Stability Theory

When a system has control, then comes the questions of whether we could make it stable under the control.

Problem Setup

A control affine system is this

$$\dot x =f(x)+\sum_i g_i(x)\bar u_i=f(x)+g(x)\bar u$$

Interpretation:

• This control system has multiple vector field, a drift field $f$ by default, and a few “Gain” field that you can control $g_i$.
• You can modulate those “Gain” fields and leverage the drift fields to navigate the space.
• Note in the degenerate case, if $f=0$ and $m=1$, then $\dot x=\bar ug(x)$, this system is driftless, and the control signal can only modulate speed / time, cannot change the track.

The equilibrium stablization problem is: For a controlled equilibrium $0$, $f(0)+g(0) u^\ast=0$, find a feedback function $u:\mathbb R^n\to\mathbb R^m$, s.t. $x=0$ is a globally asymptotic stable equilibrium.

Regular Feedback: regular feedback is a function $\bar u$ on state space $\mathbb R^n$, 1) Locally Lipschitz on $\mathbb R^n\setminus {0}$ 2) continuous at 0

• Regular feedback is continuous.

Condition for Existence

Main theorem is Arstein Sontag theorem

Arstein Sontag theorem: For $\dot x=f(x)+g(x)\bar u$, $f(0)=0$, there exists a global regular stablizer if and only if, there exist a $C1$ contrrolled lyapnov function (CLF), which satisfied small controlled property.

• Note this theorem states only continuous feedback. There are scenarios that requires in-continuous or more complex feedback like those with internal dynamics.

Controlled Lyapnov Function: A function $V:\mathbb R^n\to \mathbb R$, that is pos. def. at 0 and radially unbounded. Also satisfies the equivalent conditions

• When $x\neq 0$, and when $L_{g_1}V=L_{g_2}V=L_{g_i}V=0$, then $L_{f}V\neq 0$
• $\forall x\neq 0$, then there $\exists u\in \mathbb R^m$ s.t. $L_{f}V+\sum_i u_i L_{g_i}V<0$
  • Or $\forall x\neq 0$, $\inf_{u\in \mathbb R^m} [L_{f}V+\sum_i u_i L_{g_i}V]<0$

Interpretation:

• This C.L.F.’s property and existance is mandated by the converse Lyapnov theorems.
• There exists at least a control vector everywhere to decrease the C.L.F.
• When all the control signal cannot decrease C.L.F. (geometrically, the gradient of C.L.F. is orthogonal to all the vector fields $g_i$), then the drift field can still decrease C.L.F.

Small Control Property For a C.L.F., $\mathbb R\to\mathbb R$, $\forall \epsilon>0,\; \exists \delta >0$, such that, $\forall x\in B_\delta(0)\setminus 0$, there $\exists u\in B_\epsilon(0)$, s.t. $L_{f}V+\sum_i u_i L_{g_i}V<0$.

Interpretation: When the state deviation from equilibrium is small, you can use a small control to make it converge (decrease the C.L.F.). (the Columb repulsion is like a counter example.)

General Results

Brockett’s Necessary Condition: For system $\dot x=f(x,u)$, $f$ is locally Lipschitz w.r.t. $x$ and $u$. If it has a continuous stablizer $\bar u(x)$, $\bar u(0)=0$. then $\forall \epsilon >0, \exists\delta>0$, s.t. $B^n_\delta(0)\in f(B^n_\epsilon(0)\times B^m_\epsilon(0))$.

Interpretation:

• Any neighborhood around 0 in the tangent space, could be covered by the image of the function $f(x,u)$ on a small neighborhood of control and state.
• In a small neighborhood around 0, small control signal and the drift can create vector $\dot x$ to any direction / small amplitude! (This system is fully steerable)
  • In a sense, the velocity created is not limited to a subspace.

Corrolary for driftless affine system: If $f=0$ and $\dot x=\sum_i \bar u_i g_i(x)$, then if the $g_i$ vectors are rank deficient $rank([g_1(0),g_2(0),…,g_i(0)])< n$ then the 0 cannot be stablized continuously.

Controllability for LTI system: For linear control system, $\dot x=Ax+Bu$, then the controllabilty is determined by the Kalman rank condition: $rank\,\mathcal C=n$, $\mathcal C=[B,AB,A^2B,…,A^{n-1}B]$

Interpretation:

• This is a corrolary of Brockett theorem for LTI.
• The controllability matrix characterize the interaction between the control gain and the drift.

Passivity

Intuition: The system doesn’t generate energy, the energy increase in it is less than or equal to the energy input.

• This property is inspired by the energy property in a electrical / mechanical system.

Passivity Based Control

MIT Dynamic System and Control 2011