Nonlinear Control - 3rd Elgersburg School 2011 · Nonlinear Control 3rd Elgersburg School 2011 Linear control of nonlinear systems First pass at controlling a nonlinear system: linearize

NonlinearControl

3rd ElgersburgSchool 2011

Nonlinear Control3rd Elgersburg School 2011

Randy Freeman

Professor

Department of Electrical Engineering & Computer Science

NonlinearControl

3rd ElgersburgSchool 2011 Acknowledgments

Many thanks to the following:

Organizers:Achim IlchmannTimo ReisFabian Wirth

Sponsor:

Exercises and discussions by Dr. Robert Gregg

NonlinearControl

3rd ElgersburgSchool 2011 Linear control of nonlinear systems

First pass at controlling a nonlinear system:linearize the dynamics about an operating point,or use system identification to fit a linear model

use the linear model to design a linear controller

treat any nonlinearities as model uncertainty,to be handled with linear robust or adaptive control

This is not what this short course is about.

NonlinearControl

3rd ElgersburgSchool 2011 Linear control of nonlinear systems

First pass at controlling a nonlinear system:linearize the dynamics about an operating point,or use system identification to fit a linear model

use the linear model to design a linear controller

treat any nonlinearities as model uncertainty,to be handled with linear robust or adaptive control

This is not what this short course is about.

NonlinearControl

3rd ElgersburgSchool 2011 What is nonlinear control?

Nonlinear controlNonlinear control means design and analysis using nonlinearmodels for the plant and/or the controller.

Why nonlinear control?linear models might not capture the relevant behaviorlinear models might be valid only for small signalsnonlinear is more fun than linear!

NonlinearControl

3rd ElgersburgSchool 2011 Goals of this short course

to make you familiar with some of the fundamentalconcepts in nonlinear control

to give you some tools to allow you to try some basicnonlinear controller designs

to give you a starting point for further exploration

Not a comprehensive reviewThis is not a survey course, and we will only cover selectedtopics (some in more depth than others).

Focus of the courseThe focus will be on the mathematical theory and design andanalysis tools, not on practical implementation issues.

NonlinearControl

Nonlinearmodels

Nonlineardifferentialequations

Stability ofequilibria

Stability ofinvariant sets

Stability fortime-varyingsystems

Stability forsystems withinputs


Part I

Nonlinear models and stability

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3rd ElgersburgSchool 2011 Outline

1 Nonlinear models

2 Nonlinear differential equations

3 Stability of equilibria

4 Stability of invariant sets

5 Stability for time-varying systems

6 Stability for systems with inputs

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3rd ElgersburgSchool 2011 Linear models in continuous time

State-space models with state 𝑥(𝑡), input 𝑢(𝑡), and output 𝑦(𝑡):

Time-varying

��(𝑡) = 𝐴(𝑡)𝑥(𝑡) + 𝐵(𝑡)𝑢(𝑡)𝑦(𝑡) = 𝐶(𝑡)𝑥(𝑡) + 𝐷(𝑡)𝑢(𝑡)

Time-invariant

��(𝑡) = 𝐴𝑥(𝑡) + 𝐵𝑢(𝑡)𝑦(𝑡) = 𝐶𝑥(𝑡) + 𝐷𝑢(𝑡)

Input/output models:

Time domain

𝑦(𝑡) = ∫∞

−∞ℎ(𝑡 − 𝜏)𝑢(𝜏) 𝑑𝜏

Frequency domain

𝑌(𝑠) = 𝐻(𝑠)𝑈(𝑠)

Others: operator models, behavioral models, …

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3rd ElgersburgSchool 2011 Continuous-time state-space models

Nonlinear ordinary-differential-equation models:

Time-varying

�� = 𝑓(𝑡, 𝑥, 𝑢)𝑦 = ℎ(𝑡, 𝑥, 𝑢)

Time-invariant

�� = 𝑓(𝑥, 𝑢)𝑦 = ℎ(𝑥, 𝑢)

Control-affine time-varying

�� = 𝑓(𝑡, 𝑥) + 𝑔(𝑡, 𝑥)𝑢𝑦 = ℎ(𝑡, 𝑥) + 𝑗(𝑡, 𝑥)𝑢

Control-affine time-invariant

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢𝑦 = ℎ(𝑥) + 𝑗(𝑥)𝑢

ℝ𝑛 = vector space of 𝑛-tuples of real numbersstate 𝑥 ∈ 𝑋, where 𝑋 is an open subset of ℝ𝑛,or 𝑋 is a differentiable manifold of dimension 𝑛input 𝑢 ∈ ℝ𝑚, output 𝑦 ∈ ℝ𝑝

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3rd ElgersburgSchool 2011 Example: simple pendulum

𝜃

𝑢

ℓ

𝑚

𝑚 = mass of end weightℓ = length of rod𝑔 = acceleration due to gravity𝑏 = damping coefficient𝑢 = input torque

Equations of motion:

𝑚ℓ2 + 𝑏 + 𝑚𝑔ℓ sin( ) = 𝑢

Choose state coordinates 𝑥1 = , 𝑥2 =

��1 = 𝑥2

��2 = −𝑔ℓ

sin(𝑥1) − 𝑏𝑚ℓ2 𝑥2 + 1

𝑚ℓ2 𝑢

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3rd ElgersburgSchool 2011 Linearization about a trajectory

Given: a specific trajectory (��(𝑡), 𝑢(𝑡), 𝑦(𝑡)) with��(𝑡) = 𝑓(𝑡, ��(𝑡), 𝑢(𝑡))𝑦(𝑡) = ℎ(𝑡, ��(𝑡), 𝑢(𝑡))

Define: error variables

𝛿𝑥(𝑡) = 𝑥(𝑡) − ��(𝑡) , 𝛿𝑢(𝑡) = 𝑢(𝑡) − 𝑢(𝑡) , 𝛿𝑦(𝑡) = 𝑦(𝑡) − 𝑦(𝑡)

Compute: linear approximation from Jacobian matrices

𝛿��(𝑡) ≈ [𝜕𝑓𝜕𝑥(𝑡, ��(𝑡), 𝑢(𝑡))]𝛿𝑥(𝑡) + [

𝜕𝑓𝜕𝑢 (𝑡, ��(𝑡), 𝑢(𝑡))]𝛿𝑢(𝑡)

𝛿𝑦(𝑡) ≈ [𝜕ℎ𝜕𝑥(𝑡, ��(𝑡), 𝑢(𝑡))]𝛿𝑥(𝑡) + [

𝜕ℎ𝜕𝑢 (𝑡, ��(𝑡), 𝑢(𝑡))]𝛿𝑢(𝑡)

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Linearization about an equilibriumSimple pendulum example

An equilibrium is a constant trajectory (��, 𝑢, 𝑦), and it satisfies0 = 𝑓(𝑡, ��, 𝑢) and 𝑦 = ℎ(𝑡, ��, 𝑢) for all 𝑡.

Equilibrium trajectory with 𝑢 = 𝑚𝑔ℓ sin(��1) and ��2 = 0:

𝛿�� ≈ ⎡⎢⎣

0 1

− 𝑔ℓ cos(��1) − 𝑏

𝑚ℓ2

⎤⎥⎦

𝛿𝑥 + ⎡⎢⎣

01

𝑚ℓ2

⎤⎥⎦

𝛿𝑢

This LTI system iscontrollablestable when the equilibrium angle ��1 is in the bottom halfof the circle, that is, when |��1| < 𝜋

2unstable when ��1 is in the top half of the circle

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3rd ElgersburgSchool 2011 Notation and terminology for classes of functions

A mapping 𝑓 ∶ 𝑈 → 𝑈′, where 𝑈 ⊂ ℝ𝑛 and 𝑈′ ⊂ ℝ𝑚, is called

C0 when it is continuous

a homeomorphism when it is C0 and has a C0 inverse

C𝑟 when 𝑈 is open and 𝑓 is 𝑟 times continuouslydifferentiable (𝑟 ⩾ 1, including 𝑟 = ∞)

a C𝑟-diffeomorphism when it is C𝑟 and has a C𝑟 inverse(in this case 𝑈′ is open and 𝑛 = 𝑚)

a diffeomorphism when it is a C∞-diffeomorphism

Inverse function theoremIf 𝑓 is C𝑟 near 𝑥 ∈ 𝑈 and if its Jacobian matrix is nonsingularat 𝑥, then 𝑓 is a C𝑟-diffeomorphism near 𝑥.

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3rd ElgersburgSchool 2011 Lipschitz continuous functions

DefinitionA mapping 𝑓 ∶ 𝑈 → 𝑈′ is Lipschitz continuous when thereexists 𝐿 ⩾ 0 such that

|𝑓(𝑥1) − 𝑓(𝑥2)| ⩽ 𝐿|𝑥1 − 𝑥2| for all 𝑥1, 𝑥2 ∈ 𝑈 .

It is locally Lipschitz continuous when it is Lipschitzcontinuous on a neighborhood of every point in 𝑈.

Lipschitz continuous⇓

C1 ⟹ locally Lipschitz continuous ⟹ C0

Examples for 𝑓 ∶ ℝ → ℝ:𝑓(𝑥) = |sin(𝑥)| is Lipschitz continuous but not C1

𝑓(𝑥) = 𝑥2 is locally Lipschitz but not Lipschitz𝑓(𝑥) = √|𝑥| is C0 but not locally Lipschitz

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3rd ElgersburgSchool 2011 Coordinates: labeling the states

Local and global Cartesian coordinatesA homeomorphism from an open subset of the state space 𝑋to an open subset of ℝ𝑛 describes a local coordinate systemin which the elements of the vector in ℝ𝑛 are the 𝑛 coordinatesof the corresponding state.A coordinate mapping whose domain is the entire space 𝑋describes a global coordinate system.

Change of coordinatesIf the domains of two local coordinate systems overlap, thenthere is a homeomorphism from one set of coordinates to theother (defined on the overlapping region).The two coordinate systems are called compatible when thismapping is a diffeomorphism.

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Coordinates: labeling the statesSimple pendulum example

The state space of the pendulum is the cylinder 𝑋 = 𝑆1× ℝ:the angle lives on the circle 𝑆1

the angular velocity is a real number

the blue and yellow regions represent overlapping localcoordinate systemsthere is no global Cartesian coordinate system:𝑋 is not diffeomorphic to an open subset of ℝ2

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Coordinates: labeling the statesSome coordinates may be more convenient than others

Nonlinear system with 𝑋 = ℝ2 and 𝑥 = (𝑥1, 𝑥2):

��1 = 𝑥2 − 𝑥21

��2 = 𝑥1 + 2𝑥1(𝑥2 − 𝑥21) + 𝑢

Consider the diffeomorphism Φ ∶ 𝑋 → 𝑋 given by

𝑧 =[𝑧1𝑧2]= Φ(𝑥) =[

𝑥1𝑥2 − 𝑥2

1] , 𝑥 =[𝑥1𝑥2]= Φ−1(𝑧) =[

𝑧1𝑧2

1 + 𝑧2]

Same system in new coordinates 𝑧 = (𝑧1, 𝑧2):

𝑧1 = 𝑧2

𝑧2 = 𝑧1 + 𝑢

The system is linear when written in the new coordinates!

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1 Nonlinear models






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Solutions to ordinary differential equationsTime-invariant case

System with no input, or with some fixed feedback 𝑢 = 𝑢(𝑥):

𝑓(𝑥) is a vector field defined on 𝑋:

�� = 𝑓(𝑥) (*)

A solution of (*) from an initial condition 𝑥0 ∈ 𝑋 is a function𝑥(𝑡) defined on an open interval ℐ(𝑥0) ⊂ ℝ such that

0 ∈ ℐ(𝑥0) and 𝑥(0) = 𝑥0

𝑥(𝑡) is C1 on ℐ(𝑥0) with ��(𝑡) = 𝑓(𝑥(𝑡)) for all 𝑡 ∈ ℐ(𝑥0)

ℐ(𝑥0) is the maximal interval of existence: there is noother solution defined on a larger open interval whichagrees with 𝑥(𝑡) on ℐ(𝑥0)

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Existence and uniqueness of solutionsTime-invariant case

Sufficient conditions for local existence and uniqueness:

𝑓(𝑥) is continuous on 𝑋 ⟹ solutions exist ∀𝑥0

𝑓(𝑥) is locally Lipschitz on 𝑋 ⟹ solutions are unique

Global existence: a locally Lipschitz vector field 𝑓(𝑥) is calledcomplete when ℐ(𝑥0) = ℝ for all 𝑥0 ∈ 𝑋forward-complete when sup ℐ(𝑥0) = ∞ for all 𝑥0 ∈ 𝑋

Sufficient conditions for completeness (locally Lipschitz 𝑓):∗∗ 𝑓(𝑥) is Lipschitz on 𝑋 = ℝ𝑛 ⟹ 𝑓(𝑥) is complete

𝑋 is a compact manifold ⟹ 𝑓(𝑥) is complete

∗∗ this property depends on the choice of coordinates!

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3rd ElgersburgSchool 2011 Example: non-uniqueness of solutions

𝑓 is C0 but not locally Lipschitz continuous:

�� = 2√|𝑥| 𝑥(0) = 0 𝑋 = ℝ

A different solution for each 𝑐 ⩾ 0:

𝑥(𝑡) = ⎧⎨⎩

0 when 𝑡 ⩽ 𝑐(𝑡 − 𝑐)2 when 𝑐 ⩽ 𝑡

−2 0 2 4 6 8 10

0

20

40

𝑥(𝑡)

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Example: finite escape timesThe vector field is not forward-complete

𝑓 is locally Lipschitz continuous:

�� = 𝑥2 𝑥(0) = 1 𝑋 = ℝ

Unique solution on maximal interval ℐ(1) = (−∞, 1):

𝑥(𝑡) = 11 − 𝑡

𝑡 ∈ (−∞, 1)

−5 −4 −3 −2 −1 0 1 2

0

10

20

30

1

0

𝑥(𝑡)

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3rd ElgersburgSchool 2011 The flow of a vector field

A locally Lipschitz vector field 𝑓(𝑥) on 𝑋 admits a flow 𝜑(𝑡, 𝑥):for each fixed 𝑥 ∈ 𝑋, the mapping 𝑡 ↦ 𝜑(𝑡, 𝑥) for 𝑡 ∈ ℐ(𝑥)is the unique solution of �� = 𝑓(𝑥) starting at 𝑥:

𝜕𝜑𝜕𝑡

(𝑡, 𝑥) = 𝑓(𝜑(𝑡, 𝑥)) 𝜑(0, 𝑥) = 𝑥

the domain of 𝜑 is Dom(𝜑) = {(𝑡, 𝑥) ∈ ℝ × 𝑋 ∶ 𝑡 ∈ ℐ(𝑥)}Dom(𝜑) is open and 𝜑 is locally Lipschitz jointly in 𝑡 and 𝑥

if 𝑓(𝑥) is C𝑟 for 𝑟 ⩾ 1, then 𝜑 is C𝑟 jointly in 𝑡 and 𝑥

one consequence: if 𝑓(𝑥) ≠ 0 then there exist localcoordinates around 𝑥 under which 𝑓 is constant(the ``straightening-out theorem'')

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𝜕𝜑𝜕𝑡

(𝑡, 𝑥) = 𝑓(𝜑(𝑡, 𝑥)) 𝜑(0, 𝑥) = 𝑥




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𝜕𝜑𝜕𝑡

(𝑡, 𝑥) = 𝑓(𝜑(𝑡, 𝑥)) 𝜑(0, 𝑥) = 𝑥




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𝜕𝜑𝜕𝑡

(𝑡, 𝑥) = 𝑓(𝜑(𝑡, 𝑥)) 𝜑(0, 𝑥) = 𝑥




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𝜕𝜑𝜕𝑡

(𝑡, 𝑥) = 𝑓(𝜑(𝑡, 𝑥)) 𝜑(0, 𝑥) = 𝑥




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3rd ElgersburgSchool 2011 The space derivative of the flow

Linear time-varying system for fixed 𝑥 ∈ 𝑋:

��(𝑡) = 𝐷𝑓(𝜑(𝑡, 𝑥)) · 𝑍(𝑡) 𝑍(0) = 𝐼

𝑍(𝑡) =𝜕𝜑𝜕𝑥

(𝑡, 𝑥) for all 𝑡 ∈ ℐ(𝑥)

𝑍(𝑡) is the sensitivity matrix w. r. t. the initial state

𝐷𝑓(𝑥) denotes the Jacobian matrix of 𝑓 at 𝑥

calculate 𝑍(𝑡) at an initial state 𝑥0 ∈ 𝑋 by solving thetime-invariant augmented initial-value problem

�� = 𝑓(𝑥) 𝑥(0) = 𝑥0

�� = 𝐷𝑓(𝑥) · 𝑍 𝑍(0) = 𝐼

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Solutions to ordinary differential equationsTime-varying case

System with no input, or with some fixed input 𝑢(𝑡):

𝑓(𝑡, 𝑥) defined on 𝐼 × 𝑋 for an open interval 𝐼 ⊂ ℝ:

�� = 𝑓(𝑡, 𝑥) (**)

A solution of (**) from an initial condition (𝑡0, 𝑥0) ∈ 𝐼 × 𝑋 is afunction 𝑥(𝑡) defined on an open interval ℐ(𝑡0, 𝑥0) ⊂ 𝐼 such that

𝑡0 ∈ ℐ(𝑡0, 𝑥0) and 𝑥(𝑡0) = 𝑥0

𝑥(𝑡) is locally absolutely continuous on ℐ(𝑡0, 𝑥0)for almost all 𝑡 ∈ ℐ(𝑡0, 𝑥0), ��(𝑡) exists and equals 𝑓(𝑡, 𝑥(𝑡))ℐ(𝑡0, 𝑥0) is the maximal interval of existence: there is noother solution defined on a larger open interval whichagrees with 𝑥(𝑡) on ℐ(𝑡0, 𝑥0)

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Existence and uniqueness of solutionsTime-varying case

Sufficient conditions for the existence of solutions:𝑡 ↦ 𝑓(𝑡, 𝑥) is measurable for all fixed 𝑥 ∈ 𝑋𝑥 ↦ 𝑓(𝑡, 𝑥) is continuous for almost all fixed 𝑡 ∈ 𝐼for every compact 𝐾 ⊂ 𝐼 × 𝑋 there exists an integrablefunction 𝑚(𝑡) such that |𝑓(𝑡, 𝑥)| ⩽ 𝑚(𝑡) for all (𝑡, 𝑥) ∈ 𝐾

Sufficient conditions for existence and uniqueness:𝑡 ↦ 𝑓(𝑡, 𝑥) is measurable for all fixed 𝑥 ∈ 𝑋for every compact 𝐾 ⊂ 𝐼 × 𝑋 there exists 𝐿 ⩾ 0 such that

|𝑓(𝑡, 𝑥1) − 𝑓(𝑡, 𝑥2)| ⩽ 𝐿|𝑥1 − 𝑥2|

for all (𝑡, 𝑥1), (𝑡, 𝑥2) ∈ 𝐾(𝑓 is locally Lipschitz in 𝑥, locally uniformly in 𝑡)

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3rd ElgersburgSchool 2011 Stable sets and attractors

Given:a vector field 𝑓(𝑥) on 𝑋 with a continuous flow 𝜑(𝑡, 𝑥)an equilibrium 𝑥𝑒 ∈ 𝑋(𝑓(𝑥𝑒) = 0 and thus 𝜑(𝑡, 𝑥𝑒) = 𝑥𝑒 for all 𝑡 ∈ ℐ(𝑥𝑒) = ℝ)

The stable set of 𝑥𝑒 is the set

𝑊 𝑠(𝑥𝑒) = {𝑥 ∈ 𝑋 ∶ sup ℐ(𝑥) = ∞ and𝜑(𝑡, 𝑥) → 𝑥𝑒 as 𝑡 → ∞}

DefinitionThe equilibrium 𝑥𝑒 is a local attractor when 𝑊 𝑠(𝑥𝑒) contains aneighborhood of 𝑥𝑒. It is a global attractor when 𝑊 𝑠(𝑥𝑒) = 𝑋.

If 𝑥𝑒 is an attractor, then 𝑊 𝑠(𝑥𝑒) is open and is called the basinor region of attraction.

