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Optimal Control Design
Prof. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
Acknowledgement:
Indian Institute of Science
Founded in 1909 – more than
100 years old…
Founded by J. N. Tata (in
consultation with Swami
Vivekananda) – land was
donated by Mysore king.
Deemed University in 1958
More than 40 departments
Ranked No.1 in India for higher
education
Only institute in India among
best 100 in global ranking
20 September 2016
For further information,
please visit www.iisc.ernet.in
Prof. Radhakant Padhi, IISc-Bangalore
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Collaboration & Research Funding
Defense R&D Organisation (DRDO)
• Missile Complex (ASL, RCI, DRDL, ANURAG)
• ARDE
• CAIR
Indian Space Research Organisation (ISRO)
• VSSC
• ISAC
Air Force Research Lab (AFRL), USA
Private Aerospace Companies
• Coral Digital Technologies
• Team Indus (Axiom Research Lab)
20 September 2016
Prof. Radhakant Padhi, IISc-Bangalore
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Integrated Control Guidance and Estimation Lab (ICGEL)
& Aerospace Systems Lab (ASL)
Research Areas in ICGEL:
Guidance and Control
of Missiles
• Nonlinear, Optimal & Adaptive Control
• Dynamic Inversion & Neuro-Adaptive Designs
• Single Network Adaptive Critic (SNAC)
Guidance and control
of UAVs
• MPSP and it variants are
used to develop optimal
guidance algorithm for
better performance .
Examples:
• Impact Angle Constrained
Guidance of Tactical
Missiles
• Integrated Guidance and
Control for Missiles for
Ballistic Missile Defence
Nonlinear & NeuroAdaptive Control of
High-Perf. Aircrafts
• Model Predictive Static Programming (MPSP)
• Online Modified (OM) Design for Enhanced Robustness
• State Estimation for Feedback Guidance & Control
Formation Flying and
Attitude Control of
Satellites
• Drug is delivered as per
patient’s condition (not in
open loop) - Fast recovery
& Reduced side effects
• Demonstrated for blood
cancer, diabetes regulation
& Milk-fever of cows
A new robust nonlinear
approach is developed
for better control of high
performance (large L/D)
aircrafts, which are
unstable in nature.
• Guidance and Control for
automatic landing.
• Stereo Vision based
reactive collision avoidance
using ultra low-cost
cameras
• Nonlinear differential
geometric guidance for
collision avoidance
Contact:
Prof. Radhakant Padhi
E. mail: [email protected]
Feedback Control for
Customized Automatic
Drug Delivery
• Robust Formation flying
of satellites using online
modified real-time
optimal control
• Robust large attitude
maneuvers of satellites in
presence of significant
modelling errors
Dept. of Aerospace Engineering
Indian Institute of Science, Bangalore
Optimal Process Control
Current Team (2016)
13 Ph.D. Students, 1 Master Student
2 Project Associates, 2 Project Assistants
(many more in the past)
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Acknowledgement:
Graduated Students & Other Co-workers
Mangal Kothari (Faculty in IIT-Kanpur)
Arnab Maity (Faculty in IIT-Bombay)
Sk. Faruque Ali (Faculty in IIT-Madras)
Gurunath Gurala (Faculty in IISc-Bangalore)
Harshal Oza (Faculty in Ahmedabad Univ., Ahmedabad)
Prasiddha Nath Dwivedi (Scientist in DRDO, Hyderabad)
Prem Kumar (Scientist in DRDO, Hyderabad)
Girish Joshi (Former scientist in ISRO, doing his Ph.D. in USA)
Kapil Sachan (currently a Ph.D. student)
Avijit Banerjee (currently a Ph.D. student)
Omkar Halbe (Working in EADS)
Charu Chawla (Working in a Pvt. Company)…and many more!
20 September 2016
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore
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Outline
Lecture – 1
• Generic Overview of Optimal Control Theory
Lecture – 2
• Real-time Optimal Control using MPSP
Lecture – 3
• Solution of Challenging Practical Problems
using MPSP
Prof. Radhakant Padhi, AE Dept., IISc-Bangalore
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Lecture – 1
An Overview of Optimal Control Design
Prof. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
Why Optimal Control?
