Abstract
The use of composite linear and non-linear feedback laws for the control of constrained input discrete-time linear systems is re-examined. By making use of the delta operator formulation of a discrete-time system, an apparent restriction on the magnitude of the non-linear control law is removed, and the similarities between the continuous and discrete-time solutions to the problem are elucidated. In order to develop the results, unconstrained systems are treated initially, but it is shown that, locally, the sufficient conditions for the stabilization of such systems are actually equivalent to those for the stabilization of the corresponding constrained systems.
Highlights
Several recent papers have discussed the use of novel and/or nonlinear feedback control to improve performance in systems which are, apart from a saturation element at their input, otherwise linear
We would like to give conditions under which the stronger form of stability, namely exponential stability, can be proved for δ -operator systems. These are well-known for continuous time systems; here we present a lemma which proves the same conditions are valid for δ -operator discrete time systems
The first source of conservatism is that these ellipsoids, which are described in Theorems 3 and 5, are calculated such that the nominal ‘low gain’ linear control does not saturate (The oversaturation of De Dona et al (2000) is an alternative to this)
Summary
Several recent papers have discussed the use of novel and/or nonlinear feedback control to improve performance in systems which are, apart from a saturation element at their input, otherwise linear. One of the first of these was a piecewise linear LQ control law proposed by Wredenhagen & Belanger (1994), which used a piecewise linear state-feedback to increase the feedback gain as the system state approached the origin This was extended by De Dona et al (2000) who allowed the control law to saturate to reduce conservatism in the design. More popular, technique is that of anti-windup, where a compensator designed to counteract the phenomena of saturation becomes active when the control signal starts to saturate In addition to these methods is the low-and-high gain technique, which built upon the work of the low-gain technique (Lin & Saberi (1993)) to improve performance in saturated systems, without compromising stability. By the end of the 1990’s the low-and-high gain technique, which was introduced in Lin & Saberi (1995) for the control of an n’th order integrator chain, was well established as a technique for controlling systems subject to
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