Majid Ghorbani, Mahsan Tavakoli-Kakhki, Aleksei Tepljakov,, and Eduard Petlenkov
Abstract—The robust stability study of the classic Smith predictor-based control system for uncertain fractional-order plants with interval time delays and interval coefficients is the emphasis of this work.Interval uncertainties are a type of parametric uncertainties that cannot be avoided when modeling real-world plants.Also, in the considered Smith predictor control structure it is supposed that the controller is a fractional-order proportional integral derivative (FOPID) controller.To the best of the authors’knowledge, no method has been developed until now to analyze the robust stability of a Smith predictor based fractional-order control system in the presence of the simultaneous uncertainties in gain, time-constants, and time delay.The three primary contributions of this study are as follows: i) a set of necessary and sufficient conditions is constructed using a graphical method to examine the robust stability of a Smith predictor-based fractionalorder control system—the proposed method explicitly determines whether or not the FOPID controller can robustly stabilize the Smith predictor-based fractional-order control system; ii) an auxiliary function as a robust stability testing function is presented to reduce the computational complexity of the robust stability analysis; and iii) two auxiliary functions are proposed to achieve the control requirements on the disturbance rejection and the noise reduction.Finally, four numerical examples and an experimental verification are presented in this study to demonstrate the efficacy and significance of the suggested technique.
A.Background
IN process control, the Smith predictor control structure is a well-known dead time compensator for stable processes with large time delay [1].When the model is accurate, the closed-loop characteristic polynomial of the Smith predictor control structure will be delay free.However, in real-world applications, due to process parameter change, the plant model cannot adequately capture the dynamic behavior of the process, necessitating a robust stability study of the Smith predictor control system [2].
In [2], at first, a set of necessary and sufficient conditions for the robust stability analysis of the Smith predictor based interval integer-order plants with PI controller has been presented.Then, an application has been developed for the robust control of a first order plant with interval time delay.Moreover, some modifications of the Smith predictor have been mentioned in order to facilitate the tuning of the controller parameters in industrial applications and to improve the robustness [3]–[5].Also, in [6], the Kharitonov’s theorem has been employed to enhance the robustness of the Smith predictor based integer-order control system.Furthermore, in[7]–[9], the stability regions of PID and PI controllers have been obtained for tuning PI and PID controllers in Smith predictor control structure.
In control theory, fractional-order calculus has been widely utilized to improve the performance of control loops [10]–[16], the robustness properties [17]–[21] and to identify more precise models of physical systems [22] [23].Also, in [24], a procedure has been presented to analyze the robust stability of a class of interval fractional-order polynomials.
Recently, fractional-order controllers based on the Smith predictor structure have been used in a variety of engineering applications.In [25], a simple method has been presented to design internal model based fractional-order Smith predictor control systems for first order plus time delay plants.The main disadvantages of the method in [25] are that it cannot guarantee the robust performance and robust stability of the closed-loop control system in the presence of simultaneous uncertainties.In [26], another method has been proposed to tune the parameters of a filtered Smith predictor based fractional integral-fractional derivative controller.Also, the method in [26] cannot guarantee the robust performance and robust stability of the system in the presence of simultaneous uncertainties.In [27] and [28], the robustness analysis of fractional-order controllers combined with Smith predictor structure has been investigated in the case of the existence of time delay variations.Two main disadvantages of the method presented in [27] and [28] are that only sufficient conditions for robust stability analysis have been presented in these articles and also the simultaneous uncertainties in time delay term,model gain and time constants of the model have been not considered.Therefore, these methods cannot guarantee the robust performance and robust stability of the system having simultaneous uncertainties.Furthermore, various design methods of fractional-order controllers such as FOPI and FOPID controllers have been given in [29]–[34].However, none of the methods in the aforementioned papers have discussed the stability of the Smith predictors having uncertain models.In[35], the flat phase property was used to design FOPI controllers for a long time-delay system in order to improve the robustness of the Smith predictor, but no discussion of robust performance was provided.In [36], by benefiting from the direct synthesis, a method has been presented to design a fractional-order controller for integer-order time-delay systems without considering an uncertain model in the Smith predictor structure.It is worth mentioning that many other methods have been proposed based on gain and phase margins to design controllers in order to improve the robustness of the Smith predictor control system (see [37]–[44]).However, as it is shown in the Section V of the present paper, satisfying gain and phase margins may not be a reliable method for uncertain systems in general.
