Nonaffine Systems Research Articles

Dynamic event-triggered fault-tolerant control for nonaffine systems with asymmetric error constraint

This paper focuses on the problem of dynamic event-triggered fault-tolerant control (FTC) for nonaffine systems with unmeasurable state and asymmetric error constrain. At first, an adaptive failure compensation observer is designed to calculate unpredictable state of the nonaffine systems. Then, a dynamic event triggering strategy (DETS) is constructed to adjust the control signal and make it more flexible in transmission. Next, a new error-dependent conversion function (EDCF) is designed to confine tracing error, which can ensure that the designed Lyapunov function is always positive definite, and extend the symmetric constraint boundary to the asymmetric constraint boundary. The tracking performance and stability of closed-loop systems are proven through Lyapunov stability analysis. Finally, the availability of this method is verified by simulation experiments.

Test Mass Capture Control for Drag-Free Satellite Based on State-Dependent Riccati Equation Method

The drag-free satellite plays an important role in the space-based gravitational wave observatory. The capture control of test mass after release is a crucial technology that can affect the success of the mission. The test mass must be released to the center of the electrostatic suspension cage accurately. This paper presents a nonlinear dynamic model of drag-free satellites in Lagrange formalism. A capture control scheme for test mass release phase is proposed based on the state-dependent Riccati equation (SDRE) strategy. To deal with the actuator saturation problem, a nonlinear saturation model is introduced to the dynamics of satellite, while the SDRE strategy is applied to the non-affine system. The effectiveness of the proposed methodology is verified by the numerical simulation for the drag-free satellite.

Finite-time PID control for nonlinear nonaffine systems

Finite-time PID control for nonlinear nonaffine systems

Fuzzy Adaptive Control for Consensus Tracking in Multiagent Systems with Incommensurate Fractional-Order Dynamics: Application to Power Systems

This paper presents a novel approach to address the consensus tracking problem within a class of incommensurate fractional-order nonlinear non-affine systems. Our method employs an adaptive fuzzy technique that integrates newly developed fractional adaptive algorithms based on the Lyapunov method into the controller's design process. This method develops stability based on the global representation of the follower and leader systems, reducing assumptions on the system dynamics to address non-affinity. Additionally, it introduces a simplified approach to designing controllers for incommensurate fractional-order multiagent systems. The proposed controller effectively discerns uncertainties and external disturbances, compelling follower agents to seamlessly follow the desired trajectories set by the leader. Notably, compared to the existing literature, our method exhibits key advantages, including reduced assumptions regarding the system's non-affinity and a simpler design for controlling incommensurate systems. We demonstrate the efficacy of the proposed incommensurate fractional controller through simulations conducted using MATLAB on a fractional-order multiagent power system.

Data-Driven Dynamic Internal Model Control.

A data-driven dynamic internal model control (D3IMC) scheme is proposed for unknown nonlinear nonaffine systems bypassing modeling steps. Different from the traditional internal model constructed by either a first-principle or an identified model, a dynamic internal model (DIM) is developed in this work using I/O data where a compact form dynamic linearization approach is introduced for addressing the nonlinearity and nonaffine structure. Then, the D3IMC is proposed with both a nominal control algorithm and an uncertainty compensation control algorithm. The former can quickly respond to the feedback errors and the latter can compensate the model-plant mismatch and external disturbances. Meanwhile, the adaptive parameter updating law in the proposed D3IMC method inherits the robustness against uncertainties. A nominal D3IMC is also designed without including the compensator when there is no exogenous disturbance since the adaptive mechanism can handle system uncertainty. Further, the results are extended and a full-form dynamic linearization-based D3IMC is developed to address control of nonlinear systems with more complex dynamics. All the proposed D3IMC methods are data-driven without need of an explicit model, and thus they are significant extensions from the traditional model-based IMC. Simulation study verifies the results.

