This paper deals with the design of a DC motor speed control implemented by an embedded controller. The design is simple and brings some important changes to the traditional Ziegler–Nichols tuning. The design also includes a novel anti-windup implementation of the controller and an integrated noise-reduction filter design. The proposed tuning method considers all important aspects of the control, such as pre-processing of the measured signals and filtering (to attenuate the measurement noise), time delays of the process, modeling and identification of the process, and constraints on the control signal. Three important aspects of designing PI and PID controllers for processes with noisy output on Arduino-type embedded computers are considered. First, it deals with the integrated design of the input filter and the controller parameters, since both are interdependent. Secondly, the method of setting the controllers from step responses by Ziegler and Nichols is modified for the case of digital signal processing (without drawing the tangent), while it recommends the suitability of its modification in terms of the use of both integral and static models. Third, the most suitable anti-windup solution for the given controller structure is proposed. In summary, the paper shows that an appropriate design of the embedded controller can achieve excellent closed-loop performance even in a noisy process environment with limited control signals.

In this paper, we discuss the main features of the generalized higher-order proportional-integrative-derivative control (HO-PID) based on the integral-plus-dead-time (IPDT) plant models. It was developed by extending the traditional PI-control to include $m$ th-order derivatives and $n\geq m$ th-order binomial series filters. The HO-PID control provides two additional degrees of freedom, which allow to appropriately modify the speed of the transients and the attenuation of the measurement noise, together with the closed-loop robustness. In this way, it pursues similar goals as an alternative fractional-order PID control. A broad family of the HO-PID controllers with the included low-pass filters is employed to solve a number of new problems. Their integrated suboptimal tuning, based on explicit formulas derived by the multiple real dominant pole (MRDP) method and evaluated by a novel approach that relates the speed of transients to the excessive input and output increments, has been simplified by introducing two integrated tuning procedures (ITPs). The main new finding is that HO-PID control enables faster transients by simultaneously reducing the negative effects of measurement noise and increasing the closed-loop robustness. A brief experimental evaluation using new sensitivity measures fully confirms the excellent HO-PID characteristics and shows that commissioning remains almost as simple as with the filtered PI-control.

The article takes up the pioneering work that led to the development of mass-used industrial controllers known by the original names of automatic-reset and hyper-reset, and generalizes them to an open family of controllers with higher-order derivatives and low-pass filters of the required order. For space limitations, the controller order is limited to the maximum degree of derivatives $m=5$ . For the integrator plus dead time (IPDT) model, the tuning of the controller parameters and filter is performed by the method of multiple real dominant poles and using the equivalent filter delay. The proposed method is successfully tested on a limited controller output, where absolute stability can be demonstrated using the circle criterion. Compared with the SIMC design, illustrative examples verify applicability of derived controllers also by simulation of stable system with dominant first-order dynamics augmented with several shorter time constants and real-time temperature control. They confirm interesting properties of all the family members, acceleration of transients by increasing $m$ and show that the considered maximum $m=5$ need not be the final limit for practical application. Note to Practitioners —Every attempt to speed up transient responses ultimately leads to the need to respect the influence of control constraints. In connection with the use of parallel PI and PID controllers, the need arose for the use of various anti-windup circuits and conditioning techniques for their adjustment. The article offers an open family of controllers solving the problem more effectively by generalizing traditional automatic-reset controllers, also referred to in the literature as series PI and PID controllers. By expanding the automatic-reset with stabilizing controllers employing higher-order derivatives, the possibility of increasing the dynamics of the circuit even up to nearly-minimum-time control arises. The new solutions enable the achievement of higher dynamics of changes, excellent robustness and consideration of measurement noise attenuation without the need to deal additionally with special techniques eliminating negative impact of control signal limitations.

