3.00 State-Space Analysis of CT Systems Compact Review

Chapter 3: State-Space Representation of Continuous-Time Systems

This compact note summarizes all core concepts, matrix formulations, properties, and standard transformations of State-Space analysis. Designed as a high-density, pre-exam review card.

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1. Foundational Terminology & Definitions

*(Target: Theory Descriptive / 5-Mark Question)*

• State of a System: The absolute minimal set of variables $\{x_1(t),x_2(t),\dots ,x_n(t)\}$ such that knowledge of these variables at $t=t_0$ (initial conditions), together with the input $u(t)$ for $t\ge t_0$, uniquely and completely determines the system’s output and future behavior for all $t\ge t_0$.
• State Variables: The individual, independent dynamic variables that represent the state of the system (typically chosen as physical energy-storing quantities).
• State Vector: An $n\times 1$ column vector $\mathbf{X}(t)$ whose components are the state variables.
• State Space: The $n$-dimensional mathematical space whose coordinate axes are defined by the state variables.
• Canonic Realizations: Structured state-space representations derived from transfer functions, grouped into Control Canonic Form (CCF), Observer Canonic Form (OCF), and Diagonal/Jordan Canonic Form.

Table 3.1: Comparison: Classical Transfer Function vs. Modern State-Space

Feature	Classical Transfer Function Method	Modern State-Space Representation
Initial Conditions	Constrained strictly to zero initial conditions ($y(0)=0$).	Fully incorporates arbitrary non-zero initial conditions [$X(0)$].
System Constraints	Strictly applicable to Linear Time-Invariant (LTI) systems.	Extends seamlessly to Non-linear and Time-varying systems.
Terminal Scope	Best suited for Single-Input Single-Output (SISO) setups.	Natively handles Multiple-Input Multiple-Output (MIMO) setups.
Internal Visibility	Treats system as a “Black Box” (input-to-output only).	Unveils the complete internal dynamics and state transitions.
Computation	Frequency-domain algebraic roots.	Time-domain matrix algebra (highly optimized for computers).

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2. Standard LTI Matrix Formulations

*(Target: Numerical Solving / System Realization)*

For an LTI system with $n$ state variables (order $n$), $m$ inputs, and $p$ outputs:

2.1 State Equation (First-Order Differential Matrix Set)

$\dot{\mathbf{X}}(t)=\mathbf{AX}(t)+\mathbf{BU}(t)$

• $\dot{\mathbf{X}}(t)$ = Derivative of State Vector ($n\times 1$)
• $\mathbf{X}(t)$ = State Vector ($n\times 1$)
• $\mathbf{A}$ = State / System Matrix ($n\times n$) — Dictates natural system modes
• $\mathbf{B}$ = Input / Control Matrix ($n\times m$) — Bridges external input to state change
• $\mathbf{U}(t)$ = Input Vector ($m\times 1$)

2.2 Output Equation (Algebraic Measurement Model)

$\mathbf{Y}(t)=\mathbf{CX}(t)+\mathbf{DU}(t)$

• $\mathbf{Y}(t)$ = Output Vector ($p\times 1$)
• $\mathbf{C}$ = Output / Observation Matrix ($p\times n$) — Maps states to physical output
• $\mathbf{D}$ = Direct Transmission / Feedforward Matrix ($p\times m$) — Usually zero; non-zero if numerator order matches denominator order

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3. Selection Symmetries of State Variables

*(Target: Model Formulation / Circuit Analysis)*

3.1 Physical Variable Method (Electrical Networks)

• The Energy Rule: The number of state variables equals the number of independent energy-storing elements (inductors $L$ and capacitors $C$).
• Standard Selection Assignments:
  • State Variable $x_k(t)=v_C(t)$ (Voltage across capacitor) $⟹{\dot{x}}_k(t)=\frac{d{v}_C}{dt}=\frac{i_C(t)}{C}$
  • State Variable $x_j(t)=i_L(t)$ (Current through inductor) $⟹{\dot{x}}_j(t)=\frac{d{i}_L}{dt}=\frac{v_L(t)}{L}$
• A+ Score-Saver Trap: Avoid writing dependent loops or nodes. Verify that capacitor loop configurations and inductor node junctions do not violate linear independence, reducing matrix dimensions.

