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A geometric solution to the squaring down problem
No full textThis paper addresses the problem of the squaring down of LTI systems with the tools of the geometric control theory. More precisely, it is shown how a generic system can be turned into a square and invertible system by means of a state-feedback and an output-injection, and of two static units cascaded at the input and at the output of the given system. In this way, key system properties like phase-minimality, relative degree and infinite zero structure are preserved after the squaring down, and the additional invariant zeros introduced can be arbitrarily assigned in the complex plane
On the structure of the solution of continuous-time algebraic Riccati equations with closed-loop eigenvalues on the imaginary axis
Full text linkNew results on algorithms for the computation of output-nulling and input-containing subspaces
No full textIn this paper we present and provide a proof for a set of non-recursive formulae arising in the computation of the largest output-nulling and the smallest input-containing subspaces which have been used in a variety of contexts in the framework of the geometric approach. These expressions have been used in the literature both in the strictly proper and in the non-strictly proper case, but, to the best of our knowledge, a proof is still missing. These formulae are established here in the general possibly non-strictly proper case. Some ancillary side results of independent interest are also proposed
On the closed-form solution of the matrix Riccati differential equation for nonsign-controllable pairs
No full textIn this paper we present explicit closed form formulae for the solution of the matrix Riccati differential equation with a terminal condition. These formulae can still be employed even in the case in which the system is not sign-controllable. In such situation, the associated algebraic Riccati equation does not admit a solution in general. Therefore, the formulae presented in this paper do not directly depend on the solution of the associated algebraic Riccati equation
On the solution of the Riccati differential equation arising from the LQ optimal control problem
Full text linkIn 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
A unified approach to the finite-horizon linear quadratic optimal control problem
No full textUnder 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 H_2 preview
decoupling problem
Squaring Down LTI Systems: A Geometric Approach
Full text linkIn this paper, the problem of reducing a given LTI system into a left or right invertible one is addressed and solved with the standard tools
of the geometric control theory. First, it will be shown how an LTI system can be turned into a left invertible system, thus preserving key
system properties like stabilizability, phase minimality, right invertibility, relative degree and infinite zero structure. Moreover, the additional
invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane. By duality, the scheme
of a right inverter will be derived straightforwardly. Moreover, the squaring down problem will be addressed. In fact, when the left and right
reduction procedures are applied together, a system with an unequal number of inputs and outputs is turned into a square and invertible system.
Furthermore, as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility
of the plant in many optimization problems
A unified approach to finite-horizon generalized LQ optimal control problems for discrete-time systems
No full textA closed-form expression parameterizing the solutions of the extended symplectic difference equation over a finite time interval is given under the mild assumption of modulus-controllability. This representation is expressed in terms of the strongly unmixed solution of a discrete ARE and of an algebraic Stein equation.
The most important application of this result is a generalized version of the finite-horizon LQ regulator: In particular our framework enables different kind of boundary conditions to be treated in a unified fashion, without resorting to the Riccati difference equation for the computation of the optimal control function
Corrigendum to "H2 optimal rejection with preview in the continuous-time domain" [Automatica 41 (5) (2005) 815-821]
No full textEmploying the algebraic Riccati equation for the solution of the finite-horizon LQ problem
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