Zero Dynamics Research Articles

Evaluation of treatment effectiveness of asthma/COPD overlap in patients with and without obesity considering the polymorphic variant C646G of the NR3C1 gene (rs41423247)

The Aim. The current study evaluates the possibility of using the C646G polymorphic variant of the NR3C1 gene (rs41423247) for the personalized administration of combination therapy with inhaled glucocorticosteroids (iGCs) and long-acting β2-agonists in patients with asthma/COPD overlap (ACO) with and without obesity.Material and methods. Forty patients with ACO were included (17 patients with BMI < 30 and 23 with obesity (BMI ≥ 30) to received combined treatment with the use of prolonged β 2 -agonists and ICS. The CAT test, the asthma control questionnaire (Asthma Control Questionnaire-5 - ACQ-5) and the BODE index were used to analyze treatment outcomes, with genetic markers being considered one month after the study began. To assess the distribution of genotypes and alleles between groups, we used the two-tailed Pearson chi-square test (χ2) and the odds ratio (OR) in the 95% confidence interval (CI). Mathematical processing of the obtained data was performed using the SPSS (Statistical Package for Social Science Statistics) 16,0 programs.Results. The analysis of the obtained results revealed that a significant frequency of the GG genotype (80,00%) was observed in the group of patients who noted progressive deterioration of the symptom according to the CAT test, which significantly differed [χ²=17,293; p˂0,001] from that in the group of patients with positive CAT dynamics, in whom the GG genotype did not occur at all. In patients with no improvement on the CAT scale, the rate of the G allele was 0,96, and the rate of the C allele was 0,44; in patients with positive dynamics on the CAT scale, these numbers were 0,4 and 1, respectively [χ²=7,721; p=0,006; BP=5,000; 95% CI: 1,547-16,163]. The results of analyzing the dynamics of the number of points obtained for the ACQ-5 were comparable to those obtained when the CAT test was evaluated. The GG genotype was found in 50,00% of patients with negative or complete absence dynamics of the BODE index, compared with 12,5% of patients with positive dynamics [χ2=5,943, р=0,015; ВР=7,000 (95 % ДІ: 1,300-37,705)]. In the group of ACO patients with obesity, 94,44% of patients with negative or absent BODE index dynamics were carriers of the G allele [χ2=8,074, р=0,005], whereas in the group of ACO patients with normal/overweight body weight, all patients with the GG genotype had negative BODE index dynamics [χ2=6,679; р=0,016]. The GG genotype was found in 59,1% of patients with negative and absent dynamics of the BODE index, but not in any patient with positive dynamics of the BODE index [χ2=12,472, р˂0,001; ВР=24,556 (95 % ДІ: 2,752-219,100)]. Among the ACO patients studied, we identified 14 people with the GG genotype for the NR3C1 gene (rs41423247), 13 of whom (92,85%) had zero or negative dynamics of the condition in combined therapy with ICS and long-acting 2 agonists, according to the BODE index estimate. When the changes in the BODE index in non-obese patients (BMI˂30) were studied when the long-acting β 2 -agonists + ICS regimen was used, four of seventeen patients had negative BODE index dynamics: two with the CG genotype and two with the GG genotype [χ2=3,767, р=0,05]. In the presence of the GG genotype, patients with ACO and obesity (BMI≥30) experienced a progressive course of the disease or zero dynamics in response to the prescribed treatment. Simultaneously, according to the BODE index, there were no patients in the same group with the GG genotype for the NR3C1 (rs41423247) gene who had positive dynamics in response to treatment [χ2=5,856, р=0,016].Conclusions. The study's findings show that the number of people with the GG genotype for the NR3C1 gene (rs41423247) is significantly higher in patients with ACO and obesity than in patients with ACO and normal/overweight body weight. The GG genotype of the NR3C1 gene (rs41423247) is associated with negative dynamics in the application of combined therapy of ICS and long-acting 2-agonists, and can serve as a prognostic marker for assessing the risk of lack of effect from therapy according to this scheme prior to its initiation in patients with ACO, particularly in the presence of ACO and obesity comorbidities.

