Affine Nonlinear Control Systems Research Articles

Model-Free Optimal Control for Affine Nonlinear Systems With Convergence Analysis

In this paper, a self-learning control scheme is proposed for the infinite horizon optimal control of affine nonlinear systems based on the action dependent heuristic dynamic programming algorithm. The policy iteration technique is introduced to derive the optimal control policy with feasibility and convergence analysis. It shows that the “greedy” control action for each state is uniquely existent, the learned control policy after each policy iteration is admissible, and the optimal control policy is able to be obtained. Two three-layer perceptron neural networks are employed to implement the scheme. The critic network is trained by a novel rule to conform to the Bellman equation, and the action network is trained to yield a better control policy. Both training processes alternate until the optimal control policy is achieved. Two simulation examples are provided to validate the effectiveness of the approach.

Local nested transverse feedback linearization

We study two local feedback equivalence problems for a nonlinear control-affine system with two nested, controlled-invariant, embedded submanifolds in its state space. The first, less restrictive, result gives necessary and sufficient conditions for the dynamics of the system restricted to the larger submanifold and transversal to the smaller submanifold to be linear and controllable. This normal form facilitates designing controllers that locally stabilize the smaller set relative to the larger set. The second, more restrictive, result additionally imposes that the transversal dynamics to the larger set be linear and controllable. This result can simplify designing controllers to locally stabilize the larger submanifold. This is illustrated by sufficient conditions under which these normal forms can be used to locally solve a nested set stabilization problem.

A Unified Approach to Nonlinear Dynamic Inversion Control with Parameter Determination by Eigenvalue Assignment

This paper presents a unified approach to nonlinear dynamic inversion control algorithm with the parameters for desired dynamics determined by using an eigenvalue assignment method, which may be applied in a very straightforward and convenient way. By using this method, it is not necessary to transform the nonlinear equations into linear equations by feedback linearization before beginning control designs. The applications of this method are not limited to affine nonlinear control systems or limited to minimum phase problems if the eigenvalues of error dynamics are carefully assigned so that the desired dynamics is stable. The control design by using this method is shown to be robust to modeling uncertainties. To validate the theory, the design of a UAV control system is presented as an example. Numerical simulations show the performance of the design to be quite remarkable.

Memoryless variable structure control for affine nonlinear systems using only output information

SummaryIn this paper, local stabilisation is considered for a class of affine nonlinear control systems with uncertainties involving time‐varying delay. It is not assumed that either the dimension of the inputs is equal to the dimension of the outputs or that the nominal system is linearisable or partially linearisable. The bounds on both the matched and mismatched uncertainties are nonlinear and time delayed. A static output feedback variable structure control is synthesized to stabilise the system uniformly asymptotically. A control strategy to enforce exponential stability is also derived. The Lyapunov–Razumikhin approach is used to deal with the effect of the time delay, and the bounds on the uncertainties are used in the control design to enhance the robustness. The designed control does not depend on the time delay, and thus, it is not required that the time delay is known. Finally, the obtained results are tested on a simple mechanical system through simulation. The simulation results show that the proposed approach is effective and feasible. Copyright © 2014 John Wiley & Sons, Ltd.

Output Feedback Adaptive Neural Control Without Seeking SPR Condition

AbstractFor output‐feedback adaptive control of affine nonlinear systems based on feedback linearization and function approximation, the observation error dynamics usually should be augmented by a low‐pass filter to satisfy a strictly positive real (SPR) condition so that output feedback can be realized. Yet, this manipulation results in filtering basis functions of approximators, which makes the order of the controller dynamics very large. This paper presents a novel output‐feedback adaptive neural control (ANC) scheme to avoid seeking the SPR condition. A saturated output‐feedback control law is introduced based on a state‐feedback indirect ANC structure. An adaptive neural network (NN) observer is applied to estimate immeasurable system state variables. The output estimation error rather than the basis functions is filtered and the filter output is employed to update NNs. Under given initial conditions and sufficient control parameter constraints, it is proved that the closed‐loop system is uniformly ultimately bounded stable in the sense that both the state estimation errors and the tracking errors converge to small neighborhoods of zero. An illustrative example is provided to demonstrate the effectiveness of this approach.

