Chapter 4 Uncertainty and Robustness No mat hematical system can exactly model a physical system.For this reason we must be aware of how modeling errors might adversely affect the performance of a control system.This chapter begins with a treatment of various models of plant uncertainty.Then robust stability,stability in the face of plant uncertainty,is studied using the small-gain theorem.The final topic is robust performance,guaranteed tracking in the face of plant uncertainty. 4.1 Plant Uncertainty The basic technique is to model the plant as belonging to a set P.The reasons for doing this were presented in Chapter 1.Such a set can be either structured or unstructured. For an example of a structured set consider the plant model s2+a8+1 This is a standard second-order transfer funct ion with nat ural frequency 1 rad/s and damping ratio a/2-it could represent,for example,a mass-spring-damper setup or an R-L-C circuit.Suppose that the constant a is known only to the extent that it lies in some interval [amin,amax].Then the plant belongs to the struct ured set 1 p={32+as+:amm≤a≤om Thus one type of structured set is parametrized by a finite number of scalar parameters (one parameter,a,in this example).Another type of structured uncertainty is a discrete set of plants, not necessarily parametrized explicit ly. For us,unstructured sets are more important,for two reasons.First,we believe that all models used in feedback design should include some unstruct ured uncertainty to cover unmodeled dynam- ics,particularly at high frequency.Other types of uncertainty,though important,may or may not arise naturally in a given problem.Second,for a specific type of unstructured uncertainty,disk uncertainty,we can develop simple,general analysis met hods.Thus the basic starting point for an unstructured set is that of disk-like uncertainty.In what follows,multiplicative disk uncertainty is chosen for detailed study.This is only one type of unstructured perturbation.The important point is that we use disk uncertainty instead of a more complicated description.We do this because it greatly simplifies our analysis and lets us say some fairly precise things.The price we pay is conservativeness. 39
Chapter Uncertainty and Robustness No mathematical system can exactly model a physical system For this reason we must be aware of how modeling errors might adversely aect the performance of a control system This chapter begins with a treatment of various models of plant uncertainty Then robust stability stability in the face of plant uncertainty is studied using the smallgain theorem The nal topic is robust performance guaranteed tracking in the face of plant uncertainty Plant Uncertainty The basic technique is to model the plant as belonging to a set P The reasons for doing this were presented in Chapter Such a set can be either structured or unstructured For an example of a structured set consider the plant model s as This is a standard secondorder transfer function with natural frequency rad s and damping ratio a it could represent for example a massspringdamper setup or an RLC circuit Suppose that the constant a is known only to the extent that it lies in some interval amin amax Then the plant belongs to the structured set P s as amin a amax Thus one type of structured set is parametrized by a nite number of scalar parameters one parameter a in this example Another type of structured uncertainty is a discrete set of plants not necessarily parametrized explicitly For us unstructured sets are more important for two reasons First we believe that all models used in feedback design should include some unstructured uncertainty to cover unmodeled dynam ics particularly at high frequency Other types of uncertainty though important may or may not arise naturally in a given problem Second for a specic type of unstructured uncertainty disk uncertainty we can develop simple general analysis methods Thus the basic starting point for an unstructured set is that of disklike uncertainty In what follows multiplicative disk uncertainty is chosen for detailed study This is only one type of unstructured perturbation The important point is that we use disk uncertainty instead of a more complicated description We do this because it greatly simplies our analysis and lets us say some fairly precise things The price we pay is conservativeness
