Robust Performance Design of PID Controllers with Inverse Multiplicative Uncertainty

Transcription

1 American Control Conference on O'Farrell Street San Francisco CA USA June 9 - July Robust Performance Design of PID Controllers with Inverse Multilicative Uncertainty Tooran Emami John M Watkins Senior Member IEEE Abstract In this aer a grahical method is introduced for finding all roortional integral derivative (PID controllers that satisfy a robust erformance constraint for a given singleinut-single-outut (SISO linear time invariant (LTI transfer function of any order with time-delay This roblem can be solved by finding all achievable PID controllers that simultaneously stabilize the closed-loo characteristic olynomial and satisfy constraints defined by a set of related comlex olynomials Inverse multilicative modeling is used to describe the uncertainty of unstable erturbed system A key advantage of this rocedure is that it only deends on the frequency resonse of the system and does not require the lant transfer function coefficients If the lant transfer function is given the rocedure is still aroriate Inverse multilicative modeling often allows for designs with reduced conservativeness in the unstable ole uncertainty and it increases the size of the set of all PID controllers that robustly meet the erformance requirements P I INTRODUCTION roortional integral derivative (PID controller have been extensively used in industrial and bioengineering alications There has been a significant effort to determine the set of all PID controllers that meet secific design goals As the target of this research is to develo design methods that can be alied in industry these methods should have several key elements First they should be alicable to a broad set of lants In order for the methods to be alicable in the rocess control industry it is articularly imortant that they handle time-delays Ideally the design methods should be simle to understand and easy to imlement Methods that deend only on the frequency resonse of the system eliminate the need for a lant model which may not be available in some alications Not surrisingly most of the early work in this area sought to find all PID controllers that stabilized the nominal lant model Bhattacharyya and colleagues did much of early work in this area where knowledge of a system s rational transfer function model was assumed [] and [] Many of these results deend on generalizations of the Hermite-Biehler theorem [3] They develoed results based on theorems by Pontryagin and a generalized Nyquist criterion [4] The method introduced by Tan in [5] broke Manuscrit received Aril T Emami is with the Electrical Engineering and Comuter Science Deartment Wichita State University Wichita KS 676 USA (hone: ; txemami@wichitaedu J M Watkins is with the Electrical Engineering and Comuter Science Deartment Wichita State University Wichita KS 676 USA (hone: ; fax: ; jwatkins@ieeeorg the numerator and denominator of the lant transfer function into even and odd arts In [6] and [7] a new method which did not involve comlex mathematical derivations was used to solve the roblem of stabilizing an arbitrary order transfer function when only the frequency resonse of the lant transfer function was known This work was extended in [8] to a unified aroach involving delta oerators that found the stability region for discrete-time or continuous-time PID controllers In [7] Saeki introduced a method for finding the number of unstable oles across the boundary of PID controllers Beyond stability investigators have also looked at erformance and robustness The authors in [5] [6] and [8] found regions where the controllers were guaranteed to meet certain gain and hase margin requirements PID controllers that also satisfy gain crossover hase crossover and bandwidth requirements for double integrator systems with delay were found in [9] In [] Shafiei and Shenton found all PID controllers that laced the closed-loo oles in certain D-artitions In [] and [] the arameters of PID controller were determined using a metaheuristic algorithm In [] the metaheuristic algorithm was used to adjust the PID arameters to meet the erformance requirement for a ouring task In [3] the authors used a fractional PID controller to meet the erformance requirement for an active magnetic bearing system In this aer an adative genetic algorithm was used to determine the PID controller arameters that otimized