Time-Optimal Control Problem for a Special Class of Control Systems: Optimal Controls and Approximation in the Sense of Time Optimality

We consider the time-optimal control problem to the origin for a class of nonlinear systems, called dual-to-linear systems. We obtain the general description of possible optimal controls. In particular, we show that optimal controls take the values \(-1\), \(0\), and \(+1\) only and have a finite number of points of discontinuity. We describe a class of nonlinear affine control systems which can be approximated by dual-to-linear systems in the sense of time optimality.

On Feedback Control Techniques of Nonlinear Analytic Systems

This paper presents three approaches dealing with the feedback control of nonlinear analytic systems. The first onetreats the optimal control resolution for a control-affine nonlinear system using the State Dependant Riccati Equation(SDRE) method. It aims to solve a nonlinear optimal control problem through a Riccati equation that depends on thestate. The second approach treats a procedure of constructing an analytic expression of a nonlinear state feedbacksolution of an optimal regulation problem with the help of Kronecker tensor notations. The third one deals with theglobal asymptotic stabilization of the nonlinear polynomial systems. The designed state feedback control law stabilizesquadratically the studied systems. Our main contribution in this paper is to carry out a stability analysis for this kind ofsystems and to develop new sufficient conditions of stability. A numerical-simulation-based comparison of the threemethods is finally performed.

Adaptive Fault-Tolerant Control for a Class of Nonlinear MIMO Systems with Matched and Mismatched Uncertainties

This paper proposed a fault-tolerant control (FTC) for nonlinear control-affine uncertain MIMO systems. The proposed controller is no need for on-line fault detection and diagnosis unit, and inexpensive to compute. An adaptive FTC designed method based on Lyapunov-like approach is developed to overcome the affect of parameter uncertainties, matched and mismatched disturbance, and actuator failures. And the theoretical analysis demonstrates that asymptotical tracking error convergence would be guaranteed by the controller. Numerical simulations are provided to validate and illustrate the benefits of the proposed control scheme.

A Scenario Approach for Non-Convex Control Design

Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist randomized approaches with efficient sampling, this is not the case for non-convex problems. Approaches based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in terms of performance and require high sample complexity to achieve the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control- design problems that can be addressed via randomization, we apply our scenario approach to randomized Model Predictive Control for chance-constrained nonlinear control-affine systems.

Randomized Nonlinear MPC for Uncertain Control-Affine Systems with Bounded Closed-Loop Constraint Violations

In this paper we consider uncertain nonlinear control-affine systems with probabilistic constraints. In particular, we investigate Stochastic Model Predictive Control (SMPC) strategies for nonlinear systems subject to chance constraints. The resulting non-convex chance constrained Finite Horizon Optimal Control Problems are computationally intractable in general and hence must be approximated. We propose an approximation scheme which is based on randomization and stems from recent theoretical developments on random non-convex programs. Since numerical solvers for non-convex optimization problems can typically only reach local optima, our method is designed to provide probabilistic guarantees for any local optimum inside a set of chosen complexity. Moreover, the proposed method comes with bounds on the (time) average closed-loop constraint violation when SMPC is applied in a receding horizon fashion. Our numerical example shows that the number of constraints of the proposed random non-convex program can be up to ten times smaller than those required by existing methods.

Control-Sharing and Merging Control Lyapunov Functions

Given two control Lyapunov functions (CLFs), a “merging” is a new CLF whose gradient is a positive combination of the gradients of the two parents CLFs. The merging function is an important trade-off since this new function may, for instance, approximate one of the two parents functions close to the origin, while being close to the other far away. For nonlinear control-affine systems, some equivalence properties are shown between the control-sharing property, i.e., the existence of a single control law which makes simultaneously negative the Lyapunov derivatives of the two given CLFs, and the existence of merging CLFs. It is shown that, even for linear time-invariant systems, the control-sharing property does not always hold, with the remarkable exception of planar systems. The class of linear differential inclusions is also discussed and similar equivalence results are presented. For this class of systems, linear matrix inequalities conditions are provided to guarantee the control-sharing property. Finally, a constructive procedure, based on the recently considered “R-functions,” is defined to merge two smooth positively homogeneous CLFs.