Summary of Benefits
A variety of difficult real-life problems can be
formulated in the framework of optimal control.
State and control bounds can be incorporated
in the control design process explicitly.
Incorporation of optimal issues lead to a
variety of advantages, like minimum cost,
maximum efficiency, non-conservative design
etc.
Trajectory planning issues can be incorporated
into the guidance and control design.
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Role of Optimal Control
Question: What is R(s)? How to design it??
Unfortunately, books remain completely silent on this!
Optimization
(Optimal Control)
Optimization
(Optimal
Control)
Mission Objectives
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A Tribute to
Pioneers of Optimal Control
1700s
•
•
•
Bernoulli, Newton
Euler (Student of Bernoulli)
Lagrange
Newton
Bernoulli
Euler
....200 years later....
Lagrange
1900s
•
•
•
Pontryagin
Bellman
Kalman
Pontryagin
Bellman
Kalman
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An Interesting Observation
Euler (1726) - Lagrange - Fourier - Dirichlet Lipschitz - Klein [1A]
Euler (1726) - Lagrange - Poisson - Dirichlet Lipschitz - Klein [1B]
Gauss (1799) - Gerling - Pluecker - Klein
[2]
>> Klein - Lindeman - Hilb - Baer - Liepman Bryson - Speyer - Bala - Padhi [3]
Gauss (1799) - Bessel - Scherk - Kummer - Prym Rost - Baer - Liepman - Bryson - Speyer - Bala Padhi [4]
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Optimal control formulation:
Key components
An optimal control formulation consists of:
• Performance index that needs to be optimized
• Appropriate boundary (initial & final) conditions
• Hard constraints
• Soft constraints
• Path constraints
• System dynamics constraint (nonlinear in general)
• State constraints
• Control constraints
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Optimal Control Problem
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Meaningful Performance Index
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Meaningful Performance Index
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Optimum of a Functional
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Fundamental Theorem of
Calculus of Variations
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Fundamental Lemma
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Optimal Control Problem
Performance Index (to minimize / maximize):
Path Constraint:
Boundary Conditions:
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Necessary Conditions of
Optimality
Augmented PI
Hamiltonian
First Variation
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Necessary Conditions of
Optimality
First Variation
Individual terms
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Necessary Conditions of
Optimality
0
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Necessary Conditions of
Optimality
First Variation
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Necessary Conditions of
Optimality
First Variation
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Necessary Conditions of
Optimality: Summary
State Equation
Costate Equation
Optimal Control
Equation
Boundary Condition
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Necessary Conditions of
Optimality: Some Comments
State and Costate equations are dynamic equations. If
one is stable, the other turns out to be unstable!
Optimal control equation is a stationary equation
Boundary conditions are split: it leads to Two-PointBoundary-Value Problem (TPBVP)
State equation develops forward whereas Costate
equation develops backwards.
It is known as “Curse of Complexity” in optimal control
Traditionally, TPBVPs demand computationally-intensive
iterative numerical procedures, which lead to “open-loop”
control structure.
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General
Boundary/Transversality Condition
General condition:
Special Cases:
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Example – 1: A Toy Problem
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Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
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Example
Problem:
Solution:
Costate Eq.
Optimal control Eq.
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Example
Boundary Conditions
Define
Solution
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Example
Use the boundary condition at
Use the boundary condition at
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Example
Four equations and four unknowns:
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Example
Solution for State and Costate
Solution for Optimal Control
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Example – 2: Orbit Transfer Problem
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Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
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Example (Maximum Radius Orbit
Transfer at a Given Time)
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Example (Maximum Radius Orbit
Transfer at a Given Time)
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System Dynamics and B.C.
