Violation Of State Constraints Research Articles

A new control method for leader–follower consensus problem of uncertain constrained nonlinear multi-agent systems

The design of Adaptive Fuzzy Sliding-Mode Control (AFSMC) is discussed in this study for the distributed leader–follower consensus problem of nonlinear uncertain multi-agent systems with input and state constraints. Avoiding the critical circumstances related to the violation of input and state constraints is an important issue in the design of consensus control systems. The limitations of existing adaptive consensus controllers developed for full-state constrained systems motivated the present novel method without restrictive structural assumptions. To convert the original problem to an unconstrained consensus control problem, the proposed control system uses a barrier function-based state transformation method and the input-state linearization methodology. As a result, each agent’s adaptive fuzzy sliding-mode control input, which consists of a fuzzy system and a robust term, is constructed based on the derived representation of the system dynamics. The fuzzy system is used to approximate the dynamic equations of the agent, its neighbors, and maybe the leader in each local feedback control law structure, and the robust term is designed to compensate for the fuzzy approximation error. The magnitudes of the robust terms and the output vectors of the fuzzy systems are also determined using the proposed adaptation laws. The second Lyapunov theorem and Barbalat’s lemma are used to verify the closed-loop system’s asymptotic stability. The effectuality of the proposed methodology is confirmed by numerical simulations for several Autonomous Underwater Vehicles (AUVs). Simulation results show that by means of the proposed AFSMC, the leader–follower consensus is achieved without state constraint violation and performance degradation.

A chance‐constrained tube‐based model predictive control for tracking linear systems using data‐driven uncertainty sets

AbstractThis article presents a chance‐constrained tube‐based model predictive control (MPC) method for tracking linear time‐invariant systems based on data‐driven uncertainty sets. By defining the terminal admissible set to consider all the possible steady‐states and reformulating the stochastic tube‐based MPC framework, the proposed method can systematically hedge against the impact of uncertainties and ensure tracking for all reachable operating setpoints. To reduce the conservatism of control performance while enlarging the feasible region, a data‐driven polyhedral uncertainty set is constructed by using the principal component analysis technique, which can effectively capture correlations among uncertain variables. Since state constraint violations in a certain probability are allowed, a probability uncertainty set is constructed by using statistic limit and cutting plane methods to formulate a stochastic tube to ensure constraint satisfaction. The recursive feasibility and stability can be guaranteed if the uncertainties are bounded. The effectiveness of the proposed method is verified by numerical examples and tracking problems of a thickening process.

Robust multi‐objective model predictive control for constrained nonlinear systems with disturbances

AbstractThis work develops a new robust multi‐objective model predictive control (MoMPC) strategy for constrained non‐linear systems with bounded disturbances. The multiple objectives are always contradictory, and the presence of disturbances may result in the violation of state constraints and instability of the controlled system. Firstly, the conflict between multiple objectives is reconciled by minimizing the distance of the cost function vector to the vector of independently minimised objectives obtained by solving a set of steady‐state optimisation problems. Then the contractive state set, which takes into account the effect of disturbances on system states, is constructed to guarantee the robust satisfaction of state constraints. Finally, a stability constraint updated online by solving an auxiliary optimization problem is established to ensure the robust stability of the system under MoMPC. An isothermal chemical reactor system is employed to verify the effectiveness of the controller proposed here.

Prescribed Time Recovery From State Constraint Violation via Approximation-Free Control Approach

For systems with soft state constraints, initial violation in such constraints is acceptable if no feasible control strategy capable of maintaining such constraints exists or an excessively large amount of energy consumption is required for constraint satisfaction. However, whenever the constraint violation is detected, it is highly desirable that the states be regulated back to satisfy the constraints as soon as possible. In this paper, an approximation-free state feedback control scheme is proposed for unknown second-order nonlinear systems, capable of recovering the states from constraint violation to constraint satisfaction within the prescribed time. Besides, the implicit assumption of compatibility between different constraints in most existing works is removed, and a sufficient condition for compatibility of state constraints is established for unknown second-order systems. The proposed self-recovery control scheme has the following features: 1) the recovery time of the states can be preset by the user regardless of the initial conditions; 2) the proposed controller does not need any explicit information on system model, thus has strong robustness against model uncertainties and disturbances; and 3) no approximation technique (e.g., neural network and fuzzy system) is involved, rendering the proposed control structurally simple and computationally inexpensive.

