Multivariable Feedback Control Analysis And Design 2nd Edition Sigurd Skogestad, Ian Postlethwaite - Solutions

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2.6. Lehmann (1966) defined two random variables (X, Y), discrete or continuous, to be positively quadrant dependent if P(X P(X < x)P(Y < y) all x and y and positively likelihood-ratio dependent if their joint density satisfies f(x\,yi)f(x2,y2) > f(x\,yi)f(x2,y\)whenever x\ < X2 and y\ < yj. He
2.7. Go to sda.berkeley.edu/GSS and cross-classify the variables POLVIEWS and HAPPY for the latest survey [e.g., enter YEAR(2008) in the selection filter to obtain results for the year 2008]. Using methods of this chapter, describe the data.
3.1. Is a cumulative logit model a special case of a baseline-category logit model?Why not?
3.2. For the model, logit [P(Y < j)] —
3.3. A response scale has the categories (strongly agree, mildly agree, mildly disagree, strongly disagree, don't know). One way to model such a scale uses an ordinary binary logistic model for the probability of a don't know response and uses a separate ordinal model for the four ordered
3.4. Consider the 2 x 3 x 2 contingency table relating a binary x\ to an ordinal Y, by row, of (10, 10, 10; 0, 0, 30) at the first level of x2 and (10, 10, 10;10, 10, 10) at the second level of x2. Explain why a cumulative logit model with main effects has finite estimates for the effects of x\ and
3.5. Table 3.13 shows data cross-classifying job satisfaction and income, stratified by gender, for black Americans. The data are analyzed in Agresti (2002, Chap. 7).(a) Analyze the conditional association between job satisfaction and income, treating the variables as nominal.(b) Analyze the
3.6. Refer to Exercise 2.7. Analyze these data using methods of this chapter.
4.1. For the row effects model (4.4), show that the sufficient statistics for {ô,}are sample means computed within the rows, using the column scores( l , 2 , . . . , c ) .
4.2. If cumulative logit and adjacent-categories logit models with proportional odds structure both fit a data set well, explain why the parameter estimates from the cumulative logit model would probably be larger. Does this imply that it is easier for effects to achieve statistical significance
4.3. Prove factorization (4.13) for the multinomial distribution.
4.4. Summarize some advantages and disadvantages of the stereotype model compared to the ordinary multinomial logit model using baseline-category logits.
4.5. Analyze Table 4.1 with (a) a cumulative logit model and (b) a continuationratio logit model. Compare and contrast results to those obtained in Section 4.1.5 using adjacent-categories logit models.
4.6. Refer to Exercise 2.7. Analyze these data using methods of this chapter.
5.1. For the cumulative probit model Ö~é[Ñ(Õ < j)] = á7 — â'÷, explain why a 1-unit increase in x^ corresponds to a ßk standard deviation increase in the expected underlying latent response, controlling for other predictors.
5.2. Refer to the example in Section 1.3.3. Generate 100 observations from the given latent variable model, which has E(y*) — 20 + 0.60.x - 40.0z.(a) Plot the data and fit the model using OLS.(b) Categorize y* into y using five categories, as in that example. Now using y as the response, fit the
5.3. For cumulative link model (5.1), show that for 1 < j < k < c — 1, P(Y < k \x) z= P(Y < j | x*), where x* is obtained by increasing the ith component of x by (oik — &j)/ßi· Interpret.
5.4. A cumulative link row effects model for an r x c contingency table with a qualitative predictor is h[P(Y
5.5. Let F\(y) = 1 — exp(—ëí) for ;y>0 be a negative exponential cdf with parameter ë, and let F2{y) = 1 — exp(—ìã) for y > 0. Show the difference between the cdf s on a complementary log-log scale is identical for all y. Give implications for data analysis with an ordered categorical
5.6. For a link function h, consider the model of form presented in Section 4.2,Á[ù,-(÷)] =a, + â,·÷, where ù,· = .1 ð,- Ç Ync(a) Explain why the fit of this model is the same when fitted separately for j — 1 , . . . , c — 1 or when fitted simultaneously.(b) For the complementary
5.7. Using the logit link, fit the model (5.4) with location and dispersion terms to Table 3.8 from Section 3.5.6 in Chapter 3, for which the cumulative logit model of proportional odds form has lack of fit. Interpret.
