NUU meiling CHENModern control systems1 Lecture 01 --Introduction 1.1 Brief History 1.2 Steps to study a control system 1.3 System classification 1.4 System response

NUU meiling CHENModern control systems2 Brief history of automatic control (I) 1868 First article of control ‘on governor’s’ –by Maxwell 1877 Routh stability criterion 1892 Liapunov stability condition 1895 Hurwitz stability condition 1932 Nyquist 1945 Bode 1947 Nichols 1948 Root locus 1949 Wiener optimal control research 1955 Kalman filter and controlbility observability analysis 1956 Artificial Intelligence

NUU meiling CHENModern control systems3 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

NUU meiling CHENModern control systems4 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)

NUU meiling CHENModern control systems5 Control system analysis and design Step1: Modeling –By physical laws –By identification methods Step2: Analysis –Stability, controllability and observability Step3: Control law design –Classical, modern and post-modern control Step4: Analysis Step5: Simulation –Matlab, Fortran, simulink etc…. Step6: Implement

NUU meiling CHENModern control systems6 Time system Input signals Output signals Signals & systems

NUU meiling CHENModern control systems7 Signal Classification Continuous signal Discrete signal

NUU meiling CHENModern control systems8 System classification Finite-dimensional system (lumped-parameters system described by differential equations) –Linear systems and 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 –Antennas –Heat conduction –Optical fiber etc….

NUU meiling CHENModern control systems9 Some examples of linear system Electrical circuits with constant values of circuit passive elements Linear OPA circuits Mechanical system with constant values of k,m,b etc Heartbeat dynamic Eye movement Commercial aircraft

NUU meiling CHENModern control systems10 Linear system A system is said to be linear in terms of the system input x(t) and the system output y(t) if it satisfies the following two properties of superposition and homogeneity. Superposition ： Homogeneity ：

NUU meiling CHENModern control systems11 Example Non linear system

NUU meiling CHENModern control systems12 example The system is governed by a linear ordinary differential equation (ODE) Linear time invariant system linearity

NUU meiling CHENModern control systems13 Examples : Circuit system

NUU meiling CHENModern control systems14 Examples Discrete system Time delay

NUU meiling CHENModern control systems15 Properties of linear system : (1) (2)

NUU meiling CHENModern control systems16 Time invariance A system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time shift in the output signal. Time invariant system

NUU meiling CHENModern control systems17 Example 1.18 Time varying system

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NUU meiling CHENModern control systems19 LTI System representations 1.Order-N Ordinary Differential equation 2.Transfer function (Laplace transform) 3.State equation (Finite order-1 differential equations) ) 1.Ordinary Difference equation 2.Transfer function (Z transform) 3.State equation (Finite order-1 difference equations) Continuous-time LTI system Discrete-time LTI system

NUU meiling CHENModern control systems20 constants Order-2 ordinary differential equation Continuous-time LTI system Transfer function Linear system  initial rest

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NUU meiling CHENModern control systems22 System response: Output signals due to inputs and ICs. 1. The point of view of Mathematic: 2. The point of view of Engineer: 3. The point of view of control engineer: Homogenous solutionParticular solution ＋ ＋ ＋ Zero-state responseZero-input responseNatural responseForced response Transient responseSteady state response

NUU meiling CHENModern control systems23 Example: solve the following O.D.E (1) Particular solution:

NUU meiling CHENModern control systems24 (2) Homogenous solution: have to satisfy I.C.

NUU meiling CHENModern control systems25 (3) zero-input response: consider the original differential equation with no input. zero-input response

NUU meiling CHENModern control systems26 (4) zero-state response: consider the original differential equation but set all I.C.=0. zero-state response

NUU meiling CHENModern control systems27 (5) Laplace Method:

NUU meiling CHENModern control systems28 Complex response Zero state responseZero input response Forced response (Particular solution) Natural response (Homogeneous solution) Steady state responseTransient response