Pole assignment for uncertain systems in a specified disk by statefeedback

This paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system, the uncertainty being norm bounded. For this the “quadratic d stabilizability” concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived. This condition expressed as a parameter dependent discrete Riccati equation enables one to design the control gain matrix by solving iteratively a discrete Riccati equation     

Pole assignment for uncertain systems in a specified disk by output feedback

In this paper the problem of pole assignment in a disk by output feedback for continuous-or discrete-time uncertain systems is addressed. A necessary and sufficient condition for quadratic d stabilizability by output feedback is presented. This condition is expressed in terms of two parameter-dependent Riccati equations whose solutions satisfy two extra conditions. An output d stabilization algorithm is derived and a controller formula given.     

Pole assignment in linear time-invariant multivariable systems with constant disturbances

Necessary and sufficient conditions are derived for a minimal order realizable, linear, time invariant differential feedback control system to exist for a linear time-invariant system with constant unknown disturbances, such that the eigenvalues of the closed-loop system take on pre-assigned values in the left hand part of the complex plane and such that the outputs of the system tend to zero as t → ∞. Some numerical examples are included to illustrate the result.     

Pole assignment for linear periodic systems by memoryless outputfeedback

We consider linear periodic discrete-time systems. We are interested in the problem of placing the poles of the monodromy map by means of periodic output feedback of the same or multiple period. It is well known that, in general, the poles of time-invariant systems cannot be assigned by constant output feedback. This is in contrast with what can be obtained in the context of time-variant systems. The main contribution of this paper is that periodic output feedback suffices for pole placement of periodic systems     

Stabilization of uncertain systems via linear control

This note considers the problem of stabilizing a linear dynamical system (Σ) whose state equation includes a time-varying uncertain parameter vectorq(cdot). Given the dynamicsdot{x}(t)=A(q(t))x(t)+ B(q(t))u(t)and a bounding setQfor the valuesq(t), the objective is to choose a control lawu(t)=p(x(t))guaranteeing uniform asymptotic stability for all admissible variations ofq(cdot). Our results differ from previous work in one fundamental way; that is, we show that when working with linear controllers, it is possible to dispense with all assumptions onB(cdot)which have been made by previous authors (e.g., see [1]-[9]). This elimination of hypotheses onB(cdot)is accomplished roughly as follows: the system(Sigma) {underline {underline Delta}} (A(q), B(q))is shown to be equivalent to another system(Sigma^{+}) {underline {underline Delta}} (A^{+}(q), B^{+})as far as stabilization is concerned. SinceB^{+}is a constant matrix (independent ofq), the desired result is readily obtained.     

Pole assignment for linear time-invariant systems by periodic memoryless output feedback

It is well known that the poles of a linear time-invariant controllable and observable system can be assigned arbitrarily by state feedback. When only the output is available, pole assignment is still possible by means of dynamic output feedback. In this paper the potential of time-varying memoryless output feedback is considered. It is shown that, up to some technical conditions, it is indeed possible to allocate the poles of a linear time-invariant discrete-time system by memoryless output feedback with periodic gains. The period of the gains is (n + 1) with n the order of the system. The power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach.     

Robust fault detection of uncertain linear systems via quasi-LMIs

A novel H_∞ deconvolution filter design method is discussed for fault detection of uncertain polytopic linear systems subject to unknown inputs. The enhancement of fault sensitivity is characterized in terms of an H_- index. By means of the Projection Lemma and Congruence Transformations, a quasi-convex LMI formulation of the design problem is obtained. The effectiveness of the filter is illustrated via a numerical example.     

Robust control for linear uncertain systems via linear quadratic state feedback

By the Lyapunov stability criterion and the algebraic Riccati equation, conditions of selecting the weighting matrices in the quadratic cost function are derived so that linear quadratic state feedback can exponentially stabilize a linear uncertain system, provided the uncertainties satisfy the so-called matching conditions and within a given bounding set. Furthermore, two simple but effective algorithms are proposed for systematically selecting the weighting matrices. The main features of this approach are that the uncertain system can be exponentially stabilized with prescribed exponential rate and no precompensator is needed. Two examples are given to illustrate the results.     

