This is a repository copy of Enhanced Bounded Integral Control of Input-to-State Stable Nonlinear Systems.

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1 This is a repository copy of Enhanced Bonded Integral Control of Inpt-to-State Stable Nonlinear Systems. White Rose Research Online URL for this paper: Version: Accepted Version Proceedings Paper: Konstantopolos, G. orcid.org/ (7 Enhanced Bonded Integral Control of Inpt-to-State Stable Nonlinear Systems. In: IFAC papers online. th World Congress of the International Federation of Atomatic Control, 9/7/7-4/7/7, Tolose, France. Elsevier, pp Article available nder the terms of the CC-BY-NC-ND licence ( Rese This article is distribted nder the terms of the Creative Commons Attribtion-NonCommercial-NoDerivs (CC BY-NC-ND licence. This licence only allos yo to donload this ork and share it ith others as long as yo credit the athors, bt yo can t change the article in any ay or se it commercially. More information and the fll terms of the licence here: Takedon If yo consider content in White Rose Research Online to be in breach of UK la, please notify s by ing eprints@hiterose.ac.k inclding the URL of the record and the reason for the ithdraal reest. eprints@hiterose.ac.k

2 Enhanced Bonded Integral Control of Inpt-to-State Stable Nonlinear Systems George C. Konstantopolos Department of Atomatic Control and Systems Engineering The University of Sheffield, S 3JD, UK ( g.konstantopolos@sheffield.ac.k Abstract: The bonded integral controller (BIC as recently proposed to replace the traditional integral controller (IC for the reglation of any inpt-to-state stable (ISS nonlinear system and garantee closed-loop system stability ith a bonded control otpt. In this paper, an enhanced version of the BIC is presented to provide a better approximation of the traditional IC in the entire bonded range of the control otpt and relax the assmption on the selection of the initial conditions of the original BIC. Using Lyapnov methods, it is analytically proven that the enhanced BIC maintains the zero-gain property and garantees closed-loop stability of any nonlinear ISS plant ith a given bond at the control otpt, ithot sffering from integrator indp isses. The plant dynamics and strctre can be nkon as long as the plant is ISS. Hence, the proposed enhanced version of the BIC can replace the traditional IC in many applications here closed-loop stability cannot be proven, ithot changing the controller operation. A practical example is simlated to verify the performance of the proposed enhanced BIC compared to the original version and the traditional IC. Keyords: Integral control, nonlinear systems, inpt-to-state stability, Lyapnov methods, bonded control.. INTRODUCTION Integral control (IC has been the core control method for reglating both linear and nonlinear systems. Especially nder the presence of ncertainties and external distrbances, IC represents a key element of the closedloop system to achieve robst reglation. In this case, the integrator dynamics are added to the dynamics of the original plant and the agmented closed-loop system that reslts from this action is investigated in terms of stability Khalil (; Tsinias (989. Closed-loop stability of a nonlinear plant ith IC is not an easy task and has been an active area of research for many years. Local and semi-global stability reslts can be fond in Isidori and Byrnes (99; Hang and Rgh (99; Mahmod and Khalil (996; Isidori (997; Khalil ( especially for minimm-phase nonlinear plants ith parameter pertrbations. The reslts of Khalil ( ere frther extended in Jiang and Mareels ( ithot a priori knoledge of the bond of the ncertainties and also in Riachy et al. (6 to se mollifiers instead of high-gain observers in the feedback loop. Conditional integrators have been also introdced in Singh and Khalil (4; Li and Khalil ( to achieve asymptotic reglation of nonlinear systems and improve the transient response by applying an integral action inside a bondary layer. In Astolfi and Praly (6, it is shon that robst otpt reglation can be accomplished sing an integral action hen a stabilizing feedback controller exists for the original plant. Hence, in most of the existing IC methods, several assmptions of the plant strctre and dynamics are considered, i.e. minimm or non-minimm phase, relative degree, etc, inclding inpt-to-state stability (ISS or integral ISS properties for the distrbance dynamics Jiang et al. (4; W et al. (. The challenge of proving closed-loop stability increases hen the dynamics of the system are nknon. For example, in