a multi-objective cost function In [4] constrained ole assignment was used for design of PD controllers for a double integrator lant model with time delays or time constant As these controllers must be imlemented on real systems design methods that deal with robustness are of articular imortance In [5] [6] and [7] Saeki and colleagues looked at different methods for H controller design of PID controllers Ho used a generalization of the Hermite-Biehler theorem for H PID design [8] Tantaris Keel and Bhattacharyya looked at a similar roblem for first-order controllers [9] In [] Keel and Bhattacharyya looked at PID design given a weighted sensitivity and weighted comlementary sensitivity constraint for lants with no oles or zeros on the axis In [] Ho and Lin looked at PID controller design for robust erformance for a lant that was described by a rational transfer function Unfortunately none of these methods that deal with robustness work directly with time-delays which are revalent in the rocess control industry In [] Keel and Bhattacharyya did allow //$6 AACC 5

2 for time-delays in the nominal model when they investigated the weighted sensitivity and robust stability roblems However they did not consider the robust erformance roblem In [3] [4] [5] [6] and [7] the authors of this aers develoed techniques for finding all achievable PID controllers that simultaneously stabilize the closed-loo continuous-time system and satisfy an H sensitivity H comlementary sensitivity weighted sensitivity robust stability or robust erformance constraint resectively In [8] [9] [3] and [3] this method was extended to a unified aroach for continuous-time and discrete-time H sensitivity H comlementary sensitivity weighted sensitivity or robust stability design of PID controllers resectively In this aer we consider the robust erformance roblem for systems with inverse multilicative uncertainty Inverse multilicative modeling allows for designs where uncertain oles cross between the left-half lane and right-half lane This method is alicable for a single-inut-single-outut (SISO linear time invariant (LTI roer transfer functions of any order with time delay This method does not require the rational lant transfer function model but deends on the frequency resonse of the nominal system If the lant transfer function is known we can aly the same rocedure by first comuting the frequency resonse The remainder of this aer is organized as follows In Section II the design methodology is resented In Section III this method is alied to a numerical examle Finally the results of this aer are summarized in Section IV II DESIGN METHODOLOGY Consider a SISO LTI system shown in Figure where G ( s reresents the erturbed lant G ( s is the nominal lant and G ( s is the PID controller The reference inut c and the error signals are R( s and Z( s resectively is the sensitivity function weight W S W I is the inverse multilicative weight and I ( is the uncertain erturbation [3] The nominal lant transfer function can be written as s G ( s = G ( s e τ ( o where G ( s is an arbitrary-order roer transfer function o and τ is a constant time-delay defined as The PID controller is Ki Gc ( s = K + + Kd s ( s where K K i and derivative gains resectively K d are the roortional integral and The transfer functions in Figure can all be exressed in the frequency domain The nominal lant transfer function can be written in terms of its real and imaginary arts as G ( = R + j I (3 The PID controller is defined in the frequency domain as Ki Gc ( = K + + Kd (4 The sensitivity function weight W S and the inverse multilicative weight W I are defined in terms of their real and imaginary arts as and ( = AS + jbs (5 WI ( = AI + jbi (6 R G c Fig Block diagram of the system with inverse multilicative uncertainty The deterministic values of K K i and K d for which the closed-loo characteristic olynomial is Hurwitz stable have been found in [6] and [7] In this aer the roblem is to find all achievable PID controllers that satisfy the robust erformance constraint of the feedback system in Figure for all I ( The robust erformance constraint for the SISO system in Figure is given by ( S ( S and ω (7 where S ( = + Gc ( G ( + WI ( ( ( for ( is the sensitivity function and is a ositive scalar It can be shown that this is equivalent to - W S W I I G Z G 5