Evolution-guided value iteration for optimal tracking control

In this article, an evolution-guided value iteration (EGVI) algorithm is established to address optimal tracking problems for nonlinear nonaffine systems. Conventional adaptive dynamic programming algorithms rely on gradient information to improve the policy, which adheres to the first order necessity condition. Nonetheless, these methods encounter limitations when gradient information is intricate or system dynamics lack differentiability. In response to this challenge, evolutionary computation is leveraged by EGVI to search for the optimal policy without requiring an action network. The competition within the policy population serves as the driving force for policy improvement. Therefore, EGVI can effectively handle complex and non-differentiable systems. Additionally, this innovative method has the potential to enhance exploration efficiency and bolster the robustness of algorithms due to its population-based characteristics. Furthermore, the convergence of the algorithm and the stability of the policy are investigated based on the EGVI framework. Finally, the effectiveness of the established method is comprehensively demonstrated through two simulation experiments.

Data-Driven Adaptive Controller Based on Hyperbolic Cost Function for Non-Affine Discrete-Time Systems with Variant Control Direction

As technology evolves, more complex non-affine systems are created. These complex systems are hard to model, whereas most controllers require information on systems to be designed. This information is hard to obtain for systems with varying control directions. Therefore, this study introduces a novel data-driven estimator and controller tailored for single-input single-output non-affine discrete-time systems. This approach focuses on cases when the control direction varies over time and the mathematical model of the system is completely unknown. The estimator and controller are constructed using a Multiple-input Fuzzy Rules Emulated Network framework. The weight vectors are updated through the gradient descent optimization method, which employs a unique cost function that multiplies the error by a hyperbolic tangent. The stability analyses demonstrate that both the estimator and controller converge to uniformly ultimately bounded (UUB) functions of Lyapunov. To validate the results, we show experimental tests of force control that were executed on the z-axis of a drive-controlled 3D scanning robot. This system has a varying control direction, and we also provide comparison results with a state-of-the-art controller. The results show a mean absolute percentage tracking error smaller than one percent on the steady state and the expected variation in the system’s control direction.

Nearly optimal stabilization of unknown continuous-time nonlinear systems: A new parallel control approach

This paper develops a novel online nearly optimal control (ONOC) method for unknown continuous-time (CT) nonaffine nonlinear systems without recovering unknown systems. First, a dynamic control law is proposed for CT nonaffine nonlinear systems using parallel control. To achieve the proposed dynamic control law, an affine augmented system (AAS) is constructed according to the original system, and an augmented performance index (API) is constructed on the basis of the original performance index (OPI). Then, the stability relationship between the original system and the AAS is provided, and it is proven that, by selecting a suitable parameter in the API, optimal control of the AAS with the API is equivalent to near-optimal control of the original system with the OPI. Subsequently, based on the proposed dynamic control law, we extend integral reinforcement learning (IRL) to completely unknown CT nonaffine systems, and it is further proved that closed-loop signals are uniformly ultimately bounded (UUB) without the assumption that the input dynamics are bounded. Furthermore, the OPI can be set to an arbitrary positive-definite form, and the UUB bound for the state vector can be predetermined. Lastly, simulations are offered to exhibit the correctness of the developed ONOC method. Source code of this paper is available at: https://github.com/lujingweihh/Adaptive-dynamic-programming-algorithms/tree/main/model_free_integral_reinforcement_learning.

Neural Network Based Adaptive Inverse Optimal Control for Non-Affine Nonlinear Systems

In this paper, a novel methodology is introduced for the inverse optimal control of non-affine, nonlinear and discrete-time systems. Although inverse optimal control of affine systems is studied in detail in technical literature, there is no adequate research about its implementation on non-affine systems. here are two main contributions of this work. Firstly using the input–output data of the system to be controlled its NARMA-L2 model is obtained using a multi-layer feedforward neural network, this step provides a conversion from a non-affine to affine system model. After the affine system model is obtained, the inverse optimal control law is applied. The second contribution of this paper is the computation of the inverse optimal control signal. The selection of the P matrix in the control law is crucial since its value directly affects the control performance. Here a novel method is proposed where an adaptive and optimal P matrix is computed online using a recurrent neural network to minimize a predefined cost function. The performance of the proposed control method is evaluated by simulations performed on benchmark problems. The robustness of the method is also tested by additional simulations where noise and disturbance is imposed on the system. The obtained results justify the applicability of the proposed approach.

