Performance Assessment of Cascade Control System with Non-Gaussian Disturbance Based on Minimum Entropy

Due to the fact that cascade control can improve the single-loop’s performance well and reduce the integral error from disturbance response, it has been one of the most important control strategies in industrial production, especially in thermal power plant and chemical engineering. However, most of the existing research is based on the Gaussian system and other few studies on the non-Gaussian cascade disturbance system also have obvious defects. In this paper, an effective control loop performance assessment (CPA) of cascade control system for many non-Gaussian distributions even the unknown mixture disturbance noise has been proposed. Compared to the minimum variance control (MVC) approach, the minimum entropy control (MEC) method can obtain a more accurate estimate. In this method, like MVC, the primary loop output and secondary loop output can be represented as invariant and dependent terms, then adopted estimated distribution algorithm (EDA) is used to achieve the system model and disturbances. In order to show the effectiveness of MEC, some simulation examples based on different perturbations are given.


Introduction
As the cascade control can improve the performance better than the single loop, and reduce the deviation from the disturbance response, cascade control loop has been widely used in chemical industrial, thermal power plant, and DC motor [1,2], especially in oil refinery and high pressure drum water control.However, although the initial cascade control loop can meet the designing performance specifications, over time, the control loop's performance will deteriorate due to many factors, including the maintained controllers and the aging of the equipment, without or insufficient feed-forward compensation [3].The study that is proposed by Honeywell illustrated that nearly half of the control system were working on the poor performance situation [4].If the deterioration of the control loop still sustains without any identification, it will cause an economic loss of one hundred million dollars, even leading to the destruction of environment and threatening the safety of industrial workers.
Therefore, since the 1980s, many methods for system performance evaluation have appeared [5].Since these algorithms use statistical knowledge to analyze system performance, they are also known as statistical process control, but they still fail to give the performance of the current loop.Hence Harris proposed one performance index by using the MVC [6].It represents the starting of the research based on MVC among the CPA field.The performance of the control loop can be given only by system output whatever delay time is known or not.Then more research had been done in this field.In 1996, Harris proposed a performance evaluation method for a multiple-input multiple-output (MIMO) feedback control system [7].Although the MVC controllers can meet the control performance requirement, the system robustness perform poorly.Hence, a generalized minimum variance (GMV) concept was raised by Grimble, he used this benchmark as single-loop CPA index and designed the corresponding controller based on GMV concepts.Unfortunately, it is necessary to add errors and control weighting factors to the control loop.Unless a fixed weight value is used, it is difficult to choose the appropriate weighting factor unless a constant weight is used.Huang and Shah presented a linear quadratic Gaussian (LQG) benchmark based on the MVC method [8].Since the minimum variance of the output is only calculated compared to MVC, LQG also needs to calculate the minimum variance of the input, so for the same system, the LQG benchmark can better reflect the gap between the current actual performance and the ideal performance.However, compared to the traditional MVC benchmark, LQG is too complicated to implement in real industrial processes.In addition, many new CPA means has been put forward with further research.Such as the Hurst index is presented by Srinivasan, although it requires no prior knowledge [9,10], under the mixture noise environment this method may cause severe error.Sadasivarao tried to import the generic algorithm to cascade control loop to design optimal PID controller [11].Yu uses CPA tools to get random sample jitter to show performance impact [12].Ye presented a method about how to design the optimal cascade loop [13].
Although all these methods have been quite mature, all previous research on CPA was based on the assumption that interference interferes with Gaussian distribution.In fact, in real industrial processes, the disturbance of the control loop may be not only Gaussian noise, but also beta noise, or even a mixture of multiple unknown noises.In such a situation, for one system with non-Gaussian noise, those methods above maybe not a good benchmark for cascade control loop.In order to solve this problem, the MEC benchmark has been proposed.However, in the past publications [14][15][16], there is very little research in this area, especially for the cascade control loop.The research on cascade control loop based the MEC is presented by Jianhua Zhang [17], he proposed the Renyi Entropy to deal with systems in which disturbance is non-Gaussian.However, the method he used have been proved incorrectly [18,19], because he ignored the linear relationship when calculated the discrete entropy.In literature [19], the method is only used in single loop system.For cascade control system, the derivation of algorithm and the iteration of estimated distribution algorithm (EDA) need to be re-derived.In literature [20], Zhang proposed one improved Renyi Entropy method based on the mean value, it overcomes the mean shift of the output, but the efficiency of the algorithm reduced seriously.
Therefore, in this paper, one effective benchmark based on rational entropy is raised.It aims to establish an optimal MEC benchmark for cascade loop system with non-Gaussian disturbance.Also, the EDA is used to estimate the distribution of disturbance and identify the system ARMA model parameters, then the CPA benchmark is given.The efficiency can be improved greatly.
The structure of this paper is as follow: In Section 2, the conventional MVC is reviewed.In Section 3, the MEC will be introduced and the method of how to obtain estimated parameter vector and disturbance based on improved EDA algorithm is illustrated in detail.In Section 4, a simulation case is given.In Section 5, a conclusion is made.

