CHAPTER 5 INTERNAL MODEL CONTROL STRATEGY 5. INTRODUCTION The Internal Model Control (IMC) based aroach for PID controller design can be used to control alications in industries. It is because, for ractical alications or an actual rocess in industries PID controller algorithm is simle and robust to handle the model inaccuracies and hence using IMC-PID tuning method [36], [37] a clear trade-off between closed loo erformance and robustness to model inaccuracies is achieved with a single tuning arameter. Also the IMC-PID controller allows good set-oint tracking and gives silky disturbance resonse esecially for the rocess with a small timedelay/time-constant ratio. But, for many rocess control alications, disturbance rejection for the unstable rocesses is much more imortant than set oint tracking. Hence, controller design that emhasizes disturbance rejection rather than set oint tracking is an imortant design roblem that has to be taken into consideration. In this thesis, otimum IMC filter to design an IMC controller for better set-oint tracking and disturbance rejection in a series ower quality controller is roosed. As the IMC aroach is based on olezero cancellation, methods which comrise IMC design rinciles result in good set oint resonses. However, IMC results in a long settling time for load disturbances for lag dominant rocesses which are not
desirable in the control industry. Usually, disturbances are categorized as load disturbances, model uncertainties and noise. As the knowledge about model uncertainties and noise is unknown, their imact is not considered in thesis. The lant and model of actual rocess is considered to same. Since all the IMC-PID [38], [39] aroaches involve some kind of model reduction techniques to convert the IMC controller to the PID controller so aroximation error usually occurs. This thesis clearly illustrates an aroach to obtain otimum filter structure to IMC. Filter time constant lays a vital role to obtain better trade off between robustness and disturbance rejection and stability. Basically Internal Model Control (IMC) rincile states that Control can be achieved if and only if the control system summarizes either imlicitly or exlicitly, some reresentation of the rocess to be controlled. Internal stability and erformance characteristics (correlate to controller arameters) are the imortant asects which makes it more advantageous comared to classic feedback controller. The erfect controller can be arrived if there is no model mismatch. If disturbance rejection is not covered IMC gives sluggish resonse. The arameters of IMC controller deend on the IMC filter time constant. Increase in the filter time constant always reduces the overshoot to an accetable limit, but however reduces the disturbance rejection, desired noise suression caability. This study also rooses the rocedural method for selection of filter time constant. The concetual usefulness of the IMC lies in the fact that much concern can be ut on controller design rather than control system
stability rovided that the rocess model is a erfect reresentation of a stable rocess. If there is a comlete knowledge about the rocess being controlled, erfect control can be achieved without feedback. Feedback control is needed only when knowledge about the rocess is incomlete or inaccurate. However, rocess-model mismatch is common. Process model may not be invertible and the system is often affected by unknown disturbances. Hence oen loo control arrangement may not be able to comensate for disturbances, model uncertainties and set oint tracking whereas an IMC is able to comensate for disturbances, model uncertainties and set oint tracking. IMC must be tuned to assure the stability in model uncertainties cases. 5.2 IMC Strategy An oen loo control system is controlled directly, and only by an inut signal without the benefit of feedback. Oen loo control systems are not commonly used as closed loo control systems because of the issue of accuracy. An oen loo structure is shown in the fig 5.. Fig 5.: Oen loo structure of IMC With controller Gc () s, set to ut control on the lant G () s, then it is clear from basic linear system theory that the outut Y(s) can be modeled as the roduct of the linear blocks as follows:
Y( s) R( s) G ( s) G ( s) (5.) c If we assume there exists model of the lant with transfer function modeled as G () s such that G () s is an exact reresentation of the rocess (lant), i.e. G ( s) G ( s), then set oint tracking can be achieved by designing a controller such that: G ( s) G ( s) c (5.2) This control erformance characteristic is achieved without feedback and highlights two imortant characteristic features of this control modeling. These features are: Feedback control can be theoretically achieved if comlete characteristic features of the rocess are known or easily identifiable. Feedback control is only necessary of knowledge about the rocess is inaccurate or incomlete. This control erformance as already said has been achieved without feedback and assumed that the rocess model reresent the rocess exactly i.e. rocess model has all features of arent model. In real life alications, however, rocess models have caabilities of mismatch with the arent rocess; hence feedback control schemes are designed to counteract the effects of this mismatching. A control scheme that has gained oularity in rocess control has been formulated and known as the Internal Model Control (IMC) scheme. This design is a
