Summary of the content on the page No. 1
TM
NI MATRIXx
TM
Xmath Robust Control Module
MATRIXx Xmath Robust Control Module
April 2007
370757C-01
Summary of the content on the page No. 2
Support Worldwide Technical Support and Product Information ni.com National Instruments Corporate Headquarters 11500 North Mopac Expressway Austin, Texas 78759-3504 USA Tel: 512 683 0100 Worldwide Offices Australia 1800 300 800, Austria 43 662 457990-0, Belgium 32 (0) 2 757 0020, Brazil 55 11 3262 3599, Canada 800 433 3488, China 86 21 5050 9800, Czech Republic 420 224 235 774, Denmark 45 45 76 26 00, Finland 385 (0) 9 725 72511, France 33 (0) 1 48 14 24 24, Germany 49 89 7413130, India 91 80
Summary of the content on the page No. 3
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. National Instruments will, at its option, repair or replace software media that do not execute programming instructions if National Instruments receives notice of such defects during the warranty per
Summary of the content on the page No. 4
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 menu, select the Page Setup item, and select Options from the last dialog box. This icon denotes a note, which alerts you to important information. bold Bold text denotes items that you must select or c
Summary of the content on the page No. 5
Contents Chapter 1 Introduction Using This Manual.........................................................................................................1-1 Document Organization...................................................................................1-1 Bibliographic References ................................................................................1-2 Commonly-Used Nomenclature......................................................................1-2 Related Publications ....
Summary of the content on the page No. 6
Contents Chapter 3 System Evaluation Singular Value Bode Plots............................................................................................. 3-1 L Infinity Norm (linfnorm)............................................................................................ 3-3 linfnorm( ) ....................................................................................................... 3-4 Singular Value Bode Plots of Subsystems .........................................................
Summary of the content on the page No. 7
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 notes. It continues with an overview of the Xmath Robust Control Module (RCM) functions. Using This Manual This manual provides complete documentation for all the RCM functions along with their associated theoretical background, references, and examples. Document Organi
Summary of the content on the page No. 8
Chapter 1 Introduction 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. For example, a reference to [DoS81] corresponds to a document published by Doyle and Stein in 1981. For a table of bibliographic references, refer to Appendix A, Bibliography. Commonly-Used Nomenclature This manual uses the following general n
Summary of the content on the page No. 9
Chapter 1 Introduction � Xmath Optimization Module � Xmath Robust Control Module � Xmath Xμ Module MATRIXx Help Robust Control Module function reference information is available in the MATRIXx Help. The MATRIXx Help includes all Robust Control functions. Each topic explains a function’s inputs, outputs, and keywords in detail. Refer to Chapter 2, MATRIXx Publications, Help, and Online Support, of the MATRIXx Getting Started Guide for complete instructions on using the Help feature. Overview
Summary of the content on the page No. 10
Chapter 1 Introduction Analysis Functions smargin wcbode wcgain ssv pfscale optscale osscale Synthesis Functions hinfcontr lqgltr fslqgcomp fsesti fsregu singriccati clsys Utility Functions linfnorm perfplots 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 value calculations. The synthesis tools expand on existin
Summary of the content on the page No. 11
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 performance of this nominal system is acceptable. The goal of robustness analysis is to determine whether the performance will remain acceptable if the plant differs from the nominal plant. Modeling Uncertain Systems This section describes the method RCM uses to model an uncerta
Summary of the content on the page No. 12
Chapter 2 Robustness Analysis system, including how the uncertain transfer functions are connected to the system and the magnitude bound functions l (w). i To do this, extract the uncertain transfer functions and collect them into a k-input, k-output transfer matrix Δ, where: Δ() jω = diagonal() δ() jω ,...,δ() jω (2-2) 1 k The resulting closed-loop system can be viewed as a feedback connection of the nominal closed-loop system with transfer matrix H(jω) and the uncertain transfer matrix Δ(j
Summary of the content on the page No. 13
Chapter 2 Robustness Analysis 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 satisfy the magnitude bounds (Equation 2-1)? If so, the system is said to be robustly stable. If the magnitude bounds are small enough, the uncertainties will not destabilize the system; your system will be robustly stable. Roughly speaking, the stability margin of your
Summary of the content on the page No. 14
Chapter 2 Robustness Analysis 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 the MATRIXx Help. The approximation is exact if the number of uncertain transfer functions is less than four and scaling="OPT" (optimum scaling). In other cases, the approximation is generally considered to be extremely good. Refer to the Appr
Summary of the content on the page No. 15
+ + + + Chapter 2 Robustness Analysis reference + – error 1 x x + 1 2 + 1 1 8 –– s s + reference 2 1 K = 4 1 K = 8 2 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 the δ values will create inputs 2 through 4 and outputs 2 through 4 (r and q, respectively, in Figure 2-2). 1. The A, B, C, D matrices of the state-space system representing H are as fo
Summary of the content on the page No. 16
Chapter 2 Robustness Analysis 10 0 –20 0.1 30 100 1 Frequency, Radian/Second Figure 2-4. Bound for Sensor Uncertainty Note A value of l at one radian per second of –20 dB indicates that modeling 3 uncertainties of up to 10% (–20 dB = 0.1) are allowed. The actuator and sensor uncertainties δ and δ are bounded by –20 dB 1 2 at all frequencies. You will use these values to interpolate to obtain l . 3 First, create the bound for δ in Hz. 3 L3 = pdm([-20,-20,10,10],[0.1,1,30,100]/2/pi); 3. Now in
Summary of the content on the page No. 17
Chapter 2 Robustness Analysis 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 algorithm is type: PF smargin --> Margin computation 10% complete smargin --> Margin computation 50% complete smargin --> Margin computation 90% complete 100 Hz is a high discretization frequency for H, so the stability margin is unchanged in the discrete-time case. The new plo
Summary of the content on the page No. 18
Chapter 2 Robustness Analysis 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 qth input, w , to the pth output, z . With δ = ... = ...δ = 0, you have the q p 1 k nominal system, and this transfer function is the p,q entry of H . This is zw called the nominal transfer function. When the δ values are not zero, the transfer function from w to z
Summary of the content on the page No. 19
Chapter 2 Robustness Analysis 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 for checking a system that already has been verified to be robustly stable using smargin( ). For example, a system can have a minimum stability margin of 4 dB, so it is robustly stable. If the worst-case gain from a function input to the output it commands has a 20 dB p
Summary of the content on the page No. 20
Chapter 2 Robustness Analysis 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 advanced aspects of computing the stability margin and the related scaling algorithms. Stability Margin and Structured Singular Values (μ) The stability margin was first defined by Safonov in [Saf82]. If you let MH = diagonal() l() w , ...,l()
