1. State-Space Linear Systems 2. Block Diagrams 3. Exercises

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1 LECTURE 1 State-Space Linear Sstems This lectre introdces state-space linear sstems, which are the main focs of this book. Contents 1. State-Space Linear Sstems 2. Block Diagrams 3. Exercises 1.1 State-Space Linear Sstems A continos-time state-space linear sstem is defined b the followingtwo eqations: ẋ(t) A(t)x(t) B(t)(t), x R n, R k, (1.1a) (t) C(t)x(t) D(t)(t), R m. (1.1b) Notation. Afnction of time (either continos t [0, ) or discrete t N)is called a signal. Notation 1. We write P to mean that is one of the otpts that corresponds to, the (optional) label P specifies the sstem nder consideration. The signals :[0, ) R k, x :[0, ) R n, :[0, ) R m, are called the inpt, state, and otpt of the sstem. The first-order differential eqation (1.1a) is called the state eqation and (1.1b) is called the otpt eqation. The eqations (1.1) express an inpt-otpt relationship between theinpt signal ( ) and the otpt signal ( ). For a given inpt ( ), we need to solve the state eqation to determine the state x( ) and then replace it in the otpt eqation to obtain the otpt ( ). Attention! Forthesameinpt( ), different choices of the initial condition x(0) on the state eqation will reslt in different state trajectories x( ). Conseqentl, one inpt( ) generall corresponds to several possible otpts ( ).

2 6 LECTURE Terminolog and Notation When the inpt signal takes scalar vales (k 1) the sstem is called single-inpt (SI), otherwise it is called mltiple-inpt (MI). When the otpt signal takes scalar vales (m 1) the sstem is called single-otpt (SO), otherwise it is called mltipleotpt (MO). When there is no state eqation (n 0) and we have simpl (t) D(t)(t), R k, R m, Note. The rationale behind this terminolog is explained in Lectre 3. the sstem is called memorless. When all the matrices A(t), B(t), C(t), D(t) areconstant t 0, the sstem (1.1) is called a Linear Time-Invariant (LTI) sstem. In the general case, (1.1) is called a Linear Time-Varing (LTV) sstem to emphasize that time invariance is not being assmed. For example, Lectre 3 discsses implse responses of LTV sstems and transfer fnctions of LTI sstems. This terminolog indicates that the implse response concept applies to both LTV and LTI sstems, bt the transfer fnction concept is meaningfl onl for LTI sstems. To keep formlas short, in the following we abbreviate (1.1) to ẋ A(t)x B(t), C(t)x D(t), x R n, R k, R m (CLTV) MATLAB R Hint 1. ss(a,b,c,d) creates the continos-time LTI state-space sstem (CLTI). p. 7 and in the time-invariant case, we frther shorten this to ẋ Ax B, Cx D, x R n, R k, R m. (CLTI) Since these eqations appear in the text nmeros times, we se the special tags (CLTV) and (CLTI) to identif them Discrete-Time Case A discrete-time state-space linear sstem isdefined bthe followingtwo eqations: x(t 1) A(t)x(t) B(t)(t), x R n, R k, (1.2a) (t) C(t)x(t) D(t)(t), R m. (1.2b) Attention. One inpt generall corresponds to several otpts, becase one ma consider several initial conditions for the state eqation. All the terminolog introdced for continos-time sstems also applies to discrete time, except that now the domain of the signals is N : {0, 1, 2,...}, instead of the interval [0, ). In discrete-time sstems the state eqation is a difference eqation, instead of a first-order differential eqation. However, the relationship between inpt and otpt is analogos. For a given inpt ( ), we need to solve the state (difference)

