J. Cent. South Univ. Technol. (2010) 17: 1036-1042

DOI: 10.1007/s11771-010-0595-0

A novel PID controller tuning method based on optimization technique

LIANG  Xi-ming(梁昔明), LI Shan-chun(李山春), HASSAN A B

School of Information Science and Engineering, Central South University, Changsha 410083, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: An approach for parameter estimation of proportional-integral-derivative (PID) control system using a new nonlinear programming (NLP) algorithm was proposed. SQP/IIPM algorithm is a sequential quadratic programming (SQP) based algorithm that derives its search directions by solving quadratic programming (QP) subproblems via an infeasible interior point method (IIPM) and evaluates step length adaptively via a simple line search and/or a quadratic search algorithm depending on the termination of the IIPM solver. The task of tuning PI/PID parameters for the first- and second-order systems was modeled as constrained NLP problem. SQP/IIPM algorithm was applied to determining the optimum parameters for the PI/PID control systems. To assess the performance of the proposed method, a Matlab simulation of PID controller tuning was conducted to compare the proposed SQP/IIPM algorithm with the gain and phase margin (GPM) method and Ziegler-Nichols (ZN) method. The results reveal that, for both step and impulse response tests, the PI/PID controller using SQP/IIPM optimization algorithm consistently reduce rise time, settling-time and remarkably lower overshoot compared to GPM and ZN methods, and the proposed method improves the robustness and effectiveness of numerical optimization of PID control systems.

Key words: PID controller optimization; infeasible interior point method; sequential quadratic programming; simulation

1 Introduction

Commonly, problems in control system design are classified into two main categories: stabilization and optimization. In spite of the suitability of optimization methods in parameter estimation and optimization of PID controllers, the challenging optimization problems, which elude acceptable solutions via traditional numerical methods, often arise in control system engineering.

A significant progress has been made in improving the performance of the PI/PID control [1]. With the aim of facilitating a consistent, reliable and robust optimal solution to industrial control problems, approaches to PID controller tuning are generally focused on model estimation, system specifications and optimal tuning. For several decades, well known tuning methods have been developed to make PID controller parameters suitable for auto-tuning and adaptive control. Some use open-loop step response such as the Cohen-Coon reaction curve, while others use the knowledge of Nyquist curve, the Ziegler-Nichols (ZN), etc. [2]. Although ZN method has good load disturbance attenuation and can adequately eliminate overshoot by tuning, it suffers from poor performance with prolonged settling-time. Furthermore, gain and phase margins have served as important measures of robustness. However, in spite of the faster transient response of the gain and phase margin method (GPM), it usually suffers from severe overshoot of greater than 5% [1].

Recently, many researchers have attempted to incorporate features on the basis of the experiences of experts and random search such as genetic algorithms (GA) and fuzzy logic with regard to PID gain scheduling [3]. YAU et al [1] constructed a modified triangular membership fuzzy PID controller and used GA to fine tune the fuzzy membership function parameters to enhance the quality of the systems performance indices including the rise time, overshoot, and steady state error. In a similar approach, topology optimization of PID controllers using hybrid cellular automata (HCA) was proposed by CARLOS et al [4]. The algorithm incorporates a set of distributed control rules that minimize the error between an appropriate mechanical signal and a mechanical target derived from the Karush-Kuhn-Tucker (KKT) optimality conditions. An intelligent tuning of PID controllers via a bacterial foraging technique was proposed by HANG et al [5]. Bacterial foraging uses natural selection to eliminate entities with poor foraging strategies for locating, capturing and ingesting food. Optimization models were applied to social foraging where groups of parameters communicate to forage cooperatively. The method was applied to optimal and robust control with disturbance resistance in a motor control loop.

