--- title: "State space analysis; TF of second-order model" subject: "Control Systems" chapter: 8 key: "semester_3/control_systems:ss-adv-limit-tf-3-1" source: https://www.bctengineer.com/q/semester_3/control_systems/ss-adv-limit-tf-3-1 site: BCT Engineering --- # State space analysis; TF of second-order model Control Systems · Chapter 8: State Space Analysis - Asked in: 2076 Chaitra, 2+6 marks - Video: [State Space Analysis to Determine Transfer Function](https://www.youtube.com/watch?v=GuIvzps_4RI) — Engineering Funda - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/control_systems/ss-adv-limit-tf-3-1 - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-8 ## Question Discuss the advantages and limitations of state space analysis of control systems. Find the transfer function for the system represented by following state space model: ẋ = [−3 1; −2 0]x + [0; 1]u, y = [1 0]x. ## Short answer Advantages: handles MIMO, nonlinear, time-varying systems, includes initial conditions, reveals internal behaviour, supports modern design. Limitations: less intuitive, state model not unique, frequency-domain specs not directly visible. For the given model, G(s)=1/(s^2+3s+2)=1/[(s+1)(s+2)], poles at -1, -2, stable. ## Answer #### Advantages of state space analysis State space analysis describes a system by a set of first-order differential equations in terms of state variables, $\dot{x} = Ax + Bu$, $y = Cx + Du$. 1. Works for **MIMO** systems as easily as for single-input single-output systems. 2. Applies to **non-linear** and **time-varying** systems; the transfer function method does not. 3. Takes **non-zero initial conditions** into account. 4. Gives the **internal behaviour** (all states), not only the input-output relation; controllability and observability can be tested. 5. Suited to **computer solution** and to modern design methods (pole placement, optimal control, observers). 6. Works in the time domain directly. #### Limitations 1. Needs more mathematics (matrices, eigenvalues); less physical insight for simple systems. 2. The state model is **not unique**: different choices of states give different $A$, $B$, $C$. 3. Frequency-domain specifications (gain margin, phase margin, bandwidth) are not seen directly. 4. Some states may not be measurable, so observers are needed for feedback. #### Transfer function of the given model $$ A = \begin{bmatrix}-3 & 1\\ -2 & 0\end{bmatrix},\; B = \begin{bmatrix}0\\ 1\end{bmatrix},\; C = \begin{bmatrix}1 & 0\end{bmatrix},\; D = 0 $$ $$ G(s) = C(sI - A)^{-1}B + D $$ Step 1: $$ sI - A = \begin{bmatrix}s+3 & -1\\ 2 & s\end{bmatrix}, \qquad |sI - A| = s(s+3) + 2 = s^2 + 3s + 2 $$ Step 2: $$ (sI - A)^{-1} = \frac{1}{s^2 + 3s + 2}\begin{bmatrix}s & 1\\ -2 & s+3\end{bmatrix} $$ Step 3: $$ \begin{aligned} (sI - A)^{-1}B &= \frac{1}{s^2 + 3s + 2}\begin{bmatrix}1\\ s+3\end{bmatrix} \\ G(s) = C(sI - A)^{-1}B &= \begin{bmatrix}1 & 0\end{bmatrix}\frac{1}{s^2 + 3s + 2}\begin{bmatrix}1\\ s+3\end{bmatrix} = \frac{1}{s^2 + 3s + 2} \end{aligned} $$ **Answer:** $$ \frac{Y(s)}{U(s)} = \frac{1}{s^2 + 3s + 2} = \frac{1}{(s+1)(s+2)} $$ The poles are at $s = -1$ and $s = -2$, so the system is stable and overdamped. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which is an advantage of state space analysis? A. It takes non-zero initial conditions into account B. It requires less mathematics than transfer functions C. It shows gain and phase margin directly D. It only applies to linear time-invariant SISO systems Answer: A. State space includes initial conditions and handles MIMO, non-linear and time-varying systems. (Key point: Advantages of state space) 2. Which is a limitation of state space analysis? A. The state model is unique for a given system B. The model is not unique and frequency-domain specs are not seen directly C. It cannot handle MIMO systems D. It cannot be solved by computer Answer: B. Different state choices give different A, B, C, and gain margin or bandwidth are not directly visible. (Key point: Limitations) 3. What is |sI - A| for A = [-3 1; -2 0]? A. s^2 + 3s - 2 B. s^2 + 3s + 2 C. s^2 - 3s + 2 D. s^2 + 2s + 3 Answer: B. |sI-A| = (s+3)s - (-1)(2) = s^2 + 3s + 2. (Key point: Resolvent matrix) 4. Given C = [1 0] and B = [0; 1], what is G(s)? A. (s+3)/((s+1)(s+2)) B. s/((s+1)(s+2)) C. 1/((s+1)(s+2)) D. 1/(s^2 + 3s - 2) Answer: C. (sI-A)^-1 B = [1; s+3]/(s^2+3s+2); C = [1 0] picks the first entry, giving 1/(s^2+3s+2). (Key point: Transfer function with C = [1 0]) ## Close variants set in other exams - [State space analysis; TF and stability (−4, 10)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-adv-limit-tf-4-1) — Control Systems, chapter 8, asked 2× ## Related in Control Systems - [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 from third-order state model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-3x3-1-2-3) — Control Systems, chapter 8, asked 3× - [Transfer function from state model [0 −1; −2 −3]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-0-1-2-2) — Control Systems, chapter 8 - [Derive maximum overshoot of second order system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Amp-expression-derivation) — Control Systems, chapter 3 - [Two-inertia rotational system transfer function and T-I analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-rot-j1-j2-k) — Control Systems, chapter 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× ## Related in other subjects - [Pulse transfer function from state space form](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aderive-pulse-tf-state-space) — Digital Control System, chapter 5, asked 3× - [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× - [Transfer function of first order hold](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Afoh-transfer-function) — Digital Control System, chapter 1, asked 7× - [Models of zero, first and second order systems](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_and_measurement%3Azero-first-second-order-models) — Instrumentation and Measurement, chapter 4