Help with simulating a system model (inverted pendulum)

Question

In this documentation article, Regulate an Inverted Pendulum, the author started by creating the following system.

pendulum = 
  StateSpaceModel[
    {(M + m) x''[t] - m l Sin[θ[t]] θ'[t]^2 + m l Cos[θ[t]] θ''[t] ==  
       F[t] + d[t] Cos[θ[t]], 
     m x''[t] Cos[θ[t]] + m l θ''[t] == m g Sin[θ[t]] + d[t]}, 
    {θ[t], θ'[t], x[t], x'[t]}, 
    {F[t], d[t]}, 
    {θ[t], x[t]}, 
    t] /. {M -> 5.6, m -> 0.53, l -> 0.85, g -> 9.8};

poles = {-1 + 2 I, -1 - 2 I, -5, -7};

Then he got the feedback gains

k = 
  Join[
    StateFeedbackGains[SystemsModelExtract[pendulum, 1], poles], 
    ConstantArray[0, 4]}]    

and plotted the output response with x and θ, keeping θ close to zero, with the following

Plot[
  OutputResponse[
    SystemsModelStateFeedbackConnect[pendulum, k], 
    {0,UnitStep[t] - UnitStep[t - 1]}, 
    {t, 4}] // Evaluate, 
  {t, 0, 4}, 
  PlotRange -> All, 
  PlotStyle -> {Blue, Red}, 
  Epilog -> 
    Inset[Column[{Style["x(t)", 10, Blue], Style["θ(t)", 10, Red]}], {3, 0.004}], 
  AxesLabel -> {"t"}]

My question is how to pass the desired position and angle he wants the system to stabilize at; i.e., x = 1, θ = π/6

Answer 1

StateFeedbackGains solves a regulator problem.

You need to try AsymptoticOutputTracker to track a reference trajectory (which are constants in your case).

pendulum = AffineStateSpaceModel[{(M + m) x''[t] - 
   m l Sin[θ[t]] θ'[t]^2 + 
   m l Cos[θ[t]] θ''[t] == 
  F[t] + d[t] Cos[θ[t]], 
 m x''[t] Cos[θ[t]] + m l θ''[t] == 
  m g Sin[θ[t]] + d[t]}, {θ[t], θ'[t], x[t], 
 x'[t]}, {F[t], d[t]}, {θ[t], x[t]}, t] /. {M -> 5.6, 
m -> 0.53, l -> 0.85, g -> 9.8};

poles = {-1 + 2 I, -1 - 2 I, -5, -7};
fb = AsymptoticOutputTracker[pendulum, {π/6, 1}, poles];
csys = SystemsModelStateFeedbackConnect[pendulum, fb];

Plot[Evaluate@OutputResponse[csys, {0}, {t, 0, 4}], {t, 0, 4}, 
PlotLegends -> {θ[t], x[t]}]