Numerical computation of optimal control

The formal solution for dynamically optimized control of a process is specified in terms of a two-point boundary problem. The numerical computation then presents great difficulties and speed of convergence is vital, even with the most powerful of contemporary computers. The authors give a method for rapid final convergence in which the two-point boundary problem is avoided altogether, although it is in fact only slightly different to methods of second variations. Computing experience with two examples is then described: a) aircraft landing and b) control of a boiler, with which the speed of convergence is well illustrated.     

On finding switching times in time optimal control systems†

A system of non-linear algebraic equations is found which yields the switching times for the time optimal bang-bang control to the linear system [xdot] = Ax + bu. These equations can also yield switching curves in some instances.     

One-machine n-part-type optimal setup scheduling: analyticalcharacterization of switching surfaces

The authors consider optimal setup scheduling of a single reliable machine. Production flow of n different part types and the setup process are described by differential equations. Setup change rates are control variables. Necessary conditions on optimal setup changes are characterized analytically, and optimal setup change times are derived for a given setup change sequence. The linearization of optimal setup switching surfaces is derived, indicating the existence of attractors observed in numerical optimal solutions. The approach developed in this paper establishes a strong basis for studying multimachine production systems and for constructing tractable near-optimal numerical solution techniques     

The computation of optimal singular control

An algorithm for the solution of optimal control problems with singular subarcs is presented. The modified quasilinearization is extended to the solution of those problems where the initial conditions are updated successively. A major advantage of the quasilinearization method is its rapidity of convergence despite its simplicity in programming. It is not necessary to introduce penalty functions for the treatment of boundary conditions. An illustrative example is presented.     

Multilevel technique for the computation of optimal singular control

This paper presents an algorithm for the computation of optimal singular control. The trajectory is decomposed to singular and non-singular sub-arcs. Each sub-arc is optimized locally. The interactions between different sub-arcs are then determined by a second level controller. An illustrative example is included.     

On the computation of optimal nonlinear feedback controls

Problems of optimal control, without constraints and in fixed time, are considered. We show that the feedback law satisfies a system of quasi-linear partial differential equations which can be integrated by the method of characteristics. We illustrate the advantages of our approach by two examples; one is a realistic application in heat transfer of buildings.     

Modal and transition dwell time computation in switching systems: A set-theoretic approach

We consider a plant the dynamics of which switch among a family of systems. Each of these systems has a single stable equilibrium point. We assume that a constraint region for the state is assigned and we consider the problem of finding suitable limitations on the commutation speed in order to avoid constraints violations, even in the absence of state measurements. We introduce the concepts of modal and transition dwell times which lead to the definition of dwell time vector and dwell time graph (represented by a proper matrix), respectively. The former imposes a minimum permanence time on a discrete mode before commuting, the latter imposes the minimum permanence time on the current mode before switching to a specific new one. Both dwell time vector and dwell time graph can be computed via set-theoretic techniques. When the systems share a single equilibrium state, stability can be assured as a special case. Finally, under the assumption of affine dynamics, non-conservative values are achieved.     

Data-driven optimal switching and control of switched systems

In this paper, optimal switching and control approaches are investigated for switched systems with infinite-horizon cost functions and unknown continuous-time subsystems. At first, for switched systems with autonomous subsystems, the optimal solution based on the finite-horizon HJB equation is proposed and a data-driven optimal switching algorithm is designed. Then, for the switched systems with subsystem inputs, a data-driven optimal control approach based on the finite-horizon HJB equation is proposed. The data-driven approaches approximate the optimal solutions online by means of the system state data instead of the subsystem models. Moreover, the convergence of the two approaches is analyzed. Finally, the validity of the two approaches is demonstrated by simulation examples.     

Morphological computation: A basis for the analysis of morphology and control requirements

The fact that the morphology of a robot affects its control requirements has become increasingly evident in robotics. Not only does the morphology determine the behaviors that can be performed, but also the amount of control required for these behaviors. Particularly in systems where behavior is obtained through purely sensory-motor interactions of the body with the environment, the morphology is of prime importance. Nonetheless, even in other robotic systems, a relationship has been found to exist between morphology and control requirements, in that some morphologies yield themselves to being more easily controlled than others. This relationship was first observed and characterized by Pfeifer as the morphology and control trade-off [R. Pfeifer, C. Scheier, Understanding Intelligence, MIT Press, Cambridge, MA, 1999], but the mechanisms underlying this relationship have been unclear. However, the discovery of morphological computation,¹ [C. Paul, Investigation of Morphology and Control in Biped Locomotion, Ph.D. Thesis, Department of Computer Science, University of Zurich, Switzerland, 2004], the phenomenon that computation can be obtained through interactions of physical form, elucidates a possible mechanism underlying this relationship. The fact that simple physical interactions give rise to computation indicates the theoretical possibility for the dynamics of the morphology to play a computational role in the system, and thereby to subsume part of the role of control. Thus, it may serve to analyse the relationship between morphology and control, and guide the design of robots with reduced control requirements. The goal of the paper is to explore this possibility. The paper introduces the concept of morphological computation in the context of robot morphology, discusses its potential role in the morphology and control trade-off, and then uses it as a basis to develop a heuristic for the design of robots with reduced control requirements. The heuristic is then tested through experiments to validate its accuracy. The preliminary results are promising, and suggest that morphological computation can be a suitable framework for the analysis of morphology and control requirements.     

