Feedback Systems

Ch 4: Dynamic Behavior

Sarah Dean for CS294

Differential Equations

\( \frac{d x}{d t} = F(x) \)

With input a function of state, \( u = k(x) \)

\( \frac{d x}{d t} = f(x, u) \)                \( y = h(x, u) \)

We consider initial value problems, \(x(t_0)=x_0\)

Differential Equations: Solutions

Lipschitz continuity guarantees existence and uniqueness

\( \|F(x)-F(y)\| \leq L \|x-y\| \)

 

Sufficient condition: A uniformly bounded Jacobian \(\|\frac{\partial F}{\partial x}\|\leq L\)

\(x(t)\) is a solution on time interval \((t_0,t_f)\) if \(\frac{d}{dt}x(t) = F(x(t))\) for all \(t\)

A unique solution may not always exist for all time!

Bad Examples

Ex 4.2 Finite escape time: \(\dot x = x^2 \) with \(x(0)=1\) yields \(x(t) = \frac{1}{1-t}\), which limits to \(\infty\) as \(t\to 1\)

 

Ex 4.3 Nonunique solution: \(\dot x = 2\sqrt{x} \) with \(x(0)=0\) yields

\(x(t) = \begin{cases} 0 & 0\leq t\leq a\\ (t-a)^2 & t\geq a \end{cases},  \)

for any \(a\geq 0\)

Qualitative Analysis

Vector Fields and Phase Portraits

We can visualize solutions to 2D differential equations for many initial conditions

Vector fields plot the direction \(F(x)\) at each point.

Phase plots trace solutions for several initial conditions.

Equilibrium Points

Different types of equilibrium points

Qualitatively, we observe

movement

no movement

an equilibrium point occurs when

\(\dot x = F(x) = 0\)

Limit Cycles

Solutions can be oscillatory, i.e. \(x(t+T)=x(t)\) for some \(T>0\)

Stability

Definitions of Stability

A solution \(x(t;a)\) is stable in the sense of Lyapunov (neutrally stable) if

  • other solutions starting near \(a\), i.e. \(\|b-a\|<\delta\)
  • stay near \(x(t;a)\) for \(t>0\), i.e. \( \|x(t;b)-x(t;a)\|<\epsilon \)

A solution is asymptotically stable if it is neutrally stable and

  • \(x(t;b) \to x(t;a)\) as \(t\to \infty\) if \(b\) is close enough to \(a\)

We can define notions of local vs. global stability

Definitions of (in-)Stability

Equilibrium points that are not stable are unstable

Asymptotically Stable:

Attractor/Sink

Neutrally Stable:

Center

Unstable:

Source

Unstable: Saddle

Stability of Linear Systems

Why? If \(A = SDS^{-1}\), defining \(z = S^{-1} x\) gives

\( \dot z = D z \),

i.e. the system decouples \( \dot z_i = \lambda_i z_i\) and solutions are in the form

\(z_i(t) = e^{\lambda_i t}\)

Linear systems \(\dot x = A x\) have a single equilibrium at \(x=0\)

 

The stability is determined by the eigenvalues of \(A\):

  • stable if \(\mathrm{Re}(\lambda_i(A)) < 0\) for all \(i\)

Stability of Linear Systems

Imaginary Eigenvalues

  • come in pairs: \(\lambda = \sigma\pm j\omega\)
  • \(z(t) \propto e^{\sigma t}(\cos(\omega t) + \sin(\omega t))\)

Real Eigenvalues

  • \(z(t) \propto e^{\lambda t}\)

Q: What about discrete time linear systems?

A: stable when \(|\lambda(A)| < 1\), oscillatory when complex or negative

Stability of Linear Systems

Q: Who cares about linear systems?

A: Linearization gives information about behavior of nonlinear systems near their fixed points

\(\dot x = F(x) = \cancel{F(x_{eq})} + \frac{\partial F}{\partial x}\Big |_{x_{eq}}  (x - x_{eq}) \) + higher order terms

Limit Cycle via Linearization

Linearization can be a useful strategy to show even complex behaviors like limit cycles.