Adaptive Neural Trajectory Tracking Control for n-DOF Robotic Manipulators With State Constraints

This paper proposes an adaptive neural trajectory tracking control scheme for n-DOF robotic manipulators subjected to parameter variations, unknown functions, and time-varying external disturbances. First, the computed torque control (CTC) method is designed to reduce the system's nonlinearity. Second, radial basis function neural networks (RBFNNs) are constructed to approximate the uncertainties due to parameter variations and unknown functions. It's also important to note that the RBFNN's centers and widths are defined by state constraints. As a result of the nonlinear disturbance observer (NDO), the RBFNNs' approximation errors and disturbances are estimated to further improve tracking performance. The barrier Lyapunov function (BLF) ensures the closed-loop system's stability, guaranteeing tracking performance while preventing state constraint violation. Furthermore, sensitivity analysis provides a ranking of the importance of design parameters in influencing dynamic responses. Finally, simulations on a 7-DOF robotic manipulator are performed to validate the effectiveness of the proposed method.

Adaptive Optimized Backstepping Control-Based RL Algorithm for Stochastic Nonlinear Systems With State Constraints and Its Application

This article investigates the adaptive neural-network (NN) tracking optimal control problem for stochastic nonlinear systems, which contain state constraints and uncertain dynamics. First, to avoid the violation of state constraints in achieving optimal control, the novel barrier optimal performance index functions for subsystems are developed. Second, under the framework of the identifier-actor-critic, the virtual and actual optimal controllers are presented based on the backstepping technique, in which the unknown nonlinear dynamics are learned by the NN approximators. Moreover, the quartic barrier Lyapunov functions are constructed instead of square ones to cope with the Hessian term to ensure the stability of the systems with stochastic disturbance. The proposed optimal control strategy can guarantee the boundedness of closed-loop signals, and the output can follow the given reference signal. Meanwhile, the system states are restricted within some preselected compact sets all the while. Finally, both numerical and practical systems are carried out to further illustrate the validity of the proposed optimal control approach.

Adaptive Neural Partial State Tracking Control for Full-State-Constrained Uncertain Singularly Perturbed Nonlinear Systems and Its Applications to Electric Circuit

This paper is concerned with the adaptive neural network (NN) partial tracking control problem for a class of completely unknown multi-input multi-output (MIMO) singularly perturbed nonlinear systems possessing time-varying asymmetric state constraints. To satisfy the constraints, we utilize the state-depended transformation technique to convert the original state-constrained system to an equivalent unconstrained one, then the state constraint problem can be solved by ensuring its stability. Partial state tracking can be achieved without the violation of state constraints. The adaptive tracking controllers are designed by using singular perturbation theory and the adaptive control method, in which NNs are used to approximate unknown nonlinear functions. The ill-conditioned numerical problems lurking in the controller design process are averted and the closed-loop system stability can be guaranteed by introducing an appropriate Lyapunov function with singular perturbation parameter. Finally, a practical example is given to demonstrate the effectiveness of our proposed adaptive NN tracking control scheme.

Event-triggered adaptive neural control of fractional-order nonlinear systems with full-state constraints

In this article, we present an event-triggered scheme for fractional order nonlinear systems under full-state constraints. The neural networks are employed to approximate the continuous nonlinear unknown functions in the system. Moreover, a new adaptive event-triggered strategy is designed under the unified framework of backstepping control method, which can largely reduce the amount of communications. Since the state constraints are frequently emerged in the control procedure, the barrier Lyapunov functions is used to avoid the violation of state constraints. The stability of the closed-loop system is ensured through fractional Lyapunov stability analysis. Finally, the effectiveness of the proposed scheme is verified by simulation examples.

Decentralized finite-time neural control for time-varying state constrained nonlinear interconnected systems in pure-feedback form

This paper focuses on the problem of decentralized adaptive neural network finite-time control for pure-feedback nonlinear interconnected systems with input quantization and time-varying state constraints. Neural networks are used to model the unknown functions. To prevent the violation of state constraints, the time-varying barrier Lyapunov functions are employed in each step of the controller design. Meanwhile, combining with adaptive backstepping control technique, a decentralized adaptive neural network finite-time control strategy is raised. It is testified that the proposed control scheme can effectively ensure that all the closed-loop signals are semi-global practical finite-time stable via Lyapunov stability analysis and that the tracking errors converge to small bounded sets around the origin in finite time. Finally, some simulation results are used to verify the effectiveness of the proposed approach.

