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/ November 2011 © H T Hoang - www4.hcmut.edu.vn/~hthoang/ Chapter ANALYSIS OF CONTROL SYSTEM PERFORMANCE November 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Content Performance criteria Steady state error Transient response p The optimal performance index Relationship between frequency domain performances and time domain performances November 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Performance criteria November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ Performance criteria: Steady state error yfb(t) R(s) E(s) +_ Yfb(s) Y(s) G(s) ess r(t) e(t) H(s) ess t Error: is the difference between the set set-point point (input) and the feedback signal ⇔ e(t ) = r (t ) − y fb (t ) E ( s ) = R ( s ) − Y fb ( s ) Steady-state error: is the error when time approaching infinity ess = lim li e(t ) t →∞ November 2011 ⇔ ess = lim li sE E ( s) s →0 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Performance criteria – Percent of Overshoot (POT) Overshoot: refers to an output exceeding its steady-state steady state value value y(t) y(t) overshoot ymax yss yss ymax− yss yss No overshoot t t Percentage of Overshoot (POT) is an index to quantify the overshoot h t off a system, t POT is i calculated l l t d as: ymax − yss × 100% POT = yss November 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Performance criteria – Settling time and rise time Settling S ttli time ti (ts): ) is i the th time ti required i d for f the th response off a system t to t reach and stay within a range about the steady-state value of size specified by absolute percentage of the steady-state value (usually 2% or 5%) Rise time (tr): is the time required for the response of a system to rise from 10% to 90% of its steady steady-state state value value y(t) y(t) (1+ε)yss yss (1 )yss yss 0.9yss t ts November 2011 0.1yss t tr © H T Hồng - www4.hcmut.edu.vn/~hthoang/ Steady state error Steady November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ Steady state error Steady R( ) R(s) +_ E( ) E(s) Yfb(s) G(s) Y( ) Y(s) H(s) Error expression: R( s) E (s) = + G (s) H (s) Steady-state error: sR ( s ) ess = lim sE ( s ) = lim s →0 s →0 + G ( s ) H ( s ) Remark: Steady-state error not only depends on the structure and parameters t off the th system t but b t also l depends d d on the th input i t signal i l November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ Steady state error to step input Steady Step input: R( s ) = / s ⇒ Steady-state error: with ess = 1+ Kp K p = lim G ( s ) H ( s ) s →0 (position constant) yfb(t) yfb(t) 1 t G(s)H(s) does not have any deal integral factor November 2011 t G(s)H(s) has at least ideal integral factor © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 10 Performance indices November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 27 Integral performance indices IAE criterion (Integral of the Absolute Magnitude of the Error ) J IAE = +∞ ∫ e(t ) dt ISE criterion (Integral of the Square of the Error) +∞ J ISE = ∫ e (t )dt d ITAE criterion (Integral of Time multiplied by the Absolute Value of the Error) +∞ J ITAE = ∫ t e(t ) dt November 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 28 Optimum systems A control system is optimum when the selected performance index is minimized J IAE → when ξ → 0.707 Second order system: J ISE → i when ξ → 0.5 J ITAE → when ξ → 0.707 y(t) ξ=0.3 ξ=0.5 ξ=0.707 ξ=0.9 t Transient response of second order systems November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 29 ITAE optimal control ITAE is usually used in design of control system An n-order system is optimal according to ITAE criterion if the denominator of its transfer function has the form: November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 30 ITAE optimal control (cont’) Optimal response according to ITAE criterion y(t) 1st order d system 2nd order system 3rd order system 4th order system t November 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 31 Relationship p between frequency q y domain performances and time domain performances November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 32 Relationship between frequency response and steady state error R(s) G(s) +− Y(s) K p = lim G ( s ) H ( s ) = lim G ( jω ) H ( jω ) ω →0 s →0 K v = lim s G ( s ) H ( s ) = lim jωG ( jω ) H ( jω ) ω →0 s →0 K a = lim s G ( s ) H ( s ) = lim( jω ) G ( jω ) H ( jω ) s →0 November 2011 ω →0 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 33 Relationship between frequency response and steady state error R( ) R(s) + − G(s) Y( ) Y(s) Steady state error of the closed-loop closed loop system depends on the magnitude response of the open-loop system at low frequencies but not at high frequencies The higher the magnitude response of the open-loop system at low frequencies, the smaller the steady-state error of the closed-loop s stem system In particular, if the magnitude response of the open-loop system is infinityy as frequency q y approaching pp g zero,, then the steady-state y error of the closed-loop system to step input is zero November 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 34 Relationship between frequency response and transient response R(s) +− G(s) Y(s) In the frequency range ω then: G ( jω ) G ( jω ) Gcl ( jω ) = ≈ =1 + G ( jω ) G ( jω ) In the frequency range ω >ωc , because G ( jω ) < then: G ( jω ) G ( jω ) Gcl ( jω ) = ≈ = G ( jω ) + G ( jω ) ⇒ Bandwidth of the closed-loop closed loop system is approximate the gain crossover frequency of the open-loop system November 2011 © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 35 Relationship between frequency response and transient response Bode plot of a open-loop system November 2011 Bode plot of the corresponding closed-loop system © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 36 Relationship between frequency response and transient response R(s) +− G(s) Y(s) The higher the gain crossover frequency of open-loop open loop system, system the wider the bandwidth of closed-loop system ⇒ the faster the response of close-loop system, the shorter the settling time π 4π < tqd < ωc ωc The higher the phase margin of the open-loop system, the smaller the POT of closed-loop system In most of the cases, if the phase margin of the open-loop open loop system is larger than 600 then the POT of the closed-loop system is smaller than 10% November 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 37 Example of relationship between gain crossover frequency and settling time R(s) +− November 2011 G(s) Y(s) 10 G (s) = s(0.1s + 1)(0.08s + 1) © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 38 Example of relationship between gain crossover frequency and settling time R(s) +− November 2011 G(s) Y(s) G ( s) = © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 50 s(0.1s + 1) 39 Example of relationship between phase margin and POT R(s) +− November 2011 G(s) Y(s) G (s) = s(0.1s + 1)(0.08s + 1) © H T Hồng - www4.hcmut.edu.vn/~hthoang/ 40 Example of relationship between phase margin and POT (cont’) R(s) +− November 2011 G(s) Y(s) G ( s) = s (0.1s + 1) © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 41 ... margin of the open-loop system, the smaller the POT of closed-loop system In most of the cases, if the phase margin of the open-loop open loop system is larger than 600 then the POT of the closed-loop... www4.hcmut.edu.vn/~hthoang/ 25 Transient response of high order system High-order High order systems are the system that have more than poles poles If a high order system have a pair of poles located closer to the... higher the gain crossover frequency of open-loop open loop system, system the wider the bandwidth of closed-loop system ⇒ the faster the response of close-loop system, the shorter the settling time