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NI MA TRIXx TM Xmath ™ Control Design Module Xmath Control Design Module April 2007 370753C-01.
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Conventions The follo wing con ventions are used in this manual: < > Angle brackets that cont ain numbers s eparated by an ellipsis represent a range of v alues associated with a bit or signal name—for example, DIO<3..0>. [ ] Square brackets enclose op tional items—for example, [ response ].
© National Instruments Corporatio n v Xmath Control Design Module Contents Chapter 1 Introduction Using This Manual...................... .............. .............. .............. .............. .............. ............. 1-1 Document Organization.
Contents Xmath Control Design Module vi ni.com Chapter 3 Building System Connections Linear System Interconnection Operators ............... .............. .............. .............. ............ 3-1 Linear System Interconnection Fun ctions .......
Contents © National Instruments Corporatio n vii Xmath Control Design Module Bode Frequency Analysis ........ .............. .............. .............. .............. .............. .............. .. 5-7 bode( ) ............... .............. ....
Contents Xmath Control Design Module viii ni.com Appendix A Technical References Appendix B Technical Support and Professional Services Index.
© National Instruments Corporatio n 1-1 Xmath Control Design Module 1 Introduction The Control Design Module (CDM) is a complete library of classical and modern control design fun ctions that provides a fl exible, intuitive cont rol design framework.
Chapter 1 Introduction Xmath Control Design Modul e 1-2 ni.com particular system properti es or to change the format of a system. These topics include continuous/discret e system con v ersion, as well as fi nding equi v alent transfer f unction state-space representations.
Chapter 1 Introduction © National Instruments Corporatio n 1-3 Xmath Control Design Module • H ( s ) is used to denote t he frequency response, over some range of frequencies of a system where s is the Laplace varia ble. H ( q ) is used to indicate that the system can be continuous or discrete.
Chapter 1 Introduction Xmath Control Design Modul e 1-4 ni.com Control Design T utorial This tutorial illustrates the use of functions and commands provided in Xmath and the Xmath Control Desig n Module to solve control pro blems.
Chapter 1 Introduction © National Instruments Corporatio n 1-5 Xmath Control Design Module ssys (a state space system) = A -0.4 0 -0.01 1 0 0 -1.4 9.8 -0.
Chapter 1 Introduction Xmath Control Design Modul e 1-6 ni.com Input Names ----------- Rotor Angle Output Names ------------ Horizontal v System is continuous The system has poles and zeros in the right half of the compl ex plane and therefore is open-loop unstable.
Chapter 1 Introduction © National Instruments Corporatio n 1-7 Xmath Control Design Module One approach to stabilizing this syst em is to try to cancel th e pole at –0.656513 by adding a com pensator , K 1 ( s ), with a zero at –0.656513. Note It is important to understa nd that this is primarily an academic e xercise.
Chapter 1 Introduction Xmath Control Design Modul e 1-8 ni.com Figure 1-2. Locus of all Open-Loop and Closed-Loop Roots of Gs If you cannot move the slider so that the gain is exactly 2, click the box to the right of the slider and enter 2 . T o close the interactiv e root locus dialog box, select File»Exit .
Chapter 1 Introduction © National Instruments Corporatio n 1-9 Xmath Control Design Module Close the loop using the single-input syntax of feedback( ) , which implements direct unity-gain negati ve feedback, and obtain the system’ s step response using step( ) : Kc = 2; cl_syscomp1 = feedback(Kc*K1 s*K2s*Gs); v = step(cl_syscomp1, 0:.
Chapter 1 Introduction Xmath Control Design Module 1-10 ni.com Figure 1-4. Block Diagram of the Closed-Loop Controller This is a block diagram of the clos ed-loop contro ller with compensator K c 1 K 1 ( s ) in the feedforward path and K c 2 K 2 ( s ) in the feedback path.
Chapter 1 Introduction © National Instruments Corporatio n 1-11 Xmath Contr ol Design Module Because cl_syscomp2 contains an internal pole-zero cancellation, you can rewrite it in minimal form and th.
Chapter 1 Introduction Xmath Control Design Module 1-12 ni.com Figure 1-5. Helicopter Velocity T racking Step Input at the Rotor Y ou also can lo ok at the gain and phase margins of the system. H = bode(cl_syscomp2m, {npts = 200, !wrap}); [gm,pm] = margin(H) There are no 0 dB gain crossings.