Robust observer-based controller design for state constrained uncertain systems: attractive ellipsoid method

ABSTRACTThis study focuses in the output feedback stabilisation of constrained linear systems affected by uncertainties and noisy output measurements. The system states are restricted inside a given polytope and a classical Luenberger observer is used to reconstruct the unmeasurable states from output observations. Based on the observed states, a state feedback is proposed as the control input. The stability analysis and the control design are done using an extended version of the attractive ellipsoid method (AEM) approach. To avoid the violation of state constraints, this work proposes a barrier Lyapunov function (BLF) based analysis. The control parameters are obtained throughout the solution of some optimisation problems such that the BLF ensures an approximation of the constraints by a maximal ellipsoidal set and the AEM provides the characterisation of a minimal ultimately bounded set for the closed-loop system solutions. Numerical simulations show the advantages using the BFL-AEM methodology against classical sub-optimal controllers in academic second order and third order examples. Then, the proposed control strategy is applied over a Buck DC-DC converter. In all the cases, the method proposed here prevails over the other controllers.

Stochastic model predictive control — how does it work?

Stochastic model predictive control (SMPC) provides a probabilistic framework for MPC of systems with stochastic uncertainty. A key feature of SMPC is the inclusion of chance constraints, which enables a systematic trade-off between attainable control performance and probability of state constraint violations in a stochastic setting. This paper presents an overview of core concepts in SMPC in relation to MPC and stochastic optimal control, with numerical illustrations on a typical chemical process. Estimation of stochastic disturbances as well as the impact of estimation quality of stochastic disturbances on the SMPC performance are discussed. Some avenues for future research in SMPC are suggested.

A sampling-based stochastic optimal experiment design formulation with application to the Williams-Otto reactor

Governmental pressure and an industrially competitive climate force chemical companies to strive for a sustainable design and operation. Model-based optimization has been proven to be an indispensable tool to achieve these goals. For the development and maintenance of process models, experimental data with a high information content are required. Model-based optimal experiment design techniques have been developed to obtain experiments that yield a high information content for estimating the model parameters. Since the true parameter values are unknown, these experiment designs start from the current best guesses. In practice, this can result in a lower information content than originally expected. In addition, the experiment design can become practically infeasible (e.g., in terms of safety), due to the violation of operational constraints. In this work a sampling-based stochastic optimal experiment design formulation is employed to address these problems. If the uncertainty on the model parameters can be considered to be a priori known, the parametric uncertainty can be propagated towards the states, which makes the experiment design more robust with respect to information content. As a practical case study of this formulation, the optimal experiment design of a Williams-Otto fed-batch reactor is made robust with respect to information content and reactor temperature state constraint violations. The novelty in this contribution is twofold: (i) polynomial chaos expansion and sigma points approach are compared for stochastic experiment design and (ii) the presented approach based on polynomial chaos expansion is analyzed for robustness with respect to state constraint violations considering different parametric uncertainty distributions.

Neural Network Control-Based Adaptive Learning Design for Nonlinear Systems With Full-State Constraints.

In order to stabilize a class of uncertain nonlinear strict-feedback systems with full-state constraints, an adaptive neural network control method is investigated in this paper. The state constraints are frequently emerged in the real-life plants and how to avoid the violation of state constraints is an important task. By introducing a barrier Lyapunov function (BLF) to every step in a backstepping procedure, a novel adaptive backstepping design is well developed to ensure that the full-state constraints are not violated. At the same time, one remarkable feature is that the minimal learning parameters are employed in BLF backstepping design. By making use of Lyapunov analysis, we can prove that all the signals in the closed-loop system are semiglobal uniformly ultimately bounded and the output is well driven to follow the desired output. Finally, a simulation is given to verify the effectiveness of the method.

Stochastic MPC Framework for Controlling the Average Constraint Violation

This technical note considers linear discrete-time systems with additive, bounded, disturbances subject to hard control input bounds and a stochastic constraint on the amount of state-constraint violation averaged over time. The amount of violations is quantified by a loss function and the averaging can be weighted, corresponding to exponential forgetting of past violations. The freedom in the choice of the loss function makes this formulation highly flexible-for instance, probabilistic constraints, or integrated chance constraints, can be enforced by an appropriate choice of the loss function. For the type of constraint considered, we develop a recursively feasible receding horizon control scheme exploiting the averaged-over-time nature by explicitly taking into account the amount of past constraint violations when determining the current control input. This leads to a significant reduction in conservatism. As a simple extension of the proposed approach we show how time-varying state-constraints can be handled within our framework. The computational complexity (online as well as offline) is comparable to existing model predictive control schemes. The effectiveness of the proposed methodology is demonstrated by means of a numerical example from building climate control.

