vss2020/homepage-summer-school/... · 2019. 4. 13. · outline 1 introduction 2 generalized super...

Homogeneity Based Design of Sliding ModeControllers

Jaime A. Moreno

Universidad Nacional Autonoma de MexicoElectrica y Computacion, Instituto de Ingenierıa,

04510 Mexico D.F., Mexico, JMorenoP@ii.unam.mx

8th-12th April 2019, Rio de Janeiro, BrasilEECI International Graduate School in Control 2019, M13

www.eeci-igsc.euInternational Summer School on Sliding Mode Control -

Variable Structure Systems

Course Overview

Generalized Super-Twisting Algorithm

HOSM Control and Homogeneity

Lyapunov-Based Design of Higher-Order Sliding Mode(HOSM) Controllers

HOSM Differentiation/Observation: A Lyapunov Approach

Construction of Lyapunov Functions using GeneralizedForms

Continuous HOSM Controllers

Homogeneity Based SMC Jaime A. Moreno UNAM 2

Course Overview

Part I

Outline

1 Introduction

2 Generalized Super Twisting Algorithm (GSTA)A Quadratic Strong Lyapunov Function for the GSTAConvergence TimeGSTA with perturbations: LMI

3 The GSTA with Variable (Adaptive) Gains

4 Uniformity of\ the GSTA with Constant GainsA Non Quadratic Strong Lyapunov Function for the GSTAUniformity in the ConvergenceAn Alternative Robust Lyapunov Function

5 Output Feedback with Twisting and GSTA: A LyapunovApproach

6 Conclusions

Overview

1 Introduction

6 Conclusions

Second Order Sliding Modes

Super-Twisting Algorithm (STA)[Levant, 1993]

x1 = −k1 |x1|12 sign (x1) + x2

x2 = −k2 sign (x1) ,

Used for control to attenuate chattering,Used for exact differentiaton and observation.

Twisting Algorithm [Emel’yanov, Levant, 1985].

x1 = x2

x2 = −k1 sign (x1)− k2 sign (x2) , k1 > k2

Prescribed Convergence Law (Terminal Sliding ModeController)[Levant, 1993; Man et al., 1994]

x1 = x2

x2 = −α sign(x2 + β |x1|

12 sign (x1)

Quasi-Continuous Controller [Levant, 2006]

x1 = x2

x2 = −αx2+β|x1|12 sign(x1)

|x2|+β|x1|12

Many others: Sub-Optimal Controller, etc. [Bartolini,Ferrara, Usai, 1998]

Arbitrary Order Sliding Modes

“Arbitrary Order Robust Exact Differentiator” [Levant,2003].

x1 = −k1 |x1|n−1n sign (x1) + x2

x2 = −k2 |x1|n−2n sign (x1) + x3 ,

...xn = −kn sign (x1) ,

Used for exact differentiaton and observation.

Nested Sliding Mode Controllers [Levant, 2003].

x1 = x2

x2 = x3

x3 = −α sign

(x3 + 2

(|x2|3 + |x1|2

sign(x2 + |x1|

23 sign (x1)

for n = 3 and recursively for n > 3

Quasi-Continuous Controller [Levant, 2006]

x1 = x2

x2 = x3

x3 = −αx3+2

(|x2|+|x1|

)− 12(x2+|x1|

23 sign(x1)

)|x3|+2

(|x2|+|x1|

Many Others are possible

Analysis of HOSM Algorithms

Homogeneity

Definition

dilation: κ > 0, (positive) weights (m1, m2, · · · , mn)dκ : (x1, x2, · · · , xn)→ (κm1x1, κ

m2x2, · · · , κmnxn)homogeneous vector field (or inclusion) of degree q

F (x) = κ−qd−1κ F (dκx)

local stability = global stability (as for linear systems)If q < 0 Finite-Time stabilityRobustnessBut: It does not assure stability.

Contraction analysis using geometry to certify stability

Lyapunov analysis of HOSM I

Weak (non strict) Lyapunov Functions:

V (x) > 0 , V (x) ≤ 0 , ∀x 6= 0

They have been used for some SOSM algorithms [Orlov,2005; ....]

Super Twisting Algorithm

x1 = −k1 |x1|12 sign (x1) + x2 , V (x) = k2 |x1|+ 1

x2 = −k2 sign (x1) V (x) = −k1k2 |x1|1/2

Twisting Algorithm

x1 = x2 V (x) = k1 |x1|+ 12x2

x2 = −k1 sign (x1)− k2 sign (x2) , V (x) = −k2 |x2|

They assure stability (using generalized Lasalle’s Lemma)

Lyapunov analysis of HOSM II

They cannot assure robustness (under perturbations), andno convergence time estimation

Strong (strict) Lyapunov Functions:

V (x) > 0 , V (x) < 0 , ∀x 6= 0

It assures (robust) stabilityIf V (x) ≤ −γV p (x) , 0 ≤ p < 1, Finite-Time stability andConvergence Time Estimation

Recently some results for SOSM algorithms have been obtained:

Lyapunov analysis of HOSM III

Construction using Zubov’s Theorem [Polyakov andPoznyak, 2009, 2011]: for Twisting and Super TwistingAlgorithms, Suboptimal Controller, Terminal SMController.

Plus: systematic and general method; valid for system withperturbations; tight convergence time estimation.Difficulties: Solution of a PDE or PD Inequality; “complex”Lyapunov Fs

Quadratic Lyapunov functions for GeneralizedSupertwisting Algorithms [Moreno and Osorio, 2008;Moreno, 2009,...]

Some other results [Orlov, ....]

Why Strong Lyapunov funct. for HOSM?

Advantages of Lyapunov functions:

Simple analysis tool

It allows the estimation of convergence time

Robustness analysis tool

It does not rely on homogeneity

Several Extensions possible:

Uniform algorithms: convergence time independent of i.c.Variable (adaptive) gainsMultivariable (?)

Design tool: Control Lyapunov Functions

Addition of extra terms to improve performance

Analysis and design of interconnected systems

Some requirements to a Lyapunov method

Systematic and efficient“Simple” Lyapunov functions (recall quadratic LF forLinear Systems)

Overview

1 Introduction

6 Conclusions

Generalized Super Twisting Algorithm(GSTA)

x1 = −k1 (t)φ1 (x1) + x2 + ρ1 (t, x)x2 = −k2 (t)φ2 (x1) + ρ2 (t, x) ,

φ1 (x1) = µ1|x1|p sign (x1) + µ2|x1|q sign (x1) ,

φ2 (x1) = µ21p|x1|2p−1 sign (x1) + µ1µ2 (p+ q) |x1|p+q−1 sign (x1) +

+ µ22q|x1|2q−1 sign (x1) ,

µ1, µ2 ≥ 0 constant. Non homogeneous.

q ≥ 1 ≥ p ≥ 12 are real numbers.

φ1 (x1), φ2 (x1) monotone increasing continuous if p > 12 .

If p = 12 φ2 (x1) discontinuous at x1 = 0.

Trajectories in the sense of Filippov

Particular cases with constant gains k1, k2:

A linear Algorithm: (µ1, µ2, p, q) = (1, 0, 1, 1).

x1 = −k1x1 + x2

x2 = −k2x1 .

The classical Super-Twisting Algorithm (STA) [Levant,93]: (µ1, µ2, p, q) =

(1, 0, 1

2 , q).

x1 = −k1 |x1|12 sign (x1) + x2

x2 = −12k2 sign (x1) .

φ2 (x1) is discontinuous.

A Homogeneous Algorithm: For p ≥ 12

x1 = −k1 |x1|p sign (x1) + x2

x2 = −k2p |x1|2p−1 sign (x1) .

Generalized Super-Twisting Algorithm (GSTA) [Moreno2009]: p = 1

2 , q = 1.

Stability Properties: dependent on p, q.

Low order term High order Stability Type

p = 1 q = 1ExponentialRobust

Non UniformPractical

12 < p < 1 q = 1

Finite-TimeRobust

p = 12 q = 1

Finite-TimeExact

p = 1 q > 1ExponentialRobust

UniformPractical

12 < p < 1 q > 1

Finite-TimeRobust

UniformPractical

p = 12 q > 1

Finite-TimeExact

UniformPractical

Exact: Convergence to the origin under bounded perturbations.Homogeneity Based SMC Jaime A. Moreno UNAM 18

Outline

1 Introduction

6 Conclusions

Quadratic Lyapunov Function

Family of quadratic strong Lyapunov Functions:

VQ(x) = ζTPζ , P = P T > 0 ,

Algebraic Lyapunov Equation (ALE):

ATP + PA = −Q

[φ1 (x1)x2

], A =

[−k1 1−k2 0

Time derivative of Lyapunov Function:

VQ(x) = φ′1 (x1) ζT(ATP + PA

)ζ = −φ′1 (x1) ζTQζ

Figure : The Lyapunov function VQ(x), p = 1/2, µ2 = 0.

Stability of GSTA without Perturbations

Theorem

If µ1 > 0, k1 , k2 are constant. The statements are equivalent:

(i) x = 0 is asymptotically stable.

(ii) A Hurwitz.

(iii) k1 > 0 , k2 > 0.

(iv) ∀Q = QT > 0, ALE has a unique solution P = P T > 0.

VQ(x) is a global, strong Lyapunov function.

VQ ≤ −γ1 (Q,µ1)V3p−1

Q (x)− γ2 (Q,µ2) |x1|q−1 VQ (x) ,

γ1 (Q,µ1) , µ1p

1 pλmin Qλ

1−p2p

min Pλmax P

, γ2 (Q,µ2) , µ2qλmin Qλmax P

Remarkable: Stability of GSTA ⇔ Stability of ξ = Aξ.Homogeneity Based SMC Jaime A. Moreno UNAM 22

Outline

1 Introduction

6 Conclusions

Convergence Time

Proposition

Suppose k1 > 0 , k2 > 0, µ1 > 0.• If 1

2 ≤ p < 1: Finite-Time Convergence

T (x0) =

(1−p)γ1(Q,µ1)V1−p2p

Q (x0) µ2 = 0 or q > 1

2p(1−p)γ2(Q,µ2) ln

(1 + γ2(Q,µ2)

γ1(Q,µ1)V1−p2p

Q (x0)

)µ2 > 0 and q = 1

• If p = 1: exponential convergence.

