Integral Quadratic Cost Research Articles

Observer-based guaranteed-cost bipartite time-varying formation tracking control for multiagent systems subject to communication delays

The paper centers on the guaranteed-cost bipartite time-varying formation tracking control for multiagent systems with nonuniform time-varying communication delays under the framework of a distributed observer. An integral quadratic cost function is presented, founded on formation tracking and observation error. Firstly, distributed state observers, using the relative measurement output between neighbors, are established so that the leader and follower can accurately estimate each agent's unmeasurable state. Subsequently, a bipartite formation tracking protocol is proposed, incorporating estimation information and nonuniform time-varying communication delays. This protocol aims to achieve bipartite time-varying formation tracking for multiagent systems. Next, a new augmented Lyapunov-Krasovskii functional with delay-product and triple integral terms is constructed. By integrating the second-order Bessel-Legendre inequality technique, the delay-dependent reciprocally convex combination inequality method, and the free weight matrix technique, a new guaranteed-cost bipartite time-varying formation tracking criterion is developed, offering less conservatism. Compared to certain existing delay conditions, the stability conditions assure a greater allowable upper bound of delay. Finally, a numerical example is conducted to confirm the validity and superiority of the theoretical findings.

Improving the insulin therapy for diabetic patients using optimal impulsive disturbance rejection: Continuous time approach

The paper proposes a new model-based optimization approach to improve the clinical efficiency of compensatory insulin bolus treatment in diabetic patients, aiming to mitigate the consequences of diabetes. The most important contribution of this paper is a novel methodology for determining the optimal parameters of insulin treatment, namely the size and timing of insulin boluses, to effectively compensate for carbohydrate intake. This concept can be seen as the so-called optimal model-based bolus calculator. The presented theoretical framework deals with the problem of optimal disturbance rejection in impulsive systems by minimizing an integral quadratic cost function. The methodology considers a personalized empirical transfer function model with static gains and time constants as the only parameters assumed to be known, making the bolus calculator more straightforward to implement in clinical practice. Contrary to other techniques, the proposed methodology considers impulsive insulin administration in the form of boluses, which is more feasible than continuous infusion. In contrast to the conventional bolus calculator, the proposed algorithm allows for maximizing therapy performance by optimizing the relative time of insulin bolus administration with respect to carbohydrate intake. Another feature to highlight is that the solution of the optimization problem can be obtained analytically, hence no numerical iterative solvers are required. Additionally, the continuous-time domain approach allows for a much finer adjustments of the insulin administration timing compared to discrete-time models. The proposed approach was validated in an in-silico study, which demonstrated the importance of systematically determined insulin–carbohydrate ratio and the relative delay between disturbance and its compensation. The results showed that the proposed optimal bolus calculator outperforms the traditional suboptimal formula.

Bellman Neural Networks for the Class of Optimal Control Problems With Integral Quadratic Cost

This work introduces Bellman Neural Networks (BeNNs) and employs them to learn the optimal control actions for the class of optimal control problems (OCPs) with integral quadratic cost. BeNNs represent a particular family of Physics-Informed Neural Networks (PINNs) specifically designed and trained to tackle OCPs via applying the Bellman Principle of Optimality (BPO). The BPO provides necessary and sufficient optimality conditions, which result in a nonlinear partial differential equation known as the Hamilton-Jacobi-Bellman (HJB) equation. BeNNs learn the optimal control actions from the unknown solution of the arising HJB equation (i.e., the value function), where the unknown solution is modeled using a Neural Network. Additionally, the paper shows how to estimate the upper bounds on the generalization error of BeNNs while learning the solutions for the OCP class under consideration. The generalization error estimate is provided in terms of the choice and number of the training points as well as the training error. Numerical studies show that BeNNs can be successfully applied to learn the feedback control actions for the class of optimal control problems considered and, after the training is completed, deployed to control the system in a closed-loop fashion. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Impact Statement</i> —The proposed research improves our understanding of how to solve optimal control problems with closed-loop solutions and has potentially a countless number of applications in several different areas. The study is at the intersection between optimal control theory and artificial intelligence connected with mathematical tools for functional interpolation. This advances the ability to implement a higher level of autonomy in decision-making for practical applications with a beneficial impact on our society.

Harmonic-Based Optimal Motion Planning in Constrained Workspaces Using Reinforcement Learning

In this work, we propose a novel reinforcement learning algorithm to solve the optimal motion planning problem. Particular emphasis is given on the rigorous mathematical proof of safety, convergence as well as optimality w.r.t. to an integral quadratic cost function, while reinforcement learning is adopted to enable the cost function's approximation. Both offline and online solutions are proposed, and an implementation of the offline method is compared to a state-of-the-art RRT* approach. This novel approach inherits the strong traits from both artificial potential fields, i.e., reactivity, as well as sampling-based methods, i.e., optimality, and opens up new paths to the age-old problem of motion planning, by merging modern tools and philosophies from various corners of the field.

