Why is using small signal models of nonlinear systems for controller design effective?

Question

my question does not contain any specific maths, but rather a general question about controller design using small signal models.

I'm familiar with the fact that it is often times hard to synthesize controllers for nonlinear systems, even when the system is described well from a mathematical viewpoint. The most common way of dealing with that problem is then to linearize the nonlinear system around an operating point (i.e. equilibrium point) and use the resulting linear model to derive a controller. This whole procedure makes sense to me - as long as the behaviour of the linearized system is approximately the same for the behaviour of that system being driven into this operating point.

As an example, imagine a simplified DC-DC converter with a constant load R and control input D (D for duty cycle of some switch in the circuit). My goal is now e.g. to drive the output of the converter from initially 0V / 0A to the equilibrium of 100V / 10mA. Say we'd need a duty cycle of D = 0.7 to reach that goal. We now need to design a controller that tries to do exactly that transition 0V / 0A -> 100V / 10mA using as little time as possible. Now, why does it make sense to linearize the plant about the operating point 100V / 10mA and use that as a basis for the controller design, when actually the resulting linear system dynamics covers the real, non-linear system dynamics only in close proximity of 100V / 10mA and nowhere near the whole transition 0V / 0A -> 100V / 10mA. I.e. the dynamics of the whole transition could be completely different from that in close to the equilibrium point 100V / 10mA. That means the controller would only work well if it e.g. should control the transition 100V / 10mA -> 110V / 10mA but behave completely different for much greater changes in equilibria.

So, how can the procedure be effective for the mentioned bigger transitions? Does one just assume that the system dynamics is approximately the same?

Greetings

David

Answer 1

Do you have an actual model for this?... Speaking without expressions is a bit "philosophical".

A nonlinear system $\mathscr{S}$ can be "anything". Nonlinear means only, not linear, it do not mean it is "smooth" or "stepwise linear" or anything with a pattern you could handle. It is a mistake assuming the dynamics is "the same". Also remember the system is not the algebraic or computational models we use to describe them (!).

Hence if we dont know anything, we cannot go from state $x(0)$ into state $x(t)$ if the system $\mathscr{S}$ is "anything"; it is nonsense. You need to have some knowledge of it. As you already have. Or as you will|would.

The linearized model $\mathscr{M}_{x_i}$ can be obtained only if the system is linearizable at the state $x_i$, i.e. if the state change $\frac{dx}{dt}$ can be defined through a linear function.

If you assume your model $\mathscr{M}$ reflects your system $\mathscr{S}$, this means there exist a state change function such as $\frac{dx}{dt}=F(x,u)$ (i.e. for a continuous, invariant time, dynamic model), and the linearization condition is equivalent to prove the partial derivatives of the state change function $\frac{d}{dx}F(x,u)$, $\frac{d}{dx}F(x,u)$ exist.

At that point, a standard controller $\mathscr{K}$ should behave well under the linearized model $\mathscr{M}_{x_i}$, again, if the state $x$ keeps under some neighborhood of $x_i$ for which the linearization is valid. Outside that neighborhood, the controller would not be neccesarily enough robust to take the state back to the linearization state $x_i$.

For the simplest DC-DC converters, such as a Buck Boost Converter, and assuming our system is algebraic (can be fitted by circuits models), it is useless to normalize at a given inductor current state $j(t)$, because the device is working on a duty cycle, keeping the circuit closed up to the same value $j_1=j(t_{1i})$ is reached at each period $t_{1i}$. Hence, you must seek for the relation between the duty cycle $d$ and the obtained current $i$, which will define a curve $i=f(d)$.

In this sense, this curve is a new system, known. Hence you can apply your controller and techniques over this new system, which is nonlinear yes, but known, algebraic, continuous, derivable, static and time invariant. So, with any simple feedforward controller, you can set the current $i$ at any desired value instantaneously just by applying $d=f^{-1}(i)$ (if you are not seeking to have an optimal control penalizing the effort control over the duty cycle). Of course no feedback controller will be able to do that in zero time.

Answer 2

It would be very inappropriate to use a controller synthesized using a linearized model near the target operating point to drive the "plant" there when its initial output is very far away from such a target point. Following this approach the controller will soon bring the plant to its limits (in your case, D=1; in other case the plant can be broken or its protection can intervene limiting the input signal) that means it will be working in open loop. For such high output changes, you should try other approaches (for instance, bang-bang control) and put on line the small signal controller only near the point about which the plant was linearized.