Proportional-Integral-Derivative control system has been widely used in industrial applications. For uncertain and unstable systems, tuning controller parameters to satisfy the process requirements is very challenging. In general, the whole system’s performance strongly depends on the controller’s efficiency and hence the tuning process plays a key role in the system’s response. This paper presents a robust optimal Proportional-Integral-Derivative controller design methodology for the control of unstable delay system with parametric uncertainty using a combination of Kharitonov theorem and genetic algorithm optimization based approaches. In this study, the Generalized Kharitonov Theorem (GKT) for quasi-polynomials is employed for the purpose of designing a robust controller that can simultaneously stabilize a given unstable second-order interval plant family with time delay. Using a constructive procedure based on the Hermite-Biehler theorem, we obtain all the Proportional-Integral-Derivative gains that stabilize the uncertain and unstable second-order delay system. Genetic Algorithms (GAs) are utilized to optimize the three parameters of the PID controllers and the three parameters of the system which provide the best control that makes the system robust stable under uncertainties. Specifically, the method uses genetic algorithms to determine the optimum parameters by minimizing the integral of time-weighted absolute error ITAE, the Integral-Square-Error ISE, the integral of absolute error IAE and the integral of time-weighted Square-Error ITSE. The validity and relatively effortless application of presented theoretical concepts are demonstrated through a computation and simulation example.

Time lags occur often in various engineering systems and industry processes, such as in communication networks, chemical processes, turbojet engines, and hydraulic systems. Delays have a considerable influence on the behavior of the closed-loop systems, can generate oscillations, and even lead to instabilities [

Open-loop unstable delay systems are often encountered in process industry, and pose a more challenging problem to controller design compared to that of stable open-loop systems. The presence of an unstable pole in the system imposes a minimum limit on the control performance, which in some cases can lead to an excessive overshoot and long settling time.

Proportional-Integral-Derivative (PID) controller, though a very old design, is still one of the favorite and most widely used controllers for many industrial process control applications. This is due to its simple structure, satisfactory control performance, and acceptable robustness [

One of the well-known approaches to computing the stabilizing PID controller region is based on a generalization of the Hermite-Biehler theorem [

Robust stability of uncertain systems has become of great interest in the past few decades. Robustness is defined as the performance and stability of plants exposed to uncertainties. The Kharitonov theorem is well-known for stability analysis of interval systems. Based on the Kharitonov theorem, the edge theorem in Barmish et al. [

To determine the robust stability of a time-delay system subjected to parametric uncertainty, researchers have extended the GKT and the edge theorem to quasi-polynomials [

Prior studies have obtained some important results relating to the stabilization of interval systems. Barmish et al. [

In Ho et al. [

In this paper, we endeavor to determine the set of all PID gains that stabilize an uncertain and unstable second-order delay system, where the coefficients are subjected to perturbation within prescribed ranges. We propose an approach based on combining the background considerations presented in Section 3 and the result obtained by Farkh et al. [

The rest of the paper is organized as follows: In Section 2 we discuss the computation of all PID controllers for an unstable second-order delay system. The problem formulation is given in Section 3. Section 4 is devoted to the robust stabilization problem for an uncertain and unstable second-order system with time delay controlled via a PID controller. Section 5 is reserved for the simulation example. A description and application of the genetic algorithm (GA) is presented in Section 6, and conclusions are presented in Section 7.

In Farkh et al. [

Under the assumptions of _{0} < 0 and/or _{1} > 0, the

where

For _{i}_{d}

The parameters

where

We consider a second-order delay system described by the following transfer function:

To determine the

In this section, a procedure is proposed for robust stabilization of an unstable delay system that belongs to a linear interval plant, where the time delay,

Consider the following transfer function:

where

We can use the GKT extended for quasi-polynomials [

where

with the

In our case, we use

According to Bhattacharyya et al. [

Let

The GKT, we first need to determine the extremal set of line segments,

and

where

The extremal subset

where

Some of the subset equations may be the same, hence, the extremal subset is defined as [

The extremal subset of line segments (or generalized Kharitonov segment polynomials) is [

where

With the knowledge that

The previous results of the robust parametric approach control proved to be an efficient control design technique. In the following, they will be used for the synthesis controllers that simultaneously stabilize a given uncertain time-delay system.

