⚡ electrical engineering

State-Space Modelling and Analysis

This paper introduces the State-Space Modelling and Analysis aspect of Advanced Control Theory for Practical Applications, emphasizing the essential role of mathematical methodology in bridging theoretical knowledge with real-world engineering challenges.

Original authors: Hao Li

Published 2026-06-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hao Li

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

This paper is essentially a "how-to" guide for building the mathematical brains of machines that need to move and stay balanced. The author, Hao Li, argues that while control systems involve hardware, software, and economics, the most crucial part is the mathematical method used to understand how things move.

Here is the core of the paper, explained through everyday analogies.

1. The State-Space Map: Knowing Where You Are

Imagine you are driving a car. To drive safely, you don't just need to know where you are (your position). You also need to know how fast you are going, which way you are facing, and how quickly you are turning.

In the paper, this collection of information is called the "State."

The Analogy: Think of the "State" as a complete snapshot of a system's life at a specific moment. For a robot balancing a pole on a cart, the state isn't just "the pole is up." It's "the pole is up, leaning 5 degrees to the left, falling at 2 degrees per second, and the cart is moving forward at 3 meters per second."
The Math: The paper uses a "State Differential Equation" to describe how this snapshot changes over time. It's like a recipe that says: "If you are in this state and you push the cart this hard, here is exactly what your next state will be."

The paper shows how to write these recipes for:

Inverted Pendulums: Balancing a broomstick on your hand (or a cart).
Double Inverted Pendulums: Balancing a broomstick on top of another broomstick (much harder!).
Autonomous Vehicles: Cars and motorcycles driving themselves.

2. The "Self-Evolving" System: The System on Autopilot

Once you decide on a rule for how to control the machine (a "feedback law"), the system becomes a Self-Evolving System.

The Analogy: Imagine a marble rolling in a bowl. Once you let go, the marble doesn't need a human to push it anymore; it follows the shape of the bowl on its own. Its future path is determined entirely by where it is right now.
The Paper's Claim: When you apply a control rule, the complex machine acts like this marble. Its future is entirely determined by its current state. The paper calls this a "closed-loop feedback system."

3. Stability: Will It Fall Over?

The most important question for any robot or car is: Will it stay upright, or will it crash? This is called Stability.

The Analogy: Think of a pencil balanced on its tip. If you nudge it, it falls (unstable). Now think of a ball sitting at the bottom of a bowl. If you nudge it, it wobbles but eventually settles back at the bottom (stable).
The Paper's Tools:
- Eigenvalues (The "Falling Speed"): For simple, straight-line systems, the paper uses a mathematical tool called "eigenvalues." If these numbers are "negative" (in a specific mathematical sense), it means the system is like the ball in the bowl—it will naturally settle down. If they are positive, it's like the pencil—it will fall over.
- Routh-Hurwitz (The "Checklist"): Sometimes the math is too messy to find those numbers directly. The paper uses a "checklist" method (Routh-Hurwitz) to look at the coefficients of the equations. If the signs in the checklist flip back and forth, the system is unstable.
- Lyapunov (The "Energy" Test): For complex, non-linear systems (like a motorcycle leaning into a turn), the paper uses a method called Lyapunov Stability.
  - The Metaphor: Imagine a hill. If you can prove that a ball rolling on this hill is always losing height (energy) and never gaining it, you know it will eventually reach the bottom and stop. The paper looks for a mathematical "energy function" that always goes down. If it does, the system is safe.

4. Controllability: Can We Actually Drive It?

Before trying to build a controller, you must ask: Is it even possible to control this thing?

The Analogy: Imagine a car with no steering wheel and no brakes. No matter how hard you push the gas pedal, you cannot make it turn or stop. It is "uncontrollable."
The Paper's Claim: The paper introduces a "Controllability Matrix." This is a mathematical test to see if the inputs (like the steering wheel or cart motor) have enough power to move the system to any desired state.
The Result: The paper proves that even a Double Inverted Pendulum (the super-hard balancing act) is controllable. It's difficult, but mathematically possible. This gives engineers the confidence to try building it.

5. The Riccati Equation: The "Perfect" Control Plan

Finally, the paper touches on the Riccati Equation.

The Analogy: Imagine you are driving a car and want to reach a destination. You could slam on the brakes and swerve wildly (fast but dangerous). You could drive very slowly (safe but inefficient). The Riccati equation helps you find the perfect balance—the path that gets you there quickly but uses the least amount of fuel and causes the least wear and tear.
The Paper's Claim: This equation helps calculate the "optimal" control strategy. The paper shows how to solve this equation iteratively (step-by-step) to find that perfect balance.

Summary

This paper is a foundational guide for Advanced Control Theory. It teaches us how to:

Map a system's current condition (State-Space).
Predict how it will behave on its own (Self-Evolution).
Test if it will stay stable or crash (Stability Analysis).
Verify if we can actually steer it (Controllability).
Calculate the most efficient way to control it (Riccati Equation).

It uses real-world examples like balancing poles and driving cars to show that while the math is complex, the logic is about keeping things balanced and moving in the right direction.