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3rd ElgersburgSchool 2011 Stability in the sense of Lyapunov

Flow near an equilibrium 𝑥𝑒

𝑈

𝑈′𝑥𝑒

𝑥

Definition𝑥𝑒 is stable when for everyneighborhood 𝑈 of 𝑥𝑒 there is aneighborhood 𝑈′ of 𝑥𝑒 such that

𝑥 ∈ 𝑈′, 𝑡 ⩾ 0 ⟹ 𝜑(𝑡, 𝑥) ∈ 𝑈

𝑥𝑒 is stable when trajectories that start nearby stay nearby

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3rd ElgersburgSchool 2011 Asymptotic stability of equilibria

Definition

asymptotically stable = stable + local attractorglobally asymptotically stable = stable + global attractor

Another consequence of the continuity of the flow

If 𝑥𝑒 is stable, then its stable set 𝑊 𝑠(𝑥𝑒) is contractible(it can be continuously deformed into a point).

Simple pendulum with 𝑏 > 0 and 𝑢 = 0

The downward equilibrium ( , ) = (0, 0) is asymptoticallystable, but not globally so (a cylinder is not contractible).

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3rd ElgersburgSchool 2011 Example: an unstable global attractor

State space 𝑋 = 𝑆1, equilibrium at 𝑥 = 0:

�� = 1 − cos(𝑥)

−20 −10 0 10 20

0

2

4

6

𝑡

𝑥(𝑡) 𝑥 = 0

𝑊 𝑠(0) = 𝑋

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3rd ElgersburgSchool 2011 A first test for asymptotic stability: linearization

Let 𝑥𝑒 be an equilibrium of a C1 vector field 𝑓(𝑥):if 𝐷𝑓(𝑥𝑒) is Hurwitz (all eigenvalues have strictly negativereal parts), then 𝑥𝑒 is asymptotically stableif at least one eigenvalue of 𝐷𝑓(𝑥𝑒) has a strictly positivereal part, then 𝑥𝑒 is unstable

Simple pendulum with 𝑥𝑒 = (��1, 0), input 𝛿𝑢 = 0, damping 𝑏 > 0

𝐷𝑓(𝑥𝑒) = ⎡⎢⎣

0 1

− 𝑔ℓ cos(��1) − 𝑏

𝑚ℓ2

⎤⎥⎦

asymptotically stable when cos(��1) > 0unstable when cos(��1) < 0linearization test is inconclusive when cos(��1) = 0

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Lyapunov functionsThe energy of the simple pendulum

Simple pendulum with 𝑥1 = , 𝑥2 = , and input 𝑢 = 0

��1 = 𝑥2

��2 = −𝑔ℓ

sin(𝑥1) − 𝑏𝑚ℓ2 𝑥2

Energy (potential plus kinetic):

𝑉(𝑥1, 𝑥2) = 𝑚𝑔ℓ[1 − cos(𝑥1)] + 12 𝑚ℓ2𝑥2

2

Time derivative of energy along the flow:𝑉(𝑥1, 𝑥2) = −𝑏𝑥2

2 ⩽ 0

Energy has a unique global minimum at the equilibrium𝑥𝑒 = (0, 0) and never increases, so 𝑥𝑒 must be stable!

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Directional derivativesDerivatives along the flow of a vector field

Given:a vector field 𝑓(𝑥) on 𝑋 with flow 𝜑(𝑡, 𝑥)a C1 function 𝑉 ∶ 𝑋 → ℝ

The directional derivative or Lie derivative or timederivative of 𝑉 along 𝑓 is the function 𝐿𝑓𝑉 ∶ 𝑋 → ℝ given by

𝐿𝑓𝑉(𝑥) = 𝜕𝜕𝑡

𝑉(𝜑(𝑡, 𝑥))|||𝑡=0

The values of 𝐿𝑓𝑉 are independent of coordinates, but we cancompute them in local coordinates 𝑥 = (𝑥1, …, 𝑥𝑛) as

𝐿𝑓𝑉(𝑥) = 𝐷𝑉(𝑥) ·𝑓(𝑥) = 𝜕𝑉(𝑥)𝜕𝑥1

𝑓1(𝑥)⏟⏟⏟

��1

+ … + 𝜕𝑉(𝑥)𝜕𝑥𝑛

𝑓𝑛(𝑥)⏟⏟⏟

��𝑛

NonlinearControl

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Lyapunov functionsTime-invariant case

Suppose 𝑓(𝑥𝑒) = 0 so that 𝑥𝑒 is an equilibrium for �� = 𝑓(𝑥).

Definition

A Lyapunov function is a C1 function 𝑉 ∶ 𝑋 → ℝ such that𝑉(𝑥𝑒) = 0𝑉(𝑥) > 0 for all 𝑥 ≠ 𝑥𝑒 in a neighborhood of 𝑥𝑒

𝐿𝑓𝑉(𝑥) ⩽ 0 for all 𝑥 in a neighborhood of 𝑥𝑒

A strict Lyapunov function has the additional property𝐿𝑓𝑉(𝑥) < 0 for all 𝑥 ≠ 𝑥𝑒 in a neighborhood of 𝑥𝑒

Lyapunov stability theorems

∃ a Lyapunov function ⟹ 𝑥𝑒 is stable∃ a strict Lyapunov function ⟹ 𝑥𝑒 is asymptotically stable

NonlinearControl

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Lyapunov functionsTime-invariant case

Suppose 𝑓(𝑥𝑒) = 0 so that 𝑥𝑒 is an equilibrium for �� = 𝑓(𝑥).

Definition

A Lyapunov function is a C1 function 𝑉 ∶ 𝑋 → ℝ such that𝑉(𝑥𝑒) = 0𝑉(𝑥) > 0 for all 𝑥 ≠ 𝑥𝑒 in a neighborhood of 𝑥𝑒

𝐿𝑓𝑉(𝑥) ⩽ 0 for all 𝑥 in a neighborhood of 𝑥𝑒

A strict Lyapunov function has the additional property𝐿𝑓𝑉(𝑥) < 0 for all 𝑥 ≠ 𝑥𝑒 in a neighborhood of 𝑥𝑒

Lyapunov stability theorems

∃ a Lyapunov function ⟹ 𝑥𝑒 is stable∃ a strict Lyapunov function ⟹ 𝑥𝑒 is asymptotically stable

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3rd ElgersburgSchool 2011 Lyapunov function intuition

𝑉 is shaped like a bowl or a valley near 𝑥𝑒𝑥𝑒 is at the bottom of the valleythe flow cannot go uphill in forward timeif 𝑉 is strict, then the flow always goes downhill

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Global Lyapunov functionsTime-invariant case

𝑉 is called proper when the sublevel setsΩ𝑉(𝑐) = {𝑥 ∈ 𝑋 ∶ 𝑉(𝑥) ⩽ 𝑐}

are compact for all 𝑐 ∈ 𝑉(𝑋).

DefinitionA global Lyapunov function is a function 𝑉 ∶ 𝑋 → ℝ such that

𝑉 is C1 and proper𝑉(𝑥𝑒) = 0 and 𝑉(𝑥) > 0 for all 𝑥 ∈ 𝑋 with 𝑥 ≠ 𝑥𝑒

𝐿𝑓𝑉(𝑥) < 0 for all 𝑥 ∈ 𝑋 with 𝑥 ≠ 𝑥𝑒

Global stability theorem∃ a global Lyapunov function

⇓𝑥𝑒 is globally asymptotically stable

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Global Lyapunov functionsTime-invariant case

𝑉 is called proper when the sublevel setsΩ𝑉(𝑐) = {𝑥 ∈ 𝑋 ∶ 𝑉(𝑥) ⩽ 𝑐}

are compact for all 𝑐 ∈ 𝑉(𝑋).

DefinitionA global Lyapunov function is a function 𝑉 ∶ 𝑋 → ℝ such that

𝑉 is C1 and proper𝑉(𝑥𝑒) = 0 and 𝑉(𝑥) > 0 for all 𝑥 ∈ 𝑋 with 𝑥 ≠ 𝑥𝑒

𝐿𝑓𝑉(𝑥) < 0 for all 𝑥 ∈ 𝑋 with 𝑥 ≠ 𝑥𝑒

Global stability theorem∃ a global Lyapunov function

⇓𝑥𝑒 is globally asymptotically stable

NonlinearControl

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Global Lyapunov functionsLTI systems

LTI system with equilibrium 𝑥𝑒 = 0 and Hurwitz 𝐴

�� = 𝐴𝑥

Lyapunov matrix equationIf 𝐴 is Hurwitz, then for any 𝑄 > 0 there exists 𝑃 > 0 such that

𝐴𝑇𝑃 + 𝑃𝐴 + 𝑄 = 0

Global Lyapunov function

The quadratic function 𝑉(𝑥) = 𝑥𝑇𝑃 𝑥 has time derivative𝑉(𝑥) = −𝑥𝑇𝑄𝑥, so

𝑉 is a global Lyapunov functionthus 𝑥𝑒 = 0 is globally asymptotically stable

NonlinearControl

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3rd ElgersburgSchool 2011 Lyapunov functions for linearized systems

suppose �� = 𝑓(𝑥) with 𝑓(0) = 0 (equilibrium at 𝑥𝑒 = 0)

suppose the Jacobian matrix 𝐴 = 𝐷𝑓(0) is Hurwitz

write the system as

�� = 𝐴𝑥 + higher-order terms

choose 𝑄 > 0 and solve 𝐴𝑇𝑃 + 𝑃𝐴 + 𝑄 = 0 for 𝑃 > 0

quadratic 𝑉(𝑥) = 𝑥𝑇𝑃 𝑥 satisfies

𝐿𝑓𝑉(𝑥) = −𝑥𝑇𝑄𝑥 + higher-order terms

𝑉 is a strict Lyapunov function, so 𝑥𝑒 is asymp. stable

use 𝑉 to estimate the region of attraction 𝑊 𝑠(𝑥𝑒) …

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write the system as







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write the system as







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write the system as







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write the system as







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write the system as







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write the system as







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3rd ElgersburgSchool 2011 Estimating the region of attraction

𝐿𝑓𝑉 < 0

𝑉 = 𝑐

find largest 𝑐 > 0 such that

𝑥 ≠ 𝑥𝑒 and 𝑉(𝑥) < 𝑐 ⟹ 𝐿𝑓𝑉(𝑥) < 0

then {𝑥 ∈ 𝑋 ∶ 𝑉(𝑥) < 𝑐} ⊂ 𝑊 𝑠(𝑥𝑒)different estimates from different choices of 𝑄

NonlinearControl

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3rd ElgersburgSchool 2011 Asymptotic stability of the simple pendulum

For zero input 𝑢 = 0, the energy function

𝑉(𝑥1, 𝑥2) = 𝑚𝑔ℓ[1 − cos(𝑥1)] + 12 𝑚ℓ2𝑥2

2

satisfies𝐿𝑓𝑉(𝑥1, 𝑥2) = −𝑏𝑥2

2

and is thus a Lyapunov function for the equilibrium 𝑥𝑒 = (0, 0).

𝑉 is not a strict Lyapunov function, but 𝑥𝑒 is asymptoticallystable nonetheless when 𝑏 > 0.

To prove asymptotic stability, we canlook for a strict Lyapunov function, oruse the invariance principle …

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Invariant sets

Let 𝜑(𝑡, 𝑥) be the flow of a vector field 𝑓(𝑥) on 𝑋.

DefinitionA subset Γ ⊂ 𝑋 is called

invariant when 𝑥 ∈ Γ implies 𝜑(𝑡, 𝑥) ∈ Γ for all 𝑡 ∈ ℐ(𝑥)forward-invariant when 𝑥 ∈ Γ implies 𝜑(𝑡, 𝑥) ∈ Γ for allpositive 𝑡 ∈ ℐ(𝑥)

Invariant sets include:equilibria, stable sets of equilibria, periodic orbits(including limit cycles)the orbit 𝜑(ℐ(𝑥), 𝑥) through any 𝑥 ∈ 𝑋

Forward-invariant sets that need not be invariant include:the sublevel sets Ω𝑉(𝑐) of a Lyapunov function

NonlinearControl

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3rd ElgersburgSchool 2011 Limit points and limit sets

DefinitionA point 𝑧 ∈ 𝑋 is an 𝜔-limit point of 𝑥 ∈ 𝑋 when there existsan increasing, unbounded sequence 𝑡1, 𝑡2, … ∈ ℐ(𝑥) such that

𝜑(𝑡𝑖, 𝑥) → 𝑧 as 𝑖 → ∞

The 𝜔-limit set 𝐿𝜔(𝑥)𝐿𝜔(𝑥) = the set of all 𝜔-limit points of 𝑥

a closed subset of 𝑋an invariant subset (due to the continuity of the flow)

If 𝑥 belongs to a compact, forward-invariant set, then𝐿𝜔(𝑥) ≠ ∅ and 𝜑(𝑡, 𝑥) → 𝐿𝜔(𝑥) as 𝑡 → ∞.

NonlinearControl

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3rd ElgersburgSchool 2011 Limit points and limit sets

DefinitionA point 𝑧 ∈ 𝑋 is an 𝜔-limit point of 𝑥 ∈ 𝑋 when there existsan increasing, unbounded sequence 𝑡1, 𝑡2, … ∈ ℐ(𝑥) such that

𝜑(𝑡𝑖, 𝑥) → 𝑧 as 𝑖 → ∞

The 𝜔-limit set 𝐿𝜔(𝑥)𝐿𝜔(𝑥) = the set of all 𝜔-limit points of 𝑥

a closed subset of 𝑋an invariant subset (due to the continuity of the flow)

If 𝑥 belongs to a compact, forward-invariant set, then𝐿𝜔(𝑥) ≠ ∅ and 𝜑(𝑡, 𝑥) → 𝐿𝜔(𝑥) as 𝑡 → ∞.

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3rd ElgersburgSchool 2011 LaSalle's invariance principle

AssumeΩ ⊂ 𝑋 is compact and forward-invariantthere is a C1 function 𝑉(𝑥) on 𝑋 such that 𝐿𝑓𝑉(𝑥) ⩽ 0 on Ω

Define the set𝐸 = {𝑥 ∈ Ω ∶ 𝐿𝑓𝑉(𝑥) = 0}

Invariance principleIf 𝑥 ∈ Ω then sup ℐ(𝑥) = ∞ and 𝜑(𝑡, 𝑥) → 𝑀 as 𝑡 → ∞,where 𝑀 is largest invariant subset of 𝐸.

𝑡 ↦ 𝑉(𝜑(𝑡, 𝑥)) is bounded from below and nonincreasing,and thus approaches a limithence 𝑉 is constant on 𝐿𝜔(𝑥), which means 𝐿𝜔(𝑥) ⊂ 𝐸because 𝐿𝜔(𝑥) is invariant𝜑(𝑡, 𝑥) → 𝐿𝜔(𝑥) ⊂ 𝑀 as 𝑡 → ∞

NonlinearControl

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LaSalle's invariance principleSimple pendulum example

For zero input 𝑢 = 0, the energy function

𝑉(𝑥1, 𝑥2) = 𝑚𝑔ℓ[1 − cos(𝑥1)] + 12 𝑚ℓ2𝑥2

2

satisfies𝐿𝑓𝑉(𝑥1, 𝑥2) = −𝑏𝑥2

2

For each fixed 𝑥0 ∈ 𝑋:choose Ω = Ω𝑉(𝑉(𝑥0)) = {𝑥 ∈ 𝑋 ∶ 𝑉(𝑥) ⩽ 𝑉(𝑥0)}Ω is compact because 𝑉 is properand forward-invariant because 𝐿𝑓𝑉 ⩽ 0𝐸 = {𝑥 ∈ Ω ∶ 𝐿𝑓𝑉(𝑥) = 0} = {𝑉(𝑥) ⩽ 𝑉(𝑥0) and 𝑥2 = 0}𝑀 = {(0, 0)} if 𝑉(𝑥0) < 2𝑚𝑔ℓ, else 𝑀 = {(0, 0), (𝜋, 0)}

Every trajectory converges either to (0, 0) or to (𝜋, 0).If 𝑉(𝑥0) < 2𝑚𝑔ℓ then the trajectory converges to (0, 0).

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3rd ElgersburgSchool 2011 Stability of invariant sets

replace equilibrium 𝑥𝑒 with any compact invariant setmost stability definitions and results are unchanged

Example on 𝑋 = ℝ2

��1 = 𝜌(𝑥)[1 − 𝜌(𝑥)]𝑥1 − 𝑥2

��2 = 𝜌(𝑥)[1 − 𝜌(𝑥)]𝑥2 + 𝑥1 where 𝜌(𝑥) = 𝑥21 + 𝑥2

2

the unit circle {𝜌(𝑥) = 1} is an invariant set consisting ofthe periodic orbit (𝑥1(𝑡), 𝑥2(𝑡)) = (cos(𝑡), sin(𝑡))asymptotic stability: do trajectories which start near theunit circle stay near it and converge to it?

Try 𝑉(𝑥) = [1 − 𝜌(𝑥)]2 and apply LaSalle …

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3rd ElgersburgSchool 2011 Asymptotic stability of a periodic orbit

𝑉(𝑥) = [1 − 𝜌(𝑥)]2

𝐿𝑓𝑉(𝑥) = −4𝜌2(𝑥)[1 − 𝜌(𝑥)]2 ⩽ 0

Ω = {𝑥 ∈ 𝑋 ∶ 1𝑐 ⩽ 𝜌(𝑥) ⩽ 𝑐}

annulus Ω is compact and forward-invariant for any 𝑐 ⩾ 1

for 𝑥0 ≠ 0 choose 𝑐 ⩾ 1 so that 𝑥0 ∈ Ω

𝐸 = {𝑥 ∈ Ω ∶ 𝐿𝑓𝑉(𝑥) = 0} = {𝑥 ∈ 𝑋 ∶ 𝜌(𝑥) = 1}thus all nonzero trajectories converge to the periodic orbit

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3rd ElgersburgSchool 2011 Trajectory tracking

Time-varying system

��(𝑡) = 𝑓(𝑡, 𝑥(𝑡))

given a specific trajectory ��(𝑡) with ��(𝑡) = 𝑓(𝑡, ��(𝑡))

define the error 𝛿𝑥(𝑡) = 𝑥(𝑡) − ��(𝑡) with dynamics

𝛿��(𝑡) = 𝐹(𝑡, 𝛿𝑥(𝑡)) = 𝑓(𝑡, 𝛿𝑥(𝑡) + ��(𝑡)) − 𝑓(𝑡, ��(𝑡))

equilibrium at 𝛿𝑥𝑒 = 0 because 𝐹(𝑡, 0) ≡ 0

tracking of trajectory ��(𝑡) = stability of equilibrium 𝛿𝑥𝑒

𝐹 can depend on 𝑡 even if 𝑓 does not

We should define stability for time-varying systems …

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Comparison functionsFunctions of class 𭒫, class 𭒦, class 𭒦∞, and class 𭒦ℒ

A C0 function 𝛼∶[0, ∞) → [0, ∞) is positive definite (class 𭒫)when 𝛼(0) = 0 and 𝛼(𝑟) > 0 for all 𝑟 > 0. It is of class 𭒦 if it isof class 𭒫 and strictly increasing. It is of class 𭒦∞ if it is ofclass 𭒦 and unbounded (in which case it has an inverse in 𭒦∞).

𭒦∞ ⊂ 𭒦 ⊂ 𭒫

Examples: 𝛼(𝑟) = tanh(𝑟) ∈ 𭒦 and 𝛼(𝑟) = 𝑟2 ∈ 𭒦∞

A C0 function 𝛽 ∶ [0, ∞)2 → [0, ∞) is of class 𭒦ℒ whenfor each fixed 𝑠, the mapping 𝑟 ↦ 𝛽(𝑟, 𝑠) is of class 𭒦for each fixed 𝑟, the mapping 𝑠 ↦ 𝛽(𝑟, 𝑠) decreases to zero

Example: 𝛽(𝑟, 𝑠) = 𝑟𝑒−𝑠 ∈ 𭒦ℒ

NonlinearControl

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3rd ElgersburgSchool 2011 Uniform stability

Time-varying system with an equilibrium at zero

��(𝑡) = 𝑓(𝑡, 𝑥(𝑡)) 𝑥(𝑡0) = 𝑥0 𝑓(𝑡, 0) ≡ 0

DefinitionThe equilibrium is uniformly stable when there exists aclass-𭒦 function 𝛼 such that

|𝑥(𝑡)| ⩽ 𝛼(|𝑥0|)

for all 𝑡0, for all 𝑡 ⩾ 𝑡0, and for all 𝑥0 in a neighborhood of zero.

