Elsevier

Systems & Control Letters

Volume 53, Issue 5, December 2004, Pages 327-346
Systems & Control Letters

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

https://doi.org/10.1016/j.sysconle.2004.05.008Get rights and content

Abstract

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.

Introduction

Over the past decades, the nonlinear optimal control problem for affine control systems has received a great deal of attention in the literature. Banks and Mhana [7] provided a computationally simple and efficient nonlinear design method by extending the principles of linear-quadratic regulator (LQR) theory to control-affine nonlinear systems of the formẋ=f(x)+B(x)u=A(x)x+B(x)u,where xRn is the state, uRm is the control input and f(0)=0. Locally asymptotically stabilizing, near-optimal, nonlinear feedback controllers were designed at fixed points x=x̄ by applying the standard infinite-time horizon LQR control pointwise along the trajectory. In the literature, many authors have considered this approximation to nonlinear optimal control based on solving a Riccati equation at each point x̄ (see, for instance, [16], [18], [19], [21]), and the algorithm is often referred to as the “state-dependent Riccati equation” or SDRE feedback control.

In a recent paper [6], Banks and McCaffrey proposed a universal theory which gives general results on solutions of nonlinear differential equations. The theory introduces linear, time-varying (LTV) approximations which are arbitrarily close to the true system. This approximation theory has been applied in [4], [5], [12] to solve the finite-time horizon control-affine nonlinear optimal control problem. The proposed algorithm uses the globally converged solution of an “approximating sequence of Riccati equations” (ASRE) to explicitly construct time-varying feedback controllers for the original control-affine nonlinear problem (1). The ASRE feedback algorithm for nonlinear optimal control provides outstanding performance in many practical applications, in particular, nonlinear solitary wave motion [4], the inverted pendulum system [11] (which is stabilized from any initial state including its unstabilizable points, unlike other methods), and optimal maneuvering of super-tankers at high speeds [12]. However, optimality has not been proved.

Many existing algorithms for nonlinear optimal control, including the ones mentioned above, only handle nonlinear systems having affine control inputs (linear in the manipulated variable u), that is, systems of form (1). However, many applications of practical importance have the nonlinear structureẋ=f(x,u),which is nonaffine in the control input (nonlinear in u). Consider, for example, the initial disturbances in angle of attack of an F-8 in a level trim, unaccelerated flight at Mach=0.85 and an altitude of 30,000ft (9000m), for which the nonlinear equations of motion representing the dynamics of the aircraft become [15]ẋ1=−0.877x1+x3−0.088x1x3+0.47x12−0.019x22−x12x3+3.846x13−0.215u+0.28x12u+0.47x1u2+0.63u3,ẋ2=x3,ẋ3=−4.208x1−0.396x3−0.47x12−3.564x13−20.967u+6.265x12u+46x1u2+61.4u3,where x1 is the angle of attack (rad), x2 is the pitch angle (rad), x3 is the pitch rate (rads−1) and u is the control input (manipulated variable) provided by the tail deflection (or elevator) angle (rad). Clearly, (3) is not of form (1) since it is not affine in u, and has the control-nonaffine nonlinear structure (2). Several other well-known processes are control-nonaffine in nature, such as the temperature effect of a reacting system, since temperature enters the model via the nonlinear Arrhenius relationship, which is used as a manipulated variable [22]. Another well-known example involves active magnetic bearing systems in industry, for which the primary problem is the strong coupling in the magnetic flux between magnetic poles. Consequently, the system is strongly nonlinear, not only in the states but also in the control inputs, resulting in a nonaffine nonlinear system (see [17]).

In principal, optimal control of the general nonlinear problem (2) can be solved by the use of Lie series and infinite-dimensional bilinear systems theory (see [2], [3], [8]). However, the solution is complex and difficult to implement. Therefore, to be able to employ existing algorithms, the nonlinear nonaffine dynamics is usually approximated as being linear in the control, and often over the entire operating range. However, by acknowledging and accounting for the presence of even slight nonlinearities, superior performance improvement can be achieved over neglected dynamics. This has already been illustrated by the authors for nonlinear functions of the state, using the inverted pendulum model in [11] and a real-world super-tanker model in [11], [12]. Therefore, in this paper, the proposed ASRE synthesis approach is extended to nonlinear systems with general structure (2) so as to handle control-nonaffine dynamics. A computationally simple and systematic synthesis approach is proposed for nonlinear nonaffine controlled systems, which easily incorporates both state and control nonlinearities under very mild conditions of local Lipschitz continuity. The ASRE is used to explicitly construct a stabilizing nonlinear time-varying state-feedback control law that solves the optimal control problem for nonaffine nonlinear systems (2) with nonquadratic performance criteria (refer to [12] for the ASRE framework of the nonlinear optimal tracking control problem). The sequences will be shown to globally converge under certain conditions in an appropriate space. The optimality of the ASRE feedback algorithm will be examined by considering the full necessary equations derived from Pontryagin's maximum principle. This involves deriving the Hamiltonian function, which provides the necessary conditions for optimality. The resulting nonquadratic optimization problem with nonlinear dynamics will then be replaced with a sequence of time-varying linear-quadratic problems, which can be solved classically. This latter method is proposed to find the global optimal feedback control of nonlinear systems, in cases where such a solution exists, and will be used as a basis for comparison with the optimality of the ASRE method.