Pole assignment via the schur form

We propose an algorithm for the state-feedback pole assignment problem. The algorithm is the first of its kind, making direct use of the Schur form, and minimizing the departure from normality of the closed-loop poles for a given first Schur vector x₁. The robust pole assignment problem can then be solved via choosing x₁ optimally. Several numerical examples were presented to illustrate the feasibility of the algorithm.     

Stabilization of uncertain linear systems: an LFT approach

This paper develops machinery for control of uncertain linear systems described in terms of linear fractional transformations (LFTs) on transform variables and uncertainty blocks with primary focus on stabilization and controller parameterization. This machinery directly generalizes familiar state-space techniques. The notation of Q-stability is defined as a natural type of robust stability, and output feedback stabilizability is characterized in terms of Q-stabilizability and Q-detectability which in turn are related to full information and full control problems. Computation is in terms of convex linear matrix inequalities (LMIs), the controllers have a separation structure, and the parameterization of all stabilizing controllers is characterized as an LFT on a stable, free parameter     

Pole assignment for systems over rings

A very simple proof of the pole assignment theorem for systems over a principal ideal domain (and other rings) is given. Furthermore, an algorithm is presented. Extensions are also indicated.     

Stabilization of uncertain linear systems via observer based state feedback

A unified approach to the stabilization problem using reduced order observer based state feedback is presented for uncertain discrete and continuous systems. The main result states that, for the system class specified, the insensitive observer of Breinl and Müller (1982) may be used together with any state feedback gain matrix designed for quadratic stabilization to asymptotically stabilize the uncertain system via reduced order observer based state feedback. Intermediary results give sufficient conditions for the asymptotic stability of time-varying one-way connected subsystems.     

Robust control of uncertain multi-inventory systems via linear matrix inequality

We consider a continuous time linear multi-inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of ε-stabilising the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which ε-stabilisability is possible through a saturated linear state feedback control. All the results are based on a linear matrix inequalities approach and on some recent techniques for the modelling and analysis of polytopic systems with saturations. Numerical simulations are provided.     

Mixed filtering for uncertain linear systems: A linear matrix inequality approach

The problem of full-order mixed L₂     

Control of a class of nonlinear uncertain systems via compensatedinverse dynamics approach

The tracking control problem of a class of nonlinear systems via inverse dynamics method is addressed. The motivation for the study stems from the need to control practical systems arising from aerospace and mechanical engineering. Because modeling uncertainties and external disturbances are always present in these systems, inverse dynamics technique is not applicable directly. A compensated inverse dynamics approach is proposed to account for the effect of uncertainties. The compensation is achieved by adaptive and robust schemes. Application of the proposed strategy to the vibration suppression of a two-bay flexible truss structure is presented     

Robust stabilization of linear uncertain discrete-time systems via a limited capacity communication channel

The paper considers the problem of robust stabilization of linear uncertain discrete-time systems via limited capacity communication channels. We consider the case when the control input is to be transmitted via communication channel with a bit-rate constraint. A constructive method to design a robustly stabilizing controller is proposed.     

Control via interconnection and damping assignment of linear time-invariant systems: a tutorial

Interconnection and damping assignment is a controller design methodology that regulates the behaviour of dynamical systems assigning a desired port-Hamiltonian structure to the closed-loop. A key step for the application of the method is the solution of the so-called matching equation that, in the case of nonlinear systems, is a partial differential equation. It has recently been shown that for linear systems the problem boils down to the solution of a linear matrix inequality that, moreover, is feasible if and only if the system is stabilisable – making the method universally applicable. It has also been shown that if we narrow the class of assignable structures – e.g. to mechanical instead of the larger port-Hamiltonian – the problem is still translated to a linear matrix inequality, but now stabilisability is not sufficient to ensure its feasibility. It is additionally required that the uncontrolled modes are simple and lie on the jω axis, which is consistent with the considered scenario of mechanical systems without friction. The purpose of this article is to present these important results in a tutorial, self-contained form – invoking only basic linear algebra methods.