Modern_Control

MATLAB Modern Control Series - Concept Index

This document provides a comprehensive mapping of modern control theory concepts to specific scripts and sections where they are taught.

Table of Contents

MATLAB Functions Quick Reference

Function Purpose Primary Location Alternative Locations
eig Eigenvalues and eigenvectors S01 §1.6 S04 §4.1
inv Matrix inverse S01 §1.2 -
det Determinant S01 §1.2 -
rank Matrix rank S01 §1.2 S03 §3.1, §3.2
trace Matrix trace S01 §1.2 -
norm Vector/matrix norms S01 §1.3 -
null Null space S01 §1.5 -
orth Orthonormal basis S01 §1.5 -
dot Dot product S01 §1.5 -
linsolve Solve linear equations S01 §1.5 -
triu, tril Triangular matrices S01 §1.4 -
poly Characteristic polynomial S01 §1.6 -
jordan Jordan normal form S01 §1.8 -
expm Matrix exponential S01 §1.9 S02 §2.4
ss State-space model S02 §2.1 Throughout
tf Transfer function S02 §2.2 Throughout
ss2tf State-space to transfer function S02 §2.2 S03 §3.5
tf2ss Transfer function to state-space S02 §2.2 S03 §3.9
c2d Continuous to discrete S02 §2.6 S04 §4.10
d2c Discrete to continuous S02 §2.6 S04 §4.10
step Step response S02 §2.3 Throughout
impulse Impulse response S02 §2.3 -
initial Initial condition response S02 §2.3 -
lsim Arbitrary input response S02 §2.3 -
ode45 Numerical ODE solver S02 §2.4 -
jacobian Symbolic Jacobian S02 §2.8 S02 §2.9
laplace Laplace transform S02 §2.5 -
ilaplace Inverse Laplace S02 §2.5 -
subs Symbolic substitution S02 §2.8 -
stepinfo Step response metrics S02 §2.3 -
residue Partial fraction expansion S02 §2.5 -
ctrb Controllability matrix S03 §3.1 -
obsv Observability matrix S03 §3.2 -
gram Controllability/observability gramians S03 §3.4 -
canon Canonical forms S03 §3.6 -
ss2ss State transformation S03 §3.7 -
minreal Minimal realization S03 §3.5 S03 §3.9
balreal Balanced realization S03 §3.8 -
lyap Continuous Lyapunov equation S04 §4.2 -
dlyap Discrete Lyapunov equation S04 §4.4 -
chol Cholesky decomposition S04 §4.3 -
place Pole placement S04 §4.6 -
acker Ackermann’s formula S04 §4.7 -
lqr Linear quadratic regulator S04 §4.8 -
dlqr Discrete LQR S04 §4.10 -
care Continuous Riccati equation S04 §4.9 -
pole System poles S04 §4.5 -

Concepts by Category

1. Mathematical Foundations

Concept MATLAB Function Script Section
Vector operations Basic ops S01 §1.1
Matrix multiplication *, .* S01 §1.1
Matrix transpose .', ' S01 §1.1
Determinant det S01 §1.2
Matrix rank rank S01 §1.2
Matrix trace trace S01 §1.2
Matrix inverse inv S01 §1.2
Condition number cond S01 §1.2
Vector norms norm S01 §1.3
Matrix norms (1, 2, ∞, Fro) norm S01 §1.3
Triangular matrices triu, tril S01 §1.4
Null space null S01 §1.5
Orthogonal basis orth S01 §1.5
Dot product dot S01 §1.5
Linear equation solving \, linsolve S01 §1.5
Eigenvalues eig S01 §1.6
Eigenvectors eig S01 §1.6
Characteristic polynomial poly S01 §1.6
Similarity transformation Manual S01 §1.7
Jordan normal form jordan S01 §1.8
Matrix exponential expm S01 §1.9

2. System Modeling

Concept MATLAB Function Script Section
ODE to state-space Manual S02 §2.1
State-space object ss S02 §2.1, §2.2
Transfer function object tf S02 §2.2
SS to TF conversion ss2tf S02 §2.2
TF to SS conversion tf2ss S02 §2.2
Step response step S02 §2.3
Impulse response impulse S02 §2.3
Initial condition response initial S02 §2.3
General response lsim S02 §2.3
Step response metrics stepinfo S02 §2.3
State transition matrix expm S02 §2.4
ODE simulation ode45 S02 §2.4
Laplace transform laplace S02 §2.5
Inverse Laplace ilaplace S02 §2.5
Partial fractions residue S02 §2.5
Continuous to discrete c2d S02 §2.6
Discrete to continuous d2c S02 §2.6
Equilibrium points Symbolic S02 §2.7
Jacobian linearization jacobian S02 §2.8
Symbolic substitution subs S02 §2.8, §2.9
Physical system examples Manual S02 §2.10

