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Con ten ts 1 In tro duction 1 1 . 1 N o t a t i o n ....... ....... .......... ......... .... 1 1.2 Manual Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 How to avoid really r eading this Manual . . . . . . . . . . . .
iv CONTENTS 2.2.5 Obtaining Robust Control Mo dels for Ph ysica l Systems . . . . . . 28 2.3 H ∞ and H 2 D e s i g n M e t h o d o l o g i e s ........... ......... .. 2 9 2.3.1 H ∞ D e s i g n O v e r v i e w ..... ........... ......... 3 1 2.3.2 Assump tions for the H ∞ D e s i g n P r o b l e m .
CONTENTS v 2 . 6 M o d e l R e d u c t i o n .. ....... .......... ......... .... 6 4 2.6.1 T runcation and Residualization . . . . . . . . . . . . . . . . . . . . 65 2 . 6 . 2 B a l a n c e d T r u n c a t i o n ... .......... ......... .... 6 5 2 . 6 .
vi CONTENTS 3 . 5 S y s t e m I n t e r c o n n e c t i o n ..... .......... ......... .... 9 1 3.6 H 2 and H ∞ A n a l y s i s a n d S y n t h e s i s ......... ......... .... 9 5 3 . 6 . 1 C o n t r o l l e r S y n t h e s i s ...... ......... ...
CONTENTS vii 4.1.5 µ An alysis of the H ∞ C o n t r o l l e r ........ ......... .. 1 3 8 4.1.6 Fit ting D -scales for the D - K I t e r a t i o n .......... ...... 1 4 0 4 . 1 . 7 D e s i g n I t e r a t i o n # 2 ... .......... ......... .... 1 4 3 4.
viii CONTENTS A . 3 S y s t e m R e s p o n s e F u n c t i o n s .. ......... ....... .... 3 9 8 A . 4 S y s t e m I n t e r c o n n e c t i o n ...... ......... ......... 3 9 9 A . 5 M o d e l R e d u c t i o n ..... .......... ........... .. 3 9 9 A.
Chapter 1 In tro duction X µ is a suite of Xmath functions for the mo deling , ana lysis and syn thesis of linear robust co n trol systems. Robust con tro l theor y has developed rapidly during the last decade to the point where a useful set of computational tools can b e used to solve a wide range of control problems.
2 CHAPTER 1. INTR ODUCTION Notatio n Meaning pdm Xmath parameter dependen t matrix data ob ject Dynamic System Xmath dynam ic system data ob ject Co de examples and function names are set in t yp ewriter font to distinguish them from narrativ e text. 1.
1.3. HO W TO A VOID RE ALL Y READING THIS MANUAL 3 1.3 Ho w to a v oid really reading this M an ual The layout of the man ual pro ceeds from in tro duction to background t o syn tax detail to application descriptio ns. This may be tediously theoretical for some.
Chapter 2 Ov erview of the Underlying Theory 2.1 In tro duction The material cov ered here is taken from a v ariety of sources. The basic approach is described b y Do yle [1, 2], and further elaborated upon b y Pack ard [3]. Summaries ha ve also appeared in work by Smi th [4] and others.
6 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y several studies in volving pro cess control applications, par ticularly high pur it y distillation columns . These are detailed b y Skogestad an d Morari in [15, 16, 17, 18 ] Section 2 .2 introduces robust control p erturba tion mo dels and linear fr actiona l transform ations.
2.1. INTRODUCTION 7 Figure 2.1: The generic rob ust cont rol model stru cture T race( M ) trace of M ( P n i =1 M ii ) Block diagrams will be used to represent in terconnections of systems. Consider the example feedback in ter connection shown in Fig .
8 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y is interpreted to mean that the signa l y is the sum o f the resp o nse of system P 21 to input sig nal v and system P 22 to input signal u . In general, w e will not b e specific a b out the represen tation of the system P .
2.1. INTRODUCTION 9 Euclidean norm. Given, x = x 1 . . . x n , the E uclidean (o r 2 -norm) of x , denoted by k x k , is defined by , k x k = n X i =1 | x i | ! 1 / 2 . Man y other norms are also opti ons; more detail on the easily calculated n orms can b e found in the on-line help for the norm function.
10 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y F or p ersistent signals, where the ab ov e norm is un b ounded, w e ca n define a p ow er nor m, k x i ( t ) k = lim T →∞ 1 2 T Z T − T | x i ( t ) | 2 dt ! 1 / 2 .
2.1. INTRODUCTION 11 signals. Strictly sp eaking, sig nals in H 2 or H ⊥ 2 are not defined on the ω axis. Ho wev er w e usually consi der the m to be b y taking a lim it as we approac h the axis. A slig ht ly mo re sp ecialized set is RL 2 , the set of real rational functions in L 2 .
12 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y where σ max denotes the maxim um sing ular v alue. No t all matrix no rms are induced from vector norms. The F r o eb enius nor m (square ro ot o f the sum of the squares of all matrix elements) is one suc h example.
2.2. MOD ELING UNCER T AIN SYSTEM S 13 The set o f all s ystems with b ounded ∞ -nor m is denoted by L ∞ . W e can again split this into stable and unstable parts. H ∞ denotes the sta ble part; those systems with | P ( s ) | finite for all Re( s ) > 0.
14 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y robust control model is therefo re a set descriptio n and we hop e that so me members of this set capture some o f the uncertain or unmo deled as pects o f our physical system. F or exa mple, consider the “uncertain” mo del illustrated in Figure 2.
2.2. MOD ELING UNCER T AIN SYSTEM S 15 as sp ecifying a maximum percen tage error b etw een P nom and eve ry other elem ent of P . The system P nom ( s ) is the elemen t of P that comes fro m ∆ = 0 and is called the nominal system. In otherwords, for ∆ = 0, the input-o utput r elationship is y ( s )= P nom ( s ) u ( s ).
16 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Real 0 0.5 1 -0.5 1.5 Imaginary -0.5 0 0.5 -1 1 Figure 2.3: Nyquist diagram of the set of systems, P.
2.2. MOD ELING UNCER T AIN SYSTEM S 17 Figure 2.4: Unit y gain negativ e feedbac k for the example syst em, P 0 +∆ W a 2.2.2 Linear F ractional T ransformations A mo del is considered to be an interconnection of lumped comp onents and perturbatio n blo cks.
18 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.5 : Generic LFT mo del s tructure including p erturbatio ns,∆ A generic model s tructure, r eferred to a s a linear fractiona l transfor mation (LFT), ov ercomes the difficulties outlined above.
2.2. MOD ELING UNCER T AIN SYSTEM S 19 in an LFT format. The open-lo op system is described by , y = F u ( P olp , ∆) u, where P olp = 0 W a IP 0 . The unity ga in, nega tive feedbac k config uration, illustr ated in Figure 2.4 (and given in Equation 2.
20 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y The issue o f the inv er tibilit y o f ( I − P 11 ∆) is fundamental to the study of the stability of a system under per turbations. W e will r eturn to this question in muc h more deta il in Section 2.
2.2. MOD ELING UNCER T AIN SYSTEM S 21 Figure 2.6 : Example mo del: multiplicativ e output p er turbation with weighted output noise References to a robust co ntrol model will imply a description of the fo rm given in Equation 2. 8. As a exa mple, consider one of the most common p erturbation mo del descriptions, illustrated in Figure 2.
22 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Robust control models a re therefor e set descriptions. In the a nalysis of such models it is also assumed that the unknown inputs belo ng to some bo unded set. Several c hoices of set for the unknown signals can be made, leading to different mathematica l problems for the ana lysis.
2.2. MOD ELING UNCER T AIN SYSTEM S 23 Pac k ar d [19] discuss the implications of this a ssumption on ro bust con tr ol theor y and we briefly touch upon this in Section 2.4.6 . The most common assumption is that ∆ is an unknown, norm-bounded, linea r time-inv ar iant system.
24 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y systems. W e will now lo ok at other p ossible p erturba tion structures. F or more detail on these struct ures ( in the complex case) refer to P ack ard and Do yle [20]. Conside r a blo cks w hich are of the form scalar × iden tit y , where the scalar is unknown.
2.2. MOD ELING UNCER T AIN SYSTEM S 25 = Cz − 1 ( I − z − 1 A ) − 1 B + D = F u ( P ss ,z − 1 I ) , where P ss is the r eal v alued matr ix, P ss = AB CD , and the scalar × identit y , z − 1 I , has dimension equal to the state dimension of P ( z ).
26 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y where the transport dela y , T d , and th e comb ustion lag, T c , are appro ximately , T d = 252 v and T c = 202 v . F or the purp oses of our example we w ant to design an air /fuel ratio controller that w orks for all engi ne sp eeds in the range 2,000 to 6,000 rp m.
2.2. MOD ELING UNCER T AIN SYSTEM S 27 Putting all the pieces together gives an engine mo del in the follo wing fractional form. P ( s )= F u ( P mod , ∆) , where P mod = − 15 . 87 s 7 . 14( s − 19 . 8) s 2 141 . 5(1 + 1 .
28 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y and ∆ ∈ B∆ , with the structure defined as, ∆ = diag( δ v I 2 , ∆ 1 ) δ v ∈R , ∆ 1 ∈C . Note that this is an LFT with a rep eated real-v alued parameter, δ v ( | δ v |≤ 1), and a complex p erturba tion, ∆ 1 ( | ∆ 1 |≤ 1).
2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 29 An area of work, kno wn as iden tificat ion in H ∞ , lo oks at experimen tal iden tification techniques whic h minimize the worst case H ∞ error bet w een the ph ysical system and the mo del. The fo llowing w o rks address this issue: [27, 28, 29, 30, 31, 32, 33, 34, 35].
30 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.7: LFT confi guration for control ler synth esis, G ( s )= F l [ P ( s ) ,K ( s )] Note that the interconnection structure, P ( s ), given here, diff ers from that discussed in the previous sect ion.
2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 31 extend these approac hes t o the case where P ( s ) i s replaced b y F u ( P ( s ) , ∆), ∆ ∈ B∆ . 2.3.1 H ∞ Design Ov erview Again, reca ll from Section 2.1.2, the H ∞ is norm of G ( s )i s , k G ( s ) k ∞ =s u p ω σ max [ G ( ω )] .
32 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y 2.3.2 Assumptions for the H ∞ Design Problem There are sev eral assumpt ions required in order to achiev e a w ell-p osed design problem. The DGKF paper giv es a state-space solut ion to the H ∞ design problem and w e will use a similar notation here.
2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 33 that the eff ect of all disturban ces, w , at ev ery frequency , can b e measured b y the controller. If either of these conditions are not met then the problem could be ill-p osed. It is p ossible to violate these conditions by using pure in tegrators as design weigh ts.
34 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y W e hav e partitioned the matr ix in to tw o n × n blocks, X 1 and X 2 .I f X 1 is inv ertible, then X = X 2 X − 1 1 , is the unique, stabilizing so lution to the ARE. The ability to form X doesn’t dep end on the pa rticular choice of X 1 and X 2 .
2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 35 Cho ose γ> 0 and fo rm the following Ha miltonian matr ix, H = Aγ − 2 BB T − C T C − A T .
36 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y an y with a zero real part. In practice w e must use a tolerance t o determine what is considered as a zero real part. Finding a ba sis for the stable subspace of H in volv es either an eigenv alue o r Sch ur decomposition.
2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 37 and the component s, C 1 x and D 12 u are orthogonal . D 12 is also assumed to be normalized. This essentially means that there is no cro ss-weigh ting betw een the sta te and input penalties.
38 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y L ∞ = − Y ∞ C T 2 Z ∞ =( I − γ − 2 Y ∞ X ∞ ) − 1 ˆ A ∞ = A + γ − 2 B 1 B T 1 X ∞ + B 2 F ∞ + Z ∞ L ∞ C 2 . Actually , the ab ov e formulat ion can b e used to parametrize all stabilizing con trollers which satisfy , k G ( s ) k ∞ <γ .
2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 39 a) Choose γ ≥ γ opt b) F orm H ∞ and J ∞ c) Check that H ∞ ∈ dom(Ric) and J ∞ ∈ dom(Ric). d) Calculate X ∞ = Ric( H ∞ )a n d Y ∞ = Ric( J ∞ ) e) Check that X ∞ ≥ 0a n d Y ∞ ≥ 0 f ) Check that ρ ( X ∞ Y ∞ ) <γ 2 g) Reduce γ a n dg ot os t e pb ) .
40 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y eigen v alue of X ∞ (and Y ∞ ) is displayed. The ultimate test of the softw a re is to form the closed lo op system a nd chec k both its stability and nor m. W e str ongly sug gest that the user alwa ys perform this step .
