A novel nonlinear PID controller design with adaptive gains

Abstract

In this study, a novel nonlinear PID controller is developed for a class of second-order nonlinear systems subject to uncertainties and external disturbances. The proposed control architecture preserves the intuitive structure of the conventional PID framework while enhancing it through a systematically designed nonlinear adaptation mechanism. In particular, the controller gains are updated online via a newly constructed adaptive update rule with an innovative structural formulation, enabling improved performance under varying operating conditions. A rigorous stability analysis of the closed-loop system is carried out using Lyapunov-type methods specifically tailored for the proposed adaptive structure. It is formally proven that all closed-loop signals remain bounded, and semi-global uniform ultimate stability (SGUUS) of the tracking error is guaranteed. Unlike many adaptive or nonlinear PID formulations in the literature, the derived stability results are not limited solely to the adaptive case. It is emphasized that the proposed analytical framework is also directly applicable to classical constant-gain PID controllers. Therefore, this study additionally provides a novel stability analysis of PID control for second-order nonlinear systems, extending existing theoretical results. The practical applicability and robustness of the proposed control methodology are validated through extensive experimental studies conducted on a real-time platform. The experimental results demonstrate improved tracking performance, robustness against uncertainties, and effective disturbance rejection. The proposed controller reduces the average tracking errors by about in pitch and in roll compared with the fixed-gain PID case, thereby confirming the theoretical findings and highlighting the effectiveness of the proposed nonlinear PID design.

Introduction

The three-term control method known as PID (short for proportional–integral–derivative) was introduced about a century ago by Nicolas Minorsky^1,2 and, since then, its conventional form has been used without much change in almost all areas of industry. The main reasons why PID is so widely used lie in its simple structure and effectiveness. Due to its simple structure, it is relatively easy to understand and implement. Despite its ability to shape the transient response and reject constant disturbances, the theoretically proven applicability of the conventional PID controller is restricted to systems up to second order^3,4. Moreover, since it is a linear controller, this characteristic is often regarded as a disadvantage. In other words, the commonly accepted stability analysis of the PID controller is valid only for linear systems having at most two states^4,5. Therefore, to achieve stability results for nonlinear systems controlled by PID, researchers typically resort to linearization techniques. In that case, the nonlinear behavior of the system is ignored and the stability region becomes limited. In addition, the PID gains obtained via linear stability analysis tools may not perform well outside that region. A solution to the stabilization problem was proposed by Zhao and Lei in⁶. In their notable study, a stability analysis of a PID controller applied to second-order uncertain nonlinear systems was presented. As highlighted in their work, fully actuated mechanical systems are second-order; thus, the aforementioned study is valid for a wide range of applications. In that study, sufficient conditions on the PID gains that guarantee stability were also provided.

The conventional PID controller has constant gains and thus suffers from performance degradation, mainly because these constant gains are used for all operating conditions of the system, as well as due to parameter variations and external disturbances⁷. In the literature, this problem has been addressed by developing tuning methods for PID-type controllers. The two main approaches used for tuning PID gains are numerical-based approaches and analytical-based ones.