For Design: T depends on the gains!

Figure : Convergence time for p = 1/2, q = 1: µ2 = 0 (blue) andµ2 > 0 (green).

Outline

1 Introduction

6 Conclusions

GSTA with perturbations: LMI

x1 = −k1φ1 (x1) + x2

x2 = −k2φ2 (x1) + ρ2 (t, x) .

|ρ2 (t, x)| ≤ δ(pµ2

1 |x1|2p−1 + (p+ q)µ1µ2 |x1|p+q−1 + qµ22 |x1|2q−1

)The robust stability analysis can be performed through the LMI[

ATP + PA+ εP + δ2CTC PBBTP −1

]≤ 0 ,

[−k1 1−k2 0

], C =

], B =

Globally, robustly stable. VQ (x) = ζTPζ Lyapunovfunction.For 1 > p ≥ 1

2 , µ1 > 0 Finite-Time convergenceFor p = 1, µ1 > 0 Exponential convergence.For p = 1

2 Exact Stability.Homogeneity Based SMC Jaime A. Moreno UNAM 27

Frequency Domain Interpretation: TheCircle Criterium

Classical Circle Criterium: LMI is satisfied ⇔ Nyquist diagramof G (s) = C (sI −A)−1B is in circle with radius δ, ⇔

maxω|G (jω)| < 1

δ, G (s) =

s2 + k1s+ k2.

Figure : Nyquist Diagramm of G(jω).

maxω|G (jω)|2 =

2if k2 − 1

2k21 < 0

1(k2− 14k2

1)if k2 − 1

2k21 > 0

Figure : Robust Stability Region for the Gains k1, k2

0 0.5 1 1.5 2 2.5 30

Time (sec). p=3/2

0 0.5 1 1.5 2 2.5 30

Time (sec). p=1

0 0.5 1 1.5 2 2.5 3−0.2

Time (sec). p=3/4

0 0.5 1 1.5 2 2.5 3−0.4

−0.2

Time (sec). p=1/2

Figure : States of the GSOA without perturbation,p =

(32 , 1 , 3

), µ1 = 1, µ2 = 0. k1 = 3, k2 = 2

0 0.5 1 1.5 2 2.5 30

Time (sec). p=3/2

0 0.5 1 1.5 2 2.5 3−0.2

Time (sec). p=1

0 0.5 1 1.5 2 2.5 3−0.2

Time (sec). p=3/4

0 0.5 1 1.5 2 2.5 3−0.4

−0.2

Time (sec). p=1/2

Figure : States of the GSOA with a persistent perturbationρ2(t, x) = 1.9 sin(10t), p =

(32 , 1 , 3

), µ1 = 1, µ2 = 0.

Practical Stability: for large perturbations

|ρ1| ≤ δ1 , |ρ2| ≤ δ2 ,

1 if µ2 = 0, p = 12 , δ2 <

µ21λminQ4λmaxP then

VQ ≤ −κ(12µ

21λmin Q − 2δ2λmax P

12max P

Q (x) ,

∀ ‖ζ‖2 >µ21δ1λmax P

2 (1− κ)(12µ

21λmin Q − 2δ2λmax P

) ,⇒ Practical Stability (ISS), if δ2 is small enough.

2 If 12 < p ≤ 1 then for any 0 < κ < 1

VQ ≤ −κpµ

1 λmin Q

λ3p−12p

max PV

3p−12p

Q (x)−κµ2λmin Qλmax P

VQ (x) , ∀ ‖ζ‖2 /∈ D

D is a compact set, containing the origin ⇒ Practical Stability(ISS)

0 2 4 6 8 10 12

Time (sec). p=3/2

0 5 10 150

Time (sec). p=1

0 5 10 150

Time (sec). p=3/4

0 5 10 150

Time (sec). p=1/2

Figure : States of the Super-Twisting Algorithm with a constantperturbation ρ2(t, x) = 10, p =

(32 , 1 , 3

), µ1 = 1, µ2 = 0,

k1 = 3, k2 = 2.

Overview

1 Introduction

6 Conclusions

Variable Gain GSTA

x1 = −k1 (t, x)φ1 (x1) + x2 + ρ1 (t, x)x2 = −k2 (t, x)φ2 (x1) + ρ2 (t, x) ,

|ρ1 (t, x)| ≤ g1 (t, x) |φ1 (x1)||ρ2 (t, x)| ≤ g2 (t, x) |φ2 (x1)|

where g1 (t, x) ≥ 0, g2 (t, x) ≥ 0 are known continuous functions.If β > 0, ε > 0, δ > 0 arbitrary constants,

k1(·) =δ +1

4ε[2εg1 + g2]2 + 2εg2 + ε+ [2ε+ g1]

(β + 4ε2

k2(·) =β + 4ε2 + 2εk1(·)

x = 0 is robustly GAS.Homogeneity Based SMC Jaime A. Moreno UNAM 35

VQ (x) = ζTPζ constant, strong, robust Lyapunov function.

For 1 > p ≥ 12 , µ1 > 0 Finite-Time Convergence

T (x0) =

(1−p)γ1V

1−p2p

Q (x0) if µ2 = 0 or q > 1

2p(1−p)γ2

(1 + γ2

1−p2p

Q (x0)

)if µ2 > 0 and q = 1

γ1, γ2 some constants.

For p = 1, µ1 > 0 Exponential Convergence.

If p = 12 Exact Stability.

Useful for Adaptive Control [Shtessel 2011]

Proof: It is possible to construct a constant, robust Lyapunovfunction V (x) = ζTPζ with A(t)TP + PA(t) < 0, andζT =

[φ1 (x1) , x2

Overview

1 Introduction

6 Conclusions

Outline

1 Introduction

6 Conclusions

An Alternative Strong Lyapunov Function

Theorem

Suppose k1 > 0 , k2 > 0 constant, α = k2δ, β = 1, δ > 0.

VN (x) = α |φ1 (x1)|2−β |φ1 (x1)|1q sign (x1) |x2|

2q−1q sign (x2)+δx2

is a global, strong Lyapunov function for δ sufficiently large.

VN ≤ −1

qνmin

4δmax 1, k2

) 3q−12q

V3q−1

N (x) ,

νmin , minx1∈R

pµ1 |x1|p−q + qµ2(µ1 |x1|p−q + µ2

) q−1q

For q ≥ 1, µ2 > 0 global asymptotic stability of the origin isassured.

Outline

1 Introduction

6 Conclusions

Lyapunov function and Unif. Convergence

W (x) = VQ (x) + VN (x) strong Lyapunov func. for GSOA.Uniform Convergence: When the convergence time has anupper bound independent of the initial condition!

norm of the initial condition ||x(0)|| (logaritmic scale)

Converg

ence T

NSOSMO

GSTA with linear term

Figure : Convergence time when the initial condition grows.

Finite-Time and Uniform Convergence forp < 1 < q

Proposition

Suppose k1 > 0 , k2 > 0, µ1, µ2 > 0 and 12 ≤ p < 1 < q. Then

Finite-Time and Uniform convergence

T (x0) =2q

(q − 1)κ2

µq−12q

Wq−12q (x0)

(1− p)κ1µ

1−p2p ,

where W (x) = VQ (x) + VN (x), µ, κ1, κ2 are some constants.

T (x0) ≤ Tmax =2q

(q − 1)κ2

) q−pp(q−1)

(1− p)κ1

) q(1−p)q−p

i.e. any trajectory converges to x = 0 in a time smaller thanTmax.

Outline

1 Introduction

6 Conclusions

Robust stability for perturbations in sector

Proposition

Assume k1 > g1 , k2 >(

3q−1

) q2q−1

g3q−12q−1

2 are constant. VN (x) is

a global, robust, strong Lyapunov function.

VN ≤ −2ψminνmin

4δmax 1, k2

) 3q−12q

V3q−1

N (x) ,

with appropriate νmin, ψmin.

For q > 1, µ2 > 0 GAS.

For µ1, µ2 > 0, 12 ≤ p < 1 < q Finite-Time and uniform

stability.

For p = 12 exact stability.

Using W (x): Practical Stability for every q ≥ 1.

Overview

1 Introduction

6 Conclusions

OF Twisting Controller

Plant with relative degree 2

x1 = x2

x2 = a (t, ξ) + b (t, ξ)u ,

a, b unknown scalar functions.

0 < bm ≤ |b (t, ξ)| ≤ bM , |a (t, ξ)| ≤ a ,

Discontinuous Observer

˙x1 = −l1φ1 (x1 − x1) + x2˙x2 = −l2φ2 (x1 − x1) + 1

2 (bm + bM )u .

Output feedback

u = −k1 sign (x1)− k2 sign (x2)

LF for State Feedback: Twisting Controller

Closed Loop with State Feedback (we assume b (t, ξ) > 0)

x1 = x2

x2 = −b (t, ξ) k1 sign (x1)− b (t, ξ) k2 sign (x2) + a (t, ξ) .

(Lipschitz) Continuous (strict) Lyapunov Function

V (x) =

(π1 |x1|+

+ π2x1x2

π2 > 0 , p > 0 , π1 = p+

12π2 .

V (x) ≤ −γV23 (x) .

Under the conditions for the gains

bM> k1 −

bm> k2 >

bm− 1

)p > 0

x = 0 is a robust, finite-time, globally stable point.Convergence-Time estimation

T (x0) ≤ 3

13 (x0) .

OF Closed Loop Stability

Estimation errors e1 = x1 − x1 and e2 = x2 − x2. Closed loopdynamics

x1 = x2

x2 = −b (t, ξ) k1 sign (x1)− b (t, ξ) k2 sign (x2) + a (t, ξ) + χ (t, ξ, x2, e2) ,

e1 = −l1φ1 (e1) + e2

e2 = −l2φ2 (e1) + ρ2 (t, ξ, u) ,

χ (t, ξ, x2, e2) , b (t, ξ) k2 [sign (x2 (t))− sign (x2 (t) + e2 (t))] ,

ρ2 (t, ξ, u) , −a (t, ξ) +

2(bm + bM )− b (t, ξ)

Bounds of the perturbations

|χ (t, ξ, x2, e2)| ≤ 2bMk2 ,

|ρ2 (t, ξ, u)| ≤ 1

1 + 3µ |e1|12 + 2µ2 |e1|

g , 2a+ (bM − bm) (k1 + k2) .