Chapter 1 Introduction © National Instruments Corporatio n 1-13 Xmath Contr ol Design Module Figure 1-6. Closed-Loop Syst em Bode Plot The domain of the gain and phase ma rgin PDMs indicates the freq uency (in hertz) at which the margin occurs. So the gain can be increased by about 26.
Chapter 1 Introduction Xmath Control Design Module 1-14 ni.com Y ou can verify that your system is cont rollable, then defi ne the closed-loop poles you want and use poleplace( ) to fi nd the feedback gains required gi ven the system A and B matrices.
Chapter 1 Introduction © National Instruments Corporatio n 1-15 Xmath Contr ol Design Module Specify the observer poles at [–3 + 3j, –4] and call poleplace( ) again: op = [-3+3*jay, -4]; L = poleplace(A',C',op) L (a row vector) = 5.46645 4.
Chapter 1 Introduction Xmath Control Design Module 1-16 ni.com Y ou can choose to scale the system output here for zero steady-state error in the step response. This is accompli shed in an intuitiv e manner , dividing the system sys_cl b y the desired scaling factor .
Chapter 1 Introduction © National Instruments Corporatio n 1-17 Xmath Contr ol Design Module When you create the estimator system sys_est , you use the original A matrix for the state-update equatio n, but you pro vide a zero external input (a B matri x of zero).
Chapter 1 Introduction Xmath Control Design Module 1-18 ni.com Figure 1-10. Multiple Plots Showing Time Needed for States to be C orrectly T rac ked by Estimator , Given Incorrect Initial V alues Heli.
Chapter 1 Introduction © National Instruments Corporatio n 1-19 Xmath Contr ol Design Module Y ou can use the default exponential discretization m ethod with dt =0 . 0 1 and compare frequency responses between the original system and the discretized system: ssysd = discretize(ssys, 0.
Chapter 1 Introduction Xmath Control Design Module 1-20 ni.com Figure 1-12. Step Respon se of a Discrete System Us ing Discretized Observer -Based Controlle r As you discretize the compensator, form the closed-loop, scaled system, and simulate its response to a step in put, you must ensure that the samplin g interv al is the same ( dt = 0.
Chapter 1 Introduction © National Instruments Corporatio n 1-21 Xmath Contr ol Design Module The linearized state-space equations, including the actuator and sensor dynamics, are as follo ws: θ is the angle (in radians) the we dge makes with the v ertical axis, x is the position of the sliding mass, and u is the control input vo ltage.
Chapter 1 Introduction Xmath Control Design Module 1-22 ni.com Because this system is open-loop unstable and has fairly fast poles in both halves of the s-plane, you want to mak e sure it can bring the effect of an external disturbance (such as a sharp pu sh to the cart) to zero as quickly as possible.
Chapter 1 Introduction © National Instruments Corporatio n 1-23 Xmath Contr ol Design Module Figure 1-13. Response of Full-State Fee dback Controller to a Unit Step Disturbance Having established your regulator design, you build the estimator and simulate performance of the closed-loop system feeding back state estimates.
Chapter 1 Introduction Xmath Control Design Module 1-24 ni.com Figure 1-14. Respo nse of Observer -Based Controller to a Unit Step Disturbance.
© National Instruments Corporatio n 2-1 Xmath Control Design Module 2 Linear System Representation Xmath provides a structure for system representation called a system object . This object includes system parameters in a data structure designed to reflect the way these systems are analyzed mathemati cally.
Chapter 2 Linea r System Representation Xmath Control Design Modul e 2-2 ni.com T ransfer Function System Models One way of representing continuous -time finite-dim ensional linear time-invariant systems is with the transfer function: with num(s) and den(s) being polynomials in s .
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-3 Xmath Control Design Module form the same transfer function as that deri v ed in the preceding transfer function equat.
Chapter 2 Linea r System Representation Xmath Control Design Modul e 2-4 ni.com be used so long as a consistent choice of va riable is used for bo th numerator and denominator polyno mials.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-5 Xmath Control Design Module State-Space System Models State-space models comprise the second category of linear system representations in Xmath.
Chapter 2 Linea r System Representation Xmath Control Design Modul e 2-6 ni.com Again, you create the system using the system( ) function. This tim e you use the optional dt keyword to indicate that this system is discrete. A = [0,1;-0.75,0]; B = [1,0]'; C = [0,1]; D = 0; sys4 = system(A,B,C,D, {dt = 0.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-7 Xmath Control Design Module The system( ) function can create both the transfer-function and state-space forms of the system object. It requires four compatibly-sized matrices to create a state-space system , or a pair of polynomials to create a transfer function.