Introduction of Cascade System
Consider the general cascade control system with unity feedback in Figure 1, where the r is the set point, e(k) is the output error, u i (k), i = 1, 2; is the controller output, a i , i = 1, 2; is the system disturbance, G Li , i = 1, 2 is the disturbance filters, the effect of all unknown disturbances to the two loops are represented by them.G i , i = 1, 2; is the primary and secondary loop controllers, for G ci , i = 1, 2;, it denotes the transfer function of feedback controller, k is the sample time, and y i (k), i = 1, 2 is the system output of outer loop and inner loop.The output control loop transfer Symmetry 2019, 11, 379 3 of 15 function is r and the inner loop transfer function is (k) .The discrete shift k is substituted by the backward shift q −1 .Obviously, the output is where the primary loop transfer function is G 1 (q −1 ) = G * 1 (q −1 )q −d 1 with the time delay equal to d 1 , and G * 1 is the primary transfer model without any time delay.Similarly, the inner loop function G 2 (q −1 ) can be written as G 2 (q) = G * 2 (q −1 )q −d 2 and the inner loop time delay is d 2 .Both the two disturbance a 1 (k), a 2 (k) are assumed as a rational function with the backward shift q −1 .For simplify, r is usually set to zero.
the two loops are represented by them., 1, 2; k is substituted by the backward shift 1 q − .Obviously, the output is where the primary loop transfer function is with the time delay equal to G is the primary transfer model without any time delay.Similarly, the inner loop function r is usually set to zero.

MVC Strategy of Cascade Control System
The MVC method based on cascade control system decomposes the system output into two parts by using the Diophantine equation.For simplify and universality, make the system input equal to zero, then the Equation (1) can be decomposed as

MVC Strategy of Cascade Control System
The MVC method based on cascade control system decomposes the system output into two parts by using the Diophantine equation.For simplify and universality, make the system input equal to zero, then the Equation (1) can be decomposed as The disturbances transfer function can be replaced by following Diophantine equations, where Q 1 is the polynomial of q −1 with order of d 1 + d 2 − 1 and Q 2 , S 2 are polynomials of q −1 with order of d 2 − 1. R 1 , R 2 ,T 2 are residual polynomial coefficients of the Equations ( 4)- (6).Substituted Equations ( 4)-( 6) into Equation (1), the outer loop output y 1 (q −1 ) is given as, Feedback−Dependents (7) Specifically, the polynomials Q 1 (q −1 ), S 2 (q −1 ) are feedback invariant, they only depend on the unknown-disturbance and they can be decomposed as (8) Thus, the feedback-invariant term is represented as an AR model, the model is used as the performance benchmark of MVC.When M 1 = M 2 = 0, the optimal controller can be obtained and the MV of where Thus, MVC index for cascade control can be defined as: where σ 2 y1 mv is the optimal MV of the system output, and the σ 2 y1 is the variance of actual output.In the literature [4], Harris had noted that the index under the MVC is only valid for the noise which is subject to Gaussian distribution [21][22][23][24][25][26].However, most external disturbances in realistic industrial processes are non-Gaussian.Thus, the MVC based on CPA is not reliable.Consider the following system from literature [15], where the disturbance a 1 (q −1 ), a 2 (q −1 ) is subject to β distribution, the probability density function (PDF) of it is given by, where I (−1,1) (x) make f (x) ∈ (−1, 1).For a 1 (q −1 ), the parameter a = 9, b = 4, similarly, a 2 (q −1 ), a = 3, b = 9.According to the MVC method, the PI controller G c1 and the P controller G c2 can be used as, G c1 = (0.2848−0.2568q −1 ) and G c2 = 0.2.The normal probability plots are shown in Figure 2. From the Figure 2, we can find that the actual and estimated Freq are shift on the x-axis.The MVC algorithm is not sensitive to the means shift.When the noise is subject to Gaussian distribution which means that the mean is zero, the actual and estimated frequency will not reflect the mean shift.However, when the noise is subject to non-Gaussian distribution, its mean is non-zero.Hence, the actual frequency and estimated frequency are shift on the x-axis.This is a serious defect for performance evaluation.The corresponding distribution plots are shown in Figure 3. .The normal probability plots are shown in Figure 2. From the Figure 2, we can find that the actual and estimated Freq are shift on the x-axis.The MVC algorithm is not sensitive to the means shift.When the noise is subject to Gaussian distribution which means that the mean is zero, the actual and estimated frequency will not reflect the mean shift.However, when the noise is subject to non-Gaussian distribution, its mean is non-zero.Hence, the actual frequency and estimated frequency are shift on the x-axis.This is a serious defect for performance evaluation.The corresponding distribution plots are shown in Figure 3.    , respectively.According to this example, using the MVC method for a non-Gaussian system may result in erroneous results.Figure 3 illustrates that the disturbance is subject to non-Gaussian distribution because the plots deviates the straight line.The response with different disturbance of cascade control are shown in Figure 4, and the inner and outer loop output distribution are shown in Figure 5.As can be seen from Figures 4 and 5, the performance of the main output y 2 is significantly better than the performance of the secondary loop output y 1 , but the MVC indices of the two loops are diametrically opposed, they are η y 1 = 0.5290 and η y 1 = 0.5092, respectively.According to this example, using the MVC method for a non-Gaussian system may result in erroneous results., respectively.According to this example, using the MVC method for a non-Gaussian system may result in erroneous results.

