xili CONTENTS 15.3 Nominal Performance 401 15.3.1 Sensitivity and Complementary Sensitivity Function 401 15.3.2 H Performance Objectives 402 15.3.3 Hoo Performance Objective 404 404 15.4 Robust Stability············ l5.5 Robust Performance.,.....··,· 405 15.5.1 Sensitivity Function Approximation ..405 15.5.2 Hoo Performance Objective..... ..406 15.6 IMC Design:Step 1 (Q) .406 15.6.1 H2-Optimal Control 406 15.6.2 Setpoint Prediction.. 413 15.6.3 Intersample Rippling 413 15.6.4 Inner-Outer Factorization 416 l5.7 IMC Design:Step2(F)········· 419 15.7.1 Filter Structure.·,...···. 420 15.7.2 Robust Stability Intercounection Structure 421 15.7.3 Robust Performance Interconnection Structure 422 15.7.4 Robust H Performance ()bjective... 423 15.8 Illustration of the Design Procedure 424 15.8.1 System Description...... 424 15.8.2 Design of Q 426 15.8.3 Design of F. 428 433 15.9 Summary.... 434 l5.l0 References..·· Part V:CASE STUDY 435 16 LV-CONTROL OF A HIGH-PURITY DISTILLATION COL- 437 UMN 437 l6.1 Features.,·· 437 16.1.1 Uncertainty 16.1.2 Nonlinearity... 438 16.1.3 Logarithmic Compositions 438 16.1.4 Choice of Nominal Operating Point.. 439 16.2 The Distillation Column......... 439 16.2.1 Modelling······ 440 16.2.2 Simulations·.,···. 442 16.3 Formulation of the Control Problem 443 16.3.1 Performance and Uncertainty Specifications . 443 16.3.2 Analysis of Controllers.·.,.·· 443
NOMENCLATURE Abbreviations DIC Decentralized Integral Controllable IC Integral Controllable IM Interaction Measure IMC Internal Model Control IS Integral Stabilizable ISE Integral Squared Error LFT Linear Fractional Transformation LHP Left-Half Plane MIMO Multi-Input Multi-Output \P Minimum Phase N\P Nonminimum Phase NP Nominal Performance S Nominal Stability PFE Partial Fraction Expansion RGA Relative Gain Array RHP Right-Half Plane RHS Right Hand Side RP Robust Performance RS Robust Stability SISO Single-Input Single-Output SSE Sum of Squared Errors SSV Structured Singular Value SVD Singular Value Decomposition UC Unit Circle xix
NOMENCLATURE XX Symbols B(2) Factor to preserve system type in (and Q(=) Allpass with the open RHP(outside the UC) b,(s)(bz》 poles of p(s)(p"(z)) b.(s)(bi(z)) Allpass with the open RHP (outside the UC) poles of v(s)(v"(z)) C Field of complex numbers Cnxm Field of complex matrices of dimension n x m c(s),c()(C(s),C(z) SISO (MIMO)continuous and discrete classic feedback controllers Classic feedforward controller cr(s) Effect of d'(s)on y(s) d(s Disturbance entering pd(s) d'(s) d'(z)(d(z) =-transform of d(s)before (after) passing through Y(s) e(s)(e"(z)) Continuous(discrete)error signal:Difference between y(s)(y"(=))and r(s)(r()) Weighted error signal W2(s)e(s) e'(s) f(s),f(z)(F(s),F() SISO(MIMO)continuous and discrete IMC filters Type 1 one-parameter discrete SISO IMC filter f(s) Transfer function matrix used as argument of u in the G(s RP SSV criterion for the continuous case The equivalent of GF(s)for the discrete case: G5(s) it depends on whether u=r or v =d Gf(s)(Gif(=)) Transfer function matrix used as argument of u in the RS SSV condition for the continuous (discrete)case ho(s)(Ho(s)) SISO(MIMO)zero-order hold H2(H2) H段xm(Hnxm）forn=m=1 H(H吃) Haxm(Hinxm)for m=1 H段xm Subspace of L2xm,containing the functions with analytic continuations in the RHP (includes all rational,proper,stable transfer function matrices) H:.nxm Subspace of Lxm defined as the orthogonal complement of the Lsubspace which contains functions with analytic continuations inside the UC(Hcontains all rational,strictly proper, stable a-transfer function matrices). Im Imaginary part