In this section, we consider the problem of characterizing all PID controllers that stabilize a given unstable second-order interval plant with a time delay:

where

To obtain all PID gains that stabilize

and the compensator as follows:

The family of closed-loop characteristic quasi-polynomials

The problem of characterizing all stabilizing PID controllers requires determining all the values of

Let

Then,

where the 32 extremal plants in

The closed-loop characteristic quasi-polynomials for each of these 32 plant segments,

where

We posit the following theorem on stabilizing an unstable second-order interval plant with time delay using a PID controller.

Let

From Theorem 2, it follows that the entire family

To obtain a characterization of all PID controllers that stabilize the interval plant

We consider the plant family

The entire family

According to from

We remark here that from

We define

To compute all the stabilizing PID gains, we first determine all the

For a fixed

The intersection of these stability regions presents an overlapping area of the boundaries constituting the entire feasible controller sets that stabilize the entire family

Finally, by sweeping over

GAs are efficient stochastic search methods based on the concepts of natural selection and evolutionary genetics. GAs are communities of individuals, in which through randomizing the cycle of discovery, crossover and mutation, individuals can adjust to a specific setting. The environment offers valuable knowledge (fitness) to individuals, and the selection mechanism supports the preservation of individuals of greater quality. Therefore, during the development cycle, the overall output of the population is growing, ideally contributing to an optimal solution. GAs have been used in diverse fields and are considered as an efficient tool for global optimization. Attempts to apply GAs to control system and identification design problems have been made [

We look for the optimum system and controller parameters in the robust stability area using one of the following requirements ITAE integral of time-weighted absolute error, ISE Integral-Square-Error, IAE integral of absolute error and ITSE integral of time-weighted Square-Error defined by following relationships:

If we want to reduce the tuning energy, the ITAE and IAE criteria should be considered. Conversely, the ITAE and the IAE parameters are being considered when we want to reduce the tuning energy. If we assign preference to rising time, the ITSE criteria are adopted, while we choose the ISE criterion to guarantee the energetic tuning costs [

The following algorithm sums up the steps of the control law:

Introduction of the following parameters:

initial population

Initialization of the generation counter (

Initialization of the individual counter (

For

efficiency evaluation of j^{th} population individual

Individual counter incrementing (

If

Otherwise, application of the genetic operators (selection, crossover, and mutation) for finding a new population.

Generation counter incrementing (

If

Selecting

In the following, a GA with the generation number of 100,

We consider the uncertain unstable delay system

The robust PID stability region is shown

Criterion | ISE | IAE | ITAE | ITSE |
---|---|---|---|---|

1.9001 | 1.9002 | 1.9564 | 1.9001 | |

_{0} |
−0.4000 | −0.4023 | −0.4 | −0.6000 |

_{1} |
4.0001 | 4.0313 | 4.0008 | 5.7665 |

_{p} |
1.3280 | 1.3285 | 1.3429 | 4.2737 |

_{i} |
1.0007 | 1.0962 | 1.0022 | 1.4642 |

_{d} |
0.9605 | 0.2298 | 0.0134 | 0.3947 |

The time parameters and percentage overshoot values for unit step responses shown in

ISE | IAE | ITAE | ITSE | |
---|---|---|---|---|

Rise time | 0.9723 | 1.0076 | 0.9494 | 0.5607 |

Settling time | 34.4134 | 63.8065 | 64.5009 | 7.2038 |

Peak time | 3.9667 | 3.8094 | 2.1109 | 2.1331 |

Overshoot | 58.844 | 98.9484 | 111.0873 | 71.9221 |

This study proposed the application of the Hermite-Biehler and GKT to defining the robust PID stability area for the control an of an uncertain and unstable second-order time-delay system. In the optimization process, the optimal system and optimal PID controller parameters are calculated by using the integral performance criterion based on the error.

The authors acknowledge the support of King Saud University, Saudi Arabia.