Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Park-Ramirez Bioreactor Model n For genetically modified yeast in a fed- o i n t a o batch reactor, predict: l i u t a n m r o e u t i cell growth, t n u c u p e c l i G n A D I substrate consumption, • V = q foreign protein production, • q X = µ X - V X and foreign protein secretion. • q q S = − Y µ X + V S f - V S • q P T = f P X - V P T The optimal control problem: • q MAX (Φ) P M = φ ( P T − P M ) - V P M Φ = P M ( t f ) · V ( t f )
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Park-Ramirez Bioreactor Model n For genetically modified yeast in a fed- o i n t a o batch reactor, predict: l i u t a n m r o e u t i cell growth, t n u c u p e c l i G n A D I substrate consumption, • V = q foreign protein production, • q X = µ X - V X and foreign protein secretion. • q q S = − Y µ X + V S f - V S 3 2.5 • q P T = f P X - V P T 2 hr ) q ( L 1.5 • q 1 P M = φ ( P T − P M ) - V P M 0.5 0 0 2 4 6 8 10 12 14 Time ( hr )
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems:
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial).
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω .
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2 Similar to control vector parametrization except discretizing states too.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2 Similar to control vector parametrization except discretizing states too. Seems to be much less common than control vector parametrization.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2 Similar to control vector parametrization except discretizing states too. Seems to be much less common than control vector parametrization. Necessary conditions and Pontryagin’s Maximum Principle 3
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2 Similar to control vector parametrization except discretizing states too. Seems to be much less common than control vector parametrization. Necessary conditions and Pontryagin’s Maximum Principle 3 Can provide much more insight into the problem solution.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2 Similar to control vector parametrization except discretizing states too. Seems to be much less common than control vector parametrization. Necessary conditions and Pontryagin’s Maximum Principle 3 Can provide much more insight into the problem solution. Requires more skill/education than parametrization methods.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2 Similar to control vector parametrization except discretizing states too. Seems to be much less common than control vector parametrization. Necessary conditions and Pontryagin’s Maximum Principle 3 Can provide much more insight into the problem solution. Requires more skill/education than parametrization methods. Difficult to apply to singular control problems.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Optimal Control Solution Methods The four primary ways currently used to solve optimal control problems: Control vector parametrization (CVP) 1 Break the control u ( t ) into piecewise vectors. Each piecewise vector is a function approximation ν ( ω ) (ie. constant, linear, polynomial). Find the sensitivities ∂ Φ ∂ω . Solve for ω using a NLP solver. Collocation 2 Similar to control vector parametrization except discretizing states too. Seems to be much less common than control vector parametrization. Necessary conditions and Pontryagin’s Maximum Principle 3 Can provide much more insight into the problem solution. Requires more skill/education than parametrization methods. Difficult to apply to singular control problems. Iterative dynamic programming (IDP) 4
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Dynamic Programming What is dynamic programming? A way to solve discrete multi-stage decision problem by:
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Dynamic Programming What is dynamic programming? A way to solve discrete multi-stage decision problem by: Breaking the problem into smaller subproblems.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Dynamic Programming What is dynamic programming? A way to solve discrete multi-stage decision problem by: Breaking the problem into smaller subproblems. Remembering the best solutions to the subproblems.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Dynamic Programming What is dynamic programming? A way to solve discrete multi-stage decision problem by: Breaking the problem into smaller subproblems. Remembering the best solutions to the subproblems. Combining the solutions to the subproblems to get the overall solution.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Iterative Dynamic Programming (IDP) What is iterative dynamic programming (IDP)? A type of dynamic programming to solve the optimal control problem by:
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Iterative Dynamic Programming (IDP) What is iterative dynamic programming (IDP)? A type of dynamic programming to solve the optimal control problem by: Breaking the control u ( t ) into k stages.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Iterative Dynamic Programming (IDP) What is iterative dynamic programming (IDP)? A type of dynamic programming to solve the optimal control problem by: Breaking the control u ( t ) into k stages. Guess several profiles for the stagewise constant controls u k .