Numerical approaches employ methods such as genetic algorithms, neural networks (NN), fuzzy logic, and other heuristic optimization techniques. There are many studies that focus on investigating the effects of numerical methods on the performance of PID controllers or that utilize existing methods to improve the performance of control systems. Bidikli proposed an artificial NN-based self-tuning PID control method for multi-input multi-output (MIMO) nonlinear systems⁸. Li et al. used genetic algorithms to adjust the PID gains for a water-level control system⁹. Inci et al. proposed a saturation function–based velocity observer that estimates velocity without requiring system dynamics and automatically tunes its gains through an adaptive update law, and a self-tuned robust control for Euler Lagrange systems^10–12. Fan et al. utilized back-propagation NN to adjust PID gains for the control of a rehabilitation robot. Zheng et al. used a fuzzy inference rule-based PID controller for an electro-hydraulic position servo control system¹³. Jiang and Jiang used a fuzzy PID controller for temperature control systems¹⁴. Hasan et al. proposed an adaptive neuro-fuzzy with a nonlinear PID controller to solve the speed-tracking problem in electric vehicles with brushless DC motor¹⁵. Nguyen et al. presented an adaptive nonlinear PID radial basis function neural network for ballbots with external force¹⁶. Lu et al. used the particle swarm optimization method to tune the PID gains for a turbine engine control system¹⁷. Gupta et al. utilized the grey wolf optimizer method to tune the PID gains for the control of a grid-connected PV inverter system¹⁸. Gopi et al. proposed a particle swarm optimization method to tune the gains of a PID controller for an automatic voltage regulator system¹⁹. The works in²⁰ and²¹ proposed PID controllers in which the gains are tuned using the artificial lemming algorithm. Rad et al. presented an algorithm for the self-tuning of PID controllers that uses the Ziegler–Nichols tuning method²². Park et al. proposed an NN-based self-tuning method for vibration control systems²³. Arega et al. presented a novel approach for the trajectory tracking control of a three wheeled mobile robot using neural network based nonlinear PID controller combined with the backstepping control techniques²⁴. To optimally tune the parameters of a PID controller for a system subject to stochastic delays, Qi et al. proposed an approach based on the particle swarm optimization algorithm²⁵. Reviews of self-tuning methods for PID controllers can be found in²⁶ and²⁷. From the review of the relevant literature, it can be stated that studies utilizing numerical methods generally do not provide a rigorous stability analysis; rather, they evaluate the performance of the closed-loop control system through empirical or simulation-based studies.

Unlike numerical approaches, analytical approaches employ stability analysis methods to design update rules for tuning PID gains. Zhao et al. analyzed the fundamental limitations of classical PID control for second-order nonlinear systems and proposed robust nonlinear PID controllers with rigorous global stability proofs, offering new perspectives on nonlinear, adaptive, and gain-tunable PID designs for modern industrial applications²⁸. Izadbakhsh and Kheirkhahan proposed a model-free nonlinear PID-type controller incorporating a sector-bounded nonlinear gain and considering actuator saturation for flexible joint robot manipulators²⁹. Su et al. proposed a nonlinear PID (NPID) controller to enhance the performance of conventional PID controllers³⁰. Gambier and Nazaruddin presented results of the NPID approach in³⁰ for the pitch control system of a large wind turbine. Bouhabza and Kara proposed an adaptive nonlinear PID controller for attitude control of quadrotor system³¹. In³², a design method for NPID control with partial state knowledge was demonstrated for the general case, and a complete analysis, including necessary and sufficient conditions for the existence of a design, was provided. Shahruz and Schwartz proposed a nonlinear PI compensator for linear time-invariant single-input single-output (SISO) systems³³. Lee et al. proposed a systematic design method for tuning the gains of a PID controller based on backstepping control with time-delay estimation and nonlinear damping³⁴. Kler et al. applied an NPID controller to photovoltaic systems for maximum power point tracking³⁵. Xu et al. proposed a robust nonlinear PID controller for quadrotor attitude tracking under disturbances, employing a reduced-order extended state observer for disturbance compensation³⁶. Chen et al. presented an NPID structure fused with nonlinear norm transformation³⁷. Gupta et al. presented a fractional-order NPID control method for normalized power distribution management of an advanced heavy water reactor³⁸. In³⁹, a sliding mode-based NPID-like tracking controller was proposed to address the trajectory tracking problem of rigid robots carrying an object without exact knowledge of the payload. Yu et al. proposed an NPID controller for driving an underwater autonomous vehicle along a desired trajectory with improved accuracy⁴⁰. In⁴¹, an NPID controller based on a Lyapunov-based nonlinear model predictive control algorithm was proposed and verified via numerical simulations performed on nonlinear models of a batch reactor and quadrotor systems. Xiang et al. developed a nonlinear adaptive PID tracking control method for a class of MIMO nonlinear systems, incorporating three trimming functions that reshape or nonlinearly transform the generalized tracking errors, thereby establishing an analytical connection between fixed-gain linear PID control and nonlinear PID control⁴². Bayrak presented a sliding mode-based self-tuning NPID controller for uncertain second-order systems⁴³. These methods are mathematically rigorous and guarantee the stability of the closed-loop system. However, the resulting control structure after substituting the update rules deviates significantly from the conventional constant-gain PID controller; thus, these designs are generally classified as nonlinear PID controllers.

