Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science 1 Massachuasetts Institute of Technology 1 c � Chapter 20 Stability Robustness 20.1 Intro duction Last chapter showed how the Nyquist stability criterion provides conditions for the stability - p ossible to provide an extension of those conditions by gen robustness of a SISO system. It is eralizing the Nyquist criterion for MIMO systems. This, however, turns out to b e unnecessary derivation is p ossible through the small gain theorem, which will b e presented in and a direct chapter. this 20.2 Additive Representation of Uncertainty It is commonly the case that the nominal plant mo del is quite accurate for low frequencies b ecause of parasitics, nonlinearities and/or time- but deteriorates in the high-frequency range, varying e�ects that b ecome signi�cant at higher frequencies. These high-frequency e�ects may have b een left unmo deled b ecause the e�ort required for system identi�cation was not justi�ed by the level of p erformance that was b eing sought, or they may b e well-understo o d e�ects that were omitted from the nominal mo del b ecause they were awkward and unwieldy to carry along control design. This problem, namely the deterioration of nominal mo dels at higher during extent by the fact that almost all physical systems have frequencies, is mitigated to some prop er transfer functions, so that the system gain b egins to roll o� at high frequency. strictly In the ab ove situation, with a nominal plant mo del given by the prop er rational matrix P (s), the actual plant represented by P (s), and the di�erence P (s) ; P (s) assumed to b e 0 0 we may b e able to characterize the mo del uncertainty via a b ound of the form stable, � [ P (j ! ) ; P (j ! )] � ` (! ) (20.1) 0 a max where ( \Small" � j! j � ! c ` (! ) � (20.2) a \Bounded" � j! j � ! c This says that the resp onse of the actual plant lies in a \band" of uncertainty around that of plant. Notice that no phase information ab out the mo deling error is incorp orated the nominal into this description. For this reason, it may lead to conservative results. The preceding description suggests the following simple additive characterization of the uncertainty set: � � fP (s) j P (s) � P (s) + W (s)�(s)g (20.3) 0 where � is an arbitrary stable transfer matrix satisfying the norm condition k�k � sup � (�(j ! )) � 1 (20.4) max 1 ! and the stable prop er rational (matrix or scalar) weighting term W (s) is used to represent any information we have on how the accuracy of the nominal plant mo del varies as a function frequency. Figure 20.1 shows the additive representation of uncertainty in the context of a of servo lo op, with K denoting the comp ensator. standard When the mo deling uncertainty increases with frequency, it makes sense to use a weight- ing function W (j ! ) that lo oks like a high-pass �lter: small magnitude at low frequencies, b ounded at higher frequencies. In the case of a matrix weight, a variation increasing but � additive term W � is to use a term of the form W �W we leave you to on the use of the 1 2 how the analysis in this lecture will change if such a two-sided weighting is used. examine - - � W � � � l l - - - - - . y + + r P K 0 ; 6 Figure 20.1: Representation of the actual plant in a servo lo op via an additive p erturbation plant. of the nominal Caution: The ab ove formulation of an additive mo del p erturbation should not b e interpreted saying that the actual or p erturb ed plant is the paral lel combination of the nominal system as P (s) and a system with transfer matrix W (s)�(s). Rather, the actual plant should b e 0 b eing a minimal realization of the transfer function P (s), which happ ens to b e considered as additive form P (s) + W (s)�(s). written in the 0 Some features of the ab ove uncertainty set are worth noting: � The unstable p oles of all plants in the set are precisely those of the nominal mo del. Thus, mo deling and identi�cation e�orts are assumed to b e careful enough to accurately our p oles of the system. capture the unstable � The set includes mo dels of arbitrarily large order. Thus, if the uncertainties of ma jor were parametric uncertainties, i.e. uncertainties in the values of the concern to us parameters of a particular (e.g. state-space) mo del, then the ab ove uncertainty set would greatly overestimate the set of plants of interest to us. The control design metho ds that we shall develop will pro duce controllers that are guar- anteed to work for every member of the plant uncertainty set. Stated slightly di�erently, metho ds will treat the system as though every mo del in the uncertainty set is a p ossible our representation of the plant. To the extent that not all memb ers of the set are p ossible plant mo dels, our metho ds will b e conservative. 