3.04 State-Space Modeling of Block Diagrams & SFGs | 3.06 State Transition Matrix: Properties & Solution Methods

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3.05 Discrete-Time State-Space Representation

Core Idea

Just as continuous-time (CT) systems can be represented in state-space using derivatives, discrete-time (DT) systems described by linear constant-coefficient difference equations (LCCDEs) {equations relating discrete input and output sequences through delays} can be modeled using first-order shift-operator difference matrix equations. Instead of continuous integrators ($1/s$), DT systems utilize unit delays ($z^{-1}$) as their physical memory storage elements, where the state variables represent delayed values of the output or internal signals.

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1. The Discrete-Time State-Space Formulation

In the discrete-time domain, the state-space model is formulated as a set of coupled, first-order difference matrix equations that track state transitions from sample index $n$ to $n+1$ {the discrete sample counter representing sequential clock ticks}.

1.1 The Discrete State Equation

The state equation governs how the internal state vector at the next clock cycle ($\mathbf{X}[n+1]$) is computed from the current state vector ($\mathbf{X}[n]$) and the current input vector ($\mathbf{U}[n]$): $\mathbf{X}[n+1]=\mathbf{AX}[n]+\mathbf{BU}[n]$

1.2 The Discrete Output Equation

The output equation maps the current states and current inputs directly to the system’s measurable output terminals ($\mathbf{Y}[n]$): $\mathbf{Y}[n]=\mathbf{CX}[n]+\mathbf{DU}[n]$

Where:

• $\mathbf{X}[n]$ is the $n\times 1$ State Vector containing the internal memory states.
• $\mathbf{U}[n]$ is the $m\times 1$ Input Vector containing the external excitation sequences.
• $\mathbf{Y}[n]$ is the $p\times 1$ Output Vector containing the system outputs.
• $\mathbf{A}$ is the $n\times n$ System Matrix (governing natural dynamics and discrete poles).
• $\mathbf{B}$ is the $n\times m$ Input Matrix.
• $\mathbf{C}$ is the $p\times n$ Output Matrix.
• $\mathbf{D}$ is the $p\times m$ Direct Transmission Matrix.

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2. Phase-Variable Selection in Discrete Time

For an $N$-th order recursive LCCDE with output delays: $y[n]+a_{1}y[n-1]+a_{2}y[n-2]+\cdots +a_{N}y[n-N]=b_{0}u[n]$

To eliminate high-order delays and construct first-order state-space matrices, we apply a standard Delay-Chain Phase Variable Mapping.

Step-by-Step Selection Workflow

Step 1: Assign State Variables to Bounded Delays

We define our state variables as the output of the unit delay elements, starting from the longest delay ($y[n-N]$) up to the single-sample delay ($y[n-1]$): $x_1[n]=y[n-N]$ $x_2[n]=y[n-N+1]$ $x_3[n]=y[n-N+2]$ $⋮$ $x_N[n]=y[n-1]$

Step 2: Write First-Order Shift Equations

Advance each state variable by one sample index ($n\to n+1$) to find its state-transition behavior: $x_1[n+1]=y[n-N+1]=x_2[n]$ $x_2[n+1]=y[n-N+2]=x_3[n]$ $⋮$ $x_{N-1}[n+1]=y[n-1]=x_N[n]$

Step 3: Solve for the Final State Transition

The transition of the final state variable ($x_N[n+1]$) represents the current output $y[n]$. Isolate $y[n]$ in the original difference equation: $x_N[n+1]=y[n]=-a_{1}y[n-1]-a_{2}y[n-2]-\cdots -a_{N}y[n-N]+b_{0}u[n]$

Substitute the state variable definitions from Step 1 into this expression: $x_N[n+1]=-a_N{x}_1[n]-a_{N-1}{x}_2[n]-\cdots -a_1{x}_N[n]+b_{0}u[n]$

Step 4: Assemble the Canonical Companion Matrix

Gather the state transitions into the discrete-time companion matrix format: $\left[\begin{array}{c}{x}_1[n+1]\\ x_2[n+1]\\ ⋮\\ x_{N-1}[n+1]\\ x_N[n+1]\end{array}\right]=\left[\begin{array}{ccccc}0& 1& 0& \dots & 0\\ 0& 0& 1& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & 1\\ -a_N& -a_{N-1}& -a_{N-2}& \dots & -a_1\end{array}\right]\left[\begin{array}{c}{x}_1[n]\\ x_2[n]\\ ⋮\\ x_{N-1}[n]\\ x_N[n]\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ b_0\end{array}\right]u[n]$

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3. Comprehensive Worked Examples (The Exam Killers)

3.1 Example 1: Third-Order Recursive Difference Equation [Passage 84]

Question: Obtain the state-space model $(\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D})$ in companion form for the system described by the difference equation: $4y[n]+3y[n-1]+2y[n-2]+y[n-3]=u[n]$

Step-by-Step Solution:

1. Normalize the Difference Equation (Monic Lead Coefficient): Divide the entire difference equation by the coefficient of $y[n]$ (which is $4$): $y[n]+\frac{3}{4}y[n-1]+\frac{1}{2}y[n-2]+\frac{1}{4}y[n-3]=\frac{1}{4}u[n]$

2. Define Phase-Variable States ($N=3$): Assign our state variables to the outputs of the cascade of three delay elements: $x_1[n]=y[n-3]$ $x_2[n]=y[n-2]$ $x_3[n]=y[n-1]$

3. Formulate State Shifts: Shift each state variable forward by one step: $x_1[n+1]=y[n-2]=x_2[n]$ $x_2[n+1]=y[n-1]=x_3[n]$