2.3. H ∞ AND H 2 DESI GN METHODOLOGIES 41 framew ork. W e again assume the simplif ying assumptions used in Sect ion 2 .3.4 The H 2 design solution is obtained (at lea st conceptually) from the H ∞ design pro cedure by setting γ = ∞ and using the resulting central controller.
42 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.8: Closed loop system, G ( s ), for performance analysis initial choice of γ fo r the H ∞ design pr o cedure. W e will see later (Section 2 .5) that it can also b e used to initialize the D - K iter ation pro cedur e when an o p en-lo op H ∞ design is po orly conditioned.
2.4. µ ANAL YSIS 43 descriptions are considered, where B again denotes the unit ball. Power : BP = ( w lim T →∞ 1 2 T Z T − T | w ( t ) | 2 dt ≤ 1 ) (2.12) Energy : B L 2 = ( w k w k 2 2 = Z ∞ −∞ | w ( t ) | 2 dt ≤ 1 ) (2.
44 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2.9: Perturb ed closed lo op s ystem for stability analysis the induced norms, the reader is referred to Do y le [2]. The ma jor adv ant a ge of c ho o sing BP or B L 2 is that t he test for th e p erformance can b e considered in terms of the same norm as stability .
2.4. µ ANAL YSIS 45 Consider the case where the model ha s only one full ∆ block ( m =1 a n d q =0 i n Equation 2.9 ). This is often r eferred to as unstructured, a nd the well kno wn r esult (refer to Zames [67] and Doyle and Stein [5 ]) is g iven in the following lemma.
46 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y F u ( G ( s ) , ∆) stable fo r al l ∆ ∈ B∆ if and on ly if k µ ( G 11 ( s )) k ∞ < 1 . wher e k µ ( G 11 ( s )) k ∞ =s u p ω µ [ G 11 ( ω )] . The use of this notation masks the fact that µ is al so a function of the p erturbati o n structure, ∆ .
2.4. µ ANAL YSIS 47 if and on ly if k µ ( G ( s )) k ∞ < 1 , wher e µ is taken wi th r esp e ct to an a ugmente d structur e b ∆ , b ∆ = n diag(∆ , ˆ ∆) ∆ ∈ ∆ , ˆ ∆= C dim( w ) × dim ( e ) o .
48 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y F or the o ther extreme consider a single full blo c k ( ∆ = { ∆ | ∆ ∈C n × n } ); the definition of µ is now the same as tha t for the maximum singular v alue, ∆ = { ∆ | ∆ ∈C n × n }⇒ µ ( M )= σ max ( M ) .
2.4. µ ANAL YSIS 49 Actually , the lower bound is always equal to µ but the implied optimization ha s lo cal maxima which are not glo bal. F or the upper b ound Sa fonov and Doyle [72], hav e shown that finding the infim um is a conv ex problem and hence more easily solved.
50 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y and tw o blo ck structures, ∆ 1 (compa tible with M 11 )a n d ∆ 2 (compatible with M 22 ). There are t wo perturbed subsystems th at we can st.
2.4. µ ANAL YSIS 51 2.4.6 State-sp ace Robustness Analysis T ests W e will lo ok at some more adv anced state-space approac hes to the analysis of robust performance. Most users of the soft ware wi ll concentrat e on the more common frequency domain analysis metho ds cov ered in Section 2 .
52 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Therefore µ 1 ( A ) < 1 is equiv alen t to our system b eing stable. F urthermo re, the maximum modulus theorem for a stable system tells us t.
2.4. µ ANAL YSIS 53 Figure 2.10: P erturb ed system for st ate-space robust ness tests.
54 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Note that the nominal system is given by , G nom ( z )= F u AB 1 C 1 D 11 ,z − 1 I , and the perturbed system is, G ( z )= F u ( F l ( G, ∆) ,z − 1 I ) . W e a ssume that ∆ is an element of a unity norm b o unded blo ck structure, ∆ ∈ B∆ .
2.4. µ ANAL YSIS 55 iii ) There exists a c onstant β ∈ [0 , 1] s uch that for e ach fixe d ∆ ∈ B∆ , G ( z ) is stable and for zero initial st ate r e sp ons e, e satisfies k e k 2 ≤ β k w k 2 (r obust p erformanc e).
56 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y i ) State-spa ce upp er bound: inf D s ∈D s σ max [ D s G ss D − 1 s ] < 1; ii ) F re quency domain, constant D , upper b ound: inf D p .
2.4. µ ANAL YSIS 57 The gap betw een the state- space (or constant D ) upp er b o und and the frequency domain upp er bo und is mor e significant. In the sta te-space upper b ound, a single D scale is selected . This giv es robust performance for all ∆ satisfy ing, k v k≤k z k for all e ∈L 2 .
58 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y 2.4.7 Analysis with b oth Real and Complex Perturbations The ab ove results o nly apply to the case where ∆ is considered as a co nstant complex v alued matrix a t each frequency .
2.5. µ SYNTHESIS AND D - K ITERA TION 59 Figure 2.11: The generic in terconnecti o n structu re for synthesis k F u ( F l ( P ( s ) ,K ( s )) , ∆) k ∞ ≤ 1. This is equiv alen t to K ( s ) satisfying µ [ F l ( P ( s ) ,K ( s ))] < 1. In other words, the closed lo op system sa tisfies the ro bust performance specification.
60 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Recall that this is an upp er bo und for the µ pr oblem of in terest, implying that, µ [ F l ( P ( s ) ,K ( s )] ≤ 1 , as required.
2.5. µ SYNTHESIS AND D - K ITERA TION 61 dynamic system. This requires fitting an appro x imation to the upper b ound D -scale in the itera tion. W e will now look a t this issue mor e closely . The D - K iteration pro cedur e is illustr ated schematically in Figure 2.
62 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Figure 2. 12: D - K iterat ion pro cedure: a) Design H ∞ ( or other) con trolle r: K 0 ( s )[ s t e p i )]. b) Closed lo op p erturbed system for µ analysi s [step ii )]. c) F requency b y frequency upper bo und D ( ω ) s cale a pproximation to µ analysis [step iii )].
2.5. µ SYNTHESIS AND D - K ITERA TION 63 frequency we w ould hav e, D = d 1 I 1 . . . d m I m I e , where the identit y I e is of dimensions dim( e ) × dim( e ). The calculation of a new H ∞ con tro ller requires a state-space rea lization of D ( ω ).
64 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y Sev eral asp ects of t his pro cedure are worth noti ng. F or the µ analysis and D scale calculation, a fr equency grid must b e chosen. The range and reso lution of this g rid is a matter o f engineering judgemen t.
2.6. MODEL REDUCTION 65 2.6.1 T runcation and Residualization The simplest fo rm of mo del r eduction is state truncation. Co nsider a system, P ( s ), with a partitio ned state matrix, P ( s )= A 11 A 12 B 1 A 21 A 22 B 2 C 1 C 2 D .
66 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y The controllabilit y grammian, Y is defined a s, Y = Z ∞ 0 e At BB T e A T t dt, and the observ abilit y grammian, X , is defined as X = Z ∞ 0 e A T t C T C e At dt.
2.6. MODEL REDUCTION 67 W e will now lo ok at a particular choice of transfor mation. F or a minimal r ealizatio n, w e can always find a transformation that gives, ˆ Y = TY T T =Σ , and ˆ X = T − T XT − 1= Σ , where Σ = diag( σ 1 ,...,σ n )a n d σ i ≥ 0, i =1 ,.
68 CHAPTER 2. OVER VIE W OF THE UNDE RL YING THEO R Y and Glov er [79] independently obtained the following b ound on the error induced by balanced trunca tion. Theorem 13 Given a st able, r at ional, P ( s ) ,a n d P bal ( s ) , the b alanc e d tr uncation of or der k< n .
2.6. MODEL REDUCTION 69 Consider the problem of finding the stable, order k realization w hich minimizes the Hank el norm of the error. Defi ne, P hankel ( s ) a s the minimizing system. Then we ha ve, σ k +1 ≤k P ( s ) − P hankel ( s ) k H = inf P k ( s ) stable k P ( s ) − P k ( s ) k H .
Chapter 3 F unctional Description of X µ 3.1 In tro duction This chapter describes the X µ functions in the co n text of their in tended usage. Chapter 2 provides the reader with an idea of the theo retical basis b ehind the v arious analysis and design ca lculations.
72 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ 3.2.1 D ynamic System s Xmath has a dynamic system data ob ject which specifies a dynamic system in terms of A , B , C and D matrices.
3.2. D A T A OBJECTS 73 As above, these p olynomials can be sp ecified b y their ro ots or their coefficients. Note that w e can sp ecify the v ariable, and for con tin uous systems w e use “s”. T o create a discrete system “z” is used. # Generate the system from the numerator and denominator # coefficients.
74 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Because the dynamic system is a built-in data ob ject, the lab el info rmation will be display ed by s imply typing the ob ject name ( sys in the ab ov e) o r app ending a question mark to the statement. The core function commentof is used to read the commen t attached to a sp ecified v ariable.
3.2. D A T A OBJECTS 75 App ending and Merging Data Time functions, for cre ating simulation inputs for example, can b e cr eated by com bining pdm s. Xmath has t wo suc h core functi ons: concatseg and insertSeg The concatseg app ends the data of one pdm to a nother.
76 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Data ca n also b e extracted b y indep endent v ariable index num b er , by providing a scalar argumen t to the pdm . In the following example the fifth through tenth and the t wen tieth indep endent v ariables are specified for the smaller pdm , pdm2 .
3.2. D A T A OBJECTS 77 # index of the pdm the value 100. idxlst = indexlist([4,1,2] ) pdm1(idxlst) = 100 Op erations on the Indep endent V ariables/Domain The domain of a pdm is readily changed via the pdm co mmand. The fo llowing example illustrates a co mmon applica tion; changing a frequency do main in Her tz to radians/s econd.
78 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ # Select columns 1, 3 ,4&5a n d rows 2 & 7 from the # pdm: bigpdm. subpdm = bigpdm([1,3:5],[2 ,7]) This referen cing format can also be used to assign data to parts of a larger pdm . This is shown in the fo llowing example.
3.2. D A T A OBJECTS 79 and B . Augmentation for pdm s simp ly p erforms the augmen tation at eac h domain v ariable. The domains m ust b e the same. Diagonal augmen tation can be p erformed with the X µ functio n daug .
80 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ poles, or a zero D term. Generating rando m systems is useful for simulating systems with unknown subsystems.
3.3. MA TRIX INFORMA TION, DISPLA Y AND PLOTTING 81 # N is the decimation ratio. smallpdm = bigpdm([1:N:length(bigpdm )]) 3.2.5 Con tinuous to Discrete T ransformations Xmath has a single function, discretize , fo r calculating contin uo us to discrete transforma tions.
82 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ 3.3.2 F ormatted Displa y F unctions It is often useful to consider complex num b ers in terms of frequency and damping.
3.3. MA TRIX INFORMA TION, DISPLA Y AND PLOTTING 83 g1 = ctrlplot(sys1g, { bode } ); g1 = ctrlplot(sys2g,g1, { bo de } ); g1 = plot( { keep=g1,title = "Bode plots",... legend = ["sys1","sys2"] } )? Frequency 0.1 1 0.01 10 Magnitude 0.
84 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ # Nyquist plots g2 = ctrlplot(sys1g, { nyqui st } ); g2 = ctrlplot(sys2g,g2, { ny quist } ); g2 = ctrlplot(-1,g2, { nyqui st,marker= 1,line=0 } ); g2 = plot(g2, { projection=" orthograph ic",... legend=["sys1","sys3" ,"critical point"],.
3.4. SYSTEM RESP ONSE FUNCTIONS 85 3.4 System Resp onse F unctions 3.4.1 Creating Time Domain Signals The Signal Analysis Mo dule cont a ins several functions which are useful for building time domain signals: gcos and gsin .X µ provides gstep for the creation of stai r-step signals.
86 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ spacing in the input signa l pdm . This mea ns that the input pdm must be regularly spaced. Stair-cas e input functions with relatively few data po ints will often give erroneous results. The input signal should b e interpolated (with interp ) before being multiplied with t he D ynamic System .
3.4. SYSTEM RESP ONSE FUNCTIONS 87 [a,b,c,d] = abcd(sys) sys = system(a,b,c,d) y1 = system(sys, { X0=[-1;0] } )*u # Now plot the result g1 = ctrlplot(u, { line style=2 } ); g1 = ctrlplot(y0,g1, { line.
88 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ and intstep are the in terp olation order an d integrat ion step size. The system m ust b e con tinu ous. The int egration order and sample t ime are prespecified for discret e tim e systems making the * op erato r is suita ble for such sim ulatio ns.