Recent studies in control and networked systems have increasingly focused on advanced methodologies that integrate robustness, security, and data-driven intelligence. In particular, consensus-based control strategies for multi-agent systems have been extensively investigated under various complexities such as fractional-order dynamics, cyber-attacks, and communication constraints. For instance, sliding mode and adaptive neural network-based approaches have been proposed to ensure resilient consensus tracking in the presence of uncertainties and deception attacks^44,45, while fuzzy-based control frameworks have been developed to address quantization effects and sampled-data mechanisms in secure environments⁴⁶. In parallel, the rapid development of intelligent transportation systems has led to a growing body of work on traffic flow prediction, where both classical computational methods and deep learning-based models have been comprehensively explored to capture complex spatiotemporal dynamics^47,48. Moreover, communication and computation infrastructures such as heterogeneous 5G networks and cloud datacenters have been studied from the perspectives of quality-of-service enhancement and energy-efficient resource allocation^49,50. These studies collectively highlight the importance of robustness, adaptability, and efficiency in modern engineering systems, motivating the continued development of advanced control strategies, including self-tuning mechanisms, to address increasingly complex and uncertain environments.

In this paper, a novel nonlinear PID controller for second-order nonlinear systems with an adaptive gain tuning mechanism is proposed. The adaptive update rule is designed with an innovative structural formulation that distinguishes it from existing adaptive PID schemes. Lyapunov-based analysis is employed to guarantee that all closed-loop signals are bounded and that the tracking error is semi-globally uniformly ultimately stable. Notably, the stability framework can also be applied to classical PID controllers with constant gains, thereby providing new theoretical insight into PID control for nonlinear systems. The proposed methodology has been experimentally validated on an omnidirectional self-balancing robot, demonstrating both theoretical and practical effectiveness.

The contribution of the study to the literature are summarized below:

• A novel nonlinear PID controller design is proposed for a class of second-order nonlinear systems.

• A new adaptive gain update rule is developed for tuning the PID parameters, featuring an innovative structural formulation distinct from existing adaptive PID schemes.

• A rigorous Lyapunov-based stability analysis of the closed-loop system is provided.

• Boundedness of all closed-loop signals is formally proven.

• Semi-global uniform ultimate stability (SGUUS) of the tracking error is guaranteed.

• The presented stability analysis framework is also valid for constant-gain PID controllers, thereby extending theoretical stability results of traditional PID control to second-order nonlinear systems.

• The proposed methodology is experimentally validated on an omnidirectional self-balancing robot, demonstrating its practical effectiveness.

The remainder of the paper is organized as follows. First, the mathematical model of the considered second-order nonlinear system is introduced. Then, the open-loop error system is developed, followed by the proposed control design and the derivation of the closed-loop error dynamics. Next, the stability analysis is presented. The experimental results are then provided. Finally, the paper concludes with closing remarks and future directions.

Mathematical model

The dynamical representation of a second order SISO system is given as

1

where x(t), , and are position, velocity, and acceleration of the system, respectively, m(x) is the mass term, denotes nonlinear effects on the model and u(t) is the control input.

The dynamic model terms m(x) and are continuously differentiable up to their second order derivatives. Furthermore, their first and second derivatives with respect to their arguments remain bounded, provided that their arguments are bounded. Additionally, the mass term m(x) is positive and satisfies

2

for positive constants , for all possible x.

Open loop error system development

The main objective of this work is to design a nonlinear PID controller to ensure that the position x tracks a reference/desired time varying trajectory despite the lack of accurate dynamic model knowledge (i.e., m and f in (1) are uncertain). To ease the gain adjusting process, self-tuning the controller gains is aimed as well. Guaranteeing the overall stability of the closed loop system is also essential. The subsequent development relies on the common assumption that position x and velocity are available states.

The tracking error, denoted by , represents the discrepancy between the desired trajectory, , and the actual value, x(t), and is defined as

3

The standard assumption that the desired trajectory along with its first three time derivatives is made (i.e., , , , ). An auxiliary error term, shown with , is defined as

4

where is a positive gain. Another auxiliary error, represented with , that is specifically introduced to ease the presentation of the subsequent design and analysis, is defined as

5

where is a positive gain. It is emphasized that since position and velocity measurements are available than both e and s can be accessed, however, since r depends on acceleration sensing, it is not an available signal.

