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TOMLAB /PROPT Manual
Contents
1 PROPT Guide Overview
2 Introduction to PROPT
3 Modeling optimal control problems
4 Multi-phase optimal control
5 Scaling of optimal control problems
6 Setting solver and options
7 Solving optimal control problems
8 | | OPTIMAL CONTROL EXAMPLES | |
9 Acrobot
10 A Linear Problem with Bang Bang Control
11 Batch Fermentor
12 Batch Production
13 Batch Reactor Problem
14 The Brachistochrone Problem
15 The Brachistochrone Problem (DAE formulation)
16 Bridge Crane System
17 Bryson-Denham Problem
18 Bryson-Denham Problem (Detailed)
19 Bryson-Denham Problem (Short version)
20 Bryson-Denham Two Phase Problem
21 Bryson Maxrange
22 Catalyst Mixing
23 Catalytic Cracking of Gas Oil
24 Flow in a Channel
25 Coloumb Friction 1
26 Coloumb Friction 2
27 Continuous State Constraint Problem
28 Curve Area Maximization
29 Denbigh’s System of Reactions
30 Dielectrophoresis Particle Control
31 Disturbance Control
32 Drug Displacement Problem
33 Optimal Drug Scheduling for Cancer Chemotherapy
34 Euler Buckling Problem
35 MK2 5-Link robot
36 Flight Path Tracking
37 Food Sterilization
38 Free Floating Robot
39 Fuller Phenomenon
40 Genetic 1
41 Genetic 2
42 Global Dynamic System
43 Goddard Rocket, Maximum Ascent
44 Goddard Rocket, Maximum Ascent, Final time free, Singular solution
45 Goddard Rocket, Maximum Ascent, Final time fixed, Singular solution
46 Greenhouse Climate Control
47 Grusins Metric
48 Hang Glider Control
49 Hanging Chain
50 High Dimensional Control
51 Hyper Sensitive Optimal Control
52 Initial Value Problem
53 Isometrization of alpha pinene
54 Isoperimetric Constraint Problem
55 Jumbo Crane Container Control
56 Lee-Ramirez Bioreactor
57 Linear Tangent Steering Problem
58 Linear Gas Absorber
59 Linear Pendulum
60 Linear Problem with Bang Bang Control
61 LQR Problem
62 Marine Population Dynamics
63 Max Radius Orbit Transfer
64 Methanol to Hydrocarbons
65 Min Energy Orbit Transfer
66 Missile Intercept
67 Moonlander Example
68 Nagurka Problem
69 Nishida problem
70 Nondifferentiable system
71 Nonlinear CSTR
72 Obstacle Avoidance
73 Oil Shale Pyrolysis
74 One Dimensional Rocket Ascent
75 Parametric Sensitivity Control
76 Parallel Reactions in Tubular Reactor
77 Parameter Estimation Problem
78 Park-Ramirez bioreactor
79 Path Tracking Robot
80 Path Tracking Robot (Two-Phase)
81 Pendulum Gravity Estimation
82 Penicillin Plant
83 Plug-Flow Tubular Reactor
84 Quadratic constraint problem
85 Quadruple Integral
86 Radio telescope
87 Rayleigh Unconstrained
88 Rigid Body Rotation
89 Robot Arm Movement
90 Time-optimal Trajectories for Robot Manipulators
91 Satellite Control
92 Second Order System
93 Simple Bang Bang Problem
94 Singular Arc Problem
95 Singular CSTR
96 Singular Control 1
97 Singular Control 2
98 Singular Control 3
99 Singular Control 4
100 Singular Control 5
101 Singular Control 6
102 Spring Mass Damper (2 Degree Freedom)
103 Stirred Tank
104 Temperature Control
105 A Simple Terminal Constraint Problem
106 Third order system
107 Time Delay 1
108 Time Delay 1 (Approximate)
109 Time Delay 2
110 Time Delay 2 (Approximate)
111 Transfer Min Swing
112 Tubular Reactor
113 Turbo Generator
114 Two-Link Robot
115 Two-Link Robotic Arm
116 Two-Phase Schwartz
117 Two Stage CSTR
118 Van der Pol Oscillator
119 Zermelos problem (version 1)
120 Zermelos problem (version 2)
References
Notes
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Contents
1 PROPT Guide Overview
1.1 Installation
1.2 Foreword to the software
1.3 Initial remarks
2 Introduction to PROPT
2.1 Overview of PROPT syntax
2.2 Vector representation
2.3 Global optimality
3 Modeling optimal control problems
3.1 A simple example
3.2 Code generation
3.3 Modeling
3.3.1 Modeling notes
3.4 Independent variables, scalars and constants
3.5 State and control variables
3.6 Boundary, path, event and integral constraints
4 Multi-phase optimal control
5 Scaling of optimal control problems
6 Setting solver and options
7 Solving optimal control problems
7.1 Standard functions and operators
7.1.1 collocate — Expand a propt tomSym to all collocation points on a phase.
