3.01 Introduction to State-Space and Phase-Variable Formulat

Related Concepts: 3.02 State-Space Modeling of Transfer Functions (Canonic Forms) | 3.03 State-Space Modeling of Electrical Circuits | 3.06 State Transition Matrix: Properties & Solution Methods

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3.01 Introduction to State-Space & Phase-Variable Formulations

Core Idea

State-space analysis is a modern, time-domain mathematical model that reduces an $n$-th order linear differential equation into a system of $n$ coupled, first-order differential equations. This representation uses matrix algebra to track the internal energy states of a system alongside its inputs and outputs, providing a unified framework for multi-input multi-output (MIMO) systems and system stability analysis.

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1. The Philosophy of State-Space Representation

Classic time-domain analysis of continuous-time LTI systems relies on solving high-order linear constant-coefficient differential equations (LCCDEs) {equations relating inputs and outputs through derivatives} or converting them into $s$-domain transfer functions using Laplace transforms. While powerful, these classical methods have distinct drawbacks:

• They treat the system as a “black box,” focusing only on the input-output relationship and completely ignoring the internal states (such as capacitor voltages or inductor currents).
• They are highly tedious to apply to Multi-Input Multi-Output (MIMO) systems.
• They assume zero initial conditions when computing transfer functions, losing critical historical system state data.

State-space analysis solves these limitations by introducing a set of internal variables called state variables. By utilizing vector-matrix notation, we can analyze the internal physical states and external responses of any system simultaneously, even under non-zero initial conditions.

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2. The Core Matrix Mathematical Formulation

To mathematically represent a continuous-time system in state-space, we define three primary mathematical vectors:

1. State Vector ($\mathbf{X}(t)$): An $n\times 1$ column vector containing the $n$ state variables of the system: $\mathbf{X}(t)=\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_n(t)\end{array}\right]$
2. Input Vector ($\mathbf{U}(t)$): An $m\times 1$ column vector containing the $m$ external input signals: $\mathbf{U}(t)=\left[\begin{array}{c}{u}_1(t)\\ u_2(t)\\ ⋮\\ u_m(t)\end{array}\right]$
3. Output Vector ($\mathbf{Y}(t)$): A $p\times 1$ column vector containing the $p$ external output signals: $\mathbf{Y}(t)=\left[\begin{array}{c}{y}_1(t)\\ y_2(t)\\ ⋮\\ y_p(t)\end{array}\right]$

These vectors are coupled through two fundamental, first-order matrix differential equations:

2.1 The State Equation

The State Equation relates the first derivative of the state vector to the current states and external inputs: $\dot{\mathbf{X}}(t)=\mathbf{AX}(t)+\mathbf{BU}(t)$

2.2 The Output Equation

The Output Equation expresses the system outputs as a linear combination of the current states and external inputs: $\mathbf{Y}(t)=\mathbf{CX}(t)+\mathbf{DU}(t)$

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2.3 Structural Matrix Dimension Map

Understanding the exact dimensions of these matrices is a heavily tested conceptual topic in ECE 2107 examinations. If the system order is $n$, the input count is $m$, and the output count is $p$, the matrices must comply with the following structural dimensions:

Matrix	Physical Name	Dimensions	Role in the System
$\mathbf{A}$	System Matrix	$n\times n$	Governs the internal dynamics and natural stability of the system.
$\mathbf{B}$	Input Matrix	$n\times m$	Determines how external inputs affect the internal state transitions.
$\mathbf{C}$	Output Matrix	$p\times n$	Governs how the internal states map to the measurable output terminals.
$\mathbf{D}$	Direct Transmission Matrix	$p\times m$	Controls direct feedforward paths bypassing the system states. (Usually $\mathbf{D}=0$ in physical networks).

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3. The Phase-Variable Formulation Method

The Phase-Variable Method is a systematic, algorithmic approach used to convert a single $n$-th order continuous-time differential equation into standard state-space matrices.