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Iterative Dynamic Programming (IDP) What is iterative dynamic programming (IDP)? A type of dynamic programming to solve the optimal control problem by: Breaking the control u ( t ) into k stages. Guess several profiles for the stagewise constant controls u k . Use the guessed controls u k to calculate a guessed continuous state profiles X ( t ) , S ( t ) , etc.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Iterative Dynamic Programming (IDP) What is iterative dynamic programming (IDP)? A type of dynamic programming to solve the optimal control problem by: Breaking the control u ( t ) into k stages. Guess several profiles for the stagewise constant controls u k . Use the guessed controls u k to calculate a guessed continuous state profiles X ( t ) , S ( t ) , etc. Starting at the final time and working backward, find out which of the guessed controls was the best and remember it for the next iteration.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods Iterative Dynamic Programming (IDP) What is iterative dynamic programming (IDP)? A type of dynamic programming to solve the optimal control problem by: Breaking the control u ( t ) into k stages. Guess several profiles for the stagewise constant controls u k . Use the guessed controls u k to calculate a guessed continuous state profiles X ( t ) , S ( t ) , etc. Starting at the final time and working backward, find out which of the guessed controls was the best and remember it for the next iteration. Animation 1: Basic IDP algorithm
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why?
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc).
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc). Gradient method: fast in general, very fast near the solution.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc). Gradient method: fast in general, very fast near the solution. IDP sometimes has noisy controls.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc). Gradient method: fast in general, very fast near the solution. IDP sometimes has noisy controls. What’s good about IDP though? Stochastic method: if sufficient samples are taken, likely to find global optimum.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc). Gradient method: fast in general, very fast near the solution. IDP sometimes has noisy controls. What’s good about IDP though? Stochastic method: if sufficient samples are taken, likely to find global optimum. No sensitivity derivatives needed.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc). Gradient method: fast in general, very fast near the solution. IDP sometimes has noisy controls. What’s good about IDP though? Stochastic method: if sufficient samples are taken, likely to find global optimum. No sensitivity derivatives needed. Obtains an approximate solution very quickly.
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc). Gradient method: fast in general, very fast near the solution. IDP sometimes has noisy controls. What’s good about IDP though? Stochastic method: if sufficient samples are taken, likely to find global optimum. No sensitivity derivatives needed. Obtains an approximate solution very quickly. Very simple algorithm
Problem Introduction Park-Ramirez Model Optimal Control Solution Methods CVP vs. IDP CVP seems to be more popular than IDP . Why? Fast! At least 4x faster than IDP , often 20x faster. Easy (Matlab optimization toolbox, etc). Gradient method: fast in general, very fast near the solution. IDP sometimes has noisy controls. What’s good about IDP though? Stochastic method: if sufficient samples are taken, likely to find global optimum. No sensitivity derivatives needed. Obtains an approximate solution very quickly. Very simple algorithm (although maybe not after I’m done with it).
Speeding up IDP Smoothing IDP Speeding up IDP IDP is slow, how can we speed it up?
Speeding up IDP Smoothing IDP Speeding up IDP IDP is slow, how can we speed it up? Reduce integrator relative tolerance.
Speeding up IDP Smoothing IDP Speeding up IDP IDP is slow, how can we speed it up? CPU time (seconds) 100 Reduce integrator relative tolerance. 10 1 10 − 7 10 − 5 10 − 3 10 − 1 Integrator Relative Tolerance
Speeding up IDP Smoothing IDP Speeding up IDP IDP is slow, how can we speed it up? Reduce integrator relative tolerance. Use my new adaptive region size update methods.
Speeding up IDP Smoothing IDP Speeding up IDP IDP is slow, how can we speed it up? Reduce integrator relative tolerance. Use my new adaptive region size update methods.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls First-order Control Filter IDP often leads to noisy controls: Animation 3: Basic IDP with 50 stages
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls First-order Control Filter IDP often leads to noisy controls: Animation 3: Basic IDP with 50 stages Try a first-order control filter after every iteration.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls First-order Control Filter IDP often leads to noisy controls: Animation 3: Basic IDP with 50 stages Try a first-order control filter after every iteration. Animation 4: First-order control filter
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less?
