Window observers for linear systems

• Main Authors: Utkin Vadim, Li shengming
• Format: Article
• Language: English
• Published: Hindawi Limited 2000-01-01
• Series: Mathematical Problems in Engineering
• Subjects: Observer; Stability; Windowing function
• Online Access: http://www.hindawi.net/access/get.aspx?journal=mpe&volume=6&pii=S1024123X0000140X
• ISSN: 1024-123X; 1563-5147

<p>Given a linear system <math alttext="$dot x = Ax + Bu$"> <mrow> <mover accent="true"> <mi>x</mi> <mo>&dot;</mo> </mover> <mo>=</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>u</mi> </mrow> </math> with output <math alttext="$y = Cx$"> <mrow> <mi>y</mi> <mo>=</mo> <mi>C</mi> <mi>x</mi> </mrow> </math> and a window function <math alttext="$omega left( t ight)$"> <mrow> <mi>&omega;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </math>, <emph>i.e</emph>., <math alttext="$forall t,omega left( t ight) in $"> <mrow> <mo>&forall;</mo> <mi>t</mi> <mo>,</mo> <mi>&omega;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&isin;</mo> </mrow> </math> {0,1 }, and assuming that the window function is Lebesgue measurable, we refer to the following observer, <math alttext="$hat x = Ax + Bu + omega left( t ight)LC(x - hat x)$"> <mrow> <mover accent="true"> <mi>x</mi> <mo>&circ;</mo> </mover> <mo>=</mo> <mi>A</mi> <mi>x</mi> <mo>+</mo> <mi>B</mi> <mi>u</mi> <mo>+</mo> <mi>&omega;</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mi>L</mi> <mi>C</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>&minus;</mo> <mover accent="true"> <mi>x</mi> <mo>&circ;</mo> </mover> <mo stretchy="false">)</mo> </mrow> </math> as a window observer. The stability issue is treated in this paper. It is proven that for linear time-invariant systems, the window observer can be stabilized by an appropriate design under a very mild condition on the window functions, albeit for linear time-varying system, some regularity of the window functions is required to achieve observer designs with the asymptotic stability. The corresponding design methods are developed. An example is included to illustrate the possible applications</p>