40 CHAPTER 4.UNCERTAINTY AND ROBUSTNESS Multiplicative Perturbation Suppose that the nominal plant transfer function is P and consider perturbed plant transfer func tions of the form P =(1+AW2)P.Here W2 is a fixed stable transfer function,the weight,and A is a variable stable transfer function satisfying Alo<1.Furthermore,it is assumed that no unstable poles of P are canceled in forming P.(Thus,P and P have the same unstable poles.) Such a perturbation A is said to be allowable. The idea behind this uncertainty model is that Aw2 is the normalized plant perturbat ion away from 1: 元-1=AW2. Hence if‖△‖lo≤l,then P(jw) -1≤IW2(0w儿，w, P(jw) soW2(jw)provides the uncertainty profile.This inequality describes a disk in the complex plane: At each frequency the point P/P lies in the disk with center 1,radius W2.Typically,W2(jw) is a (roughly)increasing function of w:Uncertainty increases with increasing frequency.The main purpose of A is to account for phase uncertainty and to act as a scaling factor on the magnitude of the perturbation (i.e.,A varies between 0 and 1). Thus,this uncertainty model is characterized by a nominal plant P together with a weighting function W2.How does one get the weighting funct ion W2 in practice?This is illustrated by a few examples. Example 1 Suppose that the plant is stable and its transfer function is arrived at by means of frequency-response experiments:Magnitude and phase are measured at a number of frequencies, wi,i=1,...,m,and this experiment is repeated several,say n,times.Let the magnitude-phase measurement for frequency wi and experiment k be denoted (Mik,).Based on these data select nominal magnit ude-phase pairs (M,)for each frequency wi,and fit a nominal transfer funct ion P(s)to these data.Then fit a weighting function W2(s)so that Mikeloik |W2(0)儿，i=1,.,m;k=1,.,n. Example 2 Assume that the nominal plant transfer function is a double integrator: 1 P(s)=8 For example,a dc motor with negligible viscous damping could have such a transfer function.You can think of ot her phy sical systems with only inertia,no damping.Suppose that a more detailed model has a time delay,y ielding the transfer function P(s)=e-rs1 2 and suppose that the time delay is known only to the extent that it lies in the interval 0<<0.1. This time-delay factor exp(-Ts)can be treated as a multiplicative perturbation of the nominal plant by embedding P in the family {(1+△W2)P:‖△lo≤1
CHAPTER UNCERTAINTY AND ROBUSTNESS Multiplicative Perturbation Suppose that the nominal plant transfer function is P and consider perturbed plant transfer func tions of the form P WP Here W is a xed stable transfer function the weight and is a variable stable transfer function satisfying kk Furthermore it is assumed that no unstable poles of P are canceled in forming P Thus P and P have the same unstable poles Such a perturbation is said to be al lowable The idea behind this uncertainty model is that W is the normalized plant perturbation away from P P W Hence if kk then P j P j jWjj so jWjj provides the uncertainty prole This inequality describes a disk in the complex plane At each frequency the point P P lies in the disk with center radius jWj Typically jWjj is a roughly increasing function of Uncertainty increases with increasing frequency The main purpose of is to account for phase uncertainty and to act as a scaling factor on the magnitude of the perturbation ie jj varies between and Thus this uncertainty model is characterized by a nominal plant P together with a weighting function W How does one get the weighting function W in practice This is illustrated by a few examples Example Suppose that the plant is stable and its transfer function is arrived at by means of frequencyresponse experiments Magnitude and phase are measured at a number of frequencies i i m and this experiment is repeated several say n times Let the magnitudephase measurement for frequency i and experiment k be denoted Mik ik Based on these data select nominal magnitudephase pairs Mi i for each frequency i and t a nominal transfer function P s to these data Then t a weighting function Ws so that Mike jik Mieji jWjij i m k n Example Assume that the nominal plant transfer function is a double integrator P s s For example a dc motor with negligible viscous damping could have such a transfer function You can think of other physical systems with only inertia no damping Suppose that a more detailed model has a time delay yielding the transfer function P se s s and suppose that the time delay is known only to the extent that it lies in the interval This timedelay factor exp s can be treated as a multiplicative perturbation of the nominal plant by embedding P in the family f WP kk g
A.L.PLANT CERAINTY 41 To do ths tewelwa skuld bels so tttehornaiz taon saef P(u) P(jw) IW2(jw)l,,w,T, tkatis Wa(jw)l.,w. A littetimewithBodemanitde skws thtasuitaienstolTwis W,(= 0218 04s+i≤ FgulC4is teBodemanitideptor t wa ad RTs)1for=04,ewoRtvan(4 101 100 10 么 10-3 0 100 101 102 103 Figul4 Bodepts of W (dah ad r)1 (slid)4 To getareing forby conseRativeths is compaectar fIQucie teatauncr ta飞st withecovling disk fs:sff1.lW2(0wlg≤ Example 3 Suppsetrattepattasf(rfunction is =点
