Chapter 3 Discrete-Time Systems 3.1. Introduction Generally, a signal is a function (or distribution) with support in the time space T, and with value in the vector space E, which is defined on R. Depending on whether we have a continuous-time signal or a discrete-time signal, the time space can be identified with the set of real numbers R or with the set of integers of Z. A discrete system is a system which transforms a discrete signal, noted by u, into a discrete signal noted by y. The class of systems studied in this chapter is the class of time-invariant and linear discrete (DLTI) systems. Such systems can be described by the recurrent equations [3.1] or [3.2] 1 : += + ⎧ ⎨ =+ ⎩ (1) () () () () () xk Axk Buk yk Cxk Duk [3.1] +−=+−"" 0 () ( ) () ( ) nn yk a yk n buk buk n [3.2] where signals u, x and y are sequences with support in Z ( Ζ∈k ) and with value in m R , n R and p R respectively. They represent the input, the state and the output of the system (see the notations used in Chapters 2 and 3). ii baDCBA ,,,,, are appropriate size matrices with coefficients in R: Chapter written by Philippe CHEVREL. 1 We can show the equivalence of these two types of representations (see Chapter 2). 82 Analysis and Control of Linear Systems pxm i pxp i pxmpxnnxmnxn RbRaRDRCRBRA ∈∈∈∈∈∈ ,,,,, [3.3] If equations [3.1] and [3.2] can represent intrinsically discrete systems, such as a µ-processor or certain economic systems, they are, most often, the result of discretization of continuous processes. In fact, let us consider the block diagram of an automated process, through a computer control (see Figure 3.1). Seen from the computer, the process to control, which is supplied with its upstream digital-analog and downstream analog-digital converters (ADC), is a discrete system that converts the discrete signal u into a discrete signal y. This explains the importance of the discrete system theory and its development, which is parallel to the development of digital µ-computers. Figure 3.1. Computer control This chapter consists of three distinct parts. The analysis and manipulation of signals and discrete-time systems are presented in sections 3.2 and 3.3. The discretization of continuous-time systems and certain concepts of the sampling theory are dealt with in section 3.4. Discrete-Time Systems 83 3.2. Discrete signals: analysis and manipulation 3.2.1. Representation of a discrete signal A discrete-time signal 2 is a function (.)x with support in T = Z and with value in NnRE ∈= , n . We will talk of a scalar signal if n = 1, of a vector signal in the contrary case and of a causal signal if − ∈∀= Zkkx ,0)( . Only causal signals will be considered in what follows. There are several ways to describe them: either explicitly, through an analytic expression (or by tabulation), like in the case of elementary signals defined by equations [3.4] to [3.6], or, implicitly, as a solution of a recurrent equation (see equation [3.7]): Discrete impulse 3 : 1if 0 () 0if k k kZ δ ∗ = ⎧ ⎪ = ⎨ ∈ ⎪ ⎩ [3.4] Unit-step function: 1if () 0if kZ k kZ Γ + −∗ ⎧ ∈ ⎪ = ⎨ ∈ ⎪ ⎩ [3.5] Geometrical sequence: if g(k) 0 if k akZ kZ +∗ −∗ ⎧ ∈ ⎪ = ⎨ ∈ ⎪ ⎩ [3.6] It will be easily verified that the solution of equation [3.7] is the geometrical sequence [3.6] previously defined. Hence, the geometrical sequence has, for discrete-time signals, a role similar to the role of the exponential function for continuous-time signals. First order recurrent equation: ⎩ ⎨ ⎧ = =+ 1)0( )()1( x kaxkx [3.7] 2 Unlike a continuous-time signal, which is a function with real number support (T = R). 3 We note that if the continuous-time impulse or Dirac impulse is defined only in the distribution sense, it goes differently for the discrete impulse. 84 Analysis and Control of Linear Systems 3.2.2. Delay and lead operators The concept of an operator is interesting because it enables a compact formulation of the description of signals and systems. The manipulation of difference equations especially leads back to a purely algebraic problem. We will call “operator” the formal tool that makes it possible to univocally associate with any signal () x ⋅ with support in T another signal ()y ⋅ , itself with support in T. As an example we can mention the “lead” operator, noted by q [AST 84]. Defined by equation [3.8], it has a role similar to that of the “derived” operator for continuous-time signals. The delay operator is noted by q –1 for obvious reasons (identity operator: 1 1 qq ∆ − = D ). )1( : )( : +→ → ⇔ → → kxk ETqx kxk ETx [3.8] )1( : )( : 1 −→ → ⇔ → → − kxk ET xq kxk ETx Table 3.1. Backwards-forwards shift operators Any operator f is called linear if and only if it converts the entire sequence () () kxkx 21 λ + , R∈ λ into the sequence () () kyky 21 λ + with 11 () y fx ∆ = and 