From the structures of (4) and (5), dynamics of e and s can be obtained, respectively. In that aspect, the rest of this section will be devoted to attaining the open loop dynamics of r.

Taking the time derivative of (5), multiplying the resultant with m to which substituting second time derivative of (4), third time derivative of (3) and time derivative of (1) reveal

6

Defining an auxiliary uncertain term, represented with , in the following manner

7

allows the right hand side of (6) to be rewritten in the below compact form

8

The auxiliary term N, as introduced in (7), depends on both uncertain dynamic model terms and acceleration measurements, and will be suppressed after being segregated into two terms. Specifically, its desired form, shown with , is defined as follows

9

Since depends only on desired trajectory and its time derivatives, which are all bounded functions of time, then it can be upper bounded as

10

for some positive constant . An error-like term, denoted with , is introduced as

11

The structures of (7) and (9) can be used along with mean value theorem⁵¹ to obtain an upper bound for as

12

where is a non-decreasing function of its argument with being the combined error vector. In view of (11), (8) is manipulated to reach

13

Control design & closed loop error dynamics

The nonlinear PID controller is proposed as

14

with adaptive gain is designed as

15

where is a positive gain used to adjust the speed of the adaptation and is a constant offset that is, for the sake of the subsequent stability analysis, formed as below

16

with , , being positive constants. In (14), the constant term of “” is introduced specifically to satisfy .

The proposed controller of (14) is not in the PID from due to the presence of s rather than the tracking error e. Since s, as defined in (4), can be viewed as PD form of e. In that respect, from (14), first and third terms can be seen as PD and IP (i.e., integral of PD) forms of e, respectively, to reach a PID control structure with adaptive gains. Alternatively, substituting (4) into (14) followed by straightforward mathematical manipulations yields the below form of the controller

17

where the proportional, derivative, integral terms are separately presented along with the constant offset term.

The time derivative of the control input in (14) is obtained as

18

where the time derivative of (15) is reached as

19

where (5) was used in obtaining both expressions. Substituting (18) into (13) yields the following closed loop error dynamics for r

20

Figure 1 shows the block diagram of the proposed nonlinear PID-type controller. The control input is generated from the tracking error through the proportional, derivative, and integral terms, while the adaptive gain is updated online.

Fig. 1.

Block Diagram of the Proposed Adaptive PID-Type Controller..

Stability analysis

Theorem 1

The proposed nonlinear PID controller in (14) in conjunction with the adaptive gain designed in (15) ensures boundedness of all the signals under the closed loop operation and semi-global uniform ultimate boundedness of the tracking error provided that the constant gains are chosen to satisfy , , and is chosen big when compared to the initial conditions of the system.

Proof 1

A nonnegative Lyapunov function, represented with , is defined as follows

21

Via utilizing (2), following bounds are in order

22

where and were previously introduced in (2).

Time derivative of (21) is obtained as

23

Substituting (4), (5) and (20), respectively, for dynamics of e, s and r reveal

24

which is simplified to reach

25

After utilizing (10), (12) and the fact that along with (16), the right hand side of (25) can be upper bounded as

26

Completing the squares as of the following forms

27

28

and making use of Young’s inequality as given below

29

allow the right hand side of (26) to be further upper bounded as

30

Provided the gain conditions of , are satisfied and is chosen big when compared to the initial conditions of the system, following bound can be obtained

31

for some positive constants , . Via utilizing (22), following differential inequality is in order

