Page 2 - Support; Worldwide Technical Support and Product Information; Technical Support and Professional Services
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Page 3 - Important Information; Warranty; Patents; WARNING REGARDING USE OF NATIONAL INSTRUMENTS PRODUCTS
Important Information Warranty The media on which you receive National Instruments software are warranted not to fail to execute programming instructions, due to defects in materials and workmanship, for a period of 90 days from date of shipment, as evidenced by receipts or other documentation. Nati...
Page 4 - Conventions; response; File»Page Setup»Options; italic; monospace; monospace bold
Conventions The following conventions are used in this manual: [ ] Square brackets enclose optional items—for example, [ response ]. » The » symbol leads you through nested menu items and dialog box options to a final action. The sequence File»Page Setup»Options directs you to pull down the File men...
Page 5 - Contents; Stability Margin and Structured Singular Values (
© National Instruments Corporation v MATRIXx Xmath Robust Control Module Contents Chapter 1Introduction Using This Manual......................................................................................................... 1-1 Document Organization...................................................
Page 6 - Index
Contents MATRIXx Xmath Robust Control Module vi ni.com Chapter 3System Evaluation Singular Value Bode Plots............................................................................................. 3-1L Infinity Norm (linfnorm).........................................................................
Page 7 - Using This Manual; Document Organization; Introduction
© National Instruments Corporation 1-1 MATRIXx Xmath Robust Control Module 1 Introduction The Xmath Robust Control Module (RCM) provides 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 no...
Page 8 - Bibliographic References; Bibliography; Commonly-Used Nomenclature
Chapter 1 Introduction MATRIXx Xmath Robust Control Module 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, bibliographic references are cited with bracketed entries. ...
Page 9 - MATRIXx Help; includes all Robust Control functions.; Overview; RCM functionality is structured as shown in Figure 1-1.
Chapter 1 Introduction © National Instruments Corporation 1-3 MATRIXx Xmath Robust Control Module • Xmath Optimization Module • Xmath Robust Control Module • Xmath X μ Module MATRIXx Help Robust Control Module function reference information is available in the MATRIXx Help . The MATRIXx Help include...
Page 10 - RCM Function Structure; and frequency shaping) while adding new
Chapter 1 Introduction MATRIXx Xmath Robust Control Module 1-4 ni.com Figure 1-1. RCM Function Structure Many RCM functions are based on state-of-the-art algorithms implemented in cooperation with researchers at Stanford University. The robustness analysis functions are based on structured singular ...
Page 11 - Robustness Analysis; Modeling Uncertain Systems; known; Model of an Uncertain System; structured nonparametric uncertainties
© National Instruments Corporation 2-1 MATRIXx Xmath Robust Control Module 2 Robustness Analysis This chapter describes RCM tools used for analyzing the robustness of a closed-loop system. The chapter assumes that a controller has been designed for a nominal plant and that the closed-loop performanc...
Page 12 - feedback connection; delb; d i a g o n a l; DELB
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 2-2 ni.com system, including how the uncertain transfer functions are connected to the system and the magnitude bound functions l i ( w ). To do this, extract the uncertain transfer functions and collect them into a k -input, k -outpu...
Page 13 - robustly stable; ωΔ ≠; Note
Chapter 2 Robustness Analysis © National Instruments Corporation 2-3 MATRIXx Xmath Robust Control Module Stability Margin (smargin) Assume that the nominal closed-loop system is stable. That belief raises a question: Does the system remain stable for all possible uncertain transfer functions that sa...
Page 14 - Nominal Closed-Loop System; Creating a Nominal System
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 2-4 ni.com smargin( ) marg = smargin(SysH, delb {scaling, graph}) The smargin( ) function plots an approximation to the stability margin of the system as a function of frequency. For a full discussion of smargin( ) syntax, refer to th...
