Chapter 3: Lyapunov Stability of Autonomous Systems
Transcript of Chapter 3: Lyapunov Stability of Autonomous Systems
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Chapter 3: Lyapunov Stability of Autonomous SystemsIn this chapter we review the general stability analysis of autonomous nonlinear system, through Laypunov direct and indirect methods, and invariance principles. Furthermore, Lyapunov function generation and Lyapunov-based controller design are reviewed in detail
Nonlinear Control
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
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Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
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Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Contents
In this chapter we review the general stability analysis of autonomous nonlinear system, through Lyapunov direct and indirect methods, and invariance principles. Furthermore, Lyapunov function generation and Lyapunov-based controller design are reviewed in detail. This chapter contains the most important analysis design of nonlinear systems.
4
Linear Systems and LinearizationStability of LTI systems, Lyapunov equation, Linearization, Lyapunov indirect method. 4
Lyapunov Function GenerationKrasovskii, and generalized Krasovskii theorems, variable gradient method.5
Lyapunov-Based Controller Design Robotic manipulator stabilizing controller and regulation controller design.6
IntroductionLocal, asymptotic, global and exponential stability definitions and examples.1
Lyapunov Direct MethodThe concept, local stability theorem and proof, Lyapunov function, global stability, instability theorem.2
Invariant Set TheoremsKrasovskii-LaSalleβs theorem, local and global asymptotic stability theorems, region of attraction, attractive limit cycle.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems5
β’ Definitions Consider the closed-loop autonomous system
αΆπ₯ = π π₯ (3.1)
β’ Where π: π· β π π is a locally Lipschitz map,
β’ With eq. point @ origin.
3.1 (3.1)
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems6
β’ Stability Definitions Stability in sense of Lyapunov:
β’ The system trajectory can be kept arbitrary close to the equilibrium point.
Geometric Representation
Stable in sense
of LyapunovAsymptotically
Stable
ox
Unstable
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems
7
β’ Stability Definitions
Example: Van der Pol
β’ βπ that the trajectories diverges
β’ Unstable Eq. Point
β’ Stable Limit Cycle
Example: Pendulum
β’ βπ β βπΏ starting from inside πΏ the
trajectory remains in π
β’ Stable (not asymptotically)
7
βπ
βπ
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
β’ Stability Definitions
Exponential Stability
Global Stability
Lyapunov Stability of Autonomous Systems8
π. π
π. π
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Contents
In this chapter we review the general stability analysis of autonomous nonlinear system, through Lyapunov direct and indirect methods, and invariance principles. Furthermore, Lyapunov function generation and Lyapunov-based controller design are reviewed in detail. This chapter contains the most important analysis design of nonlinear systems.
9
Linear Systems and LinearizationStability of LTI systems, Lyapunov equation, Linearization, Lyapunov indirect method. 4
Lyapunov Function GenerationKrasovskii, and generalized Krasovskii theorems, variable gradient method.5
Lyapunov-Based Controller Design Robotic manipulator stabilizing controller and regulation controller design.6
IntroductionLocal, asymptotic, global and exponential stability definitions and examples.1
Lyapunov Direct MethodThe concept, local stability theorem and proof, Lyapunov function, global stability, instability theorem.2
Invariant Set TheoremsKrasovskii-LaSalleβs theorem, local and global asymptotic stability theorems, region of attraction, attractive limit cycle.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems10
β’ Lyapunov Direct Method: The Concept
Mathematical extension of a physical observation:
β’ If the total energy is continuously dissipating
β’ Then the system (Linear or Nonlinear) must settle down to an equilibrium point.
Example: Mass with nonlinear spring-damper
β’ Consider the system:
π α·π₯ + π αΆπ₯ αΆπ₯ + π0 + π1π₯3 = 0
β’ hardening spring + nonlinear damping
β Is the resulting motion stable?
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method11
β’ The Concept
Examine the total energy
β’ Physical observations:
β Zero energy corresponds to the equilibrium point (π₯ = 0, αΆπ₯ = 0)
β Asymptotic stability implies the convergence of the total energy to
zero
β Instability is related to the growth of total energy
β’ Stability is related to the variation of energy
αΆπ π₯ = π αΆπ₯ α·π₯ + π0π₯ + π1π₯3 αΆπ₯ = αΆπ₯ βπ αΆπ₯ αΆπ₯ = βπ αΆπ₯ 3
β The energy of the system is continuously dissipating toward zero
β The motion is converging to eq. point.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method12
β’ The Concept
The energy function has three properties:
β’ π π₯ is a scalar function.
