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Disturbance Response Dr. Kevin Craig Professor of Mechanical Engineering Rensselaer Polytechnic Institute Disturbance Response K. Craig 1 Disturbance Response • The focus up to now has been on command response. Disturbance response is also important, and in some applications, more important than command response. • Disturbance response is more difficult to measure because disturbances are more difficult to produce than are commands. • Both command response and disturbance response improve with high loop gains. – A high KP provides a higher bandwidth and better ability to reject disturbances with high frequency content. – A high KI helps the control system reject lower frequency disturbances. Disturbance Response K. Craig 2 – Setting KI high had minimal effect on the command-response Bode plots. KI is aimed at improving response to disturbances, not commands. In fact, the process of tuning KI is essentially to raise it as high as possible without significant impact on the command response. KI is not noticeable in the command response until it is high enough to cause peaking and instability. High KI provides unmistakable benefit in disturbance response. • In addition, Disturbance-Compensated Feedforward Control aids disturbance response by using measured or estimated disturbances to improve disturbance response. • Sometimes disturbance response is referred to by its inverse – disturbance rejection or dynamic stiffness. Control systems need to have high dynamic stiffness (disturbance rejection) and low disturbance response. Disturbance Response K. Craig 3 • Disturbances are undesirable inputs. We are concerned about a response to an input other than a command. • A properly placed integrator will totally reject DC (static) disturbances. High tuning gains will help the system reject dynamic disturbance inputs, but those inputs cannot be rejected entirely. Σ Σ The disturbance D(s) is applied just before the plant. Disturbance Response K. Craig 4 • The control system cannot reject the disturbance perfectly because the disturbance is detected only after it moves the output; the controller cannot react until system output has been disturbed. • Disturbance response is defined as the response of the system output C(s) to the disturbance D(s). C(s) G(s) = D(s) 1 + G c (s)G(s)H(s) • One way to improve disturbance response is to use slow-moving plants, e.g., large inertia, high capacitance, to provide low plant gains. Reduce G(s). This is a timeproven technique. Large flywheels smooth motion; large inductors and capacitors smooth voltage output. Disturbance Response K. Craig 5 • A second way to improve disturbance response is to increase the gains of the controller, GC(s). This is how integral gains grant systems perfect response to DC inputs: the gain of the ideal integrator at 0 Hz is infinite, driving up the magnitude of the transfer function denominator and, thus, driving down the disturbance response. • At other frequencies, unbounded gain is impractical, so AC disturbance response is improved with high gains but not cured entirely. • Dynamic Stiffness is the inverse of disturbance response. It is a measure of how much force is required to move a system as opposed to disturbance response, which is a measure of how much the system moves in the presence of a force. A system that is very stiff responds little to disturbances. Disturbance Response K. Craig 6 Proportional- Integral Control R(s) Σ +- 1 G (s) = Js ⎛ s + KI ⎞ GC (s ) = K P ⎜ ⎟ s ⎝ ⎠ H (s) = 1 E(s) D(s) + Gc(s) + Σ B(s) G(s) C(s) H(s) C(s) G(s) = D(s) 1 + G c (s)G(s)H(s) = s Js 2 + K P s + K P K I High-Frequency Range Low-Frequency Range Mid-Frequency Range Disturbance Response K. Craig 7 C s (s) = 2 D Js + K Ps + K P K I J =0.002 KP = 0.58 KI = 58 Increase J Increase KP Increase KI What Happens? Disturbance Response K. Craig 8 J = 0.02 KP =5.8 KI = 580 Disturbance Response K. Craig 9 • Some Observations – Increasing the value of J reduces (improves) the disturbance response in the higher frequencies. Disturbance response from the inertia improves as frequency increases (1/Js term). – In the medium frequency range the 1/KP term dominates. A larger proportional gain helps in the medium frequencies. – In