B.Motivations, Challenges and Problem Statement
This paper does a further study on robust stability analysis of the Smith predictor based fractional-order control system.Controlling time delay processes is well-known to be a major challenge in industry.This is because the system time delay makes the closed-loop control system unstable, potentially resulting in poor performance.To address the issue of large time delay compensation, the Smith predictor structures are frequently used in engineering disciplines, especially control systems engineering [45], [46].On the other hand, in real applications, because of the variation of the process parameters, the plant model can be considered as a transfer function whose denominator and numerator coefficients are all uncertain and lie in specified intervals.To the best of the authors’knowledge, no works consider the problem of analyzing the robust stability of Smith predictor based fractional-order control systems suffering from simultaneous uncertainties in the delay term, the model gain and coefficients.In the previous works, for simplicity, the robust stability analysis has been only addressed the uncertain time delay (see [28]) or the openloop gain (see [1] and [25]) and clearly such procedures cannot be employed for robust stability analysis of the Smith predictor based fractional-order control systems having simultaneous independent uncertainties in the open-loop gain, the delay term and the real uncertain coefficients.The authors of[28] exclusively discuss the robust stability analysis of a Smith predictor based control structure in the presence of time delay variations.However, in practice, simultaneous uncertainties in all model parameters, including the time delay term,model gain, numerator and denominator coefficients, are more common.Example of Section IV-B in this paper provides an illustration of such a circumstance.The findings of the identification of a level control process in the aforementioned example reveal that the transfer function of the system model contains simultaneous uncertainties in the delay term, model gain, and time-constants.As a result, the method in [28] cannot be used to analyze the robust stability of such a system.Furthermore, the approaches provided in [26]–[36] do not take into account the influence of simultaneous uncertainty in gain,time-constants, and time delay on the robust stability of Smith predictor controllers.As a result, there is a requirement to establish necessary and sufficient conditions for the robust stability analysis of a fractional-order control system in the face of simultaneous uncertainties in the delay term, the model gain, and the coefficients.The major problem is that the uncertain parameters of the characteristic function of the Smith predictor structures do not depend linearly on the polynomial terms due to the uncertain time delay.As a result, the characteristic function’s value set is non-convex.As a consequence, traditional methods in robust stability analysis, such as Kharitanov’s theorem [47] are no longer applicable.Therefore, the initial challenge lies in presenting necessary and sufficient conditions for robust stability analysis of Smith predictor based control systems.Also, the existing methods[37]–[39], [48]–[53] cannot analyze the robust stability of the Smith predictor with high-order systems.Hence, robust stability analysis of the Smith predictor with high-order systems is the second challenge.Moreover, because of the uncertain model, it is very difficult to determine the uncertainty bounds of the complementary sensitivity function and the sensitivity function.Hence, the next challenge in this literature is to achieve the control requirements on the disturbance rejection and the noise reduction in the presence of uncertainties.The current research investigates these three problems as the primary unsolved problems of Smith predictor-based fractionalorder control systems.In summary, based on the above explanations, the following control problems can be formulated as the main challenges for the Smith predictor control structure considered in this study.
Problem 1:Despite extensive research into the problem of robust stability analysis of the Smith predictor-based fractional-order control system, necessary and sufficient conditions for robust stability analysis of this type of system with simultaneous independent uncertainties in the open-loop gain,the delay term, and the time-constants appear elusive.
Problem 2:Some industrial processes are modeled with high-order transfer functions [48], and the methods such as those presented in [48]–[53] are inefficient in designing robust controllers for these processes in the presence of uncertainties.Therefore, the other challenge is robust stability analysis of high-order systems compensated by Smith predictor based fractional-order controllers.