   Solve for the final state shift $x_3[n+1]=y[n]$ by isolating $y[n]$ in the normalized LCCDE: $y[n]=-\frac{1}{4}y[n-3]-\frac{1}{2}y[n-2]-\frac{3}{4}y[n-1]+\frac{1}{4}u[n]$ $x_3[n+1]=-\frac{1}{4}{x}_1[n]-\frac{1}{2}{x}_2[n]-\frac{3}{4}{x}_3[n]+\frac{1}{4}u[n]$

4. Assemble the State Equation Matrix: $\left[\begin{array}{c}{x}_1[n+1]\\ x_2[n+1]\\ x_3[n+1]\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -\frac{1}{4}& -\frac{1}{2}& -\frac{3}{4}\end{array}\right]\left[\begin{array}{c}{x}_1[n]\\ x_2[n]\\ x_3[n]\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \frac{1}{4}\end{array}\right]u[n]$

5. Assemble the Output Equation Matrix: Since our output sequence is $y[n]=x_3[n+1]$ (the current updated output), we substitute the state transition of $x_3[n+1]$ directly to write our Output Equation strictly in terms of current states and current inputs: $y[n]=-\frac{1}{4}{x}_1[n]-\frac{1}{2}{x}_2[n]-\frac{3}{4}{x}_3[n]+\frac{1}{4}u[n]$ $y[n]=\left[\begin{array}{ccc}-\frac{1}{4}& -\frac{1}{2}& -\frac{3}{4}\end{array}\right]\left[\begin{array}{c}{x}_1[n]\\ x_2[n]\\ x_3[n]\end{array}\right]+\left[\begin{array}{c}\frac{1}{4}\end{array}\right]u[n]$

   Thus, the final discrete companion state-space matrices are: $\mathbf{A}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -\frac{1}{4}& -\frac{1}{2}& -\frac{3}{4}\end{array}\right],\mathbf{B}=\left[\begin{array}{c}0\\ 0\\ \frac{1}{4}\end{array}\right],\mathbf{C}=\left[\begin{array}{ccc}-\frac{1}{4}& -\frac{1}{2}& -\frac{3}{4}\end{array}\right],\mathbf{D}=\left[\begin{array}{c}\frac{1}{4}\end{array}\right]$

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4. Hardware Realization (Direct Form II Canonic Structure)

To visualize this discrete-time state-space system, we can map the matrices directly to a Canonic Direct Form II block diagram. Each state variable $x_i[n]$ corresponds to the output of a physical unit-delay element ($z^{-1}$).

               u[n] 
                │
                v [x 1/4]
                │
                +─────────> ( + ) <─────────────────────────────────┐
                             │                                      │
                             v                                      │
                         ┌───────┐                                  │
                         │ z^-1  │───────> x3[n] ───( x -3/4 )──────┤
                         └───────┘                                  │
                             │                                      │
                             v                                      │
                         ┌───────┐                                  │
                         │ z^-1  │───────> x2[n] ───( x -1/2 )──────┤
                         └───────┘                                  │
                             │                                      │
                             v                                      │
                         ┌───────┐                                  │
                         │ z^-1  │───────> x1[n] ───( x -1/4 )──────┘
                         └───────┘
                             │
                             +───────────> Output y[n] = x3[n+1]

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5. Common Mistakes That Cost Marks

The Monic Coefficient Oversight

Students frequently forget to normalize the lead coefficient of $y[n]$ to 1 before assigning phase-variable states. In the equation $4y[n]+3y[n-1]+\cdots =u[n]$, starting state assignments without dividing by 4 first yields feedback gains and input scaling coefficients that are off by a factor of 4, leading to a critical grading penalty.

The Current Output ($y[n]$) State Definition Trap

Defining $x_1[n]=y[n]$ as a state variable is a major error. The current output $y[n]$ is not a memory state; it is a dependent response that changes instantaneously with the current input. State variables must represent delayed storage elements (e.g., $y[n-1],y[n-2]$) to avoid algebraic loops during transition matrix calculations.

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6. Potential/Grounded Exam Questions

6.1 Discrete LCCDE Conversion

Question: Represent the second-order recursive system $y[n]-\frac{1}{6}y[n-1]-\frac{1}{6}y[n-2]=x[n]$ in state-space companion form.

• Answer Plan:
  1. Define $x_1[n]=y[n-2]$ and $x_2[n]=y[n-1]$.
  2. Write shifts: $x_1[n+1]=x_2[n]$, and $x_2[n+1]=y[n]=\frac{1}{6}{x}_1[n]+\frac{1}{6}{x}_2[n]+u[n]$.
  3. Assemble state-space matrices: $\mathbf{A}=\left[\begin{array}{cc}0& 1\\ \frac{1}{6}& \frac{1}{6}\end{array}\right],\mathbf{B}=\left[\begin{array}{c}0\\ 1\end{array}\right],\mathbf{C}=\left[\begin{array}{cc}\frac{1}{6}& \frac{1}{6}\end{array}\right],\mathbf{D}=\left[\begin{array}{c}1\end{array}\right]$

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7. Self-Check Before Moving On

• Can you write out the discrete-time State and Output matrix equations with correct variable dimensions?
• Do you know how to assign delay-chain state variables to an $N$-th order difference equation?
• Have you mastered the requirement to normalize the coefficient of $y[n]$ to 1 before formulating state companion matrices?
• Can you draw a Canonic Direct Form II block diagram from a discrete state-space model?

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Source: (k.Deergha Rao) signals and systems.pdf (Chapter 6 & 8), ECE-2107_STATE_SPACE_ANALYSIS.pdf, Rabiul sir -2309008.pdf (Class Lecture Notes).