Chapter 2 Linea r System Representation Xmath Control Design Modul e 2-8 ni.com Example 2-3 Using system( ) to Change t he Attributes of an Existing System sys4=system([0,1;-0.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-9 Xmath Control Design Module done internally to return A, B, C, and D, though the format of the variable Sys itself remains unchanged . The transfer function must be prop er .
Chapter 2 Linea r System Representation Xmath Control Design Module 2-10 ni.com numden( ) [num,den] = numden(Sys) The numden( ) function returns the numerator and deno minator polynomials comprising a sin gle-input, single-output system in transfer function form.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-11 Xmath Contr ol Design Module these polynomials into a tr ansfer-function and uses period( ) to set the sampling interval to match that of sys4 .
Chapter 2 Linea r System Representation Xmath Control Design Module 2-12 ni.com outputNames = "Velocity", stateNames = ["Torque","Angle"]}); [,,stateNames] = names(sys5)?.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-13 Xmath Contr ol Design Module check(sys, {stable}) ans (a scalar) = 0 check(sys, {discrete, ss}) ans (a scalar) = 1 [, tfsys] = check(sys, {tf, convert}) tfsys (a transfer function) = (z + 0.
Chapter 2 Linea r System Representation Xmath Control Design Module 2-14 ni.com discretization methods used based on the specification of each keyw ord are discussed in the following sections.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-15 Xmath Contr ol Design Module Pole-Zero Matching: polezero The pole-zero matching method of discretizin g a continuous.
Chapter 2 Linea r System Representation Xmath Control Design Module 2-16 ni.com The exponential ke yword assumes that the response value between samples is constant and can , therefore, be represen ted by a zero-ord er hold polynomial.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-17 Xmath Contr ol Design Module gain_z = 20*log10(abs(freq(Hd_z,F))) ; gain_e = 20*log10(abs(freq(Hd_e,F))) ; and plot it (as shown in Figure 2-1).
Chapter 2 Linea r System Representation Xmath Control Design Module 2-18 ni.com Many of the discretization techniques discussed in the Hold Equivalence Methods: exponential and first order section can be easily re vers ed to obtain a continuous equiv alent to a discrete system.
Chapter 2 Linear S ystem R epresentation © National Instruments Corporatio n 2-19 Xmath Contr ol Design Module No w con vert back to the continuous form: Hc = makecontinuous(Hd_f, {forward}) ; [num,den] = numden(Hc) num (a polynomial) = (s + 0.36) den (a polynomial) = 2 (s + 0.
© National Instruments Corporatio n 3-1 Xmath Control Design Module 3 Building System Connections Large system models are frequently built by connectin g smaller models together. You can perform different ty pe s of linear system intercon nections using the Xmath functions discussed in this chapter.
Chapter 3 Building System Connections Xmath Control Design Modul e 3-2 ni.com Sys = Sys2 * Sys1 Cascade connection of Sys 1 and Sys 2 where the output of Sys is y 2 and the input is u 1 . Sys = Sys1/Sys2 Cascade connection of Sys1 and in v erted Sys2 where Sys = Sys 1 * inv(Sys 2 ) , u=u 2 , and y = y 1 .
Chapter 3 Building System Connections © National Instruments Corporatio n 3-3 Xmath Control Design Module Dynamic systems also can be flex ibly combined with scalars and compatibly sized matrices using the operators in T able 3-1.
Chapter 3 Building System Connections Xmath Control Design Modul e 3-4 ni.com Linear System Interconnection Functions afeedback( ) , append( ) , connect( ) , and feedback( ) connect dynamic systems in state–space or transfer–function fo rm to produce a larger system in state-space form.
Chapter 3 Building System Connections © National Instruments Corporatio n 3-5 Xmath Control Design Module • By default, feedback is defined to be ne gati ve. • The number of out puts from the f irst syst em must equal the number of inputs to the second system.
Chapter 3 Building System Connections Xmath Control Design Modul e 3-6 ni.com if the conditio n estimate for either matrix is less than eps . For more information on this condition estimate, refer to the MATRIXx Help for the Xmath function rcond( ) .
Chapter 3 Building System Connections © National Instruments Corporatio n 3-7 Xmath Control Design Module is created by appending the in put s, outputs, and sta tes of Sys1 and Sys2 . A larger number of systems can be appended by appending two at a time.
Chapter 3 Building System Connections Xmath Control Design Modul e 3-8 ni.com append( ) performs a check to make sure both Sys 1 and Sys 2 have the same sample rate, and adopts this rate for the appended system . Any initial conditions on the states are also appended columnwise.
