Steady State Modelling of a RLC Circuit

INTRODUCTION TO STATE SPACE ANALYSIS:-

State space analysis is a powerful mathematical framework used to model and analyze control systems. Unlike traditional methods like transfer functions, which are limited to single-input, single-output (SISO) linear systems, state space analysis can handle multi-input, multi-output (MIMO) systems, non-linear dynamics, and time-variant processes. This flexibility makes it a vital tool in modern control engineering, offering a deeper understanding of system behavior.

In this blog, we will explore the fundamentals of state space analysis, explain its mathematical formulation, and discuss its advantages and applications.

Before diving into the mathematical details, it's essential to understand some key terms associated with state space analysis.

Definition of State Space Representation:-

State space representation describes the behavior of dynamic systems in terms of a set of first-order differential (or difference) equations. These equations relate the system's input, output, and internal state variables, providing a comprehensive view of the system's behavior at any given time.

Key Components:

• State Variables: These are variables that capture the current condition or "state" of the system. The state variables define the minimal set of information required to predict the future behavior of the system.
• State Equations: These equations describe how the state of the system evolves over time in response to the system's input.
• Output Equations: These equations define how the system's outputs are generated from the internal states and inputs.

MATHEMATICAL REPRESENTATION:-

The general form of state space representation is given by the following equations:

$x(t)=Ax(t)+Bu(t)$ 

$y(t)=Cx(t)+Du(t)$

Here's what each term represents:

• $x(t)$: The state vector, representing the system's internal state at time t.
• $u(t)$: The input vector, representing external inputs to the system.
• $y(t)$: The output vector, representing the observable outputs of the system.
• $A$: The state matrix, which defines the relationships between the current states and their derivatives.
• $B$: The input matrix, which determines how the inputs affect the states.
• $C$: The output matrix, which maps the state to the output.
• $D$: The feedthrough matrix, which directly relates inputs to outputs.

The first equation (state equation) describes the dynamics of the system, while the second (output equation) relates the state to the measurable outputs.

ADVANTAGES OF STATE SPACE ANALYSIS:-

State space analysis offers several advantages over traditional methods like transfer functions:

• Handles MIMO systems: It can easily model systems with multiple inputs and outputs, which is difficult in the transfer function approach.
• Models non-linear and time-varying systems: State space representation can describe non-linear systems by linearizing them around a specific operating point.
• Full system description: State space captures all aspects of the system's behavior, not just the input-output relationship, but also the internal state dynamics.
• Applicable to both continuous and discrete systems: State space can be used for digital control and systems with sampled inputs.

HOW TO SOLVE STATE SPACE EQUATIONS ?

To predict the future behavior of the system, we need to solve the state space equations. The general solution involves the state transition matrix and is given by:

$x(t) = e^{A t} x(0) + \int_{0}^{t} e^{A (t - \tau)} B u(\tau)\, d\tau$

Here:

• x(t) is the state at time 't'.
• \( e^{At} \) is the state transition matrix, which describes how the states evolve over time.

This solution can be found either analytically for simple systems or numerically using tools like MATLAB for more complex systems.

SOLVING STATE SPACE QUESTION STEP BY STEP:-

Example: RLC Circuit

System description

Let $i(t)$ be the series current, $V_{in}(t)$the applied voltage, and $V_C(t)$ the capacitor voltage. 

By Applying KVL,

 $V_{in}(t) = R\,i(t) + L\frac{d i(t)}{dt} + V_C(t)$

 Also

 $i(t) = C\frac{dV_C(t)}{dt}$

Step-by-step derivation of state-space representation

Step 1: Define the state variables

 Choose two states :

$ x_1(t)=V_C(t),\qquad x_2(t)=i(t)$

 So the output is $(y(t)=V_C(t)=x_1(t))$.