7.1.2 tomSym/dot
7.1.3 final — Evaluate a propt tomSym at the final point of a phase.
7.1.4 icollocate — Expand a propt tomSym to all interpolation points on a phase
7.1.5 initial — Evaluate a propt tomSym at the initial point of a phase.
7.1.6 integrate — Evaluate the integral of an expression in a phase.
7.1.7 mcollocate — Expand to all collocation points, endpoints and midpoints on a phase.
7.1.8 setPhase — Set the active phase when modeling PROPT problem.
7.1.9 tomControl — Generate a PROPT symbolic state.
7.1.10 tomControls — Create tomControl objects
7.1.11 tomPhase — Create a phase struct
7.1.12 tomState — Generate a PROPT symbolic state
7.1.13 tomStates — Create tomState objects as toms create tomSym objects
7.2 Advanced functions and operators
7.2.1 atPoints — Expand a propt tomSym to a set of points on a phase.
7.2.2 interp1p — Polynomial interpolation.
7.2.3 proptGausspoints — Generate a list of gauss points.
7.2.4 proptDiffMatrix — Differentiation matrix for interpPoly.
7.3 Screen output
7.4 Solution structure
7.5 Plotting
8 | | OPTIMAL CONTROL EXAMPLES | |
9 Acrobot
9.1 Problem Formulation
10 A Linear Problem with Bang Bang Control
10.1 Problem description
10.2 Problem setup
10.3 Solve the problem
10.4 Plot result
11 Batch Fermentor
11.1 Problem description
11.2 Solving the problem on multiple grids.
11.3 Plot result
12 Batch Production
12.1 Problem description
12.2 Problem setup
12.3 Solve the problem
12.4 Plot result
13 Batch Reactor Problem
13.1 Problem description
13.2 Problem setup
13.3 Solve the problem
13.4 Plot result
14 The Brachistochrone Problem
14.1 Problem description
14.2 Problem setup
14.3 Solve the problem
14.4 Plot the result
15 The Brachistochrone Problem (DAE formulation)
15.1 DAE formulation
15.2 Problem setup
15.3 Solve the problem
15.4 Plot the result
16 Bridge Crane System
16.1 Problem description
16.2 Problem setup
16.3 Solve the problem
16.4 Plot result
17 Bryson-Denham Problem
17.1 Problem description
17.2 Problem setup
17.3 Solve the problem
17.4 Plot result
18 Bryson-Denham Problem (Detailed)
18.1 Problem description
18.2 Define the independent variable, and phase:
18.3 A few constants.
18.4 Define a list of states
18.5 Adding the control variable and equations
18.6 Equations
18.7 Build the .m files and general TOMLAB problem
19 Bryson-Denham Problem (Short version)
19.1 Problem description
19.2 Problem setup
20 Bryson-Denham Two Phase Problem
20.1 Problem description
20.2 Problem setup
20.3 Solve the problem
20.4 Plot the result
21 Bryson Maxrange
21.1 Problem description
21.2 Problem setup
21.3 Solve the problem
21.4 Plot result
22 Catalyst Mixing
22.1 Problem formulation
22.2 Problem setup
22.3 Solve the problem
22.4 Plot result
23 Catalytic Cracking of Gas Oil
23.1 Problem Formulation
23.2 Problem setup
23.3 Solve the problem
23.4 Plot result
24 Flow in a Channel
24.1 Problem Formulation
24.2 Problem setup
24.3 Solve the problem
24.4 Plot result
25 Coloumb Friction 1
25.1 Problem Formulation
25.2 Problem setup
25.3 Solve the problem
25.4 Plot result
26 Coloumb Friction 2
26.1 Problem Formulation
26.2 Problem setup
26.3 Solve the problem
26.4 Plot result
27 Continuous State Constraint Problem
27.1 Problem description
27.2 Problem setup
27.3 Solve the problem
27.4 Plot result
28 Curve Area Maximization
28.1 Problem Description
28.2 Problem setup
28.3 Solve the problem
28.4 Plot result
29 Denbigh’s System of Reactions
29.1 Problem description
29.2 Problem setup
29.3 Solve the problem, using a successively larger number collocation points
29.4 Solve the problem
29.5 Plot result
30 Dielectrophoresis Particle Control
30.1 Problem Description