β’ π π₯ is strictly positive except @ origin.
β’ αΆπ π₯ is monotonically decreasing.
In 1892 Lyapunov showed that
β’ A certain other function could be used instead of energy
β Find a positive-definite function π π₯ in π· β π π
β whose derivative along trajectory is continuously decreasing
β Then the eq. point is asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method13
β’ Direct Method
Positive Definiteness
β’ Geometrical Representation
β’ Negative Definite
β If β π(π₯) is positive definite
β’ Positive Semi-Definite
β If π(0) = 0 and π π₯ β₯ 0 for π₯ β 0
β’ Time derivative or derivative along trajectory
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
β’ Local Stability
Proof Idea: (Full proof in next slide)
β Lyapunov level Surface: π π₯ = π for π > 0.
β If αΆπ π₯ < 0, then a trajectory crossing a Ly. S., it moves inside and can never come out again.
Lyapunov Direct Method14
3.1
(3.2)
(3.3)
(3.4)
(3.1)
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method15
β’ Proof
Given π > 0 choose π β 0, π , β
Let πΌ = min| π₯ |=π
π(π₯). Then πΌ > 0 by (3.2).
Take π½ β 0, πΌ , and let
Then, Ξ©π½ is in the interior of π΅π. Any trajectory starting from Ξ©π½ at π‘= 0, stays in Ξ©π½ for all π‘ β₯ 0, because
From existence theorem, since Ξ©π½ is a compact set, (3.1) has a unique
solution whenever π₯ 0 β Ξ©π½. As π(π₯) is continuous and π 0 = 0,
βπΏ > 0, β, Then
and
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
β’ Proof (Cont.)Therefore
which shows that the eq. point is stable. β‘
To show asymptotic stabilityAssume (3.4) holds as well. To show that lim
π‘ββπ₯ π‘ = 0, one may show that
βπ > 0, βπ < 0 β π₯ π‘ < 0 βπ‘ > π.Repeat the previous argument for every π > 0 we can choose π > 0 such that
Ξ©π β π΅π. Therefore, it is sufficient to show that limπ‘ββ
π π₯(π‘) = 0 .
Since π π₯(π‘) is monotonically decreasing and bounded from below by zero.
π π₯(π‘) β π β₯ 0, as π‘ β βTo show π = 0, use contradiction proof. Suppose π > 0. By continuity of π π₯βπ > 0 β π΅π β Ξ©π. The above limit implies that the trajectory π₯(π‘) lies outside the ball π΅π , βπ‘ β₯ 0. Let πΎ = max
πβ€ π₯ β€π
αΆπ(π₯), By (3.4) βπΎ < 0, then
Since The right hand side will eventually become negative, the inequality contradicts the assumption that π > 0.
Lyapunov Direct Method16
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method17
β’ Lyapunov function
A Continuously differentiable function π(π₯) satisfying (3.2) and (3.3) is called a Lyapunov function.
β’ The surface π π₯ = π for some π > 0, is a Lyapunov surface
β Use the following Lyapunov surfaces:
β To make the theorem intuitively clear.
β Upon the condition (3.4)
β’ The trajectories crossing a Lyapunov surface
moves inside and never come out again.
β’ Note if αΆπ π₯ β€ 0, then the trajectories may stall!
β’ It means it is stable (not going outside)
β’ But not necessarily asymptotic stable.
β If αΆπ π₯ < 0, then the Lyapunov surfaces will
shrink to origin. Implying asymptotic convergence.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method18
β’ Example 1: Pendulum without friction
β’ System:
β’ Lyapunov Candidate:
β How??! (Total Energy)
β It is positive definite in the domain
β’ Lyapunov Function?
β Derivative along trajectory:
β Eq. point is stable.
β But not asymptotically stable!
β Trajectory starting @ Ly. S. π(π₯) = π, remain on it.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method19
β’ Example 2:
Pendulum with viscous friction
β’ System:
β’ Lyapunov Candidate:
β The same as Ex1. (Total Energy)
β’ Lyapunov Function?
