# Asymptotic stability of cascade control

For many systems, it seems to be common practice to stack controllers on top of each other. For example, in a quadcopter, one first builds an attitude controller, then builds a velocity controller whose (virtual) control variable is an input to the attitude controller, then builds a position controller whose (virtual) control variable is an input to the velocity controller. I believe (correct me if I'm wrong) that this is known as cascade control.

I'm wondering if there is any theory formalizing why this approach works. To be more precise, suppose I have the following two systems:

• $x' = f(x,u)$ (e.g. a position controller with virtual control variable $u$)
• $y' = g(y,v)$ (e.g. a velocity controller with actual control variable $v$)

Furthermore, suppose I have feedback control $a(x)$ that stabilizes the first system and $b(y)$ that stabilizes the second. In other words, the following two systems are asymptotically stable:

• $x' = f(x, a(x))$
• $y' = g(y, b(y))$

Now we can form the cascade of these two systems, which I believe is formalized as:

• $x' = f(x, y)$
• $y' = g(y, b(y - a(x)))$

It seems like cascade controller designers rely on the fact that this cascade will be asymptotically stable because the two subsystems are separately asymptotically stable. However, something tells me that this is not in general true. I feel like I'm missing something. Are there conditions under which this cascade is asymptotically stable (e.g. f and g are linear)? If not, how does one design an asymptotically stable cascade controller?

I did come upon a technique called backstepping, which builds feedback control for a system using feedback control for a subsystem and a Lyapunov function for the subsystem. For example, if you have feedback control a(x) which stabilizes $x' = f(x) + g(x)*a(x)$ and you have a Lyapunov function guaranteeing this stability, then you can build a function $b(x)$ stabilizing

• $x' = f(x) + g(x)*y$
• $y' = h(y) + j(y)*b(x)$

Unfortunately, I don't think that this approach quite solves my problem. Backstepping would be a way of producing a controller that stabilizes position via acceleration, given a controller that stabilizes position via velocity. However, I want to produce a controller that stabilizes position via acceleration, given a controller that stabilizes position via velocity and a fixed controller that stabilizes velocity via acceleration. This would allow me to modularly build an attitude controller, then a velocity controller on top of that, then a position controller on top of that, etc.

• Whatever the structure of your plant is, try to analyze stability by rigorous means such as Lyapunov apparatus if you are unsure how a combined plant behaves. Sep 14 '15 at 15:15
• @ValerySaharov I'm just wondering if there is a general composition approach to analyzing stability of cascades. Backstepping is one such approach, but it's not quite the kind of composition I want. In my case, the lower level, driving controllers are fixed, while in backstepping, the driving controllers are the ones being built. Sep 14 '15 at 16:40
• There are many conceivable configurations of a cascade controller, so a general answer won't be easy to find. You could think a the complete system as a general nonlinear system and try to find Lyapunov functions. This is easier said than done, but indeed backstepping is a way to construct such Lyapunov functions. If you want to ascertain stability of the overall system using previously known stability properties of $f$ and $g$, small-gain theorems of operator-theoretic robust control theory can be useful. The key would be to identify the feedback loop errors that you want to be small.
– Pait
Sep 20 '15 at 0:58

1. Input to state stability (ISS). In this approach, one proves input to state stability of a cascade $x_1' = f(x_1,x_2), x_2' = g(x_2, u)$ by proving ISS of $x_1' = f(x_1,u)$ and ISS of $x_2' = g(x_2, u)$. I believe there are also conditions under which one only needs to prove global asymptotic stability of $x_2' = g(x_2, u)$ and then one can conclude global asymptotic stability of the cascade.
2. ISS with small gains. I don't really understand this technique, but the idea seems to be that cascade stability composes as long as the control input $u$ to the lower level system is "small enough" for some definition of small enough.
3. Singular perturbation/time-scale separation. I also don't really understand this technique, but the rough idea seems to be that if the lower level system ($x_2' = g(x_2, u)$) has significantly fast dynamics than the higher level system ($x_1' = f(x_1,x_2)$), then you can analyze stability separately and the cascade will be stable.