PLANT UNCERTAINTY To do this the weight W should be chosen so that the normalized perturbation satises P j P j jWjj that is ej jWjj A little time with Bode magnitude plots shows that a suitable rstorder weight is Ws s s Figure is the Bode magnitude plot of this W and exp s for the worst value 10-3 10-2 10-1 100 101 10-1 100 101 102 103 Figure Bode plots of W dash and exps solid To get a feeling for how conservative this is compare at a few frequencies the actual uncer tainty set P j P j ej with the covering disk fs js jjWjjg Example Suppose that the plant transfer function is P s k s
42 CHAPTER.UNCERTAINTY AND ROBUSTNESS where the gain k is uncertain but is known to lie in the interval [04110.This plant too can be embedded in a family consisting of multiplicative perturbations of a nominal plant P(S= k4 52 The weight W2 must satisfy P(jw) W2(w)1,w1k1 /P(jw) that is, 4i441sw01,w4 The lethand side is minimized by k4505,for which the left-hand side quals 445.545.In this way we get the nominal plant P(S= 505 and constant weight W2(S=405.545. The multiplicative perturbation model is not suitable for every application because the disk covering the uncertainty set is sometimes too coarse an approximation.In this case a controller designed for the multiplicativeuncertainty model would probably be too conservative for the original uncertain ty mo del. The discussion above illustrates an important point.In mo deling a plant we may arrive at a certain plant set.This set may be too awkward to cope with mathematically,so we may embed it in a larger set that is easier to handle.Conceivably,the achievable performance for the larger set may not be as good as the achievable performance for the smaller;that is,there may exist-even though we cannot find it-a controller that is better for the smaller set than the controller we design for the larger set.In this sense the latter controller is conservative for the smaller set. In this book we stick with plant uncertainty that is disk-like.It will be conservative for some problems,but the payoff is that we obtain some very nice theoretical results.The resulting theory is remarkably practical as well. Other Perturbations Other uncertainty mo dels are possible besides multiplicative perturbations,as illustrated by the following example Example 4 As at the start of this section,consider the family of plant transfer functions S+S,104≤a084 Thus 806+0里△1-1≤△≤11 so the family can be expressed as P(S 1+△W2SPS1-1≤A≤11
CHAPTER UNCERTAINTY AND ROBUSTNESS where the gain k is uncertain but is known to lie in the interval This plant too can be embedded in a family consisting of multiplicative perturbations of a nominal plant P s k s The weight W must satisfy P j P j jWjj k that is max k k k jWjj The lefthand side is minimized by k for which the lefthand side equals In this way we get the nominal plant P s s and constant weight Ws The multiplicative perturbation model is not suitable for every application because the disk covering the uncertainty set is sometimes too coarse an approximation In this case a controller designed for the multiplicative uncertainty model would probably be too conservative for the original uncertainty model The discussion above illustrates an important point In modeling a plant we may arrive at a certain plant set This set may be too awkward to cope with mathematically so we may embed it in a larger set that is easier to handle Conceivably the achievable performance for the larger set may not be as good as the achievable performance for the smaller that is there may exist even though we cannot nd it a controller that is better for the smaller set than the controller we design for the larger set In this sense the latter controller is conservative for the smaller set In this book we stick with plant uncertainty that is disklike It will be conservative for some problems but the payo is that we obtain some very nice theoretical results The resulting theory is remarkably practical as well Other Perturbations Other uncertainty models are possible besides multiplicative perturbations as illustrated by the following example Example As at the start of this section consider the family of plant transfer functions s as a Thus a so the family can be expressed as P s WsP s
4.2. REST STABILITY 43 where P():=04+w4s)=044 Note that Pis the nominal plant transfer funct ion for the value0the midpoint of the interval. The block diagram corresponding to this represent ation of the plant is shown in Figure 4.2.Thus Figure 4.2:Example 4. the original plant has been represented as a feedback uncertainty around a nominal plant. The following list summarizes the common uncertainty models: (1+△W④P P+△W4 P1(1+△W4P) P(1+△W④ Appropriate assumptions would be made on A and W4in each case.Typically,we can relax the assumption that A be stable;but then the theorems to follow would be harder to prove. 