β Derivative along trajectory:
β Positive Semi-definite: zero irrespective of π₯1β Only stable but not asymptotically stable!
β Phase portrait and linearization method Asy. Stable.
Lyapunov direct conditions are only sufficient!
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method20
β’ Example 2:
Pendulum with viscous friction
β’ Use another Lyapunov Candidate:
β’ Lyapunov Function?
β π(π₯) > 0 if
β’ Derivative along trajectory:
β If π12 = 0.5 π/π, then
becomes neg-def. over the domain
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method21
β’ Example 3:
Consider the general 1st order systemαΆπ₯ = βπ(π₯)
Where π(π₯) is locally Lipschitz on (βπ, π), and satisfies:
Like in figure. Lyapunov Candidate:
β How??! (Total Energy)
β It is positive definite in the domain π· = (βπ, π) .
β’ Lyapunov Function?
β Derivative along trajectory:
β The eq. point is Asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method22
β’ Example 4:
Consider the following system:
β’ The eq. point is @ origin.
β’ Lyapunov Candidate:
β Derivative along trajectory:
β It is negative definite in a ball:
β The eq. point is asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Scientist Bio23
Aleksandr Mikhailovich Lyapunov(June 6 1857 β November 3, 1918)
Was a Russian mathematician, and physicist. He was the son of an astronomer. Lyapunov is known for his development of the stability theory of a dynamicalsystem, as well as for his many contributions to mathematical physics and probability theory.
He studied at the University of Saint Petersburg. In 1880 Lyapunov received a gold medal for a work on hydrostatics. Lyapunov's impact was significant, and a number of different mathematical concepts therefore bear his name: Lyapunov equation, Lyapunov exponent, Lyapunov function, Lyapunov fractal, Lyapunov stability, Lyapunov's central limit theorem, and Lyapunov vector.
By the end of June 1917, Lyapunov traveled with his wife to his brother's place in Odessa. Lyapunov's wife was suffering from tuberculosis so they moved following her doctor's orders. She died on October 31, 1918. The same day, Lyapunov shot himself in the head, and three days later he died.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method24
β’ Global Stability
If the origin is asymptotically stable
β’ Define Region of Attraction (RoA)
β Let π(π‘; π₯) be the solution for (3.1), Then RoA is the set of all points π₯such that π(π‘; π₯) is defined, and βπ‘ β₯ 0, lim
π‘ββπ π‘; π₯ = 0.
β’ Analytic determination of RoA is hard or even impossible.
β’ Lyapunov functions may be used to find an estimate of RoA.β Assume a Ly. Function is negative definite in a domain D.
β Assume Ξ©π = π₯ β π π π(π₯) β€ π} is bounded and contained in π·.
β Every trajectory starting in Ξ©π converges to origin.
β Ξ©π is a (conservative) estimate of the RoA.
If RoA is π π then eq. point is globally asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method25
β’ Global Stability (Barbashin-Krasovshii)
β’ For small π the Ly. Surfaces π(π₯)= π are closed, but for large c the Ly. S.
are not closed, then the trajectory may
diverge.
Radial ly Unbounded
Radial Unboundedness
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method26
β’ Example 5:
System as in Ex3: αΆπ₯ = βπ(π₯)In which, π(0) = 0 and π₯π π₯ > 0 for π₯ β 0
β’ Lyapunov Candidate: π(π₯) = π₯2
β It is positive definite in the whole space
β It is radially unbounded
β’ Lyapunov Function?
β Derivative along trajectory: αΆπ = 2π₯ αΆπ₯ = β2π₯π(π₯)
β Hence, αΆπ < 0 as long as π₯ β 0.
β Hence, the origin is globally asymptotically stable.
β’ Typical Examples
β αΆπ₯ = βπ₯3 OR αΆπ₯ = sin2 π₯ β π₯, (sin2 π₯ β€ sin π₯ < π₯ )
β π₯π π₯ = π₯4 > 0 and π₯π π₯ = π₯2 β π₯ sin2 π₯ > π₯2 β π₯ π₯ > 0
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method27
β’ Example 6:
Consider the following system:
β’ The eq. point is @ origin.
β’ Lyapunov Candidate:
β Derivative along trajectory:
β It is negative definite everywhere,
β It is radially unbounded,
β The eq. point is globally asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method28
β’ Some Remarks:
1. Use total energy as the first Lyapunov candidate, but donβt
limit yourself to that.
