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1. Poles and Zeros of MIMO Systems Presenter name: Faraj Saleh  Mustafa Enwider Waleed Aleyan Hisham Jamal Ali

2. Contents Introduction Meaning of Poles and Zeros in MIMO Systems Importance of Finding Poles and Zeros for MIMO Methods of Determining Poles and Zeros of MIMO Systems Example  Conclusion 2

3. introduction Welcome to the presentation on poles and zeros of MIMO (Multiple Input Multiple Output) systems. MIMO systems are widely used in various fields, including telecommunications, control systems, and signal processing. Let's explore the concepts of poles and zeros and their significance in the analysis of MIMO systems. 3

4. Meaning of Poles and Zeros in MIMO Systems Poles and zeros are key concepts in the analysis of linear time-invariant systems, including MIMO systems. Poles represent the values of the system's transfer function where the output becomes infinite or unstable. Zeros are the values of the transfer function where the output becomes zero or cancels out certain input components. 4

5. Finding the poles and zeros of a MIMO system is crucial for several reasons: Stability Analysis: Poles determine the stability of a system. If any pole lies in the right half-plane, the system is unstable. Frequency Response: The distribution of poles and zeros affects the system's frequency response. They determine the system's gain and phase behavior at different frequencies. Transient Response: Poles and zeros influence the system's transient response, including settling time, overshoot, and damping. Control Design: Knowledge of poles and zeros is essential for designing controllers that stabilize and control the MIMO system effectively. 5 Importance of Finding Poles and Zeros for MIMO Systems

6. Methods of Determining Poles and Zeros of MIMO Systems Method 1: Transfer Function Analysis: The transfer function relates the input and output of a system in the frequency domain. By analyzing the transfer function, poles and zeros can be identified. Techniques such as partial fraction expansion and polynomial factorization are used to determine the poles and zeros. 6

7. Methods of Determining Poles and Zeros of MIMO Systems Method 2: State-Space Representation: State-space models provide an alternative representation of MIMO systems. Eigenvalue analysis of the state matrix reveals the poles, while the zeros can be obtained from the system's transfer function . Note:  In the next slide we ) will give an Example to Explain How the Method Works (Using Transfer Function Analysis). 7

8. Steps followed for the solution Let's consider a simple 2-input, 2-output MIMO system. The transfer function matrix is represented as H(s) = [H11(s) H12(s); H21(s) H22(s)]. To find the poles and zeros, we can perform transfer function analysis using partial fraction expansion. Let's take the transfer function H11(s) as an example. 8

9. Example: For H11(s), let's assume its transfer function can be expressed as:  H11(s) = A/(s - p) + B/(s - q), where A, B, p, and q are coefficients to be determined. By multiplying both sides by the common denominator (s - p)(s - q), we can rewrite H11(s) as: H11(s) = (A(s - q) + B(s - p))/((s - p)(s - q)) Expanding the numerator, we have: H11(s) = (As - Aq + Bs - Bp)/((s - p)(s - q)) 9

10. Example (Continued): Comparing the coefficients of the terms on the right-hand side with the original transfer function H11(s), we can equate the corresponding coefficients:                                         1 A + B = 1 (coefficient of s term) -Aq - Bp = 0 (coefficient of constant term) Solving these equations, we can determine the values of A, B, p, and q. Once we have the values of p and q, these will be the poles of H11(s). Note: The zeros can be obtained by finding the roots of the numerator polynomial,              which in this case is As - Aq + Bs - Bp. Similarly, we can perform the same analysis for the other transfer functions H12(s), H21(s), and H22(s) to determine their respective poles and zeros. 10

11. Conclusion Poles and zeros play a vital role in understanding the behavior and stability of MIMO systems. Identifying the location and distribution of poles and zeros helps in system analysis, design, and control. Various methods, including transfer function analysis and state-space representation, can be used to determine the poles and zeros of MIMO systems. 11

12. 12 Waleed  Aleyan ​ ​ Hisham Jamal Ali​​ Thank You ..! Faraj Saleh Mustafa Enwider