3 ( S( + WI ( S( ω ( WI ( S( where S( = + G ( Gc ( is the nominal sensitivity function The comlex functions in (8 can be written in terms of their magnitudes and hase angles as j W ( ( ( ( S S S e j W ( ( ( ( I S WI S e + ω j W ( ( ( ( ( ( S WI S WI S e (9 If (9 holds then for each value of ω we can write ( ( S W ( ( I S S e + WI S e ω W ( ( ( SI S WI S e ( where θ = W ( S( θ = W ( S( and S S I I θsi = ( WI ( S( for some θs [ π [ π Consequently all PID controllers that satisfy (8 must lie at the intersection of all controllers that satisfy ( for some θ [ π [ π and θ [ π SI To find this region for each value of θs [ π [ π we will find all PID controllers on the boundary of ( It is easy to show from ( that PID controllers on the boundary must satisfy P( ω θs θsi = ( where P( ω θs θsi = ( S W ( I S e + WI e + G ( Gc ( W ( ( SI S WI e By substituting (3 (4 (5 (6 e S = cosθs + j sinθs I e = cos + j sin and e SI = cosθsi + j sinθsi into ( the frequency resonse of this modified characteristic olynomial can be rewritten S (8 P( ω θs θsi = Ki + ( R + ji K + + Kd ( AS + jbs ( cosθs + j sinθs + ( AI ω + jbi ω ( + j + ( AS + jbs ( cosθsi + j sinθ SI ( A + jb ( ( cos sin I I ( Note that ( reduces to the frequency resonse of the standard closed-loo characteristic olynomial as Exanding ( in terms of its real and imaginary arts yields and R ( K + I ( Ki I ( Kd = YR ( (3 ω ω ω ω ω ω I ( K R ( Ki + R ( Kd = YI ( (4 ω ω ω ω ω ω where YR = ω + ( αs + αi + αsi ω YI = ( βs + βi + βsi αs = AS cos θs + BS sin θs αi = AI cos + BI sin ( A A B B ( A B + B A S I + S I cosθsi αsi = S I S I sinθ SI βs = AS sin θs + BS cos θs βi = AI sin + BI cos ( A B B A ( A A B B S I + S I cosθsi βsi = S I S I sinθ SI This is a three-dimensional system in terms of the controller arameters K K i and K d First the boundary of ( will be found in the ( K K i lane for a fixed value of K d After setting (3 and (4 can be rewritten as K d to the fixed value ωr I K Y R + ω I d = I ( ( ω ω R ω Ki Y I ω R d (5 d 5

4 Solving (5 for all ω θs [ π [ π and θsi [ π gives the following equations: K ( ω θ θ θ = S I SI R + ( αs + αi + αsi + I ( βs + βi + βsi G ( K ( ω θ θ θ = i S I SI ω d ωr ( βs + βi + βsi ωi + ( αs + αi + αsi + G ( (6 (7 where G ( = R + I Setting ω = in (5 we obtain X Ri ( K X Ii ( = Ki (8 and conclude that K ( θs θsi is arbitrary and Ki ( θs θsi = unless I ( = R ( = which holds only when G ( s has a zero at the origin The rocedure can be reeated in the ( K K d lane After setting rewritten as K i to a fixed value i (3 and (4 can be ωr ω I K YR I i = (9 ωi ( ( Kd YI R ( K ω ω R ω ω + i K ( ω θ θ θ = d S I SI R ( βs + βi + βsi + I + ( α + α + α ω ω ω S I SI i + G ( j ( At ω = i must be equal to zero for a solution to exist Furthermore as I ( = for all real lants Kd ( θs θsi is arbitrary and K ( αs ( + αi ( + αsi ( ( θ θ θ = R ( S I SI ( Lastly the solution is found in the ( Ki K d lane After setting K to a fixed value of (3 and (4 are rewritten as I ω I Ki YR ωr R ( ( K = d YI I ( K ω ω R ω X ω ω Id ( Although the coefficient matrix is singular a solution will exist in two cases First at ω = Kd ( θs θsi is arbitrary and Ki ( θs θsi = unless I ( = R ( = which holds only when the lant has a zero at the origin In such a case a PID comensator should be avoided as the PID ole cancels the zero at the origin and the system becomes internally unstable A second set of solutions occurs at any frequency ω i where K ( ωi θs θsi (from (6 is equal to At these frequencies Kd ( ωi θs θsi and Ki ( ωi θs θsi must satisfy the following straight line equation Solving (9 for all ω θs [ π [ π and θsi [ π gives the same exression as (6 for K ( ω θs θsi and the following equation for Kd ( ω θs θsi : ( ω θ θ θ ( ω θ θ θ = + Kd i S I SI Ki i S I SI ωi R ( ωi ( βs ( ωi + βi ( ωi + βsi ( ωi + I ( ωi + ( αs ( ωi + αi ( ωi + αsi ( ωi ω G ( i i (3 53

5 III EXAMPLE In this section a numerical examle is used to demonstrate the alication of this method Consider the second order lant transfer function where the feedback loo has an unknown ole location in a range of [ 3] The goal is to find all PID controllers that stabilize the system and satisfy the robust erformance constraint in (8 where = The closed-loo ste resonse is required to have an overshoot less than 4% and a settling time less than 8 seconds The sensitivity weight is chosen to satisfy the erformance requirement for the closed-loo system is 48( s + 6 ( s = (4 ( s + The nominal model of the system is given by K i X: 7847 Y: 9649 Nominal Stability Boundary Robust Performance Region K Fig