--- title: "State space analysis; TF and stability (−4, 10)" subject: "Control Systems" chapter: 8 key: "semester_3/control_systems:ss-adv-limit-tf-4-1" source: https://www.bctengineer.com/q/semester_3/control_systems/ss-adv-limit-tf-4-1 site: BCT Engineering --- # State space analysis; TF and stability (−4, 10) Control Systems · Chapter 8: State Space Analysis - Asked in: 2066 Jestha (old course), 10 marks - Video: [Introduction to State-Space Equations | State Space, Part 1](https://www.youtube.com/watch?v=hpeKrMG-WP0) — MATLAB - 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-4-1 - 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: [ẋ1; ẋ2] = [−4 −1; 10 0][x1; x2] + [1; 0]u, y = [1 0][x1; x2]. Also evaluate the stability of this system. ## Short answer Advantages: handles MIMO, nonlinear/time-varying systems, includes initial conditions, tests controllability/observability. Limitation: needs more computation, state variables not unique, physical insight less direct. For given model, G(s)=s/(s²+4s+10). Poles -2±j2.449 indicate stability. ## Answer #### Advantages of state-space analysis 1. Applies to **MIMO** (multi-input multi-output) systems as easily as to single-input single-output systems. 2. Can handle **non-linear and time-varying** systems; the transfer function needs linear time-invariant systems. 3. Includes **initial conditions**; the transfer function assumes zero initial conditions. 4. Gives the **internal behaviour** (all state variables), not only the input–output relation. 5. Allows tests of **controllability and observability**, and hidden (cancelled) modes are not lost. 6. Suited to **computer solution** and simulation, since it uses first-order matrix equations. 7. Forms the basis of modern and optimal control design: state feedback (pole placement), observers, Kalman filter. #### Limitations 1. Needs more computation (matrix algebra, $n$ first-order equations); physical insight is less direct than with a transfer function or Bode plot. 2. The choice of state variables is not unique, so different models describe the same system. 3. State variables may not all be measurable; an observer may be needed for state feedback. 4. Classical specifications (gain margin, phase margin, bandwidth) are not read directly. #### Transfer function $$ A = \begin{bmatrix} -4 & -1 \\ 10 & 0 \end{bmatrix},\quad B = \begin{bmatrix} 1 \\ 0 \end{bmatrix},\quad C = \begin{bmatrix} 1 & 0 \end{bmatrix} $$ $$ sI - A = \begin{bmatrix} s+4 & 1 \\ -10 & s \end{bmatrix},\qquad |sI-A| = s(s+4) + 10 = s^2 + 4s + 10 $$ $$ (sI-A)^{-1} = \frac{1}{s^2+4s+10}\begin{bmatrix} s & -1 \\ 10 & s+4 \end{bmatrix},\qquad (sI-A)^{-1}B = \frac{1}{s^2+4s+10}\begin{bmatrix} s \\ 10 \end{bmatrix} $$ $$ \frac{Y(s)}{U(s)} = \begin{bmatrix} 1 & 0 \end{bmatrix}\frac{1}{s^2+4s+10}\begin{bmatrix} s \\ 10 \end{bmatrix} = \frac{s}{s^2 + 4s + 10} $$ #### Stability Characteristic equation $s^2 + 4s + 10 = 0$: $$ s = \frac{-4 \pm \sqrt{16 - 40}}{2} = -2 \pm j2.449 $$ Both poles have negative real parts, so the system is **stable** (underdamped: $\omega_n = \sqrt{10} = 3.16$ rad/s, $\zeta = 4/(2\sqrt{10}) = 0.632$). There is a zero at $s = 0$. ## 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 needs less computation than TF B. It handles MIMO, non-linear and time-varying systems with initial conditions C. It gives gain margin directly D. It requires zero initial conditions Answer: B. TF is limited to LTI, SISO with zero initial conditions. (Key point: Advantage) 2. Which is a limitation? A. It cannot handle MIMO B. It ignores internal states C. State choice is not unique and states may not be measurable D. It cannot be solved on computers Answer: C. Different models describe the same system; observers may be needed. (Key point: Limitation) 3. What is Y/U for A = [-4 -1; 10 0], B = [1;0], C = [1 0]? A. 1/(s^2 + 4s + 10) B. 10/(s^2 + 4s + 10) C. s/(s^2 + 4s - 10) D. s/(s^2 + 4s + 10) Answer: D. (sI-A)^-1 B = [s; 10]/(s^2+4s+10). (Key point: Transfer function) 4. What are the poles and conclusion? A. -2 +/- j2.449, stable B. 2 +/- j2.449, unstable C. -2 and -10, stable D. +/- j3.16, marginal Answer: A. Negative real parts give stability. (Key point: Stability) ## Close variants set in other exams - [State space analysis; TF of second-order model](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-adv-limit-tf-3-1) — Control Systems, chapter 8, asked 2× ## Related in Control Systems - [Vertical two-mass system transfer function X1/F](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Atf-vertical-m1-m2-k1-b1) — Control Systems, chapter 2 - [Transfer function and stability from [1 0; 1 1]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-1-0-1-1) — Control Systems, chapter 8 - [Two-mass system X1/F, X2/F with F-V analogy](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afv-tf-m1-m2-k1b1-k2b2) — Control Systems, chapter 2, asked 3× - [Transfer function and stability from [0 −1; −2 −3]](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-tf-0-1-2-3) — Control Systems, chapter 8 - [Free body diagram and X1/F, X2/F of two-mass system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Afbd-m1-m2-k1k2-b1b2) — Control Systems, chapter 2 - [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× ## 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× - [Find x3[n] from X3=X1X2, x1 5-point, x2 4-point](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adft-product-x3-73s) — Digital Signal Analysis and Processing, chapter 7, asked 3× - [Practical factors in sampling; sampling interval for X1(t)X2(t)](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-practical-x1x2) — Communication Systems, chapter 4