Affine Control Systems Research Articles

Approximation of Chattering Arcs in Optimal Control Problems Governed by Mono-Input Affine Control Systems

Approximation of Chattering Arcs in Optimal Control Problems Governed by Mono-Input Affine Control Systems

Accelerated Gradient Approach For Deep Neural Network-Based Adaptive Control of Unknown Nonlinear Systems.

Recent connections in the adaptive control literature to continuous-time analogs of Nesterov's accelerated gradient method have led to the development of new real-time adaptation laws based on accelerated gradient methods. However, previous results assume that the system's uncertainties are linear-in-the-parameters (LIP). To compensate for non-LIP uncertainties, our preliminary results developed a neural network (NN)-based accelerated gradient adaptive controller to achieve trajectory tracking for nonlinear systems; however, the development and analysis only considered single-hidden-layer NNs. In this article, a generalized deep NN (DNN) architecture with an arbitrary number of hidden layers is considered, and a new DNN-based accelerated gradient adaptation scheme is developed to generate estimates of all the DNN weights in real-time. A nonsmooth Lyapunov-based analysis is used to guarantee the developed accelerated gradient-based DNN adaptation design achieves global asymptotic tracking error convergence for general nonlinear control affine systems subject to unknown (non-LIP) drift dynamics and exogenous disturbances. A comprehensive set of simulation studies are conducted on a two-state nonlinear system, a robotic manipulator, and a complex 20-D nonlinear system to demonstrate the improved performance of the developed method. Our simulation studies demonstrate enhanced tracking and function approximation performance from both DNN architectures and accelerated gradient adaptation.

Chain Recurrence and Selgrade’s Theorem for Affine Flows

AbstractAffine flows on vector bundles with chain transitive base flow are lifted to linear flows and the decomposition into exponentially separated subbundles provided by Selgrade’s theorem is determined. The results are illustrated by an application to affine control systems with bounded control range.

Existence results and stabilization of homogeneous conformable fractional order systems

In this paper, we give results on the stability of homogeneous conformable fractional order systems using assumptions on a family of compact sets and provide the stabilization of an affine control system and given an explicit homogeneous feedback control with the requirement that a control Lyapunov function exists and satisfying a homogeneous condition and we present existence and uniqueness theorems for sequential linear conformable fractional differential equations.

Global stabilization of a bounded controlled system based on the Rössler system

In this work we present a method for the global asymptotic stabilization of an affine control system based on a Rössler system (which is well known by its chaotic behavior), through regular feedback controls constrained to control value sets given by (convex) polytopes U. The proposed method for control designing is based on the control Lyapunov function (clf) theory due to Artstein and Sontag. To this aim, we construct first an explicit absorbing ballB to “trap” the global attractor of Rössler system: we show that the affine system is the feedback interconnection of two subsystems, that leads to obtain B as the level set of a Lyapunov function, V∞(x). However, since the minimum point of V∞(x) is not a rest point of Rössler system, we apply a modified solution to the “uniting clf problem” (to unite local (possibly optimal) controls with global ones, proposed in Andrieu and Prieur (2010)) in order to obtain a clfV(x) for the affine system with minimum at a desired rest point. Finally, we achieve the global asymptotic stabilization of “any” rest point of this system via bounded and regular feedback controls by using the proposed clf method, also obtaining that controls are: (i) damping controls outside B, so they collaborate with the beneficial stable free dynamics, and (ii) (possibly optimal) feedback controls inside B that stabilize the control system at “any” desired equilibrium point of the (unforced) Rössler system.

Decomposition of Equations of Nonlinear Affine Control Systems and its Application to the Synthesis of Regulators

The article deals with the decomposition of nonlinear differential equations based on the group-theoretic approach. At the beginning, the decomposition of differential equations of linear systems using a transition matrix of state is presented, and then, based on the theory of continuous groups (Lie groups), the process of decomposition of differential equations of nonlinear systems is shown. The decomposition approach is based on the isomorphism theorem of the space of vector fields and Lie derivatives, which allows us to consider vector fields as differential operators of smooth functions. A formula is derived about the adjoin representation of a Lie group in its Lie algebra, which actually determines the finding of a vector field that characterizes the interaction of two or more vector fields. The Lie algebra of derivatives makes it possible to determine the infinitesimal action of the Lie group, i.e. the linearization of this action is carried out (transformation of the points of the trajectory space of the original system in a small neighborhood). Decomposition allows, as in the linear case, to separate the finding of an action (only locally) of a group of transformations from the transformed points themselves. For linear systems, this separation is global. It is also shown that the decomposition of linear equations is a particular case of the decomposition of nonlinear equations. An algorithm of the method of model predictive control with Gramian weighting using this decomposition is presented. A practical example of decomposition and application of the model predictive control for stabilization of a nonstationary nonlinear system is considered.

