Controllability and Observability of Linear Systems PDF
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This document explains controllability and observability in linear systems. It defines controllability as the system's ability to reach any desired state from any initial state using a suitable input signal. The document further details how to check system controllability and observability using matrices and examples.
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## Controllability and Observability of Linear Systems ### Introduction This document explores the concepts of controllability and observability in linear systems. These concepts are crucial for understanding the ability to influence and monitor a system's behavior. ### Controllability **Defini...
## Controllability and Observability of Linear Systems ### Introduction This document explores the concepts of controllability and observability in linear systems. These concepts are crucial for understanding the ability to influence and monitor a system's behavior. ### Controllability **Definition:** A system is controllable if, for any initial state and final state, there exists an input sequence that can transfer the system from the initial state to the final state in finite time. **Controllability Matrix:** The controllability matrix (Uc) for a system defined by the equations * **X(K+1) = AX(K) + Bu(K)** * **Y(K) = C(X(K))** is given by: * **Uc = [B AB A²B ... A<sup>(n-1)</sup>B]** where **n** is the order of the system. **Controllability Condition:** A system is controllable if and only if the controllability matrix (Uc) has full rank, which means the determinant of Uc is non-zero. **Example:** Given the system described by: * **A= [-2 1; 1 -2]** * **B = [1; 1]** * **C = [0 1]** The controllability matrix is calculated as: * **Uc = [B AB] = [1 1; 1 -1]** The determinant of Uc is: * **|Uc| = -1 + 1 = 0** Therefore, the system is **uncontrollable**. **Transfer Function and Controllability:** The transfer function (T.F) can be expressed as: * **T.F = Y(z) / U(z) = C[zI - A]<sup>-1</sup> B** If a pole of the T.F cancels out, the system is uncontrollable. This cancellation is directly related to the controllability matrix having a zero determinant. ### Observability **Definition:** A system is observable if, given the output of the system over a finite time interval, it is possible to determine the initial state of the system. **Observability Matrix:** The observability matrix (Uo) for a system defined by the equations * **X(K+1) = AX(K) + Bu(K)** * **Y(K) = C(X(K))** is given by: * **Uo = [C CA C A<sup>2</sup> ... C A<sup>(n-1)</sup>]** where **n** is the order of the system. **Observability Condition:** A system is observable if and only if the observability matrix (Uo) has full rank, which means the determinant of Uo is non-zero. **Example:** Given the system described by: * **A= [1 2; 3 4]** * **B = [1; 1]** * **C = [1 2]** The observability matrix is calculated as: * **Uo = [C CA] = [1 2; 3 4 7 10]** The determinant of Uo is: * **|Uo| = 10 - 14 = -4 ≠ 0** Therefore, the system is **observable**. **Relationship between Controllability and Observability:** * A system can be both controllable and observable. * A system can be uncontrollable and unobservable. * A system can be controllable but unobservable. * A system can be unobservable but controllable. ### Conclusion Controllability and observability are fundamental concepts in linear systems analysis. They provide insights into the ability to influence and monitor the system's behavior. Understanding these concepts is crucial for effective system design and control.