3.2 Phase Variable Method (Higher-Order ODEs)

• The Derivative Rule: State variables are assigned as the system output $y(t)$ and its successive derivatives: $x_1(t)=y(t),x_2(t)=\dot{y}(t),x_3(t)=\ddot{y}(t),\dots ,x_n(t)=y^{(n-1)}(t)$
• Matrix Structure (Companion Form): For an $n$-th order ODE: $y^{(n)}(t)+a_1{y}^{(n-1)}(t)+\cdots +a_{n}y(t)=b_{0}u(t)$ 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & 1 \\ -a_n & -a_{n-1} & -a_{n-2} & \dots & -a_1 \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ b_0 \end{bmatrix}, \quad \mathbf{C} = \begin{bmatrix} 1 & 0 & 0 & \dots & 0 \end{bmatrix}$$

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4. Operational Block Diagram Realizations

*(Target: Block Diagram Design / Signal Flow Graphs)*

State equations are mapped directly onto active realizations using Integrators ($1/s$), Adders, and Multipliers.

• The Integrator Boundary Rule:
  • The output of the $k$-th integrator is defined as the state variable $x_k(t)$ (or $X_k(s)$).
  • The input of the $k$-th integrator represents the derivative ${\dot{x}}_k(t)$ (or $s{X}_k(s)$).
• Canonic Realization Symmetries:
  • Direct Form I: Employs separate delay/integrator chains for feedback poles and feedforward zeros.
  • Direct Form II (Canonic Form): Minimizes memory requirement by utilizing a shared integrator column for both poles and zeros.

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5. The State Transition Matrix (STM)

*(Target: Mathematical Proof / Analytical Solving)*

The state transition matrix $\mathbf{\Phi }(t)=e^{\mathbf{A}t}$ maps the initial state $\mathbf{X}(t_0)$ to a future state $\mathbf{X}(t)$ under unforced (zero-input) conditions.

5.1 Evaluation Pathways

1. S-Domain Laplace Method (Resolvent Matrix inversion): $\mathbf{\Phi }(t)={\mathcal{L}}^{-1}\left\{(s\mathbf{I}-\mathbf{A}{)}^{-1}\right\}\text{(derivation)}$
2. Time-Domain Infinite Power Series Method: $\mathbf{\Phi }(t)=e^{\mathbf{A}t}=\mathbf{I}+\mathbf{A}t+\frac{{\mathbf{A}}^2{t}^2}{2!}+\frac{{\mathbf{A}}^3{t}^3}{3!}+\dots \text{(derivation)}$

5.2 Five (5) Golden Algebraic Properties of STM

For any time parameters $t,t_0,t_1,t_2$:

1. Identity Property: $\mathbf{\Phi }(0)=\mathbf{I}$
2. Inversion Property: ${\mathbf{\Phi }}^{-1}(t)=[e^{\mathbf{A}t}{]}^{-1}=e^{-\mathbf{A}t}=\mathbf{\Phi }(-t)$
3. Cyclic Decomposition (Semi-group Property): $\mathbf{\Phi }(t_2-t_1)\mathbf{\Phi }(t_1-t_0)=\mathbf{\Phi }(t_2-t_0)$
4. Time Addition Property: $\mathbf{\Phi }(t_1+t_2)=\mathbf{\Phi }(t_1)\mathbf{\Phi }(t_2)=\mathbf{\Phi }(t_2)\mathbf{\Phi }(t_1)$
5. Power Shifting Property: ${\left[\mathbf{\Phi }(t)\right]}^k=\mathbf{\Phi }(kt)$

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6. Analytical Solution of State Equations

*(Target: Mathematical Proof / Full Numerical Solution)*

6.1 Unilateral s-Domain Resolvent Solution

Taking the Laplace transform of the state equation yields: $\mathbf{X}(s)=(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{X}(0)+(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{BU}(s)\text{(derivation)}$