If (l1, l2) and (k1, k2) are appropiately designed, then CL isfinite time stable.

What about LF for HOSM?

Second order robust exact differentiator

x1 = −k1 |x1|23 sign (x1) + x2

x2 = −k2 |x1|13 sign (x1) + x3 ,

x3 = −k3 sign (x1) + δ (t) ,

Lyapunov Function

V (x) = γ1 |x1|43 − γ12 |x1|

23 sign (x1)x2 + γ2 |x2|2 + γ13x1x3

−γ23x2 |x3|2 sign (x3) + γ3 |x3|4

V (x) = −(

3γ1k1 − γ12k2 + γ13k3

)|x1|+ 2

3γ1 − γ2k2 +

3γ12k1

13 sign (x1)x2

3γ12|x2|2

|x1|13

− (γ12 + γ13k1) |x1|23 sign (x1)x3 + γ23k2 |x1|

13 sign (x1) |x3|2 sign (x3)

− 4γ3k3 sign (x1)x33 + (2γ2 + γ13 + 2γ23k3 sign (x1x3))x2x3 − γ23 |x3|3 .

Ongoing Work

Systematic method to construct Lyapunov functions forSOSM and HOSM: Twisting Algorithm, Terminal(Prescribed Time) Controller. [Tonametl Sanchez]

Lyapunov-Based Adaptive SM Control: [Shtessel, Moreno,Fridman]

Uniformization of HOSM algorithms [Emmanuel Cruz]

SOSM for PDEs: Supertwisting Algorithm [RamonMiranda]

Robust and Robust Parameter Estimation [Eder Guzman]

Robust observation and control for Chemical andBioprocesses [A. Vargas, E. Rocha, A. Vande Wouwer, J.Alvarez...]

Overview

1 Introduction

6 Conclusions

Conclusions

Developing a Lyapunov-based methodology for HOSMseems to be important for:

To have quantitative analysis and design tools for HOSMTo integrate the HOSM to the standard Nonlinear Methods:interconections and combinations

We are at the beginning of this development: much work isneeded

To find Lyapunov functions is good but to find asystematic an efficient way of construction is better.

Thanks to colleagues and students: L. Fridman, M. Osorio,M.T. Angulo, E. Cruz-Zavala, E. Guzman, R. Miranda, T.Sanchez, ....

Part II

Outline

7 Preliminaries

8 HOSM Control Problemρ = 1, First Order Sliding Mode (FOSM) Control Problemρ = 2, Second Order Sliding Mode (SOSM) Control

9 HomogeneityClassical HomogeneityWeighted HomogeneityWeighted Homogeneity for systems with inputs(perturbations)Weighted Homogeneity and Precision under perturbationsHomogeneous Approximation/DominationExample: ”Danger” of Non Homogeneous Controllers

10 Homogeneous Design of HOSM (Levant 2005)

11 Plaidoyer for a Lyapunov-Based Framework for HOSM

Overview

7 Preliminaries

Differential Inclusions (DI)

Consider a dynamical system

x = f(x) .

We know that

If f(x) discontinuous according to Filippov we obtain a DI

x ∈ F (x) , F (x) ⊂ Rn .

If f(x) is uncertain, i.e. ‖f(x)‖ ≤ f+ we can write

x ∈[−f+, f+

]⇒ x ∈ F (x) .

In case of discontinuity or/and uncertainty we obtain aDifferential Inclusion from a Differential Equation

Filippov Differential Inclusions (DI)

x ∈ F (x)

is Filippov DI if ∀x ∈ Rn, the set-valued function F (x) ⊂ Rn is

not empty;

compact;

convex;

upper-semicontinuous, i.e.

limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0

wheredist(x, A) = inf |x− a||a ∈ A .

A solution x(t) is an absolutely continuous functionsatisfying the DI almost everywhere.

Filippov DI have the usual properties, except foruniqueness.

x ∈ F (x)

not empty;

compact;

convex;

limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0

x ∈ F (x)

not empty;

compact;

convex;

limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0

x ∈ F (x)

not empty;

compact;

convex;

limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0

x ∈ F (x)

not empty;

compact;

convex;

limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0

x ∈ F (x)

not empty;

compact;

convex;

limy→x

sup [dist(z, F (x))|z ∈ F (y)] = 0

Higher Order Sliding Mode (HOSM)

Consider a Filippov DI x ∈ F (x), with a smooth outputfunction σ = σ(x). If

1 The total time derivatives σ, σ, . . . , σ(r−1) are continuousfunctions of x

2 The setσ = σ = . . . = σ(r−1) = 0 (1)

is a nonempty integral set (i.e., consists of Filippovtrajectories)

3 The Filippov set of admissible velocities at the r-slidingpoints (??) contains more than one vector

the motion on the set (??) is said to exist in an r -sliding(rth-order sliding) mode. The set (??) is called the r-slidingset. The nonautonomous case is reduced to the one consideredabove by introducing the fictitious equation t = 1.

Overview

7 Preliminaries

HOSM Control

SISO smooth, uncertain system

z = f (t, z) + g (t, z)u, σ = h (t, z) ,

z ∈ Rn, u ∈ R, σ ∈ R: sliding variable.

f (t, z) and g (t, z) and n uncertain.

Control objective: to reach and keep σ ≡ 0 in finite time.

Relative Degree ρ w.r.t. σ is well defined, known andconstant.

Reduced (Zero) Dynamics asymptotically stable (byappropriate selection of σ).

The basic DI

Defining x = (x1, ..., xρ)T = (σ, σ, ..., σ(ρ−1))T , σ(i) = di

dtih (z, t)

The regular form

∑T :

xi = xi+1, i = 1, ..., ρ− 1,xρ = w (t, z) + b (t, z)u, x0 = x (0) ,

ζ = φ(ζ, x) ζ0 = ζ(0) ,

0 < Km ≤ b (t, z) ≤ KM , |w (t, z)| ≤ C .

Reduced Dynamics Asymptotically stable:

ζ = φ(ζ, 0) ζ0 = ζ(0) ,

The basic Differential Inclusion (DI)∑DI :

xi = xi+1, i = 1, ..., ρ− 1,xρ ∈ [−C, C] + [Km, KM ]u .

The Basic Problems

Bounded memoryless feedback controller

u = ϑρ(x1, x2, · · · , xρ) ,

Render x1 = x2 = · · · = xρ = 0 finite-time stable.

Motion on the set x = 0 is ρth-order sliding mode.

ϑρ necessarily discontinuos at x = 0 for robustness [−C, C].

Problem 1

How to design an appropriate control law ϑr ?

Problem 2

How to estimate in finite time the required derivativesx = (x1, ..., xρ)

T = (σ, σ, ..., σ(ρ−1))T ?

Outline

7 Preliminaries

ρ = 1, First Order Sliding Mode (FOSM)Control

Introduced in the mid 60’s: Fantastic results, maturetheory, lots of applications,... Utkin, Emelianov, .....Edwards, Spurgeon, Sira-Ramirez, Loukianov, Zinober, ....

σ ∈ [−C, C] + [Km, KM ]u .

u = ϑ1 = −k sign (σ), k > CKm

Robust (=insensitive) control, simple realization, finitetime convergence to the sliding manifold,...

Lyapunov design using V (σ) = 12σ

No derivative estimation required.

Mature theory including Multivariable case, adaptation,design of sliding surfaces, ...

Sliding modes

Mathematical Aspects ISliding Mode Equations (cont).

A.F. Filippov, Application of the theory of differential equations with discontinuous

right-hand sides to non-linear problems of automatic control, Proceedings of 1st IFAC

Congress in Moscow, 1960, Butterworths, London, 1961.

grad sxn

s(x)=0

0)( if

0)( if )( ),(

xsfxfxfx!

Convex

Hullfsm belongs to convex hullYu.I. Neimark, Note on A. Filippov’s paper,

1st IFAC Congress.

dx/dt=Ax+bu+dv,

u=-sign(s), v=-sign(s),

Nonuniqueness !?(f

Continuous vs. Discontinuous Control: Afirst order plant

Consider a plant

σ = α+ u, σ(0) = 1

where α ∈ (−1, 1) is a perturbation.Continuous (linear) Control

σ = α− kσ, k > 0, σ(0) = 1

Comments:

RHS of DE continuous (linear).

If α = 0 exponential (asymptotic) convergence to σ = 0.

If α 6= 0 practical convergence.

Discontinuous Control

σ = α− sign(σ), σ(0) = 1

with α ∈ (−1, 1).

σ > 0⇒ σ =< 0

σ < 0⇒ σ => 0

y σ(t) ≡ 0,∀t ≥ T .

Comments:

¿0 = α− sign(0)?

RHS of DE isdiscontinuous.

After arriving at σ = 0,sliding on σ ≡ 0.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−0.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Finite-Time convergence.

Differential Inclusion.

σ ∈ [−α, α]− sign(σ)Homogeneity Based SMC Jaime A. Moreno UNAM 69

0 5 10 15 20−1

−0.5

Lineal

Discontínuo

0 5 10 15 20−2.5

−1.5

−0.5

Lineal

Discontínuo

Notice the chattering = infinite switching of the controlvariable!

Outline

7 Preliminaries

ρ = 2, Second Order SM (SOSM) Control

Introduced in the mid 80’s: Wonderful results, maturegeometric theory, lots of applications,... Levant, Fridman,Bartolini, Ferrara, Shtessel, Usai, Feng, Man, Yu, Furuta,Spurgeon, Orlov, Perruquetti, Barbot, Floquet, Defoort, ....

∑DI :

x1 = x2,x2 ∈ [−C, C] + [Km, KM ]u .

Some controllers (see Fridman 2011):

Twisting Controller (Emelyanov, Korovin, Levant 1986).ϑ2(x) = −k1sign(x1)− k2sign(x2).Super-Twisting Algorithm (Levant 1993) (as differentiatorLevant 1998).The Sub-Optimal Algorithm (Bartolini, Ferrara, Usai 1997).Terminal Sliding Mode Control (Man, Paplinski, Wu, Yu(1994, 1997..)).

