Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018
Lesson 7: Response of LTI systems in the transform domain Laplace Transform Transform-domain response (CT) Transfer function Zeta Transform Transform-domain response (DT) Discrete Transfer function L. Giarré- Systems and Control Theory 2017-2018 - 2
Response of LTI systems - summary The solution of initial value problems (the response) for LTI systems can be written in the form: DT: y(k) =CA k x(0)+c k 1 (A k i 1 Bu(i)) + Du(k) i=0 CT: y(t) =C exp(at)x(0)+c t o exp(a(t τ))bu(τ)dτ + Du(t). L. Giarré- Systems and Control Theory 2017-2018 - 3
Response of LTI systems - summary The convolution integral (CT) and sum (DT) equation are hard to interpret, and do not offer much insight. In order to gain a better understanding, we will study the response to elementary inputs of a form that is particularly easy to analyze: the output has the same form as the input. very rich and descriptive: most signals/sequences can be written as linear combinations of such inputs. Then, using the superposition principle, we will recover the response to general inputs, written as linear combinations of the easy inputs. L. Giarré- Systems and Control Theory 2017-2018 - 4
The continuous-time case: elementary inputs Let us choose as elementary input u(t) =u 0 e st, s C. If s is real, then u is a simple exponential. If s = jω is imaginary, then the elementary input must always be accompanied by the conjugate, i.e., u(t)+u (t) =u 0 e jωt + u 0 e jωt = 2u 0 cos(ωt) if s is imaginary, then u(t) =e st must be understood as a half of a sinusoidal signal with exponentially-changing amplitude if s = σ + jω: u(t)+u (t) =u 0 (e σt e jωt + u 0 e?t e jωt ) = u 0 (e σt (e jωt + e jωt )) = 2u 0 e σt cos(ωt) L. Giarré- Systems and Control Theory 2017-2018 - 5
Output response to elementary inputs 1 Recall that y(t) =Ce (At) x(0)+c Plug in u(t) =u 0 e st t o e A(t τ) Bu(τ)dτ + Du(t). y(t) =Ce At x(0)+c =Ce At x(0)+c( t o t o e A(t τ) Bu 0 e sτ dτ + Du 0 e st e (si A)τ dτ)e At Bu 0 + Du 0 e st 0 If (si A) is invertible (i.e., s is not an eigenvalue of A), then y(t) =Ce At x(0)+c(si A) 1 e (si A)t Ie At Bu 0 + Du 0 e st. L. Giarré- Systems and Control Theory 2017-2018 - 6
Output response to elementary inputs 2 Rearranging: y(t) =Ce At x(0) C(sI A) 1 e At Bu 0 }{{} Transient response +[C(sI A) 1 B + D]u 0 e st }{{} Steady-state response If the system is asymptotically stable, e At 0, and the transient response will converge to zero. The steady state response can be written as: y ss = G(s)e st L. Giarré- Systems and Control Theory 2017-2018 - 7
Output response to elementary inputs 2 G(s) C p m G(s) =C(sI A) 1 B + D is a complex matrix. The function G : s G(s) is also known as the transfer function: it describes how the system transforms an input into the output. L. Giarré- Systems and Control Theory 2017-2018 - 8
Laplace Transform The (one-sided) Laplace transform F : C C of a sequence f : R R is defined as L[f (t)] = F (s) = + 0 f (t)e st dt for all s such that the series converges (region of convergence). Given the above definition, and recalling Laplace Transform properties, the steady state response is: Y ss (s) =G(s)U(s). G(s) is also the Laplace transform of the impulse response. L. Giarré- Systems and Control Theory 2017-2018 - 9
