Linear System Theory Wonhee Kim Lecture 1 March 7, 2018 1 / 22
Overview Course Information Prerequisites Course Outline What is Control Engineering? Examples of Control Systems Structure of Control Systems Linear Systems Nonlinear Systems 2 / 22
Course Information Instructor: Wonhee Kim Email/Office: whkim79@cau.ac.kr/room 310-439 Course Website: wonheekim.wordpress.com Textbook: Chi-Tsong Chen, Linear System Theory, 4th edition, Oxford University Press Grading Policy: HW(20%), Midterm(40%), Final Exam (40%) Check the course website for updates 3 / 22
Prerequisites linear algebra signals and systems: Laplace transform (z-transform) differential and difference equations some familiarity of classical control theory (Bode plot, root locus, Nyquist, PID, etc) and programming skills (e.g. MATLAB) would be helpful We will cover the required mathematical skills during the lectures 4 / 22
Tentative Course Topics linear algebra vector space, norm, inner product, column and null spaces, eigenvalues, basis, rank, similarity transformation, Jordan form, etc. ordinary differential equations (and difference equations) state space representation, solutions, matrix exponential, etc stability controllability, observability, decomposition linear system design state-feedback, pole-placement, observer design optimal control (if time permits) 5 / 22
What is Control Engineering? Control Engineering (Wikipedia): Control engineering is the engineering discipline that applies control theory to design systems with desired behaviors. Control Theory (Wikipedia): Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback. In this course, we will study mathematical control theory 6 / 22
Examples of Control Systems Aircraft, Vehicles, Defense, Circuits, Communication, Power Systems, Social Networks, Economics, etc Desired behavior: safety, speed, position, price, power, etc. 7 / 22
Examples of Control Systems Segway: https://www.youtube.com/watch?v=rmlg5qkusfq Space Falcon Heavy side booster landing: https://www.youtube.com/watch?v=u0-pfzkbh2k Inverted pendulum https://www.youtube.com/watch?v=jpnahkt7yy4 8 / 22
Structure of Control Systems d n r - Actuator + Plant Sensor + y u Controller y Block Plant: system that needs to be controlled (ODE or difference equations) motor, aircraft, pendulum, etc Actuator/sensor Controller: controller that controls the plant Signal Input (r): reference signal Output (y): sensor signal Noise (n) / Disturbance (d): unwanted signal (need to reduce their effect) Control (u): control signal (we need to design) Structure: Open loop / Feedback 9 / 22
Continuous-Time Linear Time-Varying (LTV) System ẋ(t) = dx(t) = A(t)x(t) + B(t)u(t) + D(t)d(t) dt y(t) = C(t)x(t) + E(t)u(t) + F (t)n(t) t 0: time x R n : state u R m : control d R l : disturbance y R p : (sensor) output n R q : noise A(t): n n system matrix B(t): n m input matrix D(t): n l disturbance matrix C(t): p n output matrix E(t): p m feedthrough matrix F (t): p q noise matrix 10 / 22
Continuous-Time Linear Time-Invariant (LTI) System ẋ(t) = dx(t) = Ax(t) + Bu(t) + Dd(t) dt y(t) = Cx(t) + Eu(t) + Fn(t) t 0: time x R n : state u R m : control d R l : disturbance y R p : (sensor) output n R q : noise A: n n system matrix B: n m input matrix D: n l disturbance matrix C: p n output matrix E: p m feedthrough matrix F : p q noise matrix 11 / 22
LTV and LTI Systems LTV system: constants are time-varying LTI system: constants are time-invariant LTI system can be converted into the transfer function via the Laplace (or z) transformation LTV and LTI systems: first-order ODE (first-order recursive equation) also called state equation or state system state x: position, velocity, acceleration, etc, which capture the behavior of the system scalar (one-dimensional) u and y: single-input-single-output (SISO) system In this course, we consider continuous-time LTV and LTI systems when D = E = F = 0 (system without disturbance, noise and feedthrough terms) 12 / 22
Discrete-Time LTV and LTI Systems discrete-time LTV system x(k + 1) = A(k)x(k) + B(k)u(k) + D(k)d(k) y(k) = C(k)x(k) + E(k)u(k) + F (k)n(k) discrete-time LTI System x(k + 1) = Ax(k) + Bu(k) + Dd(k) y(k) = Cx(k) + Eu(k) + Fn(k) tk {0, 1, 2,...} difference equation (first-order recursive equation) x, y, u are sequences sampled system: x(t) := x(kt ) (T : sampling period) 13 / 22
Nonlinear Systems continuous-time nonlinear system ẋ(t) = f (t, x(t), u(t), d(t)), y(t) = g(t, x(t), u(t), n(t)) discrete-time nonlinear system x(k + 1) = f (k, x(k), u(k), d(k)), y(k) = g(k, x(k), u(k), n(k)) Example: ẋ(t) = x 2 (t), ẋ(t) = cos(t) Linear system can be obtained by linearization of a nonlinear system next class 14 / 22
Nonlinear Systems continuous-time nonlinear system ẋ(t) = f (t, x(t), u(t), d(t)), y(t) = g(t, x(t), u(t), n(t)) discrete-time nonlinear system x(k + 1) = f (k, x(k), u(k), d(k)), y(k) = g(k, x(k), u(k), n(k)) Example: ẋ(t) = x 2 (t), ẋ(t) = cos(t) Linear system can be obtained by linearization of a nonlinear system next class 14 / 22
