On contraction analysis for non-linear systems

TLDR: These results may be viewed as generalizing the classical Krasovskii theorem, and, more loosely, linear eigenvalue analysis, and the approach is illustrated by controller and observer designs for simple physical examples.

About: This article is published in Automatica.The article was published on 1998-06-01 and is currently open access. It has received 1405 citations till now. The article focuses on the topics: Nonlinear system & Eigenvalues and eigenvectors.

Figures

Fig. 4. Underwater vehicle observer

Fig. 1. Virtual dynamics of two neighboring trajectories

Fig. 2. Convergence of two trajectories

Fig. 3. Underwater vehicle

Frequently asked questions

Q1. What have the authors contributed in "On contraction analysis for nonlinear systems analyzing stability differentially leads to a new perspective on nonlinear dynamic systems" ?

This paper derives new results in nonlinear system analysis using methods inspired from fluid mechanics and differential geometry. 

Q2. What are the properties of the Jacobian fx?

Lyapunov exponents, which are computed as numerical integrals of the eigenvalues of the symmetric part of the Jacobian ∂f∂x , depend on the chosen coordinates x and hence donot represent intrinsic properties. 

Q3. What is the simplest way to say that a system tends to converge exponentially?

Since initial conditions R(t = 0) are exponentially forgotten, the authors can also state that any trajectory converges exponentially to a ball of radius R in (15) with arbitrary initial condition R(t = 0).(viii) 

Q4. What is the simplest example of hierarchical dynamics?

Example 3.5: Using the hierarchical property, the open-loop signal generated by the brain in the biological motor control model of Example 3.2 may itself be the output of a contracting dynamics. 

Q5. What is the simplest closure property of a contracting system?

Combinations of contracting systems satisfy simple closure properties, a subset of which are reminiscent of the passivity formalism (Popov, 1973). 

Q6. What is the generalized Jacobian definition of a continuous-time system?

Furthermore global exponential convergence to the given trajectory is guaranteed if the whole state space is a contraction region with respect to the metric Mi.Most of their earlier continuous-time results have immediate discrete-time versions, as detailed in (Lohmiller and Slotine, 1997d). 

Q7. What is the simplest way to achieve the generalized Jacobian FF?

Define the observer˙̂x = A(t)x̂ + E(t) (y − c(t)x̂ + d(t)) + b(t)uSince by definition the actual state is contained in the flow field, no “openloop” term is needed, but the authors need to find a smooth coordinate transformation δx̂ = Σ(t)δẑ that leads to the generalized Jacobian FF = Σ−1 ( −Σ̇ + (A− Ec)Σ ) = 0 0 · · · 0 −ao1 0 · · · 0 −a10 1 · · · 0 −a2 ... ... . . . 0 ...0 0 · · · 1 −an−1 (26)with the desired (Hurwitz) constant characteristic coefficients ai. 

Q8. What is the convergence condition for the Jacobian fixi?

The convergence condition is equivalent to requiring that the largest singular value of the Jacobian ∂fi∂xi remain smaller than 1 uniformly. 

Q9. What is the way to explain the chaos theory?

For instance, note that chaos theory (Guckenheimer and Holmes, 1983; Strogatz, 1994) leads at best to sufficient stability results. 

Q10. What is the way to extend the result to a nonlinear system?

The result can be extended to the case where x∗ may itself depend on time, as long as it remains in an a priori bounded region.(iv) 

Q11. What is the definition of a contraction analysis?

Observer design using contraction analysis can be simplified by prior coordinate transformations similar to those used in linear reduced-order observer design (Luenberger, 1979). 

Q12. What is the difference between the observer error-dynamics and the controller dynamics?

Since the observer error-dynamics and the controller dynamics represent a hierarchical system, they can be designed separately as long as the control gain K(t) is bounded.