30.2 Problem setup
30.3 Solve the problem
30.4 Plot result
31 Disturbance Control
31.1 Problem Description
31.2 Problem setup
31.3 Solve the problem
31.4 Plot result
32 Drug Displacement Problem
32.1 Problem Formulation
32.2 Problem setup
32.3 Solve the problem
32.4 Plot result
33 Optimal Drug Scheduling for Cancer Chemotherapy
33.1 Problem description
33.2 Problem setup
33.3 Solve the problem
33.4 Plot result
34 Euler Buckling Problem
34.1 Problem description
34.2 Problem setup
34.3 Solve the problem
34.4 Plot result
34.5 Footnote
35 MK2 5-Link robot
35.1 Problem description
35.2 Problem setup
35.3 Solve the problem
36 Flight Path Tracking
36.1 Problem Description
36.2 Problem setup
36.3 Solve the problem
36.4 Plot result
37 Food Sterilization
37.1 Problem description
37.2 Problem setup
37.3 Solve the problem
38 Free Floating Robot
38.1 Problem description
38.2 Problem setup
38.3 Solve the problem
38.4 Plot result
39 Fuller Phenomenon
39.1 Problem Description
39.2 Problem setup
39.3 Solve the problem
39.4 Plot result
40 Genetic 1
40.1 Problem Formulation
40.2 Problem setup
40.3 Solve the problem
40.4 Plot result
41 Genetic 2
41.1 Problem Formulation
41.2 Problem setup
41.3 Solve the problem
41.4 Plot result
42 Global Dynamic System
42.1 Problem Description
42.2 Problem setup
42.3 Solve the problem
42.4 Plot result
43 Goddard Rocket, Maximum Ascent
43.1 Problem Formulation
43.2 Problem setup
43.3 Solve the problem, using a successively larger number collocation points
43.4 Solve the problem
43.5 Plot result
44 Goddard Rocket, Maximum Ascent, Final time free, Singular solution
44.1 Problem setup
44.2 Solve the problem, using a successively larger number collocation points
44.3 Solve the problem
44.4 Plot result
45 Goddard Rocket, Maximum Ascent, Final time fixed, Singular solution
45.1 Problem setup
45.2 Solve the problem, using a successively larger number of collocation points
45.3 Solve the problem
45.4 Plot result
46 Greenhouse Climate Control
46.1 Problem description
46.2 Problem setup
46.3 Solve the problem
47 Grusins Metric
47.1 Problem Description
47.2 Problem setup
47.3 Solve the problem
47.4 Plot result
48 Hang Glider Control
48.1 Problem Formulation
48.2 Problem setup
48.3 Solve the problem
48.4 Extract optimal states and controls from solution
48.5 Plot result
49 Hanging Chain
49.1 Problem Formulation
49.2 Problem setup
49.3 Solve the problem
49.4 Plot result
50 High Dimensional Control
50.1 Problem description
50.2 Problem setup
50.3 Solve the problem
50.4 Plot result
51 Hyper Sensitive Optimal Control
51.1 Problem Formulation
51.2 Problem setup
51.3 Solve the problem
51.4 Plot result
52 Initial Value Problem
52.1 Problem Description
52.2 Problem setup
52.3 Solve the problem
52.4 Plot result
53 Isometrization of alpha pinene
53.1 Problem Formulation
53.2 Problem setup
53.3 Solve the problem, using a successively larger number collocation points
53.4 Solve the problem
53.5 Plot result
54 Isoperimetric Constraint Problem
54.1 Problem Formulation
54.2 Problem setup
54.3 Solve the problem
54.4 Plot result
55 Jumbo Crane Container Control
55.1 Problem description
55.2 Problem setup
55.3 Solve the problem
56 Lee-Ramirez Bioreactor
56.1 Problem description
56.2 Problem setup
56.3 Solve the problem
56.4 Plot result
57 Linear Tangent Steering Problem
57.1 Problem Formulation
57.2 Problem setup
57.3 Solve the problem
57.4 Plot result
58 Linear Gas Absorber
58.1 Problem description
58.2 Problem setup
58.3 Solve the problem
58.4 Plot result
59 Linear Pendulum
59.1 Problem Description
59.2 Problem setup
59.3 Solve the problem
59.4 Plot result
60 Linear Problem with Bang Bang Control
