⚡ electrical engineering

The restrictive conditions to solve LTI Systems by Ordinary Differential Equations

This paper addresses the lack of clarity in standard engineering literature regarding the assumptions and limitations of solving linear time-invariant (LTI) systems via ordinary differential equations by formally defining the necessary input smoothness conditions and establishing a rigorous equivalence between ODEs and state space representations.

Original authors: Alexandre Sanfelici Bazanella, Tristão Garcia

Published 2026-04-10

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CC BY 4.0

Original authors: Alexandre Sanfelici Bazanella, Tristão Garcia

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the future path of a car (the output) based on how you are pressing the gas pedal (the input). In the world of engineering, we usually use a set of rules called Ordinary Differential Equations (ODEs) to make these predictions.

For decades, textbooks have taught us how to solve these equations using a mathematical "magic trick" called the Laplace Transform. It's like a translator that turns a complex, moving puzzle into a simple algebra problem you can solve with a pencil and paper.

The Problem: The "Magic Trick" Breaks Down
The problem is that this magic trick has a very strict rule: Everything must be smooth.

Imagine driving a car. If you gently press the gas, the car speeds up smoothly. The math works perfectly. But what if you slam the gas pedal from "off" to "full" in a split second? Or what if the road suddenly changes from smooth asphalt to a bumpy dirt track? In math terms, this is a discontinuity. The input signal jumps instantly.

When this happens, the standard math formula breaks. It asks for a value at the exact moment of the jump ( $t=0$ ). But at that exact instant, the value doesn't exist! It's like asking, "How fast is the car going exactly when the light turns green?" Is it 0? Is it 60? The math gets confused, and different textbooks give different, conflicting answers. Some say use the value before the jump; others say use the value after. This leaves students and engineers scratching their heads.

The Solution: The "State Space" GPS
The authors of this paper suggest a better way: stop trying to force the car to follow the broken math rules and instead look at the car's internal state.

Think of the car not just as a point moving on a map, but as a system with internal parts: the engine RPM, the gear position, the fuel pressure. This is called a State Space Representation (SSR).

Here is the magic of this new approach:

The Car is Continuous: Even if you slam the gas pedal (a discontinuous input), the car's internal parts (the engine, the gears) cannot change instantly. They have inertia. They move smoothly.
The Bridge: The authors show that you can translate the confusing "slamming gas" ODE into a smooth "State Space" model. Because the internal state is always smooth, the math never breaks. You can solve the problem easily without worrying about "undefined" moments.

The "Before" and "After" Trick
Once you have this smooth internal model, the authors give us a simple rule to handle the messy jump at the start:

The "Previous" Condition ( $t=0^-$ ): Where the car was just before you slammed the gas.
The "First" Condition ( $t=0^+$ ): Where the car is just after the gas hits, but before the car has had time to move significantly.

The paper proves a surprising fact: It doesn't matter which one you use, as long as you are consistent.

If you know where the car was before the jump, you can plug that into the formula.
If you know where the car is immediately after the jump, you can plug that in too.

The math works out the same either way, provided you don't mix them up (like using the "before" position with the "after" input).

The Big Picture
The authors are essentially saying: "Stop trying to solve the problem by staring at the confusing jump. Instead, look at the smooth, continuous machinery underneath it."

By switching from the old, rigid way of looking at equations to the State Space way, we can solve real-world problems (like switching circuits on and off, or sudden impacts) without getting stuck on mathematical technicalities. It turns a "broken" equation into a solvable one by realizing that while the input might jump, the system's memory (its state) flows smoothly through the chaos.

In a Nutshell:

Old Way: Try to calculate the speed of a car at the exact nanosecond it crashes into a wall. (Math breaks).
New Way: Look at the car's suspension and engine. They react smoothly even if the crash is sudden. Use that smooth reaction to predict the future.
Result: We can finally solve these problems clearly, without guessing which "magic number" to use at the moment of the jump.

1. Problem Statement

The paper addresses a fundamental ambiguity in the standard pedagogical treatment of Linear Time-Invariant (LTI) systems described by Ordinary Differential Equations (ODEs). Specifically, it highlights the difficulties encountered when solving Initial Value Problems (IVPs) where the input signal $u(t)$ or the output $y(t)$ is discontinuous at $t=0$ (e.g., step inputs).

The Core Issue: The standard method for solving ODEs involves applying the Laplace Transform, which relies on the property $L\{\dot{x}(t)\} = sX(s) - x(0)$ . This formulation requires well-defined initial conditions at $t=0$ . However, if the input $u(t)$ is discontinuous (like a Heaviside step function), its derivative $\dot{u}(t)$ involves Dirac delta functions, making terms like $u(0)$ and $\dot{u}(0)$ mathematically undefined or infinite within the realm of real numbers.
Literature Gap: Existing textbooks and literature are inconsistent in handling these cases. Some implicitly assume continuity (ignoring discontinuities), while others arbitrarily choose $u(0)=0$ or $u(0)=1$ without rigorous justification. This lack of uniformity creates confusion regarding which initial conditions (values at $t=0^-$ , $t=0$ , or $t=0^+$ ) should be used in the solution formulas.
Philosophical Concern: The authors question the validity of seeking real-valued solutions to equations where the right-hand side (RHS) contains singularities (infinite values) that do not exist in the real number system.

