98 CHAPTER DESIGN OF STATE FEEDBACK CONTROLLERS AND STATE OBSERVERS USING REDUCED ORDER MODEL INTRODUCTION The deign of ytem uing tate pace model for the deign i called a modern control deign and it i having an advantage of providing a complete (internal) decription of the ytem The power of tate variable method i the deign can be done with more than one control input and alo the inner variable (tate variable) are ued for feedback for atifying the deired performance Darouach et al (99) dealt with a method baed on the generalized contrained Sylveter equation for the deign of reduced order oberver for decriptor ytem with unknown input New condition for the exitence of reduced-order linear functional tate oberver for linear ytem with unknown input were preented by Trinh et al () Sytematic procedure for the ynthei of reduced-order functional oberver were given It i tated that the attractive feature of the propoed oberver wa the implicity with which the deign proce can be accomplihed Illutrative example had been given to illutrate the attractivene and implicity of the deign procedure Weiwen and Zhigiang () preented a comparion tudy of performance and characteritic of three advanced tate oberver, including the high-gain oberver, the liding-mode oberver and the extended tate
99 oberver Thee oberver were originally propoed to addre the dependence of the claical oberver, uch a the Kalman filter and the Luenberger oberver, on the accurate mathematical repreentation of the plant It wa oberved that, the extended tate oberver wa much uperior in dealing with dynamic uncertaintie, diturbance and enor noie Several novel nonlinear gain function were propoed to addre the difficulty in dealing with unknown initial condition with imulation and experimental reult In thi chapter, tate controller and oberver are deigned uing the reduced econd order model obtained uing the uggeted technique Example from chapter are conidered for the deign of tate controller and tate oberver State Feedback Controller The main objective in the deign of tate feedback control i to yield deirable cloed loop repone in term of both tranient and teady tate characteritic For the deign of tate feedback controller, pole placement i the technique which i ued to place the cloed loop pole of the ytem in pre determined location The location of pole correpond directly to the Eigen value of the ytem, which can control the characteritic of the ytem If the open loop ytem i tate controllable in nature, then an arbitrary cloed loop eigen value uing the tate feedback can be eaily achieved State Oberver In the control ytem deign, tate feedback require all the tate variable for all the time In mot of the practical ituation, the availability of all the tate variable for the meaurement i not poible and for uch cae,
if the ytem i completely obervable with a given et of output, then it i poible to determine the tate Thee tate are not directly meaured and their Eigen value can be arbitrarily aigned uing tate feedback State oberver i a device, which provide an etimate of the unknown internal tate of the ytem from the input and output of the correponding ytem DESIGN OF STATE FEEDBACK CONTROLLER FOR LTICS Conider an n th order table linear time invariant ytem decribed by the tranfer function given in Equation (): G ( ) b m n m b a b m m m m () n n n an a b where, a i ( i n) and b i ( i m) are calar contant and mn The tate model of Equation () in controllable canonical form i given a in Equation () and () x Ax Bu () y Cx Du () where, u i the input y i the output x i the tate vector
A i the tate vector a n a a a A B i the input vector B C i the output vector Auming that the pair (A, B) i controllable there exit a feedback matrix K uch that cloed loop ytem Eigen value can be placed in arbitrary location The repreentation of the ytem with tate feedback i repreented in Figure Figure Repreentation of Sytem with tate feedback
The control law i given in Equation () u Kx () where, Gain matrix n k k k K The cloed loop dynamic of the ytem i e the ytem with tate feedback controller i repreented in Equation () and (): ) ( BK A I () ) ( ) ( ) ( ) ( ) ( n a n k k a k a k a BK A () Thu the problem i to find K o that the deired characteritic polynomial of (A-BK) matche the deired characteritic polynomial Therefore the tranfer function of ytem given in Equation () with tate feedback controller i a in Equation () ) ( ) ( ) ( ) ( k a k a k a b b b b G n n n n n n n m m m m m m contr ()
Peudo Code for the Deign of State Feedback Controller for LTICS Get the higher order original ytem in tate pace model a in Equation () and () If (ytem i controllable) { If(order of the ytem>) { Convert the tate pace model into tranfer function Reduce into econd order ytem uing algorithm explained in chapter Convert the tranfer function into controllable canonical tate model } Calculate the value of the tate feedback controller gain matrix K by comparing the deired and characteritic polynomial Tune the value of K uing PSO Contruct the tranfer function for the reduced ytem with tate feedback controller From tep, derive the tranfer function of original ytem with controller Verify the repone of the ytem with and without controller } Endif
