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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.
© National Instruments Corporation v MATRIXx Xmath Robust Control Module Contents Chapter 1 Introduction Using This Manual...................... ......................... ....................... ......................... .......... 1-1 Document Organization.
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) ............
© 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.
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.
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 .
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.
© 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.
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 ).
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 .
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 .
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).
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.
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.
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 .
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 ( .
© 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.
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")/.
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.
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.
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 .
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) .
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 cont.
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 .
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 .
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.
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.
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.
© 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.
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.
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.
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 .
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.
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.
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]).
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).
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 .
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.
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.
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 ).
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}).
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.
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.
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 .
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 .
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.
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 .
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.
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.
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.
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] = lqgl.
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 .
© 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.
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.
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.
© 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.
© 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 .
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 e.
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