Neural-Network-Based Adaptive Fixed-Time Control for Nonlinear Multiagent Non-Affine Systems.

In this research, the adaptive neural network consensus control problem is addressed for a class of non-affine multiagent systems (MASs) with actuator faults and stochastic disturbances. To overcome difficulties associated with actuator faults and uncertain functions of the designed MAS, a neural network fault-tolerant control scheme is developed. Moreover, an adaptive backstepping controller is developed to solve the non-affine appearance in multiagent stochastic non-affine systems using the mean value theorem. Being different from the existing control methods, the developed adaptive fixed-time control approach can ensure that the outputs of all followers track the reference signal synchronously in the fixed time, and all signals of the controlled system are semi-globally uniformly fixed-time stable. The simulation results confirm that the presented control strategy is effective in achieving control goals.

Data-Driven Internal Model Learning Control for Nonlinear Systems.

A novel data-driven internal model learning control (DIMLC) strategy is developed for a nonlinear nonaffine system subject to unknown nonrepetitive uncertainties. At first, an iterative dynamic linearization (IDL) approach is employed for reformulating the nonlinear plant to an iterative linear data model (iLDM). Then, the nominal form of the IDL-based iLDM is used as an internal model of the nonlinear plant whose parameters are estimated by an iterative adaptive updating mechanism using only input-output (I/O) data. The equivalent feedback-principle-based internal model inversion is further applied to the subsequent controller design and analysis. The proposed DIMLC contains two parts. One is a nominal controller designed by the inversion of the internal model which achieves a perfect tracking of the target output; the other is a compensatory controller which offsets the uncertainties. The novel DIMLC is data-driven and does not require an explicit model. It can deal with model-plant mismatch and disturbances, enhancing the robustness against uncertainties. The theoretical results are verified by simulation study.

Event-Triggered Adaptive Bipartite Asymptotic Tracking Control Using Intelligent Technique for Stochastic Nonlinear Multiagent Systems

The problem of event-triggered based adaptive bipartite asymptotic tracking control for multi-agent stochastic pure-feedback nonlinear systems over sign digraph is considered. The agents classification optimization strategy (ACOS) is presented, which transforms a structurally unbalanced multi-agent topology graph into a structurally balanced multi-agent topology graph. Furthermore, how to deal with the nonaffine structure and unknown control gains for multi-agent stochastic pure-feedback nonlinear systems is a challenging issue for designing controllers. As a result, the mean value theorem is used to transfer the nonaffine systems into the affine systems, and the Nussbaum functions are used to eliminate the influence caused by the unknown control gains. In addition, the intelligent control technique is used to approximate the unknown nonlinearities, the event-triggered control strategy following the switching thresholds is introduced to save unnecessary communication resources, and the computation burden is eliminated by means of the dynamic surface technique. It is proved that all signals of the closed-loop systems are bounded in probability and the tracking errors asymptotically converge to zero in probability. Finally, the simulation results illustrate the validity of the proposed scheme. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Impact Statement</i> —In nature, there are a lot of interesting movements of multi-agents. In this paper, the multi-agent stochastic pure-feedback nonlinear systems over sign digraph is considered. How to handle the nonaffine structure and unknown control gains for the studied systems under interest is a difficult issue. Therefore, the mean value theorem and the Nussbaum functions are used to overcome the difficulty of designing controllers. The bipartition problem with the structurally unbalanced sign digraph is another design difficulty, then we propose the ACOS in the controller design process. In addition, the controllers designed in this paper, which are constructed by applying the backstepping method, event-triggered control strategy, and dynamic surface technique, can save the unnecessary communication resources and alleviate computation burden. Finally, the simulation results illustrate the validity of the proposed scheme.

