Lecture introduction to control systems chapter 3 system dynamics (dr huynh thai hoang)

Lecture Notes Introduction to Control Systems Instructor: Dr Huynh Thai Hoang Department of Automatic Control Faculty of Electrical & Electronics Engineering Ho Chi Minh City University of Technology Email: hthoang@hcmut.edu.vn huynhthaihoang@yahoo.com Homepage: www4.hcmut.edu.vn/~hthoang/ 24 September 2011 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Chapter SYSTEM DYNAMICS 24 September 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ Content The concept of system dynamics Time response q y response p Frequency Dynamics of typical components Dynamics of control systems 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ The concept of system dynamics 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ The concept of system dynamics System dynamics is the study to understanding the behaviour of complex systems over time Systems described by similar mathematical model will expose similar dynamic responses To study T t d the th dynamic d i responses, input i t signals i l are usually ll chosen h to t be basic signals such as Dirac impulse signal, step signal, or sinusoidal signal Time response Impulse response Step response Frequency response 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Impulse response U (s) G(s) Y (s) Impulse response: behavior of a system to a Dirac impulse Y ( s ) = U ( s ).G ( s ) = G ( s ) (due to U(s) = 1) y (t ) = L −1{Y ( s )} = L −1{G ( s )} = g (t ) ⇒ Impulse response is the inverse Laplace transform of the transfer function Impulse response is also referred as weighting function It is possible to calculate the response of a system to a arbitrarily input by taking convolution of the weighting function and the input t y (t ) = g (t ) * u (t ) = ∫ g (τ )u (t − τ )dτ 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Step response U (s) G(s) Y (s) Step response: behavior of a system to a step input G (s) (because U(s) = 1/s) Y ( s ) = U ( s ).G ( s ) = s t G ( s ) ⎧ ⎫ y (t ) = L −1{Y ( s )} = L −1 ⎨ ⎬ = ∫ g (τ )dτ ⎩ s ⎭ ⇒ The step response is the integral of the impulse response The step response is also referred as the transient function of the system 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Impulse and step response example Calculate the impulse response and step response of the system described by the transfer function: U (s) Y (s) s +1 G(s) = G(s) () s ( s + 5) Impulse response: ⎫ −1 −1 ⎧ s + ⎫ −1 ⎧ g (t ) = L {G ( s )} = L ⎨ ⎬=L ⎨ + ⎬ ⎩ s ( s + 5) ⎭ ⎩ 5s 5( s + 5) ⎭ ⇒ g (t ) = + e −5t 5 Step response: s +1 ⎫ 4 −1 ⎧ G ( s ) ⎫ −1 ⎧ + 2− h(t ) = L ⎨ ⎬= ⎬=L ⎨ 25( s + 5) ⎩ s ⎭ ⎩ s ( s + 5) ⎭ 25s 5s −5t ⇒ h(t ) = t − e + 25 25 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Frequency response Observe the response of a linear system at steady state when the input is a sinusoidal signal 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Frequency response definition It can be observed that, that for linear system, system if the input is a sinusoidal signal then the output signal at steady-state is also a sinusoidal signal with the same frequency as the input, but different amplitude and d phase h u (t)=Umsin (jω) U (jω) HT y (t)=Ymsin (jω+ϕ) Y (j ( ω) Definition: Frequency response of a system is the ratio between the steady-state output and the sinusoidal input F Frequency response = It is proven that: 24 September 2011 Y ( jω ) U ( jω ) Frequency response = G ( s ) s = jω = G ( jω ) © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 10 Frequency response of time delay factor Frequency response: G ( jω ) = e −Tjω M it d response: M (ω ) = Magnitude Phase response: 24 September 2011 ⇒ L (ω ) = ϕ (ω ) = −Tω © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 40 Frequency response of time delay factor (cont’) Bode diagram 24 September 2011 Nyquist plot © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 41 Dynamics of control systems 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 42 Time response of control systems Consider a control system which has the transfer function G(s): b0 s m + b1s m −1 + L + bm −1s + bm G ( s) = a0 s n + a1s n −1 + L + an −11s + an Laplace transform of the transient function: b0 s m + b1s m −1 + L + bm −1s + bm G(s) = H ( s) = s s (a0 s n + a1s n −1 + L + an −1s + an ) s 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 43 Remarks on the time response of control systems If G(s) G( ) does d nott contain t i a ideal id l integral i t l or derivative d i ti factor f t then: th Weighting function decays to ⎛ b0 s m + b1s m −1 + L + bm −1s + bm ⎞ ⎟⎟ = g (∞) = lim sG ( s ) = lim s⎜⎜ − n n s →0 s →0 ⎝ a0 s + a1s + L + an −1s + an ⎠ Transient function approaches to non-zero value at steady state: ⎛ b0 s m + b1s m −1 + L + bm −1s + bm ⎞ bm ⎟⎟ = h(∞) = lim sH ( s ) = lim s⎜⎜ ≠0 n n −1 s →0 s →0 ⎝ s a0 s + a1s + L + an −1s + an ⎠ an 24 September 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 44 Remarks on the time response of control systems (cont’) If G(s) G( ) contain t i a ideal id l integral i t l factor f t (a ( n = 0) then: th Weighting function has non-zero steady-state: ⎛ b0 s m + b1s m −1 + L + bm −1s + bm ⎞ ⎟⎟ ≠ g (∞) = lim sG ( s ) = lim s⎜⎜ n n − s →0 s →0 ⎝ a0 s + a1s + L + an −1s ⎠ Transient function approaches infinity at steady-state ⎛ b0 s m + b1s m −1 + L + bm −1s + bm ⎞ ⎟⎟ = ∞ h(∞) = lim sH ( s ) = lim s⎜⎜ n n −1 s →0 s →0 a0 s + a1s + L + an −1s ⎠ ⎝s 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 45 Remarks on the time response of control systems (cont’) If G(s) G( ) contain t i a ideal id l derivative d i ti factor f t (bm = 0) then: th Weighting function approaches zero at steady-state ⎛ b0 s m + b1s m −1 + L + bm −1s ⎞ ⎟⎟ = g (∞) = lim sG ( s ) = lim s⎜⎜ n n − s →0 s →0 ⎝ a0 s + a1s + L + an −1s + an ⎠ Transient function approaches zero at steady-state ⎛ b0 s m + b1s m −1 + L + bm −1s ⎞ ⎟⎟ = h(∞) = lim sH ( s ) = lim s⎜⎜ n n −1 s →0 s →0 ⎝ s a0 s + a1s + L + an −1s + an ⎠ 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 46 Remarks on the time response of control systems (cont’) If G(s) G( ) is i proper (m ( ≤ n)) then th h(0) = 0 ⎛ b0 s m + b1s m −1 + L + bm −1s ⎞ ⎟⎟ = h(0) = lim H ( s ) = lim⎜⎜ n n − s →∞ s →∞ s a s + a s + L + an −1s + an ⎠ ⎝ If G(s) ( ) is strictly yp proper p ((m < n)) then g( g(0)) = ⎛ b0 s m + b1s m −1 + L + bm −1s ⎞ ⎟⎟ = g (0) = lim G ( s ) = lim⎜⎜ n n − s →∞ s →∞ a s + a s + L + an −1s + an ⎠ ⎝ 24 September 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 47 Frequency response of control system Consider a control system which has the transfer function G(s) G(s) Suppose that G(s) consists of basis factors in series: l G ( s ) = ∏ Gi ( s ) i =1 l Frequency response: G ( jω ) = ∏ Gi ( jω ) i =1 l Magnitude response: M (ω ) = ∏ M i (ω ) ⇒ i =1 l L(ω ) = ∑ Li (ω ) i =1 l Phase response: ϕ (ω ) = ∑ ϕi (ω ) i =1 ⇒ The Bode diagram of a system consisting of basic factors in series equals to the summation of the Bode diagram of the basic factors 24 September 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 48 Approximation of Bode diagram S Suppose th t the that th transfer t f function f ti off the th system t is i off the th form: f G ( s ) = Ksα G1 ( s )G2 ( s )G3 ( s )K (α>0: the system has ideal derivative factor(s) α