simle build u from the ideas imlemented in the oen loo model strategy and has general structure as deicted by figure below Fig 5.2: Schematic IMC structure without disturbance Fig 5.3: Internal model control with disturbance From the fig 5.3, descrition of blocks as follows: Controller G c (s) Process G (s) Internal model G () s Disturbance d(s) The fig 5.3 shows the standard linear IMC scheme where the rocess model G () s lays an exlicit role in the control structure. This structure has some advantages over conventional feedback loo structures. For the nominal case G (s) = G () s, for instance, the
feedback is only affected by disturbance d(s) such that the system is effectively oen loo and hence no stability roblems can arise. This control structure also deicts that if the rocess G (s) is stable, which is true for most industrial rocesses, the closed loo will be stable for any stable controller G c (s). Thus, the controller G c (s) can simly be designed as a feed-forward controller in the IMC scheme. The maniulated inut Usis () introduced to both the rocess and its model. The rocess outut Ysis () comared with the outut of the model resulting in i.e. dˆ( s ) G ( s ) G ( s ) U ( s ) d ( s ) (5.3) In the above equation, if ds ( ), then ds ˆ( ) is a measure of the difference in behavior between the rocess and model and if G ( s) G ( s) then ds ˆ( ) will be unknown disturbance. Thus ds ˆ( ) can be used to imrove the control and may be treated as the missing information in the model G () s. This can be achieved by sending an error signal to the controller. The error signal incororates the model mismatch and disturbances and hels to achieve the set-oint. The resulting control signal is given by U( s) R( s) dˆ ( s) Gc ( s) R( s) G ( s) G ( s) U( s) d( s) Gc ( s) R( s) d( s) Gc ( s) Us () G ( s) G ( s) Gc ( s) (5.4) (5.5)
Since Y( s) G ( s) U( s) d( s) the closed loo transfer function for the IMC scheme is therefore given by R( s) d( s) Gc( s) G( s) Y( s) d( s) G ( s) G ( s) Gc ( s) (5.6) From the above exressions, it can be concluded that if G ( s) G ( s) c and if G ( s) G ( s), a set oint tracking and disturbance rejection is achieved. In some cases even if G ( s) G ( s), erfect disturbance rejection can still be achieved rovided G ( s) G ( s). Additionally, c minimal effects of rocess model mismatch imrove the robustness. Most of the discreancies between rocess and model behavior occur at the high frequency end of the systems frequency resonse. In general a low ass filter is used to attenuate the effects of rocess-model mismatch. Thus the internal model controller is usually designed as the inverse of the rocess model in series with a low ass filter. G ( s) G ( s) G ( s ) (5.7) imc c f Excessive differential action is usually controlled by selecting the order of the filter so as to make G () s roer. The resulting closed loo equation is given by imc Gimc ( s) G ( s) R( s) Gimc ( s) G ( s) d( s) Y( s) d( s) G ( s) G ( s) Gimc ( s) (5.8)
In IMC scheme shown by fig 5.3, the Internal Model Control loo calculates the difference between the oututs of the rocess and that of Internal Model. This difference simly reresents the effects of the disturbances and uncertainties as well as that of a mismatch of the model. Internal Model control devices have shown to have good robustness roerties against disturbances and model mismatch in the case of linear model of the rocess. A control system is generally required to regulate the controlled variables to reference commands without steady state error against unknown and immeasurable disturbance inuts. Control systems with this nature roerty are called servomechanisms or servo systems. In servomechanism system design, the internal model control rincile lays an imortant role. Hence the design of a robust servomechanism system with lant uncertainty begins with three secifications as outline below: Definition of the lant model and associated uncertainty. Secification of inuts Desired closed loo erformance. IMC theory rovides a systematic aroach in the synthesis of a robust controller for systems with secified uncertainties. This brings about two imortant advantages of alying IMC control scheme. The closed loo stability can be choosing a stable IMC controller. The closed loo erformances are related directly to the controller arameter, which makes on-line tuning of the IMC controller very convenient.
