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