Transform- Domain solution (CT) 1 Starting from the state-space model, applying the Laplace df (t) transform, recalling that if g(t) = dt, then G(s) =sf (s) f (0 ),weget This is solved to yield sx(s) x(0 ) =AX(s)+BU(s) Y (s) =CX(s)+DU(s) X(s) =(si A) 1 x(0 )+(si A) 1 BU(s) Y (s) =C(sI A) 1 x(0 )+(C(sI A) 1 B + D) U(s) }{{} transfer function L. Giarré- Systems and Control Theory 2017-2018 - 10
Transform- Domain solution (CT) 2 Note that (from properties of the State Transition matrix and Laplace transform properties) L[Φ(t)] = L[e At ]=(si A) 1 =Φ(s) X(s) =Φ(s)x(0 )+Φ(s)BU(s) Y (s) =CΦ(s)x(0 )+ (CΦ(s)B + D) U(s) }{{} transfer function L. Giarré- Systems and Control Theory 2017-2018 - 11
The discrete-time case: elementary inputs Let us choose as elementary input u(k) =u 0 z k, z C. If z is real, then u is a simple geometric sequence. Recalling y(k) =CA k x(0)+c k 1 (A k i 1 Bu(i)) + Du(k) i=0 Plug in u[k] =u 0 z k, and substitute l = k i 1: y(k) =CA k x(0)+c k 1 (A l Bu 0 z k l 1 + Du 0 z l i=0 k 1 =CA k x(0)+cz k 1 ( (Az 1) i )Bu 0 + Du 0 z k i=0 L. Giarré- Systems and Control Theory 2017-2018 - 12
Matrix geometric series Recall the formula for the sum of a geometric series: k 1 i=0 m i = 1 mk 1 m For a matrix: k 1 i=0 Mi = I + M + M 2 +...M k 1 k 1 i=0 M i (I M) =(I + M + M 2 +...M k 1 )(I M) =I M k i.e., k 1 i=0 Mi =(I M k )(I M) 1 L. Giarré- Systems and Control Theory 2017-2018 - 13
Discrete Transfer Function Using the result in the previous slide, we get y(k) =CA k x(0)+cz k 1 (I A k z k )(I Az 1 )?1 Bu 0 + Du 0 z k =CA k x(0)+c(z k I A k )(zi A)?1 Bu 0 + Du 0 z k Rearranging: y(k) =CA k x(0) (zi A) 1 Bu 0 }{{} Transient response + C(zI A) 1 B + Du 0 z k }{{} Steady-state Response If the system is asymptotically stable, the transient response will converge to zero. The steady state response can be written as: y ss (k) =G(z)z k L. Giarré- Systems and Control Theory 2017-2018 - 14
Discrete Transfer Function G(z) C G(z) =C(zI A) 1 B + D is a complex number. The function G : z G(z) is also known as the (discrete) transfer function: it describes how the system transforms an input into the output. L. Giarré- Systems and Control Theory 2017-2018 - 15
Z-Transform The (one-sided) z-transform F : C C of a sequence f : N R is defined as Z[f ]=F (z) = + k=0 f [k]z?k for all z such that the series converges (region of convergence). Given the above dfinition, and the previous discussion, Y ss (z) =G(z)U(z) G(z) is the z-transform of the impulse response, i.e., the response to the sequence u =(1, 0, 0,...). L. Giarré- Systems and Control Theory 2017-2018 - 16
Transform Domain Solution (DT) Starting from the state-space model, { applying the Zeta f (k 1) k 1 transform, recalling that if g(k) = 0 k = 0 G(z) =z 1 F (z), and that if g(k) =f (k + 1), then G(z) =z[f (z) f (0)] we get, then This is solved to yield zx(z) zx(0) =AX(z)+BU(z) Y (z) =CX(z)+DU(z) X(z) =z(zi A) 1 x(0)+(zi A) 1 BU(z) Y (z) =Cz(zI A) 1 x(0)+(c(zi A) 1 B + D) U(z) }{{} transfer function L. Giarré- Systems and Control Theory 2017-2018 - 17
Transform Domain Solution (DT) Note that (from properties of the State Transition matrix and Laplace transform properties) Φ(z)Z[Φ(k)] = Z[A k ]=z(zi A) 1 X(z) =zφ(z)x(0)+φ(z)bu(z) Y (z) =zφ(z)x(0)+ (CΦ(z)B + D) U(z) }{{} transfer function L. Giarré- Systems and Control Theory 2017-2018 - 18
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