Optimal Control Synthesis Research Articles

Finding the principal bifurcation value of the maximum control torque in the problem of optimal control synthesis for a pendulum

The principal bifurcation value in the problem of synthesis of a time-optimal control for damping the oscillations of a nonlinear pendulum is found. A one-link pendulum controlled by a torque of forces produced at the suspension point is considered. The magnitude that bounds the modulus of the control torque is the only essential parameter of the investigated system. It is supposed that an admissible control may exceed the moment of the gravitational force. By the principal bifurcation value, we mean the value of the maximum admissible control torque for which a trajectory with two switches of relay optimal control arises in the synthesis pattern. This publication is a logical complement to papers [1–3], where the mentioned synthesis patterns were obtained numerically with the use of the maximum principle. In contrast to [1–3], we formulate a system of explicit relations of integral type for the numerical construction of the boundaries of domains of the phase plane inside which the trajectories with two switches start. Based on these relations, an analytical upper estimate of the principal bifurcation value is obtained. This value is also found numerically. In a special coordinate system, the synthesis patterns are constructed that illustrate the specific features of the optimal control for large angular velocities of pendulum rotations.

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.

Distributed controller design for open water channels

In the design of an automatic controller to achieve water-level set-point regulation and off-take load-disturbance rejection for an open water channel, a key concern is an inherent trade-off between local performance and the way water-level errors propagate due to control action. Here a structured optimal controller synthesis problem is formulated to systematically manage this trade-off, using H∞ loop-shaping ideas. The loop-shaping weights can be scalably designed and the imposed structure ensures the controller only involves local information exchange. Importantly, the distributed control structure we consider confines water-level error propagation to upstream pools, with corresponding benefits in terms of water distribution efficiency. Moreover, it coincides with the interconnection structure of a channel, and so the corresponding optimal synthesis problem has a convex characterisation; detailed state-space formulae are provided. Field test data are presented to illustrate overall performance.

A unified model for state feedback of discrete event systems II: Control synthesis problems

This paper studies control synthesis problems in a new model framework for discrete event state feedback control systems. The new model framework consists of a basis model as well as concurrent models. We study relationships between the basis model and the concurrent models from the perspective of a predicate being controllable and synthesizable. We derive a linear order for the models, and moreover, show that they are equivalent under certain conditions. Based on this, three conditions are presented for synthesizing a predicate completely. Finally, the optimal control synthesis problem for both the basis model and the concurrent models is studied.

Solution of Fuller’s problem on the basis of the joint Pontryagin-Hamilton-Ostrogradskii principle

The present study is devoted to the synthesis of the optimal control of a dynamic system. It is proved that application of the method of needle variation to the invariant features of effective motion is a promising approach to increasing the efficiency with which problems involved in the synthesis of optimal control are solved. The Hamilton-Ostrogradskii action integral, to which the Pontryagin needle variation is applied, is adopted as such a feature in the study. As a result, a minium condition for the objective functional is obtained; this condition is termed the joint maximum principle.

On non-linear dynamics and an optimal control synthesis of the action potential of membranes (ideal and non-ideal cases) of the Hodgkin–Huxley (HH) mathematical model

In this paper, we have studied the plasmatic membrane behavior using an electric circuit developed by Hodgkin and Huxley in 1952 and have dealt with the variation of the amount of time related to the potassium and sodium conductances in the squid axon. They developed differential equations for the propagation of electric signals; the dynamics of the Hodgkin–Huxley model have been extensively studied both from the view point of its their biological implications and as a test bed for numerical methods, which can be applied to more complex models. Recently, an irregular chaotic movement of the action potential of the membrane was observed for a number of techniques of control with the objective to stabilize the variation of this potential. This paper analyzes the non-linear dynamics of the Hodgkin–Huxley mathematical model, and we present some modifications in the governing equations of the system in order to make it a non-ideal one (taking into account that the energy source has a limited power supply). We also developed an optimal linear control design for the action potential of membranes. Here, we discuss the conditions that allow the use of control linear feedback for this kind of non-linear system.