Problem 3:In addition to the robust stability analysis, it is necessary to ensure that some performance specifications are met for all processes of the uncertain system.For example, in this paper, the control requirements on the disturbance rejection and the noise reduction are considered as robust performance specifications.However, as it is well known, due to the uncertain model, it is very difficult to determine the bounds of the complementary sensitivity function and the sensitivity function in the presence of uncertainties.Hence, most of the papers have only satisfied the constraint on the magnitude of the nominal complementary sensitivity function and the nominal sensitivity function (i.e., they assumed that the real model and the estimated model are the same, while in practice the estimated model can never fully replicate the properties real system.For some examples in this regard, see [25] and [36]).Therefore, another difficulty is to achieve robust performance for the control requirements on the disturbance rejection and the noise reduction in the presence of uncertainties.
The above mentioned issues motivate the authors of this paper to give an efficient and exact method for robust stability analysis of the Smith predictor based control systems suffering from the simultaneous uncertainties in the delay term and the plant coefficients.
C.Main Work and Contributions
Recently, the zero exclusion principle attracts many researchers to analyze the robust stability of linear time invariant (LTI) fractional-order systems with interval time delay.Based on this principle, the value set of the characteristic function of the interval fractional-order systems is plotted for all non-negative frequencies with a subsequent check whether the value set of the characteristic function includes the origin or not [54]–[57].Three main contributions of this paper can be summarized as follows.
1) Based on a graphical method, a set of necessary and sufficient conditions is obtained to analyze the robust stability of a Smith predictor based control system.The proposed method allows to explicitly determine whether or not the FOPID controller can robustly stabilize the fractional-order Smith predictor based control system.
2) An auxiliary function as a robust stability testing function is presented to improve the computational efficiency of robust stability analysis.
3) Improving robust performance of a designed controller by two auxiliary functions.
Also, the innovations of the present paper are summarized as follows:
1) Presenting a finite frequency range for verifying the robust stability of a Smith predictor based control system (see Theorem 1).
2) Obtaining necessary and sufficient conditions for robust stability analysis of a designed Smith predictor based control system (see Theorem 2).
3) Introducing a robust stability testing function to investigate the robust stability of Smith predictor based fractionalorder control system (see Theorem 3).To the best of the authors’knowledge, no robust stability testing function has been presented up to now to analyze the robust stability of the Smith predictor based control systems.
4) Presenting sufficient robust condition of a Smith predictor for high-order systems (see Remark 5).
5) Presenting two auxiliary functions to improve the control requirements on the disturbance rejection and the noise reduction (see Remark 6).
D.Organization
This paper is organized as follows: In Section II, some required definitions are stated.The main results of the paper are presented in Section III.In Section IV, four numerical examples and an experimental verification assure the efficiency of the proposed procedure.Moreover, in Section V,some comparisons are performed about the obtained results in this paper and the existing methods.Finally, the paper is concluded in Section VI.
Fig.1.The Smith predictor controller [1], [2].
In this section, three useful theorems are proved in order to analyze the robust stability of the Smith predictor based fractional-order control system.In Theorem 1, it is proved that the origin is not included in the value set of ∆(jω,q,u) in (8) for ω＆gt;ωmax, where ωmaxis as (17).
Also, new necessary and sufficient conditions for the robust stability analysis of the closed-loop system are provided in Theorem 2.Eventually, a robust stability testing function is presented in Theorem 3 whose sign check helps to analyze the robust stability of the closed-loop control system.Moreover,an evaluation on the disturbance rejection and the noise reduction of the Smith predictor based control system in the presence of uncertainties is further studied in Remark 6.
Theorem 1:Assume that ωmaxis defined as (17).Then, the characteristic function ∆(jω,q,u) presented in (8) would not be equal to zero for ω ＆gt;ωmax.
Fig.2.Schematic of the value set ∆ a(jω) and its boundary ∂ (∆a(jω)) for(0,).
Fig.3.Schematic of the value set ∆ a(jω) and its boundary ∂ (∆a(jω)) for[ωmax] (solid line) and two radiuses R(ω) and R(ω) (dashed line).
