## Sailing in state space

Consider the following controlled (autonomous) dynamical system [1]

$\dot{x} (t) = f (x (t), u (t))$

where $x : [0, \infty) \to \mathbb{R}^n$ is the state trajectory, $u : [0,\infty) \to \mathcal{U}$ is the control input history, $\mathcal{U} \subseteq \mathbb{R}^m$ is the set of admissible control inputs, and $f : \mathbb{R}^n \times \mathcal{U} \to \mathbb{R}^n$ is a (known) vector field.

If we know the control input $u (t)$ for all $t \geq 0$, then we can integrate the ODE above with initial condition $x(0) = x_0$ and obtain the state trajectory $x : [0, \infty) \to \mathbb{R}^n$, as follows

$x (t) = x_0 + \displaystyle \int_{0}^{t} f \left(x (\tau), u (\tau) \right) \mathrm{d} \tau$

which can be represented pictorially by a single streamline in $\mathbb{R}^n$

I am (quite explicitly) alluding to fluid flow. Just like a cork on the ocean will follow a certain path depending on the velocity field (i.e., “ocean currents”), a point in state space will flow along a certain streamline.

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Let us fix the control input, i.e., $u (t) = \bar{u}$ for all $t \geq 0$, and define

$\begin{array}{rl} v^{(\bar{u})} : \mathbb{R}^n & \to \mathbb{R}^n\\ x & \mapsto f(x, \bar{u})\\\end{array}$

For the sake of simplicity, let us consider $\mathcal{U} = \{-1, +1\}$. Since $\bar{u}$ can only take two values, we have two vector fields, $v^{(+1)}(x) := f(x, +1)$ and $v^{(-1)}(x) := f(x, -1)$. Integrating the following ODEs

$\begin{array}{rl} \dot{x} (t) &= v^{(+1)} ( x(t) )\\ \dot{x} (t) &= v^{(-1)} ( x(t) )\\\end{array}$

for various initial conditions, we obtain two families of streamlines, as depicted below

For each fixed control input and collection of initial conditions, we will have a family of streamlines in state space.

Imagine that we have the following control input history

$u (t) = \begin{cases} +1, & t \in [0, t^{\star})\\ -1, & t \geq t^{\star}\\\end{cases}$

where we toggle the control input at time $t^{\star} > 0$, thus interrupting the flow along a “blue” streamline and initiating the flow along a “pink” streamline. Pictorially, we have

If the control input is fixed, say, $\bar{u} = +1$, then given an initial condition $x(0)$, we will flow along the “blue” streamline that passes through $x(0)$. This streamline is $1$-dimensional and does not allow us to “explore” the $n$-dimensional state space. However, if we switch between $\bar{u} = +1$ and $\bar{u} = -1$, then we are able to “travel” to other regions of the state space. Allowing the control input to take values in $\mathcal{U} = [-1,+1]$ would give us even more freedom.

Lastly, we arrive at a most childish idea: in some cases, controller design can be viewed as shaping the streamlines differently in different parts of the state space. This is childish because it is purely conceptual. One obviously cannot design optimal controllers by doodling with crayons.

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References

[1] Hassan K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall.