TAILIEUCHUNG - Lecture Signals, systems & inference – Lecture 7: Full modal solution, asymptotic stability, reachability and observability

The following will be discussed in this chapter: Modal solution of CT system ZIR, asymptotic stability of CT system, the DT case: linearization at an equilibrium, modal solution of driven DT system, underlying structure of LTI DT statespace system with L distinct modes, reachability and Observability,. | Lecture Signals, systems & inference – Lecture 7: Full modal solution, asymptotic stability, reachability and observability Full modal solution, asymptotic stability, reachability and observability , Spring 2018 Lec 7 1 Modal solution of CT system ZIR L X q(t) = ↵ i v i e>i t 1 with the weights {↵i }L 1 determined by the initial condition: L X q(0) = ↵i vi 1 2 Asymptotic stability of CT system In order to have q(t) ! 0 for all q(0) , we require {Re( i ) < 0}L 1 ., all eigenvalues (natural frequencies) in open left half plane 3 The DT case: linearization at an equilibrium e , x[n] = x¯ + x[n] ¯ + q[n] DT case: q[n] = q e , q[n + 1] = f (q[n], x[n]) # h @f i h @f i e + 1] ⇡ q[n e + q[n] e x[n] @q ¯ ¯ q,x @x ¯ ¯ q,x e for small perturbations q[n] e and x[n] from equilibrium 4 Modal solution of DT system ZIR Could parallel CT development, but let’s proceed di↵erently: 2 3 A1 0 0 ··· 0 6 0 A2 0 ··· 0 7 6 7 6 7 A[ v1 v2 · · · vL ] = [ v1 v2 · · · vL ] 6 7 6 . . 7 4 . . . 5 0 0 0 ··· AL or AV = V⇤ or A = V⇤V-1 or An = (V⇤V-1 ) · · · (V⇤V-1 ) = V⇤n V-1 5 q[n] = An q[0] = V⇤n V-1 q[0]

• Kĩ thuật Viễn thông
• Signals systems
• Lecture Signals systems
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• Lecture ELEC2400 signals and systems
• ELEC2400 signals and systems
• Signal types and operations
• Operations on Signals
• Types of Signals
• Signals and systems using MATLAB
• Systems using MATLAB
• From the ground up
• Continuous time signals
• Continuous time systems
• The laplace transform
• Frequency analysis
• Frequency domain analysis
• Fourier transform
• Frequency response of LTI systems
• Energy spectral density
• State Space Models
• Matrix exponential
• Transfer function
• Observers for state estimation
• State feedback
• Observer based feedback
• Probabilistic models
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• Correlation coefficient
• LTI filtering
• WSS processes
• Power Spectral Density
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• Wiener filtering illustrations
• Hypothesis testing