Chapter 3 Building System Connections © National Instruments Corporatio n 3-9 Xmath Control Design Module • By default, feedback is defined to be positi ve. T o enforce negati ve feedback, specify connect(Sys,-K) . • A “selection matrix” has a single 1 in each ro w; the rest of the ro w contains zeros.
Chapter 3 Building System Connections Xmath Control Design Module 3-10 ni.com 2 C -1.5 1.5 D 0 X0 0 0 Algorithm For the feedback system shown in Example 3-3, you can write the following system equatio.
Chapter 3 Building System Connections © National Instruments Corporatio n 3-11 Xmath Contr ol Design Module feedback( ) Sys = feedback(Sys1,{Sys2}) The feedback( ) function connects two dynamic systems together in a feedback loop as sho wn in Figure 3-4.
Chapter 3 Building System Connections Xmath Control Design Module 3-12 ni.com -1 1 D 0 X0 0 0 State Names ----------- Input Names ----------- Input 1 Output Names ------------ Output 1 System is continuous Algorithm The system used for the feedback loop, Sys2 , is optional.
Chapter 3 Building System Connections © National Instruments Corporatio n 3-13 Xmath Contr ol Design Module The single system resulting from the feedback combination of Sys1 and Sys2 has u 1 as its input, y 1 as its output, and a stat e vector consisting of the appended states of Sys1 and Sys2 .
© National Instruments Corporatio n 4-1 Xmath Control Design Module 4 System Analysis This chapter discusses time-domain solutions of the equations underlying transfer functions and state-space syst em models, and what these solutions tell us about the stabilit y of the system.
Chapter 4 System Analysis Xmath Control Design Modul e 4-2 ni.com The time-response of discrete systems i s found directly as a summ ation of the information from preceding time point s in the state and input histories.
Chapter 4 System Analysis © National Instruments Corporatio n 4-3 Xmath Control Design Module and define the zeros of S( λ ) as any v alues of λ for which the system ma trix drops rank. For single-input single-ou tput systems this is equiv alent to the polynomial zeros of the transf er-function numerator .
Chapter 4 System Analysis Xmath Control Design Modul e 4-4 ni.com directly from the roots of the transfer function numerat or . If Sys is in state-space form, the definition of its zer os arises from .
Chapter 4 System Analysis © National Instruments Corporatio n 4-5 Xmath Control Design Module ans (a column vector) = -0.98875 + 2.4773 j -0.98875 - 2.
Chapter 4 System Analysis Xmath Control Design Modul e 4-6 ni.com Example 4-3 Dynamic Response th roug h Partial Fraction Expansion To illustrate how you can examine the stability and dynamic response.
Chapter 4 System Analysis © National Instruments Corporatio n 4-7 Xmath Control Design Module 0 0 Input Names ----------- Input 1 Output Names ------------ Output 1 System is continuous Y ou can examine the stability of G cl ( s ) by representing it as a sum of partial fractions, using the residue( ) function.
Chapter 4 System Analysis Xmath Control Design Modul e 4-8 ni.com Figure 4-2. T ransient Response of the Closed- Loop System as a Function of Time Y ou also can calculate the im pul se response directly with t = [0 : 0.
Chapter 4 System Analysis © National Instruments Corporatio n 4-9 Xmath Control Design Module and orders for which the residue(s) should be found. If a user-specif ied va lu e f or pls is not actuall y a pole of the system or if the order requested is greater than the multiplicity of the pole, the correspond ing residue is returned as zero.
Chapter 4 System Analysis Xmath Control Design Module 4-10 ni.com (s + 1) (s + 0.79s + 0.16) initial integrator outputs 0 0 0 0 Input Names ----------- Input 1 Output Names ------------ Output 1 System is continuous Note G2 matches the system G where residues were computed in Example 4-4.
Chapter 4 System Analysis © National Instruments Corporatio n 4-11 Xmath Contr ol Design Module Often it is desirable to run sev eral simu lation s with different inputs. In this case, you can define a PDM whose co lumns contain the input vectors for the dif ferent simulations.
Chapter 4 System Analysis Xmath Control Design Module 4-12 ni.com Figure 4-3. System Time Response to a Series of Step Signals The (system)*( PDM) construc t for performin g time-domain sim ulation is used analogously no matter how many inputs the system has.
Chapter 4 System Analysis © National Instruments Corporatio n 4-13 Xmath Contr ol Design Module Impulse Response of a System An impulse input to a system is defined somewhat differently depending o n whether the system is discrete or continu ous.