A second order plant

x1 = x2

x2 = φ (x1, x2) + uy = x1

φ: Perturbation/uncertainty.Question: Can we just feedback the output (as for FO case)?Two alternative output controllers:

Continuous (linear) output controller (Homogeneous TimeInvariant (HTI))

x1 = x2

x2 = −k1y

Discontinuous output controller (HTI)

x1 = x2

x2 = −k1sign(y)

0 5 10 15 20 25 30 35 40−3

Linear controller

0 5 10 15 20 25 30 35 40−5

Discontinuous controller

Both Oscillate!It is impossible to stabilize a double (or triple etc) integrator bystatic output feedback!

State feedback

State feedback controllers:

Continuous (linear) state feedback controller (HTI)

x1 = x2

x2 = −k1x1 − k2x2

Exponential ConvergenceRobust, but Sensitive to perturbations: Practical stability.

Continuous HTI state feedback controller

x1 = x2

x2 = −k1 dx1c13 − k2 dx2c

d·cρ = | · |ρ sign(·)Finite Time ConvergenceRobust, but Sensitive to perturbations: Practical stability.

First Order Sliding Mode Controller

x1 = x2

x2 = −k2sign(x2 + k1x1)

Rewrite as a first order system with a stable (first order) zerodynamics: with σ = x2 + k1x1 sliding variable

x1 = −k1x1 + σσ = −k2sign(σ)

Behavior with perturbation

0 10 20 30 40−1.5

−0.5

Linear controller

0 10 20 30 40

−0.25

−0.2

−0.15

−0.1

−0.05

Linear controller stabilize exponentially and is notinsensitive to perturbation

SM control also stabilizes exponentially but is insensitive toperturbation!

The Twisting Controller

A discontinuous HTI controller able to obtain finite-timeconvergence and insensitivity to perturbations:

x1 = x2

x2 = −k1sign(x1)− k2sign(x2)

0 10 20 30 40−1.5

−0.5

Linear controller

0 10 20 30 40−3

ρ = 2, SOSM Control....

Robust (=insensitive) control, finite time convergence tothe sliding manifold,...

No Lyapunov design or analysis.

Analysis and Design is very geometric: Beautiful butdifficult to extend to ρ > 2.

Solution: Homogeneity!

Overview

7 Preliminaries

Outline

7 Preliminaries

Classical Homogeneity

Classical Homogeneity for functions. (Euler, Zubov,Hahn,...)

Let n, m be positive integers. A mapping f : Rn → Rm ishomogeneous with degree δ ∈ R iff ∀λ > 0 :

f(λx) = λδf(x).

Some examples:

Linear function: Let A ∈ Rm×n then f(x) = Ax ishomogeneous of degree δ = 1, since

f(λx) = A(λx) = λ(Ax) = λf(x) .

f(x1, x2) =x3

x21+x2

2is continuous and homogeneous of degree

δ = 1 (but not linear!), since

f(λx1, λx2) =(λx1)3 + (λx2)3

(λx1)2 + (λx2)2=λ3(x3

1 + x32)

λ2(x21 + x2

2)= λf(x1, x2) .

f(λx) = λδf(x).

Some examples:

f(λx) = A(λx) = λ(Ax) = λf(x) .

f(x1, x2) =x3

x21+x2

f(λx1, λx2) =(λx1)3 + (λx2)3

(λx1)2 + (λx2)2=λ3(x3

1 + x32)

λ2(x21 + x2

2)= λf(x1, x2) .

f(λx) = λδf(x).

Some examples:

f(λx) = A(λx) = λ(Ax) = λf(x) .

f(x1, x2) =x3

x21+x2

f(λx1, λx2) =(λx1)3 + (λx2)3

(λx1)2 + (λx2)2=λ3(x3

1 + x32)

λ2(x21 + x2

2)= λf(x1, x2) .

f(x1, x2) =

dx1c12 +dx2c

x1+x2if x1 + x2 6= 0

0 otherwise

is discontinuous and homogeneous of degree δ = −12 , since

f(λx1, λx2) =dλx1c

12 + dλx2c

λx1 + λx2= λ−

12 f(λx1, λx2)

Quadratic Form: if x ∈ Rn and P ∈ Rn×n, q(x) = xTPx ishomogeneous of degree δ = 2, since

q(λx) = (λx)TP (λx) = λ2xTPx = λ2q(x) .

Classical Form = homogeneous polynomial: if x ∈ Rn, e.g.

p(x) = α1x1x2x3 + α2x1x3x5 + α3x21x5 + α4x

32 + · · ·

is homogeneous of degree δ = 3.

f(x1, x2) =

dx1c12 +dx2c

x1+x2if x1 + x2 6= 0

0 otherwise

12 + dλx2c

λx1 + λx2= λ−

12 f(λx1, λx2)

32 + · · ·

f(x1, x2) =

dx1c12 +dx2c

x1+x2if x1 + x2 6= 0

0 otherwise

12 + dλx2c

λx1 + λx2= λ−

12 f(λx1, λx2)

32 + · · ·

Classical Homogeneity for vector fields. (Zubov, Hahn,...)

Let n be a positive integer. A vector field f : Rn → Rn ishomogeneous with degree δ ∈ R iff ∀λ > 0 :

f(λx) = λδf(x).

Associated with the vector field f(x) is the DifferentialEquation x = f(x), and it has a flow (solution) ϕ(t, x).

Homogeneity of vector field ⇒ Homogeneity of Flow

If ∀λ > 0 :

f(λx) = λδf(x)⇒ ϕ(t, λx) = λϕ(λδ−1t, x)

Classical Homogeneity for vector fields. (Zubov, Hahn,...)

Let n be a positive integer. A vector field f : Rn → Rn ishomogeneous with degree δ ∈ R iff ∀λ > 0 :

f(λx) = λδf(x).

Associated with the vector field f(x) is the DifferentialEquation x = f(x), and it has a flow (solution) ϕ(t, x).

If ∀λ > 0 :

f(λx) = λδf(x)⇒ ϕ(t, λx) = λϕ(λδ−1t, x)

Some examples:

Linear system: Let A ∈ Rn×n then x = f(x) = Ax ishomogeneous of degree δ = 1 and the flow is ϕ(t, x) = eAtx

ϕ(t, λx) = eAt(λx) = λeAtx = λϕ(λ0t, x) .

If x is scalar. System x = −sign(x) is homogeneous withdegree δ = 0, since

f(λx) = sign(λx) = sign(x) = λ0f(x) .

The flow is

ϕ(t, x) =

sign(x)(| x | −t) if 0 ≤ t ≤| x |0 if t >| x |

and it is homogeneous

ϕ(t, λx) =

sign(λx)(| λx | −t) if 0 ≤ t ≤| λx |0 if t >| λx |

= λϕ(t

λ, x) .

Some examples:

Linear system: Let A ∈ Rn×n then x = f(x) = Ax ishomogeneous of degree δ = 1 and the flow is ϕ(t, x) = eAtx

ϕ(t, λx) = eAt(λx) = λeAtx = λϕ(λ0t, x) .

If x is scalar. System x = −sign(x) is homogeneous withdegree δ = 0, since

f(λx) = sign(λx) = sign(x) = λ0f(x) .

The flow is

ϕ(t, x) =

sign(x)(| x | −t) if 0 ≤ t ≤| x |0 if t >| x |

and it is homogeneous

ϕ(t, λx) =

sign(λx)(| λx | −t) if 0 ≤ t ≤| λx |0 if t >| λx |

= λϕ(t

λ, x) .

Outline

7 Preliminaries

Weighted Homogeneity or(quasi-homogeneity) (Zubov, Hermes)

A generalized weight is a vector r = (r1, · · · , rn), withri > 0.

A dilation is the action of the group R+ \ 0 on Rn givenby

Λr : R+ \ 0 × Rn → Rn

(λ, x) → diagλrix

we will denote this for simplicity as Λrx , Λr(λ, x), λ > 0.

Weighted Homogeneity for functions. (Zubov, Hermes...)

Let n, m be positive integers. A mapping f : Rn → Rm isr-homogeneous with degree δ ∈ R iff ∀λ > 0 :

f(Λrx) = λδf(x).

Λr : R+ \ 0 × Rn → Rn

f(Λrx) = λδf(x).

Λr : R+ \ 0 × Rn → Rn

f(Λrx) = λδf(x).

Some remarks

Function f(x1, x2) = x1 + x22 is not homogeneous, but it is

(2, 1)-homogeneous of degree δ = 2 since,

f(λx) = λx1 + (λx2)2 = λx1 + λ2x22 6= λδf(x), ∀λ 6= 1 .

f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .

Classical Homogeneity = Weighted Homogeneity withr = (r1, · · · , rn) = (1, · · · , 1).Values on the unit sphere define an r-Homogeneousfunction.If f(x) is r-homogeneous of degree δ then it is(αr)-homogeneous of degree (αδ) for any α > 0Euler’s Theorem: Let V : Rn → R be differentiable. V isr-homogeneous of degree δ if and only if

n∑i=1

rixi∂V

∂xi(x) = δV (x) .

Some remarks

f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .

n∑i=1

rixi∂V

∂xi(x) = δV (x) .

Some remarks

f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .

n∑i=1

rixi∂V

∂xi(x) = δV (x) .

Some remarks

f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .

n∑i=1

rixi∂V

∂xi(x) = δV (x) .

Some remarks

f(Λrx) = λ2x1 + (λx2)2 = λ2(x1 + x22) = λ2f(x) .

n∑i=1

rixi∂V

∂xi(x) = δV (x) .

Weighted Homogeneity

Weighted Homogeneity for (set-valued) vector fields.(Zubov, Hermes, Levant, ...)

Let n be a positive integer. A vector field f : Rn → Rn (aset-valued vector field f : Rn ⇒ Rn) is r-homogeneous withdegree δ ∈ R iff ∀λ > 0 :

f(Λrx) = λδΛrf(x).

