DESIGN OF PID CONTROLLER USING REDUCED ORDER MODEL

Academic Open Internet Journal

www.acadjournal.com

Volume 15, 2005

DESIGN OF PID CONTROLLER USING REDUCED ORDER MODEL

T. Manigandan¹ N. Devarajan² S.N. Sivanandam³

1 Assistant Professor, Department of Electrical Engineering, Kongu Engineering College Perundurai, Erode-52, Tamilnadu, India – 638 052

Email: manigandan_t@yahoo.com

2 Assistant Professor, Department of Electrical Engineering, Government College of Technology, Coimbatore, Tamilnadu, India – 641 013.

Email: ndevarajan_gct@yahoo.com

3 Professor & Head, Department of Computer Science and Engineering, PSG College of Technology, Coimbatore, Tamilnadu, India – 641 004.

Email: profsivanandam@yahoo.co.in

ABSTRACT

A novel scheme is proposed to obtain a second order reduced model for stable linear time invariant continuous system. A controller is designed for the reduced second order model to meet the desired performance specifications by using pole-zero cancellation method. This controller is attached with the reduced order model and closed loop response is observed. The parameters of the controller are tuned to obtain a response with desired performance specifications. The tuned controller is attached with the original higher order system and the closed loop response is observed for stabilization process. Numerical examples are given to illustrate the proposed method.

Keywords: Model Reduction, Stable system, Unit step response, Stability,

PID controllers, Cumulative error index (J),

1. Introduction:

Stabilization is a process which controls the linear time invariant system under certain performance indices either in time domain or in frequency domain. In classical control system, the stabilization of linear time invariant system is achieved by state variable feedback technique or selection of PID controller or phase compensator. The design of controllers and compensators for higher order system involves computationally difficult and cumbersome tasks. Hence there is a need for the design of a higher order system through suitable reduced order models. In the opinion of the authors the controller designed on the basis of reduced order models should effectively control the original higher order system.

Every control system is designed for a specific application and therefore should be known as the performance criteria. The desired specifications are usually translated in the form of a rational transfer function called reference model. Given a process whose performance is unsatisfactory and the reference model having the desired performance, a controller is designed such that the performance of overall system matches the reference model.

Yeung et al [10] presented graphical design method for common continuous-time and discrete-time compensators. The method is based upon a set of Bode design charts which have been generated using appropriately normalized compensator transfer functions. Several methods for designing controllers have been developed by employing frequency response matching technique. The method proposed by Rattan et al [11] is based on complex curve fitting technique and involves the matching of frequency response of closed-loop system with that of a reference model. The complex curve-fitting method of Rattan and the dominant-data matching method of Shieh et al do not guarantee a stable controller. In the method proposed by Houpis [12], the sampled-data system is approximated by a pseudo-continuous-time control system. The approach is applicable to systems with sampling time much smaller than one second. The digital controller design method proposed by Inooka et al [13] is based on series expansions of pulse transfer function. Aguirre [14] proposed a method for the design of continuous time controllers by matching a combination of time-moments and Markov parameters of the closed-loop system with that of a reference model. The reference model need not be continuous nor in transfer function form provided their time-moments and Markov parameters are given.

The purpose of this paper is to present a simple method to get a reduced order model from the given higher-order system involving less number of computations. A controller is designed for the reduced second order model to meet the desired performance specifications. This controller is attached with the reduced order model and closed loop response is observed. The parameters of the controller are tuned to obtain a response with desired performance specifications. The tuned controller is attached with the original higher order system and the closed loop response is observed for stabilization process.

2. Statement of the Problem

Consider an n^th order stable linear time invariant single-input-single-output(SISO) system described by the transfer function.

= (1)

where a_i (0 £ i £ n-1) and b_i (0 £ i £ n) are scalar constants.

The corresponding stable k^th (k< n) order reduced model is of the form

R_k(s) = (2)

where d_i (0 £ i £ k-1) and e_i (0 £ i £ k) are scalar constants..

In this paper, assuming the original system described by equation (1), the problem is to find a reduced order model in the form of equation (2) such that the reduced order model retains the important characteristics of the original system and approximates its response as closely as possible for the same type of inputs.

3. Proposed Method of Model Reduction

The procedure for determining the reduced order model is briefly described below:

The n^th order original system given in equation (1) is equated to the k^th order reduced model with unknown parameters represented by equation (2).

