E�ective Classroom Teaching of Optimal Control andOptimization using MATLAB

Saket Adhau, Sayli Patil, Dayaram Sonawane

College of Engineering, Pune, India

MATLAB EXPO 2019

March 23, 2019

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Optimal Control

• Finding values of the variables that optimize (minimize or maximize)the objective function while satisfying the constraints

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Ingredients of Optimal Control Problem

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Methods to Solve OCP

AnalyticalMethods –Pontryagins

maximum principle

Numerical Methods

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Numerical Methods to Solve OCP

Optimal Con-trol Problem

Indirect MethodsPontryagin:Solves BVP

Direct MethodsTransformsinto NLP

Hamilton-JacobiBellman Equation:

Multiple Shooting:Controls and

node Start valuesin NLP (Simul-taneous/hybrid)

Collocation:Discretized

Controls andstate in NLP

(Simultaneous)

Single Shooting:Only discretizedControls in NLP

(Sequential)

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Optimal Control Problem Flow Chart

DAEs/ODEsz′(t) = f(t, z, y, u)0 = g(t, z, y, u)

Algebraic EquationsFk(zk+1, zk, yk, uk)

Gk(zk, yk, uk)

Large Non-Linear Programs

Path Constraints andObjective Function

umin ≤ u(k) ≤ umax ,ymin ≤ y(k) ≤ ymax,zmin ≤ z(k) ≤ zmax

Non-LinearProgram

(NLP) SolverOptimal Solution

Temporal Discretization

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Example Problem: Time Optimal Rocket Problem (TimeOptimization Problem)

minU

tf

Subject to

˙s(t) = v(t) ; ˙v(t) =(u(t)− 0.02 ∗ v(t)2)

m(t)

˙m(t) = −0.01 ∗ u(t)2 ; t ∈[0 t

]s(0) = 0; v(0) = 0; m(0) = 1;

s(tf ) = 10; v(tf ) = 0

Bounds

− 0.1 ≤ v ≤ 1.7; −1.1 ≤ u ≤ 1.1; 5 ≤ T ≤ 15

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Approach: MATLAB and Simulink Onramp

To provide a brief introduction to the MATLAB language and to give students hands-on MATLAB experience via the use of an integrated,

web-based version of MATLAB, as shown below.

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Approach: MATLAB and Simulink Onramp Report

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Classroom teaching approach: MATLAB Live Script:Interactive approach

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Teaching Optimal Control using Live Script

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Teaching Optimal Control using Live Script

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Assessing the students using MATLAB Grader

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Assessing the students using MATLAB Grader

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Assessing the students using MATLAB Grader: Report

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Case Study:

• We have a class of 20 students opting for Process Modeling andOptimization.• As an instructor, it is one challenging task to teach students MATLAB

based coding interactively and assessing them individually.• MATLAB has made this task easier than ever, by introducing MATLAB

Live script and MATLAB Grader.

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Optimal Control Problem leads to Model PredictiveControl (MPC)

inputreference

measurement

rt ut yt

xt

processoptimization

output

min

st

Use a dynamical model of the process to predict its future evolution andchoose the “best” control action

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Receding Horizon Implementation

PAST FUTURE

Measured output

Predicted output

Reference trajectory Implemented inputs

Predicted inputs

Current input

t1 t2 t3 dots tN 1

PredictionHorizon

t0

Sample

LEGEND

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Teaching students MPC using Live Editor

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Teaching students MPC using Live Editor

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Teaching students MPC using Live Editor

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Teaching students MPC using Live Editor

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Live Demo using Simulink and Arduino

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Live Demo using Simulink and Arduino

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Live Demo using Simulink and Arduino

The END

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