3.4. SYSTEM RESP ONSE FUNCTIONS 89 maximum. This can b e ha ndy when first examining a high order system. An example is giv en b elow. # Create a single-input, two-output system. sys = [1/makepoly([1,0.1,1 ],"s");... makepoly([1,1],"s")/m akepoly([1 ,10],"s") ] # The automatic point selection is used.
90 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.1 1 10 0.01 100 Magnitude 1e-05 0.0001 0.001 0.01 0.1 1 10 1e-06 100 Frequency 0.1 1 10 0.01 100 Phase (degrees) -150 -100 -50 0 50 -200 100 Note that the defa ult frequency units in Xmath are Hertz.
3.5. SYSTEM INTERCONNECTION 91 Figure 3.2: Generic Red heffer in terconnect ion structure 3.5 System In terconnection In terconnections of systems are used extensively in the design, analysis and sim ula tion calculati ons.
92 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Figure 3.3: Example in terconnecti on of subsystem s Figure 3.4: Example in terconnected syst em.
3.5. SYSTEM INTERCONNECTION 93 Using sysic to fo rm this interconnection can be considered a s four distinct o p erations. • Specify the individual subsystems. • Name and dimension the input signals. • Specify , algebra ically in terms of subsystem outputs or input signals, the output of the interconnected system.
94 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Specify the o utputs in terms of the subsystem names and the input names. Note that individual o utputs of MIMO systems and s imple ar ithmetic combinations can b e specified. Paren thesis sp ecify which output of a MIMO system is to be used.
3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 95 (rad/sec) ratio -1.0000e+00 0.0000e+00 1.0000e+00 1.0000 -5.0000e-01 6.3048e+00 6.3246e+00 0.0791 -5.0000e-01 -6.
96 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Figure 3.5: In terconnection structu re for control ler syn thesis structure contains more than just the op en lo op plant. It t ypica lly also con ta ins frequency dependent w eighting functions and specifies the structur e of t he in terconnection b etw een the ope n lo op pla nt and the controller.
3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 97 equation solutio n pro cedure o ften b ecomes po o rly conditio ned. Displaying in termedia te calculation results allows the user to fine tune several tolerances if necessary . The intermediate Hamiltonian a nd Riccati so lution details a re displayed as the bisection pro ceeds.
98 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 10 0.01 100 Open loop plant Frequency 0.01 0.1 1 10 0.001 100 Phase (degrees) -300 -200 -100 -400 0 The desired clos ed lo op configur ation is illustrated in Figure 3.
3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 99 Figure 3.7: W eight ed design in terconnect ion structure: p In order to set up the desig n problem, we consider ref as an unknown input a nd the trac king error (input to k ), and the actuator signal, u , as outputs to be minimized.
100 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.01 0.1 1 10 0.001 100 Weighting functions Wperf Wact # Form the weighted interconnection structure sysnames .
3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 101 12.500 5.2e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 6.250 5.2e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 3.125 5.1e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 1.562 5.0e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 0.781 3.9e-01 -9.
102 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ (rad/sec) ratio -1.4046e-01 -2.3161e-01 2.7087e-01 0.5186 -1.4046e-01 2.3161e-01 2.7087e-01 0.5186 -1.5863e+00 3.4754e+00 3.8203e+00 0.4152 -1.5863e+00 -3.4754e+00 3.8203e+00 0.4152 -5.2060e+00 0.0000e+00 5.
3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 103 Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.001 0.01 0.1 1 0.0001 10 Controllers Kinf K2 # Examine sensitivity functions sensinf = inv(1 + plant*Kinf) .
104 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 0.01 10 Sensitivity functions Kinf K2 Note tha t with the interconnection structure shown in Figure 3 .7, the clo sed lo op transfer fun ction from ref to e (1) is simply Wperf*sensinf .
3.6. H 2 AND H ∞ ANAL YSIS AND SYNTHESIS 105 g3 = ctrlplot(step,g3, { line style=2 } ); g3 = plot(g3, { title="Step responses",... legend=["Kinf";"K2";" input"] } )? 2468 0 10 0.2 0.4 0.6 0.8 1 0 1.2 Step responses Kinf K2 input 3.
106 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Bounds on the H ∞ norm are ret urned as out . An estimate of the frequency where the norm is achiev e d is returned a s omega .
3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 107 3.7 Structured Singular V alue ( µ ) Analysis and Syn thesis This section cov ers the functions used in the D - K i teration procedure.
108 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ The outputs of the µ functio n a re: the upper and low er bounds for µ ;t h e D matrix fo r the upp er b o und; the Q matr ix for the lower b o unds; and a sensitivity estimate for the part of the D matrix corresponding to eac h blo ck in ∆.
3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 109 c = 1/sqrt(gamma); d = -sqrt(beta/gamma) f = (1+jay)*sqrt(1/(gamma* beta)) psi1 = -pi/2; psi2 = pi U = [a,0; b,b; c,jay*c; d,f] V = [0,a; b,-b; c,-jay*c; f*exp(jay*psi1), d*exp(jay*psi2)] scl = diagonal(random(4,1) +0.
110 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ mubnds2? mubnds2 (a column vector) = 3.17155 3.17155 det(eye(4,4) - M*Delta2)? ans (a scalar) = -2.62055e-16 + 5.82345e-17 j 3.7.2 The D - K Iteration Recall from Section 2.5 that the D - K iter ation is used a s an approximation to µ syn thesis.
3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 111 (a) Cont roller and closed loop are satisfactory so stop the iterati o n. (b) The iteration has con verged and the controll er and closed lo op are not satisfactory . In this case the weighted design problem m ust be reformulated.
112 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Figure 3. 9: H ∞ controller design. Step 9 in the en umer ated pro cedure There is a ctually another p ossibility at step 5 ; numerical problems cause the itera tion to diverge. As γ approac hes its optimal v alue, the n umerical properties of the calcul ation deteri orate.
3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 113 Both the D and D − 1 systems ( Dsys and Dinvsys ) are returned. The D scale ( Dmagdata ) comes from a µ ca lculation on a closed lo op system. Howev er, Dsys and Dinvsys are requir ed to m ultiply the op en lo op system.
114 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ creating weigh ts fro m data and simple system identification. X µ provides t wo user callable functions for fitting SISO tra nsfer functions to da ta. The first is mkphase calculates the phase corresp onding to a minim um-phase stable system from magnitude data.
3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 115 plant = 1/makepoly([1,0,-0.01], "s") Wm = makepoly([1,20],"s")/ makepoly([ 1,200],"s" ) omega = logspace(0.
116 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Wperf = makepoly([0.01,1],"s")/ makepoly([ 1,0.01],"s ") Wact = 0.1* makepoly([1,1],"s")/make poly([0.05 ,1],"s") Wnoise = 0.01 Wref = makepoly([0.005,1],"s")/ makepoly([ 0.
3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 117 nms = ["plant";"Wm";"Wperf ";"Wact";" Wnoise";"W ref"] inp = ["delt";"ref";"noise ";"u"] outp = ["100*Wm"; "Wperf"; "Wact"; "Wref-0.
118 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ blk = [1,1; 2,2] [rpbnds1,D1,Dinv1,Delt a1,sens1] = mu(Gg,blk) g3 = ctrlplot(np, { log } ); # plot on a log-linear scale g3 = ctrlplot(rs,g3, { log,l ine style=3 } ); g3 = ctrlplot(rpbnds1,g3, { log,line style=[4,5] } ); g3 = plot(g3, { !grid,title= "mu analysis",y lab="Magnitude",.
3.7. STRUCTURED SINGULAR V ALUE ( µ ) ANAL YSIS AND SYNTHE SIS 119 Frequency (Hz) 0.1 1 10 0.01 100 Magnitude 0.1 1 10 0.01 100 D scale fit, block: 1 Magnitude data Previous fit, order: 2 New fit, or.
120 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ 0.1 1 10 0.01 100 Magnitude 0.5 1 1.5 2 2.5 3 3.5 0 4 mu analysis of robust performance Kmu Kinf 3.7.4 Constructing Rational Perturbations F or sim ula tion pur p oses it is useful to be able to construct a rational a pproximation to the ∆ r eturned by the µ calculat ion.
3.8. MODEL REDUCTION 121 pert = randpert(blk, { sys,sfreq, complex,pn orm } ) The user ca n specify whether the p er turbation is a dynamic system o r matrix, and whether it is real or co mplex v alued, in addition to specifying the no rm.
122 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ A g reater r ange of mo del reductio n functio ns is av a ilable in the Mo del Reduction Mo dule. Some of the functions descr ib ed here are cro ss-licensed with that mo dule. Section 2 .6 describ es the theory b ehind these functions.
3.8. MODEL REDUCTION 123 g1 = ctrlplot(sysout2g,g1, { logmagplo t,line style=4 } ); g1 = ctrlplot(residerror,g 1, { logmagp lot,line style=5 } ); g1 = ctrlplot(truncerror,g 1, { logmagp lot,line style=6 } ); g1 = plot(g1, { !grid,legend =["origina l system";.
124 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ # Displaying the Hankel singular values shows which # states are close to unobservable and uncontrollable. hsv? hsv (a column vector) = 0.0741834 0.0726887 0.0264105 0.000146401 2.7699e-07 # Compare to the errors from the previous example.
3.8. MODEL REDUCTION 125 3.8.3 Hank el Singular V alue Approximation The function ophank (also cr oss-licensed from the Mo del Reductio n Mo dule) is used to perform optimal Hank el norm appro ximation. Recall f rom Section 2.6.3 that there is an astable system achieving the lower bound.
126 CHAPTER 3. FUNCTIONAL DESCRIPTION OF X µ Frequency 0.1 1 10 0.01 100 Magnitude 1e-05 0.0001 0.001 0.01 1e-06 0.1 original system Hankel norm approximation error: ophank error: balmoore error: sre.
Chapter 4 Demonstration Examples 4.1 The Himat Example The following demo can be run by executing the f ollowing Xmath command: execute file = "$XMATH/demos/xMu/ himatdemo" 4.1.1 Problem Description The Hima t is a small scale remotely piloted a ircraft built to investigate high maneuverabilit y figh ter aircraft design.
128 CHAPTER 4. DEMONSTRA TION EXAMPLES δv Perturbations along the velo city v ector. α Angle of attack. I .e. angle betw een the veloc it y vector and the aircra ft’s longitudinal axis. q Rate-of -c hange of aircraft attitude angle. θ Aircraft at titude angle.
4.1. TH E HIMA T EXAMPLE 129 Figure 4.2: In terconnection st ructure for the himat design exam ple The inputs are the elev o n pos ition and the canard po sition. The outputs that ar e to be k ept small are angle-of-attac k ( α ) and pitch angle ( θ ).
130 CHAPTER 4. DEMONSTRA TION EXAMPLES himat (a state space system) = A -0.0226 -36.6 -18.9 -32.1 0 -1.9 0.983 0 0.0123 -11.7 -2.63 0 00 1 0 B 00 -0.414 0 -77.
4.1. TH E HIMA T EXAMPLE 131 angle-of-attack pitch angle System is continuous 4.1.3 Creating a W eigh ted In terconnection Structure for Design The multiplicativ e input p erturba tion weight, W del , is constructed as a tran sfer functi on. The weigh t is, W del = 50( s + 100) ( s + 10000) .
132 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 0.01 0.1 1 10 100 1000 0.001 10000 Magnitude 1 10 0.1 100 Weights for HIMAT Perturbation weight Performance weight The p erturbat ion w eight, W del , should actually be tw o-input, tw o -output. This is also true of the performance w eigh t, W p .
4.1. TH E HIMA T EXAMPLE 133 sysn = ["himat";"wdel";"wp"] in = ["pert(2)";"dist(2)"; "control(2 )"] out = ["wdel";"wp";&.
134 CHAPTER 4. DEMONSTRA TION EXAMPLES gamma Hx eig X eig Hy eig Y eig nrho xy p/f 6.000 2.3e-02 5.6e-05 2.3e-02 0.0e+00 0.0626 p 3.400 2.3e-02 5.7e-05 2.3e-02 0.0e+00 0.2020 p 2.100 2.3e-02 5.9e-05 2.3e-02 0.0e+00 0.5798 p 1.450 2.3e-02 6.4e-05 2.3e-02 0.
4.1. TH E HIMA T EXAMPLE 135 Zeros: real imaginary frequency damping (rad/sec) ratio -2.2516e-02 0.0000e+00 2.2516e-02 1.0000 -1.7226e+00 0.0000e+00 1.7226e+00 1.0000 -3.0272e+00 0.0000e+00 3.0272e+00 1.0000 -3.1034e+01 0.0000e+00 3.1034e+01 1.0000 -1.