Proof:The proof is given in Appendix.■
Remark4:In Theorem 2, a graphical method is presented to analyze the robust stability of a Smith predictor based fractional-order control system.According to this theorem, it is needed to depict two-dimensional graphs of the value sets in the complex plane at each positive frequency and it may be too difficult to investigate the overlap between the value sets.Accordingly, in Theorem 3 instead of depicting two-dimensional graphs of the value sets, we employ the idea of robust stability testing function based on the geometric aspects and explicit descriptions of the value sets.Therefore, the computational complexity of the robust stability conditions in Theorem 2 can be decreased significantly by the auxiliary functionΛ(ω)presented in Theorem 3.Therefore, the robust stability analysis should be performed based on the robust stability testing function Λ(ω).By Theorem 2, Theorem 3 and Remark 2, the following useful algorithm is proposed to analyze the robust stability of a Smith predictor based fractional-order control system suffering from simultaneous uncertainties in gain,time-constants, and time delay.
Algorithm 1 Robust Stability Analysis
In this section, four illustrative examples (Section IV-A)and an experimental verification (Section IV-B) are given to demonstrate the application of the proposed approach in this study to analyze the robust stability of a Smith predictor based fractional-order control system.Also,Algorithm 1is employed to analyze the robust stability of the Smith predictor structure having uncertain integer-order process, uncertain fractional-order process and high-order process.
A.Numerical Examples
Example 1:In [2], a nominal model is described by
The parametric uncertainties arek∈[0.7,0.9], τ ∈[4,6],q1∈[0.8,1.2] andq0=1.In [2], a robust PI controller has been proposed as
Therefore, the characteristic function ∆(s,q,u) in (8) is given by
Also, based on Assumption 1 the nominal characteristic function ∆∗(s) is presented as (46).
To check the stability of the nominal closed-loop system,the presented procedure in [65] is used.The polar plot ofψ1(jω)in (47) has been shown in Fig.4.As Fig.4 depicts, the nominal closed-loop system is stable.It is because that the polar plot ofψ1(jω) in (47) does not encircle the origin in the complex plane (see the first step ofAlgorithm 1).
Fig.4.The polar plot of ψ1(jω) in (47) for ω ∈[−104, 104] rad/sec.
From (17), ωmaxis obtained as 7.724 rad/sec.Therefore,based on Theorems 2 and 3, it is sufficient that the robust stability of the Smith predictor based control system is checked for ω ∈[0,7.724].To analyze the robust stability, the auxiliary function Λ(ω) in (27) is used forNθ=3.Based onvh(jω)(h=1,2) in (48),PliandPsiin (29) and (45), the robust stability testing function Λ (ω) in (27) can be plotted.
From Figs.5(a) and 5(b), it is obviously concluded that the inequality Λ(ω)＆gt;0 holds for ω ∈(0,7.724].Therefore, based on Theorem 3, the Smith predictor based control system is robust stable.
Fig.5.(a) The values of Λ (ω) in (27) for ω ∈(0,π) rad/sec; (b) The values of Λ (ω) in (27) for ω ∈[π,7.724] rad/sec.
Example 2:Consider a fractional-order plant with interval coefficients and interval time delay as
Based on Assumption 1, the nominal characteristic function∆∗(s)is presented as (53).
To check the stability of the nominal closed-loop system,the method proposed in the first step ofAlgorithm 1is used.As Fig.6 depicts, the nominal closed-loop system is stable.
Fig.6.The values of the transfer function ψ(jω) in (54) for ω ∈[−104, 104]rad/sec.
Fig.7.(a) The values of Λ(ω) in (27) for ω ∈(0,0.8976) rad/sec; (b) The values of Λ (ω) in (27) for ω ∈[0.8976,41.6017] rad/sec.
wheres=jω.Figs.7(a) and 7(b) depict the curve of Λ(ω) in(27) within ω ∈(0,41.6017] rad/sec.According to these figures, the values of Λ(ω) are positive withinω ∈(0,41.6017]rad/sec.Thus, the considered Smith predictor based control system is robust stable.It is worth mentioning that none of the methods proposed in [26]−[46] can be used to verify the robust stability of the Smith predictor control system in this example.It is because that the mentioned papers have failed to analyze the robust stability of the closed-loop control system in the presence of simultaneous uncertainties in the delay term and the plant gain and coefficients.While based on Theorems 2 and 3 of this paper, the robust stability of the closed-loop system is concluded simply.