2-2 Robust Sta bility The notion of robustness can be described as follows.Suppose that the plant transfer function P belongs to a set P,as in the preceding sect ion.Consider some characterist ic of the feedback system, for example,that it is internally stable.A controller is robust with respect to this characteristic if this characteristic holds for every plant in P.The notion of robustness therefore requires a controller,a set of plants,and some characterist ic of the system.For us,the two most important variations of this not ion are robust stability,treated in this section,and robust performance,treated in the next. Aprovides providesinabiiy for every plant in P.We might like to have a test for robust stability,a test involving Cand P.Or if P has an associated size,the maximum size such that abilizes all of P might be a useful notion of stability margin. The Nyquist plot gives information about stability margin.Note that the distance from the critical point-1 to the nearest point on the Nyquist plot of L equals 1So: distance from-1 to Nyquist plot inf/1/L(jfi) inf|1+L(f)儿 T 1 P1+L(G)I
ROBUST STABILITY where P s s s Ws s Note that P is the nominal plant transfer function for the value a the midpoint of the interval The block diagram corresponding to this representation of the plant is shown in Figure Thus P W Figure Example the original plant has been represented as a feedback uncertainty around a nominal plant The following list summarizes the common uncertainty models WP P W P WP P W Appropriate assumptions would be made on and W in each case Typically we can relax the assumption that be stable but then the theorems to follow would be harder to prove Robust Stability The notion of robustness can be described as follows Suppose that the plant transfer function P belongs to a set P as in the preceding section Consider some characteristic of the feedback system for example that it is internally stable A controller C is robust with respect to this characteristic if this characteristic holds for every plant in P The notion of robustness therefore requires a controller a set of plants and some characteristic of the system For us the two most important variations of this notion are robust stability treated in this section and robust performance treated in the next A controller C provides robust stability if it provides internal stability for every plant in P We might like to have a test for robust stability a test involving C and P Or if P has an associated size the maximum size such that C stabilizes all of P might be a useful notion of stability margin The Nyquist plot gives information about stability margin Note that the distance from the critical point to the nearest point on the Nyquist plot of L equals kSk distance from to Nyquist plot inf j Ljj inf j Ljj sup j Ljj
44 CHAPTEPL INCEERINTY AND RE SINESS =k5k2/≤ Thus if kSk2-1,the Nyquist plot comes close to the critical point,and the feedback system is nearly unstable.However,as a measure of stability margin this distance is not entirely adequate because it contains no frequency information.More precisely,if the nominal plant P is perturbed to P,having the same number of unstable poles as has P and satisfying the inequality P(jw)C(jw)1 P(jw)C(jw)j<kSki/,;w, then internal stability is preserved(the number of encirclements of the critical point by the Nyquist plot does not change).But this is usually very conservative;for instance,larger perturbations could be allowed at frequencies where P(jw)C(jw)is far from the critical point. Better stability margins are obtained by taking explicit frequency-dependent perturbat ion mod- els:for example,the mult iplicative perturbation model,P=(1+AW2)P.Fix a positive number B and consider the family of plants fp:△is stable and k△k24Bg≤ Now a controller C that achieves internal stability for the nominal plant P will stabilize this entire family if B is small enough.Denote by Baup the least upper bound on B such that C achieves internal stability for the entire family.Then Baup is a stability margin(with respect to this uncertainty model).Analogous stability margins could be defined for the other uncertainty models. We turn now to two classical measures of stability margin,gain and phase margin.Assume that the feedback system is internally stable with plant P and controller C.Now perturb the plant to kP,with k a positive real number.The Upper gain margin,denoted kmax,is the first value of k greater than 1 when the feedback system is not internally stable;that is kmax is the maximum number such that internal stability holds for 1<mo.If there is no such number,then set kmox:=fi.Similarly,the lOwer gain margin,kmin,is the least nonnegative number such that internal stability holds for min<1.These two numbers can be read off the Nyquist plot of L;for example,1 1-kmax is the point where the Nyquist plot intersects the segment (1 1,0)of the real axis,the closest point to I 1 if there are several points of intersection. Now perturb the plant to eiP,with a positive real number.The phaSe m argin,max,is the maximum number(usually expressed in