10 Transfer Matrix Models

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1 MIT EECS (FALL 26) LECTURE NOTES BY A. MEGRETSKI 1 Transfer Matrix Models So far, transfer matrices were introduced for finite order state space LTI models, in which case they serve as an important tool of visualization and analysis. However, many LTI models derived from the first principles equations are given in terms of partial differential equations, and do not allow a finite order LTI state space representation. For such systems, transfer matrices serve as an important unifying tool of representation, analysis, and approximation. A minor technical diffculty for this lecture is that the definition of a signal as an element of L m accepted earlier is not flexible enough to accomodate such models in sufficient generality: even for some relatively benign infinite dimensional CT LTI models, such as y(t) = e k f ( t 1 ), k the response y to a piecewise continuous input f does not have to be piecewise continuos. Therefore, for this lecture, the class of all m-dimensional signals will be defined as the set of locally square integrable functions. L 2,loc m 1.1 Transfer Matrices It is convenient to begin with defining transfer matrices which correspond to stable LTI systems. In the case of finite dimensional models, such transfer matrices are proper rational functions with no poles in the closed right half plane. Therefore, the entries of such matrices are analytical uniformly bounded functions on C = {s : Re(s) >. This observation is behind the following generalization of the notion of a stable transfer matrix. Definition 1.1 A complex k-by-m matrix-valued function G : C C k m is called a continuous time stable transfer matrix if it is analytical, real symmetric, and uniformly bounded. The set of all CT stable k-by-m transfer matrices is an important object in control theory and complex analysis, called the Hardy space and denoted as Hk m (H for k = m = 1) here. Example 1.1 Function H a (s) = e as, where a is a real parameter, is a CT stable transfer matrix if and only if a. Indeed, while H a is analytical on the whole complex plane, it is unbounded in every right half plane when a <. 1

2 A very general class of continuous time LTI systems can be introduced using multiplication in the Laplace transform domain by a fixed complex matrix-valued function G which is real symmetric, uniformly bounded, and analytical in C r for some r IR. However, such generality is unlikely to be useful for practical purposes, as many of resulting systems will not be stabilizable by strictly proper finite order controllers. To explain this in an informal way, consider the simple feedback interconnection shown on Figure 1.1. u d r + + K(s) P (s) + q Figure 1.1: A simple feedback control setup For the closed loop system to be stable, one expects the closed loop transfer functions L(s) = P (s) 1 + P (s)k(s), H(s) = P (s)k(s) (from d to q and from r to e = r q) to belong to the class H. Moreover, since P is to be bounded as Re(s) +, and K(s) as s, the value of H(s) converges to 1 as Re(s) converges to plus infinity. Hence P can be represented in the form P (s) = L(s) H(s) : L, H H, lim H(s) = 1. (1.1) Re(s) + In particular, since zeros of an analytical function cannot have accumulation points in its domain, P = P (s) must be meromorphic in C, which means that it can be represented by a convergent Taylor series expansion P (s) = P k (s s ) k k= q in a neigborhood of every point s C. This observation leads to the following definition of general continuous time transfer matrices. Definition 1.2 A continuous time transfer matrix G is a meromorphic complex matrixvalued function on C, each scalar component P = G il of which can be represented in the form (1.1). 2

3 It is easy to see that, according to this definition, every proper rational function is a valid transfer matrix. Indeed, if P (s) = h(s)/q(s), where n 1 q(s) = s n + q k s k, p(s) = k= then representation (1.1) takes place with L(s) = n p k s k, k= p(s) q(s), H(s) = (s + 1) n (s + 1). n Another easy but important observation is that linear combination of transfer matrices is a transfer matrix. Indeed, if P 1 = L 1 /H 1 and P 2 = L 2 /H 2 then c 1 P 1 + c 2 P 2 = L/H, where L = c 1 L 1 H 2 + c 2 L 2 H 1, H = H 1 H 2. If L i, H i H then so are L and H. Similarly, if H i (s) converge to 1 as Re(s) +, so does H. An important non-trivial observation about transfer functions is given by the following statement, given without a proof here. Theorem 1.1 If s k C are unstable poles of a transfer function (poles of multiplicity k being listed k times each) then Re(sk ) <. (1.2) 1 + s k 2 According to Theorem 1.1, unstable poles of a transfer function must approach the imaginary axis quickly enough. Example 1.2 Function G a (s) = exp((a s) 1 ), where a is a real parameter, is a CT transfer matrix if and only if a. Indeed, when a, G a (s) is analytical and bounded (by 1) in C. When a >, G a (s) has a singilarity at s = a C which is not a pole, and hence cannot be a ratio of two functions from H. Example 1.3 Function 1+k 2 a< G a (s) = 1 1 k + 1 k(s a) 1, k= where a is a real parameter, is a CT transfer matrix if and only if a <. Indeed, when a, the unstable poles of G a do not satisfy condition (1.2). When a <, function G a can be represented in the form G a = G + + G, where G + (s) = 1 k 2 (s a) 1, G (s) = 1 k 2 (s a) 1. 1+k 2 a Since G + H and G is a rational function, both G + and G are transfer functions, and hence so is G. 3