136 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 1 10 100 1000 0.1 10000 Magnitude 0.01 0.1 0.001 1 Controller: k1 Frequency 1 10 100 1000 0.1 10000 Phase (degrees) -200 -100 0 100 -300 200 Onto the closed loop, first chec king that it is stable by loo king at the po le po sitions.
4.1. TH E HIMA T EXAMPLE 137 -2.2517e-02 0.0000e+00 2.2517e-02 1.0000 -2.2600e-02 0.0000e+00 2.2600e-02 1.0000 -3.0000e-02 0.0000e+00 3.0000e-02 1.0000 -3.0000e-02 0.0000e+00 3.0000e-02 1.0000 -2.9369e+00 0.0000e+00 2.9369e+00 1.0000 -2.9974e+00 0.0000e+00 2.
138 CHAPTER 4. DEMONSTRA TION EXAMPLES 1 10 100 1000 0.1 10000 0.5 1 1.5 0 2 Singular value plot of the closed loop 4.1.5 µ A n a l y s i so ft h e H ∞ Con troller The H ∞ con trol law can b e analyzed using µ -analysis. The closed-lo o p system, g1 ,h a s4 inputs and 4 outputs.
4.1. TH E HIMA T EXAMPLE 139 complex v alued blo cks. The upp er and lower bounds of the µ function are return ed in bnds . Also returned ar e the sca ling ma trices, D and Dinv , corresponding to the upper b ound. The smallest destabilizing p ertur bation at each frequency is returned a s Delta .
140 CHAPTER 4. DEMONSTRA TION EXAMPLES 1 10 100 1000 0.1 10000 0.6 0.8 1 1.2 1.4 1.6 0.4 1.8 Max. singular value and mu comparison max. singular value mu upper bound mu lower bound Note that µ ( g1g ) is not less than one at all frequencies — we ha ve not met the design ob jectiv es.
4.1. TH E HIMA T EXAMPLE 141 The second thing to note is that the in ter connection structure has the additional control inputs and meas urement outputs. The D and Dinv systems must be augmented wit h iden tities corresponding to these additional inputs and outputs.
142 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency (Hz) 1 10 100 1000 0.1 10000 Magnitude 0.1 1 0.01 10 D scale fit, block: 1 Magnitude data Previous fit, order: 0 New fit, order: 3 Frequency (Hz) 1 10 100 1000 0.
4.1. TH E HIMA T EXAMPLE 143 comment D1sys "system approx. to D1" comment D1invsys "system approx. to D1inv" 4.1.7 Design Iteration #2 The new D scales can b e pre and p ost m ultiplied o n to the orginal in terconnection structure.
144 CHAPTER 4. DEMONSTRA TION EXAMPLES g2 = starp(himat ic,k2) comment g2 "closed loop: iteration 2" [,,nx] = size(k2) display "k2 now also has " + string(nx) + " states" k2 now also has 20 states This design pr obably resulted in a co ntrol la w which achiev ed an infinity norm of approximately 1.
4.1. TH E HIMA T EXAMPLE 145 1 10 100 1000 0.1 10000 0.6 0.8 1 1.2 1.4 1.6 0.4 1.8 Robustness analysis for the g2 system g2: mu upper bound g2: mu lower bound g1: mu upper bound g1: mu lower bound com.
146 CHAPTER 4. DEMONSTRA TION EXAMPLES g3 = starp(himat ic,k3) g3g = freq(g3,omega) [bnds3,D3,D3inv,Delta3 ,sens3] = mu(g3g,blk) A t this point we could do ano ther iteration (to get k4 ) or p erha ps run some simulations to chec k out k3 more thoroughly .
4.1. TH E HIMA T EXAMPLE 147 d = zeros(2,2) klp = system(a,b,c,d) comment klp "loop shape controller" W e will compare the designs, with no e rror or uncert aint y we igh ts, for th e nominal case and with a p erturba tion blo ck of ∆ = [0.
148 CHAPTER 4. DEMONSTRA TION EXAMPLES comment gsim mu nom "nominal closed loop sys: mu ctrl" comment gsim hinf nom "nominal closed loop sys: hinf ctrl" comment gsim lp nom "nominal closed loop sys: klp ctrl" A step disturbance is in tr o duced into the first c hannel.
4.1. TH E HIMA T EXAMPLE 149 time 0.5 1 1.5 0 2 -0.5 0 0.5 1 -1 1.5 Kmu step dist. response (nominal) time 0.5 1 1.5 0 2 -0.5 0 0.5 1 -1 1.5 Kinf step dist.
150 CHAPTER 4. DEMONSTRA TION EXAMPLES The lo opsha ping design gives a decoupled resp onse. Both the H ∞ and µ designs trade decoupling for speed of resp onse and, as w e shall see, robustness with resp ect to per turbations. The simulation is rep ea ted with a per turbation o f size 0 .
4.1. TH E HIMA T EXAMPLE 151 time 0.5 1 1.5 0 2 -0.5 0 0.5 1 -1 1.5 Kmu step dist. response (perturbed) time 0.5 1 1.5 0 2 -0.5 0 0.5 1 -1 1.5 Kinf step dist.
152 CHAPTER 4. DEMONSTRA TION EXAMPLES The lo opsha ping controller had goo d nominal perfo rmance and very p o or r obust pe rformance. This was illustra ted with a relatively small pertur bation. The difference betw een the µ and H ∞ controllers was small in b oth the nominal a nd p ertur be d cases.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 153 4.2 A Simple Flexible Structure Example The demonstration scrip t, jplphBdemo.ms , runs throu gh a D - K iteration desig n for a simple flexible structur e problem.
154 CHAPTER 4. DEMONSTRA TION EXAMPLES Figure 4.3: Sc hematic diagram of t he JPL Phase B opti cal testb ed design proble m problem is the fact that the voice-coil mirror assem bly has significant mass and its mov ement excites a mo de in the structur e.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 155 static gain. W e will include some dynamic uncertaint y in the actual design. piezo = 1 4.2.2 Creating the W eigh ted Design In terconnection Structure The weighted, open-lo op design interconnection structure is illustra ted in Figure 4.
156 CHAPTER 4. DEMONSTRA TION EXAMPLES The additive weigh t clearly provides for significant high frequency uncer taint y . A m ultiplicative w eight mo dels the low frequency uncertain ty . The v alue selected is somewhat arb itrary and can be considered as a tunable design w eigh t.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 157 F = 100 Wactp = makepoly([1/(2*pi*F),1] ,"s")/... makepoly([1/(200*pi*F) ,1],"s") Wactp = Wactp*4 The low er frequency response of the v o ice-coil system mea ns that the W actv we i g ht should begin rolling up at around 10 Hz.
158 CHAPTER 4. DEMONSTRA TION EXAMPLES Wdist,Wnoise,Wperf,W actp,Wactv ,piezo) size(P)? ans (a row vector) = 7 6 8 Now select a frequency grid for calculating the frequency resp onses. Some additiona l po ints are included near the oscillato ry mo des.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 159 Frequency 10 100 1 1000 Magnitude 0.01 0.1 1 10 100 1000 10000 0.001 100000 voice coil model Frequency 10 100 1 1000 Phase (degrees) -20 0 20 40 60 80 -4.
160 CHAPTER 4. DEMONSTRA TION EXAMPLES And examine the design weigh ts. weightsg = freq(weights,omega) gph2 = ctrlplot(weightsg, { logma gplot } ); gph2 = plot(gph2,legend=["Wperf ";"Wavoice ";"Wmvoice ";... "Wmpiezo";"Wdist";"W noise";"Wa ctp";"Wact v"],.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 161 Frequency 10 100 1 1000 Magnitude 0.01 0.1 1 10 0.001 100 Design weights Wperf Wavoice Wmvoice Wmpiezo Wdist Wnoise Wactp Wactv.
162 CHAPTER 4. DEMONSTRA TION EXAMPLES 4.2.3 Design of an H ∞ Con troller An H ∞ design is now performed. Recall that we hav e one interferometer measuremen t and tw o co nt r oller outputs. nmeas = 1 ncon = 2 glimits = [0;20] [Khinf,gamma] = hinfsyn(P,nmeas,nc on,glimit s, { tol=0.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 163 -3.5543e+02 -3.5543e+02 5.0265e+02 0.7071 -3.5543e+02 3.5543e+02 5.0265e+02 0.7071 -6.4598e+02 -3.3537e+01 6.4685e+02 0.9987 -6.4598e+02 3.3537e+01 6.4685e+02 0.9987 -3.8358e+02 -5.3777e+02 6.6055e+02 0.5807 -3.
164 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 10 100 1 1000 Magnitude 0.001 0.01 0.1 0.0001 1 Controller: Khinf Frequency 10 100 1 1000 Phase (degrees) -50 0 50 100 150 -100 200.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 165 4.2.4 Robustness Analysis The blo ck str ucture has tw o p ertur bation ∆ blo cks and a “per formance” blo ck. The t wo voice-coil per turbations are put in to a sing le 1 × 2 blo ck as they enter the system at the same poin t.
166 CHAPTER 4. DEMONSTRA TION EXAMPLES gph4 = ctrlplot(npbnds, { log,lin e style=4 } ); gph4 = ctrlplot(rsbnds,gph4, { lo g,line style=[3,5] } ); gph4 = ctrlplot(rpbnds,gph4, { lo g,line style=[1,2] } ); gph4 = plot(gph4, { !grid,legend= ["Nom perf";"Rob stab (up bnd)";.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 167 0.5 1 1.5 0 2 -0.01 -0.005 0 0.005 0.01 -0.015 0.015 perturbed closed loop: vc actuator Khinf Kmu.
168 CHAPTER 4. DEMONSTRA TION EXAMPLES 4.2.5 D - K Iteration W e will now perfo rm one D - K itera tion to generate the con tro ller Kmu . Significant robustness and p e rformance improvemen t is achiev ed with only one iteration. T ransfer f unctions are fit to t he D -scales from the previous robust performance µ test.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 169 Kmug = freq(Kmu,omega) gph5 = ctrlplot(Kmug, { bode } ); gph5 = plot(gph5, { title="Contro ller: Kmu" } )?.
170 CHAPTER 4. DEMONSTRA TION EXAMPLES Frequency 10 100 1 1000 Magnitude 0.001 0.01 0.1 0.0001 1 Controller: Kmu Frequency 10 100 1 1000 Phase (degrees) -50 0 50 100 150 -100 200.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 171 W e now examine the robustness p roperties of the new closed lo op system. W e already kno w that the robust p erformance t est will b e less than t he γ v alue fr om the H ∞ syn thesis ab ove (in this case 0.
172 CHAPTER 4. DEMONSTRA TION EXAMPLES 10 100 1 1000 0.05 0.1 0.15 0.2 0.25 0 0.3 mu analysis of Gmu Nom perf Rob stab (up bnd) Rob stab (lw bnd) Rob perf (up bnd) Rob perf (lw bnd).
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 173 4.2.6 A Sim ulation Study No w the t w o con trollers ( Khinf and Kmu ) are studied b y simulation. An unw eighted interconnection is set up with sysic and starp is used to clos e the lo op for each con troller.
174 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 -0.004 -0.002 0 0.002 0.004 -0.006 0.006 Simulation: disturbance.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 175 gph8 = ctrlplot(u(2,1)); gph8 = plot(gph8, { title="Simula tion: noise",... y lab="micrometers" } )?.
176 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 micrometers -0.002 -0.001 0 0.001 0.002 -0.003 0.003 Simulation: noise.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 177 A nominal r esp onse is calcula ted by setting ∆ = 0. T o get the op en-lo o p simulation mo del we close the unw eighted in ter connection structure with a co ntroller equa l to zero .
178 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 micrometers -5 0 5 -10 10 open loop beam length.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 179 Now w e consider the closed-lo op nominal resp onse with Khinf and Kmu . The closed-lo o p system happens to hav e a large n umber of high frequency p oles whic h do not con tribute significantly to the respo nse.
180 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 micrometers -0.2 -0.1 0 0.1 0.2 0.3 -0.3 0.4 closed loop beam length Khinf Kmu.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 181 gph11 = ctrlplot(yclphinf(2,1)) ; gph11 = ctrlplot(yclpmu(2,1),gp h11); gph11 = plot(gph11, { legend=["Kh inf";"Kmu" ],.
182 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 -0.01 -0.005 0 0.005 0.01 -0.015 0.015 closed loop: vc actuator Khinf Kmu.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 183 gph12 = ctrlplot(yclphinf(3,1)) ; gph12 = ctrlplot(yclpmu(3,1),gp h12); gph12 = plot(gph12, { legend=["Kh inf";"Kmu" ],.