On finite time stability with guaranteed cost control of uncertain linear systems

This paper deals with the design of a robust state feedback control law for a class of uncertain linear time varying systems. Uncertainties are assumed to be time varying, in one-block norm bounded form. The proposed state feedback control law guarantees finite time stability and satisfies a given bound for an integral quadratic cost function. The contribution of this paper is to provide a sufficient condition in terms of differential linear matrix inequalities for the existence and the construction of the proposed robust control law. In particular, the construction of the feedback control law is brought back to a feasibility problem which can be solved inside the convex optimization framework. The effectiveness of the proposed approach is shown by means of the results obtained on a numerical and a physical example.

Asymptotics of the solution to a singularly perturbed linear-quadratic optimal control problem

The optimization of a transient process in a linear singularly perturbed system is considered. The goal is to find an admissible control that minimizes an integral quadratic cost functional. Asymptotic approximations to optimal open-loop and feedback controls are constructed for this problem. The basic advantage of the algorithms proposed is that the original optimal control problem splits into two unperturbed problems of lower dimension.

Time-Multiplied Guaranteed Cost Control of Linear Delay Systems

This paper investigates the design of guaranteed cost controllers for a class of linear systems with a state delay using a time-multiplied linear quadratic cost function. Based on delay-dependent and delay-independent stability criteria, guaranteed cost controllers can be constructed via solutions to linear matrix inequalities (LMIs) such that the resulting closed-loop system is stable and a specified time-multiplied linear integral-quadratic cost function has an upper bound. By the cone complementary linearization method, delay-dependent state feedback controllers can be derived in terms of LMIs. A numerical example is provided to demonstrate the effectiveness of the proposed method.

An optimal linear-quadratic digital tracker for analog neutral time-delay systems

In this paper, an optimal hybrid tracking control problem for continuous neutral time-delay systems is formulated and studied. An optimal linear integral quadratic cost function that has a high-gain property is used for tracking control specification. Two interpolation methods are applied to directly convert the original analog neutral time-delay system into an equivalent digital retarded time-delay system, and meanwhile convert the continuous-time quadratic cost function into a discretized form. Then, an extended state vector is constructed for an associate extended discrete-time optimal control problem without time delay. Using the standard discrete-time linear-quadratic optimal control theory and an indirect digital redesign technique with a predictive feature, an effective digital tracker is designed for the original analog neutral time-delay system. An example is finally given for illustrating the effectiveness of the new tracker design method.

Trajectory priming with dynamic fuzzy networks in nonlinear optimal control.

Fuzzy logic systems have been recognized as a robust and attractive alternative to some classical control methods. The application of classical fuzzy logic (FL) technology to dynamic system control has been constrained by the nondynamic nature of popular FL architectures. Many difficulties include large rule bases (i.e., curse of dimensionality), long training times, etc. These problems can be overcome with a dynamic fuzzy network (DFN), a network with unconstrained connectivity and dynamic fuzzy processing units called "feurons." In this study, DFN as an optimal control trajectory priming system is considered as a nonlinear optimization with dynamic equality constraints. The overall algorithm operates as an autotrainer for DFN (a self-learning structure) and generates optimal feed-forward control trajectories in a significantly smaller number of iterations. For this, DFN encapsulates and generalizes the optimal control trajectories. By the algorithm, the time-varying optimal feedback gains are also generated along the trajectory as byproducts. This structure assists the speeding up of trajectory calculations for intelligent nonlinear optimal control. For this purpose, the direct-descent-curvature algorithm is used with some modifications [called modified-descend-controller (MDC) algorithm] for the nonlinear optimal control computations. The algorithm has numerically generated robust solutions with respect to conjugate points. The minimization of an integral quadratic cost functional subject to dynamic equality constraints (which is DFN) is considered for trajectory obtained by MDC tracking applications. The adjoint theory (whose computational complexity is significantly less than direct method) has been used in the training of DFN, which is as a quasilinear dynamic system. The updating of weights (identification of DFN parameters) are based on Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. Simulation results are given for controlling a difficult nonlinear second-order system using fully connected three-feuron DFN.

Quadratic guaranteed cost control of uncertain neutral delay‐differential systems

This paper is concerned with the robust stabilization problem for uncertain neutral delay‐differential systems. An integral quadratic cost criterion is employed to measure system performance. By checking the Hamiltonian matrix and solving an algebraic Riccati equation, a static state feedback control law is proposed. This simple state feedback controller not only stabilizes perturbed neutral systems but also guarantees an upper bound of system performance. Last, an example is included to illustrate the results developed in this paper.

Minimax controller design for a class of uncertain linear systems

This paper considers the design of minimax controllers for a class of linear systems. The system is perturbed by a known initial condition on the state and by a disturbance process constrainted by an L 2 norm bound. Structured undertainty is represented by perturbation feedback through L 2-induced norm bounded operators. The performance is measured by an integral quadratic cost function. The resultant controllers minimise the worst-case cost for all admissible distrubances and for all admissible systems. The design involves the optimal solution of a parametric algebraic Riccati equation. A new proof of minimax optimality is given using standard methods for linear systems. The multivariable optimisation problem is shown to be convex and the set of feasible parameter values is shown to be compact. Thus, the design procedure is computationally tractable.