Zero Dynamics, Pendulum Models, and Angular Momentum in Feedback Control of Bipedal Locomotion

Abstract Low-dimensional models are ubiquitous in the bipedal robotics literature. On the one hand is the community of researchers that bases feedback control design on pendulum models selected to capture the center of mass dynamics of the robot during walking. On the other hand is the community that bases feedback control design on virtual constraints, which induce an exact low-dimensional model in the closed-loop system. In the first case, the low-dimensional model is valued for its physical insight and analytical tractability. In the second case, the low-dimensional model is integral to a rigorous analysis of the stability of walking gaits in the full-dimensional model of the robot. This paper seeks to clarify the commonalities and differences in the two perspectives for using low-dimensional models. In the process of doing so, we argue that angular momentum about the contact point is a better indicator of robot state than linear velocity. Concretely, we show that an approximate (pendulum and zero dynamics) model parameterized by angular momentum provides better predictions for foot placement on a physical robot (e.g., legs with mass) than does a related approximate model parameterized in terms of linear velocity. We implement an associated angular-momentum-based controller on Cassie, a 3D robot, and demonstrate high agility and robustness in experiments.

Leader-Following Almost Output Consensus for Linear Heterogeneous Multiagent Systems With Disturbance-Affected Unstable Zero Dynamics by Output Feedback

A novel design of observer-based output feedback consensus protocols in a low-and-high gain feedback framework is proposed for a linear heterogeneous multiagent system, where the unstable zero dynamics of the follower agents is affected by disturbances. Conditions on the graph that represents the connectivity among the agents and on how the disturbances enter the zero dynamics of the follower agents are identified, under which the leader-following almost output consensus can be attained with the proposed protocols. More specifically, the proposed protocols achieve leader-following output consensus to an arbitrarily high level of accuracy, and the effect of the disturbances on the consensus errors is attenuated to an arbitrarily low level.

Disturbance-Observer-Based Feedback Linearization Control for Stabilization and Accurate Voltage Tracking of a DC–DC Boost Converter

The nonminimum-phase property of the dc–dc boost converter can create difficulties in designing a stable, robust, and fast control for the output voltage. In this article, these difficulties are reduced by modifying the state model of the boost converter in such a way that it behaves like a minimum-phase system. More specifically, the control input entering the state equation for the output voltage is treated as an unknown disturbance to make the relative degree of the system equal the system’s order. As a result, the modified model does not exhibit zero dynamics, so that it can be treated as a minimum-phase system for the control design. The modified model is, then, used to derive a dynamic compensator for the output voltage regulation. More explicitly, the dynamic compensator is designed based on combining linearizing feedback control with a disturbance observer. The latter is employed to compensate for model uncertainties and unknown load with a view to ensure asymptotic regulation under the composite controller. The asymptotic regulation is achieved due to the integral action property characterizing the disturbance observer. More interestingly, after simple algebraic manipulation, it turns out that the composite controller reduces to a dynamic state feedback control plus an antiwindup scheme to mitigate the effect of control saturation during transients. The performance of the proposed controller is verified by experimental tests. The experimental results demonstrate the ability of the proposed controller to achieve good transient and steady-state performances.

Super twisting control of torsional vibration in non-minimum phase nonlinear drill string system

In this paper, a second order sliding mode control law for regulation of rotational speed in a non-minimum phase drill string system is proposed. In this regard, first, the nonlinear dynamic model of a drill string which is in contact with non-Newtonian mud is conducted. Next, a super twisting sliding mode control is designed that ensures the asymptotic regulation of output during sliding motion in a finite time and stable zero dynamics. It is shown that the proposed super twisting controller is capable of adjusting the output on a desired value by generating a proper sliding surface in a finite time. The convergence of the proposed controller for the dynamic model with a higher number of mode shapes is also investigated, and it is found that by increasing the number of obtained modes, the desired output needs less computation time with more accuracy.