184 CHAPTER 4. DEMONSTRA TION EXAMPLES 0.5 1 1.5 0 2 -0.01 0 0.01 0.02 -0.02 0.03 closed loop: piezo actuator Khinf Kmu.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 185 Note that Kmu achiev es better performance at the expense of greater actu ator effort. W e will now repeat this sim ula tion for a pertur b ed system. A bad ∆ is chosen and scaled to ha ve norm 0.5. This is obtained fro m destabilizing ∆ c alculated for the µ lower bo und.
186 CHAPTER 4. DEMONSTRA TION EXAMPLES gph13 = ctrlplot(ybclpmu(1,1),g ph13); gph13 = plot(gph13, { legend=["Kh inf";"Kmu" ],... title="perturbed closed loop beam length",.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 187 Frequency 10 100 1 1000 Magnitude 0.001 0.01 0.1 0.0001 1 Controller: Khinf Frequency 10 100 1 1000 Phase (degrees) -50 0 50 100 150 -100 200.
188 CHAPTER 4. DEMONSTRA TION EXAMPLES gph14 = ctrlplot(ybclphinf(2,1) ); gph14 = ctrlplot(ybclpmu(2,1),g ph14); gph14 = plot(gph14, { legend=["Kh inf";"Kmu" ],.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 189 0.5 1 1.5 0 2 -0.01 -0.005 0 0.005 0.01 -0.015 0.015 perturbed closed loop: vc actuator Khinf Kmu.
190 CHAPTER 4. DEMONSTRA TION EXAMPLES gph15 = ctrlplot(ybclphinf(3,1) ); gph15 = ctrlplot(ybclpmu(3,1),g ph15); gph15 = plot(gph15, { legend=["Kh inf";"Kmu" ],.
4.2. A SIMPLE FLEX IBLE STR UCTURE EXAMPLE 191 0.5 1 1.5 0 2 -0.02 -0.01 0 0.01 0.02 -0.03 0.03 perturbed closed loop: piezo actuator Khinf Kmu.
Bibliograph y [1] J. C. Doyle, “Lecture notes on adv ances in multiv ariable control.” ONR/Honeyw ell W orkshop, Minneapolis, MN., 1984. [2] J. Doyle, “Structured uncertaint y in cont r ol system design,” in Pr o c . IEEE Cont r ol De cision Con f.
194 BIBLIOGRAPHY [11] G. J. Ba las and J. C. Doyle, “Robust con tr ol of flexible modes in the controller crossov er r egion,” in Pr o c. Amer. Control Conf.
BIBLIOGRAPHY 195 [24] D. L. Laughlin, K. G. Jordan, and M. Morari, “In ternal mo del con trol and process uncertaint y: mapping uncerta int y regions for SISO controller design,” Int. J. of Control , v ol. 44, no. 6, pp . 1675–1698, 1986. [25] R.
196 BIBLIOGRAPHY [37] R. Kosut, M . Lau, and S. Bo yd, “Param eter set ident ification of systems wit h uncertain non parametric dynam ics and distu rbances,” in Pr o c. I EEE Cont r ol De cision Con f. , v ol. 6, pp. 3162–3167, 1990. [38] G. Go o dwin, B.
BIBLIOGRAPHY 197 [50] J. M. Krause, P . P . Khargonek ar, and G. Stein , “Robust param eter adjust ment with nonparametric w eigh ted-ball-in- H ∞ uncertaint y ,” IEEE T r an s. Auto. Con tr ol , v ol. AC -35, pp. 225–229, 1990. [51] R. S. Smith a nd J.
198 BIBLIOGRAPHY [63] A. J. La ub, “A Sch ur method for solving algebr aic Riccati equatio ns,” IEEE T r ans. A uto. Co ntr ol , v ol. A C-24, pp . 913–921, 1979. [64] T. Pappas, A. J. Laub, a nd N. R. Sandell, “On the n umer ical solution o f the discrete-time algebraic Riccati equation,” IEEE T rans.
BIBLIOGRAPHY 199 [76] M. Dahleh, A . T esi , and A. V icino, “Extremal prop erties for t he parametric robust pe rformance pro blem,” T ech. Rep. UCSB-ME-9 1-4, Univ. Califo rnia, Santa Barbara, Mec h. Eng., 1991. also submitted to 30th IEEE CD C.
Chapter 6 F unction Reference 6.1 X µ F unctions The following pages contain descriptions of the X µ functions. These are also av ailable on-line via the help utility . Each description also gives an illustrative example of the function’s use. The functio ns are included in alphab etical o rder.
202 CHAPTER 6. FUNCTION RE FERENCE rifd ....... ........ ......... ........ ....... ........ ......... ........ ...... 3 3 5 ctrlplot ...... ........ ......... ........ ....... ........ ......... ........ .. 2 2 1 Time r esp onse c alculations and pdm functions gstep .
6.1. X µ FUNCTIONS 203 Mo del r e duction and state-sp ac e functions balmoore ...... ........ ......... ........ ....... ........ ......... ........ .. 2 0 5 modalstate ....... ........ ......... ........ ....... ........ ......... ...... 2 9 3 ophank .
204 CHAPTER 6. FUNCTION RE FERENCE Misc el lane ous functions conpdm ...... ........ ......... ........ ...... ......... ........ ......... .... 2 1 3 consys ...... ........ ......... ........ ...... ......... ........ ......... .... 2 1 5 csum ......
balmoore 205 balmo ore Syn t ax [SysR,HSV,T] = balmoore(Sys, { nsr,bound } ) Pa r a m e t e r Li s t Inputs: Sys Linear, stable, minima l state-spac e system nsr (optional ) If bound is used then a reduction will b e p er- formed whic h meets an error b ound specified by the v alue in nsr ,o t h e r w i s e nsr is the order of the reduced system.
206 Chapter 6. F unctio n Reference Reference B.C. Mo ore, “Prin cip al Comp onen t Analysis in Linear Systems: Control lability, Observabilit y and Mo del R ed uction ,” IEEE T rans. Auto. Ct rl., V ol. 26, N o. 1, pp. 17–32, F eb. 1981. Exam ple # Create a five state system for reduction.
balmoore 207 See Also: minimal , ophank ..
blknorm 209 blknorm Syn t ax normM = blknorm(M,blk,p,Frobeni us) Pa r a m e t e r Li s t Inputs: M Matrix (or pdm ). blk Blo ck structure . See mu sectio n of the manual for a de- scription of the syn ta x. p Scalar v alued. Specifies the Ho lder “p” norm to b e used, where 1 ≤ p ≤ inf.
210 Chapter 6. F unctio n Reference Examples A = random(3,3)-0.5*ones(3 ,3)? A (a square matrix) = 0.0618661 0.0896177 0.185398 0.390622 0.00422128 -0.150638 -0.112622 0.42229 0.448818 blknorm(A,[1,1; 1,1; 1,1]) ans (a square matrix) = 0.0618661 0.0896177 0.
blknorm 211 2.23607 3 4 7.81025 7 8 13.4536 11 12 # and compare to norm(B(1,1:2)) ans (a scalar) = 2.23607 See Also norm.
conpdm 213 conp dm Syn t ax outpdm = conpdm(mat,domain, { skipChk s } ) Pa r a m e t e r Li s t Inputs: mat constant matrix domain domain over which outsys will be defined. Keywords: skipChks Bo olea n specifying that synt a x chec king is to be skipp ed.
consys 215 consys Syn t ax outsys = consys(mat, { skipChks } ) Pa r a m e t e r Li s t Inputs: mat constant matrix Keywords: skipChks Bo olea n specifying that synt a x chec king is to be skipp ed. Outputs: outsys Dynamic System Description Creates a Dynamic System ob j ect from a constan t matrix.
csum 217 csum Syn t ax [outpdm] = csum(inpdm, { channels } ) Pa r a m e t e r Li s t Inputs: inpdm real or complex v alued pdm or constant matrix Keywords: channels Sum ov er c ha nnels. outpdm ha s the same dimensions as inpdm. Outputs: outp dm o utput pdm Description Pe rform a cum ulati ve sum o ver the row s of a matri x or a pdm .
218 Chapter 6. F unctio n Reference ans (a rectangular matrix) = 1 0.608453 2 1.46287 3 1.52714 4 2.35505 5 3.28128 6 3.848 pdmA = pdm(A,[1,2,3])? pdmA (a pdm) = domain | Col 1 Col 2 -------+-------------- --------- - 1 | Row 1 1 0.608453 | Row 2 1 0.
csum 219 ans (a pdm) = domain | Col 1 Col 2 -------+-------------- --------- 1 | Row 1 1 0.608453 | Row 2 1 0.854421 -------+-------------- --------- 2 | Row 1 2 0.672717 | Row 2 2 1.68233 -------+-------------- --------- 3 | Row 1 3 1.59895 | Row 2 3 2.
ctrlplot 221 ctrlplot Syn t ax graph = ctrlplot(pdm,old graph, { keywords } ) Pa r a m e t e r Li s t Inputs: p dm Pdm (or matrix) containing the data to be plotted. old graph (optional) Graphical ob ject to which data is added. Con- ceptually the same as plot(pdm, { keep=old graph } ) .
222 Chapter 6. F unctio n Reference linear linear domain. Default = 1 for timeresp k eyword. Defa ult = 0 for bo de k eyword. log logarith mic dom ain.
ctrlplot 223 Description This function per forms some co mmon control system related plotting. The user can use ctrlplot to set up a basic plot and per form some pr epro cessing of the data. This generates a graphical ob ject containing the data and the user ca n per form subsequen t calls to plot to add things like text , lab els, gridding etc.
224 Chapter 6. F unctio n Reference legend = ["sys1","sys2"] } )?.
ctrlplot 225 Frequency 0.1 1 0.01 10 Magnitude 0.0001 0.001 0.01 0.1 1 1e-05 10 Bode plots sys1 sys2 Frequency 0.1 1 0.01 10 Phase (degrees) -250 -200 -150 -100 -50 -300 0.
226 Chapter 6. F unctio n Reference # Nyquist plots g2 = ctrlplot(sys1g, { nyqui st } ); g2 = ctrlplot(sys2g,g2, { ny quist } ); g2 = ctrlplot(-1,g2, { nyqui st,marker= 1,line=0 } ); g2 = plot(g2, { projection=" orthograph ic",.
ctrlplot 227 Real -1 0 1 -2 2 Imaginary -1.5 -1 -0.5 0 -2 0.5 Nyquist plots sys1 sys3 critical point.
228 Chapter 6. F unctio n Reference # Create a second order lightly damped system to illustrate # time response plotting. The calculation is repeated with # a non-zero initial condition.
ctrlplot 229 2468 0 10 -2 -1 0 1 -3 2 input x0 = zero non-zero x0.
230 Chapter 6. F unctio n Reference See Also: plot ..
daug 231 daug Syn t ax out = daug (sys1,sys2,...) Pa r a m e t e r Li s t Inputs: sys1 Input systems. These can b e dynamical systems and co n- stants, or pdms and consta nt s. . . .“ Outputs: out output system. Description Diagonal augmen tation of dyn amical system/pdm/constan t, matri ces.
232 Chapter 6. F unctio n Reference ans (a square matrix) = 110 0 002 0 002 0 000I n f sys1 = randsys(1,1,2, { stable } ) sys1 = system(sys1, { statenames= "sys1state " } )? sys1 (a state space system) = A -0.886949 B 0.853282 0.012459 C 0.186754 D 0.
daug 233 -1.67106 B 0.579502 C 0.262815 0.436099 D 0.911055 0.808267 X0 1 State Names ----------- sys2state System is continuous # Note the effect of the constant in the following daug(sys1,10,sys2) ans (a state space system) = A -0.886949 0 0 -1.67106 B 0.
234 Chapter 6. F unctio n Reference D 0.492058 0.748961 0 0 001 0 0 0 0 0 0.911055 0 0 0 0.808267 X0 0 1 State Names ----------- sys1state sys2state System is continuous pdm1 = randpdm(3,2,2)? pdm1 (a pdm) = domain | Col 1 Col 2 -------+-------------- --------- ---- 0 | Row 1 0.
daug 235 0| 1 0 -------+----- 1| 1 0 -------+----- 2| 1 0 -------+----- daug(pdm1,pdm2) ans (a pdm) = domain | Col 1 Col 2 Col 3 -------+-------------- --------- ---------- - 0 | Row 1 0.810265 0.259043 0 | Row 2 0.413909 0.359993 0 | Row 3 0 0 10 -------+-------------- --------- ---------- - 1 | Row 1 0.
delsubstr 237 delsubstr Syn t ax [outstr] = delsubstr(str,charstr) Pa r a m e t e r Li s t Inputs: str String or vector of strings. charstr String Outputs: outstr Str ing or vector of strings. Description All o ccur ences of the substring, charstr , w ithin str a re deleted.