6.2 Total Time-Domain Response Formulation

Taking the inverse Laplace transform of the s-domain equation yields the unified state trajectory: $\mathbf{X}(t)=\underset{\text{Zero-Input Response (ZIR)}}{\underbrace{\mathbf{\Phi }(t)\mathbf{X}(0)}}+\underset{\text{Zero-State Response (ZSR)}}{\underbrace{{\int }_0^t\mathbf{\Phi }(t-\tau )\mathbf{BU}(\tau )d\tau }}\text{(derivation)}$

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7. State-Space to Transfer Function Conversion

*(Target: Mathematical Proof / Matrix Transformation)*

To convert a modern state-space representation back to a classical frequency-domain transfer function:

• Governing Algebraic Form (Single-Input Single-Output): $H(s)=\frac{\mathbf{Y}(s)}{\mathbf{U}(s)}=\mathbf{C}(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{B}+\mathbf{D}\text{(derivation)}$
• The Pole Identity: The poles of $H(s)$ correspond exactly to the eigenvalues of the system matrix $\mathbf{A}$, calculated by solving the characteristic equation: $\det (s\mathbf{I}-\mathbf{A})=0$

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8. Discrete-Time State-Space Modeling

*(Target: Theory Descriptive / Discrete Difference Modeling)*

For digital networks represented by difference equations:

• Unified State and Output Formulation: \mathbf{X}[n+1] &= \mathbf{A}\mathbf{X}[n] + \mathbf{B}\mathbf{U}[n] \\ \mathbf{Y}[n] &= \mathbf{C}\mathbf{X}[n] + \mathbf{D}\mathbf{U}[n] \end{aligned}$$
• Discrete Unforced Total Solution: $\mathbf{X}[n]={\mathbf{A}}^n\mathbf{X}[0]+{\sum }_{k=0}^{n-1}{\mathbf{A}}^{n-1-k}\mathbf{BU}[k]\text{(derivation)}$

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9. Common Mistakes That Cost Marks

Critical Exam Pitfalls

1. Identity Matrix ($\mathbf{I}$) Omission: Writing $(s-\mathbf{A}{)}^{-1}$ instead of $(s\mathbf{I}-\mathbf{A}{)}^{-1}$. You cannot subtract a matrix from a scalar directly! Forgetting to scale $s$ by the Identity matrix $\mathbf{I}$ completely breaks the matrix dimensions.
2. Resolvent Scaling Factor Sign Swaps: Miscalculating the inverse of $(s\mathbf{I}-\mathbf{A})$. For a $2\times 2$ matrix, remember that the matrix inversion formula is: ${\mathbf{M}}^{-1}=\frac{1}{\det (\mathbf{M})}\text{adj}(\mathbf{M})$ Double-check the signs of your cofactors and determinant value before performing algebraic division.
3. Feedback Coefficient Sign Negation (Direct Form II): Forgetting to negate the feedback coefficient terms when mapping from a differential equation to feedback multiplier blocks.
4. Initial State Time-Shift Inversion: In discrete solutions, using ${\mathbf{A}}^{n-k}$ instead of the causal shift bound ${\mathbf{A}}^{n-1-k}$ inside the convolution summation.

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10. PYQ Bank — Verbatim Questions & Answer Plans

10.1 Phase Variable Derivative Realization

KUET 2024 Section B Q. 5a [13 Marks]

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:
  1. Define three successive phase-variables as state variables: $x_1(t)=y(t)$, $x_2(t)=\dot{y}(t)$, $x_3(t)=\ddot{y}(t)$.
  2. Write the derivatives of the first two variables: ${\dot{x}}_1(t)=x_2(t)$ and ${\dot{x}}_2(t)=x_3(t)$.
  3. Re-arrange the original differential equation to express ${\dot{x}}_3(t)$: ${\dot{x}}_3(t)=-7x_1(t)-6x_2(t)-x_3(t)+2u(t)$.
  4. Compile the state equation and output equation in standard matrix Companion Form.