32

solving which yields

33

In view of (22), the tracking error is proven to be bounded as

34

and for increasing t, converges to the following ultimate bound

35

How to use this nonlinear PID controller

The proposed nonlinear PID controller can be used for different aims. The authors outlined three major possible usages of the proposed control methodology. Firstly, the proposed nonlinear PID controller can be used as is. This type of usage is recommended especially for systems having fast varying dynamics and/or systems subject to persistent external disturbances. Alternatively, if the system signals do not variate much after the system at hand reaches transient state, provided the tracking error settles in some desired envelope, the adaptive tuning mechanism can be turned off and the controller can be continued to run with the constant final values of the gains⁵². Or these two could be considered together in the sense that when the error converges inside an a priori determined desired bound, gain tuning could be turned off and then turned on, if the error goes outside. The controller parameters should be selected in accordance with the stability conditions, in particular and . Moreover, serves as the adaptation gain and can be chosen according to the dynamics of the system under consideration. In practice, larger values of may result in faster adaptation, whereas smaller values may lead to smoother gain evolution. For the above usages, the proposed methodology can be named as nonlinear PID controller. In a slightly different approach, the system at hand is operated at extreme conditions under the proposed nonlinear PID controller, then the final values of the proportional, integral, derivative gains are kept, and the system is restarted under a standard PID controller with constant gains which are set as the saved gains. In this approach, the proposed methodology can be considered as a gain tuning technique for PID controllers.

Experiment results

Experimental tests were carried out with the BallBot robot to corroborate the effectiveness and robustness of the proposed method. BallBot is an omnidirectional self-balancing mobile robot that can carry a moderate load. Unlike classical flat-shaped mobile robots, BallBot lacks static balance. Despite this drawback, the BallBot is an ideal choice for specific applications due to its high agility and remarkably small footprint. The total weight of the robot is approximately 7 kg, while the bowling ball used in the platform also weighs 7 kg. In addition, the height of the robot is 65 cm. The robot is placed on a bowling ball and is driven by three omniwheels, which are placed at 120-degree horizontal angles and 45-degree azimuth angles. This arrangement of the omniwheels allows the robot to move in any direction, making it highly maneuverable. The omniwheels are powered by stepper motors (57BHH82, CWD556 as driver) for precise position control. To obtain the Euler angles, we employ the BMX160 IMU with Kalman filter-based sensor fusion. Real-time operation is carried out on the STM32f746zg microcontroller, handling low-level tasks. Sensor reading and controller tasks run at a sampling rate of 1 kHz. For a detailed system model of BallBot, the reader is referred to the work of Inci et al.⁵³.

The proposed method is examined in the context of two important scenarios that a mobile robot may encounter. In the experimental validation, payload loading and battery-voltage depletion are practical manifestations of model complexity and parameter uncertainty, since they directly alter the effective dynamics of the platform during operation. The first scenario involves loading weight onto the already balanced robot. This scenario can be considered as an application of a system having instantly varying dynamics. In this situation, the optimal gains for the unweighted case suddenly become insufficient and require re-tuning, as expected. Tests start with no a priori model knowledge so all gains are initiated with zero. Time evolution of the gains are shown in Fig. 2 where initial training and weight loading phases are depicted with a gray background. For the initial training phase, the robot is tilted up to a limited angle with human assistance. It is observed that the gains converge to the values that can keep the robot in balance approximately after . The weight loading process is done by adding two 2.5kg masses between the 50th and 77th seconds. It is clearly seen that with the weight loading, the gains automatically converge to the new values. As can be seen from Fig. 3, the gain adaptation effectively limits the roll and pitch regulation errors back to within a degrees band. This result shows that the proposed adaptive mechanism can effectively accommodate the sudden dynamic change caused by payload addition. The improved regulation performance is achieved through online retuning of the PID gains, which enables the controller to restore balance without manual intervention.

Fig. 2.

[First scenario] Adaptation of the PID gains for the pitch and roll channels..

Fig. 3.

[First scenario] Roll and pitch tracking errors under adaptive gain adjustment. The shaded regions indicate the initial training and weight-loading intervals.

To demonstrate the contribution of the proposed method, the first scenario is repeated with the adaptation being turned off before weight loading. In this scenario, the effect of turning the adaptation algorithms off is investigated. Specifically, the adaptation was stopped approximately after the 37th second, as can be seen from Fig. 4. Similar to the first scenario, weight loading is performed between the 50th and 77th seconds of the experiment. Weight loading can be interpreted as both a step disturbance and a parameter/model variation. From Fig. 5, it is evident that the error range increases considerably with the weight loading and since there is no gain tuning, the errors are not decreased and reside in degrees band. The average and maximum tracking errors obtained for the roll and pitch angles under adaptive and fixed control cases are summarized in Table 1. For both cases, the average error values are calculated over the interval after , which corresponds to the post adaptation phase. As can be observed, the adaptive case yields lower error values for both angles compared to the fixed-gain case.

