Sarah Dean PRO
asst prof in CS at Cornell
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,
- \(V(x)\) is positive definite
- \(\frac{d}{dt} V(x)\) 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:
\(\dot x = Ax + \tilde F(x)\)
- Find quadratic Lyapunov function via \(A^\top P + PA=-Q\)
- 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?
Lotka-Volterra Models
These ecological models have been used elsewhere:
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Goodwin’s "class struggle model":
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A growth cycle. Goodwin, 1982.
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Technological adoption, stock market
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Predator-Prey: An Efficient-Markets Model of Stock Market Bubbles and the Business Cycle. Gracia, 2004.
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Analysis of the Lotka–Volterra competition equations as a technological substitution model. Morris and Pratt, 2003.
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Interpersonal cooperation/competition
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Computer simulation for exploring theories: Models of interpersonal cooperation and competition. Leik and Meeker, 1995.
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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
- Differential equation and qualitative characteristics
- Types of stability and sufficient conditions
- Complex and global behavior
Most of the discussion focused on closed-loop systems rather than control design!
Discussion points
- Equilibria vs limit cycles in social systems, limited usefulness of linear system
- Importance of parameters in dynamical systems
- Agent based interactions vs population level models
Feedback Systems Ch 4:
By Sarah Dean