Some imortant roerties of IMC scheme It rovides time delay comensation Reference signal tracking and disturbance rejection resonses can be shaed by a single filter The controller gives offset free resonses at the steady state. 5.3 SENSITIVITY AND COMPLIMENTARY SENSITIVITY FUNCTIONS Sensitivity function determines the erformance and the comlimentary sensitivity function determines the robustness. The sensitivity functions allow evaluating the controller behavior in relation to the desired attenuation constraints. The gradient of outut sensitivity function determines the dynamic behavior of the system. internal model control is the easiest way for PID tuning as it deends on selection of only one variable comared to two (PI) or three variables(rst). If () s and () s reresents the sensitivity function and comlimentary sensitivity functions, then E( s) Y( s) Gimc ( s) G ( s) () s R( s) d( s) d( s) G ( s) G ( s) Gimc ( s) (5.9) If G ( s) G ( s) then, ( s) G ( s) G ( s) (5.) imc ( s) G ( s) G ( s) (5.) imc
Internal Model Control (IMC) Algorithm Select the lant and obtain the transfer function of the lant G () s. Chose the rocess model G () s. Factorize the rocess model into minimum hase and non- minimum hase comonents. G ( s) G ( s) G ( s). This ste ensures that qs () is stable and causal. However G () s contains all Non-minimum Phase Elements (Noninvertible) in the lant model. i.e. all right half lane (RHP) zeros and time delays. The factor G () s is Minimum Phase and invertible. The controller qs () is chosen as inverse of minimum hase comonent. q( s) G ( s). If the rocess model contains only comonents which cannot be factorized but is does show stability with no right half oles (RHP) on the s-lane then the model is considered invertible. If the rocess model contains only the noninvertible comonents and with instability, the other imroved methods can be used because the IMC controller deends on the stability and invertibility of the rocess model. The noninvertibility of comonents may lead instability and realizability roblems when inverted. If the controller q(s) is imroer, then qs () is normally augmented with the otimal controller to attenuate the effects of rocessmodel mismatching and remove the higher frequency art of the
noise in the system in order to meet robust secifications. The robust comensator (filter) lays a ivotal role in the system as it combats lant uncertainties in the system design so that the designed control system can achieve the design objectives of robust stability and robust erformance. The filter transfer function f() s is to make the controller stable, causal and roer. The controller with filter is given by qs () G () s s n, (5.2) Where n is the order of the filter and is the filter time constant. The order of the filter is chosen such that G () s is roer to revent excessive differential control action. The filter arameter in the design can be chosen as a rule of thumb; hence the filter arameter values are often dictated by modeling errors, as already stated that in the design, it remains only tunable arameter. Usually from the eqn 5.2, the final form for the closed loo transfer functions characterizing the system is imc ( s) q( s) f ( s) G ( s ) (5.3) ( s) q( s) f ( s) G ( s ) (5.4) Filter time constant shall be selected so as to obtain good closed loo erformance and disturbance rejection. Internal model control arameter G imc qs () q ( s ) G ( s )
Increasing increases the closed loo time constant and slows the seed of the resonse; decreasing does the oosite. Usually the choice of the filter arameter deends on the allowable noise amlification by the controller and on modeling errors. Filter time constant avoids the excessive noise amlification and accommodate the modeling errors. To avoid excessive frequency gain of the controller is not more than 2 times its low frequency gain. For controllers that are ratios of olynomials, this criterion can be exressed as q( ) 2 (5.5) q() Higher the value of, higher is the robustness of the control system. Fig 5.4: closed loo diagram with IMC controller 5.4 THEORETICAL DESIGN The lant can be written as / LC.66* Gs () s s s R L s LC s s 2 7 n 2 2 2 2 7 2 n n ( / ) / 333.33.66*