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then for each test control do 2:
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then for each test control do 2: if test control leads to a smoother control profile then 3:
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then for each test control do 2: if test control leads to a smoother control profile then 3: � � − ∆Φ γ s = exp − R ( 0 , 1 ) 4: T · M Φ end if 5:
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then for each test control do 2: if test control leads to a smoother control profile then 3: � � − ∆Φ γ s = exp − R ( 0 , 1 ) 4: T · M Φ end if 5: end for 6: Choose the test control with the largest γ s 7: 8: end if
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then for each test control do 2: if test control leads to a smoother control profile then 3: � � − ∆Φ γ s = exp − R ( 0 , 1 ) 4: T · M Φ end if 5: end for 6: Choose the test control with the largest γ s 7: 8: end if Temperature cools with control region size and number of iterations.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then for each test control do 2: if test control leads to a smoother control profile then 3: � � − ∆Φ γ s = exp − R ( 0 , 1 ) 4: T · M Φ end if 5: end for 6: Choose the test control with the largest γ s 7: 8: end if Temperature cools with control region size and number of iterations. Animation 5: Simulated annealing filter
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Simulated Annealing Control Filter First-order control filter can over-filter. How about filtering more where Φ is hurt less? 1: if no test controls improve Φ then for each test control do 2: if test control leads to a smoother control profile then 3: � � − ∆Φ γ s = exp − R ( 0 , 1 ) 4: T · M Φ end if 5: end for 6: Choose the test control with the largest γ s 7: 8: end if Temperature cools with control region size and number of iterations. Animation 5: Simulated annealing filter Less likely to find local minima for this problem.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Control Damping How about punishing control activity directly?
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Control Damping How about punishing control activity directly? Φ ∗ = Φ − Φ d
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Control Damping How about punishing control activity directly? Φ ∗ = Φ − Φ d Φ d = discrete second derivative of controls.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Control Damping How about punishing control activity directly? Φ ∗ = Φ − Φ d Φ d = discrete second derivative of controls. Animation 6: Damping
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Control Damping How about punishing control activity directly? Φ ∗ = Φ − Φ d Φ d = discrete second derivative of controls. Animation 6: Damping Changes the problem!
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Control Damping How about punishing control activity directly? Φ ∗ = Φ − Φ d Φ d = discrete second derivative of controls. Animation 6: Damping Changes the problem! Was often harmful to solution if large enough to filter well. Good in small doses in combination with other methods.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Pivot Point Test Controls Regular test controls work backwards one stage at a time.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Pivot Point Test Controls Regular test controls work backwards one stage at a time. Why not change two stage controls at a time?
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Pivot Point Test Controls Regular test controls work backwards one stage at a time. Why not change two stage controls at a time? Animation 7: Pivot points (show every test control)
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Pivot Point Test Controls Regular test controls work backwards one stage at a time. Why not change two stage controls at a time? Animation 7: Pivot points (show every test control) Animation 8: Pivot points (show every stage)
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Pivot Point Test Controls Regular test controls work backwards one stage at a time. Why not change two stage controls at a time? Animation 7: Pivot points (show every test control) Animation 8: Pivot points (show every stage) Very fast convergence (to local minima).
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Big Problem with Smoothing All the smoothing techniques caused local minima to be found.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Big Problem with Smoothing All the smoothing techniques caused local minima to be found.
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Big Problem with Smoothing All the smoothing techniques caused local minima to be found. Solve using two-step process:
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Big Problem with Smoothing All the smoothing techniques caused local minima to be found. Solve using two-step process: Solve most of the way using basic IDP . 1
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Big Problem with Smoothing All the smoothing techniques caused local minima to be found. Solve using two-step process: Solve most of the way using basic IDP . 1 Solve some more with smoothed IDP . 2
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Big Problem with Smoothing All the smoothing techniques caused local minima to be found. Solve using two-step process: Solve most of the way using basic IDP . 1 Solve some more with smoothed IDP . 2 The two-step results?
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Big Problem with Smoothing All the smoothing techniques caused local minima to be found. Solve using two-step process: Solve most of the way using basic IDP . 1 Solve some more with smoothed IDP . 2 The two-step results?
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Stagewise Linear Continuous Controls Why not try a stagewise linear discretization of controls?
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Stagewise Linear Continuous Controls Why not try a stagewise linear discretization of controls? Animation 9: Continuous linear controls (show every stage)
First-order Filter Simulated Annealing Filter Speeding up IDP Damping Smoothing IDP Pivot Point Test Controls Linear Controls Stagewise Linear Continuous Controls Why not try a stagewise linear discretization of controls? Animation 9: Continuous linear controls (show every stage) Useful; seems to obtain equivalent Φ but smoother.
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