Why Study Linear Systems? Linear system is a special case of nonlinear systems Why do we study linear systems? If you do not understand linear systems, you cannot understand nonlinear systems Nonlinear system Existence of solution? Hard to analyze its dynamic behavior Hard to see its input/output characteristics Linear system Solution always exists System characteristics depend on coefficients of the system Computationally inexpensive Easy to implement (real-time systems) Linear algebra is the most effective tool Many applications can be represented by linear systems (circuits, aircraft, missile, communication, traffic, guidance, economics) 15 / 22
Why Study Linear Systems? Linear system is a special case of nonlinear systems Why do we study linear systems? If you do not understand linear systems, you cannot understand nonlinear systems Nonlinear system Existence of solution? Hard to analyze its dynamic behavior Hard to see its input/output characteristics Linear system Solution always exists System characteristics depend on coefficients of the system Computationally inexpensive Easy to implement (real-time systems) Linear algebra is the most effective tool Many applications can be represented by linear systems (circuits, aircraft, missile, communication, traffic, guidance, economics) 15 / 22
Continuous-Time LTI System continuous-time SISO-LTI system ẋ(t) = Ax(t), x(0) = x 0, (A R, A 0) This is a continuous-time SISO autonomous system (no input, u) Solution x(t) = e At x 0 The behavior of x(t) is determined by the value of A A > 0: x(t) as t for all x 0 R (unstable) A < 0: x(t) 0 as t for all x 0 R (stable) The scalar A determines the speed of convergence or divergence of x(t) ( system characteristics depend on coefficients of the system) What if A is a matrix? 16 / 22
Continuous-Time LTI System continuous-time SISO-LTI system ẋ(t) = Ax(t), x(0) = x 0, (A R, A 0) This is a continuous-time SISO autonomous system (no input, u) Solution x(t) = e At x 0 The behavior of x(t) is determined by the value of A A > 0: x(t) as t for all x 0 R (unstable) A < 0: x(t) 0 as t for all x 0 R (stable) The scalar A determines the speed of convergence or divergence of x(t) ( system characteristics depend on coefficients of the system) What if A is a matrix? 16 / 22
Continuous-Time LTI System continuous-time SISO-LTI system ẋ(t) = Ax(t), x(0) = x 0, (A R, A 0) This is a continuous-time SISO autonomous system (no input, u) Solution x(t) = e At x 0 The behavior of x(t) is determined by the value of A A > 0: x(t) as t for all x 0 R (unstable) A < 0: x(t) 0 as t for all x 0 R (stable) The scalar A determines the speed of convergence or divergence of x(t) ( system characteristics depend on coefficients of the system) What if A is a matrix? 16 / 22
Continuous-Time LTI System two-dimensional diagonal continuous-time LTI system ( ) a 0 ẋ(t) = Ax(t), x(0) = x 0, A =, a 0 0 a Solution x(t) = ( e at e at ) x 0 What is lim t x(t)? 17 / 22
Continuous-Time LTI System two-dimensional diagonal continuous-time LTI system ( ) a 0 ẋ(t) = Ax(t), x(0) = x 0, A =, a 0 0 a Solution x(t) = ( e at e at ) x 0 What is lim t x(t)? 17 / 22
Continuous-Time LTI System two-dimensional diagonal continuous-time LTI system ( ) a 0 ẋ(t) = Ax(t), x(0) = x 0, A =, a 0 0 a Solution x(t) = ( e at e at ) x 0 What is lim t x(t)? 17 / 22
Continuous-Time LTI System continuous-time SISO-LTI system with control u ẋ(t) = Ax(t) + Bu(t), x(0) = 0 y(t) = Cx(t) (A, B, C R, A < 0) Problem: find appropriate u so that y = h R Naive approach: consider static input and output (u, x, y constant) ẋ(t) = 0 = Ax + Bu, y = h = Cx Then (why?) u(t) = A CB h 18 / 22
Continuous-Time LTI System continuous-time SISO-LTI system with control u ẋ(t) = Ax(t) + Bu(t), x(0) = 0 y(t) = Cx(t) (A, B, C R, A < 0) Problem: find appropriate u so that y = h R Naive approach: consider static input and output (u, x, y constant) ẋ(t) = 0 = Ax + Bu, y = h = Cx Then (why?) u(t) = A CB h 18 / 22
Continuous-Time LTI System continuous-time SISO-LTI system with control u ẋ(t) = Ax(t) + Bu(t), x(0) = 0 y(t) = Cx(t) (A, B, C R, A < 0) Problem: find appropriate u so that y = h R Naive approach: consider static input and output (u, x, y constant) ẋ(t) = 0 = Ax + Bu, y = h = Cx Then (why?) u(t) = A CB h 18 / 22
Continuous-Time LTI System time-response plot when A = 1, B = C = 1, h = 0.5 MATLAB Simulink: control system toolbox 19 / 22
Continuous-Time LTI System time-response plot when A = 1, B = C = 1, h = 0.5 MATLAB Simulink: control system toolbox 20 / 22
Continuous-Time LTI System This is one simple approach to design u There are may ways of designing control u to achieve the desired control performance In this course, we will study some design techniques of u for LTI systems We will not cover classical control theory 21 / 22
Next Class classical and modern control theory LTI systems linearization modeling 22 / 22