Chapter 4 System Analysis Xmath Control Design Module 4-14 ni.com Note A continuous system and its discrete-time equiv alent (computed using the impulse-in variant z-transform) ha ve impulse responses dif fering only by a f actor of 1/ dt .
Chapter 4 System Analysis © National Instruments Corporatio n 4-15 Xmath Contr ol Design Module Figure 4-4. 15-Second Impulse Response deftimerange( ) tvec = deftimerange(Sys) deftimerange( ) compute.
Chapter 4 System Analysis Xmath Control Design Module 4-16 ni.com The maximum time, Tmax , is computed as fo llows, with vP denoting the vector of scaled pole s and dt the period: Tmax=abs(log(.
Chapter 4 System Analysis © National Instruments Corporatio n 4-17 Xmath Contr ol Design Module The simulation perfor med in initial( ) uses an input of zero for each point in the tim e vector . The output Y is a PDM where domain is the time vector .
Chapter 4 System Analysis Xmath Control Design Module 4-18 ni.com Step Response The response of a system to a unit step in put is one of the most com monly used measures of how well a given control system’s output tracks the system input.
Chapter 4 System Analysis © National Instruments Corporatio n 4-19 Xmath Contr ol Design Module From Figure 4-6 you see that the d elay time ( t d ) is about 0.5 seconds, the rise time ( t r ) is 0.8 seconds, the peak time ( t p ) is 1.6 seconds, the settling time ( t s ) is about 5.
Chapter 4 System Analysis Xmath Control Design Module 4-20 ni.com Get indices of all v alues > 0.9 * f inal v alue: gt_9_10 = find(Y > 0.9*Yf); Subtract domain v alues to get time duration: tr = domain(Y(gt_9_10(1,1)))-... domain(Y(gt_1_10(1,1))) tr (a scalar) = 0.
© National Instruments Corporatio n 5-1 Xmath Control Design Module 5 Classical Feedback Analysis The open-loop systems an alyzed in Chapter 4, System Analysis , generally perform in a satisfactory manner only if the system model is very accurate and there are no external disturbances.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Modul e 5-2 ni.com Because open-loop systems are genera lly easier to study and model than closed-loop systems, you want to design closed-loop systems based on information obtainable from the open-loop sy stem.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-3 Xmath Control Design Module rlocus( ) rlroots=rlocus(sys,K,{xmin,xmax,ymin ,ymax,pattern,graph}) rlocus(sys,{xmin,xmax,ymin,ymax,patt ern})# (interactive) The rlocus( ) function computes and draws root locus diagrams for continuous-time an d discrete-time SI SO systems.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Modul e 5-4 ni.com Figure 5-2. Root Locus of H for Gain K = 0.07 This syntax allows you to vary the root locus gain through an in teractiv e form.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-5 Xmath Control Design Module As the gain va ries, small ✱ ’ s appear on the locus indicating the closed-loop pole location for that choice of gain.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Modul e 5-6 ni.com For discrete-time state-space system s with a sampling interv al of T , the frequency response for each frequency point .
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-7 Xmath Control Design Module For an e xample of frequency response of a simple system, refer to Example 5-2.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Modul e 5-8 ni.com because they can be used to assess th e relative stability of a closed-loop system given the frequency response of the open-loop system.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-9 Xmath Control Design Module Referring to the entire closed -loop system in Figure 5-1 as G cl , the poles of G cl are the.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Module 5-10 ni.com Referring to Figure 5-4, notice the add itional lines drawn on the plots at the frequencies where the gain crosse s the 0 dB line and where the phase crosses the 180 ° line.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-11 Xmath Contr ol Design Module Figure 5-4. Bode Plot Showi ng Syst em Gain and Phase Margins These plots illustrate ho w the location of the syst em poles and zeros shapes the gain and phase curves.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Module 5-12 ni.com Each of these contrib utes a phase angle φ def ined by: with ω and p n expressed in the same units, either radians per second or Hz, and using a four-quadrant arctangent function similar to that provided b y atan2( ) in Xmath.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-13 Xmath Contr ol Design Module margin( ) loops o ver all the frequenc y points in the response and performs the following .