Uniform stability: trajectories that start nearby stay nearby,and the notion of nearby does not depend on 𝑡0.

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3rd ElgersburgSchool 2011 Uniform asymptotic stability

DefinitionThe equilibrium is uniformly asymptotically stable whenthere exists a class-𭒦ℒ function 𝛽 such that

|𝑥(𝑡)| ⩽ 𝛽(|𝑥0|, 𝑡 − 𝑡0)

for all 𝑡0, for all 𝑡 ⩾ 𝑡0, and for all 𝑥0 in a neighborhood of zero.

Uniformasymptoticstability

=uniform stability

+decay that depends only on the

elapsed time 𝑡 − 𝑡0

Global uniform asymptotic stability: above holds for all 𝑥0 ∈ 𝑋(must give global meaning to |·| if 𝑋 is not an open subset of ℝ𝑛)

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Lyapunov functionsTime-varying case

Let 𝑉 ∶ ℝ × 𝑋 → ℝ be C1 and such that for some 𝛼1, 𝛼2 ∈ 𭒦,

𝛼1(|𝑥|) ⩽ 𝑉(𝑡, 𝑥) ⩽ 𝛼2(|𝑥|) ∀𝑡 and ∀𝑥 near zero

if the time derivative of 𝑉 along trajectories satisfies

𝑉(𝑡, 𝑥) = 𝜕𝑉(𝑡, 𝑥)𝜕𝑡

+ 𝜕𝑉(𝑡, 𝑥)𝜕𝑥

𝑓(𝑡, 𝑥) ⩽ 0

∀𝑡 and ∀𝑥 near zero, then 𝑉 is a Lyapunov function

if in addition there exists 𝛼3 ∈ 𭒫 such that𝑉(𝑡, 𝑥) ⩽ −𝛼3(|𝑥|) ∀𝑡 and ∀𝑥 near zero

then 𝑉 is a strict Lyapunov function

if in addition 𝑋 = ℝ𝑛, 𝛼1 ∈ 𭒦∞, and all of these inequalitieshold for all 𝑥 ∈ 𝑋, then 𝑉 is a global Lyapunov function

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𝑉(𝑡, 𝑥) = 𝜕𝑉(𝑡, 𝑥)𝜕𝑡

+ 𝜕𝑉(𝑡, 𝑥)𝜕𝑥

𝑓(𝑡, 𝑥) ⩽ 0





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𝑉(𝑡, 𝑥) = 𝜕𝑉(𝑡, 𝑥)𝜕𝑡

+ 𝜕𝑉(𝑡, 𝑥)𝜕𝑥

𝑓(𝑡, 𝑥) ⩽ 0





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Lyapunov stability theoremsTime-varying case


��(𝑡) = 𝑓(𝑡, 𝑥(𝑡)) 𝑓(𝑡, 0) ≡ 0

∃ Lyap. func. 𝑉 ⟹ uniformly stable𝑉 is strict ⟹ uniformly asymptotically stable𝑉 is global ⟹ globally uniformly asymptotically stable

WarningLaSalle's invariance principle can fail for time-varying systems,because 𝜔-limits sets need not be invariant!

WarningPeriodic trajectories of time-invariant systems cannot be stablein this tracking sense (use invariant-set stability instead).

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��(𝑡) = 𝑓(𝑡, 𝑥(𝑡)) 𝑓(𝑡, 0) ≡ 0




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��(𝑡) = 𝑓(𝑡, 𝑥(𝑡)) 𝑓(𝑡, 0) ≡ 0




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Convergence using non-strict Lyapunov functionsTime-varying case

Given:system �� = 𝑓(𝑡, 𝑥) with 𝑓(𝑡, 0) ≡ 0 (equilibrium at zero)∃𝑐 > 0 such that ∀𝑡 and ∀𝑥 near zero, |𝑓(𝑡, 𝑥)| ⩽ 𝑐non-strict Lyapunov function 𝑉(𝑡, 𝑥)

Suppose there is a nonnegative C0 function 𝑊(𝑥) such that𝑉(𝑡, 𝑥) ⩽ −𝑊(𝑥) ∀𝑡 and ∀𝑥 near zero

by uniform stability, 𝑥(𝑡) stays near zero for small 𝑥0

thus ��(𝑡) is bounded ⟹ 𝑥(𝑡) & 𝑊(𝑥(𝑡)) are uniformly C0

∫ 𝑡𝑡0

𝑊(𝑥(𝜏)) 𝑑𝜏 ⩽ 𝑉(𝑡0, 𝑥0) for all 𝑡 ⩾ 𝑡0, and therefore:

If 𝑥0 is near zero, then

𝑊(𝑥(𝑡)) → 0 as 𝑡 → ∞ (Barbalat's lemma)

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Systems with inputsLTI example

Stable LTI system (𝐴 is Hurwitz)

�� = 𝐴𝑥 + 𝐵𝑢

The trajectories satisfy

𝑥(𝑡) = 𝑒𝐴𝑡𝑥0 + ∫𝑡

0𝑒𝐴(𝑡−𝜏)𝐵𝑢(𝜏) 𝑑𝜏

|𝑥(𝑡)| ⩽ |𝑒𝐴𝑡|⏟

decaying

|𝑥0| + ∫𝑡

0|𝑒𝐴(𝑡−𝜏)𝐵| 𝑑𝜏

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟bounded

· ess sup0⩽𝜏⩽𝑡

|𝑢(𝜏)|

We can write this bound as|𝑥(𝑡)| ⩽ 𝛽(|𝑥0|, 𝑡) + 𝛾(‖𝑢‖[0,𝑡])

where 𝛽 ∈ 𭒦ℒ and 𝛾 ∈ 𭒦 and‖𝑢‖[0,𝑡] = ess sup{|𝑢(𝜏)| ∶ 0 ⩽ 𝜏 ⩽ 𝑡} ( ess sup ≈ max )

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Input-to-state stability (ISS)

Time-varying system with inputs, 𝑋 = ℝ𝑛, 𝑢 ∈ ℝ𝑚

��(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑢(𝑡)) 𝑥(𝑡0) = 𝑥0

DefinitionThe system is input-to-state stable (ISS) when there exist𝛽 ∈ 𭒦ℒ and 𝛾 ∈ 𭒦 such that for any admissible 𝑢(𝑡),

|𝑥(𝑡)| ⩽ 𝛽(|𝑥0|, 𝑡 − 𝑡0) + 𝛾(‖𝑢‖[𝑡0,𝑡])

for all 𝑡0, for all 𝑡 ⩾ 𝑡0, and for all 𝑥0 ∈ 𝑋.

when 𝑢 ≡ 0, an ISS system has a globally uniformlyasymptotically stable equilibrium at zerothe class-𭒦 function 𝛾 is called the gain function

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Example: globally asymptotically stable but not ISS

Scalar system with single input, 𝑋 = ℝ and 𝑢 ∈ ℝ

�� = −𝑥3 + 𝑥3𝑢

the system is 0-LAS: when 𝑢 ≡ 0, the equilibrium at zerois (locally) asymptotically stable

the system is 0-GAS: when 𝑢 ≡ 0, the equilibrium at zerois globally asymptotically stable

with bounded input 𝑢 ≡ 2, the system exhibits finiteescape times from every nonzero initial state

thus the system is not ISS

NonlinearControl

Nonlinearmodels








�� = −𝑥3 + 𝑥3𝑢





NonlinearControl

Nonlinearmodels








�� = −𝑥3 + 𝑥3𝑢





NonlinearControl

Nonlinearmodels








�� = −𝑥3 + 𝑥3𝑢





NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 ISS Lyapunov functions

Definition

An ISS Lyapunov function is a C1 function 𝑉 ∶ ℝ × 𝑋 → ℝsuch that for some 𝛼1, 𝛼2, 𝛼3 ∈ 𭒦∞ and some 𝛼4 ∈ 𭒦,

𝛼1(|𝑥|) ⩽ 𝑉(𝑡, 𝑥) ⩽ 𝛼2(|𝑥|) and 𝑉(𝑡, 𝑥, 𝑢) ⩽ −𝛼3(|𝑥|) + 𝛼4(|𝑢|)

for all 𝑡, 𝑥, and 𝑢.

there exists an ISS Lyapunov function⇓

the system is ISS with gain function 𝛾 > 𝛼−11 ∘ 𝛼2 ∘ 𝛼−1

3 ∘ 𝛼4

Main idea: 𝑉 is decreasing when 𝑥 is outside some sublevel setof 𝑉 whose size depends on |𝑢|.

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Example: verifying ISS


�� = −𝑥3 + 𝑥2𝑢

Try 𝑉(𝑥) = 𝑥2 so that 𝛼1(𝑟) = 𝛼2(𝑟) = 𝑟2 and

𝑉(𝑥, 𝑢) = −2𝑥4 + 2𝑥3𝑢⩽ −2𝑥4 + 2|𝑥|3|𝑢|

|𝑥| > |𝑢| implies 𝑉(𝑥, 𝑢) < 0 (enough to conclude ISS)

can use Young's inequality to bound 𝑉 by 𝛼3 and 𝛼4 …

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Young's inequality

For any 𝜇 ∈ 𭒦∞ and any 𝑎, 𝑏 ⩾ 0,

𝑎𝑏 ⩽ ∫𝑎

0𝜇(𝑟) 𝑑𝑟 + ∫

𝑏

0𝜇−1(𝑟) 𝑑𝑟

with equality if and only if 𝑏 = 𝜇(𝑎). 0 𝑎

𝑏𝜇

Useful special case: 𝜇(𝑟) = 𝑝𝑟 𝑝−1 for any > 0 and any 𝑝 > 1

𝑎𝑏 ⩽𝑝𝑎𝑝

𝑝+ 𝑏𝑞

𝑞 𝑞1𝑝

+ 1𝑞

= 1

𝑎𝑏 ⩽ 𝑎2

2+ 𝑏2

2𝑝 = 𝑞 = 2 , replace 2 with

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Example: verifying ISS (continued)

From Young's inequality with 𝑝 = 4/3, 𝑞 = 4, = 1

|𝑥|3|𝑢| ⩽ 34 𝑥4 + 1

4 𝑢4

Thus we have the bound

𝑉(𝑥, 𝑢) ⩽ −2𝑥4 + 2|𝑥|3|𝑢|

⩽ − 12 𝑥4 + 1

2 𝑢4

so 𝛼3(𝑟) = 𝛼4(𝑟) = 12 𝑟4.

ISS with any gain function 𝛾 satisfying 𝛾(𝑟) > 𝑟 for all 𝑟 ≠ 0.

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Cascade interconnection of ISS systems

𝑢𝑥2��2 = 𝑓2 ��1 = 𝑓1

Cascade system with 𝑥2 as the input to the 𝑥1-subsystem

��1 = 𝑓1(𝑥1, 𝑥2) 𝑓1(0, 0) = 0��2 = 𝑓2(𝑥2, 𝑢) 𝑓2(0, 0) = 0

both subsystems are 0-LAS ⟹ cascade is 0-LAS

both subsystems are 0-GAS ⟹/ cascade is 0-GAS

both subsystems are ISS ⟹ cascade is ISS & 0-GAS

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Feedback interconnection of ISS systems

��2 = 𝑓2

��1 = 𝑓1𝑢1

𝑢2

𝑥2 𝑥1

Subsystem 𝑖 has inputs 𝑢𝑖 and 𝑥3−𝑖

��1 = 𝑓1(𝑥1, 𝑥2, 𝑢1) 𝑓1(0, 0, 0) = 0��2 = 𝑓2(𝑥1, 𝑥2, 𝑢2) 𝑓2(0, 0, 0) = 0

|𝑥1(𝑡)| ⩽ 𝛽1(|𝑥1(0)|, 𝑡) + 𝛾1(‖𝑥2‖[0,𝑡]) + 𝛾1(‖𝑢1‖[0,𝑡])

|𝑥2(𝑡)| ⩽ 𝛽2(|𝑥2(0)|, 𝑡) + 𝛾2(‖𝑥1‖[0,𝑡]) + 𝛾2(‖𝑢2‖[0,𝑡])

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 ISS small-gain theorem

Small-gain condition: there exists 𝜌 ∈ 𭒦∞ such that

[(𝛾1 + 𝜌) ∘ (𝛾2 + 𝜌)](𝑟) ⩽ 𝑟 for all 𝑟 ⩾ 0

Small-gain condition for linear gains 𝛾𝑖(𝑟) = 𝛾𝑖𝑟

𝛾1𝛾2 < 1

small-gain condition ⟹ interconnection is ISS

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Another non-ISS system


�� = − tanh(𝑥) + 𝑢

Try 𝑉(𝑥) = ln(cosh(𝑥)) so that 𝛼1(𝑟) = 𝛼2(𝑟) = ln(cosh(𝑟)) and

𝑉(𝑥, 𝑢) = − tanh2(𝑥) + tanh(𝑥) · 𝑢⩽ − tanh2(𝑥) + |𝑢|

𝛼3(𝑟) = tanh2(𝑟), 𝛼4(𝑟) = 𝑟

cannot conclude ISS because 𝛼3 ∈ 𭒦 but 𝛼3 ∈/ 𭒦∞(in fact it is not ISS: 𝑢 ≡ 2 produces an unbounded state)

this system is integral input-to-state stable (iISS) …

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 Integral input-to-state stability

Time-varying system with inputs, 𝑋 = ℝ𝑛, 𝑢 ∈ ℝ𝑚

��(𝑡) = 𝑓(𝑡, 𝑥(𝑡), 𝑢(𝑡)) 𝑥(𝑡0) = 𝑥0

DefinitionThe system is integral input-to-state stable (iISS) when thereexist 𝛼 ∈ 𭒦∞, 𝛽 ∈ 𭒦ℒ, and 𝛾 ∈ 𭒦 such that ∀ admissible 𝑢(𝑡),

𝛼(|𝑥(𝑡)|) ⩽ 𝛽(|𝑥0|, 𝑡 − 𝑡0) + ∫𝑡

𝑡0

𝛾(|𝑢(𝜏)|) 𝑑𝜏

for all 𝑡0, for all 𝑡 ⩾ 𝑡0, and for all 𝑥0 ∈ 𝑋.

ISS ⟹ iISS ⟹ 0-GAS

NonlinearControl

Nonlinearmodels






3rd ElgersburgSchool 2011 iISS Lyapunov functions

Definition

An iISS Lyapunov function is a C1 function 𝑉 ∶ ℝ × 𝑋 → ℝsuch that for some 𝛼1, 𝛼2 ∈ 𭒦∞, 𝛼3 ∈ 𭒫, and 𝛼4 ∈ 𭒦,

𝛼1(|𝑥|) ⩽ 𝑉(𝑡, 𝑥) ⩽ 𝛼2(|𝑥|) and 𝑉(𝑡, 𝑥, 𝑢) ⩽ −𝛼3(|𝑥|) + 𝛼4(|𝑢|)

for all 𝑡, 𝑥, and 𝑢. Note 𝛼3 ∈ 𭒫 instead of 𝛼3 ∈ 𭒦∞.

there exists an iISS Lyapunov function⇓

the system is iISS with gain function 𝛾 = 𝛼4 for any > 0

there are cascade and small-gain theorems for iISSiISS interconnections are a recent topic of research

NonlinearControl

Controllability

Feedbackequivalence

Relative degree

Feedbacklinearization

MIMO systems

PVTOL aircraft


Part II

Feedback linearization

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


7 Controllability

8 Feedback equivalence

9 Relative degree

10 Feedback linearization

11 MIMO systems

12 PVTOL aircraft

NonlinearControl

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Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 A simple unicycle

𝑥

𝑦𝜃

𝑣top view States, 𝑋 = ℝ2 × 𝑆1

(𝑥1, 𝑥2, 𝑥3) = (𝑥, 𝑦, )

Controls, 𝑢 ∈ ℝ2

𝑢1 = 𝑣 = forward velocity𝑢2 = = steering velocity

Nonlinear model

��1 = 𝑢1 cos(𝑥3)��2 = 𝑢1 sin(𝑥3)��3 = 𝑢2

�� =⎡⎢⎢⎣

cos(𝑥3)sin(𝑥3)

0

⎤⎥⎥⎦

𝑢1 +⎡⎢⎢⎣

001

⎤⎥⎥⎦

𝑢2

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Linear approximation of the unicycle model

Linearize the model about the equilibrium (𝑥, 𝑢) = (0, 0)

�� =⎡⎢⎢⎣

100

⎤⎥⎥⎦

𝑢1 +⎡⎢⎢⎣

001

⎤⎥⎥⎦

𝑢2 ��2 = 0

𝑥2 = 0

the state 𝑥2 is uncontrollable in the linearized system

the linearized unicycle is confined to the line 𝑥2 = 0

the nonlinear unicycle can leave this line

The linearized model does not capture the local behavior!

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Linear approximation of the unicycle model

Linearize the model about the equilibrium (𝑥, 𝑢) = (0, 0)

�� =⎡⎢⎢⎣

100

⎤⎥⎥⎦

𝑢1 +⎡⎢⎢⎣

001

⎤⎥⎥⎦

𝑢2 ��2 = 0

𝑥2 = 0

the state 𝑥2 is uncontrollable in the linearized system

the linearized unicycle is confined to the line 𝑥2 = 0

the nonlinear unicycle can leave this line

The linearized model does not capture the local behavior!

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Vector fields and directions of motionThe simple unicycle

Unicycle model �� = 𝑔1(𝑥)𝑢1 + 𝑔2(𝑥)𝑢2 with vector fields

𝑔1(𝑥) =⎡⎢⎢⎣

cos(𝑥3)sin(𝑥3)

0

⎤⎥⎥⎦

𝑔2(𝑥) =⎡⎢⎢⎣

001

⎤⎥⎥⎦

𝑔1(𝑥) is the forward drive direction with flow 𝜑1(𝑡, 𝑥)

𝑔2(𝑥) is the ccw steer direction with flow 𝜑2(𝑡, 𝑥)

these give two directions of instantaneous motion

𝑋 = ℝ3 (locally) and thus has three independent directions

For local controllability:

Is there a 3rd independent, instantaneous direction of motion?

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Wiggling the unicycle

Combined motion: the wiggle 𝑤(𝑡, 𝑥)

𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥))))

Start at some state 𝑥, and fix some small time 𝑡.

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥))))

Drive forward 𝑡 with (𝑢1, 𝑢2) = (1, 0).

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥))))

Steer counterclockwise 𝑡 with (𝑢1, 𝑢2) = (0, 1).

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥))))

Drive backward 𝑡 with (𝑢1, 𝑢2) = (−1, 0).

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥))))

Steer clockwise 𝑡 with (𝑢1, 𝑢2) = (0, −1).

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥))))

End at 𝑤(𝑡, 𝑥).

