National Instruments NI MATRIX Xmath Robust Control Module manual

NI MA TRIXx TM Xmath TM Robust Control Module MA TRIXx Xmath Robust Control M odule April 2007 370757C-01[...]

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Conventions The follo wing conv entions are used in this manual: [ ] Square brackets enclose op tional it ems—for example, [ response ]. » The » symbol leads you th rough nested menu items and dial og box options to a final action. The sequence File»Page Setup»Options di rects you t o pull down the File menu, select the Page Setup item, and s[...]

© National Instruments Corporation v MATRIXx Xmath Robust Control Module Contents Chapter 1 Introduction Using This Manual...................... ......................... ....................... ......................... .......... 1-1 Document Organization........... ... ...................... .......................... ..................... 1-1 [...]

Contents MATRIXx Xmath R obust Control Mo dule vi ni.com Chapter 3 System Evaluation Singular Value Bode Plots.............. ...................... .......................... ...................... ......... 3-1 L Infinity Norm (linfnorm) ............... ....................... ......................... ....................... ...... 3-3 linfnorm( [...]

© National Instruments Corporation 1-1 MA TRIXx Xmath Robust Co ntrol Module 1 Introduction The Xmath Robust Control Module (RCM) provi des a collection of analysis and synthesis tools that assist in the design of robust control systems. This chapter starts with an outline of the manual and some use notes. It continues with an overvie w of the Xma[...]

Chapter 1 Introduction MA TRIXx Xmath Robust Control Modul e 1-2 ni.com techniques. The general problem setup is explained together with known limitations; the rest is left to the references. Bibliographic References Throughout this document, biblio graphic references are cited with bracketed entries. For example, a reference to [DoS81] corresponds[...]

Chapter 1 Introduction © National Instruments Corporation 1-3 MA TRIXx Xmath Robust Co ntrol Module • Xmath Optimization Module • Xmath Robust Control Module • Xmath X μ Module MA TRIXx Help Robust Control Modul e function reference informatio n is available in the MATRIXx Help . The MATRIXx Help includes all Robust Control functions. Each [...]

Chapter 1 Introduction MA TRIXx Xmath Robust Control Modul e 1-4 ni.com Figure 1-1. RCM Function Structure Many RCM functions are based on stat e-of-the-art algorithms impl emented in cooperation with researchers at Stanford Uni versity . The robustness analysis functions are based on struct ured singular v alue calculations. The synthesis tools ex[...]