Chapter 5 Classica l Feedback Analysis Xmath Control Design Module 5-14 ni.com Note margin( ) also returns the frequencies at which the phase crosses the –180 ° line and the gain crosses the 0 dB line. These results match the gain and phase margins shown graphically in Figure 5-4.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-15 Xmath Contr ol Design Module The result is sho wn in Figure 5-5. Figure 5-5. nichols( ) Gain-Phase Plot Nyquist Stability Analysis Nyquist analysis is a frequency domain method for examining system performance of dynamic systems.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Module 5-16 ni.com plant is open-loop stable, then th ere should be no encirclements. If the plant has on e open-loop unstable p ole, there should be one negativ e (counter-clockwise) encirclement.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-17 Xmath Contr ol Design Module The Nyquist plot Xmath generates is complete only for the frequencies you specify . Ideally you would obtain a plot based on the frequency response from ω = 0 to ω = ∞.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Module 5-18 ni.com Figure 5-7. Nyquist Plot of the Open- Loop System for Frequencies from 0.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-19 Xmath Contr ol Design Module By combining th e information from the two plots, r eflecting them acr oss the real axis to.
Chapter 5 Classica l Feedback Analysis Xmath Control Design Module 5-20 ni.com -0.52263 0.00336213 + 3.75217 j 0.00336213 - 3.75217 j -11.4841 T wo of t he poles of the closed-l oop system are now unstable.
Chapter 5 Classical Feedback Analysis © National Instruments Corporatio n 5-21 Xmath Contr ol Design Module v alues at which the po wer spectral density is to be computed and where dependent matrices are the input po wer spectral dens ity matrix at each frequency .
© National Instruments Corporatio n 6-1 Xmath Control Design Module 6 State-Space Design The functions in this chapt er are generally termed “modern control” tools. They are based on the st ate-space linear system re presentation, and employ methods which are generally appli cable to both SISO and MIMO problems.
Chapter 6 State-Space Design Xmath Control Design Modul e 6-2 ni.com matrix B, then the mode of the system associated with the corresponding eigenvalue cannot be controlled with any input.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-3 Xmath Control Design Module Figure 6-1. Full-Stat e Feedback Bein g Used to Relocate the Eigenva lues of a Controllable System Ba.
Chapter 6 State-Space Design Xmath Control Design Modul e 6-4 ni.com X0 0 Input Names ----------- Input 1 Output Names ------------ Output 1 System is continuous T (a square matrix) = 2.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-5 Xmath Control Design Module Beginning with the basic state-space equations (the Du output term can be omitted without loss of gen.
Chapter 6 State-Space Design Xmath Control Design Modul e 6-6 ni.com Figure 6-2. General Observer Block Diagram If the observability matrix is nonsingular , you will be able to put the eigen v alues (pole locations) of ( A – LC ), sh own in Equation 6-4, anywhere you want.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-7 Xmath Control Design Module Example 6-2 Observ ability of a System A system is described by: A = [1,0,0.
Chapter 6 State-Space Design Xmath Control Design Modul e 6-8 ni.com minimal( ) [SysM,T,nuco] = minimal(Sys,{tol}) Because nonminimal systems are uncontrollable, unobservable, or both, you want to be able to compute the minimal realization for a giv en system.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-9 Xmath Control Design Module ans (a scalar) = 1 zeros(SysM) ans is null stair( ) [SysT,T,nc] = stair(Sys,tol) The stair( ) con verts a dynamic system to staircase fo rm.
Chapter 6 State-Space Design Xmath Control Design Module 6-10 ni.com Duality and Pole Placement The new state-update equation in the Controllabi lity section and the Observability and Est imation sect.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-11 Xmath Contr ol Design Module poleplace( ) is unusual among Xmath’ s modern control design functions in that only the A and B matr ix variables are used as input, rather than a complete system variab le.
Chapter 6 State-Space Design Xmath Control Design Module 6-12 ni.com pairs as poles( ) . For eac h pole v alue in poles( ) , pol eplace( ) forms a v ector by subt racting the pole’ s v alue from each diag onal element of S e xcept for the last element (0).
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-13 Xmath Contr ol Design Module For continuous-time systems, the quadratic performan ce index tak es the form: For the discrete cas.
Chapter 6 State-Space Design Xmath Control Design Module 6-14 ni.com The optimal estimator and regulator problems illust rate the principle of duality—that for any giv en system r ealization {A,B,C} there is a dual realization {A',C',B'} with related cont rollability and observability .
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-15 Xmath Contr ol Design Module Figure 6-4. Diagram of Plant for the Inverted Pendulum Problem Figure 6-4 shows the pendulum at φ = 0 and φ > 0.
Chapter 6 State-Space Design Xmath Control Design Module 6-16 ni.com Ruu is a scalar because you ha ve only one input for this particular model. [Kr,ev,P] = regulator(ipsys,Rxx,Ruu) ; Kr Kr (a row vector) = -348.778 -32.1056 -100 -27.3036 Note Y ou will use this regulator gain later in designing a compensator .