3 STATE-SPACE LINEAR SYSTEMS 7 Note. Since this eqation appears in the text nmeros times, we se the special tag (DLTI) to identif it. The tag (DLTV) is sed to identif the time-varing case in (1.2). eqation to determine the state x( ) and then replace it in the otpt eqation to obtain the otpt ( ). To keep formlas short to, in the seqel we abbreviate the time-invariant case of (1.2) x Ax B, Cx D, x R n, R k, R m. (DLTI) State-Space Sstems in MATLAB R MATLAB R has several commands to create and maniplate LTI sstems. The following basic command is sed to create an LTI sstem. Note. Initial conditions to LTI state-space MATLAB R sstems are specified at simlation time. MATLAB R Hint 1 (ss). The command ss_ssss(a,b,c,d) assigns to ss_ss a continos-time LTI state-space MATLAB R sstem of the form ẋ Ax B, Cx D. Optionall, one can specif the names of the inpts, otpts, and state to be sed in sbseqent plots as follows: ss_ssss(a,b,c,d,... InptName, { inpt1, inpt2,...},... OtptName,{ otpt1, otpt2,...},... StateName, { state1, state2,...}); The nmber of elements in the bracketed lists mst match the nmber of inpts, otpts, and state variables. For discrete-time sstems, one shold instead se the command ss_ssss(a,b,c,d,ts), where Ts is the sampling time, or -1 if one does not want to specif it. Note. It is common practice to denote the inpt and otpt signals of a sstem b and, respectivel. However, when dealing with interconnections, one mst se different smbols for each signal, so this convention is abandoned. 1.2 Block Diagrams It is convenient to represent sstems b block diagrams as in Figre 1.1. These diagrams generall serve as compact representations for complex eqations. The two-port block in Figre 1.1(a) represents a sstem with inpt ( )andotpt ( ), where the directions of the arrows specif which is which. Althogh not explicitl represented in the diagram, one mst keep in mind the existence of the state, which affects the otpt throgh the initial condition Interconnections Interconnections of block diagrams are especiall sefl to highlight special strctres in state-space eqations. To nderstand what is meant b this, assme that the

4 8 LECTURE 1 P 1 1 P P 2 2 (a) Single block (b) Parallel z z P 1 P 2 P 1 (c) Cascade (d) Negative feedback Figre 1.1. Block diagrams. blocks P 1 and P 2 that appear in Figre 1.1 are the two LTI sstems P 1 : ẋ 1 A 1 x 1 B 1 1, 1 C 1 x 1 D 1 1, x R n 1, R k 1, 1 R m 1, P 2 : ẋ 2 A 2 B 2 2, 2 C 2 D 2 2, x R n 2, R k 2, 2 R m 2. The general procedre to obtain the state-space for an interconnection consists of stacking the states of the individal sbsstems in a tall vector x and compting ẋ sing the state and otpt eqations of the individal blocks. The otpt eqation is also obtained from the otpt eqations of the sbsstems. In Figre 1.1(b) we have 1 2 and 1 2, which corresponds to a parallel interconnection. This figre represents the LTI sstem [ [ [ [ẋ1 A1 0 x1 B1, [ [ x C ẋ 2 0 A 2 B 1 C 1 2 (D 2 x 1 D 2 ), 2 Notation. Given a vector (or matrix) x,we denote its transpose b x. with state x : [x 1 x 2 R n 1n 2. The parallel interconnection is responsible for the block-diagonal strctre in the matrix [ A A 2. A block-diagonal strctre in this matrix indicates that the state-space sstem can be decomposed as the parallel of two state-space sstems with smaller states. In Figre 1.1(c) we have 1, 2,andz 1 2, which corresponds to a cascade interconnection. This figre represents the LTI sstem [ [ [ [ẋ1 A1 0 x1 B1, [ [ x D ẋ 2 B 2 C 1 A 2 B 2 D 2 C 1 C 1 2 D 1 D 1, 2 with state x : [x 1 x 2 R n 1n 2. The cascade interconnection is responsible for the block-trianglar strctre in the matrix [ A 1 0 B 2 C 1 A 2 and, in fact, a block-trianglar strctre in this matrix indicates that the state-space sstem can be decomposed as a cascade of two state-space sstems with smaller states.