Compensation of input-dependent hydraulic input delay for a model of a microfluidic process under Zweifach–Fung effect

We consider a model of a microfluidic process under Zweifac–Fung effect, which gives rise to a second-order nonlinear, non-affine system with control input that affects the plant both without delay and with an input-dependent delay defined implicitly through an integral of the past input values (that arises from a transport process with transport speed being the control input itself). We construct a predictor-feedback control law that exponentially stabilizes the output to a desired reference point. This is the first time that a predictor-feedback design is constructed that achieves complete input delay compensation for such a type of input delay and despite that control input affects the plant also without delay. This is attributed to the particular structure of the nonlinear system considered, which allows to deriving an implementable formula for the predictor state at the proper prediction horizon. We then identify a class of nonlinear systems with input-dependent input delay of hydraulic type for which complete delay compensation, through construction of an exact predictor state, is achievable.

Semiglobal stability of PID for uncertain nonaffine systems

Proportional–integral–derivative (PID) controller is the most widely used form of feedback control by far. However, a satisfactory theory is still lacking to explain the rationale of the widespread PID control, and the gap between theory and practice of PID needs to be filled by control scientists. It is worth noting that a recent work (Zhao and Guo, 2022) proved that the classical PID controller has the ability to globally stabilize a basic class of uncertain nonaffine nonlinear systems, while the system nonlinear function is required to have a linear growth rate with respect to the state variables. In this paper, we will consider the more general case where the system nonlinear function may have super-linear growth rates, and show that the classical PID control can always be designed to stabilize such systems in the semiglobal sense, provided that the smallest eigenvalue of the control gain matrix has a positive lower bound. Moreover, a three dimensional parameter set for the stabilizing PID parameters will be constructed. Furthermore, under some additional knowledge about the system uncertain functions, we will propose a simpler PD controller with desired compensation, and show that the proposed controllers can still be designed to achieve semiglobal stability and desired tracking targets.

Dynamic compensator-based near-optimal control for unknown nonaffine systems via integral reinforcement learning

In this paper, a dynamic compensator-based near-optimal control approach for unknown nonaffine nonlinear systems is developed by using integral reinforcement learning. Since system dynamics is unknown, it is difficult to obtain the optimal control policy via neuro-dynamic programming. To address this problem, a general dynamic compensator is introduced as the virtual control input to augment the unknown nonaffine nonlinear system as a partially unknown affine system. For the augmented system, a novel quadratic value function is designed with the system states, the actual control input and the virtual control input. The optimal control of the augmented system can be regarded as the near-optimal control for the original system since the novel optimal value function is an upper bound of the original optimal value function. In order to avoid the identification of system dynamics, the integral reinforcement learning framework is utilized to derive the optimal control based on the solution of Hamilton–Jacobi–Bellman equation via the critic-only structure. Meanwhile, the weight learning rule of the critic neural network is presented with the experience replay technique to relax the persistence of excitation condition. Moreover, the uniform ultimate boundedness of weight estimation errors and the stability of the closed-loop system are guaranteed by using the Lyapunov’s direct method. Finally, simulation results of two examples demonstrate the effectiveness of the developed dynamic compensator-based near-optimal control method.

I&I adaptive fault-tolerant control of a class of nonaffine systems with nonlinearly parameterised faults

This paper provides a novel fault-tolerant control scheme for a class of nonaffine systems with nonlinearly parameterised (NLP) faults. The nonaffine system is firstly transformed into an augmented one via dynamic feedback control. A fault-tolerant controller with Immersion and Invariance (I&I) adaptation law is designed for the transformed system through dynamic surface control. The controller avoids complexity due to the explosion of terms in backstepping design. The I&I adaptation law is developed to recover the unknown parameters of NLP faults in the system. The proposed scheme can shape the transient performance of the parameter estimation error and tracking error. In the tracking problem, all the signals and tracking error converge exponentially to a small neighbourhood of the origin. In the regulation problem, the system output converges exponentially to zero. A numerical simulation is carried out to verify the effectiveness of the proposed scheme.