238 Chapter 6. F unctio n Reference out2 (a column vector of strings) = string one aaa xy # If executed, the following would give an error # out3 = delsubstr(strvec,"a" )?.
fitsys 239 fitsys Syn t ax [sys] = fitsys(data,npoles ,nzeros,we ight, { skipchks,Hertz } ) Pa r a m e t e r Li s t Inputs: data Complex v alued data ( pdm ). npo les Order of requested fit. (optional, defa ult = 0). nzeros Num b er of zeros in t ransfer function .
240 Chapter 6. F unctio n Reference The primary use of this routine is the fitting of D scal e w eight s for mu syn thesis iterations. Cheb yshev polynomials are used as basis funct ions for b oth the n umerator and denominator p o lynomials.
fitsys 241 real imaginary frequency damping (rad/sec) ratio -1.0000e+00 0.0000e+00 1.0000e+00 1.0000 5.0000e-01 3.1225e+00 3.1623e+00 -0.1581 5.0000e-01 -3.1225e+00 3.1623e+00 -0.1581 omega = logspace(0.001,100,200) plantg = freq(plant,omega) # Use complex cepstrum to fit minimum phase equivalent # to the magnitude of the data.
242 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Phase (degrees) -300 -200 -100 0 -400 100 Data and minimum phase fit original data minimum phase fit.
fitsys 243 # Create fitting weight. 1/s works well for logspaced # data. wght = 1/makepoly([1,0],"s") # Fit new system and compare pole location with # the original. Note that it is minimum phase. nsys = fitsys(cdata,3,3,wght) rifd(nsys) Poles: real imaginary frequency damping (rad/sec) ratio -5.
244 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 10 0.01 100 Data and minimum phase fit original system minimum phase system Frequency 0.
fitsys 245 Limita tions Limited to SISO systems . See Also tfid.
gstep 247 gstep Syn t ax gPdm = gstep (ytime,timespec,valspec , { skipChks } ) Pa r a m e t e r Li s t Inputs: ytime output time v ector (seconds). timespec times for specified step data (opti onal) v alspec v alue for specified step dat a (option al) Keywords: skipChks Bo olea n specifying that synt a x chec king is to be skipp ed.
248 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 10 0.01 100 Data and minimum phase fit original system minimum phase system Frequency 0.
gstep 249 See Also randpdm , gcos , gsin , gpulse , gsawtooth , gsquarewave.
hinfnorm 251 hinfnorm Syn t ax [out,omega] = hinfnorm(sys,tol, { imag eps,max it } ) Pa r a m e t e r Li s t Inputs: sys Dynamic System , frequency response ( pdm ), or constant gain (matr ix). sys Sp ecifies the relative tolerance of the answ er when the in- put is a Dynamic System Def ault = 0.
252 Chapter 6. F unctio n Reference Stable Dynamic System norms a re calculated by an itera tive Hamiltonian metho d. In this case out is a 2 × 1 vector with upp er and low er b o unds for the no rm. Exam ple # Set up a simple closed loop problem. # This example is given in more detail in the # hinfsyn online help.
hinfnorm 253 25.000 5.2e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 12.500 5.2e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 6.250 5.2e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 3.125 5.1e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 1.562 5.0e-01 1.7e-03 1.0e-02 0.0e+00 0.0000 p 0.
h2norm 255 h2norm Syn t ax out = h2norm(sys) Pa r a m e t e r Li s t Inputs: sys Co n tin uous time Dynamic System Outputs: out H 2 no rm of the input system Description The H 2 norm of a stable, str ictly prop er system is calcula ted.
256 Chapter 6. F unctio n Reference Wperf = 100/makepoly([100,1],"s ") Wact = makepoly([0.5,0.05],"s") /makepoly( [0.05,1]," s") # Form the weighted interconnection struc.
hinfsyn 257 hinfsyn Syn t ax [k,gfin,stat] = hinfsyn(p,nmeas,ncon,gamma, { keywords } ) Pa r a m e t e r Li s t Inputs: p Generalized interconnection structur e ( Dynamic Sys- tem ) nmeas measuremen t v ector dimension. ncon con trol v ector dimension.
258 Chapter 6. F unctio n Reference Description The H ∞ (sub)optimal controller for the interconnection, p , is ca lculated. The resulting closed lo op system is illustrated b elow. p k - zw y u The v ariab les ncon and nmeas are used to sp ecify the dimensions of u and y in the ab ove diagram ( ncon = dim( u )a n d nmeas = dim( y )).
hinfsyn 259 1. ( a, b 2 ,c 2 ) is stabiliza ble and detectable 2. d 12 and d 21 ha ve full rank 3. The matri x [ a − jωI, b 2 ; c 1 ,d 12 ] has fu ll column ran k for all ω 4.
260 Chapter 6. F unctio n Reference wghtic = sysic(sysnames,sy sinp,sysou t,syscnx,p lant,... Wperf,Wact) # Design Hinf controller nctrls = 1 nmeas = 1 gmax = 25 gmin = 0 Kinf = hinfsyn(wghtic,nmeas,nct rls,[gmax; gmin]) Test bounds: 0.0000 < gamma <= 25.
hinfsyn 261 real imaginary frequency damping (rad/sec) ratio -5.0000e-02 -3.1225e-01 3.1623e-01 0.1581 -5.0000e-02 3.1225e-01 3.1623e-01 0.1581 -5.0000e+00 0.
262 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.01 0.1 1 0.001 10 Kinf Frequency 0.01 0.1 1 10 0.001 100 Phase (degrees) -80 -60 -40 -20 0 20 -100 40.
hinfsyn 263 # Use sysic to create unweighted interconnection ic = sysic("plant",["ref"; "ctrl"],[" plant";"re f-plant"] ,... "ctrl",plant) clpinf = starp(ic,Kinf) rifd(clpinf) Poles: real imaginary frequency damping (rad/sec) ratio -5.
264 Chapter 6. F unctio n Reference gph2 = ctrlplot(sensg, { logmagpl ot } ); gph2 = plot(gph2, { title="Kinf controller: sensitivity function" } )?.
hinfsyn 265 Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.1 1 0.01 10 Kinf controller: sensitivity function.
266 Chapter 6. F unctio n Reference # Examine step response step = gstep([0:0.1:10],0,1) y = clpinf*step gph3 = ctrlplot([y,step]); gph3 = plot(gph3, { title="Kinf controller: step response".
hinfsyn 267 2468 0 10 0.2 0.4 0.6 0.8 1 0 1.2 Kinf controller: step response.
268 Chapter 6. F unctio n Reference See also hinfsyn , hinfnorm , h2norm.
h2syn 269 h2syn Syn t ax k = h2syn(p,nmeas,ncon, { ke ywords } ) Pa r a m e t e r Li s t Inputs: p Generalized interconnection structur e ( Dynamic Sys- tem ) nmeas measuremen t v ector dimension.
270 Chapter 6. F unctio n Reference p k - zw y u The v ariab les ncon and nmeas are used to sp ecify the dimensions of u and y in the ab ove diagram ( ncon = dim( u )a n d nmeas = dim( y )). The ob jective is to design a sta bilizing con troller, k , whic h minimizes the H 2 norm of the closed loop system betw een w and z .
h2syn 271 Reference This function uses the state-spa ce formulae given in: “Stat e-sp ace formu lae for al l s tabilizing contr ol lers that sat isfy an H ∞ norm b ound and r elati ons to risk sensiti vity,” Keith Glov e r and John Doyle, Systems & Co nt r ol Letters 11 , pp.
272 Chapter 6. F unctio n Reference K2 = h2syn(wghtic,nmeas,nc trls) rifd(K2) Poles: real imaginary frequency damping (rad/sec) ratio -1.4046e-01 -2.3161e-01 2.7087e-01 0.5186 -1.4046e-01 2.3161e-01 2.7087e-01 0.5186 -1.5863e+00 3.4754e+00 3.8203e+00 0.
h2syn 273 Frequency 0.01 0.1 1 10 0.001 100 Magnitude 0.001 0.01 0.1 0.0001 1 K2 Frequency 0.01 0.1 1 10 0.001 100 Phase (degrees) -100 -50 0 -150 50.
274 Chapter 6. F unctio n Reference # Use sysic to create unweighted interconnection ic = sysic("plant",["ref"; "ctrl"],[" plant";"re f-plant"] ,... "ctrl",plant) clp2 = starp(ic,K2) rifd(clp2) Poles: real imaginary frequency damping (rad/sec) ratio -5.
h2syn 275 sensg = freq(sens,omega) gph2 = ctrlplot(sensg, { logmagpl ot } ); gph2 = plot(gph2, { title="K2 controller: sensitivity function" } )?.
276 Chapter 6. F unctio n Reference Frequency 0.01 0.1 1 10 0.001 100 Magnitude 1 0.1 10 K2 controller: sensitivity function.
h2syn 277 # Examine step response step = gstep([0:0.1:10],0,1) y = clp2*step gph3 = ctrlplot([y,step]); gph3 = plot(gph3, { title="K2 controller: step response" } )?.
278 Chapter 6. F unctio n Reference 2468 0 10 0.2 0.4 0.6 0.8 0 1 K2 controller: step response.
h2syn 279 See also hinfsyn , h2norm , hinfnorm.
interp 281 in terp Syn t ax outpdm = interp(inpdm,stepsize,final { keywords } ) outpdm = interp(inpdm,domspec, { keywords } ) Pa r a m e t e r Li s t Inputs: inpdm Input pdm. stepsize Incremen t in o utp dm domain. final La st v alue in outp dm doma in.
282 Chapter 6. F unctio n Reference This function differs from the interpolate function in tha t it can handle zero order hold type interp olation a nd deal with irregula rly spaced input p dms. Irreg ularly spaced output pdms can be generated wit h the domspec syn tax.
interp 283 1234 0 5 1 2 3 4 0 5 1st order interp. 0 order interp. original pdm.
284 Chapter 6. F unctio n Reference See Also interpolate.
mergeseg 285 mergeseg Syn t ax outpdm = mergeseg(pdm1,pdm2, { keywords } ) Pa r a m e t e r Li s t Inputs: pdm1 input pdm pdm2 input pdm Keyw ords: dom sort Sort the result of merging the pdm s. If !domsort then pdm2 is simply concatenated o nto pdm1 .
286 Chapter 6. F unctio n Reference Exam ple time1 = [0:0.025:1] pdm1 = gsin(time1, { frequency= 2 } ) time2 = [0.8:0.02:1.5] pdm2 = randpdm([],1,1, { dom=ti me2,zerom ean } ) outpdm = mergeseg(pdm1,pdm 2) gph1 = ctrlplot(outpdm); gph1 = plot(pdm1,gph1, { marker =1,marker style=8,.
mergeseg 287 0.5 1 0 1.5 -0.5 0 0.5 -1 1 outpdm pdm1 pdm2.
mkpert 289 mkp ert Syn t ax [pertsys] = mkpert(Delta,blk,mubnds, { f select,pn orm,Hertz } ) Pa r a m e t e r Li s t Inputs: Delta Low er b ound p ertur bation da ta fro m mu calculatio n. ( pdm ) blk blo ck structur e (refer to m u function do cumentation).
290 Chapter 6. F unctio n Reference The bo unds, m ubnds, ar e used to determine the ”worst-case” fr equency for the in terpolation. The norm of pertsys is 1/(norm(D elta( jω ))) where ω is the c hosen frequency . This is the smallest destabilizing pe rturbatio n at that frequency .
mkphase 291 mkphase Syn t ax [cdata] = mkphase(magdata, { skipchks,Hertz } ) Pa r a m e t e r Li s t Inputs: magdata Magnitude data ( pdm ) Keyw ords: skipch ks Bo olean. Skip the error ch ecking. (D efault = 0) Hertz The domain of the pdm is in Hertz.
292 Chapter 6. F unctio n Reference Limita tions Limited to SISO systems . See Also fitsys , ccepstrum.
modalstate 293 mo dalstate Syn t ax outsys = modalstate(sys, { keywords } ) Pa r a m e t e r Li s t Inputs: sys Input D ynamic System Keywords: increasing Bo olean. O rder in terms of incr easing magnitude (contin- uous) o r angle (discr ete). Default = 1 decreasing Bo olean.