The paper is organized as follows. Classical linear-quadratic optimal control theory is reviewed briefly in Section 2. In Section 3, the ASRE algorithm is presented for the general finite-time nonlinear optimal state-regulator control problem together with its proof of global convergence. The necessary conditions for optimality of the nonlinear optimal control problem are established in Section 4 from the maximum principle. In order to verify the effectiveness and optimality of the proposed ASRE control-design scheme presented in this paper, the control law is simulated against a realistic model of a real-world application. The nonaffine nonlinear equations (3), describing the longitudinal motion of the F-8 Crusader aircraft, are chosen for this study because of the readily available data, which has been repeatedly used in the literature by various authors in an attempt to solve the control problem (see [13], [15], [24]). So, in Section 5, the performance of the ASRE control system is evaluated by applying these to the F-8 Crusader to design stabilizing nonlinear feedback controllers. The ASRE solutions are compared against that achieved by the necessary conditions (optimal solution), as well as LQR control and the popular SDRE method. Concluding remarks are given in Section 6.

Section snippets

Linear-quadratic optimal control theory

Let us first revisit the classical fixed finite-time horizon LQR optimal control problem where a linear system with full-state information xRn and control input uRm, described by the equationẋ(t)=A(t)x(t)+B(t)u(t),x(t0)=x0is considered with the finite-time, linear-quadratic cost functionalJ(u)=12xT(tf)Fx(tf)+12t0tf{xT(t)Q(t)x(t)+uT(t)R(t)u(t)}dt,where F,QRn×n are positive-semidefinite, RRm×m is positive-definite, u is unconstrained, ARn×n, BRn×m, and the objective is to keep the state x(

ASRE feedback for optimal control of nonlinear systems

In this section, optimal control of general multi-input–multi-output, autonomous, nonlinear systems of form (2) is considered, where the control input u may not necessarily be linear-affine. Assuming the origin is an equilibrium point, that is f(0,0)=0, consider the general control-nonaffine nonlinear dynamics (2) represented in state-space in the factored formẋ(t)=A(x(t))x(t)+B(x(t),u(t))u(t),x(t0)=x0,where A:RnRn×n is a nonlinear matrix-valued function of x, and B:Rn×RmRn×m is a nonlinear

Global optimal feedback control of nonlinear systems

This section outlines some very general necessary conditions for the optimality of a control u of the nonquadratic optimization problem (12) subject to the control-nonaffine nonlinear dynamical constraint (11). In order to develop necessary conditions, which an optimal control must satisfy, it is assumed that an optimal control u exists. (Indeed, there may be many or none at all.) Therefore, in analogy to classical LQR optimal control theory, from the maximum principle, the Hamiltonian for the

Example: the F-8 Crusader

The objective of the automatic flight control system is to provide acceptable dynamics response over the entire range of angle of attack, which a modern high-performance aircraft may operate. At the specified flight condition, the F-8 stalls when the angle of attack is 0.41rad. In an attempt to solve this problem, Garrard and Jordan [15] presented an approach for computing a nonlinear control law, and derived second- and third-order nonlinear feedback controllers by solving a truncated version

Conclusions

In this paper, global optimal feedback control of a very general class of multi-input–multi-output nonlinear control systems has been considered, which do not necessarily need to be affine in the control inputs. A new method has been proposed to solve the general nonlinear finite-time horizon optimal state-regulator control problem associated with the continuous-time, deterministic, nonautonomous, nonlinear, and nonaffine controlled dynamic systemẋ(t)=f(x,u,t)=A(x,u,t)x(t)+B(x,u,t)u(t)and a

Acknowledgements

The corresponding author is grateful for financial support from the ORS Committee of Vice-Chancellors and Principles of the U.K. Universities for award number ORS/2000036029. We would also like to thank Prof. I.M.Y. Mareels and the anonymous reviewers for their helpful comments and suggestions, which have been invaluable in improving this paper.

References (24)

  • J. Wang et al.

    A nonlinear flight controller design for aircraft

    Control Engg. Practice

    (1995)
  • M. Athans et al.

    Optimal Control: An Introduction to the Theory and its Applications

    (1966)
  • Cited by (137)

    • Spacecraft formation flying control around L2 sun-earth libration point using on–off SDRE approach

      2021, Advances in Space Research
      Citation Excerpt :

      The performance of the SDRE depends on parameterization of SDC form while different choices may lead to different results. There have been some studies such as (Çimen and Banks, 2004a, 2004b; Topputo et al., 2015) which have considered different SDC forms to find the most suitable one using approximated sequence of Riccati equations (ASRE) approach. However, this method is quite different with the SDRE method used in this paper.

    View all citing articles on Scopus
    View full text