© National Instruments Corporation 2-1 MA TRIXx Xmath Robust Co ntrol Module 2 Robustness Analysis This chapter describes RCM tools used for analyzing the robustness of a closed-loop system. The chapter a ssumes that a controller has been designed for a nominal plant and that the closed-loop performance of this nominal system is acceptable. The go[...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-2 ni.com system, including how the un certain transfer functions are connected to the system and the magnitude bound fun ctions l i ( w ). T o do this, extract the uncertain transf er functions and co llect them into a k -input, k -outp ut transfer matrix Δ , where: (2-2) The re[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-3 MA TRIXx Xmath Robust Co ntrol Module Stability Margin (smargin) Assume that the nominal cl osed-loop system is stable . That belief raises a question: Does the system remain stab le for all possible uncertain transfer functions that satisfy the magnitude bou nds (Equation 2-1)? [...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-4 ni.com smargin( ) marg = smargin(SysH, delb {s caling, graph}) The smargin( ) function plots an approximatio n to the stability margin of the system as a function of frequency . For a full discussion of smargin( ) syntax, refer to the MATRIXx Help . The approximation is exact i[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-5 MA TRIXx Xmath Robust Co ntrol Module Figure 2-3. SI SO T racking Syst em with Three Uncertaintie s The H system will ha ve the reference i nput as input1 and the error outpu t as output1 ( w and z , respectiv ely , in Figure 2-2). Removing the δ va l u e s w i l l create inputs[...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-6 ni.com Figure 2-4. Bound for Sensor Uncertainty Note A value of l 3 at one radian per second of –20 dB indicates that modeling uncertainties of up to 10% (–20 dB = 0.1) are allowed. The actuator and sensor uncertainties δ 1 and δ 2 are bound ed by –20 dB at all frequenc[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-7 MA TRIXx Xmath Robust Co ntrol Module Figure 2-5. Stability Margin Now e xamine the effect on the st ability margin of discretizing H ( s ) at 100 Hz. dt = 0.01; Hd = discretize(H,dt); margD = smargin(Hd,delb); smargin --> Scaling algorith m is type: PF smargin --> Margin c[...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-8 ni.com W orst-Case Performance Degradation (wcbode) Even if a system is robustly stable, th e uncertain transfer functions still can have a great effect on performance. Co nsider the transfer function from the q th input, w q , to th e p th output, z p . With δ 1 = ... = ... ?[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-9 MA TRIXx Xmath Robust Co ntrol Module wcbode( ) [WCMAG, NOMMAG] = wcbode (Sy sH, delb, {input, output, graph}) The wcbode( ) function computes and plots the worst-case gain of a closed-loop transfer function. This function is useful for checking a system that already has been ver[...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-10 ni.com Figure 2-6. Performance Degradation of the SISO T rac king System Advanced T opics This section describes the theoretical background on robust ness analysis and performance degradation. Stability Margin This section discusses advanced aspect s of computing the stability[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-11 MA TRIXx Xmath Robust Control Modu le for all diagonal Δ such that where μ ( . ) is the structured singular value , introduced by Do yle in [Doy82]. Thus, the margin is the in verse of the structured singular value of H qr diagonally scaled by the magnitude bo unds. There is n[...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-12 ni.com Y ou can compare this mar g in to that of the example in the Creating a Nominal Syst em section; the following inputs produce Figure 2-7. plot ([marg,margSVD],{xlog} legend=["PF_SCALE","SVD"], ylab="Stability Margin,dB", xlab="Frequenc[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-13 MA TRIXx Xmath Robust Control Modu le of generality—so, roughly speaking, it can be solved. [SD83,SD84] discusses this optimization problem. Notice that: so you hav e the fol lowing from Equation 2-5: This inequality is tho ught to be nearly an equ ality , so that the left sid[...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-14 ni.com Comparing Scaling Algorithms Using the system from the first ex ample (Figure 2-3), you can compare the results of using the three scal ing algorithms: MARG_PF=smargin(H,delb,{scal ing="PF",!graph}); MARG_OS=smargin(H,delb,{scal ing="OS",!graph}); MA[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-15 MA TRIXx Xmath Robust Control Modu le ssv( ) [v,vD] = SSV(M, {scaling}) The ssv( ) function computes an approx imation (and gu aranteed upper bound) to the Scaled Singular V alue of a complex square matrix M , where M can be a reducible matrix. The scaled singular value v ( M ) [...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-16 ni.com VOPT=ssv(M,{scaling="OPT"}) VOPT (a scalar) = 2.43952 VSVD = max(svd(M)) VSVD (a scalar) = 2.65886 osscale( ) [v, vD] = osscale(M) The osscale( ) function scales a matrix using the Osborne Algorit hm. A diagonal scaling D OS is found that min imizes the Froben[...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-17 MA TRIXx Xmath Robust Control Modu le optscale( ) [v, vOPTD] = optscale (M, {t ol}) The optscale( ) function optimally scales a matri x. An iterativ e optimization (ellipsoid) algorithm which calculates upper and lower bounds on the left side of Equation 2-5 is used. If these bo[...]

Chapter 2 Robustness Analysis MA TRIXx Xmath Robust Control Modul e 2-18 ni.com Figure 2-10. Reduction to Separat e Systems In terms of the approximations to th e margin discussed abov e, this reducibility will manifest itself as a pro blem such as di vide-by-zero or nontermination. I t really means that the minimum of the optim ization problem is [...]