Necessary conditions for local controllability of a particular class of systems with two scalar controls

We consider control-afflne systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions for local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in L∞ or small in W1,∞. These results are illustrated with several examples.

A Robust Safety–Critical Control Framework for Control Affine Systems With Applications to AUVs

A Robust Safety–Critical Control Framework for Control Affine Systems With Applications to AUVs

A new Lagrangian approach to control affine systems with a quadratic Lagrange term

A new Lagrangian approach to control affine systems with a quadratic Lagrange term

Time optimal problems on Lie groups and applications to quantum control

<abstract><p>In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control (<sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $.</p> <p>In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup>). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by (<sup>[<xref ref-type="bibr" rid="b3">3</xref>]</sup>, <sup>[<xref ref-type="bibr" rid="b4">4</xref>]</sup>) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.</p></abstract>

LyZNet with Control: Physics-Informed Neural Network Control of Nonlinear Systems with Formal Guarantees

Optimal control for high-dimensional nonlinear systems remains a fundamental challenge. One bottleneck is that classical approaches for solving the Hamilton-Jacobi-Bellman (HJB) equation suffer from the curse of dimensionality. Recently, physics-informed neural networks have demonstrated potential in overcoming the curse of dimensionality in solving certain classes of PDEs, including special cases of HJB equations. However, one perceived limitation of neural networks is their lack of formal guarantees in the solutions they provide. To address this issue, we have built LyZNet, a Python tool that combines physics-informed learning with formal verification. The previous version of the tool demonstrated the capability for stability analysis and region of attraction estimates. In this paper, we present the tool for solving optimal control problems. We expand the functionalities of the tool to support the formulation and solving of optimal control problems for control-affine systems via physics-informed neural network policy iteration (PINN-PI). We outline the methodology that enables the learning and verification of PINN for optimal stabilization tasks. We demonstrate with a classical control example that the learned optimal controller indeed has significantly improved performance and verifiable regions of attraction.

Poisk parametrov modeli s nailuchshey lokal'noy upravlyaemost'yu

We study the problem of optimal choice of model parameters with respect to anyfunctional. Locally controllable affine systems and integral functionals depending on the programcontrol are considered. Local controllability of affine systems with nonnegative inputs isproved in the case where the columns multiplying the controls form a positive basis. For suchsystems, we introduce the local controllability coefficient and pose the problem of its maximizationdepending on the choice of model parameters. As an example, we consider a very simplifiedmodel of an underwater vehicle and study the problem of finding an arrangement of its controlpropellers in which the energy consumption of the vehicle is minimal.

On the undesired equilibria induced by control barrier function based quadratic programs

In this paper, we analyze the system behavior for general nonlinear control-affine systems when a control barrier function-induced quadratic program-based controller is employed for feedback. In particular, we characterize the existence and locations of possible equilibrium points of the closed-loop system and also provide analytical results on how design parameters affect them. Based on this analysis, a simple modification on the existing quadratic program-based controller is provided, which, without any assumptions other than those taken in the original program, inherits the safety set forward invariance property, and further guarantees the complete elimination of undesired equilibrium points in the interior of the safety set as well as one type of boundary equilibrium points, and local asymptotic stability of the origin. Numerical examples are given alongside the theoretical discussions.

Motion planning and stabilization of nonholonomic systems using gradient flow approximations

Nonlinear control-affine systems described by ordinary differential equations with time-varying vector fields are considered in the paper. We propose a unified control design scheme with oscillating inputs for solving the trajectory tracking and stabilization problems under the bracket-generating condition. This methodology is based on the approximation of a gradient-like dynamics by trajectories of the designed closed-loop system. As an intermediate outcome, we characterize the asymptotic behavior of solutions of the considered class of nonlinear control systems with oscillating inputs under rather general assumptions on the generating potential function. These results are applied to examples of nonholonomic trajectory tracking and obstacle avoidance.