Conversely, TOSCANO [6] applied a numerical optimization algorithm to solving model predictive control problems with nonlinear models and the standard SQP algorithm was used in robust PID controller design [7-9]. Since most of optimal control problems are of lower order type [10-11], and the SQP/IIPM algorithm has been proved to outperform the standard SQP algorithm in both efficiency and robustness for small to medium sized constrained NLPs [12], applying this algorithm to establishing a more efficient and stable PID parameter optimization method was the primary goal of this work. The formulation and solution of PID parameter tuning via optimization principles were presented. As a case study, the first- and second-order systems were controlled via PI and PID controllers, respectively. Comparison on the performance of the SQP/IIPM algorithm to GPM and ZN methods was made.

2 PID structure and tuning rules

Popularly known as the three-term controller, PID controller is a robust and simple system that can provide excellent control performance despite the varied dynamic characteristics of process plants [13-15]. The most studied PID structure is the ideal PID controller structure that has implementation restrictions. There are other algorithms that are widely used by different manufacturers, which are fundamental to recognize these structures and their differences with respect to the ideal algorithm for tuning parameters. The most commonly used structures are the parallel ideal PID structure, non-interacting ideal PID structure and the interacting PID structure [16-18]. A parallel ideal PID controller structure is shown in Fig.1, where k_p, k_i and k_d are the proportional gain, integral gain and derivative gain, respectively; G(s) is the process transfer function; and U(s), E(s), R(s) and Y(s) are process input signal, error signal, input signal and output signal, respectively.

Fig.1 Parallel PID controller structure

PID tuning rules recommend the mode of the PID controller and suggest the values to adjust the controller parameters, depending on the variable to be controlled. Although the success of every tuning method does not usually require the exact knowledge of the process dynamics, knowledge of the inherent characteristics of the individual components of the PID controllers is of particular importance.

It can be seen that, from the transfer function of controller,

                       (1)

proportional controller k_p will have the effect of reducing the rise time and will reduce, but never eliminate, the steady-state (SS) error; integral controller k_i will have the effect of eliminating the SS error, but it may make the transient response worse; and derivative controller k_d will have the effect of increasing the stability of the system, reducing the overshoot, and improving the transient response. Worth noting, the cumulative effect of these controllers critically relies on each other.

3 Proposed optimal PID tuning method

3.1 SQP/IIPM optimization algorithm

As proposed and implemented by LIANG et al [12], SQP/IIPM algorithm is designed to solve constrained NLP (Eq.(2)),

                                   (2)

g_i(x)≤                  

where f(x) is the objective function to be minimized; h_i(x) and g_i(x) are the equality and inequality constraint functions, respectively. SQP/IIPM algorithm uses an outer loop to sequentially linearize the NLP and an inner loop to derive its search directions d ^k by solving the following QP subproblem via an infeasible interior point method (IIPM),

                      (3)

≤ 

where k stands for the kth iteration; T stands for the transposition operation on a vector or a matrix;  and  are Hessian matrix. A modified Hessian update procedure Broyden-Fletcher- Goldfarb-Shanno (BFGS) is used at every outer iteration k. The algorithm adaptively invokes line search and/or quadratic search to evaluate step length. The scheme of the SQP/IIPM algorithm is shown in Fig.2.

3.2 Proposed optimization technique for PID tuning

The problem of parameter estimation in PID

Fig.2 Flow chart of SQP/IIPM algorithm

controllers can be formulated and solved as an optimization problem [16-17]. For the controlled system shown in Fig.1, the first- and the second-order plus dead time plants were considered and the effect of using PI and PID control systems was observed.