3.1 General Workflow

Consider a general $n$-th order differential equation without input derivatives: $\frac{d^{n}y(t)}{d{t}^n}+a_{n-1}\frac{d^{n-1}y(t)}{d{t}^{n-1}}+\cdots +a_1\frac{dy(t)}{dt}+a_{0}y(t)=b_{0}u(t)$

Step 1: Assign State Variables

We assign the system output $y(t)$ and its successive derivatives as our phase variables: $x_1(t)=y(t)$ $x_2(t)=\dot{y}(t)={\dot{x}}_1(t)$ $x_3(t)=\ddot{y}(t)={\dot{x}}_2(t)$ $⋮$ $x_n(t)=y^{(n-1)}(t)={\dot{x}}_{n-1}(t)$

Step 2: Formulate First-Order State Derivatives

Express the derivative of each state variable in terms of the other states: ${\dot{x}}_1(t)=x_2(t)$ ${\dot{x}}_2(t)=x_3(t)$ $⋮$ ${\dot{x}}_{n-1}(t)=x_n(t)$

To find the final derivative ${\dot{x}}_n(t)$, we substitute our state variable definitions back into the original $n$-th order differential equation: ${\dot{x}}_n(t)=y^{(n)}(t)=-a_0{x}_1(t)-a_1{x}_2(t)-\cdots -a_{n-1}{x}_n(t)+b_{0}u(t)$

Step 3: Construct the Companion Matrix Form

Now, we gather these first-order differential equations directly into the canonical matrix format: $\left[\begin{array}{c}{\dot{x}}_1(t)\\ {\dot{x}}_2(t)\\ ⋮\\ {\dot{x}}_{n-1}(t)\\ {\dot{x}}_n(t)\end{array}\right]=\left[\begin{array}{ccccc}0& 1& 0& \dots & 0\\ 0& 0& 1& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 1\\ -a_0& -a_1& -a_2& \dots & -a_{n-1}\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_{n-1}(t)\\ x_n(t)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ b_0\end{array}\right]u(t)$

Since our output is $y(t)=x_1(t)$, the corresponding Output Equation is: $y(t)=\left[\begin{array}{ccccc}1& 0& 0& \dots & 0\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ ⋮\\ x_n(t)\end{array}\right]+\left[\begin{array}{c}0\end{array}\right]u(t)$

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4. Comprehensive Worked Examples (The Exam Killers)

4.1 Example 1: Third-Order System Formulation [PYQ 2024]

Question: Represent the following differential equation in a state model: $\frac{d^{3}y(t)}{d{t}^3}+\frac{d^{2}y(t)}{d{t}^2}+6\frac{dy(t)}{dt}+7y(t)=2u(t)$

Step-by-Step Solution:

1. Define Phase Variables ($n=3$): Let: $x_1(t)=y(t)$ $x_2(t)=\dot{y}(t)={\dot{x}}_1(t)$ $x_3(t)=\ddot{y}(t)={\dot{x}}_2(t)$

2. Formulate State Derivatives: From our phase variable definitions: ${\dot{x}}_1(t)=x_2(t)$ ${\dot{x}}_2(t)=x_3(t)$

   To find ${\dot{x}}_3(t)$, isolate the highest-order derivative ($\stackrel{...}{y}(t)$) in the original differential equation: $\stackrel{...}{y}(t)=-\ddot{y}(t)-6\dot{y}(t)-7y(t)+2u(t)$

   Substitute our defined state variables into this equation: ${\dot{x}}_3(t)=-7x_1(t)-6x_2(t)-x_3(t)+2u(t)$

3. Construct the State Equation: Gather the equations into vector-matrix form $\dot{\mathbf{X}}(t)=\mathbf{AX}(t)+\mathbf{BU}(t)$: $\left[\begin{array}{c}{\dot{x}}_1(t)\\ {\dot{x}}_2(t)\\ {\dot{x}}_3(t)\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -7& -6& -1\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 2\end{array}\right]u(t)$

4. Construct the Output Equation: Since $y(t)=x_1(t)$, map it to standard format $\mathbf{Y}(t)=\mathbf{CX}(t)+\mathbf{DU}(t)$: $y(t)=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]+\left[\begin{array}{c}0\end{array}\right]u(t)$

   Thus, the final state-space matrices are: $\mathbf{A}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -7& -6& -1\end{array}\right],\mathbf{B}=\left[\begin{array}{c}0\\ 0\\ 2\end{array}\right],\mathbf{C}=\left[\begin{array}{ccc}1& 0& 0\end{array}\right],\mathbf{D}=\left[\begin{array}{c}0\end{array}\right]$

5. State-Space Block Diagram Realization: To ensure maximum exam marks, you must be ready to sketch the physical realization of your phase variable system.

