Design of a Robust Optimal Decentralized PI Controller Based on Nonlinear Constraint Optimization For Level Regulation: An Experimental Study

Soumya Ranjan Mahapatro,, Bidyadhar Subudhi,, and Sandip Ghosh,

Abstract—This paper presents the development of a new robust optimal decentralized PI controller based on nonlinear optimization for liquid level control in a coupled tank system. The proposed controller maximizes the closed-loop bandwidth for specified gain and phase margins, with constraints on the overshoot ratio to achieve both closed-loop performance and robustness. In the proposed work, a frequency response fitting model reduction technique is initially employed to obtain a first order plus dead time (FOPDT) model of each higher order subsystem. Furthermore, based on the reduced order model, a proposed controller is designed. The stability and performance of the proposed controller are verified by considering multiplicative input and output uncertainties. The performance of the proposed optimal robust decentralized control scheme has been compared with that of a decentralized PI controller. The proposed controller is implemented in real-time on a coupled tank system. From the obtained results,it is shown that the proposed optimal decentralized PI controller exhibits superior control performance to maintain the desired level, for both the nominal as well as the perturbed case as compared to a decentralized PI controller.

NOMENCLATURE

ACross sectional area of the tanks.

a12Cross-sectional area of the pipe from tank 1 to tank 2.

h1water level in Tank 1

h2Water level in Tank 2

a1Outlet area of tank 1

a2Outlet area of tank 2

gGravitational constant

η Constant relating the control input with water inflow from the pumps

ωciPhase crossover frequency

τiiOverall time constant

AmGain margin

φmPhase margin

PTMaximum closed loop amplitude ratio

om*Upper bound of maximum closed loop amplitude ratio

I. INTRODUCTION

AUTOMATIC regulation of precise liquid level is an important issue in many process industries such as pharmaceutical, chemical, and food processing industries. In general,most industrial processes are multi input multi output (MIMO)systems. The coupled tank system (CTS) is a special class of an unstable non-minimum phase system. Control of a CTS is one of the many challenging problems in this area owing to the complexities in controlling a nonlinear MIMO system. In practice, implementing a controller for a multivariable system is more complex than implementing one for a single input single output (SISO) system because of variations in process dynamics which arise due to changes in operating points and coupling effects. In multivariable control design, often, a decoupler is designed to disintegrate a multiloop control system into a set of single individual loops [1]-[3]. Due to the presence of coupling effects between inputs and outputs in a MIMO system, many well developed closed-loop control methods for a SISO system with time delays can hardly be extended to a MIMO system with delay. To handle coupling effect in a multivariable systems, usually two types of control strategies are adopted i.e. a decentralized controller or a centralized controller. Out of these two, decentralized control structure is widely preferred due to its superior performance,simplicity, and robustness.

Over the past few decades, many methods have been reported for developing multiloop controller design including the autotuning method [4], [5], independent loop tuning method [3], [4]and detuning method [6]. Although, the detuning process is simple, the loop performance and stability of the closed loop system cannot be adequately defined through this approach. In the independent loop tuning method, the response may be inadequate, as the exact information about controller dynamics of the other loops are not used. Besides these multiple loop control design approaches, many other decoupling approaches have been reported in literature i.e. decentralized control[7]-[9], adaptive neural decentralized control [10], and adaptive decoupling control [11]-[13]. Despite the availability of several other advanced controllers, SISO PID controllers with decouplers are the preferred controllers to use in industry owing to their simple structure and ease in implementation.Despite there are extensive applications of PID controller in industry, it can still be demanding to find an adequate tuning method to determine the appropriate PID controller parameters.A considerable amount of research work has been carried out to find out an efficient tuning formula for the PID controller in[4], [14]-[17]. Robustness is an important aspect of controller design, because the process model used is often an approximation of actual industrial process. Robustness plays a vital role in the control design and is often measured by the gain margin (GM) and phase margin (PM) [18]-[20]. Developing a PID controller to meet GM and PM specifications is an important design approach. However, control design to satisfy GM and PM criteria approach is not new. The solution is usually achieved by numerical methods or graphically by using a Bode plot [21], [22]. In [1], [15], [23], a decentralized PID controller has been proposed based on gain and phase margin (GPM)specifications PID for MIMO system. In general, PI control design based on GM and PM specifications normally leads to four equations with four unknowns ( ωg, ωp,Kcland τi). To solve this problem, numerical methods can be directly employed, or an approximation of arctan function can be adopted to obtain these parameters [24], [25]. In [26], a GPM specification PID controller incorporates an internal model control (IMC) algorithm which significantly simplifies the problem as there is only one tuning parameter.