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥))))

What direction is the wiggle in the limit as 𝑡 → 0?Straight down, with no rotation …

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


A third direction of motionThe simple unicycle

Taylor expansion of the wiggle 𝑤(𝑡, 𝑥) about 𝑡 = 0

𝑤(𝑡, 𝑥) = 𝑥 +⎡⎢⎢⎣

sin(𝑥3)− cos(𝑥3)

0

⎤⎥⎥⎦⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑔3(𝑥)

𝑡2 + … (no linear term!)

the new direction 𝑔3(𝑥) is independent of 𝑔1(𝑥) and 𝑔2(𝑥)(it is the direction ⊥ to the forward motion of the wheel)

can move instantaneously in any direction in ℝ3 by takinglinear combinations (local controllability)

the 𝑔3 direction is slower than 𝑔1 and 𝑔2 because it appearsin the 𝑡2 term of the Taylor expansion, not the linear term

NonlinearControl

Controllability

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Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 The Lie bracket of vector fields

Define wiggle for arbitrary C∞ vector fields 𝑔1(𝑥) and 𝑔2(𝑥)

𝑤(𝑡, 𝑥) = 𝜑2(−𝑡, 𝜑1(−𝑡, 𝜑2(𝑡, 𝜑1(𝑡, 𝑥)))) = 𝑥 + 𝑔3(𝑥)𝑡2 + …

The wiggle vector field 𝑔3(𝑥) turns out to be

𝑔3(𝑥) = 𝐷𝑔2(𝑥) · 𝑔1(𝑥) − 𝐷𝑔1(𝑥) · 𝑔2(𝑥) (in local coordinates)

DefinitionThe Lie bracket [𝑔1, 𝑔2] of C∞ vector fields 𝑔1(𝑥) and 𝑔2(𝑥) isthe associated wiggle vector field 𝑔3(𝑥):

[𝑔1, 𝑔2] = 𝐷𝑔2 · 𝑔1 − 𝐷𝑔1 · 𝑔2 (in local coordinates)

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

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NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft








NonlinearControl

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Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Some properties of the Lie bracket

Let 𝑓(𝑥), 𝑔(𝑥), and ℎ(𝑥) be C∞ vector fields on 𝑋[𝑓, 𝑔] = −[𝑔, 𝑓] (skew symmetry) ⇒ [𝑓, 𝑓] = 0

[𝑓, 𝛼𝑔 + 𝛽ℎ] = 𝛼[𝑓, 𝑔] + 𝛽[𝑓, ℎ] for 𝛼, 𝛽 ∈ ℝ (bilinearity)

[𝑓, [𝑔, ℎ]] + [ℎ, [𝑓, 𝑔]] + [𝑔, [ℎ, 𝑓]] = 0 (Jacobi identity)

for any C∞ function 𝜑(𝑥) on 𝑋,

𝐿[𝑓,𝑔]𝜑(𝑥) = 𝐿𝑓𝐿𝑔𝜑(𝑥) − 𝐿𝑔𝐿𝑓𝜑(𝑥)

Additional notationThe mapping ad𝑓(·) takes a vector field 𝑔 and produces thevector field ad𝑓(𝑔) = [𝑓, 𝑔] so that

ad0𝑓(𝑔) = 𝑔 ad1

𝑓(𝑔) = [𝑓, 𝑔] ad2𝑓(𝑔) = [𝑓, [𝑓, 𝑔]] …

NonlinearControl

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Relative degree


MIMO systems

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𝑓(𝑔) = [𝑓, 𝑔] ad2𝑓(𝑔) = [𝑓, [𝑓, 𝑔]] …

NonlinearControl

Controllability

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Relative degree


MIMO systems

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𝑓(𝑔) = [𝑓, 𝑔] ad2𝑓(𝑔) = [𝑓, [𝑓, 𝑔]] …

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

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𝑓(𝑔) = [𝑓, 𝑔] ad2𝑓(𝑔) = [𝑓, [𝑓, 𝑔]] …

NonlinearControl

Controllability

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Relative degree


MIMO systems

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𝑓(𝑔) = [𝑓, 𝑔] ad2𝑓(𝑔) = [𝑓, [𝑓, 𝑔]] …

NonlinearControl

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Relative degree


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3rd ElgersburgSchool 2011 Lie algebras of vector fields

𭒱(𝑋) = set of all C∞ vector fields on 𝑋

𭒱(𝑋) is a real vector space under pointwise operations

a subspace of 𭒱(𝑋) is called a Lie algebra when it isclosed under Lie brackets

given a collection {𝑓𝑖} of vector fields in 𭒱(𝑋),Lie{𝑓𝑖} = the smallest Lie algebra containing all the 𝑓𝑖

given 𝑥 ∈ 𝑋, the set

span{𝑓(𝑥) ∶ 𝑓 ∈ Lie{𝑓𝑖}}

is a subspace of ℝ𝑛 (or the tangent space of 𝑋 at 𝑥)

NonlinearControl

Controllability

Feedbackequivalence

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3rd ElgersburgSchool 2011 The accessibility rank condition

Control-affine nonlinear system

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢 = 𝑓(𝑥) + 𝑔1(𝑥)𝑢1 + … + 𝑔𝑚(𝑥)𝑢𝑚 (*)

DefinitionThe system (*) satisfies the accessibility rank condition(ARC) at a point 𝑥 ∈ 𝑋 when

dim span{ℎ(𝑥) ∶ ℎ ∈ Lie{𝑓, 𝑔1, …, 𝑔𝑚}⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟the control Lie algebra

} = 𝑛

ARC(𝑥) =can generate all directions at 𝑥 by taking linearcombinations and Lie brackets of vector fieldsin {𝑓, 𝑔1, …, 𝑔𝑚} (including drift 𝑓)

What does ARC imply?

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3rd ElgersburgSchool 2011 The accessibility rank condition

Control-affine nonlinear system

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢 = 𝑓(𝑥) + 𝑔1(𝑥)𝑢1 + … + 𝑔𝑚(𝑥)𝑢𝑚 (*)

DefinitionThe system (*) satisfies the accessibility rank condition(ARC) at a point 𝑥 ∈ 𝑋 when

dim span{ℎ(𝑥) ∶ ℎ ∈ Lie{𝑓, 𝑔1, …, 𝑔𝑚}⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟the control Lie algebra

} = 𝑛

ARC(𝑥) =can generate all directions at 𝑥 by taking linearcombinations and Lie brackets of vector fieldsin {𝑓, 𝑔1, …, 𝑔𝑚} (including drift 𝑓)


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Accessibility rank conditionLTI case

suppose 𝑓(𝑥) = 𝐴𝑥 and each 𝑔𝑖(𝑥) = 𝑔𝑖 = constant

because [𝑓, ℎ] = −𝐴ℎ for any constant ℎ, we have

dim span{ℎ(𝑥) ∶ ℎ ∈ Lie{𝑓, 𝑔1, …, 𝑔𝑚}}= rank [𝐴𝑥 𝐵 𝐴𝐵 𝐴2𝐵 … 𝐴𝑛−1𝐵]

where 𝐵 = [𝑔1 … 𝑔𝑚]

ARC(0) is the LTI controllability rank condition

if 𝑥 ≠ 0 then ARC(𝑥) may be weaker than the LTIcontrollability rank condition


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The reachable setSmall-time local accessibility and controllability

Given 𝑥 ∈ 𝑋, an open neighborhood 𝑈 of 𝑥, and a time 𝑇 > 0:

reachable set 𝑅𝑇𝑈(𝑥) =

all states that can be reachedwithin time 𝑇 starting from 𝑥and without leaving 𝑈

DefinitionThe system is small-time accessible (STA) at 𝑥 when

int 𝑅𝑇𝑈(𝑥) ≠ ∅ ∀𝑈 ∀𝑇 > 0

It is small-time locally controllable (STLC) at 𝑥 when

𝑥 ∈ int 𝑅𝑇𝑈(𝑥) ∀𝑈 ∀𝑇 > 0

Warning: inconsistent terminology in the literatureSometimes accessibility is called weak controllability.Sometimes STLC is defined with 𝑈 = 𝑋 instead of ∀𝑈.

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The reachable setSmall-time local accessibility and controllability

Given 𝑥 ∈ 𝑋, an open neighborhood 𝑈 of 𝑥, and a time 𝑇 > 0:

reachable set 𝑅𝑇𝑈(𝑥) =

all states that can be reachedwithin time 𝑇 starting from 𝑥and without leaving 𝑈

DefinitionThe system is small-time accessible (STA) at 𝑥 when

int 𝑅𝑇𝑈(𝑥) ≠ ∅ ∀𝑈 ∀𝑇 > 0

It is small-time locally controllable (STLC) at 𝑥 when

𝑥 ∈ int 𝑅𝑇𝑈(𝑥) ∀𝑈 ∀𝑇 > 0

Warning: inconsistent terminology in the literatureSometimes accessibility is called weak controllability.Sometimes STLC is defined with 𝑈 = 𝑋 instead of ∀𝑈.

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3rd ElgersburgSchool 2011 What does ARC imply?

Sufficient conditions

ARC(𝑥) ⟹ STA at 𝑥

Necessary and sufficient conditionsARC holds on an open dense subset of 𝑋

⇕the system is STA on an open dense subset of 𝑋

Real analytic systemsARC holds on all of of 𝑋

⇕the system is STA at every 𝑥 ∈ 𝑋

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What does ARC imply?Systems without drift

For systems without drift (𝑓 ≡ 0)

STA at 𝑥 ⟺ STLC at 𝑥

ARC(𝑥) ⟹ STLC at 𝑥

ARC holds on an open dense subset of 𝑋⇕

the system is STLC on an open dense subset of 𝑋

the system is completely controllable⇑ (𝑋 is connected)

ARC holds on all of of 𝑋⇓ ⇑ (system is real analytic)

the system is STLC at every 𝑥 ∈ 𝑋

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What does ARC imply?Systems without drift

For systems without drift (𝑓 ≡ 0)

STA at 𝑥 ⟺ STLC at 𝑥

ARC(𝑥) ⟹ STLC at 𝑥

ARC holds on an open dense subset of 𝑋⇕

the system is STLC on an open dense subset of 𝑋

the system is completely controllable⇑ (𝑋 is connected)

ARC holds on all of of 𝑋⇓ ⇑ (system is real analytic)

the system is STLC at every 𝑥 ∈ 𝑋

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7 Controllability


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Feedback equivalenceStatic state feedback

start with a system�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢

apply static state feedback

𝑢 = 𝛼(𝑥) + 𝛽(𝑥)𝑣

to get a system with a new input 𝑣

�� = 𝑓(𝑥) + 𝑔(𝑥)𝛼(𝑥) + 𝑔(𝑥)𝛽(𝑥)𝑣

change coordinates 𝑧 = Φ(𝑥) to obtain

𝑧 = 𝐹(𝑧) + 𝐺(𝑧)𝑣where

𝐹(Φ(𝑥)) = 𝐷Φ(𝑥)[𝑓(𝑥) + 𝑔(𝑥)𝛼(𝑥)]𝐺(Φ(𝑥)) = 𝐷Φ(𝑥)𝑔(𝑥)𝛽(𝑥)

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𝑢 = 𝛼(𝑥) + 𝛽(𝑥)𝑣


�� = 𝑓(𝑥) + 𝑔(𝑥)𝛼(𝑥) + 𝑔(𝑥)𝛽(𝑥)𝑣




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𝑢 = 𝛼(𝑥) + 𝛽(𝑥)𝑣


�� = 𝑓(𝑥) + 𝑔(𝑥)𝛼(𝑥) + 𝑔(𝑥)𝛽(𝑥)𝑣




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𝛼(𝑥) + 𝛽(𝑥)𝑣 �� = 𝑓(𝑥) + 𝑔(𝑥)𝑢 Φ 𝑧𝑣𝑥𝑢

𝑧 = 𝐹(𝑧) + 𝐺(𝑧)𝑣

Reversible feedback transformation on an open set 𝑈 ⊂ 𝑋C∞ functions 𝛼 ∶ 𝑈 → ℝ𝑚 and 𝛽 ∶ 𝑈 → ℝ𝑚×𝑚

𝛽(𝑥) is nonsingular on 𝑈Φ ∶ 𝑈 → 𝑈′ is a diffeomorphism

Why transform? The new system might be easier to use …

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Feedback equivalenceStatic state feedback example

System with three states 𝑋 = ℝ3 and two inputs 𝑢 ∈ ℝ2

��1 = 𝑥21 + 𝑥2

��2 = 𝑥1𝑥2 + 𝑥23𝑢1 + 𝑢2

��3 = 𝑥1𝑥3 + 𝑢1

Static state feedback and global coordinate change

[𝑢1𝑢2] = [

−𝑥1𝑥3𝑥1𝑥3

3 − 2𝑥31 − 3𝑥1𝑥2]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝛼(𝑥)

+ [0 11 −𝑥2

3]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝛽(𝑥)

[𝑣1𝑣2]

⎡⎢⎢⎣

𝑧1𝑧2𝑧3

⎤⎥⎥⎦

= Φ(𝑥) =⎡⎢⎢⎣

𝑥1𝑥2

1 + 𝑥2𝑥3

⎤⎥⎥⎦

,⎡⎢⎢⎣

𝑥1𝑥2𝑥3

⎤⎥⎥⎦

= Φ−1(𝑧) =⎡⎢⎢⎣

𝑧1−𝑧2

1 + 𝑧2𝑧3

⎤⎥⎥⎦

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The transformed system is LTI

𝑧 =⎡⎢⎢⎣

𝑧1𝑧2𝑧3

⎤⎥⎥⎦

=⎡⎢⎢⎣

𝑧2𝑣1𝑣2

⎤⎥⎥⎦

=⎡⎢⎢⎣

0 1 00 0 00 0 0

⎤⎥⎥⎦

𝑧

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝐹(𝑧)

+⎡⎢⎢⎣

0 01 00 1

⎤⎥⎥⎦⏟⏟⏟⏟⏟

𝐺(𝑧)

𝑣

The original nonlinear system is feedback equivalent to acontrollable LTI system. This suggests a design strategy:

Feedback linearizationdesign a controller for 𝑣 using linear methods applied tothe LTI transformed systemimplement 𝑢 = 𝛼(𝑥) + 𝛽(𝑥)𝑣 on the original system

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The transformed system is LTI

𝑧 =⎡⎢⎢⎣

𝑧1𝑧2𝑧3

⎤⎥⎥⎦

=⎡⎢⎢⎣

𝑧2𝑣1𝑣2

⎤⎥⎥⎦

=⎡⎢⎢⎣

0 1 00 0 00 0 0

⎤⎥⎥⎦

𝑧

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝐹(𝑧)

+⎡⎢⎢⎣

0 01 00 1

⎤⎥⎥⎦⏟⏟⏟⏟⏟

𝐺(𝑧)

𝑣

The original nonlinear system is feedback equivalent to acontrollable LTI system. This suggests a design strategy:

Feedback linearizationdesign a controller for 𝑣 using linear methods applied tothe LTI transformed systemimplement 𝑢 = 𝛼(𝑥) + 𝛽(𝑥)𝑣 on the original system

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3rd ElgersburgSchool 2011 Some questions about feedback linearization

is a given system feedback equivalent to an LTI system?

if so, what is the feedback and the coordinate change?

if not, would it help to consider dynamic feedback?

what if the model nonlinearities are uncertain?

what if there are control constraints?

does the linearizing feedback lead to a good design?

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7 Controllability


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Relative degreeSISO case

Control-affine system with 𝑢, 𝑦 ∈ ℝ

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢𝑦 = ℎ(𝑥) + 𝑗(𝑥)𝑢

DefinitionThe system has relative degree 0 at 𝑥0 ∈ 𝑋 when 𝑗(𝑥0) ≠ 0.

DefinitionThe system has relative degree 𝑟 ⩾ 1 at 𝑥0 ∈ 𝑋 when thereexists a neighborhood 𝑈 of 𝑥0 such that

𝑗(𝑥) = 0 for 𝑥 ∈ 𝑈

𝐿𝑔𝐿𝑟−1𝑓 ℎ(𝑥0) ≠ 0

if 𝑟 ⩾ 2 then 𝐿𝑔𝐿𝑘𝑓 ℎ(𝑥) = 0 for 𝑥 ∈ 𝑈 and 𝑘 = 0, …, 𝑟 − 2

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Relative degree and output derivativesSISO case

relative degree = number of time-derivatives of theoutput 𝑦 until the input 𝑢 appears

Example

��1 = 𝑥21 + 𝑥2

��2 = 𝑥3 + 𝑢��3 = (𝑥2 + 1)𝑥3 + 𝑥2𝑢

Relative degree 𝑟 at 𝑥0 = 0?

𝑦 = 𝑥1 ⟹ 𝑟 = 2𝑦 = 𝑥1𝑥3 + 𝑥2 ⟹ 𝑟 = 1𝑦 = 𝑥3 ⟹ 𝑟 = ??

The relative degree might not exist at every 𝑥0 ∈ 𝑋.

For SISO LTI systems, the relative degree is the pole/zeroexcess of the transfer function (#poles − #zeros).

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Partial feedback linearization (I/O linearization)Normal form

If a system has relative degree 𝑟 ⩾ 1 at 𝑥0 ∈ 𝑋 then it is locally(in a neighborhood of 𝑥0) feedback equivalent to

= 𝑞( , 𝜉) ∈ ℝ𝑛−𝑟

𝜉1 = 𝜉2 𝜉 = (𝜉1, …, 𝜉𝑟) ∈ ℝ𝑟

𝜉2 = 𝜉3 𝑧 = ( , 𝜉) = Φ(𝑥)⋮

𝜉𝑟 = 𝑣 𝑣 = new input𝑦 = 𝜉1

The input/output map is the LTI system 𝑦(𝑟) = 𝑣.

Alternative form: = 𝑞( , 𝜉) + 𝑝( , 𝜉)𝑣(can be easier to calculate the coordinates)

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3rd ElgersburgSchool 2011 Finding the normal form

Example

��1 = 𝑥21 + 𝑥2

��2 = 𝑥3 + 𝑢��3 = (𝑥2 + 1)𝑥3 + 𝑥2𝑢𝑦 = 𝑥1

Relative degree 𝑟 = 2

𝜉1 = 𝑦 = 𝑥1

𝜉2 = 𝑦 = 𝑥21 + 𝑥2

= ??

How do we choose the third coordinate ?(𝑥1, 𝑥2, 𝑥3) ↦ ( , 𝜉1, 𝜉2) must be a diffeomorphism(optionally) make independent of 𝑢: = − 1

2 𝑥22 + 𝑥3

= + 12 (𝜉2 − 𝜉2

1)2

𝜉1 = 𝜉2 𝑦 = 𝜉1 , 𝑦 = 𝑣𝜉2 = 𝑣 𝑢 = −2𝑥1(𝑥2

1 + 𝑥2) − 𝑥3 + 𝑣

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3rd ElgersburgSchool 2011 Tracking via I/O linearization

in this example, the input/output map is 𝑦 = 𝑣

thus we can design a linear controller 𝑣 for tracking areference output 𝑦ref(𝑡) having known derivatives:

𝑣 = 𝑦ref − 𝑘1(𝜉1 − 𝑦ref) − 𝑘2(𝜉2 − 𝑦ref)

this controller guarantees 𝑦(𝑡) → 𝑦ref(𝑡) as 𝑡 → ∞

this controller does not take into account the dynamics

= + 12 (𝜉2 − 𝜉2

1)2

can grow unbounded, resulting in an unbounded 𝑥 and 𝑢⟹ the zero dynamics = are unstable

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= + 12 (𝜉2 − 𝜉2

1)2


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= + 12 (𝜉2 − 𝜉2

1)2


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= + 12 (𝜉2 − 𝜉2

1)2


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= + 12 (𝜉2 − 𝜉2

1)2


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3rd ElgersburgSchool 2011 Zero dynamics

Zero dynamicsThe zero dynamics are the internal state trajectories that arisewhen the output is held at zero, 𝑦(𝑡) ≡ 0. In the normal form,we set 𝜉 and 𝑣 to zero to obtain the zero dynamics

= 𝑞( , 0)

the system is minimum-phase when = 0 is anasymptotically stable equilibrium of the zero dynamics

the minimum-phase condition guarantees that the trackingcontrollers will work locally for sufficiently small 𝑦ref, 𝑦ref, …

stronger stability conditions on the zero dynamics, such asbeing ISS w.r.t. (𝜉, 𝑣), can guarantee internal boundednessfor larger initial conditions and reference signals

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Zero dynamicsLTI case, frequency domain

LTI system with relative degree 𝑟 (𝑏0 ≠ 0)

𝑌(𝑠) = 𝑏(𝑠)𝑎(𝑠)

𝑈(𝑠) =𝑏0𝑠𝑛−𝑟 + 𝑏1𝑠𝑛−𝑟−1 + … + 𝑏𝑛−𝑟

𝑠𝑛 + 𝑎1𝑠𝑛−1 + … + 𝑎𝑛𝑈(𝑠)

normal form is 𝑌(𝑠) = 𝑠−𝑟𝑉(𝑠) for the new input 𝑉(𝑠), so

𝑈(𝑠) = 𝑎(𝑠)𝑠𝑟𝑏(𝑠)

𝑉(𝑠) (inverted plant)

poles at roots of 𝑏(𝑠) are rendered unobservable from 𝑦

these poles are at the plant zeros, hence zero dynamics

feedback transformation into normal form effectivelycancels plant zeros with controller poles

the simple tracking controller may lead to internalunboundedness if the plant has unstable zero dynamics

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𝑈(𝑠) =𝑏0𝑠𝑛−𝑟 + 𝑏1𝑠𝑛−𝑟−1 + … + 𝑏𝑛−𝑟

𝑠𝑛 + 𝑎1𝑠𝑛−1 + … + 𝑎𝑛𝑈(𝑠)








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𝑈(𝑠) =𝑏0𝑠𝑛−𝑟 + 𝑏1𝑠𝑛−𝑟−1 + … + 𝑏𝑛−𝑟

𝑠𝑛 + 𝑎1𝑠𝑛−1 + … + 𝑎𝑛𝑈(𝑠)








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𝑈(𝑠) =𝑏0𝑠𝑛−𝑟 + 𝑏1𝑠𝑛−𝑟−1 + … + 𝑏𝑛−𝑟

𝑠𝑛 + 𝑎1𝑠𝑛−1 + … + 𝑎𝑛𝑈(𝑠)








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𝑈(𝑠) =𝑏0𝑠𝑛−𝑟 + 𝑏1𝑠𝑛−𝑟−1 + … + 𝑏𝑛−𝑟

𝑠𝑛 + 𝑎1𝑠𝑛−1 + … + 𝑎𝑛𝑈(𝑠)








NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


7 Controllability


9 Relative degree


11 MIMO systems

12 PVTOL aircraft

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Feedback linearization

What if the relative degree is 𝑛?the resulting normal form is the LTI system 𝑦(𝑛) = 𝑣there are no zero dynamics (vacuously minimum-phase)the system is feedback equivalent to 𝑦(𝑛) = 𝑣the system is feedback linearizable

the system is feedback equivalent to a controllable LTI system⇕

there exists an output function 𝑦 = ℎ(𝑥)for which the system has relative degree 𝑛

When does such an output function ℎ(𝑥) exist?

