ClemensZeile at 13:44, 14 October 2019

2019-10-14T13:44:09Z

← Older revision		Revision as of 13:44, 14 October 2019
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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.]).		-->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
	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

ClemensZeile at 13:43, 14 October 2019

2019-10-14T13:43:36Z

← Older revision		Revision as of 13:43, 14 October 2019
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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		-->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}
	\displaystyle \min_{x, w~~, s~~} & ~~x_2~~(t_f) + ~~10s~~ \\[1.5ex]		\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.}
	& \dot{x}_0 & = & ~~x_0 - x_0 x_1 - \; c_0 x_0 \; w~~, \\		& \dot{x}_1 & = & x_2(t), \\
	& \dot{x}_1 & = & ~~- x_1~~ + ~~x_0 x_1 -~~ \~~; c_1 x_1~~ \~~; w~~, \\		& \dot{x}_2 & = & g \sin( x_5(t)) + w_1(t)u(t)\frac{\sin(x_5(t))}{M}, \\
	& \dot{x}_2 & = & (~~x_0~~ - 1)~~^2 +~~ (~~x_1 - 1~~)^2, \\~~[1.5ex]~~		& \dot{x}_3 & = & x_4(t), \\
	& ~~x_0~~ & ~~\geq~~ & ~~1.1~~ - s, \\		& \dot{x}_4 & = & g \cos( x_5(t))-g+ w_1(t)u(t)\frac{\cos(x_5(t))}{M}, \\
	& x(0) &=& (0.5, 0.7, 0)^T, \\		& \dot{x}_5 & = & x_6(t), \\
	& w(t) &\in& \{0, 1\}, \\		& \dot{x}_6 & = & -w_2(t)L \frac{u(t)}{I}+w_3(t)L \frac{u(t)}{I} \\[1.5ex]
	& s & \geq & 0.		& 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}		\end{array}
	</math>		</math>
	</p>		</p>

	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}
	[t_0, t_f] &=& [0, 12],\\		[t_0, t_f] &=& [0, 7.5],\\
	(~~c_{0}~~, ~~c_{1}~~) &=& (0.4, 0.2),		(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 ==

	~~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~~.		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.

	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>.

	~~<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">~~
	~~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>.~~
	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.
	~~</gallery>~~

			== 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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	[[Category:Chattering]]		[[Category:Chattering]]
	[[Category:Sensitivity-seeking arcs]]		[[Category:Sensitivity-seeking arcs]]
	~~[[Category:Population dynamics]]~~

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Quadrotor helicopter control problem - Revision history

ClemensZeile at 13:44, 14 October 2019

ClemensZeile at 13:43, 14 October 2019

ClemensZeile: Created page with "{{Dimensions |nd = 1 |nx = 6 |nu = 1 |nw = 3 |nre = 6 |nrc = 3 }}