CDS 101/110: Lecture 3.1 Linear Systems Goals for Today: • Revist and motivate linear time-invariant system models: • Summarize properties, examples, and tools − Convolution equation describing solution in response to an input − Step response, impulse response − Frequency response • Characterize performance of linear systems Reading: • Åström and Murray, FBS-2e, Ch 6.1-6.3 1 Why are Linear Systems Important? u(t) Many important examples q2 q1 • Mechanical Systems m1 m2 k1 k2 k3 c m x f M 2 Why are Linear Systems Important? 3 Why are Linear Systems Important? Many important examples • Electronic circuits − Especially true after feedback − Frequency response is key performance specification 4 Why are Linear Systems Important? Many important examples Many important tools • Frequency and step response, – Traditional tools of control theory – Developed in 1930’s at Bell Labs CDS 110 • Classical control design toolbox – Nyquist plots, gain/phase margin – Loop shaping • Optimal control and estimators CDS – u(t) Linear quadratic regulators 112 q2 – Kalman estimators q1 m1 m2 • Robust control design CDS k1 k2 k3 – H1 control design 212/ c – µ analysis for structured 213 uncertainty 5 Solutions of Linear Time Invariant Systems: “Modes” Linear Time Invariant (LTI) System: • If Linear System, input u(t) leads to output y(t) • If u(t+T) leads to output y(t+T), the system is time invariant • Matrix LTI system, with no input • Let and be eigenvalue/eigenvector of . Then: ^2 = + + + = + + + 1! 22! 1! 2! 2 2 = ⋯ ⋯ • If n distinct eigenvalues, then 0 = + + + , and 0 = + 1 1 +2 2 + Sum of ⋯ “modes” 1 2 1 1 2 2 ⋯ 6 The Convolution Integral: Step 1 Let denote the response of a LTI system to a unit step input at t=0. – Assuming the system starts at Equilibrium The response to the steps are: – First step input at time t=0: – Second step input at time : ( ) − 0 0 – Third step input at time : ( ) 1 − 1 1 − 0 By linearity, we can add the response2 − 2 2 − 1 = + + … = + ( ) − 0 0 − 1 1 − 0 + … 0 1 0 1 2 1 = 0< − − − ( )( − −) − +1 − − − +1 � +1 0 − Taking the =limit0 as − +1 = − → ′ 7 �0 − Impulse Response homogeneous • What is the “impulse response” due to u(t)=δ(t)? take limit as dt 0 but keep unit area u → 1/dt 0 < 0 t = = 1/ 0 < = lim ( ) dt 0 � ≤ →0 • Apply this unit impulse to the system (with x(0)=0): ≥ = 8 Response to inputs: Convolution homogeneous At • Impulse response, h(t) = Ce B – Response to input “impulse” – Equivalent to “Green’s function” • Linearity ⇒ compose response to arbitrary u(t) using convolution – Decompose input into “sum” of shifted impulse functions – Compute impulse response for each – “Sum” impulse response to find y(t) – Take limit as dt→ 0 • Complete solution: use integral instead of “sum” • linear with respect to initial condition and input • 2X input ⇒ 2X output when x(0) = 0 Convolution Theorem 9 MATLAB Tools for Linear Systems Initial Condition Results A = [-1 1; 0 -1]; B = [0; 1]; 1 C = [1 0]; D = [0]; 0.8 x0 = [1; 0.5]; 0.6 To: Y(1) 0.4 Amplitude sys = ss(A,B,C,D); 0.2 0 initial(sys, x0); 0 1 2 3 4 5 6 7 impulse(sys); Linear Simulation Results 1 t = 0:0.1:10; 0.5 u = 0.2*sin(5*t) + cos(2*t); To: Y(1) Amplitude 0 lsim(sys, u, t, x0); -0.5 0 1 2 3 4 5 6 7 8 9 10 • Other MATLAB commands Time (sec.) – gensig, square, sawtooth – produce signals of diff. types ltiview – linear time invariant – step, impulse, initial, lsim – time domain analysis system plots – bode, freqresp, evalfr – frequency domain analysis 10 MATLAB Tools for Phase Space System Equations MATLAB CODE = 2 [x1, x2]=meshgrid(-0.5:0.05:0.5, -0.5:0.05:0.5); = 2 x1dot=-x1 - 2*x2*x1^2 + x2; ̇1 −1 − 21 2 x2dot=-x1-x2; ̇2 −1 − 2 quiver(x1,x2,x1dot,x2dot); x 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 11 MATLAB Tools for Phase Space function my_phases(IC) hold on [~,X]=ode45(@EOM,[0,50],IC); u=X(:,1); 0.6 w=X(:,2); plot(u,w,'r'); 0.4 xlabel('u') ylabel('w') 0.2 grid 0 end w function dX=EOM(t,X) -0.2 dX=zeros(2,1); x1=X(1); -0.4 x2=X(2); -0.6 x1dot=-x1 - 2*x2*x1^2 + x2; -0.6 -0.4 -0.2 0 0.2 0.4 0.6 x2dot=-x1-x2; u dX=[x1dot;x2dot]; end 12 Input/Output Performance • How does system respond to changes in input values? – Transient response: – Steady state response: • Characterize response in terms of – Impulse response Transient Steady State – Step response – Frequency response Mass spring system (L1.2) Response to I.C.s, Response to inputs, Transient response Steady State (if constant inputs) 13 Step Response • Output characteristics in response to a “step” input – Rise time: time required to move from 5% to 95% of final value – Overshoot: ratio between amplitude of first peak and steady state value – Settling time: time required to remain w/in p% (usually 2%) of final value – Steady state value: final value at t = 1 1.5 Overshoot 1 Steady state value Amplitude 0.5 Rise time Settling time 0 0 5 10 15 20 25 Time (sec.) 14 Frequency Response • Measure steady state response of system to sinusoidal input – Example: audio amplifier – would like consistent (“flat”) amplification between 20 Hz & 20,000 Hz – Individual sinusoids are good test signals for measuring performance in many systems • Approach: plot input and output, measure relative amplitude and phase – Use MATLAB or SIMULINK to generate 1.5 u response of system to sinusoidal output 1 Au y A – Gain = Ay/Au 0.5 y – Phase = 2π ∙ ΔT/T 0 T Amplitude -0.5 Δ • May not work for nonlinear systems -1 T -1.5 – System nonlinearities can cause 0 5 10 15 20 harmonics to appear in the output Time (sec.) – Amplitude and phase may not be well-defined – For linear systems, frequency response is always well defined 15 Computing Frequency Responses • Technique #1: plot input and output, measure relative amplitude and phase – Generate response of system to 1.5 u sinusoidal output 1 Au y 0.5 Ay – Gain = Ay/Au – Phase = 2π ∙ ΔT/T 0 – For linear system, gain and phase -0.5 ΔT T don’t depend on the input amplitude -1 -1.5 0 5 10 15 20 • Technique #2 (linear systems): use bode (or freqresp) command – Assumes linear dynamics in state 20 10 bode(ss(A,B,C,D)) space form: 0 -10 -20 -30 -40 Magnitude (dB) -50 -60 – Gain plotted on log-log scale 0 -90 • dB = 20 log10 (gain) – Phase plotted on linear-log scale -180 -270 Phase (deg) -360 0.1 1 10 Frequency (rad/sec) 16 Calculating Frequency Response from convolution equation • Convolution equation describes response to any input; use this to look at response to sinusoidal input: u(t) Transient (decays if stable) Ratio of response/input “Frequency response” 17 Second Order Systems • Many important examples: • Insight to response for higher orders (eigenvalues of A are either real or complex) – Exception is non-diagonalizable A (non-trivial Jordan form) 2 1.8 1.6 T=2π/ω0 1.4 1.2 y 1 0.8 0.6 ζ 0.4 0.2 0 0 5 10 15 20 25 time (sec) – Analytical formulas exist for overshoot, rise time, settling time, etc – Will study more next week 18 Frequency Spring Mass System response: C(jωI-A)-1B+D Eigenvalues of A: u(t) • For zero damping, jω1 and jω2 q 2 • ω ω q1 1 and 2 correspond frequency m1 m2 response peaks • The eigenvectors for these eigenvalues k 1 k2 k3 give the mode shape: – In-phase motion for lower freq. c – Out-of phase motion for higher freq. Frequency Response 10 1 0.1 0.01 Gain (log scale) 0.001 0 -90 -180 -270 Phase [deg] -360 0.1 Frequency1 [rad/sec] 19 10 Summary: Linear Systems u y 1 1 0 0 -1 -1 + 0 5 10 + 0 5 10 1 0.5 0 0 -1 -0.5 0 5 10 0 5 10 • Properties of linear systems 2 2 – Linearity with respect to initial condition 0 0 and inputs -2 -2 0 5 10 0 5 10 – Stability characterized by eigenvalues – Many applications and tools available – Provide local description for nonlinear systems 20 Example: Inverted Pendulum on a Cart m State: Input: u = F x Output: y = x Linearize according to previous f formula around = 0 M 22 Second Order Systems • Many important examples: • Response of 1st and 2nd order systems -> insight to response for higher orders (eigenvalues of A are either real or complex) – Exception is non-diagonalizable A (non-trivial Jordan form) 2 1.8 1.6 T=2π/ω0 1.4 1.2 1 y 1 0.8 0<ζ<1 0.5 0.6 ζ 0.4 ) ζ λ >1 0.2 0 0 Imag( 0 5 10 15 20 25 -0.5 time (sec) 0<ζ<1 – Analytical formulas exist for overshoot, rise time, settling time, etc -1 -2 -1.5 -1 -0.5 0 – Will study more next week Real(λ) 23 Second Order Systems Important class of systems in many applications areas 2 1.8 1.6 T=2/ 0 1.4 1.2 y 1 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 time (sec) Analytical formulas exist for overshoot, rise time, settling time, etc Will study second order systems characteristics in more detail next week 24.