Page 15 - SISO Tracking System with Three Uncertainties
Chapter 2 Robustness Analysis © National Instruments Corporation 2-5 MATRIXx Xmath Robust Control Module Figure 2-3. SISO Tracking System with Three Uncertainties The H system will have the reference input as input1 and the error output as output1 ( w and z , respectively, in Figure 2-2). Removing t...
Page 16 - Bound for Sensor Uncertainty
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 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 ...
Page 17 - Stability Margin; Now examine the effect on the stability margin of discretizing
Chapter 2 Robustness Analysis © National Instruments Corporation 2-7 MATRIXx Xmath Robust Control Module Figure 2-5. Stability Margin Now examine the effect on the stability margin of discretizing H ( s ) at 100 Hz. dt = 0.01; Hd = discretize(H,dt); margD = smargin(Hd,delb); smargin --> Scaling a...
Page 18 - nominal transfer function
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 2-8 ni.com Worst-Case Performance Degradation (wcbode) Even if a system is robustly stable, the uncertain transfer functions still can have a great effect on performance. Consider the transfer function from the q th input, w q , to th...
Page 19 - Using wcbode( ) to Analyze Performance Degradation
Chapter 2 Robustness Analysis © National Instruments Corporation 2-9 MATRIXx Xmath Robust Control Module wcbode( ) [WCMAG, NOMMAG] = wcbode (SysH, delb, {input, output, graph}) The wcbode( ) function computes and plots the worst-case gain of a closed-loop transfer function. This function is useful f...
Page 20 - Performance Degradation of the SISO Tracking System; Advanced Topics; then you can express the margin at frequency; d i a g o n a l l
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 2-10 ni.com Figure 2-6. Performance Degradation of the SISO Tracking System Advanced Topics This section describes the theoretical background on robustness analysis and performance degradation. Stability Margin This section discusses ...
Page 21 - structured singular value; Stability Margin Bounds Using Singular Values; Use of Scaling Example; margin
Chapter 2 Robustness Analysis © National Instruments Corporation 2-11 MATRIXx Xmath Robust Control Module for all diagonal Δ such that where μ ( . ) is the structured singular value , introduced by Doyle in [Doy82]. Thus, the margin is the inverse of the structured singular value of H qr diagonally ...
Page 22 - You can compare this margin to that of the example in the; Approximation with Scaled Singular Values; , an assumption that can be made without loss; D M D
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 2-12 ni.com You can compare this margin to that of the example in the Creating a Nominal System section; the following inputs produce Figure 2-7. plot ([marg,margSVD],{xlog} legend=["PF_SCALE","SVD"], ylab="Stabili...
Page 23 - so you have the following from Equation 2-5:
Chapter 2 Robustness Analysis © National Instruments Corporation 2-13 MATRIXx Xmath Robust Control Module of generality—so, roughly speaking, it can be solved. [SD83,SD84] discusses this optimization problem. Notice that: so you have the following from Equation 2-5: This inequality is thought to be ...
Page 24 - Comparing Scaling Algorithms; Results of Scaling Algorithm Options
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 2-14 ni.com Comparing Scaling Algorithms Using the system from the first example (Figure 2-3), you can compare the results of using the three scaling algorithms: MARG_PF=smargin(H,delb,{scaling="PF",!graph}); MARG_OS=smargin(H...
Page 25 - Osborne
Chapter 2 Robustness Analysis © National Instruments Corporation 2-15 MATRIXx Xmath Robust Control Module ssv( ) [v,vD] = SSV(M, {scaling}) The ssv( ) function computes an approximation (and guaranteed upper bound) to the Scaled Singular Value of a complex square matrix M , where M can be a reducibl...
Page 26 - M D; M p
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 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 Algorithm. A diagonal sca...
Page 27 - Reducibility; Non-Interacting Uncertain Transfer Functions
Chapter 2 Robustness Analysis © National Instruments Corporation 2-17 MATRIXx Xmath Robust Control Module optscale( ) [v, vOPTD] = optscale (M, {tol}) The optscale( ) function optimally scales a matrix. An iterative optimization (ellipsoid) algorithm which calculates upper and lower bounds on the le...