Although PID controller tuning always requires a good closed loop performance controller along with robustness,there is a tradeoff between performance and robustness. The maximum closed-loop amplitude ratio (PT) and bandwidth( ωc) are an essential closed-loop performance criteria. In general, a largerPTleads to a faster response. However, it provides a larger overshoot with less robustness. Thus, it should be properly bounded. The bandwidth ( ωc) can be made as large as possible, since it results in a faster closed loop response and lowers the settling time. Even though several well-developed GPM specifications based PID controllers have been proposed [1], [11], [20] and [27], there is no record found on directly imposing constraints on the performance criteriaPTand ωc, along with robustness criteria on GPM PID tuning approach or any experimental validation for the TITO system. Therefore, the motivation of the proposed work is to design an optimal robust PID controller while imposing constraints on both performance and robustness criteria. In this proposed work, constraints onPTbounds peak overshoot and constraints on GPM assure robustness. The settling time is thus minimized as the response is made as fast as possible through maximization of the bandwidth. The contribution of this work lies in the design of an optimal robust PID controller based on nonlinear optimization to precisely regulate the level in a coupled tank system in the face of disturbance as well as uncertainties. Furthermore, the performance of the proposed controller is validated experimentally on a coupled-tank liquid level MIMO system to verify the efficacy of the proposed controller.

The rest of the paper is organized as follows. In Section II,modeling of the coupled tank system is presented. Decoupler design is presented in Section II. In Section III, a frequency response fitting based model reduction technique is presented to obtain a first order plus dead time (FOPDT) model of each higher order decoupled subsystem. The proposed controller design is described in Section IV. Section V presents robustness analysis of the proposed controller. Simulation as well as experimental results with analysis are presented in Section VI.Finally, conclusions are drawn in Section VII.

II. MODELING OF A COUPLED TANK SYSTEM

A schematic diagram of a coupled tank system (CTS) is shown in Fig 1. The objective of controlling the CTS is to maintain the desired liquid level in the individual tanks. By using the mass balance equation and Bernoulli’s principle, the model of the CTS can be written as

Fig. 1. Schematic diagram of a coupled tank system.

Some parameters of the CTS such as η andAare not known exactly. Hence, there is a need to obtain an accurate dynamic model of the CTS. So, in this work, FOPDT model is obtained for the CTS from the set of experimental input-output data by employing the system identification technique [28]. A set of input output data of 2500 samples were collected from open loop experiments on the coupled tank system with a sampling time 0.1 s.

Fig. 2. Model validation based on residual analysis.

We consider pump control voltage as the input and water level as the output for conducting system identification. By employing the system identification tool box, we obtain a model of the CTS as given in (28). The identified model is validated by using residual analysis. The residual analysis can be performed by evaluating the autocorrelation of output residuals and calculating the cross correlation between the input of the process and to the residual output. The parameters of the identified model in (28) are estimated with prediction error method (PEM). The corresponding results are presented in Figs. 2(a)-2(d). From Figs. 2(a)-2(d), it is observed that the correlation functions remain inside the confidence interval.Hence it is confirmed that, the identified model is suitable for the decoupler design. The obtained identified model is given by

The above model given in (1) is used for designing the decoupling controllers. In the next section, we present the design of the decoupling controller.

III. DECOUPLING CONTROLLER DESIGN

Fig. 3 depicts the block diagram of the control structure of a TITO system. In this structure, two PID controllers are used and the outputs of the PID controllers are fed as inputs to the decouplers. Also, the outputs from the decoupler are used as the two inputs to the CTS. Therefore, in this structure, the control problem is separated into two parts, namely the first part, which deals with decoupling and the second part, which deals with the control of the decoupled loops. Fig. 4 represents the schematic of real-time controller implementation of proposed algorithm. The objective of designing a decoupler is to reduce control loop interactions. In general, two cases are considered for design of decouplers [14] as follows

Case 1: Off diagonal elements of (1) have no right hand pole (RHP) poles and diagonal elements do not have RHP zeros. Then a decoupling control can be designed as follows.

Fig. 3. Schematic decentralized control structure for TITO system.

wherep1(s),p2(s),d12(s) andd21(s) are defined as

Case 2: Diagonal elements of (1) have no RHP pole and off diagonal elements do not have RHP zeros.

The decoupled processH(s) with processGp(S) and decouplerD(s) is given by

The decoupled elementshii(s) are to be controlled by the proposed decentralized controllerCii(s) where i = 1, 2.

A. Model Order Reduction

As elementshii(s) of (2) consist of large time delays, it is quite complicated to implement and design a decoupling controller. Therefore, a model reduction approach is considered which is further simplified as (4). In this section, a FOPDT modelPii(s) of each elementshii(s) ofH(s) is obtained as follows [29].

