Chapter 5 Robustness Analysis for Multivariable Systems In this chapter, st ability and performance for multivariable sy stems with uncert ainty will be considered. Consider a general multivariable system as depicted in Figure 5. 1. All signals will in general be vectors, and G(s)and K(s will be transfer matrices. d(s) is an output distur bance signal and n(s)represents measurement noise. Disturbances related to the input u(s) have been neglected but can easily be incuded if necessary. In Section 5.1.2, the performance problem is formulated as a 2X 2 problem. To facilit ate this reformulation, a sign convention is introduced such that the minus usually included in the loop, instead is incorporated in measurement noise, however, the error can not be seen directly in the block diagram. ce of the compensator K(s). The error is defined as e(s)=y(s-r(s). due to the preser eols K(s) G(s) m(8)(∑ 72(s) Figure 5. 1: General feedback configuration The design of the comp ensator K(s in general has to satisfy the following four ob jectives Nominal closed loop st ability Robust stability. ● Robust performance
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Page 45 of 92 In order for disturb ances u(s) and measurement noise n(s)not to deteriorate the input u(s), it is seen from (5.4), that the control sensitivity M(s)has to be small It still needs to be quantified, however, what 'small'means in this context systematic manner, the norms for signals and transfer functions introduced in Chapter 3 have to be extended to multivariable systems. 1.1 Norms of signals for Multivariable Systems In Chapter 3, the following norms for scalar signals u(t)were introduced The 1-norm:‖a lu(t)ld v-(t)da (5.12) The oo-norm: vo=sup u(t) (5.13) power'norm': vp=pow(u)= lim m0(x7/02()d (5.14) The power ' vlp is actually not a norm, but rather a semi-norm. With some abuse of terms it will refered to in these notes, nevertheless, as the power norm. Now, assuming u(t) is a vector rather than a scalar, the power(semi-)norm is defined as P lu(t)2dt (5.15) where the integrand o(t)2 is the vector 2-norm, introduced in Section 4.3.1 on page 38 lv(t)‖2 v(t)|2=y/rt)(t) (5.16) Here, v(t)denotes the transpose of u(t ). Note, that the not ation ull denotes a signal norm for the signal u(t), whereas u(t) denotes a vector norm for the value of the signal at time t In addition to the vector 2-norm, in Section 4.3. 1 on page 38 also the vector 1- and the vector oo-norms were introduced 1-normen: vi=∑|() (5.17) O-hormen lv(t)‖l∞= max jvi(t) (5.18)
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Now, combining the scalar 1, 2, oo signal norms with the 1, 2, oo vector norms leads to nine different multivariable signal norms Doll u(t)1dt ∑|()|t (5.19) u(t (5.20) v(t)loo dt max| u(tle (5.21) (t)idt ()|2dt ul(t)u(t)dt (5.23) v(t)2 maxus(t)1dt (5.24) )1=sup∑|(t) (5.25) vlloo, 2=sup lu(t)2=supvuT(t)u(t) (5.26) sup‖(t)‖o= sup max|v;(t) (5.27) i.e.,vlx.w denotes the norm obt ained by applying the vector y-norm at every time instance and by applying the signal norm as the temporal norm The nal norms can now be used with the power norm ollp to define a number of induced norms for multivariable transfer functions. To that end, let g(t)be the impulse response matrix associated with the transfer matrix G(s). The output y(t)is then given by y(t)=g(t)*() (5.28) where g(t)*u(t) denotes the convolution integral of g(t) and u(t). Now, an induced norm for can be ll(t)(un, y)-(yay)sup iym yv=sup juy (5.29) where can be chosen ly combination 2,∞ o and p. Hence, induced norm la(t )( a, u,)-(ya,W w)measure the maximal 'gain of the sy stem when the ir is measured by the norm u and the output by the norm y In this way, than a hundred different induced norms can be defined for g(t). Roughly this corresponds to more than a hundred different ways to measure the gain of multivariable system, although only a few of these have any practical significance Example 5.1(Nominal Performance) nsider a bjective of minim ressource. Thus, it would be reas onable to meas ure the control error e(t) by the norm ell