Associated with the (set-valued) vector field f(x) is theDifferential Equation x = f(x) (DI x ∈ f(x)), and it has a flow(solution) ϕ(t, x).

r-Homogeneity of vector field ⇒ r-Homogeneity of Flow

If ∀λ > 0 :

f(Λrx) = λδΛrf(x)⇒ ϕ(t, Λrx) = Λrϕ(λδt, x)

Weighted Homogeneity

Weighted Homogeneity for (set-valued) vector fields.(Zubov, Hermes, Levant, ...)

Let n be a positive integer. A vector field f : Rn → Rn (aset-valued vector field f : Rn ⇒ Rn) is r-homogeneous withdegree δ ∈ R iff ∀λ > 0 :

f(Λrx) = λδΛrf(x).

Associated with the (set-valued) vector field f(x) is theDifferential Equation x = f(x) (DI x ∈ f(x)), and it has a flow(solution) ϕ(t, x).

r-Homogeneity of vector field ⇒ r-Homogeneity of Flow

If ∀λ > 0 :

f(Λrx) = λδΛrf(x)⇒ ϕ(t, Λrx) = Λrϕ(λδt, x)

Some examples:

Super-Twisting (ST) Algorithm:

x1 = −k1 dx1c12 + x2

x2 ∈ −k2 dx1c0 + [−1, 1] .

is (2, 1)-homogeneous of degree −1, since

⌈λ2x1

⌋ 12 + λx2 = λ2−1(−k1 dx1c

12 + x2)

⌈λ2x1

⌋0+ [−1, 1] = λ1−1(−k2 dx1c0 + [−1, 1]) .

Twisting Algorithm:

x1 = x2

x2 ∈ −k1 dx1c0 − k2 dx2c0 + [−1, 1] .

is (2, 1)-homogeneous of degree −1, since

λx2 = λ2−1(x2)

− k1

⌈λ2x1

⌋0 − k2 dλx2c0 + [−1, 1] =

λ1−1(−k1 dx1c0 − k2 dx2c0 + [−1, 1]) .

Two important examples:

Levant’s arbitrary order differentiator:

xi = −ki dx1 − f(t)cn−in + xi+1

xn ∈ −kn dx1 − f(t)c0 .

is (n, n− 1, · · · , 1)-homogeneous of degree −1.

Homogeneous Controller of a chain of integrators:

x1 = x2

xn = −n∑i=1

ki dxicαi

is r-homogeneous of degree δ ∈ [−1, 0] with

ri = 1 + (i− n)δ, αi =1 + δ

1 + (i− n)δ.

Two important examples:

Levant’s arbitrary order differentiator:

xi = −ki dx1 − f(t)cn−in + xi+1

xn ∈ −kn dx1 − f(t)c0 .

is (n, n− 1, · · · , 1)-homogeneous of degree −1.

Homogeneous Controller of a chain of integrators:

x1 = x2

xn = −n∑i=1

ki dxicαi

is r-homogeneous of degree δ ∈ [−1, 0] with

ri = 1 + (i− n)δ, αi =1 + δ

1 + (i− n)δ.

Dynamic interpretation of r-homogeneity

System x = f(x) is r-homogeneous of degree δ, i.e.f(Λrx) = λδΛrf(x).

State Transformation z = Λrx

z = Λrx = Λrf(x) = λ−δf(Λrx)

and thereforedz

d(λδt)= f(z)

System x = f(x) is invariant under the transformation

Gλ : (t, x) 7→ (λ−δt, Λrx) .

and thereforedz

d(λδt)= f(z)

Gλ : (t, x) 7→ (λ−δt, Λrx) .

and thereforedz

d(λδt)= f(z)

Gλ : (t, x) 7→ (λ−δt, Λrx) .

Properties of Homogeneous Systems

Zubov, Hahn, Hermes, Kawski, Rosier, Aeyels, Sepulchre,Grune, Praly, Perruquetti, Efimov, Polyakov,..... Levant,Orlov, Bernuau et al. 2013

If x = 0 Locally Attractive (LA) ⇔ GloballyAsymptotically Stable (GAS)

local contraction ⇒ global contraction ⇒ globalasymptotic stability

If x = 0 GAS and δ < 0 ⇔ x = 0 Finite Time Stable

If x = 0 GAS and δ = 0 ⇔ x = 0 Exponentially Stable (e.g.LTI systems)

If x = 0 GAS and δ > 0 ⇔ x = 0 Asymptotically Stable

If x = 0 GAS ⇔ It exists a Homogeneous LyapunovFunction

Outline

7 Preliminaries

Weighted Homogeneity with inputs

Consider a System with inputs u ∈ Rm

x = f(x, u) .

State and input weight vectors r = (r1, · · · , rn), ri > 0,ρ = (ρ1, · · · , ρm), ρi > 0

State and input dilations Λr and Λρ.

Weighted Homogeneity for (set-valued) vector fields withinputs.

A vector field f : Rn × Rm → Rn (a set-valued vector fieldf : Rn × Rm ⇒ Rn) is (r, ρ)-homogeneous with degree δ ∈ R iff∀λ > 0 :

f(Λrx, Λρu) = λδΛrf(x, u).

x = f(x, u) .

Associated with the (set-valued) vector field f(x, u) is theDifferential Equation x = f(x, u) (DI x ∈ f(x, u)), and it has aflow (solution) ϕ(t, x, u).

If f is homogeneous then ∀λ > 0 :

ϕ(t, Λrx, Λρu(λδ·)) = Λrϕ(λδt, x, u(·))

Dynamic interpretation of homogeneity

System x = f(x, u) is homogeneous of degree δ, i.e.f(Λrx, Λρu) = λδΛrf(x, u).

State and input Transformation z = Λrx, w = Λρu

z = Λrx = Λrf(x, u) = λ−δf(Λrx, Λρu)

and thereforedz

d(λδt)= f(z, w)

Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .

Internal stability ⇒ external stability (iISS, ISS) [Bernuauet al. 2013]

and thereforedz

d(λδt)= f(z, w)

Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .

and thereforedz

d(λδt)= f(z, w)

Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .

and thereforedz

d(λδt)= f(z, w)

Gλ : (t, x, u) 7→ (λ−δt, Λrx, Λρu) .

Outline

7 Preliminaries

Weighted Homogeneity and Precision

Consider a System with a scalar and constant input u ∈ R

x = f(x, u) .

so thatϕ(t, Λrx, λ

ρu) = Λrϕ(λδt, x, u) .

Suppose that asymptotically or after a finite time for some u0

limt→∞|ϕi(t, x, u0)| = |ϕ∞i(u0)| ≤ ai .

Therefore (using λ = ( uu0)

|ϕ∞i(u)| = |ϕ∞i(λρu0)| = λri |ϕ∞i(u0)| ≤ νiuriρ ,

with νi = aiu− riρ

Outline

7 Preliminaries

Homogeneous Approximation

First Lyapunov’s Theorem: x = 0 is LAS for x = f(x) if

x = 0 is AS for linearized system x = Ax, where A = ∂f(0)∂x .

This is not true for Taylor Approximations. Example fromBacciotti & Rosier, 2005:

x1 = −x31 + x3

x2 = −x1 + x52 .

x = 0 is GAS for approximation of order 3 (black), but it isnot AS with perturbation (red) of higher order 5!But true for homogeneous approximations (Not unique!)

x1 = −x31 + x2 ,

x2 = −x51 + x2

x = 0 is GAS for homogeneous approximationr1 = 1, r2 = 3, δ = 2 (black), and it is still AS withperturbation (red), which is homogeneous of degree 3.

Homogeneous Domination

Typical control (similar for observation) problem:

x1 = x2+f1(x1) ,

x2 = x3+f2(x1, x2) ,

xn = u+ fn(x) .

Homogeneous Approximation (Black):

x1 = x2 ,...

xn = u = φ(x) .

It is homogeneous of degree δ = −1, 0, +1 and weightsr = (r1 + δ, r1 + 2δ, · · · , r1 + nδ): NOT UNIQUE! ⇒ Fixed byselection of the control law φ(x) ⇒ Domination of other termsfi.

Outline

7 Preliminaries

Alternative Integral + state feedback controllers for System

x1 = x2

x2 = u+ ρ (t) ,

Linear Integral + state feedback controller (Homogeneous)

u = −k1x1 − k2x2 + x3

x3 = −k3x1

Linear state feedback + Discontinuous Integral controller(Not Homogeneous)

u = −k1x1 − k2x2 + x3

x3 = −k3sign(x1)

Discontinuous I-Controller (Extended Super-Twisting)(Homogeneous)

u = −k1|x1|13 sign(x1)− k2|x2|

12 sign(x2) + x3

x3 = −k3sign(x1)

Controller without perturbation

0 10 20 30−10

Linear Integral controller

0 10 20 30−10

Linear+Discontinuous Integrator

0 10 20 30−7

Super−Twisting controller

Controller with perturbation

0 5 10 15 20 25 30−10

0 5 10 15 20 25 30−7

Overview

7 Preliminaries

Homogeneous Design of HOSM [?]

∑DI :

u = ϑr(x1, x2, · · · , xρ) ,

ϑr homogeneous of degree 0 (discontinuous at x = 0) withweights rs = (ρ, ρ− 1, ..., 1)

ϑr(ερx1, ε

ρ−1x2, . . . , εxρ)

= ϑr (x1, x2, . . . , xρ) ∀ε > 0

Local boundedness ⇒ global boundedness

System∑

DI is homogeneous of degree −1 with weights rs

Local contractive ⇔ Global, uniformly Finite-Time stabilityRobustness of stability ⇒ Accuracy with respect tohomogeneous perturbations |xi| ≤ γiτρ+1−i = O

(τρ+1−i),

γi constants.

Concrete Homogeneous HOSM Controllers

Notation: dxcp = |x|psign (x).

Nested Sliding Controllers (NSC)

u2L = −k2

⌈x2 + β1 dx1c

u3L = −k3

⌈x3 + β2(|x2|3 + |x1|2)

⌈x2 + β1 dx1c

⌋0⌋0

Quasi-Continuous Sliding Controllers (QCSC)

u2C = −k2(x2+β1dx1c1/2)|x2|+β1|x1|1/2

u3C = −k3x3+β2(|x2|+β1|x1|3/2)

−1/2(x2+β1dx1c3/2)

|x3|+β2(|x2|+β1|x1|3/2)1/2 .

