Question about translating input/output ODE to state space representation

Question

It is well known that a differential equation (that models an input/output behavior of single input) for an LTI system, of the form

$$b_{n-1}\frac{d^{n-1} y(t)}{d t^n} + ... + b_1\frac{d y(t)}{d t} + b_0y(t) = u(t)$$

can be translated into the state space:

$$ \mathbf{\dot{x}}(t) = A\mathbf{x}(t) + B\mathbf{u}(t)$$

$$ y(t) = C\mathbf{x}(t) + D\mathbf{u}(t)$$

with $A$,$B$, $C$, $D$, real matrices of appropriate sizes

however my question is

If I have a more general equation such as:

$$b_{n-1}\frac{d^{n-1} y(t)}{d t^n} + ... + b_1\frac{d y(t)}{d t} + b_0y(t) = a_{m-1}\frac{d^{m-1} u(t)}{d t^{m-1}} + ... + a_1\frac{d u(t)}{d t} + a_0u(t)$$

¿Can I translate this into the state space equation?

If so, could you point me to bibliography where I can see this.

The reason why I'm asking this, is because I asked Chat GPT what was the most general differential equation that models an LTI system and it's answer was this:

OF course I then asked why was that the most general equation and where did it found it, it told me that it can't say what was it's source. I've been looking at books on control theory and I can't find that general form that GPT said, only the first one.

Answer 1

Since you ask for the general solution and an explicit reference: This problem was solved by Ülle Kotta in the paper

Kotta, Ü. "Removing input derivatives in generalized state-space systems: a linear algebraic approach." 1998 4th International Conference on Actual Problems of Electronic Instrument Engineering Proceedings. APEIE-98 (Cat. No. 98EX179). IEEE, 1998.

There, a general nonlinear system with finite but arbitrary number of input derivatives is considered and necessary and sufficient conditions are given for the existence of a generalized state transformation such that the transformed system depends only on (new) state variables and inputs without derivative.

A corrolary of this result is given as Corollary 2 in the paper for linear systems of the form

$$ \dot{x}=Fx+G_1u+\dots+G_{\alpha+1}u^{(\alpha)} $$

where $u^{(\alpha)}$ is the $\alpha$-th derivative of $u$. Then there always exists a state transformation of the form

$$ z=Px+R_1u+\dots+R_{\alpha}u^{(\alpha-1)} $$

where $P$ is an invertible matrix such that

$$ \dot{z}=\widetilde{F}z+\widetilde{G}u $$

See also the references given in that paper, i.e.

• Delaleau, E., & Respondek, W. (1992, December). Removing input derivatives and lowering their orders in generalized state-space representations. In [1992] Proceedings of the 31st IEEE Conference on Decision and Control (pp. 3663-3668). IEEE.
• Di Benedetto, M. D., Grizzle, J. W., & Moog, C. H. (1989). Rank invariants of nonlinear systems. SIAM journal on control and optimization, 27(3), 658-672.
• Fliess, M. (1990). Some basic structural properties of generalized linear systems. Systems & Control Letters, 15(5), 391-396.
• Pottle, C. (1966). State-space techniques for general active network analysis. System Analysis by Digital Computers.