Inverse Optimal Control Problem Research Articles

Stabilizing controllers, Lyapunov functions, and the inverse problem of optimal control

Abstract We explore the connections of Margreta Kuijper's parametrization of stabilizing controllers, with some issues arising in inverse optimal control and in Lyapunov stability theory for higher-order linear time-invariant differential systems.

Forme standard de passage. Une alternative à la synthèse multi-objectif

This chapter presents a generalization of the Cross Standard Form (CSF) for a given plant and a given controller of arbitrary order (even, a two-degree of freedom compensator). The CSF is a canonical augmented standard plant whose H∞ or H2 unique optimal controller is a given controller, that is a solution to the general inverse optimal control problem. From the control design point of view, the general idea is to apply the CSF to a first controller in order to initialize an H∞ design procedure to handle complementary frequency-domain constraints or robustness specifications. An academic example and an industrial application in the field of aeronautics engineering are given.

Cross standard form: a solution to improve a given controller with H 2 or H ∞ specifications

This paper introduces in cross standard form (CSF) as a solution to the inverse optimal control problem. That is, the CSF is a canonical standard problem whose unique H ∞ or H 2 optimal controller is a given controller. From the control design point of view, the general idea is to apply the CSF to a given controller in order to set up a standard problem which can be completed to handle frequency domain H 2 or H ∞ specification. The analytical formulation of the CSF proposed in this paper can be applied to reduced-, full- or augmented-order compensators or two-degree of freedom compensations. Numerical and academic examples are given.

IMPROVING A GIVEN CONTROLLER WITH H2 OR Hx SPECIFICATIONS

In this paper, the Cross Standard Form (CSF) is introduced as a solution to the inverse optimal control problem. That is, the CSF is a canonical augmented standard plant whose unique H∞ or H2 optimal controller is a given controller. From the control design point of view, the general idea is to apply the CSF to a given controller in order to set up a standard problem which can be completed to handle frequency-domain H2 or H∞ specifications. The analytical formulation of the CSF proposed in this paper can be applied to reduced-, full- or augmented-order compensators. An academic example is given.

A New Approach to Multi-Objective Control Design from the Viewpoint of the Inverse Optimal Control Problem

This paper introduces the Cross Standard Form (CSF) as a solution to the generalized inverse optimal control problem. The CSF is a canonical augmented standard plant whose H∞ or H2 optimal controller is a given controller. The definition of the CSF relies on the possibility to construct minimal observer-based realization of a given controller for a given model. From the multi-objective or robust control point of view, the general idea is to apply the CSF to a first controller satisfying nominal performances in order to initialize an H∞ design procedure to handle frequency-domain constraints or robustness specifications.

A minimal dissipation type-based classification in irreversible thermodynamics and microeconomics

We formulate the problem of finding classes of kinetic dependencies in irreversible thermodynamic and microeconomic systems for which minimal dissipation processes belong to the same type. We show that this problem is an inverse optimal control problem and solve it. The commonality of this problem in irreversible thermodynamics and microeconomics is emphasized.

Quasi-Newton-Type Continuation Method for Nonlinear Receding Horizon Control

A fast computational method of nonlinear receding horizon control is proposed, based on the continuation method combined with the quasi-Newton method. Jacobians in the differential equation of the unknown costate are replaced by their estimates, and the quasi-Newton method is employed to determine time derivatives of those estimates. Several modieed algorithms are obtained by imposing different conditions. The proposed method leads to considerable reduction of computational load compared to the conventional algorithms. Numerical examples of a two-wheeled car and a tethered satellite demonstrate computational time, accuracy, and robustness against noises of the proposed method. controlLyapunovfunctionstostabilizeaclassofnonlinearsystems. A controller designed by backstepping can also be interpreted asan optimal controller in terms of the inverse optimal control problem. 2 However, backstepping does not give a synthesis method to min- imize a performance index specie ed by a designer. Therefore, di- rect optimization methods for a given performance index should be explored. In the direct optimization of a controller, one has to solve the Hamilton- Jacobi-Bellman equation (HJBE). There have been nu- merous efforts to solve the HJBE. Because the HJBE is a partial differential equation for the optimal cost function, its solution has to be given over a some region in the state space. The structure of the solution is usually assumed for the HJBE, for example, in the form of power series, 3 interpolation, 4 or expansion with basis functions. 5 There is also a possibility of an approximate solution technique that does not assume the structure of the solution in the framework of genetic programming. 6 However, those methods can- not avoid the explosive growth of data storage or number of terms for high-dimensional systems and are dife cult to implement for a system whose dimension is higher than two or three. Ontheotherhand, iftheoptimalcontrolinputiscalculatedonline onlyfor thecurrentstate,ahugeamountofdata storageisnotneces- sary, even for a high-dimensional system. In this case, anopen-loop optimalcontrolproblem,whichleadstoatwo-pointboundary-value problem (TPBVP), is solved in place of the HJBE. Receding hori- zon control (model predictive control) is one of such control tech- niques. In receding horizon control, a e nite horizon performance index with a moving initial time and a moving terminal time ismin- imized. Receding horizon control is attractive from the theoretical point of view because it guarantees closed-loop stability, ifthe state isconstrained tobe zero at the end of the horizon, 7;8 or ifother con- ditions are satise ed. 9i13 Those characteristics concerning stability give guidelines for problem setting and selection of a performance index.

An Inverse Optimal Control Problem for a Class of Controlled Systems

A class of optimal control problems having an optimal solution of a specified property is investigated. Examples from irreversible thermodynamics and microeconomics are given. Such a formulation, as has been shown, can be regarded as an inverse optimal control problem.

What system is optimal for given control law and criterion ?