Gaussian Process Based Modeling and Control of Affine Systems with Control Saturation Constraints

Gaussian Process Based Modeling and Control of Affine Systems with Control Saturation Constraints

A new theorem on finite‐time stability of stochastic homogeneous systems and its application

AbstractThis paper provides a novel Lyapunov theorem on finite‐time stability of stochastic homogeneous systems. Different from the existing results, the differential operator in this paper is not required to be negative definite and could be negative semidefinite. The obtained condition is somehow similar to the LaSalle's condition. Moreover, the existing theorem of finite‐time stability for stochastic homogeneous systems is a special case of our result. As an application of the obtained theorem, stochastic finite‐time stabilization is considered for stochastic homogeneous affine control systems. Some examples and simulations are provided to show the effectiveness of the results.

A Learning-Based Method for Computing Control Barrier Functions of Nonlinear Systems With Control Constraints

Verifying the safety of states and designing a safety controller are very important for safety critical systems (for example, robotic and automotive systems). Since the reachable set of a state is hard to calculate online, it is difficult to determine whether the current state will enter the unsafe set in the future. Thus, the safe set of a system is usually obtained by calculating the forward invariant set offline. For control affine systems, the control barrier function provides a linear safety constraint on the control, which helps us obtain safety control online. However, the computation of control barrier functions is recognized as difficult, and existing methods make over-approximations or are only applicable to problems without control constraints. In this paper, we propose a learning-based method to compute the control barrier function for nonlinear control affine systems with control constraints. The method consists of a constraint satisfaction (CS) algorithm and a set expansion (SE) algorithm. The CS algorithm computes the control barrier function that satisfies the constraints, and the SE algorithm expands the volume of the safe set. The numerical results demonstrate that our algorithm outperforms existing methods and can improve existing control barrier functions.

Robust Control Barrier Functions under high relative degree and input constraints for satellite trajectories

This paper presents methodologies for constructing Control Barrier Functions (CBFs) for nonlinear, control-affine systems, in the presence of input constraints and bounded disturbances. More specifically, given a constraint function with high-relative-degree with respect to the system dynamics, the paper considers three methodologies, two for relative-degree 2 and one for higher relative-degrees, for creating CBFs whose zero sublevel sets are subsets of the constraint function’s zero sublevel set Three special forms of Robust CBFs (RCBFs) are developed as functions of the input constraints, system dynamics, and disturbance bounds, such that the resultant RCBF condition on the control input is always feasible for states in the RCBF zero sublevel set. The RCBF condition is then enforced in a switched fashion, which allows the system to operate safely without enforcing the RCBF condition when far from the safe set boundary and allows tuning of how closely trajectories approach the safe set boundary The proposed methods are verified in simulations demonstrating the developed RCBFs in an asteroid flyby scenario for a satellite with low-thrust actuators.

Compactly Restrictable Metric Policy Optimization Problems

We study policy optimization problems for deterministic Markov decision processes (MDPs) with metric state and action spaces, which we refer to as Metric Policy Optimization Problems (MPOPs). Our goal is to establish theoretical results on the well-posedness of MPOPs that can characterize practically relevant continuous control systems. To do so, we define a special class of MPOPs called Compactly Restrictable MPOPs (CR-MPOPs), which are flexible enough to capture the complex behavior of robotic systems but specific enough to admit solutions using dynamic programming methods such as value iteration. We show how to arrive at CR-MPOPs using forward-invariance. We further show that our theoretical results on CR-MPOPs can be used to characterize feedback linearizable control affine systems.

Safety-Critical Control for Control Affine Systems under Spatio-Temporal and Input Constraints

Safety-critical control is a type of modern control task where potentially conflicting stability, safety, and input constraints coexist. In this paper, the Prescribed-Time Zeroing Control Barrier Function (PT-ZCBF) is introduced, which can be applied as a prescribed-time stability constraint in safety-critical control tasks. Furthermore, we formulate a PT-ZCBF-based Quadratic Program (QP), which is able to mediate the potentially conflicting constraints of safety-critical control. The solution of the newly designed QP, acting as the control input of a safety-critical system, can drive the closed-loop trajectories to converge in a user-defined prescribed time period while observing the safety and input constraints. Finally, we use the Adaptive Cruise Control (ACC) problem as an example of numerical simulation to evaluate the performance of the QP-based method.