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Lecture 06 Analysis (II) Controllability and Observability. 6.1 Controllability and Observability 6.2 Kalman Canonical Decomposition 6.3 Pole-zero Cancellation in Transfer Function 6.4 Minimum Realization. Motivation1. uncontrollable. controllable.
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Lecture 06 Analysis (II) Controllability and Observability 6.1 Controllability and Observability 6.2 Kalman Canonical Decomposition 6.3 Pole-zero Cancellation in Transfer Function 6.4 Minimum Realization Modern Contral Systems
Motivation1 uncontrollable controllable Modern Contral Systems
A system is said to be (state) controllable at time , if there exists a finite such for any and any , there exist an input that will transfer the state to the state at time , otherwise the system is said to be uncontrollable at time . Controllability and Observability Plant: Definition of Controllability Modern Contral Systems
※ State is uncontrollable. Controllability Matrix Example: An Uncontrollable System Modern Contral Systems
Proof of controllability matrix Initial condition Modern Contral Systems
Motivation2 observable unobservable Modern Contral Systems
A system is said to be (completely state) observable at time , if there exists a finite such that for any at time , the knowledge of the input and the output over the time interval suffices to determine the state , otherwise the system is said to be unobservable at . Definition of Observability Modern Contral Systems
※ State is unobservable. Observability Matrix Example: An Unobservable System Modern Contral Systems
Proof of observability matrix Inputs & outputs Modern Contral Systems
Example Plant: Hence the system is both controllable and observable. Modern Contral Systems
Theorem I Controllable canonical form Controllable Theorem II Observable canonical form Observable Modern Contral Systems
example Controllable canonical form Observable canonical form Modern Contral Systems
Linear system 1. Analysis Theorem III Jordan form Jordan block Least row has no zero row First column has no zero column Modern Contral Systems
If uncontrollable If unobservable Example Modern Contral Systems
controllable observable In the previous example controllable unobservable Modern Contral Systems
Example L.I. L.I. L.D. L.I. Modern Contral Systems
Kalman Canonical Decomposition Diagonalization: All the Eigenvalues of A are distinct, i.e. such that There exists a coordinate transform (See Sec. 4.4) System in z-coordinate becomes Homogeneous solution of the above state equation is Modern Contral Systems
How to construct coordinate transformation matrix for diagonalization All the Eigenvalues of A are distinct, i.e. The coordinate matrix for diagonalization Consider diagonalized system Modern Contral Systems
∑ Transfer function is H(s) has pole-zero cancellation. Modern Contral Systems
Kalman Canonical Decomposition Modern Contral Systems
Kalman Canonical Decomposition: State Space Equation (5.X) Modern Contral Systems
Example Plant: is uncontrollable. is unobservable. The same reasoning may be applied to mode 1 and 2. Modern Contral Systems
Pole-zero Cancellation in Transfer Function From Sec. 5.2, state equation may be transformed to Hence, the T.F. represents the controllable and observable parts of the state variable equation. Modern Contral Systems
Example Plant: Transfer Function Modern Contral Systems
Example 5.6 Plant: Transfer Function Modern Contral Systems
Minimum Realization Realization: Realize a transfer function via a state space equation. Example Realization of the T.F. Method 1: Method 2: ※There is infinity number of realizations for a given T.F. . Modern Contral Systems
Minimum realization: Realize a transfer function via a state space equation with elimination of its uncontrollable and unobservable parts. Example 5.8 Realization of the T.F. Modern Contral Systems
