Abstract In this paper, the coefficient diagram method (CDM) is used to design a controller for stable processes with a time delay to achieve high performance. For this,3 first order plus time delay (FOPTD) and second order plus time delay (SOPTD) models are used. The explicit tuning formulae of the controller using the polynomial representation are presented based on closed-loop poleallocation strategy according to the FOPTD and SOPTD plant models. The results are compared with other commonly available tuning methods. It is shown that the CDM control system is more successful in view of the stability, time response, disturbance rejection and robustness properties of the closed loop system.
The existence of a time delay in input-output relations is an important consideration in process modelling and control. Time delay, which is also known as the dead time, can originate from the process itself, from the use of process outcomes again as the process input, or from the impossibility of the synchronous measurements of the input and output signals. Undesired effects may occur in the stability and transient characteristics of the process as a result of the delay effect [1]. Therefore it is usually inappropriate to use the usual control methods in which the time delay is not considered for the design.
In the literature, the problem of controlling time delay processes has been a subject of considerable interest. Many proponents of the associated theoretical developments have been mentioned by Astrom and Hagglund [2], Corripiou [3] and Kaya [4] for different methods ranking from simple PI/PID/PI-PD control systems to Smith predictor structures. In all methods, the main purpose of the designer is to design a controller for the process such that the control system achieves the desired performance. The coefficient diagram method (CDM), recently developed and introduced by Manabe [5], can be extended to the time delay processes. CDM is an algebraic approach that uses polynomial representation. The most important properties of the method are 1) Adaptation of the polynomial representation for both the plant and the controller, 2) Use of a two degrees of freedom (2DOF) control system structure, 3) Non-existence (or minimal existence) of overshoot in the step response of the closed loop system, 4) Determination of the settling time at the start and to continue the design accordingly, 5) Robustness of the control system with respect to the parameter changes, and 6) Sufficient gain and phase margins for the closed-loop system [6].
In this paper, a procedure based on approximation of the time delay is proposed for stable processes. First order plus time delay (FOPTD) and second order plus time delay (SOPTD) plant models are used for the control of the processes. The most important advantages of CDM for the time delay processes can be listed as follows:
1. The design procedure is easily understandable, systematic and useful. Therefore, the coefficients of the CDM controller polynomials can be determined more easily than those of the PID or other types of controller. This creates the possibility of an easy realisation for a new designer to control any kind of system.
2. There are explicit relations between the performance parameters specified before the design and the coef- ficients of the controller polynomials as described in Sect. 2.1. For this reason, the designer can easily realise many control systems having different performance properties for a given control problem with a wide range of freedom.
3. The development of different tuning methods is required for time delay processes of different properties in PID control. But it is sufficient to use the single design procedure given in Sect. 3 in the CDM technique. This is an outstanding advantage [7, 8].
4. It is particularly hard to design robust controllers realising the desired performance properties for oscillatory processes having poles near the imaginary axis. It has been reported that successful designs can be achieved even in these cases using CDM [9].
5. It is theoretically proven that CDM design is equivalent to LQ design with proper state augmentation. Thus, CDM can be considered an ‘‘improved LQG’’, because the order of the controller is smaller and weight selection rules are also given [10].
It is usually required that the controller for a given plant should be designed under some practical limitations. The controller is desired to be of minimum degree, minimum phase (if possible) and stable. It must have enough bandwidth and power rating limitations. If the controller is designed without considering these limitations, the robustness property will be very poor, even though the stability and time response requirements are met. CDM controllers designed while considering all these problems is of the lowest degree, has a convenient bandwidth and results with a unit step time response without an overshoot. These properties guarantee the robustness, the sufficient damping of the disturbance effects and the low economic property [11].
Although the main principles of CDM have been known since the 1950s [12,13,14], the first systematic method was proposed by Shunji Manabe [15]. He developed a new method that easily builds a target characteristic polynomial to meet the desired time response. CDM is an algebraic approach combining classical and modern control theories and uses polynomial representation in the mathematical expression. The advantages of the classical and modern control techniques are integrated with the basic principles of this method, which is derived by making use of the previous experience and knowledge of the controller design. Thus, an efficient and fertile control method has appeared as a tool with which control systems can be designed without needing much experience and without confronting many problems [5].
Many control systems have been designed successfully using CDM [9, 16, 17]. It is very easy to design a controller under the conditions of stability, time domain performance and robustness. The close relations between these conditions and coefficients of the characteristic polynomial can be simply determined. This means that CDM is effective not only for control system design but also for controller parameters tuning.
CDM uses a ‘‘simultaneous approach’’ [5] to obtain the controller and closed loop transfer function. In this approach, the type and degree of the controller polynomials and the characteristic polynomial of the closedloop system are defined at the beginning. Considering the design specifications, coefficients of the polynomials are found later in the design procedure. Because of simultaneous design structure, the designer is able to keep a good balance between the rigor of the requirements and the complexity of the controller.
The experimental identification of these models using many techniques are well described [2, 20]. The Padeґ approximation has widespread use in process control. Especially if the ratio of the time delay to time constant is small, a first order or second order Padeґ approximation can successfully be used for the time delay. The first order Padeґ approximation is sufficient because its higher number leads to a higher order of the approximate transfer function of a controlled system and consequently to more complex resulting controllers. The results obtained in the next section show that the first order approximation for small and also long time delay is acceptable and gives good results.
Since CDM is a polynomial-based method, the transfer function of the plant is thought to be two independent polynomials.For a good performance, the degrees of the controller polynomials to be chosen get importance. The most important fact that affects the degrees is the existence of a disturbing signal and its type. It is advised that the minimum degree polynomials are chosen depending on the type of disturbance. In this paper, the controller polynomials are chosen for the step disturbance signal.
The pole-placement method is a straightforward design method often used in control engineering basically as a way to compute controller polynomials in CDM. A feedback controller is chosen by the pole-placement technique and then, a feedforward controller is determined so as to match the steady-state gain of closed loop system. According to this, the controller polynomials, which are determined by Eq. 14, are replaced in Eq. 2. Hence, a polynomial depending on the parameters ki and li is obtained. Later, the controller parameters are simply obtained by solving this linear algebraic equation.