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-17 Xmath Contr ol Design Module Figure 6-5. Diagram of the Estimator Representat ion estimator( ) inputs include the dynamic syst em Sys , and the noise intensity matrices Q xx , Q yy , or Q xy .
Chapter 6 State-Space Design Xmath Control Design Module 6-18 ni.com numerical diff iculties are encounte red, the algorithm will attempt to determine whether or not the prob lem is well posed. Checks are made to determine the reachability a nd the positi v e definiteness or semipositi ve-def initeness of the co v ariance matrices.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-19 Xmath Contr ol Design Module The discrete-time estimator follo ws fr om a similar system description, using the discrete-time dif ference e quation representation of the system, as sho wn in the following equations.
Chapter 6 State-Space Design Xmath Control Design Module 6-20 ni.com this measurement up date, deriv ed in [Kal60], are shown in the follo wing equations. Substituting the system and noise matr ices for the steady-state case, you solve the discrete Riccati equation to obtain P and thence K e , as sho wn in Equation 6-7 and Equation 6-8.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-21 Xmath Contr ol Design Module If you want the closed-loop system eigen v alues, compute them as the eigen v alues of A – K e C . F or an example of ho w to design a state estimator for the in verted pendulum problem, refer to Example 6-6.
Chapter 6 State-Space Design Xmath Control Design Module 6-22 ni.com a compensator is shown in Figure 6-6. This figure combines full-state regulator with gain K r and state estimator with gain K e .
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-23 Xmath Contr ol Design Module lqgcomp( ) SysC = lqgcomp(Sys,Kr,Ke,{direct}) The lqgcomp( ) function creates a dynamic compensator given a dynamic system having at least one state and the regulator and estimator gain matrices.
Chapter 6 State-Space Design Xmath Control Design Module 6-24 ni.com simulate the system’ s response to a slo w sine input, starting with the cart at rest and the pendulum initially held in the upright ( φ = 0) po sition to obtain Figure 6-7: t = 0:0.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-25 Xmath Contr ol Design Module Figure 6-7. f and x as a Function of Time, Starting from Zero, as a Result of a Sinusoidal Force Ap.
Chapter 6 State-Space Design Xmath Control Design Module 6-26 ni.com continuous-time Riccati equation , which is used if B and S are not specified, is: Note The meaning of R is qu ite different in this case. riccati( ) [P,resid, Kr, ev] = riccati(A,Q,R,{B ,tol,S,d}) Here, A can be either a matrix or a system object.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-27 Xmath Contr ol Design Module 0 0 0 -4 B 0 -2 0 1 C 0.211325 0.756044 0.000221135 0.330327 D 0 X0 0 0 0 0 System is continuous [P,resid] = riccati(A,Q,R); norm(A'*P+P*A-P*R*P+Q,1) ans (a scalar) = 2.
Chapter 6 State-Space Design Xmath Control Design Module 6-28 ni.com 0 1 0 B 1 0 0 C 1 0 0 D 0 X0 0 0 0 System is discrete, sampling at 1 se conds. [P,resid]=riccati(Sys,Q,RD,B); norm(A'*P*A-P-A'*P*B*inv(RD+B'*P*B)* B'*P*A+Q,1) resid (a scalar) = 7.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-29 Xmath Contr ol Design Module all the eigen values of the system A matrix are neg ati ve. Th e discrete L yapun ov equation is: (6-12) Analogously , the preceding equa tion has a unique solution X when λ i (A) λ j (A) ≠ 1 for all i and j .
Chapter 6 State-Space Design Xmath Control Design Module 6-30 ni.com or the follo wing for the discrete case: (6-16) These results deri ve from the L y apunov method of stability analysis for linear systems. Steady state means that at some poi nt the states no longer change.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-31 Xmath Contr ol Design Module of the diagonal elements of the Sc hur-decomposed A and B matrices sum to zero, a warning is given that the continuo us equation solution may not be unique.
Chapter 6 State-Space Design Xmath Control Design Module 6-32 ni.com Special L yapunov Equation A = [-4,10;2,7]; C = [.3,6;2,9]; X = lyapunov(A,C) X (a square matrix) = 1.1816 -0.209028 1.12431 -0.773611 A*X + X*A' + C; norm(A*X + X*A' + C,1) ans (a scalar) = 5.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-33 Xmath Contr ol Design Module Example 6-11 rms( ) Response Sys = system([-2.3,0.