294 Chapter 6. F unctio n Reference -8.1602e-01 1.3353e+00 1.5649e+00 0.5215 -8.1602e-01 -1.3353e+00 1.5649e+00 0.5215 -4.6142e+00 -1.6648e+01 1.7275e+01 0.2671 -4.6142e+00 1.6648e+01 1.7275e+01 0.2671 -2.2478e+01 0.0000e+00 2.2478e+01 1.0000 -3.9525e+01 0.
mu 295 mu Syn t ax [mubnds,D,Dinv,Delta, sens] = mu(M,blk) Pa r a m e t e r Li s t Inputs: M Matrix or p dm. blk Blo ck structure defined b y a matrix o f dimension: num b er of blo cks × 2. If the ith blo ck has c outputs and r inputs, then blk(i,:) = [r,c].
296 Chapter 6. F unctio n Reference # that mu is not equal to its upper bound for # more than three full blocks. gamma = 3 + sqrt(3); beta = sqrt(3) -1 a = sqrt(2/gamma); b = 1/sqrt(gamma) c = 1/sqrt(.
mu 297 ans (a scalar) = 2.81264 mubnds2? mubnds2 (a column vector) = 2.81264 2.81264 det(eye(4,4) - M*Delta2)? ans (a scalar) = -2.53156e-16 + 4.3828e-17 j F or an example of how m u is used for system robustness analysis, refer to the on-line help for m usynfit (p age 299).
musynfit 299 m usynfit Syn t ax [Dsys,Dinvsys] = musynfit(Dmag,blk,nmeas,nc trls,.. weight,M,order, { keyw ords } ) Pa r a m e t e r Li s t Inputs: Dmag New D matrix fr om mu calcula tion (magnitude data o nly). The domain is assumed t o be in Hertz.
300 Chapter 6. F unctio n Reference Keyw ords: He rtz Boolean. This ke yword i s mandatory as th e funct ion m ust know whether the do main is in Her tz or ra dians/second (specified by !Hertz) to fit correctly . Note that the Xmath function freq assumes that the frequency range is sp ecified in Hertz.
musynfit 301 P = 1/makepoly([1,0,-0.01] ,"s") W = makepoly([1,20],"s")/m akepoly([1 ,200],"s") # Set up an unweighted interconnection structure # for unity gain negative feedback.
302 Chapter 6. F unctio n Reference 1.562 6.8e-01 8.0e-04 7.7e-03 0.0e+00 0.6811 p 1.406 6.7e-01 8.1e-04 7.0e-03 -1.1e-18 2.0900 f 1.484 6.7e-01 8.0e-04 7.4e-03 0.0e+00 1.0394 f 1.523 6.7e-01 8.0e-04 7.5e-03 -5.0e-19 0.8249 p Gamma value achieved: 1.5234 G = starp(wghtic,Kinf) omega = logspace(0.
musynfit 303 0.1 1 10 0.01 100 0.2 0.4 0.6 0.8 1 0 1.2 mu analysis nominal perf. robust stab. robust perf.(upper) robust perf. (lower).
304 Chapter 6. F unctio n Reference # Fit transfer functions to D1 & Dinv1 for a mu # synthesis iteration [Ds,Dinvs] = musynfit(D1,blk,nmeas,ncnt rls,sens1 ,Gg, { Hertz } ).
musynfit 305 Frequency (Hz) 0.1 1 10 0.01 100 Magnitude 10 100 1000 1 10000 D scale fit, block: 1 Magnitude data Previous fit, order: 0 New fit, order: 3 Frequency (Hz) 0.1 1 10 0.01 100 Magnitude 0.01 0.1 1 10 0.001 100 Upper bound comparison, block: 1 Data based bound Previous fit bound, order: 0 New fit bound, order: 3 Frequency (Hz) 0.
306 Chapter 6. F unctio n Reference # Apply the D scales to another H infinity design Kmu = hinfsyn(Ds*wghtic*Di nvs,nmeas, ncntrls,[g min;gmax] ) Test bounds: 0.0000 < gamma <= 10.0000 gamma Hx eig X eig Hy eig Y eig nrho xy p/f 10.000 6.5e-01 5.
musynfit 307 0.1 1 10 0.01 100 0.2 0.4 0.6 0.8 1 0 1.2 Kmu & Kinf mu analysis Kmu: robust perf. Kinf: robust perf..
308 Chapter 6. F unctio n Reference # Look at the worst case perturbations for each of # the Kinf and Kmu controllers. Compare also a # random perturbation for each controller. In all cases the # perturbation is of size 0.5 and we choose a perturbation # which is bad at 1Hz.
musynfit 309 ymunom(1,1),ymupert(1 ,1)]); gph4 = plot(gph4, { legend=["inpu t step";"Kinf nominal";... "Kinf pert.";"Kmu nominal";"Kmu pert.
310 Chapter 6. F unctio n Reference 123 0 4 0.2 0.4 0.6 0.8 1 1.2 0 1.4 input step Kinf nominal Kinf pert. Kmu nominal Kmu pert..
musynfit 311 # Compare with a random perturbation yinfrandp = infrpert*step ymurandp = murpert*step gph5 = ctrlplot([step,yinfnom(1 ,1),yinfra ndp(1,1),. .. ymunom(1,1),ymurandp(1 ,1)]); gph5 = plot(gph5, { legend=["inpu t step";"Kinf nominal";.
312 Chapter 6. F unctio n Reference 123 0 4 0.2 0.4 0.6 0.8 1 1.2 0 1.4 input step Kinf nominal Kinf rand. pert. Kmu nominal Kmu rand. pert..
musynfit 313 See Also: mu , hinfsyn , mkpert , hinfnorm ..
ophank 315 ophank Syn t ax [SysR,SysU,HSV] = ophank(Sys, { nsr,o nepass } ) Pa r a m e t e r Li s t Inputs: Sys Linear , stable, state-space system (contin uo us) nsr (option al) Order of the reduced system. I f not sp ecified, the user will b e prompted for its v a lue after the Hankel singular v alues a re display e d.
316 Chapter 6. F unctio n Reference Uses additional s ubroutines ophiter , ophred , ophmult and stable . This function is cross-licensed from the Model Reduction Mo dule. Exam ple # Create a five state system for reduction. a = daug(-0.891334,[-1.208 57,0.
ophank 317 Frequency 0.1 1 10 0.01 100 Magnitude 0.001 0.01 0.0001 0.1 original system reduced: ophank reduced: sresidualize ophank error sresidualize error.
318 Chapter 6. F unctio n Reference See Also minimal , balmoore ..
orderstate 319 orderstate Syn t ax outsys = orderstate(sys,indx) Pa r a m e t e r Li s t Inputs: sys Input D ynamic System indx Lists the desired order of the states in outsys. Outputs: outsys o utput dynamic system. Description Reorder th e s tates according to the index argument .
320 Chapter 6. F unctio n Reference 0.710595 0.688873 C 0.659532 0.181512 0.390497 D 0.15869 X0 0.1 0.2 0.3 State Names ----------- s1 s2 s3 System is continuous sys2 = orderstate(sys1,[2,1,3]) ? sys2 (a state space system) = A -5.41526 0.65518 9.0643 3.
orderstate 321 0.2 0.1 0.3 State Names ----------- s2 s1 s3 Input Names ----------- Input 1 Output Names ------------ Output 1 System is continuous.
randpdm 323 randp dm Syn t ax pdmout = randpdm (ndomain,nrows,ncolumns, { keywords } ) Pa r a m e t e r Li s t Inputs: ndomain length o f the domain nrows num b er of rows in p dmout ncolumns num b er of columns in p dmo ut Keywords: complex Bo olean.
324 Chapter 6. F unctio n Reference Examples pdm0 = randpdm(3,1,2, { zeromean } )? pdm0 (a pdm) = domain | Col 1 Col 2 -------+-------------- --------- -- 0 | -0.043917 -0.0789571 -------+-------------- --------- -- 1 | 0.37878 -0.457265 -------+-------------- --------- -- 2 | 0.
randpdm 325 10.0323 | 0.922528 ---------+------------ 18.3191 | 0.81113 ---------+------------ pdm3 = randpdm(0,3,2, { complex,z eromean } )? pdm3 (a rectangular matrix) = -0.237052 - 0.10845 j -0.250958 - 0.392096 j 0.097563 + 0.0140395 j -0.255142 - 0.
randpert 327 randp ert Syn t ax [pert] = randpert(blk, { sys,sfreq,complex,pnorm } ) Pa r a m e t e r Li s t Inputs: blk block structur e (refer to mu function do cumentation). (matri x) Keywords: sys Bo olean. Sp ecifies that p ert is a dynamic system.
328 Chapter 6. F unctio n Reference Exam ple The use of randpert is studied in co ntext in the on-line help fo r musynfit and the man ual do cumen tation f or musynfit (page 299).
randsys 329 randsys Syn t ax sys = randsys (nstates,noutputs,ninput s, { keywor ds } ) Pa r a m e t e r Li s t Inputs: nstates n um be r of sta tes in sys noutputs num b er o f outputs in sys ninputs num b er o f inputs in s ys Keywords: stable Bo o lean.
330 Chapter 6. F unctio n Reference Description A rando m system, with user sp ecified state, input and output dimension, is generated. Several additional features can b e specified b y keyw ords: w.
randsys 331 real imaginary frequency damping (rad/sec) ratio -1.0408e+00 0.0000e+00 1.0408e+00 1.0000 -1.1390e+00 0.0000e+00 1.1390e+00 1.0000 -2.8050e-01 -1.4858e+00 1.5120e+00 0.1855 -2.8050e-01 1.4858e+00 1.5120e+00 0.1855 -1.5337e+00 0.0000e+00 1.
332 Chapter 6. F unctio n Reference Frequency 0.1 1 10 0.01 100 Magnitude 0.01 0.1 1 0.001 10 sys1 (input 1) sys1 (input 2) sys2 Frequency 0.1 1 10 0.01 100 Phase (degrees) -80 -60 -40 -20 0 -100 20.
randsys 333 sys3 = randsys(6,1,1, { discrete, dt=5 } ) rifd(sys3)? Poles: radius angle (radians) 0.1652 0.0000 0.4580 0.0000 0.7872 -0.0394 0.7872 0.0394 0.9307 -0.2163 0.9307 0.2163 Zeros: radius angle (radians) 0.1623 0.0000 0.7981 0.0000 0.6658 0.0000 0.
rifd 335 rifd Syn t ax [stat] = rifd(vec, { discrete,Hertz,d egrees } ) Pa r a m e t e r Li s t Inputs: vec complex v a lued vector (or Dynamic System - see b elow). Keyw ords: discret e Bo olean. V ector is to be in terpreted in the z domain, rather than the s domain.
336 Chapter 6. F unctio n Reference Examples sys1 = randsys(4,3,2, { stable } ) rifd(sys1) Poles: real imaginary frequency damping (rad/sec) ratio -1.6847e+00 0.0000e+00 1.6847e+00 1.0000 -2.4383e+00 0.0000e+00 2.4383e+00 1.0000 -8.7457e+00 0.0000e+00 8.
rifd 337 sys2 = randsys(3,3,2, { stable,di screte } ) rifd(sys2) Poles: radius angle (radians) 0.9641 0.0000 0.2985 -0.5348 0.2985 0.5348 Zeros:.
sdtrsp 339 sdtrsp Syn t ax [v,y,u] = sdtrsp(Sys,dSys,w,tfinal,. .. { ord,intstep,cdelay } ) Pa r a m e t e r Li s t Inputs: Sys Con tinuous dynamic system. This is the upper system in the LFT. The initial states are used in the sim ula tion. dSys Digita l dynamic system.
340 Chapter 6. F unctio n Reference Description Time doma in simulation of a sa mpled data interconnection. The applica ble closed lo o p system is illustra ted be low. Sys dSys - vw y u This is co nceptually the equiv alent of: v = starp ( Sys , dSys ) ∗ w .
sdtrsp 341 triangle ho ld equiv alent is used. This is the same a linea rly connecting the samples a t the input to the hold. Some care is needed in the choice of cdelay and intstep. The default for intstep is ba sed on the contin uo us system eigenv alues and the minimum time spa cing in the input vector, w.
342 Chapter 6. F unctio n Reference inps = "ref" outs = "digP" cnx = ["digC"; ... # input to digP "[ ref;digP]"] # input to digC digclp = sysic(snm,inps,ou ts,c.
sdtrsp 343 0.2 0.4 0.6 0.8 0 1 -1 0 1 2 3 -2 4 sampled data calc. discrete calc..
344 Chapter 6. F unctio n Reference See Also trsp.
simtransform 345 sim transform Syn t ax out = simtransform(sys,X) Pa r a m e t e r Li s t Inputs: sys Input system. This ma y b e a state-space system, pdm or constant matrix. X Similarit y transf orm. X must be inv ertible Outputs: out output system - in the sa me class as the input.