Page 28 - Reduction to Separate Systems; Conversion to a Stability Margin Problem; performance loop
Chapter 2 Robustness Analysis MATRIXx Xmath Robust Control Module 2-18 ni.com Figure 2-10. Reduction to Separate Systems In terms of the approximations to the margin discussed above, this reducibility will manifest itself as a problem such as divide-by-zero or nontermination. It really means that th...
Page 29 - Xmath Help
Chapter 2 Robustness Analysis © National Instruments Corporation 2-19 MATRIXx Xmath Robust Control Module Using this relation and any of the previously discussed approximations for μ ( . ), you can compute an approximation to wcgain( ) . Because the approximations to μ ( . ) are upper bounds, the re...
Page 30 - System Evaluation; Singular Value Bode Plots
© National Instruments Corporation 3-1 MATRIXx Xmath Robust Control Module 3 System Evaluation This chapter describes system analysis functions that create singular value Bode plots, performance plots, and calculate the L ∞ norm of a linear system. Singular Value Bode Plots The singular value Bode p...
Page 32 - Singular Value Plot; s u p
Chapter 3 System Evaluation © National Instruments Corporation 3-3 MATRIXx Xmath Robust Control Module Figure 3-1. Singular Value Plot L Infinity Norm (linfnorm) The L ∞ norm of a stable transfer matrix H is defined as: where is the maximum singular value and H ( j ω ) is the transfer matrix under c...
Page 33 - Sys; tol; H j
Chapter 3 System Evaluation MATRIXx Xmath Robust Control Module 3-4 ni.com factor by which the RMS value of a signal flowing through H can be increased. By comparison, the H 2 norm is defined as: This norm can be interpreted as the RMS value of the output when the input is unit intensity white noise...
Page 34 - sigma
Chapter 3 System Evaluation © National Instruments Corporation 3-5 MATRIXx Xmath Robust Control Module • If A has an imaginary eigenvalue at j ω 0 , linfnorm( ) returns: vOMEGA = SIGMA = Infinity where ω 0 is one of the imaginary eigenvalues of A . • Even if H is unstable, linfnorm( ) returns its ma...
Page 35 - Singular Values of; The
Chapter 3 System Evaluation MATRIXx Xmath Robust Control Module 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...
Page 36 - Singular Value Bode Plots of Subsystems; Typical System with Plant and Controller
Chapter 3 System Evaluation © National Instruments Corporation 3-7 MATRIXx Xmath Robust Control Module Singular Value Bode Plots of Subsystems To evaluate the performance achieved by a given controller rapidly, it is useful to check four basic maximum singular value plots—for example, the transfer m...
Page 37 - regulation
Chapter 3 System Evaluation MATRIXx Xmath Robust Control Module 3-8 ni.com The four transfer matrices are labeled e / d , e / n , u / d , and u / n in the final plot. The plots in the top row, consisting of e / d and e / n , show the regulation or tracking achieved by the controller. If both these q...
Page 38 - System is continuous; endfor
Chapter 3 System Evaluation © National Instruments Corporation 3-9 MATRIXx Xmath Robust Control Module The system matrix can be calculated using the afeedback( ) function for different values of K . Consider two cases: K = 1 and K = 5 . P = 1/makepoly([1,0],"s") P (a transfer function) = 1 -...
Page 39 - SysCL; closed-loop system from; Closed Loop System from w to z
Chapter 3 System Evaluation MATRIXx Xmath Robust Control Module 3-10 ni.com Figure 3-5. Perfplots( ) for K = 1 and K = 5 clsys( ) SysCL = clsys( Sys, SysC ) The clsys( ) function computes the state-space realization SysCL , of the closed-loop system from w to z as shown in Figure 3-6. Figure 3-6. Cl...
