State Inequality Constraints Research Articles

New method for stability of probabilistic Boolean networks with inequality constraints

This paper studies the stability of probabilistic Boolean networks (PBNs) with state inequality constraints by establishing a new algebraic representation. On one hand, combining with the semi-tensor product (STP) of matrices and the law of total expectation, the algebraic representation of PBNs is established. On the other hand, based on the STP of matrices, an algorithm is proposed to determine the solution set of the state inequality constraints. Then, based on the algebraic representation and the inequality constrained solution set, a new form of transition probability matrix is constructed for PBNs. Finally, by the proposed transition probability matrix, new criteria are proposed for the finite-time stability with positive probability of PBNs. Several simulation examples are used to illustrate the effectiveness of the obtained results.

Concurrent learning for adaptive pontryagin's maximum principle of nonlinear systems with inequality constraints

AbstractIn this article, a finite‐horizon adaptive Pontryagin's maximum principle is presented for nonlinear systems with state inequality constraints. Concurrent learning (CL) technique is adopted to identify the unknown parameters of the dynamic systems. Based on the identification model, a novel adaptive iterative algorithm under the Pontryagin's framework is introduced to learn the finite‐horizon optimal control solution. Convergence analysis of the algorithm is provided by showing that the cost function sequence is monotonically decreasing. Furthermore, we extend the adaptive iterative algorithm to time‐varying nonlinear systems. The new algorithm overcomes the technical obstacles of the existing adaptive/approximate dynamic programming (ADP) approaches to deal with the time‐varying characteristic of Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE), especially when state constraints exist. Simulation examples are carried out to validate the effectiveness of the theoretical results.

The applicability of Zermelo's equation to indirect and direct optimization of commercial aircraft flights

This paper focuses on the applicability of Zermelo's equation within the context of 3D commercial aircraft trajectory optimization problems. The associated optimal control problem includes two singular controls, specifically aerodynamic path angle and throttle setting, and one regular control, specifically heading angle. Using Pontryagin's maximum principle and the direct adjoining method, we show that the optimal heading angle is defined by Zermelo's equation. The significance of the presented analysis is that Zermelo's equation holds for 3D commercial aircraft flights, even in the presence of standard state-inequality constraints and a more general objective function. With the help of Zermelo's equation, the control problem is initially analyzed for a 3D time-optimal climb-phase scenario, which is solved by an indirect approach. Through the analysis of the switching function, we demonstrate the dependency of the optimal aerodynamic path angle on the optimal heading angle. Next, by tackling a 3D time-fuel-optimal free-routing flight, solved by a direct approach, we show that Zermelo's equation can help reduce the overall dimension of the associated nonlinear programming. To successfully handle the initial guess in both indirect and direct problems, it is proposed a diminutive nonlinear programming technique as a fast and robust initializer. This simple-yet-effective initializer provides sufficient information about the optimal controls and state dynamics required for initializing a direct optimization, as well as the optimal switching times and co-states required for initializing an indirect optimization.

Bound‐based control design for permanent magnet linear motor servo system with inequality constraints and uncertainties

AbstractThe permanent magnet linear motor (PMLM) systems are widely applied in various engineering fields. However, the state inequality constraints and uncertainties of parameters and unknown external disturbances need to be further explored. In this article, first, a state transformation method is proposed for converting bounded states to unbounded states. As a result, the output state of system is limited within the expected boundary constraint. Then, a new control strategy is constructed. This strategy can ensure that the controlled system with uncertainties is uniformly bounded and uniformly ultimately bounded. Finally, the effectiveness of the controller and the state transformation method are verified through simulation of PMLM system. The inequality constraints of the output state are satisfied, and the system performance is improved.