Substituting the filter values in the equation and from the equation, the values of daming factor and natural frequency are.4 and 482.4 rad/sec Since the lant contains oles on the left hand lane, the system / LC is a minimum hase system. Hence G () s 2 s ( R / L) s / LC 2 s ( R / L) s / LC q( s) G ( s), From the equation, it is evident / LC that q(s) is imroer and needs to be roer for realization, so with adding the filter 2 G() s s ( R / L) s / LC qs () n s (/ LC)* s n becomes roer. Considering the order of the filter same as the lant (n=2) and λ as. based on equation s qs () 333.33_.667* 2 7 2 (.).2s (5.6) G imc () s 2 7 q( s) 6( s 333.33s.667* ) 2 q( s) G( s) s.2s Table 5.: Test arameters Parameters Suuly Voltage Values kv Filter Caacitance 2µF Filter Inductance Filter resistance 3mH IMC filter time constant. Load ower factor 45deg lagging
magnitude The simulink diagram and results are shown in figs 5.5-5.6. Ste inut is given to the system with magnitude of at sec. Fig 5.5: Simulink diagram for IMC controller without disturbance 2 8 6 4 2 2 3 4 5 6 7 8 9 Time (secs) Fig 5.6: IMC control with ste inut without disturbance To test the erformance of the roosed controller, a ste disturbance is added at the outut in the test system. The simulink diagram of the
test system with disturbance is deicted in fig 5.7. The corresonding result is deicted in fig 5.8 which clearly indicates the effectiveness of the roosed controller in mitigating the disturbance. The outut is nearly same as the inut reference signal tracking nature of the controller in the resence of the disturbance. The transient arameters for the ste resonse with internal model controller are Rise time =.348secs Settling time =.589 Peak overshoot =.55* -3 % Peak time =.6secs. Fig 5.7: IMC Design: Ste Inut disturbance at the outut of lant
magnitude 2 8 6 4 2 2 3 4 5 6 7 8 9 Time (secs) Fig 5.8: Ste resonse of IMC control disturbance at the outut The transient arameters namely rise time, settling time and eak overshoot deends on the filter time constant. The lower is the filter time constant, lower is the rise time and eak overshoot. Moreover the robustness in terms of stability is affected by the filter time constant. Smaller filter time constant leads to more robust system but decreases the disturbance rejection caability. Hence an otimum value of filter time constant is very much required. Particle Swarm Otimization for obtaining Filter Time constant [4]. Randomly initialized osition and velocity of the articles: X i () and V i () 2. Evaluate the fitness function for the article X i.
3. Position of the article becomes article s best ( best ) and global best ( g best ). 4. for i = to number of articles 5. Evaluate the fitness:= f i, f i error 6. For each article, comare the article s value with best. If the current value is better than the best value, than set this value as the best and current article s osition, X i as i 7. Identify the article that has the best fitness value. The value of its fitness function is identified as g best and its osition as g. 8. Udate the osition and velocities of all articles 9. X ( t) X ( t ) v ( t) and i i i v ( t) v ( t ). rand ( X ( t ). rand ( X ( t ). i i i i 2 2 g i. Adat velocity of the article using equations (); 2. Udate the osition of the article; 3. increase i 4. Reeat stes 2-5 until a stoing criterion is met (either maximum number of iterations or sufficiently good fitness value) The outut sensitivity function rovides the measure of dynamic behavior of the system. The dynamic behavior is quantified by modulus margin and delay margin; measure for robustness of modeling uncertainties.