Zero-Sum Two-Player Game Theoretic Formulation of Affine Nonlinear Discrete-Time Systems Using Neural Networks

In this paper, the nearly optimal solution for discrete-time (DT) affine nonlinear control systems in the presence of partially unknown internal system dynamics and disturbances is considered. The approach is based on successive approximate solution of the Hamilton-Jacobi-Isaacs (HJI) equation, which appears in optimal control. Successive approximation approach for updating control and disturbance inputs for DT nonlinear affine systems are proposed. Moreover, sufficient conditions for the convergence of the approximate HJI solution to the saddle point are derived, and an iterative approach to approximate the HJI equation using a neural network (NN) is presented. Then, the requirement of full knowledge of the internal dynamics of the nonlinear DT system is relaxed by using a second NN online approximator. The result is a closed-loop optimal NN controller via offline learning. A numerical example is provided illustrating the effectiveness of the approach.

Non-Lipschitz points and the $${\textit{SBV}}$$ SBV regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function \(T\) under controllability conditions which do not imply the Lipschitz continuity of \(T\). We consider first the case of normal linear control systems with constant coefficients in \({\mathbb {R}}^N\). We characterize points around which \(T\) is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call \(\mathcal {S}\). Furthermore, we show that \(\mathcal {S}\) is countably \(\mathcal {H}^{N-1}\)-rectifiable with positive \(\mathcal {H}^{N-1}\)-measure. Second, we consider a class of control-affine planar nonlinear systems satisfying a second order controllability condition: we characterize the set \(\mathcal {S}\) in a neighborhood of the origin in a similar way and prove the \(\mathcal {H}^1\)-rectifiability of \(\mathcal {S}\) and that \(\mathcal {H}^1(\mathcal {S})>0\). In both cases, \(T\) is known to have epigraph with positive reach, hence to be a locally \(BV\) function (see Colombo et al.: SIAM J Control Optim 44:2285–2299, 2006; Colombo and Nguyen.: Math Control Relat 3: 51–82, 2013). Since the Cantor part of \(DT\) must be concentrated in \(\mathcal {S}\), our analysis yields that \(T\) is locally \(SBV\), i.e., the Cantor part of \(DT\) vanishes. Our results imply also that \(T\) is differentiable outside a \(\mathcal {H}^{N-1}\)-rectifiable set. With small changes, our results are valid also in the case of multiple control input.

Concurrent learning-based approximate feedback-Nash equilibrium solution of N-player nonzero-sum differential games

This paper presents a concurrent learning-based actor-critic-identifier architecture to obtain an approximate feedback-Nash equilibrium solution to an infinite horizon N-player nonzero-sum differential game. The solution is obtained online for a nonlinear control-affine system with uncertain linearly parameterized drift dynamics. It is shown that under a condition milder than persistence of excitation (PE), uniformly ultimately bounded convergence of the developed control policies to the feedback-Nash equilibrium policies can be established. Simulation results are presented to demonstrate the performance of the developed technique without an added excitation signal.

Adaptive neural control and learning of affine nonlinear systems

This paper presents deterministic learning from adaptive neural network control of affine nonlinear systems with completely unknown system dynamics. Thanks to the learning capability of radial basis function, neural network (NN), stable adaptive NN controller is designed for the unknown affine nonlinear systems. The designed adaptive NN controller is rigorously shown that learning of the unknown closed-loop system dynamics can be achieved during the stable control process because partial persistent excitation condition of some internal signals in the closed-loop system is satisfied. Subsequently, neural learning controller using the knowledge obtained from deterministic learning is constructed to achieve closed-loop stability and improve control performance. Numerical simulation is provided to show the effectiveness of the proposed control scheme.