60.1 Problem description
60.2 Problem setup
60.3 Solve the problem
60.4 Plot result
61 LQR Problem
61.1 Problem Description
61.2 Problem setup
61.3 Solve the problem
61.4 Plot result
62 Marine Population Dynamics
62.1 Problem Formulation
62.2 Problem setup
62.3 Solve the problem, using a successively larger number collocation points
62.4 Solve the problem
62.5 Plot result
63 Max Radius Orbit Transfer
63.1 Problem description
63.2 Problem setup
63.3 Solve the problem
64 Methanol to Hydrocarbons
64.1 Problem Formulation
64.2 Problem setup
64.3 Solve the problem, using a successively larger number collocation points
64.4 Solve the problem
64.5 Plot result
65 Min Energy Orbit Transfer
65.1 Problem description
65.2 Problem setup
65.3 Solve the problem
66 Missile Intercept
66.1 Problem Description
66.2 Problem setup
66.3 Solve the problem
66.4 Plot result
67 Moonlander Example
67.1 Problem description
67.2 Problem setup
67.3 Solve the problem
67.4 Plot result
68 Nagurka Problem
68.1 Problem description
68.2 Problem setup
68.3 Solve the problem
68.4 Plot result
69 Nishida problem
69.1 Problem description
69.2 Problem setup
69.3 Solve the problem
69.4 Plot result
70 Nondifferentiable system
70.1 Problem description
70.2 Problem setup
70.3 Solve the problem
70.4 Plot result
71 Nonlinear CSTR
71.1 Problem description
71.2 Problem setup
71.3 Solve the problem, using a successively larger number collocation points
71.4 Solve the problem
71.5 Plot result
72 Obstacle Avoidance
72.1 Problem Formulation
72.2 Solve the problem, using a successively larger number collocation points
72.3 Solve the problem
72.4 Plot result
73 Oil Shale Pyrolysis
73.1 Problem description
73.2 Problem setup
73.3 Solve the problem
73.4 Plot result
74 One Dimensional Rocket Ascent
74.1 Problem Formulation
74.2 Problem setup
74.3 Solve the problem
74.4 Plot result
75 Parametric Sensitivity Control
75.1 Problem description
75.2 Problem setup
75.3 Solve the problem
76 Parallel Reactions in Tubular Reactor
76.1 Problem description
76.2 Problem setup
76.3 Solve the problem
76.4 Plot result
77 Parameter Estimation Problem
77.1 Problem description
77.2 Problem setup
77.3 Solve the problem, using a successively larger number collocation points
77.4 Solve the problem
77.5 Plot result
78 Park-Ramirez bioreactor
78.1 Problem description
78.2 Problem setup
78.3 Solve the problem, using a successively larger number collocation points
78.4 Solve the problem
78.5 Plot result
79 Path Tracking Robot
79.1 Problem Formulation
79.2 Problem setup
79.3 Solve the problem
79.4 Plot result
80 Path Tracking Robot (Two-Phase)
80.1 Problem Formulation
80.2 Problem setup
80.3 Solve the problem
80.4 Plot result
81 Pendulum Gravity Estimation
81.1 Problem Formulation
81.2 Problem setup
81.3 Solve the problem
81.4 Show result
82 Penicillin Plant
82.1 Problem description
82.2 Problem setup
82.3 Plot result
83 Plug-Flow Tubular Reactor
83.1 Problem description
83.2 Problem setup
83.3 Solve the problem
83.4 Plot result
84 Quadratic constraint problem
84.1 Problem Formulation
84.2 Problem setup
84.3 Solve the problem
84.4 Plot result
85 Quadruple Integral
85.1 Problem Formulation
85.2 Problem setup
85.3 Solve the problem
85.4 Plot result
86 Radio telescope
86.1 Problem description
86.2 Problem setup
86.3 Solve the problem
87 Rayleigh Unconstrained
87.1 Problem Formulation
87.2 Problem setup
87.3 Solve the problem
87.4 Plot result
88 Rigid Body Rotation
88.1 Problem Description
88.2 Problem setup
88.3 Solve the problem
88.4 Plot result
89 Robot Arm Movement
89.1 Problem Formulation
89.2 Problem setup
89.3 Solve the problem
89.4 Plot result