Technical Summary: State-Space Modelling and Analysis

Problem Statement
Control science is identified as a cornerstone of modern civilization, yet its practical application requires a rigorous methodological foundation beyond hardware, software, or economic considerations. The paper addresses the need for a systematic approach to State-Space Modelling and Analysis, a core aspect of Advanced Control Theory. While classical control (dominant during the mid-20th century) relies on $s$ -domain analysis and is generally limited to single-input-single-output (SISO) problems, modern control utilizes state-space analysis to handle complex, multi-variable systems (MIMO). The paper seeks to bridge the gap between theoretical mathematical frameworks and practical engineering applications by providing a unified treatment of modelling, stability analysis, and controllability for diverse dynamic systems.

Methodology
The paper employs a structured mathematical approach, progressing from non-linear state differential equations to linearized approximations, and finally to stability and controllability analysis.

State-Space Modelling:
- Non-linear Formulation: The dynamics of various systems are initially modelled using generic non-linear state differential equations of the form $\dot{x} = f(x, u)$ $\overset{x}{˙} = f (x, u)$ . Specific applications include:
  - Inverted Pendulums: Single and double inverted pendulums on carts, deriving complex non-linear equations involving trigonometric functions of angles and accelerations.
  - Autonomous Vehicles: Low-speed (kinematic bicycle model) and high-speed (dynamic bicycle model incorporating tyre cornering stiffness and side-slip angles) vehicle navigation.
  - Motorcycles/Bicycles: Modelling lateral and vertical (roll) dynamics, highlighting the coupling between steering and balance.
- Linearization: For systems operating near an equilibrium point (e.g., small angles), the non-linear equations are linearized to form the standard linear state-space representation: $\dot{x} = Ax + Bu$ .
Self-Evolutionary Systems and Feedback:
- The paper defines closed-loop feedback systems as self-evolutionary (autonomous) systems where the control input $u$ is a function of the state $x$ (i.e., $u = g(x)$ ), resulting in $\dot{x} = f_c(x)$ .
- It establishes that the stability of a linear self-evolutionary system is determined by the eigenvalues of the state transition matrix $A_c$ .
Stability Analysis:
- Linear Systems: Stability is assessed via the eigenvalue criterion (all eigenvalues must have negative real parts). The paper connects this to the Routh-Hurwitz criterion for determining root locations without explicit eigenvalue computation.
- Non-Linear Systems: The Lyapunov strategy is introduced. Stability is proven by finding a scalar function $V(x)$ that is positive definite with a negative definite (or semi-definite) time derivative.
- Matrix Equations: The paper details the Lyapunov equation ( $PX + X^TP = Y$ ) and the Riccati equation (in three formalisms: $P A + A^T P - P B R^{-1} B^T P + Q = 0$ , etc.). It provides iterative algorithms and Hamiltonian matrix methods for solving these equations, linking them to optimal control and stability verification.
Controllability Analysis:
- The paper defines controllability as the ability to steer a system from any initial state to any desired state.
- It introduces the Controllability Matrix ( $C_{A,B} = [B, AB, \dots, A^{n-1}B]$ ) and establishes that a linear system is controllable if and only if this matrix has full rank.

Key Contributions and Results

Unified Modelling Framework: The paper provides explicit derivations for the state differential equations of complex systems, including the double inverted pendulum (both single-input and multi-input variants), autonomous vehicles (accounting for tyre side-slip at high speeds), and motorcycles (incorporating roll dynamics).
Stability Verification:
- Demonstrates that P-lateral control for low-speed autonomous vehicles is inherently unstable using the Routh-Hurwitz criterion, regardless of the proportional gain $P$ .
- Verifies the stability of a damped pendulum and a rotating disk using Lyapunov functions and eigenvalue analysis.
- Proves the stability of a rotating disk under P-control using the Lyapunov inequality ( $PA_c + A_c^T P < 0$ ).
Controllability Validation:
- Confirms that the double inverted pendulum is controllable (under linear assumptions), countering intuitive skepticism about its feasibility.
- Confirms the controllability of low-speed autonomous vehicle lateral control.
Algorithmic Solutions: The paper presents specific algorithms and MATLAB code snippets for:
- Solving Lyapunov equations via vectorization.
- Iteratively solving Riccati equations to find stabilizing gain matrices.
- Computing controllability matrices symbolically.
Theoretical Unification: It explicitly demonstrates the equivalence between linear state-space modelling and system modelling via Laplace transforms, showing that the characteristic polynomial of the state matrix $A$ is identical to the denominator of the system's transfer function.

Significance
The paper positions itself as a foundational chapter (Chapter 1) of a broader work on "Advanced Control Theory for Practical Applications." Its significance lies in:

Practical Applicability: It emphasizes that control theory is not merely abstract mathematics but is deeply rooted in practical applications, offering a "know-why" and "know-how" fusion.
Methodological Rigor: It provides a systematic, step-by-step methodology for moving from physical system descriptions to state-space models, linearization, and rigorous stability/controllability checks.
Educational Value: By working through complex examples like the double inverted pendulum and high-speed vehicle dynamics, it illustrates how theoretical tools (Lyapunov, Routh-Hurwitz, Controllability matrices) can resolve doubts about the feasibility of controlling inherently unstable or complex systems.
Foundation for Advanced Control: The paper serves as a prerequisite for understanding more advanced topics (such as optimal control and robust control) by establishing the necessary mathematical formalisms (Riccati equations, controllability) required for those fields.

The author maintains a modest tone, noting that while theoretical analysis is indispensable, it complements rather than replaces practical experience, and that the provided models are approximations suitable for specific operating regimes (e.g., low-speed vs. high-speed, equilibrium vs. large deviations).