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Conceptual framework of implementing explicit optimal control policy in a f...
Published Online: September 20, 2022
Fig. 1 Conceptual framework of implementing explicit optimal control policy in a feedback controlled system, where u is the vector of control actions, ξ is the vector of state variables, κ ( ⋅ ) is the function of control law, and (·)* denotes the optimum. The surface on the top i... More
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Flowchart of developing the explicit optimal control policy by sp-NLPC. The...
Published Online: September 20, 2022
Fig. 2 Flowchart of developing the explicit optimal control policy by sp-NLPC. The surface at the bottom represents an example of the explicit visualization for the approximate optimal control action as a function of state variables. More
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Schematics of inverted pendulums on a cart with generalized coordinates ( a...
Published Online: September 20, 2022
Fig. 3 Schematics of inverted pendulums on a cart with generalized coordinates ( a ) single inverted pendulum on a cart and ( b ) double inverted pendulum on a cart More
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Phase portraits for the single inverted pendulum on a cart in ( a )    (  x...
Published Online: September 20, 2022
Fig. 4 Phase portraits for the single inverted pendulum on a cart in ( a ) ( x , x ˙ ) -domain and ( b ) ( θ , θ ˙ ) -domain More
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Comparison of the control trajectories for Test 1 by MPC and sp-NLPC. The z...
Published Online: September 20, 2022
Fig. 5 Comparison of the control trajectories for Test 1 by MPC and sp-NLPC. The zoom-in image shows that the tremendous changes and the discontinuity from one instant to another by MPC, whereas the trajectory for sp-NLPC is comparably smooth and continuous. More
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Phase portraits for the double inverted pendulum on a cart with three tests...
Published Online: September 20, 2022
Fig. 6 Phase portraits for the double inverted pendulum on a cart with three tests: ( a ) Test 1, ( b ) Test 2, and ( c ) Test 3. The arrows are the directions for the system dynamics, the black circles denote the start points, and the stars represent the terminal points. More
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A planar motion { x ( t ),  y ( t ),  θ ( t )} with  M  and  F  denoting th...
Published Online: September 20, 2022
Fig. 1 A planar motion { x ( t ), y ( t ), θ ( t )} with M and F denoting the moving and fixed frame, respectively More
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( a ) An ellipse formed by two consecutive circles  h  1  and  h  −1 , ( b ...
Published Online: September 20, 2022
Fig. 2 ( a ) An ellipse formed by two consecutive circles h 1 and h −1 , ( b ) a belt-coupled 2R dyad, ( c ) a four-bar mechanism, and ( d ) a six-bar mechanism More
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A planar rigid body and an arbitrary point  p  on it.  a  denotes the point...
Published Online: September 20, 2022
Fig. 3 A planar rigid body and an arbitrary point p on it. a denotes the point which traces the simple Fourier trajectory on the motion More
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An illustrative example to show the comparison metric: ( a ) distances cove...
Published Online: September 20, 2022
Fig. 4 An illustrative example to show the comparison metric: ( a ) distances covered by d ^ − 2 , ( b ) distances covered by d ^ − 2 − 3 , ( c ) distances covered by d ^ − 2 1 , ( d ) distances covered by d ^ 2 , ( e ) distances co... More
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