A hybrid approach to optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the state x(t), the control u(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.The approach taken is a sequence of two-phase processes or cycles, each composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functions x(t), u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable.The technique employed is of the hybrid type, in an attempt to combine some of the best features of both the indirect and direct approaches. While a predetermined number and sequence of subarcs are assumed (a feature of direct methods), enforcement of the state inequality constraint is obtained through a Valentine-type representation (a feature of indirect methods).By properly choosing the analytical form of certain non-differential constraints to be satisfied by the augmented control along each subarc composing the extremal arc (these nondifferential constraints are arrived at through the Valentine-type transformation), one ensures satisfaction of the state inequality constraint everywhere. Specifically, strict inequality is enforced for the subarcs internal to the state boundary and strict equality is enforced for the subarcs lying on the state boundary.To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized time t, defined in such a way that each subarc composing the extremal arc has a normalized time length Δt = 1. In this way, variable-time corner conditions and variable-time terminal conditions are transformed into fixed-time corner conditions and fixed-time terminal conditions. The actual times θ₁, θ₂, τ at which (i) the state boundary is entered, (ii) the state boundary is exited, and (iii) the terminal manifold is reached are regarded to be components of the parameter π being optimized.The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.     

A differential operator approach to multidimensional optimal control

We use recent improvements in the parametrizations of controllable linear multidimensional systems to show how to transform the study of a linear quadratic optimal problem into that of a variational problem without constraints. We give formal conditions on the differential module defined by the linear control system to pass from the Pontryagin approach to a purely Euler–Lagrange variational problem. This formal approach uses the cost function in order to link the locally exact sequence formed by the controllable system and its parametrizations with the sequence formed by their formal adjoint operators. In the case of partial differential equations, this scheme is typical for any problem of linear elasticity theory and electromagnetism.     

Game theory approach to optimal control problem with multi-channel control

In this paper, we mainly study the optimal control problem with multi-channel control. The main contribution is to treat the optimal control problem as a leader-follower problem with multi-hierarchy decision makers in game theory. According to the sequence of the decision makers, the optimal controller is derived by the maximum principle. It is worth mentioning that the optimal solution is in the feedback form related to two symmetric Riccati equations with same dimension as the original state. Furthermore, the obtained solution is sufficient and necessary.     

A non-linear optimal control approach to central sewer network flow control

The problem of optimal water distribution to several retention reservoirs in an urban sewer network during rainfall is considered. The goal of the control actions is the minimization of overflows and eventually the reduction of their polluting impact on receiving waters. To this end, a non-linear optimal control approach is used and the numerical solution of the control problem is effectuated by use of a feasible direction algorithm. A detailed study of the central control problem for a particular large sewer network using this method is presented. Results demonstrate the efficiency and the real-time feasibility of the developed methodology.     

A new approach to fuzzy estimation of Takagi-Sugeno model and its applications to optimal control for nonlinear systems

An efficient approach is presented to improve the local and global approximation and modelling capability of Takagi-Sugeno (T-S) fuzzy model. The main aim is obtaining high function approximation accuracy. The main problem is that T-S identification method cannot be applied when the membership functions are overlapped by pairs. This restricts the use of the T-S method because this type of membership function has been widely used during the last two decades in the stability, controller design and are popular in industrial control applications. The approach developed here can be considered as a generalized version of T-S method with optimized performance in approximating nonlinear functions. A simple approach with few computational effort, based on the well known parameters’ weighting method is suggested for tuning T-S parameters to improve the choice of the performance index and minimize it. A global fuzzy controller (FC) based Linear Quadratic Regulator (LQR) is proposed in order to show the effectiveness of the estimation method developed here in control applications. Illustrative examples of an inverted pendulum and Van der Pol system are chosen to evaluate the robustness and remarkable performance of the proposed method and the high accuracy obtained in approximating nonlinear and unstable systems locally and globally in comparison with the original T-S model. Simulation results indicate the potential, simplicity and generality of the algorithm.     

