CDS 101/110: Lecture 2.1 System Modeling. Model-Based Analysis of Feedback Systems

Transcription

1 CDS 11/11: Leture 21 System Modeling Rihard M Murray 5 Otober 215 Goals:! Define a model and its use in answering questions about a system! Introdue the onepts of state, dynamis, inputs and outputs! Review modeling using ordinary differential equations (ODEs) Reading:! Åström and Murray, Feedbak Systems, 2e, Se 31 33, 41 [4 min] Model-Based Analysis of Feedbak Systems Analysis and design based on models A model provides a predition of how the system will behave Feedbak an give ounter-intuitive behavior; models help sort out what is going on For ontrol design, models don t have to be exat: feedbak provides robustness Weather Foreasting Control-oriented models: inputs and outputs The model you use depends on the questions you want to answer A single system may have many models Time and spatial sale must be hosen to suit the questions you want to answer Formulate questions before building a model Question 1: how muh will it rain tomorrow? Question 2: will it rain in the next 5-1 days? Question 3: will we have a drought next summer? Different questions different models 2

2 Example #1: Spring Mass System Appliations Flexible strutures (many apps) Suspension systems (eg, Bob ) Moleular and quantum dynamis Questions we want to answer How muh do masses move as a funtion of the foring frequeny? What happens if I hange the values of the masses? Will Bob fly into the air if I take that speed bump at 25 mph? Modeling assumptions Mass, spring, and damper onstants are fixed and known Springs satisfy Hooke s law Damper is (linear) visous fore, proportional to veloity 3 Modeling a Spring Mass System Model: rigid body physis (Ph 1) Sum of fores = mass * aeleration Hooke s law: F = k(x x rest) Visous frition: F = v = ( ) Converting models to state spae form! Construt a vetor of the variables that are required to speify the evolution of the system! Write dynamis as a system of first order differential equations: = f(x, u) x Rn, u R p y = h(x) y R q d = (u ) ( ) = y = m (u ) m ( ) m m ( ) State spae form m q CDS 11/11, 5 Ot 215 Rihard M Murray, Calteh CDS 4

3 x = More General Forms of Differential Equations State spae form = f(x, u) y = h(x, u) General form Higher order, linear ODE x 1 x 2 x n 1 x n d n 1 q = Ax + Bu y = Cx + Du Linear system d n q n + a 1 n a nq = u d n 1 q y = b 1 n b n 1 q + b n q d n 1 q/ n 1 d n 2 q/ n 2 = dq/ q d x 1 x 2 x 3 x n = x R n, u R p x = state; nth order u = input; will usually set p = 1 y = output; will usually set q = 1 5 y R q a 1 a 2 a n 1 a n y = b 1 b 2 b n x x + 1 u Phase [deg] Gain (log sale) CDS 11/11, 5 Ot 215 Simulation of a Mass Spring System Frequeny Response Frequeny [rad/se] Rihard M Murray, Calteh CDS Steady state frequeny response Fore the system with a sinusoid Plot the steady state response, after transients have died out Plot relative magnitude and phase of output versus input (more later) Matlab simulation (see handout) funtion dy = f(t, y, ) u = 315*os(omega*t); dy = [ y(3); y(4); -(k1+k2)/m1*y(1) + k2/m1*y(2); k2/m2*y(1) - (k2+k3)/m2*y(2) - /m2*y(4) + k3/m2*u ]; [t,y] = ode45(dy,tspan,y,[], k1, k2, k3, m1, m2,, omega); 6

4 State aptures effets of the past independent physial quantities that determines future evolution (absent external exitation) Modeling Terminology Inputs desribe external exitation Inputs are extrinsi to the system dynamis (externally speified) Dynamis desribes state evolution update rule for system state funtion of urrent state and any external inputs Outputs desribe measured quantities Outputs are funtion of state and inputs not independent variables Outputs are often subset of state Example: spring mass system! State: position and veloities of eah mass:! Input: position of spring at right end of hain:! Dynamis: basi mehanis! Output: measured positions of the masses: 7 Choie of state is not unique There may be many hoies of variables that an at as the state Trivial example: different hoies of units (saling fator) Less trivial example: sums and differenes of the mass positions Modeling Properties Choie of inputs, outputs depends on point of view Inputs: what fators are external to the model that you are building - Inputs in one model might be outputs of another model (eg, the output of a ruise ontroller provides the input to the vehile model) Outputs: what physial variables (often states) an you measure - Choie of outputs depends on what you an sense and what parts of the omponent model interat with other omponent models e u r + C(s) + P(s) y d -1 Can also have different types of models Ordinary differential equations for rigid body mehanis Finite state mahines for manufaturing, Internet, information flow Partial differential equations for fluid flow, solid mehanis, et 8

5 Differene Equations Differene equations model disrete transitions between ontinuous variables Disrete time desription (loked transitions) New state is funtion of urrent state + inputs State is represented as a ontinuous variable Example: predator prey dynamis Hare Lynx x[k + 1] = f(x[k],u[k]) y[k] =h(x[k]) Questions we want to answer Given the urrent population of hares and lynxes, what will it be next year? If we hunt down lots of lynx in a given year, how will the populations be affeted? How do long term hanges in the amount of food available affet the populations? Modeling assumptions Trak population annual (disrete time) The predator speies is totally dependent on the prey speies as its only food supply The prey speies has an external food supply and no threat to its growth other than the speifi predator 9 Example #2: Predator Prey Modeling Disrete Lotka-Volterra model State - H[k] # of rabbits in period k - L[k] # of foxes in period k Inputs (optional) - u[k] amount of rabbit food Outputs: # of rabbits and foxes Dynamis: Lotka-Volterra eqs H[k + 1] = H[k]+b r (u)h[k] L[k + 1] = L[k]+L[k]H[k] al[k]h[k], d f L[k], Parameters/funtions - b r (u) hare birth rate (per period); depends on food supply - d f lynx mortality rate (per period) - a, interation terms Population MATLAB simulation (see handout)! Disrete time model, simulated through repeated addition Comparison with data hares lynxes Year Hare Lynx