Phase (deg) Magnitude (db) Bode Diagram - -2-3 -4 9 45 2 3 4 Frequency (rad/sec) Fig 5.9: Bode lot for outut sensitivity function with IMC The fig 5.9 illustrates the ste resonse of outut sensitivity function. The transient arameters are: Rise time =.336secs Peak time = secs The eak magnitude of the outut sensitivity function is observed to be.23db. This eak magnitude clearly indicates the robustness caability of IMC controller.
Phase (deg) Magnitude (db) Bode Diagram -5 - -5-2 -45-9 -35-8 2 3 4 5 6 7 8 9 Frequency (rad/sec) Fig 5.: Bode lot of closed loo system with IMC The fig 5. reresents the bode lot of the closed loo system. Comared to oen loo system stability are observed to be increased. Gain margin: inf Phase margin: 8 Gain margin indicates that there is a large scoe of adding a gain at hase crossover frequency to bring the system to verge of instability. Phase margin indicates that the maximum of 8 angle can be added to the system at the gain crossover frequency to bring the system to verge of instability. Since both gain margin and hase margin are large, system is more robust to the disturbances. Since the stability margins
are increased the system may be treated as more robust to disturbances. 5.5 SIMULATION RESULTS The test system is described briefly in chater 3. The test system includes distribution system with medium voltage level. Voltage sag and interrution are considered as the ower quality issues. These disturbances are created in the test system by varying the fault resistances. The fault resistance for voltage sag is.66ω and for voltage interrution is.ω. The DVR is modeled with Internal Model Control (IMC) for the generation of control angle δ. This control angle δ is used for generation of reference signal. The various case studies are resented in the thesis to verify the erformance of the controller. The first case study includes distribution system emloying DVR feeding to RL load. DVR oerates only during the eriod of voltage sag and interrution. Voltage sag is mitigated with IMC based DVR. The fig 5. deicts the load voltage with IMC controller in DVR. It can be seen very clearly that DVR is able to maintain the load voltage at 98%. The tame taken by the DVR to resond to voltage sag is less than 4ms. The corresonding Total Harmonic Distortion (THD) of load voltage is observed to be.6%. The THD is measured for the fault duration only comrising of 22 cycles. Case : Voltage sag mitigation with IMC based DVR
Mag (% of Fundamental) Vc(V) Vb(V) Vca(V) -.2.4.6.8 -.2.4.6.8 -.2.4.6.8 Time Fig 5.: Load voltage with IMC controller comensating voltage sag Selected signal: 55.34 cycles. FFT window (in red): 22 cycles -.2.4.6.8 Time (s) Fundamental (5Hz) =.935, THD=.6% 5 5 2 3 4 5 6 7 8 9 Harmonic order Fig 5.2: Total harmonic distortion of load voltage with IMC Case 2: DVR with rectifier load for mitigation of voltage sag
Vc(V) Vb(V) Va(V) The second case study refers to test system involving distribution system feeding to a rectifier load. The non-linearity nature of the rectifier load distorts the load voltage waveform. The voltage sag is created as described in the first case study. The erformance of the controller is verified by incororating it in DVR, used for mitigating voltage sag. The fig 5.3 reresents the load voltage waveform with DVR conducting during the eriod of voltage sag. The recovery time for restoration of load voltage to normal is less than 4ms. It is very clearly evident that the injected voltage by DVR is free from harmonics. The Total Harmonic Distortion (THD) is found to be 2.2% which is within the standards. -.2.4.6.8 -.2.4.6.8 -.2.4.6.8 Time Fig 5.3: load voltage after comensation of voltage sag In utility system with rectifier load
Vc(V) Vb(V) Va(V) Mag (% of Fundamental) Selected signal: 5.94 cycles. FFT window (in red): 22 cycles -.2.4.6.8 Time (s) Fundamental (5Hz) =.94, THD= 2.2% 5 5 2 4 6 8 Harmonic order Fig 5.4: Total harmonic distortion of load voltage Case 3: Mitigation of voltage interrution with IMC based DVR -.2.4.6.8.2.4 -.2.4.6.8.2.4 -.2.4.6.8.2.4 Time Fig 5.5: Load voltage with IMC