Output tracking control of non‐minimum phase systems using on‐line output redefinition approach based on barrier function method

This paper considers the output tracking problem of non-minimum phase non-linear systems using the inverse dynamics control which is impossible in conventional methods due to unstable zero dynamics. The proposed method is based on the output redefinition approach defined as a parametric function. The tracking error is decomposed into two terms. The first one named redefined output tracking error is the difference between the redefined output and the reference signal, and the second one is the distance between the redefined output and the system output. For the first term, due to minimum phase property of the system with the redefined output, an inverse dynamics control is used to guarantee the convergence of the redefined output tracking error to zero. To minimize the second term, the parameters of the redefined output are set by solving an optimization problem whose constraints include minimum phase conditions. To solve the optimization problem which satisfies all constraints, the barrier function method as an interior point method is adopted. The optimization parameters are set using steepest descent algorithm. Due to on-line updating the redefined output parameters, the related zero dynamics is time varying. Therefore, sufficient conditions for exponential stability of the resulting time-varying zero dynamics are obtained.

Nonlinear Control Design for Regenerative and Hybrid Antilock Braking in Electric Vehicles

The use of regenerative braking and recent electrohydraulic or electromechanical brake-by-wire systems with enhanced dynamics enables improved antilock braking control performance in electric vehicles. However, the possible benefit is often limited by suboptimal control architectures with separate distributed controllers for electric motor (EM) and friction brake torque requests. Furthermore, oversimplifications in synthesis models for brake control design constrain the achievable performance. Instead, to date, threshold-based algorithms are widely used in modern production vehicles despite having been designed for slower hydraulic brake systems originally. To exploit the full potential, this work proposes a continuous nonlinear control design using input–output linearization for robust wheel speed tracking control. The design uses a unified controller for the redundant actuators. In addition, it explicitly considers drivetrain oscillations induced by regenerative braking using onboard EMs in the synthesis model. The stability for the resulting zero dynamics is shown through the Lyapunov analysis. The nonlinear controller is augmented by a subsequent control allocation for splitting the control effort on the redundant actuators. The allocation design ensures consistent control performance for both regenerative braking and hybrid braking while simultaneously aiming for high-energy recovery. The overall antilock braking control design is implemented in an electric test vehicle. Its tracking performance, disturbance attenuation, and robustness are experimentally validated through various emergency braking maneuvers on different road surfaces.

Parallel compensator for zero dynamics stabilization in fluidized bed spray granulation

This contribution is concerned with the stabilization of continuous fluidized bed spray granulation processes with external sieve mill cycle. To keep the overall mass of particles inside the granulation chamber in well-defined bounds these processes are in general operated with a mass controller. This configuration is well-known to become unstable for certain ranges of the milling diameter. Here, the instability results in a nonlinear limit cycle of the particle size distribution. As has been shown in a previous contribution the stability problem is not inherent to the process, which is open-loop stable over a wide range of operating conditions. In fact, the root cause for the instability is the mass controller, which interacts with the unstable zero dynamics of the granulation process. To overcome this problem a parallel compensator will be designed, which allows to stabilize the process zero dynamics and thus solves the problem of control induced instabilities.

Global stabilization via output feedback for a class of uncertainty nonlinear systems with time-varying delay and zero dynamics

This paper proposes a new control strategy to deal with three different types of uncertainties for a class of time-varying delay nonlinear systems. Quite different from related celebrated works, the considered systems allow the existence of parameter uncertainties in the output function and nonlinearities, dynamic uncertainties of the unmeasurable state, and continuous external disturbances, which invokes the motivation of the paper. The objective is to construct a continuous output feedback controller by utilizing a new restructured integral function and the double-domination method. The novelty control design is the capability of tackling zero-dynamic and unknown output function simultaneously, thereby guaranteeing the state convergence of the closed-loop system. Finally, the proposed scheme is applied to a practical example and a numerical one simultaneously to demonstrate the effectiveness and the superiority.