Algorithm to Deign State Feedback Controller for LTICS Step : Get the higher order original ytem in tate pace model a in Equation () and () Step : Check for the controllability of the given ytem Step : Convert the tate pace model into tranfer function model uing the Equation (8): G ( ) C ( I A ) B (8) Step : Uing the propoed method of model reduction explained in chapter, convert the higher order tranfer function given in (8) into econd order model Step : Obtain the controllable canonical form (phae variable form) of the econd order reduced model a written in Equation () and () Step : Uing the tate model of reduced ytem, obtain the characteritic polynomial of the given ytem uing the Equation (9) I A (9) +a + a = () where a and a are the coefficient of the polynomial Step : Get the deired pecification ettling time and percentage overhoot Uing the pecification, calculate the deired damping ratio () and natural frequency of ocillation ( n ) Step 8: Uing Equation (), the deired characteritic polynomial i obtained a in Equation ()
+ n + n = () where i the damping ratio and n i the natural frequency of ocillation S + = () where, and are the coefficient of the polynomial Step 9: The tate feedback gain matrix i calculated uing Equation () and () a: K=[( -a ) ( -a )] () Step : The value of gain matrix K are tuned uing PSO in uch a way that the deired pecification are met Step : Uing the calculated gain matrix K in Equation (), the controllable canonical form of the reduced model with tate feedback controller R contr tranfer function i alo calculated i contructed and the correponding Step : From the tranfer function of reduced model with tate feedback controller obtained in tep, the tranfer function of the original ytem with tate feedback controller revering the procedure of model reduction G contr i contructed by Step : By comparing original higher order ytem hown in Equation () and G contr, controller gain matrix i calculated Step : Verify the tep repone of the G contr, for the deired pecification
DESIGN OF STATE FEEDBACK CONTROLLER FOR LTIDS and Equation () The tate model for the dicrete ytem i given in Equation () x( k ) Ax( k) Bu( k) () y( k) Cx( k) Du( k) () where, x i the tate vector A i the tate vector of ize nxn B i the input vector of ize nx C i the output vector of ize xn D i the tate vector of ize nxn For the ytem to be completely controllable, the control law i given in Equation () u( k) Kx () where, Gain matrix K k k k n The cloed loop dynamic of the ytem i e the ytem with tate feedback controller i hown in Equation () and ()
Peudo Code for the Deign of State Feedback Controller for LTIDS Get the higher order original ytem in tate pace model a in Equation () and () If (ytem i controllable) { If (order of the ytem>) { Convert the tate pace model into tranfer function Reduce into econd order ytem uing algorithm given in Chapter Convert the tranfer function into controllable canonical tate model } Calculate the value of the tate feedback controller gain matrix K by comparing the deired and characteritic polynomial Tune the value of K uing PSO Contruct the tranfer function for the reduced ytem with tate feedback controller From above tep derive the tranfer function of original ytem with controller Verify the repone of the ytem with and without controller } Endif
8 Algorithm to Deign State Feedback Controller for LTIDS Step : Get the higher order original ytem in tate pace model a in Equation () and () Step : Check for the controllability of the given ytem uing Ackerman formula Step : Obtain the tranfer function model of the given dicrete ytem uing () and check for the pecification G ( z ) C ( zi A) B () Step : If the pecification are not met, convert into continuou domain uing tranformation technique and obtain the reduced model in continuou domain Step : Uing the algorithm explained in ection, deign tate feedback controller and obtain the gain matrix K for reduced model The value are tuned uing PSO in uch a way that the required pecification are met Step : Convert the continuou model into dicrete model uing invere tranformation technique Step : Contruct the higher order dicrete time tranfer function uing the invere procedure of model reduction Step 8: Verify the deired pecification
9 ILLUSTRATIONS Example Conider a higher order ytem in the context of Palaniwami et al () hown in tate pace a: 8 9 9 9 A ; B 8 8 8 8 8 89 98 C ; D The tranfer function of the given ytem i obtained uing Equation (8) a: 9 9 9 8 98 89 8 8 8 8 8 ) ( 8 S G
The econd order model uing the propoed method of model reduction explained in Chapter ection i: 9 98 R ( ) 99 9 On rearranging the above equation, 99 899 R ( ) (8) 8 9 The controllable canonical form of the reduced model i derived in Equation (9) uing Equation () and (): A ; 9 8 B ; C 899 988] ; D (9) For a choice of deigner pecification hown in ection, the characteritic polynomial in Equation () i obtained uing Equation () a: +8+9= () with ettling time t ec percentage overhoot% damping ratio =9 and natural frequency of ocillation n = 8 The deired characteritic polynomial in Equation () i obtained uing Equation () ++9= ()