Hence,

G_n (s) = R_k (s)

= (3)

On cross multiplying and rearranging the equation (3)

= (4)

Equating the coefficients of the corresponding terms in the equation (4), the following relations are obtained:

. . .

The unknown parameters are determined by taking any positive values for d₀ or e_0, for simplification, choosing d₀= 1 or e₀ = 1,and using the above relations, the other unknown parameters are evaluated. The reduced order models are tested for stability using Routh array.

4. Numerical Example

Example 1:

Consider the fourth order system transfer function given in [1, 6]:

G_n(s) = (5)

Consider a second order reduced model represented by

(6)

where d₀, d₁, e₀, e₁and e₂are unknown parameters.

Equating equations (5) and (6) and cross multiplying, we obtain

2400e₀ + (2400e₁ + 1800e₀) s + (2400e₂ + 1800e₁ + 496e₀) s²+ (1800e₂ + 496e₁ + 28 e₀) s³+ (496e₂ + 28e₁) s⁴ + 28e₂s⁵= 240d₀+ (240d₁ + 360d₀) s + (360d₁ + 204d₀) s² + (36d₀ + 204d₁) s³ + (36d₁ + 2d₀) s⁴ + 2d₁ s⁵ (7)

Comparing the like terms in equation (7), we obtain the following:

2400 e₀= 240 d₀ (8)

2400e₁ + 1800e₀ = 240d₁+360d₀ (9)

2400e₂+1800e₁+496e₀= 360d₁+204d₀ (10)

1800e₂+496e₁+28e₀ = 204d₁+36d₀ (11)

496e₂+28e₁ = 36d₁+2d₀ (12)

28e₂ = 2d₁ (13)

Choosing either d₀=1 or e₀ =1, and using the equations (8 – 13), the unknown parameters are evaluated and the second order reduced model is obtained as

R₂(s) = (14)

For comparing the proposed method with various reduced order models [3, 9], an error criterion ‘J’ is used as given below [7] :

J = (15)

where Y(t_i) and Y_r (t_i) are the outputs of the original and the reduced order systems respectively at the i^th sampling instant t_i and N is the number of sampling periods. Here the cumulative error ‘J’ is calculated for 10 sec with a sampling time increment of 1 sec. The unit step responses of the reduced models [3, 9] with that of the original system are shown in Fig.1, while the comparison of various order reduction methods is given in table 1.

Table 1. Comparison of Methods

Method of Reduction	Reduced Model	Cumulative error index ‘J’ at 10 sec
1. Prasad method 1 [9] with t=1 and m=1		81.3602
2. Prasad method 2 [9] with t=2 and m=0		27.4459
3. Routh-Hurwitz array method [3]		12.0783
4. Proposed method		4.3794

Fig. 1 Comparison of unit step responses

5. DESIGN OF PID CONTROLLER

In general, series controllers are preferred over feedback controllers because for higher order systems, a large number of state variables would require large number transducers to sense during feedback. This makes the use series controllers very common. The controller is attached with the reduced order model and closed loop response is observed. The parameters of the controller are tuned to get a response, meeting the desired specifications. The tuned parameters are introduced into the higher order system for stabilization processes.

The transfer function of PID controller is written for a continuous system as [15, 16]

(16)

The design involves the determination of the values of the constants K_p, K_i and K_d, meeting the required performance specifications. The general block diagram of the PID controller is shown in Fig.2.

Fig.2 Feedback Control System with PID Controller

The proportional control action multiplies the error signal with a constant to improve the overall gain of the system.

5.1 Performance Specifications

Performance specifications are considered with respect to the closed loop response of the compensated system to unit step input. The specifications are chosen as:

(i) Overshoot ≤ 3%

(ii) Settling time ≤ 3 seconds

(iii) Steady state error ≤ 2%

Note: The above values are left to the choice of the designer.

5.2 System Description

The closed loop transfer function of a unity feedback system with G(s) as the open loop transfer function is given in equation (17) as:

(17)

If the system output response does not satisfy the specifications, the PID controller is added to the forward path and the closed loop transfer function of the system is given in equation (18) as:

(18)

where G_c(s) is the transfer function of the controller.

5.3 General Algorithm for Design of Controller Using Reduced Order Model

Step 1 : Read the open loop transfer function of the given higher order system.

Step 2 : Form the closed loop transfer function.

Step 3 : Obtain the step response of closed system.

Step 4 : Check the response for the required specifications.

Step 5 : If the specifications are not met, obtain a reduced order model using the

proposed scheme and design a controller for the reduced order model.