5.8. Table 5.8 summarizes observations of passengers in autos and light trucks involved in accidents in Maine in one year. The table classifies passengers TABLE 5.8. Passenger Observations Gender Female Male Location Urban Rural Urban Rural Seat Belt Use No Yes No Yes No Yes No Yes 17,287 11,587
5.5. Let F\(y) = 1 — exp(—ëí) for ;y>0 be a negative exponential cdf with parameter ë, and let F2{y) = 1 — exp(—ìã) for y > 0. Show the difference between the cdf s on a complementary log-log scale is identical for all y. Give implications for data analysis with an ordered categorical
5.6. For a link function h, consider the model of form presented in Section 4.2,Á[ù,-(÷)] =a, + â,·÷, where ù,· = .1 ð,- Ç Ync(a) Explain why the fit of this model is the same when fitted separately for j — 1 , . . . , c — 1 or when fitted simultaneously.(b) For the complementary
5.7. Using the logit link, fit the model (5.4) with location and dispersion terms to Table 3.8 from Section 3.5.6 in Chapter 3, for which the cumulative logit model of proportional odds form has lack of fit. Interpret.
5.8. Table 5.8 summarizes observations of passengers in autos and light trucks involved in accidents in Maine in one year. The table classifies passengers TABLE 5.8. Passenger Observations Gender Female Male Location Urban Rural Urban Rural Seat Belt Use No Yes No Yes No Yes No Yes 17,287 11,587
5.9. Refer to Exercise 2.7. Analyze these data using methods of this chapter.
6.1. Consider the L x L model (6.2), scaling row scores and column scores to have means of zero and standard deviations of 1. Express the model as a probability function for the cell probabilities {ðí;}, and demonstrate the similarity of this function with the bivariate normal density having unit
6.2. For score parameters {ì,} and {v/}, explain why the model log ßijk = k + kf + kYj + kz k + ßßi Vj + kfk z + k)k zis a homogeneous association version of the RC model (6.13). What complications are there in using it to test XY conditional independence?
6.3. An alternative loglinear model for the ordinal-ordinal table, having fixed scores as well as unknown row and column effects, is log ßij = k + kf + kY + ì,· vj + Ui Vj.With equally spaced {«,} and {VJ}, Goodman (1979a, 1981a) referred to it as the R + C model because of the additivity in
6.4. Consider a cross-classification of two ordinal variables with fixed row and column scores.(a) Describe the association pattern represented by the model log ßij = ë + ë* + ë] + ß\UiVj + ß2U2 iVj.For a 3xc table, explain why this model is equivalent to the row effects model. [More
6.5. For the homogeneous linear-by-linear association model (6.9), show that the likelihood equation corresponding to ß isÓIÓÓ"'"^'* I = Ó I ÓÓÌ
6.6. Refer to Exercise 2.7. Analyze these data using methods of this chapter.
6.7. Refer to Table 9.1 in Chapter 9. Fit the uniform global odds ratio model for each gender. Show how to compare the associations inferentially.
7.1. Show that the ordinal odds ratio nc/Tld (Agresti 1980) simplifies for 2 x c tables to È = P(Y2 > Y\)/P(Y\ > Y2), presented in Section 2.1.4, and for 2 x 2 tables to the odds ratio.