A Differential-Flatness-Based Approach for Autonomous Helicopter Shipboard Landing

This paper presents a differential flatness-based approach to trajectory planning and tracking for helicopters and demonstrates this strategy for time-optimal autonomous helicopter shipboard landing. For planning and outer-loop tracking control, helicopter dynamics are captured by a simplified nonlinear model that is shown to be differentially flat, which enables the transformation of the original nonlinear system dynamics into an equivalent linear form with endogenous nonlinear constraints. For time-optimal trajectory planning, this equivalent model is used to construct a computationally-efficient optimization problem for free end-time end-state problems, where the decision-space is that of the trajectories of the flat outputs and their derivatives. While the attitude dynamics are stabilized by an inner loop controller, for tracking the planned trajectory a flatness-based outer loop control law is designed to stabilize the unstable zero dynamics. Further, analytical guarantees that relate inner loop tracking error to outer loop robust stability and tracking performance are derived. Simulations are performed on a high-fidelity numerical UH-60 helicopter model under ground effect and ship airwake disturbances, together with ship motion profiles collected from a naval environment to validate the proposed approach for realistic landing scenarios.

Hybrid Zero Dynamics Control for Gait Guidance of a Novel Adjustable Pediatric Lower-Limb Exoskeleton.

Exoskeleton technology has undergone significant developments for the adult population but is still lacking for the pediatric population. This paper presents the design of a hip–knee exoskeleton for children 6 to 11 years old with gait abnormalities. The actuators are housed in an adjustable exoskeleton frame where the thigh part can adjust in length and the hip cradle can adjust in the medial-lateral and posterior-anterior directions concurrently. Proper control of exoskeletons to follow nominal healthy gait patterns in a time-invariant manner is important for ease of use and user acceptance. In this paper, a hybrid zero dynamics (HZD) controller was designed for gait guidance by defining the zero dynamics manifold to resemble healthy gait patterns. HZD control utilizes a time-invariant feedback controller to create dynamically stable gaits in robotic systems with hybrid models containing both discrete and continuous dynamics. The effectiveness of the controller on the novel pediatric exoskeleton was demonstrated via simulation. The presented preliminary results suggest that HZD control provides a viable method to control the pediatric exoskeleton for gait guidance.

Suboptimal output consensus of a group of discrete-time heterogeneous linear non-minimum phase systems

This paper formulates the suboptimal output consensus problem for a discrete-time heterogeneous linear multi-agent system with unstable zero dynamics, where each agent possesses its own objective function and the sum of all these private objective functions, called the overall objective function, is to be minimized. Mild assumptions on the communication topology and the agent dynamics are made under which a parameterized distributed consensus protocol based on low gain feedback is proposed for each agent. The multi-agent system is shown to achieve suboptimal output consensus under the proposed protocols in the sense that the states of all agents remain bounded while their outputs converge to a pre-specified arbitrarily small neighborhood of the optimal point as long as the design parameter is chosen small enough. Simulation results are presented that demonstrate the proposed design.

Anti-Windup Robust Backstepping Control for an Underactuated Reusable Launch Vehicle

The attitude control of an underactuated reusable launch vehicle (RLV) in the reentry phase involving nonminimum phase problem and control input constraints is investigated in this article. To address the nonminimum phase problem, an approach combining output redefinition and robust backstepping is proposed, where a synthetic output is constructed using the combination of the original output and the internal states to obtain stable zero dynamics, and then robust backstepping is performed on the new output. Besides, the ideal internal dynamics are obtained by using optimal bounded inversion, which are incorporated into the controller as the reference trajectories for the internal states to improve the output tracking accuracy. To cope with the control input constraints, a simple and useful anti-windup strategy is proposed by using feedback error clipping, which is shown to be very effective in mitigating control input saturation. Numerical simulations are given to validate the effectiveness of the proposed method.

