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lpvpuiobsv

Given a Polytopic Linear Parameter Varying Descriptor System, this function computes a bank of observers in a Generalized observer scheme (GOS) by a full order observer with unknown inputs [1,2]

Calling Sequence

[N,G1,L,H2,Bo1]=lpvgosbank(E,A,B,C,R,alpha,flag)

Parameters

E, A, B, C, R:

Continuos matrices of descriptor system. E singular. Where A=A(theta), B=B(theta), R= R(theta) and C=C. Theta is varying parameter.

alpha:

For the stabilize the observer in a stable LMI region (see [1])

flag:

flag='s' for sensor faults and 'a' for actuator faults. By default is programed for sensor faults

N, G1 L, H2:

Gains matrices of the observer. This is obtained like N(j) where (j) is for the 1,2... j model of the polytopic observer

Bo1

Auxiliar matrices for simulation, this can be obtained from qrrse transformation

Description

Given a polytopic LPV Descriptor system in the form
M
dxEdt= Σmu(theta)[A(theta) x(t) + B(theta) u(t) +R(theta) d (t)]
i=1
y   =C x(t) + D u(t) 

This compute the observer gains 
M
dz/dt= Σmu(theta) [Nz(t) + G1u(t) + L(yo(t))]
i=1
M
xe(t)= Σmu(theta) [z(t) +H2(yo(t))]
i=1
where yo(t) results from QR r.s.e transformation
yo(t)=[-B1u(t); y(t)];  y(t)= C x(t)
alpha put the observer in the stability LMI region

Example Gos for actuator faults (AFD)

u inputs (in)         outputs
__________ 
-------| LPV model|-------------
------ |__________|------------

__________
in2-------| LPV Obsv1-------------
in2------|__________|-------------
inm------

__________
in1-------| LPV obsv2|-------------
in3-------|__________|-------------
inm--------     
__________
in1    ----| LPV obsvm  |-------------
in2 -------|__________|-------------
in(m-1)----

Examples

// parameter varying
theta1=[-0.05 0.05]; //[ ]
theta2=[-0.1 0.1 ];

// genering a matrix with the two parameter varying
n=0;
for k=1:2
    for j=1:2
        n=n+1;
        theta(n,:)=[theta1(k) theta2(j) ];
    end
end

// descriptor system, constant matrices
E=[1 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 0];

C=[1 0 1 0 ; 0 1 0 1; 1 0  0 0];
// Building the parameter varying matrices
for j=1:4
    
A(:,:,j)=[0 1 0 0; -1  theta(j,1) 0 0; 0 -1 -1+theta(j,1) 0; 0 0 0 -1+theta(j,2)];
    
B(:,:,j)=[1+theta(j,1) 1;1 theta(j,2); 0 0 ;0.5-theta(j,2) 0];
    
R(:,:,j)=[1 theta(j,2) 0 0]';
end
alpha=3;
flag= 's' // for sensor faults

// computing the gains of the observer
[N,G1,L,H2,Bo1]=lpvgosbank(E,A,B,C,R,alpha,flag)
// subindex (j) indicates for the 1,2,.. jth model

See also

Author

Bibliography

[1] M., Hamdi., Rodrigues, M., Mechmeche, C., Theilliol, D., Braiek, N. B., & Tunisie, E. P. D. (2009). State Estimation for Polytopic LPV Descriptor Systems : Application to Fault Diagnosis. Convergence (pp. 438-443).

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