This reduced order FOPDT model in (4) is derived by employing the frequency response fitting at two points i.e.ω=0 , ω=ωcito obtain the FOPDT model ofhii(s). where ωcidenotes the phase crossover frequency [30].

By considering the above conditions, the parameters of the FOPDT model given in (4) can be calculated as follows:

Further, this reduced order decoupled model is used in controller design. The proposed control design approach is presented in the next section.

IV. PROPOSED OPTIMAL DECENTRALIZED PID CONTROLLER DESIGN

The proposed PI controller tuning method is designed based on nonlinear optimization. In this work, the overshoot is bounded by the constraint onPTand simultaneously, the robustness is guaranteed by the constraint on GPM. The proposed approach given below is employed to obtain the control laws for the reduced order decoupled subsystem as given in (4), i.e.,P11(s) andP22(s). Often, a FOPDT model is used in industry because it reasonably represents the process gain, time constant, and deadtime of a higher order process.One can write the transfer function of a FOPDT process as follows.

The transfer function of the PID controller can be written as

Further, from (6) and (7), the open loop transfer function can be obtained as

With frequency analysis of each term of (8), the amplitude ratioAROand phase change φOare written as

where

From the open loop systemGO, one can obtained the closed loop transfer function as follows:

The amplitude ratio of closed loop system can be calculated as

The amplitude ratio of the closed loop system can be obtained from open loop amplitude ratioAROand φO. Further,ωccan be obtained by solving (12)

The maximum closed loop amplitude ratioPTcan be obtained as

using definitions of GPM one can write the following set of equations

Fig. 4. Real-time controller implementation of proposed control.

where

By substituting (9) and (10) into (14)-(15), one obtains

In (16)-(19), there are five unknowns ( ωg, ωp,Kcl, τiand τd).Hence, these equations can not be solved directly. However,the following lemma is defined to formulate an optimization problem by imposing constraints on gain margin, phase margin, and maximum closed loop bandwidth.

Lemma 1:The optimization problem is defined as follows:

whereom*is the upper bound of maximum amplitude ratio.

Remark 1:

1) The maximum closed loop amplitude ratio is often related to gain and phase margin bounds [30]

Equation (20) gives the relation betweenPTand GPM bounds and it is identified that ifom*is low, then the actual value ofAmandom*may be higher thanAm*andom*. Thus, a more robust and less aggressive response may be achieved.

3) In the proposed controller, to solve the unknown variables in (16)-(19) the “fmincon” function is used from the MATLAB optimization tool box.

V. ROBUSTNESS AND STABILITY ANALYSIS

In general, there usually exists an unmodeled process dynamics in real-time systems. Therefore, robustness analysis of the designed controller is an important aspect of the control system. The dynamics of the real-time system often has many sources of uncertainties which may cause poor performance and even degrades the stability of the system. Therefore, in this paper to analyze the stability of the decoupled control system multiplicative input and multiplicative output,uncertainties are incorporated into the nominal model system.The corresponding schematic structure of the multiplicative input and multiplicative output uncertainty and itsT-Δ form is shown in Fig. 5 and Fig. 6. By representing the perturbed system in the form ofT-Δ structure for stability analysis, the transfer function from the output to input i.e. ‘u’ to ‘v’ can be written as

According to the small gain theorem in [30], a perturbed system with uncertainty holds robust stability well if and only if the mentioned robust constraints conditions below are satisfied, i.e.

Fig. 5. Schematic diagram of multiplicative input uncertainty and itsT-Δ structure.

Fig. 6. Schematic diagram of multiplicative output uncertainty and itsT-Δ structure.

But the stability constraints shown in (23) and (24) results in more time needed to evaluate the norm. Hence, to overcome this issue, an equivalent relationship is developed using the small gain theorem and the spectral radius stability[30] which is as follows

With the aforementioned stability constraints, (25) can be further reformulated as

VI. RESULTS AND DISCUSSION

A. Simulation Results

In this section, simulation results are incorporated to verify the efficacy and performances of the proposed controller. The results of the proposed controller is compared with that of [1].The proposed controller is simulated on the identified model of CTS which is given in Section II. The decoupler for the identified model is obtained by using Case 1 of the decoupling controller design that is described in Section III. The obtained identified model for the CTS that is described in Section II as follows.

The decouplerD(s) is obtained as

The resulting diagonal decoupled subsystems are as follows.

For controller realization,h11(s) andh22(s) should be expressed as FOPDT process. Hence, using (4), the FOPDT models for ofh11(s) andh22(s) are determined as follows.

