Controllability and Observability 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

Modern Contral Systems Motivation1 uncontrollable controllable Modern Contral Systems

Modern Contral Systems Controllability and Observability Plant: Definition of Controllability 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 . Modern Contral Systems

Modern Contral Systems Controllability Matrix Example: An Uncontrollable System ※ State is uncontrollable. Modern Contral Systems

Modern Contral Systems Proof of controllability matrix Initial condition Modern Contral Systems

Modern Contral Systems Motivation2 observable unobservable Modern Contral Systems

Modern Contral Systems Definition of Observability 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 . Modern Contral Systems

Modern Contral Systems Observability Matrix Example: An Unobservable System ※ State is unobservable. Modern Contral Systems

Modern Contral Systems Proof of observability matrix Inputs & outputs Modern Contral Systems

Modern Contral Systems Example Plant: Hence the system is both controllable and observable. Modern Contral Systems

Modern Contral Systems Theorem I Controllable canonical form Controllable Theorem II Observable canonical form Observable Modern Contral Systems

Modern Contral Systems example Controllable canonical form Observable canonical form Modern Contral Systems

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

Modern Contral Systems Example If uncontrollable unobservable Modern Contral Systems

Modern Contral Systems

Modern Contral Systems controllable observable In the previous example controllable unobservable Modern Contral Systems

Modern Contral Systems Example L.I. L.I. L.D. L.I. Modern Contral Systems

Modern Contral Systems Kalman Canonical Decomposition Diagonalization: All the Eigenvalues of A are distinct, i.e. There exists a coordinate transform (See Sec. 4.4) such that System in z-coordinate becomes Homogeneous solution of the above state equation is Modern Contral Systems

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

Modern Contral Systems Transfer function is H(s) has pole-zero cancellation. ∑ Modern Contral Systems

Modern Contral Systems Kalman Canonical Decomposition Modern Contral Systems

Modern Contral Systems Kalman Canonical Decomposition: State Space Equation (5.X) Modern Contral Systems

Modern Contral Systems Example Plant: is uncontrollable. is unobservable. The same reasoning may be applied to mode 1 and 2. Modern Contral Systems

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

Modern Contral Systems Example Plant: Transfer Function Modern Contral Systems

Modern Contral Systems Example 5.6 Plant: Transfer Function Modern Contral Systems

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

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