can be rewritten as

(28)

Now, according to Eq. (26) and considering that the last term () is feedback invariant, the controller that minimizes the cost function is obtained by making the remain terms in right hand side of Eq. (28) to zero, resulting in:

(29)

or

(30)

Applying controller (30) to system (28), yields , thus we have

(31)

According to Eq. (25) and definition of , we can write , where and are the interactor-filtered and real optimal process output under control of the distributed MV controller (30), respectively. Thus substituting Eq. (31) into Eq. (30), yields:

(32)

Theorem 1 is thus proved.

Remark 1 – Decentralized MV benchmark is achieved when setting .

Remark 2 – Centralized MV benchmark is achieved when setting .

Remark 3 – As multiplication of two unitary matrices is still a unitary matrix, instead of calculating two unitary transfer function matrices and separately, a new unitary transfer function matrix (we name it distributed interactor matrix) can be directly calculated by . To doing this, the two equations (2) and (22) are combined as

(33)

where Ddis is given by:

(34)

Remark 4 – Comparing Eq. (27) and (6) shows has all terms of , thus we can conclude the distributed MV benchmark is greater than or equal to centralized MV benchmark, i.e.

(35)

III.b) Case of unequal communication delays:

Now, consider the control framework of Fig. (1) in which each local regulator communicates with each other with un-equal delays. Further, without loss of generality, assume that and , where max stands for the maximization operator.

In this section, we introduce another unitary transfer function matrix D2 which converts the centralized controller expression (14) to the one compatible with the distributed control structure shown in (17), when the communication delays are un-equal. Therefore, in this section, the optimal values for defined at Eq. (17) are to be determined.

The vector form of Eq. (17) can be written as

(36)

where,

(37)

where and are matrices whose elements are and , respectively. By comparing Eq. (36) with Eq. (19), we see that and are new terms added because of unequal input and output communication delays. They will be zero when input and also output communication delays are identical (see Eq. (19)).

Theorem2 – For the linear system which is described by model (1) with the communication line structure shown in Fig. (1) (with un-equal communication delay), the optimal distributed MV benchmark is

(38)

and the distributed MV control law with achievable structure (36) is

(39)

where , , , , and are defined in the following proof process.

According to Theorem 2 and comparing Eq. (39) with Eq. (36), the optimal parameters for the achievable MV controller (36) are , and .

Similar to proof of Theorem 1, Theorem 2 can be proven in two steps

Step 1: Considering effect of the input communication delay on the process model- First, we define a new unitary transfer function matrix D2, which satisfies.

(40)

where is a new term added because of un-equal input communication delay and defined as

where is a matrix with elements

where denotes the element located at row k and column j of matrix .

Appendix B illustrates how D2 is determined.

Now, multiplying D2 to both sides of Eq. (5) and considering Eq. (40), yields:

(41)

Step 2: Considering impact of the output communication delay on disturbance model- Term in Eq. (41) can be divided into three parts: future term, output communication delay dependent term and the remaining term, i.e.

(42)

where

The transfer function matrix and the remaining term in Eq. (42) can be determined using the Diophantine equation. It is worth to note that the number of terms in is more than the terms in classic centralized MV benchmark , defined in Eq. (6), due to output communication delay dy. Substituting Eq. (42), equation (41) can be rewritten as

(43)

Distributed MV benchmark term is invariant with respect to the control signalas it represents future disturbances. The distributed MV controller can be obtained by setting the remaining terms in Eq. (43) to zero as

(44)

Now, substituting Eq. (44) in Eq. (43), and considering , yields

(45)

where and are the output and filtered process output (by ) under control of the distributed MV controller (44), respectively. Substituting Eq. (45) in Eq. (44), gives:

(46)

Comparing it to the achievable structure of MV controller (36), the control signal in Eq. (46) is still not achievable yet (compare last terms of both equations). Therefore, there is still need to revisit the last terms in Eq. (45) and Eq. (44). Now, we define another unitary transfer function matrix which satisfies:

(47)

where is a matrix with elements

where denotes element located at row k and column j of matrix .

Appendix C explains how to calculate D4 which satisfies Eq. (47).

Utilize D4 to eliminate the terms of the output signal which are unavailable at sampling time t because of output communication delay. To show this, Eq. (42) is updated as:

(48)

where .

Now, substituting Eq. (48) in Eq. (41), gives:

(49)

Repeating the steps from Eq. (44) to (46) on Eq. (49), gives

(50)

Substituting Eq. (47), in Eq. (50) gives:

(51)

Comparing Eq. (51) with Eq. (36), it is clear that the distributed MV controller Eq. (51) is achievable now, where , , and .

Considering the remaining term after substituting the controller (51) in Eq. (49), it is ; therefore, the distributed MV benchmark is

(52)

Proof of Theorem 2 is now completed.

It must be noted that since D4 is a unitary transfer function matrix, we have

(53)

Therefore, steps from Eq. (47) to (52) are required to determine the distributed MV controller. However, in order to estimate the distributed MV benchmark, can be utilized directly and there is no need to calculate the unitary transfer function matrix D4.