Hermite series approach to optimal control

A general expression for the operational matrix of integration, P, in the case of Hermite polynomials is derived. The structure of P is simple and it is therefore computationally attractive. Using this P, the optimal control with quadratic index problem and the singularly perturbed optimal control problem are studied. Some numerical examples are also included.     

An optimal control approach to robust control design

We propose an optimal control approach to robust control design. Our goal is to design a state feedback to stabilize a system under uncertainty. We translate this robust control problem into an optimal control problem of minimizing a cost. Because the uncertainty bound is reflected in the cost, the solution to the optimal control problem is a solution to the robust control problem. Our approach can deal with both linear and non-linear systems. Furthermore it can handle both matched and unmatched uncertainties. It can also handle uncertainty in the control input matrix.     

A gradient flow approach to decentralised output feedback optimal control

This paper visits the quadratic optimal control problem of decentralised control systems via static output feedback. A gradient flow approach is introduced as a tool to compute the optimal output feedback gain. Several nice properties are revealed concerning the convergence of the gain matrix along the trajectory of an ordinary differential equation obtained from the gradient of objective cost, i.e. the objective cost is decreasing along this trajectory. If the equilibrium points are isolated, the convergence can be guaranteed. A simulation example is given to illustrate the effectiveness of this approach.     

Geometric approach to non-linear optimal control problems

Minimum energy control problems are investigated for affine non-linear systems evolving on manifolds, in the framework of the geometric theory of hamiltonian systems. Some insights into the problems are gained via this machinery.     

Detailed wheel/rail geometry processing with the conformal contact approach

Multibody System Dynamics - This paper proposes a new way of considering wheel–rail contact in multibody systems simulation that goes beyond the traditional planar constraint and elastic...     

A game theoretic approach to the optimal control of uncertain systems

The issue of performance robustness is considered in a dynamic games framework where the goal is to find a feedback controller that stabilizes the system under all plant perturbations and that minimizes the value of a quadratic performance index under the worst perturbations. To find a solution to this hard-constrained problem, an auxiliary soft-constrained dynamic game problem is formulated. It is shown that the solution to the auxiliary problem provides the solution to the original problem, under the assumption that the solution to the original problem occurs on the boundary of the perturbations set. The approach, however, can still be applied, with modifications, to the case when this assumption fails.     

A new approach to Lipschitz continuity in state constrained optimal control

For a linear-quadratic state constrained optimal control problem, it is proved in [11] that under an independence condition for the active constraints, the optimal control is Lipschitz continuous. We now give a new proof of this result based on an analysis of the Euler discretization given in [9]. There we exploit the Lipschitz continuity of the control to estimate the error in the Euler discretization. Here we show that the theory developed for the Euler discretization can be used to derive the Lipschitz continuity of the optimal control.     

A combined genetic algorithms-shooting method approach to solving optimal control problems

In this paper, an alternative method for solving optimal control problems is presented. By applying calculus of variations, the optimal control problem can be reduced to solving a two-point boundary value problem. Here, the solution is generated with a combination of two methods genetic algorithms (GA) and the shooting method. An estimate of the optimal solution is first obtained using GA. This solution is in turn used as the initial guess for the shooting method. This combined method is applied to an optimal missile guidance problem. The performances of the combined method and GA are evaluated by simulation and compared. The results clearly show that the proposed combined method is able to locate the optimal solution more efficiently than GA. The results also show that the combined method never fails to correctly determine the optimal solution. Therefore, it proves to be more robust than the shooting method whose convergence is not always guaranteed.     