6 Finite State Mahines Finite state mahines model disrete transitions between finite # of states Represent eah onfiguration of system as a state Model transition between states using a graph Inputs fore transition between states Example: Traffi light logi Car arrives on E-W St " Timer expires Timer expires " Car arrives on N-S St State: Inputs: Outputs: urrent pattern of lights that are on + internal timers presene of ar at intersetions urrent pattern of lights that are on 11 Summary: System Modeling Model = state, inputs, outputs, dynamis = f(x, u) y = h(x) x[k + 1] = f(x[k],u[k]) y[k] =h(x[k]) Priniple: Choie of model depends on the questions you want to answer funtion dy = f(t,y, k1, k2, k3, m1, m2,, omega) u = 315*os(omega*t); dy = [ y(3); y(4); -(k1+k2)/m1*y(1) + k2/m1*y(2); k2/m2*y(1) - (k2+k3)/m2*y(2) - b/m2*y(4) + k3/m2*u ]; 12

7 Sep 28, 8 13:45 L1_2_modelinglst Page 1/2 L1_2_modelingm Leture 12 MATLAB alulations RMM, 6 Ot 3 Spring mass system Spring mass system parameters m = 25; m1=m; m2=m; masses (all equal) k = 5; k1=k; k2=k; k3=k; spring onstants b = 1; damping A = 315; omega = 75; foring funtion Call ode45 routine (MATLAB 6 format; help ode45 for details) tspan=[ 5]; time range for simulation y = [; ; ; ]; initial onditions [t,y] = ode45(@springmass, tspan, y, [], k1, k2, k3, m1, m2, b, A, omega); Plot the input and outputs over entire period figure(1); lf plot(t, A*os(omega*t), t, y(:,1), t, y(:,2)); Now plot the data for the final 1 (assuming this is long enough) endlen = round(length(t)/1); last 1 of data reord range = [length(t) endlen:length(t)] ; reate vetor of indies (note ) tend = t(range); figure(2); lf plot(tend, A*os(omega*tend), tend, y(range,1), tend, y(range,2)); Compute the relative phase and amplitude of the signals We make use of the fat that we have a sinusoid in steady state, as well as its derivative This allows us to ompute the magnitude of the sinusoid using simple trigonometry ( sin^2 + os^2 = 1) u = A*os(omega*tend); udot = A*omega*sin(omega*tend); ampu = mean( sqrt((u * u) + (udot/omega * udot/omega)) ); fprintf(1, Amplitude = 5e m, ampu*1); end; Store the final population Ha(duration) = H(duration*nperiods+1); La(duration) = L(duration*nperiods+1); Plot the populations of rabbits and foxes versus time figure(3); lf; plot( [1:duration], Ha,, [1:duration], La, ); Adjust the parameters of the plot axis([ ]); xlabel( Year ); ylabel( Population ); Now reset the parameters to look like we want lgh = legend(ga, hares, lynxes, Loation, NorthEast, Orientation, Horizontal ); legend(lgh, boxoff ); springmassm ODE45 funtion for a spring mass system RMM, 6 Ot 3 This file ontains the differential equation that desribes the mass spring system used as an example in CDS 11 It allows individual mass and spring values, plus sinusoidal foring The state is stored in the vetor y The values for y are y(1) = q1, position of first mass y(2) = q2, position of seond mass y(3) = q1dot, veloity of first mass y(4) = q2dot, veloity of seond mass Printed by Rihard Murray Sep 28, 8 13:45 L1_2_modelinglst Page 2/2 Predator prey system Set up the initial state lear H L year H(1) = 1; L(1) = 1; For simpliity, keep trak of the year as well year(1) = 1845; Set up parameters (note that = a in the model below) br = 6; df = 7; a = 14; nperiods = 365; simulate eah day duration = 9; number of years for simulation funtion dy = springmass(t, y, k1, k2, k3, m1, m2, b, A, omega) ompute the input to drive the system u = A*os(omega*t); ompute the time derivative of the state vetor dy = [ y(3); y(4); (k1+k2)/m1*y(1) + k2/m1*y(2); k2/m2*y(1) (k2+k3)/m2*y(2) b/m2*y(4) + k3/m2*u ]; Iterate the model for k = 1:duration*nperiods b = br; onstant food supply b = br*(1+5*sin(2*pi*k/(4*nperiods))); varying food supply (try it!) H(k+1) = H(k) + (b*h(k) a*l(k)*h(k))/nperiods; L(k+1) = L(k) + (a*l(k)*h(k) df*l(k))/nperiods; year(k+1) = year(k) + 1/nperiods; if (mod(k, nperiods) == 1) Store the annual population Ha((k 1)/nperiods + 1) = H(k); La((k 1)/nperiods + 1) = L(k); end; Sunday September 28, 28 L1_2_modelinglst 1/1