Transfer function approach Dynamic response Summary Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis Paweł Malczyk Division of Theory of Machines and Robots Institute of Aeronautics and Applied Mechanics Faculty of Power and Aeronautical Engineering Warsaw University of Technology October 17, 2019 © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 1 / 31 Transfer function approach Dynamic response Summary Outline 1 Transfer function approach 2 Dynamic response 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 2 / 31 Transfer function approach Dynamic response Summary Transfer function approach 1 Transfer function approach SISO system Definition Poles and zeros Transfer function for multivariable system Properties 2 Dynamic response 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 3 / 31 Transfer function approach Dynamic response Summary SISO system Fig. 1: Block diagram of a single input single output (SISO) system Consider the continuous, linear time-invariant (LTI) system defined by linear constant coefficient ordinary differential equation (LCCODE): dny dn−1y + − + ··· + _ + = an n an 1 n−1 a1y a0y dt dt (1) dmu dm−1u = b + b − + ··· + b u_ + b u m dtm m 1 dtm−1 1 0 initial conditions y(0), y_(0),..., y(n−1)(0), and u(0),..., u(m−1)(0) given, u(t) – input signal, y(t) – output signal, ai – real constants for i = 1, ··· , n, and bj – real constants for j = 1, ··· , m. How do I find the LCCODE (1)? . How do I find the coefficients ai and bj? © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 4 / 31 Transfer function approach Dynamic response Summary Definition Definition 1 (Transfer function) The transfer function G(s) of a linear, time- invariant differential equation system is defined as the ratio of the Laplace transform of the output Y(s) (response function) to the Laplace transform of the input U(s) (driving Fig. 2: Block diagram of a single input single output (SISO) system function) under the assumption that all initial conditions are zero. Let us transform Eq. (1): n n−1 a s + a − s + ··· + a s + a Y(s) = n n 1 1 0 m m−1 (2) = bms + bm−1s + ··· + b1s + b0 U(s) then P − m k ( ) m + − m 1 + ··· + + Y s bms bm 1s b1s b0 Pk=0 bks G(s) = = − = n (3) n − n 1 ··· k U(s) ans + an 1s + + a1s + a0 k=0 aks Comment 1 For causal systems m ≤ n (proper transfer functions). © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 5 / 31 Transfer function approach Dynamic response Summary Poles and zeros Definition 2 (Characteristic equation) The characteristic equation of a system is defined as the equation obtained by setting the characteristic polynomial of a transfer function G(s) in Eq. (3) to zero, i.e.: n n−1 N(s) = ans + an−1s + ··· + a1s + a0 = 0 (4) Definition 3 (Pole) The roots of the characteristic equation N(s) = 0 are called poles of a transfer function G(s). Definition 4 (Zero) m m−1 The roots of the polynomial M(s) = bms +bm−1s +···+b1s+b0 are called zeros of a transfer function G(s). © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 6 / 31 Transfer function approach Dynamic response Summary Transfer function for multivariable system Fig. 3: Block diagram of a multiple input multiple output (MIMO) system Definition 5 (Transfer function for MIMO systems) th th The transfer function between the j input uj(t) (j = 1, 2, ··· , p) and the i output yi(t) (i = 1, 2, ··· , q) is defined as Yi(s) Gij(s) = (5) Uj(s) with Uk(s) = 0 for k = 1, 2, ··· , p, k ≠ j under the assumption that all initial conditions are zero. When all the p inputs are in action, the ith output transform is written Yi(s) = Gi1(s)U1(s) + Gi2(s)U2(s) + ··· + Gip(s)Up(s) (6) © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 7 / 31 Transfer function approach Dynamic response Summary Transfer matrix It is convenient to express Eq. (6) in matrix-vector form: Y(s) = G(s)U(s) (7) where U (s) Y (s) 1 1 U2(s) Y2(s) U(s) = ··· , Y(s) = ··· (8) Up(s) Yq(s) are the transformed p×1 input vector and the transformed q×1 output vector, whereas G (s) G (s) ··· G (s) 11 12 1p G21(s) G22(s) ··· G2p(s) G(s) = (9) . ··· . Gq1(s) Gq2(s) ··· Gqp(s) is the q × p