Affine Systems Research Articles

Machine learning model‐based optimal tracking control of nonlinear affine systems with safety constraints

AbstractThis work focuses on the development of a machine learning (ML) model‐based framework for safe optimal tracking control of a class of nonlinear control‐affine systems to ensure simultaneous closed‐loop stability and safety. Specifically, a novel multilayer feedforward neural network (FNN) with a control‐affine architecture is designed to model nonlinear dynamic systems. Subsequently, a model‐based reinforcement learning (RL) framework is presented, utilizing a novel cost function with Control Lyapunov‐Barrier Function (CLBF) properties, to learn both the control policy and the optimal value function for an infinite‐horizon optimal tracking control problem for nonlinear systems with safety constraints. The efficacy of the proposed methodology is demonstrated through simulations of a one‐link robot manipulator and a chemical process example.

H ∞ bumpless transfer control for switched affine systems with its application to turbofan engine model

Switched affine systems (SASs) are a special class of switched systems. The existing works about SASs predominantly focuses on steady-state performance. There exist limited attention given to transient performance, particularly the transient performance induced by switchings. The inappropriate switchings can induce severe signal jumps, degrade the steady-state performance of the system, and even lead to instability. The bumpless transfer control (BTC) represents an effective strategy for mitigating signal jumps. However, the existing BTC methods are primarily designed for general switched systems, and these approaches are often not applicable to SASs. This article is focussed on the issue of H ∞ BTC for SASs under unmodeled disturbances. Firstly, an affine term (AT) related to H ∞ bumpless transfer (BT) performance definition is introduced for the SAS to characterise the alleviation level of the bumps in the control input signal at switching points. Secondly, a criterion is established to enable the solvability of the H ∞ BTC problem for the SAS. Thirdly, an H ∞ BT controller and a switching signal are co-designed to drive the SAS to satisfy the H ∞ BT performance which may not be shared by the subsystems. Finally, a turbofan engine model example is implemented to illustrate the availability and rationality of the suggested H ∞ BTC scheme.

Equivariant oriented homology of the affine Grassmannian

We generalize the property of small-torus equivariant K-homology of the affine Grassmannian to general oriented (co)homology theory in the sense of Levine and Morel. The main tool we use is the formal affine Demazure algebra associated to the affine root system. More precisely, we prove that the small-torus equivariant oriented cohomology of the affine Grassmannian satisfies the Goresky-Kottwitz-MacPherson (GKM) condition. We also show that its dual, the small-torus equivariant homology, is isomorphic to the centralizer of the equivariant oriented cohomology of a point in the formal affine Demazure algebra.

Finite-Time Stability Analysis and Stabilization of Switched Affine Systems via an Event-Triggered Strategy.

This article investigates the finite-time control problem of the switched affine systems via an event-triggered strategy. It is well known that the existence of affine terms brings great difficulties in analysis of the finite-time property of such systems. Furthermore, the design of the globally feasible event-triggered mechanism (ETM) under a finite-time control framework is challenging. Thus, a two-step hybrid control scheme is proposed in this article. The first step focuses on the event-triggered finite-time control for practical stability, while the second step aims to achieve finite-time stabilization. Particularly, in step one, by constructing the intersection between the affine term's threshold and feasible state region of the established ETM, it is verified that the Zeno behavior can be excluded. Thereafter, an affine state-dependent switching law and sufficient conditions are provided for achieving practical stability. Meanwhile, an estimation for the practical settling time to enter the bounded set is provided. In step two, the criteria for finite-time stabilization of the considered systems are further presented, and an overall settling-time upper bound is derived. Finally, a numerical example is illustrated to demonstrate the effectiveness of our proposed method.

Adaptive Inverse Nonlinear Optimal Control Based on Finite-Time Concurrent Learning and Semidefinite Programming.

This article investigates the problem of inverse optimal control (IOC) for a class of nonlinear affine systems. An adaptive IOC approach is proposed to recover the cost functional using only the system state data, which integrates the finite-time concurrent learning (FTCL) technique and the semidefinite programming (SDP) technique. First, an identifier neural network (NN) is employed to approximate the unknown nonlinear control policy, and an FTCL-based update law is proposed to estimate the weights of the identifier NN online, which removes the traditional persistent excitation (PE) condition. Moreover, the finite-time convergence as well as the uniformly ultimately boundness (UUB) of estimation error of the identifier NN weights are analysed according to whether or not there exists the identifier NN approximation error. Then, with the help of a value NN for approximating the value function, an SDP problem with a quadratic objective function can be set up for determining the weighting matrices of the cost functional. Finally, simulation results are presented to validate the proposed method.