degrees)such that internal stability holds for 0< You can see that mox is the angle through which the Ny quist plot must be rotated until it passes through the critical point,1 1;or,in radians,mo equals the arc length along the unit circle from the Ny quist plot to the critical point. Thus gain and phase margins measure the distance from the critical point to the Nyquist plot in certain specific direct ions.Gain and phase margins have tradit ionally been important measures of stability robustness:if either is small,the system is close to instability.Notice,however,that the gain and phase margins can be relatively large and yet the Nyquist plot of L can pass close to the critical point;that is,Sm Utan eOUs small changes in gain and phase could cause internal instability.We return to these margins in Chapter 11. Now we look at a typical robust stability test,one for the multiplicative perturbation model. Assume that the nominal feedback system (i.e.,with A =0)is internally stable for controller C. Bring in again the complementary sensitivity function L PC T=11S=1+乙=1+PC≤ Theorem 1 -Multiph cat ve Uh certain tym Olel)C provi des rchust sta blityi2 kW2Tk2 1
CHAPTER UNCERTAINTY AND ROBUSTNESS kSk Thus if kSk the Nyquist plot comes close to the critical point and the feedback system is nearly unstable However as a measure of stability margin this distance is not entirely adequate because it contains no frequency information More precisely if the nominal plant P is perturbed to P having the same number of unstable poles as has P and satisfying the inequality jP jCj P jCjj kSk then internal stability is preserved the number of encirclements of the critical point by the Nyquist plot does not change But this is usually very conservative for instance larger perturbations could be allowed at frequencies where P jCj is far from the critical point Better stability margins are obtained by taking explicit frequencydependent perturbation mod els for example the multiplicative perturbation model P WP Fix a positive number and consider the family of plants fP is stable and kk g Now a controller C that achieves internal stability for the nominal plant P will stabilize this entire family if is small enough Denote by sup the least upper bound on such that C achieves internal stability for the entire family Then sup is a stability margin with respect to this uncertainty model Analogous stability margins could be dened for the other uncertainty models We turn now to two classical measures of stability margin gain and phase margin Assume that the feedback system is internally stable with plant P and controller C Now perturb the plant to kP with k a positive real number The upper gain margin denoted kmax is the rst value of k greater than when the feedback system is not internally stable that is kmax is the maximum number such that internal stability holds for k kmax If there is no such number then set kmax Similarly the lower gain margin kmin is the least nonnegative number such that internal stability holds for kmin k These two numbers can be read o the Nyquist plot of L for example kmax is the point where the Nyquist plot intersects the segment of the real axis the closest point to if there are several points of intersection Now perturb the plant to ejP with a positive real number The phase margin max is the maximum number usually expressed in degrees such that internal stability holds for max You can see that max is the angle through which the Nyquist plot must be rotated until it passes through the critical point or in radians max equals the arc length along the unit circle from the Nyquist plot to the critical point Thus gain and phase margins measure the distance from the critical point to the Nyquist plot in certain specic directions Gain and phase margins have traditionally been important measures of stability robustness if either is small the system is close to instability Notice however that the gain and phase margins can be relatively large and yet the Nyquist plot of L can pass close to the critical point that is simultaneous small changes in gain and phase could cause internal instability We return to these margins in Chapter Now we look at a typical robust stability test one for the multiplicative perturbation model Assume that the nominal feedback system ie with is internally stable for controller C Bring in again the complementary sensitivity function T S L L P C P C Theorem Multiplicative uncertainty model C provides robust stability i kWT k
