WIP, Description in progress
In this section, we treat the problem of designing a full-order state observer for system such that the effect of the disturbance w ( t ) {\displaystyle w(t)} to the estimate error is
prohibited to a desired level.
System Setting
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The system is following
x ˙ ( t ) = A x ( t ) + B 1 u ( t ) + B 2 w ( t ) , x ( 0 ) = x 0 , {\displaystyle {\dot {x}}(t)=Ax(t)+B_{1}u(t)+B_{2}w(t),x(0)=x_{0},}
y ( t ) = C 1 x ( t ) + D 1 u ( t ) + D 2 w ( t ) , {\displaystyle y(t)=C_{1}x(t)+D_{1}u(t)+D_{2}w(t),}
z ( t ) = C 2 x ( t ) , {\displaystyle z(t)=C_{2}x(t),}
where
x ∈ R n , y ∈ R l , z ∈ R m {\displaystyle x\in \mathbb {R} ^{n},\ y\in \mathbb {R} ^{l},\ z\in \mathbb {R} ^{m}} are respectively the state vector, the measured
output vector, and the output vector of interests.
w ∈ R p {\displaystyle w\in \mathbb {R} ^{p}} are the disturbance vector and control vector , respectively.
A , B 1 , B 2 , C 1 , C 2 , D 1 , and D 2 {\displaystyle A,\ B_{1},\ B_{2},\ C_{1},\ C_{2},\ D_{1},{\text{ and }}D_{2}} are the system coefficient matrices of
appropriate dimensions.
Problem Formulation
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For the system, we introduce a full-order state observer in the following form:
x ^ ˙ = ( A + L C 1 ) x ^ − L y + ( B 1 + L D 1 ) u {\displaystyle {\dot {\hat {x}}}=(A+LC_{1}){\hat {x}}-Ly+(B_{1}+LD_{1})u}
where x ^ {\displaystyle {\hat {x}}} is the state observation vector and L ∈ R n × m {\displaystyle L\in \mathbb {R} ^{n\times m}} is the observer gain. Obviously,
the estimate of the interested output is given by
z ^ ( t ) = C 2 x ^ ( t ) {\displaystyle {\hat {z}}(t)=C_{2}{\hat {x}}(t)}
which is desired to have as little affection as possible from the disturbance w ( t ) {\displaystyle w(t)} .
Using system dynamics,
x ˙ ( t ) = A x ( t ) + B 1 u ( t ) + B 2 w ( t ) = A x ( t ) + L y − L y + B 1 u ( t ) + B 2 w ( t ) = ( A + L C 1 ) x ( t ) − L y ( t ) + ( B 1 + L D 1 ) u ( t ) + ( B 2 + L D 2 ) w ( t ) {\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}u(t)+B_{2}w(t)\\&=Ax(t)+Ly-Ly+B_{1}u(t)+B_{2}w(t)\\&=(A+LC_{1})x(t)-Ly(t)+(B_{1}+LD_{1})u(t)+(B_{2}+LD_{2})w(t)\end{aligned}}}
Denoting
e ˙ = ( A + L C 1 ) e + ( B 2 + L D 2 ) w {\displaystyle {\dot {e}}=(A+LC_{1})e+(B_{2}+LD_{2})w}
z ~ ( t ) = C 2 e {\displaystyle {\tilde {z}}(t)=C_{2}e} .
The transfer function of the system is clearly given by
G z ~ w ( s ) = C 2 ( s I − A − L C 1 ) − 1 ( B 2 + L D 2 ) {\displaystyle G_{{\tilde {z}}w}(s)=C_{2}(sI-A-LC_{1})^{-1}(B_{2}+LD_{2})} .
With the aforementioned preparation, the problems of H ∞ and H 2 {\displaystyle {\mathcal {H}}_{\infty }{\text{ and }}{\mathcal {H}}_{2}} state
observer designs can be stated as follows.
Problem 1
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(H ∞ {\displaystyle {\mathcal {H}}_{\infty }} state observers) Given system (9.22) and a positive scalar γ {\displaystyle \gamma } ,
find a matrix L {\displaystyle L} such that
| | G z ~ w ( s ) | | ∞ < γ {\displaystyle ||G_{{\tilde {z}}w}(s)||_{\infty }<\gamma } .
Problem 2
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(H 2 {\displaystyle {\mathcal {H}}_{2}} state observers) Given system (9.22) and a positive scalar γ {\displaystyle \gamma } ,
find a matrix L {\displaystyle L} such that
| | G z ~ w ( s ) | | 2 < γ {\displaystyle ||G_{{\tilde {z}}w}(s)||_{2}<\gamma }
As a consequence of the requirements in the previous problems, the error system
is asymptotically stable, and hence we have
e ( t ) = x ( t ) − x ^ ( t ) → 0 , as t → ∞ {\displaystyle e(t)=x(t)-{\hat {x}}(t)\rightarrow 0,{\text{ as }}t\rightarrow \ \infty }
This states that x ^ ( t ) {\displaystyle {\hat {x}}(t)} is an asymptotic estimate of x ( t ) {\displaystyle x(t)} .
Solution/Theorem
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Regarding the solution to the problem of H∞ state observers design, we have the
following theorem.
