Observer synthesis for switched linear systems results in switched observers with state jumps.
![{\displaystyle {\begin{aligned}{\dot {x}}(t)&=A_{\sigma (t)}x(t)+Bu(t),\\y(t)&=Cx(t)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd9920c89d930859c87bd3249d43eec6fcf2978)
where
,
,
and
is the index function in discrete state given by
deciding which one of the linear vector fields is active at a certain time instant.
- The matrices
are system matrices of appropriate dimensions and are known.
- The unknown variables of the observer synthesis LMI are
and
.
Given a State-space representation of a system given as above. The dynamics of the continuous time observer is defined as:
![{\displaystyle {\begin{aligned}{\dot {\hat {x}}}&=A_{{\hat {\sigma }}(t){\hat {x}}}+Bu+K_{\hat {\sigma (t)}}(y-{\hat {y}})\\{\hat {y}}&=C{\hat {x}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9473a66e629fd03ee72d7006a48cbb30f5910e79)
where
is the state estimate of the vector field
,
is the observer gains,
is the index function, and
is the output of the mode location observer.
The observer is divided into two parts, the mode location observer estimating the active dynamics and the continuous-time observer estimating the continuous state of the switched system.
The estimated state jumps will be updated according to
![{\displaystyle {\hat {x}}^{+}=T_{1}x(t)+T_{2}y(t)\quad t\in \mathbb {T} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e79bf9762cfc3e66bb96341df92e8ff0cd741a)
where
is the set of times when the mode location observer switches mode, which are the times when
changes value.
The following are equivalent:
(a)There exists
and
such that
![{\displaystyle \alpha I\leq P_{i}\leq \beta I,\qquad i\in I_{N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77fc2a43fa5a0ea6a17c18337a27592e809d7b2d)
![{\displaystyle \Gamma _{i,j}={\begin{bmatrix}\Gamma _{i,j}^{11}&\Gamma _{i,j}^{12}\\(\Gamma _{i,j}^{12})^{T}&\Gamma _{ij}^{22}\\\end{bmatrix}}\leq 0,\qquad (i,j)\in I_{s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8026eca8609b077a3eed7925f6d702e6414f66d1)
![{\displaystyle P_{j}=P_{i}+d_{i,j}^{T}C+C^{T}d_{i,j},\qquad (i,j)\in I_{s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/521e3c577daa576f33d78dd1e02e6635ff66e7f2)
![{\displaystyle {\begin{bmatrix}\lambda _{i}^{2}I_{p\times p}&W_{i}^{T}\\W_{i}&I_{n\times n}\\\end{bmatrix}}\geq 0,\qquad i\in I_{N}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeea0d813267866d81562b87314aae32be86f9d3)
where
![{\displaystyle {\begin{aligned}\Gamma _{i,j}^{11}&=(A_{i}-K_{i}C)^{T}P_{i}+P_{i}(A_{i}-K_{i}C)+\gamma I\\\Gamma _{i,j}^{12}&=P_{i}(A_{j}-A_{i})\\\Gamma _{i,j}^{22}&=\mu _{i,j}Q_{j}-\gamma \epsilon ^{2}I\\\epsilon \geq 0,\alpha >0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62e2115277cefa920815be1119eb647133c76aa4)
and the states of the hybrid observer is updated according to
![{\displaystyle {\hat {x}}^{+}=(I-R_{i}^{-1}(CR_{i}^{-1})^{\dagger }C){\hat {x}}+R_{i}^{-1}(CR_{i}^{-1})^{\dagger }y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dee17b398c8fab0200156c9eb9ce2a3d795a9d6d)
(b) If for some
![{\displaystyle \sup _{t>T_{0}}\|x(t)\|\leq x_{m}ax}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46640ee3b777b23cbfaa1d5ad2860125ca9c1ef5)
then
![{\displaystyle \lim _{t\rightarrow \infty }\sup \|{\tilde {x}}\|\leq \epsilon x_{m}ax{\sqrt {\frac {\beta }{\alpha }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b1ba16d6e0aa28c1917d905186946d23831184)
Using multiple Lyapunov functions and properly updating the continuous estimated states when the mode changes occur, an observer can be synthesized by solving a linear matrix inequality problem above.
A list of references documenting and validating the LMI.
- S. Pettersson and B. Lennartson. Hybrid system stability and robustness verification using linear matrix inequalities. International Journal of Control, 75(16-17):1335–1355, 2002.
- Stefan Pettersson. Designing switched observers for switched systems using multiple Lyapunov functions and dwell-time switching. IFAC Proceedings Volumes, 39(5):18–23, 2006. 2nd IFAC Conference on Analysis and Design of Hybrid Systems.
- Stefan Pettersson. Switched state jump observers for switched systems. IFAC Proceedings Volumes, 38(1):127–132, 2005. 16th IFAC World Congress.