7.2. Let m = min(r,c) in an r x c table. Show that the maximum attainable value of n c — Ud is (m — l)/m, which occurs when the probability is uniformly distributed on m cells in a longest diagonal of the table. Hence, m(nc - Ud)ô€ =m — 1 can equal 1.0 in absolute value for any table size
7.3. Refer to formula (7.4) for the null variance of C — D. For large marginal totals, show that it equals approximately _2 v-EiPlw-EjPlj)"3°C-D = g ·(For Table 7.1, aC-D = 43, 941 compared to aC-D = 44,022 .) Show that the estimated null variance of (fic — Ð^) is approximately 4(1 ~ Ó,·
7.4. In an ordinal r x c table, we could assign ordered scores and use a linear trend model, E(Y | X = «,)= a + ßty. With the least squares estimator b of jö, show that the Pearson statistic for testing independence partitions into X2(I)=X2(L) + X2(I|L), where X2(I | L) = b2/Var(b). Here X2(L)
7.5. For sample mean ridit scores A\ and Aj for two groups, show that the Wilcoxon test statistic (7.6) equals _ Äi -0.50 _ _ Ä 2 - 0 . 5 0 VVar(Äi) v/Var(A2)'where n2(n]+n2 + l) ç2Õß](Þ-ß])Var(Ai) = 12ni(ni + n2)2 12m(ni +n2)3(m + n 2 - 1)'and Var(A2) is the formula for Var(Ai) but with n\
7.6. For drug dosage (low, medium, high) and the effect on a patient's condition(negative, none, positive), the conditional distributions on the effect are expected to exhibit the pattern:Dosage Negative None Positive Low Medium High 0.30 0.10 0.30 0.40 0.30 0.40 0.30 0.60 0.30 Indicate why it is
7.7. Using the multivariate version of the delta method (Bishop et al. 1975, p.493), show that the sample mean ridits Ä = ( Ä i , . . . , Är_i)' in a r x c table have asymptotic covariance matrix ¼'¸¼/ç, where Ó/ç is the re x re multinomial-based covariance matrix for the sample
7.8. In testing Ho: independence for an r x r table, suppose that «,,· = 1 for all i and tiij — 0 for all ( ö j .(a) Show that an exact test that uses the Pearson X2 to order all tables having the given margins has P-value = 1.0.(b) Explain why a two-sided test using an ordinal criterion such
7.9. Go to sda.berkeley.edu/GSS and cross-classify the variables POLVIEWS and HAPPY for the latest survey. Using the methods of this chapter, describe the association.
8.1. Suppose that (1º, K2) has bivariate normal probability density function/ ( y i , y2), with E(Y{) = ì, Å(Õ2) = ì + Á, Var(y,) = Var(y2) - ó2, and Corr(r,, Y2) = p. Show that f(yu y2)//(y2, yi) has form 8y''y2 for some constant S. Thus, under the assumption of an underlying bivariate
8.2. For matched pairs, consider the special case (8.18) of the loglinear ordinal quasi-symmetry model (8.15), namely log ßij = ë + ki + Xj + ß\u, + ß2Uj + pi+j.For equally spaced scores, explain why this model has the formßij = otiOijSi-j8*+j, which is in a class of models proposed by Goodman
8.3. Consider the conditional symmetry (CS) model (8.9).(a) Show that this model has the loglinear representation logMiy = Xmjn(;j),max(i,7·) + ßl{i < j), where /(·) is an indicator (see also Bishop et al. 1975, pp. 285-286).(b) Show that conditional symmetry + marginal homogeneity =
8.4. Formulate a marginal model using adjacent-categories logits or continuation-ratio logits that is analogous to cumulative logit model (8.11). Interpret parameters.
8.5. Table 8.11, from a general social survey, was analyzed with the OQS model by Agresti (1995). Subjects gave their opinions regarding government spending on (1) the environment, (2) health, (3) assistance to big cities, and (4) law enforcement. Analyze the data using an alternative model such as
8.6. For weighted kappa with {éõ,7 — 1 — (i — j)2/(c — l)2}, show that two tables that have the same marginal distributions have the same values of KW whenever they have the same correlations, with row and column numbers as the scores (Graham and Jackson 1993).
9.1. A study investigates home Internet use, measured as (none, 1 hour a day), for subjects in families living in a rural location and families living in an urban area. Each cluster consists of the people in a particular family. For this study, give an example of a (a) within-cluster effect and
9.2. In Section 8.2.2 we used ML to fit a marginal cumulative logit model to GSS responses on how well the government provides health care for the sick and protects the environment. Obtain GEE estimates, compare to the ML estimates, and interpret.
9.3. Analyze the crossover data in Table 8.5 using the GEE method with a marginal cumulative logit model. Compare {â,} and their SE values to those obtained using ML in Section 8.4.3.