Entropy Information Analysis
According to information theory, entropy is the probability of an uncertain variable [17], and entropy is determined by the probability distribution function of a random variable.Zhang [17] proposed a cascaded control CPA method based on minimum entropy.According to his method, the parameterization formula of Renyi entropy is, where the α is a continuous parameter and Thus, the author gave the entropy calculate of cascade control system, ( ) H y H Q q a q S q a q q M a q M a q H Q q a q q M a q H S q a q q M a q Hence,

Entropy Information Analysis
According to information theory, entropy is the probability of an uncertain variable [17], and entropy is determined by the probability distribution function of a random variable.Zhang [17] proposed a cascaded control CPA method based on minimum entropy.According to his method, the parameterization formula of Renyi entropy is, where the α is a continuous parameter and X = {x 1 , . . . ,x n } is the discrete random variable.
Lemma 1 [17].H(cx) = H(x), where x is a random variable and c is a constant.
Lemma 2 [17].The entropy of joint distribution can be represented by conditional entropy which can be expressed as H , where X and Y are random variables.
However the conclusion in Equation ( 15) is wrong because the above linear relationship cannot been ignored.For example, considering the following model, where a t is the disturbance and t is the discrete time.According to the above lemma, the entropy is Then the density plot of output y t with different coefficients is shown in Figure 6.
Symmetry 2019, 11, x FOR PEER REVIEW 9 of 17 Lemma 3 [17].On a common probability space, ≥ while X and Y are the random variables.
However the conclusion in Equation ( 15) is wrong because the above linear relationship cannot been ignored.For example, considering the following model, where t a is the disturbance and t is the discrete time.According to the above lemma, the entropy is Then the density plot of output t y with different coefficients is shown in Figure 6.It is obviously that different coefficient will lead to different density distributions, hence the theory in literature [17] is wrong.In fact, considering continuous case, where the , , f f f is the PDF of the continuous variables , , x y z , and assume that the PDF Z X Y = + .According to Lemma 2, there is the equation ( ) . However, this equation is not set up according to the Equation (18).It seems that entropy is not suitable as a benchmark.Fortunately, a Renyi entropy (RE) is proposed in literature [19] which is calculated as, ( ) ( )log 1 ( ) where the x is a random variable whatever it is discrete or continuous and ( ) x γ is its PDF.This type entropy can prevent the index becoming negative of Inf [19].In addition, it still has most properties of Shannon entropy.In this paper, RE is used to calculate the performance index.It is obviously that different coefficient will lead to different density distributions, hence the theory in literature [17] is wrong.In fact, considering continuous case,

Minimum Entropy Based on EDA
where the f X , f Y , f Z is the PDF of the continuous variables x, y, z, and assume that the PDF Z = X + Y.
According to Lemma 2, there is the equation H(Z) = H(X) + H(Y).However, this equation is not set up according to the Equation (18).It seems that entropy is not suitable as a benchmark.Fortunately, a Renyi entropy (RE) is proposed in literature [19] which is calculated as,   Assume that the disturbance is unknown, we can use the EDA to estimate the system model and disturbance distribution.Routinely, we set the seed selection criteria is ε = 0.1 and stopping criteria is H (l) − H (l−1) < 0.01 which means achieving optimal residual and smallest entropy.Thus, the estimated disturbances are shown in Figure 9.The variance of estimated disturbance is var(a 1 ) = 5.0069, var(a 2 ) = 0.0494.The performance assessment of this system based on MVC and MEC is as follows.