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Distributions

If {𝑓𝑖} is a collection of vector fields on 𝑋, then

Δ = span{𝑓𝑖}

represents a mapping which takes a point 𝑥 ∈ 𝑋 to a subspaceΔ(𝑥) of ℝ𝑛 (or the tangent space of 𝑋 at 𝑥).

Δ is called a distribution on 𝑋

if the 𝑓𝑖 are smooth (C∞), then Δ is a smooth distribution

a vector field 𝑓 belongs to Δ, written 𝑓 ∈ Δ, when𝑓(𝑥) ∈ Δ(𝑥) for all 𝑥 ∈ 𝑋 (pointwise membership)

a distribution Δ is nonsingular at 𝑥 ∈ 𝑋 when Δ(·) hasconstant dimension in a neighborhood of 𝑥

A distribution is like a multi-directional vector field.

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Frobenius theoremA multi-directional straightening-out theorem

DefinitionA distribution Δ is involutive when [Δ, Δ] ⊂ Δ, that is, when𝑓, 𝑔 ∈ Δ implies [𝑓, 𝑔] ∈ Δ.

If Δ = span{𝑓𝑖} is nonsingular, then it is involutive if and only if[𝑓𝑖, 𝑓𝑗] ∈ Δ for all 𝑖 and 𝑗 (a verifiable condition).

Frobenius theoremSuppose a distribution Δ is nonsingular at some 𝑥 ∈ 𝑋. ThenΔ is involutive in a neighborhood of 𝑥 if and only if it iscompletely integrable, that is, there exist local coordinatesaround 𝑥 in which Δ is the span of constant vector fields.

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Frobenius theoremA multi-directional straightening-out theorem

DefinitionA distribution Δ is involutive when [Δ, Δ] ⊂ Δ, that is, when𝑓, 𝑔 ∈ Δ implies [𝑓, 𝑔] ∈ Δ.

If Δ = span{𝑓𝑖} is nonsingular, then it is involutive if and only if[𝑓𝑖, 𝑓𝑗] ∈ Δ for all 𝑖 and 𝑗 (a verifiable condition).

Frobenius theoremSuppose a distribution Δ is nonsingular at some 𝑥 ∈ 𝑋. ThenΔ is involutive in a neighborhood of 𝑥 if and only if it iscompletely integrable, that is, there exist local coordinatesaround 𝑥 in which Δ is the span of constant vector fields.

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Frobenius theorem in action

Control-affine system with 𝑋 = ℝ3 and 𝑢 ∈ ℝ2

�� =⎡⎢⎢⎣

𝑥2𝑥3

𝑥22 + 𝑥1𝑥3 + 𝑥3 − 𝑥1𝑥2

⎤⎥⎥⎦⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑓(𝑥)

+⎡⎢⎢⎣

10𝑥2

⎤⎥⎥⎦⏟⏟⏟

𝑔1(𝑥)

𝑢1 +⎡⎢⎢⎣

01𝑥1

⎤⎥⎥⎦⏟⏟⏟

𝑔2(𝑥)

𝑢2

define Δ = span{𝑔1, 𝑔2} (nonsingular with dim Δ = 2)[𝑔1, 𝑔2] = 0 ∈ Δ, so Δ is involutive[𝑓, 𝑔1] = −𝑥2𝑔2 ∈ Δ and [𝑓, 𝑔2] = −𝑔1 − 𝑥1𝑔2 ∈ Δ⟹ [𝑓, Δ] ⊂ Δ (and we say that Δ is invariant under 𝑓 )Frobenius: there exist local coordinates 𝑧 = Φ(𝑥) in which

Δ = span{⎡⎢⎢⎣

100

⎤⎥⎥⎦

,⎡⎢⎢⎣

010

⎤⎥⎥⎦}

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



�� =⎡⎢⎢⎣

𝑥2𝑥3

𝑥22 + 𝑥1𝑥3 + 𝑥3 − 𝑥1𝑥2

⎤⎥⎥⎦⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑓(𝑥)

+⎡⎢⎢⎣

10𝑥2

⎤⎥⎥⎦⏟⏟⏟

𝑔1(𝑥)

𝑢1 +⎡⎢⎢⎣

01𝑥1

⎤⎥⎥⎦⏟⏟⏟

𝑔2(𝑥)

𝑢2


Δ = span{⎡⎢⎢⎣

100

⎤⎥⎥⎦

,⎡⎢⎢⎣

010

⎤⎥⎥⎦}

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



�� =⎡⎢⎢⎣

𝑥2𝑥3

𝑥22 + 𝑥1𝑥3 + 𝑥3 − 𝑥1𝑥2

⎤⎥⎥⎦⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑓(𝑥)

+⎡⎢⎢⎣

10𝑥2

⎤⎥⎥⎦⏟⏟⏟

𝑔1(𝑥)

𝑢1 +⎡⎢⎢⎣

01𝑥1

⎤⎥⎥⎦⏟⏟⏟

𝑔2(𝑥)

𝑢2


Δ = span{⎡⎢⎢⎣

100

⎤⎥⎥⎦

,⎡⎢⎢⎣

010

⎤⎥⎥⎦}

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft



�� =⎡⎢⎢⎣

𝑥2𝑥3

𝑥22 + 𝑥1𝑥3 + 𝑥3 − 𝑥1𝑥2

⎤⎥⎥⎦⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝑓(𝑥)

+⎡⎢⎢⎣

10𝑥2

⎤⎥⎥⎦⏟⏟⏟

𝑔1(𝑥)

𝑢1 +⎡⎢⎢⎣

01𝑥1

⎤⎥⎥⎦⏟⏟⏟

𝑔2(𝑥)

𝑢2


Δ = span{⎡⎢⎢⎣

100

⎤⎥⎥⎦

,⎡⎢⎢⎣

010

⎤⎥⎥⎦}

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Frobenius theorem in actionExample continued

Because [𝑓, Δ] ⊂ Δ, we have (in the new coordinates)

[𝑓,⎡⎢⎢⎣

100

⎤⎥⎥⎦] = −

𝜕𝑓𝜕𝑧1

∈ Δ ⟹𝜕𝑓3𝜕𝑧1

≡ 0

[𝑓,⎡⎢⎢⎣

010

⎤⎥⎥⎦] = −

𝜕𝑓𝜕𝑧2

∈ Δ ⟹𝜕𝑓3𝜕𝑧2

≡ 0

Thus the third coordinate 𝑧3 has dynamics of the form

𝑧3 = 𝑓3(𝑧3)

This is an uncontrollable subsystem.

Frobenius theorem does not tell ushow to calculate these special coordinates.

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Feedback linearizationNecessary and sufficient conditions

Single-input, control-affine system �� = 𝑓(𝑥) + 𝑔(𝑥)𝑢Given 𝑥0 ∈ 𝑋, there exists an output 𝑦 = ℎ(𝑥) with respect towhich the system has relative degree 𝑛 at 𝑥0 if and only if

the matrix [𝑔(𝑥0) ad𝑓 𝑔(𝑥0) … ad𝑛−1𝑓 𝑔(𝑥0)] has rank 𝑛

Δ = span{𝑔, ad𝑓 𝑔, … , ad𝑛−2𝑓 𝑔} is involutive near 𝑥0

if 𝑓(𝑥0) = 0 (so that 𝑥0 is an equilibrium when 𝑢 = 0), thenthe first condition is equivalent to the controllability of thelinear approximation of the system at 𝑥0

these conditions are straightforward to check, but it maystill be difficult to calculate the output function ℎ(𝑥)

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


7 Controllability


9 Relative degree


11 MIMO systems

12 PVTOL aircraft

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Vector relative degreeSquare systems

Control-affine system with 𝑢, 𝑦 ∈ ℝ𝑚

�� = 𝑓(𝑥) + 𝑔1(𝑥)𝑢1 + … + 𝑔𝑚(𝑥)𝑢𝑚

𝑦 = [ℎ1(𝑥) … ℎ𝑚(𝑥)]𝑇

The system has vector relative degree (𝑟1, …, 𝑟𝑚) at 𝑥0 ∈ 𝑋when there exists a neighborhood 𝑈 of 𝑥0 such that

the 𝑚 × 𝑚 matrix with 𝐿𝑔𝑗𝐿𝑟𝑖−1

𝑓 ℎ𝑖(𝑥) in row 𝑖 and column 𝑗is nonsingular at 𝑥0

for each 𝑖 = 1, …, 𝑚, if 𝑟𝑖 ⩾ 2 then 𝐿𝑔𝑗𝐿𝑘

𝑓 ℎ𝑖(𝑥) = 0for all 𝑥 ∈ 𝑈 and 𝑗 = 1, …, 𝑚 and 𝑘 = 0, …, 𝑟𝑖 − 2

Necessary for the existence of a vector relative degree at 𝑥0:

rank [𝑔1(𝑥0) … 𝑔𝑚(𝑥0)] = 𝑚

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Finding the vector relative degreeDifferentiating the outputs

��1 = 𝑥21 + 𝑥3 𝑦1 = 𝑥1

��2 = 𝑥1𝑥3 + 𝑢1 𝑦2 = 𝑥2

��3 = 𝑥2 + 𝑥3𝑢1 + 𝑢2

[𝑦1𝑦2] = [

2𝑥31 + 2𝑥1𝑥3 + 𝑥2

𝑥1𝑥3 ] + [𝑥3 11 0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟nonsingular

[𝑢1𝑢2] = [

𝑣1𝑣2]

the system has vector relative degree (𝑟1, 𝑟2) = (2, 1)after feedback transformation, the input/output mapbecomes 𝑦1 = 𝑣1, 𝑦2 = 𝑣2

in general, vector relative degree means feedbackequivalent to 𝑦(𝑟𝑖) = 𝑣𝑖 for 𝑖 = 1, …, 𝑚,plus zero dynamics if 𝑟1 + … + 𝑟𝑚 < 𝑛

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Feedback linearizationNecessary and sufficient conditions, MIMO case

fix 𝑥0 ∈ 𝑋 and assume rank [𝑔1(𝑥0) … 𝑔𝑚(𝑥0)] = 𝑚define the distributions

Δ0 = span{𝑔1, …, 𝑔𝑚}Δ1 = span{𝑔1, …, 𝑔𝑚, ad𝑓 𝑔1, …, ad𝑓 𝑔𝑚}

⋮Δ𝑖 = span{ad𝑘

𝑓 𝑔𝑗 ∶ 0 ⩽ 𝑘 ⩽ 𝑖, 1 ⩽ 𝑗 ⩽ 𝑚}

There exist outputs 𝑦𝑖 = ℎ𝑖(𝑥) with respect to which the systemhas vector relative degree (𝑟1, …, 𝑟𝑚) at 𝑥0 with 𝑟1 + … + 𝑟𝑚 = 𝑛(no zero dynamics) if and only if

Δ𝑖 is nonsingular at 𝑥0 for 0 ⩽ 𝑖 ⩽ 𝑛 − 1dim Δ𝑛−1 = 𝑛Δ𝑖 is involutive for 0 ⩽ 𝑖 ⩽ 𝑛 − 2

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Checking the feedback linearization conditions


�� =⎡⎢⎢⎢⎣

𝑥20𝑥40

⎤⎥⎥⎥⎦⏟⏟⏟

𝑓(𝑥)

+⎡⎢⎢⎢⎣

01𝑥10

⎤⎥⎥⎥⎦⏟⏟⏟

𝑔1(𝑥)

𝑢1 +⎡⎢⎢⎢⎣

0001

⎤⎥⎥⎥⎦⏟⏟⏟

𝑔2(𝑥)

𝑢2

dim Δ0 = 2[𝑔1, 𝑔2] = 0 so Δ0 is involutivedim Δ1 = 4 so Δ𝑖 is nonsingular with dim Δ𝑖 = 4(and thus Δ𝑖 is involutive) for 𝑖 ⩾ 1

The linearizing outputs 𝑦1 = ℎ1(𝑥) and 𝑦2 = ℎ2(𝑥) exist …

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft

3rd ElgersburgSchool 2011 Finding the linearizing outputs

try 𝑦1 = 𝑥1 and 𝑦2 = 𝑥3 − 𝑥1𝑥2 :

[𝑦1𝑦2] = [

1 0−2𝑥2 1]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟nonsingular

[𝑢1𝑢2]

vector relative degree (2, 2) with no zero dynamics

what if we instead want to track 𝑦1 = 𝑥1 and 𝑦2 = 𝑥3 ?

[𝑦1𝑦2] = [

0𝑥4] + [

1 0𝑥1 0]

⏟⏟⏟⏟⏟⏟⏟⏟⏟singular

[𝑢1𝑢2]

no vector relative degree; try dynamic extension …

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


try 𝑦1 = 𝑥1 and 𝑦2 = 𝑥3 − 𝑥1𝑥2 :

[𝑦1𝑦2] = [

1 0−2𝑥2 1]


[𝑢1𝑢2]



[𝑦1𝑦2] = [

0𝑥4] + [

1 0𝑥1 0]


[𝑢1𝑢2]


NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


try 𝑦1 = 𝑥1 and 𝑦2 = 𝑥3 − 𝑥1𝑥2 :

[𝑦1𝑦2] = [

1 0−2𝑥2 1]


[𝑢1𝑢2]



[𝑦1𝑦2] = [

0𝑥4] + [

1 0𝑥1 0]


[𝑢1𝑢2]


NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


try 𝑦1 = 𝑥1 and 𝑦2 = 𝑥3 − 𝑥1𝑥2 :

[𝑦1𝑦2] = [

1 0−2𝑥2 1]


[𝑢1𝑢2]



[𝑦1𝑦2] = [

0𝑥4] + [

1 0𝑥1 0]


[𝑢1𝑢2]


NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


Dynamic extensionAdding integrators to achieve relative degree

introduce a controller state 𝑥5 with dynamics ��5 = 𝑢′1,

where 𝑢′1 is a new input

set 𝑢1 = 𝑥5 to achieve

[𝑦1𝑦2] = [

𝑥5𝑥4 + 𝑥1𝑥5]

take another derivative to get

[𝑦1𝑦2] = [

0𝑥2𝑥5] + [

1 0𝑥1 1]


[𝑢′

1𝑢2]


Dynamic feedback can be useful for achievingvector relative degree in MIMO systems.

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft






[𝑦1𝑦2] = [

𝑥5𝑥4 + 𝑥1𝑥5]


[𝑦1𝑦2] = [

0𝑥2𝑥5] + [

1 0𝑥1 1]


[𝑢′

1𝑢2]



NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft






[𝑦1𝑦2] = [

𝑥5𝑥4 + 𝑥1𝑥5]


[𝑦1𝑦2] = [

0𝑥2𝑥5] + [

1 0𝑥1 1]


[𝑢′

1𝑢2]



NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft






[𝑦1𝑦2] = [

𝑥5𝑥4 + 𝑥1𝑥5]


[𝑦1𝑦2] = [

0𝑥2𝑥5] + [

1 0𝑥1 1]


[𝑢′

1𝑢2]



NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft






[𝑦1𝑦2] = [

𝑥5𝑥4 + 𝑥1𝑥5]


[𝑦1𝑦2] = [

0𝑥2𝑥5] + [

1 0𝑥1 1]


[𝑢′

1𝑢2]



NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


7 Controllability


9 Relative degree


11 MIMO systems

12 PVTOL aircraft

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


PVTOL aircraftPlanar vertical take-off and landing aircraft

𝑥

𝑧

𝜃

𝑇

(𝑥, 𝑧) = center of mass

States, 𝑋 = ℝ4 × 𝑆1 × ℝ

(𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6)= (𝑥, ��, 𝑧, 𝑧, , )

Controls, 𝑢 ∈ ℝ2

𝑢1 = 𝑇 = vertical thrust𝑢2 = = rotational torque

Normalized dynamics, = input coupling parameter

�� = −𝑢1 sin( ) + 𝑢2 cos( )𝑧 = 𝑢1 cos( ) + 𝑢2 sin( ) − 1 = 𝑢2

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


PVTOL zero dynamicsTracking the center of mass

With the output as the center of mass, (𝑦1, 𝑦2) = (𝑥, 𝑧),

[𝑦1𝑦2] = [

0−1] + [

− sin( ) cos( )cos( ) sin( )]

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟nonsingular

[𝑢1𝑢2] = [

𝑣1𝑣2]

vector relative degree (2, 2)setting 𝑣1 = 𝑣2 = 0 yields 𝑢2 = 1

𝜀 sin( ),so the zero dynamics are

= 1𝜀 sin( )

these are the simple inverted pendulum dynamics( = 0 represents the unstable position)nonminimum-phase system;aircraft will flip over under the simple tracking control

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft




[𝑦1𝑦2] = [

0−1] + [


⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟nonsingular

[𝑢1𝑢2] = [

𝑣1𝑣2]



= 1𝜀 sin( )


NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft




[𝑦1𝑦2] = [

0−1] + [


⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟nonsingular

[𝑢1𝑢2] = [

𝑣1𝑣2]



= 1𝜀 sin( )


NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft




[𝑦1𝑦2] = [

0−1] + [


⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟nonsingular

[𝑢1𝑢2] = [

𝑣1𝑣2]



= 1𝜀 sin( )


NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


PVTOL feedback linearizationTracking the center of oscillation

Choose a different output, the center of oscillation:

𝑦1 = 𝑥 − sin( ) 𝑦2 = 𝑧 + cos( )

(a fixed point on the aircraft above the center of mass)

𝑦1 = −(𝑢1 − 2) sin( ) 𝑦2 = (𝑢1 − 2) cos( ) − 1

No vector relative degree, so try dynamic extension:introduce controller states 𝑥7, 𝑥8 with dynamics

��7 = 𝑥8 , ��8 = 𝑢′1

choose feedback 𝑢1 = 𝑥7 + 2 to obtain

𝑦1 = −𝑥7 sin( ) 𝑦2 = 𝑥7 cos( ) − 1

both controls 𝑢′1 and 𝑢2 will appear

after two more differentiations …

NonlinearControl

Controllability

Feedbackequivalence

Relative degree


MIMO systems

PVTOL aircraft


PVTOL feedback linearizationFinding the new relative degree

⎡⎢⎣𝑦(4)

1𝑦(4)

2

⎤⎥⎦

= − [cos( ) − sin( )sin( ) cos( )] [

2𝑥8 𝑥7

2]

− [sin( ) 𝑥7 cos( )

− cos( ) 𝑥7 sin( )] [𝑢′

1𝑢2]

vector relative degree (4, 4) defined except where 𝑥7 = 0⟹ except where 𝑦2

1 + ( 𝑦2 + 1)2 = 0

no zero dynamics

simple tracking controller will work if the trajectories avoidthe controller singularity at 𝑦2

1 + ( 𝑦2 + 1)2 = 0

for tracking a different output, we can perform open-looppath planning to translate a given reference signal to areference signal for the center of oscillation

NonlinearControl

Feedbackstabilization

Optimalstabilization

ControlLyapunovfunctions

Backstepping

Dissipativity

Forwarding


Part III

Stabilization

NonlinearControl




Backstepping

Dissipativity

Forwarding


13 Feedback stabilization

14 Optimal stabilization

15 Control Lyapunov functions

16 Backstepping

17 Dissipativity

18 Forwarding

NonlinearControl




Backstepping

Dissipativity

Forwarding


Stabilization via continuous state feedbackStatic feedback

Time-invariant system, 𝑥 ∈ 𝑋, 𝑢 ∈ ℝ𝑚, equilibrium at 𝑥𝑒 = 0

�� = 𝑓(𝑥, 𝑢) 𝑓(0, 0) = 0 (*)

Static state feedback

The system (*) is stabilizable via C0 static state feedback whenthere exists a C0 function 𝑘 ∶ 𝑋 → ℝ𝑚 with 𝑘(0) = 0 such that𝑥𝑒 = 0 is an asymptotically stable equilibrium of the system

�� = 𝑓(𝑥, 𝑘(𝑥))

global stabilizability when 𝑥𝑒 = 0 is GAS

time-varying feedback when 𝑢 = 𝑘(𝑡, 𝑥)

NonlinearControl




Backstepping

Dissipativity

Forwarding


Stabilization via continuous state feedbackDynamic feedback

Dynamic state feedback

The system (*) is stabilizable via C0 dynamic state feedbackwhen for some 𝑞 ⩾ 1, the augmented system

�� = 𝑓(𝑥, 𝑢) = 𝑣

with state (𝑥, ) ∈ 𝑋 × ℝ𝑞 and control (𝑢, 𝑣) ∈ ℝ𝑚 × ℝ𝑞 isstabilizable via C0 static state feedback.

the states are the internal controller states

stabilizability via dynamic state feedback does not implystabilizability via static state feedback

NonlinearControl




Backstepping

Dissipativity

Forwarding


Brockett's conditionA necessary condition for continuous stabilizability

Consider the system (*) with continuous 𝑓(·, ·)

If the system is stabilizable via C0 static or dynamic feedback,then 𝑓(·, ·) maps every neighborhood of (0, 0) ∈ 𝑋 × ℝ𝑚 to aneighborhood of 0 ∈ 𝑋.

