ZigangPan/LinearQuadraticControl

ZigangPan`LinearQuadraticControl`

LEQGcost

[system,θ]

calculates the LEQG cost of the LTI

system

with all the disturbance inputs as independent Wiener processes and all of the control inputs are ignored; the measurement outputs are ignored and the controlled outputs as the signal whose long term average of exponentiated energy (with risk-sensitivity parameter θ) is the cost function. If

system

is open-loop unstable, then the program returns 'error'. If the

system

is open-loop stable but optimal cost is ∞, then the program returns {∞,θstar}, where θstar is the supremum of θ's such that the optimal cost is finite. If the

system

is open-loop stable and the optimal cost is finite, then the program returns the optimal cost.

Examples

(1)

Basic Examples

(1)

In[1]:=

system1={{x1,x2,x3},{u1,u2,w1,w2,w3},{y1,y2,z1,z2,z3,z4},{{-1,1,-1,0,0,1,-1,1},{-2,0,1,1,-1,0,0,1},{-3,0,-2,1,1,1,1,0},{1,0,0,0,0,1,0,0},{0,0,1,0,0,0,1,0},{1,0,0,0,0,0,0,0},{0,0,1,0,0,0,0,0},{0,0,0,1,2,0,0,0},{0,0,0,2,1,0,0,0}},{1,2},{1,2},{1,2},{3,4,5},{1,2},{3,4,5,6}};systemcheck[system1]

Out[1]=

True

In[2]:=

stablizabilityQ[system1〚4〛〚1;;3,1;;3〛,system1〚4〛〚1;;3,4;;5〛]

Out[2]=

True

In[3]:=

detectabilityQ[system1〚4〛〚1;;3,1;;3〛,system1〚4〛〚4;;5,1;;3〛]

Out[3]=

True

In[4]:=

{z,controller}=

LQGcontrol

[system1]

Out[4]=

{8.91665,{{x1chk,x2chk,x3chk},{y1,y2},{u1,u2},{{-2.32238,1.,0.234439,1.32238,-1.23444},{-3.0763,-1.30056,1.36451,0.224204,-0.0261179},{-3.58715,0.0375991,-3.40869,0.765561,1.28946},{-0.336843,-0.631483,0.109582,0,0},{0.515253,0.669082,-0.228809,0,0}},{1,2},{1,2},{1,2},{},{1,2},{}}}

In[5]:=