90 Time-optimal Trajectories for Robot Manipulators
90.1 Problem Formulation
90.2 Problem setup
90.3 Solve the problem, using a successively larger number collocation points
90.4 Solve the problem
90.5 Plot result
91 Satellite Control
91.1 A satellite control problem
91.2 Problem setup
91.3 Problem 7b and 7c modifications
91.4 Solve the problem
91.5 Plot result
92 Second Order System
92.1 Problem Formulation
92.2 Problem setup
92.3 Solve the problem
92.4 Plot result
93 Simple Bang Bang Problem
93.1 Problem Description
93.2 Problem setup
93.3 Solve the problem
93.4 Plot result
94 Singular Arc Problem
94.1 Problem Formulation
94.2 Problem setup
94.3 Solve the problem
94.4 Plot result
95 Singular CSTR
95.1 Problem Formulation
95.2 Problem setup
95.3 Solve the problem
95.4 Plot result
96 Singular Control 1
96.1 Problem Formulation
96.2 Problem setup
96.3 Solve the problem
96.4 Plot result
97 Singular Control 2
97.1 Problem Formulation
97.2 Problem setup
97.3 Solve the problem
97.4 Plot result
98 Singular Control 3
98.1 Problem Formulation
98.2 Problem setup
98.3 Solve the problem
98.4 Plot result
99 Singular Control 4
99.1 Problem Formulation
99.2 Problem setup
99.3 Solve the problem
99.4 Plot result
100 Singular Control 5
100.1 Problem Formulation
100.2 Problem setup
100.3 Solve the problem
100.4 Plot result
101 Singular Control 6
101.1 Problem Description
101.2 Problem setup
101.3 Solve the problem
101.4 Plot result
102 Spring Mass Damper (2 Degree Freedom)
102.1 Problem Formulation
102.2 Problem setup
102.3 Solve the problem
102.4 Plot result
103 Stirred Tank
103.1 Problem Description
103.2 Problem setup
103.3 Solve the problem
103.4 Plot result
104 Temperature Control
104.1 Problem Description
104.2 Problem setup
104.3 Solve the problem
104.4 Plot result
105 A Simple Terminal Constraint Problem
105.1 Problem Description
105.2 Problem setup
105.3 Solve the problem
105.4 Plot result
106 Third order system
106.1 Problem description
106.2 Problem setup
106.3 Solve the problem
107 Time Delay 1
107.1 Problem Formulation
107.2 Problem setup
107.3 Solve the problem
107.4 Plot result
108 Time Delay 1 (Approximate)
108.1 Problem Formulation
108.2 Problem setup
108.3 Solve the problem
108.4 Plot result
109 Time Delay 2
109.1 Problem Formulation
109.2 Problem setup
109.3 Solve the problem
109.4 Plot result
110 Time Delay 2 (Approximate)
110.1 Problem Formulation
110.2 Problem setup
110.3 Solve the problem
110.4 Plot result
111 Transfer Min Swing
111.1 Problem Formulation
111.2 Problem setup
111.3 Solve the problem, using a successively larger number collocation points
111.4 Solve the problem
111.5 Plot result
112 Tubular Reactor
112.1 Problem Description
112.2 Problem setup
112.3 Plot result
113 Turbo Generator
113.1 Problem Formulation
113.2 Problem setup
113.3 Solve the problem
113.4 Plot result
114 Two-Link Robot
114.1 Problem description
114.2 Problem setup
114.3 Solve the problem
115 Two-Link Robotic Arm
115.1 Problem Formulation
115.2 Problem setup
115.3 Solve the problem
115.4 Plot result
116 Two-Phase Schwartz
116.1 Problem Formulation
116.2 Problem setup
116.3 Solve the problem
116.4 Plot result
117 Two Stage CSTR
117.1 Problem Description
117.2 Problem setup
117.3 Solve the problem
117.4 Plot result
118 Van der Pol Oscillator
118.1 Problem Description
118.2 Problem setup
118.3 Solve the problem
118.4 Plot result
119 Zermelos problem (version 1)
119.1 Problem description
119.2 Problem setup
119.3 Solve the problem
120 Zermelos problem (version 2)
120.1 Problem description
120.2 Problem setup
120.3 Solve the problem
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