Example 3:Consider the following process transfer function:
Fig.8.The values of the transfer function ψ(jω) in (54) for ω ∈[−104, 104]rad/sec.
Fig.9.(a) The values of Λ(ω) in (27) for ω ∈(0,2π) rad/sec; (b) The values of Λ (ω) in (27) for ω ∈[2π,20.5919] rad/sec.
Also, based on Theorem 1, ωmaxis obtained as 20.5919 rad/sec.The values of the robust stability testing function Λ(ω)have been reported in Figs.9(a) and 9(b).From Figs.9(a) and 9(b), it can be seen that the inequality Λ (ω)＆gt;0 holds for 0＆lt;ω ≤20.5919 rad/sec.Accordingly, Theorem 3 ensures the robust stability of the closed-loop system.Whereas the references [48]–[53] cannot ensure the robust stability of the closed-loop control system.
Example 4:In [28], the robust fractional-order controllerC(s) in (60) has been designed for the plantP(s) in (61).
Fig.10.(a) The values of Λ (ω) in (27) for ω ∈(0,) rad/sec; (b) The values of Λ (ω) in (27) for ω ∈[,1] rad/sec.
It is worth mentioning that the method presented in [28]cannot be used for the robust stability analysis of the Smith predictor based control system for the interval plant with simultaneous uncertainties in gain, time delay and time constants as (62).But, as it is seen from this example the procedure presented in this paper can analyze robust stability of the Smith predictor based control system for the interval plantP(s)in (62).
B.Application to A Level Control Process
To show the applicability of the results of the paper results in a practical setting, in this example a robust FOPI controller is designed for the level control process GUNT- RT512 (see Fig.11).Then, the robust stability of the designed Smith predictor based control system is analyzed according to the proposed procedure in this paper and the results of the practical implementation of the designed controller are provided.
The results of the identification of the level control process GUNT-RT512 for 13 iterations are provided in Table I.According to Table I, the identified model can be represented as follows:
wherek∈[2.94,4.59], τ ∈[4.8,14.4],q0=1 andq1∈[26.4,35.6].For the nominal closed-loop system, the FOPI controllerC(s) in (64) is designed according to the procedure proposed in [66].
Fig.11.Level control process (GUNT-RT512) [72].
TABLE I IDENTIFIED PARAMETERS OF THE PLANT
The FOPI controller (64) is implemented by leveraging the approximation method presented in Remark 3.To this end, the order and the frequency range of the approximation filter are taken asN=3 and ω ∈[0.01,100] rad/sec, respectively.Also,the applied disturbance is considered as an additive step disturbance at the time instantt=2200 sec.
Fig.12.(a) The values of Λ(ω) in (27) for ω ∈(0,0.6545) rad/sec; (b) The values of Λ (ω) in (27) for ω ∈[0.6545,1] rad/sec.
Fig.13.(a) The values of the auxiliary functions Ξ T(ω) in (35) andΞS(ω)in (39) for ω ∈(0,0.6545) rad/sec; (b) The values of the auxiliary functions ΞT(ω) in (35) and Ξ S(ω) in (39) for ω ∈[0.6545,20] rad/sec.
Fig.14.(a) The level output by applying the robust FOPI controller (64) in the Smith predictor control structure; (b) The control signal by applying the robust FOPI controller (64).