Gay (6, a proportional-integral controller ith an extremm-seeking method is proposed to stabilize a nonlinear plant to an nknon optimal eilibrim. In addition, since in many practical examples the IC is applied in plant systems ith inpt constraints, either actator or stability constraints (e.g. locally ISS systems, the controller otpt is often reired to be bonded in a given range. Althogh the traditional IC can be combined ith a satration nit to maintain a given bond for the control otpt, the closed-loop system may still sffer from instability de to the integrator indp phenomenon. In these cases, several anti-indp technies can be introdced in the control design and have been extensively stdied in the literatre, see for example Zaccarian and Teel (, 4; Peng et al. (996. The motivation for designing IC for nonlinear systems and garanteeing closed-loop stability comes from a series of applications inclding motor control Choi and Lee (9, nmanned aerial vehicles Ngyen and Damm (, poer systems Zhong and Weiss (; Zhong et al. (6, etc. A recent approach that introdces a satrating integrator to overcome the indp problem and garantee closed-loop stability for a ide class of nonlinear systems is provided in Weiss and Natarajan (6. Hoever, most of the traditional anti-indp methods cannot garantee a rigoros closed-loop stability and modern anti-indp technies

3 reire knoledge of the plant dynamics, strctre or several properties. The present paper investigates nonlinear plant systems hich may have nknon strctre, dynamics and parameters. The only information is that the plant is ISS ith respect to the control inpt and the main task is to reglate the plant and garantee closed-loop stability. For simplicity, the investigation is restricted to single-inpt systems. For these type of systems, the bonded integral control (BIC as recently proposed in Konstantopolos et al. (6 to achieve a desired reglation scenario and garantee closed-loop stability. The BIC introdces a zerogain property, i.e. the control otpt is bonded independently from the inpt signal; ths proving stability for any ISS plant. The main prpose as to replace the traditional IC in applications here the IC as sed ithot a rigoros proof of stability. In this paper, an enhanced version of the BIC is proposed to overcome to main drabacks of the original BIC: i provide a better approximation of the traditional IC in the entire bonded range of the control otpt and ii relax the assmption regarding the selection of the initial condition of the controller states. It is shon that a sitable modification of the BIC dynamics reslts in a more generic strctre here the initial conditions of the BIC states can be chosen inside a ider area. Using Lyapnov methods, it is proven that the proposed enhanced BIC maintains that zero-gain property and the given bond for the control otpt ithot sffering from indp isses, leading to the closed-loop stability in the sense of bondedness for any ISS plant. This is achieved ithot additional satration nits and the continostime strctre of the proposed controller allos the investigation of stability for the entire range of the plant inpt. Closed-loop stability in the sense of bondedness is proven sing the generalized small-gain theorem Jiang et al. (994 and represents the main task of the paper. For conditions of convergence to a desired eilibrim point, the reader is referred to the original BIC in Konstantopolos et al. (6. In order to verify the properties of the enhanced BIC, its performance is compared to the original BIC and the traditional IC ith a satration nit for a practical example of a dc/dc bck-boost converter system. The rest of the paper is organized as follos. In Section, an overvie of the original BIC is provided to explain the main concept and nderline its limitations. In Section 3, the enhanced version of the BIC is proposed and analyzed. The zero-gain property and the main tasks that distingish the proposed controller ith the original one are explained. In Section 4, a practical example of a poer converter is simlated to investigate the performance of the proposed approach and finally, the conclsions of the paper are presented in Section 5.. OVERVIEW OF THE BOUNDED INTEGRAL CONTROL Consider the nonlinear plant system of the form ẋ = f(x,, ( ith x, representing the state vector and inpt vector, respectively, f : D D R n being locally Lipschitz in x, and D, D being open neighborhoods of the origin. To simplify the analysis, the investigation ill be focsed on single-inpt systems here the control task is the reglation of a scalar fnction