A performance oriented multi-loop constrained adaptive robust tracking control of one-degree-of-freedom mechanical systems: Theory and experiments

A performance oriented multi-loop approach to the adaptive robust tracking control of one-degree-of-freedom mechanical systems with input saturation, state constraints, parametric uncertainties and input disturbances is presented. The control system contains three loops. In the outer loop, constrained optimization algorithms are developed to generate a replanned trajectory on-line at a low sampling rate so that the converging speed of the overall system response to the desired target is maximized while not causing input saturation and the violation of state constraints. In the inner loop, a constrained adaptive robust control (ARC) law is synthesized and implemented at high sampling rate to achieve the required robust tracking performances with respect to the replanned trajectory even with various types of uncertainties and input saturation. In the middle loop, a set-membership identification (SMI) algorithm is implemented to obtain a tighter estimate of the upper bound of the inertia so that more aggressive replanned trajectory could be used to further improve the overall system response speed. Interaction of the three loops is explicitly characterized by a set of inequalities that the design variables of each loop have to satisfy. It is theoretically shown that the resulting closed-loop system can track feasible desired trajectories with a guaranteed converging time and steady-state tracking accuracy without violating the state constraints. Experiments have been carried out on a linear motor driven industrial positioning system to compare the proposed multi-loop constrained ARC algorithm with some of the traditional control algorithms. Comparative experimental results obtained confirm the superior performance of the proposed algorithm over existing ones.

Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data

Estimates on the distance of a given process from the set of processes that satisfy a specified state constraint in terms of the state constraint violation are important analytical tools in state constrained optimal control theory; they have been employed to ensure the validity of the Maximum Principle in normal form, to establish regularity properties of the value function, to justify interpreting the value function as a unique solution of the Hamilton-Jacobi equation, and for other purposes. A range of estimates are required, which differ according the metrics used to measure the `distance' and the modulus $\theta(h)$ of state constraint violation $h$ in terms of which the estimates are expressed. Recent research has shown that simple linear estimates are valid when the state constraint set $A$ has smooth boundary, but do not generalize to a setting in which the boundary of $A$ has corners. Indeed, for a velocity set $F$ which does not depend on $(t,x)$ and for state constraints taking the form of the intersection of two closed spaces (the simplest case of a boundary with corners), the best distance estimates we can hope for, involving the $W^{1,1,}$ metric on state trajectories, is a super-linear estimate expressed in terms of the $h|\log(h)|$ modulus. But, distance estimates involving the $h|\log (h)|$ modulus are not in general valid when the velocity set $F(.,x)$ is required merely to be continuous, while not even distance estimates involving the weaker, Holder modulus $h^{\alpha}$ (with $\alpha$ arbitrarily small) are in general valid, when $F(.,x)$ is allowed to be discontinuous. This paper concerns the validity of distance estimates when the velocity set $F(t,x)$ is $(t,x)$-dependent and satisfy standard hypotheses on the velocity set (linear growth, Lipschitz $x$-dependence and an inward pointing condition). Hypotheses are identified for the validity of distance estimates, involving both the $h|\log(h)|$ and linear moduli, within the framework of control systems described by a controlled differential equation and state constraint sets having a functional inequality representation.

Trajectories Satisfying a State Constraint: Improved Estimates and New Non-Degeneracy Conditions

For a state-constrained control system described by a differential inclusion and a single functional inequality state constraint, it is known that, under an “inward pointing condition”, the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,1</sup> distance of an arbitrary state trajectory to the set of state trajectories, which have the same left endpoint and which satisfy the state constraint, is linearly related to the state constraint violation. In this paper we show that, in situations where the state-constrained control system is described instead by a controlled differential equation, this estimate can be improved by replacing the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,1</sup> distance on state trajectories by the Ekeland metric of the distance of the control functions. A counter-example reveals that a refinement of this nature is not in general valid for state constrained differential inclusions. Finally we show how the refined estimates may be used to establish new conditions for non-degeneracy of the state constrained Maximum Principle, in circumstances when the data depends discontinuously on the control variable.