Lecture 19- Solution of State Equations

Lecture 19- Solution of State Equations - Lecture 19...

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1 Lecture 19 – Solution of State Equations ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 19: Solution of State Equations Reading: Boulet - Chapter 10 Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 19 – Solution of State Equations ECSE304 Signals and Systems II Course Outline • Discrete-Time Fourier Series and DT Fourier Transform • The Z – Transform • Time and Frequency Analysis of DT Signals and Systems • Sampling Systems – Lecture 12: The Sampling Theorem – Lecture 13: Discrete Time Processing of Continuous Time Signals – Lecture 14: Sampling of Discrete Time Signals • Application to Communications Systems – Lecture 15: Amplitude Modulation – Lecture 16: Single Sideband and Pulse Amplitude Modulation – Lecture 17: Frequency and Time Division Multiplexing and Angle Modulation • State Models of Continuous Time LTI Systems – Lecture 18: State Space Analysis – Lecture 19: Solution of State Equations – Lecture 20: Observability and Controllability • Linear Feedback Systems
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3 Lecture 19 – Solution of State Equations ECSE304 Signals and Systems II • Laplace Transform Domain Solution of State Equations • System Stability • Time Domain Solution of State Equations Outline 4 Lecture 19 – Solution of State Equations ECSE304 Signals and Systems II • Last time we obtained normal form equations: N simultaneous first-order state equations for a causal continuous-time state- space system … where we assumed a single input single output system: • We can solve for the state vector using Laplace transforms – Start with i th state equation with N states and M inputs: – Take the unilateral Laplace transform by letting Laplace Transform Solution of State Equations x Ax Bu y Cx Du = + =+ ± x 11 ... ... i i iN N i iM M xa x a x b u b u = ++ + ± () ( ) (0 ) ii i xt s s x ⇔− X ± s X ut s U , () n yt ∈∈ RR R
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5 Lecture 19 – Solution of State Equations ECSE304 Signals and Systems II • The unilateral Laplace transform of the i th state equation is … and the unilateral LT of the normal form equations is • Solve for unilateral LT of the state vector: … where is the N x N identity matrix
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Lecture 19- Solution of State Equations - Lecture 19...

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