Chapter 2 Robustness Analysis © National Instruments Corporation 2-19 MA TRIXx Xmath Robust Control Modu le Using this relation and any of the previously discussed appro ximations for μ ( . ), you can compute an approximation to wcgain( ) . Because the approximations to μ ( . ) are up per bounds, the resulting approxim ations to wcgain( ) also a[...]

© National Instruments Corporation 3-1 MA TRIXx Xmath Robust Co ntrol Module 3 System Evaluation This chapter describes system analysis functions that create singu lar value Bode plots, performance plots, and calculate the L ∞ norm of a linear system. Singular V alue Bode Plots The singular value Bode plot is a MIMO general ization of the bode( [...]

Chapter 3 System E valuation MA TRIXx Xmath Robust Control Modul e 3-2 ni.com Refer to [BoB91 ] in Appendix A, Bibliography . Example 3-1 Creating a Singular Value Plot 1. Let a system H be a 2-input/2-output system: tf=makepoly([1,2],"s")/... polynomial([0,-2.334,-12],"s ") tf (a transfer function) = s + 2 -------------------- [...]

Chapter 3 System Ev aluation © National Instruments Corporation 3-3 MA TRIXx Xmath Robust Co ntrol Module Figure 3-1. Sing ular Value Plot L Infinity Norm (linfnorm) The L ∞ norm of a stable transfer matrix H i s defined as: where is the maximu m singular value and H ( j ω ) is the transfer matr ix under consideration. The L ∞ norm of a stabl[...]

Chapter 3 System E valuation MA TRIXx Xmath Robust Control Modul e 3-4 ni.com factor b y wh ich the RMS value of a signal flo wing th rough H can be increased. By comparison, the H 2 norm is defined as: This norm can be interpreted as th e RMS value of the output when the input is unit intensity whit e noise. It can be compu ted in Xmath using the [...]

Chapter 3 System Ev aluation © National Instruments Corporation 3-5 MA TRIXx Xmath Robust Co ntrol Module •I f A has an imaginary eigen value at j ω 0 , linfno rm( ) retu rns: vOMEGA = SIGMA = Infinity where ω 0 is one of the imaginary eigen values of A . •E v e n i f H is unstable, linfnorm( ) returns i ts maximum sing ular v alue on the j [...]

Chapter 3 System E valuation MA TRIXx Xmath Robust Control Modul e 3-6 ni.com Figure 3-2. Singular Values of H ( j ω) as a Function of ω Note sv is returned in dBs. Check that sigma is within 0.01 (the default value of tol ) of 10**(max(sv,{channels})/20) . [sigma,10^(max(sv,{channels} )/20)] ans (a row vector) = 5.07322 4.98731 The linfnorm( ) f[...]

Chapter 3 System Ev aluation © National Instruments Corporation 3-7 MA TRIXx Xmath Robust Co ntrol Module Singular V alue Bode Plots of Subsystems To evaluate the performance achieved by a given controller rapidly, it is useful to check four basic maximum si ngular value plots—for example, the transfer matrices from process and se nsor noises to[...]

Chapter 3 System E valuation MA TRIXx Xmath Robust Control Modul e 3-8 ni.com The four transfer matrices are labeled e / d , e / n , u / d , and u / n in the f inal plot. The plots in the top ro w , consisting of e / d and e / n , show the regulation or tracking achie ved by the controller . If both these quantities are small, then the disturbance [...]

Chapter 3 System Ev aluation © National Instruments Corporation 3-9 MA TRIXx Xmath Robust Co ntrol Module The system matrix can be calculated using the afeedback( ) function for different v alues of K . Consider two cases: K=1 and K=5 . P = 1/makepoly([1,0],"s") P (a transfer function) = 1 -- s System is continuous K1= 1/makepoly(1,"[...]