346 Chapter 6. F unctio n Reference -1.9737e+01 0.0000e+00 1.9737e+01 1.0000 -9.7569e+00 -2.9129e+01 3.0720e+01 0.3176 -9.7569e+00 2.9129e+01 3.0720e+01 0.3176 Zeros: real imaginary frequency damping (rad/sec) ratio -2.1990e+01 0.0000e+00 2.1990e+01 1.
simtransform 347 X0 0 0 0 Input Names ----------- Input 1 Output Names ------------ Output 1 System is continuous.
spectrad 349 sp ectrad Syn t ax out = spectrad(mat) Pa r a m e t e r Li s t Inputs: ma t Square matr ix or pdm . Outputs: out sp ectr al radius o f the input. Description Calculates the sp ectral radius (magnitude of the maximum eigen v alue) of the input matrix.
350 Chapter 6. F unctio n Reference eig(pdm1)? ans (a pdm) = domain | -------+-------------- --------- -------- 0 | Row 1 0.013657 + 0.137601 j | Row 2 0.013657 - 0.137601 j -------+-------------- --------- -------- 1 | Row 1 0.0871876 + 0.409031 j | Row 2 0.
sresidualize 351 sresidualize Syn t ax sysout = sresidualize(sysin,ord) Pa r a m e t e r Li s t Inputs: sysin Input D ynamic System ord Order of sysout Outputs: sysout output Dynamic System Description Residualize the states o f sysin to the num b er sp ecified by ord .
352 Chapter 6. F unctio n Reference sysout1 = sresidualize(sys1,3) sysout2 = truncate(sys1,3) fHz = logspace(0.01,100,10 0) sys1g = freq(sys1,fHz) sysout1g = freq(sysout1,fHz) sysout2g = freq(sysout2,fHz) residerror = sys1g - sysout1g truncerror = sys1g - sysout2g gph1 = ctrlplot([sys1g,sysout1g ,sysout2g, residerror ,.
sresidualize 353 Frequency 0.1 1 10 0.01 100 Magnitude 0.0001 0.001 0.01 1e-05 0.1 original system residualized system truncated system residualization error truncation error.
354 Chapter 6. F unctio n Reference See also rifd , simtransform , orderstate modalstate , truncate ..
starp 355 starp Syn t ax out = starp (upper,lower,dim1,dim 2,skipChks ) Pa r a m e t e r Li s t Inputs: upper Upper ob ject ( D ynamic System , constan t or pdm )i nt h e interconnection low er Low er ob ject in the in terconnection. dim1 The n umber o f outputs of upp er to b e connected as inputs to lower.
356 Chapter 6. F unctio n Reference specified. This is equiv alent to: dim1 = min ( upper output dim , lower input dim ) and dim2 = min ( upper input dim , lower output dim ) .
starp 357 # structure for closing control loops. The following is # the structure for a simple unity gain negative feedback # system with plant P. P = 1/makepoly([1,1],"s") M = consys([0,1;1.
358 Chapter 6. F unctio n Reference -1.0000e+01 0.0000e+00 1.0000e+01 1.0000 # Now consider the system with an output multiplicative # perturbation (of 10%) W = consys(0.1) G = daug(W,1,1)*consys([0, 0,1;1,0,1; -1,1,-1])* daug(1,1, P) # The nominal system is constructed by closing the # upper loop with 0.
starp 359 Poles: real imaginary frequency damping (rad/sec) ratio -5.0000e+00 -8.0623e+00 9.4868e+00 0.5270 -5.0000e+00 8.0623e+00 9.4868e+00 0.5270 Zeros: real imaginary frequency damping (rad/sec) ratio -1.
substr 361 substr Syn t ax littlestring = substr(bigstring,charinde x, { skipChk s } ) Pa r a m e t e r Li s t Inputs: bigstring Input string (a 1 × 1 s tring ma trix) ch arindex vector indexing the c haracters t o b e returned in the output . Keywords: skipChks Boolean specifying that syntax chec king is to be skipp ed.
sysic 363 sysic Syn t ax [sys] = sysic(sysNames,sysInput s,sysOutp uts,sysCon nects,... subsys1,subsys2,...) Pa r a m e t e r Li s t Inputs: sysNa mes A vector of strings (of the sa me length as the num b er of subsystems) naming the subsystems. sysInputs A vector of strings naming the exo genous inputs to the final system.
364 Chapter 6. F unctio n Reference Matrices can a lso be included in the interconnection and are consider ed to b e consta nt gains when connected with Dynamic System s and con sidered to be constan t f or all domain v alues when connected with pdm s.
trsp 365 trsp Syn t ax [y,uint] = trsp(Sys,u,tfinal, ord,intstep) Pa r a m e t e r Li s t Inputs: Sys Con tinuous dynamic system. The initial states ar e used in the simulation. u PDM. Input signal. tfinal Final time in the simulation (optiona l). Default = ma x t i m es p e c i fi e di nu .
366 Chapter 6. F unctio n Reference necessary . If int step is not specified the in tegration stepsize is d etermined from t he system eigen v alues and the minim um spac ing in the input signal, u . A zero or fi rst order discrete equiv alen t of the systems can b e sp ecified.
trsp 367 0.5 1 1.5 0 2 -1 -0.5 0 0.5 1 -1.5 1.5 output input.
368 Chapter 6. F unctio n Reference # Compare trsp calculation to standard [ytrsp,uint] = trsp(P,u) gph2 = ctrlplot(ytrsp); gph2 = plot(y,gph2, { line style=4 } ); gph2 = plot(u,gph2, { line=0,mark er=1 } ); gph2 = plot(uint,gph2, { line style=3,legend=["trsp calc.
trsp 369 0.5 1 1.5 0 2 -1 -0.5 0 0.5 1 -1.5 1.5 Time response calculation comparisons trsp calc. * calc. input interpolated input.
370 Chapter 6. F unctio n Reference # Now look at 1st order interpolation [y1trsp,u1int] = trsp(P,u, { ord=1 } ) gph3 = ctrlplot([y1trsp,u1int]) ; gph3 = plot(u,gph3, { line=0,mark er=1,legen d=... ["1st order interpolation";... "interpolated input";"input data"],title=.
trsp 371 0.5 1 1.5 0 2 -1 -0.5 0 0.5 1 -1.5 1.5 Time response calculation comparisons 1st order interpolation interpolated input input data.
sresidualize 373 truncate Syn t ax sysout = truncate(sysin,ord) Pa r a m e t e r Li s t Inputs: sysin Input D ynamic System ord Or der of the truncated system: s ysout Outputs: sysout T runcated outpu.
374 Chapter 6. F unctio n Reference Exam ple This example is iden tica l to tha t describ ed for sresidualize and compares the tw o m et hod s of m ode l re du ct i on . # Create a five state system for reduction. a = daug(-0.891334,[-1.208 57,0.79904 2;-0.
sresidualize 375 Frequency 0.1 1 10 0.01 100 Magnitude 0.0001 0.001 0.01 1e-05 0.1 original system residualized system truncated system residualization error truncation error.
376 Chapter 6. F unctio n Reference See also rifd , simtransform , sresidualize , orderstate modalstate ..
6.2. X µ SUBR OUTINES AND UTILITIES 377 6.2 X µ Subroutines and Utilities Several subroutines ma y also b e of interest to the user. These subroutines typically perform self cont ained part s of a calculat ion.
blkbal 379 blkbal Syn t ax d = blkbal(M) Description Balances a square ma trix assuming only scalar blo cks. The Osb or ne metho d (growth rate: n 2 ) is used for large systems and the P erron method (gro wth rate: n 3 ) for smaller systems. The Perron method will exactl y calculate m u for p ositiv e matrices.
hinfcalc 381 hinfcalc Syn t ax [X,Y,f,h,Ric fail,HX,HY,HXmin,HYmin ] = ... hinfcalc(p,nmeas,nco n,g,epr, { keyword s } ).
382 Chapter 6. F unctio n Reference Pa r a m e t e r Li s t Inputs: p Generalized interconnection structur e ( Dynamic Sys- tem ) nmeas measuremen t v ector dimension. ncon con trol v ector dimension. g H ∞ norm of sub o ptimal controller to b e ca lculated.
hinfcalc 383 Description F orm and solve th e Riccati equations for the H ∞ con trol problem. X and Y are the resulting Riccati solutio ns. THIS FUNCTION IS INTENDED ONL Y AS A SUBR OUTINE CALLED BY THE HINFSYN FUNCTION.
powermu 385 po w e r m u Syn t ax [lbnd,delta,errstat] = powermu(M,blk,rp,cp) Description Low er bound p ow er algorithm based on th e work of Andy Pac k ard.
riccati eig 387 riccati eig Syn t ax [x1,x2,stat,Heig min] = riccati eig(H,epp) Pa r a m e t e r Li s t Inputs: H Hamiltonian matrix. epp T olera nce for detecting proximit y of eigenv alues to the jω axis. Outputs: x1,x2 Basis v ector s for stable subspace.
388 Chapter 6. F unctio n Reference v ariab le, X = x 2 x − 1 1 , is the stabilizing solution to the Ricca ti equation. If H has jω axis eigenv alues then no stabilizing solution exists and the function retur ns a failure status.
riccati schur 389 riccati sc h ur Syn t ax [x1,x2,stat,Heig min] = riccati schur(H,epp) Pa r a m e t e r Li s t Inputs: H Hamiltonian matrix. epp T olera nce for detecting proximit y of eigenv alues to the jω axis. Outputs: x1,x2 Basis v ector s for stable subspace.
390 Chapter 6. F unctio n Reference v ariab le, X = x 2 x − 1 1 , is the stabilizing solution to the Ricca ti equation. If H has jω axis eigenv alues then no stabilizing solution exists and the function retur ns a failure status.
App endices A T ranslation Bet ween Ma tlab µ -T o ols and X µ This app endix outlines the functional equiv alences betw een the Ma tlab µ -T o ols and Xmath X µ . The ob ject ive is t o pro vide a smooth transiti on for users mo ving from µ -T o ols to X µ .
392 APPEND ICES The functionally equiv alen t commands will b e listed, in each sub-section, for conv enience. A more detailed discussion is given to illustrate the mor e subtle differences in the mu and D - K iteration asp ects. Mor e imp ortantly , the Hima t demo is av ailable in bot h µ -T oo ls and X µ .
A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 393 Description µ -T o ols F unction Xmath/X µ equiv alen t form system pck system decompose system unpck abcd form system nd2sys transfer f unct.
394 APPEND ICES Subblo cks: s electing input & outputs In µ -T o ols the function sel selects rows and columns fro m a va r y i n g matrix or inputs and outputs from a system matrix. In Xma th these can b e o btained by specifying row and column indexes.
A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 395 Note that the transpose and conjugate tran sp ose op erators are defined differen tly f or Ma tl ab and Xmat h. system / Dynamic System F unctions The following functions p erfo rm useful manipula tions to, or information a b out, the sta te of a system or Dynamic System .
396 APPEND ICES Description µ -T o ols F unction Xmath/X µ equiv alen t peak norm pkvnorm norm , max absolute v alue vabs abs diagonal m atrix vdiag diagonal round down wards vfloor round round upw .
A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 397 Miscellaneou s Utilities Several utilities a re pr ovided in µ -T o ols. These are subroutines used b y other µ -T ools funct ions wh ich ma y be of more general use.
398 APPEND ICES A.3 System R esp onse F unctions Creating Ti me Domain Signal s The Xmath pdm data ob ject allows the crea tion of time domain signals via standard and operators.
A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 399 information. A.4 System In terconnection Simple interconnection has already been outlined in the a ugmentation section above. The mo re complicated in ter connection functions are a lmost identical in µ -T o o ls and X µ .
400 APPEND ICES Description µ -T o ols function Xma th/X µ equiv alen t H 2 norm calcula tion h2norm h2norm H ∞ norm calcula tion hinfnorm hinfnorm H 2 controller syn thesis h2syn h2syn H ∞ controller syn thesis hinfsyn hinfsyn The ma jor syntactical diff erence is that the X µ fu nction s do not return the closed loop system.
A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 401 Description µ -T o ols function Xma th/X µ equiv alen t structured singula r v alue mu mu D scale deco ding unwrapd not required pe rturbatio.
402 APPEND ICES g1g = freq(g1,omega) [mubnds,Dmagdata] = mu(g1g,blk) F rom this, frequency do main D -scales a re obtained, a nd a ra tional appr oximation is obtained via musynfit .
A. TRANSLA TION BETWE EN MA TLA B µ -TOOLS AND X µ 403 The adv an tage of this is that in order to restart, or reproduce, an iterat ion, one need only sav e the previous con troller. The µ -T o ols approach requ ires saving the rational approximation to the previous D -scales.
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