Therefore, the gain matrix K i calculated a: K=[988 8] The value of the K matrix are tuned uing PSO and the tuned value of K are obtained a: K= [ 8] The tranfer function of reduced model with tate feedback controller i given in Equation 99 899 R contr ( ) () 9 By revering the procedure of model reduction, the tranfer function of the original ytem with tate controller i calculated a in Equation (): G contr ( S ) 89 8 8 9 8 8 89 98 9 8 8 9 8 89 The gain of the original ytem i, () K= [ 98 99 989 8 99 ] The tep repone of the original ytem and reduced ytem with the deigned tate feedback controller are hown in Figure and
8 Step Repone of Original Sytem without and with State Controller Original ytem Original ytem with State Controller Output 8 8 8 Time (econd) Figure Comparion of tep repone of original ytem with and without tate controller of Example 8 Step Repone of Reduced Sytem without and with State Controller Reduced ytem Reduced ytem with State Controller Output 8 8 8 Time (econd) Figure Comparion of tep repone of reduced ytem with and without tate controller of Example
From Figure and, it i oberved that propoed reduced ytem with tate feedback controller follow the original ytem with tate feedback controller Both the repone indicate that the overhoot preent in the original ytem i completely eliminated Here the deign complexity i reduced by uing the econd order model Example Conider the given third order ytem: A ; B C D The tranfer function model for the given ytem in Equation i obtained uing Equation (8) G ( ) () By uing the propoed method of model reduction, the reduced model of Equation () i obtained a in Equation () R ( ) () 8 8
On rearranging the above equation, 8 R ( ) () Equation () The characteritic polynomial from Equation () i written a in ++= () By comparing Equation () with deired characteritic polynomial repreented in Equation (), the gain matrix i calculated a: K= [ 9] The value are tuned uing PSO and the tuned value are: K= [ ] With the tate feedback controller, the tranfer function of the reduced ytem i obtained a in Equation (8): 8 R contr ( ) (8) 8 The tranfer function of original ytem with controller i obtained by revering the procedure of model reduction The tranfer function i obtained a in Equation (9) G contr ( ) (9) 9 9 The tep repone of original and reduced ytem with and without tate feedback controller are given in Figure () and ()
Step Repone of Original Sytem without and with State Controller Original ytem Original ytem with State Controller Output 8 8 Time (econd) Figure Comparion of tep repone of original ytem with and without tate controller of Example Step Repone of Reduced Sytem without and with State Controller Reduced ytem Reduced ytem with State Controller Output 8 8 Time (econd) Figure Comparion of tep repone of reduced ytem with and without tate controller of Example
From the tep repone of propoed reduced ytem with tate feedback controller and original ytem with tate feedback controller hown in Figure and, it i oberved that it i poible to deign a tate feedback controller only with the reduced econd order model to atify the deigner pecification Both the repone indicate that the maximum peak overhoot preent in the original ytem i minimied Example Conider eighth order dicrete ytem from Ravichandran et al (), 8 8 88 9 98 A ; B 9 9 8 C ;
The tranfer function of the given ytem i obtained a: ( 8z z z z z z z 8 G z) 8 8z z 8z z z 8z 8z z 8 () The eighth order original ytem i tranformed into G () a in Equation () by uing Bilinear tranformation, z 8 9 G S) 8 9 8 8 8 ( () Uing the propoed algorithm of model reduction, the reduced model i obtained a in Equation (): 99 R ( ) () 8 On rearranging the above Equation () 8 R ( ) () 9 8 Equation () The controllable form tate model of Equation () i given in A ; B 8 9 ; C 8 ; D ()
8 Equation The characteritic polynomial of Equation () i given in +9+8= () For the choice of performance pecification given in ection, the deired characteritic polynomial i obtained a in Equation () ++9= () Therefore the gain matrix K i calculated uing Equation () and () a: K=[ 988] Uing PSO, the value of the K matrix are tuned and the tuned value of controller gain matrix K are K= [ ] Therefore, the tranfer function of reduced ytem with tate feedback controller i given in Equation 8 R contr ( ) () 8 By applying the invere tranformation technique, the reduced domain model of Equation () i converted into dicrete model a in Equation (8) z 8 R contr ( z) (8) z 8z The controller gain matrix in dicrete domain i: K= [-9 ]
9 The tranfer function of given higher order dicrete ytem Equation (9) i obtained by applying invere reduction procedure to Equation (8) G contr 8z z z z z z z 8 ( z) 8 8z z 8z z z 8z 8z z 8 (9) The comparion of tep repone of original and reduced ytem with tate controller i hown in Figure () and () It i depicted that the ytem with tate feedback controller produce minimum rie time, ettling time and peak overhoot Comparion of Step Repone of Original Sytem with and without tate controller 8 Original ytem Original ytem with tate controller Output 8 8 9 Sampling intant Figure Comparion of tep repone of reduced ytem with and without tate controller of Example
Comparion of Step Repone of Reduced Sytem with and without tate controller 8 Reduced ytem Reduced ytem with tate controller Output 8 8 9 Sampling intant Figure Comparion of tep repone of reduced ytem with and without tate controller of Example DESIGN OF STATE OBSERVER In the deign of tate oberver, the oberver gain G i elected in uch a way that the continuou error dynamic converge to zero aymptotically Conider linear time invariant ytem hown in Equation () and Equation () The error dynamic i given by (A - GC) If thi i table in nature, the error vector will converge to zero for any initial error i e the etimated value reache the original value, where G i the oberver gain matrix