Synthesis of optimal control of dynamic systems with infinite aftereffect, a small parameter, and poisson perturbations

A methodology is obtained that makes it possible to synthesize an optimal control for stochastic systems of functional-differential equations with the entire prehistory and a small parameter and takes into account continuous Wiener-type and discontinuous Poisson-type perturbations. It is proved that the sought-for control can be found as an optimal control of some auxiliary problem.

OPTIMAL DISCRETE CONTROLLER SYNTHESIS FOR MODELING FAULT-TOLERANT DISTRIBUTED SYSTEMS

We propose a safe design method for safe execution systems, based on fault-tolerance techniques: it uses optimal discrete controller synthesis (DCS) to generate a correct-by-construction fault-tolerant system. The properties enforced concern consistent execution, functionality fulfillment (whatever the faults, under some failure hypothesis), and several optimizations (of the tasks’ execution time). We propose an algorithm for optimal DCS on bounded paths. We propose model patterns for a set of periodic tasks with checkpoints, a set of distributed, heterogeneous and fail-silent processors, and an environment model that expresses potential fault patterns. The implementation is illustrated using the Sigali symbolic DCS tool and the Mode Automata programming language.

Solution to the variation problem for information path functional of a controlled random process

The paper introduces a new approach to dynamic modeling, using the variation principle, applied to a functional on trajectories of a controlled random process, and its connection to the process' information functional. In [V.S. Lerner, Dynamic approximation of a random information functional, J. Math. Anal. Appl. 327 (1) (2007) 494–514, available online 5-24-06], we presented the information path functional with the Lagrangian, determined by the parameters of a controlled stochastic equation. In this paper, the solution to the path functional's variation problem provides both a dynamic model of a random process and the model's optimal control, which allows us to build a two-level information model with a random process at the microlevel and a dynamic process at the macrolevel. A wide class of random objects, modeled by the Markov diffusion process and a common structure of the process' information functional, leads to a universal information structure of the dynamic model, which is specified and identified on a particular object with the applied optimal control functions. The developed mathematical formalism, based on classical methods, is aimed toward the solution of problems identification, combined with an optimal control synthesis, which is practically implemented and also demonstrated in the paper's example.

A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems

Even though dynamic programming offers an optimal control solution in a state feedback form, the method is overwhelmed by computational and storage requirements. Approximate dynamic programming implemented with an Adaptive Critic (AC) neural network structure has evolved as a powerful alternative technique that obviates the need for excessive computations and storage requirements in solving optimal control problems. In this paper, an improvement to the AC architecture, called the "Single Network Adaptive Critic (SNAC)" is presented. This approach is applicable to a wide class of nonlinear systems where the optimal control (stationary) equation can be explicitly expressed in terms of the state and costate variables. The selection of this terminology is guided by the fact that it eliminates the use of one neural network (namely the action network) that is part of a typical dual network AC setup. As a consequence, the SNAC architecture offers three potential advantages: a simpler architecture, lesser computational load and elimination of the approximation error associated with the eliminated network. In order to demonstrate these benefits and the control synthesis technique using SNAC, two problems have been solved with the AC and SNAC approaches and their computational performances are compared. One of these problems is a real-life Micro-Electro-Mechanical-system (MEMS) problem, which demonstrates that the SNAC technique is applicable to complex engineering systems.

Controller Design for Discrete-Time Stochastic Processes With Nonquadratic Loss

The problem of optimal controller synthesis for discrete-time stochastic nonlinear processes is examined. When the objective for design is specified as the expected value of a nonsymmetric, nonquadratic loss functional, it is necessary to relate the closed-loop process dynamics to the stationary probability density function (pdf). Because explicit relationships are available for only a few special cases, a general solution to the problem is not possible. In this work a technique for developing approximations of the optimal control law by application of a dynamics–pdf approximation is presented. Numerical simulations illustrate application of the proposed technique.

Design of /spl lscr//sub 1/-optimal controllers with flexible disturbance rejection level

This note presents a new design methodology that allows for flexible management of the tradeoff between the ability of a system to attenuate disturbance signals versus its expected worst peak-to-peak amplification. The proposed strategy applies to linear time-invariant systems which are subject to persistent disturbance signals and combines a recently developed quasi-robust linear programming concept with a well-known /spl lscr//sub 1/-optimal controller synthesis approach. The benefit of the resulting technique is demonstrated using an example.