3. Structural Properties

Concept MATLAB Function Script Section
Controllability matrix ctrb S03 §3.1
Controllability rank test rank(ctrb(...)) S03 §3.1
Observability matrix obsv S03 §3.2
Observability rank test rank(obsv(...)) S03 §3.2
PBH controllability test Manual S03 §3.3
PBH observability test Manual S03 §3.3
Controllability gramian gram(..., 'c') S03 §3.4
Observability gramian gram(..., 'o') S03 §3.4
Gramian positive definiteness eig S03 §3.4
Kalman decomposition Manual/minreal S03 §3.5
Minimal realization minreal S03 §3.5, §3.9
Controllable canonical form canon(...,'companion') S03 §3.6
Observable canonical form Dual transform S03 §3.6
Modal canonical form canon(...,'modal') S03 §3.6
State transformation ss2ss S03 §3.7
Balanced realization balreal S03 §3.8
Hankel singular values balreal S03 §3.8
Model reduction balreal + truncation S03 §3.8

4. Stability and Control

Concept MATLAB Function Script Section
Eigenvalue stability (CT) eig S04 §4.1
Eigenvalue stability (DT) eig S04 §4.1
Lyapunov stability theory Theory S04 §4.2
Lyapunov equation (CT) lyap S04 §4.2
Lyapunov equation (DT) dlyap S04 §4.4
Matrix definiteness chol, eig S04 §4.3
Cholesky decomposition chol S04 §4.3
BIBO stability pole S04 §4.5
State feedback Theory S04 §4.6
Pole placement place S04 §4.6
Ackermann’s formula acker S04 §4.7
LQR optimal control lqr S04 §4.8
Cost function tuning (Q, R) Experimentation S04 §4.8
Continuous Riccati (CARE) care S04 §4.9
Discrete Riccati (DARE) dare S04 §4.9
Digital controller design dlqr S04 §4.10
Controller discretization c2d S04 §4.10

Detailed Concept Mapping

Comprehensive Concept Coverage

Below is the complete mapping of all required concepts from the syllabus:

Concept MATLAB Function/Tool Script Section Notes
Vector/matrix operations Basic ops, *, .*, .^ S01 §1.1 Element-wise and matrix operations
Eigenvalues eig S01 §1.6 Eigenvalues and eigenvectors
Similarity transformation inv, manual S01 §1.7 $B = T^{-1}AT$
Linear equation solutions \, linsolve S01 §1.5 Ax = b
Basis and orthonormality null, orth, dot, norm S01 §1.5 Null space, orthogonal basis
State-space modeling (CT) ss S02 §2.1 Continuous-time systems
State-space modeling (DT) ss, c2d S02 §2.6 Discrete-time systems
Transfer function ↔ SS ss2tf, tf2ss S02 §2.2 Conversion between representations
State trajectory expm, ode45, initial S02 §2.4 Time evolution of states
Controllability ctrb, rank S03 §3.1 Rank test and PBH test
Observability obsv, rank S03 §3.2 Rank test and PBH test
Kalman decomposition Custom/minreal S03 §3.5 Structural decomposition
Minimal realization minreal, balreal S03 §3.5, §3.8 Remove uncontrollable/unobservable modes
Canonical forms canon, ss2ss S03 §3.6 Controllable, observable, modal forms
Stability (matrix) eig S04 §4.1 Eigenvalue criterion
Lyapunov stability lyap, dlyap S04 §4.2, §4.4 Lyapunov equation
BIBO stability pole S04 §4.5 Transfer function poles
Definiteness tests chol, eig S04 §4.3 Positive definite matrices
State feedback Theory S04 §4.6 u = -Kx
Pole placement place, acker S04 §4.6, §4.7 Eigenvalue assignment
LQR optimal control lqr S04 §4.8 Minimize J with Q, R
Riccati equation care, dare S04 §4.9 Algebraic Riccati equation
Simulation ode45, lsim, sim S02 §2.4 Simulate control systems
Discretization c2d, d2c S02, S04 §2.6, §4.10 ZOH, Tustin, matched
Digital control dlqr, c2d S04 §4.10 Discrete controller design
Model realization tf2ss, minreal, canon S03 §3.6, §3.9 Different realizations
Gain tuning lqr, experiments S04 §4.8 Q, R matrix selection
Physical examples Scripts S02 §2.10 Pendulum, motor, etc.

Quick Lookup by Application

For Specific Tasks

Task Script Functions Notes
Model a system from physics S02 ss, jacobian Derive state-space from ODEs
Check if controllable S03 ctrb, rank Rank test
Check if observable S03 obsv, rank Rank test
Simplify a model S03 minreal, balreal Remove extra states
Check stability S04 eig, lyap Multiple methods
Design controller S04 place, lqr Pole placement or optimal
Implement digitally S04 c2d, dlqr Discretize for computer

Cross-Reference by MATLAB Function

Function Primary Use Example Call Where Taught
eig(A) Eigenvalues of A lambda = eig(A) S01 §1.6
expm(A*t) Matrix exponential Phi = expm(A*t) S01 §1.9
ss(A,B,C,D) Create state-space sys = ss(A,B,C,D) S02 §2.1
ctrb(A,B) Controllability matrix Co = ctrb(A,B); rank(Co) S03 §3.1
obsv(A,C) Observability matrix Ob = obsv(A,C); rank(Ob) S03 §3.2
place(A,B,p) Pole placement K = place(A,B,poles) S04 §4.6
lqr(A,B,Q,R) LQR controller [K,S,e] = lqr(A,B,Q,R) S04 §4.8
lyap(A,Q) Solve Lyapunov eqn P = lyap(A',Q) S04 §4.2

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