Nominal stability boundary and robust erformance region in the ( K K i lane 5 + ( ( + σ ( + s s G s = e s s (5 To verify the results an arbitrary controller from this region is chosen giving us the PID controller where σ = has been selected to be the mean value of the uncertain ole The inverse multilicative weight WI ( s = (6 s + is chosen to bound the inverse multilicative errors Note inverse multilicative modeling allows for designs with unstable ole uncertainty Equations (6 and (7 are used in the ( K K i lane for a fixed value of K d = 5 As discussed reviously the PID stability boundary of the nominal system can be found by setting = in (6 and (7 All PID controllers that satisfy the robust erformance constraint in (8 are found by setting = in (6 and (7 for some θs [ π [ π and then finding the intersection of all regions The region that satisfies the robust erformance constraint and the nominal stability boundary is shown in Figure The intersection of all regions inside the nominal stability boundary of the ( K K i lane is the robust erformance region 9 Gc ( s = s (7 s Substituting (4 (5 (6 and (7 into (8 gives ( S( + WI ( S( 98 As the ( WI ( S( magnitude of robust erformance system is less than one the design goal is met The second method uses (6 and ( in the ( K K d lane for a fixed value of i which follows the same rocedure as the ( K K i lane The third method is alied in the ( Ki K d lane for a fixed value of K = 5 Plots of K ( ω θs θsi at = and K ( ω θs θsi (from (6 for values of θs [ π [ π are shown in Figure 3 For each curve the ωi s are the frequencies at which the chosen value for K ( ωi θs θsi = K = 5 Each ω i for this chosen constant coefficient of is substituted into (3 to find the required boundaries In addition we have the boundary at Ki ( θs θsi = 54

6 seconds and the maximum ercent overshoot is 7% which corresond to the stable ole location at -3 and the unstable ole location at resectively 4 4 System: - Peak amlitude: 7 Overshoot (%: 7 At time (sec: 6 Ste Resonse 3 K-Fixed Amlitude System: 3 Settling Time (sec: Fig 3 Plots of K ( ω θs θsi versus ω - -4 The region that satisfied the robust erformance constraint and the nominal stability boundary is shown in Figure 4 The intersection of all regions inside the nominal stability boundary of the ( Ki K d lane is the robust erformance region To verify the results an arbitrary controller from this region is chosen giving us the PID controller 3 Gc ( s = s (8 s K d 4 3 X: 336 Y: 439 Nominal Stability Boundary Time (sec Fig 5 Ste resonse of the closed loo system for various ole locations IV CONCLUSION A grahical technique was introduced for finding all achievable PID controllers that satisfy the robust erformance constraint of a single-inut-single-outut (SISO linear time invariant (LTI transfer function of an arbitrary-order with time-delay Inverse multilicative modeling was used to describe the uncertainty of unstable erturbed system This method required only the frequency resonse of the nominal system A numerical examle with time-delay and an unknown ole that may be in the righthalf or left-half lane was resented to demonstrate the alication of this method Inverse multilicative modeling allowed for designs where uncertain oles cross between the left-half lane and right-half lane - Robust Performance Region K i Fig 4 Nominal stability boundary and robust erformance region in the ( K K lane i d Substituting (4 (5 (6 and (8 into (8 gives ( S( + WI ( S( 93 As the ( WI ( S( magnitude of robust erformance system is less than one the design goal is met Ste resonses of the closed-loo system with the PID controller in (8 and various ole between [ 3] are shown in Figure 5 As can be seen the closed-loo ste resonses all have an overshoot less than 4% and a setting time less than 8 seconds The maximum setting time is REFERENCES [] S P Bhattacharyya H Chaellat and L H Keel Robust Control: The Parametric Aroach Uer Saddle River NJ: Prentice-Hall 995 [] G J Silva A Datta and S P Bhattacharyya PID Controllers for Time-Delay Systems Boston: Birkhäuser 5 [3] K W Ho A Datta and S P Bhattacharyya Generalizations of the Hermite-Biehler theorem Linear Algebra and its Alications Vol [4] H Xu A Datta and S P Bhattacharyya PID stabilization of LTI lants with time-delay Proceedings of the 4 nd IEEE Conference on Decision and Control Maui Hawaii [5] N Tan Comutation of stabilizing PI and PID controllers for rocesses with time delay ISE Transactions Vol [6] S Sujoldžić and J M Watkins Stabilization of an arbitrary order transfer function with time delay using PID controller Proceedings of 45 th IEEE Conference on Decision and Control San Diego CA 6 [7] M Saeki Proerties of stabilizing PID gain set in arameter sace IEEE Transactions on Automatic Control Vol 5 No [8] T Lee J M Watkins T Emami and S Sujoldžić A unified aroach for stabilization of arbitrary order continuous-time and discrete-time transfer functions with time delay using a PID 55