Design of performance robustness for uncertain linear systems with state and control delays

The linear systems considered in this paper are subject to uncertain perturbations of norm-bounded time-varying parameters and multiple time delays in system state and control. The time delays are uncertain, independent of each other, and allowed to be time-varying. The integral quadratic cost criterion is employed to measure system performance. Using solutions of Lyapunov and Riccati equations, a linear state feedback control law is proposed to stabilize the perturbed system and to guarantee an upper bound of system performance, which is applicable to arbitrary time delays.

Optimality of extremals for linear systems with quadratic costs

In this paper we consider linear control systems on R n with integral quadratic cost functionals and investigate optimality properties of extremals. In this context, extremals are curves which satisfy the Pontryagin maximum principle. We extend the Jacobi necessary condition from the calculus of variations to the optimal control setting and describe a relationship between optimality of extremals and the existence of a solution for a certain Riccati differential equation.

A control operator and some of its applications

The conventional function space algorithms for solving minimization of penalized cost functional for optimal control problem characterized by linear-system integral quadratic cost due to Di Pillo and others, though falling within the framework pertinent to the conjugate gradient method algorithm, is difficult to apply computationally. The difficulty arises principally because there exists in the algorithm a number of stringent requirements imposed on the minimization procedures to facilitate its convergence. Incidentally, such computations are very cumbersome to carry out numerically. To circumvent this major numerical draw back, we construct here a control operator associated with this class of problems and use our explicit knowledge of the operator to devise an extended conjugate gradient method algorithm for solving this family of problems. Furthermore, the establishment of some functional inequalities which are obtained using the knowledge of the control operator is discussed.

Distributed Wiener-Poisson control †

A one-dimensional Wiener plus independent Poisson control problem, with the state governed by a partial differential equation, has an integrated discounted quadratic cost function and asymmetric bounds on the control, which is a function of the current state. A Bellman equation and the maximum principle for partial differential equations are used to obtain the optimal closed-loop control in bang-bang form. The finite and infinite integral quadratic cost functions are treated separately.

Optimal output feedback by minimizing ||K(F)||&amp;lt;inf&amp;gt;2&amp;lt;/inf&amp;gt;

Consider <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LQP</tex> with output feedback <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</tex> , initial condition x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , and integral quadratic cost value given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x'_{0}K(F)x_{0}</tex> . Choice of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</tex> to minimize <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\parallel K(F) \parallel_{2}</tex> instead of the Levine-Athans cost, trace [ <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(F)</tex> ], is considered. An algorithm is presented.

A double integral quadratic cost and tolerance of feedback nonlinearities

A double integral quadratic cost with an associated integral constraint on state trajectories is shown to result in a stable feedback control law for linear time-varying differential systems. The gain matrix for this control is obtained by integrating a Riccati-type matrix differential equation over a finite time interval and is shown to allow for a large class of nonlinearities in the feedback loop without destroying its asymptotically stable property. The class of nonlinearities is larger than that which is generally permitted for the standard steady-state linear optimal regulator, and a phase margin of 90 ° can be approached by the closed-loop system in the case of time-invarlant systems.

Optimal output feedback without trace

For a linear system (C,A,B) with integral quadratic cost, an optimal control problem is presented which has as its solution an output feedback control. The output feedback chosen ensures that the closed-loop cost is not worse than the open-loop cost for any initial condition, which is not guaranteed by the standard optimization method for finding output feedback (optimization with respect to the feedback matrix of an average over initial conditions of the closed-loop cost). The most severe restriction involved is thatker[C]⊂ R[B]. Finite- and infinite-time cases are discussed.

Adaptive nuclear reactor control for integral quadratic cost functions

The problem of optimally controlling the power level changes of a nuclear reactor is considered. The model of an existing power plant is used, which is a ninth-order nonlinear system, having time-varying parameters. A closed form solution of the optimal feedback controller is not known for such a system, and numerical solutions require prohibitive amounts of computer time and/or storage for on-line applications. The approach in the present work is, therefore, to approximate the response of the reactor by that of a low-order linear model in a piece-wise manner. The model parameters are chosen to minimize the deviations, in any desired sense, between the system and model responses. The optimal feedback parameters are readily computed for the model, and the same controller results in suboptimal system performance. The difference, however, has been found negligible in the examples considered. Since the model parameters are updated to reflect the plant nonlinearities as well as changes in the plant parameters, the resultant control system is adaptive. It has been found, that using a second-order model, optimal performance could be approached to within practical limits, and the necessary computer time is realizable for on-line applications.

Reduction of the Sensitivity of Optimal Control Systems by Using Two Degrees of Freedom

This paper considers the implementation of optimal control using two degrees of freedom, i.e., a combination of the open- and closed-loop configurations. It has been shown that with this arrangement, one can minimize the integral state sensitivity, while at the same time, either the performance sensitivity, or the terminal-state sensitivity may be reduced to zero. In practice, some compromise must be accepted between the reductions in the various sensitivities. An example of a linear system with an integral quadratic cost function is used to illustrate the procedure.