A Unified Approach to the Finite-Horizon Linear Quadratic Optimal Control Problem*

doi:10.3166/ejc.13.473-488

European Journal of Control

Volume 13, Issue 5, 2007, Pages 473-488

https://doi.org/10.3166/ejc.13.473-488 Get rights and content

Under the mild assumption of sign-controllability, a closed-form expression parameterizing all the solutions of the Hamiltonian differential equation over a finite time interval is presented in terms of a strongly unmixed solution of an algebraic Riccati equation (ARE) and of the solution of an algebraic Lyapunov equation. This result is employed for the solution of a generalized version of the finite-horizon linear quadratic (LQ) problem, encompassing the case of fixed end-point. Furthermore, it is shown how this method can be applied to the H₂ preview decoupling problem.

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Cited by (14)

Free finite horizon LQR: A bilevel perspective and its application to model predictive control
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Citation Excerpt :
Hence, we restrict ourselves to this case, in which the initial state of the system is given and an affine constraint on the state might be imposed at the final time as in (2). However, the analysis can be easily extended to deal with the free final time version of more general finite-horizon LQR problems, e.g. Ferrante and Ntogramatzidis (2007). Problem 1 can be casted into an equivalent bilevel optimization problem, whose upper level reads:
We consider linear-quadratic optimal control problems with free final time and terminal state constraints and propose a solution procedure that is particularly useful for online feedback control in a model-predictive control (MPC) framework. The procedure avoids the standard time transformation, which transforms the problem into an equivalent but non-convex optimal control problem on a fixed time horizon. The transformed problem typically suffers from many local minima, which might cause instabilities in online optimization tasks like LQR or model-predictive control. To avoid this drawback of the time transformation we develop a method from the viewpoint of bilevel optimal control, which is beneficial especially in online control tasks. The novelty of the approach is the optimal final time tracking procedure, for which we exploit a property of the Hamiltonian function. To this end we show that within the continuous-time closed-loop controlled system the optimal final time linearly decreases in time, as one could intuitively expect. Finally, numerical experiments support the effectiveness of the proposed algorithm.
On the exact solution of the matrix Riccati differential equation
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Citation Excerpt :
Moreover, sign-controllability is also the weakest known assumption for which the associated algebraic Riccati equation is guaranteed to admit a symmetric solution. Under this assumption, a formula parameterising in finite terms all the trajectories originating from the Hamiltonian differential equation, introduced by the same authors in [18], is exploited to derive a non-recursive formula for the solution of the Riccati differential equation. This parameterisation of the trajectories of the Hamiltonian differential equation generalises those proposed in [19,20] for controllable and stabilisable systems, respectively; see also [21,22] for the discrete-time counterpart.
In this paper we consider the matrix Riccati differential equation (RDE) that arises from linear-quadratic (LQ) optimal control problems. In particular, we establish explicit closed formulae for the solution of the RDE with a terminal condition using particular solutions of the associated algebraic Riccati equation. We discuss how these formulae change as assumptions are progressively weakened. An application to LQ optimal control is briefly analysed.
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^*: Partially supported by the Ministry of Higher Education of Italy (MIUR) under the project Identification and Control of Industrial Systems any by the Australian Research Council (DP0664789).

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A Unified Approach to the Finite-Horizon Linear Quadratic Optimal Control Problem**

Syst Control Lett

Automatica

Automatica

Automatica

Linear Algebr Appl

Matrix Riccati equations in control and systems theory

Optimal control: an introduction to the theory and its applications

Optimal control theory

On discrete-time H∞ fixed-lag smoothing

IEEE Trans Signal Process

Guaranteed H∞ bounds for Wiener filtering and prediction

Int J Robust Nonlinear Control

The Riccati equation solution of the linear quadratic problem with constrained terminal state

IEEE Trans Autom Control

Convergence of the time-invariant Riccati differential equation and LQ-problem: mechanisms of attraction

Int J Control

A J-spectral factorization approach for H∞ estimation problems in discrete-time

IEEE Trans Autom Control

Algebraic Riccati equation and Jspectral factorization for H∞ smoothing and deconvolution

SIAM J Control and Optim

A Unified Approach to the Finite-Horizon Linear Quadratic Optimal Control Problem^**

On discrete-time H_∞ fixed-lag smoothing

Guaranteed H_∞ bounds for Wiener filtering and prediction

A J-spectral factorization approach for H_∞ estimation problems in discrete-time

Algebraic Riccati equation and Jspectral factorization for H_∞ smoothing and deconvolution