Scaling the gains: again homogeneity!

Example: 2nd Order Nested Controller

x1 = x2

x2 = −k2

⌈x2 + β1 dx1c

Linear change of coordinates: z = λ2x, 0 < λ ∈ R

z1 = λ2x2 = z2

z2 = −λ2k2

λ2+ β1

⌈ z1

⌋ 12

= −λ2k2

λ+ β1 dz1c

Preserves stability!

Example: 3rd Order Nested Controller

x1 = x2, x2 = x3,

x3 = −k3

⌈x3 + β2(|x2|3 + |x1|2)

⌈x2 + β1 dx1c

⌋0⌋0

z = λ3x, 0 < λ ∈ R⇒ Preserves stability!

z1 = z2, z2 = z3,

z3 = −λ3k3

λ3+ β2

(∣∣∣ z2

∣∣∣3 +∣∣∣ z1

∣∣∣2) 16⌈z2

λ3+ β1

⌈ z1

⌋ 23

⌋0⌋0

= −λ3k3

λ2+ β2

(∣∣∣z2

∣∣∣3 + |z1|2) 1

6 ⌈z2

λ+ β1 dz1c

⌋0⌋0

Balance

Advantages:

Beautiful and powerful theory: more qualitative thanquantitative

Simple: local = global, convergence = finite-time = robust

scaling the gains (for nested controllers) for convergenceacceleration

u = λρϑr

λ, . . . ,

xρλρ−1

), λ > 1

Balance....

Limitations:

Beyond homogeneity unclear how to design ϑr!

It does not provide quantitative results, e.g.

Stabilizing GainsConvergence Time estimationAccuracy GainsPerformance quantities

Behavior with respect to non homogeneous perturbations

Behavior under interconnection

Design for performance

Due to Limitations we require other methods, e.g. Lyapunov(but not exclusively)

Overview

7 Preliminaries

Motivation

Lyapunov based methods

Standard methods in nonlinear control theory

Design for robustness, optimality, performance, etc.

Gain tuning methods

Lyapunov Function for Internal Stability and Lyapunov-likeFunctions for External Stability (e.g. ISS, iISS,...)

LF for design: Control Lyapunov Functions (CLF)

Interconnection analysis is possible

Robustness analysis: noise, uncertainties, perturbations

Objective:

A Lyapunov Based framework for HOSM

Belief: Combination of Homogeneity and LF ⇒ powerful tool!

Main Problem: Construction of LF

Existence of LF for f(x) Filippov Differential Inclusion

It exists a smooth p.d. V (x) [Clarke et al. 1998]

f(x) homogeneous ⇒ V (x) homogeneous [Nakamura et al.2002, Bernuau et al. 2014]

There are basically two issues:

What is the form or structure of the LF?

How to decide if V (x) and W (x) are positive definite(p.d.)?

There are many works on these general topics. But there arefew for HOSM algorithms and taking advantage of thehomogeneity properties.

State of the art

Orlov, 2005 Weak LF for Twisting Algorithm. MechanicalEnergy.

Moreno and Osorio, 2008 Strong non smooth LF forSuper–Twisting. No method.

Polyakov and Poznyak, 2009 Strong non smooth LF forTwisting and ST. Zubov’s Method.

Santiesteban et al., 2010 Strong LF for Twisting with linearterms. No method.

Polyakov and Poznyak, 2012 Strong LF for Twisting, Terminaland Suboptimal. Zubov’s Method.

Sanchez and Moreno, 2012 Strong non smooth LF for Twisting,Terminal and a sign controller. TrajectoryIntegration method.

Some attempts (in our group)

First steps: the quadratic form approach

Young’s inequality and extensions

Trajectory integration

Reduction method

Generalized Forms approach

Homogenous Control Lyapunov Functions

I will not talk about Lille’s Group Implicit Lyapunov Functions(ILF) approach (Polyakov, Efimov, Perruquetti,...)!

Part III

Lyapunov-Based Design of

Higher-Order Sliding Mode (HOSM)

Controllers

Outline

12 Control Lyapunov Functions and Families of HOSM CLFs

13 HOSM Controllers

14 HOSM Controllers: Some Examples

15 Gain Calculation

16 Simulation Example

Robust Stabilization Problem (E. Cruz)

Perturbed nonlinear system

x ∈ F (x) + g (x) ξ (x)u ,

x ∈ Rn, u ∈ Rg(x) known vector field

F (x) set-vector field, ξ multivalued ⇒ Uncertainties.

Assumptions

0 < Km ≤ ξ (x) ≤ KM

F, g are r-homogeneous of degree l.

Overview

13 HOSM Controllers

15 Gain Calculation

Control Lyapunov Function (CLF)

(Homogeneous) CLF

V (x) ∈ C1, p.d., r-homogeneous of degree m > −l

∂V (x)

∂xg(x) = 0⇒ sup

v∈F (x)

LvV < 0, ∀x ∈ Rn \ 0 .

r-homogeneous of degree 0 (discontinuous) Controllers

u = −kϕ1 (x) = −k⌈Lg(x)V (x)

u = −kϕ2 (x) = −kLg(x)V (x)

‖x‖l+mr,p

If k ≥ k∗, x = 0 is GAS, and if l < 0 it is Finite-Time Stable.

‖x‖r,p =(|x1|

pr1 + ·+ |xn|

, p ≥ max ri, homogeneous norm .

HOSM design based on CLF

The Basic Uncertain System∑DI :

The Design is reduced to find a CLF.

By a Back-Stepping-like procedure construct a CLF

Define

xTi = [x1, · · · , xi] ,

r = (r1, r2, ..., rρ) = (ρ, ρ− 1, ..., 1) ,

αρ ≥ αρ−1 ≥ · · · ≥ α1 ≥ ρ ,

m ≥ ri + αi−1 .

Family of CLFs

CLF r-homogeneous of degree m: ∀γi > 0, ∃ki > 0

V (x) = γρWρ (xρ) + · · ·+ γiWi (xi) + · · ·+ γ1ρ

Wi (xi) =rim|xi|

mri − dνi−1c

m−riri xi +

(1− ri

)|νi−1|

νi (xi) = −ki dσicri+1αi , σ1 = dx1c

σi(xi) = dxicαiri − dνi−1c

αiri = dxic

αiri + k

αirii−1 dσi−1c

αiαi−1 ,

V (x) is a continuously differentiable and r-homogeneous CLF ofdegree m.

Overview

13 HOSM Controllers

15 Gain Calculation

HOSM Controllers

HOSM Discontinuous Controller

Discontinuity at σρ(x) = 0

uD (x) = −kρ dσρ (x)c0 , kρ 0 ,

HOSM Quasi-Continuous Controller

Discontinuity only at x = 0

uQ (x) = −kρσρ (x)

M(x), kρ 0 ,

M (x) is any continuous r-homogeneous positive definitefunction of degree αρ.

ρ-th order sliding mode x = 0 is established in Finite-Time.

Convergence Time Estimation

T (x0) ≤ mτρV1mρ (x0) ,

where τρ is a function of the gains (k1, ..., kρ), Km and C.

Variable-Gain HOSM Controller

If C = C + Θ (t, z), with constant C and time-varyingΘ (t, z) ≥ 0 known.

Variable-Gain Controller

The Discontinuous Variable-Gain HOSM Controller

uD (x) = − (K (t, z) + kρ) dσρ (x)c0 , kρ 0 ,

and the Quasi-Continuous Variable-Gain HOSM Controller

uQ (x) = − (K (t, z) + kρ)σρ (x)

M(x), kρ 0 ,

stabilize the origin x = 0 in Finite-Time if kρ 0 andKmK (t, z) ≥ Θ (t, z).

Overview

13 HOSM Controllers

15 Gain Calculation

Discontinuous Nested HOSM Controllers

For αρ ≥ · · · ≥ α1 ≥ ρ, the discontinuous Controllers of ordersρ = 2, 3, 4 are given by

u2D = −k2

⌈dx2cα2 + kα2

1 dx1cα22

u3D = −k3

⌈dx3cα3 + kα3

⌈dx2c

α22 + k

1 dx1cα23

⌋α3α2

u4D = −k4

dx4cα4+kα43

dx3cα32 +k

⌈dx2c

α23 +k

1 dx1cα24

⌋α3α2

α4α3

and are, in general, of the type of the Nested Sliding Controllers(Levant 2005).

Discontinuous Relay Polynomial HOSMControllers

Cruz-Zavala & Moreno (2014, 2016) [?, ?, ?]; Ding, Levant & Li(2015,2016) [?, ?].For αρ = αρ−1 = · · · = α1 = α ≥ ρ discontinuous ”relaypolynomial” controllers

u2R = −k2sign(dx2cα + k1 dx1c

u3R = −k3sign(dx3cα + k2 dx2c

α2 + k1 dx1c

)u4R = −k4sign

(dx4cα + k3 dx3c

α2 + k2 dx2c

α3 + k1 dx1c

)where for ρ = 2, k1 = kα1 ; for ρ = 3, k1 = kα2 k

α21 , k2 = kα2 ; and

for general ρ, ki =∏ρ−1j=i k

αρ−jj .

Quasi-Continuous Nested or RelayPolynomial HOSM Controllers

For arbitrary βi > 0

u2Q = −k2dx2cα2 + kα2

1 dx1cα22

|x2|α2 + β1 |x1|α22

u3Q = −k3

dx3cα3 + kα32

⌈dx2c

α22 + k

1 dx1cα23

⌋α3α2

|x3|α3 + β2 |x2|α32 + β1 |x1|

u4Q = −k4

dx4cα4+kα43

dx3cα32 +k

⌈dx2c

α23 +k

1 dx1cα24

⌋α3α2

α4α3

|x4|α4 + β3 |x3|α42 + β2 |x2|

α43 + β1 |x1|

Overview

13 HOSM Controllers

15 Gain Calculation

Numerical Gain Calculation

ki, for i = 1, · · · , ρ− 1, selected to render V (x) a CLF,

kρ to obtain V < 0.