10.2 Transfer Function Matrix Resolution

KUET 2019 Section B Q. 7b [10 Marks]

A state variable description of a system is given by the matrix equation: $\dot{\mathbf{X}}=\left[\begin{array}{cc}-1& 0\\ 1& -2\end{array}\right]\mathbf{X}+\left[\begin{array}{c}1\\ 0\end{array}\right]u,y=\left[\begin{array}{cc}1& 1\end{array}\right]\mathbf{X}$ Find: (i) The Transfer function, (ii) The State transition matrix, (iii) State diagram.

• Answer Plan:
  1. Identify matrices: $\mathbf{A}=\left[\begin{array}{cc}-1& 0\\ 1& -2\end{array}\right]$, $\mathbf{B}=\left[\begin{array}{c}1\\ 0\end{array}\right]$, $\mathbf{C}=\left[\begin{array}{cc}1& 1\end{array}\right]$, $\mathbf{D}=0$.
  2. Solve for $(s\mathbf{I}-\mathbf{A})=\left[\begin{array}{cc}s+1& 0\\ -1& s+2\end{array}\right]$ and evaluate its inverse resolvent matrix $(s\mathbf{I}-\mathbf{A}{)}^{-1}=\frac{1}{(s+1)(s+2)}\left[\begin{array}{cc}s+2& 0\\ 1& s+1\end{array}\right]$.
  3. Calculate the transfer function $H(s)=\mathbf{C}(s\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{B}=\frac{s+3}{(s+1)(s+2)}$.
  4. Determine the State Transition Matrix $\mathbf{\Phi }(t)={\mathcal{L}}^{-1}\{(s\mathbf{I}-\mathbf{A}{)}^{-1}\}=\left[\begin{array}{cc}{e}^{-t}& 0\\ e^{-t}-e^{-2t}& e^{-2t}\end{array}\right]$.
  5. Draw the state diagram illustrating feedback nodes and summing junctions based on individual state equations.

10.3 Discrete Time Delay Formulations

KUET 2022/2021 Section B Q. 8a [13 Marks]

Obtain the state-space representation of a discrete time system described by the following difference equation: $y(n-3)+2y(n-2)+3y(n-1)+4y(n)=U(n)$

• Answer Plan:
  1. Convert the difference equation into causal form by shifting indices: $y[n]+\frac{3}{4}y[n-1]+\frac{1}{2}y[n-2]+\frac{1}{4}y[n-3]=\frac{1}{4}u[n]$.
  2. Assign state variables to past delays: $x_1[n]=y[n-3]$, $x_2[n]=y[n-2]$, $x_3[n]=y[n-1]$.
  3. Formulate the state updating vector: $x_1[n+1]=x_2[n]$, $x_2[n+1]=x_3[n]$, $x_3[n+1]=-\frac{1}{4}{x}_1[n]-\frac{1}{2}{x}_2[n]-\frac{3}{4}{x}_3[n]+\frac{1}{4}u[n]$.
  4. Express the output equation: $y[n]=x_3[n+1]=-\frac{1}{4}{x}_1[n]-\frac{1}{2}{x}_2[n]-\frac{3}{4}{x}_3[n]+\frac{1}{4}u[n]$.
  5. Compile variables in discrete standard matrix form.

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11. Self-Check Before Moving On

• Can you define “State of a System” verbatim with its prerequisite mathematical boundaries?
• Can you write the complete s-domain state-space-to-transfer-function conversion equation and prove its eigenvalue-pole identity?
• Can you state the five algebraic properties of the State Transition Matrix ($\mathbf{\Phi }(t)$)?
• Do you know how to choose physical state variables for series/parallel capacitor-inductor networks using KVL/KCL?
• Can you draw a canonic Direct Form II realization layout from a third-order transfer function?

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Source: (k.Deergha Rao) signals and systems.pdf, ECE-2107_STATE_SPACE_ANALYSIS.pdf, Rabiul sir class notes, KUET Past Year Question Bank.