Cascade Control System with Non-Gaussian Disturbance
We use the same noise as the beta noise mentioned in the Section 2. The MVC performance assessment has been given in Section 2. For MEC, the estimated disturbances are given in Figure 10, The performance assessment of this system based on MVC and MEC is as follows.

Cascade Control System with Non-Gaussian Disturbance
We use the same noise as the beta noise mentioned in the Section 2. The MVC performance assessment has been given in Section 2. For MEC, the estimated disturbances are given in Figure 10.According this new MEC method, the residual + can be obtained based the identification model by using Diophantine equation.And according to simulation in the Section 2, the index based on different methods can be given.
From the Tables 1 and 2, we can find the method which is proposed in this paper performs well on Gaussian and non-Gaussian disturbance.

Index
Estimated Index Theoretical Index According this new MEC method, the residual F â(q −1 ) = Q 1 (q −1 )a 1 (q −1 ) + S 2 (q −1 )a 2 (q −1 ) can be obtained based the identification model by using Diophantine equation.And according to simulation in the Section 2, the index based on different methods can be given.
From the Tables 1 and 2, we can find the method which is proposed in this paper performs well on Gaussian and non-Gaussian disturbance.

Conclusions
To improve the cascade control loop performance, we need to identify its model.In the past, many CPA algorithms had been proposed, but most of these are based on second order statistics and the disturbance is assumed as Gaussian noise.However, in the actual industry, the disturbance of the cascade system is usually a mixture of several unknown disturbances, and is generally subject to a non-Gaussian distribution.Since the minimum variance control is not suitable for non-Gaussian cascade systems, this paper proposes a reliable performance evaluation method based on minimum entropy, which can be used for cascade control loops with non-Gaussian disturbances.The MEC method is proposed based on rational entropy.In order to obtain a more accurate estimation of parameter vector and perturbation, an improved EDA algorithm is adopted.The effectiveness of this approach has been demonstrated in previous simulations.Thus, it can be concluded that these methods can be used for cascade control system whether the disturbance is in a Gaussian distribution or not.In addition, further research is needed to improve the efficiency of EDA.
, it denotes the transfer function of feedback controller, k is the sample time, and ( ), 1, 2 i y k i = is the system output of outer loop and inner loop.The output control loop transfer function is

2 d
. Both the two disturbance 12 ( ), ( ) a k a k are assumed as a rational function with the backward shift 1 q − .For simplify,

Figure 2 .
Figure 2. The actual and estimated distribution of (a) 1 a and (b) 2 a which are non-Gaussian

Figure 2 .Figure 2 .Figure 3 .
Figure 2.The actual and estimated distribution of (a) a 1 and (b) a 2 which are non-Gaussian disturbances.

Figure 3
Figure 3 illustrates that the disturbance is subject to non-Gaussian distribution because the plots deviates the straight line.The response with different disturbance of cascade control are shown in Figure 4, and the inner and outer loop output distribution are shown in Figure 5.As can be seen from Figures 4 and 5, the performance of the main output 2 y is significantly better than the performance

Figure 3 .
Figure 3. Normal probability plots of the actual disturbance (a) a 1 and (b) a 2 which are non-Gaussian disturbance.

Figure 4 ,
Figure4, and the inner and outer loop output distribution are shown in Figure5.As can be seen from Figures4 and 5, the performance of the main output 2 y is significantly better than the performance

Figure 6 .
Figure 6.The density plot of output with different coefficient.

Figure 6 .
Figure 6.The density plot of output with different coefficient.

Figure 7 .
Figure 7.The normal probability of two disturbances (a) 1 a and (b) 2 a .

Figure 7 .Figure 8 .
Figure 7.The normal probability of two disturbances (a) a 1 and (b) a 2 .

Figure 8 .
Figure 8.The actual and estimated distribution plots from different disturbances (a) a 1 and (b) a 2 based on minimum variance control (MVC).

Figure 9 .
Figure 9.The distribution plots of two different disturbances (a) 1 a and (b) 2 a based on minimum

Figure 9 .
Figure 9.The distribution plots of two different disturbances (a) a 1 and (b) a 2 based on minimum entropy control (MEC).

Figure 10 .
Figure 10.The actual and estimated distribution plots of two disturbances (a) 1 a and (b) 2 a based

Figure 10 .
Figure 10.The actual and estimated distribution plots of two disturbances (a) a 1 and (b) a 2 based on MEC.

Table 1 .
The index by MVC and MEC benchmarks of primary loop (Gaussian).

Table 2 .
The index by MVC and MEC benchmarks of primary loop (non-Gaussian).

Table 1 .
The index by MVC and MEC benchmarks of primary loop (Gaussian).

Table 2 .
The index by MVC and MEC benchmarks of primary loop (non-Gaussian).