Step 2: Write the state equations

 Use the capacitor relation:

 $\dot x_1(t)=\frac{dV_C}{dt}=\frac{1}{C}i(t)=\frac{1}{C}x_2(t)$

 From KVL:

 $L\frac{d i(t)}{dt}=V_{in}(t)-R,i(t)-V_C(t)$

so

$ \dot x_2(t)=\frac{1}{L}\bigl(V_{in}(t)-R,x_2(t)-x_1(t)\bigr)$

Step 3: State-space (matrix) form

 Stacking the two state equations:

$\begin{bmatrix}\dot{x}_1(t) \\\dot{x}_2(t)\end{bmatrix}=\underbrace{\begin{bmatrix}0 & \dfrac{1}{C} \\-\dfrac{1}{L} & -\dfrac{R}{L}\end{bmatrix}}_{A}\begin{bmatrix}x_1(t) \\x_2(t)\end{bmatrix}+\underbrace{\begin{bmatrix}0 \\\dfrac{1}{L}\end{bmatrix}}_{B}V_{in}(t)$

Step 4: Output equation

 The output is the capacitor voltage:

$y(t) = x_1(t)= \underbrace{\begin{bmatrix}1 & 0\end{bmatrix}}_{C}\begin{bmatrix}x_1(t) \\x_2(t)\end{bmatrix}+\underbrace{0}_{D}V_{in}(t)$

So the full state-space model is

$$\boxed{\dot{\mathbf{x}}(t)=\begin{bmatrix}0 & \dfrac{1}{C} \\[6pt]-\dfrac{1}{L} & -\dfrac{R}{L}\end{bmatrix}\mathbf{x}(t)+\begin{bmatrix}0 \\[6pt]\dfrac{1}{L}\end{bmatrix}V_{in}(t),\; y(t)=\begin{bmatrix}1 & 0\end{bmatrix}\mathbf{x}(t)}$$

with $\mathbf x(t)=\begin{bmatrix}x_1(t)\ x_2(t)\end{bmatrix}=\begin{bmatrix}V_C(t)\ i(t)\end{bmatrix}$

TRASFER FUNCTION: 

$G(s)=\dfrac{V_C(s)}{V_{in}(s)}$

Using state-space formula:

 $G(s)=\frac{\dfrac{1}{C s}}{R + L s + \dfrac{1}{C s}}= \frac{1}{L C s² + R C s + 1}$

Write in standard second-order form by dividing by (LC):

$$G(s)=\frac{1}{LC}\cdot\frac{1}{s^{2} + \dfrac{R}{L}s + \dfrac{1}{LC}}\; \text{or more compactly} \;G(s)=\frac{1}{L C s^{2} + R C s + 1}$$

Natural Frequency & Damping:

From the denominator $(s² + \dfrac{R}{L}s + \dfrac{1}{LC}):$

$\omega_n=\frac{1}{\sqrt{LC}},\qquad\zeta=\frac{R}{2}\sqrt{\frac{C}{L}}$

Poles are at$(s=-\zeta\omega_n\pm j\omega_n\sqrt{1-\zeta²}) (depending  on (\zeta))$.

Remarks (Stability / Controllability / Observability)

• Poles depend on (R,L,C). For (R>0) the system is damped; if $(\zeta>0)$ the real parts of poles are negative ⇒ Stable.
• Controllability: the controllability matrix $[B,\, AB]$ is full rank for (R,L,C>0) ⇒ controllable.
• Observability: since $C = [1\, 0]$ and system order 2, the observability matrix is full rank ⇒ observable (you measure the capacitor voltage).

APPLICATION OF STATE SPACE ANALSIS:-

State space analysis is used in various fields for both system modeling and control design:

• Robotics: In designing robots, especially with complex dynamics, state space models are used for trajectory planning and feedback control.
• Aerospace Engineering: Used in the design of flight control systems and spacecraft navigation.
• Electrical Engineering: Essential in designing control systems for power electronics, signal processing, and communication systems.

COMPARISION WITH TRANSFER FUNTION APPROACH:-

Here’s a comparison of state space analysis and the transfer function method:

CONCLUSION:-

State space analysis is a versatile and powerful tool that extends beyond the limitations of transfer functions. It enables engineers to model complex systems, design advanced controllers, and understand system behavior in greater depth. Whether you’re working on a robotics project or analyzing electrical circuits, state space analysis offers the flexibility needed for modern control applications.

To dive deeper, try implementing state space models in software tools like MATLAB or Python for simulations and further analysis.

Presented By:-

1. Pradyut Manjarekar
2. Aayush Chaudhari
3. Aryan Chaudhary

Special Thanks To:

Prof. Najib Ghatte

Prof. Narendra Bhagat