The Indirect Bang-Singular Algorithm (IBSA) for singular control problems with state-inequality constraints

In this study, we present the Indirect Bang-Singular Algorithm (IBSA), a straightforward computational framework developed to solve a wide range of Singular Optimal Control Problems (SOCP) with state-inequality constraints. The algorithm is a type of arc-classification technique which reformulates the SOCP as a nonlinear programming problem over the switching times, and possibly some other parameters such as the co-states’ initial values at the entry time to a singular interval. We derive the singular control feedback using the Pontryagin’s maximum principle and analyze the possibility of an interval where multiple controls are simultaneously singular. Furthermore, we incorporate the state-inequality constraints using the direct-adjoining method. Owing to the linear property of the co-state dynamics, the co-state variables and consequently, the singular controls are computed automatically using MATLAB’s symbolic platform. The nonlinear programming is constructed in a manner to circumvent the challenges posed by state-inequality constraints in more intricate scenarios involving singular controls expressed in terms of incomplete state-feedback functions. We also present several theorems that are integral to devising a straightforward computational approach for solving SOCPs. To assess the effectiveness of the proposed algorithm, we solve the following novel problems: (1) time–fuel-optimal commercial aircraft cruise flight in a vertical plane (i.e., with state-inequality constraint, a scalar singular control, and wind shear effects), and (2) the free-routing time–fuel-optimal commercial aircraft flight in a vertical plane (i.e., with state-inequality constraint, a dual-entry singular control, and wind shear effects). Notably, the optimality of the graphed results has been carefully inspected through first and second-order optimality conditions.

Optimality conditions for bilevel optimal control problems with non-convex quasi-variational inequalities

We establish Pontryagin optimality conditions for a generalized bilevel optimal control problem in which the leader is subject to a pure state inequality constraint, while the follower is governed by a non-convex quasi-variational inequality parameterized by the final state. To simplify the problem at hand, we convert it into a single-level optimal control problem by mapping the solution set of the quasi-variational inequality to a parametric optimization problem and employing the value function reformulation. Furthermore, we introduce certain regularity conditions to ensure that the derived maximum principle remains non-degenerate. Finally, we provide an illustrative example to elucidate our research findings.

Adaptive Concurrent Learning Algorithm Based on Pontryagin's Maximum Principle for Nonlinear System Optimal Tracking Control with State Inequality Constraints

Adaptive Concurrent Learning Algorithm Based on Pontryagin's Maximum Principle for Nonlinear System Optimal Tracking Control with State Inequality Constraints

Pontryagin optimality conditions for generalized bilevel optimal control problems with pure state inequality constraints

Pontryagin optimality conditions for generalized bilevel optimal control problems with pure state inequality constraints

Multi-control commercial aircraft trajectory optimization in a vertical plane with state-inequality constraints via singular control theory

We present a computational framework based on singular control theory to optimize commercial aircraft flights in a vertical plane. The optimization problem involves the lift coefficient (a regular control), and throttle setting (a singular control). However, given a small aerodynamic path angle typically associated with commercial aircraft flights and the significant time-scale separation in flight dynamics, it is customary to treat the aerodynamic path angle as a singular control and replace the lift coefficient as a function of states. This results in a model that is singular on both controls. This research sheds light on several key challenges that have either been unexplored, such as the possibility of switches in both controls even during a specified flight phase, or underexplored, such as state-inequality constraints and wind shear, in relation to the pure singular model. Adhering to the Pontryagin’s maximum principle, we develop a switching-point algorithm to solve the pure singular model for the direct operative cost. Extensive analysis is conducted to understand the impact of various relaxed parameters. The simulation results obtained from the pure singular model are validated through first and second order optimality conditions. Finally, we compare the solutions obtained from the pure singular model with those of the original model, ultimately concluding that the pure singular model yields exceptionally accurate results. In conclusion, the paper highlights the computational time and accuracy advantages of the proposed optimization framework. Furthermore, it underscores the noteworthy observation that the sensitivity of the associated Hamiltonian system with respect to aircraft mass is marginal.

Solvability of the Quaternary Continuous Classical Boundary Optimal Control Dominated by Quaternary Parabolic System

The purpose of this paper is to study the solvability of the quaternary continuous classical boundary optimal control vector problem dominated by quaternary nonlinear parabolic boundary value problem with state constraints. The existence theorem for a quaternary continuous classical boundary optimal control vector with equality and inequality state constraints is stated and demonstrated under suitable assumptions. The mathematical formulation of the quaternary adjoint eqs. associated with the quaternary nonlinear parabolic boundary value problem with state constraints is discovered. The Fréchet derivative of the cost function and the constraints functions are derived. The necessary and sufficient theorems (conditions) for optimality of the quaternary continuous classical boundary optimal control vector problem are stated and demonstrated under suitable assumptions.