Periodic Event-Triggered Control for a Class of Nonminimum-Phase Nonlinear Systems Using Dynamic Triggering Mechanism

In this paper, the output feedback based periodic event-triggered control problem is considered for a class of nonminimum-phase nonlinear systems via dynamic event-triggering mechanism. When only the sampled-data output is known, a new periodic event-triggered control method is proposed via output feedback and the control input updates based on a discrete-time dynamic event-triggering condition. Despite the unstable zero dynamics, the proposed control method can asymptotically stabilize the hybrid control systems by closing the loop only when it is necessary. In contrast to the continuous-time static one, the proposed dynamic triggering condition has several advantages including the easier digital implementation and the larger average inter-event time interval. The delicate analysis gives the explicit expression of the maximum allowable sampling period, and the global asymptotic stability can be achieved for the hybrid control systems. Finally, the effectiveness of the proposed periodic event-triggered control method is verified by a numerical simulation.

Reinforcement Learning-Based Cascade Motion Policy Design for Robust 3D Bipedal Locomotion

This paper presents a novel reinforcement learning (RL) framework to design cascade feedback control policies for 3D bipedal locomotion. Existing RL algorithms are often trained in an end-to-end manner or rely on prior knowledge of some reference joint or task space trajectories. Unlike these studies, we propose a policy structure that decouples the bipedal locomotion problem into two modules that incorporate the physical insights from the nature of the walking dynamics and the well-established Hybrid Zero Dynamics approach for 3D bipedal walking. As a result, the overall RL framework has several key advantages, including lightweight network structure, sample efficiency, and less dependence on prior knowledge. The proposed solution learns stable and robust walking gaits from scratch and allows the controller to realize omnidirectional walking with accurate tracking of the desired velocity and heading angle. The learned policies also perform robustly against various adversarial forces applied to the torso and walking blindly on a series of challenging and unstructured terrains. These results demonstrate that the proposed cascade feedback control policy is suitable for navigation of 3D bipedal robots in indoor and outdoor environments.

Generalized Swing-Up Control of Underactuated Mechanical Systems

Time-reversal symmetry, which is one of the main characteristics of mechanical systems without frictional forces, is applied to underactuated mechanical systems to generate desired trajectories for the generalized swing-up control. A novel necessary and sufficient condition is provided to examine whether a system trajectory is time-reversible. A desired trajectory for the generalized swing-up control is experimentally generated by a simple swing-down controller. The time-reverse of the swing-down trajectory is used as the desired trajectory for the generalized swing-up control. With the help of this method, desired swing-up trajectories can be generated also for the popular underactuated mechanical systems with pendulum-like zero dynamics such as the Acrobot, the Pendubot, the Furuta pendulum or the cart-pendulum system. Moreover, it is argued heuristically that a planar robot with an arbitrary number of revolute joints can always be swung up by only one driven joint. The theoretical results are supported by simulations with a planar robot with three revolute joints whose only one joint is actuated.

Design and Stability Analysis of a Robust-Adaptive Sliding Mode Control Applied on a Robot Arm with Flexible Links