7 controller Proceedings of 46 th IEEE Conference on Decision and Control New Orleans LA 7-5 [9] J Watkins and G Pier Investigating the effects of cross-link delays on sacecraft formation control Journal of the Astronautical Science Vol 53 No [] Z Shafiei and A T Shenton Tuning of PID- tye controllers for stable and unstable system with time delay Automatica Vol 3 No [] M P Tzamtzi F N Koumboulis and M G Skaretis On the modeling and controller design for the outut hase of ouring rocess EAS Transactions on System and Control Vol 4 No 9 - [] F N Koumboulis and MP Tzamtzi A metaheuristics aroach for controller design of multivariable rocess th IEEE Conference on Emerging Technologies and Factory Automation Patras Greece [3] L Y Chang and H C Chen Tuning of fractional PID controllers using adative genetic algorithm for active magnetic bearing system EAS Transactions on System and Control Vol 8 No [4] K Žáková One tye of controller design for delayed double integrator system EAS Transactions on System and Control Vol No [5] M Saeki and J Kimura Design method of robust PID controller and CAD system th IFAC Symosium on System Identification Vol [6] M Saeki and K Aimoto PID controller otimization for H control by linear rogramming International Journal of Robust and Nonlinear Control : [7] M Saeki Fixed structure PID controller design for standard H control roblem Automatica Vol [8] M T Ho Synthesis of H PID controllers: a arametric aroach Automatica Vol [9] R N Tantaris L H Keel and S P Bhattacharyya H Design with first order controllers IEEE Transactions on Automatic Control Vol 5 No [] L H Keel and S P Bhattacharyya PID Controller synthesis free of analytical methods IFAC 6 th Triennial World Congress Prague Czech Reublic [] M T Ho and CY Lin PID controller design for robust erformance IEEE Transactions on Automatic Control Vol 48 No [] L H Keel and S P Bhattacharyya Controller synthesis free of analytical models: three term controllers IEEE Transactions on Automatic Control Vol 53 No [3] T Emami and J M Watkins Sensitivity design of PID controllers for arbitrary-order transfer functions with time-delay alied to a DC motor with communication delay Proc of the IEEE Multi Conf on Systems and Control San Antonio Texas Setember 8 [4] T Emami and J M Watkins Comlementary sensitivity design of PID controller for arbitrary-order transfer function with time delay Proc of the 8 ASME Dynamic Systems and Control Conf Ann Arbor Michigan October 8 [5] T Emami and J M Watkins Weighted sensitivity design of PID controller for arbitrary-order transfer function with time delay Proc of the Eleventh IASTED International Conf on Intelligent Systems and Control -5 Orlando Florida November 8 [6] T Emami and J M Watkins Robust stability design of PID controllers for arbitrary-order transfer functions with uncertain time delay Proc of the 4st Southeastern Symosium on System Theory Tullahoma TN 9 [7] T Emami and J M Watkins Robust erformance characterization of PID controllers in the frequency domain EAS Trans on Systems and Control vol 4 no [8] T Emami and J M Watkins A unified aroach for sensitivity design of PID controllers in the frequency domain EAS Trans on Systems and Control vol 4 no [9] T Emami and J M Watkins A Unified aroach for H comlementary sensitivity design of PID controllers alied to a DC motor with communication delay Proc of the 9 ASME International Mechanical Engineering Congress & Exosition Lake Buena Vista Florida November 9 [3] T Emami and J M Watkins A unified aroach for weighted sensitivity design of PID controllers Proc of the 9 ASME Dynamic Systems and Control Conf Hollywood California October 9 [3] T Emami and J M Watkins A unified aroach for robust stability design of PID controllers Proceedings of the American Control Conference Baltimore Maryland June [3] S Skogestad and I Postlethwaite Multivariable Feedback Control Analysis and Design John Wiley & Sons NJ 5 56