22 () y fx ∆ = . It is called stationary if it converts any entire delayed or advanced sequence )( rkx − , Z r ∈ into the sequence )( rky − , with () xfy ∆ = (formally, )()( xfqxqf rr −− = ). The gain of the operator is induced by the standard used in the space of the signals considered (for example, L 2 or L ∞ ). The gain of the lead operator is unitary. These definitions will be useful in section 3.3. Except for the lead operator, operator T q T 1 1 − − = ∆ δ and operator )1()1( 1 −+= − ∆ qqw D will be used sometimes. Discrete-Time Systems 85 3.2.3. z-transform 3.2.3.1. Definition The z-transform represents one of the main tools for the analysis of signals and discrete systems. It is the discrete-time counterpart of the Laplace transform. The z- transform of the sequence {()}xk , noted by )(zX , is the bound, when it exists, of the sequence: () ∑ ∞ ∞ − ∆ = - )( k zkxzX where z is a variable belonging to the complex plan. For a causal signal, the z-transform is given by [3.9] and we can define the convergence radius R of the sequence (the sequence is assumed to be entirely convergent for R>z ). ∆∆ ∞ − == ∑ 1 0 () {()} ()Xz xk xkzZ R>z [3.9] )(zX is the function that generates the numeric sequence {()}xk . We will easily prove the results of Table 3.2. )(kx )(zX R )(kx )(zX R )(k δ 1 ∞ )(ka k Γ az z − a )(kΓ 1− z z 1 ω Γsin( ) ( ) k akk ω ω −+ 22 sin (2 cos ) z z aza a )(kk Γ 2 )1( −z z 1 ω Γcos( ) ( ) k akk ω ω − −+ 2 22 cos (2 cos ) zz z aza a Table 3.2. Table of transforms 3.2.3.2. Inverse transform The inverse transform of )(zX , which is a rational fraction in z, can be obtained for the simple forms by simply reading through the table. In more complicated cases, a previous decomposition into simple elements is necessary. We can also calculate the sequence development of )(zX by polynomial division according to 86 Analysis and Control of Linear Systems the decreasing powers of 1− z or apply the method of deviations, starting from the definition of the inverse transform: ∫ == − C k dzzzX j zXZkx )( 2 1 ))(()( 1 π [3.10] where C is a circle centered on 0 including the poles of ).(zX 3.2.3.3. Properties of the z-transform 4 We will also show, with no difficulties (as an exercise), the various properties of the z-transform that can be found below. The convergence rays of the different sequences are mentioned. We note by x R the convergence ray of the sequence associated with the causal sequence )(kx . P1: z-transform is linear ( ),(max yxbyax RRR = + ) +=+ ∀∈({ () ()}) () (), ,Zaxk byk aXz bYz ab R P2: delay theorem ( x xq r RR = − ) +−− ∈∀= ZrzXzkxqZ rr ),()})({( P3: lead theorem ( x xq n RR = ) + − − ∈∀ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −= ∑ ZnzkxzXzkxqZ n knn ,)()()})({( 1 0 In particular: )0()()})1({( xzzXkxZ −=+ P4: initial value theorem If )(kx has )(zX as a transform and if )(lim zX z ∞→ exists, then: )(lim)0( zXx z ∞→ = 4 Note: the various manipulated signals are assumed to be causal. Discrete-Time Systems 87 P5: final value theorem If )(lim kx k ∞→ exists, then: )()1(lim)(lim 1 1 zXzkx zk − →∞→ −= P6: discrete convolution theorem ( ),( 2121 max xxxx RRR = ∗ ) Let us consider two causal signals 1 () x k and 2 () x k and their convolution integral )()()()()( 2 0 12121 kxknxkxknxnxx n kk −=−=∗ ∑∑ = +∞ −∞= . We have: ∗= 12 1 2 ({ ()}) () ()Zx xn Xz Xz P7: multiplication by k ( xkx RR = ) =− () ({ ( )}) dX z Zkxk z dz P8: multiplication by k a ( x xa a k RR = ) )()})({( 1 zaXkxaZ k − = 3.2.3.4. Relations between the Fourier-Laplace transforms and the z-transform The aim of this section is not to describe in detail the theory pertaining to the Fourier transform. More information on this theory can be found in [ROU 92]. Only the definitions are mentioned here, that enable us to make the comparison between the various transforms. Continuous signal: x a (t) Discrete signal: x(k) Fourier transform ∫ ∞ ∞− Ω− =Ω dtetxX tj aF )()( ∑ +∞ −∞= − = k kj F ekxX ω ω )()( Laplace transform/ z-transform Cp dtetxpX pt aa ∈ = ∫ ∞ ∞− − )()( Cz zkxzX k k ∈ = ∑ +∞ −∞= − )()( Table 3.3. Synthesis of the various transforms 88 Analysis and Control of Linear Systems Hence, if we suppose that )(zX exists for ω j ez = , the signal discrete Fourier transform )(kx is given by )()( ω ω j F eXX = , whereas in the continuous case, ω is a homogenous impulse at a time inverse, the discrete impulse d ω (also called reduced impulse) is adimensional. The relations between the two transforms will become more obvious in section 3.4 where the discrete signal is obtained through the sampling of the continuous signal. 