3.2.1 Numerical optimization of the first-order PI controlled plant

For a PI controlled plant, the sensitivity function can be defined as follows [6]:

                       (4)

where L(s) is the open loop transfer function. Let the plant and controller transfer functions respectively be:

                                  (5)

where t₀ is the initial time delay; and τ is the time constant. If taking  and setting T_i=1/k_i, then the closed loop transfer function T(s) can be written as

         (6)

T(s) yields a system with the third-order characteristic equation, and hence, its control parameters can be determined by comparing its coefficients with that of the following standard third-order system:

                 (7)

where K is the system gain; and ζ and ω₀ are the damping factor and natural frequency of oscillations of the plant, respectively. Therefore, it yields:

                            (8a)

                     (8b)

                                (8c)

The necessary and sufficient conditions to ensure the closed loop system stable are that, all the poles must lie on the negative side of s-plane [19]. Therefore, the following constraints can be deduced by setting ζ=ζ_m,

, b＞2,, and from Eq.(8a):

, a＞0                       (9)

Recall from definitions in Eqs.(4) and (6), the maximum of the sensitivity function and its transfer function can be defined in frequency domain as

                          (10)

where M_S and M_T are the measures of robustness for variations of the controlled process. According to the study of TOSCANO [6] and VILANOVA and ARRIETA [20], M_S shows the disturbances influenced by feedback control system and M_T represents the first overshoot of the step response. Small M_S yields a system that can stand larger variation of the process. A smaller M_T results in a more robust system with a smaller overshoot. Furthermore, 1/|S(jω)| is equal to |1+L(jω)|, which represents the distance between the Nyquist curve of open loop transfer function L(s) and the critical point at the frequency of 2π/ω, the minimum of the distance represents a good measure of stability margin [21].

Thus, we formulate a min-max numerical optimization problem that ensures improvement in transient response, stability margin, robustness and load disturbance rejection by first minimizing 1/|S(jω)| over ω and then maximizing the obtained result subjected to the following constraints and bound.

                      (11)

b＞2, a＞2  

3.2.2 Numerical optimization of second-order PID controlled plant

For second-order plant G(s), we employ a PID controller. Let G(s) be a second-order system with time delay and K(s) be the controller as follows:

                        (12)

where a₁ and a₀ are known constants. Similarly, taking  and setting T_i=1/k_i give the closed loop transfer function T(s):

                        (13)

Hence, the characteristic equation in system (13) is now compared with the general fourth-order characteristic equation   to yield the following equality constraints:

               (14)

To ensure stability, parameter a＞0. Taking ζ=ζ_m follows that

b＞1

Then, similar to the previous section, formulate the PID tuning problem as a min-max optimization as follows:

                       (15)

 b＞1, a＞0.

4 Simulation of optimal PID tuning

This section investigates formulations (11) and (15) to derive the controller parameters. The quality of the PID parameters obtained with SQP/IIPM algorithm was compared with that obtained using the classical gain and GPM and ZN method. The step and impulse responses for the first- and second-order plus dead time plants controlled with PI and PID controllers were observed.

4.1 Case 1 (first-order PI controlled plant)

Consider the following first-order PI controlled system:

                             (16)

Comparing Eq.(16) with Eq.(5) yields t₀=1, τ=1 and k=1. Let ζ_m=0.75 for stable transient response, then the PI tuning problem is

                       (17)

 b＞2, a＞0

Notice that, problem (17) is a “min-max” optimization type. The solution is obtained by firstly minimizing |1+L(jω)| over natural frequency ω. The obtained result is then maximized, subjected to the constraints to obtain the PI parameters.

4.2 Case 2 (second-order PID controlled plant)

Consider a second-order plant G(s) with a dead-time lag and a PID controller system K(s).

                        (18)

Comparing Eq.(18) with Eq.(12) yields: t₀=1.58, a₁=2, a₀=1 and k=1. Let ζ_m=0.76 for stable transient response, then, the PID tuning problem is

      (19)

b＞1, a＞0       

4.3 Simulation results

Table 1 shows the PI/PID controller parameters obtained by the SQP/IIPM algorithm, GPM and ZN methods, where A_m and Φ_m are the maximum gain and phase margins, respectively. Table 2 lists the performance measures (rise-time, settling-time and overshoot) for the three methods. As can be seen from Table 2, for both the PI and PID controlled plants, SQP/ IIPM algorithm significantly reduces the required settling-

time as compared with the GPM and ZN methods.

