Optimal Control Design - iitk.ac.in techniques/ppt/Padhi Lecture... · 2 Project Associates, 2...

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Optimal Control DesignOptimal Control DesignOptimal Control DesignOptimal Control Design

Prof. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant PadhiProf. Radhakant Padhi

Dept. of Aerospace Engineering

Indian Institute of Science - Bangalore

Prof. Radhakant Padhi, IISc-Bangalore2

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

For further information,

please visit www.iisc.ernet.in

20 September 2016

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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)

Prof. Radhakant Padhi, IISc-Bangalore 320 September 2016

Integrated Control Guidance and Estimation Lab (ICGEL)

& Aerospace Systems Lab (ASL)

Research Areas in ICGEL: • Nonlinear, Optimal & Adaptive Control

• Dynamic Inversion & Neuro-Adaptive Designs

• Single Network Adaptive Critic (SNAC)

Guidance and Control

of Missiles

Guidance and control

of UAVs

Feedback Control for

Customized Automatic

Drug Delivery


Indian Institute of Science, Bangalore

Contact:

Prof. Radhakant Padhi

E. mail: [email protected]

• 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

• Guidance and Control for

automatic landing.

• Stereo Vision based

reactive collision avoidance

using ultra low-cost

cameras

• Nonlinear differential

geometric guidance for

collision avoidance

• Model Predictive Static Programming (MPSP)

• Online Modified (OM) Design for Enhanced Robustness

• State Estimation for Feedback Guidance & Control

Nonlinear & Neuro-

Adaptive Control of

High-Perf. Aircrafts

A new robust nonlinear

approach is developed

for better control of high

performance (large L/D)

aircrafts, which are

unstable in nature.

• Robust Formation flying

of satellites using online

modified real-time

optimal control

• Robust large attitude

maneuvers of satellites in

presence of significant

modelling errors

• 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

Formation Flying and

Attitude Control of

Satellites

Current Team (2016)

13 Ph.D. Students, 1 Master Student

2 Project Associates, 2 Project Assistants

(many more in the past)

Optimal Process Control

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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 5

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 6

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Lecture Lecture Lecture Lecture –––– 1111

An Overview of Optimal Control DesignAn Overview of Optimal Control DesignAn Overview of Optimal Control DesignAn Overview of Optimal Control Design




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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.

Prof. Radhakant Padhi, AE Dept., IISc-Bangalore

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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


A Tribute to

Pioneers of Optimal Control

� 1700s

• Bernoulli, Newton

• Euler (Student of Bernoulli)

• Lagrange

....200 years later....

� 1900s

• Pontryagin

• Bellman

• Kalman

Bernoulli

Euler

Lagrange

Pontryagin

BellmanKalman

Newton

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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


Meaningful Performance Index

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Meaningful Performance Index


Optimum of a Functional

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Fundamental Theorem of

Calculus of Variations


Fundamental Lemma

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� Performance Index (to minimize / maximize):

� Path Constraint:

� Boundary Conditions:


Necessary Conditions of

Optimality

� Augmented PI

� Hamiltonian

� First Variation

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Optimality

� First Variation

� Individual terms



Optimality

0

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Optimality

� First Variation



Optimality

� First Variation

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Optimality: Summary

� State Equation

� Costate Equation

� Optimal Control Equation

� Boundary Condition



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-Point-Boundary-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:

Example Example Example Example –––– 1: A Toy Problem1: A Toy Problem1: A Toy Problem1: A Toy Problem




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Example

Problem:

Solution:

Costate Eq.

Optimal control Eq.


Example

Boundary Conditions

Define

Solution

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Example

Use the boundary condition at

Use the boundary condition at


Example

Four equations and four unknowns:

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Example

� Solution for State and Costate

� Solution for Optimal Control

Example Example Example Example –––– 2: Orbit Transfer Problem2: Orbit Transfer Problem2: Orbit Transfer Problem2: Orbit Transfer Problem




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Example (Maximum Radius Orbit

Transfer at a Given Time)


Example (Maximum Radius Orbit

Transfer at a Given Time)

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System Dynamics and B.C.

System dynamics Boundary conditions


Performance index

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Necessary Condition


Necessary Condition

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A Classical Numerical Approach for Solving A Classical Numerical Approach for Solving A Classical Numerical Approach for Solving A Classical Numerical Approach for Solving

Optimal Control Problems: Gradient MethodOptimal Control Problems: Gradient MethodOptimal Control Problems: Gradient MethodOptimal Control Problems: Gradient Method





Gradient Method

� Assumptions:

• State equation satisfied

• Costate equation satisfied

• Boundary conditions satisfied

� Strategy:

• Satisfy the optimal control equation

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Gradient Method


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,


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

Dynamic Programming and Dynamic Programming and Dynamic Programming and Dynamic Programming and

HamiltonHamiltonHamiltonHamilton––––JacobiJacobiJacobiJacobi––––Bellman (HJB) TheoryBellman (HJB) TheoryBellman (HJB) TheoryBellman (HJB) Theory




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Fundamental Philosophy

Fundamental Theorem

Any part of an optimal trajectory is an optimal trajectory!

Motivation / Objective

To obtain a “state feedback” optimal control solution

A

C

B

Non-optimal path

Optimal path



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Hamilton–Jacobi–Bellman (HJB)

Equation



Equation…contd.

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Equation…contd.



Equation…contd.

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Equation…contd.



Equation…contd.

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Equation…contd.


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



Some Relevant Results

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Example: A Benchmark Toy ProblemExample: A Benchmark Toy ProblemExample: A Benchmark Toy ProblemExample: A Benchmark Toy Problem





Example

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Example


Example

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Example-2



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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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”.


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.


Thanks for the Attention….!!

Optimal Control Design - iitk.ac.in techniques/ppt/Padhi Lecture... · 2 Project Associates, 2...

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