4.2.ROBUST STABILITY 45 Proof (Assume that W2Tlloo 1.Construct the Nyquist plot of L,indenting D to the left around poles on the imaginary axis.Since the nominal feedback system is internally stable, we know this from the Nyquist criterion:The Nyquist plot of L does not pass through-1 and its number of counterclockwise encirclements equals the number of poles of P in Res >0 plus the number of poles of C in Res >0. Fix an allowable A.Construct the Nyquist plot of PC =(1+AW2)L.No additional inden tations are required since AW2 introduces no additional imaginary axis poles.We have to show that the Nyquist plot of (1+AW2)L does not pass through-1 and its number of counterclockwise encirclements equals the number of poles of(1+AW2)P in Re s >0 plus the number of poles of C in Re s >0;equivalently,the Nyquist plot of(1+AW2)L does not pass through-1 and encircles it exactly as many times as does the Nyquist plot of L.We must show,in other words,that the perturbation does not change the number of encirclements. The key equation is 1+(1+△W2)L=(1+)(1+△W2T) (4.1) Since ‖△W2TI‖o≤IW2Tlo0,constructing an admissible A takes a little more work;the details are omitted. The theorem can be used effectively to find the stability margin Bsup defined previously.The simple scaling technique {P=(1+△W2)P:‖△I‖l≤3}={P=(1+B-1△6W2)P:IB-1△‖≤1} ={P=(1+△13W2)P:‖△l≤1} toget her with the theorem shows that Fsup=sup{B:IBW2Tl‖lo<1}=1/川W2Tlo The condition W2T<1 also has a nice graphical interpretat ion.Note that lW2Tlo<1÷ W2(jw)L(jw) <1,w 1+L(w) 台IW2(w)L(w)川<|1+L(w),w
ROBUST STABILITY Proof Assume that kWT k Construct the Nyquist plot of L indenting D to the left around poles on the imaginary axis Since the nominal feedback system is internally stable we know this from the Nyquist criterion The Nyquist plot of L does not pass through and its number of counterclockwise encirclements equals the number of poles of P in Res plus the number of poles of C in Res Fix an allowable Construct the Nyquist plot of P C WL No additional inden tations are required since W introduces no additional imaginary axis poles We have to show that the Nyquist plot of WL does not pass through and its number of counterclockwise encirclements equals the number of poles of WP in Re s plus the number of poles of C in Re s equivalently the Nyquist plot of WL does not pass through and encircles it exactly as many times as does the Nyquist plot of L We must show in other words that the perturbation does not change the number of encirclements The key equation is WL L WT Since kWT k kWT k the point WT always lies in some closed disk with center radius for all points s on D Thus from as s goes once around D the net change in the angle of WL equals the net change in the angle of L This gives the desired result Suppose that kWT k We will construct an allowable that destabilizes the feedback system Since T is strictly proper at some frequency jWjT jj Suppose that Then WT is a real number either or If WT then is allowable and WT From the Nyquist plot of WL passes through the critical point so the perturbed feedback system is not internally stable If constructing an admissible takes a little more work the details are omitted The theorem can be used eectively to nd the stability margin sup dened previously The simple scaling technique fP WP kk g fP WP k k g fP WP kk g together with the theorem shows that sup supf k WT k g kWT k The condition kWT k also has a nice graphical interpretation Note that kWT k WjLj Lj jWjLjj j Ljj
46 CHAPTER 4.UN ERTAINTY AND ROBUSTNESS r W2Lj 2 Figure 4.3:Robust stability graphically. W2 Figure 4.4:Perturbed feedback system The last inequality says that at every frequency,the critical point,-1,lies outside the disk of center L(),radius 2()L()j (Figure 4.3). There is a simple way io see the relevance of the condition kW2Tk4 1.First,draw the blodk diagram of the perturbed feedback system,but ignoring inputs(Figure 44).The transfer function from the output of A around to the input of A equals-W2T,so the block diagram collapses to the configuration shown in Figure 4.5.The maximum bop gain in Figure 4.5 equals k-AW2Tk4, W2T Figure 4.5:Collapsed block diagram. which is 1 for all allowable As iff the small-gain condition kW2Tk4 1 holds. The fdregoing discussion is related to the small-gain theorem,a special case of which is this:If L is stable and kLk4 1,then (1+L)/is stable too.An easy proof jises the Ny quist criterion
CHAPTER UNCERTAINTY AND ROBUSTNESS r r jWLj L Figure Robust stability graphically C P W Figure Perturbed feedback system The last inequality says that at every frequency the critical point lies outside the disk of center Lj radius jWjLjj Figure There is a simple way to see the relevance of the condition kWT k First draw the block diagram of the perturbed feedback system but ignoring inputs Figure The transfer function from the output of around to the input of equals WT so the block diagram collapses to the conguration shown in Figure The maximum loop gain in Figure equals k WT k WT Figure Collapsed block diagram which is for all allowable s i the smallgain condition kWT k holds The foregoing discussion is related to the smal lgain theorem a special case of which is this If L is stable and kLk then L is stable too An easy proof uses the Nyquist criterion