2. Many Lyapunov functions exist for a system. If π(π₯) is a
Lyapunov function, so is π1 = πππΌ. .
3. Lyapunov theorems are sufficient theorems, if a Lyapunov
candidate doesnβt work, search for another one!
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method29
β’ Instability Theorem: (Chetaev)
Let π: π· β π be a continuously differentiable function on Domain π· β π π.
Let π· contain the origin, and π(0) = 0.
For a point π₯0 arbitrary close to origin π(π₯0) > 0.
Choose π > 0 such that the ball π΅π = π₯ β π π π₯ β€ π is contained in π·.
Let U be a nonempty set in Br, such that
Its boundary is the surface π(π₯) = 0.
Example: π π₯ =1
2(π₯1
2 β π₯22)
β π 0 = 0, π π₯0 > 0 in the hatched area:
β π π₯ = 0, at the boundaries of π₯1 = Β±π₯2β The region U is the hatched area
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method30
β’ Instability Theorem: (Chetaev)
Proof: As long as π₯(π‘) is inside π, π π₯ π‘ β₯ π, since αΆπ(π₯) > 0.
β’ Let
β’ Then πΎ > 0 and
β’ This shows that π₯(π‘) have to leave π, because π(π₯) is bounded on π.
β’ π₯(π‘) cannot leave from the surface boundary since V π₯ = 0 while π π₯ π‘ β₯ π.
β’ Therefore it shall leave from the sphere π₯ = π.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Direct Method31
β’ Instability Theorem: (Chetaev)
Consider the system:
β’ Where π1, π2 are locally Lipschitz, and satisfy the following in a region π·.
β’ These imply that π1(0) = π2(0) = 0. Hence the origin is the eq. point.
β’ Consider the function π(π₯) =1
2(π₯1
2 β π₯22). π π₯ > 0 on the line π₯2 = 0.
β’ The set U is set as before as shown in here.
β’ while
β’ Hence,
β’ Choose π < 1/(2π), Then αΆπ becomes positive, and the origin is unstable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Scientist Bio32
Nikolay Gur'yevich Chetaev(23 November 1902 β 17 October 1959)
Was a Russian Soviet mechanician and mathematician. He is renowned
from his work on elliptic partial differential equations. He belongs to the
Kazan school of mathematics. From 1930 to 1940 N. G. Chetaev was a
professor of Kazan University where he created a scientific school of
the mathematical theory of stability of motion, where N. Krosovskii was
a Ph.D. student.
Chetaev made a number of significant contributions to Mathematical
Theory of Stability, Analytical Mechanics and Mathematical Physics. His
major scientific achievements relates to as: The PoincarΓ© equations;
Lagrangeβs theorem of stability of an equilibrium, PoincarΓ©βLyapunov
theorem on a periodic motion & Chetaev's theorems; Chetaevβs method
of constructing Lyapunov functions as a coupling (combination) of first
integrals; D'AlembertβLagrange and Gauss Principles.
A representative region for
Chetaev instability Theorem
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Contents
In this chapter we review the general stability analysis of autonomous nonlinear system, through Lyapunov direct and indirect methods, and invariance principles. Furthermore, Lyapunov function generation and Lyapunov-based controller design are reviewed in detail. This chapter contains the most important analysis design of nonlinear systems.
33
Linear Systems and LinearizationStability of LTI systems, Lyapunov equation, Linearization, Lyapunov indirect method. 4
Lyapunov Function GenerationKrasovskii, and generalized Krasovskii theorems, variable gradient method.5
Lyapunov-Based Controller Design Robotic manipulator stabilizing controller and regulation controller design.6
IntroductionLocal, asymptotic, global and exponential stability definitions and examples.1
Lyapunov Direct MethodThe concept, local stability theorem and proof, Lyapunov function, global stability, instability theorem.2
Invariant Set TheoremsKrasovskii-LaSalleβs theorem, local and global asymptotic stability theorems, region of attraction, attractive limit cycle.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
34
Lyapunov Stability of Autonomous Systems
β’ Invariant Set Theorems:
Asymptotic stability needs αΆπ π₯ < 0β’ In many systems we may reach only to αΆπ π₯ β€ 0β’ Use invariant set to prove asymptotic stability
A set M is an invariant set with respect to (3.1) if
π₯ 0 β π β π₯ π‘ β π, βπ‘ β π A set M is an positively invariant set with respect to (3.1) if
π₯ 0 β π β π₯ π‘ β π, βπ‘ β₯ 0β’ Examples of invariant sets
β An equilibrium point
β A limit cycle
β Any trajectory
β The RoA of an eq. point or a limit cycle
β The whole state space
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem35
β’ Barbashin - Krasovskii - Lasalleβs Theorem:
β’ It is not directly stated that π(π₯) > 0, But
β’ The function V is continuous on the compact set Ξ©
β It is bounded from below (somehow positive)
β The set Ξ©l is called a compact set.