Mag (% of Fundamental) Selected signal: 7 cycles. FFT window (in red): 3 cycles -.2.4.6.8.2.4 Time (s) Fundamental (5Hz) =.7, THD= 3.8% 6 4 2 2 3 4 5 6 7 8 9 Harmonic order Fig 5.6: Total Harmonic Distortion of load voltage Third case study illustrates the robustness of the controller in mitigating the voltage interrution in a distribution system feeding RL Load. DVR with closed loo control can mitigate the voltage fluctuation uto 5% only. Research work involves oen loo control to increase the caability of DVR in mitigating deeer voltage fluctuations like interrutions. In this case study, DVR is fed from indeendent DC voltage source and the magnitude of DC voltage required to mitigate the voltage interrution gives the measure of robustness of IMC controller. The fig 5.5 shows the load voltage waveform with IMC controller based DVR injecting voltage during the eriod of interrution. Since, the DVR has to inject a large voltage, a small delay is observed at the time of switching on of DVR. This delay is due to slow resonse of filter and PWM controller for large voltage error. The corresonding THD is observed to be 4.67% as shown in the fig 5.6
Mag (% of Fundamental) Vc(V) Vb(V) Va(V) Case 4: DVR with rectifier load for mitigation of voltage interrution -.2.4.6.8 -.2.4.6.8 -.2.4.6.8 Time Fig 5.7: Load voltage with rectifier load Selected signal: 65.2 cycles. FFT window (in red): 25 cycles -.2.4.6.8.2 Time (s) Fundamental (5Hz) =.77, THD= 4.67% 6 4 2 2 4 6 8 Harmonic order Fig 5.8: Total harmonic distortion of load voltage Fourth case study illustrates the erformance of the IMC controller in generating switching ulses for the multilevel inverter which injects the missing voltage at the load end. The rectifier load is considered in
this case study. The figs 5.7 & 5.8 reresent the load voltage and THD at the load side. To inject missing voltage of kv, DC voltage magnitude of.8kv is sufficient with IMC controller. This DC voltage magnitude indicates effectiveness of IMC controller in rejecting the disturbance and reducing the stress on PWM controller. However, the PWM controller and filter introduced a small delay which is very clearly seen in the fig 5.7. The THD is observed to be 4.67%. 5.6 SUMMARY The IMC is rocess model deendant method i.e the control is ossible only when there is no mismatch between the lant and rocess model. Hence, the selection of rocess model is very imortant in IMC based controller design. IMC technique involves the ole cancellation rocess and selection of filter time constant. The algorithm for IMC based controller is described in this chater with mathematical calculations of the roosed controller. The selection of only one variable (filter time constant) makes the design of controller very easy. However, a balance is required between the good voltage regulation and disturbance rejection in selection of filter time constant. Particle swarm otimization technique is described for the selection of filter time constant. The roosed controller is a feedback controller with only one degree of freedom.
IMC based controller is able to reject the disturbance to some extent but not comletely. Hence, IMC is able to reduce the DC voltage magnitude to 2kV for mitigation of voltage sag of 2%. However, two degree of freedom is required to rocess the inut and outut signals effectively. Four case studies are resented to validate the erformance of IMC based controller in DVR for mitigation of voltage sag and interrution with RL and rectifier loads. Finally, IMC based controller is better than PI controller but still unable to reduce the DC voltage magnitude effectively. The controller is effective in reducing the Total Harmonic Distortion (THD) and mitigation voltage sag and interrution at the utility end.