Chapter 3 System E valuation MA TRIXx Xmath Robust Control Modul e 3-10 ni.com Figure 3-5. Per fplots( ) for K = 1 and K =5 clsys( ) SysCL = clsys( Sys, SysC ) The clsys( ) function computes th e state-space realization SysCL , of the closed-loop system from w to z as sho wn in Figu re 3-6. Figure 3-6. Closed Loop System from w to z Sys w u y z Sys[...]

Chapter 3 System Ev aluation © National Instruments Corporation 3-11 MA TRIXx Xmath Robust Control Modu le Where SysC=system(Ac,Bc,Cc,Dc ) , Sys=system(A,B,C,D) , and nz is the dimension of z and nw is the dimension of w : Given th e ab ove, SysCL is calculated as shown in Figure 3-7. Figure 3-7. Calculation of the Clos ed Loop System (SysCL) The [...]

Chapter 3 System E valuation MA TRIXx Xmath Robust Control Modul e 3-12 ni.com Figure 3-8. I ll-Posed Feedback System Example 3-4 Example of Closed-Loop System a = 1; b = [1,0,1]; c = b'; d = [0,0,0;0,0,1;0,1,0]; Sys = SYSTEM(a,b,c,d); SysC = SYSTEM(-40,2.7,-40,0) ; SysCL = clsys(Sys,SysC) SysCL (a state space system) = A 1 -40 2.7 -40 B 1 0 0[...]