Joint Screening and Identification of Potential Targets of Nitazoxanide by Affinity Chromatography and Label-Free Techniques.

Nitazoxanide not only exhibits a broad spectrum of activities against various pathogens infecting animals and humans but also induces cellular autophagy. Currently, the pattern of action and subcellular targets of nitazoxanide-induced cellular autophagy are still unclear. To identify potential targets of nitazoxanide in mammalian cells, we developed an af-finity chromatography system using tizoxanide, a deacetyl derivative of nitazoxanide, as a ligand. Affinity chromatography was performed using VERO cell extracts on tizoxanide-biotin, and the isolated binding proteins were identified by mass spectrometry. Candidate target proteins ob-tained using affinity chromatography were co-analysed with the drug affinity response target sta-bility method. Fluorescent probes obtained by coupling rhodamine B to nitazoxanide were used for intracellular localisation of the binding targets. Solvent-induced protein precipitation profiling and thermal proteome profiling were used to further validate the binding proteins. The joint analysis of the drug affinity response target stability method and affinity chro-matography resulted in the screening of six possible candidate target proteins. Fluorescent probes localised the nitazoxanide-binding protein around the nuclear membrane. Molecular docking re-vealed that the binding proteins mainly formed hydrogen bonds with the nitro group of nitazoxa-nide. Solvent-induced protein precipitation profiling and thermal proteome profiling further vali-dated SEC61A, PSMD12, and PRKAG1 as potential target proteins of nitazoxanide. The data supports the idea that nitazoxanide is a multifunctional compound with multiple targets.

Data-Driven Control Design for Power Converters Approximated as Switched Affine Systems and Experimental Validation

Data-Driven Control Design for Power Converters Approximated as Switched Affine Systems and Experimental Validation

A Computationally Efficient Trajectory Prediction in MPC for Piecewise Affine Systems

A Computationally Efficient Trajectory Prediction in MPC for Piecewise Affine Systems

Bifurcation analysis of a two-neuron central pattern generator model for both oscillatory and convergent neuronal activities.

The neural oscillator model proposed by Matsuoka is a piecewise affine system that exhibits distinctive periodic solutions. Although such typical oscillation patterns have been widely studied, little is understood about the dynamics of convergence to certain fixed points and bifurcations between the periodic orbits and fixed points in this model. We performed fixed point analysis on a two-neuron version of the Matsuoka oscillator model, the result of which explains the mechanism of oscillation and the discontinuity-induced bifurcations such as subcritical/supercritical Hopf-like, homoclinic-like and grazing bifurcations. Furthermore, it provided theoretical predictions concerning a logarithmic oscillation-period scaling law and noise-induced oscillations observed around those bifurcations. These results are expected to underpin further investigations into oscillatory and transient neuronal activities concerning central pattern generators.

Intelligent cooperative control for flexible landing on small celestial bodies

In view of the potential landing risks in small celestial body landing missions, the concept of flexible landing has been proposed in recent years. By employing a flexible lander with structural compliance to complex surface topography, the risks of rebound or overturning upon touchdown can be reduced. During flexible landing, the overall motion of the flexible lander is controlled by several nodes embedded in the flexible structure. Due to the flexible deformation, the motion among the nodes is intricately coupled, posing challenges to the control algorithm design. To address this problem, an intelligent cooperative control method for flexible landing is proposed in this paper. By decomposing the flexible internal force into a low-order principal term with specific physical interpretation and an unmodeled high-order residual term, the coupling relationship among the nodes is characterized. On this basis, the low-order dynamics system is reconstructed into a piecewise affine system, upon which a multi-node cooperative optimal controller is designed. A long-short-term memory recurrent neural network is further established to compensate for the unmodeled high-order term. Through the combination of the low-order principal control and the high-order intelligent compensation, a feasible approach for achieving flexible landing is obtained. Finally, the effectiveness of the proposed control method is verified via 433 Eros-based flexible landing simulations.