5 STATE-SPACE LINEAR SYSTEMS 9 z z P 3 P 4 (a) ẋ x, x (b) LTI sstem in (1.4) Figre 1.2. Block diagram representation sstems. Note. Howtoarriveat eqation (1.3)? Hint: Start with the otpt eqation, note that C 1 x 1 D 1 ( ), and solve for. MATLAB R Hint 2. To avoid ill-posed feedback interconnections, MATLAB R warns abot algebraic loops when one attempts to close feedback loops arond sstems like P 1 with nonzero D 1 matrices (even when I D 1 is invertible). In Figre 1.1(d) we have 1 1 and 1, which corresponds to a negativefeedback interconnection. This figre represents the following LTI sstem ẋ 1 (A 1 B 1 (I D 1 ) 1 C 1 )x 1 B 1 (I (I D 1 ) 1 D 1 ), (I D 1 ) 1 C 1 x 1 (I D 1 ) 1 D 1, (1.3a) (1.3b) with state x 1 R n 1. Sometimes feedback interconnections are ill-posed. In this example, this wold happen if the matrix I D 1 were singlar. The basic interconnections in Figre 1.1 can be combined to form arbitraril complex diagrams. The general procedre to obtain the final state-space sstem remains the same: Stack the states of all sbsstems in a tall vector x and compte ẋ sing the state and otpt eqations of the individal blocks Sstem Decomposition MATLAB R Hint 3. This tpe of decomposition is especiall sefl to bild sstems in Simlink R. Note. The pper trianglar form in (1.4) gives awa that this cold be decomposable as a cascade. Note. In general, this tpe of decomposition is not niqe, since there ma be man was to represent a sstem as the interconnection of simpler sstems. Block diagrams are also sefl to represent complex sstems as the interconnection of simple blocks. This can be seen throgh the following two examples: 1. The LTI sstem ẋ x, x can be viewed as a feedback connection in Figre 1.2(a), where the integrator sstem maps each inpt z to the soltions of 2. Consider the LTI sstem [ẋ1 ẋ 2 [ Writing these eqations as [ x1 ẏ z. [ 1, 5 [ 1 0 [ x 1. (1.4) ẋ 2 3 5, 2 (1.5)

6 10 LECTURE 1 and ẋ 1 x 1 z, x 1, (1.6) where z :, weconcldethat(1.4)canbeviewedastheblockdiagram in Figre 1.2(b), where P 3 corresponds to the LTI sstem (1.5) with inpt and otpt 2 and P 4 corresponds to the LTI sstems (1.6) with inpt z and otpt Sstem Interconnections with MATLAB R Attention! Note the different order in which ss1 and ss2 appear in the two forms of this command. MATLAB R Hint 4 (series) The command ssseries(ss1,ss2) or, alternativel, ssss2*ss1 creates a sstem ss from the cascade connection of the sstem ss1 whose otpt is connected to the inpt of ss2. For MIMO sstems, one can se ssseries(ss1,ss2,otpts1, inpts2), whereotpts1 and inpts2 are vectors that specif the otpts of ss1 andinptsofss2, respectivel, that shold be connected. These two vectors shold have the same size and contain integer indexes starting at 1. MATLAB R Hint 5 (parallel) The command ssparallel(ss1,ss2) or, alternativel, ssss1ss2 creates a sstem ss from the parallel connection of the sstems ss1 and ss2. For MIMO sstems, one can se ssparallel(ss1,ss2,inpts1, inpts2,otpts1,otpts2),whereinpts1 and inpts2 specif which inpts shold be connected and otpts1 and otpts2 specif which otpts shold be added. All for vectors shold contain integer indexes starting at 1. MATLAB R Hint 6 (append) The command ssappend(ss1,ss2,...,ssn) creates a sstem ss whose inpts are the nion of the inpts of all the sstems ss1, ss2,...,ssn and whose otpts are the nion of the otpts of all the same sstems. The dnamics are maintained decopled. MATLAB R Hint 7 (feedback) The command ssfeedback(ss1,ss2) creates a sstem ss from the negative feedback interconnection of the sstem ss1 in the forward loop, with the sstem ss2 in the backward loop. A positive feedback interconnection can be obtained sing ssfeedback(ss1,ss2,1). For MIMO sstems, one can se ssfeedback(ss1,ss2,feedinpts, feedotpts,sign), where feedinpts specif which inpts of the forwardloop sstem ss1 receive feedback from ss2, feedotpts specif which otpts of the forward-loop sstem ss1 are feedback to ss2, and sign { 1, 1} specifies whether a negative or positive feedback configration shold be sed. More details can be obtained b sing help feedback.

7 1.3 Exercises Copright, Princeton Universit Press. No part of this book ma be STATE-SPACE LINEAR SYSTEMS (Block diagram decomposition). Consider a sstem P 1 that maps each inpt to the soltions of [ẋ1 ẋ 2 [ [ x1 [ 4, [ 1 3 [ x 1. 1 Represent this sstem in terms of a block diagram consisting onl of integrator sstems, represented b the smbol,thatmaptheirinpt( ) R to the soltion ( ) R of ẏ ; smmation blocks, represented b the smbol, that map their inpt vector ( ) R k to the scalar otpt (t) k i1 i(t), t 0; and gain memorless sstems, represented b the smbol g, thatmaptheirinpt ( ) R to the otpt (t) g(t) R, t 0 for some g R.