20.3 Multiplicative Representation of Uncertainty Another simple means of representing uncertainty that has some nice analytical prop erties is multiplicative perturbation, which can b e written in the form the � � fP j P � P (I + W �)� k�k � 1g: (20.5) 0 1 - - � W � � � m - . - - + P 0 Figure 20.2: Representation of uncertainty as multiplicative p erturbation at the plant input. An alternative to this input-side representation of the uncertainty is the following output- representation: side � � � fP j P � (I + W �)P k�k � 1g: (20.6) 0 1 In b oth the multiplicative cases ab ove, W and � are stable. As with the additive represen- tation, mo dels of arbitrarily large order are included in the ab ove sets. Still other variations may b e imagined� in the case of matrix weights, for instance, the term W � can b e replaced . by W �W 1 2 The caution mentioned in connection with the additive p erturbation b ears rep eating ab ove multiplicative characterizations should not b e interpreted as saying that the here: the plant is the cascade combination of the nominal system P and a system I + W �. actual 0 plant should b e considered as b eing a minimal realization of the transfer Rather, the actual (s), which happ ens to b e written in the multiplicative form. function P Any unstable p oles of P are p oles of the nominal plant, but not necessarily the other way, b ecause unstable p oles of P may b e cancelled by zeros of I + W �. In other words, 0 plant is allowed to have fewer unstable p oles than the nominal plant, but all its the actual p oles are con�ned to the same lo cations as in the nominal mo del. In view of the unstable such cancellations do not corresp ond to unstable hidden caution in the previous paragraph, mo des, and are therefore not of concern. 20.4 More General Representation of Uncertainty Consider a nominal interconnected system obtained by interconnecting various (reachable and observable) nominal subsystems. In general, our representation of the uncertainty regarding any nominal subsystem mo del such as P involves taking the signal � at the input or output 0 \uncertainty blo ck" with transfer function of the nominal subsystem, feeding it through an , where W �W each factor is stable and k�k � 1, and then adding the output W � or 1 2 1 � of this uncertainty blo ck to either the input or output of the nominal subsystem. The additive and two multiplicative representations describ ed earlier are sp ecial cases of this one p ossibilities with construction, but the construction actually yields a total of three additional given uncertainty blo ck. Sp eci�cally, if the uncertainty blo ck is W �, we get the following a feedback representations of uncertainty: additional ;1 � P � P (I ; W �P ) � 0 0 ;1 � P � P (I ; W �) � 0 ;1 . � P � (I ; W �) P 0 uncertainty representations itemized ab ove is that the unstable A useful feature of the three p oles of the actual plant P are not constrained to b e (a subset of ) those of the nominal plant . P 0 Note that in all six representations of the p erturb ed or actual system, the signals � and � b ecome internal to the actual subsystem mo del. This is b ecause it is the combination of P with the uncertainty mo del that constitutes the representation of the actual mo del P , and 0 mo del is only accessed at its (overall) input and output. the actual In summary, then, p erturbations of the ab ove form can b e used to represent many typ es of uncertainty, for example: high-frequency unmo deled dynamics, unmo deled delays, unmo deled variations. sensor and/or actuator dynamics, small nonlinearities, parametric 20.5 A Linear Fractional Description , and a We start with a given a nominal plant mo del P feedback controller K that stabilizes 0 . The robust P stability question is then: under what conditions will the controller stabilize 0 al l P 2 �� More generally, we assume we have an interconnected system that is nominally internally stable, by which we mean that the transfer function from an input added in at any subsystem input to the output observed at any subsystem output is always stable in stability question is then: under what conditions will the the nominal system.