Construction of control Lyapunov functions for damping stabilization of control affine systems

This paper considers the stabilization of nonlinear control affine systems that satisfy Jurdjevic–Quinn conditions. We first obtain a differential one-form associated to the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a locally-defined dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we compute a control Lyapunov function for the closed-loop system. Under Jurdjevic–Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics. The technique is also applied to construct damping feedback controllers for the stabilization of periodic orbits. Examples are presented to illustrate the proposed method.

On the use of gradual dense–sparse discretizations in receding horizon control

SUMMARYA key factor to success in implementations of real time optimal control, such as receding horizon control (RHC), is making efficient use of computational resources. The main trade‐off is then between efficiency and accuracy of each RHC iteration, and the resulting overall optimality properties of the concatenated iterations, that is, how closely this represents a solution to the underlying infinite time optimal control problem (OCP). Both these issues can be addressed by adapting the RHC solution strategy to the expected form of the solution. Using gradual dense–sparse (GDS) node distributions in direct transcription formulations of the finite time OCP solved in each RHC iteration is a way of adapting the node distribution of this OCP to the fact that it is actually part of an RHC scheme. We have previously argued that this is reasonable, because the near future plan must be implemented now, but the far future plan can and will be revised later. In this paper, we investigate RHC applications where the asymptotic qualitative behavior of the OCP solution can be analyzed in advance. For some classes of systems, explicit exponential convergence rates of the solutions can be computed. We establish such convergence rates for a class of control affine nonlinear systems with a locally quadratic cost and propose to use versions of GDS node distributions for such systems because they will (eventually) be better adapted to the form of the solution. The advantages of the GDS approach in such settings is illustrated with simulations. Copyright © 2013 John Wiley & Sons, Ltd.

Finite-horizon control-constrained nonlinear optimal control using single network adaptive critics.

To synthesize fixed-final-time control-constrained optimal controllers for discrete-time nonlinear control-affine systems, a single neural network (NN)-based controller called the Finite-horizon Single Network Adaptive Critic is developed in this paper. Inputs to the NN are the current system states and the time-to-go, and the network outputs are the costates that are used to compute optimal feedback control. Control constraints are handled through a nonquadratic cost function. Convergence proofs of: 1) the reinforcement learning-based training method to the optimal solution; 2) the training error; and 3) the network weights are provided. The resulting controller is shown to solve the associated time-varying Hamilton-Jacobi-Bellman equation and provide the fixed-final-time optimal solution. Performance of the new synthesis technique is demonstrated through different examples including an attitude control problem wherein a rigid spacecraft performs a finite-time attitude maneuver subject to control bounds. The new formulation has great potential for implementation since it consists of only one NN with single set of weights and it provides comprehensive feedback solutions online, though it is trained offline.

Guaranteed cost control of affine nonlinear systems via partition of unity method

We consider the problem of guaranteed cost control (GCC) of affine nonlinear systems in this paper. Firstly, the general affine nonlinear system with the origin being its equilibrium point is represented as a linear-like structure with state-dependent coefficient matrices. Secondly, partition of unity method is used to approximate the coefficient matrices, as a result of which the original affine nonlinear system is equivalently converted into a linear-like system with modeling error. A GCC law is then synthesized based on the equivalent model in the presence of modeling error under certain error condition. The control law ensures that the system under control is asymptotically stable as well as that a given cost function is upper-bounded. A suboptimal GCC law can be obtained via solving an optimization problem in terms of linear matrix inequality (LMI), in stead of state-dependent Riccati equation (SDRE) or Hamilton–Jacobi equations that are usually required in solving nonlinear optimal control problems. Finally, a numerical example is provided to illustrate the validity of the proposed method.

Finite-Time H ∞ Inverse Optimal Control of Affine Nonlinear Systems

This paper is focused on the problem of the finite-time H ∞ inverse optimal control for affine nonlinear systems. Based on the finite-time control Lyapunov function, we derive a sufficient condition for the existence of time-invariant, continuous, finite-time stabilizing and inverse optimal state feedback control law, and propose a universal formula for constructing the finite-time H ∞ inverse optimal control law. We investigate the relationship between the finite-time stabilization and the finite-time H ∞ inverse optimal control. Finally, some examples are given to illustrate the effectiveness of the presented results.