Solving (20) forAm= 3 and φm38°, the proposed decentralized PI controller is obtained as

The suggested ranges of GM and PM are 2-5 and 3 0°-60°,(Am) and PM (φm) to be 3 and 38°respectively. The proposed respectively [31]. To pursue the controller design, we set GM controller parameters are obtained by solving (16)-(20) by using “fmincon” function from MATLAB optimization toolbox. The obtained controller parameters are given in Table I.

TABLE I PROPOSED CONTROLLER PARAMETER FOR SIMULATION

To verify the efficacy of the proposed optimal GPM PI controller, its performance is compared to a GPM PI controller reported in [1]. In simulation, first a step change inh1is given att= 0 s andh2demand is set tot= 500 s respectively. Similarly, to illustrate the disturbance rejection performance, a step disturbanced= 0.8 att= 1700 s is introduced to both loops. The corresponding results of both set-point tracking and disturbance rejection are given in Figs. 7-8. From the above simulation results, it is observed that both tanks attain the desired level smoothly without affecting other tank’s levels. However, GPM PI [1] needs more control action to archive steady state and also yields steady state error when there is a load disturbance as compared to the proposed optimal GPM PI controller.Figs. 9-10 show the set-point tracking and disturbance attenuation responses of the proposed optimal GPM PI with a set of differentom*values. From Figs. 9-10, it is observed that, smallerom*values lead to better set-point tracking as well as disturbance attenuation performances.

Fig. 7. Set-point tracking for nominal system.

Fig. 8. Disturbance rejection for nominal system.

Fig. 9. Set point tracking with different om* value.

Fig. 10. Disturbance rejection with different om* value.

To study the robustness of the proposed controller in face of parameter uncertainty, the delay, gain, and time constants are changed by ±10%. The corresponding results are presented in Figs. 11-13. To show the practicality and feasibility of the proposed controller, both the performance of the proposed optimal GPM PI and GPM PI [1] are also investigated based on the integral absolute error (IAE), total variance (TV), and RMS tracking error (eti). These are presented in Table II.Apart from this, the robust stability of the proposed controller is analyzed while considering the multiplicative input and multiplicative output uncertainties in Section V. The corresponding magnitude plot of the spectral radius of(26)-(27) are shown in Fig. 14. From Fig. 14, it is observed that the proposed controller also provides robust stability.

Fig. 11. Set-point tracking for perturbed system with 10% parameter uncertainty.

Fig. 12. Set-point tracking for perturbed system with -10% parameter uncertainty.

Fig. 13. Simulation results of disturbance rejection for perturbed system.

Fig. 14. Magnitude plots of spectral radius.

TABLE II PERFORMANCE ANALYSIS

B. Experimental Results

In order to demonstrate practicality of the proposed decoupled optimal GPM PI controller described in Section IV,it is validated in real-time. Fig. 15 depicts the sketch of the coupled tank system that is used for real-time implementation.In this set-up, each tank is fitted with an outlet pipe to transmit overflow water into the storage tank. Also, there are two submersible pumps to drive water from bottom to top of the tanks. An air pressure sensor is attached to the base of each container to measure the water level of the corresponding tank. A PC with an Advantech card and MATLAB/SIMULINK is used to implement the algorithms. The PSUPA (Power Supply Power Amplify) unit amplifies the water pressurelevel signals and passes them as an analogue signal to the PCI1711 DAQ card. The pumps control signal can be sent from the PC through the DAQ (PCI1711) card and PSUPA unit. The control signals, which are between 0 V-5 V, are transferred to the PSUPA unit where these are transformed into 24 V PWM signals to drive the pumps. The water level information is also sent to the PC via the same PSUPA unit.The model of the CTS is obtained based on the set of experimental input-output data of 2000 samples. Each sample is then collected from the open loop experiment on the CTS with a sampling time of 0.1 s by using the System Identification Tool Box in MATLAB/SIMULINK. The response of model output and actual output using pump 1(pump 2 off) is shown in Fig. 16(a) for tank 1 and Fig. 16(b)for tank 2. Similarly, using pump 2 (pump 1 off) is shown in Fig. 17(a) for tank 2 and Fig. 17(b) for tank 1. Figs. 18-19 exhibit the set-point tracking and disturbance rejection results of the proposed controller. From Fig. 19, it is observed that,the proposed controller exhibits good disturbance rejection capabilities when a load disturbance is applied at 1200 s by opening of valve 4 and valve 5 for 100 s as shown in Fig. 15.Fig. 20 depicts the efficacy of the proposed controller in the face of uncertainty, which is introduced by 50 % opening of the coupling valve whenever both tanks are at their steady state. The corresponding control signals are shown in Figs.21-22. In practice, the opening of the coupling valve not only affects the level interaction, but also alters the operating point of the system. In general, the changes introduces parametric uncertainty. However, from Fig. 20 it is observed that, the proposed controller provides appropriate control action to reduce the interaction for achieving the desired level in the face of uncertainty.