NOMENCLATURE xxi ea(s)(LA(s)) .(w)(a(w) Additive SISO (MIMO)uncertainty Bound on the additive uncertainty for the continuous (discrete)case LE(s)(eE(w)) (Bound on the)MIMO inverse LI(s)(Er(w)) multiplicative output uncertainty (Bound on the)MIMO input multiplicative uncertainty lm(s） (w)((w)) Multiplicative SISO uncertainty Bound on the multiplicative SISO Lo(s)(lo(w)) uncertainty for the continuous (discrete)case (Bound on the)MIMO output multiplicative uncertainty L2(L) L(L") L3xm(L2nxm）form=n=1 L好xm(L空nxm) Laxm(Linxm)for m=1 Space of n x m matrix valued functions that p(s)(P(s)) are square integrable on the imaginary axis(UC) SISO(MIMO)plant;also denotes the process model p(s)(P(s)) in sections where p=p(P=P)is assumed SISO (MIMO)process model p*(z),p(z(P*(z),P(z) Zero-order hold discrete equivalent of p(s)(P(s)) p(2,p,(z(P(z),P(z) without and with y(s)(r(s))included,respectively Zero-order hold discrete equivalents of p(s)(P(s)) pA(s),pA(z)(PA(s),Pa(z)) without and with y(s)(T(s))included,respectively Allpass with the NMP elements of p(s)and p(:)(P(s）andp(:)）respectively pa(s),pd(s） True and model of the disturbance effect plant transfer function Pm(s),pm(s) True and model of the measurement device transfer function px(s),pi(=)(Pu(s),Pi(=)) MIP factors of p(s)and p*(z) 9(s),q(Q(s),Q(z) (P(s)and P(=))respectively SISO (MIIMO)continuous and discrete IMC controllers s,(:(Q(s,Q(s) SISO (MIMIO)continuous ga(s).ga(s)(Gr(s),qr(s)) and discrete 1st step IMC controllers Disturbance rejection(setpoint tracking) Ist step and overall IMC controllers of the two-degree-of-freedom structure 9r(s) Feedforward IMC controller
vxii NOMENCLATURE 4(:)(QH(z) SISO (MIMO)H;-optimal IMC controller 1() Factor included in g(z)and Q(z)to avoid intersample rippling R Field of real numbers Res Residue e Real part r(s)(r"(z)) Continuous(discrete)setpoint ' Sampling time (s)(u(z)) Continuous(discrete)input to the plant "(s)(u'(s) External input r(s)or d(s)(normalized to impulse) () z-transform of v(s） '(s)(V(z) Square matrix whose columns consist of external inputs v(s)(v(z)) (y) Set of inputs v(s)(v'(s))or v"(z) A(s),vA(2)(VA(s),VA(z)) Allpass with the NMP elements of v(s),v(z)(V(s),V(=))respectively vu(s),vu(Vu(s),Vu(2)) MP factors of v(s),v"(=)(V(s),V(2)) respectively w(s) Performance weight for the SISO case wa(s)(wr(s)) Performance weight for disturbance rejection (setpoint tracking)in the two-degree-of-freedom structure IV(s)(W2(s)) Input (Output)weights for the MIMO case (s)((s) Continuous plant (model)output (),((),() 3-transform of y(s)(i(s))before and after passing through y(s)respectively im(s) Output of measurement device Greek Characters (s)(I(s)) SISO (MIMO)anti-aliasing prefilter ⊥(s(△() Block diagonal matrix containing the uncertainty for the continuous (discrete)case △ Set of possible 's 」(s(⊥u(之) Uncertainty block in A(s)(A(=)) 」p(s) RP block in⊥(s) (s),E(s)((s),E(s) (Nominal)sensitivity function for the continuous SISO and MIMO cases,respectively (s) first step IMC nominal sensitivity function