g(x to zero, hich incldes the most common reglation scenario of one of the system states x i to a reference vale, i.e. g(x = x ref x i. The most idely sed technie to achieve this task is to apply a traditional integral controller (IC as = ( ẇ = k I g(x, (3 ith k I being a positive integral gain, and is fond in many engineering examples. Hoever, the stability of the closedloop system cannot be garanteed in general, especially in the case here the dynamics of the plant are nknon. In addition, in several practical applications it is often reired for the control otpt to be bonded in a given range, i.e. [, ], >. In these cases, satration nits and additional anti-indp mechanisms are being placed, hich may still reire information of the plant dynamics or frther complicate the stability analysis. To this end, based solely on the assmption that the plant is ISS ith respect to the inpt and ithot any frther knoledge on the plant, it has been shon in Konstantopolos et al. (6 that closed-loop stability can be garanteed if the IC is replaced by the bonded integral controller (BIC given in the form = (4 [ẇ ] k I g(x = ( c ẇ k I g(x k c + [ ], (5 here k > and c (, are constant gains. By considering the Lyapnov fnction candidate W = its derivative becomes ( Ẇ = +, (6 + k (7 hich implies that if the initial conditions of the BIC states and are defined on the ellipse } W =, R : + =, (8 then the trajectory of the BIC states response on the plane ill be restricted on the ellipse W for all ftre time, i.e. [, ] (see Fig.. This holds independently from the fnction g(x, ths reslting in a zero-gain property of the controller, i.e. the controller states and are bonded independently from the inpt g(x. In fact, this is a crcial property for garanteeing closed-loop stability of the feedback interconnection of any ISS plant ( ith the BIC (4-(5. Looking at the first eation of the BIC (4, it can expressed as ẇ = k I g(x. (9 c Given a desired eilibrim point ( e, e = ( e, e, corresponding to a point on the ellipse W, according to

4 θ ɺ e m= m= m=5 m= W e Figre. Original BIC states on plane the analysis in Konstantopolos et al. (6, then near the desired eilibrim (9 becomes ẇ k I g(x e. ( c This indicates that if c is chosen eal to the steadystate vale of the inpt e (or close to this vale, then ( reslts in ẇ k I g(x and the BIC approximates the traditional IC. Hoever, this approximation holds only near the eilibrim point and not in the entire range of the inpt [, ]. Similarly, by selecting a different vale for c, the BIC approximates the IC near a different operating point. In addition, it is clear from (9 that the integral gain of the BIC is k I and can change c significantly hile and operate on the desired ellipse W becase changes. Frthermore, the initial conditions of and shold be defined on the ellipse W, i.e. they shold satisfy + = ith a common choice of = and =. Hoever, this can be a restrictive argment in a real environment here the system can be distrbed, leading to an inaccrate definition of the initial conditions. Therefore, a better approximation of the traditional IC in the entire operating range of the inpt [, ] and increased robstness of the controller in terms of the selection of the initial conditions are to important isses that need to be addressed. These shold be accomplished ithot changing the zero-gain property of the BIC that garantees the stability of any ISS nonlinear system ith nknon dynamics and parameters. An enhanced version of the BIC that achieves these tasks is analyzed in the seel. 3. ENHANCED VERSION OF THE BOUNDED INTEGRAL CONTROL In order to overcome the limitations of the original BIC, the dynamics of the original controller (4 are modified to take the form [ẇ ] = ẇ ( k + m k I g(x k I g(x m [ (, k + m ] ( here m N and k is a large positive constant gain. To analyze system (, consider the Lyapnov fnction candidate V m = + m m. ( Figre. Enhanced BIC states on plane Taking the time derivative of V m and sbstitting from (, it yields V m = ẇ ( = k + m ẇ k I g(x m ( = k + m ( k ( + m + k I g(x m + m + m m, (3 hich shos that V m negative otside of the closed crve } W m =, R : + m = (4 positive inside, except from the origin here V m is zero. Hence, W m acts as an attractive closed crve since every trajectory ((t, (t starting from any point (, on the plane except from the origin ill be attracted on W m. Note that the larger the vale of k, the faster the attraction of and on W m. For m =, then the invariant set for and is defined by the nion of the origin and the ellipse + =, and the strctre of ( becomes similar to the original BIC in (5. The closed crve W m for different vales of m N is shon in Fig.. It shold be nderlined that V m < for any and otside W m independently from the integral fnction g(x. Note that for m =, then the BIC states and ill be attracted on W and the system ( ill operate similarly to the original BIC (4. Hence the proposed enhanced version represents a generalized strctre of the original BIC. Given the Lyapnov fnction candidate V m defined in ( and the fact that V m < otside of W m, then one can find a positive vale s > and a set Ω m = V m (, s}, otside of hich V m <. By selecting s = then Ω m =, R : } + m m. (5