This paper discusses optimal design of the series proportional–integral–derivative–accelerative (PIDA) controller for integral-plus-dead-time (IPDT) plants. The article starts with the design of disturbance reconstruction and compensation based on proportional-derivative-accelerative (PDA) stabilizing controllers. It shows that by introducing positive feedback by a low-pass filter from the (limited) output of the stabilizing PDA controller, one gets disturbance observer (DOB) for the reconstruction and compensation of input disturbances. Thereby, the DOB functionality is based on evaluating steady-state controller output. This DOB interpretation is in full agreement with the results of the analysis of the optimal setting of the stabilizing PDA controller and of its expanded PIDA version with positive feedback from the controller output. By using the multiple real dominant pole (MRDP) method, it confirms that the low-pass filter time constant in positive feedback must be much longer than the dominant time constant of the stabilized loop. This paper also shows that the constrained PIDA controller with the MRDP setting leads to transient responses with input and output overshoots. Experimentally, such a constrained series PIDA controller can be shown as equivalent to a constrained MRDP tuned parallel PIDA controller in anti-windup connection using conditional integration. Next, the article explores the possibility of removing overshoots of the output and input of the process achieved for MRDP tuning by interchanging the parameters of the controller transfer function, which was proven as very effective in the case of the series PID controller. It shows that such a modification of the controller can only be implemented approximately, when the factorization of the controller numerator, which gives complex conjugate zeros, will be replaced by a double real zero. Neglecting the imaginary part and specifying the feedback time constant with a smaller approximative time constant results in the removal of overshoots, but the resulting dynamics will not be faster than for the previously mentioned solutions. A significant improvement in the closed-loop performance can finally be achieved by the optimal setting of the constrained series PIDA controller calculated using the performance portrait method. This article also points out the terminologically incorrect designation of the proposed structure as series PIDA controller, because it does not contain any explicit integral action. Instead, it proposes a more thorough revision of the interpretation of controllers based on automatic reset from the controller output, which do not contain any integrator, but at the same time represent the core of the most used industrial automation. In the end, constrained structures using automatic reset of the stabilizing controller output can ensure a higher performance of transient responses than the usually preferred solutions based on parallel controllers with integral action that, in order to respect the control signal limitation, must be supplemented with anti-windup circuitry. The excellent properties of the constrained series PIDA controller are demonstrated by an example of controlling a thermal process and proven by the circle criterion of absolute stability.

The paper extends the earlier work entitled “Making the PI and PID Controller Tuning Inspired by Ziegler and Nichols Precise and Reliable”, to higher-order controllers and a broader range of experiments. The original series PI and PID controllers, based on automatic reset calculated by filtered controller outputs, are now augmented by higher-order output derivatives. This increases the number of degrees of freedom that can be used to modify the resulting dynamics, accelerates transient responses, and increases robustness to unmodeled dynamics and uncertainties. The fourth order noise attenuation filter used in the original work allows for the addition of an acceleration feedback signal, thus resulting in a series PIDA controller or even a jerk feedback that leads to a PIDAJ series controller. Such a design can further use the original process and filter approximation of the step responses through the integral-plus-dead-time (IPDT) model, while allowing experimentation with disturbance and setpoint step responses of the series PI, PID, PIDA and PIDAJ controllers, and thus, evaluating the role of output derivatives and noise attenuation from a broader perspective. All controllers considered are tuned using the Multiple Real Dominant Pole (MRDP) method, which is complemented by a factorization of the controller transfer functions to achieve the smallest possible time constant for automatic reset. The smallest time constant is chosen to improve the constrained transient response of the considered controller types. The obtained excellent performance and robustness allow the proposed controllers to be applied to a wider range of systems with dominant first-order dynamics. The proposed design is illustrated on a real-time speed control of a stable direct-current (DC) motor, which is approximated (together with a noise attenuation filter) by an IPDT model. The transient responses obtained are nearly time-optimal, with control signal limitations active for most setpoint step responses. Four controllers with different degrees of derivative with generalized automatic reset were used for comparison. It was found that controllers with higher-order derivatives may significantly improve the disturbance performance and virtually eliminate overshoots in the setpoint step responses in constrained velocity control.

The new modular approach to constrained control of higher-order processes with dominant first-order dynamics using generalized controllers with automatic resets (ARCs) is addressed. The controller design is based on the multiple real dominant pole (MRDP) method for the integrator plus dead time (IPDT) process models. The controller output constraints are taken into account by inserting the smallest numerator time constant of the controller transfer function into the positive feedback loop representing the automatic reset (integral) term. In the series realization of the proportional–integral–derivative–acceleration (PIDA) controller (and other controllers with even derivative degree), the time constant mentioned is complex, so only the real part of the time constant has been used so far. Other possible conversions of a complex number to a real number, such as the absolute value (modulus), can be covered by introducing a tuning parameter that modifies the calculated real time constant and generalizes the mentioned conversion when designing controllers with constraints. In this article, the impact of the tuning parameter on the overall dynamics of the control loop is studied by simulation. In addition, an evaluation of the stability of the closed-loop control system is performed using the circle criterion in the frequency domain. The analysis has shown that the approximation of the complex zero by its real part and modulus leads to a near optimal response to the set point tracking. The disturbance rejection can be significantly improved by increasing the tuning parameter by nearly 50%. In general, the tuning parameter can be used to find a compromise between servo and regulatory control. The robustness and applicability of the proposed controller is evaluated using a time-delayed process with first-order dominant dynamics when the actual transfer function is much more complicated than the IPDT model. A comparison of the proposed MRDP-PIDA controller with series PI, PID and PIDA controllers based on a modified SIMC method has shown that the MRDP-PIDA controller performs better than the SIMC method, although the SIMC uses a more complex process model.