Theorem 1
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The H ∞ {\displaystyle {\mathcal {H}}_{\infty }} state observers problem 1 has a solution
if and only if there exist a matrix W {\displaystyle W} and a symmetric positive definite matrix P {\displaystyle P}
such that
[ A T P + C 1 T W T + P A + W C 1 P B 2 + W D 2 C 2 T ( P B 2 + W D 2 ) T − γ I 0 C 2 0 − γ I ] < 0 {\displaystyle {\begin{bmatrix}A^{T}P+C_{1}^{T}W^{T}+PA+WC_{1}&PB_{2}+WD_{2}&C_{2}^{T}\\(PB_{2}+WD_{2})^{T}&-\gamma I&0\\C_{2}&0&-\gamma I\end{bmatrix}}<0}
When such a pair of matrices W and P are found, a solution to the problem is
given as
L = P − 1 W {\displaystyle L=P^{-1}W}
With a prescribed attenuation level, the problem of H∞ state observers design is
turned into an LMI feasibility problem in the form problem stated before. The problem with a
minimal attenuation level γ {\displaystyle \gamma } can be sought via the following optimization problem:
min γ {\displaystyle \gamma }
s.t. P > 0 [ A T P + C 1 T W T + P A + W C 1 P B 2 + W D 2 C 2 T ( P B 2 + W D 2 ) T − γ I 0 C 2 0 − γ I ] < 0 {\displaystyle {\begin{aligned}P&>0\\{\begin{bmatrix}A^{T}P+C_{1}^{T}W^{T}+PA+WC_{1}&PB_{2}+WD_{2}&C_{2}^{T}\\(PB_{2}+WD_{2})^{T}&-\gamma I&0\\C_{2}&0&-\gamma I\end{bmatrix}}&<0\end{aligned}}}
Theorem 2
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The H 2 {\displaystyle {\mathcal {H}}_{2}} state observers problem 2 has a solution
the following 2 conclusions hold.
1.It has a solution if and only if there exists a matrix W, a symmetric matrix Q,
and a symmetric matrix X such that
[ X A + W C 1 + ( X A + W C 1 ) T X B 2 + W D 2 ( X B 2 + W D 2 ) T − I ] < 0 {\displaystyle {\begin{bmatrix}XA+WC_{1}+(XA+WC_{1})^{T}&XB_{2}+WD_{2}\\(XB_{2}+WD_{2})^{T}&-I\end{bmatrix}}<0} ,
[ − Q C 2 C 2 T − X ] < 0 {\displaystyle {\begin{bmatrix}-Q&C_{2}\\C_{2}^{T}&-X\end{bmatrix}}<0} ,
trace ( Q ) < γ 2 {\displaystyle {\text{trace}}(Q)<\gamma ^{2}} .
When such a triple of matrices are obtained, a solution to the problem is
given as
L = X − 1 W {\displaystyle L=X^{-1}W} .
2. It has a solution if and only if there exists a matrix V, a symmetric matrix Z,
and a symmetric matrix Y such that
A T Y + C 1 T V T + Y A + V C 1 + C 2 T C 2 < 0 {\displaystyle A^{T}Y+C_{1}^{T}V^{T}+YA+VC_{1}+C_{2}^{T}C_{2}<0} ,
[ − Z ( Y B 2 + V D 2 ) T Y B 2 + V D 2 − Y ] < 0 {\displaystyle {\begin{bmatrix}-Z&(YB_{2}+VD_{2})^{T}\\YB_{2}+VD_{2}&-Y\end{bmatrix}}<0} ,
trace( Z ) < γ 2 {\displaystyle (Z)<\gamma ^{2}} .
When such a triple of matrices are obtained, a solution to the problem is
given as
L = Y − 1 V {\displaystyle L=Y^{-1}V} .
In applications, we are often concerned
with the problem of finding the minimal attenuation level γ {\displaystyle \gamma } . This problem can be
solved via the optimization
min ρ {\displaystyle \rho }
s.t. [ X A + W C 1 + ( X A + W C 1 ) T X B 2 + W D 2 ( X B 2 + W D 2 ) T − I ] < 0 {\displaystyle {\begin{bmatrix}XA+WC_{1}+(XA+WC_{1})^{T}&XB_{2}+WD_{2}\\(XB_{2}+WD_{2})^{T}&-I\end{bmatrix}}<0} ,
[ − Q C 2 C 2 T − X ] < 0 {\displaystyle {\begin{bmatrix}-Q&C_{2}\\C_{2}^{T}&-X\end{bmatrix}}<0} ,
trace ( Q ) < γ 2 {\displaystyle {\text{trace}}(Q)<\gamma ^{2}} ,
or
min ρ {\displaystyle \rho }
A T Y + C 1 T V T + Y A + V C 1 + C 2 T C 2 < 0 {\displaystyle A^{T}Y+C_{1}^{T}V^{T}+YA+VC_{1}+C_{2}^{T}C_{2}<0}
[ − Z ( Y B 2 + V D 2 ) T Y B 2 + V D 2 − Y ] < 0 {\displaystyle {\begin{bmatrix}-Z&(YB_{2}+VD_{2})^{T}\\YB_{2}+VD_{2}&-Y\end{bmatrix}}<0}
trace( Z ) < γ 2 {\displaystyle (Z)<\gamma ^{2}}
When a minimal ρ is obtained, the minimal attenuation level is γ = ρ {\displaystyle \gamma ={\sqrt {\rho }}} .
WIP, additional references to be added
External Links
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A list of references documenting and validating the LMI.
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