© National Instruments Corporation 4-1 MA TRIXx Xmath Robust Co ntrol Module 4 Controller Synthesis This chapter discusses synthe sis tools in two categories, H ∞ and H 2 . This chapter does not explain all of the theory of H ∞ , LQG/LTR, and frequency shaped LQG design techniques. The general problem setup is explained together with k nown li[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-2 ni.com The functio n hinfcontr( ) can be used to find an optimal H ∞ controller K that is arbitrarily close to solving: (4-2) The hinfcontr( ) function description in the hinfcontr( ) section describes how the optimum can be found manually by decreasing γ until an error co[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-3 MA TRIXx Xmath Robust Co ntrol Module Equi valently , as a transfer matrix: T o enter the extended system , you must kno w the sizes of e and w shown in Figure 4-1. The extended plant P can be cons tructed using the Xmath interconnection functio ns, as shown in Example 4-1. Buil[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-4 ni.com The transfer matrix G can be viewed as a model of the underlying system dynamics with v and u as generalized forces that produce ef fects in the performance signals z and measured signals y . The weight W in is used to model the exogenous input v by v = W in w . Simila[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-5 MA TRIXx Xmath Robust Co ntrol Module here the weighting matrices are tran sfer matrices, whereas in the LQG setup they are constants. A description of the plant in Figure 4-3 is as follows: • Dynamical system G dyn : • Measured v ariables y = y sens + n : • Input weight W[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-6 ni.com Selecting these weigh ts has much the same effect he re. Specif ically , let H zv be the closed-loop transfer matr ix (with u = K γ ) from inputs: to outputs: Thus, Suppose that the controller u = K y approximates Eq uation 4-2. Thus, In many ca ses, this means that t[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-7 MA TRIXx Xmath Robust Co ntrol Module where and The weights also can be viewed as “design knobs” (for example, [ONR84]). In this view , the weights are not directly related to specific disturbance or performance models but ra ther are used as a vehicle to obtain a closed-loo[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-8 ni.com •F o r a l l ω ≥ 0, • Condition 1 is a standard cond ition to ensure the existence of a stabilizing controller . Condition 2 en sures that the control signal u is contained in the normalized error v ector e (refer to Figure 4-3). Conv ersely , conditio n 3 ensur[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-9 MA TRIXx Xmath Robust Co ntrol Module If no error message occurs, then is guaranteed. Ho wever , this does not preclude the po ssibility that either or that . For the former c ase, there are two checks: •U s e t h e linfnorm( ) function to compute . • Compute the grap h vers[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-10 ni.com Suppose the i nput/output we ights are as follows: 2. Create the four weights: Wdist = 1/makepoly([1,1],"s" ) Wdist (a transfer function) = 1 ----- s + 1 Wnoise = 0.1; Wreg=1/makepoly(1,"s"); Wact = 0.1; 3. Combine the weights in W in and W out (re[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-11 MA TRIXx Xmath Robust Control Modu le 4. For this e xample, you will start with gamma=1 as the initial guess and enter: [K,Hew] = hinfcontr(P,1,2,2) ; No error messages are reported. This means that a stabilizing controller has been found such that Equation 4-1 holds. That is, [...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-12 ni.com Figure 4-5. Perfplots for H ew It also is useful to perform perfplots( ) on the unweighted closed-loop system, H zv , wh ich in this case is the closed-loop transfer matrix fr om ( d , n ) into ( x , u ). The following function calls produce Figure 4-6: Hzv=clsys(G,K)[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-13 MA TRIXx Xmath Robust Control Modu le Figure 4-6. Per fplots for H zv singriccati( ) [P, solstat] = singriccati(A ,Q,R {method}) The singriccati( ) function solv es the Indefinite Algebraic Riccati Equation (ARE): The ARE is solv ed by deco mposing the Hamiltonian: The required[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-14 ni.com Linear -Quadratic-Gaussian Control Synthesis The H 2 Linear-Quadratic-Gaussi an (LQG) control design methods are based on minimizing a quadratic functi on of stat e variables and control inputs. Conventionally, the prob lem is specified in the time domain. By converti[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-15 MA TRIXx Xmath Robust Control Modu le This expression can be con verted into the following form [Gu80]: If R ( j ω ) is not a funct ion of frequency , then C 12 = 0 and D = I . Note The system has a new input v and the old input u is now the output of the system. This structur[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-16 ni.com fsesti( ) [SysF, vEV] = fsesti(SysA, n s, QWWA, QVVA, {QWVA}) The fsesti( ) function computes a freque ncy-shaped state estimator . The estimatio n problem is stated as follo ws: The frequen cy-shaped f ilter design pr oblem is to minimize, which can be written as: or[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-17 MA TRIXx Xmath Robust Control Modu le fslqgcomp( ) [SysCC, vEV] = fslqgcomp(Sys F, SysC) The fslqgcomp( ) function combines f ilter an d control law to compute a controller from a control law and an estimator . For more info rmation on the fslqgcomp( ) syntax, refer to the Xmat[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-18 ni.com -0.500025 + 0.866011 j -0.500025 - 0.866011 j 5. T ry t he LQG compensator w ith the full-or der system: [Syscl_fo]=feedback(Sys,Sysc ); poles(Syscl_fo) ans (a column vector) = -0.401519 + 0.864869 j -0.401519 - 0.864869 j -0.638796 + 0.855861 j -0.638796 - 0.855861 j[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-19 MA TRIXx Xmath Robust Control Modu le 0 0 0 1 0 0 0 0 B 0 0 0 1 C 0 0 1 0 D 0 X0 0 0 0 0 System is continuous 7. Frequency-weight the control signal. T ransfer the weight on U from RUU to the third di agonal entry in RXXA . Note In Equation 4-3, u is the third state of the augm[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-20 ni.com System is continuous fs_evr (a column vector) = -0.645263 + 0.587929 j -0.645263 - 0.587929 j -0.347592 + 1.09155 j -0.347592 - 1.09155 j 8. Calculate the frequency-shaped estimator: Sysaf=system(ar,br,cr,0);qww a=qxx;qvva=quu; [Sysfs_se,fs_eve]=fsesti(Sys af,2,qwwa,q[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-21 MA TRIXx Xmath Robust Control Modu le 9. Design the LQG compensator . [Sysfs_sc,fs_evc]=fslqgcomp( Sysfs_se,Sysfs_sr) Sysfs_sc (a state space syst em) = A 0 1 0 0 -1 -1.00005 1 0 0 0 0 1 0.951712 -0.228069 -1.95171 -1.97571 B 5.52357e-17 0.99005 0 0 C 0 0 1 0 D 0 X0 0 0 0 0 Sys[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-22 ni.com 10. Compute the closed-loop system fo r the reduced order plant and the frequency-shaped compensator: [Sysfs_scl]=feedback(Sysr,Sy sfs_sc); poles(Sysfs_scl) ans (a column vector) = -0.645263 + 0.587929 j -0.645263 - 0.587929 j -0.500025 + 0.866011 j -0.500025 - 0.8660[...]