Analysis and design of control systems via parameter‐based approach

Analysis and design of control systems via parameter‐based approach

Two reduced non-smooth Newton's methods for discretised state constrained optimal control problem governed by advection-diffusion equation

We are interested in solving numerically an optimal control problem governed by advection-diffusion equation with state inequality constraints. This problem can be implemented to modelise the minimisation of air pollution. First the infinite dimensional problem is discretised by application of the upwind symmetric interior penalty galerkin (SIPG) method. Then, we write the optimality conditions for the discretised problem. We reformulate these conditions into an equivalent system of nonlinear and non-smooth equations by the use of the Fischer-Burmeister function. Then, two effective algorithms that combine the non-smooth Newton's method and an active (index) set technique or pseudo inverse method are applied to solve this system. At the end, numerical examples are presented to illustrate the performance of our methods.

Time-Fuel-Optimal Navigation of a Commercial Aircraft in Cruise With Heading and Throttle Controls Using Pontryagin’s Maximum Principle

In this research, we consider the commercial aircraft trajectory optimization problem for a general cruise model with arbitrary spatial wind fields to be solved using the Pontryagin’s maximum principle. The model features two fundamental controls, namely “throttle setting” (appearing as a singular control) and “heading angle” (appearing as a regular control). For a problem with state-inequality constraints and minimum time-fuel objective, we show that the optimal “heading angle” can be described through the classic Zermelo’s navigation identity. We also demonstrate, by analyzing the switching function, that the singular “throttle setting” can be characterized through a feedback function that relies on both the optimal states and “heading angle”. The switching-point algorithm is employed to solve a case study where we inspect the optimality conditions and graph the optimal controls together with the optimal state and co-state variables.

On the Optimality Condition for Optimal Control of Caputo Fractional Differential Equations with State Constraints

We consider the fractional optimal control problem with state constraints. The fractional calculus of derivatives and integrals can be viewed as generalizations of their classical ones to any arbitrary real order. In our problem setup, the dynamic constraint is captured by the Caputo fractional differential equation with order α ∈ (0, 1), and the objective functional is formulated by the left Riemann-Liouville fractional integral with order β ≥ 1. In addition, there are terminal and running state constraints; while the former is described by initial and final states within a convex set, the latter is given by an explicit instantaneous inequality state constraint. We obtain the maximum principle, the first-order necessary optimality condition, for the problem of this paper. Due to the inherent complex nature of the fractional control problem, the presence of the terminal and running state constraints, and the generalized standing assumptions, the maximum principle of this paper is new in the optimal control problem context, and its proof requires to develop new variational and duality analysis using fractional calculus and functional analysis, together with the Ekeland variational principle and the spike variation.

Optimization Scheme for Online Viewpoint Planning of Active Optical Navigation System in Orthopedic Surgeries

Optical tracking system (OTS) is widely used in robot orthopedic surgeries. During the surgery, the locomotion of the surgical instrument along different osteotomy planes and the adjustment of the patient’s body will lead to obvious variations on the spatial relationship among the positioning tools. Usually, the OTS is manually placed at a fixed viewpoint. As the size of the measurement volume (MV) of the OTS is limited, some positioning tools might move towards the boundaries of the MV, increasing the risk of navigation failure. This paper proposes a two-phase optimization scheme for the active online adjusting of the OTS viewpoint to guarantee the consistency of navigation. A novel optimization structure is proposed with the consideration of continuous state inequality constraints. The dynamic accuracy of the OTS during the active adjusting is also demonstrated to be reliable. Ultimately, simulations and experiments are performed on an active navigation system to verify the effectiveness of the proposed two-phase optimization scheme. Experimental results show that the OTS can be moved to the optimal viewpoint smoothly when confronted with significant variations on the spatial relationship among the positioning tools, guaranteeing the consistency of navigation. The navigation robot successfully keeps the positioning tools around the central region of the MV at different surgical phases.