Let us compare our results with the existing works [26]–[46] and [48]–[53].In the present paper, we have proposed two different methods, i.e., Theorems 2 and 3 to check the robust stability of the Smith predictor based fractional-order control systems.Theorem 2 presents necessary and sufficient conditions for robust stability analysis of the Smith predictor based fractional-order control systems having simultaneous uncertainties in gain, time constants and time delay.In Theorem 3, an auxiliary function as a robust stability testing function is presented to facilitate the robust stability analysis of the Smith predictor based fractional-order control systems suffering from simultaneous uncertainties.According to the uncertainty structure, the robust stability analysis of the Smith predictor based control systems can be classified into eight categories shown in Table II.References [25] and [39] guarantee the robust stability of the Smith predictor based fractionalorder control systems having only uncertain gain and [25]–[28] can also ensure the robust stability of the Smith predictor based fractional-order control systems having only uncertain time delay.This implies that none of [25]–[28] and [39] can analyze the robust stability of the Smith predictor based fractional-order control systems having simultaneous uncertainties in gain, time constants and time delays (such as Examples 2–4).On the other hand, in [29]–[36], [38], [41], [42], some design methods have been proposed based on gain and phase margins to improve the robustness of the Smith predictor based control system.However, gain and phase margins may lead to unreliable results.For example, in [35], the FOPI controllerC(s)=5.1709+7.6428/s0.4587has been designed for the systemP0(s)=e−9s/(12s+1).Assume that the actual plant contains simultaneous uncertainties as
TABLE II REVIEW OF THE SMITH PREDICTOR BASED INTEGER-ORDER CONTROL SYSTEMS (SPIOCS) AND THE SMITH PREDICTOR BASED FRACTIONAL-ORDER CONTROL SYSTEMS (SPFOCS)
Now, Theorem 2 can be employed to investigate the robust stability of the system.As shown in Fig.15, the value sets of χ1(jω) (solid line) in (22) and χ2(jω) (dotted line) in (23)have an overlap at ω=1.143 rad/sec in the complex plane.Therefore, this system would not be robust stable based on Theorem 2.This implies that there is at least one unstable member in the uncertain space.Let us check the stability of the characteristic function in (66) which is a member of∆(s,q,u).
Fig.15.The overlap between the value sets of χ1(jω) (solid line) and χ2(jω) (dotted line) at ω =1.143 rad/sec.
To check the stability of ∆ˆ(s) in (66), the method proposed in [65] can be used as stated inAlgorithm 1.Based on this method, the polar plot ofψ(jω) in (67) is plotted in the complex plane for − ∞＆lt;ω＆lt;∞.
Moreover, none of [26]–[46] can be used to design a robust controller in Example 2.Also, as shown in Examples 3 and 4,a robust FOPID controller can be designed for high-order systems in the presence of uncertainties.Whereas, [48]–[53] have failed to design a robust controller for uncertain processes.
Fig.16.(a) The polar plot of ψ(jω) in (67); (b) A better view around the origin.
In this paper, the robust stability analysis of the Smith predictor based control system with an FOPID controller for an interval delayed fractional-order plant was investigated.Necessary and sufficient conditions were proposed to analyze the robust stability of the closed-loop system.Also, a robust stability testing function was offered to facilitate the procedure of the robust stability analysis.The robust stability analysis method proposed in this paper can be used for the both integer-order and fractional-order Smith predictor based control systems.Moreover, two auxiliary functions were provided to achieve the control requirements related to the desired properties of the disturbance rejection and the noise reduction.Four numerical examples were offered to demonstrate the method’s applicability and efficiency.Finally, the method was applied to a real-life plant—a level control process.Following that, a model with interval uncertainty was identified using process data.The proposed method was also effectively applied to this model, demonstrating the technology’s potential for realworld applications.
Fig.17.Schematic of the value set ∆a(jω), the verticesPli(jω)(i=0,1,...,2Nθ) and Psi(jω)(i=0,1,...,2Nθ) for 0 ＆lt;ω＆lt;.
Fig.18.The edge of the three consecutive vertices Pli(jω),Pl(i+1)(jω) andPl(i+2)(jω) for i=0,2,...,2Nθ−2.
Fig.19.The edge of the three consecutive vertices Psi(jω),Ps(i+1)(jω) andPs(i+2)(jω) for i=0,2,...,2Nθ−2.
Fig.20.A view of δ (jω) in (72).
Fig.21.A view of the triangle inequality between the vertex Pg(jω) and the edge e(−Nˆ(jω)vh(jω),−Nˆ(jω)vh+1(jω)).
can prove the theorem easily.■