Ex. 4.8 \( \dot x_1 = x_2 + x_1(1-x_1^2 - x_2^2),~~ \dot x_2 = -x_1 + x_2(1-x_1^2-x_2^2) \)

Polar coordinates: \(x_1 = r \cos \theta,~x_2 = r \sin \theta\) yields decoupled

\( \dot r =  r(1-r^2),~~\dot \theta = -1\)

 

Then we can show that \(r=1\) is asymptotically stable and \(\theta(t) = \theta(0) - t\)

Lyapunov Stability Analysis

Lyapunov Functions

A Lyapunov function is an energy-like function that is

  • nonnegative and
  • decreases along all trajectories

Lyapunov Stability Theorem:

If \(V\) is a nonnegative function such that for \(x\) in some ball,

  1. \(V(x)\) is positive definite
  2. \(\frac{d}{dt} V(x)\) is negative semidefinite (definite)

then \(x=0\) is locally (asymptotically) stable

Synthesizing Lyapunov Functions for Linear Systems

If \(\dot x = Ax\), then \(V(x) = x^\top P x\) is a Lyapunov function iff \(P\succ 0\) and

  • \(A^\top P + PA = -Q\) for some \(Q\succ 0\)

It is always possible to find a quadratic Lyapunov function for stable \(A\)

The Lyapunov function will be valid for all \(x\), meaning global stability

linear equation

Synthesizing Lyapunov Functions

This idea applies to more than just linear systems:

\(\dot x = Ax + \tilde F(x)\)

  1. Find quadratic Lyapunov function via \(A^\top P + PA=-Q\)
  2. Show that \( \dot V\) is negative definite for small \(x\)

 

\(\implies \)Then we have shown local asymptotic stability for this equilibrium of the nonlinear system

This strategy will always work when

\(\|\tilde F(x) \|/\|x\| \to 0\) as \(x\to 0\)

which follows from Taylor expansion (higher order terms)

Krasovski–Lasalle Invariance Principle

This principle allows us to show asymptotic stability even for positive semidefinite \( V\)

Synthesizing Lyapunov Functions

More complex forms of Lyapunov functions can be computed through sum-of-squares methods, but complexity is limited by computation

Nonlocal Behavior

Regions of Attraction

Lyapunov functions give inner approximations to regions of attraction

The region of attraction for a stable equilibrium point is the set of initial conditions that converge to it

Parametric Dependence

Qualitative system behavior depends on parameters of dynamic model

Predator-Prey

Predator-prey interactions occur stochastically and at an individual level.

 

A simple ODE describes qualitative behavior.

 

What other phenomena can be modeled in this way?

 

http://www.ahahah.eu/trucs/pp/

Lotka-Volterra Models

These ecological models have been used elsewhere:

  • Goodwin’s "class struggle model":

    • A growth cycle. Goodwin, 1982.

  • Technological adoption, stock market

    • Predator-Prey: An Efficient-Markets Model of Stock Market Bubbles and the Business Cycle. Gracia, 2004.

    • Analysis of the Lotka–Volterra competition equations as a technological substitution model. Morris and Pratt, 2003.

  • Interpersonal cooperation/competition

    • Computer simulation for exploring theories: Models of interpersonal cooperation and competition. Leik and Meeker, 1995.

Nonlinear Feedback Design

Noise cancelling headphones: adaptively update filter parameters via Lyanpunov stability analysis:

\(\dot a = -\alpha w (w-z)\) and \(\dot b = -\alpha n (w-z)\)

SUMMARY

Today we talked about

  1. Differential equation and qualitative characteristics
  2. Types of stability and sufficient conditions
  3. Complex and global behavior

Most of the discussion focused on closed-loop systems rather than control design!

 

Discussion points

  1. Equilibria vs limit cycles in social systems, limited usefulness of linear system
  2. Importance of parameters in dynamical systems
  3. Agent based interactions vs population level models

Feedback Systems Ch 4:

By Sarah Dean

Feedback Systems Ch 4:

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