Fix m, ρ, αi and γi. Set k1 > 0, for i = 2, · · · , ρ

ki > maxxi∈Si

Φi (xi) =: Gi (k1, · · · , ki−1) , (2)

Maximization feasible since1 Φ is r-homogeneous of degree 0:achieves all its values on the

homogeneous unit sphere Si = xi ∈ Ri : ‖xi‖r,p = 1, and2 Φ is upper-semicontinuous ⇒ it has a maximum on Si.

Parametrization

ki = µikρ

ρ−(i−1)

1 , kρ >1Km

(µρkρ1 + C) , (3)

for some positive constants µi independent of k1.

Parametrization can be used for all controllers, but kρdifferent for discontinuous and quasi-continuous controllers.

Analytical Gain calculation

It is possible (but cumbersome) to provide for any order ananalytical estimation of the values of the gains using classicalinequalities.

The simplest case with ρ = 3, u3D, u3R and u3Q

For any values of α3 ≥ α2 ≥ r1 = 3, m ≥ r2 + α2, γ1 > 0,0 < η < 1 and k1 > 0,

k2 >r22

m−2r2α2

m− 1

(m− r1)2α2−r2α2

m− 1

m−r1r2

(γ1 + m−r2

r1kmr21

)m−1r2

(ηγ1k1)m−r1r2

Remarks

By homogeneity the gain scaling with any L ≥ 1

kT = (k1, · · · , kρ)→ kTL = (L1ρk1, · · · , L

1ρ+1−iki, · · · , Lkρ)

preserves the stability for any αj .

Convergence will be accelerated for L > 1, or the size ofthe allowable perturbation C will be incremented to LC.

The gains obtained by means of the LF can be very largefor practical applications, so that a simulation-based gaindesign is eventually necessary (see [?]).

The gain design problem is an important and unexploredone.

Overview

13 HOSM Controllers

15 Gain Calculation

Example

Kinematic model of a car [?]

z1 = v cos (z3) , z2 = v sin (z3) , z3 =(vL

)tan (z4) , z4 = u,

z1, z2: cartesian coordinates of the rear-axle middle point,

z3: the orientation angle,

z4: the steering angle, (Actual control)

v: the longitudinal velocity (v = 10 m.s−1),

L: distance between the two axles (L = 5 m), and

u: the control input.

u is used as a new control input in order to avoid discontinuitieson z4.

Control Objective

Control Task: steer the car from a given initial position tothe trajectory z2ref = 10 sin (0.05z1) + 5 in finite time.

Turn on the controllers after Observer converged (0.5 sec.).

Sliding variable σ = z2 − z2ref, relative degree ρ = 3, Model

x1 = x2, x2 = x3, x3 = φ(·) + γ(·)u,

where x =[σ σ σ

Simulations: Euler’s method, sampling time τ = 0.0005.

Bounds: |φ| ≤ C0 = 49.63, Km = 6.38 ≤ γ ≤ KM = 46.77.

Controllers: (i) Levant’s Discontinuous Controller (L3)with β1 = 1, β2 = 2 and k3 = 20; (ii) Levant’sQuasi-Continuous Controller (Q3) with β1 = 1, β2 = 2 andk3 = 24.5; and (iii) Proposed Discontinuous Controller(E3) u3D = −k3dσ3c0 with k1 = 1, k2 = 1.5, k3 = 20.

Simulations

0 10 20 30−5

Time [s]

0 10 20 30−5

Time [s]

0 10 20 30−5

Time [s]

0 10 20 30

Time [s]

0 10 20 30

Time [s]x

0 10 20 30

Time [s]

0 10 20 30−0.4

−0.2

Time [s]

0 10 20 30−0.4

−0.2

Time [s]

0 10 20 30−0.4

−0.2

Time [s]

Figure : Left column: Levant’s Discontinuous Controller (L3), MiddleColumn: Levant’s Quasi-Continuous Controller (Q3); Right Column:Proposed Discontinuous Controller (E3).

Simulations

9 11 13 15−0.35

−0.25

−0.15

−0.05

Time [s]

Accura

17 21 25 29 33 35−0.35

−0.25

−0.15

−0.05

0.35(b)

Time [s]

Accura

9 11 13 15−0.35

−0.25

−0.15

−0.05

0.35(c)

Time [s]

Accura

10 20 30−0.35

−0.25

−0.15

−0.05

0.35(d)

Time [s]A

Figure : Accuracy: (a) with (L3); (b) with (Q3); (c) with (E3); (d)with (Q3) (k3 = 70).

Advantage of (E3): combines fast convergence rate of (L3)with smooth transient response of (Q3).

Resume

New: Methodological approach to design HOSM controllersusing CLF.

Different alternatives to find CLFs: Back-stepping,Polynomial methods, ...

It can be extended to design controllers with Fixed-Timeconvergence.

Drawback: Calculation of gains ki needs maximization of0-degree homogeneous functions.

Part IV

HOSM Differentiation/Observation: A

Lyapunov Approach

Outline

17 Basic Observation Problem

18 Super-Twisting Observer

19 Generalized Super-Twisting Observers

20 Lyapunov Approach for Second-Order Sliding ModesStability AnalysisGSTA with perturbations: ARI

21 Example: Reaction rate estimation in Bioreactors

22 The arbitrary order HOSM DifferentiatorDifferentiation ProblemSome known DifferentiatorsA family of Homogeneous DifferentiatorsExamples

23 Conclusions

Overview

23 Conclusions

Basic Observation Problem

Variations of the observation problem: with unknown inputs,practical observers, robust observers, stochastic framework todeal with noises, ....

An important Property: Observability

Consider a nonlinear system without inputs, x ∈ Rn, y ∈ R

x (t) = f (x (t)) , x (t0) = x0

y (t) = h (x (t))

Differentiating the output

y (t) = h (x (t))

y (t) =d

dth (x (t)) =

∂h (x)

∂xx (t) =

∂h (x)

∂xf (x) := Lfh (x)

y (t) =∂Lfh (x)

∂xx (t) =

∂Lfh (x)

∂xf (x) := L2

fh (x)

y(n−1) (t) =∂Ln−2

f h (x)

∂xx (t) =

∂Ln−2f h (x)

∂xf (x) := Ln−1

f h (x)

where Lkfh (x) are Lie’s derivatives of h along f .

Evaluating at t = 0y (0)y (0)y (0)...

y(k) (0)

h (x0)Lfh (x0)L2fh (x0)

...Lkfh (x0)

:= On (x0)

On (x): Observability map

Theorem

If On (x) is injective (invertible) → The NL system isobservable.

Observability Form

In the coordinates of the output and its derivatives

z = On (x) , x = O−1n (z)

the system takes the (observability) form

z1 = z2

z2 = z3...

zn = φ (z1, z2, . . . , zn)y = z1

So we can consider a system in this form as a basic structure.

A Simple Observer and its Properties

Plant: x1 = x2 , x2 = w(t)Observer: ˙x1 = −l1 (x1 − x1) + x2 , ˙x2 = −l2 (x1 − x1)Estimation Error: e1 = x1 − x1, e2 = x2 − x2

e1 = −l1e1 + e2 , e2 = −l2e1 − w (t)

Figure : Linear Plant with an unknown input and a Linear Observer.

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer Linear Observer

Figure : Behavior of Plant and the Linear Observer without unknowninput.

0 20 40 600

Time (sec)S

0 20 40 600.5

Time (sec)

0 20 40 60−2

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Figure : Behavior of Plant and the Linear Observer with unknowninput.

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Figure : Behavior of Plant and the Linear Observer without UI withlarge initial conditions.

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Figure : Behavior of Plant and the Linear Observer without UI withvery large initial conditions.

Recapitulation.

Linear Observer for Linear Plant

If no unknown inputs/Uncertainties: it convergesexponentially fast.

If there are unknown inputs/Uncertainties: no convergence.At best bounded error.

Convergence time depends on the initial conditions of theobserver

Is it possible to alleviate these drawbacks?

Recapitulation.

Sliding Mode Observer (SMO)

Figure : Linear Plant with an unknown input and a SM Observer.Homogeneity Based SMC Jaime A. Moreno UNAM 158

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear ObserverLinear Observer

Nonlinear Observer

Figure : Behavior of Plant and the SM Observer without unknowninput.

0 20 40 600

Time (sec)

10 20 40 60

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear ObserverLinear Observer

Nonlinear Observer

Figure : Behavior of Plant and the SM Observer with unknown input.

Recapitulation.

Sliding Mode Observer for Linear Plant

If no unknown inputs/Uncertainties: e1 converges in finitetime, and e2 converges exponentially fast.

If there are unknown inputs/Uncertainties: no convergence.At best bounded error. Only e1 converges in finite time!

It is not the solution we expected! None of the objectiveshas been achieved!

Recapitulation.

Overview

23 Conclusions

Super-Twisting Algorithm (STA)

Plant:x1 = x2 ,x2 = w(t)

Observer:˙x1 = −l1 |e1|

12 sign (e1) + x2 ,

˙x2 = −l2 sign (e1)

Estimation Error: e1 = x1 − x1, e2 = x2 − x2

e1 = −l1 |e1|12 sign (e1) + e2

e2 = −l2 sign (e1)− w (t) ,

Solutions in the sense of Filippov.

Figure : Linear Plant with an unknown input and a SOSM Observer.

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withoutunknown input.

0 20 40 600

Time (sec)S

0 20 40 600.5

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withunknown input.

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer without UIwith large initial conditions.

Recapitulation.

Super-Twisting Observer for Linear Plant

If no unknown inputs/Uncertainties: e1 and e2 converge infinite-time!

If there are unknown inputs/Uncertainties: e1 and e2

converge in finite-time! Observer is insensitive toperturbation/uncertainty!

Convergence time depends on the initial conditions of theobserver. This objective is not achieved!

Recapitulation.

Overview

23 Conclusions

Generalized Super-Twisting Algorithm(GSTA)

Plant:x1 = x2 ,x2 = w(t)

Observer:˙x1 = −l1φ1 (e1) + x2 ,˙x2 = −l2φ2 (e1)

Estimation Error: e1 = x1 − x1, e2 = x2 − x2

e1 = −l1φ1 (e1) + e2

e2 = −l2φ2 (e1)− w (t) ,

φ1 (e1) = µ1 |e1|12 sign (e1) + µ2 |e1|

32 sign (e1) , µ1 , µ2 ≥ 0 ,

φ2 (e1) =µ2

2sign (e1) + 2µ1µ2e1 +

2 |e1|2 sign (e1) ,Homogeneity Based SMC Jaime A. Moreno UNAM 170

Figure : Linear Plant with an unknown input and a Non LinearObserver.

