Difference between revisions of "Quadrotor helicopter control problem"

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}}<!-- Do not insert line break here or Dimensions Box moves up in the layout...
 
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-->The mixed-integer optimal control problem of a quadrotor helicopter in two dimensions is taken from [7] and [8]. The evolution
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-->The mixed-integer optimal control problem of a quadrotor helicopter in two dimensions is taken from (Link: [https://pdfs.semanticscholar.org/75a0/211476ddc21363cfb3262c04d18794ad06ef.pdf Gillula et al.]) and from (Link: [http://epubs.siam.org/doi/pdf/10.1137/120901507 Vasudevan et al.]). The evolution
 
of the quadrotor can be defined with respect to a fixed two dimensional reference
 
of the quadrotor can be defined with respect to a fixed two dimensional reference
 
frame using six dimensions, where the first three dimensions represent the position
 
frame using six dimensions, where the first three dimensions represent the position
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<math>
 
<math>
 
\begin{array}{llclr}
 
\begin{array}{llclr}
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  \displaystyle \min_{x, w, s} & x_2(t_f) + 10s   \\[1.5ex]
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  \displaystyle \min_{x,u, w} & 5(x_1(t_f)-6)^2&+&5(x_3(t_f)-1)^2+(\sin(x_5(t_f)0.5))^2 +\int\limits_{t_0}^{t_f} 5u(\tau)^2 \ d \tau   \\[1.5ex]
 
  \mbox{s.t.}  
 
  \mbox{s.t.}  
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  & \dot{x}_0 & = &  x_0 - x_0 x_1 - \; c_0 x_0 \; w, \\
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  & \dot{x}_1 & = &  x_2(t), \\
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  & \dot{x}_1 & = & - x_1 + x_0 x_1 - \; c_1 x_1 \; w, \\
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  & \dot{x}_2 & = & g \sin( x_5(t)) + w_1(t)u(t)\frac{\sin(x_5(t))}{M},   \\
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  & \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex]
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  & \dot{x}_3 & = & x_4(t),  \\
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  & x_0 & \geq & 1.1 - s,  \\
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& \dot{x}_4 & = & g \cos( x_5(t))-g+ w_1(t)u(t)\frac{\cos(x_5(t))}{M},   \\
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  & x(0) &=& (0.5, 0.7, 0)^T, \\
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  & \dot{x}_5 & = & x_6(t),   \\
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  & w(t) &\in&  \{0, 1\}, \\
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  & \dot{x}_6 & = & -w_2(t)L \frac{u(t)}{I}+w_3(t)L \frac{u(t)}{I}  \\[1.5ex]
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& s & \geq & 0.
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  & x(0) &=& (0, 0, 1, 0 , 0, 0)^T, \\
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  & w_i(t) &\in&  \{0, 1\}, i=1,\ldots,3 \\
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& \sum\limits_{i=1}^{3}w_i(t) &=& 1, \\
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& u(t) & \in& [0,0.001], \quad t\in[t_0,t_f],\\
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& x_3(t) & \geq & 0, \quad t\in[t_0,t_f].
 
\end{array}  
 
\end{array}  
 
</math>
 
</math>
 
</p>
 
</p>
  
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Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant penalizes a biomass x(0) that is below 1.1 at the end of the time horizon.
 
  
 
== Parameters ==
 
== Parameters ==
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<math>
 
<math>
 
\begin{array}{rcl}
 
\begin{array}{rcl}
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[t_0, t_f] &=& [0, 12],\\
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[t_0, t_f] &=& [0, 7.5],\\
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(c_{0}, c_{1}) &=& (0.4, 0.2),
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(g, M, L, I) &=& (9.8, 1.3, 0.305, 0.0605),
 
\end{array}
 
\end{array}
 
</math>
 
</math>
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== Reference Solutions ==
 
== Reference Solutions ==
  
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If the problem is relaxed, i.e., we demand that <math>w(t)</math> is in the continuous interval <math>[0, 1]</math> rather than being binary, the optimal solution can be determined by means of direct optimal control.  
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A reference solution can be found in [http://epubs.siam.org/doi/pdf/10.1137/120901507 Vasudevan et al.] based on the embedding transformation technique for switched systems.
  
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The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=200  </math> is <math>x_2(t_f) =1.36548113</math>. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is <math>x_2(t_f) =1.38756111</math>. 
 
  
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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
 
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Image:Lotka_term_ineq_Relaxed_12000_200.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=200</math>.
 
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Image:Lotka_term_ineq_CIA_12000_200.pdf| Differential states and binary cotnrol determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=200</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
 
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</gallery>
 
  
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== Variants ==
  
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There are several alternative formulations and variants of the above problem, in particular
  
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* [[Quadrotor (binary variant)]]: The quadrotor helicoptor problem, where the continuous control is replaced via partial outer convexification by binary controls.
  
  
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[[Category:Chattering]]
 
[[Category:Chattering]]
 
[[Category:Sensitivity-seeking arcs]]
 
[[Category:Sensitivity-seeking arcs]]
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[[Category:Population dynamics]]
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Latest revision as of 14:44, 14 October 2019

Quadrotor helicopter control problem
State dimension: 1
Differential states: 6
Continuous control functions: 1
Discrete control functions: 3
Interior point equalities: 6

The mixed-integer optimal control problem of a quadrotor helicopter in two dimensions is taken from (Link: Gillula et al.) and from (Link: Vasudevan et al.). The evolution of the quadrotor can be defined with respect to a fixed two dimensional reference frame using six dimensions, where the first three dimensions represent the position along a horizontal axis, the position along the vertical axis, and the roll angle of the helicopter, respectively, and the last three dimensions represent the time derivative of the first three dimensions.

Mathematical formulation

The mixed-integer optimal control problem is given by

\begin{array}{llclr} \displaystyle \min_{x,u, w} & 5(x_1(t_f)-6)^2&+&5(x_3(t_f)-1)^2+(\sin(x_5(t_f)0.5))^2 +\int\limits_{t_0}^{t_f} 5u(\tau)^2 \ d \tau \\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = & x_2(t), \\ & \dot{x}_2 & = & g \sin( x_5(t)) + w_1(t)u(t)\frac{\sin(x_5(t))}{M}, \\ & \dot{x}_3 & = & x_4(t), \\ & \dot{x}_4 & = & g \cos( x_5(t))-g+ w_1(t)u(t)\frac{\cos(x_5(t))}{M}, \\ & \dot{x}_5 & = & x_6(t), \\ & \dot{x}_6 & = & -w_2(t)L \frac{u(t)}{I}+w_3(t)L \frac{u(t)}{I} \\[1.5ex] & x(0) &=& (0, 0, 1, 0 , 0, 0)^T, \\ & w_i(t) &\in& \{0, 1\}, i=1,\ldots,3 \\ & \sum\limits_{i=1}^{3}w_i(t) &=& 1, \\ & u(t) & \in& [0,0.001], \quad t\in[t_0,t_f],\\ & x_3(t) & \geq & 0, \quad t\in[t_0,t_f]. \end{array}


Parameters

These fixed values are used within the model.

\begin{array}{rcl} [t_0, t_f] &=& [0, 7.5],\\ (g, M, L, I) &=& (9.8, 1.3, 0.305, 0.0605), \end{array}

Reference Solutions

A reference solution can be found in Vasudevan et al. based on the embedding transformation technique for switched systems.


Variants

There are several alternative formulations and variants of the above problem, in particular

  • Quadrotor (binary variant): The quadrotor helicoptor problem, where the continuous control is replaced via partial outer convexification by binary controls.
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