transfer-function matrix. © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 8 / 31 Transfer function approach Dynamic response Summary Transfer functions of physical systems Electrical network transfer functions V(s) Component Voltage-current Voltage-charge G(s) = I(s) Z t 1 1 1 v(t) = C i(τ)dτ v(t) = C q(t) Cs 0 dq(t) v(t) = Ri(t) v(t) = R dt R di(t) d2q(t) v(t) = L dt v(t) = L dt2 Ls Mechanical system transfer functions F(s) Component Force-velocity Force-displacement G(s) = X(s) Z t f(t) = k v(τ)dτ f(t) = kx(t) k 0 dx(t) f(t) = bv(t) f(t) = b dt bs 2 dv(t) d x(t) 2 f(t) = m dt f(t) = m dt2 ms © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 9 / 31 Transfer function approach Dynamic response Summary Analogies between electrical and mechanical systems Electrical network transfer functions Mechanical system transfer functions V(s) F(s) Component Voltage-current Voltage-charge G(s) = I(s) Component Force-velocity Force-displacement G(s) = X(s) Z Z t t 1 1 1 v(t) = C i(τ)dτ v(t) = C q(t) Cs f(t) = k v(τ)dτ f(t) = kx(t) k 0 0 dq(t) dx(t) v(t) = Ri(t) v(t) = R dt R f(t) = bv(t) f(t) = b dt bs 2 2 di(t) d q(t) ( ) = dv(t) ( ) = d x(t) 2 v(t) = L dt v(t) = L dt2 Ls f t m dt f t m dt2 ms The equations of motion and the behavior of systems involving various physical media are found to be analogous. Force is analogous to voltage, and velocity to current. Velocity is analogous to voltage, and force to current. Different analogies exist. © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 10 / 31 Transfer function approach Dynamic response Summary Example Example 1 Consider the mechanical system (Fig. 4). We assume that the system is linear. The external force u(t) is the input signal, and the displacement y(t) of the mass is the output. The displacement y(t) is measured from the equilibrium position in the absence of external force. Write the equations of motion and find the transfer function for the system. The equations of motion for the system are as follows m¨y + by_ + ky = u The Laplace transform of the eq. (y(0) = 0, y_(0) = 0): Fig. 4: Mechanical ms2Y(s) + bsY(s) + kY(s) = U(s) system Y(s) 1 G(s) = = Transfer function U(s) ms2 + bs + k Comment 2 The characteristic eq. N(s) = ms2 + bs + k has two poles. Comment 3 The transfer function conveys useful information: Y(s) ≈ 1 G(s) = U(s) k for small s (spring drive by a force) Y(s) ≈ 1 G(s) = U(s) ms2 for large s (mass driven by a force) © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 11 / 31 Transfer function approach Dynamic response Summary Properties of a transfer function 1 The transfer function of a system is a mathematical model of that system, in that it is an operational method of expressing the differential equation that relates the output variable to the input variable. 2 The transfer function is a property of a system itself, unrelated to the magnitude and nature of the input or driving function. 3 The transfer function includes the units necessary to relate the input to the output; however, it does not provide any information concerning the physical structure of the system. 4 If the transfer function of a system is known, the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system. 5 If the transfer function of a system is unknown, it may be established experimentally by introducing known inputs and studying the output of the system (identification). © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 12 / 31 Transfer function approach Dynamic response Summary Dynamic response 1 Transfer function approach 2 Dynamic response Time response Unit impulse response Response by convolution Unit step response Transient and steady state response Total response – example Example of a heat-flow model 3 Summary © Paweł Malczyk. Basics of Automation and Control I Lecture 3: Transfer function and dynamic response analysis 13 / 31 Transfer function approach Dynamic response Summary Time response Definition 6 (Time response) The time response represents how the state and output of a dynamic system with nonzero initial conditions changes in time when subjected to a particular input.