Linear system Brief history of automatic control I

Linear system Brief history of automatic control (I) • • • 1868 1877 1892 1895 1932 1945 1947 1948 1949 1955 1956 first article of control ‘on governor’s’ –by Maxwell Routh stability criterien Liapunov stability condition Hurwitz stability condition Nyquist Bode Nichols Root locus Wiener optimal control research Kalman filter and controlbility observability analysis Artificial Intelligence

Linear system Brief history of automatic control (II) • • 1957 Bellman optimal and adaptive control 1962 Pontryagin optimal control 1965 Fuzzy set 1972 Vidyasagar multi-variable optimal control and Robust control • 1981 Doyle Robust control theory • 1990 Neuro-Fuzzy

Linear system three eras of control • Classical control : 1950 before – Transfer function based methods • Time-domain design & analysis • Frequency-domain design & analysis • Modern control : 1950~1960 – State-space-based methods • Optimal control • Adaptive control • Post modern control : 1980 after – H∞ control – Robust control (uncertain system)

Linear system Control system analysis and design • Step 1: Modeling – By physical laws – By identification methods • Step 2: Analysis – Stability, controllability and observability • Step 3: Control law design – Classical, modern and post-modern control • Step 4: Analysis • Step 5: Simulation – Matlab, Fortran, simulink etc…. • Step 6: Implement

Linear system Signals & systems Output signals Input signals Time system

Linear system Signal Classification • Continuous signal • Discrete signal

Linear system System classification • Finite-dimensional system (lumped-parameters system described by differential equations) – – Linear systems Nonlinear systems Continuous time and discrete time systems Time-invariant and time varying systems • Infinite-dimensional system (distributed parameters system described by partial differential equations) – – Power transmission line Antenas Heat conduction Optical fiber etc….

Linear system Some examples of linear system • Electrical circuits with constant values of circuit passive elements see figure 1. 10 • Linear OPA circuits • Mechanical system with constant values of k, m, b etc… see figure 1. 11 • Heartbeat dynamic see Page 22 • Eye movement see Page 23 • Commercial aircraft see Page 24

Linear system Definition: A system is linear if superposition principle is satisfied Linearity : (a) homogeneous principle (b) addition principle (c) superposition principle => (a) +(b)

Linear system Continuous-time system ---characterized by differential equations Definition: input and output of the system are continuous functions of the continuous variable time. Discrete-time system ---characterized by different equations Definition: input and output of the system change at only discrete instants of time.

Linear system Linear time invariant (LTI)--continuous Described by a linear differential equation in time domain can be transferred to linear algebra form by using Laplace transform.