β’ Largest invariant set means the union of all invariant sets.
β’ This theorem introduces the notion of Region of Attraction.
β’ Can be used for Eq. point, limit cycle, or any invariant set.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem36
β’ Lasalleβs Corollaries:
Local Asymptotic Stability
Global Asymptotic Stability
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem37
Example: Mass with nonlinear spring-damper
β’ Consider the system:
π α·π₯ + π αΆπ₯ αΆπ₯ + π0 + π1π₯3 = 0
β’ Lyapunov candidate
β π π₯ =1
2π αΆπ₯2 +
1
2π0π₯
2 +1
4π1π₯
4
β αΆπ π₯ = π αΆπ₯ α·π₯ + π0π₯ + π1π₯3 αΆπ₯ = αΆπ₯ βπ αΆπ₯ αΆπ₯ = βπ αΆπ₯ 3
β The set R where αΆπ π₯ = 0 is π = π₯, αΆπ₯ αΆπ₯ = 0}
β Is the largest invariant set in π ,π = { 0, 0 }?
Suppose any arbitrary point of R, such as (π₯1, 0) is also in M.
Any trajectory passing through this point must satisfy:
α·π₯ = βπ0
ππ₯ β
π1
ππ₯3 β 0, hence, the trajectory moves out from R.
β The equilibrium point is asymptotically stable.
x
x
1x
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem38
Example 2:
β’ System dynamics:
β In which,
β’ Eq. point @ origin.
β’ Lyapunov Candidate:
β In the domain is positive definite.
β’ Lyapunov function derivative:
β Positive semi-definite, needs invariant set Theorem.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem39
Example 2: (cont.)
β’ Characterize the Set π where:
β’ Hence, π = π₯1, π₯2 π₯2 = 0}
β’ Show that M includes only origin:
β Suppose x(t) is a trajectory belonging to R, then
β Hence, the solution to this trajectory is only the origin.
β’ The equilibrium point in asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem40
Example 3: Region of Attraction
β’ System dynamics:
β’ Eq. point is @ origin.
β’ Lyapunov candidate:
β for π = 2, the region Ξ©π is defined by π(π₯) < 2 is a compact set.
β’ Derivative along trajectory:
β For the set Ξ©π the derivative is always negative except @ origin.
β’ The set π includes only the origin.
β Invariant Set Theorem conditions hold.
β The eq. point is locally asymptotically stable.
β The Region of Attraction is estimated by Ξ©π a circle with radius π = 2.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem41
Example 4: Attractive Limit Cycle
β’ System dynamics
β’ There exist an invariant set:
β Since, its derivative is zero on the set.
β’ On the invariant set:
β Simplified system dynamics
β Invariant set is a limit cycle
β’ Is the limit cycle attractive?
β Lyapunov candidate:
β Physical insight: distance to the limit cycle.
0
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem42
Example 4: Attractive Limit Cycle (cont.)
β’ For any π > 0,
β the region Ξ©π defined by π(π₯) < π is a compact set.
β’ Lyapunov function derivative:
β from before,
β αΆπ π₯ < 0 everywhere except at
β’ π₯14 + 2π₯2
2 = 10 The limit Cycle.
β’ π₯110 + 3π₯2
6 = 0 The Eq. point @ origin.
β The eq. point at origin is unstable.
β’ From invariant set theorem, all the trajectories converge to
the limit cycle.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Invariant Set Theorem43
Example 4: Attractive Limit Cycle (cont.)