Guaranteeing Global Stability for Neuro-Adaptive Control of Unknown Pure-Feedback Nonaffine Systems via Barrier Functions.

Most existing approximation-based adaptive control (AAC) approaches for unknown pure-feedback nonaffine systems retain a dilemma that all closed-loop signals are semiglobally uniformly bounded (SGUB) rather than globally uniformly bounded (GUB). To achieve the GUB stability result, this article presents a neuro-adaptive backstepping control approach by blending the mean value theorem (MVT), the barrier Lyapunov functions (BLFs), and the technique of neural approximation. Specifically, we first resort the MVT to acquire the intermediate and actual control inputs from the nonaffine structures directly. Then, neural networks (NNs) are adopted to approximate the unknown nonlinear functions, in which the compact sets for maintaining the approximation capabilities of NNs are predetermined actively through the BLFs. It is shown that, with the developed neuro-adaptive control scheme, global stability of the resulting closed-loop system is ensured. Simulations are conducted to verify and clarify the developed approach.

Adaptive Critic for Event-Triggered Unknown Nonlinear Optimal Tracking Design With Wastewater Treatment Applications.

In this article, an event-based near-optimal tracking control algorithm is developed for a class of nonaffine systems. First, in order to gain the tracking control strategy, the costate function is established through the iterative dual heuristic dynamic programming (DHP) algorithm. Then, the event-based control method is employed to improve the utilization efficiency of resources and ensure that the closed-loop system has an excellent control performance. Meanwhile, the input-to-state stability (ISS) is proven for the event-based tracking plant. In addition, three kinds of neural networks are used in the event-based DHP algorithm, which aims to identify the nonaffine nonlinear system, estimate the costate function, and approximate the tracking control law. Finally, a numerical experimental simulation is conducted to verify the effectiveness of the proposed scheme. Moreover, in order to further validate the feasibility, the algorithm is applied to the wastewater treatment plant to effectively control the concentrations of dissolved oxygen and nitrate nitrogen.

Adaptive dynamic programming-based hierarchical decision-making of non-affine systems

In this paper, the problem of multiplayer hierarchical decision-making problem for non-affine systems is solved by adaptive dynamic programming. Firstly, the control dynamics are obtained according to the theory of dynamic feedback and combined with the original system dynamics to construct the affine augmented system. Thus, the non-affine multiplayer system is transformed into a general affine form. Then, the hierarchical decision problem is modeled as a Stackelberg game. In the Stackelberg game, the leader makes a decision based on the information of all followers, whereas the followers do not know each other’s information and only obtain their optimal control strategy based on the leader’s decision. Then, the augmented system is reconstructed by a neural network (NN) using input–output data. Moreover, a single critic NN is used to approximate the value function to obtain the optimal control strategy for each player. An extra term added to the weight update law makes the initial admissible control law no longer needed. According to the Lyapunov theory, the state of the system and the error of the weights of the NN are both uniformly ultimately bounded. Finally, the feasibility and validity of the algorithm are confirmed by simulation.

Performance recovery of uncertain nonaffine systems by active disturbance rejection control

This paper studies performance recovery problem of a class of uncertain nonaffine strict-feedback systems with mismatched uncertainties. First, the strict-feedback nonaffine system with mismatched system dynamics is transformed into an integrator-chain nonaffine system by a diffeomorphism. Second, the transformed system is converted to an affine one by constructing an auxiliary control input term, and dynamical uncertainties and external disturbances are viewed as total disturbance of the affine system. Then, a two-time-scale active disturbance rejection controller (ADRC) is designed for the transformed affine system. Under some mild assumptions, it can be proved that the proposed ADRC-based closed-loop system can achieve performance recovery and weak performance recovery in different cases of mismatched system dynamics, respectively. Experimental results on a magnetic levitation ball system demonstrate the effectiveness of the proposed control scheme.