Simple unicycle

��1 = 𝑢1 cos(𝑥3)��2 = 𝑢1 sin(𝑥3)��3 = 𝑢2

For small > 0, solve= 𝑢1 cos(𝑥3)= 𝑢1 sin(𝑥3)

0 = 𝑢2

no solution for |𝑥3| < 𝜋4

not stabilizable via C0 static or dynamic feedback

need either discontinuous or time-varying feedback

NonlinearControl




Backstepping

Dissipativity

Forwarding





16 Backstepping

17 Dissipativity

18 Forwarding

NonlinearControl




Backstepping

Dissipativity

Forwarding


Optimal stabilizationCost function

Cost function with continuous 𝐿(·, ·) ⩾ 0, 𝐿(0, 0) = 0

𝐽(𝑥0, 𝑢(·)) = ∫∞

0𝐿(𝑥(𝜏), 𝑢(𝜏)) 𝑑𝜏

the integral is along trajectories of (*) under control 𝑢(·)starting at 𝑥0 (existence & uniqueness ∀𝑡 ⩾ 0 is assumed)

𝐽 may be infinite for some 𝑥0 and 𝑢(·)

𝐽 = 0 when 𝑥0 = 0 and 𝑢(·) ≡ 0 (sitting at the equilibrium)

Example: a quadratic cost function for 𝑋 = ℝ3 and 𝑢 ∈ ℝ2

𝐿(𝑥, 𝑢) = 𝑥21 + 2𝑥2

2 + 5𝑥23 + 𝑢2

1 + 10𝑢22

NonlinearControl




Backstepping

Dissipativity

Forwarding


Optimal stabilizationOptimal state feedback law

Locally optimal stabilizing feedback 𝑢 = 𝑘(𝑥)

𝑘 is C0 with 𝑘(0) = 0𝑘 renders the equilibrium 𝑥𝑒 = 0 asymptotically stable∀𝑥0 ≈ 0, 𝑘 minimizes 𝐽 over all 𝑢(·) that steer 𝑥0 to zero

Globally optimal stabilizing feedback 𝑢 = 𝑘(𝑥)

𝑘 is C0 with 𝑘(0) = 0𝑘 renders the closed-loop system GAS∀𝑥0 ∈ 𝑋, 𝑘 minimizes 𝐽 over all 𝑢(·) that steer 𝑥0 to zero

NonlinearControl




Backstepping

Dissipativity

Forwarding


Hamilton-Jacobi-Bellman equationA sufficient condition for optimal stabilization

suppose ∃ a C1 nonnegative function 𝑉(𝑥) with 𝑉(0) = 0which solves the Hamilton-Jacobi-Bellman equation

min𝑢∈ℝ𝑚[𝐿(𝑥, 𝑢) + 𝐷𝑉(𝑥) ·𝑓(𝑥, 𝑢)] ≡ 0

suppose there is a C0 minimizer

𝑘(𝑥) = arg min𝑢∈ℝ𝑚[𝐿(𝑥, 𝑢) + 𝐷𝑉(𝑥) ·𝑓(𝑥, 𝑢)] , 𝑘(0) = 0

which renders the system �� = 𝑓(𝑥, 𝑘(𝑥)) GAS

then 𝑢 = 𝑘(𝑥) is a globally optimal stabilizing feedback

𝑉 is called the value function

NonlinearControl




Backstepping

Dissipativity

Forwarding










NonlinearControl




Backstepping

Dissipativity

Forwarding










NonlinearControl




Backstepping

Dissipativity

Forwarding










NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Proof of optimality

if 𝑢(·) drives 𝑥0 to zero, then 𝑉(𝑥(𝑡)) → 0 as 𝑡 → ∞ and

𝐽 = 𝑉(𝑥0) + ∫∞

0[𝐿(𝑥(𝜏), 𝑢(𝜏)) + 𝐷𝑉(𝑥(𝜏)) ·𝑓(𝑥(𝜏), 𝑢(𝜏))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑉(𝜏)

] 𝑑𝜏

HJB equation: integrand is ⩾ 0, and is ≡ 0 when 𝑢 = 𝑘(𝑥)

because 𝑢 = 𝑘(𝑥) is asymptotically stabilizing(by assumption), it minimizes 𝐽

the minimum value of 𝐽 is the value function 𝑉(𝑥0)

usually 𝑉 is also a Lyapunov function that verifies theasymptotic stability of the closed-loop system under thefeedback 𝑢 = 𝑘(𝑥) (note that 𝑉 = −𝐿 ⩽ 0)

NonlinearControl




Backstepping

Dissipativity

Forwarding



𝐽 = 𝑉(𝑥0) + ∫∞

0[𝐿(𝑥(𝜏), 𝑢(𝜏)) + 𝐷𝑉(𝑥(𝜏)) ·𝑓(𝑥(𝜏), 𝑢(𝜏))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑉(𝜏)

] 𝑑𝜏





NonlinearControl




Backstepping

Dissipativity

Forwarding



𝐽 = 𝑉(𝑥0) + ∫∞

0[𝐿(𝑥(𝜏), 𝑢(𝜏)) + 𝐷𝑉(𝑥(𝜏)) ·𝑓(𝑥(𝜏), 𝑢(𝜏))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑉(𝜏)

] 𝑑𝜏





NonlinearControl




Backstepping

Dissipativity

Forwarding



𝐽 = 𝑉(𝑥0) + ∫∞

0[𝐿(𝑥(𝜏), 𝑢(𝜏)) + 𝐷𝑉(𝑥(𝜏)) ·𝑓(𝑥(𝜏), 𝑢(𝜏))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑉(𝜏)

] 𝑑𝜏





NonlinearControl




Backstepping

Dissipativity

Forwarding



𝐽 = 𝑉(𝑥0) + ∫∞

0[𝐿(𝑥(𝜏), 𝑢(𝜏)) + 𝐷𝑉(𝑥(𝜏)) ·𝑓(𝑥(𝜏), 𝑢(𝜏))

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑉(𝜏)

] 𝑑𝜏





NonlinearControl




Backstepping

Dissipativity

Forwarding


Hamilton-Jacobi-Bellman equationControl-affine system with control-quadratic cost

Special case 𝑢 ∈ ℝ𝑚, 𝑔(𝑥) = [𝑔1(𝑥) … 𝑔𝑚(𝑥)]

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢 𝐿(𝑥, 𝑢) = 𝑞(𝑥) + 𝑢𝑇𝑅(𝑥)𝑢

𝑞(𝑥) ⩾ 0 and 𝑅(𝑥) = 𝑅𝑇(𝑥) > 0 for all 𝑥

with 𝐿𝑔𝑉 = [𝐿𝑔1𝑉 … 𝐿𝑔𝑚

𝑉 ], the HJB equation becomes

𝑞(𝑥) + 𝐿𝑓𝑉(𝑥) − 14 𝐿𝑔𝑉(𝑥) · 𝑅−1(𝑥) · [𝐿𝑔𝑉(𝑥)]

𝑇 = 0

the minimizing feedback law is

𝑘(𝑥) = − 12 𝑅−1(𝑥) · [𝐿𝑔𝑉(𝑥)]

𝑇

this controller has an infinite gain margin (in theory):for any 𝜌 ∈ ( 1

2 , ∞) the controller 𝑢 = 𝜌𝑘(𝑥) still stabilizesthe system

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Problems with the optimal stabilization approach

the HJB equation is a first-order nonlinear PDE, and can bevery difficult to solve

classical C1 solutions 𝑉(𝑥) may not even exist

even if the optimal feedback control cannot be found, acost function may still be useful in evaluating controllerperformance

if there is flexibility in the choice of 𝐿(𝑥, 𝑢), it may beeasier to claim the optimality of a feedback law(so-called inverse optimal design)

Nevertheless, there is an important special case in which wecan readily solve the HJB equation …

NonlinearControl




Backstepping

Dissipativity

Forwarding


Linear-Quadratic Regulator (LQR)LTI system, quadratic cost

Special case, 𝑄 = 𝑄𝑇 ⩾ 0, 𝑅 = 𝑅𝑇 > 0

�� = 𝐴𝑥 + 𝐵𝑢 𝐿(𝑥, 𝑢) = 𝑥𝑇𝑄𝑥 + 𝑢𝑇𝑅𝑢

if (𝐴, 𝐵) is stabilizable and (𝐴, 𝑄) is detectable, then thereis a unique positive semidefinite solution 𝑃 to thealgebraic Riccati equation

𝐴𝑇𝑃 + 𝑃𝐴 − 𝑃𝐵𝑅−1𝐵𝑇𝑃 + 𝑄 = 0

the quadratic function 𝑉(𝑥) = 𝑥𝑇𝑃 𝑥 solves HJB, and𝑘(𝑥) = −𝑅−1𝐵𝑇𝑃 𝑥

is the corresponding globally optimal stabilizing feedback

For nonlinear systems with a stabilizable linearizationWe can solve the LQR problem to discover how a nonlinearfeedback law should look near 𝑥 = 0 for local optimality.

NonlinearControl




Backstepping

Dissipativity

Forwarding


Linear-Quadratic Regulator (LQR)LTI system, quadratic cost

Special case, 𝑄 = 𝑄𝑇 ⩾ 0, 𝑅 = 𝑅𝑇 > 0

�� = 𝐴𝑥 + 𝐵𝑢 𝐿(𝑥, 𝑢) = 𝑥𝑇𝑄𝑥 + 𝑢𝑇𝑅𝑢

if (𝐴, 𝐵) is stabilizable and (𝐴, 𝑄) is detectable, then thereis a unique positive semidefinite solution 𝑃 to thealgebraic Riccati equation

𝐴𝑇𝑃 + 𝑃𝐴 − 𝑃𝐵𝑅−1𝐵𝑇𝑃 + 𝑄 = 0

the quadratic function 𝑉(𝑥) = 𝑥𝑇𝑃 𝑥 solves HJB, and𝑘(𝑥) = −𝑅−1𝐵𝑇𝑃 𝑥

is the corresponding globally optimal stabilizing feedback

For nonlinear systems with a stabilizable linearizationWe can solve the LQR problem to discover how a nonlinearfeedback law should look near 𝑥 = 0 for local optimality.

NonlinearControl




Backstepping

Dissipativity

Forwarding





16 Backstepping

17 Dissipativity

18 Forwarding

NonlinearControl




Backstepping

Dissipativity

Forwarding


Control Lyapunov functionsLyapunov functions for systems with control inputs

�� = 𝑓(𝑥)asymptotic stability

⇕there exists a

Lyapunov function

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢asymptotic stabilizability

⇕there exists a control

Lyapunov function (CLF)

Control-affine system, 𝑥 ∈ 𝑋, 𝑢 ∈ ℝ𝑚, equilibrium at 𝑥𝑒 = 0

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢 𝑓(0) = 0 (*)

Definition

A control Lyapunov function (CLF) for the system (*) is a C1,positive definite, proper function 𝑉(𝑥) such that

[ 𝑥 ≠ 0 and 𝐿𝑔𝑉(𝑥) = 0 ] ⟹ 𝐿𝑓𝑉(𝑥) < 0

NonlinearControl




Backstepping

Dissipativity

Forwarding


Control Lyapunov functionsIntuition and Artstein's theorem

Time derivative of the CLF𝑉(𝑥, 𝑢) = 𝐿𝑓𝑉(𝑥) + 𝐿𝑔𝑉(𝑥)𝑢

if 𝐿𝑔𝑉(𝑥) ≠ 0, we can choose a value of 𝑢 to make 𝑉 < 0if 𝐿𝑔𝑉(𝑥) = 0 then 𝑉 is independent of 𝑢 and should thussatisfy 𝑉(𝑥) = 𝐿𝑓𝑉(𝑥) < 0

Artstein's theorem (a corollary of a more general result)∃ a feedback law 𝑢 = 𝑘(𝑥), continuous everywhere except

possibly at 𝑥 = 0, which renders the closed-loop system GAS⇕

∃ a CLF

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Sontag's formula

If 𝑉(𝑥) is a CLF, then the feedback law defined as

𝑘(𝑥) = −𝐿𝑓𝑉(𝑥) + √[𝐿𝑓𝑉(𝑥)]

2 + ||𝐿𝑔𝑉(𝑥)||4

||𝐿𝑔𝑉(𝑥)||2 [𝐿𝑔𝑉(𝑥)]

𝑇

when 𝐿𝑔𝑉(𝑥) ≠ 0 (and 𝑘(𝑥) = 0 otherwise) is such that𝑘 renders the closed-loop system GAS𝑘 is as smooth as 𝐿𝑓𝑉 and 𝐿𝑔𝑉, except possibly at 𝑥 = 0if 𝑉 satisfies the small control property (scp)then 𝑘 is continuous at 𝑥 = 0

Small control property: we can choose small 𝑢 for small 𝑥𝑉 satisfies the scp when ∀ > 0 ∃𝛿 > 0 such that if 𝑥 ≠ 0 and|𝑥| ⩽ 𝛿 then there exists |𝑢| ⩽ such that 𝐿𝑓𝑉(𝑥) + 𝐿𝑔𝑉(𝑥)𝑢 < 0.

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 A CLF approach to stabilization

Key observationThe CLF and scp properties are preserved under feedbacktransformations (feedback plus coordinate change).

if the system is feedback equivalent to a simpler systemfor which we can find a CLF, then we have a CLF for theoriginal systemwe no longer need to implement the feedbacktransformation (e.g., use Sontag's formula instead)

An alternative stabilizer for feedback linearizable systemsfind a CLF for the transformed LTI system(solve the linear Lyapunov matrix equation)use Sontag's formula instead of the linearizing feedback

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 A modification of Sontag's formula

Choose a function 𝑞(𝑥) such that𝑞(·) ⩾ 0 and 𝑞(0) = 0if a bounded solution of �� = 𝑓(𝑥) is such that 𝑞(𝑥(𝑡)) ≡ 0,then 𝑥(𝑡) → 0 as 𝑡 → ∞ (e.g., true if 𝑞(𝑥) > 0 ∀𝑥 ≠ 0)

Given a CLF 𝑉(𝑥), consider the feedback law defined as

𝑘(𝑥) = −𝐿𝑓𝑉(𝑥) + √[𝐿𝑓𝑉(𝑥)]

2 + 𝑞(𝑥)||𝐿𝑔𝑉(𝑥)||2

||𝐿𝑔𝑉(𝑥)||2 [𝐿𝑔𝑉(𝑥)]

𝑇

when 𝐿𝑔𝑉(𝑥) ≠ 0 (and 𝑘(𝑥) = 0 otherwise).

the special choice 𝑞(𝑥) = ||𝐿𝑔𝑉(𝑥)||2 satisfies the above

conditions and results in Sontag's formula

many other choices for 𝑞 are possible: flexibility!What are the properties of this modified feedback law?

NonlinearControl




Backstepping

Dissipativity

Forwarding


A modification of Sontag's formulaProperties of the feedback law

𝑘 renders the closed-loop system GAS

𝑘 is as smooth as 𝐿𝑓𝑉, 𝐿𝑔𝑉, and 𝑞,except possibly where 𝑞(𝑥) = 𝐿𝑓𝑉(𝑥) = 0

if 𝐿𝑓𝑉, 𝐿𝑔𝑉, and 𝑞 are locally Lipschitz continuous,then so is 𝑘, except possibly at 𝑥 = 0

if 𝑉 satisfies the scp then 𝑘 is continuous at 𝑥 = 0

if 𝑉 has the same level sets as the HJB solution associatedwith the cost function

𝐽 = ∫∞

0[𝑞(𝑥(𝜏)) + |𝑢(𝜏)|2] 𝑑𝜏

then 𝑘 is globally optimal!

use this fact to design locally optimal feedback laws

NonlinearControl




Backstepping

Dissipativity

Forwarding








𝐽 = ∫∞

0[𝑞(𝑥(𝜏)) + |𝑢(𝜏)|2] 𝑑𝜏



NonlinearControl




Backstepping

Dissipativity

Forwarding








𝐽 = ∫∞

0[𝑞(𝑥(𝜏)) + |𝑢(𝜏)|2] 𝑑𝜏



NonlinearControl




Backstepping

Dissipativity

Forwarding








𝐽 = ∫∞

0[𝑞(𝑥(𝜏)) + |𝑢(𝜏)|2] 𝑑𝜏



NonlinearControl




Backstepping

Dissipativity

Forwarding








𝐽 = ∫∞

0[𝑞(𝑥(𝜏)) + |𝑢(𝜏)|2] 𝑑𝜏



NonlinearControl




Backstepping

Dissipativity

Forwarding








𝐽 = ∫∞

0[𝑞(𝑥(𝜏)) + |𝑢(𝜏)|2] 𝑑𝜏



NonlinearControl




Backstepping

Dissipativity

Forwarding





16 Backstepping

17 Dissipativity

18 Forwarding

NonlinearControl




Backstepping

Dissipativity

Forwarding


Backstepping: adding an integratorConstructing a CLF for an extended system

Suppose the system is feedback equivalent to

��1 = 𝐹(𝑥1, 𝑥2) 𝑥1 ∈ ℝ𝑛−𝑚 𝐹(0, 0) = 0��2 = 𝑢 𝑥2, 𝑢 ∈ ℝ𝑚 𝐹 ∈ C1

Suppose we can stabilize the reduced system

��1 = 𝐹(𝑥1, 𝑣) 𝑣 = virtual control

with a 𝐶1, proper, positive definite Lyapunov function 𝑉1(𝑥)and a 𝐶1 feedback law 𝑣 = 𝑘(𝑥1) with 𝑘(0) = 0 such that

𝐷𝑉1(𝑥1) ·𝐹(𝑥1, 𝑘(𝑥1)) < 0 for 𝑥1 ≠ 0

Then we can construct a CLF for the full system …

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 A CLF for the full system

A simple choice

𝑉2(𝑥1, 𝑥2) = 𝑉1(𝑥1) + ||𝑥2 − 𝑘(𝑥1)||2

A more flexible choice (but not the most flexible)

𝑉2(𝑥1, 𝑥2) = ∫𝑉1(𝑥1)

0𝛾(𝑟) 𝑑𝑟 + (𝑥2 − 𝑘(𝑥1))𝑇 Γ(𝑥1)(𝑥2 − 𝑘(𝑥1))

𝛾 ∶ [0, ∞) → [0, ∞) is C0 with 𝛾(𝑟) > 0 for 𝑟 > 0and ∫ ∞

0 𝛾(𝑟) 𝑑𝑟 = ∞ (simple choice 𝛾 ≡ 1)

Γ is C1 with Γ(𝑥1) = Γ𝑇(𝑥1) > 0 ∀𝑥1 (simple choice Γ ≡ 𝐼)

this makes 𝑉2 a C1, positive definite, and proper function

now verify the CLF property …

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Verifying the CLF property

We compute

𝑉2 = 𝛾(𝑉1(𝑥1)) · 𝐷𝑉1(𝑥1) ·𝐹(𝑥1, 𝑥2)

+ (𝑥2 − 𝑘(𝑥1))𝑇 ⟨𝐷Γ(𝑥1), 𝐹(𝑥1, 𝑥2)⟩(𝑥2 − 𝑘(𝑥1))

+ 2(𝑥2 − 𝑘(𝑥1))𝑇 Γ(𝑥1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐿𝑔𝑉2(𝑥1,𝑥2)

[𝑢 − 𝐷𝑘(𝑥1) ·𝐹(𝑥1, 𝑥2)]

If 𝐿𝑔𝑉2(𝑥1, 𝑥2) = 0 then 𝑥2 = 𝑘(𝑥1) and

𝑉2 = 𝛾(𝑉1(𝑥1)) · 𝐷𝑉1(𝑥1) ·𝐹(𝑥1, 𝑘(𝑥1)) < 0 for 𝑥1 ≠ 0

Furthermore, 𝑉2 satisfies the scp, so there is a C0 globallystabilizing feedback (e.g., from Sontag's formula).

