LST_lecture_4.pdf - HYU ELE7030 Linear System Spring 2018...

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HYU ELE7030 Linear System, Spring 2018 S.-H. Lee Lecture 4 Controllability and Observability Goals of this lecture: · Controllability and Observability · Controllability and Observability Grammians · Controllability and Observability Matrices · Controllability and Observability for LTI Systems · Controllability and Observability for Discrete-time Systems · Controllability and Observability Tests · Controllable and Observable Subspaces · Controllable and Observable Decomposition
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HYU ELE7030 Linear System, Spring 2018 S.-H. Lee Consider a system u 2 U y 2 Y x 2 S x o ( ˙ x = f ( t, x, u ) x ( t o ) = x o y = g ( t, x, u ) Σ = ( U , S , Y , φ s , φ ys ) : dynamic system φ s ( t, t o , x o , u [ t o ,t ] ) : the solution corresponding to x ( t o ) = x o and the input u ( t ) φ ys ( t, x, u [ t o ,t ] ) : output response Given an initial state x ( t o ) , to what state x ( t ) can I steer the system by varying the inputs u ( t ) ? Given u ( t ) and y ( t ) , t o t T , can I uniquely determine the initial condition x ( t o ) = x o ? Ctrb. & Obsv. IV-1
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HYU ELE7030 Linear System, Spring 2018 S.-H. Lee Good to recall: Inverse and Adjoint of Linear Continuous Operation: L : U → V Inverse L - 1 : When R ( L ) = V and N ( L ) = { 0 } hold, u = L - 1 ( v ) s.t. L ( L - 1 ( v )) = v and L - 1 ( L ( u )) = u u ∈ U , v ∈ V Adjoint L * : For Hilbert spaces, i.e., complete inner product spaces ( U , F , , ·i U ) and ( V , F , , ·i V ) , the adjoint of L : U → V is defined as the map L * : V → U , s.t. hL u, v i V = h u, L * v i U u ∈ U , v ∈ V e.g. hL u, v i = ( L u ) T v = u T L T v = u, L T v Note : L : U → V is continuous iff a real number γ s.t. kL u k ≤ γ k u k ∀ u ∈ U . Ctrb. & Obsv. IV-2
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HYU ELE7030 Linear System, Spring 2018 S.-H. Lee Good to recall: Domain and Codomain, Range (Image), Kernel (Null space) 0 V U = R ( L ¤ ) ? L N ( L ) V = R ( L ) ? L N ( L ¤ ) 0 U L R ( L ¤ ) N ( L ) R ( L ) N ( L ¤ ) L ¤ L : U → V L * : V → U R ( LL * ) = R ( L ) N ( LL * ) = N ( L * ) R ( L * L ) = R ( L * ) N ( L * L ) = N ( L ) Note : V = R ( L ) L N ( L * ) = R ( LL * ) L N ( L * ) = R ( LL * ) L N ( LL * ) Ctrb. & Obsv. IV-3
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HYU ELE7030 Linear System, Spring 2018 S.-H. Lee For Hilbert spaces ( U , F , , ·i U ) and ( V , F , , ·i V ) , let L : U → V be a continuous linear map with adjoint L * : V → U . Under these conditions LL * : V → V L * L : U → U are continuous linear maps with LL * and L * L self-adjoint. U = R ( L * ) L N ( L ) , V = R ( L ) L N ( L * ) The restriction L| R ( L * ) is a bijection of R ( L * ) onto R ( L ) and N ( LL * ) = N ( L * ) R ( LL * ) = R ( L ) The restriction L * | R ( L ) is a bijection of R ( L ) onto R ( L * ) and N ( L * L ) = N ( L ) R ( L * L ) = R ( L * ) Ctrb. & Obsv. IV-4
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HYU ELE7030 Linear System, Spring 2018 S.-H. Lee Controllability, Reachability, Stabilizability: Can we steer the state, via the control input, to certain locations in the state space? Controllability: can initial state be driven back to origin? Reachability: can a certain state be reached from origin? Stabilizability: can all states be taken back to origin (equil. pt)? Ctrb. & Obsv. IV-5
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HYU ELE7030 Linear System, Spring 2018 S.-H. Lee Good to note: Controllability and Reachability For continuous-time, linear time-invariant systems: Completely controllable ⇐⇒ Completely reachable In discrete-time, x ( k + 1) = 0 x ( k ) completely controllable but not completely reachable (no non-zero state is reachable) Note : By ‘completely’, we mean all states are ...
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