A new therapeutic parameter and computer control to approach optimal haemodialysis

A two-compartmental (intracellular and extracellular volumes) patient model is being employed to derive the time variation of dialysis rate, which is required to meet the criterion that the intercompartmental concentration difference should not exceed the maximum value the patient could tolerate during dialysis treatment. Dialysate flow rate is the operational variable selected for controlling the dialysis rate. The maximum intercompartmental solute concentration difference allowable for individual patients is suggested as a new therapeutic parameter. The model algorithm and a personal computer can be linked with the delivery pump to monitor the dialysis treatment so as to avoid the disequilibrium syndrome and thus rationalise the entire procedure of dialysis.     

The PMSS approach to large-scale optimal control problems

Large-scale optimal control problems arise in several different contexts from a variety of applications. Many general computational algorithms have been proposed for the solution of these problems and a large literature on the subject abounds. To assimilate this material, it is necessary to overcome several difficulties. For example, the large-scale nature of the problems induces a complicated notation and specialized jargon that sometimes confuses the basic issues. Further, the ad hoc nature of the development of many algorithms tends to add an air of mystery that is difficult to dispel. Consequently, it is not always apparent that there are simple concepts and principles that underlie most, if not all, of these algorithms. In this paper, the general taxonomic scheme proposed by Geoffrion is exploited to provide a systematic framework for viewing computational algorithms for solving large-scale, optimal control problems. To demonstrate the utility of the viewpoint, a new algorithm for solving large-scale, discrete-time nonlinear optimal control problems is presented. The development is directed by the simple concepts and principles that underlie the proposed taxonomy.     

The optimal control approach to generalized multiprocessor scheduling

In this paper we present several new results in the theory of homogeneous multiprocessor scheduling. We start with some assumptions about the behavior of tasks, with associated precedence constraints, as processor power is applied. We assume that as more processors are applied to a task, the time taken to compute it decreases, yielding some speedup. Because of communication, synchronization, and task scheduling overhead, this speedup increases less than linearly with the number of processors applied. We also assume that the number of processors which can be assigned to a task is a continuous variable, with a view to exploiting continuous mathematics. The optimal scheduling problem is to determine the number of processors assigned to each task, and task sequencing, to minimize the finishing time.These assumptions allow us to recast the optimal scheduling problem in a form which can be addressed by optimal control theory. Various theorems can be proven which characterize the optimal scheduling solution. Most importantly, for the special case where the speedup function of each task isp , wherep is the amount of processing power applied to the task, we can directly solve our equations for the optimal solution. In this case, for task graphs formed from parallel and series connections, the solution can be derived by inspection. The solution can also be shown to be shortest path from the initial to the final state, as measured by anl _1/ distance metric, subject to obstacle constraints imposed by the precedence constraints.This research has been funded in part by the Advanced Research Project Agency monitored by ONR under Grant No. N00014-89-J-1489, in part by Draper Laboratory, in part by DARPA Contract No. N00014-87-K-0825, and in part by NSF Grant No. MIP-9012773. The first author is now with AT&T Bell Laboratories and the second author is with BBN Incorporated.     

A game theoretic approach to optimal control in the presence of uncertainty

The optimal control problem, in the presence of uncertainty in the plant, is formulated as a game between the uncertainty and the control variables. This approach gives an optimal control strategy which is effective even under the "worst" conditions of uncertainty. The optimal control of a second-order plant with uncertainty in frequency brings out several interesting features. In particular, the existence of a barrier demarcating the controllable and uncontrollable regions in the phase plane is revealed, which is absent in the corresponding one-sided optimization.     

An optimal control based approach to designing plantwide control system architectures

An approach to designing decentralized plantwide control system architectures is presented. The approach is based on splitting the optimal controller gain matrix that results from solving an output optimal control problem into feedback and feedforward parts. These two parts are then used to design and evaluate decentralized control systems. Results for the application of the methodology to a realistic, 4 by 4 reactor with recycle process are given. For this system, the optimal control based approach suggests feedback pairings that are significantly different than those suggested by the steady state RGA. The approach presented can give an indication if MPC is preferred over a decentralized approach to plantwide control. Comparison of the results produced by the best decentralized plantwide system and a model predictive control system are presented.