If all data are smooth, then there is a smooth feedback.

NonlinearControl




Backstepping

Dissipativity

Forwarding


Backstepping exampleChoosing a virtual feedback

System with 𝑋 = ℝ2 and 𝑢 ∈ ℝ

��1 = 𝑥31 + (𝑥2

1 + 𝑥2)3

��2 = 𝑢

unstable (with finite escape times) when 𝑢 = 0

the linear approximation about 𝑥 = 0 is not stabilizable

Replace 𝑥2 with a virtual control 𝑣 in the 𝑥1-subsystem

��1 = 𝑥31 + (𝑥2

1 + 𝑣)3

𝑣 = 𝑘(𝑥1) = −2𝑥1 − 𝑥21 yields the GAS system ��1 = −7𝑥3

1with Lyapunov function 𝑉(𝑥1) = 𝑥2

1

NonlinearControl




Backstepping

Dissipativity

Forwarding


Backstepping exampleConstructing a CLF

A simple choice

𝑉2(𝑥1, 𝑥2) = 𝑉1(𝑥1) + (𝑥2 − 𝑘(𝑥1))2 = 𝑥21 + (𝑥2 + 2𝑥1 + 𝑥2

1)2

we can use Sontag's formula, or

there are polynomial feedback laws that render 𝑉2 < 0

What about performance?there are many choices for the virtual control law

there are many choices for the CLF

there are many choices for the stabilizing feedback

An open challengeHow do we exploit these design flexibilities to achieve goodclosed-loop performance?

NonlinearControl




Backstepping

Dissipativity

Forwarding


Should we choose an optimal virtual control law?Not necessarily

Apply backstepping to the linear system

��1 = −2𝑥1 + 𝑥2 (virtual subsystem ��1 = −2𝑥1 + 𝑣)��2 = −2𝑥1 + 𝑢

a linear virtual control law 𝑣 = −𝑐𝑥1 leads to a quadraticbackstepping CLF of the form

𝑉2(𝑥1, 𝑥2) = 𝑎𝑥21 + 𝑏(𝑥2 + 𝑐𝑥1)2

for positive 𝑎 and 𝑏

the value function for the cost 𝐽 = ∫ ∞0 (𝑥2

1 + 𝑥22 + 𝑢2) 𝑑𝜏

is of the same quadratic form with 𝑐 = −0.29

the best choice for the virtual control is 𝑣 = 0.29𝑥1, whichis not optimal for the virtual subsystem for any cost thatpenalizes both 𝑥1 and 𝑣 !

NonlinearControl




Backstepping

Dissipativity

Forwarding


Nonlinear dampingHandling unmeasured disturbances

System with 𝑢 ∈ ℝ𝑚 and matched disturbance input 𝑑(𝑡) ∈ ℝ𝑞

�� = 𝑓(𝑥) + 𝑔(𝑥)[𝑢 + 𝑆(𝑥)𝑑] 𝑆(𝑥) ∈ ℝ𝑚×𝑞

suppose we know a CLF 𝑉(𝑥) with an associated stabilizingfeedback law 𝑘(𝑥) for the system with 𝑑 ≡ 0:

𝐿𝑓𝑉(𝑥) + 𝐿𝑔𝑉(𝑥)𝑘(𝑥) < 0 for 𝑥 ≠ 0

assume 𝐿𝑓𝑉(𝑥) + 𝐿𝑔𝑉(𝑥)𝑘(𝑥) ⩽ −𝛼(|𝑥|) for some 𝛼 ∈ 𭒦∞(if this is not the case, we can make it so by replacing 𝑉with 𝜌(𝑉 ) for an appropriate class-𭒦∞ function 𝜌)

add nonlinear damping to the feedback law:

𝑢 = 𝑘(𝑥) − 𝑐𝑆(𝑥)𝑆 𝑇(𝑥)[𝐿𝑔𝑉(𝑥)]𝑇

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟nonlinear damping

𝑐 > 0

NonlinearControl




Backstepping

Dissipativity

Forwarding




�� = 𝑓(𝑥) + 𝑔(𝑥)[𝑢 + 𝑆(𝑥)𝑑] 𝑆(𝑥) ∈ ℝ𝑚×𝑞







𝑐 > 0

NonlinearControl




Backstepping

Dissipativity

Forwarding




�� = 𝑓(𝑥) + 𝑔(𝑥)[𝑢 + 𝑆(𝑥)𝑑] 𝑆(𝑥) ∈ ℝ𝑚×𝑞







𝑐 > 0

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 What does nonlinear damping accomplish?

The new Lyapunov derivative becomes

𝑉 = 𝐿𝑓𝑉(𝑥) + 𝐿𝑔𝑉(𝑥)𝑘(𝑥) − 𝑐||𝐿𝑔𝑉(𝑥)𝑆(𝑥)||2 + 𝐿𝑔𝑉(𝑥)𝑆(𝑥)𝑑

⩽ 𝐿𝑓𝑉(𝑥) + 𝐿𝑔𝑉(𝑥)𝑘(𝑥)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

⩽−𝛼(|𝑥|)

+ 14𝑐 |𝑑|2 (Young's inequality)

Therefore the closed-loop system is ISS with respect to 𝑑(𝑡).

Nonlinear damping generates ISS systemsAdding a nonlinear damping term to a stabilizing feedback lawwill render the closed-loop system ISS with respect to amatched disturbance.

What if the disturbance is not matched with the control input?

NonlinearControl




Backstepping

Dissipativity

Forwarding


Backstepping with nonlinear dampingAchieving ISS with respect to unmatched disturbances

Example with 𝑋 = ℝ2, 𝑢 ∈ ℝ, 𝑑 ∈ ℝ2

��1 = 𝑥2 + 𝑥21𝑑1 (virtual subsystem ��1 = 𝑣 + 𝑥2

1𝑑1)��2 = 𝑢 + 𝑥1𝑥2𝑑2

Choose 𝑉1(𝑥1) = 𝑥21 and design 𝑣 for the virtual subsystem:

𝑣 = −𝑥1 − 𝑥51⏟⏟⏟

nonlinear damping

𝑉1||𝑥2=𝑣 ⩽ −2𝑥21 + 1

2 𝑑21

Now make a simple choice for the CLF 𝑉2:

𝑉2(𝑥) = 𝑉1(𝑥1) + (𝑥2 + 𝑥1 + 𝑥51)

2

𝑉2 = 𝑉1||𝑥2=𝑣 + 2𝑥1(𝑥2 + 𝑥1 + 𝑥51)

+ 2(𝑥2 + 𝑥1 + 𝑥51)[𝑢 + 𝑥1𝑥2𝑑2 + (1 + 5𝑥4

1)(𝑥2 + 𝑥21𝑑1)]

NonlinearControl




Backstepping

Dissipativity

Forwarding


Backstepping with nonlinear dampingExample continued

To simplify, first choose

𝑢 = −𝑥1 − (1 + 5𝑥41)𝑥2 − (𝑥2 + 𝑥1 + 𝑥5

1) + 𝑢nd

𝑢nd = nonlinear damping term to be designed

and plug in to obtain𝑉2 ⩽ −2𝑉2(𝑥) + 1

2 𝑑21

+ 2(𝑥2 + 𝑥1 + 𝑥51)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐿𝑔𝑉2(𝑥)

[𝑢nd + [(1 + 5𝑥41)𝑥2

1 𝑥1𝑥2]⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑆(𝑥)

𝑑]

Choose 𝑢nd = − 𝑐𝑆(𝑥)𝑆 𝑇(𝑥)𝐿𝑔𝑉2(𝑥) (with 𝑥171 !!) to obtain

𝑉2 ⩽ −2𝑉2(𝑥) + 12 𝑑2

1 + 14𝑐 (𝑑2

1 + 𝑑22)

The closed-loop system is ISS with respect to 𝑑 = (𝑑1, 𝑑2).

NonlinearControl




Backstepping

Dissipativity

Forwarding


Recursive backsteppingStrict feedback form

Block-triangular system with 𝑥𝑖, 𝑢 ∈ ℝ𝑚 and disturbance 𝑑 ∈ ℝ𝑞

��1 = 𝑓1(𝑥1) + 𝑔1(𝑥1)[𝑥2 + 𝑆1(𝑥1)𝑑]��2 = 𝑓2(𝑥1, 𝑥2) + 𝑔2(𝑥1, 𝑥2)[𝑥3 + 𝑆2(𝑥1, 𝑥2)𝑑]��3 = 𝑓3(𝑥1, 𝑥2, 𝑥3) + 𝑔3(𝑥1, 𝑥2, 𝑥3)[𝑥4 + 𝑆3(𝑥1, 𝑥2, 𝑥3)𝑑]

⋮��𝑟 = 𝑓𝑟(𝑥) + 𝑔𝑟(𝑥)[𝑢 + 𝑆𝑟(𝑥)𝑑]

assume each 𝑔𝑖(·) ∈ ℝ𝑚×𝑚 is invertible everywhereapply the backstepping procedure 𝑟 − 1 times in a recursivemanner, each time replacing 𝑥𝑖 with a virtual control lawuntil the actual control 𝑢 appears at the last stepinclude a nonlinear damping term in each virtual controllaw (and in the final actual control law)the resulting closed-loop system will be ISS w.r.t. 𝑑(𝑡)

NonlinearControl




Backstepping

Dissipativity

Forwarding





16 Backstepping

17 Dissipativity

18 Forwarding

NonlinearControl




Backstepping

Dissipativity

Forwarding


DissipativityGeneral concepts

In a nutshella dissipative system cannot produce

an infinite amount of work,—OR—

the change in its internal energynever exceeds the work done on it

Key pointEnergy and work are abstractly defined and may have norelation to physical quantities …

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Classical dissipativity

Given:a system with state 𝑥(𝑡), input 𝑢(𝑡), and output 𝑦(𝑡)a supply rate 𝜔(𝑢, 𝑦, 𝑡)

Available storage 𝜑𝑎

𝜑𝑎(𝑥0, 𝑡0) = − inf ∫𝑡

𝑡0

𝜔(𝑢(𝜏), 𝑦(𝜏), 𝜏) 𝑑𝜏

where the infimum is taken over all 𝑡 ⩾ 𝑡0 and all systemtrajectories having state 𝑥0 at time 𝑡0

DefinitionThe system is dissipative with respect to 𝜔 when

𝜑𝑎(𝑥0, 𝑡0) < ∞ for all 𝑥0 and 𝑡0

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Dissipation inequality

TheoremA system is dissipative if and only if there is a nonnegativestorage function 𝑉 which satisfies the dissipation inequality

𝑉(𝑥(𝑡), 𝑡)⏟⏟⏟⏟⏟⏟⏟⏟⏟

current energy

⩽ 𝑉(𝑥(𝑡0), 𝑡0)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟initial energy

+ ∫𝑡

𝑡0

𝜔(𝑢(𝜏), 𝑦(𝜏), 𝜏) 𝑑𝜏⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

work done on system

along all system trajectories for all 𝑡 ⩾ 𝑡0.

In a dissipative system, the available storage 𝜑𝑎 is a storagefunction which bounds every other storage function from below.

Differential version: 𝑉(𝑡) ⩽ 𝜔(𝑢, 𝑦, 𝑡)

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 System properties characterized by dissipativity

Classical properties:passivity: 𝜔 = 𝑦𝑇𝑢

ℒ𝑝-gain: 𝜔 = 𝛾𝑝|𝑢|𝑝 − |𝑦|𝑝 (1 ⩽ 𝑝 < ∞)

output strict passivity: 𝜔 = 𝑦𝑇𝑢 − |𝑦|2 (implies ℒ2-gain)

input strict passivity: 𝜔 = 𝑦𝑇𝑢 − 𝛿|𝑢|2

More recent connections to dissipativity:ISS and iISS

minimum-phase property

ℓ∞-gain for discrete-time systems

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Why is dissipativity useful?

Divide and conquerThe interconnection of dissipative systems with complementarysupply rates is again dissipative!

Classical examples:small-gain theorem: if two systems have ℒ𝑝-gains whoseproduct is less than one, then their feedbackinterconnection is stablepassivity theorem: the negative feedbackinterconnection of passive systems is stable/passive

NonlinearControl




Backstepping

Dissipativity

Forwarding


An example of a passive systemA robot manipulator

Dynamics of a robot manipulator

𝑀(𝑞) 𝑞 + 𝐶(𝑞, 𝑞) 𝑞 + 𝑔(𝑞) = 𝜏

𝑞 is the vector of configuration variables𝜏 is the vector of applied joint torques𝑀(𝑞) is the mass matrix𝐶(𝑞, 𝑞) characterizes the Coriolis and centrifugal forces𝑔(𝑞) represents the gravitational forces

Model properties:𝑔(·) is the gradient of a potential field:

𝑔𝑇(𝑞) = 𝐷𝐺(𝑞) for some nonnegative gravity field 𝐺(𝑞)𝑞𝑇[��(𝑞, 𝑞) − 2𝐶(𝑞, 𝑞)] 𝑞 ≡ 0 (fictitious forces do no work)

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 A storage function for the robot manipulator

Look at total energy as a candidate storage function

𝑉(𝑞, 𝑞) = 12 𝑞𝑇𝑀(𝑞) 𝑞⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

kinetic

+ 𝐺(𝑞)⏟⏟⏟

potential

This storage function satisfies𝑉 = 𝑞𝑇𝜏

and thus the system is passive from the applied torques 𝑢 = 𝜏to the joint velocities 𝑦 = 𝑞.

This passivity is the basis for many robot control strategies.

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Passivity-based stabilization

Control-affine system

�� = 𝑓(𝑥) + 𝑔(𝑥)𝑢 𝑓(0) = 0

suppose the equilibrium of �� = 𝑓(𝑥) at 𝑥𝑒 = 0 is stable,but not necessarily asymptotically stable

suppose we know a Lyapunov function 𝑉(𝑥) for �� = 𝑓(𝑥)such that 𝐿𝑓𝑉(𝑥) ⩽ 0 for all 𝑥

this Lyapunov function satisfies𝑉 = 𝐿𝑓𝑉(𝑥) + 𝐿𝑔𝑉(𝑥)𝑢

the system is passive from 𝑢 to the output 𝑦 = [𝐿𝑔𝑉(𝑥)]𝑇

because 𝑉 ⩽ 𝑦𝑇𝑢

NonlinearControl




Backstepping

Dissipativity

Forwarding


Passivity-based stabilizationUsing LaSalle's theorem to prove asymptotic stability

Choose 𝑢 = −𝜎(𝑦) where 𝑦𝑇𝜎(𝑦) is positive definite in 𝑦:

𝑉 = 𝐿𝑓𝑉(𝑥) − 𝑦𝑇𝜎(𝑦)

(for example, 𝑢 = −𝐾𝑦 for 𝐾 > 0). Then 𝑥 converges to thelargest invariant set contained in the set

{𝑥 ∈ 𝑋 ∶ 𝐿𝑓𝑉(𝑥) = 𝐿𝑔𝑉(𝑥) = 0}

if this largest invariant set contains just the equilibrium𝑥𝑒 = 0, then we have asymptotic stability

this will hold in particular when the system �� = 𝑓(𝑥) iszero-state observable through the output 𝑦 = [𝐿𝑔𝑉(𝑥)]𝑇

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Feedback passivation

Suppose a system is feedback equivalent to the cascade

��1 = 𝑓(𝑥1) + 𝑔(𝑥1)𝑥2 𝑓(0) = 0��2 = 𝑢

suppose we know a Lyapunov function 𝑉1(𝑥1) for��1 = 𝑓(𝑥1) such that 𝐿𝑓𝑉1(𝑥1) ⩽ 0 for all 𝑥1

the system has relative degree (1, …, 1) and isweakly minimum-phase with respect to the output 𝑦 = 𝑥2

consider 𝑉2(𝑥1, 𝑥2) = 𝑉1(𝑥1) + 12 |𝑦|2 with derivative

𝑉2 = 𝐿𝑓𝑉1(𝑥1) + 𝐿𝑔𝑉1(𝑥1)𝑦 + 𝑦𝑇𝑢

choose 𝑢 = −[𝐿𝑔𝑉1(𝑥1)]𝑇 + 𝑣 for a new input 𝑣:𝑉2 = 𝐿𝑓𝑉1(𝑥1) + 𝑦𝑇𝑣

passivity from 𝑣 to 𝑦, stabilization 𝑣 = −𝜎(𝑦) as before

NonlinearControl




Backstepping

Dissipativity

Forwarding



��1 = 𝑓(𝑥1) + 𝑔(𝑥1)𝑥2 𝑓(0) = 0��2 = 𝑢







NonlinearControl




Backstepping

Dissipativity

Forwarding



��1 = 𝑓(𝑥1) + 𝑔(𝑥1)𝑥2 𝑓(0) = 0��2 = 𝑢







NonlinearControl




Backstepping

Dissipativity

Forwarding



��1 = 𝑓(𝑥1) + 𝑔(𝑥1)𝑥2 𝑓(0) = 0��2 = 𝑢







NonlinearControl




Backstepping

Dissipativity

Forwarding



��1 = 𝑓(𝑥1) + 𝑔(𝑥1)𝑥2 𝑓(0) = 0��2 = 𝑢







NonlinearControl




Backstepping

Dissipativity

Forwarding


Feedback passivationExample

Cascade system, relative degree 1 and weakly minimum-phase

��1 = −𝑥2 + 𝑥21𝑥3

��2 = 𝑥1 + 𝑥1𝑥2𝑥3

��3 = 𝑢 𝑦 = 𝑥3

choose 𝑉1(𝑥1, 𝑥2) = 𝑥21 + 𝑥2

2 so that 𝐿𝑓𝑉1(𝑥1, 𝑥2) ≡ 0

then 𝐿𝑔𝑉1(𝑥1, 𝑥2) = 2𝑥1(𝑥21 + 𝑥2

2)

choose 𝑢 = −𝐿𝑔𝑉1(𝑥1, 𝑥2) − 𝑦 = −2𝑥1(𝑥21 + 𝑥2

2) − 𝑥3

asymptotic stability because 𝑦 ≡ 0 implies 𝑢 ≡ 0 implies𝐿𝑔𝑉1(𝑥1, 𝑥2) ≡ 0 implies 𝑥1 ≡ 𝑥2 ≡ 0 (then use LaSalle)

NonlinearControl




Backstepping

Dissipativity

Forwarding





16 Backstepping

17 Dissipativity

18 Forwarding

NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Finding a Lyapunov function for passivity design

Example with 𝑋 = ℝ2 and 𝑢 ∈ ℝ

��1 = 𝑥2 − 𝑥22 + 𝑥2

1𝑢��2 = −𝑥2 + 𝑢

with 𝑢 = 0, the equilibrium 𝑥𝑒 = 0 is stablebut not asymptotically stable

to perform passivity-based stabilization,we need to know 𝑉(𝑥) such that 𝐿𝑓𝑉(𝑥) ⩽ 0

propose a candidate 𝑉(𝑥) of the form

𝑉(𝑥) = 𝑥21 + Ψ(𝑥1, 𝑥2) + 𝑥2

2

what should the cross term Ψ(𝑥1, 𝑥2) be?