9.4. Refer to Exercise 8.5. Fit a marginal model to these data about government spending, using ML or GEE. Interpret the results.
9.5. Analyze the soft-drink comparison data of Table 8.8 in Section 8.6.3 using methods of this chapter that treat the three evaluations by a subject as potentially correlated. Interpret the results.
9.6. Refer to Table 9.5 on the clinical trial for insomnia patients.(a) To compare effects while controlling for the initial response, add an interaction term to model (9.2). Summarize how the estimated treatment effect varies according to the initial responses by showing that the estimated
10.1. Use a random effects model to estimate the effect in the subject-specific model that Section 8.2.5 considered for opinions about government performance in providing health care and in protecting the environment.Compare the results to the ones shown there.
10.2. Refer to Exercise 8.5 with Table 8.11 and Example 10.5.2 on opinions about government spending.(a) Formulate a cumulative logit random intercept model. Show that the ML estimates are as Section 10.5.2 shows. Interpret.(b) Fit the corresponding marginal cumulative logit model, using ML or GEE.
10.3. Explain how you could use a random effects model to analyze effects of explanatory variables on severity of retinopathy for the study described in Section 9.1.3, by using different variance components for the random effects for females and for males to handle the different association between
10.4. Refer to Table 9.5 from the insomnia study.(a) Analyze these data with a random effects model. Report and interpret the effects.(b) Fit the simpler model forcing ó„ — 0. Compare models, using either a likelihood-ratio test or AIC. What do you conclude?(c) Analyze the data with a marginal
11.1. Outline how you would conduct a Bayesian analysis for an ordinal measure of association that is not necessarily connected with a model, such as (a)gamma and Kendall's tau-b, and (b) a global odds ratio.
11.2. For an ordinal 2xc contingency table, develop a Bayesian analog of the Wilcoxon test and Bayesian inference for the related measures of stochastic superiority, Ä anda. For Table 11.4, find a posterior tail probability using one of these measures that is analogous to the frequentist exact
11.3. Refer to the example on government spending on the arts in Section 11.2.2.The corresponding counts for the sampled males in the 18-21 age range were(6, 4, 10, 5, 2). Using methods of this chapter, (a) report Bayesian estimates of the corresponding cell probabilities, (b) report Bayesian
11.4. Try to replicate the results shown for modeling the ulcer data in Sections 11.3.4 and 11.3.5. Obtain results for the corresponding continuationratio logit model for these data.
11.5. Refer to the opinions about teen sex, premarital sex, and extramarital sex shown in Agresti and Lang (1993a) and Congdon (2005, p. 262). Analyze these data with (a) a marginal model, (b) a random effects model, and (c) a Bayesian model. Compare results and interpretations.
Exercise Consider the RPproblem with weighted sensitivity and multiplicative output uncertainty Derive the interconnection matrix N for the conventional case with b diagf P g and the stacked case when b P Use this to prove
Exercise Consider the plant Gs in which is illconditioned withG at all frequencies but note that the RGAelements of G are all about With an inversebased controller Ks s Gs compute for RP with both diagonal and fullblock input uncertainty using the
Exercise If were true for any structure of then it would imply AB A B Show by a counterexample that this is not true
Exercise Assume A and B are square matrices Show by a counterexample that AB is not in general equal to BA Under what conditions is AB BA Hint Recall
Exercise Let a b c and d be complex scalars Show that fordiagf g ab ad bc cd ab ab cd cd jabj jcdj Does this hold when is scalar times identity or when is full Answers Yes and No
Exercise Let c be a complex scalar Show that for diagf g M MM M M cMcM M
Exercise For M in and a diagonal show that M jaj jbj using the upper bound M minD DMD which is exact in this case since D has two blocksSolution Use D diagfd g Since DMD is a singular matrix we have fromA thatDMD a dad b b pjaj jdaj
Exercise For M in and a diagonal show that M jaj jbj using the lower bound M maxU MU which is always exact Hint Use U diagfej g the blocks in U are unitary scalars and we may x one of them equal to
Exercise Consider the same example and check for robust stability with fullblock multiplicative output uncertainty of the same magnitude Solution RS satis ed