Modelling errors and robust stabilization/tracking problems under parameter and model uncertainties complicate the control of the flexible underactuated systems. Chattering-free sliding-mode-based input-output control law realizes robustness against the structured and unstructured uncertainties in the system dynamics and avoids the excitation of unmodeled dynamics. The main purpose of this paper was to propose a robust adaptive solution for stabilizing and tracking direct-drive (DD) flexible robot arms under parameter and model uncertainties, as well as external disturbances. A lightweight robot arm subject to external and internal dynamic effects was taken into consideration. The challenges were compensating actuator dynamics with the inverter switching effects and torque ripples, stabilizing the zero dynamics under parameter/model uncertainties and disturbances while precisely tracking the predefined reference position. The precise control of this kind of system demands an accurate system model and knowledge of all sources that excite unmodeled dynamics. For this purpose, equations of motion for a flexible robot arm were derived and formulated for the large motion via Lagrange’s method. The goals were determined to achieve high-speed, precise position control, and satisfied accuracy by compensating the unwanted torque ripple and friction that degrades performance through an adaptive robust control approach. The actuator dynamics and their effect on the torque output were investigated due to the transmitted torque to the load side. The high-performance goals, precision and robustness issues, and stability concerns were satisfied by using robust-adaptive input-output linearization-based control law combining chattering-free sliding mode control (SMC) while avoiding the excitation of unmodeled dynamics. The following highlights are covered: A 2-DOF flexible robot arm considering actuator dynamics was modelled; the theoretical implication of the chattering-free sliding mode-adaptive linearizing algorithm, which ensures robust stabilization and precise tracking control, was designed based on the full system model including actuator dynamics with computer simulations. Stability analysis of the zero dynamics originated from the Lyapunov theorem was performed. The conceptual design necessity of nonlinear observers for the estimation of immeasurable variables and parameters required for the control algorithms was emphasized.

Structural Analysis of 8/6 Switched Reluctance Motor Linear and Non-linear Models

This work presents the process of obtaining the simplified model of a switched reluctance motor (SRM) 8/6. Subsequently, the structure of the single-phase model is analyzed, obtaining an exact linearization and zero dynamics of the system. Finally, the model is linearized at an operating point set at 2000 rpm The model includes Coulomb plus viscous friction nonlinearity and an ideal inverter circuit based on bridge converter topology. The simplified and linear models are simulated and compared in the Matlab®/Simulink software in order to validate the design of a classic controller using the linear model.

Fault-Tolerant Funnel Control for Uncertain Linear Systems

We study adaptive fault-tolerant tracking control for uncertain linear systems. Based on recent results in funnel control, and the time-varying Byrnes-Isidori form, we develop a low-complexity model-free controller that achieves prescribed performance of the tracking error for any given sufficiently smooth reference signal. Within the considered system class, we allow for more inputs than outputs as long as a certain redundancy of the actuators is satisfied. An important role in the controller design is played by the controller weight matrix. This is a rectangular input transformation chosen such that in the resulting system, the zero dynamics, which are assumed to be uniformly exponentially stable, are independent of the new input. We illustrate the fault-tolerant funnel controller by an example of a linearized model for the lateral motion of a Boeing 737 aircraft.

Identification of the inertial parameters of underactuated Cable-Driven Parallel Robots

Cable-Driven Parallel Robots (CDPR) employ extendable cables to control the pose of an end-effector (EE). If the number of cables is smaller than the degrees of freedom of the EE, and cables have no special arrangement reducing the EE freedoms, the robot is underactuated, and the EE is underconstrained: as a consequence, the EE preserves some freedoms even when all actuators are locked, which may lead to undesirable free motions. This paper proposes a novel methodology for the identification of the EE inertial parameters of these robots. Inertial parameters are useful, for example, in the application of feedforward control techniques. The main merit of our approach is that it does not require force or torque measurements, and only a subset of the robot kinematic variables needs to be measured. The method consists in the application of the EE internal-dynamics equations along a free-motion trajectory, also referred to as self-motion zero dynamics. This results in an over-determined system of equations that are linear in the EE inertial parameters (the Free-motion Internal-Dynamics Identification Model, FIDIM); the said system is solved according to the Total-Least-Square technique. Free-motion trajectories that are optimal for identification purposes are investigated and experimentally tested on a 4-cable robot. FIDIM is then applied, statistical analysis is performed, and the experimental results are cross-validated against additional free-motion trajectories.