5 - Ω m S m W m- Ω c Figre 3. Maximm bond of the enhanced BIC states on plane Note that the bond of Ω m, i.e. S m =, R : } + m m = (6 has to common points (, and (, ith W m for any m and the to sets coincide for m =. For m, this can be proven by combining (4 and (6, from hich it yields that m m m =. Since m, then = hich corresponds to points (, or (, on the } plane. By defining Ω c =, R : + m then Ω c Ω m as shon in Fig. 3. Since V m < otside the closed set Ω m, then for any set of initial conditions, defined in Ω m, every trajectory starting from point (, on the plane ill remain in Ω m for all t. This means that = [, ], t. Since the control otpt is proven to remain inside the reired bonded range for any set of initial conditions (, Ω m, opposed to the original BIC (4 here and mst be specifically defined on the ellipse W, the enhanced BIC shos increased robstness in the selection of the initial conditions compared to the original strctre. It is therefore proven that for (, Ω m, then [, ] independently from the fnction g(x that needs to be reglated to zero according to the analysis in (3. Hence the enhanced BIC maintains the zero-gain property of the original BIC, i.e. the controller states and are bonded independently from the controller inpt g(x. As a reslt, considering the plant ( satisfying the ISS property ith respect to and ith a gain γ plant >, and taking into accont that the enhanced BIC (, ( is also ISS ith zero gain (γ control =, then according to the generalized small-gain theorem Jiang et al. (994, the closed-loop system represented by the feedback interconnection of the plant and the proposed controller is stable in the sense of bondedness. It is nderlined that this holds independently from the dynamics of the plant, i.e. fnction f(x,, nmber of states, system parameters or strctre, leading to the closed-loop stability in the sense of bondedness for any nknon nonlinear plant ( satisfying the ISS property. As already mentioned, for a sfficiently large vale of the controller parameter k, the BIC states ill be attracted ickly to the crve W m and remain very close to it for all ftre time. For any set of initial conditions (, Ω m ith >, the controller states ill remain very close to the pper crve of W m, i.e. above the horizontal axis, since as soon as they try to approach the horizontal axis, then and from ( there is ±, hich means that the states slo don and approach one of the points (, or (, independently from g(x. This provides and inherent anti-indp strategy for the enhanced BIC becase the integration smoothly slos don near the pper and loer limit of the control otpt. Additionally, since and remain close to W m for a large k, the controller state dynamics can be approximated by ẇ k I g(x m (7 ẇ k I g(x, (8 since the diagonal terms of the matrix in ( become very close to zero. According to Fig., for a high vale of m, the controller state remains close to almost in the entire area of the control otpt = [, ]. In this case, from (8, the rate of change of increases since the gain k I is larger for a ider range of vales of the control otpt. Hence, ill change faster and the enhanced BIC ill provide a better approximation of the traditional IC almost in the entire range of = [, ]. As a reslt, the enhanced BIC can replace the traditional IC to reglate a fnction g(x to zero and i garantee a bonded control otpt [, ] ithot additional satration nits, ii approximate the IC almost in the entire range of the control otpt, iii garantee nonlinear stability in the sense of bondedness for any nonlinear plant in the form of ( that is ISS ith respect to, and iv ill not sffer from integrator indp. The fact that the proposed enhanced BIC provides a better approximation of the IC in the entire range and inherits an anti-indp property is verified in the simlation reslts that follo. 4. EXAMPLE In order to evalate the performance of the enhanced BIC, the same practical example investigated in Konstantopolos et al. (6 is tested again ith the same parameters. The system represents a dc/dc bck-boost converter ith dynamics L di dt = ( v + E (9 C dv dt = ( i v R, ( here L is the converter indctance, C is the otpt capacitor, E is a constant inpt voltage (ncontrollable and R is the load resistor, as shon in Fig. 4. The system states are the indctor crrent i and the otpt voltage v, hile the control inpt is defined as and represents the

