--- title: "Transfer function from third-order state model" subject: "Control Systems" chapter: 8 key: "semester_3/control_systems:ss-tf-3x3-1-2-3" source: https://www.bctengineer.com/q/semester_3/control_systems/ss-tf-3x3-1-2-3 site: BCT Engineering --- # Transfer function from third-order state model Control Systems · Chapter 8: State Space Analysis - Asked in: 2078 Bhadra, 6+2 marks; 2071 Shrawan, 8 marks (asked 2 times) - Video: [Transfer Function Derivation from State Model Statement Problem No 4 - Control Systems](https://www.youtube.com/watch?v=n_spUERHzP8) — Ekeeda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/ss-tf-3x3-1-2-3 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8 ## Question For given state equation and output equation, find transfer function Y(s)/U(s). Ẋ = [0 1 0; 0 0 1; −1 −2 −3]X + [10; 0; 0]u and y = [1 0 0]X ## Short answer The transfer function is Y(s)/U(s) = 10(s^2+3s+2)/(s^3+3s^2+2s+1), found from C adj(sI-A)B / det(sI-A). ## Answer For $\dot{x} = Ax + Bu$, $y = Cx + Du$: $$ \frac{Y(s)}{U(s)} = C(sI - A)^{-1}B + D = \frac{C\,\operatorname{adj}(sI-A)\,B}{|sI-A|} + D $$ $$ A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & -3 \end{bmatrix},\quad B = \begin{bmatrix} 10 \\ 0 \\ 0 \end{bmatrix},\quad C = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix},\quad D = 0 $$ #### Step 1: $sI - A$ and its determinant $$ sI - A = \begin{bmatrix} s & -1 & 0 \\ 0 & s & -1 \\ 1 & 2 & s+3 \end{bmatrix} $$ $$ \begin{aligned} |sI - A| &= s\left[s(s+3) + 2\right] - (-1)\left[0\cdot(s+3) - (-1)(1)\right] + 0 \\ &= s^3 + 3s^2 + 2s + 1 \end{aligned} $$ #### Step 2: Required part of the adjoint Because $B$ has only its first element non-zero and $C$ picks only the first row, only the (1,1) element of the adjoint is needed: $$ \operatorname{adj}(sI-A)_{11} = \begin{vmatrix} s & -1 \\ 2 & s+3 \end{vmatrix} = s^2 + 3s + 2 $$ Full first column of the adjoint: $\begin{bmatrix} s^2+3s+2 \\ -1 \\ -s \end{bmatrix}$, so $$ \operatorname{adj}(sI-A)B = 10\begin{bmatrix} s^2+3s+2 \\ -1 \\ -s \end{bmatrix} $$ #### Step 3: Transfer function $$ C\,\operatorname{adj}(sI-A)B = 10(s^2 + 3s + 2) $$ $$ \frac{Y(s)}{U(s)} = \frac{10(s^2+3s+2)}{s^3+3s^2+2s+1} = \frac{10(s+1)(s+2)}{s^3+3s^2+2s+1} $$ #### Stability check Routh array for $s^3 + 3s^2 + 2s + 1$: | Row | | | |---|---|---| | $s^3$ | 1 | 2 | | $s^2$ | 3 | 1 | | $s^1$ | $\frac{3\times2 - 1}{3} = 1.67$ | 0 | | $s^0$ | 1 | | No sign change, so the system is stable. The poles are $-2.325$ and $-0.338 \pm j0.562$; the zeros are $-1$ and $-2$. **Answer:** $\dfrac{Y(s)}{U(s)} = \dfrac{10(s^2+3s+2)}{s^3+3s^2+2s+1}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In Y/U = C adj(sI - A) B / |sI - A| + D, what does the denominator represent? A. The determinant of B B. The characteristic polynomial |sI - A| C. The product C B D. The adjoint of A Answer: B. The denominator is |sI - A|, whose roots are the poles. (Key point: Transfer function from a state model) 2. For the companion matrix with last row [-1 -2 -3], what is |sI - A|? A. s^3 - 3s^2 - 2s - 1 B. s^3 + s^2 + 2s + 3 C. s^3 + 3s^2 + 2s + 1 D. s^3 + 3s^2 + 2s - 1 Answer: C. For a companion form the last row gives the coefficients directly: s^3 + 3s^2 + 2s + 1. (Key point: Determinant of 3x3) 3. Why is only adj(sI - A)_11 needed here? A. Because the matrix is symmetric B. Because D = 0 removes other elements C. Because the other elements are always zero D. B has only its first element non-zero and C picks only the first row Answer: D. C adj B reduces to C1 * adj_11 * B1 when only the first entries of C and B are non-zero. (Key point: Which adjoint element is needed) 4. What is Y(s)/U(s)? A. 10(s^2 + 3s + 2)/(s^3 + 3s^2 + 2s + 1) B. 10/(s^3 + 3s^2 + 2s + 1) C. (s^2 + 3s + 2)/(s^3 + 3s^2 + 2s + 1) D. 10 s/(s^3 + 3s^2 + 2s + 1) Answer: A. adj_11 = s^2 + 3s + 2 and B1 = 10 give numerator 10(s^2 + 3s + 2). (Key point: Final transfer function) ## Close variants set in other exams - [Transfer function, poles and zeros from 3×3 state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-poles-zeros-3x3) — Control Systems, chapter 8, asked 3× ## Related in Control Systems - [Transfer function from state equations, output x1](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-from-de-3x1-x2) — Control Systems, chapter 8, asked 2× - [Transfer function from state equations, output x2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-from-de-3x1-x2-y) — Control Systems, chapter 8, asked 2× - [Transfer function from state model [0 1; −5 −6]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-0-1-5-6) — Control Systems, chapter 8, asked 3× - [Transfer function from symbolic mass-spring-damper state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-mbk-symbolic) — Control Systems, chapter 8 - [Second order model from step response (peak 1.25)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Asecond-order-from-step-125) — Control Systems, chapter 3 - [Transfer function of op-amp with Rs ∥ C feedback](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-opamp-rs-c-r1-r2) — Control Systems, chapter 2 ## Related in other subjects - [Series programming of a third-order transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aseries-programming-z-plus-1) — Digital Control System, chapter 4, asked 2× - [Ladder programming of a third-order transfer function](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aladder-z3-4z2) — Digital Control System, chapter 4, asked 2× - [3rd order Butterworth HPF from LPF transfer function](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Athird-order-hpf-from-lpf-tf) — Filter Design, chapter 5 - [Transfer function of zero order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Azoh-transfer-function) — Digital Control System, chapter 1, asked 7×