THE UNIVERSITY OF SCIENCE AND TECHNOLOGY LIAONING Abstract
THE UNIVERSITY OF SCIENCE AND TECHNOLOGY LIAONING Abstract of Linear Systems Theory and Design Group: Four X. Zhao, SQ. Deng, YB. Xi, X. liu, YH. LU 1/3/2022
Chapter 1 State Space Description of Linear Systems 1. State Space Description State Variable A minimum set of variables to describe the system behavior in time domain. State Space An n-dimensional space composed by state variables State Space Description Complete description of system dynamic process 2022/1/3
Chapter 1 State Space Description of Linear Systems 2. State Space Description from I/O Description Linear Differential Equation Transfer Function Transform State Space Description Block Diagram 2022/1/3
Chapter 1 State Space Description of Linear Systems 3. Linear Transformation of the LTI System State The select of state variables is not exclusive. The number of system state variables is exclusive. Different set of state variables can transform into each other by nonsingular transformation. 2022/1/3
Chapter 2 Linear Systems Motion Analysis 1. State-Transition Matrix & Free Motion In zero input condition Initial State X 0 Uniquely Determining System State at Time t State-Transition Matrix 2022/1/3
Chapter 2 Linear Systems Motion Analysis 2. Controlled Motion In controlled condition Motion controlled by input. Follow superposition principle. Free motion caused by initial state. 2022/1/3
Chapter 2 Linear Systems Motion Analysis 3. Transition Matrix and Its Solution l Definition l Inverse Laplace Transform Matrix Exponential l Hamilton-Cayley Theorem 2022/1/3
Chapter 3 System Stability 1. Balanced State & the Stability in Sense of Lyapunov The study of stability in sense of Lyapunov is to study the stability of system balanced point stability. l Balanced State ： Where, Xe is the system balanced piont stability. l Lyapunov’s First Method: Discriminate the system stability based on the system eigenvalue. l Lyapunov’s Second Method: Construct an scalar system scalar function V(X). 2022/1/3
Chapter 3 System Stability 2. Lyapunov Stability Lyapunov Stable Asymptotically Stable Instable 2022/1/3
Chapter 3 System Stability 3. Lyapunov Second Method Systems Quadric Form (Experience, mathematical techniques and etc. ) (Derivation) (Decision rule) • • • Lyapunov Stable Asymptotically Stable Instable For LTI systems Asymptotically Stable 2022/1/3
Chapter 4 the Controllability and Observability of LTI 1. Criterion For LTI systems with n dimension state Criterion Controllability Observability Form 1 Form 2 Form 3 B does not contain a row with all 0 element in diagonal canonical form SISO system dose not have pole-zero cancellation C does not contain a column with all 0 element in diagonal canonical form SISO system dose not have pole-zero cancellation 2022/1/3
Chapter 4 the Controllability and Observability of LTI 2. Transform System to Controllable Canonical Form l Calculate matrix l Calculate invert of Qc l Set l Calculate l Observable canonical form 2022/1/3
Chapter 4 the Controllability and Observability of LTI 3. Transform System to Observable Canonical Form l Calculate matrix l Calculate invert of Qg l Set l Calculate l Observable canonical form 2022/1/3
Chapter 5 State Feedback and State Observer 1. State Feedback State feedback takes system state as part of system input to improve system performance. Original System State Feedback + Closed-loop System = 2022/1/3
Chapter 5 State Feedback and State Observer 2. State Reconstruction State reconstruction is a process building a new utilizing system when design a state observer for original system. Original System State Reconstruction from original State Observer u, y 2022/1/3
Chapter 5 State Feedback and State Observer 3. the Closed-loop System with State Observer State Feedback Using Original System + Closed-loop System = 2022/1/3
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