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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)...
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