Stabilization of aperiodic sampled-data switched affine systems to hybrid limit cycles

This paper deals with the stabilization of aperiodic sampled-data switched affine systems to a predetermined hybrid limit cycle using a hybrid dynamical system representation and a control Lyapunov function approach. Some preliminaries on the hybrid dynamical system formalism provide the framework for modeling switched affine systems followed by definitions on hybrid limit cycles and related notions. The main result, based on simple Linear Matrix Inequalities (LMI), guarantees that the solutions to the closed-loop system converge to a hybrid limit cycle defined by the states, functioning modes with their corresponding dwell times. The theoretical results are evaluated on academic examples and demonstrate the potential and the originality of the method over the recent literature.

Sliding Homoclinic Bifurcations in a Class of Three-Dimensional Piecewise Affine Systems

This paper studies sliding homoclinic bifurcations in a class of symmetric three-zone three-dimensional piecewise affine systems. The systems have one parameter and the unperturbed systems have a pair of sliding homoclinic orbits to a saddle. Based on the analysis of the one-dimensional Poincaré maps, two types of sliding cycles are obtained from the sliding homoclinic bifurcations of the systems. In addition, two examples of sliding homoclinic orbits and sliding cycles are provided with simulations to illustrate the effectiveness of the theorems.

H∞ Anti-Disturbance Switching Control for Switched Affine Systems With Its Application to Turbofan Engine Model

H_∞ Anti-Disturbance Switching Control for Switched Affine Systems With Its Application to Turbofan Engine Model

Twin Strep-Tag Modified CPT1A Mitochondrial Membrane Chromatography in Screening Lipid Metabolism Regulators.

Mitochondrial Membrane Chromatography (MMC) is a bioaffinity chromatography technique developed to study the interaction between target proteins embedded in the mitochondrial membrane and their ligand compounds. However, the MMC stationary phases (MMSP) prepared by chemical immobilization are prone to nonspecific binding in candidate agent screening inevitably. To address these challenges, Twin Strep-Tag/Strep Tactin was employed to establish a specific affinity system in the present study. We prepared a carnitine palmitoyltransferase 1A (CPT1A) MMSP by specifically linking Strep-tactin-modified silica gel with the Twin Strep-Tag on the CPT1A-oriented mitochondrial membrane. This Twin Strep-Tag/Strep Tactin modified CPT1A/MMC method exhibited remarkably better retention behavior, longer stationary phase lifespan, and higher screening specificity compared with previous MMC systems with glutaraldehyde immobilization. We adopted the CPT1A-specific MMC system in screening CPT1A ligands from traditional Chinese medicines, and successfully identified novel candidate ligands: ononin, isoliquiritigenin, and aloe-emodin, from Glycyrrhiza uralensis Fisch and Senna tora (L.) Roxb extracts. Biological assessments illustrated that the compounds screened promote CPT1A enzyme activity without affecting CPT1A protein expression, as well as effectively reduce the lipid droplets and triglyceride levels in the high fat induction HepG2 cells. The results suggest that we have developed an MMC system, which is promising for studying the bioaffinity of mitochondrial membrane proteins to candidate compounds. This system provides a platform for a key step in mitochondrial medicine discovery, especially for bioactive molecule screening from complex herbal extracts.

Coexistence of singular cycles in a class of three-dimensional piecewise affine systems

Coexistence of singular cycles in a class of three-dimensional piecewise affine systems

Stable predictive control of continuous stirred-tank reactors using deep learning

In this work, a Lyapunov-based predictive control method utilizing deep learning techniques is proposed for driving continuous-time nonlinear processes towards the desired equilibrium point. Initially, a neural ordinary differential equation model is developed to learn the continuous-time dynamics from discrete-time sampling data. Subsequently, the theoretical conditions that the control Lyapunov function (CLF) developed on the learned continuous-time system is feasible to act on the actual system through a zero-order-holder are analyzed. Building upon the theoretical analysis and the constructed continuous-time dynamics, a deep neural network control Lyapunov function (DNN-CLF) controller is developed. Importantly, the DNN-CLF is independent of the affine system formulation, effectively addressing the challenge of designing the CLF. Benefiting from the stability guarantee provided by the established DNN-CLF, it is integrated into the model predictive control (MPC) framework to develop Lyapunov-based MPC to further improve the control performance. To ensure safety, considering error propagation and associated risks with enlarging the feasible region of the CLF, tubes are constructed and integrated into the MPC scheme. Finally, a numerical simulation of a chemical process is conducted to validate the efficiency of the proposed method.