Chapter 6 State-Space Design Xmath Control Design Module 6-34 ni.com For discrete-time systems, the integrals in the W c and W o equations are replaced by summation signs and th e grammians are obtain.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-35 Xmath Contr ol Design Module and σ 1 2 through σ n 2 are the singular values of the matrix H satisfying Σ 2 = H'H. They are termed the Hank el singular v alues. The σ k 2 terms are ordered so that σ 1 2 ≥ σ 2 2 ≥ … ≥ σ n 2 ≥ 0.
Chapter 6 State-Space Design Xmath Control Design Module 6-36 ni.com T is the transforma tion relating the states of the original system to the states of the balanced system T ransforming to balanced .
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-37 Xmath Contr ol Design Module and compare the condition numbers of the balanced system’ s grammians: WcB=lyapunov(Ab,Bb*Bb'); WoB=lyapunov(Ab',Cb'*Cb); condition(WcB) ans (a scalar) = 12.
Chapter 6 State-Space Design Xmath Control Design Module 6-38 ni.com accurate. Gi ven a v ariable Sys b uilt from the matrices {A,B,C,D}, the modal decomposition SysMod is built from T –1 AT , T –1 B , CT , an d D , where T is the transformation matr ix to modal form.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-39 Xmath Contr ol Design Module the deri v ati v e of x 2 is set to zero, resulting in reduced-order state equations of the form: I.
Chapter 6 State-Space Design Xmath Control Design Module 6-40 ni.com 0 0 0 0.924711 B -0.00116788 0.00272531 0.00334243 -0.00162497 C 1 0.866186 -0.848754 -1.0118 D 0 X0 0 0 0 0 State Names ----------- State 1 State 2 State 3 State 4 Input Names ----------- Input 1 Output Names ------------ Output 1 System is discrete, sampling at 0.
Chapter 6 State-Space D esign © National Instruments Corporatio n 6-41 Xmath Contr ol Design Module 0 0.52 1.94289e-16 6.66134e-16 0 0 0.665289 4.44089e-16 0 0 0 0.924711 SysMR = mreduce(SysM, [1,2,4]) SysMR (a state space system) = A 0.37 0 0 0 0.52 0 0 0 0.
Chapter 6 State-Space Design Xmath Control Design Module 6-42 ni.com Figure 6-8. Modal System and Reduced Modal Syst em.
© National Instruments Corporatio n A -1 Xmath Co ntrol Design Modu le A T echnical References [BeV88] T . Beelen, P . V an D ooren, “ A n impr oved algorithm for the computation of Kronecker’ s canonical form of a singular pencil, ” Linear Algebra and Applications , 105, pages 9–65, 1988.
Appendix A T echnical References Xmath Control Design Modul e A-2 ni.com [Kai80] T . Kailath, Linear Systems , Prentice-Hall, Inc., Engle wood Clif fs, New Jerse y , 1980. [Kai81] T . Kailath, Lectures on Wiener and Kalman Filtering , S pringer-V erlag , New Y ork, 1981.
© National Instruments Corporatio n B -1 Xmath Co ntrol Design Modu le 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.
of fice W eb sites, which provide up-to-date contact informati on, support phone numbers, email addre sses, and current e vents..
© National Instruments Corporatio n I-1 Xmath Control Design Module Index A abcd, 1-16, 2-8 adjoint system, 3-3 afeedback, 3-4 append, 3-6 appending dynamic systems, 3-6 autocorrelation function, 5 -.
Index Xmath Control Design Modul e I-2 ni.com converting to continuous equivalent, 2-19 from a continuous system (example) , 1-18 discrete-time Riccati equation, 6-25 discretize, 2-13 discretizing a s.
Index © National Instruments Corporatio n I-3 Xmath Control Design Module I impulse, 4-13 input, 4-13 continuous time, 4-13 discrete time, 4-13 response, 4-13, 6-33 initial, 4-16 initial conditions, .
Index Xmath Control Design Modul e I-4 ni.com steady-state response to white noise, 6-29 white process, 6-23 nomenclature, 1-2 numden, 2-10 numerical integration method s, 2-14 Nyquist contour, 5-19 p.
Index © National Instruments Corporatio n I-5 Xmath Control Design Module rms, 6-32 root locus, plotting, 5 -3 root-mean-square response, 6-32 S sample period, extract with period, 2-10 sample rate, .
Index Xmath Control Design Modul e I-6 ni.com T technical support, B-1 time domain simulation, general, 4-10 update, 6-19 training and certification (NI resources), B-1 transfer function checking for,.
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