NonlinearControl




Backstepping

Dissipativity

Forwarding



��1 = 𝑥2 − 𝑥22 + 𝑥2

1𝑢��2 = −𝑥2 + 𝑢




𝑉(𝑥) = 𝑥21 + Ψ(𝑥1, 𝑥2) + 𝑥2

2


NonlinearControl




Backstepping

Dissipativity

Forwarding



��1 = 𝑥2 − 𝑥22 + 𝑥2

1𝑢��2 = −𝑥2 + 𝑢




𝑉(𝑥) = 𝑥21 + Ψ(𝑥1, 𝑥2) + 𝑥2

2


NonlinearControl




Backstepping

Dissipativity

Forwarding



��1 = 𝑥2 − 𝑥22 + 𝑥2

1𝑢��2 = −𝑥2 + 𝑢




𝑉(𝑥) = 𝑥21 + Ψ(𝑥1, 𝑥2) + 𝑥2

2


NonlinearControl




Backstepping

Dissipativity

Forwarding

3rd ElgersburgSchool 2011 Defining the cross term

when 𝑢 = 0 we have𝑉 = 𝐿𝑓𝑉(𝑥) = 2𝑥1(𝑥2 − 𝑥2

2) + Ψ − 2𝑥22

if Ψ = −2𝑥1(𝑥2 − 𝑥22) then 𝐿𝑓𝑉(𝑥) = −2𝑥2

2 ⩽ 0

thus choose

Ψ(𝑥1, 𝑥2) = Ψ(𝑥) =∫∞

02𝜑1(𝜏, 𝑥)[𝜑2(𝜏, 𝑥) − 𝜑2

2(𝜏, 𝑥)] 𝑑𝜏

where 𝜑(𝑡, 𝑥) = (𝜑1, 𝜑2) denotes the flow of 𝑓

we can find 𝜑(𝑡, 𝑥) and Ψ(𝑥) in closed form:

Ψ(𝑥1, 𝑥2) = 𝑥1(2𝑥2 − 𝑥22) + 𝑥2

2 − 𝑥32 + 1

4 𝑥42

the resulting 𝑉(𝑥) is positive definite and proper,and we proceed with the passivity design …

NonlinearControl




Backstepping

Dissipativity

Forwarding



2) + Ψ − 2𝑥22


2 ⩽ 0

thus choose

Ψ(𝑥1, 𝑥2) = Ψ(𝑥) =∫∞

02𝜑1(𝜏, 𝑥)[𝜑2(𝜏, 𝑥) − 𝜑2

2(𝜏, 𝑥)] 𝑑𝜏



Ψ(𝑥1, 𝑥2) = 𝑥1(2𝑥2 − 𝑥22) + 𝑥2

2 − 𝑥32 + 1

4 𝑥42


NonlinearControl




Backstepping

Dissipativity

Forwarding



2) + Ψ − 2𝑥22


2 ⩽ 0

thus choose

Ψ(𝑥1, 𝑥2) = Ψ(𝑥) =∫∞

02𝜑1(𝜏, 𝑥)[𝜑2(𝜏, 𝑥) − 𝜑2

2(𝜏, 𝑥)] 𝑑𝜏



Ψ(𝑥1, 𝑥2) = 𝑥1(2𝑥2 − 𝑥22) + 𝑥2

2 − 𝑥32 + 1

4 𝑥42


NonlinearControl




Backstepping

Dissipativity

Forwarding



2) + Ψ − 2𝑥22


2 ⩽ 0

thus choose

Ψ(𝑥1, 𝑥2) = Ψ(𝑥) =∫∞

02𝜑1(𝜏, 𝑥)[𝜑2(𝜏, 𝑥) − 𝜑2

2(𝜏, 𝑥)] 𝑑𝜏



Ψ(𝑥1, 𝑥2) = 𝑥1(2𝑥2 − 𝑥22) + 𝑥2

2 − 𝑥32 + 1

4 𝑥42


NonlinearControl




Backstepping

Dissipativity

Forwarding



2) + Ψ − 2𝑥22


2 ⩽ 0

thus choose

Ψ(𝑥1, 𝑥2) = Ψ(𝑥) =∫∞

02𝜑1(𝜏, 𝑥)[𝜑2(𝜏, 𝑥) − 𝜑2

2(𝜏, 𝑥)] 𝑑𝜏



Ψ(𝑥1, 𝑥2) = 𝑥1(2𝑥2 − 𝑥22) + 𝑥2

2 − 𝑥32 + 1

4 𝑥42


NonlinearControl




Backstepping

Dissipativity

Forwarding


Finishing the controller designBounded control with arbitrarily small bound

Now we compute the passive output 𝑦 = 𝐿𝑔𝑉(𝑥):

𝐿𝑔𝑉(𝑥) = 2𝑥1 + 4𝑥2 − 2𝑥1𝑥2 − 3𝑥22 + 2𝑥3

1 + 2𝑥21𝑥2 + 𝑥3

2 − 𝑥21𝑥2

2

For a bounded control we can choose

𝑢 = 𝑘(𝑥1, 𝑥2) = − tanh(1𝜀 𝐿𝑔𝑉(𝑥)) for any > 0

Finally, confirm that the equilibrium 𝑥𝑒 = 0 is the largestinvariant set satisfying 𝐿𝑓𝑉(𝑥) ≡ 𝐿𝑔𝑉(𝑥) ≡ 0 (yes, because𝐿𝑓𝑉(𝑥) = −2𝑥2

2 which means 𝑥2 ≡ 0 which means𝐿𝑔𝑉(𝑥) = 2𝑥1(1 + 𝑥2

1) ≡ 0 which means 𝑥1 ≡ 0).

The closed-loop system is GAS and locally exponentially stable(the linearization at 𝑥𝑒 = 0 is �� = 𝐴𝑥 with a Hurwitz 𝐴).

NonlinearControl




Backstepping

Dissipativity

Forwarding


Forwarding: adding another partRepeating the passivity design for the new system

Add a third state 𝑥0 ∈ ℝ

��0 = 𝑓(𝑥0, 𝑥1, 𝑥2) + 𝑔(𝑥0, 𝑥1, 𝑥2)𝑢 𝑓(𝑥0, 0, 0) ≡ 0��1 = 𝑥2 − 𝑥2

2 + 𝑥21𝑢 𝑓, 𝑔 glob. Lip. in 𝑥0

��2 = −𝑥2 + 𝑢

add a new term to the controller, 𝑢 = 𝑘(𝑥1, 𝑥2) + 𝑣when 𝑣 = 0, the equilibrium 𝑥𝑒 = 0 is stablebut not asymptotically stable, just like beforepropose a candidate Lyapunov function 𝑉0 of the form

𝑉0(𝑥0, 𝑥1, 𝑥2) = 𝑥20 + Ψ0(𝑥0, 𝑥1, 𝑥2) + 𝑉(𝑥1, 𝑥2)

compute the cross term Ψ0 as before (numerically?)apply passivity design for the new control 𝑣to achieve GAS for the new system

NonlinearControl




Backstepping

Dissipativity

Forwarding




��0 = 𝑓(𝑥0, 𝑥1, 𝑥2) + 𝑔(𝑥0, 𝑥1, 𝑥2)𝑢 𝑓(𝑥0, 0, 0) ≡ 0��1 = 𝑥2 − 𝑥2


��2 = −𝑥2 + 𝑢


𝑉0(𝑥0, 𝑥1, 𝑥2) = 𝑥20 + Ψ0(𝑥0, 𝑥1, 𝑥2) + 𝑉(𝑥1, 𝑥2)


NonlinearControl




Backstepping

Dissipativity

Forwarding




��0 = 𝑓(𝑥0, 𝑥1, 𝑥2) + 𝑔(𝑥0, 𝑥1, 𝑥2)𝑢 𝑓(𝑥0, 0, 0) ≡ 0��1 = 𝑥2 − 𝑥2


��2 = −𝑥2 + 𝑢


𝑉0(𝑥0, 𝑥1, 𝑥2) = 𝑥20 + Ψ0(𝑥0, 𝑥1, 𝑥2) + 𝑉(𝑥1, 𝑥2)


NonlinearControl




Backstepping

Dissipativity

Forwarding




��0 = 𝑓(𝑥0, 𝑥1, 𝑥2) + 𝑔(𝑥0, 𝑥1, 𝑥2)𝑢 𝑓(𝑥0, 0, 0) ≡ 0��1 = 𝑥2 − 𝑥2


��2 = −𝑥2 + 𝑢


𝑉0(𝑥0, 𝑥1, 𝑥2) = 𝑥20 + Ψ0(𝑥0, 𝑥1, 𝑥2) + 𝑉(𝑥1, 𝑥2)


NonlinearControl




Backstepping

Dissipativity

Forwarding




��0 = 𝑓(𝑥0, 𝑥1, 𝑥2) + 𝑔(𝑥0, 𝑥1, 𝑥2)𝑢 𝑓(𝑥0, 0, 0) ≡ 0��1 = 𝑥2 − 𝑥2


��2 = −𝑥2 + 𝑢


𝑉0(𝑥0, 𝑥1, 𝑥2) = 𝑥20 + Ψ0(𝑥0, 𝑥1, 𝑥2) + 𝑉(𝑥1, 𝑥2)


NonlinearControl

Unconstrainedcontrol

Constrainedcontrol


Part IV

Optimal control

NonlinearControl


Constrainedcontrol


Finite-horizon optimal controlCost function

System with finite-horizon cost function

�� = 𝑓(𝑥, 𝑢) 𝑥(0) = 𝑥0

𝐽(𝑥0, 𝑢(·)) = 𝜑(𝑥(𝑡𝑓)) + ∫𝑡𝑓

0𝐿(𝑥(𝜏), 𝑢(𝜏)) 𝑑𝜏

𝜑 is the terminal cost(the penalty on the the state 𝑥(𝑡𝑓) at the final time 𝑡𝑓)

𝐿 is the Lagrangian cost

assume 𝑓, 𝜑, and 𝐿 are C1, and assume 𝑥0 and 𝑡𝑓 are fixed

GoalDerive necessary conditions for an open-loop control 𝑢(𝑡) tominimize the cost over all possible controls.

NonlinearControl


Constrainedcontrol


19 Unconstrained control

20 Constrained control

NonlinearControl


Constrainedcontrol


The method of Lagrange multipliersConstrained optimization

Constrained minimization problem

Given C1 functions 𝐹 ∶ ℝ𝑛 → ℝ and 𝐺 ∶ ℝ𝑛 → ℝ𝑟, solve

min𝑥∈ℝ𝑛

𝐹(𝑥) subject to 𝐺(𝑥) = 0 .

Lagrange multiplier theorem

If 𝑥∗ is a solution and rank 𝐷𝐺(𝑥∗) = 𝑟, then there exists avector 𝜆 ∈ ℝ𝑟 of Lagrange multipliers such that

𝐷𝐹(𝑥∗) + 𝜆𝑇𝐷𝐺(𝑥∗) = 0

(the function Ξ = 𝐹 + 𝜆𝑇𝐺 has a critical point at 𝑥∗).

for us, we have a functional 𝐽 instead of a function 𝐹our constraints 𝐺 are differential equations …

NonlinearControl


Constrainedcontrol


The method of Lagrange multipliersConstrained optimization

Constrained minimization problem

Given C1 functions 𝐹 ∶ ℝ𝑛 → ℝ and 𝐺 ∶ ℝ𝑛 → ℝ𝑟, solve

min𝑥∈ℝ𝑛

𝐹(𝑥) subject to 𝐺(𝑥) = 0 .

Lagrange multiplier theorem

If 𝑥∗ is a solution and rank 𝐷𝐺(𝑥∗) = 𝑟, then there exists avector 𝜆 ∈ ℝ𝑟 of Lagrange multipliers such that

𝐷𝐹(𝑥∗) + 𝜆𝑇𝐷𝐺(𝑥∗) = 0

(the function Ξ = 𝐹 + 𝜆𝑇𝐺 has a critical point at 𝑥∗).

for us, we have a functional 𝐽 instead of a function 𝐹our constraints 𝐺 are differential equations …

NonlinearControl


Constrainedcontrol


Adding a time-varying Lagrange multiplierAn informal derivation

The augmented cost Ξ with Lagrange multiplier 𝜆(𝑡) ∈ ℝ𝑛

Ξ = 𝜑(𝑥(𝑡𝑓)) + ∫𝑡𝑓

0[𝐿(𝑥(𝜏), 𝑢(𝜏)) + 𝜆𝑇(𝜏)(𝑓(𝑥(𝜏), 𝑢(𝜏)) − ��(𝜏))]𝑑𝜏

The Hamiltonian

𝐻(𝑥, 𝑢, 𝜆) = 𝐿(𝑥, 𝑢) + 𝜆𝑇𝑓(𝑥, 𝑢)

Integrate by parts to get

Ξ = 𝜑(𝑥(𝑡𝑓)) + ∫𝑡𝑓

0[𝐻(𝑥(𝜏), 𝑢(𝜏), 𝜆(𝜏)) + 𝜆𝑇(𝜏)𝑥(𝜏)]𝑑𝜏

+ 𝜆𝑇(0)𝑥0 − 𝜆𝑇(𝑡𝑓)𝑥(𝑡𝑓)

Now set derivatives of Ξ with respect to 𝑥 and 𝑢 to zero …

NonlinearControl


Constrainedcontrol

3rd ElgersburgSchool 2011 The derivative with respect to 𝑥(𝑡)

Informally,

𝜕Ξ𝜕𝑥

= ∫𝑡𝑓

0[

𝜕𝐻𝜕𝑥

(𝑥(𝜏), 𝑢(𝜏), 𝜆(𝜏)) + 𝜆𝑇(𝜏)]𝑑𝜏

To make this zero, we set

𝜆 = −𝜕𝐻𝑇

𝜕𝑥(𝑥, 𝑢, 𝜆)

If 𝑥(𝑡𝑓) is not specified, then we set

0 = 𝜕Ξ𝜕𝑥(𝑡𝑓)

= 𝐷𝜑(𝑥(𝑡𝑓)) − 𝜆𝑇(𝑡𝑓)

If 𝑥(𝑡𝑓) is free (not specified)

𝜆(𝑡𝑓) = [𝐷𝜑(𝑥(𝑡𝑓))]𝑇

NonlinearControl


Constrainedcontrol

3rd ElgersburgSchool 2011 The derivative with respect to 𝑢(𝑡)

Informally,

𝜕Ξ𝜕𝑢

= ∫𝑡𝑓

0

𝜕𝐻𝜕𝑢

(𝑥(𝜏), 𝑢(𝜏), 𝜆(𝜏)) 𝑑𝜏

This should be zero so we set

𝜕𝐻𝜕𝑢

(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡)) = 0 for almost all 𝑡

As a consequence, we observe that

�� = 𝜕𝐻𝜕𝑥

⋅ ��⏟

𝑓(𝑥,𝑢)

+ 𝜕𝐻𝜕𝜆⏟

𝑓𝑇(𝑥,𝑢)

⋅ 𝜆⏟

−𝜕𝐻𝑇/𝜕𝑥

= 0 .

The Hamiltonian is constant under an optimal trajectory.

NonlinearControl


Constrainedcontrol


Putting it all togetherNecessary conditions for optimality

If 𝑢(𝑡) is an optimal trajectory, then

��(𝑡) = 𝑓(𝑥(𝑡), 𝑢(𝑡)) = 𝜕𝐻𝑇

𝜕𝜆(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡))

𝜆(𝑡) = −𝜕𝐻𝑇

𝜕𝑥(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡))

0 = 𝜕𝐻𝜕𝑢

(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡)) for almost all 𝑡

Boundary conditions if 𝑥(𝑡𝑓) is free:

𝑥(0) = 𝑥0 𝜆(𝑡𝑓) = [𝐷𝜑(𝑥(𝑡𝑓))]𝑇

Boundary conditions if 𝑥(𝑡𝑓) is specified:

𝑥(0) = 𝑥0 𝑥(𝑡𝑓) = 𝑥𝑓

NonlinearControl


Constrainedcontrol

3rd ElgersburgSchool 2011 Computing an optimal control

first set 𝜕𝐻/𝜕𝑢 = 0 and solve for 𝑢 in terms of 𝑥 and 𝜆

plug this 𝑢 into the Hamiltonian ODE

��(𝑡) = 𝑓(𝑥(𝑡), 𝑢(𝑡)) = 𝜕𝐻𝑇

𝜕𝜆(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡))

𝜆(𝑡) = −𝜕𝐻𝑇

𝜕𝑥(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡))

which has mixed boundary conditions at 𝑡 = 0 and 𝑡 = 𝑡𝑓solve the resulting two-point boundary-value problem(in MATLAB you might use bvp4c or bvp5c)

if there are multiple solutions,choose the ones having the smallest cost

use other methods to verify that the solution is optimal(for example, check second derivatives at the solution)

NonlinearControl


Constrainedcontrol


Computing an optimal controlAn example

Fixed initial and final states with 𝑡𝑓 = 1

��1 = 𝑥22 𝑥(0) = (0, 0)

��2 = 𝑢 𝑥(1) = (1, 0)

𝐽 = ∫1

0𝑢2(𝜏) 𝑑𝜏

𝐻(𝑥, 𝑢, 𝜆) = 𝑢2 + 𝜆1𝑥22 + 𝜆2𝑢 𝜕𝐻

𝜕𝑢= 2𝑢 + 𝜆2

𝑢(𝑡) = − 12 𝜆2(𝑡) 𝜕𝐻

𝜕𝑥= [0 2𝜆1𝑥2]

NonlinearControl


Constrainedcontrol


Computing an optimal controlAn example

Solve the boundary-value problem

��1 = 𝑥22 𝑥(0) = (0, 0)

��2 = − 12 𝜆2 𝑥(1) = (1, 0)

𝜆1 = 0𝜆2 = −2𝜆1𝑥2

Many solutions:

𝜔 = 𝑘𝜋 , 𝑘 = 1, 2, … 𝜆1(𝑡) ≡ −𝜔2

𝑥1(𝑡) = 𝑡 − 12𝜔 sin(2𝜔𝑡) 𝜆2(𝑡) = ∓𝜔2√2 cos(𝜔𝑡)

𝑥2(𝑡) = ±√2 sin(𝜔𝑡) 𝑢(𝑡) = ±𝜔√2 cos(𝜔𝑡)

The solutions with 𝜔 = 𝜋 have the smallest cost.

NonlinearControl


Constrainedcontrol


19 Unconstrained control

20 Constrained control

NonlinearControl


Constrainedcontrol

3rd ElgersburgSchool 2011 The Pontryagin maximum (or minimum) principle

suppose the controls can only take values in 𝒰 ⊂ ℝ𝑚

suppose 𝒰 is not open (e.g., 𝒰 is compact)

then 𝜕𝐻/𝜕𝑢 = 0 is no longer necessaryalong the optimal trajectory

instead we must have

𝐻(𝑥(𝑡), 𝑢(𝑡), 𝜆(𝑡)) = min𝔲∈𝒰

𝐻(𝑥(𝑡), 𝔲, 𝜆(𝑡))

for almost all 𝑡

NonlinearControl


Constrainedcontrol







𝐻(𝑥(𝑡), 𝔲, 𝜆(𝑡))

for almost all 𝑡

NonlinearControl


Constrainedcontrol







𝐻(𝑥(𝑡), 𝔲, 𝜆(𝑡))

for almost all 𝑡

NonlinearControl


Constrainedcontrol







𝐻(𝑥(𝑡), 𝔲, 𝜆(𝑡))

for almost all 𝑡

Nonlinear Control - 3rd Elgersburg School 2011 · Nonlinear Control 3rd Elgersburg School 2011 Linear control of nonlinear systems First pass at controlling a nonlinear system: linearize

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