β’ Phase portrait:
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Scientist Bio44
Nikolay Nikolayevich Krasovskii
(7 September 1924 β 4 April 2012)
Was a prominent Russian mathematician who worked in the
mathematical theory of control, the theory of dynamical systems, and the
theory of differential games. He was the author of Krasovskii-LaSalle
principle and the chief of the Ural scientific school in mathematical theory of
control and the theory of differential games. In 1963 Stanford University
Press published a translation of his book Stability of Motion: applications of Lyapunov's second method to differential systems and equations with delay that
had been prepared by Joel Lee Brenner. Krasovskii received many honours
for his contributions. He was elected a corresponding member of the USSR
Academy of Sciences in 1964 and became a full member in 1968. He was
awarded the M V Lomonosov Gold Medal of the Russian Academy of
Sciences, the A M Lyapunov Gold Medal, the Demidov Prize in physics and
mathematics, and the 'Triumph' Prize which is awarded to the leading scientists
for their contribution to Russian and world science as a whole.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Scientist Bio45
Joseph P. LaSalle (28 May 1916 - 7 July 1983)
Was an American mathematician, specialising in dynamical
systems and responsible for important contributions
to stability theory, such as LaSalle's invariance
principle which bears his name. Joseph LaSalle defended
his Ph.D. thesis on β³Pseudo-Normed Linear Sets over Valued Ringsβ³ at the California Institute of Technology in
1941. During a visit to Princeton in 1947β1948, LaSalle
developed a deep interest in differential equations through
his interaction with Solomon Lefschetz and Richard
Bellman. he worked closely with Lefschetz and in 1960
published his extension of Lyapunov stability
theory,[5] known today as LaSalle's invariance principle.
For the proof of LaSalleβs
invariance principle
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Contents
In this chapter we review the general stability analysis of autonomous nonlinear system, through Lyapunov direct and indirect methods, and invariance principles. Furthermore, Lyapunov function generation and Lyapunov-based controller design are reviewed in detail. This chapter contains the most important analysis design of nonlinear systems.
46
Linear Systems and LinearizationStability of LTI systems, Lyapunov equation, Linearization, Lyapunov indirect method. 4
Lyapunov Function GenerationKrasovskii, and generalized Krasovskii theorems, variable gradient method.5
Lyapunov-Based Controller Design Robotic manipulator stabilizing controller and regulation controller design.6
IntroductionLocal, asymptotic, global and exponential stability definitions and examples.1
Lyapunov Direct MethodThe concept, local stability theorem and proof, Lyapunov function, global stability, instability theorem.2
Invariant Set TheoremsKrasovskii-LaSalleβs theorem, local and global asymptotic stability theorems, region of attraction, attractive limit cycle.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems
β’ Linear Systems and Linearization
Consider the linear time invariant system
αΆπ₯ = π΄π₯ (3.9)
β’ That has an eq. point @ origin.
β’ The eq. point is isolated if det π΄ β 0.
β’ The solution for a given initial state π₯(0):π₯ π‘ = exp π΄π‘ π₯ 0
47
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Linear Systems and Linearization
β’ Stability of LTI system
When all eigenvalues of π΄ satisfy Reππ < 0,
β’ π΄ is called Hurwitz matrix.
For stability analysis consider π π₯ = π₯πππ₯
β’ Where π is a real, symmetric and positive definite matrix.
β’ Then
β’ Where π is a positive definite matrix defined by
β’ This is called Lyapunov Equation.
48
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Linear Systems and Linearization
β’ Stability of LTI system
Proof:
β’ Sufficiency follows from Theorem 3.1
β’ For Necessity assume all eigenvalues of π΄ satisfy Reππ < 0, and
β’ P exists and is proved to be positive definite by contradiction :
49
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Linear Systems and Linearization
β’ Stability of LTI system
Proof: (Cont.)
β’ Since exp(π΄π‘) is nonsingular, this results in contradiction.
β Substitute (3.13) in (3.12)
β This shows that P is a solution of (3.12).
β Prove it is a unique solution by contradiction
β Pre- post multiply
by exp π΄ππ‘
β Hence,
β Since, exp π΄0 = I:
β Therefore, ΰ·¨π = π.
50
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Linear Systems and Linearization
β’ Linearization of nonlinear system
Consider αΆπ₯ = π π₯ , and 0,0 its eq. point.