Dynamic output feedback based fault tolerant control for continuous-time switched affine systems via reduced-order observer

In this study, the issue on dynamic output feedback fault-tolerant controller design is addressed for a class of continuous-time switched affine systems in the presence of actuator faults. The existence of affine terms gives rise to the lack of a common equilibrium point and further makes it more difficult to propose the desired output-feedback-based fault-tolerant control scheme. Firstly, the actuator fault estimation is achieved through a reduced-order observer design approach. Then, a kind of dynamic output feedback controller is constructed to perform the fault tolerance goal by utilizing fault estimation information. Moreover, the mode-dependent average dwell time switching constraint is borrowed to avoid the chattering owing to the high-frequency switching. The sufficient conditions with less conservative are derived to guarantee the practical exponential stability of closed-loop system with a prescribed L∞ gain. Finally, both the effectiveness and validity of the designed fault-tolerant control strategy are illustrated via a case study of a buck-boost DC-DC converter.

A Sparse Neural Network-Based Control Method for Saturated Nonlinear Affine Systems

Saturated nonlinear affine systems are widely encountered in many engineering fields. Currently, most control methods on saturated nonlinear affine systems are not specifically designed based on sparsity-based control methodologies, and they might require sparse identification at the beginning stage and applying tracking control afterwards. In this paper, a sparse neural network (SNN)-based control method from an lp-norm (1≤p<2) optimization perspective is proposed for saturated nonlinear affine systems by taking advantage of the nice properties of primal dual neural networks for optimization. In particular, when p=1, a new alternative controller based on SNN is derived, encountering computational difficulties distinct from those of another solution set in the basic dual neural network. The convergence properties of such SNN-based controllers are investigated and analyzed to find a control solution. Five illustrative examples further are shown to demonstrate the efficiency of the proposed SNN-based control method for tracking the desired references of saturated nonlinear affine systems. In the practical application scenario involving the UR5 robot control, the trajectory’s average errors are consistently confined to a minimal magnitude of 10−4 m. These findings substantiate the efficacy of the SNN-based control approach proposed for precise tracking control in saturated nonlinear affine systems.

Global Stabilization of a Bounded Controlled Lorenz System

In this work, we present a method for the Global Asymptotic Stabilization (GAS) of an affine control chaotic Lorenz system, via admissible (bounded and regular) feedback controls, where the control bounds are given by a class of (convex) polytopes. The proposed control design method is based on the control Lyapunov function (CLF) theory introduced in [Artstein, 1983; Sontag, 1998]. Hence, we first recall, with parameters including those in [Lorenz, 1963], that these equations are point-dissipative, i.e. there is an explicit absorbing ball [Formula: see text] given by the level set of a certain Lyapunov function, [Formula: see text]. However, since the minimum point of [Formula: see text] does not coincide with any rest point of Lorenz system, we apply a modified solution to the “uniting CLF problem” (to unify local (possibly optimal) controls with global ones, proposed in [Andrieu & Prieur, 2010]) in order to obtain a CLF [Formula: see text] for the affine system with minimum at a desired equilibrium point. Finally, we achieve the GAS of “any” rest point of this system via bounded and regular feedback controls by using the proposed CLF method, which also contains the following controllers: (i) damping controls outside [Formula: see text], so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) feedback controls inside [Formula: see text] that stabilize the control system at “any” desired rest point of the (unforced) Lorenz system.

Optimal quadratic full state feedback control of nonlinear affine systems

The solution of the optimal quadratic full-state feedback closed-loop control of a nonlinear affine system problem is presented and solved. This solution uses the State-Dependent Coefficient (SDC) form of a nonlinear system and the Calculus of Variations. Further, it is shown that this solution enables the computation of the value function associated with the respective Hamilton-Jacobi-Bellman equation thus satisfying the necessary and sufficient conditions for optimality. The use of the SDC form representation in the optimal solution for a nonlinear affine system results in an explicit and exact full-state feedback closed-loop control implementation as opposed to the existing open-loop solutions of the optimal control. This solution, like for the linear case, is non-causal and thus cannot be solved in real-time. The closed-loop application involves precomputing a time-varying full-state feedback matrix. The optimal feedback is a linear combination of the current state. The time-varying gain matrix is the backward solution of a non-symmetric matrix Riccati equation. Conditions for the stability of the full-state feedback are presented. This new representation of the optimal solution gives a well-understandable algorithm that enables implementation in the closed loop of the optimal feedback control of a non-linear system which is not possible with the existing optimal open loop representations.