4 1.2 Transfer Matrix Models A k-by-m CT transfer matrix G defines naturally an i/o model S = S [G] = {(f, y) L 2,loc m L 2,loc k : y = G f} formed by those pairs (f, y) for which y is the result of multiplying f by G in the Laplace transform domain. This definition has a unique output y = G f L 2,loc k assigned to each input f L 2,loc m. As it was shown in the previous lecture, for LTI state space models Laplace transform domain multiplication by the transfer matrix describes the zero initial condition response. While it is not always convenient to use state space modeling for LTI systems defined by non-rational transfer matrices, one can still interpret S [G] as describing the zero state response. Note that, formally speaking, S [G] is not a time invariant system model whenever G is not a constant, as the time instance t = is treated in a special way in the definition of S[G]. To cover the effect of non-zero initial conditions in LTI models, and to make the behavior time invariant, let S lti [G] = {D τ (f + f, y + y ) : f L 2,loc m, y = G f, f L 2 m, y L 2 k}, where L 2 n stands for the subset of all elements x L 2,loc n which have finite energy, i.e. f (t) 2 dt <, y (t) 2 dt <. The finite energy signals f, y, introduced in the definition of S e [G] as input and output disturbances (see Figure 1.2) are used to emulate system response to the initial conditions which are not reachable in finite time. f G(s) f y y Figure 1.2: Interpretation of S e [G] Theorem 1.2 If G is a k-by-m CT tranfer matrix then the set S lti [G] defines a linear, causal, and time invariant input-output model. Moreover, if G = G[A, B, C, D] is a minimal realization of a rational transfer matrix G then S lti [G] contains the set B CT [A, B, C, D] of all possible input output pairs defined by the state space model S CT [A, B, C, D]. 4

5 1.2.1 Example: One Dimensional Heat Equation In applications, most LTI systems with non-rational transfer matrices are defined in terms of systems of partial differential equations (PDE). Rigorous handling of such systems can present significant mathematical challenges, addressing which goes beyond the scope of this class. The example in this section, based on a very simple setup, is aimed at showing the main technical points of converting a PDE description into a transfer matrix model Modeling Objectives Consider the evolution of temperature distribution along a thin wire of unit length, assuming no heat exchange with the outside environment except at the ends of the wire. The temperature at one end is being kept at a constant level, while at the other end it is controlled by the input signal f = f(t). The system equations have the form dv(t, x) dt = d2 v(t, x) dx 2, v(t, ) =, v(t, 1) = u(t), ü(t) = f(t), y(t) = v(t, a), (1.3) where v : IR + [, 1] IR is the wire temperature distribution as a function of time t and space x [, 1], and a (, 1) is a parameter defining the output sensor location. It is assumed that the initial temperature distribution v (x) = v(, x) is three times continuously differentiable. The partical derivatives in (1.3) are understood in a strict sense: only those functions v : IR + [, 1] IR which are continuously differentiable with respect to the first arument and two times continuously differentiable with respect to the second argument are considered as satisfying (1.3). Let S be the set of all pairs (f, y) L 2,loc L 2,loc for which there exists a function v satisfying (1.3). It can be shown that S is an input-output model, i.e. a solution exists for all inputs f L 2,loc. Our objective is to derive a transfer function model S lti [G] which contains S Finding The Transfer Function To find an explicit expression for G = G(s), it is useful to search for a solution of (1.3) of the form v(t, x) = e st V (s, x), y(t) = e st Y (s), f(t) = e st. (1.4) The approach is motivated by the fact that, for a state space LTI model ẋ = Ax + Bf, y = Cx + Df with a scalar input f, a solution x(t) = e st X(s), f(t) = e st, y(t) = e st Y (s) 5

6 would exist for almost all values of s C, and will define the transfer matrix according to Y (s) = D + C(sI A) 1 B = G[A, B, C, D](s). Substituting the expressions from (1.4) into (1.3) yields which implies sv (s, x) = d2 V (s, x) dx 2, V (s, ) =, s 2 V (s, 1) = 1, V (s, x) = 1 e sx e sx s 2 e s e, s where s stays for the main branch of the square root, defined for s C. This suggests that G(s) def = Y (s) = 1 e sa e sa s 2 e s e s is the transfer function of interest. Note that G a = L/H, where H(s) = s 2 (s + 1), L(s) = 1 e sa e sa 2 (s + 1) 2 e s e s are elements of H, and H(s) 1 as Re(s) +. Hence G a is a transfer matrix Uniqueness of Solutions Once a transfer function is known, it is now necessary to verify the inclusion S S lti [G]. This can be done by constructing explicitly zero state and zero input solutions of (1.3). The validity of such arguments depends on establishing uniqueness of solutions once the initial conditions (in this case, the function v (x) = v(, x)) and input f L 2,loc are fixed. Note that, by definition, u() = v (1), u() = v (1), and hence the initial conditions of the double integrator ü = f are determined by v ( ). Since the equations are linear, it is sufficient to show that v(t, x) is the only solution of (1.3) with v(, x) and v(t, 1). To prove this, let E(t) = 1 v(t, x) 2 dx. 6