The results from Figs.3 and 4 show that, optimization of PI/PID controller parameters via SQP/ IIPM algorithm yields an obvious improvement in both the step and impulse responses of the controlled plants. Critical observation of Fig.3 reveals that, the step response from SQP/IIPM_PI controller is faster and more stable (i.e., it has a reduced overshoot, settling-time and steady-state error) than that of ZN method. However, although more stable, the speed of SQP/IIPM_PI algorithm is slower than that of its GPM_PI counterpart, which is a direct consequence of our choice of high damping factor ζ_m=0.75. In essence, the foregoing simulation reveals that, the choice of high damping factor has the effect of improving system steady-state response at the expense of a slow initial response and vice versa. Thus, it is obvious that, for all PI controlled systems, there is a trade-off in speed and stable transient response.

On the other hand, an observation of the PID controlled plant shown in Fig.4 reveals that, the incorporation of a derivative controller to the PI system

Table 1 PI/PID controller parameters obtained by three tuning methods

Table 2 Performance comparison of PI/PID tuning methods

Fig.3 Simulation results for case 1: (a) Comparison of unit step response of SQP/IIPM algorithm, GPM and ZN methods for PI controlled plant; (b) Comparison of impulse response of SQP/IIPM algorithm, GPM and ZN methods for PI controlled plant

Fig.4 Simulation results for case 2: (a) Comparison of unit step response of SQP/IIPM algorithm, GPM and ZN methods for PID controlled plant; (b) Comparison of impulse response of SQP/IIPM algorithm, GPM and ZN methods for PID controlled plant

has effectively damped the system’s overshoot, thereby improving its settling-time. Moreover, this has little or no effect on the plant’s rise-time and steady-state error. The disturbance in the early stages of the response from ZN method in Fig.4(a) is due to the presence of complex pole that has a larger negative real part than the dominant pole(s) and is expected to decay quickly, but it exists long enough and gives rise to a hump near the beginning. This problem is common with ZN method especially when the system tries to eliminate overshoot.

Another worthwhile observation is that, although the use of PI controller ensures improvement in rise-time and steady-state error, Fig.3 reveals that, it tends to increase the amount of overshoot encountered in both the step and impulse responses. This consequently leads to an increase in the required settling-time. Thus, the use of this type of controller is recommended for systems that can withstand poor initial transient responses.

Ultimately, the excellent response from the SQP/IIPM optimal PID controller not only shows its competitive advantage over GPM and ZN methods, but also highlights the superiority of PID controller system over its PI counterpart.

5 Conclusions

(1) The step responses yielded by the SQP/IIPM algorithm have equal or shorter rise-time compared with its counterparts, which signifies its suitability for systems demanding faster initial response.

(2) The SQP/IIPM algorithm settles in a very short time compared with GPM and ZN methods, which can be applied to the system requiring shorter transient response.

(3) The SQP/IIPM algorithm maintains an acceptable level of overshoot of less than 2% for both the PI and PID controlled systems. Besides, an overshoot of nearly 1% is recorded by the proposed method.

(4) The simulation quality of the PID controller tuned by SQP/IIPM algorithm is much better than that obtained via GPM and ZN methods. The use of SQP/IIPM algorithm is a good way in determining the optimal PID controller parameters and the goal of establishing a robust optimization of PID tuning is achieved.

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(Edited by LIU Hua-sen)

Foundation item: Project(60874070) supported by the National Natural Science Foundation of China; Project(20070533131) supported by the National Research Foundation for the Doctoral Program of Higher Education of China; Project supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Ministry of Education of China

Received date: 2009-12-28; Accepted date: 2010-03-11

Corresponding author: LIANG Xi-ming, PhD, Professor, Tel: +86-731-88879628; E-mail: ananxml@126.com