   [DIAGRAM: Continuous 3-integrator block diagram (Direct Form II style) realizing the state-space equations. Left side: input u(t) multiplied by 2 feeding the final adder. Center: a cascade of three 1/s integrator blocks whose outputs represent states x3(t), x2(t), and x1(t) from left to right. Bottom: feedback lines with multipliers -1, -6, and -7 routing from x3, x2, and x1 back into the input adder. Right side: Output y(t) tapped directly from state x1(t) - source: Textbook Sec 3.1]

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4.2 Example 2: Classroom Classical Realization

Question: Convert the following third-order system into state matrices and sketch its integrator block diagram: $\stackrel{...}{y}(t)+6\ddot{y}(t)+11\dot{y}(t)+6y(t)=u(t)$

Step-by-Step Solution:

1. Define Phase Variables: Let: $x_1(t)=y(t)$ $x_2(t)=\dot{y}(t)={\dot{x}}_1(t)$ $x_3(t)=\ddot{y}(t)={\dot{x}}_2(t)$

2. State Derivatives: ${\dot{x}}_1(t)=x_2(t)$ ${\dot{x}}_2(t)=x_3(t)$ ${\dot{x}}_3(t)=-6x_1(t)-11x_2(t)-6x_3(t)+u(t)$

3. Matrix Realization: $\left[\begin{array}{c}{\dot{x}}_1(t)\\ {\dot{x}}_2(t)\\ {\dot{x}}_3(t)\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -6& -11& -6\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]u(t)$ $y(t)=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1(t)\\ x_2(t)\\ x_3(t)\end{array}\right]+\left[\begin{array}{c}0\end{array}\right]u(t)$

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5. Common Mistakes That Cost Marks

The Zero-Coefficient Trap

If your differential equation skips a derivative term (for example, missing the $\dot{y}(t)$ term), do not skip the state variable! Keep the variable and write its coefficient as zero in your matrices. Skipping a state variable collapses the matrix order and results in a zero-mark evaluation.

The Non-Unity Coefficient Danger

If your differential equation starts with a non-unity coefficient on the highest derivative (e.g., $3\stackrel{...}{y}(t)+\ddot{y}(t)+\cdots =u(t)$), you must divide the entire equation by 3 before starting the phase variable assignments. Neglecting this division step scales all of your feedback and input matrix coefficients incorrectly.

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6. PYQ Bank — Verbatim Questions & Answer Plans

6.1 PYQ 2024 Question 2c [13 Marks]

Question: Represent the following differential equation given below in a state model: $\frac{d^{3}y(t)}{d{t}^3}+\frac{d^{2}y(t)}{d{t}^2}+6\frac{dy(t)}{dt}+7y(t)=2u(t)$

• Answer Plan: Follow the exact step-by-step mathematical formulation shown in Section 4.1. Define phase states $x_1,x_2,x_3$, calculate the derivatives, write the coupled matrices, state the dimensions, and draw the matching 3-integrator block diagram to lock in all 13 marks.

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7. Self-Check Before Moving On

• Can you define State, State Vector, State Equation, and Output Equation mathematically? [9.01, 9.02]
• Do you know how to determine the exact matrix dimensions of $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$ for any $n$-th order MIMO system? [9.02]
• Can you convert any continuous-time differential equation with constant coefficients into companion matrix form using phase variables? [9.03]
• Have you memorized the warning to divide by the highest-order coefficient before building state matrices? [9.05]

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Source: (k.Deergha Rao) signals and systems.pdf (Chapter 3), ECE-2107_STATE_SPACE_ANALYSIS.pdf, Azmat Sir-2309008.pdf (Class Lecture Slides).