the lowest frequency range, the s/KIKP term dominates. Larger proportional gain improves the low-frequency disturbance response, as does larger integral gain. – Remember that larger proportional gain allows larger integral gains. So increasing Kp improves medium- and low-frequency disturbance response directly and also indirectly helps lowfrequency disturbance response by allowing larger KI. – Raising J improves the high-frequency disturbance response directly, but sometimes improves the rest of the frequency spectrum indirectly by allowing a larger value of Kp (noise, resolution, and resonance may limit improvement). Disturbance Response K. Craig 10 Open-Loop Input-Compensated Feedforward Control: Plant Disturbance Input Disturbance-Compensated Disturbance Com pensation Flow of Energy and/or Material Control Effector Disturbance Sensor Controlled Variable Plant Plant Manipulated Input • Measure (or estimate) the disturbance • Estimate the effect of the disturbance on the controlled variable and compensate for it Disturbance Response Control Director Desired Value of Controlled Variable K. Craig 11 • Disturbance-Compensated Feedforward Control • Basic Idea: Measure an input disturbance to the plant and take corrective action (adjust the manipulated variable) before it upsets the process (causes the controlled variable to deviate from its set point). This measurement provides an early warning that the controlled variable will be upset some time in the future. • In contrast, a feedback controller does not take corrective action until after the disturbance has upset the process and generated an error signal. • This controller does not use feedback! However, it is usually combined with feedback control so that the important features of feedback are retained in the overall strategy. Disturbance Response K. Craig 12 • There are several disadvantages to disturbancecompensated feedforward control: • The disturbance must be measured on line. In many applications, this is not feasible. • The quality of the feedforward control depends on the accuracy of the process model; one needs to know how the controlled variable responds to changes in both the disturbance and manipulated variables. • Ideal feedforward controllers that are theoretically capable of achieving perfect control may not be physically realizable. Fortunately, practical approximations of these ideal controllers often provide very effective control. Disturbance Response K. Craig 13 Feedback / Disturbance-Feedforward Temperature Control Example Objective: Ensure that TC remains at or near the set point. G i = dm i/dt T U1 T U2 TC V, M TH Disturbance Response Q T CM G o = dm o /dt Continuous Stirred-Tank Heater Electrical Heater K. Craig 14 • Physical Model Simplifying Assumptions – No change of phase occurs in the tank fluid. – Volume of fluid in the tank is constant. – Perfect mixing is assumed in the tank, therefore the temperature of the fluid in the tank is uniform and equal to the exit temperature. – Liquid in the tank has constant density, mass M , and specific heat C . – Heater is an electrical resistance heater that follows input voltage instantly with heater gain Kh. – Heater mass, specific heat, and heat transfer area are Mh, Ch, and Ah and the heat transfer coefficient to liquid is Uh. Disturbance Response K. Craig 15 – Tank wall is assumed to be pure thermal resistance 1/UtAt (no energy storage) – Assume an initial equilibrium operating point and take all variables as perturbations. – Temperature sensor (assumed negligible dynamics) is located in the pipeline downstream of the tank, due to tank vibration from stirrer and electrical noise from heater, resulting in a dead time τDT. – Tank inlet temperature TU1 is uncontrolled and thus is a disturbance input – Ambient temperature TU2 is uncontrolled and thus is a disturbance input. Disturbance Response K. Craig 16 • Background on Mathematical Modeling: Basic Equations – Conservation of Mass: Tank ∂ 0= ρdV + ∫ ρvidA ∫ ∂t CV CS • • d Assume constant density. ( ρt Vt ) = ρt V ti − ρt V to dt d ( Vt ) = 0 Assume constant volume in the tank. dt • • V ti = V to Disturbance Response volume flow rate in = volume flow rate out K. Craig 17 – Conservation of Energy: Tank • • ∂ Q+ W = eρdV + ∫ eρvidA ∫ ∂t CV CS v2 e = u + + gz 2 • For most processes where there are thermal effects, kinetic and potential energy terms can be neglected because their contribution is generally at least two orders of magnitude less than that of the internal energy term. • Rate of work done on Control Volume by surroundings: • • • W = W shaft + W normal • W normal = ∫σ nn vidA σ nn ≈ − p CS Disturbance Response K. Craig 18 • • ⎛ p⎞ ∂ uρdV + ∫ ⎜ u + ⎟ ρvidA = ∫ ∂t CV ρ⎠ CS ⎝ • • • • dU • = + V t ρ t ( h to − h ti ) dt d ( H − pV ) • = + V t ρ t ( h to − h ti ) dt Q+ W shaft Q+ W shaft Q+ W shaft • But since the volume of the tank is constant and the mean pressure change is negligible (good assumption for liquids provided the pressure change is not too large), we have: d ( pV ) dt Disturbance Response dp dV =V +p =0 dt dt K. Craig 19 • • dH • = V t ρ t ( h ti − h to ) + Q+ W shaft dt – Neglect the work done by the mixing impeller and assume single phase and a constant heat capacity: h = c p ΔT • dTt • ρt Vt c pt = V t ρ t c pt ( Tti − Tto ) + Q dt • • dTt V t Q = ( Tti − Tto ) + ρt Vt c pt dt Vt Disturbance Response K. Craig 20 • Mathematical Model – Conservation of Mass • Mass flow rate in = Mass flow rate out – Conservation of Energy applied to the Heater dTh Q − U h A h ( Th − TC ) = M h C h dt M h C h dTh Q + Th = + TC U h A h dt Uh Ah – Conservation of Energy applied to the Tank dTC U h A h ( Th − TC ) + GC TU1 − GC TC − U t A t ( TC − TU 2 ) = M C dt dTC MC + ( U h A h + GC + U t A t ) TC = U h A h Th + GC TU1 + U t A t TU 2 dt Disturbance Response K. Craig 21 T U1 Open-Loop System Block Diagram Q K4 + Σ + 1 Th ( τ as + 1) K1 + Σ + K2 + K3 + T U2 Σ TC 1 ( τ bs + 1) TC GC = 0.0694 K1 = U h A h + GC + U t A t GC = 0.0154 Ut At = 0.2775 K2 = U h A h + GC + U t A t U h A h = 0.145 M h C h = 13.05 U t A t = 0.0616 Uh Ah = 0.6532 K3 = U h A h + GC + U t A t M h Ch τa = = 90 Uh Ah 1 = 6.90 K4 = Uh Ah MC τb = = 138.7 U h A h + GC + U t A t Disturbance Response M C = 30.8 K. Craig 22 – Elimination of Th leads to: ( τ1D + 1)( τ2 D + 1) TC = K Q Q + K U1 ( τ3D + 1) TU1 + K U 2 ( τ3D + 1) TU 2 M C τ3 τ1τ2 = GC + U t A t τ1 + τ2 = M h Ch τ3 = Uh Ah τ3 ( U h A h + GC + U t A t ) + M C GC + U t A t 1 KQ = GC + U t A t K U2 Disturbance Response GC K U1 = GC + U t A t Ut At = GC + U t A t GC = 0.0154 M h C h = 13.05 U h A h = 0.145 U t A t = 0.0616 M C = 30.8 τ1 = 600, τ2 = 60 τ3 = 90, K Q = 12.99 K U1 = 0.2, K U 2 = 0.8 K. Craig 23 T U1 K U1 + Σ + τ 3s + 1 Q + KQ + Σ K U2 T U2 Open-Loop System Block Diagram TC 1 ( τ 1s + 1 )( τ 2 s + 1 ) ( τ1D + 1)( τ2 D + 1) TC = K Q Q + K U1 ( τ3D + 1) TU1 + K U 2 ( τ3D + 1) TU 2 Disturbance Response K. Craig 24 T U1 K U1 + Σ + K U2 T U2 τ 3s + 1 TV eR Ka + eE Σ Kh - + Q KQ + Σ TC 1 ( τ 1s + 1 )( τ 2 s + 1 ) eB Ka e − τ DT s Proportional Control of Temperature: Block Diagram Disturbance Response K. Craig 25 • Dead-Time Approximations – The simplest dead-time approximation can be obtained graphically or by taking the first two terms of the Taylor series expansion of the Laplace transfer function of a dead-time element, τDT. Qo ( s ) = e−τDTs ≈ 1 − τDTs Qi q o ( t ) ≈ q i ( t ) − τDT dq i dt – The accuracy of this approximation depends on the dead time being sufficiently small relative to the rate of change of the slope of qi(t). If qi(t) were a ramp (constant slope), the approximation would be perfect for any value of τDT. When the slope of qi(t) varies rapidly, only small τDT's will give a good approximation. – A frequency-response viewpoint gives a more general accuracy criterion; if the amplitude ratio and the phase of the approximation are sufficiently close to the exact frequency response curves of for the range of frequencies present in qi(t), then the approximation is valid. Disturbance Response K. Craig 26 Dead-Time Graphical Approximation qi τDT q o = q i ( t − τDT ) tangent line qi(t) q o = q i ( t ) − τDT dq i dt t Disturbance Response K. Craig 27 – Use this simple dead-time approximation to simplify the closed-loop system equation and facilitate gain setting: −τDTs e ≈ −τDT D + 1 – The closed-loop system equation is: ⎛ D 2 2ζ D ⎞ K U1 K + 1⎟ TC = TV + ( τ3D + 1) TU1 ⎜ 2 + ωn K +1 K +1 ⎝ ωn ⎠ K U2 + ( τ3D + 1) TU 2 K +1 K +1 ωn = τ1τ2 Disturbance Response ζ= τ1 + τ2 − KτDT 2 τ1τ2 ( K + 1) K = Ka Kh KQ K. Craig 28 • Parameter Values – – – – – – – – – Disturbance Response Ka = 0.05 Kh = 10.0 KQ = 12.99 KU1 = 0.2 KU2 = 0.8 τ3 = 90 τ1 = 600 τ2 = 60 τDT = 5, 50 K. Craig 29 • Disturbance Feedforward Compensation – Since both disturbances TU1 and TU2 are measurable, the feedback system can be augmented with a feedforward system. – Ideally the feedforward dynamics should be: τ3s + 1 – However, the derivative signal might accentuate noise, so we include a low-pass filter, even though this prevents perfect dynamic compensation: 1 1 = τ4s + 1 0.1τ3s + 1 – KU1FF = 0.2 – KU2FF = 0.8 – τ4 = 0.1τ3 Disturbance Response K. Craig 30 K U1FF τ 3s + 1 T U1 K h K Q ( τ 4s + 1) Σ + K U1 + Σ K U 2 FF + + K U2 T U2 τ 3s + 1 TV eR Ka + Feedback + Feedforward Control - eE Σ Kh - + Q KQ + Σ TC 1 ( τ 1s + 1 )( τ 2 s + 1 ) eB Disturbance Response Ka e − τ DT s K. Craig 31 • Important Issues in Feedforward Control – The feedforward controller must be physically realizable and stable. – The disturbance time delay must be greater than the process time delay for perfect feedforward compensation. If the process time delay is greater than the disturbance time delay, the feedforward controller will have no time-delay compensation, and perfect control cannot be achieved. – If the process has a RHP zero, it must be factored out before designing the feedforward controller. Otherwise, the feedforward controller will be unstable. – If the process is higher order than the disturbance, fast time constants probably must be neglected in designing the feedforward controller. Disturbance Response K. Craig 32 – Often a static feedforward controller will have good performance, particularly if the disturbance dynamics are the same time scale as the process dynamics. – A feedforward controller does not change the closedloop stability of the feedback system, assuming that the feedforward controller is stable. Also, a feedforward controller does not change the set point response of a closed-loop system. – Feedforward control can be implemented with either classical feedback (PID-type) or Internal Model Control structure. Disturbance Response K. Craig 33 • Numerical Values τ1 = 600 sec τ2 = 60 sec τDT = 5 sec – If we design for ζ = 0.6, we get K = 6.48 and ωn = 0.0144 rad/sec (period is 435 sec/cycle). • Simulations – Run numerical simulations, using both approximate and exact dead-time models, to check these results and get additional information about system behavior. – Ka = 0.05 – Kh = 10.0 – KQ = 12.96 Disturbance Response K. Craig 34 – – – – – – – – KU1 = 0.2 KU2 = 0.8 TU1 = 0, initially, then a unit step TU2 = 0, initially, then a unit step τ3 = 90 τ1 = 600 τ2 = 60 τDT = 5 (then increase to 50 sec, to see decrease in accuracy of approximation) – Command Input: Unit Step, initially, then zero Disturbance Response K. Craig 35 – Since both disturbances TU1 and TU2 are measurable, the feedback system can be augmented with a feedforward system. – Ideally the feedforward dynamics should be: τ3s + 1 – However, the derivative signal might accentuate noise, so we include a low-pass filter, even though this prevents perfect dynamic compensation: 1 1 = τ4s + 1 0.1τ3s + 1 – KU1FF = 0.2 – KU2FF = 0.8 – τ4 = 0.1τ3 Disturbance Response K. Craig 36 K U1FF τ 3s + 1 K h K Q ( τ 4s + 1) T U1 Σ + K U1 + Σ K U 2 FF + + K U2 T U2 τ 3s + 1 TV eR Ka + Feedback + Feedforward Control - e E Σ Kh - + Q KQ + Σ TC 1 ( τ 1s + 1 )( τ 2 s + 1 ) eB Disturbance Response Ka e − τ DT s K. Craig 37 – Investigate both momentary and sustained disturbances: T U1 8.0 250 T U2 350 tim e 2.0 50 Disturbance Response tim e K. Craig 38 • Compare the feedback with the feedback + feedforward system. – What do you observe about transient errors? – What do you observe about steady-state errors? – The total elimination of error depends on a perfect match of numerical parameter values in the feedforward and main system signal paths, a condition we can only approximate in practice. – The transient errors remaining in the simulation can be reduced by making τ4 smaller. Let τ4 = 1.0 sec and observe the results. Whether a practical system could use such a small time constant value depends on the noisiness of the signal to the feedforward controller. Disturbance Response K. Craig 39