Synthesis of optimal controllers for piecewise affine systems with sampled-data switching

This paper discusses the optimal control problem of the continuous-time piecewise affine (PWA) systems with sampled-data switching, where the switching action is executed based upon a condition on the state at each sampling time. First, an algebraic characterization for the problem to be feasible is derived. Next, an optimal continuous-time controller is derived for a general class of PWA systems with sampled-data switching, for which the optimal control problem is feasible but whose subsystems in some modes may be uncontrollable in the usual sense. Finally, as an application of the proposed approach, the high-speed and energy-saving control problem of the CPU processing is formulated, and the validity of the proposed methods is shown by numerical simulations.

Optimal persistent disturbance attenuation control for linear hybrid systems

In this paper, the disturbance attenuation properties for a class of linear hybrid systems are investigated, and a hybrid optimal persistent disturbance attenuation control problem is studied. First, a procedure is developed to determine the minimal l ∞ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are discussed. The termination of the procedure in a finite number of steps is shown for a subclass of hybrid systems with simplified discrete event dynamics, called switched linear systems. Finally, the optimal persistent disturbance attenuation controller synthesis problem is studied. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law, and a systematic approach is proposed to design such feedback control.

Optimal Control Synthesis in Systems with an Unbounded Velocity Set

Optimal Control Synthesis in Systems with an Unbounded Velocity Set

Efficient Solution of Optimal Control Problems Using Hybrid Systems

We consider the synthesis of optimal controls for continuous feedback systems by recasting the problem to a hybrid optimal control problem: synthesize optimal enabling conditions for switching between locations in which the control is constant. An algorithmic solution is obtained by translating the hybrid automaton to a finite automaton using a bisimulation and formulating a dynamic programming problem with extra conditions to ensure non-Zenoness of trajectories. We show that the discrete value function converges to the viscosity solution of the Hamilton--Jacobi--Bellman equation as a discretization parameter tends to zero.

OPTIMAL CONTROL SYNTHESIS IN INTERVAL DESCRIPTOR SYSTEMS APPLICATION TO TIME STREAM EVENT GRAPHS

This paper presents a new class of model in the field of topical algebra whose time evolution belongs to intervals. The lower and upper bounds depend on the maximization, minimization and addition operations, simultaneously. A motivation for the current study is the modelling of Time Petri nets like Time Stream Petri Nets and P-time Petri nets which extend Timed Petri Nets and generalize the semantic of its synchronization. The final objective is to make optimal control synthesis on this new model and to generalize the classical “backward” equations applied to Timed Event Graphs.

EXPLICIT HYBRID OPTIMAL CONTROLLER FOR DISTURBANCE ATTENUATION IN LINEAR HYBRID SYSTEMS

In this paper, the disturbance attenuation properties for classes of linear hybrid systems perturbed by exterior disturbances are investigated, and a hybrid l1 robust optimal control problem is studied. First, a procedure is developed to determine the minimal l∞ induced gain of linear hybrid systems. However, for general hybrid systems, the termination of the procedure is not guaranteed. Then, the decidability issues are briefly discussed. Finally, we study the robust l1 optimal controller synthesis problem. It is shown that the optimal performance level can be achieved by a piecewise linear state feedback control law for the linear hybrid systems, and a systematic approach to design such feedback control is proposed.

Approximate schemes of optimal control synthesis

There are considered recommendations based on experience of investigating the applied problems on practical aspects of the approximation for optimal control sythesis, the choice of good initial approximations for iterative processes to reach the global optimum, etc., which cannot be studied in a purely theoretical way.

Optimal supervisory control of finite state automata

This paper presents optimal supervisory control of dynamical systems that can be represented by deterministic finite state automaton (DFSA) models. The performance index for the optimal policy is obtained by combining a measure of the supervised plant language with (possible) penalty on disabling of controllable events. The signed real measure quantifies the behaviour of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in earlier publications. Synthesis of the optimal control policy requires at most n iterations, where n is the number of states of the DFSA model generated from the unsupervised plant language. The computational complexity of the optimal control synthesis is polynomial in n. Syntheses of the control algorithms are illustrated with two application examples.