Techniques for stabilization of linear descriptor systems by state-derivative feedback are proposed. The methods are based on Linear Matrix Inequalities (LMIs) and assume that the plant is a controllable system with poles different from zero. They can include design constraints such as: decay rate, bounds on output peak and bounds on the state-derivative feedback matrix

The Linear Matrix Inequalities (LMIs) formulation has emerged recently as a useful tool for solving a great number of practical control problems [

Recently, LMI has been used for the study of descriptor systems [

There exist few researches using only derivative feedback (

Structural failures appear naturally in systems, for physical wear of the equipment, or for short circuit of electronic components. Recent researches on structural failures (or faults), have been presented in LMI framework [

In this paper, we will show that it is possible to extend the presented results in [

Consider a controllable linear descriptor system described by

Find a constant matrix

the closed-loop system (

In [

Unfortunately, there exist several practical problems that not satisfy (

Assuming that

Necessary and sufficient conditions for asymptotic stability of standard linear system (

Assuming that (

Observe that for any nonsymmetric matrix

System (

Note that from (

Equations (

Usually, only the stability of the control systems is insufficient to obtain a suitable performance. In the design of control systems, the specification of the decay rate can also be very useful.

Consider, for instance, the controlled system (

Assuming that (

Stability corresponds to positive decay rate,

The next section shows that it is possible to extend the presented results, for the case where there exist polytopic uncertainties or structural failures in the plant. A fault-tolerant design is proposed.

In this work, structural failure is defined as a permanent interruption of the system's ability to perform a required function under specified operating conditions [

Consider the linear time-invariant uncertain polytopic descriptor system, with or without structural failures, described as convex combinations of the polytope vertices:

A sufficient condition for the solution of Problem

From (

A sufficient condition for the decay rate of the robust closed-loop system given by (

It follows directly from the proofs of Theorems

Due to limitations imposed in the practical applications of control systems, many times it should be considered output constraints in the design.

Consider that the output of the system (

An interesting method for specification of bounds on the state-derivative feedback matrix

Given a constant

See [

In the following section, Example

The effectiveness of the proposed LMI designs is demonstrated by simulation results.

A simple electrical circuit, can be represented by the linear descriptor system below [

Suppose the output of the system is given by

Note that, as discussed before, the obtained solution

For the initial condition

Note that the absolute values of the entries of

This procedure can also be applied to the control design of uncertain systems subject to failures.

Stability | Stability with decay rate ( |
---|---|

The response of the signal

The response of the signal

Consider the linear uncertain descriptor system represented by matrices:

A fail in the actuator is described by:

And the example was solved considering stability with decay rate. It was specified a lower bound for the decay rate equal to

The eigenvalues location of the vertices from robust controlled uncertain system (

The response of the signal

Necessary and sufficient stability conditions based on LMI for state-derivative feedback of linear descriptor systems, were proposed. We can include in the LMI-based control design, the specification of the decay rate, bounds on output peak, and bound on the state-derivative feedback matrix

The authors gratefully acknowledge the financial support by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) from Brazil.