Chapter 4 Controller Synthesis © National Instruments Corporation 4-23 MA TRIXx Xmath Robust Control Modu le Figure 4-8. LQG Feedbac k System for Loop T ransfer Recovery lqgltr( ) [SysC,EV,Kr] = lqgltr(Sys,Wx ,Wy,K,rho,{keywords}) The lqgltr( ) function designs an estimator or regulator which recovers loop transfer robustness through the design pa[...]

Chapter 4 Controller Synthe sis MA TRIXx Xmath Robust Control Modul e 4-24 ni.com Then ρ is increased so that pointwise in s : Regulator reco very is only guaranteed if G ( s ) is minimum-phase and there are at least as many control signals u as measurements y . If recover="estimator" , the loop-transfer is recov ered by designing an est[...]

© National Instruments Corporation A -1 MA TRIXx Xmath Robust Control M odule A Bibliography [BBK88] S. Boyd, V . Balakrishnan , and P . Kaba mba. “ A bisectio n method for com puting the L ∞ norm of a transfer matrix and related problems. ” Mathematical Control Sign als, Systems V ol. 2, No. 3, pp 207–219, 1989. [BeP79] A. Berman and R.J.[...]

Appendix A B ibliograph y MA TRIXx Xmath Robust Control Modul e A-2 ni.com [FaT88] M.K. Fan and A.L. Tits. “m-form Nu merical Range and the Computation of the Structured Singular V alue. ” IEEE Transactions on Auto matic Control , V ol. 33, pp 284 –289, March 198 8. [FaT86] M.K. Fan and A.L. T its. “Character ization and Eff icient Computat[...]

Appendix A B ibliograph y © National Instruments Corporation A -3 MA TRIXx Xmath Robust Control M odule [SA88] G. Stein and M. At hans. “The LQG/ L TR Procedure for Multivariable Control Design. ” IEEE Transacti ons on Automatic Control , V ol. A C-32 , No. 2, pp 105–114 , February 1987. [Za81] G. Zames. “Feedback and optimal sens iti vity[...]

© National Instruments Corporation B -1 MA TRIXx Xmath Robust Control M odule B T echnical Support and Professional Ser vices Visit the following sections of the National Instruments Web site at ni.com for technical support an d professional services: • Support —Online technical support resources at ni.com/support include the following: – Se[...]

© National Instruments Corporation I-1 MA TRIXx Xmath Robust Co ntrol Module Index A Algebraic Riccati Equation (ARE), 4-13 C clsys( ), 3-10 conventions used in the manual, iv D diagnostic tools (NI resources), B-1 documentation conventions used in the manual, iv NI resources, B-1 drivers (NI resources), B-1 E examples (NI resources), B-1 extended[...]

Index MA TRIXx Xmath Robust Control Modul e I-2 ni.com nominal transfer function, 2-8 norm H 2 , 3-4 L ∞ , 3-3 O optscale( ), 2-17 osscale( ), 2-16 P perfplots( ), 3-7 pfscale( ), 2-16 programming examples (NI resources), B-1 R reducibility, 2-17 robust stability, 2-3 S scaled singular values, 2-11 scaling Optimal (Boyd), 2-15 Osborne, 2-15 Perro[...]