Page 40 - Calculation of the Closed Loop System (SysCL); is; nw; nz
Chapter 3 System Evaluation © National Instruments Corporation 3-11 MATRIXx Xmath Robust Control Module 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 the above, SysCL is calculated as shown in Figure 3-7. Figure 3-7. Calculat...
Page 41 - Ill-Posed Feedback System; Example of Closed-Loop System
Chapter 3 System Evaluation MATRIXx Xmath Robust Control Module 3-12 ni.com Figure 3-8. Ill-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...
Page 42 - Controller Synthesis; H-Infinity Control Synthesis; Problem Definition; Closed-Loop System with Plant P and Controller K
© National Instruments Corporation 4-1 MATRIXx Xmath Robust Control Module 4 Controller Synthesis This chapter discusses synthesis 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 ...
Page 43 - Restrictions on the Extended Plant; Extended Transfer Matrix; extended
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 4-2 ni.com The function 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...
Page 44 - Building the Plant Model; Construction of Plant; P s
Chapter 4 Controller Synthesis © National Instruments Corporation 4-3 MATRIXx Xmath Robust Control Module Equivalently, as a transfer matrix: To enter the extended system, you must know the sizes of e and w shown in Figure 4-1. The extended plant P can be constructed using the Xmath interconnection ...
Page 45 - Typical Plant Configuration
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 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 effects in the performance signals z and measured signals y . The weight W in is used to mode...
Page 46 - Weight Selection; A x B
Chapter 4 Controller Synthesis © National Instruments Corporation 4-5 MATRIXx Xmath Robust Control Module here the weighting matrices are transfer 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 vari...
Page 47 - j w
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 4-6 ni.com Selecting these weights has much the same effect here. Specifically, let H zv be the closed-loop transfer matrix (with u = K γ ) from inputs: to outputs: Thus, Suppose that the controller u = K y approximates Equation 4-2....
Page 48 - GW
Chapter 4 Controller Synthesis © National Instruments Corporation 4-7 MATRIXx Xmath Robust Control 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 rather are...
Page 49 - hinfcontr –>No stabilizing controller meets the spec!
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 4-8 ni.com • For all ω ≥ 0, • Condition 1 is a standard condition to ensure the existence of a stabilizing controller. Condition 2 ensures that the control signal u is contained in the normalized error vector e (refer to Figure 4-3)....
Page 50 - gamma
Chapter 4 Controller Synthesis © National Instruments Corporation 4-9 MATRIXx Xmath Robust Control Module If no error message occurs, then is guaranteed. However, this does not preclude the possibility that either or that . For the former case, there are two checks: • Use the linfnorm( ) function to...
Page 52 - The result is that on this first iteration:; perfplots
Chapter 4 Controller Synthesis © National Instruments Corporation 4-11 MATRIXx Xmath Robust Control Module 4. For this example, 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 ...
Page 53 - Perfplots for; It also is useful to perform; , which in this case is the closed-loop transfer
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 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 , which in this case is the closed-loop transfer matrix from ( d , n ) into ( x , u ). The following func...
Page 54 - PRP
Chapter 4 Controller Synthesis © National Instruments Corporation 4-13 MATRIXx Xmath Robust Control Module Figure 4-6. Perfplots for H zv singriccati( ) [P, solstat] = singriccati(A,Q,R {method}) The singriccati( ) function solves the Indefinite Algebraic Riccati Equation (ARE): The ARE is solved by...
Page 55 - Linear-Quadratic-Gaussian Control Synthesis; LQG Frequency Shaping; A x B u
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 4-14 ni.com Linear-Quadratic-Gaussian Control Synthesis The H 2 Linear-Quadratic-Gaussian (LQG) control design methods are based on minimizing a quadratic function of state variables and control inputs. Conventionally, the problem is...
Page 58 - Frequency-Shaped Control Design Commands; Create a reduced order system by selecting only the first mode:
Chapter 4 Controller Synthesis © National Instruments Corporation 4-17 MATRIXx Xmath Robust Control Module fslqgcomp( ) [SysCC, vEV] = fslqgcomp(SysF, SysC) The fslqgcomp( ) function combines filter and control law to compute a controller from a control law and an estimator. For more information on ...