6 = = 8 IC+sat original BIC enhanced BIC E + - i L C v R - + v/v 6 4 Figre 4. Schematic of a dc/dc bck-boost converter dty ratio of the converter restricted in the range [, ]. The continos-time model (9-( is obtained sing average analysis on the actal sitching converter model hich is controlled by the sitching element ū =, }. By sitably controlling the dty ratio, the main task is to reglate the otpt voltage v to a reference vale v ref. The plant is obviosly a nonlinear system, de to the mltiplication of the control inpt ith the system states, and for any bonded control inpt in the range [, γ], for < γ, the system states remain bonded. Hoever, hen =, it is clear from (9 that the indctor crrent ill escape to infinity leading to instability. This is hy in practice the dty ratio is desired to be limited in a loer range, e.g. in [,.8], and is often achieved ith additional satration nits. In order to reglate the otpt voltage v to a reference vale v ref, the traditional IC can be sed ith integral fnction k I g(x = k I (v ref v. Since the dty ratio shold be limited in the range [,.8], a satration nit can be added at the otpt of the IC. Similarly, the original and the enhanced BIC can be applied to achieve the same reglation. Given that [,.8] shold hold tre, then the control inpt can be defined as = and the controller state is limited in the range [.8,.8], i.e. =.8 for the BIC. In order to have a direct comparison, the IC ith the satration nit is applied ith = and dynamics ẇ = k I g(x, here k I =.. For the original BIC, the rest of the parameters are chosen as k = and c =.6 and for the enhanced BIC, there is k = and m =. The plant parameters are L = mh, C = 3µF, R = 5Ω and E = 5V. The simlation has been performed sing Matlab/Simlink. Starting from zero initial conditions for the plant states, the reference otpt voltage changes to v ref = 3 V at t =.5 s. As it is demonstrated in Fig. 5(a, the otpt voltage of the converter is reglated at the desired vale for all three cases inclding the IC ith satration and both versions of the BIC. The indctor crrent response is shon in Fig. 5(b. At t = s, the reference voltage increases to 7 V hich reires a high dty ratio for the converter leading the control inpt to its imm vale of.8 (Fig. 5(c. The IC ith the satration nit forces the control inpt to satrate to the pper limit hile both versions of the BIC smoothly converge to the pper limit. In this case, the otpt voltage is reglated to a loer vale. After s, the reference voltage v ref retrns to 3 V, and all three controllers reglate the otpt voltage to the desired vale after a short transient. From Fig. 5, it is clear that the response of the enhanced BIC almost overlaps ith the response of the IC ith a satration nit for the first 3 s, opposed to the original BIC. The difference is clear hen v ref changes to 5 V at the time i/a Time/s (a otpt voltage v IC+sat original BIC enhanced BIC Time/s (b indctor crrent i IC+sat original BIC enhanced BIC Time/s (c dty-ratio Figre 5. Simlation reslts of the bck-boost converter comparing the IC ith a satration nit, ith the original and the enhanced BIC instant t = 3 s, here the IC ith satration sffers from integrator indp, hile the enhanced BIC can reglate the otpt voltage to the desired vale. This is better observed in the response of the inpt arond t = 3 s in Fig. 5(c, here the enhanced BIC does not satrate bt slos don near the pper limit and does not introdce an integrator indp. The theoretical analysis of the BIC states and presented in the paper is verified in Fig. 6, shoing the controller state trajectory on the plane, here it is clear that and take vales on W and W 99, for the cases of the original and the enhanced BIC, respectively. Therefore, it becomes clear that the enhanced BIC can garantee the crcial properties of the original BIC, i.e. zero-gain property, given bond for the control otpt, no indp isses, and additionally provides a better approximation of the traditional IC almost in the entire range of the control otpt. Hence, the enhanced BIC can easily replace the traditional IC ith a satration nit in many engineering applications and offer an inherent anti-