7 Using (1.3), integration by part, and condition v(t, 1), 1 Ė(t) = 2 = 2 = 1 1 v t (t, x)v(t, x)dx v xx (t, x)v(t, x)dx v x (t, x) 2 dx. Since E() = and by construction E(t) for all t, it follows that E(t) = for all t, i.e. v(t, x) Zero State Response This step verifies that for every input f L 2,loc equations (1.3) have a solution v = v(t, x) for which v(, x), u() =, and y = G f. Such solution can be constructed explicitly in terms of inverse Laplace transform integrals. The main technical difficulty concerns establishing the required smoothness of v. Fix h >. For every T > let f T L 2 be defined by { f(t), t < T, f T (t) =, t T. For every x [, 1] define v T (t, x) as the inverse Laplace transform of V (s, x)f T (s), where F T (s) is the Laplace transform of f T (converges everywhere in C). In other words, v T (t, x) = 1 2π V (jω + h, x)e (jω+h)t F T (jω + h)dω. (1.5) Since for Re(s) >, and sv (s, x) = d 2 V (s, x) dx 2 1 s the upper bound for 1 F T (jω + h) 2 dω = 2π ρ(ω) = F T (jω + h) jω + h f T (t)e ht 2 <, (jω + h)v (jω + h, x)f T (jω + h) = d 2 V (jω + h, x) F dx 2 T (jω + h) 7

8 is absolutely integrable. Hence the integral in (1.5) can be differentiated with respect to t once, or twice with respect to x, to get dv T (t, x) dt = 1 2π = 1 2π = d2 v T (t, x) dx 2. (jω + h)v (jω + h, x)e (jω+h)t F T (jω + h)dω d 2 V (s, x) e (jω+h)t F dx 2 T (jω + h)dω Since (1.5) also implies v T (t, ) = and d 2 v T (t, 1)/dt 2 = f T (t), function v(t, x) = v T (t, x) satisfies (1.3) for all x [, 1] and t < T. On the other hand, since v T (, x) is obtained by multiplying f T by V (, x) in the Laplace transform domain, the values of v T (t, x) for t < T do not depend on T. Hence the identity v(t, x) def = v t+1 (t, x) defines a solution of (1.3) of required smoothness. To complete the proof, note that, according to the properties of Laplace transform, the expression for v T (t, x) in (1.5) must produce zero for t <. Since the continuity with respect to t has been established, this implies that v T (, x) Zero Input Response This step verifies that for every three times continuously differentiable function v : [, 1] IR such that v () = there exist a solution of (1.3) such that v(, x) = v (x), f L 2, y L 2. The solution v = v(t, x) is to be constructed as a linear combination of the terms β k (t, x) = e π2 k 2t sin(πkx), α k (t, x) = e π2 k 2t cos(πkx). First, note that there exist real constants c 1, c 2, c 3, c 4 such that the function satisfies the condition v c (x) = c k cos(πkx) (1.6) v c () = v () =, v c () = v (), v c (1) = v (1), v c (1) = v (1). (1.7) Indeed, (1.7) subject to (1.6) is a non-singular system of four linear equations with four unknowns. Equalities (1.7) imply that the 2-periodic function v s : IR IR, defined by { v (x 2k) v v s (x) = c (x 2k), 2k x 2k + 1, k ZZ, v (2k x) v c (2k x), 2k 1 x 2k, k ZZ, 8

9 is tree times continuously differentiable, odd, and 2-periodic. Hence v s can be represented by a Fourier series expansion v s (x) = b k sin(πkx) x IR, where, due to the three times continuous differentiability, k 3 b k 2 <. (1.8) Since v s (x) = v (x) v c (x) for x [, 1], function v (x) is now represented in the form v (x) = c k cos(πkx) + b k sin(πkx). Let v(t, x) = c k e k2 π 2t cos(πkx) + Since, by assumption, (1.8) implies that k 2 b k = b k e k2 π 2t sin(πkx). ( k 3 b k 1 ) 1/2 ( k ) 1/2 k 3 b k 2 1 <, k 2 the function v = v(t, x) is twice continuously differentiable with respect to x, continuously differentiable with respect to t, and solves (1.3) with v(, x) = v (x), f(t) = k 2 c k e k2 π 2t cos(πk), y(t) = c k e k2 π 2t cos(πka) + b k e k2 π 2t sin(πka). To conclude the proof, note that f has finite energy, as a finite sum of decreasing exponents, and y has finite energy due to (1.8) Conclusions Let us examine the meaning of the transfer matrix models S [G] and S lti [G] in this setup. The model S [G] turns out to be the set of all input output pairs generated with v(, x). As expected, S [G] is an exact representation of the zero state response of S. The model S lti [G] contains the set of all input output pairs generated by those initial temperature distributions which are smooth enough. In addition, S lti [G] would contain input-output pairs corresponding to generalized (or weak) solutions of (1.3), i.e. those for which the derivatives involved exist only in a certain generalized sense. 9

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