3.3. Discrete systems (DLTI) A discrete system is a system that converts an incoming data sequence )(ku into an outgoing sequence )(ky . Formally, we can assign an operator f that transforms the signal u into a signal y ( () ()() , y k f u k k Z=∀∈). The system is called linear if the operator assigned is linear. It is stationary or time-invariant if f is stationary (see section 3.2). It is causal if the output at instant nk = depends only on the inputs at previous instants nk ≤ . It is called BIBO-stable if for any bound-input corresponds a bound-output and this, irrespective of the initial conditions. Formally: ( ∞<⇒∞< ))((sup)(sup kfuku kk ). In this chapter we will consider only time- invariant linear discrete systems. Different types of representations can be envisaged. 3.3.1. External representation The representation of a system with the help of relations between its only inputs and outputs is called external. 3.3.1.1. Systems defined by a difference equation Discrete systems can be described by difference equations, which, for a DLTI system, have the form: )()()()( 0 nkubkubnkyaky nn −+=−+ "" [3.11] We will verify, without difficulty, that such a system is linear and time-invariant (see the definition below). The coefficient in y(k) is chosen as unitary in order to ensure for the system the property of causality (only the past and present inputs affect the output at instant k). The order of the system is the order of the difference equation, i.e. the number of past output samples necessary for the calculation of the present output sample. From the initial conditions −−"(1), ,( )yyn, it is easy to recursively calculate the output of the system at instant k. Discrete-Time Systems 89 3.3.1.2. Representation using the impulse response Any signal ⋅()u can be decomposed into a sum of impulses suitably weighted and shifted: ∑ ∞ −∞= −= i ikiuku )()()( δ On the other hand, let ⋅()h be the signal that represents the impulse response of the system (formally: δ = ()hf ). The response of the system to signal δ i q − is hq i− due to the property of stationarity. Hence, linearity leads to the following relation: ∑∑ ∞ −∞= ∞ −∞= =−=−= ii kuhikuihikhiuky )(*)()()()()( [3.12] The output of the system is expressed thus as the convolution integral of the impulse response h and of the input signal u. We can easily show that the system is causal if and only if 0,0)( <∀= kkh . In addition, it is BIBO-stable if and only if ∑ ∞ = ∞< 0 )( i ih . 3.3.2. Internal representation In section 3.3.1.1 we saw that a difference equation of order n would require n initial conditions in order to be resolved. In other words, these initial conditions characterize the initial state of the system. In general, the instantaneous state ∈() n xk R sums up the past of the system and makes it possible to predict its future. From the point of view of simulation, the size of ()xk is also the number of variables to memorize for each iteration. Based on the recurrent equation [3.11], the state vector can be constituted from the past input and output samples. For example, let us define the i th component of ()xk , () i xk, through the relation: = =−+−−−+− ∑ () [ ( 1) ( 1)] n ij j ji xk buk j i ayk j i [3.13] 90 Analysis and Control of Linear Systems Then we verify that the state vector satisfies the recurrent relation of first order [3.14a] called equation of state and that the system output is obtained from the observation equation [3.14b]: N ⎛⎞ − ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ += + ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ − ⎝⎠ ⎝⎠ # #% # # "  1 1 100 00 (1) () () 001 00 n n B A b a xk xk uk b a [3.14a] N =+"    0 () (1 0 0) () () D C yk x k b uk [3.14b] We note that the iterative calculation of y(k) requires only the initial state () 0 0 xx ∆ = (obtained according to [3.13] from C.I. () () { } 1, ,yyn−−" ) and the past input samples { } (), 0ui i k≤< . As in the continuous case, this state representation 5 is defined only for a basis change and the choice of its parameterization is not without incidence on the number of calculations to perform. In addition, the characterization of structural properties introduced in the context of continuous-time systems (see Chapters 2 and 4), such as controllability or observability, are valid here. The evolution of the system output according to the input applied and initial conditions is simply obtained by solving [3.14]: () () ()   ky k i ik ky k f l kDuiBuCAxCAky ∑ − = −− ++= 1 0 1 0 )()( [3.15] () l yk and () f yk designate respectively the free response and the forced response of the system. Unlike the continuous case, the solution involves a sum, and not an integration, of powers of A and not a matrix exponential function. Each component () i xk of the free response can be expressed as a linear combination of terms, such as ρ λ () k ii k , where ρ ⋅() i is a polynomial of an order equal to 1− i n , where i n is the multiplicity order of i λ and i th the eigenvalue of A. 5 A canonical form called controllable companion. [...]