System dynamics
Boundary conditions
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Performance index
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Necessary Condition
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Necessary Condition
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A Classical Numerical Approach for Solving
Optimal Control Problems: Gradient Method
Prof. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
Gradient Method
Assumptions:
• State equation satisfied
• Costate equation satisfied
• Boundary conditions satisfied
Strategy:
• Satisfy the optimal control equation
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Gradient Method
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Gradient Method
After satisfying the state & costate equations
and boundary conditions, we have
Select
This leads to
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Gradient Method
We select
This lead to
Note:
Eventually,
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Gradient Method: Procedure
Assume a control history (not a trivial task)
Integrate the state equation forward
Integrate the costate equation backward
Update the control solution
•
This can either be done at each step while integrating
the costate equation backward or after the integration
of the costate equation is complete
Repeat the procedure until convergence
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Gradient Method: Selection of
Select
so that it leads to a certain
percentage reduction of
Let the percentage be
Then
This leads to
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Dynamic Programming and
Hamilton–
Hamilton–Jacobi–
Jacobi–Bellman (HJB) Theory
Prof. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
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Fundamental Philosophy
Motivation / Objective
To obtain a “state feedback” optimal control solution
Fundamental Theorem
Any part of an optimal trajectory is an optimal trajectory!
B
Optimal path
C
A
Non-optimal path
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Optimal Control Problem
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Hamilton–Jacobi–Bellman (HJB)
Equation
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Hamilton–Jacobi–Bellman (HJB)
Equation…contd.
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Hamilton–Jacobi–Bellman (HJB)
Equation…contd.
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Hamilton–Jacobi–Bellman (HJB)
Equation…contd.
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Hamilton–Jacobi–Bellman (HJB)
Equation…contd.
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Hamilton–Jacobi–Bellman (HJB)
Equation…contd.
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Hamilton–Jacobi–Bellman (HJB)
Equation…contd.
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Summary of HJB Equation
Define optimized cost
function V as:
Then V(t) must satisfy:
HJB equation
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Dynamic Programming:
Some Relevant Results
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Dynamic Programming:
Some Relevant Results
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Example: A Benchmark Toy Problem
Prof. Radhakant Padhi
Dept. of Aerospace Engineering
Indian Institute of Science - Bangalore
Example
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Example
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Example
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Example-2
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Dynamic Programming:
Some Important Facts
Dynamic programming is a powerful technique
in the sense that if the HJB equation is solved,
it leads to a “state feedback form” of optimal
control solution.
HJB equation is both necessary and sufficient
for the optimal cost function.
At least one of the control solutions that results
from the solution of the HJB equation is
guaranteed to be stabilizing.
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Dynamic Programming:
Some Important Facts
The resulting PDE of the HJB equation is
extremely difficult to solve in general.
Dynamic Programming runs into a “huge”
Computational and storage requirements for
reasonably higher dimensional problems. This
is a severe restriction of dynamic programming
technique, which Bellman termed as “curse of
dimensionality”.
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Books on Optimal Control Design
R. Padhi, Applied Optimal Control, Wiley, Manuscript
Under Preparation (expected in 2018).
D. S. Naidu, Optimal Control Systems, CRC Press,
2002.
D. E. Kirk, Optimal Control Theory: An Introduction,
Prentice Hall, 1970.
A. E. Bryson and Y-C Ho, Applied Optimal Control,
Taylor and Francis, 1975.
A. P. Sage and C. C. White III, Optimum Systems
Control (2nd Ed.), Prentice Hall, 1977.
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Survey Papers on Classical
Methods for Optimal Control Design
H. J. Pesch (1994), “A Practical Guide to the
Solution of Real-Life Optimal Control Problems”,
Control and Cybernetics, Vol.23, No.1/2, 1994,
pp.7-60.
R. E. Larson (1967), “A Survey of Dynamic
Programming Computational Procedures”, IEEE
Transactions on Automatic Control, December, pp.
767-774.
M. Athans (1966), “The Status of Optimal Control
Theory and Applications for Deterministic Systems”,
IEEE Trans. on Automatic Control, Vol. AC-11, July
1966, pp.580-596.
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Thanks for the Attention….!!
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