Classical low-order controllers such as propotional-integral-derivative(PID) and First-order controllers account for more than 95% of the industrial controllers although many modern control theories have been developed. Because PID controller is simple to implement and allows us to design without a mathematical model of the process, related studies are still being published.
The design methods of controllers can be classified into model-based and data-based design. In the 2000s, a very remarkable result on PID design was suggested by S.P. Bhattacharyya et al. For a given transfer function plant model with a delay, they have showed that all the stabilizing set of PID controllers can be determined by solving linear programming. Once a stabilizing PID set is obtained, a subset of controllers satisfying different performance specifications can be selected from the set. This method provides almost all possible PID type controllers as the conventional methods give only one controller under the same design conditions.
Data-based PID design methods sometimes referred to as PID tuning methods have been improved for a long time since the Ziegler-Nichols rule. According to several surveys, it was revealed that only 20% of actual control loops worked well but the rest gave poor performance due to poor controller tuning, sensor/control valve problems, and control system design problem. Then it becomes obvious that the PID controller with an automatic tuner is attractive for better control. There are many vendors that provide commercial PID autotuners. All these automatic tuners inculde the plant identification based on open or closed-loop step tests, ramp tests, relay feedback tests, and pseudo-random binary signal(PRBS). Most identification are to fit a finite set of input and output data to a model equation. Hence, modeling accuracy may decrease in the presence of the measurement noise. Also, there is hardly a way to tune the controller so that the closed-loop system satisfies the desired time-response specifications, such as overshoot and settling time.
Main problem in this thesis is to propose a new closed-loop tuning rule of discrete-time lower-order controllers with time-response specifications which is carried out by only a single pulse response data. The controller parameters are calculated so that the moments of a reference model from 0~ to the kth match the moments of the closed loop system including the controller to be tuned. We assume that the unknown discrete-time process be a stable Linear Time Invariant(LTI) system. We also consider two kinds of controller structures: cascade and two-parameter configurations. The type of the controller is considered to be either a PID controller or a general contrller less than or equal to the second-order. That is, these are PID, PI, second-order, first-order controllers in cascade structure, and PID, PI, second-order, first-order controllers in two-parameter structure. It should be emphasized that one pulse test can determine the parameters of the eight controllers in the two structures. The tuning algorithm consists of 3 steps. In this approach, we first obtain 0~kth moments of the unknown sampled plant from a set of pulse responses. For this procedure, a stabilizable initial controller is arbitrarily selected and the sampled output to a simple pulse input like a rectangular pulse is collected. Then the kth moment of the unknown plant can be computed using input and output moments. This formula has been proved. The Second step is to generate a discrete-time reference model satisfying the desired time responses. The moments of the reference model from 0th to kth can be calculated direcly from its transfer function. In the third step, the parameters of the eight controllers are determined by several closed-form formula, which are derived under the matching conditions between the closed-loop system and the reference model moments.
If the cascade controller structure is chosen, it is not avoidable that the zeros of the controller appear in the zeros of the overall system, which makes it difficult to meet the time response. To resolve this problem, we introduce a constraint on the zeros of the controller.
The remaining problem is how to determine the closed-loop stability of the tuned controllers. A simple but not exact method for stability check is proposed. Fortunately, a low-order model of the plant can be estimated using the moments of the plant. Assume that, the estimated low-order model is equal to a plant model. Then applying Nyquist stability criterion, the stability of the tuned controller can be examined. If the tuned controller makes the overall system unstable, it is necessary to change the performance specifications, the order of controller, or structure of controller.
Through several examples, we have showed that although the proposed tuning algorithm is simple, it produces very good results.