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withoutunknown input and large initial conditions.

0 20 40 600

Time (sec)S

0 20 40 600

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer withoutunknown input and very large initial conditions.

0 20 40 600

Time (sec)S

0 20 40 600.5

Time (sec)

0 20 40 60−1

Time (sec)

0 20 40 60−1.5

−0.5

Time (sec)

Linear Observer

Nonlinear Observer

Linear Observer

Nonlinear Observer

Figure : Behavior of Plant and the Non Linear Observer with UI withlarge initial conditions.

Effect: Convergence time independent ofI.C.

norm of the initial condition ||x(0)|| (logaritmic scale)

NSOSMO

GSTA with linear term

Figure : Convergence time when the initial condition grows.

Recapitulation.

Generalized Super-Twisting Observer for Linear Plant

Convergence time is independent of the initial conditions ofthe observer!.

All objectives were achieved!

How to proof these properties?

Recapitulation.

What have we achieved?

An algorithm

Robust: it converges despite of unknowninputs/uncertainties

Exact: it converges in finite-time

The convergence time can be preassigned for any arbitraryinitial condition.

But there is no free lunch!

It is useful for

Observation

Estimation of perturbations/uncertainties

Control: Nonlinear PI-Control

in practice?

Some Generalizations are available but Still a lot is missing

Overview

23 Conclusions

Lyapunov functions:

1 We propose a Family of strong Lyapunov functions, thatare Quadratic-like

2 This family allows the estimation of convergence time,

3 It allows to study the robustness of the algorithm todifferent kinds of perturbations,

4 All results are obtained in a Linear-Like framework, knownfrom classical control,

5 The analysis can be obtained in the same manner for alinear algorithm, the classical ST algorithm and acombination of both algorithms (GSTA), that is nonhomogeneous.

Generalized STA

x1 = −k1φ1 (x1) + x2

x2 = −k2φ2 (x1) ,(4)

φ1 (e1) = µ1 |e1|12 sign (e1) + µ2 |e1|q sign (e1) , µ1 , µ2 ≥ 0 , q ≥ 1 ,

φ2 (e1) =µ2

2sign (e1) +

)µ1µ2 |e1|q−

12 sign (e1) +

+ qµ22 |e1|2q−1 sign (e1) ,

Standard STA: µ1 = 1, µ2 = 0

Linear Algorithm: µ1 = 0, µ2 > 0, q = 1.

GSTA: µ1 = 1, µ2 > 0, q > 1.

Outline

23 Conclusions

Quadratic-like Lyapunov Functions

System can be written as:

ζ = φ′1 (x1)Aζ , ζ =

[φ1 (x1)x2

], A =

[−k1 1−k2 0

Family of strong Lyapunov Functions:

V (x) = ζTPζ , P = P T > 0 .

Time derivative of Lyapunov Function:

V (x) = φ′1 (x1) ζT(ATP + PA

)ζ = −φ′1 (x1) ζTQζ

Algebraic Lyapunov Equation (ALE):

ATP + PA = −Q

Figure : The Lyapunov function.

Lyapunov Function

Proposition

If A Hurwitz then x = 0 Finite-Time stable (if µ1 = 1) andfor every Q = QT > 0, V (x) = ζTPζ is a global, strongLyapunov function, with P = P T > 0 solution of the ALE,and

V ≤ −γ1 (Q,µ1)V12 (x)− γ2 (Q,µ2)V (x) ,

γ1 (Q,µ1) , µ1λmin Qλ

12minP

2λmax P, γ2 (Q,µ2) , µ2

λmin Qλmax P

If A is not Hurwitz then x = 0 unstable.

Convergence Time

Proposition

If k1 > 0 , k2 > 0, and µ2 ≥ 0 a trajectory of the GSTA startingat x0 ∈ R2 converges to the origin in finite time if µ1 = 1, andit reaches that point at most after a time

γ1(Q,µ1)V12 (x0) if µ2 = 0

2γ2(Q,µ2) ln

(γ2(Q,µ2)γ1(Q,µ1)V

12 (x0) + 1

)if µ2 > 0

When µ1 = 0 the convergence is exponential.

For Design: T depends on the gains!

Outline

23 Conclusions

GSTA with perturbations: ARI

GSTA with time-varying and/or nonlinear perturbations

x1 = −k1φ1 (x1) + x2

x2 = −k2φ2 (x1) + ρ (t, x) .

Assume2 |ρ (t, x)| ≤ δ

Analysis: The construction of Robust Lyapunov Functions canbe done with the classical method of solving an AlgebraicRicatti Inequality (ARI), or equivalently, solving the LMI[

ATP + PA+ εP + δ2CTC PBBTP −1

]≤ 0 ,

[−k1 1−k2 0

], C =

], B =

Overview

23 Conclusions

General Model of a Bioreactor

ξ = Kϕ(ξ, t)−Dξ −Q(ξ) + F

Notationξ ∈ Rn State vector,concentrations

ϕ ∈ Rq Vector of reaction rates

K ∈ Rn×q Matrix of yield coefficients

D Dilution rate, D = FV

F Vector of supply rates

Q Removal rates components

Main Issue: ϕ(ξ, t) uncertain

Observers and estimators

Main approaches (state-of-the-art)

Detailed reaction rates ϕ

Extended Kalman Observer

High Gain Observer

Extended LuenbergerObserver

Linear Observers

Partial or no knowledge ofreaction rates

Interval observers

Dissipative approach

Asymptotic Observer

High Gain Observer

GSTO(generalized super-twisting observer)

Objective

Reaction rate estimation using GSTO´s

Generalized STA

x1 = −k1φ1 (x1) + x2 + ρ1 (t, x)x2 = −k2φ2 (x1) + ρ2 (t, x) ,

φ1 (x1) = m1|x1|12 sign (x1) +m2 |x1|q sign (x1) , m1 , m2 ≥ 0 , q >

φ2 (x1) =m2

2sign (x1) +

2q + 1

2m1m2 |x1|

2q−12 sign (x1) +

+m22q |x1|2q−1 sign (x1) ,

Standard STA: m1 = 1, m2 = 0

Linear Algorithm: m1 = 0, m2 > 0, q = 1In particular High Gain Observer (HGO)

GSTA: m1 = 1, m2 > 0, q > 12

Generalized STA

φ2 (x1) =m2

2sign (x1) +

2q + 1

2m1m2 |x1|

2q−12 sign (x1) +

+m22q |x1|2q−1 sign (x1) ,

GSTA: m1 = 1, m2 > 0, q > 12

Generalized STA

φ2 (x1) =m2

2sign (x1) +

2q + 1

2m1m2 |x1|

2q−12 sign (x1) +

+m22q |x1|2q−1 sign (x1) ,

GSTA: m1 = 1, m2 > 0, q > 12

Basic Idea: A Simple Microbial growth

Considering the simple bioprocess model1

X = µX −DXS = −kµX +D(Sin − S)

X and S are concentrations of biomass and substrate, µ is thespecific growth rate, k yield coefficient, D dilution rate (controlinput), Sin substrate concentration in the input. For simulationpurposes

µ =µmaxS

ks + S

where µmax is the maximal value in the specific growth rate, ksis the saturation constant.

1M. Farza, M. Nadri, H. Hammouri, Nonlinear observation of specificgrowth rate in aerobic fermentation process

Specific growth rate: GSTO

Considering that ϕ = µX, then the reduced model

X = ϕ−DXϕ = δ(t)

where ϕ is the reaction rate. The objective is estimate µ, formake this consider

˙X = ϕ−DX − l1φ1(X)˙ϕ = −l2φ2(X)

Ω1 (8)

Thus, an estimation of µ is given by

µΩ1 =ϕ

Specific growth rate: finite time parameterestimator

In this case, the model is

X = µX −DXµ = δµ(t)

But, in this case we apply an estimator to reconstruct thevariant time parameter

Super-Twisting VariableEstimator (STVE)

˙X = µX −DX − l1φ1(X)

˙µ = −Xl2φ2(X)

Avoid the singularity of

µΩ1 =ϕ

Simulation

Initial ConditionsX(0) = 1.41(g/l)S(0) = 2.17(g/l)Parametersk = 2, ks = 5.0(g/l)µmax = 0.33(h−1), Sin = 5(g/l)Dilution rate D

Observers Ω1,Ω2Initial conditions and parametersX(0) = 1.41(g/L)µ(0) = 0.15(h−1)S(0) = 4(g/L)

q = 1, l1 = 2, l2 = 1Ω1 m1 = 0.2,m2 = 1Ω2 m1 = 0.3,m2 = 1

Specific growth rate, estimation with the proposed observers

Observer Ω2 and HGO

˙X = µX1 −DX − 2θ(X)˙µ = − θ2

X(X), θ = 3

Despite noise in X

Enhanced velocity and noiserejection in the Ω2

Non measurable state S

Recall, the original model

X = µX −DXS = −kµX +D(Sin − S)

An AO is designed, definingZ = kX + S, such that

˙Z = −DZ +DSinS = Z − kX algebraic

and the error is given by

˙Z = −D(Z − Z)

With µ provided by ESTV, we have

˙S = −kµX +D(Sin − S)

the error dynamic is

˙S = −kµX −DS

Estimation of S despite noise

Because in the ESTV-case the dynamic of S is included

Overview

23 Conclusions

Outline

23 Conclusions

Differentiation

Signal f (t) is a Lebesgue-measurable function on [0,∞).

f(t) = f0(t) + v(t): unknown

f0(t), unknown base signal, n-times differentiable,|f0|(n)(t) ≤ L, L known|v(t)| ≤ η uniformly bounded noise signal.

Using: ς1 = f0 (t) , . . . , ςi+1 = f(i)0 (t) , di

dtif0 (t), i = 1, ..., n,

state representation of the base signal