Math Problem Statement
solve it classic form step by step
Solution
To solve the problem step by step and find the state-space representation of the given transfer function , let us proceed systematically.
(a) Find a state-space representation:
The given transfer function is:
\begin{bmatrix} \frac{3}{(s^2 + 2s + 10)(s+5)} & \frac{2s}{(s+5)(s+1)} \\ \frac{1}{s+1} & \frac{s+5}{(s+5)(s+1)} \end{bmatrix}.$$ #### Step 1: Break the system into subcomponents We aim to represent each term in $$H(s)$$ using state-space matrices. --- 1. **For $$\frac{3}{(s^2 + 2s + 10)(s+5)}$$:** - The denominator is a third-order polynomial: $$(s^2 + 2s + 10)(s+5)$$. Let us write its state-space representation. - First, consider the second-order system $$\frac{1}{s^2 + 2s + 10}$$. Using standard canonical form: - The state-space representation for this subsystem can be written as: $$\dot{x}_1 = \begin{bmatrix} 0 & 1 \\ -10 & -2 \end{bmatrix} x_1 + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u_1,$$ $$y_1 = \begin{bmatrix} 3 & 0 \end{bmatrix} x_1.$$ - Next, include the $$(s+5)$$ pole. Add another state variable $$x_2$$: $$\dot{x}_2 = -5 x_2 + u_2,$$ $$y = y_1 x_2.$$ Combining these gives the complete state-space representation for $$\frac{3}{(s^2 + 2s + 10)(s+5)}$$. --- 2. **For $$\frac{2s}{(s+5)(s+1)}$$:** - The denominator $$(s+5)(s+1)$$ corresponds to two first-order poles. - The numerator $$2s$$ implies a coupling of state variables. - In canonical form, write the system: $$\dot{x}_1 = -5x_1 + u,$$ $$\dot{x}_2 = -x_2 + x_1,$$ $$y = \begin{bmatrix} 0 & 2 \end{bmatrix} x.$$ --- 3. **For $$\frac{1}{s+1}$$:** - This is a simple first-order system. State-space representation: $$\dot{x} = -x + u, \quad y = x.$$ --- 4. **For $$\frac{s+5}{(s+5)(s+1)}$$:** - Split the terms: $$\frac{s+5}{(s+5)(s+1)} = \frac{1}{s+1}.$$ - State-space is the same as the previous subsystem. --- #### Step 2: Combine all subcomponents into a single state-space representation. Let the combined state vector be: $$X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \dots \end{bmatrix}.$$ Then, the full system can be represented as: $$\dot{X} = AX + BU,$$ $$Y = CX + DU.$$ The matrices $$A, B, C, D$$ are constructed by placing the individual subsystems along the diagonal. --- ### (b) Analyze the irreducibility of the system A system is **irreducible** if it cannot be decomposed into smaller subsystems with no coupling between them. To test irreducibility: 1. Compute the controllability matrix $$\mathcal{C} = [B \ AB \ A^2B \ \dots \]$$. 2. Compute the observability matrix \( \mathcal{O} = [C^T \ A^TC^T \ (A^2)^TC^T \ \dots \]. If both matrices have full rank, the system is irreducible. --- Do you want me to compute specific state-space matrices explicitly, or do you have any questions? Let me know! Here are some follow-ups: 1. How to simplify transfer functions for state-space conversion? 2. What is the significance of canonical forms in state-space analysis? 3. How to verify controllability and observability of a given system? 4. Can irreducible systems be simplified further while preserving properties? 5. How to handle coupled systems in state-space modeling? **Tip:** Always convert higher-order systems step-by-step, starting with their pole-zero decomposition.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Control Systems
State-Space Representation
Transfer Functions
System Dynamics
Formulas
State-space representation: \( \dot{X} = AX + BU \), \( Y = CX + DU \)
Transfer function decomposition
Theorems
Controllability and Observability Theorems
Canonical Forms for State-Space Systems
Suitable Grade Level
Undergraduate Engineering (Control Systems Course)
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