Page 59 - Try the LQG compensator with the full-order system:
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 4-18 ni.com -0.500025 + 0.866011 j -0.500025 - 0.866011 j 5. Try the LQG compensator with the full-order system: [Syscl_fo]=feedback(Sys,Sysc); poles(Syscl_fo) ans (a column vector) = -0.401519 + 0.864869 j -0.401519 - 0.864869 j -0....
Page 60 - RUU
Chapter 4 Controller Synthesis © National Instruments Corporation 4-19 MATRIXx Xmath Robust Control Module 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. Transfer the weight on U from RUU to the third diagonal entry in RXXA . Note In E...
Page 61 - Frequency-Shaped Compensator
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 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);qwwa=qxx;qvva...
Page 62 - Design the LQG compensator.
Chapter 4 Controller Synthesis © National Instruments Corporation 4-21 MATRIXx Xmath Robust Control Module 9. Design the LQG compensator. [Sysfs_sc,fs_evc]=fslqgcomp(Sysfs_se,Sysfs_sr) Sysfs_sc (a state space system) = 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...
Page 63 - Compute the closed-loop system for the full-order plant and the; rho; ) can be manipulated by the user to obtain loop; lqrltr
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 4-22 ni.com 10. Compute the closed-loop system for the reduced order plant and the frequency-shaped compensator: [Sysfs_scl]=feedback(Sysr,Sysfs_sc); poles(Sysfs_scl) ans (a column vector) = -0.645263 + 0.587929 j -0.645263 - 0.58792...
Page 64 - LQG Feedback System for Loop Transfer Recovery; recovers; Xmath; recover; A x w
Chapter 4 Controller Synthesis © National Instruments Corporation 4-23 MATRIXx Xmath Robust Control Module Figure 4-8. LQG Feedback System for Loop Transfer Recovery lqgltr( ) [SysC,EV,Kr] = lqgltr(Sys,Wx,Wy,K,rho,{keywords}) The lqgltr( ) function designs an estimator or regulator which recovers lo...
Page 65 - graph
Chapter 4 Controller Synthesis MATRIXx Xmath Robust Control Module 4-24 ni.com Then ρ is increased so that pointwise in s : Regulator recovery 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...
Page 66 - norm of a transfer matrix and related problems.”
© National Instruments Corporation A-1 MATRIXx Xmath Robust Control Module A Bibliography [BBK88] S. Boyd, V. Balakrishnan, and P. Kabamba. “A bisection method for computing the L ∞ norm of a transfer matrix and related problems.” Mathematical Control Signals, Systems Vol. 2, No. 3, pp 207–219, 1989...
Page 67 - Control
Appendix A Bibliography MATRIXx Xmath Robust Control Module A-2 ni.com [FaT88] M.K. Fan and A.L. Tits. “m-form Numerical Range and the Computation of the Structured Singular Value.” IEEE Transactions on Automatic Control , Vol. 33, pp 284 –289, March 1988. [FaT86] M.K. Fan and A.L. Tits. “Characteri...
Page 68 - IEEE Transactions on Automatic Control
Appendix A Bibliography © National Instruments Corporation A-3 MATRIXx Xmath Robust Control Module [SA88] G. Stein and M. Athans. “The LQG/LTR Procedure for Multivariable Control Design.” IEEE Transactions on Automatic Control , Vol. AC-32, No. 2, pp 105–114, February 1987. [Za81] G. Zames. “Feedbac...
Page 70 - iv
© National Instruments Corporation I-1 MATRIXx Xmath Robust Control Module Index A Algebraic Riccati Equation (ARE), 4-13 C clsys( ), 3-10conventions used in the manual, iv D diagnostic tools (NI resources), B-1documentation conventions used in the manual, iv NI resources, B-1 drivers (NI resources)...