Robust suboptimal feedback control for a fed-batch nonlinear time-delayed switched system

The optimal control strategy constructed in the form of a state feedback is effective for small state perturbations caused by changes in modeling uncertainty. In this paper, we investigate a robust suboptimal feedback control (RSPFC) problem governed by a nonlinear time-delayed switched system with uncertain time delay arising in a 1,3-propanediol (1,3-PD) microbial fed-batch process. The feedback control strategy is designed based on the radial basis function to balance the two (possibly competing) objectives: (i) the system performance (concentration of 1,3-PD at the terminal time of the fermentation) is to be optimal; and (ii) the system sensitivity (the system performance with respect to the uncertainty of the time-delay) is to be minimized. The RSPFC problem is subject to the continuous state inequality constraints. An exact penalty method and a novel time scaling transformation approach are used to transform the RSPFC problem into the one subject only to box constraints. The resulting problem is solved by a hybrid optimization algorithm based on a filled function method and a gradient-based algorithm. Numerical results are given to verify the effectiveness of the developed hybrid optimization algorithm.

Time-optimal Dubins trajectory for moving obstacle avoidance

In this paper, the problem of computing the global time-optimal Dubins trajectory between two configurations (two points with prescribed heading angles) while avoiding a moving obstacle by a specified safe distance is discussed. The obstacle is allowed to move on any arbitrary trajectory as long as its trajectory satisfies some regularity constraints. By Pontryagin’s maximum principle, the fundamental segments of the time-optimal trajectory are proposed. Along each segment, the control input is either constant or is expressed in state-feedback form. Thereafter, using the necessary conditions for state-inequality constraints, the underlying geometric and analytical properties that govern the concatenation of these fundamental segments are revealed. All these properties exhibit solely state-dependent relations. This has two-fold applications: (1) such relations are essential in ruling out spurious extremals obtained by numerical optimal control methods, and (2) they enable the formulation of a set of nonlinear equations whose solution gives rise to the exact time-optimal trajectory. Numerical examples are presented, validating and highlighting the application of these properties.

The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems with State Vector Constraints

This paper aims to study the quaternary classical continuous optimal control problem consisting of the quaternary nonlinear parabolic boundary value problem, the cost function, and the equality and inequality constraints on the state and the control. Under appropriate hypotheses, it is demonstrated that the quaternary classical continuous optimal control ruling by the quaternary nonlinear parabolic boundary value problem has a quaternary classical continuous optimal control vector that satisfies the equality constraint and inequality state and control constraint. Moreover, mathematical formulation of the quaternary adjoint equations related to the quaternary state equations is discovered, and then the weak form of the quaternary adjoint equations is obtained. Lastly, both the necessary conditions for optimality and sufficient conditions for optimality of the proposed problem are stated and proved. The derivation for the Fréchet derivative of the Hamiltonian is attained.

Robust parameter identification of a nonlinear impulsive time-delay system in microbial fed-batch process

In this paper, taking the feeding process as an impulsive form and considering the time-delay in fermentation process, we propose a nonlinear impulsive time-delay system to formulate the fed-batch culture in microbial 1,3-propanediol production. For this system, we discuss some important properties of the solution. To determine the optimal time-delay and kinetic parameters in the system, we then present a robust parameter identification model subject to continuous state inequality constraints. By the constraint transcription technique and a penalty method, we transform the robust parameter identification problem to a sequence of optimization problems with only box constraints. On this basis, we propose a parallel Particle Swarm Optimization algorithm to solve the resulting problem. Finally, the numerical results show that the impulsive time-delay system can well describe the fed-batch process and the developed algorithm is highly effective.

Optimal control of nonlinear Markov jump systems by control parametrisation technique

This paper considers an optimal control problem of nonlinear Markov jump systems with continuous state inequality constraints. Due to the presence of continuous-time Markov chain, no existing computation method is available to solve such an optimal control problem. In this paper, a derandomisation technique is introduced to transform the nonlinear Markov jump system into a deterministic system, which simultaneously gives rise to an equivalent deterministic dynamic optimisation problem. The control parametrisation technique is then used to partition the time horizon into a sequence of subintervals such that the control function is approximated by a piecewise constant function consistent with the partition. The heights of the piecewise constant function on the corresponding subintervals are taken as decision variables to be optimised. In this way, the approximate dynamic optimisation problem is an optimal parameter selection problem, which can be viewed as a finite dimensional optimisation problem. To solve it using a gradient-based optimisation method, the gradient formulas of the cost function and the constraint functions are derived. Finally, a real-world practical problem involving a bioreactor tank model is solved using the method proposed.