7 original BIC enhanced BIC W Figre 6. Controller states response on the plane for the original and the enhanced BIC indp property ithot changing the control operation in the entire bonded range, maintaining a continos-time strctre that allos the investigation of the closed-loop stability. 5. CONCLUSIONS An enhanced version of the recently proposed bonded integral controller as presented in this paper. It as analytically proven that the enhanced BIC maintains the crcial zero-gain property hich garantees the stability of the closed-loop system for any ISS plant, even ith nknon dynamics and strctre. Additionally, the enhanced BIC relaxes the assmption of the initial conditions of the controller states being defined on a specific ellipse, introdced by the original BIC, reslting in a larger set of possible initial conditions. It as also shon that the enhanced BIC provides a better approximation of the traditional IC in the entire range of the control otpt, providing a more sitable soltion for replacing the IC. The theoretical contribtion of the proposed method as verified sing a practical example of a dc/dc bck-boost converter. REFERENCES Astolfi, D. and Praly, L. (6. Integral action in otpt feedback for mlti-inpt mlti-otpt nonlinear systems. IEEE Trans. Atom. Control. To appear. Choi, J.W. and Lee, S.C. (9. Antiindp strategy for pi-type speed controller. IEEE Trans. Ind. Electron., 56(6, Gay, M. (6. A pertrbation-based proportional integral extremm-seeking control approach. IEEE Trans. Atom. Control, 6(, Hang, J. and Rgh, W.J. (99. On a nonlinear mltivariable servomechanism problem. Atomatica, 6(6, Isidori, A. (997. A remark on the problem of semiglobal nonlinear otpt reglation. IEEE Trans. Atom. Control, 4(, Isidori, A. and Byrnes, C.I. (99. Otpt reglation of nonlinear systems. IEEE Trans. Atom. Control, 35(, 3 4. Jiang, Z.P., Mareels, I., J., H.D., and Hang, J. (4. A nifying frameork for global reglation via nonlinear W otpt feedback: from iss to iiss. IEEE Trans. Atom. Control, 49(4, Jiang, Z.P., Teel, A.R., and Praly, L. (994. Small-gain theorem for ISS systems and applications. Mathematics of Control, Signals and Systems, 7(, 95. Jiang, Z.P. and Mareels, I. (. Robst nonlinear integral control. IEEE Trans. Atom. Control, 46(8, Khalil, H.K. (. Universal integral controllers for minimm-phase nonlinear systems. IEEE Trans. Atom. Control, 45(3, Khalil, H.K. (. Nonlinear Systems. Prentice Hall. Konstantopolos, G.C., Zhong, Q.C., Ren, B., and Krstic, M. (6. Bonded integral control of inpt-to-state practically stable nonlinear systems to garantee closedloop stability. IEEE Trans. Atom. Control, 6(, doi:.9/tac Li, R. and Khalil, H.K. (. Conditional integrator for non-minimm phase nonlinear systems. In 5st IEEE Conference on Decision and Control (CDC, Mahmod, N.A. and Khalil, H.K. (996. Asymptotic reglation of minimm phase nonlinear systems sing otpt feedback. IEEE Trans. Atom. Control, 4(, 4 4. Ngyen, V.C. and Damm, G. (. Modeling and conditional integrator control of an nmanned aerial vehicle for airlanch. In American Control Conference (ACC, Peng, Y., Vrancic, D., and Hans, R. (996. Anti-indp, bmpless, and conditioned transfer technies for PID controllers. IEEE Control Syst. Mag., 6(4, Riachy, S., Efimov, D., and Mbop, M. (6. Universal integral control: An approach based on mollifiers. IEEE Trans. Atom. Control, 6(, 4 9. Singh, A. and Khalil, H.K. (4. State feedback reglation of nonlinear systems sing conditional integrators. In 43rd IEEE Conference on Decision and Control (CDC, volme 5, IEEE. Tsinias, J. (989. Sfficient Lyapnov-like conditions for stabilization. Mathematics of control, Signals and Systems, (4, Weiss, G. and Natarajan, V. (6. Stability of the integral control of stable nonlinear systems. In 6 ICSEE International Conference on the Science of Electrical Engineering, 5. W, Y.Q., Y, J.B., and Zhao, Y. (. Frther reslts on global asymptotic reglation control for a class of nonlinear systems ith iiss inverse dynamics. IEEE Trans. Atom. Control, 56(4, Zaccarian, L. and Teel, A.R. (. A common frameork for anti-indp, bmpless transfer and reliable designs. Atomatica, 38(, Zaccarian, L. and Teel, A.R. (4. Nonlinear schedled anti-indp design for linear systems. IEEE Trans. Atom. Control, 49(, Zhong, Q.C., Konstantopolos, G.C., Ren, B., and Krstic, M. (6. Improved synchronverters ith bonded freency and voltage for smart grid integration. IEEE Trans. Smart Grid. doi:.9/tsg Zhong, Q.C. and Weiss, G. (. Synchronverters: Inverters that mimic synchronos generators. IEEE Trans. Ind. Electron., 58(4,