4.3.ROBUST PERFORMANCE 47 Summary of Robust Stability Tests Table 4.1 summarizes the robust stability tests for the cther uncertainty models. Pert urbation Condition (1+△W2)P W2T<1 P+△W2 W2CS]loo 1 P(1+△W2P) W2PSIloo <1 P1+△W2) W2Slloo 1 Table 4.1 Note that we get the same four transfer functions-T,CS,PS,S-as we did in Section 3.4.This should not be too surprising since (up to sign)these are the only closed-locp transfer functions for a unity feedback SISO sy stem. .3 Robust Performance Now we look into performance of the perturbed plant.Suppose that the plant transfer function belongs to a set P.The general notion of robust performance is that internal stability and per formance,of a specified type,should hold for all plants in P.Again we focus on multiplicative pert urbat ions. Recall that when the nominal feedback system is internally stable,the nominal performance condition is WiSlloo 1 and the robust stability condition is W2Tlloo<1.If P is pert urbed to (1+AW2)P,S is perturbed to 1 1+(1+△W2)Z=1+△wW27T≤ Clearly,the robust performance condition should therefore be WiS W2Tlloo 1 and 1+△W2T <1,△≤ Here A must be allowable.The next theorem gives a test for robust performance in terms of the funct ion sHW1(s)S(s)j+jW2(s)T(s) which is denoted W Sj+2Tj Theorem 2 A necessary and sufficient condition for robust performance is W1Sj+JW2T<1≤ (4.3) Proof ()Assume (4.3),or equivalently, W2Tllo and <1≤ (4.4)
ROBUST PERFORMANCE Summary of Robust Stability Tests Table summarizes the robust stability tests for the other uncertainty models Perturbation Condition WP kWT k P W kWCSk P WP kWP Sk P W kWSk Table Note that we get the same four transfer functions T CS P S S as we did in Section This should not be too surprising since up to sign these are the only closedloop transfer functions for a unity feedback SISO system Robust Performance Now we look into performance of the perturbed plant Suppose that the plant transfer function belongs to a set P The general notion of robust performance is that internal stability and per formance of a specied type should hold for all plants in P Again we focus on multiplicative perturbations Recall that when the nominal feedback system is internally stable the nominal performance condition is kWSk and the robust stability condition is kWT k If P is perturbed to WP S is perturbed to WL S WT Clearly the robust performance condition should therefore be kWT k and WS WT Here must be allowable The next theorem gives a test for robust performance in terms of the function s jWsSsj jWsT sj which is denoted jWSj jWT j Theorem A necessary and sucient condition for robust performance is kjWSj jWT jk Proof Assume or equivalently kWT k and WS jWT j
48 CHAPTER.UNCERTAINTY AND ROBUSTNESS Fix A4In what follows,functions are evaluated at an arbitrary point w,but this is suppressed to simplify notation4 We have 1=I1+△W2T1△W2T|≤|1+△W2T|+IW2T and t herefore 11IW2T\≤|1+△W2T|≤ This implies that This and (444)yield 1+△W2T <1≤ (→)Assume that W2Tlloo 1 and WiS 1+△W2T <1H△≤ (45) Pick a frequency w where WiS 11 W2T is maximum4 Now pick A so that 11IW2T=|1+△W2T≤ The idea here is that A(w)should rotate W2(w)T(w)so that A(w)W2(w)T(w)is negative real4 The details of how to construct such an allowable A are omitted4 Now we have W S 11W2T WiS 1+△W2T WiS 1+△W2T So from this and (445)there follows (44)4 Test (443)also has a nice graphical interpretation4 For each frequency w,construct two closed disks:one with center 11,radius Wi(w);the other with center L(w),radius W2(w)L(w)4 Then (443)holds i.for each w these two disks are disjoint (Figure 446)4 The robust performance condition says that the robust performance level 1 is achieved4 More generally,let's say that robust performance level a is achieved if WiS W2Tlloo 1 and 1+△W2T <aH△≤ Noting that at every frequency WiS WiS max 1+△W2T 11W2T
CHAPTER UNCERTAINTY AND ROBUSTNESS Fix In what follows functions are evaluated at an arbitrary point j but this is suppressed to simplify notation We have jWT WT jjWT j jWT j and therefore jWT jjWT j This implies that WS jWT j WS WT This and yield WS WT Assume that kWT k and WS WT Pick a frequency where jWSj jWT j is maximum Now pick so that jWT j jWT j The idea here is that j should rotate WjT j so that jWjT j is negative real The details of how to construct such an allowable are omitted Now we have WS jWT j jWSj jWT j jWSj jWT j WS WT So from this and there follows Test also has a nice graphical interpretation For each frequency construct two closed disks one with center radius jWjj the other with center Lj radius jWjLjj Then holds i for each these two disks are disjoint Figure The robust performance condition says that the robust performance level is achieved More generally let s say that robust performance level is achieved if kWT k and WS WT Noting that at every frequency max jj WS WT jWSj jWT j