S. Boyd EE102Lecture 2Systems• meaning & notation• common examples & block diagram representations• electronic realizations• linearity• interconnected systems• differential equations2–1Systems• a system transforms input signals into output signals• a system is a function mapping input signals into output signalswe concentrate on systems with one input and one output signal, i.e.,single-input, single-output (SISO) systemsnotation:• y = Su or y = S(u) means the system S acts on input signal u toproduce output signal y• y = Su does not (in general) mean multiplication!Systems 2–2Block diagramssystems often denoted by block diagram:PSfrag replacementsuyS• lines with arrows denote signals (not wires)• boxes denote systems; arrows show inputs & outputs• special symbols for some systemsSystems 2–3Examples(with input signal u and output signal y)scaling system: y(t) = au(t)• called an amplifier if |a| > 1• called an attenuator if |a| < 1• called inverting if a < 0• a is called the gain or scale factorsometimes denoted by triangle or circle in block diagram:PSfrag replacementsuuyyaaSystems 2–4differentiator: y(t) = u0(t)integrator: y(t) =Ztau(τ) dτ (a is often 0 or −∞)common notation for integrator:PSfrag replacementsuyZrunning average system: y(t) =1tZt0u(τ) dτtime shift system: y(t) = u(t − T )• called a delay system if T > 0• called a predictor system if T < 0Systems 2–5sign detector or 1-bit limiter system:y(t) = sgn(u(t)) =½1, u(t) ≥ 0−1, u(t) < 0PSfrag replacementsuyconvolution system:y(t) =Zu(t − τ)h(τ) dτ,where h is a given function (you’ll be hearing much more about this!)Systems 2–6Examples with multiple inputs(with inputs u1, u2, and output y)• summing system: y(t) = u1(t) + u2(t)PSfrag replacementsu1u2y• difference system: y(t) = u1(t) − u2(t)PSfrag replacementsu1u2y• multiplier system: y(t) = u1(t)u2(t)PSfrag replacementsu1u2y• comparator system: y(t) =½1, u1(t) ≥ u2(t)−1, u1(t) < u2(t)PSfrag replacementsu1u2ySystems 2–7Electronic realizationsthe systems described above can be realized as electronic circuits, e.g.,with op-ampsscaling: y(t) = (1 + R2/R1)u(t)PSfrag replacementsuyR1R2difference: y(t) = u1(t) − u2(t)PSfrag replacementsu1u2yR RRRSystems 2–8integrator: y(t) = −1/(RC)Ztu(τ) dτPSfrag replacementsuyRC• these are circuit schematics, not block diagrams• signals are represented by voltages (which is common but not universal)Systems 2–9Linearitya system F is linear if the following two properties hold:1. homogeneity: if u is any signal and a is any scalar,F (au) = aF (u)2. superposition: if u and ˜u are any two signals,F (u + ˜u) = F u + F ˜u(watch out — just a few symbols here express a very complex meaning)in words, linearity means:• scaling before or after the system is the same• summing before or after the system is the sameSystems 2–10linearity means the following pairs of block diagrams are equivalent, i.e.,have the same output for any input(s)PSfrag replacementsuuyyyyFFFF Faau1u1u2u2examples of linear systems: scaling system, differentiator, integrator,running average, time shift, convolution, summer, difference systemsexamples of nonlinear systems: sign detector, multiplier, comparatorSystems 2–11Interconnections of systemswe can interconnect systems to form new systems, e.g.,cascade (or series): y = G(F u) = GF uPSfrag replacementsuyFG(note that block diagrams and algebra are reversed)sum (or parallel): y = F u + GuPSfrag replacementsuyFGSystems 2–12feedback: y = F (u − Gy)PSfrag replacementsuyFG• the minus sign is just a tradition, and often isn’t there• we’ll study this arrangement laterin general,• block diagrams are just a symbolic way to describe a connection ofsystems• we can just as well write out the equations relating the signalsSystems 2–13Example: Two-stage amplifierPSfrag replacementsuyn1n2a1a2• input signal u, output signal y• noise signals n1, n2• first stage gain a1, second stage gain a2y = a2(a1(u + n1) + n2) = (a1a2)u + (a1a2)n1+ (a2)n2• input to first amplifier is u + n1• output of first amplifier is a1(u + n1)• input to second amplifier is a1(u + n1) + n2• output of second amplifier is a2(a1(u + n1) + n2)Systems 2–14Example: Integrator with feedbackPSfrag replacementsuyZainput to integrator is u − ay, soZt(u(τ) − ay(τ)) dτ = y(t)(soon we’ll be able to give an explicit expression for y in terms of u)another (useful) method: the input to an integrator is the derivative of itsoutput, so we haveu − ay = y0(of course, same as above)Systems 2–15Systems described by differential equationsmany systems are described by a linear constant coefficient ordinarydifferential equation (LCCODE):any(n)+ · · · + a2y00+ a1y0+ a0y = bmu(m)+ · · · + b1u00+ b1u0+ b0uwith given initial conditionsy(n−1)(0), y(n−2), . . . , y0(0), y(0)(which fixes y, given u)• n is called the order of the system• b0, . . . , bm, a0, . . . , anare the coefficients of the system• when initial conditions are all zero, LCCODE systems are linearan LCCODE gives an implicit description of a system; soon we’ll be able toexplicitly express y in terms of uSystems 2–16Examplessimple examples• scaling system (a0= 1, b0= a)• integrator (a1= 1, b0= 1)• differentiator (a0= 1, b1= 1)• integrator with feedback (page 2–15)RC circuitPSfrag replacementsuyRCcurrent flowing into capacitor is Cy0(t) =u(t) − y(t)Rrewrite as first-order LCCODE: RCy0(t) + y(t) = u(t)Systems 2–17second-order RC circuitPSfrag replacementsuyv1R1R2C1C2• current into C2is C2y0=v1− yR2• current into C1is C1v01=u − v1R1−v1− yR2using v1= y + R2C2y0in the 2nd equation yields:C1(y + R2C2y0)0=uR1+yR2−µ1R1+1R2¶(y + R2C2y0)rewrite (eventually) as second-order LCCODE(R1C1R2C2)y00+ (R1C1+ R1C2+ R2C2)y0+ y = uSystems 2–18mechanical system (mass-spring-damper)PSfrag replacementsuymk b(can represent suspension system, building during earthquake, . . . )• u(t) is displacement of base; y(t) is displacement of mass• spring force is k(u − y); damping force is b(u − y)0• Newton’s equation is my00= b(u − y)0+ k(u − y)rewrite as second-order LCCODEmy00+ by0+ ky = bu0+ kuSystems