--- title: "Jordan canonical form of 5/((z+1)²(z+2))" subject: "Digital Control System" chapter: 5 key: "semester_6/digital_control_system:jordan-form-5-over" source: https://www.bctengineer.com/q/semester_6/digital_control_system/jordan-form-5-over site: BCT Engineering --- # Jordan canonical form of 5/((z+1)²(z+2)) Digital Control System · Chapter 5: Discrete Time State Equations - Asked in: 2082 Chaitra (new course), 4 marks; 2074 Bhadra, 4 marks; 2073 Bhadra, 8 marks (asked 3 times) - Video: [Jodran Canoncial Form and State Space Representation](https://www.youtube.com/watch?v=Ie37sB5i91Y) — Electric Fund - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/digital_control_system/jordan-form-5-over - Whole chapter: https://www.bctengineer.com/md/semester_6/digital_control_system/chapter-5 ## Question Obtain the state space representation of the system shown below in Jordan canonical form: Y(z)/U(z)=5/((z+1)²(z+2)). ## Short answer For Y(z)/U(z)=5/((z+1)²(z+2)), partial fractions give 5/(z+1)² - 5/(z+1) + 5/(z+2). Jordan canonical form: G=[[-1,1,0],[0,-1,0],[0,0,-2]], H=[0;1;1], C=[5,-5,5], D=0. The repeated pole at -1 creates a Jordan block with 1 on the superdiagonal. ## Answer When the denominator has a repeated pole, the diagonal form is not possible, and the Jordan canonical form is used. The repeated pole gives a Jordan block with 1 on the superdiagonal. #### Partial fractions $$ \frac{Y(z)}{U(z)} = \frac{5}{(z+1)^2(z+2)} = \frac{c_1}{(z+1)^2} + \frac{c_2}{z+1} + \frac{c_3}{z+2} $$ $$ \begin{aligned} c_1 &= \left[\frac{5}{z+2}\right]_{z=-1} = 5\\ c_2 &= \frac{d}{dz}\left[\frac{5}{z+2}\right]_{z=-1} = \left[\frac{-5}{(z+2)^2}\right]_{z=-1} = -5\\ c_3 &= \left[\frac{5}{(z+1)^2}\right]_{z=-2} = 5 \end{aligned} $$ So $\dfrac{Y(z)}{U(z)} = \dfrac{5}{(z+1)^2} - \dfrac{5}{z+1} + \dfrac{5}{z+2}$. #### Choice of states $$ X_3(z) = \frac{U(z)}{z+2},\qquad X_2(z) = \frac{U(z)}{z+1},\qquad X_1(z) = \frac{X_2(z)}{z+1} = \frac{U(z)}{(z+1)^2} $$ This gives $$ \begin{aligned} x_1(k+1) &= -x_1(k) + x_2(k)\\ x_2(k+1) &= -x_2(k) + u(k)\\ x_3(k+1) &= -2x_3(k) + u(k)\\ y(k) &= 5x_1(k) - 5x_2(k) + 5x_3(k) \end{aligned} $$ #### Jordan canonical form $$ \begin{bmatrix} x_1(k+1) \\ x_2(k+1) \\ x_3(k+1) \end{bmatrix} = \begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -2 \end{bmatrix}\begin{bmatrix} x_1(k) \\ x_2(k) \\ x_3(k) \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}u(k) $$ $$ y(k) = \begin{bmatrix} 5 & -5 & 5 \end{bmatrix}\mathbf{x}(k) + 0\cdot u(k) $$ Check: $\mathbf{C}(z\mathbf{I}-\mathbf{G})^{-1}\mathbf{H} = \dfrac{5}{z^3+4z^2+5z+2} = \dfrac{5}{(z+1)^2(z+2)}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why is the diagonal form not used for 5/((z+1)^2(z+2))? A. The system is unstable B. The numerator is constant C. The denominator has a repeated pole at z = -1 D. The poles are positive Answer: C. A repeated pole needs a Jordan block with 1 on the superdiagonal. (Key point: Why Jordan form) 2. The partial fraction expansion is: A. 5/(z+1)^2 + 5/(z+1) + 5/(z+2) B. 5/(z+1)^2 - 5/(z+1) + 5/(z+2) C. -5/(z+1)^2 + 5/(z+1) - 5/(z+2) D. 5/(z+1)^2 - 5/(z+2) Answer: B. c1 = 5, c2 = d/dz[5/(z+2)] at -1 = -5, c3 = 5/(z+1)^2 at -2 = 5. (Key point: Partial fraction coefficients) 3. The G matrix is: A. diag(-1, -1, -2) with no off-diagonal term B. [-1 0 0; 1 -1 0; 0 0 -2] C. [1 1 0; 0 1 0; 0 0 2] D. [-1 1 0; 0 -1 0; 0 0 -2] Answer: D. The block [-1 1; 0 -1] is for the repeated pole and -2 is separate. (Key point: Jordan block in G) 4. The output matrix C is: A. [5 -5 5] B. [1 1 1] C. [5 5 5] D. [0 0 5] Answer: A. Coefficients of the partial fractions c1, c2, c3 give C = [5 -5 5]. (Key point: Output matrix) ## Close variants set in other exams - [Jordan canonical form of (2z³+6z+5)/((z+2)²(z+3))](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Ajordan-form-2z3-6z-5) — Digital Control System, chapter 5, asked 5× ## Related in Digital Control System - [Why state representation; controllable canonical form](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Astate-representation-why-ccf) — Digital Control System, chapter 5, asked 2× - [Controllable canonical form of a third-order system](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-1-3z-5z-4z) — Digital Control System, chapter 5, asked 14× - [Controllable canonical form of (z⁻¹+2z⁻²)/(1+4z⁻¹+3z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-z1-2z2-4z1-3z2) — Digital Control System, chapter 5, asked 15× - [Controllable canonical form of (4z³+3z²+5z+4)/(2z³+5z²+2z+3)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Accf-4z3-3z2) — Digital Control System, chapter 5, asked 14× - [Observable canonical form of (0.368z⁻¹+0.264z⁻²)/(1−1.368z⁻¹+0.368z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Aocf-0-368-0-264) — Digital Control System, chapter 5, asked 13× - [Diagonal canonical form of (z⁻¹+2z⁻²)/(1+0.7z⁻¹+0.12z⁻²)](https://www.bctengineer.com/question.md?key=semester_6%2Fdigital_control_system%3Adiagonal-form-0-7-0-12) — Digital Control System, chapter 5, asked 13× ## Related in other subjects - [State space model of vertical two-mass system](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-mech-m1-m2-k-b) — Control Systems, chapter 8 - [Space partitioning representation](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Aspace-partitioning-representation) — Computer Graphics, chapter 6 - [SOP/POS forms and canonical form of A+BC+ABC](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Asop-pos-canonical-form) — Digital Logic, chapter 3 - [Canonical form of A(B'+C)+BD; minterm/maxterm](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Acanonical-form-minterm-maxterm) — Digital Logic, chapter 3