Constrained control of switching systems: a positive invariant approach

This paper presents conditions for the stabilization of switching discrete-time linear systems with constrained control by using a positive invariance approach. A state feedback control law is used to construct the stabilizing controller. A numerical example is presented to illustrate the technique.     

Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

In model-based nonlinear optimal control switching decisions that can be optimized often play an important role. Prominent examples of such hybrid systems are gear switches for transport vehicles or on/off valves in chemical engineering. Optimization algorithms need to take the discrete nature of the variables that model these switching decisions into account. Unnecessarily, for many applications still an equidistant time discretization and either rounding or standard mixed-integer solvers are used. In this article we survey recent progress in theoretical bounds, reformulations, and algorithms for this problem class and show how process control can benefit from them. We propose a comprehensive algorithm based on the solution of a sequence of purely continuous problems and simulations, and provide a new and more compact proof for its well-posedness. Instead of focusing on a particular application, we classify different solution behaviors in the applications section. We provide references to respective case studies with prototype character and cite newly emerging benchmark libraries. We conclude by pointing out future challenges for process control with switching decisions.     

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.     

On centralized optimal control

It can be argued that the Holy Grail of control theory is the determination of the optimal feedback control law or simply the feedback control law. This is understandable given the huge success of the linear quadratic Gaussian (LQG) theory and applications for the past half-century. It is not an exaggeration to say that the entire aerospace industry, from the Apollo moon landing to the latest global positioning system (GPS), owe a debt to this control-theoretic development in the late 1950s and early 1960s. As a result, the curse of dimensionality notwithstanding, finding the optimal control law for more general dynamic systems remains an idealized goal for all problem solvers. We continue to hope that with each advance in computer hardware and mathematical theory, we will move one step closer to this ultimate goal. Efforts such as feedback linearization and multimode adaptive control can be viewed as such successful attempts. It is the thesis of this note to argue that this idealized goal of control theory is somewhat misplaced. We have been seduced by our early successes with the LQG theory and its extensions. The simple but often not emphasized fact is this: It is extremely difficult to specify and impossible to implement a general multivariable function even if the function is known.     

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.     

On the orthogonal collocation and mathematical programming approach for state constrained optimal control problems

The orthogonal collocation approach is now well known to solve, effectively, the state constrained optimal control problems. Mathematical programming technique was also used as an effective tool to construct the optimal trajectories. In this paper, a study is done on the efficiency and accuracy requirements of the combined orthogonal collocation and mathematical programming approach, as regarding the employed optimization algorithm, and the number of orthogonal collocation points. It is shown, by experimentation with numerical examples that Fletcher-Powell optimization algorithm is much more faster to produce convergence than Fletcher-Reeves algorithm. The efficiency can be a ratio of six-to-one. The results are compared with an alternative approach to solve the same problem. It is shown that the present algorithm is less costly than the alternative approach, although requiring more computation time. The choice is then a compromise one. As the number of orthogonal points increases, the resulting solutions are more accurate, but the convergence speed decreases. Experimentation with N, shows a save of five-to-one in computing time can be achieved with almost the same cost function. Finally, it is shown, by a numerical example, that uniformly distributed collocation points result in non-optimal solutions, which also violate the problem constraints. It is a numerical proof of the superiority of the orthogonal collocation approach.     

On the comparison of optimal control systems

This correspondence presents conditions which permit the comparison of the optimal performance of control systems which have the same criterion but different dynamics. These conditions are applied to linear quadratic problems and to establishing performance bounds for suboptimal systems.     

On the inverse problem of optimal control

If a state feedback control taw u=–Kx is applied to an asymptotically stable system x = Ax+bu, if the absolute value of the return difference is greater than one for every s=jω and if the pair (K, A) is observable, then to every quadratic performance index that admits K as a solution of the optimal control problem, the uniqueness of this solution is proved.     

On the optimal control of nonlinear systems

The Lie algebra of tensors on a Hilbert space is used to obtain optimal controls for a class of nonlinear systems.