Lorenzo Ntogramatzidis

A geometric theory for 2-D systems including notions of stabilisability

In this paper we consider the problem of internally and externally stabilising controlled invariant and output-nulling subspaces for two-dimensional (2-D) Fornasini–Marchesini models, via static feedback. A numerically tractable procedure for computing a stabilising feedback matrix is developed via linear matrix inequality techniques. This is subsequently applied to solve, for the first time, various 2-D disturbance decoupling problems subject to a closed-loop stability constraint.

Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case

In this paper, a new methodology is developed for the closed-form solution of a generalized version of the finite-horizon linear-quadratic regulator problem for LTI discrete-time systems. The problem considered herein encompasses the classical version of the LQ problem with assigned initial state and weighted terminal state, as well as the so-called fixed-end point version, in which both the initial and the terminal states are sharply assigned. The present approach is based on a parametrization of all the solutions of the extended symplectic system. In this way, closed-form expressions for the optimal state trajectory and control law may be determined in terms of the boundary conditions. By taking advantage of standard software routines for the solution of the algebraic Riccati and Stein equations, our results lead to a simple and computationally attractive approach for the solution of the considered optimal control problem without the need of iterating the Riccati difference equation.

Brief paper: A new approach to the cheap LQ regulator exploiting the geometric properties of the Hamiltonian system

The cheap LQ regulator is reinterpreted as an output nulling problem which is a basic problem of the geometric control theory. In fact, solving the LQ regulator problem is equivalent to keep the output of the related Hamiltonian system identically zero. The solution lies on a controlled invariant subspace whose dimension is characterized in terms of the minimal conditioned invariant of the original system, and the optimal feedback gain is computed as the friend matrix of the resolving subspace. This study yields a new computational framework for the cheap LQ regulator, relying only on the very basic and simple tools of the geometric approach, namely the algorithms for controlled and conditioned invariant subspaces and invariant zeros.

A unified approach to the finite-horizon linear quadratic optimal control problem

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 H 2 preview decoupling problem.

Self-Bounded Subspaces for Nonstrictly Proper Systems and Their Application to the Disturbance Decoupling With Direct Feedthrough Matrices

In this paper, the concept of self-bounded controlled invariance is extended to nonstrictly proper systems. Moreover, its use in connection with the disturbance decoupling problem with internal stability is investigated in the case where the feedthrough matrices from the control input and from the disturbance to the output to be decoupled are possibly nonzero.

A parametrization of the solutions of the Hamiltonian system for stabilizable pairs

In this note, a simple method is proposed for the solution of the finite-horizon LQ problem, which does not require the integration of the Riccati differential equation. Precisely, the problem is tackled by parametrizing the set of trajectories solving the Hamiltonian system in finite terms, under the assumption of stabilizability of the underlying system. In this way, it is possible to determine closed-form expressions for the state and control functions, as well as for the optimal cost in terms of the assigned state at the end-points.

Employing the algebraic Riccati equation for the solution of the finite-horizon LQ problem

A new approach to the study of the finite-horizon linear quadratic regulator problem is presented for linear, time-invariant controllable systems. This simple and computationally attractive procedure is based on the solutions of an algebraic Riccati equation and of a Lyapunov equation, that enable all the state-costate functions that satisfy the Hamiltonian system to be parametrized in a closed form. In this way, it is possible to easily solve in a unified framework optimal control problems in which either both the initial and the terminal states are assigned, or one of them is weighted in the performance index. In this way the optimal state trajectory and input control law are efficiently obtained without resorting to the integration of a Riccati differential equation. The optimal value of the cost is also explicitly parametrized.

On the Partial Realization of Noncausal 2-D Linear Systems

The problem of partial realization is to construct a latent variable model which matches a specified input-output behavior over a bounded frame of interest. In this paper, an algorithm is proposed for constructing a partial realization from the Toeplitz kernel of a possibly noncausal 2D linear system. By construction, the resulting latent variable model and corresponding boundary conditions comprise four components, each with recursively computable structure.

Squaring down LTI systems: A geometric approach

In 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.

Discussion on : "Measurable signal decoupling with dynamic feedforward compensation and unknown-input observation for systems with direct feedthrough"

Several feedforward decoupling and estimation problems are treated here in a unified setting, and their exact geometric solution is extended to the general case where the direct feedthrough matrices of all the systems involved are possibly non-zero. To this end, the concepts of selfboundedness and self-hiddenness are generalised and investigated within the general context of non-strictly proper systems. Then, for each problem considered, solvability conditions are provided as well as the explicit structure of the solving compensator or observer.