β’ Let π: π· β π π be a continuously differentiable vector field
β’ By mean value theorem
β’ Where π§π lies on the line segment from 0 to π₯
β’ Since ππ 0 = 0, write
51
0 π§π π₯
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Linear Systems and Linearization
β’ Linearization of nonlinear system
β’ The function ππ π₯ satisfies:
β’ By continuity
β’ This suggests that in a small neighborhood of the origin we can
approximate αΆπ₯ = π(π₯) by its linearization:
β’ How about the stability of the origin? Any Condition?
52
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
β’ Stability Analysis by Linearization
β’ Lyapunov indirect or linearization method
β’ If eq. point is non-hyperbolic Inconclusive!
Linear Systems and Linearization53
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Linear Systems and Linearization54
β’ Stability Analysis by Linearization
Example 1:
β’ Consider the system αΆπ₯ = ππ₯3
β The eigenvalue is on imaginary axis Inconclusive!
β’ Example 2: Pendulum
β The eq. points are @ (0,0) and (Ο,0).
β Jacobian:
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Linear Systems and Linearization55
β’ Stability Analysis by Linearization
β’ Example 2: Pendulum (cont.)
β For (0,0) Eq. point :
β’ All Eigenvalues Hurwitz Asymptotically Stable
β For (π, 0) eq. point.
β’ Change variables π§1 = π₯1 β π, π§2 = π₯2β’ Chack Jacobian @ π§ = 0
β’ One of the eigenvalues is not Hurwitz Unstable
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Contents
In this chapter we review the general stability analysis of autonomous nonlinear system, through Lyapunov direct and indirect methods, and invariance principles. Furthermore, Lyapunov function generation and Lyapunov-based controller design are reviewed in detail. This chapter contains the most important analysis design of nonlinear systems.
56
Linear Systems and LinearizationStability of LTI systems, Lyapunov equation, Linearization, Lyapunov indirect method. 4
Lyapunov Function GenerationKrasovskii, and generalized Krasovskii theorems, variable gradient method.5
Lyapunov-Based Controller Design Robotic manipulator stabilizing controller and regulation controller design.6
IntroductionLocal, asymptotic, global and exponential stability definitions and examples.1
Lyapunov Direct MethodThe concept, local stability theorem and proof, Lyapunov function, global stability, instability theorem.2
Invariant Set TheoremsKrasovskii-LaSalleβs theorem, local and global asymptotic stability theorems, region of attraction, attractive limit cycle.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems57
β’ Lyapunov Function Generation
Krasovskii Method
: 3.1 ,
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Function Generation58
β’ Krasovskii Method
β’ Example: Consider the system
β The Jacobian:
β F is negative definite for the whole space.
β Lyapunov Function
β It is Radially unbounded
β The Eq. point is globally asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
β’ Lyapunov Function Generation
Krasovskii Method
β’ Proof Idea:
β If πΉ < 0 and π > 0, then αΆπ π₯ < 0.
Lyapunov Function Generation59
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Function Generation60
β’ Variable Gradient Method
Search backward, start with αΆπ π₯ < 0, then find π(π₯).
β’ Procedure:
β Suppose π(π₯) is the gradient of π(π₯):
β Derivative of π(π₯) along trajectory:
β Choose π(π₯) such that αΆπ π₯ < 0, while π π₯ > 0.
β For g(x) to be gradient of a scalar function:
β Under this constraint choose π(π₯) such that ππ π₯ π π₯ < 0
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Function Generation61
β’ Variable Gradient Method
β’ Procedure (cont.):
β Then generate π(π₯) by integration
β The integration can be taken along any path, but usually it is taken
along the principal axes:
β Leave some parameters of π(π₯) undetermined, and try to choose
them to ensure that π(π₯) positive.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Function Generation62
β’ Variable Gradient Method
Example 1:
β’ Consider the system:
β where, π > 0, β 0 = 0 πππ π¦β π¦ > 0 , βπ¦ β (βπ , π ) .
β’ To ensure αΆπ π₯ < 0 β ππ π₯ π π₯ < 0
β’ The Lypunov function is:
β’ Let us try
β’ Gradient condition
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Function Generation63
β’ Variable Gradient Method
Example 1: (cont.)
β’ Derivative of Ly. f.
β To cancel cross terms
β Therefore,
β To simplify assign π½, πΎ, and πΏ to be constant but keep πΌ(π₯)
β’ From gradient condition
β Ξ±(x) = Ξ±(x1) and Ξ² = Ξ³.