Peudo Code for State Oberver Deign Get the higher order original ytem in tate pace model a in Equation () and Equation () If ( ytem i LTIDS) {If (ytem i Obervable) {If(order of the ytem>) { Convert the tate pace model into tranfer function Reduce into econd order ytem uing the propoed algorithm explained in chapter ection Convert the tranfer function into obervable form of tate model } Calculate the value of the tate oberver gain matrix by comparing the deired and characteritic polynomial Tune the value of Oberver gain matrix uing PSO and obtain optimized value Contruct the tranfer function for the reduced ytem with tate oberver Derive the tranfer function of original ytem with oberver Convert LTICS into LTIDS Verify the repone of the ytem with and without oberver Endif } Endif
Algorithm for State oberver Deign Step : Step : Step : Step : Step : Step : Step : Get the higher order ytem Check for the obervability of the ytem If the ytem i obervable, convert the tate model into tranfer function model by uing Equation (8) Check for the deired pecification If the pecification are not met, derive the reduced model for the given ytem uing the propoed method of model reduction a in Equation () and obtain the obervable phae variable form of it Obtain the characteritic polynomial of reduced ytem a in Equation () From the deired characteritic, derive the deired polynomial a in Equation () From Equation () and () determine the oberver gain matrix a: a G a Step 8: Step 9: The value of G are tuned uing PSO to get the deired pecification Uing the tune value of G, obtain the tate pace model of reduced ytem with oberver Step : The tranfer function of reduced ytem with tate oberver i obtained from tate pace model Step : The tranfer function of the original ytem with tate oberver i obtained from reduced ytem with tate oberver by applying invere procedure of model reduction
ILLUSTRATIONS Example (): Conider the eighth order ytem function from Palaniwami et al ( 8 8 8 8 8 89 98 G S) 8 8 9 9 9 Uing the propoed method explained in Chapter ection, the econd order model i obtained a: R 9 98 ) 99 9 ( On rearranging the above equation, 99 899 R ( ) () 8 9 By conidering the deired characteritic polynomial given in Equation () and characteritic polynomial from Equation (), the oberver gain matrix i obtained a: G The value are tuned uing Particle Swarm Optimiation The parameter for the tuning proce are elected from ection to get the deired performance The Gain matrix with tuned value i, G
Therefore the tranfer function of reduced ytem with tate oberver i obtained in Equation () 99 89 R ob ( ) () 8 By applying the revere procedure of model reduction, the tranfer function of original ytem with tate oberver i obtained in Equation () G ob 8 8 8 8 8 8998 (S) 8 888 9 99 9 8 899 8 8 () The comparion of tep repone i given in Figure 8 and 9 The tate oberver deigned uing the propoed econd order ytem atifie atifying the deired pecification of repone without overhoot Hence the deign complexity i reduced in the proce deigning tate oberver for a higher order ytem 8 Step Repone of Original Sytem without and With tate oberver Original ytem Original ytem with tate oberver Output 8 Figure 8 8 8 Time Comparion of tep repone of original ytem with and without tate oberver of Example
8 Step Repone of Reduced Sytem without and With tate oberver Reduced ytem Reduced ytem with tate oberver Output 8 Figure 9 8 8 Time Comparion of tep repone of reduced ytem with and without tate oberver of Example Deign of State controller from State oberver: The characteritic polynomial i obtained from Equation () a: 8 The value of Gain matrix i obtained a: K=[9 -] The value are tuned uing PSO and the tuned value are: K=[ -88] Uing the tuned gain matrix the tranfer function of the reduced order ytem with tate controller i given in Equation () 99 89 R con ( ) () 9 8 By applying the revere procedure of model reduction, the tranfer function of original ytem with tate controller i obtained a in Equation ()
G( S) 8 8 8 8 8 89 98 8 8 88 88 8 998 () The Figure () how the comparion of tep repone of original ytem with and without tate controller 8 Step Repone of original Sytem without and With tate Controller Original ytem original ytem with tate controller Output 8 8 8 Time (econd) Figure Comparion of tep repone of original ytem with and without tate controller of Example SUMMARY The deign of tate feedback controller and oberver uing propoed econd order model i explained through illutrative example elected from Chapter From the tep repone of the propoed reduced ytem with tate feedback controller and original ytem with tate feedback controller for variou Example, it i oberved that the deign complexity can be reduced by uing imple econd order model The tate oberver deigned uing the propoed econd order ytem alo atifie the deired pecification of repone without overhoot Hence the deign complexity i reduced in the proce deigning tate oberver for a higher order ytem