... inverse order, etc The system is stable if and only if the first coefficients of the odd rows of the table ( a0 , b0 , c0 , etc.) are all strictly positive 94 Analysis and Control of Linear Systems NOTE 3. 3.– the class of rational systems that can be described by [3. 16] or [3. 18] is a sub-class of DLTI systems To be certain of this, let us consider the system characterized by the irrational transfer:... since the frequency response of the discretized system is periodic, contrary to the one of the continuous system 14 If A is non-singular, we also have 15 TU: Time Unit BT = A −1 ( e AT − I) B 104 Analysis and Control of Linear Systems Figure 3. 5 Bode diagram of the continuous system and its discretization 3. 4.5 The problem of sub-sampling Let us consider the case of a “standardized” pendulum subjected... jω d ) and we will draw H ⎜ ⎟ ⎝ 1 − w ⎠ w = jω d Figure 3. 3 Bode diagram Let us consider the case of a first order system given by its transfer function 1− a , a < 1 The bandwidth of the Bode diagram drawn in Figure 3. 3 is H ( z) = z−a more important if a is “small” This result can be linked to the time response of this same system studied in the next section 98 Analysis and Control of Linear Systems. .. theory for the discrete-time systems The next result is close to the result for continuoustime systems in Chapter 2 THEOREM 3. 1.– the system described by the recurrence x (k + 1) = Ax (k ), x (0) = x0 is asymptotically stable if and only if: ∃Q = Q T > 0 and ∃P = PT > 0 solution of equation6: AT PA − P = Q 3. 3 .3 Representation in terms of operator The description and manipulation of systems as well as... After briefly analyzing the behavior of basic discrete systems, we presented in short the issue of sampling passage from continuous-time signals and systems to discrete-time signals and systems Our goal was to provide the basics that will make it possible to deal with (Chapters 8, 12 and 13) the digital simulation of continuous systems and their control by the computer We deliberately ignored certain... illustrated in Figure 3. 2 In general, we can define an algebra of diagrams which makes it possible to reduce the complexity of a defined system from interconnected sub -systems Figure 3. 2 Interconnected systems 7 The system is causal Discrete-Time Systems 93 NOTE 3. 1.– acknowledging the initial conditions, which is natural in the state formalism and more suitable to the requirements of the control engineer,... jωd − 1 ⎞ ⎟ ⎜ τ ⎟ ⎝ ⎠ κ (ω ) = Hδ ⎜ ⎛ e jωd − 1 ⎞ and ϕ (ωd ) = arg Hδ ⎜ ⎟ ⎜ τ ⎟ ⎝ ⎠ 3. 3.5 Time response of basic systems A DLTI system of an arbitrarily high order can be decomposed into serialization or parallelization of first and second order systems (see Chapter 1) Hence, it is interesting to outline the characteristics of these two basic systems 3. 3.5.1 First order system Let us consider the first... generally, and according to the situation of the poles, there are various types of unit-step responses (see Figure 3. 4), which are stable or unstable depending on whether the poles belong or not to the unit disc Figure 3. 4 Relation between the poles and the second order unit-step response 3. 4 Discretization of continuous-time systems The diagram in Figure 3. 1, process excluded, represents a typical chain of. .. obtain HT ( z) from the z-transform of this impulse response 102 Analysis and Control of Linear Systems Due to this relation and Table 3. 5, we will be able to easily obtain the transfer function of the discretized system and, consequently, its frequency response It is also shown (see next section) that if pc is a pole of the continuous system, then pd = eTpc , whereas the poles of the discretized system... A, B, C , D of the q representation based on equation [3. 24] We note that the condition of reversibility of matrix ( A + I ) is required and that this condition is always satisfied if the discretized system is the result of the discretization of a continuous system (see section 3. 4 .3) 2 Aw = ( A − I ) ( A + I )−1 2 Bw = ( A + I )−1 B Cw = C Dw = D τ τ [3. 24] Representations [3. 20] and [3. 23] have certain . parts. The analysis and manipulation of signals and discrete-time systems are presented in sections 3. 2 and 3. 3. The discretization of continuous-time systems and certain concepts of the sampling. equivalence of these two types of representations (see Chapter 2). 82 Analysis and Control of Linear Systems pxm i pxp i pxmpxnnxmnxn RbRaRDRCRBRA ∈∈∈∈∈∈ ,,,,, [3. 3] If equations [3. 1] and [3. 2]. of the odd rows of the table () 000 , , , etc.abc are all strictly positive. 94 Analysis and Control of Linear Systems NOTE 3. 3.– the class of rational systems that can be described by [3. 16]