Fig. 4.

[First scenario] Time evolution of gains with adaptation turned off after initial training.

Fig. 5.

[First scenario] Roll and pitch tracking errors with fixed gains. The shaded regions indicate the initial training and weight-loading intervals, and the vertical line marks the instant when adaptation is stopped.

Table 1.

Average and maximum tracking errors for roll and pitch angles.

Control type	Error metric	Pitch	Roll
Adaptive	Average error	0.2336	0.2617
Adaptive	Maximum error	0.6177	1.0442
Fixed	Average error	0.8457	1.1581
Fixed	Maximum error	1.8503	2.1242

The second scenario is that the preset gains are not suitable due to changes in the battery voltage, which is commonly encountered in mobile robots. This scenario may be considered as a system subject to persistent nonlinear effects. Specifically, the robot uses a six-cell lithium battery that can vary between 25V and 18V. Battery voltage level being lower than 18V causes the system to be automatically turned off. In addition, as a characteristic feature of lithium batteries, the voltage drop, especially below the rated voltage, is non-linear. To show that controller gains need to be readjusted under battery voltage variation, a 26 minutes long testing was performed with a semi-charged battery. Battery discharging can be interpreted as a parameter/model variation. Figure 6 shows the battery voltage dropping from 21 V to 18 V and eventually automatically turning the battery off and thus the system. As can be seen from Fig. 7, the pitch and roll errors remain within the degree band even when the battery voltage changes which is obtained by increasing the gains as shown in Fig. 8. This result shows that online gain adjustment helps preserve regulation performance under battery-voltage depletion.

Fig. 6.

[Second scenario] Time evolution of the BallBot battery voltage. The inset provides a magnified view of the gradual voltage drop over the experiment..

Fig. 7.

[Second scenario] BallBot pitch and roll tracking errors under the proposed adaptive controller during the battery-voltage drop experiment.

Fig. 8.

[Second scenario] Adaptation of the PID gains for the pitch and roll channels during the experiment with decreasing battery voltage..

Conclusions

In this study, a novel nonlinear PID control framework with automatically tuned gains has been proposed for second-order nonlinear systems. The proposed design allows the controller to function either as a self-tuning PID mechanism or as a gain adaptation algorithm for classical constant-gain PID controllers, bridging the gap between conventional linear PID control and modern nonlinear adaptive control. A rigorous Lyapunov-based stability analysis has been developed, establishing uniform ultimate boundedness of the tracking error and guaranteeing boundedness of all closed-loop signals. Notably, the analysis extends to classical PID controllers with constant gains, providing new theoretical insight into the stability properties of PID control for nonlinear systems; a contribution that strengthens the foundational understanding of PID theory. Extensive experiments on an unstable omnidirectional self-balancing robot under multiple operating scenarios demonstrated that the proposed controller achieves stable tracking, robust disturbance rejection, and improved transient performance without manual gain retuning. These results confirm both the practical applicability and the theoretical predictions of the proposed methodology. Future research will focus on extending the stability framework to nonlinear systems with relative degree higher than two and to underactuated mechanical systems⁵⁴. Further investigations may also explore fusing the proposed framework with observer-based designs to eliminate velocity sensing, and with adaptive, optimal, reinforcement learning^55,56, RISE-based^57–59, or prescribed-time⁶⁰ control strategies to improve robustness and performance, thereby further advancing the theory and application of PID controllers in complex nonlinear systems.

Author contributions

A.S.O.: Writing—original draft, Writing—review & editing, Validation, Software, Visualization, Investigation, Formal analysis. A.B.: Writing—review & editing, Writing—original draft, Visualization, Software, Investigation, Formal analysis. E.S.: Writing—review & editing, Writing—original draft, Project administration, Validation, Investigation. E.T.: Writing—review & editing, Writing—original draft, Supervision, Project administration, Methodology, Investigation, Formal analysis, Conceptualization.

Data availability

The data supporting this study is securely stored in a cloud application managed by the first author (A.S.O.) and is available upon request. Interested parties may contact the first author for details on access.

Declarations

Competing interests

The authors declare that they have no competing interests.