Abstract This paper presents Optimal System Problem, that is what system is optimal when both control law and control criterion are given. Unlike optimal control and the inverse optimal control which have been well researched, Optimal System Problem has not been considered untill now. In this paper, only the case that the criterion is quadratic and the input is single is considered. A necessary and sufficient condition of existence of optimal system which makes closed-loop system stable by given control feedback control law is presented. Also a computation method to get an existing optimal system is discussed. At the last part of this paper, an example is presented to compare optimal control, inverse optimal control and optimal system problem.

Partial pole placement by LQ regulators: an inverse problem approach

This paper gives a necessary and sufficient condition under which a state feedback control law places part of the closed-loop poles exactly at specified points and, at the same time, is linear quadratic optimal for some quadratic weightings. This is made possible by means of a solution to the inverse problem of optimal control. A design example is given to illustrate the result.

Optimality and robustness of linear quadratic control for nonlinear systems

The paper is concerned with an inverse optimal control problem, which is to find the conditions for optimality of a nonlinear time-varying plant driven by a Linear Quadratic (LQ) control. The control law is obtained by the standard quadratic optimization of the linear part of the plant. Once optimality of the nonlinear closed-loop system is established, the robustness of stability with respect to bounded variation in open-loop dynamics is assured in terms of the gain and phase margins. Of particular interest are the extensions of the optimality and robustness results of unconstrained feedback to LQ control of large systems with decentralized information structure constraints. In particular, we will show that optimality of decentralized LQ control guarantees gain and phase margins at each control input channel, as well as tolerance to nonlinear time-varying distortions of the feedback control laws.

The relationship between the stability and the optimality of linear systems (II)

Different from the inverse problem put forward by R.E.Kalman, another kind of inverse problem of linear optimal control is proposed and discussed in [1] as follows: Given an asymptotically stable linear constant system and a nonnegative quadratic performance index, when can a state-feedback be separated from the stable system so that this state-feedback control law is optimal for the given index? In this paper this problem is extended. Similar conclusions are obtained for linear discrete systems and linear time-variable systems. According to these conclusions we can say that the correspondence between the asymptotically stable system and the optimal feedback system is the inherent character of all kinds of linear systems.

A new solution to the inverse problem of optimal regulator control

AbstractIn chemical process control, the application of an optimal control strategy, dictated by the theory of the linear quadratic problem, does not always guarantee the best process performance. The main problem is that the unspecified weighting matrices in the quadratic performance index have a pronounced effect on the optimal control law derived from linear quadratic theory. In the past, values of the weighting matrices have been selected by experimenting with the process in much the same manner as a PID controller is tuned. The objective of the present study is to resolve the inverse problem of optimal regulator control, limiting the performance index in the quadratic form with two independent weighting matrices which are investigated. A new criterion has been developed such that the computation of determining those matrices can be carried out off‐line in advance, and then stored in a computer for real‐time use.

Optimal pole allocation and weighting matrix selection

Abstract The Levy-Hadamard's theorem and Bendixson's theorem are applied to derived algebraic relations which will set bounds on the real and imaginary parts of the eigenvalues of the closed-loop matrix. It is shown that the required feedback matrix can be found very simply. Further, the inverse problem of optimal control, which characterizes the cost function based on the given feedback matrix, is considered. Gutman and Jury's (1981) result is used to derive some algebraic relations which cluster the feedback system poles inside a second-order curve in the complex frequency s-plane. It is also shown that once these relations are obtained, a proposed algorithm can be performed readily to determine the quadratic performance index of linear optimal control problems.

The inverse problem of linear optimal control for constant disturbance

This paper presents the solution to the inverse problem of linear optimal control systems to achieve pole placement and at the same time minimize the cost function and accommodate a constant but unknown disturbance. The sufficient conditions for an optimal feedback law to have a positive R and positive semi-definite Q are found. The relationships between the open-loop poles and the closed-loop poles that guarantee a positive R and Q ≥ 0 are also obtained. These relations offer a rule for selection of the appropriate closed-loop poles which will guarantee the required feedback properties.

Trajectory Properties and Cost Invariance in Discrete Linear Quadratic Problems

A linear discrete time-invariant system with a quadratic cost is considered and conditions are found for existence of non - optimal control laws which generate the optimal input and state trajectories. The problem of invariance of the quadratic cost on a set of non-optimal control laws is studied and its connection with the inverse optimal control problem is discussed.

A Complete Optimality Condition in the Inverse Problem of Optimal Control

A complete optimality condition in the standard inverse problem of optimal control for the multi-input case is established. The optimality of a feedback control law is characterized completely in terms of a new geometric condition as well as the well-known return difference condition. The new condition not only provides a better insight into the well-known sensitivity reduction property of optimal control for the multi-input case, but also indicates an essential difference between the solutions of the single- and multi-input inverse problem.

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 receding horizon feedback control

Receding horizon feedback control (RHFC) was originally introduced as an easy method for designing stable state-feedback controllers for linear systems. Here those results are generalized to the control of nonlinear autonomous systems, and we develop a performance index which is minimized by the RHFC (inverse optimal control problem). Previous results for linear systems have shown that desirable nonlinear controllers can be developed by making the RHFC horizon distance a function of the state. That functional dependence was implicit and difficult to implement on-line. Here we develop similar controllers for which the horizon distance is an easily computed explicit function of the state.

Robust inverse optimal control for flexible structures

An inverse optimal control problem is formulated to develop robust control laws for purely oscillatory systems. The optimal control solution requires output feedback with specified constraints, leading to robustness with respect to unmodeled modes and a large class of parameter variations. The robustness properties are proved directly from known properties of control laws resulting from quadratic performance indices. The control laws are useful for poorly damped flexible structures.