Similar to remark 4, we have remark 5.

Remark 5 – Comparing Eq. (42) and (6) shows contains all terms of , thus we can conclude the distributed MV benchmark with un-equal communication delays is greater than or equal to centralized MV benchmark.

The required steps to estimate MV benchmark as well as design MV controller considering input and output communication delays are as follows.

Algorithm-1: Steps to calculate MV benchmark and controller with communication delay

Identify the model of the systems Eq. (1)

Calculate the interactor matrix based on the model in Eq. (2)

Calculate unitary transfer function matrix D2 (see Appendix B)

Calculate using Diophantine Eq. (42)

Calculate the distributed MV benchmark ()

To design the MV controller, calculate unitary transfer function matrix D4 using Eq. (47) (see Appendix C)

Determine the distributed MV controller using (51)

Numerical Example

In this section, we will use a numerical example to demonstrate the calculation procedure of the proposed distributed MV benchmark and the MV controller, and then compare it with classical centralized (regular) MV.

Consider a 2 × 2 minimum-phase multivariable process with the open-loop transfer function matrix and disturbance transfer function matrix given by (in this test d=4):

(54)

where, the white noise, d(t), is a normal-distributed white noise of dimension 2 with zero mean and covariance ?d = I.

The interactor matrix without considering any communication delay can be calculated as 17

(55)

The centralized MV benchmark for this system (when d=4) is 4.24 17. Now, suppose that the unequal input and output communication delay matrices are

(56)

Following the proposed algorithm-1, the unitary transfer function matrix D1 which converts the process delay free model (54) (i.e. ) to the equal communication delay structure Eq. (22) is:

(57)

The resulting Kdes as defined in Eq. (22) is

(58)

Therefore, using Appendix B, D2 can be determined as:

(59)

Multiplying D2 to the delay free , results in delay-free form as:

(60)

Now, in the second step, the noise sequence is decomposed to three parts: future term, output communication delay dependent term and remaining term. Using Eq. (42), the feedback invariant term is

(61)

Then the distributed MV benchmark can be calculated from Eq. (61) and (53) as 4.8038.

According to step six of Algorithm-1, to design the distributed MV controller, D4 should be calculated according to the procedure shown in Appendix C. According to Eq. (60), , and solving Eq. (47) yields,

(62)

Finally, the distributed MV controller for the process with communication delays given in (56), is

(63)

Figure 2 shows response of the distributed MV controller (63). The obtained results for the first and second outputs variances are 1.006 and 3.815, respectively, therefore the controller (63) achieves the performance 4.815 in terms of cost function (9), which is close to the optimal MV benchmark 4.8038.

Fig.2- Shows response of the distributed MV controller (63).

The distributed MV benchmark can be calculated for different equal input and also equal output communication delays. Figure 3 shows the distributed MV benchmark estimation for system (54) with various equal communication delays. For this special case, based on Fig. 3, the output communication delay has higher impact on distributed MV benchmark than the input communication delay. In addition, the distributed MV benchmark converges by increasing both output and input communication delays.

Fig.3- Impact of input and output communication delays on the distributed MV benchmark

Conclusion

Compared to centralized control, distributed control has limited overall control performance because of communication delays, as well as because of the limitations in obtaining global information within each subsystem. Though there is a plethora of literatures available for assessing control performance for centralized control systems, they cannot be directly applied to the performance assessment of distributed control systems. Our focus in this study was therefore on the performance assessment of distributed control system. Distributed MV benchmark is derived in this work to assess the limit of control performance caused by distributed control structure. In this process, four unitary transfer function matrices were derived. Applying them to the system model yields a structure that can be used to determine the benchmark for distributed control systems. Further, the distributed minimum variance controllers are also derived in this work.

Appendix A

To determine the unitary transfer function matrix D1, consider that the Markov parameters representation of a transfer function matrix can be written as:

(A-1)

Then to satisfy Eq. (23), we have

(A-2)

Thus, we have

(A-3)

The above algebraic equation can be further written in matrix form as:

(A-4)

To calculate the unitary transfer function matrix D1 which satisfies equation (A-4), the procedure proposed in 14,15,17 can be utilized.

Appendix B

we illustrate how can be determined. First, we find a unitary transfer function matrix D3 as

(B-1)

where is a constant matrix.

The unitary transfer function matrix can be calculated in a way similar to calculating D simply by replacing instead of in Eq. (2).

Now multiplying to both sides of Eq. (40) and setting

(B-2)

yields

(B-3)

Therefore, considering and comparing Eq. (B-3) and Eq. (22), we can conclude:

(B-4)

Eq. (B-4) is utilized to calculate D2.

Figure B-1 illustrates the conversion between three forms: centralized, equal and un-equal communication delay structures using unitary transfer function matrices D1, D2 and D3.

Fig B-1. Transition between defined distributed control structures

Appendix C

Here, we show how D4 can be calculated from Eq. (47). Multiplying both sides of Eq. (47) to gives

(C-1)

Now, the calculation of is similar to the calculation of D2 in Eq. (40); we use in place of and use in place of in Eq. (40).