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ECE504: Lecture 2 ECE504: Lecture 2 ECE504: Lecture 2 D. Richard Brown III Worcester Polytechnic Institute 15-Sep-2009 Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 1 / 46 ECE504: Lecture 2 Lecture 2 Major Topics We are still in Part I of ECE504: Mathematical description of systems model mathematical description → You should be reading Chen chapters 2-3 now. 1. Advantages and disadvantages of different mathematical descriptions 2. CT and DT transfer functions review 3. Relationships between mathematical descriptions Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 2 / 46 ECE504: Lecture 2 Preliminary Definition: Relaxed Systems Definition A system is said to be “relaxed” at time t = t0 if the output y(t) for all t t is excited exclusively by the ≥ 0 input u(t) for t t . ≥ 0 Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 3 / 46 ECE504: Lecture 2 Input-Output DE Description: Capabilities and Limitations Example: by˙(t) ay(t)+ = du(t)+ eu˙(t) cy¨(t) + Can describe memoryless, lumped, or distributed systems. + Can describe causal or non-causal systems. + Can describe linear or non-linear systems. + Can describe time-invariant or time-varying systems. + Can describe relaxed or non-relaxed systems (non-zero initial conditions). — No explicit access to internal behavior of systems, e.g. doesn’t directly to apply to systems like “sharks and sardines”. — Difficult to analyze directly (differential equations). Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 4 / 46 ECE504: Lecture 2 State-Space Description: Capabilities and Limitations Example: x˙ (t) = Ax(t)+ Bu(t) y(t) = Cx(t)+ Du(t) — Can’t describe distributed systems. Only memoryless or lumped systems. — Can’t describe non-causal systems. Only causal systems. + Can describe linear or non-linear systems (the example is linear). + Can describe time-invariant or time-varying systems. + Can describe relaxed or non-relaxed systems (non-zero initial conditions). + Explicit description of internal system behavior, e.g. we can analyze the properties of signals internal to a system. + Abundance of analysis techniques. Linear state-space descriptions are analyzed with linear algebra, not calculus. Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 5 / 46 ECE504: Lecture 2 Transfer Function Description: Capabilities and Limitations Example: as2 + bs + c gˆ(s) = ds3 +1 + Can describe memoryless, lumped, and some distributed systems. — Can’t describe non-causal systems. Only causal systems. — Can’t describe non-linear systems. Only linear systems. — Can’t describe time-varying systems. Only time-invariant systems. — No explicit access to internal behavior of systems. — Can’t describe systems with non-zero initial conditions. Implicitly assumes that system is relaxed. + Abundance of analysis techniques. Systems are usually analyzed with basic algebra, not calculus. Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 6 / 46 ECE504: Lecture 2 Impulse Response Description: Capabilities and Limitations ◮ Recall that the impulse response of a system is the output of the system given an input u(t)= δ(t) and relaxed initial conditions. ◮ Example: βe−αt t 0 g(t) = ≥ (0 t< 0 ◮ What are the capabilities and limitations of the impulse-response description? Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 7 / 46 ECE504: Lecture 2 Impulse Response Matrix Suppose you have a linear system with p inputs and q outputs. Rather than a simple impulse response function, you now need an impulse response matrix: g11(t,τ) ... g1p(t,τ) . G(t,τ) = . g (t,τ) ... g (t,τ) q1 qp th where gkℓ(t,τ) is the response at the k output from an impulse at the ℓth input at time τ. The vector output can be computed as ∞ y(t) = G(t,τ)u(τ) dτ −∞ Z ◮ Sanity check: What are the dimensions of everything here? ◮ How do we integrate the vector G(t,τ)u(τ)? Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 8 / 46 ECE504: Lecture 2 The One-Sided Laplace Transform ◮ Suppose f (t) : R Rq×p is a matrix valued function of t 0, i.e. + 7→ ≥ f11(t) ... f1p(t) . f(t) = . f (t) ... f (t) q1 qp ◮ Define the Laplace transform of f (t) ∞ − fˆ (s)= e stf (t) dt Z0 where the integral of a matrix is done element by element. ◮ The notation here is consistent with your textbook: − fˆ (s)= [f (t)] f (t)= 1 fˆ (s) f (t) fˆ (s) L L ↔ ◮ See your textbook for more details onh the convergencei of the one-sided Laplace transform. Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 9 / 46 ECE504: Lecture 2 The Inverse Laplace Transform ∞ 1 σ+j f(t) = estfˆ(s) ds 2πj − ∞ Zσ j where j := √ 1 and σ is any point in the region of absolute convergence − of fˆ(s). ◮ This is complex integration on a path in the complex plane. ◮ In general, this integral is usually not easy to compute. ◮ Whenever possible, use tables instead. Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 10 / 46 ECE504: Lecture 2 Continuous-Time Transfer Function Matrix Definition Given a causal, linear, time-invariant system with p input terminals, q output terminals, and relaxed initial conditions at time t = 0 x(0) = 0 y(t), t 0 u(t), t 0 → ≥ ≥ then the transfer function matrix is defined as gˆ11(s) . gˆ1p(s) . yˆk(s) gˆ(s) := . where gˆkℓ(s) := uˆℓ(s) gˆ (s) . gˆ (s) q1 qp for k = 1,...,q and ℓ = 1,...,p. Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 11 / 46 ECE504: Lecture 2 Continuous-Time Transfer Function Matrix: Remarks ◮ By the relaxed assumption, the transfer function describes the zero-state response of the system. ◮ Since the transfer function must be the same for any input/output combination, most textbooks (including Chen) define it as the Laplace transform of the impulse response of the system. ◮ What is the Laplace transform of δ(t)? ◮ Hence, given a causal, linear, time-invariant system with p input terminals, q output terminals, and relaxed initial conditions x(0) = 0 y(t), t 0 uℓ(t)= δ(t) and um(t) = 0 for all m = ℓ, t 0 → ≥ 6 ≥ then yˆ1(s) gˆ1ℓ(s) uˆℓ(s) yˆ1(s) . . = . = . yˆ (s) gˆqℓ(s) q yˆq(s) uˆℓ(s) Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 12 / 46 ECE504: Lecture 2 Rational Transfer Functions and Degree Theorem (stated as a fact, Chen p.14) If a continuous-time, linear, time-invariant system is lumped, each transfer function in the transfer function matrix is a rational function of s. The proof for this theorem can be found in some of the other textbooks mentioned in Lecture 1. In any case, this result means that m m− N(s) b s + b − s 1 + + b s + b g s m m 1 1 0 ˆkℓ( )= = n n−1 · · · D(s) ans + an− s + + a s + a 1 · · · 1 0 where N(s) and D(s) are polynomials of s. Definition The degree of a polynomial in s is the highest power of s in the polynomial. Example: deg(0 s4 + 2 s3 +6) = . · · Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 13 / 46 ECE504: Lecture 2 The Three Most Important Laplace Transforms: #1 1 eat ↔ s a − ◮ Easy to show directly by integrating according to the definition. ◮ Application: For a continuous-time, linear, time-invariant, lumped system, we have m m− b s + b − s 1 + + b s + b g s m m 1 1 0 ˆ( )= n n−1 · · · ans + an− s + + a s + a 1 · · · 1 0 Compute partial fraction expansion ... c c gˆ(s)= 1 + + n s d · · · s dn − 1 − Then what can we say about g(t)? ◮ Note that this is slightly more complicated if the roots of the denominator are repeated. Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 14 / 46 ECE504: Lecture 2 The Three Most Important Laplace Transforms: #2 n (n) dg (t) Notation: g (t) := dtn . − − − g(n)(t) sngˆ(s) sn 1g(0) sn 2g˙(0) g(n 1)(0) ↔ − − −···− ◮ n = 1 case is especially useful: g˙(t) sgˆ(s) g(0). ↔ − ◮ General relationship can be shown inductively using the definition and integration by parts. Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 15 / 46 ECE504: Lecture 2 The Three Most Important Laplace Transforms: #2 Application #1: Given the input-output differential equation description of a continuous-time, linear, time-invariant, lumped system, we can easily compute the transfer function. (n) (n−1) any (t)+ an−1y (t)+ + a1y˙(t)+ a0y(t)= (m) (m−1) · · · bmu (t)+ bm− u (t)+ + b u˙(t)+ b u(t) 1 · · · 1 0 Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 16 / 46 ECE504: Lecture 2 The Three Most Important Laplace Transforms: #2 Application #2: Given the state space description of a continuous-time, linear, time-invariant, lumped system, we can easily compute the transfer function. x˙ (t) = Ax(t)+ Bu(t) y(t) = Cx(t)+ Du(t) Worcester Polytechnic Institute D. Richard Brown III 15-Sep-2009 17 / 46 ECE504: Lecture 2 The Three Most Important Laplace Transforms: #3 Notation (convolution assuming a causal relaxed system): t t f(t) g(t) := f(t τ)g(τ) dτ = f(τ)g(t τ) dτ. ∗ − − Z0 Z0 It isn’t too hard to show that f(t) g(t) fˆ(s)ˆg(s) ∗ ↔ ◮ Application #1: By definition of the transfer function of a linear, time-invariant, causal system, yˆ(s) =g ˆ(s)ˆu(s). This implies that t t y(t)= g(t τ)u(τ) dτ = g(τ)u(t τ) dτ.
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