β
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Function Generation64
β’ Variable Gradient Method
Example 1: (cont.)
β’ Integrate g(x) to get the Ly. function
β in which,
β’ Choose πΏ > 0, and 0 < πΎ < πΌπΏ to ensure π π₯ > 0, and αΆπ π₯ < 0β For example πΎ = πππΏ for 0 < π < 1
β Then,
β π π₯ > 0, and αΆπ π₯ < 0 for π· = {π₯ β π 2| β π < π₯1 < π}
β The eq. point is asymptotically stable.
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Contents
In this chapter we review the general stability analysis of autonomous nonlinear system, through Lyapunov direct and indirect methods, and invariance principles. Furthermore, Lyapunov function generation and Lyapunov-based controller design are reviewed in detail. This chapter contains the most important analysis design of nonlinear systems.
65
Linear Systems and LinearizationStability of LTI systems, Lyapunov equation, Linearization, Lyapunov indirect method. 4
Lyapunov Function GenerationKrasovskii, and generalized Krasovskii theorems, variable gradient method.5
Lyapunov-Based Controller Design Robotic manipulator stabilizing controller and regulation controller design.6
IntroductionLocal, asymptotic, global and exponential stability definitions and examples.1
Lyapunov Direct MethodThe concept, local stability theorem and proof, Lyapunov function, global stability, instability theorem.2
Invariant Set TheoremsKrasovskii-LaSalleβs theorem, local and global asymptotic stability theorems, region of attraction, attractive limit cycle.3
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems66
β’ Lyapunov Based Controller Design
Example: Robotic Manipulator
β’ Physically derived Lyapunov function
β’ System dynamics
β’ Controller
β’ Lyapunov Candidate
β Total Energy
β’ Lyapunov Function Derivative
β Power of the external forces
β Used control law
β Lasalle: Global Asymptotically stable
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems67
β’ Lyapunov Based Controller Design
Design Idea:
β’ Consider a Lyapunov candidate
β’ Stability:
β Design the control law as a nonlinear function to ensure negative definiteness of the Ly. F. Derivative.
β’ Performance:
β Rate of decay is related to the time performance.
β’ Base of many nonlinear controller designs:
β Back-stepping
β Sliding mode control
β Lyapunov redesign
β . . .
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Lyapunov Stability of Autonomous Systems68
β’ Lyapunov Based Controller Design
Example 2: Regulation
β’ System dynamics:
β’ Objective
β Push the trajectories toward origin.
β’ Consider the controller as:
β’ Lyapunov Candidate:
β’ Derivative:
β’ Design u such that :
β For example,
β Stability: Lasalle asymptotically stable eq. point.
β Performance: increase K to have faster response.
β Controller is not unique.
( , )u u x x2 21/ 2( )V x x
3 2( )V x x x x u
( ) 0V x 2 2 3V K x u x x K x x
3 2x x x u
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Chapter 3: Lyapunov Stability of Autonomous Systems
To read more and see the course videos visit our course website:
http://aras.kntu.ac.ir/arascourses/nonlinear-control/
Thank You
Nonlinear Control
Nonlinear Control K. N. Toosi University of Technology, Faculty of Electrical Engineering,
Prof. Hamid D. Taghirad Department of Systems and Control, Advanced Robotics and Automated Systems October 7, 2020
Hamid D. Taghirad has received his B.Sc. degree in mechanical engineering
from Sharif University of Technology, Tehran, Iran, in 1989, his M.Sc. in mechanical
engineering in 1993, and his Ph.D. in electrical engineering in 1997, both
from McGill University, Montreal, Canada. He is currently the University Vice-
Chancellor for Global strategies and International Affairs, Professor and the Director
of the Advanced Robotics and Automated System (ARAS), Department of Systems
and Control, Faculty of Electrical Engineering, K. N. Toosi University of Technology,
Tehran, Iran. He is a senior member of IEEE, and Editorial board of International
Journal of Robotics: Theory and Application, and International Journal of Advanced
Robotic Systems. His research interest is robust and nonlinear control applied to
robotic systems. His publications include five books, and more than 250 papers in
international Journals and conference proceedings.
About Hamid D. Taghirad
Hamid D. TaghiradProfessor