Step 6 : Obtain the initial values of the parameters K_p, K_i and K_d by pole-zero

Cancellation method.

Step 7 : Attach the controller with the reduced order model and get the closed loop

response with the initial values of the controller parameters.

Step 8 : Find the optimum values for the controller parameters which satisfy the

required specifications.

Step 9: With the optimum values, attach this controller with the original system.

Step 10: Obtain the closed loop response of the system with the controller.

Step 11: If the specifications are met, exit; else tune the parameters of the controller till it

meets the required specifications.

6. Numerical Examples

Example 1

Consider the fourth order system transfer function given in [1, 6]:

G_n(s) = (19)

By applying the proposed scheme (as detailed in section 3), the reduced order model is:

R₂(s) = (20)

Applying pole-zero cancellation method to the reduced model, the initial values of K_p, K_i and K_d are obtained as [17]:

K_p= 29.5897, K_i = 41.0256 and K_d = 1

Using the simulation procedure, initial parameters are tuned to get unit response of the compensated system to meet the required specifications.

The tuned values obtained are as follows:

K_p= 29.5897, K_i = 41.0256 and K_d = 0.001

The designed PID controller G_c(s) is attached with the original higher order system. The closed response is found to meet the required specifications as given in section 5.1. Fig.3 shows the step response of the original system. Fig.4 and Fig.5 respectively show the step response of the reduced and higher order system with PID controller.

Fig. 3 Step Response of Original System

Fig. 4 Step Response of Reduced Order Model with PID Controller

Fig. 5 Step Response of Original System with PID Controller

Example 2

Consider the eighth order system transfer function from [18]:

G_n(s) = (21)

By applying the proposed scheme (as detailed in section 3), the second order reduced model is obtained as:

R₂(s) = (22)

Applying pole-zero cancellation method to the reduced model, the initial values of K_p, K_i and K_d are obtained as [17]:

K_p= 1.436, K_i = 36.63, K_d = 1

Using the simulation procedure, initial parameters are tuned to get unit response of the compensated system to meet the required specifications.

The tuned values obtained are as follows:

K_p= 9, K_i = 30, K_d = 0.0001

The designed PID controller G_c(s) is attached with the original higher order system. The closed response is found to meet the required specifications as given in section 5.1. Fig.6 shows the step response of the original system. Fig.7 and Fig.8 respectively show the step response of the reduced and higher order system with PID controller.

Fig. 6 Step Response of Original System

Fig. 7 Step Response of Reduced Order Model with PID Controller

Fig. 8 Step Response of Original System with PID Controller

7. Discussion and Conclusion

The proposed method of model reduction is computationally simple and does not involve much procedure. It gives the stable reduced system if the original system is stable. The given examples show that the second order reduced models are obtained in a straight forward manner. The proposed scheme provides good approximation and preserves the stability of higher order system. The PID controller has been designed for the reduced order model to meet the desired performance specifications. The tuned controller is attached with the original higher order system and the closed loop response is observed for stabilization process. The method is mathematically simple, so that the PID controllers of different complexities may be tried to achieve best results. The proposed method can also be extended to the design of compensators and sub-optimal controllers for continuous and discrete systems.

8. References

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[10] K.S. Yeung, K.Q. Chaid and T.X. Dinh, “Bode design charts for continuous-time and discrete-time compensators,” IEEE Transaction on Automatic Control, vol. AC- 38 (3), pp. 252-257, 1995.

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[12] C.H. Houpis, “Refined design method for sampled data control systems: The pseudo-continuous-time (PCT) Control System,” Proc. IEE, Pt. D. 132 (2), pp. 69-74, 1985.

[13] H. Inooka, G. Obinata and M. Takeshima, “Design of Digital controller based on series expansions of pulse transfer functions,” Trans. ASME, J. Dyn. Sys. Meas. Control, 105, pp. 204-208, 1983.

[14] L.A. Auguirre, “New Algorithm for closed-loop matching,” Electronic Letters, 27(24), pp. 2260-2262, 1991.

[15] B.C. Kuo., “Automatic Control Systems,” Prentice Hall of India, New Delhi, India, 1982.

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[17] I.J. Nagarath and M. Gopal, “Control Systems Engineering,” New Age International (P) Ltd., New Delhi, India, 1997.

[18] S. Palaniswami and S.N. Sivanandam, “Design of PID Controller using Lower Order Model,” National Symposium on Intelligent Measurement and Control, NSMIC-2000, MIT, Chennai, India, pp. 290-296, February 2000.

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