In the proposed approach, to determine the unknown variables (ωg, ωp,Kcl, τi, τd), a nonlinear optimization margin, phase margin, and maximum amplitude ratio (om*).problem is formulated, imposing constraints on the gain This nonlinear optimization problem is solved by using Lemma 1 with the “fmincon” function in MATLAB. In the proposed controller, the value ofom*influences the dynamic response (both set-point tracking and disturbance rejection) of the closed loop system as seen from Figs. 9-10, in which, the effect on the closed loop performance of the system with variation ofom*are studied. Thus, the solution of the above optimization problem with the “fmincon” function, shows that the value ofom*plays an essential role in obtaining appropriate parameters ( ωg, ωp,Kcl, τi, τd) of the PI controller. From the obtained results shown in Figs. 9-10, it is clear that selection of an appropriateom*is a trade-off between the closed loop performance and robustness. From a computational view point, it is observed that the proposed controller involves less computational complexity for the choice of an appropriate value forom*. However, as compared to the existing GPM PI controller [1], the proposed optimal GPM PI controller shows a trade-off between good closed loop dynamic performance and robustness.

VII. CONCLUSIONS

In this paper, a robust optimal decentralized PID controller is proposed based on nonlinear constraint optimization to simultaneously meet both robustness and loop performance requirements. The maximum closed loop amplitude ratio (PT)and bandwidth ( ωc) are considered as the closed-loop performance criteria. On the other hand, GPM serves as a robustness criterion. Furthermore, the unique advantage of the proposed controller, is the flexibility that is brought by imposing constraints on the maximum closed-loop amplitude.With proper tuning of the bound onPT, one can easily obtain the parameters of the PID to achieve good set-point tracking as well as a good load disturbance attenuation response. To investigate the performance of the proposed controller, the simulation results and different performance indices are compared in Table II with the work reported in [1]. To illustrate the robustness of the proposed controller,±10%parametric uncertainty is added to gain and time delays. Also,robust stability is analysed while considering multiplicative input and multiplicative output uncertainties. The obtained results reveal that the proposed controller exhibits improved performance with minimal interaction and also provides robust performance in face of uncertainty as compared to [1].To represent the practical applicability of the proposed controller, an experiment is conducted with a real-time coupled-tank liquid level system in the laboratory. From the obtained simulation and experimental results, it is determined that the proposed controller provides robust stability as well as excellent loop performance.

Fig. 15. Sketch of the experimental set-up.

Fig. 16. The step response of model from input 1 to output 1 and output 2.

Fig. 17. The step response of model from input 2 to output 2 and output 1.

Fig. 18. Experimental response of set-point tracking.

Fig. 19. Experimental response of disturbance rejection.

Fig. 20. Experimental response of both tanks for 50% opening of coupling valve.

Fig. 21. Experimental response of control input of both tanks for 50% opening of coupling valve.

Fig. 22. Experimental response of control input of both tanks for set-point tracking.

APPENDIX

The open loop transfer function of a FOPDT system that is given in (8) is rewritten here

In general the solution and closed-loop analysis of a third order system is complex. Therefore, the Routh stability criterion model reduction method [32] is adopted for simplification of the obtained model (34). The reduced order transfer function is determined directly from the elements of the higher order denominator and numerator of the Routh Array. The obtained reduced model is yielded as follows

where κ1and κ2are defined as

Now considering the characteristics equation by substituting(35) in (36) gives

where

Equation (37) is in the form ofs2+2ξωns+ωn2. After simplification of (37), one can write the following set of equations

After model order reduction, the model given in (34) is approximated as a second order model. Therefore, one can write the closed loop resonant frequency, resonant peak,maximum overshoot, and bandwidth as follows:

The damping ratio and phase margin relation can be written as

From (40), it is clear that, bothMrandMpare the functions of ξ. Also from Fig. 23, it is seen that as ξ increase, bothMrandMpdecrease. Fig. 24 depicts the frequency domain performance specification of a second order system. Also from Fig. 23 it is observed that when ξ decreases, bothMrandMpincrease. Thus, whenMrincreases, the range of frequency with gain becomes more than 0.707, i.e., the bandwidth also increases.

Fig. 23. Response of ξ vs bandwith, Mp and Mr.

Fig. 24. Frequency domain performance specification.