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lqe
linear quadratic estimator (Kalman Filter)
Syntax
[K,X]=lqe(Pw)
[K,X]=lqe(P,Qww,Rvv [,Swv])
Arguments
- Pw
A state space representation of a linear dynamical system (see syslin)
- P
A state space representation of a linear dynamical system (
nu
inputs,ny
outputs,nx
states) (see syslin)- Qww
Real
nx
bynx
symmetric matrix, the process noise variance- Rvv
full rank real
ny
byny
symmetric matrix, the measurement noise variance.- Swv
real
nx
byny
matrix, the process noise vs measurement noise covariance. The default value is zeros(nx,ny).- K
a real matrix, the optimal gain.
- X
a real symmetric matrix, the stabilizing solution of the Riccati equation.
Description
This function computes the linear optimal LQ estimator gain of the state estimator for a detectable (see dt_ility) linear dynamical system and the variance matrices for the process and the measurement noises.
Syntax [K,X]=lqe(P,Qww,Rvv [,Swv])
Computes the linear optimal LQ estimator gain K for the dynamical system:
orWhere are white noises such as
CautionThe values of and are not taken into account. |
Standard formThis covariance matrix can be factored using full-rank factorization (see fuffrf) as And consequently the initial dynamical system is equivalent to or Where is now a white noise such as |
Syntax [K,X]=lqe(Pw)
Computes the linear optimal LQ estimator gain K for the dynamical system
Where is a white noise with unit covariance.
Properties
is stable.
the state estimator is given by the dynamical system:
or
It minimizes the covariance of the steady state error .For discrete time systems the state estimator is such that: (one-step predicted ).
Algorithm
let , and
For the continuous time case K is given by
where
X
is the solution of the stabilizing Riccati equationFor the discrete time case K is given by
where
X
is the solution of the stabilizing Riccati equation
Examples
Assume the dynamical system formed by two masses connected by a spring and a damper:
A force (where is a noise) is applied to the big one, the deviations from equilibrium positions and of the masses are measured. These measures are also subject to an additionnal noise .A continuous time state space representation of this system is:
and and are discrete time white noises such asThe instructions below can be used to compute a LQ state observer of the discretized version of this dynamical system.
// Form the state space model M = 1; m = 0.2; k = 0.1; b = 0.004; A = [ 0 1 0 0 -k/M -b/M k/M b/M 0 0 0 1 k/m b/m -k/m -b/m]; B = [0; 1/M; 0; 0]; C = [1 0 0 0 //dy1 0 0 1 0];//dy2 //inputs u and e; outputs dy1 and dy2 P = syslin("c",A, B, C); // Discretize it dt=0.5; Pd=dscr(P, dt); // Set the noise covariance matrices Q_e=0.01; //additive input noise variance R_vv=0.001*eye(2,2); //measurement noise variance Q_ww=Pd.B*Q_e*Pd.B'; //input noise adds to regular input u //----syntax [K,X]=lqe(P,Qww,Rvv [,Swv])--- Ko=lqe(Pd,Q_ww,R_vv); //observer gain //----syntax [K,X]=lqe(P21)--- bigR =sysdiag(Q_ww, R_vv); [W,Wt]=fullrf(bigR); Bw=W(1:size(Pd.B,1),:); Dw=W(($+1-size(Pd.C,1)):$,:); Pw=syslin(Pd.dt,Pd.A,Bw,Pd.C,Dw); Ko1=lqe(Pw); //same observer gain //Check gains equality norm(Ko-Ko1,1) // Form the observer O=observer(Pd,Ko); //check stability and(abs(spec(O.A))<1) // Check by simulation // Modify Pd to make it return the state Pdx=Pd;Pdx.C=eye(4,4);Pdx.D=zeros(4,1); t=0:dt:30; u=zeros(t); x=flts(u,Pdx,[1;0;0;0]);//state evolution y=Pd.C*x; // simulate the observer x_hat=flts([u+0.01*rand(u);y+0.0001*rand(y)],O); clf; subplot(2,2,1) plot2d(t',[x(1,:);x_hat(1,:)]'), e=gce();e.children.polyline_style=2; L=legend('$x_1=dy_1$', '$\hat{x_1}$');L.font_size=3; xlabel('Time (s)') subplot(2,2,2) plot2d(t',[x(2,:);x_hat(2,:)]') e=gce();e.children.polyline_style=2; L=legend('$x_2=dy_1^+$', '$\hat{x_2}$');L.font_size=3; xlabel('Time (s)') subplot(2,2,3) plot2d(t',[x(3,:);x_hat(3,:)]') e=gce();e.children.polyline_style=2; L=legend('$x_3=dy_2$', '$\hat{x_3}$');L.font_size=3; xlabel('Time (s)') subplot(2,2,4) plot2d(t',[x(4,:);x_hat(4,:)]') e=gce();e.children.polyline_style=2; L=legend('$x_4=dy_2^+$', '$\hat{x_4}$');L.font_size=3; xlabel('Time (s)')
Reference
Engineering and Scientific Computing with Scilab, Claude Gomez and al.,Springer Science+Business Media, LLC,1999, ISNB:978-1-4612-7204-5
See also
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