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Avvy. n cgi degi hi edhhf5ft l gn afvcyhpit env =y pw cgnfh nt nh w fv hi v, t i pan i v unso Tgnt f=dd fv i 1,, a. ri vyn c h=xo Tgfv. rit v cgi d cgn e, t dh s, pjiedm ei t pn chi vsi da ft i. ft f. fbi c, t, ccn li ft ft chhllnda i v cgn c ss uft I t, h f sf1(1 (5.30) agnhn )(x fv cgn f. lysin Invl, t vn. i chf2 c h cgn dit vah ct ed, t so)ch. =)d so) At first glance, all these induced norms might seem overwhelming. It can be shown, however, that several of them always equal either zero or infinity, and therefore are without any prac- tical significance. Current research activities include the development of compensator design pro cedures related to a number of the norms mentioned above In the ' robust robust control theory, only one single induced norm has been consid ered: the'pure' 2-norm, i. e, the norm induced by f of dd on f of dd u2(xu(xg f gf fg* uf dd (5.31) “FPfd“e( As the f f dd norm measures the energy of a signal, fgf dde Ai de represent the largest possible energy gain for a sy stem. The reason why this norm has been of particular interest is in part that it is applicable in practice in many cases, and that it leads to simple conditions both for robust st ability and for nominal performance An important property for f of dd is obtained from Parseval's Theorem u2(x)(x=x= ju)ulw) (5.32) where u(ju) denotes the LAPlace transform of u(xg. Hence, also: ogsa gf ce d∫/u(ju)u(j)= (5.34) u=.jw)u(ju)=w (),((j)(()(i)= (5.35) G((u) (5.36) (5.37) To recapitulate Thd i. yscrihfi psn wvd pn anvehfpna pw fcv ci t vah. i chf2( Oo fic cgn ft lyd u(xod gn vwc. ita ign, ydlyd uxd in p, cg. ni vyhna pw gn 2-t, h f of dd cgn. i 2f. i sli ft c cgn vwd. fv I fnt pw cgn H t,h. ,c(O) f(f ((ju) f uf dd (538)
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Gs Ad ny ue yr se input multiplicative uncertainty Cp aCec Che inverse output multiplicative uncert ainty: CpGee Ch?. Ch In Chapter I similar- inert ainty mo dels were considered for scalar sy stems, and the size of the perturbation c Lvas bounded by the absolute value of its frequency response eCkMan In this chapter, however, e chs a matrix. Hence, it has to be decided which norm to be used to bound the - size of g un In modern robust control, mainly the matrix anorm used to bound c infor two main reasons. First, the corresponding perturb ation model is appropriate to describe high frequent unmodeled dynamics, time delays, and phenomena arising from sy stems with distributed parameters. Second, this choice leads to mat hematically simple conditions for robust stability. Thus, it is assumed that e Chs a complex matrix, which is unknown but bounded in amplitude. n gIapo @Cluth fnr (5.57) Usually, two diagonal weight matrices Wa Chnd Wa hre introduced such tha e Cpwack chac (5.58) with n gImpo CAIm frm (5.5) The input weight w used for scaling, if e.g. the signals are measured in different units The 沿h 5.2.1 The Small gain Theorem Below, the famous Small Gain Theorem will be presented and applied to derive conditions for robust st ability with respec system in Figure 5. 1 and let P i⑦mtmm 51(Th 7 n the closed loop system is stable if the9edmp○Qhy a stahle transfer matriz Proof of Theorem 5.1(By contradiction) Assume that the spectral radius p(PGw))<1 and that the closed loop sy stem is unst able. According to the generalize d Ny quist Theorem, instability implies that the im age curve of det(I+ P(s)encircles the origin as s traverses the Nyquist d contour As the Ny quist d contour is closed, so will its im age curve be also. Hence, there exists an E E 0, 1 (60) 入1(I+cP(ju)=0 (5.61) 分1+cA(P(u*)=0 for some given i (5.62) A:(P(u*) →|A:(Pu*)≥1 (5.64) which is a contradiction as we assumed that p(PGw))<1,odw
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The Small Gain Theorem states, that for an open loop stable sy stem, a sufficient condition or closed loop st ability is that the open loop gain measured by p(Pgw ss than unity. Fortunately, this a just a(potentially very conservative) sufficient condition for closed loop st ability. Otherwise it could be impossible to satisfy the usual performance requirement for high gains in the low frequent range Below, the Small Gain Theorem will be applied to determine the st ability of the closed loop system for the uncert ainty models intro duced above. This application of the Small gain Theorem is classical in robust control. Assume, e. g, that the perturbed system can be described by an additive perturb ation G△(8)=G(8)+W2(s)△()Wu1(s) (5.65) where a(AGu))<1. This can be represented in block diagram form as shown in Figure 5.4 Now, let P(s) denote the transfer matrix as'seen'from the terminals of A(s). It can easily be verified that P(8)=W1(s)K(s)(I-G(s)K(8)wa2(8) Wu1(8)M(s)W2(s) (5.67) The following result is obt ained G(s w(s Figure 5.4: Clos ed loop system with additive perturbation. P(s) is the transfer matric as seen the terminals of△(s) Theorem 5.2(Robust Stabilitet Assume that the sys tem P(s)is stable, and that the perturbation A(s) is such that the perturbed clos ed loop system is stable if and only f the image curve of det(I-P(sA(s)), as s traverses the Nyquist d contour, does not encircle closed s tem in Figure 5.4 is stable for all F(A(u))<1, f and only f one of the following equivalent conditions is satisfied (I-P(ju)△(ju)≠ u,V△(ju):7(△(ju)≤ (5.68) p(P(u)△(ju) vu,V△(ju):7(△(ju)≤1(5.69) 7(P(j)<1 P(s)川n
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