Exercise Consider combined multiplicative and inverse multiplicative uncertainty at the output Gp I iOWiOI OWOG where we choose to normbound the combined uncertainty k iO O k Make a block diagram of the uncertain plant and derive a necessary and sucient condition for
Exercise M structure for combined input and output uncertain ties Consider the block diagram in Figure where we have both input and output multiplicative uncertainty blocks The set of possible plants is given by Gp I WO OWOGI WI IWI where k Ik and k Ok Collect
Exercise Find the transfer function M N for studying robust stability for the uncertain plant Gp in
Exercise Find the interconnection matrix N for the uncertain system in Figure What is M
Exercise Find P for the uncertain plant Gp in when w r and z y r
Exercise Find P for the uncertain system in Figure
Exercise Derive P and N for the case when the multiplicative uncertainty is at the output rather than the input
Exercise Derive N in from P in using the lower LFT in You will note that the algebra is quite tedious and that it is much simpler to derive N directly from the block diagram as described above
Exercise For the system in Figure we see easily that the uncertain transfer function from w to z is F WP I GI WI I K Show that this is identical to FuN evaluated using where from we have N WITI N WIKS N WPSG and N WPS
Exercise Show in detail how P in is derived
Exercise The uncertain plant in may be represented in the additive uncertainty form Gp GW AW where A is a single scalar perturbationFind W and W
Exercise Neglected dynamics Assume we have derived the following detailed model Gdetails s s s and we want to use the simpli ed nominal model Gs s with multiplicative uncertainty Plot lI and approximate it by a rational transfer function wI s
Exercise Consider again the system in Figure What kind of uncertainty might wu and u represent
Exercise Parametric zero uncertainty in zero formConsider the following alternative form of parametric zero uncertainty Gps s zpGs zmin zp zmax which caters for zeros crossing from the LHP to the RHP through the origincorresponding to a sign change in the steadystate
Exercise Derive and Hint The gain variation in can be written exactly as kp ki rk Show that the form in does not allow for kp even with large and discuss why this may be a possible advantage
Exercise Also derive from jwpSjjwITj the following necessary bounds for RP which must be satis edjLj jwPj jwI j for jwP j and jwI j jLj jwP j jwIj for jwP j and jwI j
Exercise Derive the loopshaping bounds in and which are sucient for jwP Sj jwITj RP Hint Start from the RPcondition in the form jwP jjwILj j Lj and use the facts that j Lj jLj and j Lj jLj
Exercise Suppose that the nominal model of a plant is Gs s and the uncertainty in the model is parameterized by multiplicative uncertainty with the weight wI s s s Call the resulting set Now nd the extreme parameter values in each of the plants ag below so
Exercise Controllability of an FCC process Consider the following plantyyy Gsuuu fs s s Gs fs s s s s s s s s s s It turns out that this
Exercise Find the poles and zeros and analyze the inputoutput controllability for Gs c ss c ss
Exercise Analyze the inputoutput controllability for the following three plants with inputs and output Gs gs gsa gs gs ss b gs ss gs ss c gs ss gs ss Design controllers and perform closedloop simulations of
Exercise Analyze the inputoutput controllability for Gs gds ss gd ss Which disturbance is the worst
Exercise Analyze the inputoutput controllability for Gs sssssss
Exercise Order the following three plants in terms of their expected ease of controllability Gs Gs es es G s es Remember to also consider the sensitivity to input gain uncertainty
Exercise Consider the following two plants Do you expect any control problems Could decentralized or inversebased control be used What pairing would you use for decentralized controlGas s s s s s Gbs s s s s s s
Exercise Let A B IC D a Perform a controllability analysis of Gsb Let x" Ax Bu d and consider a unit disturbance d z z T Which direction value of zz gives a disturbance that is most dicult to reject consider both RHPzeros
Exercise Analyze the inputoutput controllability for Gs s s s s where consider two casesa and b Remark This is a simple twomixingtank model of a heat exchanger where u hTin Tin i y hTout Tout i and is the number of heat transfer

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Multivariable Feedback Control Analysis And Design 2nd Edition Sigurd Skogestad, Ian Postlethwaite - Solutions

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