2. Methodology

The authors employ a rigorous theoretical approach combining classical ODE analysis with State Space Representation (SSR) theory to resolve these ambiguities.

Formal Definition of Continuity: They define the conditions under which traditional ODE solutions are valid. They introduce Theorem 1, stating that the output vector $Y(t)$ (containing $y(t)$ and its derivatives) is continuous at $t=0$ if and only if the input $u(t)$ and its first $m-1$ derivatives are continuous at $t=0$ (where $m$ is the relative degree of the system).
State Space Equivalence: To handle cases where Theorem 1 is violated (discontinuous inputs), the authors propose defining the solution of the ODE as the solution of an equivalent State Space Representation (SSR).
- They establish a formal definition of equivalence: An SSR is equivalent to an ODE if they share the same input-output response and the same response to initial conditions.
- They derive the relationship between the ODE initial conditions $Y(0)$ and the SSR state vector $x(0)$ using the observability matrix ( $O$ ) and the Markov parameter matrix ( $M$ ).
Reformulation of IVPs: The authors redefine the IVP to focus on $t > 0$ $t > 0$ , distinguishing between:
- Previous Conditions ( $0^-$ ): Values just before the input change.
- First Conditions ( $0^+$ ): Values just after the input change.
Derivation of the Jump Condition: Using the continuity of the state vector $x(t)$ in SSRs (which holds for any finite input), they derive a mapping equation (Eq. 22) to calculate the "first conditions" $Y(0^+)$ from the "previous conditions" $Y(0^-)$ and the jump in the input $U(0^+) - U(0^-)$ .
Proof of Invariance: They prove Theorem 2, demonstrating that the standard Laplace solution formula yields the exact same result whether one substitutes the "previous conditions" ( $0^-$ ) or the "first conditions" ( $0^+$ ), provided the substitution is consistent (i.e., one does not mix $0^-$ and $0^+$ values).

3. Key Contributions

Identification of Restrictive Assumptions: The paper explicitly identifies that standard textbook solutions for ODEs are only valid under strict smoothness assumptions (continuous inputs) that are rarely met in practical engineering applications (e.g., step responses).
Rigorous Equivalence Definition: The authors provide a formal definition of equivalence between ODEs and SSRs that encompasses both input-output behavior and initial condition responses, a nuance often missing in standard realization theory.
Resolution of the "Discontinuity Paradox": By defining the ODE solution via its equivalent SSR, the authors bypass the need for singular functions (distributions/Dirac deltas) to solve the equation. Since the state $x(t)$ in an SSR is always continuous for finite inputs, the solution remains within the realm of real numbers.
Practical Solution Formula: They derive a simple, robust formula (Eq. 22) to convert "previous conditions" (known system states before an event) to "first conditions" (required for standard solution methods) based on the instantaneous change in the input.
Unification of Approaches: The paper proves that engineers can use either $t=0^-$ or $t=0^+$ values in standard Laplace formulas without error, as long as they are consistent and aware of the input's history.

4. Results

Theorem 1: Confirms that discontinuities in the output derivatives occur only when the input derivatives exceed the system's relative degree.
Theorem 2: Proves that the standard ODE solution formula is invariant to the choice of $0^-$ or $0^+$ initial conditions, provided the input history is accounted for correctly.
Example Validation: Through Examples 1, 2, and 3, the authors demonstrate that calculating the solution using the "first conditions" derived from the SSR yields the same result as using "previous conditions" directly in the Laplace domain formula.
State Continuity: The analysis confirms that while $y(t)$ or its derivatives may jump at $t=0$ , the underlying state vector $x(t)$ of the equivalent SSR remains continuous, validating the SSR as a robust mathematical framework for these problems.

5. Significance

Pedagogical Clarity: The paper resolves long-standing confusion in engineering education regarding how to handle initial conditions for discontinuous inputs. It provides a clear, theoretically sound alternative to the "guesswork" often found in textbooks.
Theoretical Robustness: It removes the reliance on "psychedelic" mathematical objects (singular distributions) for solving standard engineering problems, grounding the solution in the continuous dynamics of state space models.
Practical Applicability: The derived mapping (Eq. 22) allows engineers to correctly model systems where inputs switch abruptly (common in control and circuit theory) without needing advanced distribution theory.
Bridging Disciplines: The work effectively bridges the gap between classical ODE theory and modern State Space theory, arguing that SSRs are not just an alternative representation but a necessary tool for rigorously defining solutions in the presence of discontinuities.

In conclusion, the authors argue that the State Space Representation is not merely a computational tool but a fundamental framework that expands the validity of LTI system solutions to include discontinuous inputs, providing a clear, consistent, and mathematically sound method for solving Initial Value Problems in engineering.

1. Problem Statement

2. Methodology

3. Key Contributions

4. Results

5. Significance

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